# Heat transfer and flow field in the entrance region of a symmetric wavy-channel with constant wall heat flux density.

Abstract

Two-dimensional laminar flow and heat transfer in the entrance region of a symmetric wavy-channel with constant wall heat flux density which includes fourteen undulations are presented. Numerical solutions are obtained using the control-volume finite-difference method. Thermal and flow fields, Nusselt number and shear stress are presented for different flow rates (100 [less than or equal to] Re [less than or equal to] 1500), amplitude of wavy surface (0 [less than or equal to] A [less than or equal to] 0.5) and Prandtl number values. The results show that the wavy-wall generates vortices in the hollow region, which grow in magnitude and spatial flow coverage with increasing Re and/or A. The shear stress and the Nusselt number increase with increasing the Reynolds number and have the highest magnitude in the entrance region. The increase of average Nusselt number is found limited with the increase of the amplitude.

Keywords: Forced convection, laminar flow, entrance region, wavy channel, heat transfer.

Introduction

The exchanger's plate concepts go back to almost one century. These exchangers have been originally studied to meet the needs of industry, and then used in different branches. For a fully developed flow, the plate grooves aim to increase vortices and augment heat transfer coefficients. The channel with a periodically converging-diverging cross section is one of several devices employed for enhancing the heat and mass transfer efficiency due to turbulence promotion. A numerical scheme has been used by G. Wang and S.P. Vanka [1] to study the heat transfer rates for a flow in periodical wavy passages.

In steady flow, average Nusselt numbers for the undulated wall channel were slightly superior to those obtained for a parallel wall channel. However, in the transitional-flow regime, the enhancement of heat transfer was about a factor of 2.5. Wang, Zhang and Metwally [2, 3, 4] analyzed the laminar forced convection in wavy-wall channels. Results show that the size of the vortices and the skin-friction coefficient increases as Reynolds number and/or amplitude. The mixing of the flow and the acceleration of the core flow results in an increase of the heat transfer. Sarma et al. [5] suggested a new method to estimate the average heat transfer coefficients in the entrance region of a short length of tube under developing laminar flow conditions. Fontes et al. [6] have investigated experimentally the developing laminar non-isothermal and non-Newtonian flow in the entrance region of short tubes with heating and cooling conditions. M. Rahmena et al. [7] present a numerical study of a forced laminar flow through an array of parallel flat plates with finite thickness. Results show that the shear stress depends on the wall thickness, takes the value corresponding to a fully developed flow, and is constant far from the inlet of the channel. Magno et al. [8] employed the generalized integral transform technique in non-Newtonian fluids within a parallel plates channel.

It was verified that for values of power-law indices greater than unity, lower values for the Nusselt numbers in the entrance region are obtained. Bazdidi et al. [9] investigated numerically the laminar fluid flow and heat transfer in the entrance region of a two dimensional horizontal channel with in-line ribs. The results show that at a fixed Reynolds number increasing the rib spacing causes a marginal increase in the local Nusselt number and a decrease in the friction factor. Fabbri et al. [10] presented the analysis of the heat transfer of a Newtonian fluid in the entrance region of a corrugated channel. The analysis of the local equivalent Nusselt number has shown that the global heat transfer is effectively enhanced as the amplitude of the corrugated profile and the Reynolds number increase.

The absolute values of the Nusselt number in the fully developed region of the channel are of the same order of magnitude as that of a smooth channel. A novel approach to the study of the Graetz Brinkman problem in a plane-parallel channel has been presented by Bartella et al.[11]. Comparison between results obtained with his method and results obtained by employing the traditional procedure, with reference to the axial evolution of both the local Nusselt number and the temperature field. Strong discrepancies between the two approaches have been revealed in the initial part of the thermal entrance region. Therefore, the bibliographic survey shows that in most investigations either numerical or experimental, the flow is studied in the part of the channel, far from the entrance and the exit, where it can be considered to be fully developed or periodic. However, in real situations the flow field in the channel includes not only the fully developed region but also the entrance region. The survey of these real situations made few publications. The aim of the present paper is to study the laminar forced convection in the entrance region of a wavy-channel which includes fourteen undulations; the walls are submitted to a density of constant heat flux. Transport equations are solved using the control-volume finite-difference method and Thomas algorithm. The linkage between the velocity and pressure variables is handled by the PISO algorithm. The effects of Prandtl number, Reynolds number and the amplitude of the sinusoidal profile of the wall on the transfers in the entrance region of this channel are studied.

Numerical solution

The laminar forced convection in wavy-plate-fin channel with sinusoidal wall corrugations is numerically simulated. Walls of the channel are submitted each to a flux of heat whose density is constant, but the smooth parts to the exit of the channel are adiabatic. The fluid flow in the channel is supposed to be two-dimensional; hence the width of plates is supposed larger in relation to their spacing. The flow is governed by the Navier--Stokes equations. In addition the retained simplifying hypotheses, we disregarded the influence of the gravity in the momentum equation, effects of the viscous dissipation and the thermal expansion in the heat equation. The system can be written in Cartesian coordinates in the following dimensionless form:

[partial derivative]U / [partial derivative]X + [partial derivative]V / [partial derivative]Y = 0 (1)

U [partial derivative]U / [partial derivative]X + V [partial derivative]U / [partial derivative]Y = - [partial derivative]P / [partial derivative]X + 1 / Re ([[partial derivative].sup.2]U / [partial derivative][X.sup.2] + [[partial derivative].sup.2]U / [partial derivative][Y.sup.2]) (2a)

