Pipeline Design and Management

## **Chapter 5** Flexible Pipes

*Minggang Tang*

#### **Abstract**

An unbonded flexible pipe is one of the most important equipment in offshore engineering, transporting oil and gas between the floater on the sea and the well located on the seabed. Flexible pipes consist of several metallic helical-reinforced layers and internal and external polymer sheaths, and relative slip between the layers is allowed, so that the structure show high axial stiffness and radial stiffness associated with relatively low bending stiffness. During the operation and installation, the flexible pipe will be subjected to complex and coupled loads such as tension, internal pressure, external pressure, torsion and bending, which lead to multiple structural failures. This chapter will present the current theoretical models and research progress to effectively evaluate the response of such composite structure, providing reference ideas for the engineering design of the flexible pipes.

**Keywords:** offshore flexible pipe, helical wire, complex loads, interlayered pressure, structural response

#### **1. Introduction**

Pipelines are the "lifeline" equipment for the production and development of offshore oil and gas resources and mineral resources. The traditional steel pipeline technology is mature, but the limitations in bendability, corrosion protection, installation and laying speed have significantly increased its comprehensive cost, so the concept of offshore flexible pipeline came into being. It can be bent into a small bending radius without reducing the mechanical bearing capacity, and can be installed and recycled by the reel method, so the comprehensive cost is relatively low;Meanwhile, in some medium water depth environments, using offshore flexible pipes as risers is the only solution. Therefore, since the 80s of last century, flexible pipelines have gradually been paid attention to and applied in the development of offshore oil and gas resources, and flexible pipelines account for more than 80% of the marine risers currently in service [1].

Each reinforcement layer is specifically designed to accommodate a specific load. Under the axisymmetric loads, such as tension, internal pressure, external pressure, and torsion, the bearing capability of the reinforced layer usually depends not only on the structural design of the layer itself, but also on the interaction with the adjacent layers [2], so that the pipe wall is formed by the armor layers. While, under the combined non-axisymmetric loads such as bending loads, the interlayered interaction becomes more complicated and makes the mechanical behavior of the reinforced

components show the strong nonlinearity. These bring challenges into accurately evaluating the structural strength and fatigue performance of flexible pipes.

The objective of the current study is to comprehensively establish the latest theoretical models of structural responses of offshore flexible pipes under various typical loads, and analyze the coupling relationship between the structural deformation and interlayered interaction under different loads. Especially for the tension armor wires, the calculation method of nonlinear responses under combined tension and bending is developed, and the theoretical models are verified by one case. These can not only help readers clearly understand the bearing principles and the response rules of flexible pipelines, but also provide a comprehensive, convenient and effective tool for carrying out the quantitative design and evaluation of offshore flexible pipeline in practical engineering.

This chapter is organized as follows. Section 2 puts forward the design requirements of flexible pipelines from the perspective of engineering applications; Section 3 describes the mechanical principles and structural types of flexible pipes; Section 4 introduces theoretical models of pipeline structural responses for axisymmetric loads such as tension, internal pressure, external pressure and torsion; The mechanical responses of pipes under non-axisymmetric loads such as bending and complex loads are discussed in Section 5; The last section takes one 8 inches internal diameter pipeline in practical engineering as example to quantitatively describe the mechanical responses of flexible pipeline structure under various loads.

#### **2. Offshore flexible pipe design requirements**


#### **Figure 1.**

*Schematic diagram of the flexible pipe laying- J lay.*

**Figure 2.** *Schematic diagram of flexible riser in operation.*

subjected to huge axial tension (when the flexible riser of an 8-inch inner diameter is used in water depth of 1500 meters, the axial tension can generally reach more than 100 tons). Therefore, the pipeline should have the low bending stiffness associated with the high axial stiffness and radial stiffness.

6.During the installation using reeling method, flexible pipes are subjected to repeated axial torsion. And during normal operation, the pipeline itself will also twist due to the movement of the floater and dynamic environment. Therefore, the torsion resistance in the clockwise/counterclockwise direction of flexible pipes needs to be considered during the design process, and the torsional equilibrium is also required, i.e. the pipeline does not twist freely under other loads.

7.Owing to its self-weight coupled with the floater motion and dynamic environment, the flexible riser at the interface with the floater (generally within 50 m) will encounter relatively alternative curvatures as well as high axial loads and internal pressure in operation, and stress fatigue failure is more likely to occur on the tension armor layer in the pipeline [4]. Currently, the service life of Flexible pipes is designed to reach 25 years at least, so accurately assessing the fatigue resistance of pipelines is a constant need for oil companies and owners.

In summary, the flexible pipeline structure needs to have excellent bending performance and corrosion resistance, and at the same time, it must have good resistance in internal pressure, external pressure, tension, torsion and fatigue. That is, while giving full play to the advantages of flexibility, it can meet the safety and reliability of the structure.

#### **3. Mechanical principle and structural type of offshore flexible pipeline**

#### **3.1 Mechanical principle**

According to the design requirements of flexible pipes, it is no longer feasible to use homogeneous materials for the entire pipe wall. Considering the need to seal inside and outside the pipe, the innermost and outermost layers of the pipe are usually made of polymer materials of a certain thickness. In order to meet the bearing capacity needs in all directions at the same time, the metal reinforcement structure in the form of a spiral is required. Considering that the single-layer helical structure is difficult to balance and achieve the design resistances, it is necessary to increase the spiral reinforcement layer according to different bearing capacity requirements, and maintain an unbonded form between the layers to reduce the minimum bending radius of the pipeline. The following describes the strengthening principle of pipeline structure according to different load resistance requirements.


**Figure 3.** *Typical cross-section of the pressure armour layer.*


It can be seen that in addition to the inner and outer polymer sheaths, the flexible pipe can simultaneously realize the overall flexibility and the resistance to loads in other directions by setting different types of metal spirals in the annulus.

#### **3.2 Typical structure**

According to the mechanical principle, the typical structure of the offshore flexible pipeline used in deep water is shown in **Figure 4**. From the inside to the outside, they are respectively:

• Carcass layer: interlocked section wire helically wound, mainly used to resist uniform external pressure from seawater, and avoid the erosion of the inner

**Figure 4.** *Schematic diagram of typical structure of flexible pipeline.*

sheath from the medium. The helical angle between the direction of the wire and the axial direction of the pipeline is close to 90 degrees, and the material is usually stainless steel to avoid corrosion caused by the internal medium.


#### *Flexible Pipes DOI: http://dx.doi.org/10.5772/intechopen.109504*

• Outer sheath: cylindrical polymer sheath formed by continuous extrusion, and mainly used to seal the external sea water while transferring the external pressure. Its common materials are HDPE, PA, etc.

In addition, according to the specific needs from users, layers can be added or subtracted on the basis of the above basic structure. The composite pipe wall is unbonded, and the whole is formed through the interaction between the layers, so it is of great significance to accurately understand the structural responses under different loads for designing and evaluating the safety of the pipeline.

#### **4. Mechanical behavior of pipelines under symmetrical loads**

According to the symmetry of the load on the pipe relative to the central axis, the loads including axial tension, internal pressure, external pressure and torsion are collectively referred to as the axisymmetric loads. Under this type of loads, the adjacent layers in the pipe wall contact and squeeze with each other.

#### **4.1 Resistance to internal pressure**

Flexible pipes are usually designed using the pressure armor layer to independently bear the internal pressure. The internal pressure is transferred through the inner sheath layer to the pressure armor layer. Considering that the cross-sectional area of the armor wire has a direct impact on its internal pressure resistance, the specialshaped section can be simplified to a rectangular section with the same thickness and material and certain voids in the design stage. Based on the assumption of line elasticity and small deformation, an analytical model is established for the general mechanical behavior of the spiral steel strip with rectangular section under internal pressure, in which the axial stress along the direction of the steel wire is concerned.

In case of the helical angle of pressure armor wire 90°, a closed plane ring is acted with radial pressure *P* along circumference of cylinder core, as shown in **Figure 5a**.

Referring to the most flexible pipe structures, the thickness of wires is much smaller than the diameter of the internal core. Then the classic Lame formula of the plane problem [7] can be used and equilibrium can be written as:

$$P = \frac{\sigma\_0 h\_0}{R} \tag{1}$$

In which, *R* means the helical radius of the pressure armor layer, *σ*<sup>0</sup> denotes the circumferential stress, that is the axial stress of the wire, *h*<sup>0</sup> is the equivalent thickness of the wire.

In case of the helical angle of steel wire not 90°, that is, the helical wire is pressured in the radial direction illustrated in **Figure 5b**. Let *f* be the axial force along the wire and *σ* represents axial stress, the component force in hoop direction from the axial force can be expressed as *f* <sup>0</sup> ¼ *f* � sin *α*. And the mapping sectional area in hoop direction of the wire is *a* ¼ *A=* sin *α*. Then the stress of the wire in hoop direction is:

$$\sigma = \frac{f}{A} = \frac{f\_0 / \sin a}{a \sin a} = \frac{f\_0}{a \sin^2 a} = \frac{\sigma\_0}{\sin^2 a} \tag{2}$$

Substituting Eq. (2) into (1), the relationship between radial pressure from single armor layer and axial strain along the wire can be obtained as [8]:

$$P = \frac{h\_0 \sigma \sin^2 a}{R} \tag{3}$$

It can be seen that as the internal pressure increases, the tensile stress along the direction of the wire axis increases. Considering the thickness, helical angle and radial radius of the wire unchanged, the stress along the wire direction and internal pressure basically show a linear relationship.

#### **4.2 Resistance to external pressure**

Flexible pipes are usually designed using the innermost carcass layer to independently bear the external pressure. Although the sea water pressure generally acts on the outer sheath, the outer sheath leaks in extreme cases and the seawater passes through the annulus and directly acts on the outside of the inner sheath. In the design phase, the interlocked carcass layer can be equivalent to a homogeneous cylinder with a certain thickness. Then the problem can be simplified to analyze the buckling collapse of the cylinder under external pressure. This section establishes an analytical model of the general mechanical behavior of a homogeneous ring (cylinder) under external pressure based on the theory of elastic stability.

Timoshenko and Gere [9] first gave the deflection of a thin bar with a circular cross-section as shown in Eq. (4), for the problem of the elastic buckling of a ring or tube. It was assumed that radial displacement is small and the displacement in the tangential direction can be ignored.

$$\frac{d^2\alpha}{d\theta^2} + \alpha = -\frac{1}{EI}qR^3(\alpha + \alpha\_1 \cos 2\theta) \tag{4}$$

where *ω* denotes radial deformation of a thin bar, *θ* is the angle in the hoop direction, *M* is the bending moment loading on the bar, *EI* is the bending stiffness of the bar cross section, *R* is the mean radius of curvature, *ω*<sup>1</sup> is the maximum initial radial deviation, and *q* means a uniform external pressure on the plane ring.

Considering the continuity condition on points A, B, C and D on the plane ring in **Figure 6**, the analytical solution of Eq. (4) is:

**Figure 6.** *Schematic diagram of a plane ring with an initial imperfection (ovality).*

$$\boldsymbol{\alpha} = \frac{\boldsymbol{\alpha}\_1 \mathbf{q}}{\mathbf{q}\_{\rm cr} - \mathbf{q}} \cos 2\boldsymbol{\theta} \tag{5}$$

Eq. (5) gives the analytical expression for the elastic buckling of a plane ring under uniform external pressure, where the critical pressure *qcr* of the ring is given by:

$$\mathbf{q}\_{\rm crr} = \frac{\mathbf{3EI}}{\mathbf{R}^3} \tag{6}$$

In the case of a rectangular ring section and the plane strain condition, the inertia moment *I* per unit length can be written as *t* <sup>3</sup>*=*12 and *<sup>E</sup>* is converted to *<sup>E</sup><sup>=</sup>* <sup>1</sup> � *<sup>ν</sup>*<sup>2</sup> ð Þ. Then the critical pressure value under the plane strain conditions can be expressed as follows:

$$\mathbf{q}\_{\rm cr} = \frac{\mathbf{E}}{4(\mathbf{1} - \nu^2)} \left(\frac{\mathbf{t}}{\mathbf{R}}\right)^3 \tag{7}$$

where *t* represents the equivalent thickness of the carcass layer resulted from equivalent methods. It can be seen that the critical pressure and the equivalent thickness of the carcass layer basically show a cubic relationship. It should be noted that in the design stage, it is critical to use conservative and effective equivalent methods to determine thickness. Current equivalent methods include area equivalence [10], bending stiffness equivalence per unit length [11], bending stiffness equivalence per unit area and strain energy equivalence methods [12]. Through the experimental verification, the equivalent thickness obtained by the strain energy equivalent method can keep the critical pressure in a conservative state, which is convenient for engineering applications.

#### **4.3 Resistance to axial tension**

The double tension armor layers of flexible pipes are designed mainly to resist the axial tension, meanwhile the structure of internal core is designed to provide a support

**Figure 7.** *Mechanical model for tensile stiffness of flexible pipeline.*

in the radial direction. To capture structural character, a simplified model that a column is wound by helical steel wires is established to analyze the tensile stiffness as illustrated in **Figure 7**.

To simplify the deducing of the tensile stiffness, the following assumptions are made in the model: (i) Helical wires in any layer are equally spaced around the circumference of the flexible pipe. (ii) Only the axial deformation of the wires is considered during tension and the bend stiffness and torsional stiffness of wires are neglected. (iii) The internal core is modeled as a cylinder which has radial deformation under pressure from armor wires. (iv) All elements meet the assumption of small deformation and all possible frictions are neglected. (v) The helical wires and cylinder core element are homogeneous, isotropic and linearly elastic.

The analysis will be carried out on the plane for convenience by unfolding the cylinder with a helical pitch length as shown in **Figure 8a**. Considering axial tension *F* on the pipe model and the axial strain *ε*, we analyze the two situations below [13].

**Figure 8.** *Deformation of steel wire under axial tension.*

#### *Flexible Pipes DOI: http://dx.doi.org/10.5772/intechopen.109504*

1. If only the axial deformation of the pipeline is considered, Δ*l* is defined as axial deformation just as shown in **Figure 8b**. According to the geometrical relationship, the elongation along wire axial direction can be expressed as Δ*l* � cos *α*, in which *α* means the angle between umbilical axial direction and wire axial direction. Considering the length of the helix wire before deformation *l*<sup>0</sup> ¼ *l=* cos *α*, we can obtain the strain in the wire axial directions as follows:

$$
\varepsilon = \frac{\Delta l \cdot \cos a}{l\_0} = \frac{\Delta l}{l} \cos^2 a \tag{8}
$$

2. If we only consider the radial deformation of pipeline and let Δ*R* be radial elongation as shown in **Figure 8c**, the elongation along wire axial direction is Δ*R* � sin *α*. If we define the length of the helix wire before deformation *R*<sup>0</sup> ¼ *R=* sin *α*, where *R* means radius of the helix wires, the strain of wire can be expressed as eq. (9).

$$
\varepsilon = \frac{\Delta R \cdot \sin a}{R\_0} = \frac{\Delta R}{R} \sin^2 a \tag{9}
$$

However, the actual deformation is the synthetization of the axial deformation from tension and the radial deformation from the contraction of cylinder core caused by the pressure of the outer tension armor layers as shown in **Figure 8d**. Add the eq. (8) and (9), the synthetization strain can be written as:

$$
\varepsilon = \frac{\Delta l}{l} \cos^2 a - \frac{\Delta R}{R} \sin^2 a \tag{10}
$$

From eq. (10), the axial tension of all the armor wires can be expressed as:

$$\begin{split} F &= \sum\_{i=1}^{m} EA e\_i \cos a\_i \\ &= \frac{\Delta l}{l} \left( \sum\_{i=1}^{m} EA \cos^3 \alpha\_i \right) - \Delta R \left( \sum\_{i=1}^{m} \frac{EA \sin^2 \alpha\_i \cos a\_i}{R\_i} \right) \\ &= \Theta\_1 \frac{\Delta l}{l} - \Theta\_2 \Delta R \end{split} \tag{11}$$

where *E* is Young modulus of steel wire, *A* is cross-sectional area of the wire, *m* identifies the amount of armor layers and *i* means layer number, *ni* is the wire number in *i*th layer, Θ<sup>1</sup> and Θ<sup>2</sup> are the representative symbols of the corresponding analytical algebraic formulas.

From eq. (3), the relationship between radial pressure from single armor layer and axial strain along the wire can be extended as [14]:

$$P = \frac{nEA\epsilon\sin^2a}{2\pi R^2\cos a} \tag{12}$$

There are usually two or more armor layers for common flexible pipes. Then the radial pressure on cylinder core should be added by the pressure from all the armor layers. By substituting (10) into (12), the equilibrium equation about *P* and *ε* is further written as:

$$\begin{split} P &= \sum\_{i=1}^{m} \frac{n\_i E A e\_i \sin^2 \alpha\_i}{2 \pi R\_i^2 \cos \alpha\_i} \\ &= \frac{\Delta l}{l} \left( \sum\_{i=1}^{m} \frac{n\_i E A \sin^2 \alpha\_i \cos \alpha\_i}{2 \pi R\_i^2} \right) - \Delta R \left( \sum\_{i=1}^{m} \frac{n\_i E A \sin^4 \alpha\_i}{2 \pi R\_i^3 \cos \alpha\_i} \right) \\ &= \Psi\_1 \frac{\Delta l}{l} - \Psi\_2 \Delta R \end{split} \tag{13}$$

where Ψ<sup>1</sup> and Ψ<sup>2</sup> are the representative symbols of the corresponding analytical algebraic formulas. When the cylinder core is pressed by armor steel wires, it comes to the radial contraction. In order to describe the mechanical phenomenon, the radial stiffness Ω of the core is introduced and defined as:

$$
\Omega = P / \Delta R \tag{14}
$$

Eqs. (11), (13) and (14) together form a close equation set (15), which leads to the equilibrium of the steel wires with tensions considering the compressible deformation of the internal core.

*F* ¼ Θ<sup>1</sup> Δ*l <sup>l</sup>* <sup>þ</sup> <sup>Θ</sup>2Δ*<sup>R</sup> P* ¼ Ψ<sup>1</sup> Δ*l <sup>l</sup>* <sup>þ</sup> <sup>Ψ</sup>2Δ*<sup>R</sup>* <sup>Ω</sup> <sup>¼</sup> *<sup>P</sup>* Δ*R* 8 >>>>>>< >>>>>>: (15)

By eliminating *P* and Δ*R*, the relation of the tension and axial strain can be obtained as:

$$F = \left(\Theta\_1 + \frac{\Theta\_2 \Psi\_1}{\Omega - \Psi\_2}\right) \frac{\Delta l}{l} \tag{16}$$

The relationship between the axial tension and the generated interlayer pressure can be analytically expressed as:

$$F = \frac{\Theta\_1 \Omega + \Theta\_2 \Psi\_1 - \Theta\_1 \Psi\_2}{\Psi\_1 \Omega} P \tag{17}$$

If the radial contraction of the internal cylindrical core is not considered, that is, the radial stiffness Ω tends to infinity, Eq. (17) can be further simplified to obtain the explicit expression of axial tension and radial pressure:

$$P = \frac{\Psi\_1}{\Theta\_1} F = \frac{n \tan^2 \alpha}{2\pi R^2} F \tag{18}$$

It can be found that as the axial tension increases, both the tensile stress along the direction of the wire and the radial pressure on the internal cylindrical core increases, and the interlayer pressure and axial tension basically show a linear relationship. It is precisely because the radial stiffness provided by the pressure armor layer is relatively large, the axial tensile stiffness of the pipeline is sufficient, and the helical armor wire can better withstand the tension. As the radial stiffness of the internal cylindrical components gradually decreases, the deformation of the helical wire will no longer be a small geometric deformation, and each wire will tend to have the independent spring deformation, therefore the tension resistance will be greatly reduced.

#### **4.4 Resistance to torsion**

Flexible pipes are primarily designed to resist torque in the clockwise/counterclockwise direction through the double tension armor layers [15]. The basic theory and assumptions are the same as those in the previous section, and this section focuses on the relationship among the torque, the radial pressure and the axial stress along the wire helical. It can be seen from the eq. (10) that the circumferential component of the axial force along the wire direction resists the overall torque of the pipeline, so the relationship between the torque on the single tension armor layer of the pipeline and the deformation along the wire direction can be expressed as:

$$Q = nREAe\sin a\tag{19}$$

Combined with eq. (12), the analytical relationship between interlayered pressure and torque without consideration of the radial contraction can be written as:

$$P = \frac{\tan a}{2\pi \text{R}^3} Q \tag{20}$$

It can be noticed that as the torque increases, both the tensile stress along the direction of the wire and the radial pressure on the internal cylindrical core increases, and the interlayer pressure and torque also show a linear relationship. Considering both the axial and radial deformation, the derivation process is similar to the previous section, resulting in a more complete expression for eq. (20). It is worth noted that due to the different helical directions of tension armor layers, the direction of each layer's resistance to torque is different. The neighboring tension armor layers will tend to squeeze or separate with each other due to the different torsion direction.

#### **5. Mechanical behavior of pipelines under complicated loads**

When the pipeline is subjected to non-axisymmetric loads, such as bending, the relative sliding between neighboring layers occurs due to the unbonded condition, so that the pipeline has excellent flexibility. When the pipeline is subjected to both nonaxisymmetric loads and axisymmetric loads at the same time, the tangential friction is generated between the layers due to radial pressure and relative slippage. Especially for tension armor wire, the stress state becomes very complicated as the pipeline curvature changing, which will firstly cause the fatigue failure.

#### **5.1 Bending performance**

When the flexible pipe is only subject to bending, the interlayered interaction is weak, and both the deformation and relative slippage exist for each layer. Since the minimum bending radius of the inner and outer sheath and the interlocked armor layer can be obtained by the classical material mechanics, this section focuses on the mechanical behavior of tension armor wire bending under weak interlayered interaction conditions, which provides a theoretical basis to evaluate the bending performance of pipelines.

