## 1. Introduction

Rectangular ducts are widely used in heat transfer devices, for instance, in compact heat exchangers, gas turbine cooling systems, cooling channels in combustion chambers and nuclear reactors. Forced turbulent heat convection in a square or rectangular duct is one of the fundamental problems in the thermal science and fluid mechanics. Recently, Qin and Fletcher [1] showed that Prandtl's secondary flow of the second kind has a significant effect in the transport of heat and momentum, as revealed by the recent Large Eddy Simulation (LES) technique. Several experimental and numerical studies have been conducted on turbulent flow though of non-circular ducts: Nikuradse [2]; Gessner and Emery [3]; Gessner and Po [4]; Melling and Whitelaw [5]; Nakayama et al. [6]; Myon and Kobayashi [7]; Assato [8]; Assato and De Lemos [9]; Home et al. [10]; Luo et al. [11]; Ergin et al. [12] Launder and Ying [13]; Emery et al. [14]; Hirota et al.[15]; Rokni [16]; Hongxing [17]; Yang and Hwang [18]; Park [19];Zhang et al. [20]; Zheng et al. [21]; Su and Da Silva Neto [22]; Saidi and Sundén [23]; Rokni [24]; Valencia [25]; Sharatchandra and Rhode [26]; Campo et al. [27]; Rokni and Sundén [28]; Yang and Ebadian [29] and others. The Melling and Whitelaw´s [5] experimental work shows characteristics of turbulent flow in a rectangular duct where they have been used a laser-Doppler anemometer in which report the axial development mean velocity, secondary mean velocity, etc. Nakayama et al. [6] show the analysis of the fully developed flow field in rectangular and trapezoidal cross-section ducts; finite difference method was implemented and the model of Launder and Ying [13] has been used. On the other hand, Hirota et al. [15] present an experimental work in turbulent heat transfer in square ducts; they show details of turbulent flows and temperature fields. Likewise, Rokni [16] carried out a comparison of four different turbulence models for predicting the turbulent Reynolds stresses, and three turbulent heat flux models for square ducts. The literature presents various turbulence modeling in which confirm Linear Eddy Viscosity Models (LEVM) can give inaccurate predictions for Reynolds normal stresses: it does not have ability to predict secondary flows in non-circular ducts due to its isotropic treatment. In spite of that, they is one of the most popular model in the engineering due to its simplicity, good numerical stability in which can be applied in a wide variety of flows. Thus, the Nonlinear Eddy Viscosity Model (NLEVM) represents a progress of the classical LEVM in which this last one gives inequality treatment of the Reynolds normal stresses, needs of conditions for calculating turbulence-driven secondary flow in non-circular ducts and it has relatively high cost for solving the necessary two-equation formulation. The Reynolds Stress Model (RSM), also called second order or second moment closure model, is very accurate in the calculation of mean flow properties and Reynolds stresses, for simple to more complex flows including wall jets, asymmetric channel, non-circular duct and curved flows. However, RSM has some disadvantages, such as, very large computing costs. For calculating the turbulent heat fluxes, the Simple Eddy Diffusivity (SED) and Generalized Gradient Diffusion Hypothesis (GGDH) models have been adopted and investigated. Most of the works presented in the literature show results assuming constant temperature on the wall. However, in many engineering applications the heat fluxes and surface temperatures are non-constants around the duct, therefore becoming important the knowledge of the variation of the conductance around the duct, according to Kays and Crawford [30]. According to Garcia´s developments [31], it is possible to carry out analysis with non-constant wall temperature boundary conditions. In this case, it is necessary to define a value that represents the mean wall temperatures in a given cross section, in which he has named T_{Wm}. Following this treatment, for the present paper, important results have been computed and they are here being presented for rectangular cross-section ducts. Fluids such as air and water were analyzed under the influence of non-constant wall temperature distributions, with consequent presentations of resulted about turbulent convective heat exchanges and flow temperature profiles.

## 2. Mathematical formulation

### 2.1. Governing equations

The Reynolds Averaged Navier Stokes (RANS) equation system is composed of: continuity equation (1), momentum equation (2), and energy equation (3).

For analyses of fully developed turbulent flow and heat transfer, the following hypothesis has been adopted: steady state, condition of non-slip on the wall and fluid with constant properties. The turbulent Reynolds stress

### 2.2. Turbulence models for reynolds stresses

#### 2.2.1. Nonlinear Eddy Viscosity Model (NLEVM)

The NLEVM Model to reproduce the tensions of Reynolds, it is necessary to include non-linear terms in the basic constitutive equations. This is done by attempting to capture the sensitivity of the curvatures of the stream lines. This model is based on the initial proposal of Speziale [35]. The Reynolds average equations, Equations (1) to (3), are applied for the device presents in the Figure 1(a) and (b).