U [partial derivative]V / [partial derivative]X + V [partial derivative]V / [partial derivative]Y = [partial derivative]P / [partial derivative]Y + 1 / Re ([[partial derivative].sup.2]V / [partial derivative][X.sup.2] + [[partial derivative].sup.2]V / [partial derivative][Y.sup.2]) (2b)

U [partial derivative][theta] / [partial derivative]X + V[partial derivative][theta] / [partial derivative]Y = 1 / Re x Pr ([[partial derivative].sup.2][theta] / [partial derivative][X.sup.2] + [[partial derivative].sup.2][theta] / [partial derivative][Y.sup.2]) (3)

Where dimensionless variables are defined as:

X = x / c, Y = y / c, [theta] = (T - [T.sub.0] / [[phi].sub.n.sup. x c] / [lambda]) U = u / [U.sub.e], V = v / [U.sub.e], P = p / [rho][.U.sup.2.sub.e], [R.sub.e] = rho [U.sub.e.sup. x c] / [mu], [P.sub.r] = [mu] [C.sub.p] / [lambda] (4)

Meanwhile, the equations (2) and (3) have a similar form; they can also be reduced to a simple equation of the convective- diffusive conservation of the form:

[partial derivative] / [partial derivative][X.sub.i] [[rho][U.sub.i][PHI] - [[GAMMA].sub.[PHI]] [partial derivative][PHI] / [partial derivative][X.sub.i]] = [S.sub.[PHI]] (5)

Where [PHI] characterizes the dependent variable, [[GAMMA].sub.[PHI]] the effective diffusion and [S.sub.[PHI]] the source term; such as:

For momentum equation:

[GAMMA] = 1 / Re ; S = [partial derivative]P / [partial derivative][X.sub.i] (6a-b)

For heat equation

[GAMMA] = 1 / Re x Pr ; S = 0 (7a-b)

As shown in Fig.1, where OX is the axis of symmetry of the channel, the upper wall profile in the region 0 [less than or equal to] X [less than or equal to] [L.sub.0] is given by the equation:

F(X) = 1 / 2 + A x Sin([pi] x c / L X) (8)

[FIGURE 1 OMITTED]

Where [L.sub.0] as the total length of the sinusoidal surface of the channel, [L.sub.1] the length of the smooth surface, A the amplitude and N the normal. These variables are defined as:

A = a / c, [L.sub.0] = [l.sub.0] / c, [L.sub.1] = [l.sub.1] / c, N = n / c (9)

Where c as the average separation distance between wavy walls.

Thus, the corresponding boundary conditions are considered as follows: At entrance channel (X = 0, 0 < Y < 1):

U = 1. ; V = 0 and [theta] = q (10a-c)

At upper wavy surface:

U(X, F(X)) = V(X, F(X)) = 0; [[bar.[phi].sub.N] = [partial derivative][theta] / [partial derivative]N =1 (11a-c)

Where [[bar.[phi].sub.N] represents the dimensionless heat flux density.

For [L.sub.0] < X [less than or equal to] [L.sub.1]:

U(X, 1/2) = V(X, 1/2) = 0; [partial derivative][theta] / [partial derivative]Y = 0 (12a-c)

At the exit of the smooth part of the channel

(X = [L.sub.0] + [L.sub.1]): [partial derivative]U / [partial derivative]X = 0 ; [partial derivative]V / [partial derivative]X = 0 ; [partial derivative][theta] / [partial derivative]X = 0 (13a-c)

Although the boundary condition (13a-c) is strictly valid only when it is fully developed, its use in other flow conditions is allowed for calculation convenience, too, if this condition is situated: in a region where the flow is in the downstream direction and far enough from the region of study downstream [7].

The local Nusselt number at the wavy wall surface of the channel is defined by the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14a-b)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent respectively angles formed by the normal and OX and OY axes.

Also, the average Nusselt number is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15a-b)

With S is the surface of wall exchange.

Integrals of equations (15a) are calculated by the Simpson method [15].

The shear stress at the surface is given by the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with the viscous stress tensor in 2D is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16a-b)

To determine the flow dynamic field, a simulation by a fluid dynamic calculation has been used. The transfer equation (5) and the boundaries conditions (10-13) are solved using the finite Alternating Direction Implicit method (ADI) [12]. This last is semi-iterative method in which the equations are solved at each step by maintaining full implicitness in one direction at a time, while relaxing the requirement in the other direction; for each direction a tri-diagonal linear system results that can be solved very efficiently. The algorithm PISO (Pressure- Implicit with Splitting of Operators) [13] was applied to solve the coupled system of governing equations. This algorithm employs a series of sequential operations at each time step which the discretized momentum and pressure-based continuity equations are solved in an alternating "predictor-corrector" fashion. This approach offers an advantage over many other schemes. The convergence criteria represented by the relative magnitude of error [epsilon], was grouped, respectively by velocity fields, pressure and temperature by the following relations.

[epsilon] = [absolute value of [[PHI].sup.n+1] - [[PHI].sup.n] / [[PHI].sup.n+1]] < [10.sup.-5] (17)

The grid sharpness precision test has been made to establish an independent numerical solution grid. A sensitive test to the mesh size is made by comparing the effect of the grid size on the shear stress and the axial velocity. According to the Fig.2, we note that the mesh size of 431x21 lead to results very close to those obtained by using a mesh size of 565x31. Consequently, we retained in our calculus the mesh size of 431x21.