One helical wire with a rectangular cross-section wrapped around a cylindrical shell with radius *r* is considered to represent the initial condition of the tension armor. When the model is bent (the radius of curvature is *ρ* ¼ 1*=κ*), a curve is created on the toroid, as illustrated in **Figure 9a**. The angular coordinate *θ* located along the torus radius and the arc length coordinate *u* located along the torus centerline are chosen as parameters of the torus surface. The space vector **R** on the surface can be expressed in ð Þ *θ*, *u* -coordinates instead of in Cartesian coordinates as [16]:

$$\mathbf{R}(\mathbf{u},\theta) = \begin{bmatrix} \left(\frac{\mathbf{1}}{\kappa} + \mathbf{r} \cdot \cos\theta\right) \cos(\kappa \mathbf{u}) - \frac{\mathbf{1}}{\kappa} \\\\ \left(\frac{\mathbf{1}}{\kappa} + \mathbf{r} \cdot \cos\theta\right) \sin(\kappa \mathbf{u}) \\\ \mathbf{r} \cdot \sin\theta \end{bmatrix} \tag{21}$$

Correspondingly, ð Þ *x*2, *x*<sup>3</sup> -rectangular coordinates can be built in the cutting plane perpendicular to the centerline of the wire (see **Figure 9b**), where *x***<sup>2</sup>** is perpendicular to the local unit wire normal **n** and is located along the rectangular width, and *x***<sup>3</sup>** points to the local unit wire binormal **b** and is located along the thickness.

When the cylindrical shell is bent with a specified curvature, the initial equilibrium state of the helical wire is broken, and the wire is forced to slip on the torus surface. Correspondingly, the changed space vector of the wire can be expressed as **R**<sup>0</sup> *θ*<sup>0</sup> , *u*<sup>0</sup> ð Þ. Based on differential geometry theory, the variation of curvature components of the bending wire can be defined as:

$$\begin{cases} \Delta \mathbf{\kappa}\_{\mathbf{n}} = \mathbf{\kappa}\_{\mathbf{n}} - \mathbf{\kappa}\_{\mathbf{n}}{}^{0} \\ \Delta \mathbf{\kappa}\_{\mathbf{b}} = \mathbf{\kappa}\_{\mathbf{b}} - \mathbf{\kappa}\_{\mathbf{b}}{}^{0} \\ \Delta \mathbf{\tau} = \mathbf{\tau} - \mathbf{\tau}^{0} \end{cases} \tag{22}$$

In which, the stress on the wire cross section caused by the normal curvature Δκ<sup>n</sup> and the binormal curvature Δκ<sup>b</sup> is the positive stress and along the wire direction; the stress on the wire cross section caused by the torsional curvature Δτ is the shear stress

**Figure 9.** *Wire geometry on a toroid under the curvature- (a) overall view, (b) cross-section A-A view.*

and perpendicular to the wire direction. During the wire bending, the fatigue failure usually occurs firstly at the corner point of the wire rectangular section due to high local stress, which needs to be concerned. Since the shear stress at the corner point is zero, the stress is in the uniaxial stress state and can be expressed as:

$$
\sigma^\text{y} = \sigma\_\text{n} + \sigma\_\text{b} = \left(\Delta\kappa\_\text{n} \times \mathbf{x}\_{2\text{y}}\right) \cdot \mathbf{E} + \left(\Delta\kappa\_\text{b} \times \mathbf{x}\_{3\text{y}}\right) \cdot \mathbf{E} \tag{23}
$$

where y represents the corner point number.

Based on the existing experimental results and theoretical models, the bending behavior of the helical wire considering the weak interlayered interaction is closer to that of a spring [4]. Therefore, according to the classical principle of elasticity, the curvature variation in the three directions of the helical wire can be obtained as eq. (24) when the pipeline has a certain curvature *κ*.

$$\begin{cases} \Delta \kappa\_{\text{n}} = \frac{\text{Bcos}\theta}{\text{EI}\_{\text{n}}} \kappa \\\\ \Delta \kappa\_{\text{b}} = -\frac{\text{Bsin}\theta \cos \alpha}{\text{EI}\_{\text{b}}} \kappa \\\\ \Delta \tau = \frac{\text{Bsin}\theta \sin \alpha}{\text{G} \text{J}} \kappa \end{cases} \tag{24}$$

Use of Eq. (24) in Eq. (23) leads to the stress on a corner point of the helical wire:

$$\sigma^{\circ} = \frac{\text{Bcos}\theta}{\text{EI}\_{\text{n}}} \text{\textkappaE} \cdot \text{x}\_{2\text{y}} - \frac{\text{Bsin}\theta \cos\alpha}{\text{EI}\_{\text{b}}} \text{\textkappaE} \cdot \text{x}\_{3\text{y}} \tag{25}$$

in which *B* represents the bending stiffness of the spring and is expressed as 2*cosα=* <sup>1</sup> *EIn* <sup>þ</sup> cos <sup>2</sup>*<sup>α</sup> EIb* <sup>þ</sup> sin <sup>2</sup>*<sup>α</sup> GJ* � �, *EIn* and *EIb* represent the bending stiffness of the rectangular cross-section with respect to the *x*3, *x*<sup>2</sup> axis, respectively, and *GJ* indicates the torsion stiffness of wire section.

It can be seen that under the condition of weak interlayered interaction, the bending behavior of flexible pipes can be evaluated layer by layer, and then the minimum bending radius (MBR) can be comprehensively obtained. In general, the MBR of the pipeline is first determined by the stress at corner points of tension armor wires.

#### **5.2 Structural response under complicated loads**

When the pipeline is subjected to complicated multiaxial loads, the friction will be generated between the layers, which brings challenges to the evaluation of the overall mechanical responses of the pipeline. This section focuses on the mechanical behavior of tension armor wires under complicated loads, that is, considering the large interlayered pressure caused by the axial tension, internal and external pressure or torsion, and as the curvature increases from zero, the interlayered friction will force the steel wire on the surface of internal cylindrical core to go through three stages [17] of no-slip, stick–slip and full-slip, as shown in **Figure 10**. The theoretical model is based on the common assumptions: i) The lay angle of the wire remains constant during helical wire tension and bending, which yields a loxodromic curve on a torus surface; ii) The size of wire cross section is very small compared to the helical radius;

**Figure 10.** *Schematic diagram of nonlinear bending behavior of offshore flexible pipes.*

iii) Small deformations occur in the linear elastic range of the material; iv) The helical wires are evenly distributed along the circumference and there is no mutual influence between neighboring wires.

#### *5.2.1 No-slip stage (O-A)*

When the curvature is very small, the helical wire remains firmly attached to the bent cylinder without any relative movement due to the interlayered static friction. Let R be the helical radius of the wire, α is the helical angle, θ represents the phase angle in the circumferential direction, which varies from zero to π*=*2. Considering the pipeline bent to a small curvature and the helical angle of the wire unchanged, the spatial three-dimensional coordinates κ x<sup>0</sup> , y<sup>0</sup> , z<sup>0</sup> � � of the deformed helical wire can be described parametrically as eq. (26) based on the differential geometry [18].

$$\begin{cases} \mathbf{x}' = \text{Rcos}\Theta\\ \mathbf{y}' = \sin\left(\frac{\mathbf{R}}{\tan\alpha}\Theta\kappa\right)\frac{\mathbf{1}}{\mathbf{x}} - \text{Rsin}\Theta\sin\left(\frac{\mathbf{R}}{\tan\alpha}\Theta\kappa\right) \\\ \mathbf{z}' = \left(\mathbf{1} - \cos\left(\frac{\mathbf{R}}{\tan\alpha}\Theta\kappa\right)\right)\frac{\mathbf{1}}{\mathbf{x}} + \text{Rsin}\Theta\left(\frac{\mathbf{R}}{\tan\alpha}\Theta\kappa\right) \end{cases} \tag{26}$$

The corresponding axial strain along the bent helical strip is given by:

$$\varepsilon(\phi) = \frac{\mathrm{d}s'}{\mathrm{d}s} - \mathbf{1} = \sqrt{\frac{\mathrm{d}\mathbf{x}'^2 + \mathrm{d}\mathbf{y}'^2 + \mathrm{d}\mathbf{z}'^2}{\mathrm{d}\mathbf{x}^2 + \mathrm{d}\mathbf{y}^2 + \mathrm{d}\mathbf{z}^2}} - \mathbf{1} \tag{27}$$

where s represents the arch length along the helical wire. By using eq. (26) in eq. (27), and assuming small deflections, a linear expression for the axial strain can be obtained by eliminating second and higher order terms. The linearized axial strain is given by:

*Flexible Pipes DOI: http://dx.doi.org/10.5772/intechopen.109504*

$$
\varepsilon(\phi) = -\mathbb{R}\cos^2\mathfrak{a}\sin\mathfrak{R}\tag{28}
$$

It can be seen that the maximum stress of the wire in the no-slip stage located at the intrados and extrados, and can be expressed as:

$$\sigma\_{\text{f}-\text{ns}}^{\text{max}} = \text{RE}\cos^2\alpha\text{s} \tag{29}$$

*5.2.2 Stick–slip stage (A-B)*

In the stick–slip stage, the axial force of the helical wire and the interlayered friction between are balanced with each other. However, as the curvature increases, the axial force on the wire section increases, and when the maximum static friction is exceeded, the wire begins to slide relative to internal components. Thus, the slipping condition can be written as [19]:

$$\frac{\text{dN}}{\text{ds}} > f \tag{30}$$

in which, N is the internal force along the wire direction, *f* means the friction force on the unit length of the wire, *s* and denotes the length along the wire axis. According to the eq. (28) and taking into account the geometric relationship *θ* ¼ *s* � sinα*=R*, the critical curvature (Point A in **Figure 10**) corresponding to the wire starting to slip can be derived as:

$$\kappa\_0 = \frac{\text{f}}{-\text{EA}\cos^2\text{a}\text{sin}\text{ar}\cos\theta} \tag{31}$$

It can be found that when *θ* equals to *kπ* ð Þ *k* ¼ 0, 1, 2, … , the critical curvature is minimized. That implies the helical wire at the neutral axis first beginning to slip as the curvature increasing. Assuming that the contact pressure on the inner and outer surfaces of the wire layer are separately qi and qo, and the corresponding static friction coefficients are μ<sup>i</sup> and μo, the critical curvature of the wire can be further expressed as:

$$\mathbf{x}\_0 = \frac{\mathbf{q}\_i \mu\_i + \mathbf{q}\_o \mu\_o}{-\mathbf{E} \mathbf{A} \cos^2 \alpha \sin \alpha} \tag{32}$$

When the curvature is greater than the critical curvature, the slipping area of the wire rapidly extends from the neutral axis to both ends, during which one part of the wire is slipping while the other part still in the stick stage. According to the equilibrium relationship of the axial force and interlayered friction in the stick area, the initial slipping curvature at the phase angle *θ* of the wire can be obtained:

$$\kappa\_{\rm f} = \left(\frac{\rm f}{-EA\cos^2\alpha \sin\alpha}\right)\frac{\theta}{\sin\theta} = \kappa\_0 \frac{\theta}{\sin\theta} \tag{33}$$

If *θ* ¼ π*=*2 in the above equation, the curvature when all parts of the wire enter the slipping state (Point B in **Figure 10**) can be described as:

$$\mathbf{\kappa}\_{\mathbf{f}} = \frac{\boldsymbol{\pi}}{2} \mathbf{\kappa}\_0 \tag{34}$$

From point A to point B, the wire completes the transition from the no-slip state to the full-slip state. As the axial stress of the wire is gradually released, the overall bending stiffness of the layer decreases.

#### *5.2.3 Full-slip stage (B-)*

Assuming that the friction along the wire direction dominates, the friction on the inner and outer surfaces of the rectangular wire produces the axial stress evenly distributed by the wire section. According to the eq. (33), the axial stress generated by the total friction F0 of the slipping wire can be written as:

$$\sigma\_{\text{f}-\text{fs}} = \frac{\mathbf{F}\_0}{\mathbf{A}} = \left[\frac{\mathbf{R}\theta}{\mathbf{A}\sin\alpha}\right] \mathbf{f} \tag{35}$$

where A is the cross-sectional area of the wire. It can be seen that the friction varies with the phase angle, but the stress at the four corner points of the wire at the same phase angle is the same. Therefore, the maximum axial stress of the wire generated by the total friction in the full-slip state is further obtained:

$$
\sigma\_{\rm f-fs}^{\rm max} = \frac{\rm R\pi}{2\rm Asin\alpha} \left( \mathbf{q}\_i \mu\_i + \mathbf{q}\_o \mu\_o \right) \tag{36}
$$

In summary, the theoretical model for calculating the maximum axial stress of the helical wire at each stage during the curvature increasing is established considering the interlayered friction. For conservative consideration in practical engineering, the above three stages are usually instead of two stages: one is the no-slip stage where the curvature varies from zero to κf*:*, and the other is the full-slip section where the curvature is greater than κ<sup>f</sup>

Additionally, due to the complicated loads, it is also necessary to consider the axial stress *σ<sup>T</sup>* (as shown in Eq. (11)) of the wire caused by the tension and the axial stress *σκ* (as shown in Eq. (25)) caused by the helical "spring" itself during bending in the full-slip stage. Then the nonlinear stress of the helical wire under complicated loads with the consideration of the interlayered interaction can be expressed as Eq. (37) and shown as **Figure 11**.

$$\boldsymbol{\sigma} = \begin{cases} \boldsymbol{\sigma}\_{\mathrm{T}} + \boldsymbol{\sigma}\_{\mathrm{f}-\mathrm{ns}}, \kappa \le \kappa\_{\mathrm{f}} \\ \boldsymbol{\sigma}\_{\mathrm{T}} + \boldsymbol{\sigma}\_{\mathrm{x}} + \boldsymbol{\sigma}\_{\mathrm{f}-\mathrm{fs}}, \kappa > \kappa\_{\mathrm{f}} \end{cases} \tag{37}$$

In order to perform the fatigue evaluation [20], it is necessary to calculate the maximum alternating stress amplitude Kcð Þκ of offshore flexible risers as the curvature changes. Removing the average stress term σ<sup>T</sup> and taking into account the maximum stress caused by the curvature and interlayered friction, the maximum alternating stress can be described as:

$$
\Delta \sigma^{\text{max}} = \begin{cases}
\sigma\_{\text{f}-\text{ns}}^{\text{max}}, \kappa \le \kappa\_{\text{f}} \\
\sigma\_{\text{\kappa}}^{\text{max}} + \sigma\_{\text{f}-\text{fs}}^{\text{max}}, \kappa > \kappa\_{\text{f}}
\end{cases} \tag{38}
$$

It can be seen that the alternating stress exhibits a nonlinear relation with the curvature. Therefore, *Kc*ð Þ*κ* is not a constant value and cannot be directly used in the existing commercial software to calculate the fatigue life. In this case, it is necessary to

**Figure 11.** *Schematic diagram of nonlinear stress at the corner point of the wire cross-section.*

carry out the secondary development in order to input the alternating stress with curvature, and then the fatigue life calculation can be performed by the damage accumulation using the Miner formula.

#### **6. A case study**

#### **6.1 Description of the pipeline structure**

In order to quantitatively illustrate the relation between interlayer action and the overall performance of offshore flexible pipelines, an 8-inch inner diameter flexible riser actually applied in a water depth of 1500 meters is taken in this section as an example, and the mechanical properties of pipelines are given in **Table 1**.

The pipeline structure is designed to have two tension armor layers, of which material elastic modulus is 206Gpa, the Poisson's ratio is 0.3 and the yield strength is 800Mpa. The rectangular cross-sectional dimensions of the steel wire in the two tension armor layers are the same (4 mm 12 mm) and the helical angles are separately positive and negative 30 degrees. According to the configuration design, functional requirements and hydrodynamic analysis of the flexible riser, the design loads can be determined as the internal pressure of 20 Mpa, the external pressure of 15 Mpa, the tension of 170 tons, the MBR of 2.5 meters and the fatigue life of 25 years.

#### **6.2 Analysis and discussion of structural responses**

Using the theoretical models for calculating the interlayered interaction and structural responses proposed in this chapter, this section quantitively gives the


#### **Table 1.**

*Material and dimensional parameters of flexible pipelines.*

interlayered pressure and the material utilization of the pipeline under axisymmetric and non-axisymmetric design loads, and the results are shown as in **Table 2**.

It can be clearly seen that the pipeline structure meets the design requirements, and still have a large safety margin under various independent loads. Those illustrate that the theoretical models for the structural responses proposed in this chapter could provide effective tools to design and evaluate the flexible pipelines in actual engineering. For the tension armor wires, achieving the tensile resistance of 170 tons requires the internal cylindrical component to provide a radial supporting pressure of nearly 6Mpa. Therefore, the internal pressure will be partially compensated by the pressure generated by axial tension, so that the material utilization of the tension armor wire is further reduced during actual operation of the pipeline.


#### **Table 2.**

*Interlayered pressure and material utilization of the pipeline under different design loads.*

#### *Flexible Pipes DOI: http://dx.doi.org/10.5772/intechopen.109504*

Since the fatigue failure occurs first on the helical tension armor wire of offshore flexible riser under alternative loads, the nonlinear stress at the corner points of the wire with the curvature is calculated by using Eq. (38) under combining the axial tension and bending curvature, as shown by the red dot line in **Figure 12**. The figure also shows the linear stress (blue dot line) of the wire due to friction only in the no-slip state and the linear stress (black line) of the slipping wire without consideration of the interlayered friction. It can be clearly seen that the nonlinear wire stress considering interlayer friction is between the two linear stress distribution curves, so the corresponding distributions of the fatigue life should also have the similar tendency.

It should be noted that the interlayered contact and friction cause the stress of helical wires obvious nonlinear, so their accurate prediction is of great significance for the calculation of fatigue life. However, for the current analyzing theory of fatigue stress, there is a lack of accurate description of the wire behavior in the stick–slip stage [21], which has a decisive effect on the fatigue life when the riser is subjected to large axisymmetric loads and small bending curvature. In addition, after the wire enters the full-slip stage, it is worth further exploring whether the bending stress of the wire under axial tension can be simulated through the spring theory [22]. Moreover, despite the anti-wear layer, long-term repeated interlayered friction will still cause wear [23], which impact on the stress variation of the wire and the fatigue life cannot be ignored. All of those discussion items bring big challenges for evaluating the fatigue life of flexible pipes, which will promote the further development of the theoretical and experimental methods.

**Figure 12.** *Calculation results of nonlinear stress on the helical wire.*

#### **7. Conclusion**

This chapter systematically introduces the mechanical principles of offshore flexible pipes, and gives the theoretical methods for evaluating the structural responses under axisymmetric loads, bending and combined complicated loads. It can be seen from a case study that although different strengthening layers in the pipeline are separately designed to resist against different loads, the overall resistance of the pipeline strongly depends on the interaction between the layers due to the use of metal helical armor wires, and then the unbonded pipe wall forms a whole. The contact pressure dominates the interlayered interaction under the axisymmetric loads such as axial tension, internal pressure, external pressure and torsion. And while the nonaxisymmetric bending load is introduced, interlayered friction and relative slippage occur, resulting in the obvious stick–slip nonlinear behavior of pipeline components, which has an important impact on the subsequent fatigue life analysis.

#### **Acknowledgements**

This work was supported by the Program of NSFC (grant number 5170924); the Key Technologies Research and Development Program (grant number 2021YFC2801600); and the High-tech Ship Scientific Research Program (grant number CY05N20). Their support is gratefully appreciated.

### **Author details**

Minggang Tang1,2

1 China Ship Science Research Center, Wuxi, China

2 Taihu Laboratory of DeepSea Technological Science, Wuxi, China

\*Address all correspondence to: tangmg@cssrc.com.cn

© 2023 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Zhang Y, Chen B, Qiu L, et al. Analytical tools optimize unbonded flexible pipes for Deepwater environments. Journal of Petroleum Technology. 2004;**56**(5):48-58

[2] International Organization for Standardization (ISO). Unbonded Flexible Pipe Systems for Subsea and Marine Applications. ISO-13628-2; (Switzerland) 2006

[3] De Sousa JRM, Lima ECP, Ellwanger GB, et al. Local mechanical behaviour of flexible pipes subjected to installation loads. In: Proceedings of 20th International Conference on Offshore Mechanics and Arctic Engineering. Rio de Janeiro, Brazil: ASME; 2001. pp. 219-227

[4] Tang M, Yang C, Yan J, et al. Validity and limitation of analytical models for the bending stress of a helical wire in unbonded flexible pipes. Applied Ocean Research. 2015;**50**:58-68

[5] Yue Q J, Lu QZ, Yan J, et al. Tension behavior prediction of flexible pipelines in shallow water. Ocean Engineering. 2013;**58**(15):201-207

[6] Knapp RH. Derivation of a new stiffness matrix for helically armoured cables considering tension and torsion. International Journal for Numerical Methods in Engineering. 1979;**14**(4): 515-529

[7] Gere JM, Goodno BJ. Mechanics of Materials. Toronto: Cengage Learning; 2009

[8] Zhu XK, Leis BN. Theoretical and numerical predictions of burst pressure of pipelines. Journal of Pressure Vessel Technology. 2007; **129**(4):644-652

[9] Timoshenko SP, Gere JM. Theory of Elastic Stability. Russia: St. Petersburg; 1961

[10] Zhang Y, Chen B, Qiu L, et al. State of the art analytical tools improve optimization of unbonded flexible pipes for Deepwater environments. In: Proceedings of the Offshore Technology Conference. Houston, Texas, USA: OTC; 2003

[11] Gay NA, de Arruda MC. A co MParative wet collapse buckling study for the carcass layer of flexible pipes. Journal of Offshore Mechanics and Arctic Engineering. 2012;**134**(3):031701

[12] Tang M, Lu QZ, Yan J, et al. Buckling collapse study for the carcass layer of flexible pipes using a strain energy equivalence method. Ocean Engineering. 2016;**2016**(111):209-217

[13] Tang MG, Yan J, Wang Y, et al. Tensile stiffness analysis on ocean dynamic power umbilical. China Ocean Engineering. 2014;**28**(2):259-270

[14] Roberto RJ, Pesce CP. A consistent analytical model to predict the structural behavior of flexible risers subjected to combined loads. Journal of Offshore Mechanics and Arctic Engineering. 2004;**126**(2):141-146

[15] Ramos R, Martins CA, Pesce CP, et al. Some further studies on the axialtorsional behavior of flexible risers. Journal of Offshore Mechanics and Arctic Engineering. 2014;**136**(1):011701

[16] Østergaard NH, Lyckegaard A, Andreasen JH. A method for prediction of the equilibrium state of a long and slender wire on a frictionless toroid applied for analysis of flexible pipe

structures. Engineering Structures. 2012; **34**:391-399

[17] Witz J, Tan Z. On the flexural structural behaviour of flexible pipes, umbilicals and marine cables. Marine Structures. 1992;**5**(2):229-249

[18] Knapp R. Helical wire stresses in bent cables. Journal of Offshore Mechanics and Arctic Engineering. 1988; **110**:55-61

[19] Kraincanic I, Kebadze E. Slip initiation and progression in helical armouring layers of unbonded flexible pipes and its effect on pipe bending behaviour. The Journal of Strain Analysis for Engineering Design. 2001;**36**(3): 265-275

[20] Sævik S. Theoretical and experimental studies of stresses in flexible pipes. Computers & Structures. 2011;**89**(23–24):2273-2291

[21] Tang M, Li SP, Zhang H, et al. Monitoring the slip of helical wires in a flexible riser under combined tension and bending. Ocean Engineering. 2022; **2022**(256):111512

[22] Zhang Y, Qiu L. Numerical model to simulate tensile wire behavior in unbonded flexible pipe during bending. In: Proceedings of the 26th International Conference on Offshore Mechanics and Arctic Engineering. San Diego, CA, United states: ASME; 2007. pp. 17-29

[23] Ye N, Sævik S, Zhou C. Anti-Wear tape and bending/fatigue of flexible risers. In: Proceedings of the 24th International Offshore and Polar Engineering Conference. Busan, Korea: ASME; 2014

#### **Chapter 6**

## Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration

*Mukund A. Patil and Ravikiran Kadoli*

#### **Abstract**

Knowledge of natural frequency of pipeline conveying fluid has relevance to designer to avoid failure of pipeline due to resonance. The damping characteristics of pipe material can be increased by using smart materials like magnetostrictive namely, TERFENOL-D. The objective of the present chapter is to investigate vibration and instability characteristics of functionally graded Terfenol-D layered fluid conveying pipe utilizing Terfenol-D layer as an actuator. First, the divergence of fluid conveying pipe is investigated without feedback control gain and thermal loading. Subsequently, the eigenvalue diagrams are studied to examine methodically the vibrational characteristics and possible flutter and bifurcation instabilities eventuate in different vibrational modes. Actuation of Terfenol-D layer shows improved stability condition of fluid conveying pipe with variation in feedback control gain and thermal loading. Differential quadrature and differential transform procedures are used to solve equation of motion of the problem derived based on Euler-Bernoulli beam theory. Finally, the effects of important parameters including the feedback control gain, thermal loading, inner radius of pipe and density of fluid on vibration behavior of fluid conveying pipe, are explored and presented in numerical results.

