**4. The model under investigation**

The calculations were made for a one-cycle heat exchanger, Fig. 1, whose model has been developed on the basis of the technical documentation of the Py-100-020 decarbonized water heater.

The elements that were taken into account in the 3D model of the heat exchanger are as follows: perforated plates (1), bottoms (2), a jacket (3), heating cartridge pipes (8), heating cartridge gaskets (10) shown in Fig.1. Perforated plates are fixed to bottoms with screw fasteners (9). Connector pipes (6) and (7) supply and take off the steam. Detail B shows dimensions of the hole in the perforated bottom before rolling out and a view of the pipeperforated bottom connection after rolling out. The perforated plates have a hexagonal system of holes that make perforations. The materials used in the structure are listed in Table 1.

Numerical Analysis of the Structural Stability of Heat Exchangers – The FEM Approach 171

For the strength analysis in the elastic-plastic range, stresses σ versus strains ε at the maximum operating temperature were used as the basic model of the material the perforated plate and the heating cartridge pipes were made of. A physical model of the St41K steel is presented in Fig. 2, whereas the physical model of the I-K10 steel is shown in Fig. 3. In the calculations, the material consolidation factor λ=0.99 was assumed, according

The pressure acting on the gasket was calculated on the basis of the required value of the initial stress of screws in the bottom–gasket–perforated bottom–gasket–jacket connection

A shell-solid model (Fig. 4) was built for the numerical calculations. The calculations were conducted for the plate thickness decreased by approx. 15% up to 50 mm and by 50% up to 30 mm, the remaining dimensions were unaltered. The numerical model was divided into elements of the SOLID 45 type (perforated bottoms) and Shell 43 (heating cartridge pipes and the jacket). The numerical calculations were conducted to calculate initial stresses and strains occurring in the heat exchanger with respect to a stability loss, whereas the thickness of perforated bottoms was altered. The calculations were carried out for the emergency

Fig. 2. Physical model of the St 41K steel. (perforated bottom plate).

Table 2. Parameters of the heat exchanger operation.

given in the technical documentation.

to Table 1.

state.

Parameters Steam chamber Pressure p0=1.17 MPa Temperature T0 =523 K

Fig. 1. Schematic view of the heat exchanger.


Table 1. Strength properties of the materials used in the structure

#### **5. Numerical calculations**

The numerical calculations were conducted for the case when the water inflow and the outflow of were closed, whereas the steam was still supplied to the heat exchanger. The data presented in Table 2 were assumed in the calculations on the basis of the technical documentation:


Table 2. Parameters of the heat exchanger operation.

170 Heat Exchangers – Basics Design Applications

Fig. 1. Schematic view of the heat exchanger.

**5. Numerical calculations** 

documentation:

I - K10 PN-74/H-74252 boiler pipe

tensile strength Rm=345440 MPa Rm =400 <sup>490</sup>

Table 1. Strength properties of the materials used in the structure

yield point Re =235 MPa Re =255 MPa Re=196 MPa

Young' s modulus E =2٠105 MPa E =2٠105 MPa E =2٠105 MPa consolidation factor =0.99 =0.99 =0.99 Poisson's ratio =0.3 =0.3 =0.3 friction coefficient = 0.23 - -

The numerical calculations were conducted for the case when the water inflow and the outflow of were closed, whereas the steam was still supplied to the heat exchanger. The data presented in Table 2 were assumed in the calculations on the basis of the technical

St41K PN-75/H-92123 boiler sheet

MPa -

St 36K PN-75/H-92123 boiler sheet

For the strength analysis in the elastic-plastic range, stresses σ versus strains ε at the maximum operating temperature were used as the basic model of the material the perforated plate and the heating cartridge pipes were made of. A physical model of the St41K steel is presented in Fig. 2, whereas the physical model of the I-K10 steel is shown in Fig. 3. In the calculations, the material consolidation factor λ=0.99 was assumed, according to Table 1.

The pressure acting on the gasket was calculated on the basis of the required value of the initial stress of screws in the bottom–gasket–perforated bottom–gasket–jacket connection given in the technical documentation.

A shell-solid model (Fig. 4) was built for the numerical calculations. The calculations were conducted for the plate thickness decreased by approx. 15% up to 50 mm and by 50% up to 30 mm, the remaining dimensions were unaltered. The numerical model was divided into elements of the SOLID 45 type (perforated bottoms) and Shell 43 (heating cartridge pipes and the jacket). The numerical calculations were conducted to calculate initial stresses and strains occurring in the heat exchanger with respect to a stability loss, whereas the thickness of perforated bottoms was altered. The calculations were carried out for the emergency state.