[FIGURE 2 OMITTED]

Results and discussion

Numerical results in the entrance region for development of the hydrodynamic, thermal fields, Nusselt number and shear stress are presented for different flow rates (100 [less than or equal to] RE [less than or equal to] 1500), wall corrugations severity (0 [less than or equal to] A [less than or equal to] 0.5) and two different values of Prandtl number Pr = 6.69 and 0.708.

Flow distribution and shear stress

The velocity profile supposed to be uniform at the entry of the channel is modified by the effect of the wall roughness of the channel. As the flow rate is constant in a cross-section of the channel, the parabolic velocity profile is not influenced by the wall roughness. Consequently, the fluid velocity is maximal at the summit of the wavy wall. Besides, we note that the maximum velocity for a smooth channel is according to [14] [U.sub.max.] =1.5 (Fig.2b). Variations in the amplitude of the sinusoidal plate-surface and the flow rate have a significant effect on the dynamic behaviour of the flow field. The flow field is found to be strongly influenced by A and Re, and it displays two distinct regimes (Fig.3): * Undisturbed laminar-flow regime for low Re or A, the flow behavior is very similar to that in fully developed straight-duct flows with no cross-stream disturbance.

[FIGURE 3 OMITTED]

Vortex-flow regime for high Re or A, flow separation and reattachment in the hollows generates transverse vortex cells that grow spatially with Re and A. For Re = 800, the separation starts to occur at A = 0.4. Besides, the intensity and flow area coverage of counter-rotating lateral vortex grows with the increase of the amplitude of the sinusoidal profile, which in turn tends to smaller momentum transfer. These vortices increase as the Reynolds number increases until occupied completely the hollows for A = 0.5 and Re = 1500 (Fig. (3b)). However, for small amplitude (A [less than or equal to] 0.2), the Reynolds number has no effect on the structure of the flow, and the fluid moves undisturbed through the channel by simply adopting the wavy passage shape. In the entrance region of the channel, the shear stress is characterized by a very fast decrease (Figs. 4 and 5).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In the first wave, the shear stress has the highest magnitude and decrease quickly; the decreasing becomes smaller in the divergent section of the channel with the diminution of the Reynolds number. But far from regions of entry and exit of the channel, the shear stress increases as the Reynolds number increases. It is noticed that as the Reynolds number is decreased from 1500 to 100, the minimum shear stress of each undulation increases.

[FIGURE 6 OMITTED]

This indicates that the diffusion effects are big compared with the convection effects for the very weak Reynolds numbers. Far from the entrance region, where the heat transfer becomes periodic, with the increase of the Reynolds number, the maximum values of shear stress increase in each convergent section, but the minimum values become more and more constants along the divergent section. These results agree with those of Wang et al [2]. We notice that the shear stress for a smooth channel, far from the entrance and the exit of the channel, is independent of the Reynolds number and remains constant along the channel (Figs. 4 and 5). The evolution of the shear stress is represented by a harmonic curve, this last has the same frequency that of the symmetric wavy surface.

Temperature distribution and Nusselt numbers

The isotherms spread along the wall and the highest temperature is observed in the hollows (Fig.6). In these zones, the heat is transferred to the fluid essentially by conduction. The representative curve of the local Nusselt number has a shape of saw tooth. The amplitudes of peaks decrease far from the entrance region. This curve presents a fluctuation for L/4 <X <L that is attenuated while moving away of the entrance region (Figs.7 and 8).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

In the first half period, the local Nusselt number is characterized by a very fast decrease; and takes its higher value in the convergent part. This is because at the channel entrance the thermal boundary layer is thinner than under fully developed conditions, and the Nusselt number is therefore higher.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Also, for all flow rates and all amplitude of wall waviness, the heat transfer coefficient is higher near the channel entrance. The local Nusselt number undergoes again another very fast decrease in the first half period of the second undulation. From the third undulation we notice a displacement of the maximum, particularly for higher amplitudes (A = 0.5), of X=3L/4 to X = L; it is visualized by a discontinuous line in fig.7. In the diverging section of the channel, flows re-circulations occur and the velocity assumes the minimum values. In the converging regions and for the same value of the Reynolds number, the local Nusselt number increases as the value of the wavy profile decreases. This result agrees with the one of Fabbri et al [10].

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

The average Nusselt number shows that for Pr = 6.69 and 400<Re <1300, the smooth wall channel leads to heat transfers superior to those obtained for a wavy wall where the amplitude is superior or equal to 0.3 (Fig.9). But for Pr = 0.708 and Re < 1500, the heat transfer decreases as the amplitude of the channel increases (Fig.10). The average Nusselt number increases with the Reynolds number especially for Pr = 6.69 (Fig.11). Generally, there is a significant decrease of heat transfer when the amplitude of the channel increases for Pr = 0.708 (Fig.12).

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

There is only a slight increase for Re = 1500 and the maximum is obtained for A somewhere in the interval [0, 0.1], around A = 0.07. Further there is a strong decrease with increasing A also for this Reynolds number. For all other Reynolds numbers, there is a strong decrease. Indeed, the average velocity [U.sub.m] close to walls for higher amplitudes is characterized by the smaller values; for instance for Re = 500: [U.sub.m](A=0.1) = 0.276, [U.sub.m](A=0.3) = 0.204 and [U.sub.m](A=0.5) = 0.166; for Re=1500: [U.sub.m](A=0.1) = 0.278, [U.sub.m](A=0.3) = 0.195 and [U.sub.m](A=0.5) = 0.157. Therefore, the weak velocities induce a less intense heat transfer in wavy-wall channels to higher amplitudes. The maximal value of the average Nusselt number is obtained for a wavy-wall channel of amplitude equal to 0.1 for Pr = 6.69 with Re>100 and for Pr = 0.708 with Re=1500. Curves of Nusselt numbers relative to C = 2L possess the same character that in the case where C = L. The increase of C is accompanied by an elevation of values of the local Nusselt number (Fig.7). In addition, for the two Prandtl numbers considered the average Nusselt number increases with C (Figs.13-16). The transfers by diffusion are more significant in water (Pr = 6.69) than in air (Pr = 0.708) (Fig.7). The improved convective behaviour with increasing of the Prandtl number is clearly evident (Fig.7), where the higher temperature gradients at the wall are obtained.

