**2. Material and methods**

#### **2.1 Formulation**

An enclosure with 2D geometry under forced convection have been studied. This model corresponds to enclosures where the inlet and outlet openings are as large as the third dimension. In these cases, the 3D effects of flow are negligible compared with the effects that occur in the 2D plane [5]. In a 2D enclosure under forced convection with only one inlet and one outlet, three distinct typologies occur: typology 1 includes an inlet and outlet that are located on opposite walls; typology 2 includes an inlet and outlet that are located on adjacent walls; and typology 3, includes an inlet and outlet that are located on the same wall. The present work

focuses on typology 1, which is the most common in construction, although the developed methodology is valid for typologies 2 and 3 as well. **Figure 1** shows the geometric variables of the case being studied, including the dimensions of the enclosure, the positions and dimensions of the inlet and outlet openings and the temperature of the entry and walls.

For the formulation of the problem working with dimensionless numbers, the most important being the following:

$$Gr = \frac{\text{g} \cdot \beta \cdot (T\_s - T\_\infty) L\_c^3}{\nu^2} \tag{1}$$

$$Pr = \frac{\nu \cdot \rho \cdot Cp}{k} \tag{2}$$

$$Ra\_L = Gr \cdot Pr \tag{3}$$

$$Re\_{in} = \frac{V\_{in} \cdot W\_{in}}{\nu} \tag{4}$$

$$Nu\_i = \frac{h\_i \cdot \mathcal{W}\_{in}}{k} \tag{5}$$

$$Ri = \frac{Gr}{Re^2} \tag{6}$$

$$
\eta^{+} = \overline{\nu} \cdot \frac{\underline{u}\_{\tau}}{\nu} \tag{7}
$$

The Richardson number indicates the relative importance of natural convection with respect to forced convection in mixed convection processes. This number determines the processes that are more important for convection: For low Richardson numbers (Ri < <1), the Reynolds number is larger than the Grashof number, and forced convection is predominant. In the opposite case (Ri > > 1), the effects of forced convection are negligible compared with that of natural convection. However, if the Richardson number lies somewhere between these two limits, both effects are important, and convection is considered mixed. The present work studies cases of predominant forced convection and does not consider the effects of natural convection.

#### **2.2 CFD governing equations**

To calculate the heat-transfer coefficients, the velocity, temperature and pressure fields of the enclosure must be determined. Thus, we employed the

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

Navier–Stokes equations, which describe the fluid motion for a given set of boundary conditions. These equations along with the turbulence model and energy equation are solved at each node of the mesh.

The turbulence model employed here is the realisable k-ε model. This model differs from the standard k-ε model through a new formulation of turbulent viscosity and transport equation for ε. The equations for the 3D model are provided in tensor notation, where *xi* represents the variables X, Y and Z and *ui* represents the corresponding velocity components.

The continuity equation is as follows:

$$\frac{\partial \rho}{\partial t} + \frac{\partial (\rho u\_i)}{\partial \mathbf{x}\_i} = \mathbf{0} \tag{8}$$

The equation for conservation of momentum is as follows:

$$\frac{\partial(\rho u\_i)}{\partial t} + \frac{\partial(\rho u\_i u\_j)}{\partial \mathbf{x}\_j} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \frac{\partial}{\partial \mathbf{x}\_j} \left[ \mu \left( \frac{\partial u\_i}{\partial \mathbf{x}\_j} + \frac{\partial u\_j}{\partial \mathbf{x}\_i} - \frac{2}{3} \delta\_{ij} \frac{\partial u\_k}{\partial \mathbf{x}\_k} \right) \right] \tag{9}$$

The energy equation is as follows:

$$\frac{\partial(\rho T)}{\partial t} + \frac{\partial(\rho u\_i T)}{\partial \mathbf{x}\_i} = \frac{\partial}{\partial \mathbf{x}\_i} \left( \left( \frac{\mu}{Pr} + \frac{\mu\_t}{Pr\_T} \right) \frac{\partial T}{\partial \mathbf{x}\_i} \right) \tag{10}$$

