Numerical Simulation of Turbulent Forced Convection

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[22] G.A. Rivas Ronceros, Simulação numérica da convecção forçada turbulenta acoplada à condução de calor em dutos retangulares,. Doctor degree thesis ...
International Review of Mechanical Engineering (I.RE.M.E.), Vol. 8, N. 3 ISSN 1970 - 8734 May 2014

Numerical Simulation of Turbulent Forced Convection Coupled to Heat Conduction in Square Ducts Gustavo Adolfo Ronceros Rivas1, Ézio Castejon Garcia2, Marcelo Assato3 Abstract – In the present work, the numerical simulation was adopted to resolve the problem of the turbulent forced convection coupled to heat conduction in a square cross-section duct. The governing equations for turbulent convection are the continuity, momentum, and energy equations. These equations are coupled to heat conduction comings through of four plates situated around of the channel flow. The plates among itself with thermal resistance of ideal contact are coupled. Two turbulence models to resolve the momentum equations and one model to resolve the energy equation were used. To determine the profiles of velocity, the models of turbulence, k-ε Non Linear Eddy Viscosity Model (NLEVM) and Reynolds Stress Model (RSM) were adopted. The fluid temperature field was determined from the model Simple Eddy Diffusivity (SED). The dimensionless energy equation was developed in a code of programming FORTRAN. The models have been validated in base the experimental and numerical results of literature. Finally, the results of this investigation allow evaluating the fluid temperature field for different square crosssectional sections throughout of the main flow direction, which is influenced mainly by the temperature distribution at the wall. Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved.

Keywords: Numerical Simulation, Turbulent Forced Convection, Square Ducts, Coupled Solid Fluid, Thermal Systems

  μ μt  t

Nomenclature cp D Dh dP/dz f, Cf kf L Nu Pe Pk Re T Tb TWm T1, T2, T3 and T4 Ub U, V and W y+

Specific heat at constant pressure Duct height Hydraulic diameter, Dh  4  A Pe Pressure gradient at z direction (longitudinal axis) Factor of Moody´s friction, and Fanning´s friction coefficient, respectively Fluid thermal conductivity Duct width Nusselt number Perimeter Turbulence production term Reynolds number Temperature Internal flow bulk temperature Wall mean temperature Temperature distributions at duct wall (bottom, right side, top and left side, respectively) Internal flow bulk velocity Average velocity in the Cartesian plane in the direction x, y and z, respectively Dimensionless wall distance

Distance normal to the wall Dimensionless temperature distribution Dynamic viscosity Turbulent viscosity Density Turbulent Prandtl number

I.

Introduction

Square ducts are widely used in heat transfer devices. For example, in compact heat exchangers, gas turbine cooling systems (secondary flows), cooling channels in combustion chambers, nuclear reactors. The forced turbulent heat convection in a square duct is one of the fundamental problems in the thermal science and fluid mechanics. Recently, in [1] is showed that the Prandtl’s secondary flow of the second kind has a significant effect on the transport of heat and momentum as revealed by the recent large eddy simulation (LES). Several experimental and numerical studies have been conducted on turbulent flow though a non-circular duct, namely, [2]-[8]. Similarly important works in the turbulent heat convection were developed as can be found in [9]-[16]. The experimental work of [5] shows detailed characteristics of turbulent flow in a rectangular duct where they used a laser-Doppler anemometer to report the axial development of the mean velocity, secondary mean velocity, etc. In reference [6], it shows the analysis

Greek Symbols α Thermal diffusivity, α = kf / (ρ·Cp)