Conclusions

Numerical solutions for laminar (100 [less than or equal to] Re [less than or equal to] 1500), incompressible, constant property and forced convection in the entrance region of wavy-wall channels maintained at constant wall heat flux density are obtained. We analyse the effect of the sinusoidal profile (0 [less than or equal to] A [less than or equal to] 0.5) on the Nusselt numbers and shear stress profiles for air (Pr = 0.708) and water (Pr = 6.69). Results show that the flow field is strongly influenced by the amplitude of channel and Reynolds number, and displays two regimes. For a low number Reynolds or amplitude, the regime is undisturbed laminar-flow, and for a high number Reynolds or amplitude the regime is vortex-flow. The wavy-wall channel induces vortices in the hollow region, which grow and spatial flow coverage with increasing Reynolds number and/or amplitude of the wavy-wall. But for small amplitude, the Reynolds number has no effect on the structure of the flow, and the fluid moves undisturbed through the channel with any re-circulation by simply adopting the wavy passage shape. The harmonic curve for the shear stress in the channel has the same frequency as that of the wavy surface, and the maximum and minimum values occur at the locations of minimum and maximum cross-section of the symmetric wavy-channel. But the representative curve of the local Nusselt number has a shape of saw tooth where the intensity of peaks decreases far from the entrance region. The curves of Nusselt number and shear stress are characterized by a very fast decrease at the entry of the channel; tend towards a constant value far from the entry, and they have the highest value in the first half period of the wavy-wall channel. For water, the smooth wall channel leads to heat transfers superior to those obtained for a wavy wall where the amplitude is superior or equal to 0.3 and 400<Re <1300. But for air, the heat transfer decreases as the amplitude of the wavy-wall channel increases when Re < 1500. The heat transfer increases with Reynolds number and the average separation distance between wavy walls, but the enhancement is limited when the amplitude continues to increase. The wave amplitude of walls equal to 0.1 for Pr=6.69 with Re>100 and for Pr=0.708 with Re=1500 is found to significantly enhance the heat transfer.

References

[1] Wang G., and Vanka S.P., 1995, "Convective heat transfer in periodic wavy passages," Int. J. Heat Mass Transfer, 38(38), pp. 3219-3230.

[2] Wang C.-C., and Chen C.-K., 2002, "Forced convection in a wavy-wall channel," Int. J. Heat Mass Transfer, 45, pp. 2587-2795.

[3] Zhang J., Kundu J., Manglik R.M., 2004, "Effect of waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores," Int. J. of Heat and Mass Transfer, 47, pp.1719-1730.

[4] Metwally H.M., and Manglik R.M., 2004, "Enhanced heat transfer due to curvature-induced lateral vortices in laminar flows in sinusoidal corrugatedplate channels," Int. J. of Heat and Mass Transfer, 47, pp. 2283-2292.

[5] Sarma P.K., Kishore P.S., Subrahmanyam T., Rao V.D., 2004, "A novel method to determine laminar convective heat transfer in the entry region of a tube," Int. J. of thermal sciences, 43, pp. 555-559.

[6] Fontes S.R., and Gasparetto C.A., 2001, "Convective laminar flow of food additives solutions in circular entrance region," Int. Comm. Heat Mass Transfer, 28(5), pp. 693-702.

[7] Rahmena M., Yaghoubi M.A., and Kazeminejad H., 1997, "A numerical study of convective heat transfer from a array of parallel blunt plates," Int. J. of Heat and Fluid Flow, 18, pp. 430-436.

[8] Magno R.N.O., Macedo E.N., and Quaresma J.N.N., 2002, "Solutions for the internal boundary layer equations in the simultaneously developing flow of power-law fluids within parallel plates channels," Chemical Engineering Journal, 87, pp. 339-350.

[9] Bazdidi-Tehrani F., and Naderi-Abadi M., 2004, "Numerical analysis of laminar transfer in entrance region of a horizontal channel with transverse fins," Int. Comm. in Heat Mass Transfer, 31(2), pp. 211-220.

[10] G. Fabbri and R. Rossi, 2005, "Analysis of the heat transfer in the entrance region of optimised corrugated wall channel," Int. Comm. in Heat Mass Transfer, 32, pp. 902-912.

[11] Barletta A., and Magyari E., 2006, "The Graetz-Brinkman problem in a plane-parallel channel with adiabatic-to-isothermal entrance", Int. Comm. In Heat Mass Transfer, 33, pp. 677-685.

[12] S.V. Patankar, Numerical heat transfer and fluid flow, Hemisphere Publishing Corporation, New York 1980.

[13] Issa R.I., 1986, "Solution of the implicitly discretised fluid flow equations by operator-splitting," J. Computational physics, 62(1), pp. 40-65.

[14] Taine J., and Petit J.-P., 1989, "Transferts thermiques," Bordas, Paris.

[15] Peaceman, D.W. and H.H. Radford, 1995, "The numerical solution of parabolic and elliptic differential equations," J. of the Industrial Applied Mathematics, 3, pp. 28-41.

