**6. CFD Simulation Results and Nusselt Number Correlations**

The CFD simulation results of average mass flow temperatures behind the tube rows are shown in Table 2.


**Table 2.** CFD simulation data.

Presented correlations for air-side Nusselt numbers relate to Reynolds numbers. The Reynolds number (Equation (5) is related to the hydraulic diameter, which is calculated using the definition proposed by Kays and London [23], where *wmax* is given by Equation (6). Parameter *wmax* has been calculated for the minimum airflow cross-section between tubes and it can exist in a different place in case of different PFTHE construction. The hydraulic diameter has been calculated by assumption dividing the volume through which air flows in one row by the surface area in contact with the air. Most often, this hydraulic diameter in the case of PFTHE is a bit smaller than double the fin pitch, and using geometry from Figure 1, equals 5.35 mm.

Reynolds numbers (*Redh*,*a*) (Equation (5) and maximal velocities (*wmax*) (Equation (6)) have been calculated for each row separately, due to a different maximum air velocity caused by higher air temperature causing increased air volume. The calculated parameters such as Reynolds numbers, HTCs, Nusselt numbers, and Colburn factors are presented in Tables 3–6.

$$Re\_{d\_{\hbar},a} = \frac{w\_{\text{max}} \cdot d\_{\hbar}}{\nu\_a} \tag{5}$$


**Table 3.** Calculated thermal parameters for the first row of PFTHE.

**Table 4.** Calculated thermal parameters for the second row of PFTHE.


**Table 5.** Calculated thermal parameters for the third row of PFTHE.



**Table 6.** Calculated thermal parameters for the first row of PFTHE.

The following designations are used in Equation (5): *Re dh*,*<sup>a</sup>* is the Reynolds number in case of air hydraulic diameter; *dh* is the air hydraulic diameter; *ν<sup>a</sup>* is the air kinematic viscosity.

$$w\_{\text{max}} = \frac{(s \cdot p\_l)}{\left(s - \delta\_f\right) \cdot (p\_l - d\_{o,\text{min}})} \cdot \frac{\left(\overline{T}\_a^i\right)}{\left(\overline{T}\_{a,o}\right)} \cdot w\_o \tag{6}$$

The symbol *wmax* designates the air velocity in the least cross-section of the air flow. The symbol *s* denotes fin pitch. Other symbols represent: *pl* the longitudinal fin pitch *δ<sup>f</sup>* the fin thickness, *do*,*min* the minimal dimension between tubes, *<sup>T</sup><sup>i</sup> <sup>a</sup>* the mass average air temperature in *i*-th row of PFTHE, and *Ta*,*o* the inlet mass average air temperature.

The approximation function was determined using the least-squares method. Using this function, the Nusselt number was approximated, depending on the Reynolds and Prandtl numbers, according to Equation (7). The range of Reynolds numbers considered is factored in the present study, and is also present in Equation (7).

$$Nu\_a = \varkappa\_1 \cdot Re^{\varkappa\_2} \cdot Pr^{\frac{1}{3}} \qquad \quad 140 \; < \; Re \; < \; 1500 \tag{7}$$

The heat transfer correlation for the entire heat exchanger is shown in Figure 6. The Nusselt numbers for the first and second rows of tubes as a function of the Reynolds number, calculated separately for each row, are illustrated in Figure 7a,b. The heat transfer correlations for the third and fourth rows of tubes are shown in Figure 8a,b. Figures 6–8 also show the confidence intervals for the Nusselt number calculated using Equation (7). The legend of each figure shows the corresponding correlation, either for the whole exchanger (Figure 6) or for a given row of tubes (Figures 7 and 8). Each figure also shows the limits of the 95% confidence intervals for the Nusselt number on a given tube row, where the coefficients *x*<sup>1</sup> and *x*<sup>2</sup> are determined by the least-squares method. For the 95% confidence interval, the values of the Nusselt number calculated for a given Reynolds number (Equation (7)) differ by +/−2 *σ*, where the symbol *σ* denotes the mean standard deviation of the Nusselt numbers obtained by the CFD modelling. The process of determining 95% confidence intervals for correlations per Nusselt number obtained by least-squares is detailed in Chapter 11 of Taler's book [24].

**Figure 6.** The average value of the Nusselt number as a function of the Reynolds number for the entire PFTHE.

**Figure 7.** The Nusselt number as a function of the Reynolds number: (**a**) for the first row of PFTHE; (**b**) for the second row of PFTHE.

**Figure 8.** The Nusselt number as a function of the Reynolds number: (**a**) for the third row of PFTHE; (**b**) for the fourth row of PFTHE.
