Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915 www.springerlink.com/content/1738-494x

DOI 10.1007/s12206-014-0338-5

The forced convection flow over a flat plate with finite length with a constant convective boundary condition† Asterios Pantokratoras* School of Engineering, Democritus University of Thrace, 67100 Xanthi, Greece (Manuscript Received May 20, 2013; Revised December 2, 2013; Accepted December 24, 2013) ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Abstract The flow of a fluid past a flat plate of finite length and infinite width (two-dimensional flow) is considered. The plate is heated by convection from a fluid with constant temperature T f with a constant heat transfer coefficient h f . In all previous works, the problem was considered using boundary layer theory whereas, in the present work, the solution is based on the full Navier-Stokes equations. The problem is investigated numerically with a finite volume method using the commercial code ANSYS FLUENT. The governing parameters are the Reynolds number, the new heat transfer parameter, and the Prandtl number. In addition, the influence of these three parameters on the temperature field is investigated. It is found that high Reynolds and high Prandtl numbers the wall temperature increases along the plate. They reach a maximum near the trailing edge then decrease. The same occurs as the heat transfer parameter increases. When the Reynolds and Prandtl numbers are low, the plate temperature tends to become symmetric, with a maximum at the middle of the plate. The temperature profiles become thicker as the Reynolds number and the Prandtl number is reduced while the temperature profiles become thicker as the heat transfer parameter increases. Keywords: Convective boundary condition; Flat plate; Forced convection; Wake ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

1. Introduction The heat transfer through a wall with a convective boundary condition is important due to its various applications in engineering and various industries such as in the design of cooling systems for electronic devices, in the field of solar energy collection, heat exchangers, thermal insulation, nuclear reactors cooling, and around buildings. The problem of laminar hydrodynamic and thermal boundary layers over a flat plate in a uniform stream of fluid is a thoroughly researched problem in fluid mechanics [1]. The velocity distribution in the hydrodynamic boundary layer is given by the well-known Blasius similarity solution. The similarity solution for the thermal boundary layer for the case of constant surface temperature and constant heat flux at the plate is also well established and widely quoted in heat transfer textbooks. Aziz [1] was probably the first to treat the case of convective heat transfer on a plate, proving that a similarity solution is possible if the convective heat transfer of the fluid heating the plate on its lower surface is proportional to x -1/ 2 . If the convective heat transfer coefficient is a constant, then the problem does not admit a similarity solution.

The appearance of the above paper provoked the creation of a large number of subsequent papers concerning different boundary layer problems with convective boundary conditions. Example studies include boundary layer flow with variable viscosity [2], with entropy generation [3], in power-law fluids [4], in micropolar fluids [5], in mixed convection [6], in nanofluids [7], in Jeffrey fluids [8], in porous media [9], in variable fluid properties [10], and analytical solutions [11]. All works mentioned above concern a convective heat transfer coefficient as a function x such that the problems accept similarity solutions. In contrast to this hypothesis, Merkin and Pop [12] presented a solution of to Blasius flow with convective boundary condition using a constant heat transfer coefficient. Considering that this problem is non-similar, the authors presented a solution that depends on the distance along the plate. All works presented until now, either similar or nonsimilar, including Ref. [12], are based on boundary layer theory, which is valid for the case of an infinite Reynolds number, that is, a plate with infinite length. In the present paper, we treat heat transfer along a plate with finite length and constant convective heat transfer coefficient for the first time in the literature.

*

Corresponding author. Tel.: +30 25410 79618, Fax.: +30 25410 79616 E-mail address: [email protected] † Recommended by Associate Editor Ji Hwan Jeong © KSME & Springer 2014

2. Mathematical model and numerical code Consider the flow of a Newtonian fluid past a horizontal

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A. Pantokratoras / Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915

Pr =

u

(7)

a

where L is the plate length. Merkin and Pop [12] used the following non-dimensional axial and transverse variables in the boundary layer theory approach.

x= h=

plate as it is shown in Fig. 1. The full equations of this flow are the following: ¶u ¶v + =0 ¶x ¶y

x-momentum equation: ¶u ¶u ¶ 2u ¶ 2u 1 ¶p +v =+u ( 2 + 2 ) u r ¶x ¶x ¶y ¶x ¶y y-momentum equation: ¶v ¶v ¶ 2v ¶ 2v 1 ¶p +v = +u ( 2 + 2 ) u r ¶y ¶x ¶y ¶x ¶y

u¥ k 2

(8)

x

hf / k (2x )1/ 2

(9)

y.