Fig. 2. Physical model of the St 41K steel. (perforated bottom plate).

Numerical Analysis of the Structural Stability of Heat Exchangers – The FEM Approach 173

Q function of stress termed potential (which determines the direction of plastic

The following boundary conditions were assumed in the calculations: the rear perforated plate Ux=Uy=Uz=0, the front perforated plate Ux=Uy=0. Such boudary conditions were

plastic multiplier (which determines the amount of plastic straining)

assumed on the circumference of both plates (Fig.4).

Fig. 4. Numerical model of the heat exchanger under investigation.

In heat exchangers, the analysis of a complex influence of perforated bottoms on the jacket connected to them, as well as a distribution of stresses in the jacket, pipes and perforated bottoms is essential. The substitutive stress that decides basically about strains occurring in steel structures was calculated according to the Huber-Misses hypothesis. The calculations of the heat exchanger were conducted on the assumption of high strains in the structural

The Finite Element Method - ANSYS 12.0 - was used for the numerical calculations. The results of the numerical calculations have been presented in the form of maps of stresses and strains. The results shown in Figs. 5-12 concern the heat exchanger in which the thickness of

straining)

**5.1 Results** 

elements.

 *<sup>e</sup>* equivalent stress {*S*} deviatoric stress

Fig. 3. Physical model of the I-K10 steel (heating cartridge pipes).

The mathematical model of the numerical code is described by the following equations from numerical program ANSYS

$$\mathbb{E}\left\{\boldsymbol{\varepsilon}^{p\boldsymbol{i}}\right\} = \left[\boldsymbol{B}\right] \{\boldsymbol{\mu}\} - \left\{\boldsymbol{\varepsilon}^{t\boldsymbol{h}}\right\} \text{ - strain} \tag{1}$$

$$\{\sigma\} = \begin{bmatrix} D \end{bmatrix} \begin{Bmatrix} \varepsilon^{cl} \end{Bmatrix} \text{ - stress} \tag{2}$$

$$
\sigma\_e = \left[\frac{3}{2} \{\mathbf{S}\}^T \left[T\right] \{\mathbf{S}\}\right]^{\frac{1}{2}} \text{ - equivalent stress} \tag{3}
$$

the flow rule determines the direction of plastic straining

$$
\left\{\mathbf{d}\varepsilon^{\rm pl}\right\} = \mathcal{L}\left\{\frac{\delta\mathbf{Q}}{\delta\sigma}\right\}\tag{4}
$$

where:

$$
\begin{array}{ll}
\{\boldsymbol{x}^{\operatorname{pi}}\} & \text{structures that cause stresses} \\
\{\boldsymbol{B}\} & \text{strain-displacement matrix evaluated at integration point} \\
\{\boldsymbol{\mu}\} & \text{nodal displacement vector} \\
\{\boldsymbol{x}^{\operatorname{th}}\} & \text{thermal strain vector} \\
\{\boldsymbol{\sigma}\} & \text{stress vector} \\
\{\boldsymbol{D}\} & \text{elasticity matrix}
\end{array}
$$

*<sup>e</sup>* equivalent stress

172 Heat Exchangers – Basics Design Applications

The mathematical model of the numerical code is described by the following equations from



(4)

*<sup>e</sup> S TS* - equivalent stress (3)

*pi th*

*B u*

*el*

*T*

3 <sup>2</sup>

1

pl <sup>δ</sup><sup>Q</sup> <sup>d</sup><sup>ε</sup>

δ 

Fig. 3. Physical model of the I-K10 steel (heating cartridge pipes).

2

[*B*] strain-displacement matrix evaluated at integration point

the flow rule determines the direction of plastic straining

strains that cause stresses

{*u*} nodal displacement vector

thermal strain vector

 stress vector [*D*] elasticity matrix *D*

numerical program ANSYS

where: *pi* 

 *th* 

 {*S*} deviatoric stress

plastic multiplier (which determines the amount of plastic straining)

Q function of stress termed potential (which determines the direction of plastic straining)

The following boundary conditions were assumed in the calculations: the rear perforated plate Ux=Uy=Uz=0, the front perforated plate Ux=Uy=0. Such boudary conditions were assumed on the circumference of both plates (Fig.4).

Fig. 4. Numerical model of the heat exchanger under investigation.