Manuscript received and revised April 2014, accepted May 2014

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645

G. A. R. Rivas, E. C. Garcia, M. Assato

the fully developed flow field in ducts of rectangular and trapezoidal cross-sections using a finite-difference method with the model of [9]. On the other hand, in [11] presents an experimental work on the turbulent heat transfer in a square duct, shows detailed characteristics of turbulent flow and temperature field. Likewise, in the doctoral thesis, [17], carries out a comparison of four different turbulence models for predicting the turbulent Reynolds Stresses and three turbulent heat fluxes models for ducts square. In the turbulence model it is well nown that Linear Eddy Viscosity Models (LEVM) can give rise to inaccurate predictions for the Reynolds normal stresses and so that not have the ability to predict secondary flows of the second kind. In spite of that, they are one of the most popular models in the engineering due to its simplicity, good numerical stability and it can be applied to a wide variety of flows. Thus, NLEVM represents a progress of the classical LEVM which permits inequality of the Reynolds normal stresses, a necessary condition for calculating turbulence-driven secondary flow in noncircular ducts within the relative cost of a two-equation formulation. The model RSM, also called the second order or second moment closure model is very accurate in the calculation of mean flow properties and all Reynolds stresses for many simple and more complex flows including wall jets, asymmetric channel and non-circular duct flows and curved flows, also present, disadvantages, just as, very large computing costs. The SED models for calculated turbulent heat flux have been adopted and studied. The bibliographical revision shows that the majority of the cited works had been previously developed for constant temperatures in the contour. However, in many applications the heat flux and surface temperature are non-uniform around the duct, becoming important the knowledge of the variation of the conductance around of the duct, in according to [18]. According to developments performed by [19], it is possible to carry out 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, “TWm”, in such away, in the present work intended to give a small contribution with respect to the influence the non-uniform wall temperature on the fluid temperature field considering internal flow with fluids air.

II.

 U J   0 xJ  1 P 1    U i U j U iU j       x j  xi  x j   x j xi 1     u'i u'j  x j



      (2)







  1    T f U jT f     u'j t'   x j  x j  Pr x j 

(3)

  Tw   ks 0 xi  xi 

(4)









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



turbulent heat flux   u'j t'

 u u  ' ' i j

and the

 were modeled and solved

by algebraic and/or differential expressions. II.2.

Turbulence Models for Reynolds Stresses

II.2.1. Non Linear 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 [20]. The Reynolds average equations, Eqs. (1) to (3), are applied for the device presents in Figs. 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 , respectively, they are given by:

Ui

Mathematical Formulation II.1.

(1)

Governing Equation

Ui

The Reynolds Averaged Navier Stokes (RANS) equation system is composed of: continuity equation, Eq. (1); momentum equation, Eq. (2); energy equation, Eq. (3); and heat conduction equation, Eq. (4):

k   t k     xi xi   k xi   Pk      t      xi xi    xi 

(5)

(6)

 2  c1 Pk  c2 f 2 k k

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International Review of Mechanical Engineering, Vol. 8, N. 3

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G. A. R. Rivas, E. C. Garcia, M. Assato

added the original constitutive relation. This quadratic term represents the degree of anisotropy between the normal tensions of Reynolds, which makes it possible to predict the presence of the secondary flow in noncircular ducts. The value of c1NL proposed by [20] is equal to 1.68. Here, c1NL will be analyzed and will adopt values for the formulation of Low Reynolds Numbers. The tensions of Reynolds, normal and shear, are presented in Eq. (9) and Eq. (10), are expressed as: (a)

 xx

 yy

 1  W  2 2  W 2          3  x  3  y   2 2 k  1  W  2  W     c1NL t       3  y  3  x    

(9)

k  W W  ;   x y  W W  t ;  yz  t x y

(10)

k  c1NL t 

 xy  c1NL t (b)

 xz

Figs. 1. (a) Fully developed turbulent flows in rectangular ducts (b) Rectangular duct: reference system and transversal section

The following differences for the normal tensions of Reynolds are presented and have been observed for this type of flow:

The symbols Pk and t , represent the rate of the turbulent kinetic energy production and the turbulent viscosity, respectively, and thus, we have:



U i k2 Pk   ij ;  t  c f   x j 

(7)



observed in Eq. (6) and Eq. (7) and shown in the thesis of [8]. These functions and the constant C1 and C2 have been used together with the equations κ-ε, the subscribed letter P refers to the nodal point near to the wall. Thus U P and k P are the values of the velocity and kinetic energy in this point, respectively. The constants c , c1 ,