The plate has a finite length of L. Taking this fact into account, the following non-dimensional heat transfer parameter, which includes this finite length, is introduced.

Fig. 1. The flow configuration and coordinate system.

continuity equation:

uhf 2

(1)

(2)

Ht =

uhf 2 u¥ k 2

(10)

L

which is the third non-dimensional parameter that governs this problem (the other two are the Reynolds and Prandtl numbers). The non-dimensional vertical coordinate and the nondimensional temperature are the following: y L T - T¥ J= . T f - T¥

(11)

Y=

(12)

(3) In addition, the local and average Nusselt numbers are:

energy equation: ¶T ¶T ¶ 2T ¶ 2T u +v = a( 2 + 2 ) ¶x ¶y ¶x ¶y

(4)

Nu = -

é ¶T ù L ê ú (Tw - T¥ ) ë ¶y û y = 0 L

Nuavg = -

subject to boundary conditions at the plate: u = v = 0, - k

¶T = h f (T f - T ) on y = 0 ¶y

u ® u¥ , T ® T¥ as y ® ¥ .

(5)

where x is the horizontal coordinate, y is the vertical coordinate, u is the horizontal velocity, v is the vertical velocity, p is the pressure, ρ is a reference density, u is the fluid kinematic viscosity, a is the fluid thermal diffusivity, k is the fluid thermal conductivity, and T is the fluid temperature. The plate is assumed to be heated by convection from a fluid with constant temperature T f with a heat transfer coefficient h f . The two classical non-dimensional parameters of this problem are the Reynolds and Prandtl numbers, which are defined as Re =

u¥ L

u

(13)

(6)

é ¶T ù 1 ê ú x dx . L(Tw - T¥ ) ë ¶y û y = 0 0

ò

(14)

The present numerical study has been carried out using ANSYS FLUENT (version 12.0). The two-dimensional, steady, laminar solver was used with the third-order scheme for the convective terms in the momentum equation. The coupled scheme was used for pressure-velocity coupling. Double precision accuracy was used and a convergence criterion of 10-10 was used for the x- and y- velocity components as well as for the residuals of the continuity and energy equation. The proposed code is well known and is a robust CFD (computational fluid dynamics) solver that has been used extensively in the literature. Therefore, no further information about this code will be given here. The applied boundary conditions (Fig. 1) are the following according to the ANSYS FLUENT code. The boundary AG was defined as a “velocity inlet,” where the horizontal velocity is constant and the vertical velocity is zero whereas the tem-

A. Pantokratoras / Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915

perature is set equal to T¥ . The boundary EF was defined as “pressure outlet” where the static pressure is placed equal to ambient pressure and all other flow quantities are extrapolated from the interior. The boundaries AB, DE, and FG were defined as “symmetry,” where the velocity and temperature gradients in the vertical direction are forced to be zero. The plate was defined as “wall,” where both the horizontal and vertical velocities are zero and the heat transfer coefficient h f and T f were taken as constant values. The thickness of a tangential flow above a horizontal plate is very large when the Reynolds number is small (creeping flow, Re < 1) whereas the thickness is very small when the Reynolds number is large (boundary layer approximation). In the present work, a wide Reynolds number range that extends from very small ( Re = 0.01) to large ( Re = 1000) values is considered. To achieve domain-independent results for low Reynolds numbers, all boundaries were placed far away from the plate (at distance 5000 L where L is the plate length). All calculations were carried out within this large field so that the results become domain-independent for all Reynolds numbers. The grid near the surface of the plate was made sufficiently fine to adequately resolve the flow characteristics near the plate. To refine the grid, the grid-adaptation function of ANSYS FLUENT was used as follows. The program starts with an initial number of grid points and, after a first run, the velocity gradients in the x and y directions are calculated automatically. If the velocity gradients are high, the program automatically adds more grid points at the regions with high gradients. This procedure is repeated many times until the program stops to add more points. Thus, a sufficient number of grid points are used and placed at the proper places. For the present problem, the number of nodes was about 100000 and most of them were placed near the plate.

