**4.2 Number of tubes**

The number of tubes (*Nt*) can be calculated as follows:

$$N\_t = \frac{4\ \dot{m\_t}}{\rho\_t \nu\_t \pi d\_i^2} \tag{32}$$

Where *ṁ<sup>t</sup>* is the flow rate of fluid inside the tube, *ρ<sup>t</sup>* is the density of the fluid inside the tube, *ν<sup>t</sup>* is the velocity of the fluid inside the tube, *At* is the cross-sectional area of the tube, and *di* is the tube inside diameter.

**Figure 5.** *Schematic of a shell and tube heat exchanger a) fixed-tube b) floating-head c) removable U-tube [18].*

### **4.3 Tube-side heat transfer coefficient**

The heat transfer coefficient for the tube side (*ht*) is calculated as follows:

$$h\_t = Nu\_t \frac{k\_t}{d\_i} \tag{33}$$

Where *Nut* is the Nusselt number for the tube-side fluid and *kt* is the thermal conductivity of the tube-side fluid. The *Nut* is a function of Reynolds number (*Re*) and Prandtl number (*Pr*). *Re* and *Pr* can be obtained by the following:

$$\text{Re}\_t = \frac{\rho\_t \nu\_t d\_t}{\mu\_t} \tag{34}$$

$$\text{Pr}\_t = \frac{\text{C}\_p \mu\_t}{K} \tag{35}$$

Where *μ<sup>t</sup>* is the dynamic viscosity of the tube-side fluid, *K* is the heat conductivity coefficient, and *Cp* is the heat capacity of the tube-side fluid. The *Nut* can be calculated according to the type of flow as follows:

$$Nu\_t = \frac{\left(f\_t/2\right) \text{Re}\_t \text{Pr}\_t}{1.07 + 12.7 \left(f\_t/2\right)^{1/2} \left(\text{Pr}\_t^{2/3} - 1\right)}; \text{for} : 10^4 < \text{Re} < 5 \times 10^6 \&\ 0.5 < \text{Pr} < 2000$$

$$\text{(36)}$$

$$Nu\_l = 1.86 \left( \frac{\text{Re}\_l \text{Pr}\_l d\_l}{L} \right)^{1/3}; \text{ for}: \ \left( \frac{\text{Re}\_l \text{Pr}\_l d\_l}{L} \right)^{1/3} > 2 \ \& \ 0.48 < \text{Pr} < 16700 \quad (37)$$

Where *L* is the length of the tube and *ft* is the friction factor of the tube side, which can be calculated from

$$f\_t = (\mathbf{1.58} \ln \text{Re}\_t - \mathbf{3.28})^{-2} \tag{38}$$

The convection heat transfer coefficient in the tube is obtained based on the value of the Ret from [19]:

$$h\_t = \frac{k\_t}{d\_i} \left[ 3.657 + \frac{0.0677 \left( \mathrm{Re}\_t \mathrm{Pr}\_t \frac{d\_i}{L} \right)^{1.3}}{1 + 0.1 \mathrm{Pr}\_t \left( \mathrm{Re}\_t + \frac{d\_i}{L} \right)^{0.3}} \right]; \text{ for } \,\mathrm{Re}\_t < 2300 \tag{39}$$

$$h\_{l} = \frac{k\_{l}}{d\_{i}} \left[ \frac{\frac{1}{8} (\text{Re}\_{l} - 1000) \text{Pr}\_{l}}{1 + 12.7 \sqrt{\frac{1}{8} (\text{Pr}\_{l}^{0.67} - 1)}} \left( 1 + \left( \frac{d\_{i}}{L} \right)^{0.67} \right) \right]; \text{ for } 2300 < \text{Re}\_{l} < 10000 \text{ }^{\circ} \text{C} \tag{40}$$

$$h\_t = \frac{k\_t}{d\_i} 0.027 \operatorname{Re}\_t^{0.8} \text{Pr}\_t^{\frac{1}{2}} \left(\frac{\mu\_t}{\mu\_{w,t}}\right)^{0.14}; \text{ for } \text{Re}\_t > 10000 \tag{41}$$

Where *μw,t* is the dynamic viscosity of the tube-side fluid at the wall temperature and λ is the Darcy friction coefficient which can be defined as [19]:

$$
\lambda = \left(\mathbf{1.82}\,\log\_{10}\log\_{10}\mathbf{Re}\_l - \mathbf{1.64}\right)^2 \tag{42}
$$

The tube-side pressure drop is calculated by the following:

$$
\Delta P\_t = \left( 4f\_t \frac{LN\_p}{d\_i} + 4N\_p \right) \frac{\rho\_t \mu\_t^2}{2} \tag{43}
$$

Where *Np* is the tube passes.