Pk   xz

W W   yz x y

(12)

II.2.2. Reynolds Stress Model (RSM) The most complex turbulence is the Reynolds Stress Model (RSM), also called of second order, it involves calculations of the tensions of Reynolds in an individual

c2 ,  k and   are adopted the values of 0.09; 1.5; 1.9; 1.4 e 1.3; respectively. The new constitutive relation for the tensions of Reynolds in the model NLEVM, assumed in the thesis of [8], is given by:

form,  u'i u'j , used for this differential equations of transport. The individual Reynolds tensions are utilized to close the average Reynolds equations of the momentum. This model has shown superiority regarding the models of two equations in complex flows that involve swirl, rotation, etc. The exact equations of transport for the Reynolds

L

 ij   t Sij   NL

(11)

In such a way in Eq. (7), including the derivatives above the tensions of Reynolds, the turbulence production term, is expressed as:

In the present work for NLEVM, the formulations of Low Reynolds Number will be assumed for wall treatment. The damping functions f 2 and f  is

  k  Sik Skj    c1NL t  1      3 Skl Skl  ij    

2

2  W        x  

 yy   xx   c1NL t k  Wy 

(8)

tensions,  u'i u'j , can be written as follows:

This expression shows that the second term of the right side of the Eq. (8), represents the nonlinear term

Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Mechanical Engineering, Vol. 8, N. 3

647

G. A. R. Rivas, E. C. Garcia, M. Assato

  u'i u'j t

' ' k i j

k

  u' u' u'   i j k  ' '  p  kj ui   ik u j

  xk

assumes values of 0.89, independent on the wall proximity effect:

  a   x  u u u  b  







  c    

u jt  

   u'i u'j   d     x k   u j  u     u'i u'k  u'j u'k i   e   xk xk   

 xk

 



  gi u'j  g j ui'   u' u'j  p i   x j xi 

In reference [22] presents the following formulation to the turbulent heat flux:

(13)

 u j t    Ct

'  u' u j   g   2 i h   xk xk 



where the respective letters represent: (a) local derivative of the time; (b) Cij  convection; (c) DT ,ij  turbulent diffusion; (d) DL,ij 

molecular diffusion; (e) Pij 

production

tensions;

of

production term; (g) ij 

(f) Gij 

T f  k  u j uk   xk 

(15)

The constant Ct , assuming the value of 0.3, is adopted. The main advantage of this model is that it considers the anisotropic behavior of the fluid heat transport in ducts.

2  k u'j u'm  ikm  ui' u'm  jkm  i   S  j 

term

(14)

II.3.2. Generalized Gradient Diffusion Hypothesis (GGDH)

 f  



t T f  T x j

III.3.3. Dimensionless Energy Equation for SED and GGDH Models

buoyancy

For a given cross section of area “A”, it is possible to define a mean velocity “Ub” and a bulk temperature “Tb”, express as:

term of pressure-tension

(redistribution); (h)  ij  term of dissipation; (i) Fij  term production for the rotation system; (j) S j  source

Ub 

term. The terms of the exact equations, presented previously, Cij , DL,ij , Pij and Fij do not require

 W  T f  dA Tb 

modeling. However, the terms DT ,ij , Gij , ij and  ij need to be modeled to close the equations. For the present analysis, the model LRR presented in [9] is chosen, which assumes that the correlation velocity pressure is a linear function of the anisotropy tensor LRR in the phenomenology of the redistribution, ij . For the

Turbulence Models for Turbulent Heat Flux II.3.1.

A

Ub  A



 W  dx  dy 1 A U b

(16)

 W  T f  dx  dy

(17)

In Ref [18] was developed an applicable formulation to rectangular cross section ducts. They considered the boundary conditions with prescribed uniform wall temperatures at the cross section, and along the duct length.According to developments performed by [19], it is possible to carry out 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, “TWm”, given as:

treatment of the wall, it is also assumed the Low Reynolds numbers and the periodic conditions, described in [21]. This model had been simulated in the commercial code Fluent 6.3. II.3.