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Fig. 2. Non-dimensional temperature profiles calculated using boundary layer theory and the solutions of complete, elliptic equations for Prandtl = l.

3. Results and discussion

Fig. 3. Wall temperature along the plate for Pr = 1, Ht = 1, and different Reynolds number values.

The above continuity and momentum Eqs. (1)-(3) have been already solved by the present author in Ref. [13], where extensive validation tests have been applied to check the accuracy of the results (for more information, see Ref. [13]). Consequently, only the validation of the solution to the energy equation is required. However, finding this validation is difficult because only results for boundary layer approximations are available in the literature only. When the Reynolds number becomes large, the present problem approaches the boundary layer state but never reaches it due to its finite plate length, the existence of the trailing edge, and the wake in the rear part of the plate (see Fig. 1). In Fig. 2, the present results and those by Ref. [12] are compared. In this figure, two non-dimensional temperature profiles are shown by Ref. [12] for distance x = 0.124 and by the present work for the same distance and Re = 1000. Taking into account that the results of Ref. [12] are based on boundary layer theory (parabolic problem) and the present results on the solution of the full equations (elliptic problem), the comparison is considered satisfactory.

In the following sections, the results of the present work are presented. In Fig. 3, the variation of the wall temperature along the plate is shown for six different Reynolds numbers as well as Pr = 1 and Ht = 1 . The following conclusions can be drawn from this figure. First, at high Reynolds numbers, the wall temperature is low at the plate leading edge and increases as the plate trailing edge is approached. The plate temperature reaches a maximum near the trailing edge and then decreases. This behavior is different from the prediction of boundary layer theory, where the plate temperature increases monotonically (see Fig. 3(a) in Ref. [12]). Second, as the Reynolds number decreases (Stokes flow), the wall temperature along the plate tends to become symmetric, that is, the temperature tends to become equal at the plate leading and trailing edges with a maximum at the middle of the plate. Third, the plate temperature depends on the Reynolds number, in contrast to boundary layer theory, and the temperature increases as Re increases. The explanation for the above behavior is the fol-

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Fig. 4. Wall temperature along the plate for Re = 100, Pr = 1, and different values of the heat transfer parameter.

lowing: It is known in fluid mechanics that the Navier-Stokes equations highly non-linear due to existence of convection terms (left hand side of Eqs. (2) and (3)). As the Reynolds number decreases, tending to zero, the convection terms tend to zero and the equations tend to become linear. These flows are called Stokes flows or creeping flows. The Stokes flows are reversible in the sense that the reverse flow is also a Stokes flow. Thus, the velocity pattern of a symmetrical object must be symmetrical. The downstream flow has the same streamline pattern and velocity magnitude as the upstream pattern (Ref. [14], page 574 and Fig. 21.1 on the same page). The flow field becomes perfectly symmetrical when the Reynolds number becomes zero (Ref. [15], page 201 and Figs. 3-50 on the same page). Taking into account that the energy in Eq. (4) is uncoupled from the momentum equations (the energy equation does not influence the momentum equations), assuming that the temperature field around a symmetrical body will be symmetrical in the Stokes flow similar to that of the velocity field is reasonable. The plate of the present problem is a symmetrical body around the plate middle point (point C in Fig. 1). The above arguments explain why the wall temperature along the plate tends to become symmetrical as the Reynolds number decreases. When the Reynolds number increases, the flow changes from linear (Stokes type) to non-linear (NavierStokes), and the symmetry is lost. In this case, the plate temperature increases because it is heated through convection by the fluid below the plate. However, the temperature of the plate cannot increase continuously because it has finite length. In addition, at the trailing edge (point D in Fig. 1), the plate temperature should be compatible with the ambient temperature T¥ , which is lower than T f . Thus, at some point before the trailing edge, the plate temperature starts to decrease so that it can adjust to the low ambient temperature. As a result, the temperature maximum emerges. The influence of the heat transfer parameter on the wall temperature is illustrated in Fig. 4. The behavior is similar to

