3. Heat and mass transfer at outward convex corrugated tube heat exchangers

#### 3.1 Numerical methods

#### 3.1.1 Physical model and meshing system

The first task to accomplish in a numerical simulation is the definition of the geometry followed by the mesh generation. The geometry of the design needs to be

#### Heat and Mass Transfer in Outward Convex Corrugated Tube Heat Exchangers DOI: http://dx.doi.org/10.5772/intechopen.85494

created from the initial design. Any modeling software can be used for modeling and shifted to other simulation software for analysis purpose.

Figure 7 shows a schematic view of the structural parameters for corrugated tube investigated in this chapter, which include inner diameter (D), tube length (L), corrugation height (H), corrugation pitch (P), corrugation crest radius (R), and corrugation trough radius (r). Since the investigated corrugated tubes are used in tube-shell type heat exchanger, the flow region inside of tube is named "tube side" and out of tube is named "shell side."

Mesh generation is the process of subdividing a region to be modeled into a set of small control volumes. In general, a control volume model is defined by a mesh network, which is made up of the geometric arrangement of control volumes and nodes. Nodes represent points at which features such as displacements are calculated. Control volumes are bounded by set of nodes and also defined by the number of mesh. One or more values of dependent flow variable (e.g. velocity, pressure, temperature, etc.) will be contained in each control volume. Usually, these represent some type of locally averaged values. Numerical algorithms representing approximation to the conservation law of mass, momentum, and energy are then used to compute these variables in each control volume.

Mesh generation is often considered as the most important and most time consuming part of CFD simulation [13]. The quality of the mesh plays a direct role on the quality of the analysis, regardless of the flow solver used. In this work, a 3D nonuniform mesh system of hexahedral elements was established via the professional mesh generation software ICEM to accurately control the size and number of cells in the domain, as illustrated in Figure 8. The near-wall vicinity should be present

Figure 7. Structure parameters of outward convex corrugated tube.

Figure 8. Schematic diagram of meshing system for the simulated corrugated tube.

drastic velocity and temperature gradients, so a high density of gradient elements was applied in this region. Nevertheless, the remaining domain was modeled with relatively sparse elements. The first layer of thickness should satisfy y <sup>+</sup>≈ 1.

### 3.1.2 Mathematical model

Mathematical model should be constructed to numerically describe flow and heat transfer of corrugated tube. The Navier-Stokes equations generally are adopted to describe the laminar and turbulent flows, which could be solved by various kinds of simulation model including DNS, LES, and RANS. The direct numerical simulation (DNS) can solve accurately the turbulent fluctuation, but these models require huge computing power, which is many orders of magnitude higher than other models. Reynolds-averaged Navier-Stokes (RANS) is a high efficient model that can be used to approximate turbulence by time-averaged turbulent fluctuation, but the accuracy of the models is much less than DNS. The accuracy and efficient of LES are between the DNS and RANS.

The k-ε (k-epsilon) model is one of the most prominent RANS models, which has been implemented in most CFD codes and is considered the most common industry model. The stability and robustness of the models have a well-established regime of predictive capability, satisfying general purpose simulation by offering a comparative good accuracy. In our research work for outward convex corrugated tube, we use standard k-ε model for numerical simulation research.

The governing equations in a RANS (Reynolds Averaged Navier-Stokes) manner are given below.

Continuity equation:

$$\frac{\partial(\rho u\_i)}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{12}$$

Momentum equation:

$$\frac{\partial}{\partial \mathbf{x}\_j} \left( \rho u\_i u\_j \right) = -\frac{\partial P}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} - \frac{2}{3} \delta\_{ij} \frac{\partial u\_k}{\partial \mathbf{x}\_k} \right) \right] + \frac{\partial}{\partial \mathbf{x}\_j} \left( -\rho \overline{u\_i' u\_j'} \right) \tag{13}$$

Heat and Mass Transfer in Outward Convex Corrugated Tube Heat Exchangers DOI: http://dx.doi.org/10.5772/intechopen.85494

Energy equation:

$$\frac{\partial}{\partial \mathbf{x}\_{i}} \left[ \boldsymbol{\mu}\_{i} (\rho \boldsymbol{E} + \boldsymbol{P}) \right] = \frac{\partial}{\partial \mathbf{x}\_{j}} \left[ \left( \boldsymbol{\lambda} + \frac{\mathbf{c}\_{p} \mu\_{\mathrm{t}}}{Pr\_{\mathrm{t}}} \right) \frac{\partial T}{\partial \mathbf{x}\_{j}} + \mu\_{\mathrm{eff}} \boldsymbol{u}\_{i} \left( \frac{\partial \boldsymbol{u}\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial \boldsymbol{u}\_{j}}{\partial \mathbf{x}\_{i}} - \frac{2}{3} \delta\_{ij} \frac{\partial \boldsymbol{u}\_{k}}{\partial \mathbf{x}\_{k}} \right) \right] \tag{14}$$