The shear viscosity equation is as follows:

$$
\mu\_t = \rho \mathbf{C}\_\mu \mathbb{k}^2 / \varepsilon \tag{11}
$$

The variable *C<sup>μ</sup>* is calculated as follows:

$$C\_{\mu} = \frac{1}{4.04 + A\_s \frac{kU^\*}{c}} \tag{12}$$

where

$$\mathcal{U}^{\*} \equiv \sqrt{\mathcal{S}\_{\vec{\eta}} \mathcal{S}\_{\vec{\eta}} + \tilde{\mathcal{Q}}\_{\vec{\eta}} \tilde{\mathcal{Q}}\_{\vec{\eta}}};\\\tilde{\Omega}\_{\vec{\eta}} = \Omega\_{\vec{\eta}} - 2\varepsilon\_{\vec{\eta}k} o\nu\_{k};\ \mathcal{A}\_{\vec{\iota}} = \sqrt{\mathsf{f}}\cos\phi;\ \mathcal{S}\_{\vec{\eta}} = \frac{1}{2} \left(\frac{\partial u\_{j}}{\partial \mathbf{x}\_{i}} + \frac{\partial u\_{i}}{\partial \mathbf{x}\_{j}}\right) \tag{13}$$

The "k" transport equation in the turbulence model is as follows:

$$\frac{\partial(\rho k)}{\partial t} + \frac{\partial(\rho k u\_j)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_T}{\sigma\_k} \right) \frac{\partial k}{\partial \mathbf{x}\_j} \right] + G\_k + G\_b - \rho e + \mathbf{S}\_k \tag{14}$$

where Gk represents the production of turbulent kinetic energy, which is common to all k-*ε* turbulence models and is given by

$$G\_k = -\overline{u'\_i u'\_j} \frac{\partial u\_j}{\partial \mathbf{x}\_i} \tag{15}$$

The term Gb represents the generation of turbulent kinetic energy because of buoyant forces when the system is under a gravitational field, and it is calculated as follows:

*Applications of Computational Fluid Dynamics Simulation and Modeling*

$$\mathbf{G}\_b = \beta \mathbf{g}\_i \frac{\mu\_T}{Pr\_t} \frac{\partial T}{\partial \mathbf{x}\_i} \tag{16}$$

where Prt = 0.72 is the Prandtl number for energy and β is the thermal expansion coefficient, which is calculated as follows:

$$\beta = -\frac{1}{\rho} \left( \frac{\partial \rho}{\partial \mathbf{x}\_i} \right) \tag{17}$$

The transport equation for *ε* from the turbulence model is as follows:

$$\frac{\partial(\varepsilon)}{\partial t} + \frac{\partial(\rho \varepsilon u\_j)}{\partial \mathbf{x}\_j} = \frac{\partial}{\partial \mathbf{x}\_j} \left[ \left( \mu + \frac{\mu\_T}{\sigma\_\varepsilon} \right) \frac{\partial \varepsilon}{\partial \mathbf{x}\_j} \right] + \rho \mathbf{C}\_1 \mathbf{S} \mathbf{e} - \rho \mathbf{C}\_2 \frac{\varepsilon^2}{k + \sqrt{\nu \varepsilon}} + \mathbf{C}\_{1\varepsilon} \frac{\varepsilon}{k} \mathbf{C}\_{3\varepsilon} \mathbf{G}\_b + \mathbf{S}\_\varepsilon \mathbf{e} \tag{18}$$

The source terms *S<sup>ε</sup>* and *Sk* can be defined for each case and are optional. The coefficient *C*3*<sup>ε</sup>* is calculated as follows:

$$\mathbf{C}\_{3\varepsilon} = \tanh\left|\frac{v}{u}\right|\tag{19}$$

The constants used in the realisable k- *ε* model are as follows:

$$
\sigma\_k = 1.0, \sigma\_\varepsilon = 1.3, \mathbf{C}\_{1\varepsilon} = 1.44, \mathbf{C}\_{2\varepsilon} = 1.92, \mathbf{C}\_2 = 0.43, \mathbf{C}\_2 = 1.9 \tag{20}
$$