**Keywords:** control gain, isothermal load, flutter, bifurcation instability, differential quadrature and differential transform method

#### **1. Introduction**

Composite fluid-conveying pipes have become a practicable substitute to metallic pipes in several engineering applications such as oil and gas transport lines, hydraulic and pneumatic systems, thermal power plants, heat transfer equipment, petroleum and chemical process industries, underground refueling pipelines in airports, hospitals, medical devices, municipal sewage and drainage, corporation water supply and many more. Divergence and flutter instabilities are illustrious in fluid-conveying pipe due to fluid–structure interaction. One type of instability encountered in cantilever fluid-conveying pipes is called bifurcation, when the imaginary portion of the

complex frequency disappears and the real portion splits into two branches. Fundamental concepts and early development in fluid structure interaction of fluid conveying pipes have been complied and studied by [1] systematically. A few more specialized topics are briefly discussed and well documented in Ref. [2–4]. Remarkable contributions in the area of fluid-conveying pipe vibrations also include the works of Chen [5].

In the meantime, performing a review on literature, it can be seen that a few studies have been carried out in the several field of vibrations such as in-depth nonlinear dynamics [6–10], vibration control [11–18], microtubes or nanotubes in microfluidic devices [19–22], and pipes using functionally graded materials [23–26].

The pseudo excitation method in conjunction with the complex mode superposition method was deduced to solve dynamic equation of Timoshenko pipeline conveying fluid [6]. The post-buckling and closed-form solutions to nonlinear frequency and response [8] of a FG fluid-conveying pipe have been investigated using analytical homotopy analysis method. Natural frequencies and critical flow velocities has been obtained for free vibration problem of pipes conveying fluid with several typical boundary conditions using DTM [11]. Dynamics and pull-in instability of pipes conveying fuid with nonlinear magnetic force have been investigated by [13], for clamped-clamped and clamped-free boundary conditions. The conclusion of investigation is that, location of magnets has a great impact on the static deflection and stability of the pipe. Wavelet based FEM has been used to examine the effect of internal surface damage [14] on free vibration behavior of fluid-conveying pipe. The natural frequencies of pipe conveying fluid has been determined by [15], using Muller's bisection method.

Failure due to filament wound with consideration of production process inconsistencies have been assessed by Rafiee et al. [16]. Vibration and instability response of magnetostrictive sandwich cantilever fluid-conveying micro-pipes is investigated utilizing smart magnetostrictive layers as actuators by [18].

Nonlinear vibration of a carbon nanotube conveying fluid with piezoelectric layer lying on Winkler-Pasternak foundation under the influence of thermal effect [21] and magnetic field [22] have been investigated using Galerkin and multiple scale method. The in-plane free vibration frequency of a zirconia-aluminum functionally graded curved pipe conveying fluid have been explored by the complex mode method [23]. The effect of axial variations of elastic modulus and density on dynamical behavior of an axially functionally graded cantilevered pipe conveying fluid has been analyzed by [24]. Dai et al. [25] studied the thermo-elastic vibration of axially functionally graded pipe conveying fluid considering temperature changes. Heshmati [26] studied the stability and vibration behaviors of functionally graded pipes conveying fluid considering the the effect of eccentricity imperfection induced by improper manufacturing processes. Xu Liang et al. [27] have used differential quadrature method (DQM) and the Laplace transform and its inverse, to analyze the dynamic behavior of a fluid-conveying pipe with different pipe boundary conditions. Huang Yi-min et al. [28] used the separation of variables method and the derived method from Ferrari's method to decouple the the natural frequency and the critical flow velocity equations of fluid-conveying pipe with both ends supported. Planar and spatial curved fluid-conveying pipe [29] have been investigated for their free vibration behavior with Timoshenko beam model and B-spline function used as the shape function in Galerkin method.

There are few investigations in the literature on fluid-conveying pipes containing Terfenol-D layers. Certainly, a study on the mechanical behavior of functionally graded Terfenol-D layered fluid conveying pipe will contribute to the understanding *Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

for future design engineers, hence an attempt on the vibration and stability of functionally graded Terfenol-D layered fluid conveying pipe. Inherent features of the Terfenol-D layer to regulate the vibration instabilities and critical flow velocity of a FGMT pipe are attempted numerically. Terfenol-D is a popular magnetostrictive material exhibiting force output for a corresponding magnetic field input and produces magnetic field for mechanical force as an input. Every term in Terfenol-D has a meaning (see **Figure 1**), for example, Ter means Terbium, Fe signifies chemical symbol for iron, Nol stands for Naval Ordnance Laboratory, and D stands for Dysprosium [30]. Terfenol-D has numerous distinguish characteristics, including a high electromechanical coupling coefficient (0.73), a high magnetostrictive strain (800– 1600 ppm), a fast response, a high energy density, and a large output force. The total stiffness of the pipe is affected by actuation of the Terfenol-D layer due to the creation of tensile forces with a change in feedback control gain and temperature change in the fluid-conveying pipe. The governing equation of motion for FGMT fluid-conveying pipe is derived based on Euler-Bernoulli's theory. Differential quadrature and differential transform approaches are used to obtain the frequency of boundary value problem. Critical velocities of the FGMT pipe are also determined for various boundary conditions, feedback control gain, and thermal loading. Validation of frequencies and critical velocities is accomplished using accessible analytical relations.

#### **2. Functionally graded fluid conveying pipe**

Powder metallurgy is considered as manufacturing process for present functionally graded Terfenol-D layered fluid-conveying pipe. The functioanlly graded pipe is assumed to compose of aluminum (as metallic) and aluminum oxide (as ceramic). In between the graded composition of aluminum and aluminum oxide Terfenol-D layer is included. The material properties, volume fraction and expression for calculation of properties is given in [31]. **Figure 2** shows the layout of FGMT fluid-conveying pipe.

#### **2.1 Derivation of governing equation**

Considering the FGMT fluid-conveying pipe as an Euler-Bernoulli beam, the equation for the motion of the pipe can be derived using Hamilton's principle. The

**Figure 1.** *Schematic for meaning of Terfenol-D.*

**Figure 2.** *Physical model of simply supported FGMT fluid-conveying pipe.*

kinetic energy of the internal fluid is appended to the kinetic energy of the pipe to obtain total kinetic energy of FGMT pipe, and is described by the equation

$$J = J\_p + J\_f \tag{1}$$

Where, *Jp* and *Jf* signify the kinetic energy of the composite FGMT fluidconveying pipe and the kinetic energy of the fluid flowing through the pipe, respectively. The elements of kinetic energy (*J*) as defined in Eq.(1) can be expressed as:

$$J\_p = \frac{1}{2} \int\_0^l m\_p \left(\frac{\partial w}{\partial t}\right)^2 \mathrm{d}x \tag{2}$$

$$J\_f = \frac{1}{2} \int\_0^l m\_f \left( \left( v \frac{\partial w}{\partial \mathbf{x}} + \frac{\partial w}{\partial t} \right)^2 + v^2 \right) \mathbf{dx} \tag{3}$$

Where, *w* symbolize for the displacement in the vertical direction, *v* symbolize for the fluid velocity, The flow of liquid, water, oil, and similar liquid flowing through the pipe are assumed to have a flat velocity profile at every section of the flow (i.e. popularly called as plug flow). *mp* and *mf* respectively denote the mass per unit length of the pipe and the internal fluid. The strain energy *U* of the fluid-conveying pipe can be defined as:

$$U = \frac{1}{2} \int\_0^l E\_p I\_p \left(\frac{\partial^2 w}{\partial \mathbf{x}^2}\right)^2 d\mathbf{x} \tag{4}$$

Where *EpIp* is the flexural rigidity of the FGMT fluid-conveying pipe. Constitutive relation for a magnetostrictive beam type structure [32] could be written as:

$$
\sigma\_{\text{xx}}^T = \mathbf{C}\_{11}\mathbf{e}\_{\text{xx}} - \mathbf{e}\_{\text{31}}\mathbf{H}\_x \tag{5}
$$

where *σ<sup>T</sup> xx*, ϵ*xx* signifies axial stress and strain of the Terfenol-D layer. In addition, C11 and e31 are elastic stiffness coefficient and magnetostrictive constant, respectively. The subscript 31 indicates that, the magnetic field is applied in the 3(z) direction and mechanical response obtained in the 1(x) direction. The strength of the magnetic field *Hz* may now be stated as follows.

$$H\_x = k\_c C(t) \frac{\partial w}{\partial t} \tag{6}$$

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

Where, *kc*, *C t*ð Þ and *<sup>∂</sup><sup>w</sup> <sup>∂</sup><sup>t</sup>* denotes the coil constant, feedback control gain and transverse displacement of fluid conveying pipe with respect to time, respectively. The strain energy of the Terfenol-D layer is given as:

$$U\_T = \int\_0^l \int\_A \sigma\_{\text{xx}}^T \epsilon\_{\text{xx}} \mathbf{d}A d\mathbf{x} \tag{7}$$

Also, the axial moment produced by Terfenol-D layer is,

$$M\_{\rm xx} = \int\_{A} \sigma\_{\rm xx}^{T} z \mathbf{d}A \tag{8}$$

Applying the Hamilton's principle, one can write the functional of FGMT pipe as,

$$\int\_{t\_1}^{t\_2} \delta (\mathbf{J} - \mathbf{U} - \mathbf{U}\_T) \mathbf{d}t + \int\_{t\_1}^{t\_2} \delta \mathbf{W}\_{for \epsilon} \mathbf{d}t = \mathbf{0} \tag{9}$$

Where, *J* is the total kinetic energy of the system; *U* is the deformation energy of the system; *Wforce* denotes the work of the non-conservative force. Therefore, the equation of motion for the free vibration of FGMT composite pipe conveying fluid can be written as:

$$\underbrace{E\_p I\_p \frac{\partial^4 w}{\partial x^4}}\_{\text{Elastic}} + \underbrace{m\_f v^2 \frac{\partial^2 w}{\partial x^2}}\_{\text{Cenrffugal}} + \underbrace{2m\_f v \frac{\partial^2 w}{\partial x \partial t}}\_{\text{Coriolis}} + \underbrace{\varepsilon \frac{\partial^2 w}{\partial x \partial t}}\_{\text{Magnetic-static-Moment}} + \underbrace{\left(m\_p + m\_f\right) \frac{\partial^2 w}{\partial t^2}}\_{\text{Inertia}} = 0 \quad \text{(10)}$$

The governing equation for FGMT fluid-conveying pipe with thermal loading making use of Ref. [33] can be obtained as:

$$E\_p I\_p \frac{\partial^4 w}{\partial \mathbf{x}^4} + \left(m\_f v^2 + A \gamma(\Delta T)\right) \frac{\partial^2 w}{\partial \mathbf{x}^2} + 2m\_f v \frac{\partial^2 w}{\partial \mathbf{x} \partial t} + \varepsilon \frac{\partial^2 w}{\partial \mathbf{x} \partial t} + \left(m\_p + m\_f\right) \frac{\partial^2 w}{\partial t^2} = \mathbf{0} \tag{11}$$

Where,

$$m\_p = \sum\_{j=1}^{n} \pi \rho\_j \left(r\_{j+1}^2 - r\_j^2\right) \tag{12}$$

$$E\_p I\_p = A\_{11} r^3 - D\_{11} r \qquad r = \frac{d\_o + d\_i}{4} \tag{13}$$

$$A\_{11} = \sum\_{j=1}^{n} Q\_{11}(r\_{j+1} - r\_j) \tag{14}$$

$$D\_{11} = \frac{1}{3} \sum\_{j=1}^{n} Q\_{11} \left( r\_{j+1}^3 - r\_j^3 \right) \tag{15}$$

$$\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}\_{31} \boldsymbol{k}\_{\boldsymbol{c}} \mathbf{C}(t) \left( \boldsymbol{r}\_{j+1}^{2} - \boldsymbol{r}\_{j}^{2} \right) \tag{16}$$

$$
\gamma(\Delta T) = E a \Delta T \tag{17}
$$

Where, *α* indicates the thermal expansion coefficient of the fluid conveying pipe material, Δ*T* is the temperature change in the layers, *E* is the Young's modulus of the fluid conveying pipe and *γ*ð Þ Δ*T* symbolize the linear elastic stress–temperature coefficient.

#### **3. Transformation of PDE into a sets of ODEs**

Authors used the differential quadrature method to solve the free vibration equation of FGMT fluid-conveying pipe as given in Eq. (9). Here, the Eq. (9) is transformed into sets of ordinary differential equations. The standard eigenvalue form [34, 35] of the Eq. (9) can be obtained by assuming:

$$
\omega\_0 = W\_0 e^{\Lambda t} \tag{18}
$$

*W*<sup>0</sup> is the mode shape of transverse motion and Λ is the frequency of the FGMT fluid-conveying pipe. Substitute the Eq. (12) in Eq. (9), accordingly Eq. (9) re-reads as follows:

$$\begin{split} E\_p I\_p \frac{\partial^4}{\partial \mathbf{x}^4} \left( W\_0 e^{\mathbf{A}^\mathbf{t}} \right) + m\_f v^2 \frac{\partial^2}{\partial \mathbf{x}^2} \left( W\_0 e^{\mathbf{A}^\mathbf{t}} \right) + 2m\_f v \frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial}{\partial t} W\_0 e^{\mathbf{A}^\mathbf{t}} \right) + \varepsilon \frac{\partial}{\partial \mathbf{x}} \left( \frac{\partial}{\partial t} W\_0 e^{\mathbf{A}^\mathbf{t}} \right) \\ + \left( m\_p + m\_f \right) \frac{\partial^2}{\partial \mathbf{t}^2} \left( W\_0 e^{\mathbf{A}^\mathbf{t}} \right) = \mathbf{0} \end{split} \tag{19}$$

$$\begin{aligned} \left(E\_p I\_p \frac{d^4 \mathcal{W}\_0}{dx^4} e^{\Lambda t}\right) &+ \left(m\_f v^2 \frac{d^2 \mathcal{W}\_0}{dx^2} e^{\Lambda t}\right) + \left(2m\_f v \frac{d \mathcal{W}\_0}{dx} e^{\Lambda t}\right) \Lambda + \left(\epsilon \frac{d \mathcal{W}\_0}{dx} e^{\Lambda t}\right) \Lambda \\ &+ \left(\left(m\_p + m\_f\right) \mathcal{W}\_0 e^{\Lambda t}\right) \Lambda^2 = 0 \end{aligned} \tag{20}$$

$$\left(E\_p I\_p \frac{d^4 \mathcal{W}\_0}{d\mathbf{x}^4}\right) + \left(m\_f v^2 \frac{d^2 \mathcal{W}\_0}{d\mathbf{x}^2}\right) + \left(\left(2m\_f v^2 + \epsilon\right) \frac{d \mathcal{W}\_0}{d\mathbf{x}}\right) \Lambda + \left(\left(m\_p + m\_f\right) \mathcal{W}\_0\right) \Lambda^2 = 0. \tag{21}$$

Now, substitute the analog form of differential quadrature for respective derivative (first, second, third and fourth) such as:

$$\frac{d^4 \mathcal{W}\_0}{d\mathbf{x}^4} = \sum\_{j=1}^N A^{(4)}\_{\vec{\eta}} \mathcal{W}\_j, \qquad \frac{d^2 \mathcal{W}\_0}{d\mathbf{x}^2} = \sum\_{j=1}^N A^{(2)}\_{\vec{\eta}} \mathcal{W}\_j, \qquad \frac{d \mathcal{W}\_0}{d\mathbf{x}} = \sum\_{j=1}^N A^{(1)}\_{\vec{\eta}} \mathcal{W}\_j \tag{22}$$

Now, Eq.15 becomes,

$$\begin{split} \mathbf{E}\_{p}\mathbf{I}\_{p} \sum\_{j=1}^{N} \mathbf{A}\_{ij}^{(\boldsymbol{\alpha})} \mathbf{W}\_{j} + m\_{f}\boldsymbol{\nu}^{2} \sum\_{j=1}^{N} \mathbf{A}\_{ij}^{(2)} \mathbf{W}\_{j} + \left( \left( 2m\_{f}\boldsymbol{\nu}^{2} + \varepsilon \right) \sum\_{j=1}^{N} \mathbf{A}\_{ij}^{(1)} \mathbf{W}\_{j} \right) \boldsymbol{\Lambda} + \left( \left( m\_{p} + m\_{f} \right) \mathbf{W}\_{i} \right) \boldsymbol{\Lambda}^{2} \\ = \mathbf{0} \end{split} \tag{23}$$

Now separate the terms associated with Λ and Λ<sup>2</sup> to prepare the damping and mass matrices, respectively as shown in Eq. 18.

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

$$\left\{-\left[M\right]\Lambda^{2}\right\}\{d\} + \left\{\left[\Gamma\right]\Lambda\right\}\{d\} + \left[K\right]\{d\} = \mathbf{0} \tag{24}$$

Where,

$$\begin{bmatrix} \Gamma \end{bmatrix} = \begin{bmatrix} \mathbf{C}\_{dd} \end{bmatrix} - \begin{bmatrix} \mathbf{C}\_{db} \end{bmatrix} \begin{bmatrix} \mathbf{S}\_{bb} \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{S}\_{bd} \end{bmatrix} \tag{25}$$

$$\begin{bmatrix} \mathbf{K} \end{bmatrix} = \begin{bmatrix} \mathbf{S}\_{dd} \end{bmatrix} - \begin{bmatrix} \mathbf{S}\_{db} \end{bmatrix} \begin{bmatrix} \mathbf{S}\_{bb} \end{bmatrix}^{-1} \begin{bmatrix} \mathbf{S}\_{bd} \end{bmatrix} \tag{26}$$

Where, *Cdd* and *Cdb* are the damping sub matrices which includes the domaindomain and domain-boundary elements of damping. Similarly, *Sbb*, *Sbd*, *Sdb* and *Sdd* are the stiffness sub matrices which includes the boundary-boundary, boundarydomain, domain-boundary and domain-domain elements, respectively. The standard form of eigenvalue can be obtained from Eq. (18) as:

$$\left\{ \begin{bmatrix} \mathbf{0} & I \\ \Gamma & K \end{bmatrix} - \begin{bmatrix} I & \mathbf{0} \\ \mathbf{0} & M \end{bmatrix} \Lambda \right\} \begin{Bmatrix} d \\ \Lambda d \end{Bmatrix} = \mathbf{0} \tag{27}$$

Where *I*, [*K*], [Γ] and [*M*] denote the identity, structural stiffness, damping and mass matrix, respectively. One can obtain the two sets of eigenvalues. The eigenvalue obtained can be written as Λ ¼ �*α* � *iωd*.