1 A

Simple Eddy Diffusivity (SED)

This method is based on the Boussinesq viscosity model. The turbulent diffusivity for the energy equation can be expressed as:  t  t /   t  , where the

TWm

1 L   1    L

L

1

D



0 T1  0, y   dy  D  0 T2  x,0   dx   L

0

1  D 2 L  D

T3  D, y   dy 

D

0

 T4  x,L   dx   (18)

It is possible to develop a formula similar in according to [18], the new expression for the turbulent energy equation, is presented as:

turbulent Prandtl number,  t , needs to be given. The SED model assumes that the turbulent Prandtl number is constant in the entire region, for the air  t it

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International Review of Mechanical Engineering, Vol. 8, N. 3

648

G. A. R. Rivas, E. C. Garcia, M. Assato

   T f    ut         x  x   T f T f T f    T f  U V W     vt     0 (19) x y z  y  y       T f    wt         z  z

Tb  TWm 



x Dh

(20)

Y

y Dh

(21)

  TWm  T f



 dT  U b  Dh2   b   dz 

  W      Dh  U V    Y  U B  B   X

(22)

q f    U b  A  c p 



Tb  TWm 

Dh2  U b dTb   b  dz

P    e    Nu  z1  z2   2 A  Ub

(32)

From Eq. (32), the bulk temperature longitudinal (zaxis) variation “Tb” is obtained. It is done by “cutting” the duct into a lot of segments and applying the numerical method to find “Tb” at each finite cross section. For a given bulk temperature at the duct inlet section (Tb1), after solving the equation system, duct outlet bulk temperature (Tb2) is computed from that expression, Eq. (32). The Dimensionless boundary conditions are given by the following equations:

(24)

  0,Y  

  TWm  T1    TWm  T2  ,  X , 0   (33)  dT   dT  U b  Dh2   b  U b  Dh2   b   dz   dz 

(25)

    TWm  T3    D D ,Y   h  U  D 2   dTb   b h    dz 

(26)

 X, L  D h 

From Eq. (21), the Eq. (26) is obtained, and applying this in Eq. (17), one gets Eq. (28):

 dT    U b  Dh2   b   dz  T f  TWm  



Tbz 2  TWm  TWm  Tbz1  e

(23)

The fluid temperature field “Tf” can be replaced by “Tb” and the Eq. (22) can be expressed as:

  TWm  Tb   dT  U b  Dh2   b   dz 

(31)

2

 

b 

dTb dz

when Eq. (30) is made equal to Eq. (31), integrating two cross sections (inlet, namely z1, and outlet, namely z2), it is possible to develop the resulting expression, Eq. (32).

        ex      ey X  X  Y  Y    W      Dh  U V    Y  U B  B   X         Ct X  Ct Y    X  Y  Y  X 

(28)

It is possible to compute the heat transfer rate per unit length on the wall surface, “ q’ ”, in Eq. (30). It depends on values for “TWm”, “Tb”, and on the average heat convection coefficient, “ h ”. From fluid enthalpy derivative gradient [dhb = cp.dTb], the heat transfer rate per unit length in the fluid, “ q’f ”, can be expressed by Eq. (31): q ,  Pe  h  TWm  Tb  (30)

Replacing Eq. (14), Eq. (15), Eqs. (20)-(22) in Eq. (19), dimensionless energy equation for SED and GGDH becomes, respectively:

        t       t     X  X  Y  Y 

 W    dx  dy

Replacing Eq. (27) in Eq. (24), and using also Eqs. (19) and (20), the dimensionless bulk temperature is obtained: Dh2 (29) b   W    dX  dY A  U b 

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

X 

Dh2 dTb     A dz

   TWm  T4     U b  Dh2   dTb   dz   

(34)

when considering uniform wall temperature, Eq. (33) and Eq. (34) are equal to zero, and for these particular conditions, it is possible to notice that these boundary conditions are not functions of “ dTb dz ”.