The standard k-ε model is adopted here to close governing equations:

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho k u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + G\_k - \rho \varepsilon \tag{15}$$

$$\frac{\partial}{\partial \mathbf{x}\_i} (\rho e u\_i) = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_t}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} (\mathbf{G}\_k) - \mathbf{C}\_{2\varepsilon} \rho \frac{\varepsilon^2}{k} \tag{16}$$

where μt is the turbulent or eddy viscosity, and Gk represents the generation of turbulent kinetic energy due to the mean velocity gradients. The model constants C1ε, C2ε, Cμ, σk, and σε are chosen for the default values 1.44, 1.92, 0.09, 1.0, and 1.30, respectively.

#### 3.1.3 Boundary and initial condition

The next step in preprocessing is setting up the boundary conditions. Boundary conditions refer to the conditions that the solution of the equations should satisfy at the boundary of the moving fluid. Boundary condition will be different for each type of problem. In our research work, the initial and boundary conditions of the outward convex corrugated tube heat exchangers are shown as follows:


#### 3.1.4 Numerical procedure

The final step in preprocessing is setting up the numerical procedure, which includes solver, discretization, and convergence criterion. In our work, the governing equations are discretized by the finite volume method and solved by the steady-state implicit format. The SIMPLE algorithm is used to couple the velocity and pressure fields. The second-order upwind scheme is applied herein. The convergence criterion for energy is set to be 10�7 relative error and 10�4 relative error for other variables.

#### 3.2 Heat and mass transfer in tube and shell side of corrugated tube

The variable distribution exhibits the opposite similar tendency at the shell side compared with that at the tube side. In this chapter, we mainly analyze the distribution of velocity, temperature, and turbulence kinetic energy.

#### 3.2.1 Flow structure

Figure 9(a) shows the velocity vector distribution in the tube side of outward convex corrugation tube. As shown from this figure, when fluid flow starts to cross the corrugation section bended from the straight segment, the flow boundary layer separates into two parts: one is the wall boundary layer developed at the near wall region; the other is shear layer associated with an inflection point of large velocity gradient developed away from the wall, which moves away from the surface at the separation point and forms a free shear layer. When the fluid flows through the upstream of the corrugation, the flow velocity decreases and the pressure increases due to the narrowing of the flow cross section. The fluid layer near the wall is gradually difficult to overcome the rising pressure due to the small amount of momentum, resulting in a reflow of the original flow direction. The recirculating zone between the separating streamline and the free boundary streamline is generated at the upstream of the corrugation.

Figure 9(b) indicates the velocity vector distribution in the shell side of outward convex corrugation tube. As shown in this figure, the upstream side boundary of the corrugation is influenced by the accelerating outer-flow, that is, a favorable gradient. As the boundary layer thickens, instabilities occur when the near-wall fluid begins to decelerate as shown in Figure 7. The flow separates at the downgrade of the corrugation crest, which is associated with an inflection point of the large velocity gradient developed away the wall.

#### 3.2.2 Temperature distribution

Figure 10(a) shows the temperature distribution in the tube side of outward convex corrugation tube. As shown in Figure 8, the wall velocity boundary layer becomes thicker at the upstream side of the corrugation accompany gradually, while the temperature boundary layer gets thicker along the flow direction, due to the eddy generating. Then it goes into thinner at the downstream side of the corrugation with the velocity boundary layer getting thinner, due to the scouring action of the fluid.

Figure 10(b) shows the temperature distribution in the tube side of outward convex corrugation tube. As shown in Figure 9, the wall velocity boundary layer becomes thicker at the downstream side of the corrugation accompany gradually, while the temperature boundary layer gets thicker along the flow direction, due to the eddy generating. Then, it goes into thinner at the upstream side of the corrugation with the velocity boundary layer getting thinner, due to the scouring action of the fluid. The thinnest temperature boundary layer occurs at the corrugation crest.

Figure 9. Velocity vector distribution at tube side and at shell side. (a) tube side; (b) shell side.

Heat and Mass Transfer in Outward Convex Corrugated Tube Heat Exchangers DOI: http://dx.doi.org/10.5772/intechopen.85494

Figure 10.

Temperature distribution at tube side and at shell side. (a) tube side; (b) shell side.

Figure 11. TKE distribution at tube side and at shell side. (a) tube side; (b) shell side.