#### **4. Application of differential transform method to FGMT fluid-conveying pipe**

Differential transform technique (DTM) may be used to solve integral equations, ordinary partial differential equations, and differential equation systems. Using this approach, a polynomial solution to differential equations may be derived analytically. For large orders, the Taylor series approach is computationally time-consuming. This method is appropriate for linear and nonlinear ODEs since it does not need linearization, discretization, or perturbation. It is also possible to significantly reduce the amount of computing labour required while still precisely delivering the series solution and rapidly converging. The DTM has several disadvantages, though. Using the DTM, a truncated series solution may be obtained. This truncated solution does not display the actual behavior of the problem, but in the vast majority of situations it offers a good approximation of the actual solution in a relatively limited area. Solutions are expressed as convergent series with components that may be readily computed using the differential transform technique. The linear equation of motion for free vibration of FGMT fluid-conveying pipe is given by,

$$\left(E\_p I\_p \frac{d^4 \mathcal{W}\_0}{dx^4}\right) + \left(m\_f v^2 \frac{d^2 \mathcal{W}\_0}{dx^2}\right) + \left(\left(2m\_f v^2 + \varepsilon\right) \frac{d \mathcal{W}\_0}{dx}\right) \Lambda + \left(\left(m\_p + m\_f\right) \mathcal{W}\_0\right) \Lambda^2 = 0\tag{28}$$

The differential transformation form of Eq. (22) can be written as

$$\begin{aligned} E\_p I\_P((i+1)(i+2)(i+3)(i+4)\mathcal{W}(i+4)) + m\_f v^2 ((i+1)(i+2)\mathcal{W}(i+2)) \\ + \left(2m\_f v^2 + \varepsilon\right)((i+1)\mathcal{W}(i+1)) + \left(m\_p + m\_f\right)\mathcal{W}(i) = 0 \end{aligned} \tag{29}$$


**Table 1.**

*Transformed form of boundary condition for differential transform method.*

Rearranging Eq. (23), one will get a simple recurrence relation as:

$$\mathcal{W}(i+4) = -\frac{\left(m\_f v^2 (i+1)(i+2)\mathcal{W}(i+2) + \left(2m\_f v^2 + \varepsilon\right)(i+1)\mathcal{W}(i+1) + \left(m\_p + m\_f\right)\mathcal{W}(i)\right)}{E\_p I\_p (i+1)(i+2)(i+3)(i+4)} \tag{30}$$

Similarly, analogous form of original boundary conditions for the differential transformation can be done using **Table 1**, where *x* ¼ 0 and *x* ¼ 1 represents the boundary points. It can be seen that *W i*ð Þ, (*i* ¼ 4,5, … ,*N*) is a linear function of *W*ð Þ2 and *W*ð Þ3 . Thus, *W*ð Þ2 and *W*ð Þ3 are considered as unknown parameters and taken as *W*ð Þ¼ 2 *b*1, *W*ð Þ¼ 3 *b*<sup>2</sup> for clamped-clamped boundary conditions. With Eq. (23), *W i*ð Þ can be calculated via an iterative procedure. Substituting *W i*ð Þ into boundary conditions at other end of FGMT pipe, the two equations (Substituting all *W i*ð Þ terms into boundary condition expressions) can be written as matrix form,

$$
\begin{bmatrix} R\_{11} & R\_{12} \\ R\_{21} & R\_{22} \end{bmatrix} \begin{bmatrix} b\_1 \\ b\_2 \end{bmatrix} = \mathbf{0} \tag{31}
$$

Where *Rij* are associated with the eigenvalues *ω*, *b*<sup>1</sup> and *b*<sup>2</sup> are the constants and other parameters of the FGMT pipe system, corresponding to *N*. To obtain a nontrivial solution of Eq. (25), it is required that the determinant of the coefficient matrix vanishes, namely

$$
\begin{vmatrix}
\ R\_{11} & \ R\_{12} \\
\ R\_{21} & \ R\_{22}
\end{vmatrix} = \mathbf{0} \tag{32}
$$

Therefore, the eigenvalues *ω* can be computed numerically from Eq. (26). Generally, *ω* is a complex number.

#### **5. Results and discussion**

In the following section, the numerical results are proposed to investigate the free vibration behavior of FGMT fluid-conveying pipe subjected to control gain and thermal loading. Since there is no published research on the subject of free vibration of FGMT fluid-conveying pipes in the open literature, a differential quadrature and

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

differential transform approach is used to conduct a condensed analysis of the current study. The imaginary component (ℑm) of the complex frequency [Ω ¼ ℜeð Þ� Ω ℑmð Þ Ω ] denotes the energy stored in either mass or strain energy in the fluid conveying pipe. The accumulated strain energy is linked to the failure behavior of the fluid conveying pipe. Furthermore, the real element (ℜe) of the complex frequency represents damping and the energy that will be transformed to heat or other energy by friction or other molecular actions.

#### **5.1 Validation of present study**

The current MATLAB code for the differential quadrature and transform technique is validated using Ref. [28], as shown in **Table 2**. The validation for FGMT fluid conveying pipe is also given by the author in Ref. [36]. Furthermore, the solution obtained using the differential quadrature approach corresponds well with the solution acquired using the differential transform method.

It has been identified that, the differential transform method requires the 58 number of terms to get the converged solution whereas 19 grid points used to obtain the convergence solutions shown. The natural frequencies of pipes conveying fluid depend on the fluid velocity *v:* The physical parameters of FGMT fluid-conveying pipe are calculated as: *mp* ¼ 1*:*0670 kg � m, *mf* ¼ 0*:*23562 kg � m, *EpIp* <sup>¼</sup> <sup>5</sup>*:*1620 N � <sup>m</sup>2, *<sup>L</sup>* <sup>¼</sup> 1 m. In order to calculate these physical parameters, authors have used Eq. 2.1. MATLAB software is used to create a package that performed the foregoing computations. The correctness of the results are shown by the comparison of the results of differential transform method in **Table 3** under different boundary conditions for *v* ¼ 0*:*5 m*=*s. The number of grid points was modified from 7 to 19 to reach the converged solution. From the **Table 3**, it can be concluded that the imaginary component of the damped frequency calculated using DQM and DTM coincides rather well.

One of the key concerns for fluid conveyance pipes to be of significant importance is stability. The natural frequencies decrease with higher flow rates for pipelines with supported ends. The system destabilizes by diverging (buckling) when the natural frequencies fall to zero, and the resulting flow velocity is known as the critical flow velocity. In the case of *v* 6¼ 0, **Figures 3**–**10** represent the natural frequencies of fluidconveying FGMT pipe with different boundary conditions. The first three natural


#### **Table 2.**

*Validation of simply-supported natural frequencies (rad/sec) of fluid conveying pipe (Parameters used:* EI <sup>¼</sup> 100 Nm<sup>2</sup>*,* <sup>m</sup>*<sup>f</sup>* <sup>¼</sup> 2 kg*=*m*,* <sup>m</sup>*<sup>p</sup>* <sup>¼</sup> 2 kg*=*<sup>m</sup> *and L = 1 m).*


#### **Table 3.**

*Convergence of imaginary component of damped frequency for different boundary conditions when v* ¼ 5 m*=*s*.*

frequencies of the C-F fluid-conveying FGMT pipe with 0 ≤*v*≤50 are depicted in **Figures 3** and **4**. The critical velocity of the pipe is *v* ¼ 42 m*=*s, and the third mode appears flutter instability. The findings of the differential quadrature method were utilized to plot the results presented in the **Figures 3**–**10**.

The first four natural frequencies of the simply supported fluid-conveying FGMT pipe with 0 ≤*v*≤50 are plotted in **Figures 5** and **6**. The first mode appears divergence instability when the critical velocity of the FGMT pipe is *v* ¼ 15 m*=*s, and Paidoussis coupled mode flutter instability appears when the critical velocity is *v* ¼ 31. Real component (ℜe) of the complex frequency is almost zero during the first mode divergence instability. By increasing the flow velocity 30 m/s, the imaginary part of combination of first and second modes becomes zero, while the real part is non-zero, and the non-zero frequency and damping of first and second mode at the same values are coupled, then the system will be unstable again. This sort of instability, caused by

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

**Figure 3.** *Effect of fluid velocity v on imaginary component of clamped-free damped frequency.*

**Figure 4.**

*Effect of fluid velocity v on real compoent of clamped-free damped frequency.*

the interaction of two modes, is known as flutter instability, and its amplitude develops exponentially as a function of time.

**Figures 7** and **8** shows the first four natural frequencies of the C-C fluid-conveying pipe with 0≤ *v*≤ 50. The critical velocity of the FGMT pipe is *v* ¼ 30 m*=*s and 43, and corresponds to divergence instability in the first mode and couple-mode flutter instability. Bifurcation critical flow velocity is the term used to describe the flow velocity at which the bifurcation instability occurs. It should be noted that the system enters an

**Figure 5.** *Effect of fluid velocity v on imaginary component of simply supported damped frequency.*

**Figure 6.** *Effect of fluid velocity v on real component of simply supported damped frequency.*

over-damping mode, which prevents the FGMT pipe from vibrating, when the working fluid velocity surpasses its critical value.

**Figures 9** and **10** presents the first four natural frequency of the S-C fluidconveying FGMT pipe with 0 ≤*v*≤50. It is obvious that the first mode appears divergence instability when fluid velocity *v* ¼ 22 m*=*s, and coupled-mode flutter instability appears when fluid velocity reaches to *v* ¼ 37 m*=*s. The specific critical velocities under different boundary conditions are listed in **Table 4**. The critical velocity for the *Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

**Figure 7.** *Effect of fluid velocity v on imaginary component of clamped-clamped damped frequency.*

**Figure 8.** *Effect of fluid velocity v on real component of clamped-clamped damped frequency.*

simply supported-simply supported and clamped-clamped boundary conditions are validated using Navier solution given by [37].

The relationships between the imaginary component of frequency of the FGMT pipe and the fluid density for different boundary conditions are plotted in **Figure 11**. Because the inertial and Coriolis forces were stronger with increasing fluid density, it was more simpler for the pipe to lose its stability. This led to a lower natural frequency. The changes of imaginary component of frequency with inner radius of the

**Figure 9.** *Effect of fluid velocity v on imaginary component of simply supported-clamped damped frequency.*

**Figure 10.** *Effect of fluid velocity v on real component of simply supported-clamped damped frequency.*

FGMT pipe for different boundary conditions are shown in **Figure 12**. For very small values of the inner radius, an increase in the inner radius has a considerable impact on frequency; nevertheless, when the inner radius value is near to the outer radius, the frequency increases. In the boundary conditions clamped-clamped, simply supportedsimply supported, and simply supported-clamped, the imaginary component of frequency drops as the feedback control gain rises. Imaginary component of the eigenvalue for a clamped-free frequency becomes zero for 3000 feedback control gain, *r* ¼ 0*:*005 m and *v* ¼ 5 m*=*s shown in **Figure 13**.

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*


#### **Table 4.**

*Critical velocities for FGMT pipe with different boundary conditions.*

**Figure 11.** *Variation in fundamental natural frequency of FGMT pipe with changes in fluid density.*

It is worth pointing out that the important aspect of present research work is maneuvering the use of Terfenol-D layers attached on the top FGMT fluid-conveying pipe to control the critical flow velocity and also improve the stability region. When Terfenol-D layer actuates tensile forces are generated in FGMT fluid-conveying pipe which affects the stiffness of fluid-conveying pipe. In order to evaluate this objective, **Figure 14** shows the real part (ℜe) of clamped-free first mode frequency with flow velocity for 0, 1000 and 1500 feedback control gain. It is observed that, 30, 28 and 9 m/s are the critical flutter velocity for 0, 1000 and 1500 feedback control gain, respectively. Therefore, one can make fluid-conveying pipe more stable by varying the feedback control gain. **Figure 15** shows the variation of analytical nonlinear

**Figure 12.** *Variation of fundamental frequency with changes in inner radius of FGMT pipe for different boundary conditions.*

*Variation of imaginary component of the frequency with changes in feedback control gain at r* ¼ 0*:*005 *m and v* ¼ 5 m*=*s*.*

frequency of FGMT fluid conveying pipe calculated based on relations published by [38] for simply supported boundary condition. It has been shown that when fluid velocity rises, the nonlinear frequency falls.

**Figure 16** depicts the coupled effect of feedback control gain along with thermal loading. It is inferred that, there is decreasing effect of critical flow velocity as thermal loading increases. The reduction in overall stiffness of pipe is the reason for instability of FGMT pipe at lower flow velocity with thermal loading. Therefore, critical flow

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

**Figure 14.** *Variation of clamped-free fundamental frequency with changes in control gain and fluid velocity.*

**Figure 15.** *Variation of nonlinear simply supported frequency with changes in fluid velocity.*

velocity condition under thermal loading can be amplified through imposing higher feedback control gain. The control gain varies between 0 and 2000 as the temperature of the fluid conveying pipe changes. It is inferred that, with a zero control gain and 1<sup>∘</sup> C and 0<sup>∘</sup> C, the instability state of the fluid conveying pipe reduces from a fluid velocity from 27 to 25.2 m/s. Additionally, with a control gain of 1000, the fluid conveying pipe's unstable condition decreases from 29 to 24 m/s. Similar to this, with the control gain of 2000, the fluid conveying pipe's unstable condition decreases from 30 to 22 m/s.

**Figure 16.** *Variation of S-S fundamental frequency with changes in control gain, thermal loading and fluid velocity.*

#### **6. Concluding remarks**

In this chapter, the differential quadrature and differential transform method is applied to analyze the free vibration of FGMT pipes conveying fluid with different boundary conditions. Boundary value problem of FGMT fluid-conveying pipe is solved straightforwardly using DQM and DTM. Close agreement is established for critical velocity and frequencies results generated by DQM, DTM with those of Navier and Galerkin solution. Eigenvalue diagrams are detailed enough to shows the illustration about the effects of feedback control gain, density of fluid, inner radius of pipe and thermal loading on the vibrational and instability characteristics. To attenuate the amplitude of vibration or displacement, inherent damping property of the material cannot be sufficient. To dampen out large amplitude vibration during resonance, special techniques have been explored, like using sandwich pipes namely, viscoelastic layer placed between two layers of the parent pipe material. This approach is called passive damping. Viscoelastic materials like, natural rubber, and synthetic rubber like nitrile butadine rubber and styrene butadine rubber, silicone rubber can be proposed. Sophisticated technique is the active vibration. This method involves use of materials like, piezoelectric, magnetostrictive, magnetorehology, electrostricitve and shape memory alloys. Magnetostrictive material presented in this chapter works on the ability of the material to respond mechanically to the presence of magnetic field. The magnetic field is produced using a coil with passage of time dependent current. A magnetostrictive material responds with a force, hence magnetostrictive actuator. The force produced should be used to counteract the forces due to vibration. Thus, damping is introduced. The idea of incorporting Terfenol-D layer facilitates the best control of the fluid conveying FGMT pipe to avoid the bifurcation and flutter instabilities and achieve more adaptive and efficient system. Additionally increasing or decreasing effect of feedback control gain and thermal loading on critical flow velocity and instabilities have been addressed.

*Terfenol-D Layer in a Functionally Graded Pipe Transporting Fluid for Free Vibration DOI: http://dx.doi.org/10.5772/intechopen.108227*

#### **Conflict of interest**

The authors declare no conflict of interest.

### **Abbreviations**


#### **Author details**

Mukund A. Patil1† and Ravikiran Kadoli<sup>2</sup> \*†

1 Department of Mechanical Engineering, G.H. Raisoni Institute of Engineering and Business Management, Jalgaon, Maharashtra, India

2 Department of Mechanical Engineering, National Institute of Technology Karnataka, Surathkal, Mangalore, Karnataka, India

\*Address all correspondence to: rkkadoli@nitk.edu.in

† These authors contributed equally.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 7**

## Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT© Platform for Online Simulations

*Mariano Alarcón, Manuel Seco-Nicolás, Juan Pedro Luna-Abad and Alfonso P. Ramallo-González*

#### **Abstract**

This chapter studies the fluid flow within pipes subjected to thermal asymmetrical boundary conditions. The phenomenon at hand takes place in many real-world industrial situations, such as solar thermal devices, aerial pipelines. A steady-state analysis of laminar forced-convection heat transfer for an incompressible Newtonian fluid is studied. The fluid is considered to flow through a straight round pipe provided with straight fins. For the case studied, axial heat conduction in the fluid has been considered and the effects of the forced convection have been considered to be dominant. A known uniform temperature field is applied at the upper external surface of the assembly. The 3D assembly has been created combining cylindrical and Cartesian coordinates. The governing differential equation system is solved numerically through suitable discretization in a set of different finite volume elements. The results are shown through the thermal profiles in respect of longitudinal and radial-azimuthal coordinates and the problem characteristic length. To facilitate the resolution of this phenomenon, an open computing platform called HEATT©, based on this model, has been developed, and it is also shown here. The platform is now being built and is expected to be freely available at the end of year 2022.

**Keywords:** straight round pipe, asymmetrical boundary conditions, 3D simulation, conjugate-extended Graetz problem, HEATT platform

#### **1. Introduction**

Laminar forced convection mechanisms take place in many industrial installations where the processes involve the use of fluids in pipes. Numerous applications exist of such flows, including flat solar thermal collectors [1–4], solar trough devices [5], nanofluids [4, 6], mini and micro channels [7] and a wide variety of heat exchangers [8]. Currently, the interest on the studies of thermal behaviour in oil or gas pipelines is growing due to the international context, as well as the study of hydrogen through liquefied petroleum gases (LPG) pipelines that are already installed [9–12].

From the point of view of the analysis of fluids' heat exchange on pipelines, it is technically relevant to consider that flow and heat processes occur simultaneously, i.e. they are coupled, which increases the complexity of the process. Understanding flow behaviour under these conditions is key to pipeline design and device efficiency.

The fundamentals of the thermal mechanisms involved on fluid flow have been extensively studied [13–15]. The complexity in their analysis and the geometry of practical applications has made common its resolution using experimental and numerical studies, examples of them are [16, 17].

One of the more significant models of fluid behaviour within a pipe is the one known as Graetz Problem (GP), stated by Graetz in 1882 [18], where in a given point of the pipe a fluid flowing in laminar forced flow is subject to a sudden change in its external boundary conditions, either temperature or heat flow. Graetz proposed a bidimensional approach, neither considering pipe nor axial fluid conduction, which was analytically solved. More than a 100 years after his work, Graetz's problem continues to receive the attention of researchers. In the present century, some researchers have extended this known problem to take into account both the physical presence of the pipe and the axial fluid conduction (*conjugate extended Graetz problem*), finding their results by analytical [19] or numerical procedures [20, 21], also some have included transient process [22]. Other researchers have studied the flow with periodically varying inlet temperature in pipes of different shapes [23], pipes subjected to a sudden [24] or periodical change [25] in external heat or ambient temperature, etc. Also significant is the formulation of the concept of characteristic length of the process, carried out by discriminated dimensional analysis by Seco-Nicolás et al. [26]. All these studies, based on radial symmetry, assume the 2D hypothesis. However, in certain cases such relatively simplified models do not provide results as accurate as those obtained through tri-dimensional numerical models.

This chapter faces the problem of fluid flow within pipes subjected to thermal asymmetrical boundary conditions which take place in many real industrial situations such as those related to solar thermal devices, aerial pipelines subjected to external temperatures, etc. Other examples and attempts to solve this problem can be seen on: [27–31]. Despite the asymmetry of the problem, much simpler bi-dimensional models are currently used for pipe design purposes, ignoring the important consequences of the asymmetry that it exists.

This work presents a steady-state analysis of the laminar forced-convection heat transfer process for a liquid flowing through a straight round pipe when radially asymmetrical external conditions are applied to the tube's external surface (a known uniform temperature to the upper surface and adiabatic condition to the lower) and taking into consideration axial heat conduction in the fluid.

A governing differential equation system is coupled to the Laplace equation for the solid and is solved numerically through suitable discretisation in a set of different finite volume elements, considering the axial heat conduction in the fluid, but neglecting the heat generation by viscous dissipation, the buoyancy effects or the variation of the thermal properties of the materials.

Many techniques have been developed to simulate convective flows using finite element techniques [32], finite difference solvers [33], method of lines [34] and many others. In the present case to evaluate the proposed model Network Simulation Method (NSM) [35–37], a powerful numerical methodology has been chosen. This

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

method, based on a finite-difference scheme, can virtually solve any ordinary and partial differential equation. The method starts from suitable discretisation of the problem [37–39]; after which an equivalent electric circuit of the process is built, including boundary conditions, the so-called network model. Finally, appropriate software is used to solve the circuit, from whose results the thermal response is obtained. NSM has been chosen in this case because it yields results that have been seen to be as accurate as those obtained with CFD software using significantly fewer computational resources. Open-source software NGSpice [40], originally created for electric circuit analysis, has been used in our case to solving this problem; also proprietary software such as Pspice© [41] can be used for this purpose. The model has been validated by experimental results obtained in an experimental solar thermal plant [42].

Related to this model, an open computing platform called HEATT©, based on this model and solved using the NTM, is now being built, which will allow online calculation of flow within pipes subject to complex thermal conditions. The platform is expected to be freely available to the public before the end of 2022.

#### **2. Physical and mathematical model**

The set-up studied here is composed by a flat plate welded to a round duct as shown in **Figure 1**. The flat plate acts as a fin. Its addition to the model is due to the fact that many of the pipes subjected to thermal stress incorporate fins to improve heat exchange.

Consider an isotropic fluid at uniform temperature, T1, and a certain velocity profile inside a tube whose whole external surface is also at temperature T1. At *z* = 0 the fluid enters a long duct, whose upper half surface is maintained from this point at a constant temperature T2 (T2 > T1), while its lower surface is thermally isolated (**Figure 1**); these conditions are maintained along the whole duct length (z > 0). The plate is maintained at a constant temperature T2 on its upper surface, and it is insulated on its lower surface and extremes in the same conditions of the pipe. Heat is transferred throughout the plate to the duct by conduction. Therefore, the plate acts as a fin.

Consequently, at *z* = 0, the fluid suffers a sudden change in its boundary conditions and heat is transferred through the tube by conduction, and then by convection to the fluid, in which conduction takes place in both radial and axial directions.

The fluid working conditions are similar to those of the bi-dimensional problem stated by Graetz [18], except for the radial asymmetry of the temperature boundary condition caused by the radially asymmetric thermal condition in the external surface

**Figure 1.** *Simulation model outline.*

of the pipe and the presence of the fin. These boundary conditions make it necessary to formulate a 3D model.

The geometry of the problem requires incorporating both cylindrical and Cartesian coordinate systems. Regarding cylindrical coordinates for the round duct, direction *z* is located parallel to the axis of the pipe, the *r* vector is normal to it, and the third dimension is described by angle *φ*. Regarding Cartesian coordinate system for the fin, direction *z* is also located parallel to the axis of the pipe, and directions *x* and *y* are orthogonal to it, as can be seen in **Figure 2**.

As regards the material parts, the assembly consists of a round pipe, with *ks* being the constant conductivity and *ep* the constant thickness. The pipe is assumed straight and non-deformable. Heat generation by viscous dissipation or other sources is not considered. As regards the time domain, the model is considered to be at stationary state.

It is assumed that the fluid flows in laminar-forced convection in stationary regime, while its thermal properties (density, ρf, specific heat, cf, and thermal conductivity, kf) remain unchanged in their values at temperature T1.

#### **Figure 2.**

*3D Duct problem geometry. (a) Section planes outline. (b) 2D longitudinal section β including velocity profile. (c) 2D transversal section α.*

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

For the velocity profile, a polynomic profile of grade 10 has been chosen. This kind of profiles can be found in certain devices, such as inclined tubes [43]. The velocity will be considered invariable along the whole pipe (**Figure 2b**).