(27)

Copyright © 2014 Praise Worthy Prize S.r.l. - All rights reserved

International Review of Mechanical Engineering, Vol. 8, N. 3

649

G. A. R. Rivas, E. C. Garcia, M. Assato

That simplification becomes equal to the one studied by [23]. Eqs. (25), (26) and (27), as well as the boundary conditions from Eqs. (33) and (34), form a set of differential equations, in which “” and “dTb/dz” parameters are unknown. When that equation system is solved, it is possible to obtain “Tf”. II.4.

updated (Step 3) to obtain a solution for the new temperature field (Step 4), until convergence is obtained (dTb/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 “Øb” (dimensionless bulk temperature), “dTb /dz" and “Ø” (dimensionless temperature field). The above procedure is applied for contours of variable temperatures.

Additional Equations

Additional equations were utilized for the calculation of the factor of friction Moody, f ; coefficient of friction of Fanning, C f ; Prandtl law; local Nusselt number for the Low Reynolds formulation ([12]), Correlation of the Nu equations are given by:

f 

f

IV.

and

  dP / dz   Dh

U B2



 2 log Re

Nu xp  Dh

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

(35)

2

f 4

Results and Discussion

IV.1. Fluid Flow and Heat Transfer Field

[22] , respectively; these

Cf  1

Nu xp

(36)



f  0.8

Tw  TP   Tw  Tb 

(37)

(38) (a)

  f / 8 Re 1000  Pr  Nu    2  1  12.7  f / 8  1 2  Pr 3  1          

(39)

III. 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 ([19]): Step 1: To define the function value of the nonuniform temperatures in the walls of the duct TWm  f T1  0, y  ,T2  x, 0  ,T3  D, y  ,T4  x,L   , what that

(b) Figs. 2. (a) Grid 120x120 for numerical simulation (b) Secondary flow contours comparisons between the axial mean velocity measured in [5] for water and numerical results obtained by NLEVM Model

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 Figs. 3(a) and 3(b), respectively. Fig. 4(a): comparisons of the Results (RSM-SED) numerical with the experimental for temperature profile (wall constant temperature) TWm  T f / TWm  TC  with

can be expressed by a Fourier expansion; Step 2: To obtain velocity field and estimated values for “ U b ”“TWm” and “dTb/dz”; Step 3: Equations for the boundary conditions are evaluated (Eqs. (33) and (34)); Step 4: Dimensionless energy equation (Temperature field, “Ø”) in Eq. (24) is solved and “Øb” is computed according to Eq. (30), until convergence is obtained (Øb < tolerance). This is the end of the first iterative loop; Step 5: A value for “dTb/dz” is computed in accordance with Eq. (25). Boundary conditions are





fluid air and Re=65000 ([11]) are shown, the Fig. 4(b) shows the variation of the temperature profile with nonuniform wall Temperature: south=400K, north=373K, east=393K, west=353K; it called of Case I.

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10

 nx  1  north= 395  10  K  nxmáx  1 

9

Numerical Simulation

8

Model Non Linear k-e Model RSM

7

 ny  1  east= 405 - 10  K  nymáx  1 

Cf (x10-3)

6

5

 ny  1  west= 415  10   K  nymáx  1 

4

Methods Experimental and Correlations 3

Prandtl Friction Law Harnett et al. Leutheusser Launder e Ying Lund

where some results for rectangular ducts will be shown in the Table I. In work of [19] was analyzed the laminar flow with the coupled of the conduction and radiation in rectangular ducts and concluded that as increases the aspect ratio, the Nusselt number found for the coupling (non uniform temperature), differs from that found for ducts with constant temperature imposed around the perimeter of the section. That shows that would be admissible to make a mistake in the case of using literature results without calculating the energy equation. In the present work, 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 I.