Nabou Mohamed (a), Bouhadef Khedidja (b), Slimani Abdelkader (a) and Zeghmati Belkacem (c)

(a) Laboratory of Thermodynamic and Energetic, University of Bechar B.P. 417, Bechar, Algeria

(b) Laboratory of Multiphase Transport and Porous Media, Faculty of Mechanical, University of Sciences and Technology- Houari Boumediene, Algiers, Algeria

(c) Laboratory of Mathematics and System Physics--Group of Energetic Mechanic University of Perpignan, 52 avenue de Villeneuve-66860 Perpignan, France

Two-dimensional laminar flow and heat transfer in the entrance region of a symmetric wavy-channel with constant wall heat flux density which includes fourteen undulations are presented. Numerical solutions are obtained using the control-volume finite-difference method. Thermal and flow fields, Nusselt number and shear stress are presented for different flow rates (100 [less than or equal to] Re [less than or equal to] 1500), amplitude of wavy surface (0 [less than or equal to] A [less than or equal to] 0.5) and Prandtl number values. The results show that the wavy-wall generates vortices in the hollow region, which grow in magnitude and spatial flow coverage with increasing Re and/or A. The shear stress and the Nusselt number increase with increasing the Reynolds number and have the highest magnitude in the entrance region. The increase of average Nusselt number is found limited with the increase of the amplitude.

Keywords: Forced convection, laminar flow, entrance region, wavy channel, heat transfer.

Introduction

The exchanger's plate concepts go back to almost one century. These exchangers have been originally studied to meet the needs of industry, and then used in different branches. For a fully developed flow, the plate grooves aim to increase vortices and augment heat transfer coefficients. The channel with a periodically converging-diverging cross section is one of several devices employed for enhancing the heat and mass transfer efficiency due to turbulence promotion. A numerical scheme has been used by G. Wang and S.P. Vanka [1] to study the heat transfer rates for a flow in periodical wavy passages.

In steady flow, average Nusselt numbers for the undulated wall channel were slightly superior to those obtained for a parallel wall channel. However, in the transitional-flow regime, the enhancement of heat transfer was about a factor of 2.5. Wang, Zhang and Metwally [2, 3, 4] analyzed the laminar forced convection in wavy-wall channels. Results show that the size of the vortices and the skin-friction coefficient increases as Reynolds number and/or amplitude. The mixing of the flow and the acceleration of the core flow results in an increase of the heat transfer. Sarma et al. [5] suggested a new method to estimate the average heat transfer coefficients in the entrance region of a short length of tube under developing laminar flow conditions. Fontes et al. [6] have investigated experimentally the developing laminar non-isothermal and non-Newtonian flow in the entrance region of short tubes with heating and cooling conditions. M. Rahmena et al. [7] present a numerical study of a forced laminar flow through an array of parallel flat plates with finite thickness. Results show that the shear stress depends on the wall thickness, takes the value corresponding to a fully developed flow, and is constant far from the inlet of the channel. Magno et al. [8] employed the generalized integral transform technique in non-Newtonian fluids within a parallel plates channel.

It was verified that for values of power-law indices greater than unity, lower values for the Nusselt numbers in the entrance region are obtained. Bazdidi et al. [9] investigated numerically the laminar fluid flow and heat transfer in the entrance region of a two dimensional horizontal channel with in-line ribs. The results show that at a fixed Reynolds number increasing the rib spacing causes a marginal increase in the local Nusselt number and a decrease in the friction factor. Fabbri et al. [10] presented the analysis of the heat transfer of a Newtonian fluid in the entrance region of a corrugated channel. The analysis of the local equivalent Nusselt number has shown that the global heat transfer is effectively enhanced as the amplitude of the corrugated profile and the Reynolds number increase.

The absolute values of the Nusselt number in the fully developed region of the channel are of the same order of magnitude as that of a smooth channel. A novel approach to the study of the Graetz Brinkman problem in a plane-parallel channel has been presented by Bartella et al.[11]. Comparison between results obtained with his method and results obtained by employing the traditional procedure, with reference to the axial evolution of both the local Nusselt number and the temperature field. Strong discrepancies between the two approaches have been revealed in the initial part of the thermal entrance region. Therefore, the bibliographic survey shows that in most investigations either numerical or experimental, the flow is studied in the part of the channel, far from the entrance and the exit, where it can be considered to be fully developed or periodic. However, in real situations the flow field in the channel includes not only the fully developed region but also the entrance region. The survey of these real situations made few publications. The aim of the present paper is to study the laminar forced convection in the entrance region of a wavy-channel which includes fourteen undulations; the walls are submitted to a density of constant heat flux. Transport equations are solved using the control-volume finite-difference method and Thomas algorithm. The linkage between the velocity and pressure variables is handled by the PISO algorithm. The effects of Prandtl number, Reynolds number and the amplitude of the sinusoidal profile of the wall on the transfers in the entrance region of this channel are studied.