The viscous buoyancy-driven heat transfer is considered negligible due to dominant effects of the studied forced convection in a laminar and incompressible Newtonian fluid flow, which is the case of water and other fluids in certain conditions [44].

Under these conditions, the tri-dimensional equations that govern the coupled system are (on cylindrical coordinates) [45, 46]:

Equation of the solid (pipe) region. Cylindrical coordinates:

$$\frac{1}{r} \left[ \frac{\partial}{\partial r} \left( r k\_s \frac{\partial T\_s}{\partial r} \right) \right] + \frac{1}{r^2} \frac{\partial}{\partial \rho} \left( k\_s \frac{\partial T\_s}{\partial \rho} \right) + \frac{\partial}{\partial \mathbf{z}} \left( k\_s \frac{\partial T\_s}{\partial \mathbf{z}} \right) = \left( \rho \mathbf{c} \right)\_s \frac{\partial T\_s}{\partial t} \tag{1}$$

Equation of the fluid (inside pipe) region. Cylindrical coordinates:

$$\frac{1}{r} \left[ \frac{\partial}{\partial r} \left( r k\_f \frac{\partial T\_f}{\partial r} \right) \right] + \frac{1}{r^2} \frac{\partial}{\partial \rho} \left( k\_f \frac{\partial T\_f}{\partial \rho} \right) + \frac{\partial}{\partial z} \left( k\_f \frac{\partial T\_f}{\partial z} \right) = (\rho u\_x c)\_f \frac{\partial T\_f}{\partial z} + (\rho c)\_f \frac{\partial T\_f}{\partial t} \tag{2}$$

Equation of the fin. Cartesian coordinates:

$$\frac{\partial}{\partial \mathbf{x}} \left( k\_a \frac{\partial T\_a}{\partial \mathbf{x}} \right) + \frac{\partial}{\partial \mathbf{y}} \left( k\_a \frac{\partial T\_a}{\partial \mathbf{y}} \right) + \frac{\partial}{\partial \mathbf{z}} \left( k\_a \frac{\partial T\_a}{\partial \mathbf{z}} \right) = \left( \rho c \right)\_a \frac{\partial T\_a}{\partial \mathbf{t}} \tag{3}$$

This model can be considered as an extension of the classical conjugate-extended Graetz problem and has been widely used in the literature [20–22].

Time-dependent terms will be omitted in the current model due to fact that the study focuses on the stationary phenomenon, and conduction coefficients are considered equal in all directions as the fluid is considered isotropic and with invariable thermal properties. The rest of boundary conditions that define the problem are detailed in **Table 1**.


**Table 1.**

*Analytical and dimensional boundary conditions.*

#### **3. Numerical model**

Numerical solutions of coupled governing Eqs. (1)–(3) under boundary conditions (4)–(11) have been reached using Network Simulation Method (NSM) and the circuit solver NGSpice.

#### **3.1 The network model**

In NSM terminology, the equivalent, or analogous, electrical circuit of the considered process is called the *network model*. The construction of the network model begins with the discretisation of the mentioned system of differential equations in a finite volume elements mesh which transforms the governing partial differential equations of balance Eqs. (1)-(3) into the set of algebraic Eqs. (12)–(14).

Discretisation of the equation of the solid (duct) region. Cylindrical coordinates:

$$\begin{split} &\frac{1}{r} \cdot \left[ \frac{T\_{j+\Delta r/2} - T\_{j-\Delta r/2}}{\Delta r} \right] + \frac{\mathbf{1}}{\Delta r} \cdot \left[ \frac{T\_{j+\Delta r/2} - T\_{j-\Delta r/2}}{\Delta r} \right] + \frac{\mathbf{1}}{r^2 \varrho} \left[ \left( T\_{\varrho+\Delta \varrho/2} - T\_{\varrho-\Delta \varrho/2} \right) / \Delta \varrho \right] \\ &+ \mathbf{1} / \Delta \mathbf{z} \cdot \left[ \left( T\_{i+\Delta \mathbf{z}/2} - T\_{i-\Delta \mathbf{z}/2} \right) / \Delta \mathbf{z} \right] \\ &= \mathbf{0} \end{split}$$

Discretisation of the equation of the fluid (duct) region. Cylindrical coordinates:

(12)

$$\begin{aligned} &\mathbf{1}/r \cdot \left[ \left( T\_{j+\Delta r/2} - T\_{j-\Delta r/2} \right) / \Delta r \right] + \mathbf{1}/\Delta r \cdot \left[ \left( T\_{j+\Delta r/2} - T\_{j-\Delta r/2} \right) / \Delta r \right] \\ &+ \mathbf{1}/r^2 \mathbf{1}/\rho \cdot \left[ \left( T\_{\varphi+\Delta \rho/2} - T\_{\varphi-\Delta \rho/2} \right) / \Delta \rho \right] + \mathbf{1}/\Delta \mathbf{z} \cdot \left[ \left( T\_{i+\Delta \mathbf{z}/2} - T\_{i-\Delta \mathbf{z}/2} \right) / \Delta \mathbf{z} \right] \\ &- \left( (\rho \cdot \mathbf{c} \cdot \mathbf{u}\_{\mathbf{z}}) / \mathbf{k} \right)\_{\mathbf{f}} \left[ \left( T\_{i+\Delta \mathbf{z}/2} - T\_{i-\Delta \mathbf{z}/2} \right) / \Delta \mathbf{z} \right] \\ &= \mathbf{0} \end{aligned} \tag{13}$$

Discretisation of the equation of the fin. Cartesian coordinates:

$$\begin{aligned} &\mathbf{1}/\Delta \mathbf{x} \cdot \left[ \left( T\_{i+\Delta x/2} - T\_{i-\Delta x/2} \right) / \Delta \mathbf{x} \right] + \mathbf{1}/\Delta \mathbf{y} \cdot \left[ \left( T\_{i+\Delta y/2} - T\_{i-\Delta y/2} \right) / \Delta \mathbf{y} \right] \\ &+ \mathbf{1}/\Delta \mathbf{z} \cdot \left[ \left( T\_{i+\Delta x/2} - T\_{i-\Delta x/2} \right) / \Delta \mathbf{z} \right] \\ &= \mathbf{0} \end{aligned} \tag{14}$$

Based on the governing equations and boundary conditions, an electrical circuit for each equation has been created (**Figure 3**). Each term in these equations becomes an

**Figure 3.** *Electrical circuit scheme of the fluid basic cell.*

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

electric component. Consequently, each finite volume element (elementary cell) is composed of a set of electrical elements according to the thermo-electric analogy corresponding to the different terms of the previous equations. In this temperature is equivalent to voltage, and heat fluxes (∂T/∂x, ∂T/∂y, ∂T/∂z, ∂T/∂r and ∂T/∂*φ*) are equivalent to electric currents. More details of the fundamentals of the NSM can be found in Ref. [42].

From the point of view of the NSM, the terms contained in each of the above equations can be considered as currents according to the currents Kirchhoff's law (because its summation over a node needs to be zero). As an example, the term 1*=r* � *Tj*þΔ*r=*<sup>2</sup> � *Tj*�Δ*r=*<sup>2</sup> *=*Δ*r* introduces heat conduction in cylindrical co-ordinates both on the fluid and on the pipe equation. The step-by-step description of the elementary cell building (**Figure 3**) has been detailed in the Appendix of this chapter.

Eq. (13) describes the three-dimensional flow behaviour which is dominated by the axial velocity *uz*, assumed as a velocity function of order 10, profile equation of which is done by Eq. (15):

$$\begin{aligned} u\_x &= 1.06 \cdot 10^{-2} + \left(\frac{r}{R}\right) \cdot 4.59 \cdot 10^{-1} + 3.75 \cdot \left(\frac{r}{R}\right)^2 + 19.4 \cdot \left(\frac{r}{R}\right)^3 + 15.6 \cdot \left(\frac{r}{R}\right)^4 \\ &- 92.1 \cdot \left(\frac{r}{R}\right)^5 + 40.6 \cdot \left(\frac{r}{R}\right)^6 + 843 \cdot \left(\frac{r}{R}\right)^7 + 577 \cdot \left(\frac{r}{R}\right)^8 - \left(\frac{r}{R}\right)^9 \cdot 3.26 \cdot 10^3 \\ &- \left(\frac{r}{R}\right)^{10} \cdot 4.93 \cdot 10^3 \end{aligned}$$
 
$$\text{(15)}$$

In total, the entire system has been discretised using a three-dimensional mesh of identical cells in a manner that the square-sectioned straight fin is divided into 200 cells in z direction, 10 cells in the *x* direction and 1 cell in the *y* direction (**Figure 4**). The pipe is also symmetrically divided into 200 cells in the *z* direction, 4 cells in *φ* direction and 7 cells in the *r* direction where 5 belongs to the fluid and 2 to the pipe thickness (**Figure 4**). This mesh has been considered to be sufficient for the purpose of this work.

**Figure 5** shows the five main planes of the pipeline (named similarly to the cardinal points) that will be referred to in Section 4 to identify the different points of the pipeline's cross section. Although we have pointed out the asymmetry of the physical model, it should be noted that the pipe-fin assembly is radially asymmetric, but it presents symmetry with respect to the N-S plane.

Once the elementary circuit has been built, the boundary conditions must be implemented. In this case, some of the most relevant are voltage sources than fit the constant external temperature T2 in the upper side of the assembly and infinite (very

**Figure 4.** *Numerical model discretisation mesh.*

**Figure 5.** *Longitudinal pipe section planes screened in simulations.*

high) resistance in its lower side. Finally, elementary cells and devices corresponding to the boundary conditions are assembled building an electric circuit (network model), which can be solved using an appropriate software, such as NGSpice.

A complete description of the circuit construction as well as computing details of the simulation can be found in the Appendix and reference [42]. In order to know the numerical value of the different electrical components forming out network (resistors, voltage and current sources, etc.), it is necessary to build the circuit and to obtain realistic values of parameters and boundary conditions corresponding to the case study. In our case, a solar thermal collector. The parameters are listed on **Table 2**. As stated above, solar thermal devices are one of the practical applications of this model, with obvious similarities in their geometry.

In this case, when Reynolds number is high (41989) and Rayleigh number remains low (3.92<sup>10</sup><sup>3</sup> ), the fluid flows can be considered in a laminar forced-convection regime, and in consequence, the buoyancy effects are negligible [44]. The minimum pipe length in which the thermal phenomena are developed came from the use of the Nusselt number approximations [47].


About the length of the pipe, it has been used the concept of characteristic length, l\*, defined as the length needed for the fluid to fully develop the thermal process [26], in

**Table 2.** *Simulated model magnitudes.*

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

this case to achieve the external temperature T2. Therefore, the simulated pipe should be long enough (L > l\*), to ensure that the thermal phenomena due to the sudden change in temperature at *z* = 0 has ended. This *hidden parameter* must be previously determined. In this case, the preliminary simulation with a 2D model guided the determination of the characteristic length (afterwards checked). So, we found that the corresponding characteristic length in this case would be 2 m if the whole exterior pipe surface was kept isothermally surrounded. This is not exactly the case, because in the 3D model presented in this chapter, only upper half surface is at constant temperature and the rest is insulated; consequently, 4 m length duct has been simulated.

#### **3.2 Model validation**

The 3D numerical model built following these rules and using the parameters mentioned in the previous section was validated by comparing its results to literature and experimental data [42].

The case modelled in the present chapter is a complete innovation; therefore, no references for comparison can be found in the literature. Furthermore, no references of the 3D configuration of the conjugate extended Graetz problem could be found in the literature previously to the work of reference [42] of the same authors of this chapter. In comparison to the finned tube model of the present chapter, the above cited reference modelled a bare tube (without fins) with a parabolic velocity profile, both being models analogous in the rest of characteristics.

**Figure 6** shows the comparison of the fluid temperature field at 0.5 m of the entrance in both models. Main differences are a consequence of the different fluid velocity and the presence of fins. **Figure 6a** depicts the isothermal lines of the finned tube, where a slight tendency towards horizontality can be appreciated in the surroundings of the tube-fin junction compared with that of the **Figure 6b** of the bare tube. Nevertheless, the bulk fluid is little affected by the presence of fins. This was to be expected as in the finned tube, the temperature of the fins is the same as that of the upper half-pipe in both models. Consequently, it can be said that the fluid temperature field of the finned tube model agrees with that of the bare tube, and it can be considered that the validation procedure carried out in Ref. [42] applies to the present model.

**Figure 6.** *Comparison of the cross-section temperature maps at 0.5 m in the finned tube (a) and in a bare tube (b).*

In respect to the code, the results of the 3D model "acting as" a 2D model (radially symmetric boundary conditions) were compared with the 2D published results [21, 22], obtaining relative errors along the whole tube length below 2% and mean of 0.98%, the error standard deviation [42] being 0.56%. Analysis of the typified residuals of the 2D and 3D simulation results, error variance and a regression analysis were carried out. The coefficient of skewness (nearly 0) and kurtosis showed that the error data set was normally distributed. Finally, the relative error of the 3D model acting as a 2D model, and the 2D model was found to be 0.98 7.74E-2% at a 95% confidence level.

On the other hand, the external temperatures (tube) of the cross section of the 3D simulation at different lengths were compared with the experimental measures yielded in a solar thermal experimental rig [42]. An analogous error study was conducted between numerical and experimental temperature data, concluding that the 3D numerical results were sufficiently close to those measured experimentally, relative errors being of 3.40 0.601%, at 95% of confidence. More details of experimental rig, measured data and error study can be found in [42].

Consequently, those results confirmed the accuracy of the bare tube 3D model, and this conclusion can be extended to the finned tube model of this chapter, substantially equivalent to that of reference [42], especially as regards the fluid, and which can be considered as a continuation of the bare tube model.

#### **4. Results and discussion**

Typical fluid flow in laminar regime shows a parabolic profile. Nevertheless, there are situations in which velocity has a more complex profile [43]. Because of that a polynomial function of grade 10 (**Figure 7**) has been used in this work. Eq. (15) is the formula corresponding to this curve, and it is the value for *uz* in Eqs. (2) and (13). Eq. (15) has been yielded from experimental results, and it corresponds to the case of the flow in tilted solar collectors [43].

#### **4.1 Axial temperature profile and characteristic length of the process**

**Figure 8** shows the profile of the fluid temperature along the pipe in the different planes of the pipe (see **Figure 4**). In this case, the temperature rises faster than in the case of the parabolic profile of the velocity, reaching the external temperature T2 even

**Figure 7.** *Axial velocity profile following [43].*

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

#### **Figure 8.**

*Temperature profiles for uz given by Eq. (15).*

in the lower sections of the tube, i.e. flows with this speed profile mix much faster than with the laminar parabolic profile, yielding a higher convective coefficient.

If characteristic length, l\*, is defined as the one where the temperature in the middle of the pipe (red line) reaches 90% of the total temperature jump, in this case 65.45 + 0.9�(100–65.45) = 96.55°C, from **Figure 7** gives l\* = 3.6 m. Beyond this point, the temperature of the fluid does not increase substantially, which indicates that the rest of the pipe is not efficient for heat transfer purposes, so the length could be reduced.

#### **4.2 Fluid temperature maps in a cross section**

**Figure 9** shows the temperature field of the cross section of the fluid (perpendicular to the axis) and of some vertical and horizontal sections of it at 13 cm from the inlet. As expected, the temperature curves show symmetry about the vertical axis.

**Figure 9a** represents the thermal map of the cross section. It can be observed that the fluid initially (*z* = 0) at T1 (65.45°C) acquires at 13 cm from the entrance temperatures close to T2 (100°C) in the layers near the top of the pipe (dark brown layers between the N (φ = π/2) and NE (φ = π/4) planes and nearby areas). At this point of the pipe, the upper half of the fluid shows almost parallel isotherms following the circular curve of the pipe, whose values decrease from plane N, (at the top) to the horizontal plane (plane E, φ = 0) and even up to almost plane SE (φ = –π/4). Sections A-A' and D-D<sup>0</sup> (**Figure 8b** and **e**, respectively) clearly show this situation, presenting a minimum at the N-S axis as a consequence of the shape of the isotherms. Compared with the case of velocity parabolic profile [42], it can be said that in the case of polynomial profile, the temperature increases much faster than in the former, reaching the external temperature T2 in the lower sections of the tube relatively close to the entrance, i.e. flows with this speed profile mix much faster than with the laminar parabolic profile, yielding a higher convective coefficient. A consequence of this is the shorter characteristic length shown by the polynomial velocity process compared with that of parabolic velocity profile.

In contrast, near the south (S) and southeast (SE) planes, the isothermal curves take a U-shape, except near the N-S axis, where a loop is formed at 68°C approximately in the centre of the bottom half of the tube. This is due to the fact that the fluid is being heated from the top half of the tube (whose temperature has been imposed at a uniform value of T2), while the lower half of the tube, which is externally insulated,

**Figure 9.**

*Fluid temperature map of the cross section (a) and temperature profiles for different sections (b–e) at 13 cm from the entrance of the pipe.*

is warmed by heat conduction from the solid pipe due to the fact that kf > > ks. As a result, in the S plane (φ = � π/2), the fluid near the tube is hotter than in the middle of the tube. This fact can also be detected in Section B-B<sup>0</sup> , which corresponds to the N-S plane (**Figure 9c**), where the minimum (66.98°C) is not found in the lower part of the section; you can see that the temperature profile is roughly flat along the bottom quarter of the section. This suggests that the thermal process is more dependent on pipe-wall conduction effects than on the velocity profile, even with such a complex and irregular profile, due to the low conductivity of the fluid and the laminar flux. It is relevant to note that the classic Graetz Problem does not consider the pipe through which the fluid flows, thus making it impossible to detect this temperature distortion. *Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

Another issue is the influence of the fin on fluid temperatures. The fin-tube junction takes place in plane E, which corresponds to section A-A' (**Figure 9b**). A small distortion of the isotherms can be observed at this point compared with the case of a bare duct [42]. In this case, this distortion is not very important because the external temperature T2 is uniform, both in the pipe and in the fin, and a very thin fin with little heat conduction capacity has been modelled. Otherwise, it could be very relevant, as in the case of the heat boundary condition.

Although not studied in this work, consequences from the non-uniform temperature field, which affects most of the thermo-fluid properties (density, viscosity, etc.) among others, can be drawn. This behaviour could impact on the operation of the pipeline and the related equipment.

#### **5. The HEATT© platform**

The methodology explained above has been used to develop a 3D model to simulate a laminar flow under conditions of forced convection subject to asymmetric boundary conditions, such as those found in the tube grid of a low-temperature solar thermal collector, in which the upper half of each tube receives energy from the sun, while the bottom half of the tube remains embedded in a layer of insulation [42]. The simulated results could be compared with the previous bibliography and with the experimental results obtained from the Solar Laboratory of the University of Murcia under real operating conditions, obtaining great differences between the results obtained using 2D models versus 3D models. The model roughly coincides with the one presented in this chapter (except for the fact of considering tube without fins).

This model required the creation of a three-dimensional mesh with nearly 17,000 cells, in which an electrical equivalent circuit of approximately 140,000 elements was implemented. The model includes resistors, voltage and current sources and capacitors.

Once the problem has been solved for a sufficiently variety of real cases, the tool has been created for solving the problem using a web as a service, providing free service to any professional or researcher anywhere in the world without the need to acquire expensive software or to instal any application that becomes obsolete with the evolution of the Operating Systems.

The resulting web application has been named HEATT ©, acronym for Pipeline Thermal Analysis and Assessment Tool (in Spanish: Herramienta de Evaluación y Análisis Térmico de Tuberías).

This simple and friendly platform is currently being launched in its version 1.0 as a Proof of Concept, to be released "as a service" for general use and for researchers and professionals from all over the world to send their opinions and improvement proposals to gradually make it growing up.

#### **6. Conclusions**

A numerical physical–mathematical model is presented for a laminar forcedconvection fluid flow within a finned round duct subjected to constant temperature on its upper side and insulated on its lower side. The governing heat transfer and fluid flow 3D partial differential equations, combining cylindrical and Cartesian coordinates, are solved for steady-state conditions. The numerical model is run using the accurate contrasted Network Simulation Method, a low-workload computational method.

In the present chapter, the Graetz Problem is extended to incorporate axial fluid conduction, 3D coordinates, wall thickness with attached longitudinal flat-fins, radially asymmetrical boundary conditions and highly non-linear velocity profiles. Realistic conditions corresponding to the flow in flat plate solar collectors have been used in the simulations. A high non-linearity velocity profile has been evaluated, confirming that solid conductivity and thickness effects are not negligible for the studied thermal phenomena.

Temperature evolution across the fluid is analysed in detail. Different temperature values were found for different angles within every plane, due to the radial asymmetry of the geometry. Temperature fluid maps were obtained at different distances from the entrance of the pipe. Isotherms show parallel-like shapes on the top half of the tube, which become distorted in the lower half, where some loops appear due to the conduction effects of the studied pipe wall thickness, illustrating the nonuniformity of the temperatures within the fluid.

In addition, 3D simulation reveals that, in cases of asymmetry, the thermal phenomena require much more length to completely develop the flow than the length yielded by the 2D radially symmetric model. This is relevant because the 3D simulation reveals that the pipe needs up to six times more length than that predicted by 2D model.

When the characteristic length of the problem was considered (a virtual dimension equivalent to the distance at which the thermal phenomena are fully developed), the 2D approach was found to be no longer valid. That occurs in most of the studied real cases, when the pipe is subjected to asymmetrical boundary conditions.

All those changes enhance the solution of the Graetz problem and bring it nearer to real pipe conditions. From this, a basis for future works, including heat boundary conditions, different union thermal resistances or a variety of complex velocity functions, among many other possibilities, may be mentioned in order to encourage future optimisation studies. Meanwhile, the findings of the present study have applications in solar energy collectors, thermal heat dissipators, oillines and heat exchangers among many other facilities.

In addition, an open computing platform called HEATT©, based on this model, is now being built. The platform is expected to be freely available to the public before the end of 2022.

#### **Acknowledgements**

This project has been possible thanks to the funds obtained in public concurrence in the 2021 call for Proofs of Concept from the Seneca Foundation – Agency for Science and Technology of the Region of Murcia (Spain). Ph.D, Ramallo-González would like to thanks the European Commission for their funding of project PHOENIX grant number 893079.