The heat-transfer coefficient between the surface of the wall and fluid in motion at different temperatures is provided by Newton's law of cooling and is dependent on the total heat of the wall, the transfer area and the temperature difference between the surface and unperturbed fluid:

$$\overline{h}\_{i} = \frac{Q\_{i}}{A\_{i}(T\_{si} - T\_{\infty})} \tag{21}$$

The total heat flow that is transferred between the fluid and the wall is calculated from the temperature gradient produced inside of the thermal boundary layer of the fluid through an integration of Fourier's law at each wall. Because the heat flow depends on the temperature gradient, it is solved using CFD. Thus, the heat flow at a wall is obtained by integrating the temperature gradient along the wall, and it is affected by the fluid's conductivity, as shown in Eqs. (22) and (23):

$$Q\_{\rm CFD} \Big|\_{X} = \int\_{0}^{L} k \frac{\partial T}{\partial \mathbf{y}} d\mathbf{x} \Big|\_{\mathbf{y}=\mathbf{0}} \tag{22}$$

$$Q\_{CFD}\Big|\_{Y} = \int\_{0}^{H} k \frac{\partial T}{\partial \mathbf{x}} d\mathbf{y} \Big|\_{\mathbf{x}=\mathbf{0}} \tag{23}$$

Therefore, at the fluid–solid interface, the heat transfer by conduction equals the heat transfer by convection, and the average heat-transfer coefficients are as follows:

$$\overline{h}\_{\rm ix} = \frac{\int\_0^L k \frac{\partial T}{\partial \gamma} d\boldsymbol{\omega}}{A\_i (T\_{si} - T\_{\infty})} \tag{24}$$

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

$$\overline{h}\_{\dot{\nu}} = \frac{\int\_0^H k \frac{\partial T}{\partial \mathbf{x}} dy}{A\_i (T\_{\dot{\kappa}} - T\_{\infty})} \tag{25}$$

Eqs. (24) and (25) show that the temperature difference between the wall (Tsi) and free fluid (T∞) is proportional to the temperature gradient of the thermal boundary layer. Thus, increases of the temperature transferred between the wall and fluid correspond to increases of the gradient within the boundary layer, which maintains a constant ratio and indicates that the heat-transfer coefficient is temperature independent. This finding is valid for flat plates where the fluid is not perturbed by other walls and for enclosures with walls whose temperatures are all equal. However, this finding is not valid for cases in which the walls have different temperatures.

#### **2.3 Computational model**

The computational model presented here considers a steady flow, 2D geometry and incompressible Newtonian fluid. All of the fluid properties remain constant, and it behaves as an ideal gas. All of the properties are evaluated at the fluid's average temperature within the enclosure. The phenomenon under study is forced convection; therefore, the velocities are large enough for all buoyancy effects to be negligible and for gravity to be ignored. The CFD results are obtained by solving the Navier–Stokes equations and energy equation through the finite volume method using the commercial software package Fluent V14 [20]. The SIMPLE (Semi-Implicit Method for Pressure Linked Equations) numerical algorithm developed by Patankar and Spalding [21] is also used. The equations for mass, momentum and energy are solved iteratively for the corresponding boundary conditions using numerical methods until convergence is reached for the variables of interest (velocity, temperature, pressure and heat flow at the walls).

One of the most important aspects to consider when using CFD tools is the construction of the mesh for the computational environment. The mesh utilised in the simulation is built from rectangular elements, and it enlarges from the walls toward the centre of the enclosure in a uniform fashion. This type of mesh produces a better convergence because there is a higher density of elements within the thermal boundary layer where the heat-transfer coefficient is calculated. To ensure the correct solution within the boundary layer, nodes must be placed within the viscous sub-layer. Thus, the first element must be located at a maximum distance of 1 mm from the wall, and the growth rate toward the centre of the enclosure must be between 10% and 15% [22].

The parameter that controls the correct solution of the viscous sub-layer is y+. This dimensionless parameter depends on the turbulence model [Eq. (26)]. Thus, for k-ε turbulence with "enhanced wall treatment," the parameter y + must have a value of approximately one. **Figure 2** shows an example of an enclosure represented by a mesh with an aspect ratio of one.

$$
\lambda \mathbf{y}^+ = \mathbf{y} \cdot \frac{\mathbf{u}\_\mathbf{r}}{\nu} \tag{26}
$$

#### **2.4 Validation of the CFD methodology**

To validate the CFD methodology employed in 2D enclosures under forced convection, the following problem with a known solution is used as a reference: the case of a flat plate. By solving this problem using the CFD method and comparing

#### **Figure 2.**

*Uniform-growth mesh for a 2D enclosure under forced convection.*

the results with those obtained by correlating the flat plate in both the laminar and turbulent regimes, the CFD methodology is validated. The problem to be solved is that of a horizontal plate of length L with an incident air flow at velocity vin in the parallel direction. The circulating air is at temperature T∞, whereas the flat plate is at temperature Ts. The computational domain is large enough so that it does not influence the solution. It is necessary to ensure that the number of nodes and type of mesh do not depend on the solution. The correct solution depends on the parameter y+, which must have a value of approximately 1. Therefore, the number of mesh elements and elements located within the limiting thermal layer for each case are adjusted to obtain y + ≈1.