Numerical solution

The laminar forced convection in wavy-plate-fin channel with sinusoidal wall corrugations is numerically simulated. Walls of the channel are submitted each to a flux of heat whose density is constant, but the smooth parts to the exit of the channel are adiabatic. The fluid flow in the channel is supposed to be two-dimensional; hence the width of plates is supposed larger in relation to their spacing. The flow is governed by the Navier--Stokes equations. In addition the retained simplifying hypotheses, we disregarded the influence of the gravity in the momentum equation, effects of the viscous dissipation and the thermal expansion in the heat equation. The system can be written in Cartesian coordinates in the following dimensionless form:

[partial derivative]U / [partial derivative]X + [partial derivative]V / [partial derivative]Y = 0 (1)

U [partial derivative]U / [partial derivative]X + V [partial derivative]U / [partial derivative]Y = - [partial derivative]P / [partial derivative]X + 1 / Re ([[partial derivative].sup.2]U / [partial derivative][X.sup.2] + [[partial derivative].sup.2]U / [partial derivative][Y.sup.2]) (2a)

U [partial derivative]V / [partial derivative]X + V [partial derivative]V / [partial derivative]Y = [partial derivative]P / [partial derivative]Y + 1 / Re ([[partial derivative].sup.2]V / [partial derivative][X.sup.2] + [[partial derivative].sup.2]V / [partial derivative][Y.sup.2]) (2b)

U [partial derivative][theta] / [partial derivative]X + V[partial derivative][theta] / [partial derivative]Y = 1 / Re x Pr ([[partial derivative].sup.2][theta] / [partial derivative][X.sup.2] + [[partial derivative].sup.2][theta] / [partial derivative][Y.sup.2]) (3)

Where dimensionless variables are defined as:

X = x / c, Y = y / c, [theta] = (T - [T.sub.0] / [[phi].sub.n.sup. x c] / [lambda]) U = u / [U.sub.e], V = v / [U.sub.e], P = p / [rho][.U.sup.2.sub.e], [R.sub.e] = rho [U.sub.e.sup. x c] / [mu], [P.sub.r] = [mu] [C.sub.p] / [lambda] (4)

Meanwhile, the equations (2) and (3) have a similar form; they can also be reduced to a simple equation of the convective- diffusive conservation of the form:

[partial derivative] / [partial derivative][X.sub.i] [[rho][U.sub.i][PHI] - [[GAMMA].sub.[PHI]] [partial derivative][PHI] / [partial derivative][X.sub.i]] = [S.sub.[PHI]] (5)

Where [PHI] characterizes the dependent variable, [[GAMMA].sub.[PHI]] the effective diffusion and [S.sub.[PHI]] the source term; such as:

For momentum equation:

[GAMMA] = 1 / Re ; S = [partial derivative]P / [partial derivative][X.sub.i] (6a-b)

For heat equation

[GAMMA] = 1 / Re x Pr ; S = 0 (7a-b)

As shown in Fig.1, where OX is the axis of symmetry of the channel, the upper wall profile in the region 0 [less than or equal to] X [less than or equal to] [L.sub.0] is given by the equation:

F(X) = 1 / 2 + A x Sin([pi] x c / L X) (8)

[FIGURE 1 OMITTED]

Where [L.sub.0] as the total length of the sinusoidal surface of the channel, [L.sub.1] the length of the smooth surface, A the amplitude and N the normal. These variables are defined as:

A = a / c, [L.sub.0] = [l.sub.0] / c, [L.sub.1] = [l.sub.1] / c, N = n / c (9)

Where c as the average separation distance between wavy walls.

Thus, the corresponding boundary conditions are considered as follows: At entrance channel (X = 0, 0 < Y < 1):

U = 1. ; V = 0 and [theta] = q (10a-c)

At upper wavy surface:

U(X, F(X)) = V(X, F(X)) = 0; [[bar.[phi].sub.N] = [partial derivative][theta] / [partial derivative]N =1 (11a-c)

Where [[bar.[phi].sub.N] represents the dimensionless heat flux density.

For [L.sub.0] < X [less than or equal to] [L.sub.1]:

U(X, 1/2) = V(X, 1/2) = 0; [partial derivative][theta] / [partial derivative]Y = 0 (12a-c)

At the exit of the smooth part of the channel

(X = [L.sub.0] + [L.sub.1]): [partial derivative]U / [partial derivative]X = 0 ; [partial derivative]V / [partial derivative]X = 0 ; [partial derivative][theta] / [partial derivative]X = 0 (13a-c)

Although the boundary condition (13a-c) is strictly valid only when it is fully developed, its use in other flow conditions is allowed for calculation convenience, too, if this condition is situated: in a region where the flow is in the downstream direction and far enough from the region of study downstream [7].

The local Nusselt number at the wavy wall surface of the channel is defined by the relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14a-b)

Where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] represent respectively angles formed by the normal and OX and OY axes.

Also, the average Nusselt number is defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15a-b)

With S is the surface of wall exchange.

Integrals of equations (15a) are calculated by the Simpson method [15].

The shear stress at the surface is given by the following relation:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], with the viscous stress tensor in 2D is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16a-b)

To determine the flow dynamic field, a simulation by a fluid dynamic calculation has been used. The transfer equation (5) and the boundaries conditions (10-13) are solved using the finite Alternating Direction Implicit method (ADI) [12]. This last is semi-iterative method in which the equations are solved at each step by maintaining full implicitness in one direction at a time, while relaxing the requirement in the other direction; for each direction a tri-diagonal linear system results that can be solved very efficiently. The algorithm PISO (Pressure- Implicit with Splitting of Operators) [13] was applied to solve the coupled system of governing equations. This algorithm employs a series of sequential operations at each time step which the discretized momentum and pressure-based continuity equations are solved in an alternating "predictor-corrector" fashion. This approach offers an advantage over many other schemes. The convergence criteria represented by the relative magnitude of error [epsilon], was grouped, respectively by velocity fields, pressure and temperature by the following relations.