2 2

3

4

5

6

7

8

9 10

20

30

Re (x104)

(a) 10

3

Numerical Simulation

Nu/Pr0.4

SED Model GGDH Model

10

2

1

1 0.65

Methods experimental - Correlations

0.65

0.8

Lowdermilk et al. Gnielinsky Brundrett & Burroughs

0.70 0.75 0.80

0.70 0.75 0.80

0.6

0.4

10

4

5

10

0.90 0.4

0.95

0.95

6

10

Re

0.6

0.85

0.85 0.90

101

0.8

0.2

0.2

(b) 0

0 -1

Figs. 3. (a) Friction coefficient for fully developed flow, (b) Nusselt number dependence on Reynolds number for fully developed flow.

-0.8

-0.6

-0.4

-0.2

0

Numerical Prediction

0.2

0.4

0.6

0.8

1

Experimental

(a)

Already the Fig. 5(a) shows: The variation of the temperature profile with non-uniform wall temperature, it is 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-/2) and i, j (points number of the grid in the direction x and y, respectively). Figure 5(b) represent the behavior of the “Tb” and “DTb/dz” for different square cross-sectional sections along of the direction of the main flow, according to Eq. (31). Figs. 6(a) and (b) shown the temperature distribution for a rectangular duct of the aspect ratio (1:2) represented also by means of functions sine (Case II). Exist a third case denominated Case III, it is represented by:

(b) Figs. 4. (a) Results (RSM-SED) numerical and experimental ([11]) for mean temperature (uniform wall temperature) (b) Fluid temperature with non-uniform wall Temperature (Case I)

 nx  1  south= 405  10  K  nxmáx  1 

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TABLE I NUMERICAL RESULTS OBTAINED THROUGH RSM-SED MODEL ([22]), FOR THE AVERAGED NUSSELT NUMBER IN A RECTANGULAR DUCT WITH ASPECT OF RATIO (1:2) Nusselt Correlation Cases Reynolds Number Dittus Boelter Analyzed Number Calculated (Nu) Temperature 142,89 65000 145,910 Constant Case II 65000 139,682 Case III 65000 145,059 Temperature 77,1 28853 79,101 Constant Case III 28853 76,769 -

380

18

370

16

360

14

350

12

340

10

Case II

330

8

Twm Tb(Z) (DTb/dz)

320

IV.2. Heat Conduction Coupled to Turbulent Forced Convection Fig. 7 shows, the grid non-uniform utilized for the coupled solid-fluid. DTb/dz

0.06

0.05 6

Case Constant Tw

310

0.04

4

Twm Tb(Z) (DTb/dz)

300

2

290

0.03

0 0

0.5

1

1.5

2

2.5

3

3.5

y

T Bulk (K)

(a)

4

0.02

Z (m)

(b) 0.01

Figs. 5. (a) Fluid temperature with non-uniform wall temperature (Case II) (b) Behavior “Tb” for different square cross-sectional sections and cases throughout of the direction of the main flow

0

-0.01 0

0.02

0.04

0.06

x

0.0006

0.0004

y

0.0002

0

(a)

-0.0002

-0.0004

-0.0006

-0.0005

0

0.0005

x

Fig. 7. Grid non-uniform coupled solid-fluid

Figs. 8 show below, different cases for the temperatures prescribed in the external contour of the plates coupled (solid), the case (a) shows constant temperatures in the all external contour of 373K, (b) different temperatures in the face North=600K, south=500K, east and west=373K and the case (c) shows a example qualitative representing the versatility of the program to work with different thermal conductivities, ([22]).

(b) Figs. 6. (a) Rectangular duct of aspect ratio (1:2) case II, Re=65000 with Tb=300 K, utilizing the SED model (b) Retangular duct of aspect ratio (1:2) with constant temperature in the perimeter Twm=373 K, Re=65000 with Tb=300 K, utilizing the SED model

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correlation of the literature, (Figs. 3(a), (b)) using the turbulent convective heat transfer proposed. Figs. 4(b) and 5(a) show new results investigated in present work, note a distortion of the temperatures field and as consequence the 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 ([18]). Fig. 5(b) shows the comparisons of the behavior of the curves “Tb” and “DTb/dz” to the long of the main direction of the 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. The coupled solid fluid represents good qualitative results.

