**3. Result and discussion**

All of the cases defined in the previous section are solved using CFD by calculating the heat flow at every wall and the average temperature of the air in the enclosure. The solutions to all of the cases are considered to have converged when the convergence criteria adopted in the CFD simulations are met. In our case, the criteria are two-fold: the residues of the equations must be under 10�<sup>8</sup> ; however, the monitored variable (heat flow at the wall) must converge to a constant value within at least 1000 iterations. With these constraints in the simulation software, most of the cases converge within 5000 iterations.

#### **3.1 Proposed correlations for enclosures with walls at equal temperatures**

To obtain a correlation that calculates the average heat-transfer coefficients of walls of a 2D enclosure based on geometric and hydrodynamic parameters, a mathematical treatment of the data is necessary to determine the function that best fits the simulation results. Additionally, the correlation must have the same form as the flat plate under forced convection. That is, it must depend on the following: the Reynolds number at the inlet to the power n; the Prandtl number for air (0.72); and a constant C that depends on all of the geometric parameters.

$$Nu\_i = \mathbf{C} \cdot \operatorname{Re}\_{in}^n Pr \tag{33}$$

$$\mathbf{C} = f\left(\frac{L}{H}, \frac{\mathcal{W}\_{in}}{H}, \frac{\mathcal{W}\_{out}}{H}, \frac{H\_{in}}{H}, \frac{H\_{out}}{H}\right) \tag{34}$$

Once the simulation results are obtained, mathematical optimisation techniques are applied to obtain the correlation coefficients that best fit the results from the CFD simulations. The form of the correlation that best fits the CFD results is

Eq. (35), which is obtained in dimensionless form through the Nusselt number and Eq. (36). The correlation coefficients for the walls are shown in **Table 4**.

$$Nu\_i = \left[ a \frac{W\_{in}}{H} + b \frac{W\_{out}}{H} + c \frac{H\_{in}}{H} + d \frac{H\_{out}}{H} + e \frac{L}{H} + f \right] \cdot Re\_{in}^n \tag{35}$$

$$h\_i = \frac{Nu\_i k}{W\_{in}}\tag{36}$$

**Figures 6** and **7** show a comparison between the results from correlations and those obtained from the simulations. A mean error of approximately 15% was observed; however, the correlation describes the phenomenon with sufficient precision despite the complexity of the cases being studied.

### **3.2 Proposed correlations for enclosures with walls at different temperatures**

In cases where the walls are at different temperatures, the heat-transfer coefficient is obtained from the corresponding coefficient through correlations for walls at the same temperature with the correcting factor. This factor depends on the flow and the positions of the inlet and outlet. **Figure 8** shows a correlation of the correction factors for each wall of an enclosure of typology 2, where a variety of cases have been simulated, and the temperatures of the walls and air at the entry are


#### **Table 4.**

*Correlation coefficients for 2D enclosures under forced convection.*

*A New Forced Convection Heat Transfer Correlation for 2D Enclosures DOI: http://dx.doi.org/10.5772/intechopen.99375*

**Figure 7.** *Comparison of Nu3 and Nu4 obtained through correlations and simulations.*

**Figure 8.** *Correlation between the correcting factor with T*∞*<sup>i</sup> and Tin. Enclosure of typology 2.*

varied. As shown in **Figure 8**, the fit and function that relates the correcting factor obtained through *T*∞*<sup>i</sup>* to that obtained through *Tin* are good. To appreciate the quality of the fit, **Figure 9** shows the heat-transfer coefficient predicted using the correcting factor from CFD for each of the walls. The precision is high, which demonstrates that this factor must be employed to precisely calculate the heattransfer coefficients when the walls have different temperatures. We proceed in the same fashion for the remaining typologies. **Table 5** shows the coefficients of the correlation for the correction factor for each wall and for the five typologies. Eq. (37) provides the heat-transfer coefficient when the wall temperatures are different.

#### **Figure 9.**

*Heat-transfer coefficient calculated using the correcting factor compared with the heat-transfer coefficient calculated using CFD.*


**Table 5.** *Correcting factors for all typologies.* *A New Forced Convection Heat Transfer Correlation for 2D Enclosures DOI: http://dx.doi.org/10.5772/intechopen.99375*

$$h\_i^{T\_i \neq} = h\_i^{corr} \cdot \left( a\_i \left( \frac{Ts\_i - T\_{in}}{Ts\_i - T\_a} \right) + b\_i \right) \tag{37}$$