[epsilon] = [absolute value of [[PHI].sup.n+1] - [[PHI].sup.n] / [[PHI].sup.n+1]] < [10.sup.-5] (17)

The grid sharpness precision test has been made to establish an independent numerical solution grid. A sensitive test to the mesh size is made by comparing the effect of the grid size on the shear stress and the axial velocity. According to the Fig.2, we note that the mesh size of 431x21 lead to results very close to those obtained by using a mesh size of 565x31. Consequently, we retained in our calculus the mesh size of 431x21.

[FIGURE 2 OMITTED]

Results and discussion

Numerical results in the entrance region for development of the hydrodynamic, thermal fields, Nusselt number and shear stress are presented for different flow rates (100 [less than or equal to] RE [less than or equal to] 1500), wall corrugations severity (0 [less than or equal to] A [less than or equal to] 0.5) and two different values of Prandtl number Pr = 6.69 and 0.708.

Flow distribution and shear stress

The velocity profile supposed to be uniform at the entry of the channel is modified by the effect of the wall roughness of the channel. As the flow rate is constant in a cross-section of the channel, the parabolic velocity profile is not influenced by the wall roughness. Consequently, the fluid velocity is maximal at the summit of the wavy wall. Besides, we note that the maximum velocity for a smooth channel is according to [14] [U.sub.max.] =1.5 (Fig.2b). Variations in the amplitude of the sinusoidal plate-surface and the flow rate have a significant effect on the dynamic behaviour of the flow field. The flow field is found to be strongly influenced by A and Re, and it displays two distinct regimes (Fig.3): * Undisturbed laminar-flow regime for low Re or A, the flow behavior is very similar to that in fully developed straight-duct flows with no cross-stream disturbance.

[FIGURE 3 OMITTED]

Vortex-flow regime for high Re or A, flow separation and reattachment in the hollows generates transverse vortex cells that grow spatially with Re and A. For Re = 800, the separation starts to occur at A = 0.4. Besides, the intensity and flow area coverage of counter-rotating lateral vortex grows with the increase of the amplitude of the sinusoidal profile, which in turn tends to smaller momentum transfer. These vortices increase as the Reynolds number increases until occupied completely the hollows for A = 0.5 and Re = 1500 (Fig. (3b)). However, for small amplitude (A [less than or equal to] 0.2), the Reynolds number has no effect on the structure of the flow, and the fluid moves undisturbed through the channel by simply adopting the wavy passage shape. In the entrance region of the channel, the shear stress is characterized by a very fast decrease (Figs. 4 and 5).

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In the first wave, the shear stress has the highest magnitude and decrease quickly; the decreasing becomes smaller in the divergent section of the channel with the diminution of the Reynolds number. But far from regions of entry and exit of the channel, the shear stress increases as the Reynolds number increases. It is noticed that as the Reynolds number is decreased from 1500 to 100, the minimum shear stress of each undulation increases.

[FIGURE 6 OMITTED]

This indicates that the diffusion effects are big compared with the convection effects for the very weak Reynolds numbers. Far from the entrance region, where the heat transfer becomes periodic, with the increase of the Reynolds number, the maximum values of shear stress increase in each convergent section, but the minimum values become more and more constants along the divergent section. These results agree with those of Wang et al [2]. We notice that the shear stress for a smooth channel, far from the entrance and the exit of the channel, is independent of the Reynolds number and remains constant along the channel (Figs. 4 and 5). The evolution of the shear stress is represented by a harmonic curve, this last has the same frequency that of the symmetric wavy surface.

Temperature distribution and Nusselt numbers

The isotherms spread along the wall and the highest temperature is observed in the hollows (Fig.6). In these zones, the heat is transferred to the fluid essentially by conduction. The representative curve of the local Nusselt number has a shape of saw tooth. The amplitudes of peaks decrease far from the entrance region. This curve presents a fluctuation for L/4 <X <L that is attenuated while moving away of the entrance region (Figs.7 and 8).

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

In the first half period, the local Nusselt number is characterized by a very fast decrease; and takes its higher value in the convergent part. This is because at the channel entrance the thermal boundary layer is thinner than under fully developed conditions, and the Nusselt number is therefore higher.

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Also, for all flow rates and all amplitude of wall waviness, the heat transfer coefficient is higher near the channel entrance. The local Nusselt number undergoes again another very fast decrease in the first half period of the second undulation. From the third undulation we notice a displacement of the maximum, particularly for higher amplitudes (A = 0.5), of X=3L/4 to X = L; it is visualized by a discontinuous line in fig.7. In the diverging section of the channel, flows re-circulations occur and the velocity assumes the minimum values. In the converging regions and for the same value of the Reynolds number, the local Nusselt number increases as the value of the wavy profile decreases. This result agrees with the one of Fabbri et al [10].

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

The average Nusselt number shows that for Pr = 6.69 and 400<Re <1300, the smooth wall channel leads to heat transfers superior to those obtained for a wavy wall where the amplitude is superior or equal to 0.3 (Fig.9). But for Pr = 0.708 and Re < 1500, the heat transfer decreases as the amplitude of the channel increases (Fig.10). The average Nusselt number increases with the Reynolds number especially for Pr = 6.69 (Fig.11). Generally, there is a significant decrease of heat transfer when the amplitude of the channel increases for Pr = 0.708 (Fig.12).

