**4.4 Shell diameter**

Inside sell diameter (*Ds*) is calculated as follows:

$$D\_s = \sqrt{\frac{4AN\_t}{(CTP)\pi}}\tag{44}$$

Where *A* is the projected area of the tube layout expressed as an area corresponding to one tube and can be obtained from Eq. (45). Also, *Pt* is tube pitch and *CL* is the tube layout constant. **Figure 6** depicts two common tube layouts, square pitch and triangular pitch. The *CTP* is the tube count calculation constant that accounts for the incomplete coverage of the shell diameter by the tubes, due to necessary clearances between the shell and the outer tube circle and tube omissions due to tube pass lanes for multitude pass design [15]. Eq. (46) shows the *CTP* for different tube passes.

$$\begin{aligned} A &= P\_t^2(CL) \\ CL &= 1 \rightarrow for\ square-pitch\text{ }layer \\ CL &= 0.866 \rightarrow for\ triangular-pitch\text{ }layer \end{aligned} \tag{45}$$

$$\begin{aligned} CT &= 0.93 \rightarrow for\ one-tube\ pass \\ CT &= 0.9 \rightarrow for\ two-tube\ pass \\ CT &= 0.85 \rightarrow for\ three-tube\ pass \end{aligned} \tag{46}$$

Combining Eq. (44) with Eq. (45) as well as defining tube pitch ratio as Pr, one gets:

$$D\_s = \sqrt{\frac{4\left(P\_r d\_o\right)^2 (CL) N\_t}{(CTP)\pi}}\tag{47}$$

**Figure 6.** *(a) Square-pitch (b) triangular-pitch layout [20].*

$$P\_r = \frac{P\_t}{d\_o} \tag{48}$$

Where *do* is tube outside the diameter. Eq. (47) can be written as follows:

$$D\_s = \sqrt{\frac{4\left(P\_r^2\right)(CL)A\_o d\_o}{(CTP)\pi^2L}}\tag{49}$$

Where *Ao* is the outside heat transfer surface area based on the outside diameter of the tube and can be calculated from:

$$A\_o = \pi d\_o N\_t L \tag{50}$$

The shell side flow direction is partially along the tube length and partially across to tube length or heat exchanger axis. The inside shell diameter can be obtained based on the cross-flow direction and the equivalent diameter (*De*) is calculated along the long axes of the shell. The equivalent diameter is given as follows:

$$D\_{\epsilon} = \frac{4 \times free - flow\ area}{wetted\ perimeter} \tag{51}$$

From **Figure 6** the equivalent diameter for the square pitch and triangular pitch layouts are as below:

$$D\_{\epsilon} = \frac{4\left(P\_t^2 - \frac{\pi d\_o^2}{4}\right)}{\pi d\_o} \; ; \; for \; square-pitch \; tube \tag{52}$$

$$D\_{\epsilon} = \frac{4\left(\frac{\rho\_{\epsilon}^{2}\sqrt{3}}{4} - \frac{\pi d\_{o}^{2}}{8}\right)}{\frac{\pi d\_{o}}{2}} \; ; \; for \; triangular-pitch \; tube \tag{53}$$

Reynolds number for the shell-side (*Res*) can be obtained as follows:

$$\mathrm{Re}\_{\mathfrak{s}} = \left(\frac{\dot{m}\_{\mathfrak{s}}}{A\_{\mathfrak{s}}}\right) \frac{D\_{\mathfrak{e}}}{\mu\_{\mathfrak{s}}} \tag{54}$$

Where *ṁ<sup>s</sup>* is the flow rate of shell-side fluid, *μ<sup>s</sup>* is the viscosity of the shell-side fluid, and *As* is the cross-flow area at the shell diameter which can be obtained as below:

$$A\_s = \frac{D\_s}{P\_t} (B \times \mathbf{C}\_t) \tag{55}$$

Where *B* is the baffle spacing and *Ct* is the clearance between adjacent tubes. According to **Figure 6** *Ct* is expressed as follows:

$$\mathbf{C}\_{t} = P\_{t} - d\_{o} \tag{56}$$

The shell-side mass flow rate (*Gs*) is found with:

$$G\_s = \frac{\dot{m}\_s}{A\_s} \tag{57}$$

*Heat Exchanger Design and Optimization DOI: http://dx.doi.org/10.5772/intechopen.100450*

In Kern's method, the heat transfer coefficient for the shell-side (*hs*) is estimated from the following:

$$\begin{aligned} h\_{\mathfrak{s}} &= \frac{0.36 \mathfrak{k}\_{\mathfrak{s}}}{D\_{\mathfrak{e}}} \operatorname{Re}\_{\mathfrak{s}}^{0.55} \operatorname{Pr}\_{\mathfrak{s}}^{1/3} \\ &\text{for } 2 \times 10^3 < \operatorname{Re}\_{\mathfrak{s}} = \frac{G\_{\mathfrak{s}} D\_{\mathfrak{e}}}{\mu\_{\mathfrak{s}}} < 1 \times 10^6 \end{aligned} \tag{58}$$

Where *ks* is the thermal conductivity of the shell-side fluid. The tube-side pressure drop is calculated by the following:

$$
\Delta P = \frac{f\_s G\_s^2 (N\_b + 1) D\_s}{2 \rho\_s D\_e \left(\frac{\mu\_b}{\mu\_{w\_s}}\right)^{0.14}} \tag{59}
$$