To generalise the studied cases, the Reynolds number is set between 3 x<sup>10</sup><sup>3</sup> and 7 x<sup>10</sup><sup>6</sup> along the length of the plate in the direction of the air velocity. Thus, both the laminar and turbulent regimes are explored. The studied cases are presented in **Table 1**. To validate the methodology for the case of a flat plate, the CFD results are represented in terms of the average Nusselt number along with the results obtained through the flat plate correlations by Pohlhausen [23] and Reynolds analogy. As shown in **Figure 3**, the match between the CFD results and correlations is good and presents a relative error of less than 3%. Thus, the methodology employed for the mesh, convergence criterion and turbulence model are correct and can be used in the solution of 2D enclosures under forced convection.

#### **2.5 Case studies**

#### *2.5.1 Enclosures with walls at the same temperature*

In 2D enclosures under forced convection where all of the walls are at the same temperature, which is generally different from the air temperature at entry, the variables can be varied continuously. For the present work, only three values are considered for each variable. The typical enclosure height employed for construction is 3 m; therefore, this variable is kept fixed and used to rescale all other variables and make them dimensionless. **Table 1** shows the values used for each dimensionless variable. To solve the problem, a factorial experiment is conducted with six factors and four responses [24]. The variables are labelled as (Xj ), and the variable of interest is labelled as a response Yi. The relationship between factor and response is provided by the function Yi = f(Xi).

For the 2D case being studied, there are four responses: the average heat-transfer coefficients at each wall (Nu1, Nu2, Nu3 and Nu4) and six factors or independent variables (Vin/H, L/H, Win/H, Wout/H, Hin/H and Hout/H). **Table 2** shows the


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

#### **Table 1.**

*Validation cases for a flat plate and forced convection.*

**Figure 3.** *Validation of the CFD methodology for a flat plate and forced convection.*

### *Applications of Computational Fluid Dynamics Simulation and Modeling*


#### **Table 2.**

*Range of application of the variables (dimensionless).*

#### **Figure 4.**

*Study cases for the positions of the inlet and outlet openings for typology 1.*

numerical value of each variable.. The goal is to determine the function f that best fits the results from the simulation, and it is therefore necessary to solve the full range of cases to obtain the most general correlation possible.

To obtain enough data and the best fit to the results, all of the variables must be combined with their three possible values so that any possible combination within the variable applicability range is contemplated. Thus, the six variables can take three values each, which produces a total of 3<sup>6</sup> = 729 cases. However, because the mechanism being studied is forced convection, gravity does not influence the motion of the fluid, which means that the fluid motion is independent of the spatial orientation of the enclosure. Therefore, we can infer that among all possible cases, certain cases will be symmetric with respect to the positions of the inlet and outlet (Hin and Hout). **Figure 4** shows all of the possible cases when only the positions of the inlet and outlet are varied.

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

**Figure 4** shows that there are nine possible typologies as a function of the positions of the inlet and outlet openings. It is only necessary to study five out of these nine because the remaining typologies are symmetric cases because of forced convection. In these symmetric cases, the heat-transfer coefficients at the inlet and outlet (walls one and three, respectively) are equal; the coefficient of wall two is equal to that of wall four in its symmetric case, and wall four is equal to that of wall two in its symmetric case. Therefore, symmetry considerations reduce the number of cases to be studied from nine to five, thus affecting the values used for the variables Hin and Hout. The corresponding number of cases is reduced from 729 down to 405 (3x 3x 3x 3x 5). To avoid convergence problems and inconsistent results, all of the cases where the inlet opening is much larger than the outlet opening (Win> > Wout) are eliminated. Thus, out of the nine possible combinations of Win and Wout, three are eliminated. This reduced the number of cases from 405 to 270 (3x 3x 6x 5).