#### **Conflict of interest**

The authors declare no conflict of interest.

#### **Nomenclature**

```
a absorber width (m)
```

```
c specific heat (Jkg1
                      K1
                          )
```
*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*


#### **Superscripts**

<sup>0</sup> dimensionless variable

#### **Subscripts**


#### **Greek characters**


#### **A. Appendix. Procedure of building the numerical model**

The procedure that needs to be followed to reproduce the simulation results of this chapter is described below.

#### **A.1 Discretisation of the spatial domain**

In this case, we can account three different regions or sub-domains, as stated in the physical–mathematical model, every one described by its corresponding equation of thermal behaviour: solid (pipe) region, Eq. (1), fluid (inside pipe) region, Eq. (2) and the fin, Eq. (3). Each region is divided into an adequate number of bi-dimensional cells (the third dimension is the axial coordinate, which is common for all cells). In this problem, the number of discrete dimensions are:


The third dimension, i.e. z-dimension, has been divided into 200 parts (**Figure 4**) and is common for all the bidimensional cells previously accounted for.

In summary, an overall number of 38 divisions (8 for the pipe, 20 for the fluid and 10 for the fin) of the cross section have been done, which gives a total of 38x200 = 7600 cells for the whole domain. In this case, the results have shown that the discretisation is adequate.

#### **A.2 Building the numerical model**

This step is divided into different minor steps, each one has to be applied to the corresponding region. Due to the fact that the most complex region is that of the fluid, and in order not to be repetitive, the process of construction of its elementary cell. The starting point is the set of discretised Eqs. (12)–(14) of the problem.

#### **A.3 Building the elementary cell**

The Network Simulation Method is based in the well-known thermal-electrical analogy [35]. In accordance to this, temperature is equivalent to voltage, and ΔT/Δz, ΔT/Δr and ΔT/(rΔφ), which are related with heat fluxes, are equivalent to electric currents. Each of the Eqs. (12)-(14) represents the energy balance in the corresponding cell. These cells, at steady state must have a zero summation of currents. Consequently, each term of the differential equation must be converted into a current of the elementary circuit, which as a whole fulfils Kirchhoff's Laws.

In order to arrange the equations in a more convenient way, Eqs. (12) and (13) are multiplied by ΔzΔrΔφ; Eq. (14) is multiplied by ΔxΔyΔz. Note that the elementary cell (**Figure 3**) is in fact divided in two parts, being the node at the centre of the cell.

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

**Tables 3**, **4** and **5** contain the formulae of the different devices that integrate the elementary cells of solid pipe, fluid and fin, respectively. As the most complex, the fluid region elementary circuit is explained. In this case, five terms are found:

a. J1 is a non-lineal term which comes from the increasing cell area with the radius. It must be implemented by a voltage-dependent current source, Gr,f (**Figure 3**), value of which is expressed by Eq. (15), as it can be seen in **Table 4**.

$$J\_1 = \frac{\Delta x \cdot \Delta \rho}{r} \left[ \left( T\_{j + \frac{\Delta \rho}{2}} - T\_{j - \frac{\Delta \rho}{2}} \right) \right] \tag{16}$$

b. The second term of the Eq. (13) is a current which corresponds to the heat flux in r-direction, Tj + <sup>Δ</sup>r/2 and Tj-Δr/2 being the temperatures at both sides of the elementary j-cell in this direction, which are made equivalent to the external voltages of the elementary circuit, Eq. (16).

$$J\_2 = \Delta \mathbf{z} \cdot \Delta \rho \left[ \frac{\left( T\_{j + \frac{\mathbf{a} \cdot \mathbf{e}}{2}} - T\_{j - \frac{\mathbf{a} \cdot \mathbf{e}}{2}} \right)}{\left( \Delta r / 2 \right)} \right] \tag{17}$$

Consequently, the thermal resistance is represented by two electric resistances of value *Rr*,*<sup>f</sup>* <sup>¼</sup> <sup>Δ</sup>*<sup>r</sup>* <sup>2</sup>Δ*z*�Δ*<sup>φ</sup>*, as can be seen in **Table 4**. Similarly, the third and fourth terms become J3 and J4 currents corresponding to heat flux in the φ and z directions. The value of the thermal resistances Rφ,f and Rz,f yielded from these terms can be found in **Table 4**.

c. Finally, the fifth term represents the axial heat conduction due to the velocity field and has been implemented by a voltage-dependent current source, Gz,f, whose value is given by Eq. (17):

$$J\_5 = \Delta \mathbf{z} \cdot \Delta \rho ((\rho \cdot \mathbf{c} \cdot \mathbf{u}\_x)/k)\_f \cdot \left(T\_{i + \Delta x/2} - T\_{i - \Delta x/2}\right) \tag{18}$$

The location and connections of the different electric devices in the fluid elementary cell can be seen in **Figure 3**; plus or minus symbols in the different resistors distinguish both sides of the cell in respect to the central node.

This procedure must be followed for the different regions of the assembly, i.e. the fluid, the pipe and the fin. The pipe equation, Eq. (12) has only four terms, all of them coincide with those of the fluid except for the velocity term, and the fin, Eq. (14), has


**Table 3.**

*Formulae and analogous electrical devices yielded from solid Eq. (12).*


#### **Table 4.**

*Formulae and analogous electrical devices yielded from fluid Eq. (13).*


**Table 5.**

*Formulae and analogous electrical devices yielded from fin Eq. (14).*

only three terms, one for each Cartesian direction. **Tables 3**–**5** summarise the devices used in the elementary circuits.

#### **A.4 Building the network model or whole equivalent circuit**

Once the elementary circuits of the different parts of the system have been built, the equivalent circuit of each region is assembled by adding as many cells as needed according to the discretisation carried out; then the circuits of the regions are joined according to the topology.

Finally, the problem boundary conditions, Eqs. (4)–(11), must be implemented. In this case, the most relevant are voltage sources for Eqs. (4), (7) and (9), very large resistance resistors are used to emulate the adiabatic conditions, Eqs. (4), (5), (8), (10) and (11) or just electrical continuity, Eqs. (6) and (7bis).

#### **A.5 Writing the code**

One of the advantages of NSM is the use of well-known and reliable software for the analysis of electronic and electrical circuits (NGSpice, PSpice or others), which are also very easy to program. Naturally, for the programming of the elementary (and complete) circuit, we direct the reader to the corresponding websites [40] and literature [41], where the documentation can be found.

*Forced Laminar Flow in Pipes Subjected to Asymmetric External Conditions: The HEATT©… DOI: http://dx.doi.org/10.5772/intechopen.107215*

#### **B. Simulating the electric circuits**

The whole electric circuit (network model) is solved using appropriate software, such as NGSpice, PSpice or others.

### **B.1 Obtaining results**

The results obtained must be interpreted taking into account that in the thermoelectric analogy, voltages and currents are analogous to temperature and heat flow, respectively. Both temperature profile along the pipe-fin and temperature maps of the cross section are plotted using suitable software from the voltages at the appropriate points along the pipeline or the selected cross section using the software.

Formulae and analogous electrical devices yielded from Eqs. (12)–(14).

### **Author details**

Mariano Alarcón<sup>1</sup> \*, Manuel Seco-Nicolás<sup>1</sup> , Juan Pedro Luna-Abad<sup>2</sup> and Alfonso P. Ramallo-González<sup>3</sup>

1 Electromagnetism and Electronics Department, International Campus of Excellence in the European context (CEIR) Campus Mare Nostrum, University of Murcia, Murcia, Spain

2 Thermal and Fluid Engineering Department, International Campus of Excellence in the European context (CEIR) Campus Mare Nostrum, Technical University of Cartagena, Cartagena, Spain

3 Information and Communication Engineering Department, International Campus of Excellence in the European context (CEIR) Campus Mare Nostrum, University of Murcia, Spain

\*Address all correspondence to: mariano@um.es

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 8**

## Review on the Stability of the Nanofluids

*Sumit Kumar Singh*

#### **Abstract**

Both mono and hybrid nanofluids, the engineered colloidal mixture made of the base fluid and nanoparticles, have shown many interesting properties and become a high potential next-generation heat transfer fluid in various engineering applications. The present review focuses on improving the stability of the nanofluids. For this, the present review briefly summarizes the impact of nanofluid preparation on the stability of various nanofluids and described in the following classification; (a) Nanofluid constituent, (b) Nanomaterial synthesis, and (c) Nanofluid synthesis techniques which are well-grouped and thoroughly discussed. Physical mechanisms for heat transfer enhancement using nanofluids are explored as well. Most of the studies reveal that there are significant improvements in the stability of the nanofluids. Hence, there is an excellent opportunity to use stabled nanofluids in various engineering applications. Finally, some useful recommendations are also provided.

**Keywords:** nanofluid, stability, nanomaterial synthesis, surfactant

#### **1. Introduction**

Nanofluids are engineered by dispersing nanoparticles, having average sizes below 100 nm, in conventional heat transfer fluids. Proper and stable dispersion of even a negligible fraction of particles in nanofluids can offer significant enhancement in the heat transfer properties. Various types of nanoparticles like metals, metal oxides, alloys, allotropes of carbon, ceramics, phase change materials, and metal carbides are being used for preparing nanofluids. In addition to nanofluids, hybrid nanofluids have also gained attention recently due to significant improvement in heat transfer characteristics and stability may be caused by the synergistic effect of hybridization. Heat exchangers that use tubes or pipes often have a circular, rectangular, or elliptical cross-section and are easier to design. Tubular heat exchangers are fairly prevalent in pipeline engineering applications. These heat exchangers might be built to handle fluids under high pressure or to handle pressure differentials between cold and hot fluids. Double-pipe and shell-tube heat exchangers are additional categories that apply to these heat exchangers. Modifying the fluids' characteristics can also increase the heat exchange rate of a heat exchanger. Due to the fact that stable nanofluids have significantly improved heat transfer characteristics, particularly in terms of thermal conductivity, slip mechanisms, and the nanofin effect, they may be employed in tubular heat exchangers to increase energy efficiency. For preparing mono or

hybrid nanofluids, the two-step method is generally used where firstly different nanoparticles or nanocomposites are prepared. Then they are mixed in the base fluid through magnetic or mechanical stirring. After that, the solution is sonicated and then characterized using different techniques to assure the proper (homogeneous) mixing and stability of the hybrid nanofluids. Both mono and hybrid nanofluids are thus prepared to provide improved heat transfer characteristics due to an increase in thermal conductivity, Brownian motion, proper dispersion, agglomeration, solid/ liquid interface layering, thermophoresis, the improved thermal network between the solid nanoparticle and fluid molecules, nanofin and nanoporous effects at the heat transfer surface. The reason behind this improvement can be summarized as: (i) More heat transfer surface between nanoparticles and fluid, (ii) Collision between the nanoparticles, (iii) Increment in the thermal conductivity due to the interactive effect of different nanoparticles, and (iv) Proper dispersion of the nanoparticles in the base fluid, creating micro turbulences. Therefore, in hybrid nanofluids, both nanoparticles compromise their properties and provide better thermo-physical, chemical, and rheological properties within the low cost that makes it preferable over nanofluids for different applications. Stability is the main key factor for the performance of nanofluids in various engineering applications. All the thermo-physical properties of nanofluid are dependent on its stability. The unstability of nanofluid can inhibit its performance in several applications such as heat exchangers, chemical industry applications, enhanced oil recovery etc. The unstability of nanofluid is caused due to the propensity of nanoparticles to form a cluster in the fluid. The nanofluids may be broadly categorized into three groups based on the nanoparticle composition, namely: (i) mono-nanofluids (made from one type of nanoparticles), (ii) hybrid nanofluids containing different nanoparticles, and (iii) hybrid nanofluids consisting one solid covered by a layer of another solid (composite nanoparticles).

The current review emphasizes the impact of nanofluid preparation on the stability of various nanofluids and is described in the following classification; (a) Nanofluid constituent, (b) Nanomaterial synthesis, and (c) Nanofluid synthesis techniques.

#### **2. Literature review**

The available literature on the preparation, characterization, and stability of mono/ hybrid nanofluids are discussed in three sections. In all section, it summarizes the impact of nanofluid preparation on the stability of various nanofluids and described in the following classification; (a) Nanofluid constituent, (b) Nanomaterial synthesis, and (c) Nanofluid synthesis techniques. Exclusive reviews on the heat transfer, pressure drop characteristics, and energy performance of both double-tube and shell-tube heat exchangers using nanofluids are presented in the third and fourth sections.

#### **2.1 Impact of nanofluid preparation**

There are two main approaches to synthesize nanofluids: the single-step method and the two-step method. In the one-step method, nanofluid is prepared directly by dispersing nanoparticles in the base fluid without the requirement of numerous steps such as particle drying, storage, etc. Using this method, the stability of nanofluid exhibits most superior compared to the two-step method. But this technique is not beneficial for large scale because of its high production cost. Therefore, the two-step

#### *Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

method is the more effective and generally common method of nanofluid preparation. The foremost disadvantage of this process is the control of particle agglomeration tendency. The common application of wide ultrasonication and stirring is the most frequently used method to control agglomeration. Several forces such as Van der Waal attractive force, gravitational force, buoyancy force, and electrostatic repulsive force are acted which lead to destabilization and form sediments. The Van der Waal attractive force and gravitational force work against the stability of any colloidal suspension. Stability is the main key factor for the performance of nanofluids in various engineering applications. All the thermo-physical properties of nanofluid are dependent on its stability. The unstability of nanofluid can inhibit its performance in several applications such as heat exchangers, chemical industry applications, enhanced oil recovery etc. The unstability of nanofluid is caused due to the propensity of nanoparticles to form a cluster in the fluid. For considering a stable nanofluid, agglomeration propensity has to be removed. Some stability evaluation methods are used in literature i.e., sedimentation and centrifugation method, zeta potential measurement, spectral absorbance and transmittance measurement, and dynamic light scattering. Numerous efforts have been made to prepare long-time stable and homogenous nanofluids using various techniques. The current review emphasizes the impact of nanofluid preparation on the stability of various nanofluids and is described in the following classification; (a) Nanofluid constituent, (b) Nanomaterial synthesis, and (c) Nanofluid synthesis techniques.

#### *2.1.1 Nanofluid constituent*

#### *2.1.1.1 Nanomaterial type*

There are several types of nanofluids: metallic nanofluids (Al, Ag, Cu, Fe, Au), metal oxide nanofluids (Al2O3, CuO, Fe3O4, SiO2, TiO2, ZnO, etc.), and non-metallic nanofluids (SiC, TiC, graphite, diamond, SWCNT/MWCNT, graphene, etc.). Several studies on the impact of the nanofluid constituents on its stability are shown in **Table 1**. Xu et al. [4] prepared hybrid nanofluids with nanoparticles of different masses added with a small amount of SDBS and PEG into DW and observed that 25% Al2O3 + 75% TiO2 hybrid nanofluid shows good suspension stability. The zeta potential value for the 25% Al2O3 + 75% TiO2 hybrid nanofluid is found 42.6 mV indicating high stability. Zeta potential means electrostatic repulsion force between nanoparticles and base fluid. High repulsion force indicates high stability of nanofluid, whereby 30mV is generally considered as a benchmark for a stable nanofluid and excellent nanofluid stability may exceed 60 mV. Some studies investigated the impact of functionalizing the nanoparticles surface which reduces aggregation and improves dispersion. Said et al. [5] studied the stability of Carbon nanofiber (CNF), Functionalized Carbon nanofiber (F-CNF), Reduced graphene oxide (rGO), and F-CNF/rGO nanofluids. The results indicated that hybrid (FCNF/rGO) nanofluid shows higher stability than as compared to CNF, F-CNF, and rGO nanofluids. Also, the sample of CNF almost completely sedimented on 2nd day as shown in the **Figure 1**. It is due to the low charge density on the surface of the CNF nanoparticle which leads to the tendency of agglomeration. Said et al. [15] used acid treatment of CNF to examine the stability. The zeta potential of 0.02 vol. % F-CNF nanofluid was −42.9 and − 41.8 mV after 2 and 90 days which indicates that the stability was improved while the zeta potential of CNF was −16.3 and − 15.5 mV, indicating a relatively unstable dispersion. One way to achieve long-term stability is to adjust




#### **Table 1.**

*Synopsis of the investigations about the impact of the nanofluid constituents on its stability.*

the nanofluid pH, away from the isoelectric point (IEP). Thus, IEP differs from one sample to another. These values were prepared in acidic and alkaline ranges using HCl and NaOH solutions and adjusted by pH meter. Kazemi et al. [6] used two different nanoparticles (GnP, SiO2) with the same base fluid (water) as well as different pH values (3,6,9, and 12) to study the stability of the nanofluids. The results found that SiO2/ Water nanofluids have good stability at all pH values, especially for samples with pH >3 and GnP/water achieve better stability at higher pH values. Akhgar and Toghraie [9] examined the stability of water-based MWCNT and TiO2 nanofluid at different pH (3, 6,9, and 12). The results observed that the nanofluid containing water/ TiO2 with pH = 9 had more stability than the rest of the samples. On the other hand, MWCNT particles are not dispersed in water and are not stable in any pH without any surfactant. Kazemi et al. [21] compared the stability of three types of nanofluids,

*Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

#### **Figure 1.**

*Photographs of vials showing the stability of nanofluids for: (a) 1 st day, (b) 2nd day, (c) 30 days, (d) 45 days, (e) 60 days, and (f)180 days [5].*

G/Water, SiO2/Water, and G-SiO2/Water and found that SiO2/Water nanofluid shows excellent stability at all pH values while G/Water sustainability is poor in lower pH value. Due to better stability in higher pH values, the CMC surfactant can be used to increase pH by creating a negative charge surface for graphene nanoparticles and developing functional groups. Siddiqui et al. [8] performed a stability study with metal (Cu), oxide (Al2O3), and meta-oxide Cu- Al2O3) nanofluid containing the same base fluid (DI water). Al2O3 nanofluid exhibits better stability between 0 and 6 h followed by good stability with little particle settling between 6 and 240 h. Cu nanofluid shows poor dispersion stability after 1 hour of preparation. In case of hybrid nanofluid, nanofluid with optimum mixing ratio exhibits relatively better stability. Muthoka et al. [14] used PCM-DI water as the base fluid and two different nanoparticles (MgO and MWCNT) to examine the stability. They observed that the stability of MgO and 24 wt% base fluid without surfactant showed poor stability after only 24 h while the functionalized MWCNT nanofluid showed no separation after 24 h. Also, it was concluded that the stability of the nanofluid at low temperatures is increased by the use of surfactant. Alawi et al. [16] synthesized PEG-GnP, PEG-TGr, Al2O3, and SiO2 water-based nanofluids. They observed the dispersion stabilities of carbon-based nanofluids and metallic oxides nanofluids for 30days, and the results showed the higher dispersibility of the PEG-GnP, PEG-TGr nanofluids in an aqueous media with very low sedimentation. Akbari and Saidi [17] observed TiO2/DW nanofluid shows good stability as compared to GnP/DW nanofluid. Since graphene is an inherently hydrophobic material and the stability of graphene/water nanofluid is not favorable without any surfactants. Boroomandpour et al. [22] studied the stability of ternary hybrid nanofluids containing MWCNT-TiO2-ZnO/DW-EG (80:20) as well as binary and mono nanofluids. They found that all nanofluids have good stability up to 48 h after fabrication and the addition of CTAB surfactant lead to better stability.

From the literature reviews on the preparation of nanofluids with different particles, it is found that the stability of water mono/hybrid nanofluid is strongly dependent on the particle shape and size. It is found that the propensity of aggregation

is increased with the reduction in particle size and isoelectric point (pH value) decreases with the decrease in particle size. Therefore, the agglomeration process moves toward lesser pH value. The cylindrical-shaped particles sediment faster than spherical and platelet-shaped particles. High aspect ratio nanoparticles are more susceptible to agglomeration.

#### *2.1.1.2 Surfactant type*

Addition of different surfactants such as: Anionic (Sodium Dodecyl Sulfate (SDS), Sodium Dodecyl Benzene Sulfonate (SDBS)), Cationic (Cetyltrimethylammonum Bromide (CTAB)), Non-ionic (Span 80, Tween 20) and polymer (Polyvinyl Pyrrolidone (PVP), Poly Vinyl Alcohol (PVA), Gum Arabic (GA)) during nanofluid preparation is an additional way of controlling particle aggregation. A negatively charged suspension may be obtained by using anionic surfactants (SDC, SDS, and SDBS) while a cationic surfactant (CTAB) may contribute a positive charge. The augmentation in stability will achieve by the coating of surfactant on nanoparticles, which leads to a dominating electrostatic repulsion over the van der Waals force and thus prevent nanoparticles from agglomerating. Also, the stability of the nanofluid can be improved by decreasing the sedimentation velocity of the nanoparticles. According to Stokes law, the sedimentation velocity can be reduced by using nanoparticles with smaller diameters. However, when the nanoparticles diameter decreases, the surface energy will be increased which leads increase possibility of agglomeration. The best way to suppress the agglomeration without disturbing the sedimentation velocity is the usage of surfactants. While surfactant addition is an active way to improve the stability of the nanofluids but surfactants may lead to cause some problems. Surfactants may contaminate the heat transfer media. Surfactants may produce foams while heating and cooling are regular processes in heat exchange systems. Additionally, surfactant molecules attributed to the surfaces of nanoparticles might increase the thermal resistance between the nanoparticles and the base fluid, which may hinder the augmentation of the thermal conductivity.