Where *Nb* is the number of baffles, *ρ<sup>s</sup>* is the density of the shell-side fluid, *μ<sup>b</sup>* is the viscosity of the shell-side fluid at bulk temperature, and *μw,s* is the viscosity of the tube-side fluid at wall temperature. The *fs* is the friction factor for the shell and can be obtained as follows:

$$f\_s = \exp\left[0.576 - 0.19 \ln\left(\text{Re}\_s\right)\right];\ for\ 400,\ \text{Re}\_s < 1 \times 10^6\tag{60}$$

The wall temperature can be calculated as follows:

$$T\_w = \frac{1}{2} \left( \frac{T\_{H,in} + T\_{H,out}}{2} + \frac{T\_{C,in} + T\_{C,out}}{2} \right) \tag{61}$$

According to Eq. (21), the heat transfer surface area (*A*) of the shell and tube heat exchanger is obtained by the following:

$$A = \frac{Q}{U\_m F \Delta T\_{LMTD}}\tag{62}$$

The required length of the heat exchanger can be calculated based on the heat transfer surface area as follows:

$$L = \frac{A}{\pi d\_o N\_t} \tag{63}$$

#### **5. Optimization of heat exchangers**

The applications of heat exchangers are very different. Therefore, they are optimized based on their application. The most common criteria for optimizing heat exchangers are the minimum initial cost, minimum operating cost, maximum effectiveness, minimum pressure drop, minimum heat transfer area, minimum weight or material, etc. These criteria can be optimized individually or in combination. It is clear from the above that the optimal design of heat exchangers is based on many geometrical and operational parameters with high complexity. So it is difficult to design a cheap and effective heat exchanger. The optimization techniques are usually applied to ensure the best performance as well as lower the cost of the heat exchanger. The optimization is carried out using different techniques. Traditional techniques such as linear and dynamic programming as well as steepest descent usually fail to solve nonlinear large-scale problems. The need for gradient information is another drawback of traditional techniques. Therefore, it is not possible to solve non-differentiable functions using these methods. To overcome these difficulties, advanced optimization algorithms are developed which are gradient-free. Several advanced optimization methods, such as genetic algorithm (GA) [21], non-nominated sorting GA (NSGA-II) [22], bio-geographybased optimization (BBO) [23], particle swarm optimization (PSO) [24], Jaya algorithm, and teaching–learning-based optimization (TLBO) [25], had been used for the optimization of heat exchangers by many researchers each of which has its advantages and disadvantages. Using GA, it is possible to solve all optimization problems, which can be described with the chromosome encoding and solves problems with multiple solutions. But in order to use GA, it is necessary to set a number of specific algorithmic parameters such as jump probability, selection operator, cross probability. NSGA-II has an explicit diversity preservation mechanism and elitism prevents an already found Pareto optimal solution from being removed, but crowded comparison can limit the convergence and it needs the tuning of algorithmic-specific parameters including mutation probability, crossover probability, etc. The optimization using BBO is also effective and it inhibits the degradation of the solutions, but poor exploiting the solutions is the main drawback of this method. The PSO is a heuristic and derivative-free technique that has the character of memory but it needs the tuning of algorithmic specific parameters and plurality of the population is not enough to achieve the global optimal solution. Similarly, TLBO and Jaya need the tuning of their own algorithmic-specific parameters [26].

Generally, an optimization design starts by selecting criteria (quantitatively) to minimize or maximize, which is called an objective function. In an optimization design, the requirements of a particular design such as required heat transfer, allowable pressure drop, limitations on height, width and/or length of the exchanger are called constraints. Several design variables such as operating mass flow rates and operating temperatures can also participate in an optimization design [15]. The single target optimization can be expressed as [27]:

$$\begin{array}{ll}\min & f(\mathbf{x})\\ \mathbf{g}\_{i}(\mathbf{x}) \ge \mathbf{0}, \ j = \mathbf{1}, \ 2, \ \dots, \ J\\ h\_{k}(\mathbf{x}) \ge \mathbf{0}, \ k = \mathbf{1}, \ \mathbf{2}, \ \dots, \ K\end{array} \tag{64}$$

Where *f(x)* is the objective function, *gi(x)* ≥ 0 is the inequality constraint, and *hi(x)* ≥ 0 is the equality constraint. Multi-objective combination optimization can be indicated as [27]:

$$\begin{aligned} \min \, \, f(\mathbf{x}) &= \left( \, f\_1(\mathbf{x}), \, \, f\_2(\mathbf{x}), \dots, \, f\_m(\mathbf{x}) \right) \\ \, \, g\_i(\mathbf{x}) &\ge 0, \, \, j = 1, \, \, 2, \quad \dots, \, \, J \\ \, \, h\_k(\mathbf{x}) &\ge 0, \, \, k = 1, \, \, 2, \quad \dots, \, \, K \end{aligned} \tag{65}$$

Using the data modeling, the optimization of a heat exchanger can be transformed into a constrained optimization problem and then solved by modern optimization algorithms. In this chapter, the focus is on GA and PSO because many researchers mentioned that these algorithms lead to remarkable savings in computational time and have an advantage over other methods in obtaining multiple solutions of the same quality. So it gives more flexibility to the designer [2].