The velocity components *U and V* represent the secondary flow, and the axial velocity component *W*, the velocity of the main flow. The transport equations in tensorial form for the turbulent kinetic energy, κ, and the rate of dissipation ε, respectively, they are given by:

The symbols

In the present work for NLEVM, the formulations of Low Reynolds Number will be assumed for wall treatment. The damping functions*P* refers to the nodal point near to the wall. Thus

This expression shows that the second term of the right side in Equation (7) represents the nonlinear relation added the original constitutive relation. This quadratic term represents the degree of anisotropy between the normal tensions of Reynolds responsible for predicting the secondary flow in non circular ducts. The values of

The following differences for the normal tensions of Reynolds are presented and used in order to predict the anisotropy in turbulent flow at non circular ducts,

Therefore, the Equation (6), including the tensions of Reynolds given in Equation (9), the turbulence production term is expressed as:

#### 2.2.2. Reynolds Stress Model (RSM)

The most complex turbulence model is the Reynolds Stress Model (RSM), also known as second order model. It involves calculations of Reynolds stresses to an individual form,

(12) |

Where the letters represent: (a) Local derivative of the time; (b)

### 2.3. Turbulence models for turbulent heat flux

#### 2.3.1. Simple Eddy Diffusivity (SED)

This method is based on the Boussinesq viscosity model. The turbulent diffusivity for the energy equation can be expressed as:

#### 2.3.2. Generalized Gradient Diffusion Hypothesis (GGDH)

Daly and Harlow [38] present the following formulation to the turbulent heat flux:

The constant

#### 2.3.3. Dimensionless energy equation for SED and GGDH models

For a given cross section of area “*A*”, it is possible to define a mean velocity “*U*_{b}” and a bulk temperature “*T*_{b}*”*, expressed as:

and

Kays and Crawford [30] developed a formulation to rectangular cross section ducts. They considered the boundary conditions with prescribed uniform wall temperatures at the cross section, and at the duct length. According to Garcia [31], it is possible to carry out an analysis with non-uniform wall temperature boundary conditions. In this case, it is necessary to define a value that represents the mean wall temperatures in a given cross section, “*T*_{Wm}”, given as:

It is possible to develop a formula similar to Kays and Crawford [30], and a new expression for the turbulent energy equation can be presented as:

The following considerations are applied to obtain the variables in dimensionless form:

and

Replacing the Equations (13) or (14), (19)-(21) in Equation (18), dimensionless energy equations for SED and GGDH are obtained, respectively, expressed as:

The fluid temperature field “*T*_{f}” can be replaced by “*T*_{b}” and the Equation (21) can be expressed as:

and

From Equation (21), the Equation (26) is obtained, and applying this in Equation (16), the bulk temperature is obtained and expressed in Equation (27):

and

Replacing Equation (27) in Equation (24), and using Equations (19) and (20), the dimensionless bulk temperature is given as:

It is possible to compute the heat transfer rate per unit length on the wall surface, “ *q’* ”, as shown in Equation (29) in function of “*T*_{Wm}”, “*T*_{b}”, and the average heat convection coefficient, “*dh*_{b}
*= c*_{p}*.dT*_{b}], the heat transfer rate per unit length in the fluid, “ *q’*_{f} ”, can be expressed by Equation (30).

and

Equation (30) can be integrated to two cross sections (inlet, *z*_{1}*,* and outlet, *z*_{2}), thus, the following expression is obtained,

From Equation (31), the bulk temperature longitudinal (*z-axis*) variation “*ΔT*_{b}” is obtained. It is done by “cutting” the duct into a lot of segments and applying the numerical method to find *“ΔT*_{b}*”* at each finite cross section. For a given bulk temperature at the duct inlet section (*T*_{b1}), after solving the equation system, duct outlet bulk temperature (*T*_{b2}) is calculated from Equation (31). The Dimensionless boundary conditions are given by the following equations:

and

When considering uniform wall temperature, the Equations (32) and (33) are equal to zero, and for these particular conditions, it is possible to notice that these boundary conditions are not functions of “*ϕ”* and “*dT*_{b}*/dz”* parameters are unknown. When that equation system is solved, it is possible to obtain “*T*_{f}*”.*

#### 2.3.4. Additional equations

Additional equations were utilized for the calculation of the factor of friction Moody,

## 4. Numerical implementation

After applying the method of finite differences to the algebraic equations, to obtain the temperature fields, the following five steps indicate the developed methodology in the numerical solution. (Garcia [31]):

Step 1. To define the function value of the non uniform temperatures in the walls of the duct

Step 2. To obtain velocity field and estimated values for “*“T*_{Wm}*” and* “*dT*_{b}*/dz*”;

Step 3. Equations for the boundary conditions are evaluated (Equations 32 and 33);

Step 4. Dimensionless energy equation (Temperature field, *“Ø”*) the Equation (23) is solved and “*Ø*_{b}*”* is computed according to Equation (28), until convergence is obtained (*Ø*_{b} < tolerance). This is the end of the first iterative loop;

Step 5. A value for “*dT*_{b}*/dz*” is computed in accordance with Equation (25). Boundary conditions are updated (step 3) to obtain a solution for the new temperature field (step 4), until convergence is obtained (*dT*_{b}*/dz* < tolerance). This is the end of the second iterative loop;

For all steps, “tolerance of 10^{-7}” is the value to be accomplished by the convergence criteria, which is applicable to *“Ø*_{b}*”* (dimensionless bulk temperature), “*dT*_{b}
*/dz*" and “*Ø”* (dimensionless temperature field*)*.