(a)

Acknowledgements The authors would like to acknowledge the CNPq for financial support during this work.

(b)

References [1]

[2]

[3]

[4]

[5] [6]

(c) Figs. 8. Coupled solid-fluid with prescribed conditions (a) external temperatures Text= 373K; (b) different temperatures in the face North=600K, south=500K, east and west 373K, (c) example qualitative representing the versatility of the program to work with different thermal conductivities

[7]

[8]

V.

Conclusion

The results presented for the friction factor and Nusselt number in function of a large range of the Reynolds number for uniform wall temperature present good agreement with the experimental works and

[9]

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Z. H. Qin, R. H. Plecther, Large eddy simulation of turbulent heat transfer in a rotating square duct, International Journal Heat Fluid Flow, 27, pp. 371-390, 2006. J. Nikuradse, Untersuchung uber die geschwindigkeitsverteilung in turbulenten stromungen, Diss. Göttingen, VDI – forschungsheft, 281, 1926. F. B. Gessner, A. F. Emery, A Reynolds stress model for turbulent corner flows – Part I: Development of the model, Journal Fluids Eng., 98, pp. 261-268, 1976. F. B. Gessner, J. K. Po, A Reynolds stress model for turbulent corner flows – Part II: Comparison between theory and experiment, Journal Fluids Eng., 98, pp. 269-277, 1976. A. Melling, J. H. Whitelaw, Turbulent flow in a rectangular duct, Journal Fluid Mechanical, 78, pp. 289-315, 1976. A. Nakayama, W. L. Chow, D. Sharma, Calculation of fully development turbulent flows in ducts of arbitrary cross-section, Journal Fluid Mechanical, 128, pp. 199-217, 1983. H. K. Myon, T. Kobayashi, Numerical Simulation Of Three Dimensional Developing Turbulent Flow in a Square Duct with the Anisotropic κ-ε Model, Advances in Numerical Simulation of Turbulent Flows ASME, Fluids Engineering Conference, Vol.117, Portland, United States of America, 1991, pp. 17-23. M. Assato, Análise numérica do escoamento turbulento em geometrias complexas usando uma formulação implícita, Doctoral Thesis, Departamento de Engenharia Mecânica, Instituto Tecnológico de Aeronáutica - ITA, São José dos campos - SP, Brazil, 2001. B. E. Launder, W. M. Ying, Prediction of flow and heat transfer in ducts of square cross section, Proc. Inst. Mech. Eng., 187, 1973, pp. 455-461,

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[10] F. Emery, P. K. Neighbors, F. B. Gessner, The numerical prediction of developing turbulent flow and heat transfer in a square duct, Journal Heat Transfer, 102, pp. 51–57, 1980. [11] M. Hirota, H. Fujita, H. Yokosawa, H. Nakai, H. Itoh, Turbulent heat transfer in a square duct, International Journal Heat and Fluid Flow, 18, pp. 170-180, 1997. [12] M. Rokni, A new low-Reynolds version of an explicit algebraic stress model for turbulent convective heat transfer in ducts, Numerical Heat Transfer – Part B: Fundamentals, 37 (3), pp. 331-363, 2000. [13] Y. Hongxing, Numerical study of forced turbulent heat convection in a straight square duct, International Journal of Heat and Mass Transfer, 52, pp. 3128-3136, 2009. [14] Gawali, B.S., Kamble, D.A., Optimization of rectangular microchannel depth under forced convection heat transfer condition, (2013) International Review of Mechanical Engineering (IREME), 7 (2), pp. 379-383. [15] Bourabaa, A., Saighi, M., Fekih, M., Belal, B., Study on the heat transfer of the rectangular fin with dehumidification: Temperature distribution and fin efficiency, (2013) International Review of Mechanical Engineering (IREME), 7 (5), pp. 857-863. [16] Brinda, R., Joseph Daniel, R., Sumangala, K., Influence of cross sectional shape on the heat transfer characteristics of ladder type micro channel heat sinks for ULSI, (2013) International Review of Mechanical Engineering (IREME), 7 (6), pp. 1053-1061. [17] M. Rokni, Numerical investigation of turbulent fluid flow and heat transfer in complex duct, Doctoral Thesis, Department of Heat and Power Engineering. Lund Institute of Technology, Sweden, 1998. [18] W. M. Kays, M. Crawford, Convective Heat and Mass Transfer, McGraw-Hill, New York, USA, pp. 250-252, 1980. [19] E.C. Garcia, Condução, convecção e radiação acopladas em coletores e radiadores solares, Doctor degree thesis, ITA Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brasil. 1996. [20] C. G. Speziale, On non linear k-l and k-e models of turbulence, Journal Fluid Mechanical, Vol. 178, pp. 459-475, 1987. [21] M. Assato, M. J. S. de Lemos, Turbulent flow in wavy channels simulated with nonlinear models and a new implicit formulation, Numerical Heat Transfer – Part A: Applications, 56 (4), pp. 301324, 2009. [22] G.A. Rivas Ronceros, Simulação numérica da convecção forçada turbulenta acoplada à condução de calor em dutos retangulares, Doctor degree thesis, ITA - Instituto Tecnológico de Aeronáutica, São José dos Campos, SP, Brasil, 2010. [23] S. V. Patankar, Computation of Conduction and Duct Flow Heat Transfer, Innovative Research, Maple Grove, USA, 1991.