Xian et al. [1] used three different surfactants, i.e., SDS, CTAB, and SDBS to stabilize the COOH-TiO2 hybrid nanofluid. They observed that COOH-TiO2 hybrid nanofluid with CTAB surfactant exhibited the best surfactant to stabilize this hybrid nanofluid. The visual inspection of sedimentation of nanofluids with different surfactants after 40 days is shown in **Figure 2**. Almanassra et al. [2] compared the effect of different types of surfactants on the stability of CNT/water nanofluids. They investigated with three types of surfactants namely, GA, PVP, and SDS and found that the nanofluids with GA as well as PVP surfactants were more stable for more than 6 months. Gum Arabic can be a promising surfactant for stabilizing the CNT in water-based nanofluids. Cacua et al. [3] found Al2O3 nanofluid with SDBS at 1 CMC and CTAB at 0.5 CMC were the most stable and unstable nanofluids, respectively. Anionic SDBS provides high repulsive forces between nanoparticles. Ouikhalfan et al. [7] prepared surface-modified TiO2 nanofluid with two different surfactants (SDS and CTAB). The quick sedimentation was found in non-treated TiO2 nanofluid after 24 hours of the preparation as shown in **Figure 3**. TiO2 nanofluid with CTAB showed better stability up to several days while the nanofluid with SDS surfactant shows less but overall better dispersion compared to nanofluid with non-treated TiO2. Choi et al. [10] studied the effect of various surfactants as well as the temperature on the stability of water-based MWCNT nanofluid. They prepared nanofluid with four different surfactants, i.e., SDBS, CTAB, SDS, and TX-100 between the

#### *Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

#### **Figure 2.**

*Visual inspection of sedimentation of nanofluids with different surfactants and ultra-sonication time after 40 days. [1].*

#### **Figure 3.**

*Sediment photograph capturing of the nanofluid with (1) nontreated TiO2, (2) CTAB-treated TiO2 nanofluid, and (3) SDS-treated TiO2 nanofluid [7].*

temperatures 10°C–80°C. It was observed that for short-term time period (3 h), nanofluids prepared with SDBS, CTAB, and TX-100 show better stability while for long-term time period (1 month), the SDBS and TX-100 nanofluids have the highest suspension stability. On the account of temperature, TX-100, CTAB, and SDS are not suitable surfactants for nanofluids operating from 10 to 85°C. Das et al. [11] found TiO2 (Anatase) with SDS and CTAB show excellent stabilization (stable for exceeding 12 h and 24 h) as compared with nanofluid with SDS and acetic acid surfactant. Kuang et al. [13] prepared nanofluids by dispersing three nanoparticles (i.e., SiOx, Al2O3, and TiO2) and five different chemical agents i.e., oleic acid (OA), polyacrylic acid (PAA), a cationic, an anionic, and a nonionic surfactant) in base brine solutions. Nanofluids made with the anionic surfactant made the surface slightly more water wet. The results revealed that SiOx nanofluids exhibit stability in all cases while Al2O3 + PAA and Al2O3 + cationic surfactant show the most stability. In case of TiO2 nanofluid, TiO2 + PAA show the most stability among all surfactants. Cacua et al. [18] used UV–vis spectroscopy to examine the stability of Al2O3 with two different surfactants (SDBS and CTAB). The outcome reveals that the nanofluid with SDBS at 1 CMC and that with CTAB at 0.5 CMC achieved the lowest and highest absorbance variation, respectively. Low absorbance variation over time indicates high nanofluid stability. Etedali et al. [19] investigated the stability of SiO2 nanofluids with different surfactants, i.e., SLS, CTAB, and Ps 20 through the Zeta potentials test. The results of the Zeta-potential test found that the maximum surface charge for the nanofluids with SLS, CTAB, and Ps20 surfactants were − 87.4, 74.2, and − 97.9, respectively, confirming the stability conditions.

#### *2.1.1.3 Base fluid type*

Gao et al. [12] prepared GNP nanofluid with three different base fluids namely, EG, DW, and EG/DW and reported that the stability of nanofluid with EG base fluid is better than that of DW-based nanofluid. **Figure 1** shows the visual observation of GNP nanofluid with different base fluids. Giwa et al. [20] used two-step method to prepare Al2O3-Fe2O3 hybrid nanofluid with two type of base fluid viz., DW and EG/ DW. SDS and NaDBS were used as a surfactant. Using UV-visible spectrophotometer, they found DW-based Al2O3-Fe2O3 were relatively more stable than the EG–DW Al2O3-Fe2O3 hybrid nanofluid. The absorbance value of the DW-based Al2O3-Fe2O3 displayed better horizontal straight lines than those of the EG–DW Al2O3-Fe2O3.

#### **2.2 Nanomaterial synthesis**

Ding et al. [23] prepared the functionalized graphene (ESfG) by adding the graphite powder into the milling jar with steel balls of smaller diameter and the system was filled with SO3 gas. After removing metallic impurities, the samples were then freeze-dried for 36 hours at −120°C to yield black powder as the final ESfG. The prepared ESfG was stable for several months in water. The sulfonic-acid groups can bond with carbon atoms at the edge of graphite which tends to enhance the stability of ESfG water-based nanofluids. Gul and Firdous [24] synthesized the graphene oxide nanosheet by the oxidation of graphite using the Hummers method as shown in **Figure 4**. In this method the graphite powder was mixed with NaNO3, H2SO4, and KMnO4 and stirred in an ice bath for about 30 min. Finally, the mixture was sonicated and added H2O2 and HCl to quench the reaction and get light yellow graphite oxide. The results found that the highly dispersible nature of GO in water which is fruitful for the preparation of GO nanofluid for multipurpose applications. Li et al. [25] introduced the β-cyclodextrin(β-CD) onto the surface of MWCNTs by a simple chemical synthesis method. It was found that the introduction of β-CD onto the surface of MWCNTs exhibited better stability of nanofluids. The possibility of aggregation between CD-CNTs is significantly decreased due to the Vander Waals force or steric interrupts between β-CD. Rahimi et al. [26] treated the hydrophilization of MWCNTs with different concentrated acids. They added the raw MWCNTs into the mixture of H2SO4 and HNO3 and the mixture was refluxed for 3 hours. The acidtreated MWCNTs were obtained after washing with DI water and dried for 4 hours. Acid-treated MWCNTs suspensions display good stability in water. This is due to

*Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

**Figure 4.** *Synthetic route of graphene oxide by hummers method [24].*

the generation of hydroxyl groups on nanotube surfaces. Vozniakovskii et al. [27] synthesized a hybrid nanomaterial composed of nanodiamonds-multi-walled carbon nanotubes (DND-CNT) using a catalyst chemical vapor deposition (CCVD) method. The results showed that DND-CNT hybrid suspension was stable up to 100 hours while the initial DND began to precipitate after 1 hour. The stability of DND-CNT hybrid particles in water is explained by the opening of a previously closed surface covered with groups with a labile proton, which ensures the stability of the particles of the hybrid material in water.

#### **2.3 Nanofluid synthesis technique**

Numerous nanofluid stabilization techniques are used for reducing the cluster size of nanoparticles i.e. ultrasonic vibration and ball milling etc. The role of ultrasonication is to break the nanoparticle cluster and create a homogenous mixture. Ultrasonic vibration can be employed in two ways; (a) indirect method (ultrasonic bath), and (b) direct method (probe sonicator). Among these two methods, the probe sonicator offers better results in terms of breaking the particle cluster and lowering the average cluster size. Several studies on the impact of the nanofluid synthesis technique on its stability are shown in **Table 2**. Asadi et al. [28] used two-step method to prepare TiO2-CuO hybrid nanofluid. They applied a magnetic stirrer for 1 hour in order to distribute the nanomaterial in the base fluid. Moreover, for breaking the clusters and uniformly distributing the nanoparticles in the base fluid, a probed ultrasonic device was applied for 1 hour. The DLS results ensured the nanoparticles exist in the base fluid, and the phenomenon of agglomeration did not happen. Chen et al. [29] investigated the impact of sonication time on the stability of the Al2O3/liquid paraffin nanofluid. They used two-step method with varying the magnetic stirrer time from 10 to 40 minutes and sonication time from 1 to 4 hours. It was found that nanofluids prepared using shorter sonication times show stability for a minimum of 1 month. When increase in sonication time, it breaks the bond between the nano additives and



#### *Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*



*Synopsis of the investigations about the impact of the nanofluid synthesis technique on its stability.*

#### *Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

the surfactant which leads to be unstable. Asadi et al. [30] varied the sonication time from 10 to 80 minutes to measure the stability of MWCNT/water nanofluid. They reported that after the 30th day, the samples subjected to 10, 20, 40, and 60 minutes of ultrasonication showed good stability while the samples subjected to longer time ultrasonication showed the amount of sedimentation leads to having agglomerated particles. Ranjbarzadeh et al. [31] used magnetic stirrer for 1 hour to mix the SiO2 nanoparticles in the base fluid and then sonicated for 60 minutes. By visual observation, the result found that no sediments were formed after 6 months. Aberoumand and Jafarimoghaddam [32] prepared Ag-WO3/Transformer oil nanofluid using the first step method. They applied Electrical Explosion Wire (EEW) to prepare the nanofluid. The Zeta potential of applied nanofluids in three different concentrations of 1%, 2%, and 4% was measured. The results indicate the excellent stability of applied hybrid nanofluids. Using the same EEW method, Aberoumand et al. [34] prepared Ag/water nanofluid and found that with EEW method, the nanofluid maintained their stability for a long time. Dalkılıç et al. [33] prepared CNT-SiO2/DW using two-step methods and the mixture was sonicated for 3 hours. It was found that the sedimentation was not observed up to 48 hours. The raw CNT particles showed poor dispersion stability in the base fluid. SiO2 particles support and increase the stability of CNTs particles in water. Tests showed that CNT particles exhibit less stability in water without SiO2 particles and surfactants. Kakavandi and Akbari [35] used DLS test to examine the distribution of the MWCNT and SiC nanoparticles in the hybrid nanofluids. The results indicated acceptable stability of nanofluids. The hybrid nanofluid was magnetically stirred for 1 hour and then sonicated for 45 minutes. Keyvani et al. [36] used Ce2O/EG nanofluid to examine the stability. The nanofluid was stirred and then exposed to ultrasonic waves for 2 and 7 hours, respectively. The sedimentation of particles was found after 2 weeks. Nanofluid with a higher concentration of particles, nanoparticles led to agglomerate; therefore, the stability of the nanofluid weakened. It was also reported that the stability of the prepared nanofluid with a lower volume fraction of nanoparticles was stable for a longer period of time compared to the nanofluids with a higher volume fraction [39].

Liu et al. [37] prepared rGO by the reduction of graphene oxide with L-ascorbic acid as a reductant in an aqueous solution. To prepare rGO, the graphene oxide solution was dispersed in DI water and ultrasonicated for 1 hour. NH3-water was then added to control the pH to 10 with sonication for 30 minutes. L-ascorbic acid was added and the mixture was maintained at 95°C for 3 hours for the completion of the reaction. The rGO solution was filtered to obtain rGO on the filter paper. Finally, rGO nanofluids were prepared by sonicating the filtered powder in a certain amount of DI water. The whole process is shown in **Figure 5**. The rGO nanofluids exhibited good stability for 10 day without the addition of other dispersants. Ranjbarzadeh et al. [38] conducted a test to study of pH effects on the stability in acid and alkaline spectrums for GO-SiO2/Water hybrid nanofluids. The results observed that the nanofluid, due to the presence of functional groups on the surface of its nanoparticles, shows acceptable stability in all spectrums; however, in the long term, nanofluids with pH > 7 showed better stability. Zeng and Xuan [40] sonicated the MWCNT-SiO2/Ag binary nanofluids for 1 hour and reported that the stability of the binary nanofluid sustained the dispersion stability for 7 days. Gulzar et al. [41] dispersed hybrid nanopowder (Al2O3-TiO2) in Therminol-55 oil and the mixture was subjected to high shear stirring at 2500 rpm using a magnetic stirrer for 4 hours. The mixture was then sonicated for 2 hours using a high energy probe sonicator. Oleic acid was used as a surfactant as of

*Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

**Figure 5.** *Schematic graph of rGO nanofluids preparation.*

better miscibility with Therminol-55 oil. They observed that the value of zeta potential declines with the rise in concentration which may cause agglomeration adequately after a long time. The surfactant which changes the surface charge and increases the repulsive forces between the nanoparticles also contributes to improved stability. Same way, Alarifi et al. [42] used magnetic stirring for 2 hours and sonicated for 1 hour to prepare a long-term stable MWCNT-TiO2/oil nanofluid. The stability of the prepared hybrid nanofluid was observed over 14 days and no sedimentation was found. Akram et al. [43] checked the stability of CGNP–water nanofluid by zeta potential at different pH values. They prepared this nanofluid after the sonication for 60 minutes and observed that the CGNP nanofluid had high negative values (− 4.42 mV to −49.5 mV) within the pH variations from 1.84 to 10.55. The zeta potential values for the CGNP nanofluids are far from the isoelectric point (i.e., point of zero charges), which indicates that this pH range (2.8–10.55) results in strong electric repulsion forces. Sharafeldin and Grof [44] did sonication of WO3/water nanofluid continuously for about 75 minutes to break the agglomeration between the nanoparticles which leads to well disperse the particles into water. The mean zeta potential value for WO3/water nanofluid was −43.12 and a little decrease in the values was observed over the period of 7 days. MWCNT nanofluids suffer from low dispersion and short-time stability which inhibit their practical application. Chen et al. [45] used a novel method, i.e., a one-pot method by stirred media mill technique. In this method, raw MWCNTs nanoparticles were treated by ball milling to change their morphology, length, and specific surface first. After centrifuging, dry nanoparticles were purified by acid treatment to improve their dispersion in the solution. Thus, the resulting powder was dispersed again in base fluids by ultrasonication and meanwhile, surfactant was added to improve dispersion. The results showed that the milling-treated MWCNT nanofluid exhibited better stability as compared to raw MWCNT and Acid treated MWCNT nanofluid. Ali et al. [46] investigated the

stability of dispersed Al nanoparticles in base fluid (water) prepared by the conventional and the controlled bath temperature two-step methods. The sonication process was taken the same for 4 hours in the range of 10–60°C. The results revealed that the sedimentation behavior of the nanofluids prepared through the controlled bath temperatures of less than 30°C was of dispersed sedimentation type, while those produced by the conventional method and the fixed temperatures of 30°C and higher were of flocculated sedimentation type. Furthermore, increasing the controlled sonication temperature led to an increase in the settling process of the sediments. Also, the rise in nanoparticle concentration was seen to reduce the variation in sedimentation height ratio between the fixed temperature samples. A comparison between the two preparation methods was shown that the 30°C nanofluids had better short- and long-term stability than the conventionally produced suspensions. Mahbubul et al. [47] varied the sonication time from 1 to 5 hours to study the effect of sonication time on the stability of the 0.5 vol% Al2O3 nanofluids. They observed that with low sonication time or no sonication, the sedimentation rate is higher. It can be concluded that longer ultrasonication reduces the sedimentation of nanoparticles and hence, increases the stability of nanofluids. Mahyari et al. [48] used probe-type ultrasonicator to achieve the stability of GO-SiC/water-EG hybrid nanofluid. DLS test results with different patterns approved acceptable stability of the nanofluid. Chen et al. [49] prepared the saline water based magnetic MWCNT nanofluids at different mass concentration from 0 to 0.04 wt% by two-step method. A mechanical stirrer was used at 500 rpm continuously for 30 minutes to mix nanoparticles and water and then the mixture was sonicated thoroughly for 2 hours. Magnetic MWCNTs nanofluids showed high stability in 1000 ppm saline water, and when the solution salinity increased, the original colloidal structure would be destroyed by charge ion. Therefore, the salt-resisting surfactant was added to reinforce the double-layer repulsion and remained the system stable. Okonkwo et al. [50] prepared the Al2O3-Fe/ Water using two-step methods and measured the stability of nanofluid through the Zeta potential test. Hybrid nanofluid was found significantly more stable at pH values of 12 when compared at any other pH value. Terueal et al. [51] performed using the liquid phase exfoliation technique starting with bulk MoSe2 to prepare stable nanofluids. Triton X-100 was used as a surfactant. The suspension underwent sonication in an ultrasound bath for 4 hours with two different frequencies: 80 kHz and 130 kHz. The samples were then centrifuged at 1000 rpm for 10 minutes and again at 4000 rpm for 10 minutes. The results showed that the nanofluid prepared with the frequency of 80 kHz and 130 kHz show the highest extinction coefficient values after 30 days. Higher extinction coefficient values mean the highest amount of nanomaterial in suspension. Li et al. [52] analyzed the stability of SiO2-oleic acid/liquid paraffin nanofluid through the Zeta potential test. The nanofluid was prepared with two-step methods (magnetic stirrer for 30 minute and then sonicated for 1 hour). It was found that the large numbers of SiO2 nano-sized particles possess maximum value for the total count at values less than −40 mV indicating high stability of SiO2 nano-sized particles in liquid paraffin. Geng et al. [53] used the DLS test to study the stability of ZnO-MWCNT/Oil nanofluid and found that the nanoparticles are in the nanoscale after the preparation of nano-oil. Li et al. [54] produced SiO2/EG nanofluids by the two-step method with a mass fraction of 0.005–5%. The zeta potential value of the nanofluid was found −56.28 mV and claimed that the nanofluid is stable. Greater the number of particles with a smaller diameter, the higher the probability of stability. Nanofluid cluster formation may lead to the larger diameter of the nanoparticles. As the number of clusters increases, the fluid stability will decrease.

*Review on the Stability of the Nanofluids DOI: http://dx.doi.org/10.5772/intechopen.107154*

#### **3. Conclusion**

From the literature, it can be concluded that the stability of suspension of nanoparticles in the base fluid is improved when the nanofluid is synthesized by the one-step method as compared to the two-step method but the preparation of nanofluids by one-step method is difficult and expensive relative than two-step method. The literature also reveals that with low sonication time or no sonication, the sedimentation rate is higher. It can be concluded that longer ultrasonication reduces sedimentation of nanoparticles and hence, increases the stability of nanofluids. There are major tasks, which need to be focused on for selection of mono/hybrid nanofluids and their fabrication process, the stability of hybrid nanofluids. The stabilized nanofluids and their characteristics can increase the heat exchange rate of heat exchangers which are generally used in pipeline engineering. In order to help newcomers and researchers in this field recognize the potential research gap, this review study seeks to provide the latest research and development on stable nanofluids and their applications in pipeline engineering. Due to the lack of understanding of the mechanism of nanofluid at the atomic level, many experimental studies are needed to consider several important issues such as particle migration, agglomeration, and stability.

### **Author details**

Sumit Kumar Singh Department of Mechanical Engineering, Indian Institute of Technology (B.H.U.)*,* Varanasi*,* India

\*Address all correspondence to: sumitkrs.rs.mec15@itbhu.ac.in

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 9**

## The Importance of Government Support for Pipeline Network Construction

*Satoru Hashimoto*

#### **Abstract**

This chapter describes that, focusing on economics theory, a country needs its government support to construct a pipeline network throughout the country. In Japan, before deregulation, vertical integrated companies (gas utilities) provided natural gas to customers in their monopoly area respectively. When the companies transport gas into their own areas, the companies choose to construct pipelines or to use LNG tank trucks from the sight of their strategies. Focusing on long-term uncertainty, short-term uncertainty, and locations, this chapter analyzed the factors of pipeline constructions. The results indicate that if there is another gas utility near a company, then the company construct a pipeline to the gas utility to transport gas. In contrast, if there are no neighbor utilities, the company tends to purchase gas via LNG tank truck. This means gas companies do not construct a pipeline network, or do not try to do it, but construct point to point pipelines. Therefore, without government supports, a pipeline network would not be constructed throughout the country.

**Keywords:** natural gas, transaction cost economics, Probit model, pipeline networks, LNG tank trucks

#### **1. Introduction**

This chapter considers the importance of government support for pipeline network construction focusing on Japan's natural gas industry from the perspective of economics, in particular, transaction cost economics and the organizational forms of gas local distribution utilities.

#### **1.1 Overview of natural gas import**

Before proceeding to outline this study, this section provides an overview of the domestic natural gas supply chain. Nearly all of Japan's natural gas requirements have been imported from overseas via LNG tankers. In 2020, 92.02% of all-natural gas was imported as LNG, 4.36% was produced from gas domestic fields, and the remainder was generated from imported petroleum-based gas (Agency for Natural Resources and Energy, Gas Market Division, 2021). Tokyo Gas and Tokyo Electric Power

**Figure 1.**

*Natural gas use in Japan (source: Ministry of Economics, trade, and industry).*

Company (TEPCO) began to import liquefied natural gas (LNG) in 1969. Since then, increasing gas consumption has resulted in increasing imports of LNG. **Figure 1** shows domestical natural gas use, and indicates that approximately 60% of LNG is consumed for power generation, and "Town Gas" is basically used as cooking and heating by end users.

In recent years, many companies have started to import LNG, for example, the three major incumbents, Tokyo Gas, Osaka Gas, Toho Gas, and following incumbents with LNG terminals (Saibu Gas, Shizuoka Gas, and Hokkaido Gas). In addition, both upstream companies and power generation companies also import LNG. Here, the term 'upstream company' refers to a company that specializes in the production and transportation of energy, such as Japan Petroleum Exploration Company Limited (JAPEX) and INPEX Corporation, while power generation companies include Tokyo Electric Power Corporation (TEPCO), Kansai Electric Power Corporation (KEPCO), and Chubu Electric Power Corporation (CEPCO).

#### **1.2 Retail markets and pipeline networks**

The gas retail market has two main features. First, the companies are classified into two categories based on ownership structure. As of March 2021, 173 utilities were private companies while 20 were municipality-owned companies. Of the 173 privately owned utilities, 12 were listed companies. Second, the size of these firms varies significantly. As can be seen in **Table 1**, the maximum revenue is USD 12,586,010,000 (Tokyo Gas), while the minimum is USD 294,740. The large and medium-sized incumbents are involved in production (import), transmission, and distribution, that is, they are vertically integrated utilities. Meanwhile, there are medium- and smallsized incumbents that only sell gas to consumers. These incumbents purchase natural gas from upstream companies via pipelines or LNG tank trucks.

One of the reasons for this situation is that a regional monopoly policy has previously been enforced in the natural gas industry. As far as we can see the natural gas

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*


*(1USD = 100JPY)*

**Table 1.**

*Basic information relating to gas distribution utilities (2015).*

industry after 1945, gas utilities had never been integrated politically, whereas electric utilities were integrated into 10 groups. Also, the spread of LPG use (liquified petroleum gas) since 1945 had greatly influenced the establishment of small sized natural gas utilities. Therefore, the natural gas industry has many utilities (incumbents) and large difference of the largest and smallest utilities. This policy has also affected the characteristics of the natural gas distribution network. **Figure 2**, in which the trunk pipeline networks<sup>1</sup> are depicted, shows that the trunk pipeline networks are quite poor, that is, the coverage area of networks is narrow and they are not enough connected, although the natural gas consumption in 2016 was 111.2 billion m<sup>3</sup> which represented about 3.1% of global consumption, and Japan is the largest LNG importing country in the world<sup>2</sup> .

