**3. Numerical model to evaluate and to optimize the wire and tube heat exchanger performance**

Numerical model in this chapter will be discussed based on the two approaches, that is, the finite element method (FEM) whose program was developed and run using a MATLAB program [24, 25] and the FEM using ANSYS Fluent for the computational fluid dynamics (CFD) approach [25–27]. For the FEM developed in MATLAB, the finite element was modeled as wire-tube element which is shown in **Figure 3**. Each modeled element is comprised of a tube whose length is equal to the wire pitch (pw) and the wire, which acts as a fin was set to have a length as long as the tube spacing or pitch (pt). The thermophysical properties of the fluid, which includes mass flow, temperature, enthalpy, and heat, were spatially calculated at the position of *x* and *x + dx* for each element, where dx = pw.

The heat transfer in each element from the working fluid inside the tube to the surrounding air followed:

$$\mathbf{Q}\_{\rm el} = \mathbf{U} \mathbf{A}\_{\rm el} \left( T\_f - T\_{\rm os} \right)\_{\rm el} \tag{7}$$

where the conductance variable UAel of each element was equal to <sup>1</sup> UAel ¼ *Ri* þ *Rt* þ *Ro*, and the thermal resistance of each wire-tube element could be expressed:

$$R\_{\text{w\&T}} = \left(\frac{\mathbf{1}}{h\_i A\_i} + \frac{\ln\left(r\_o/r\_i\right)}{2\pi k \Delta z} + \frac{\mathbf{1}}{h\_o A\_o}\right)\_{\text{el}}\tag{8}$$

**Figure 3.** *The schematic of the wire-tube element to build finite element model for wire and tube heat exchanger.*

The thermal resistance formula was used as a basis to calculate the convective heat transfer from the fluid to the tube wall, conduction inside the tube wall, and convection from the tube surface to the surrounding air. The area of each element was then determined as *Ao* ¼ *At* þ *Aw* ¼ *π:dt*,*o*�*pw* þ 2*:π:dw:pt* . As each element of the heat exchanger was extended by a wire-based fin, the wire efficiency could be calculated as follows:

$$\eta\_w = \frac{\left[\tanh\left(\frac{m \times p\_t}{2}\right)\right]}{\left[\left(\frac{m \times p\_t}{2}\right)\right]}\tag{9}$$

in which it required convection and conduction heat transfer coefficient data to determine *m* ¼ ffiffiffiffiffiffiffiffi 4*hw kwdw* q . For initial calculation of the wire heat transfer coefficient, hw was set to obtain *η<sup>w</sup>* ¼ 0*:*9. Assuming that the heat transfer coefficient is constant along with the wire element and the difference between the fluid temperature and the tube temperature is 0.5°C, the wire temperature was determined as follows:

$$T\_w = \eta\_w (T\_{\text{t,o}} - T\_{\text{os}}) + T\_{\text{os}} \tag{10}$$

The average external (outer surface) temperature of each element could then be calculated as follows:

$$T\_{\rm ex} = \frac{(T\_{\rm to} + GP.\eta\_w.(T\_{t,\rho} - T\_{\rm os}) + GP.T\_{\rm os})}{(1 + GP)}\tag{11}$$

where GP is the geometrical parameter, *GP* <sup>¼</sup> <sup>2</sup> *pt <sup>d</sup>*t,o � � *dw pw* � �. The heat transfer coefficient from the outer surface of the wire-tube element was determined from both free convection and radiation *ho* ¼ *hc* þ *hr* where the radiative heat transfer coefficient was defined as follows:

$$h\_r = \varepsilon . \sigma. \frac{\left(T^4\_{\text{t,o}} - T^4\_{\text{\textinfty}}\right)}{\left(T\_{\text{t,o}} - T\_{\text{\textinfty}}\right)} \tag{12}$$