Gustavo Adolfo Ronceros Rivas is currently Professor at Federal University of Latin American Integration, Brazil. He graduated in Fluid Mechanics and Thermo-Fluid Engineering in 1999 at Universidad Nacional Mayor de San Marcos Lima – Perú. He finished his M.Sc. (2005) and PhD (2010) in Aeronautics and Mechanics Engineering at Aeronautical Technological Institute, Brazil. His expertises are in areas of mechanicals engineering, computational fluid dynamics and renewable energy. He has experiences in project of Satellite thermal control and internal air system design for cooling of gas turbine components. Ézio Castejon Garcia is graduated in mechanical engineering from the Federal University of Uberlândia (UFU, 1983), M.Sc. degree in aeronautical and mechanical engineering at Brazilian Aeronautical Technological Institute (ITA) in 1987, and also D.Sc. by the same Institute, in 1996. Currently works as associate professor and deputy head of the aeronautics/mechanical engineering division at ITA, with experience in mechanical and aerospace engineering, thermodynamics, heat transfer and satellite thermal control, teaching courses on these areas. Also, coordinates the thermal control subsystem of the ITASAT satellite program. Marcelo Assato is currently Researcher at Aeronautical and Space Institute (IAE), Brazil. He graduated in Mechanical Engineering and finished his M.Sc. in computational heat transfer and fluid flow at Federal University of Itajubá in 1994 and 1997, respectively. He obtained his PhD degree in 2001 at Aeronautical Technological Institute in area of computational fluid dynamics (CFD) - turbulence models. He worked with experimental aerodynamics (wind tunnel tests – turbulence measurements) from 2002 to 2006. He has experience in gas turbine performance analysis since 2006. His research interests are development of CFD numerical code, performance of turbo machines (gas turbine) and thermal plants using alternative fuels.

Authors’ information 1

Federal University of Latin American Integration (UNILA), department of Renewable Energy Engineering, Av. Tancredo Neves, 6731 – Bloco 4, CEP 85867-970, Foz do Iguaçu – Paraná – Brazil. E-mail: [email protected] 2

Aeronautical Technological Institute (ITA), Division of Mechanical and Aeronautical Engineering, department of Energy, Praça Marechal do Ar Eduardo Gomes, 50 – Vila das Acácias, CEP 12228-900, São José dos Campos – São Paulo – Brazil. E-mail: [email protected] 3

Aeronautical and Space Institute IAE), Division of Aeronautical Propulsion (APA), Praça Marechal do Ar Eduardo Gomes, 50 – Vila das Acácias, CEP 12228-900, São José dos Campos – São Paulo – Brazil. E-mail: [email protected]

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