Fig. 5. Wall temperature along the plate for Re = 100, Ht = 1 and different values of the Prandtl number.

that of high Reynolds numbers. The wall temperature increases from the leading edge towards the trailing edge, reaches maximum near the trailing edge, and then decreases. In addition, as Ht increases, the plate temperature tends to 1. The explanation for the appearance of the temperature maximum is the same as that previously. The plate temperature increases due to heating from the fluid below. However, near the trailing edge, the plate temperature starts to decrease to reach the ambient fluid temperature T¥ . When the heat transfer parameter in Eq. (10) is very large (Ht = 1000), much heat is transferred from the lower fluid to the plate. Consequently, the plate temperature becomes very high and approaches temperature T f . In this case, the plate temperature J (0) = (Tw - T¥ ) / (T f - T¥ ) tends to 1. In Fig. 5, the influence of the Prandtl number is shown. When the Pr numbers are high, the usual trend appears, that is, the plate temperature increases along the plate, reaches maximum near the trailing edge, and then decreases. However, the behavior is different at low Pr numbers. As the Pr number decreases, the wall temperature tends to become symmetric along the plate with a maximum at the middle of the plate. In addition, as the Pr number decreases, the plate temperature tends to 1. The reduction of the Reynolds number and the Prandtl number has the same qualitative effect on the plate temperature, forcing the plate temperature to become symmetric along the plate. In Fig. 5, the Reynolds number is relatively high (Re = 100) and the temperature along the plate is expected to be asymmetrical. However, at Pr = 0.001, the temperature is almost symmetrical. The explanation is based on the Peclet number (Pe = PrRe), which is also used in heat transfer problems in the Stokes flow [16]. Although the Reynolds number is relatively high, when Pr = 0.001, the Peclet number is low (Pe = 0.1). Thus, in this case, symmetry appears. In Fig. 9, the temperature gradient ¶T / ¶y at the plate decreases as the Pr number decreases. From boundary condi-

A. Pantokratoras / Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915

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Table 1. Calculated values of average Nusselt number at the plate. Pr = 0.1 Ht

Re = 1000 Re = 100 Re = 10 Re = 1 Re = 0.1 Re = 0.01

0.01

11.76

3.49

1.231

0.666

0.450

0.3392

1

10.44

3.32

1.242

0.668

0.451

0.3392

10

9.80

3.23

1.249

0.671

0.451

0.3394

100

9.44

3.17

1.253

0.675

0.453

0.3398

Pr = 1 Ht

Re = 1000 Re = 100 Re = 10 Re = 1 Re = 0.1 Re = 0.01

0.01

28.90

8.99

2.773

1.112

0.638

0.4406

1

26.59

8.43

2.743

1.117

0.639

0.4407

10

24.68

7.96

2.704

1.124

0.639

0.4409

100

23.20

7.61

2.666

1.132

0.644

0.4416

Pr = 10 Ht

Fig. 6. Local Nusselt number along the plate for Pr = 1, Ht = 1, and different values of the Reynolds number.

Re = 1000 Re = 100 Re = 10 Re = 1 Re = 0.1 Re = 0.01

0.01

64.12

20.79

6.605

2.198

1.003

0.6076

1

61.03

19.87

6.421

2.198

1.004

0.6078

10

57.18

18.73

6.182

2.197

1.007

0.6083

100

53.08

17.51

5.922

2.189

1.014

0.6096

tion Eq. (5), we have - k (¶T / ¶y ) = h f (T f - T ). Thus, as ¶T / ¶y decreases, the term (T f - T ) decreases, and the plate temperature tends to T f . Consequently, plate temperature J (0) = (Tw - T¥ ) / (T f - T¥ ) tends to 1 in Fig. 5. In Fig. 6, the variation of the local Nusselt number along the plate is shown for six different Reynolds numbers as well as Pr = 1 and Ht = 1 . The following conclusions can be drawn from this figure. First, at high Reynolds numbers, the Nusselt number is high at the plate leading edge and decreases as the plate trailing edge is approached. The Nusselt number reaches a minimum near the trailing edge and then increases. The second conclusion is that as the Reynolds numbers decreases (Stokes flow) the Nusselt number along the plate tends to become symmetric, that is, the Nusselt number tends to become equal at the plate leading and trailing edges with a minimum at the middle of the plate. Except of the local Nusselt number, the average Nusselt number were calculated and the results are shown in Table 1 for different values of the three governing parameters (Pr, Re, and Ht). The average Nusselt number increases as the Reynolds number increases. The same happens as the Prandtl number increases. The influence of the heat transfer parameter is as follows. When the Reynolds numbers are high, Ht significantly influences the average Nusselt number. However, as the Reynolds number decreases, the influence tends to disappear and the average Nusselt number remains almost constant (at each Reynolds number) and independent of the heat transfer parameter. In Fig. 7, non-dimensional temperature profiles are shown along the vertical middle axis of the plate (see point C in Fig.