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

There is only a slight increase for Re = 1500 and the maximum is obtained for A somewhere in the interval [0, 0.1], around A = 0.07. Further there is a strong decrease with increasing A also for this Reynolds number. For all other Reynolds numbers, there is a strong decrease. Indeed, the average velocity [U.sub.m] close to walls for higher amplitudes is characterized by the smaller values; for instance for Re = 500: [U.sub.m](A=0.1) = 0.276, [U.sub.m](A=0.3) = 0.204 and [U.sub.m](A=0.5) = 0.166; for Re=1500: [U.sub.m](A=0.1) = 0.278, [U.sub.m](A=0.3) = 0.195 and [U.sub.m](A=0.5) = 0.157. Therefore, the weak velocities induce a less intense heat transfer in wavy-wall channels to higher amplitudes. The maximal value of the average Nusselt number is obtained for a wavy-wall channel of amplitude equal to 0.1 for Pr = 6.69 with Re>100 and for Pr = 0.708 with Re=1500. Curves of Nusselt numbers relative to C = 2L possess the same character that in the case where C = L. The increase of C is accompanied by an elevation of values of the local Nusselt number (Fig.7). In addition, for the two Prandtl numbers considered the average Nusselt number increases with C (Figs.13-16). The transfers by diffusion are more significant in water (Pr = 6.69) than in air (Pr = 0.708) (Fig.7). The improved convective behaviour with increasing of the Prandtl number is clearly evident (Fig.7), where the higher temperature gradients at the wall are obtained.

Conclusions

Numerical solutions for laminar (100 [less than or equal to] Re [less than or equal to] 1500), incompressible, constant property and forced convection in the entrance region of wavy-wall channels maintained at constant wall heat flux density are obtained. We analyse the effect of the sinusoidal profile (0 [less than or equal to] A [less than or equal to] 0.5) on the Nusselt numbers and shear stress profiles for air (Pr = 0.708) and water (Pr = 6.69). Results show that the flow field is strongly influenced by the amplitude of channel and Reynolds number, and displays two regimes. For a low number Reynolds or amplitude, the regime is undisturbed laminar-flow, and for a high number Reynolds or amplitude the regime is vortex-flow. The wavy-wall channel induces vortices in the hollow region, which grow and spatial flow coverage with increasing Reynolds number and/or amplitude of the wavy-wall. But for small amplitude, the Reynolds number has no effect on the structure of the flow, and the fluid moves undisturbed through the channel with any re-circulation by simply adopting the wavy passage shape. The harmonic curve for the shear stress in the channel has the same frequency as that of the wavy surface, and the maximum and minimum values occur at the locations of minimum and maximum cross-section of the symmetric wavy-channel. But the representative curve of the local Nusselt number has a shape of saw tooth where the intensity of peaks decreases far from the entrance region. The curves of Nusselt number and shear stress are characterized by a very fast decrease at the entry of the channel; tend towards a constant value far from the entry, and they have the highest value in the first half period of the wavy-wall channel. For water, the smooth wall channel leads to heat transfers superior to those obtained for a wavy wall where the amplitude is superior or equal to 0.3 and 400<Re <1300. But for air, the heat transfer decreases as the amplitude of the wavy-wall channel increases when Re < 1500. The heat transfer increases with Reynolds number and the average separation distance between wavy walls, but the enhancement is limited when the amplitude continues to increase. The wave amplitude of walls equal to 0.1 for Pr=6.69 with Re>100 and for Pr=0.708 with Re=1500 is found to significantly enhance the heat transfer.

Nomenclature a, A amplitude of wavy surface, dimensional and dimensionless c ,C average separation distance between wavy walls, dimensional and dimensionless [C.sub.p] specific heat of fluid at constant pressure L wavelength of the wavy wall [l.sub.0],[L.sub.0] total length of the sinusoidal surface, dimensional and dimensionless [l.sub.1],[L.sub.1] length of the smooth surface of the channel dimensional and dimensionless [Nu.sub.x], [Nu.sub.m] local and average Nusselt number p, P pressure, dimensional and dimensionless Pr Prandtl number Re Reynolds number [U.sub.m] the average velocity close to walls Greek symbols [mu] dynamic viscosity [theta] dimensionless temperature [phi] heat flux density [rho] density of the fluid [tau] shear stress [lambda] thermal conductivity Subscripts m average, mechanic n normal p pressure x local w surface conditions

References

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[2] Wang C.-C., and Chen C.-K., 2002, "Forced convection in a wavy-wall channel," Int. J. Heat Mass Transfer, 45, pp. 2587-2795.

[3] Zhang J., Kundu J., Manglik R.M., 2004, "Effect of waviness and spacing on the lateral vortex structure and laminar heat transfer in wavy-plate-fin cores," Int. J. of Heat and Mass Transfer, 47, pp.1719-1730.

[4] Metwally H.M., and Manglik R.M., 2004, "Enhanced heat transfer due to curvature-induced lateral vortices in laminar flows in sinusoidal corrugatedplate channels," Int. J. of Heat and Mass Transfer, 47, pp. 2283-2292.

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Nabou Mohamed (a), Bouhadef Khedidja (b), Slimani Abdelkader (a) and Zeghmati Belkacem (c)

(a) Laboratory of Thermodynamic and Energetic, University of Bechar B.P. 417, Bechar, Algeria

(b) Laboratory of Multiphase Transport and Porous Media, Faculty of Mechanical, University of Sciences and Technology- Houari Boumediene, Algiers, Algeria

(c) Laboratory of Mathematics and System Physics--Group of Energetic Mechanic University of Perpignan, 52 avenue de Villeneuve-66860 Perpignan, France

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Author: | Mohamed, Nabou; Khedidja, Bouhadef; Abdelkader, Slimani; Belkacem, Zeghmati |
---|---|

Publication: | International Journal of Dynamics of Fluids |

Article Type: | Report |

Geographic Code: | 6ALGE |

Date: | Jun 1, 2007 |

Words: | 4278 |

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