Through this analysis, we have reduced the number of study cases as well as the simulation time and work involved in elaborating the meshes. Therefore, the parametric study is feasible, with 270 cases to be solved, and statistical methods are not required to reduce the range of study cases.

#### *2.5.2 Enclosures with walls at different temperatures*

In the cases where the wall temperatures are all different, the heat-transfer coefficient depends on the temperature distribution of the enclosure walls regardless of the occurrence of forced convection. **Table 3** shows a comparison of the results for walls at equal temperatures with those for walls at different temperatures to demonstrate this influence.

**Table 3** shows that the heat-transfer coefficients obtained for different wall temperatures do not match those for walls at equal temperatures because the fluid temperature considered in Newton's law of cooling is the average enclosure temperature (Ta), which is different from that of unperturbed fluid (T∞). In cases where the wall temperatures are equal, both temperatures (Ta and T∞) are consistent, and the heat-transfer coefficients are equivalent for all cases. However, when the walls temperatures are not equivalent, the mean temperature of enclosure Ta is not consistent with the unperturbed fluid temperature T∞, which leads to different heat-transfer coefficients.


**Table 3.**

*Casos estudiados en 2D con convección forzada y temperaturas de las Paredes distintas.*

**Figure 5.** *Relationship between Tin and T*∞*<sup>i</sup> for each wall.*

In these cases, a correction factor for temperature must be obtained that considers this phenomenon. This factor (Fi) relates the heat-transfer coefficient in the case of different wall temperatures (hTi6¼) to the corresponding value for equal wall temperatures (hcorr) as follows:

$$h\_i^{T\_i \neq} = h\_i^{corr} \cdot F\_i \tag{27}$$

Therefore, for a given *T*∞*<sup>i</sup>* for cases solved using CFD, a correcting factor for the temperature jump ð Þ *Tsi* � *Ta* must be obtained so that the heat-transfer coefficient and the average air temperature *Ta* can be calculated from this correcting factor. Newton's law of cooling and the heat-transfer coefficient *hcorr <sup>i</sup>* for the case of walls with equal temperatures are applied, and the correcting factor is provided by Eq. (30):

$$Q\_{wall\\_i}^{\text{CFD}} = A\_i h\_i^{corr} (T\mathfrak{s}\_i - T\_{\text{osi}}) \tag{28}$$

$$\mathbf{Q}\_{wall\\_i}^{\rm CFD} = A\_i h\_i^{corr} (T\mathbf{s}\_i - T\_a) \cdot F\_i \tag{29}$$

$$F\_i = \frac{(T\mathfrak{s}\_i - T\_{\infty i})}{(T\mathfrak{s}\_i - T\_a)}\tag{30}$$

Eq. (30) shows that the correction factor Fi depends on T∞i, which is calculated using CFD through Newton's cooling law as well as through the heat flow and heat-transfer coefficient obtained from the following correlation:

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

$$T\_{\text{osi}} = \frac{Q\_{wall\\_i}^{CFD}}{A\_i h\_i^{corr}} \tag{31}$$

However, the heat flow is only known for CFD cases; therefore, a correction factor must be calculated that does not depend on T<sup>∞</sup> but depends on a known variable, such as Tin. This temperature was employed by Novoselac [10, 11] to obtain the correction factor in his experiments. **Figure 5** shows the temperature difference using Tin and T∞i, which produces an excellent match; thus, the variable Tin can be used in the correction factor.

We conclude that the correction factor can be written as a function of Tin, Ta and Tsi in the following equation:

$$F\_i = \frac{T\mathfrak{s}\_i - T\_{osi}}{T\mathfrak{s}\_i - T\_a} = f\left(\frac{T\mathfrak{s}\_i - T\_{in}}{T\mathfrak{s}\_i - T\_a}\right) \tag{32}$$

This correcting factor depends on the flux and positions of the inlet and outlet openings. Thus, we have demonstrated how to calculate this factor and must determine a correlation expression for the factor for each typology of the enclosure. For that purpose, 100 cases are simulated using CFD, with 20 cases for each of the five enclosure typologies where the temperatures of the wall and inlet are varied randomly. Thus, a correlation of the correction factor can be obtained for each typology and for each wall of the enclosure.