Fig. 7. Temperature profiles at the middle of the plate for Pr = 1, Ht = 1 and different values of the Reynolds number.

1) for different Reynolds numbers as well as Pr = 1 and Ht = 1 . Clearly, the plate temperature (Y = 0) decreases as the Reynolds number decreases and the temperature profiles become wider (low plate temperature, large profile width). In contrast to the above behavior, the temperature profiles in Fig. 8 shows that the plate temperature (Y = 0) increases as the heat transfer parameter increases and the profiles become thicker (large plate temperature, large profile width). Finally, the temperature profiles as a function of the Pr number are shown in Fig. 9. The figure shows that a large plate temperature corresponds to a large profile width. In Fig. 10, five temperature profiles are illustrated along the plate at five points (plate leading edge, middle of the plate, plate trailing edge, and two points beyond the plate inside the wake). The plate temperature is low at the leading edge (point 1), reaches maximum at the middle (point 2), is reduced again at the trailing edge (point 3), and is reduced even more inside the wake (points 4 and 5).

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A. Pantokratoras / Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915

Fig. 8. Temperature profiles at the middle of the plate for Re = 100, Pr = 1, and different values of the heat transfer parameter.

Fig. 9. Temperature profiles at the middle of the plate for Re = 100, Ht = 1. and different values of the Prandtl number.

4. Summary and conclusions The heat transfer problem past a flat plate with finite length with a convective boundary condition was investigated in the present paper. The governing parameters are the Reynolds and Prandtl numbers and the heat transfer parameter. The main conclusions that may be drawn from the present study are summarized below: (1) At high Reynolds numbers, the wall temperature increases along the plate, reaches a maximum near the trailing edge, and then decreases. At low Reynolds numbers, the wall temperature tends to become symmetric, with a maximum at the middle of the plate. The temperature profiles become thicker as the Reynolds number decreases. (2) The influence of the heat transfer parameter on the wall temperature is similar to that of high Reynolds numbers. The wall temperature increases from the leading edge towards the trailing edge, becomes maximum near the trailing edge, and

Fig. 10. Temperature profiles at five points along the plate for Pr = 1, Ht = 1, and Re = 1.

then decreases. In addition, as Ht increases, the plate temperature tends to 1. The temperature profiles become thicker as the heat transfer parameter increases. (3) At high Pr numbers, the usual trend appears, that is, the plate temperature increases along the plate, acquires a maximum near the trailing edge, and then decreases. At low Pr numbers, the wall temperature tends to become symmetric along the plate with a maximum at the middle of the plate. In addition, as Pr number decreases, the plate temperature tends to 1. The temperature profiles become thicker as the Prandtl number decreases. (4) At high Reynolds numbers, the local Nusselt number decreases along the plate, reaches a minimum near the trailing edge, and then increases. At low Reynolds numbers, the local Nusselt number tends to become symmetric, with a minimum at the middle of the plate. (5) At low Reynolds numbers, the average Nusselt number approaches a constant value and becomes independent of the heat transfer parameter.

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A. Pantokratoras / Journal of Mechanical Science and Technology 28 (5) (2014) 1909~1915

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A. Pantokratoras received his Ph.D. in Civil Engineering from the University of Thrace, Greece in 1989. Currently, he is Professor at the Department of Civil Engineering at Democritus University of Thrace, Xanthi, Greece. His research interests include fluid mechanics, water supply, sewerage networks, and drinking water treatment.