## 5. Results and discussion

### 5.1. Fluid flow and heat transfer field

The Figure 2(a) shows the utilized grid (120X120) in the numerical simulation for the formulations of Low Reynolds, the Figure 2(b) it represents the secondary flow contours and comparisons of the velocity profile (NLEVM, Assato [8]) with the experimental work of the Melling and Whitelaw [5] for fluid water and Re=42000.

The predicted distributions of the friction coefficient (NLEVM and RSM) and Nusselt number (SED and GGDH) dependence on Reynolds number for fully developed flow and heat transfer in a square duct is shown in Figure 3(a) and 3(b), respectively.

Figure 4(a): comparisons of the Results (RSM-SED) numerical with the experimental for temperature profile (wall constant temperature) *Re*=65000 (Hirota [15]) are shown, the figure 4(b) shows the variation of the temperature profile with non-uniform wall Temperature: south=400K, north=373K, east=393K, west=353K; presented as Case I.

Already the Figure 5(a) shows: The variation of the temperature profile with non-uniform wall temperature, represented by means of functions sine (Case II), south=(350-20Sin(ζ))K, north=(400-50Sin(ζ))K, east=(330+20Sin(ζ))K, west=(350+50Sin(ζ))K. where ζ is function of the radians (0-π/2) and *i*,*j* (points number of the grid in the direction x and y, respectively). The Figure 5(b) shows the behavior of the “*T*_{b}” and “*DTb/dz*” for different square cross sections in the direction of the main flow, according to Equation (31).

The Figures 6 (a) and (b), shown the temperature distribution for a rectangular duct aspect ratio (1:2) represented by means of a function sine (Case II). A third case denominated Case III is represented by: south=*K*; North=*K*; east=*K;* west=*K. S*ome results for rectangular ducts are shown in the Table 1.

In the doctoral thesis Garcia (1996) was analyzed the laminar flow coupled to the conduction and radiation in rectangular ducts and concluded that as increases the aspect ratio, the Nusselt number found in the coupling, differs from that found for ducts with constant temperature imposed around the perimeter of the section. which shows that could be making a mistake to consider the literature results without calculating the energy equation.

In the present study, the variations of the average Nusselt number for a square duct and different cases analyzed (uniform and non uniform temperature in the perimeter) are minimal. Already in the case of rectangular duct with an aspect ratio (1:2), the variations should be taken into account as shown in Table 1.

## 5. Conclusions

The results shown what for the friction factor and Nusselt number in a wide range of the Reynolds number with uniform wall temperature have a reasonable approach with the experimental works and correlation of the literature, (Figure 3(a), (b)). The Figures 4(b) and 5(a) shown new results investigated in present study, which is observed a distortion of the temperatures field and as consequence a variation of the Nusselt number caused mainly by the distribution of the non-uniform wall temperature (Case I and II, with fluid air and *Re*=65000, respectively). Most applications can be approximated by the functions sine and cosine in the wall, but we are able to resolve by means of the methodology presented, any peripheral heat flux variation that can be expressed by a Fourier expansion (Kays and Crawford [30]). The Figure 5(b) shows the comparisons of the behavior of the curves “*T*_{b}” and “*DTb/dz*” in the direction of the main flow for Case II and Case uniform wall temperature. The variations of the average Nusselt number for a square duct and different cases analyzed (uniform and non uniform temperature in the perimeter) are minimal. Already in the case of duct with an aspect ratio (1:2) the variations should be taken into account. These results can be helpful in the project of thermal devices as in heat transfer and secondary flows in cavities, seals, channel of gas turbines and others.

### Nomenclature

*c*_{p} specific heat at constant pressure

*D* duct height

*D*_{h} hydraulic diameter,

*dP/dz* pressure gradient at *z* direction (longitudinal axis)

*f, C*_{f} factor of Moody´s friction, and Fanning´s friction coefficient, respectively.

*k*_{f} fluid thermal conductivity

*L* duct width

*Nu* Nusselt number

*P*_{e} perimeter

*P*_{k} turbulence production term

*Re* Reynolds number

*T* temperature

*T*_{b} internal flow bulk temperature

*T*_{Wm} wall mean temperature

*T*_{1}*, T*_{2}*, T*_{3} and *T*_{4} temperature distributions at duct wall (bottom, right side, top and left side)

*U*_{b} internal flow bulk velocity

*U, V and W* average velocity in the direction x, y and z; respectively

y^{+} dimensionless wall distance

### Greek Symbols

*Α* thermal diffusivity, α = k_{f} / (ρ. c_{p})

Μ dynamic viscosity

μ_{t} turbulent viscosity

ρ density