As is well known, LNG is transformed into natural gas by regasification facilities at or close to LNG terminals. Each large incumbent typically constructed LNG terminals at a sea port close to cities with large populations in its monopolistic supply area and constructed new pipelines after estimating the profits that would be generated by the additional investment. Meanwhile, middle or small incumbents purchase LNG via tank truck or natural gas via pipelines from upstream companies. In the former case, incumbents constructed gasified facilities, and in the latter case, incumbents constructed trunk pipelines to neighbor suppliers. According to the Gas Business Annual Report (2015), total transportation volumes via pipelines and tank trucks were approximately 1735 billion MJ and 1324 billion MJ, respectively, in 2015. Even Tokyo Gas, the largest gas company, uses LNG tank trucks to haul natural gas in a part of its supply area. Consequently, pipeline networks radiated outwards from the 35 existing LNG terminals, and becomes narrow networks. Hence, the pipeline networks terminate in each region, and there are insufficient trunk pipelines connecting the various regions. Incumbents with a vertically integrated structure would not have sufficient incentive to connect their trunk pipelines with those of other incumbents. In fact, there was no pipeline connecting Tokyo and Osaka, a total distance of approximately 2450 km, until December 2015 (Source: Gas Business Annual Report, 2015).

<sup>1</sup> The Japan Gas Association and regulatory authority do not classify pipelines into transmission and distribution pipelines. Instead, they classifies them into high-, medium-, and low-pressure pipelines. Trunk pipelines in **Figure 2** mean high-pressure pipelines.

<sup>2</sup> See *BP* Statistical *Review of World Energy, June 2017*.

**Figure 3** shows the LNG terminals and pipeline networks of Osaka Gas, with the supply area shaded red. It is easy to see that high-pressure (red) and mediumpressure (green) pipelines spread outward from the two LNG terminals. **Figure 3** also shows that the Japanese pipeline network is sparse. The supply area for Gojo Gas is shaded yellow. Although Gojo city in the Nara prefecture is located approximately 41.8 km southeast of Osaka city, there are no pipelines connecting the Osaka Gas and Gojo Gas.

Sadorsky (2001) [1] indicates that it is difficult to introduce product differentiation in relation to natural gas, therefore, gas suppliers (utilities) are most likely to face price competition. Weir (1999) [2] describes that if an incumbent owns both transport and distribution facilities, it would be a barrier to entry for entrants that only have third-party access to pipelines. Hence, the UK government introduced unbundling regulations to increase the number of new entrants and improve competition in the UK retail market.

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*

**Figure 3.**

*Osaka gas pipeline network and LNG terminals (source: Ministry of Economics, trade, and industry).*

#### **1.3 Natural gas supply chains**

The natural gas supply chain varies according to their historical and geographical characteristics.

Sailer et al. (2009) [3] define a natural gas supply chain as following six stages: exploration, extraction, production, transportation, storage, and distribution. In this study, the supply chain is simplified three stages: import, gasification, and distribution (**Figure 4**). The "import" activities include transportation from overseas, exploration, extraction, and production because it focuses on the process of delivering imported gas to end users. "gasification" activities involve procedures to gasify LNG into natural gas and to transport gas from upstream companies to distribution utilities. "distribution" activities include both storage and distribution into end users. In general, Upstream companies operate "import" activities while downstream companies operate "distribution" activities. Regarding "gasification" activities, in some cases, the upstreams operate, and in the other cases the downstreams do. Basically, no transportation companies with pipelines or regasification facilities that are independent of

**Figure 4.** *The natural gas supply chain.*

**Figure 5.** *Organizational structures.*

the upstreams and downstreams exist in Japan (**Figure 5**). Thus, either an upstream or a downstream company needs to shoulder the responsibility for regasification activities to re-gasify LNG and transportation.

Unlike in the United States and EU countries, the Japanese Government (regulatory authority) had never enforced unbundling regulations that prohibit management of both transportation (including import) and distribution activities until 2020, but since April of 2022, the government has introduced the regulations into three largest companies. Besides, almost all upstream companies basically own gas storage, while local distribution utilities (downstream companies) have to undertake the responsibility for stable gas supply to end users. Also, the Government had authorized local distribution utilities to provide natural gas on the principle of a natural monopoly, permitting the utility a business license in the permitted area. These utilities are obligated to provide natural gas to end users in their own area safely and continually, however, the government had not imposed any relevant regulations on gasification activities.

Many upstream companies have both gasification facilities and trunk pipelines to provide into local distribution utilities with natural gas through pipelines or with LNG using tank trucks (**Figure 5**). Because raw commodities such as natural gas is impossible with product differentiation, the best performing natural resource companies are generally the lowest cost producers (Sadorsky 2001) [1]. Therefore, taking into account cost minimization and management strategies, the utilities purchase natural gas directly by joining a pipeline from its own facility to trunk pipelines owned by upstream companies<sup>3</sup> , or purchase LNG by tank trucks. If distribution utilities purchase LNG directly, then they need to construct in-house gasification facilities to provide natural gas into end users. As a result, some downstream utilities have in-house gasification facilities, while the others do not (**Figure 5**).

<sup>3</sup> Many distribution utilities can purchase natural gas via pipelines from not only upstream companies but neighboring distribution utilities (downstream companies).

#### **1.4 Government policy and utility's decision**

A utility's decision on whether or not it needs to establish gasification facilities is critical to that utility's attempts to manage its economic performance. Besides, three historical or crucial circumstances might affect this decision: the Integrated Gas Family 21 plan (IGF 21) issued by the Ministry of International Trade and Industry in 1990<sup>4</sup> , official network plans by the government, and managerial uncertainties.

Next, regarding pipeline network plans, the government has not yet made official plans for a pipeline network, nor has it provided financial aid for its construction, although both retail prices and supply areas have been regulated for a long time. When incumbents implement construction of their own pipelines, they need to raise the money for its construction, and, the decision to implement it depends upon longterm demand and managerial efficiencies. In the case where a massive amount of gas is transported, pipelines are superior to LNG tank truck, but, the former option requires huge capital expenditure to build pipeline facilities. When an incumbent encounters large uncertainties related to the weather conditions (meteorological conditions) or a volatile industrial demand that is affected by the economic conditions, it tends to refrain from an investment in a trunk pipeline even if large demand is expected. In these cases, the incumbent selects to purchase LNG via tank trucks. Thus, the government has never been strongly concerned with pipeline construction.

Also, managerial uncertainties might affect the vertical integration choice. Stable procurement is an indispensable part of distribution utilities. However, it might be difficult to implement an obligation to sustain security of supply for a long time. This is because even if managerial uncertainties are large, the distribution utilities have to continue providing natural gas in constant and sufficient quantities for a long time. To decrease the uncertainties, some utilities strive to purchase gas from plural wholesalers to maintain multiple supply chains, while others set up multiple natural gas storage tanks. Hence, pipeline construction would be affected by political issues, uncertainties, stable procurement. As a manner to explore pipeline construction factors, this study, focusing on a transaction cost economics theory, estimates the transaction cost empirically, and then considers the importance of comprehensive construction policies.

#### **2. Theoretical background**

#### **2.1 Transaction cost economics**

Here, the context of Transaction Cost Economics (TCE) is defined.

Coase (1937) [4] predicted existence in external costs between two firms, and for a single firm, internal costs exist between its divisions. The concept of external and internal costs was considered to be one of the significant factors when an entrepreneur determines a firm's boundaries. When engaging in business transactions, a firm has a strong incentive to integrate with another firm that has significantly high external costs. In contrast, if the external costs between the two firms are not very high, then the former firm does not have a strong incentive to integrate with the latter firm, though would continue to do business with it.

<sup>4</sup> See The Japan Gas Association, https://www.gas.or.jp.

The external and internal cost concept defined by Coase (1937) [4] was developed into transaction cost economics by Williamson (1975, 1985, 1995) [5–7], who defined the term "invisible costs" as "transaction costs", and explained the origin of transaction costs based on three factors: (a) uncertainty, (b) relationship-specific assets, and (c) frequency (Williamson, 1985) [6].

There are several notable papers based on transaction cost views of vertical integration in the manufacturing industry. First, Monteverde and Teece (1982) [8] analyze asset specificity of GM and Ford, and found that the probability of vertical integration between a parent company and a subsidiary might rise because the manufacture of parts which needs advanced technologies in automobiles tends to become relationship specific assets. Second, Masten et al. (1989) [9], separating specific assets into human assets and physical assets, insist that human assets affect vertical integration more than physical assets. Third, Walker and Weber (1984,1987) [10, 11] show that, focusing on the uncertainty, when the uncertainty to get manufacturing parts becomes higher the probability for vertical integration also becomes higher.

Regarding empirical analyses, Levy (1985) [12] estimates the boundaries of firm by using 67firms' data (37industries), and puts asset specificity as R&D investment, and moreover puts uncertainty as variance of sales. As a seminal work for transaction cost economics of power generation industry, Joskow (1985, 1988) [13, 14] found that power generation plants incline to be constructed close to mining pits, and that vertical integration between plants and pits, and long-term contracts, were widely practiced. Crocker and Masten (1996) [15] investigated the organizational forms of public utilities in the United States.

Shelanski and Klein (1995) [16] reviewed many empirical literatures related to transaction cost economics theory, and then concluded that they have conspicuously consistent with predictions from the theory. In contrast, David and Han (2004) [17] and Carter and Hodgson (2006) [18], from traditional literature surveys, found that asset specificity and uncertainty have received considerable scrutiny or commonly examined, whereas frequency has not. Hence, they concluded that some literatures have produced results that the transaction cost framework would not predict.

Sheravani et al. (2007) [19] noted the importance of the relationship between transaction costs and market power. They suggested that high market power appears to provide safeguards to a firm using nonintegrated channels not envisioned in predictions from transaction cost economics. Furthermore, they argued that firms with high market power are likely to have significant monitoring and surveillance capabilities, can exercise legitimate authority, and offer diverse incentives to associated channel members5 .

#### **2.2 Organizations and LNG supply chains**

This section introduces several related literatures on LNG supply chains. Xunpeng, (2016) [20] points out that almost all the incumbent gas companies in Asian have vertically integrated supply chains. Lee et al. (1999) [21] found that KOGAS (the

<sup>5</sup> To evaluate market power, this study measured the Hyfindal Hussuman Index (HHI) based on sales volume. In 2010, the HHI value of gas distribution utilities was 2037, which indicated that the market power of the industry was not very high. However, Tokyo Gas, the largest gas utility, had 30% of the market share, and Osaka Gas, the second largest utility, had a 20% market share. In addition, these utilities each had three types of operations activity, from import to distribution. Therefore, this study estimated the transaction costs excluding these two largest utilities.

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*

Korean national firm) have a lower productivity level compared to firms with acquiring their gas through pipelines because it depends on LNG import, which requires additional capital facilities for shipping, storage, and regasification. As been described by Vivoda (2014a) [22], a number of international LNG trade are dominated by longterm contracts. This is because the trading companies need the huge capital costs including liquefaction and regasification facilities and the inherent inflexibility in the value chain required contractual arrangements to protect both the suppliers and the buyers. In contrast, Cabalu (2010) [23] and Hartley (2013) [24] described that technological innovations make LNG transportation costs decrease significantly and LNG import and export volume were gradually increasing. Also, Gkonis and Psaraftis (2009) [25] indicate that LNG shipping markets are basically oligopolistic, and then suggested that competing companies have to consider a transportation capacity in the LNG shipping market. Vivoda (2014b) [26] points out that it is important for Japan and Korea to elaborate diverse LNG strategies to import LNG smoothly.

Turner and Johnson (2017) [27] denote that LNG trade is superior to pipeline transportation, and then point out that importers and exporters can easily send and receive gas to any locations with liquefaction and regasification facilities when LNG trade is possible. Xunpeng (2016) [20] and Hashimoto (2020) [28] describe the prospect in Asian LNG spot and hub markets.

#### **3. Methodology and data**

Here, this section describes the application of transaction cost economics theory into gas utilities. According to this theory, transaction costs consist of external and invisible costs between two firms, while internal costs are composed from invisible costs between two divisions in a single firm (**Figure 6**). A firm basically assumes to determine the choice whether or not it selects vertical integration, based on the transaction cost economics theory. **Figure 6** illustrates the application of transaction cost economics. When transaction costs exceed internal costs, then firm B would merge with firm A or acquire it. This study defines this type of consolidation as a vertical integration. In contrast, when internal costs exceed transaction costs, firm B would not merge with firm A.

**Figure 6.** *Schematic of application of transaction costs and internal costs.*

Regarding empirical estimations, this theory has two prudent treatments. First, Williamson (1985, p.20) [6] indicates the existence of ex ante and ex post transaction costs, and Monteverde and Teece (1982) [8] investigated ex ante transaction costs. This study also attempts to investigate ex ante transaction costs. Second, although both transaction costs and internal costs should be observed directly, it would be impossible to measure internal costs precisely. However, because transaction cost economics theory expects a positive correlation between transaction costs and incentive for vertical integration this study estimates only transaction costs to analyze the incentive.

Williamson (1985) [6] explains uncertainty, relationship-specific assets, and frequency as transaction costs. Many empirical analyses treat with uncertainty and relationship-specific assets [8–12]. Frequency is not analyzed because of the difficulty in estimation. This study also adopts the two components; uncertainty and relationship-specific assets. Regarding uncertainty, this study divides it into long-term and short-term components for its importance to the natural gas industry. In distribution utilities, because growth rate and demand fluctuation would be the main components of uncertainty, this study defined sales volume, number of customers, and average revenue growth rate as sources of long-term uncertainty. Also, Gas demand and underpinning sales would be affected by seasonal factors, which are defined as monthly sales variance and the inventory rate. These are components of short-term uncertainty in this analysis<sup>6</sup> .

Meanwhile, relationship-specific assets are classified into the assets by site specificity, physical asset specificity, human asset specificity, and dedicated assets by Williamson (1985) [6]. This study employs physical and site specificities, and does not analyze physical assets, human assets, and dedicated assets. For physical asset specificity, gas utilities comprise public and private administrations. Public gas utilities are expected to receive aid from municipalities when they face bankruptcy. Therefore, they may decrease transaction costs for related to relationship-specific assets.

To scrutinize long-term uncertainty, short-term uncertainty, and site specificity, the Integration equation is assumed by means of the methodology of Levy (1985) [12] and Wang and Mogi (2017) [29];

$$\text{Integration} = f(LU, \text{SU}, \text{SS}) \tag{1}$$

where LU, SU, and SS are long-term uncertainty, short-term uncertainty, and site specificity, respectively. If transaction costs increase, the value of Integration would become high. Hence, the high value means high incentive to integrate.

The dependent variable, "vertical integration (VI)", represents whether or not a utility has gasification facilities. When a distribution utility has gasification facilities, it means that transaction costs exceed internal costs. Hence, "vertical integration (VI)" is adopted as the dependent variable. The dependent variable (x) represents whether or not a utility has gasification facilities.

**Table 2** shows the definitions of dependent and independent variables, including the expected sign. RGR, SAL, CUS, ASS, and PRO are defined as components of longterm uncertainty, and SDR, SDM, SVV, AVI, and HHR are defined as components of short-term uncertainty. SSD and PUD are adopted as components of site specificity. The data sources are gas business annual reports, and SSD was obtained from the natural gas supply area map (agency for natural resources and energy).

In this study, the integration eq. (1) was defined as:

<sup>6</sup> More detailed information is described by Hashimoto (2021) [30].

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*


#### **Table 2.**

*Definition of variables (source; Hashimoto, 2021 [30]).*

$$e^{\mathbf{x}} = a \prod\_{l} LU\_{l}^{b\_{l}} \cdot \prod\_{m} SU\_{m}^{c\_{m}} \cdot \prod\_{n} \text{SS}\_{n}^{d\_{n}}$$

$$\ln e^{\mathbf{x}} = \ln a \prod\_{l} LU\_{l}^{b\_{l}} \cdot \prod\_{m} SU\_{m}^{c\_{m}} \cdot \prod\_{n} \text{SS}\_{n}^{d\_{n}} \tag{2}$$

$$\infty = \beta + \sum\_{l} b\_{l} \ln LU\_{l} + \sum\_{m} c\_{m} \ln SU\_{m} + \sum\_{n} d\_{n} \ln \text{SS}\_{n},$$

where x means the probability of integration. LUl, SUm, and SSn respectively mean the l-th long-term uncertainty, m-th short-term uncertainty, and n-th site specificity, while α and β are constants. The high value of x means high incentive to integrate, while the low value means low incentive. **Table 3** shows the descriptive statistics.

#### **4. Results**

**Table 4** shows the Probit model results. The results of model 1–6 indicate robustness. Because strong correlations among independent variables of long-term uncertainty (LU) might be shown, these variables were not applied simultaneously<sup>7</sup> .

The three types of transaction costs will be discussed<sup>8</sup> . First, for long-term uncertainty, the coefficients of PRO, ASS, SAL, and CUS are not significant at 10% level.

<sup>7</sup> In general, strong correlations among variables are neither necessary nor sufficient to cause multicollinearity. More detailed information is described by Hashimoto (2021) [30].

<sup>8</sup> More detailed information and discussion is described in Hashimoto (2021) [30].


#### **Table 3.**

*Descriptive statistics (source; Hashimoto, 2021 [30]).*


#### **Table 4.**

*Probit model results (source; Hashimoto, 2021 [30]).*

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*

RGR is significant at 10% in all models, but the sign was negative, not being consistent with the expected sign. Hence, the component of long-term uncertainty is required careful interpretation<sup>9</sup> .

Second, for short-term uncertainty, AVI was significant at 1% in all models, and standard deviation of SVV was also significant at 5% or 10%. HHR, SDR and SDM were not significant at the 10% level. Regarding coefficients for AVI, the consideration for a causal relation is required carefully because it would be possible to interpret that utilities with possessing regasification facilities increase their inventory volume. This study concluded that short-term uncertainty cannot be strongly supported.

Third, the coefficients of SSD were significant at the 1% level in all models, and also, those of PUD were significant at 5% or 10% level in some models. Therefore, the existence of assets and site specificities are strongly recognized.

#### **5. Conclusions**

This chapter, to confirm the importance of comprehensive pipeline network plans by the government, examined whether or not local distribution utilities integrated gasification activities, in terms of transaction cost economics theory, and then found that the utilities prefer to purchase natural gas via pipelines when there are wholesalers or neighboring utilities that provide gas to end users, or there are natural gas fields in the vicinity. As a result of the utility's behavior, a broad pipeline network would not be built throughout the country.

Pipeline construction depending on distribution utilities affects the growth of broad pipeline networks. Because a utility prefers to construct a point-to-point pipeline between itself and a neighbor wholesaler the utility by no means construct a pipeline network by itself. Even if a number of utilities invest the point-to-point pipelines, a nationwide pipeline network cannot be built. Consequently, as seen in **Figure 2**, a broad pipeline network infrastructure has not been constructed throughout Japan. Therefore, this study concluded that we need the government support for pipeline network construction.

**Table 5** shows the cases of pipeline investment. **Table 5** indicates that companies invest comparatively small distance pipelines that are under 100 km, and moreover, joint ownership is adopted by the two related firms such as distribution utilities and wholesalers10. The former would reinforce this study that distribution utilities tend to purchase gas from those companies when there are wholesalers or neighbor utilities in the vicinity. Meanwhile, the latter might bear out the necessity of government supports for nationwide pipeline networks. The government basically has not supported the companies by means of pipeline network planning, comprehensive infrastructure policies, or financial support. The reason why joint-ownership is adopted might be without any governmental supports. In other words, this might be indicating that the

<sup>9</sup> In fact, the relationship between firm size or sales volume and the risk of purchasing natural gas is unclear. More detailed discussion is described by Hashimoto (2021) [30].

<sup>10</sup> In **Table 5**, Ogijima-Kawasaki line [1], Minamifuji Line [3], Setouchi line [4], Koriyama line [5], Shizuhama line [6], Isewan Odan pipeline [8], and Mie-shiga line [9] are adopted a form of jointownership. Meanwhile, Chiba-Kashima line [7], Saito line [12], Ibaragi-Tochigi line [13], Furukawa-Maoka line [14], and Ibaragi line [15] are the projects of Tokyo Gas, and Himeji-Okayama line [10] and Amagasaki-Kugayama line [16] are the projects of Osaka Gas. Those projects have been invested by a single company because those pipelines are all in its own monopolistic area.


#### **Table 5.**

*The cases of pipeline investment.*

government need to have some supports to build a nationwide pipeline network throughout the country.

In addition, the government decided to unbundle natural gas companies. The three largest incumbents (Tokyo gas, Osaka gas, and Toho Gas) were unbundled in April, 2022. While the unbundling regulation is expected to promote competition, it might also discourage pipeline investment. Hence, incentives for investment after the introduction of unbundling regulation might need to be considered.

*The Importance of Government Support for Pipeline Network Construction DOI: http://dx.doi.org/10.5772/intechopen.108841*

Finally, this study explored the determinants of vertical integration and the importance of pipeline network plans. However, this study has not considered or investigated the characteristics of network externalities, a natural monopoly, economies of scale, and management strategies that cannot be evaluated by transaction cost economics11. They will be the focus on future work.

#### **Author details**

Satoru Hashimoto Faculty of Economics, Teikyo University, Tokyo, Japan

\*Address all correspondence to: s-hashi@main.teikyo-u.ac.jp

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

<sup>11</sup> See Baumol and Oates (1975) [31] and Sharkey (1982) [32].

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### *Edited by Sayeed Rushd and Mohamed Anwar Ismail*

All around the world, pipelines ensure the economic transmission of essential fluids to different industries and residential buildings. The discipline of pipeline engineering covers a wide range of topics, including design, construction, operation, instrumentation, maintenance, integrity, management, corrosion, and failure. Probably the most significant subjects are design, failure, and management, as these specialties have direct impacts on all other aspects of pipeline engineering. This book focuses on some recent evidence-based developments in these fields. The chapters include experiment-, simulation-, and analysis-based studies. The contributing authors come from diverse geographical locations with strong experience in their respective fields. The technological aspects examined here would definitely reinforce a pipeline engineer's decision-making process.

Published in London, UK © 2023 IntechOpen © PhonlamaiPhoto / iStock

Pipeline Engineering - Design, Failure, and Management

Pipeline Engineering

Design, Failure, and Management

*Edited by Sayeed Rushd* 

*and Mohamed Anwar Ismail*