To obtain the convective heat transfer coefficient *hc* <sup>¼</sup> Nu�<sup>k</sup> *<sup>H</sup>* , some dimensionless parameters, for example, Nusselt number (Nu) and Rayleigh number (Ra) have to be calculated:

$$Nu = 0.66 \left( \frac{Ra.H}{d\_{t, \rho}} \right)^{0.25} \left\{ 1 - \left[ 1 - 0.45 \left( \frac{d\_{t, \rho}}{H} \right)^{0.25} \right] \exp\left(\frac{-\varsigma\_w}{\phi}\right) \right\} \tag{13}$$

*Design, Performance, and Optimization of the Wire and Tube Heat Exchanger DOI: http://dx.doi.org/10.5772/intechopen.100817*

$$Ra = \left(\frac{\beta \rho^2 cp}{\mu k}\right)\_a \lg(T\_{t,o} - T\_{\infty}) \, H^3 \tag{14}$$

$$\phi = \left(\frac{28.2}{H}\right)^{0.4} s\_w \,^{0.9} s\_t^{-1.0} + \left(\frac{28.2}{H}\right)^{0.4} \left[\frac{262}{(T\_{t,o} - T\_\infty)}\right]^{0.5} s\_w \,^{-1.5} s\_t^{-0.5} p \tag{15}$$

$$s\_t = \frac{\left(p\_t - d\_t\right)}{d\_t} \tag{16}$$

$$s\_w = \frac{\left(p\_w - d\_w\right)}{d\_{w.}}\tag{17}$$

Once the ho was obtained, the ho was compared to hw. If the difference between ho and hw > 0.1 W m�<sup>2</sup> K�<sup>1</sup> , the ho was set equal to hw, and the calculation of hi, ho, and Q el was run until the convergence criteria were satisfied. At the steady-state condition, the heat transfer from the fluid to the surrounding air was equal to the heat transfer from the fluid to the outer surface of the tube. Thus, the outer surface temperature of the tube, Tt,o, was determined as follows:

$$T\_{\rm t,o} = T\_f - Q\_{\rm el} \left( \frac{1}{h\_i A\_i} + \frac{\ln \left( r\_o / r\_i \right)}{2 \pi k l} \right)\_{\rm el} \tag{18}$$

The calculated Tt,o was then compared to the initial Tt,o. If the error was more than 0.05°C, new Tt,o was substituted into Eq. (11) and the program was run to satisfy the convergence criteria. The calculation resulted in the Q el. For each element, the outlet fluid temperature was determined as follows:

$$T\_{\rm out} = \frac{Q\_{\rm el}}{\dot{m}} + T\_{\rm in} \tag{19}$$

where *m*\_ is the mass flow rate of the working fluid. The output of the i-th element will be used for the calculation for the (i + 1)-th element until the last element of the wire and tube heat exchanger. The summation of all elements yielded Q tot and the heat exchanger capacity was calculated using the following:

$$C = \frac{Q\_{\text{tot}}}{\dot{m}} \tag{20}$$

The heat exchanger efficiency was determined from the ratio between the actual heat transfer rate to the heat transfer rate if all wire temperature = tube temperature. For heat exchanger temperature equal to the tube temperature, the heat exchanger efficiency followed:

$$\eta\_{\text{tot}} = \frac{(\eta\_w A\_w + A\_t)}{A\_o} \tag{21}$$

For detailed simulation, the heat transfer process that was coupled with the fluid dynamics was modeled and solved using CFD Software package of ANSYS Fluent. Details of the step-by-step procedure of CFD can be found elsewhere [26, 27]. In this simulation, preprocessing, solving, and post-processing steps were respectively executed. The geometry of the wire and tube heat exchanger was modeled in a 1:1 geometric scale and meshed to crease finite elements. Then, the boundary type and boundary condition of modeled geometry were defined. To solve the governing equations (the conservation of energy, mass conservation, and momentum), the initial condition of each boundary condition was inputted. The turbulence model used was Reynold-Average Navier-Stokes (RANS) k-ω SST (shear-stress transport). This k-ω SST was used since it has a high stability in numerical calculation and could predict accurately the flow on adverse pressure gradient in boundary layer area [25].
