Free convection flow and heat transfer between two vertical parallel

Acta Technica 56 (2011), 103–113

c 2011 Institute of Thermomechanics AS CR, v.v.i.

Free convection flow and heat transfer between two vertical parallel plates with variable temperature at one boundary Marneni Narahari 1 , Binay K. Dutta 2

Abstract. An exact solution to the problem of unsteady free convection flow and heat transfer in a viscous incompressible fluid between two long vertical parallel plates is presented. The temperature at one of the plates increases linearly with time while that at the other plate remains constant at the initial fluid temperature. The dimensionless governing equations are solved using Laplace transform technique. The solutions are obtained in the forms of rapidly converging infinite series. Computed velocity and temperature profiles as well as the shear stress and heat flux at the plates are presented. The parametric and temporal variations of the relevant quantities are discussed. Key words. Natural convection, parallel vertical plates, variable temperature, parallel plate channel, skin friction, Nusselt number.

1. Introduction Free convection flow and heat transfer in vertical parallel plate channels were extensively investigated during the last a few decades. There are many situations of practical importance that conform to this phenomenon. Typical examples are cooling of electronic elements, printed circuit boards and electronic packaging [1], some solar collectors and air heaters [2], [3]. A variety of physical conditions described by surface temperature or heat flux at a plate have been considered for the purpose of analysis of this class of problems. The study of fully developed free convection flow and steady state heat transfer between two parallel plates at constant temperature was initiated by 1 Fundamental

and Applied Sciences Department, Universiti Teknologi PETRONAS, 31750 Tronoh, Bandar Seri Iskandar, Perak Darul Ridzuan, Malaysia; e-mail: [email protected] 2 Chemical Engineering Program, The Petroleum Institute, Abu Dhabi, UAE; e-mail: [email protected]

http://journal.it.cas.cz

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M. NARAHARI, B. K. DUTTA

Ostrach [4]. Combined natural and forced convection laminar flow and heat transfer with linear wall temperature profile were also studied by Ostrach [5]. The first exact solution for free convection in a vertical parallel plate channel with asymmetric heating for a fluid of constant properties was presented by Aung [6]. A number of studies that followed dealt with variations of the basic problem pertaining to diverse thermal boundary conditions of theoretical as well as practical interest. Many of the early works have been reviewed by Manca et al. [7]. Recent contributions on this problem include the numerical investigation of natural convection flow by Lee and Yan [8] in a vertical channel with unheated entry and unheated exit sections. The study revealed interesting results on the induced volumetric flow rate of the fluid and the Nusselt number. Lee [9] investigated laminar natural convection heat and mass transfer between two vertical parallel plates with a heated section preceded and followed by unheated entry and exit that represent a discrete heating condition. The presence of unheated zones was found to have a strong influence on the transport processes. Recently, Campo et al. [10] considered natural convection for heated iso-flux boundaries of the channel containing a low-Prandtl number fluid. Pantokratoras [11] studied the fully developed free convection flow between a pair of asymmetrically heated vertical parallel plates for a fluid of varying thermophysical properties. However, all the above studies are restricted to fully developed steady state flows. Very few papers deal with unsteady flow situations in vertical parallel plate channels. Transient free convection flow between two long vertical parallel plates maintained at constant but unequal temperatures was studied by Singh et al. [12]. Jha et al. [13] extended the problem to consider symmetric heating of the channel walls. Narahari et al. [14] analyzed the transient free convection flow between two long vertical parallel plates with constant heat flux at one boundary, the other being maintained at a constant temperature. Recently, Singh and Paul [15] presented an analysis of the transient free convective flow of a viscous incompressible fluid between two parallel vertical walls occurring as a result of asymmetric heating/cooling of the walls. However, practically important problems on the transient free convection between two parallel plates in which the plate temperature is a function of time have not been addressed in the literature. Such a situation occurs in a parallel plate solar heater or collector in which the plate temperatures are functions of time depending upon the irradiance of a plate which is liable to variation during a day. In the present work we have addressed this problem and analyzed the transient free convection flow and heat transfer in a parallel plate channel in which the temperature of one of the plates is a linear function of time. Closed form analytical solutions to velocity distribution, viscous drag, temperature distribution and heat flux have been arrived at and typical computed results have been presented.

FREE CONVECTION FLOW AND HEAT TRANSFER

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Fig. 1. Schematic diagram of the physical model

2. Governing equations and their solutions Consider an unsteady free convection flow of an incompressible viscous fluid between two vertical parallel plates, the distance between them being d. Initially both the plates and the enclosed fluid are at a temperature Td0 , and the fluid is stagnant. At time t0 > 0, the surface temperature of plate-1 at y 0 = 0 starts increasing linearly with time while that of plate-2 at y 0 = d continues to remain at the initial temperature Td0 . The x0 -axis is taken along plate-1 in the vertically upward direction and the y 0 -axis is normal to the plates as shown in Fig. 1. Then the free convection flow of the fluid can be shown to be governed by the following equations under usual Boussinesq’s approximations [12]–[15]: ∂u0 ∂ 2 u0 0 0 = gβ (T − T ) + ν , d ∂t0 ∂y 02 ∂2T 0 ∂T 0 ρ Cp 0 = k 02 , ∂t ∂y

(1) (2)

where u0 and T 0 are velocity and temperature of the fluid, respectively, g is acceleration due to gravity, β is thermal expansion coefficient, ν is kinematic viscosity, ρ is density, Cp is specific heat at constant pressure and k is thermal conductivity. The following initial and boundary conditions are specified:  t0 ≤ 0 : u0 = 0 , T 0 = Td0 for 0 ≤ y 0 ≤ d ,  0 u = 0 , T 0 = Td0 + (Tw0 − Td0 )At0 at y 0 = 0 , (3)  t0 > 0 : 0 0 0 0 u = 0 , T = Td at y = d , where Tw0 denotes the temperature of the plate at y 0 = 0 and A = ν/d2 .

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It is assumed that the upper and lower ends of the channel between two parallel surfaces are open to the ambient medium at temperature Td0 . The following non-dimensional quantities are defined:  t0 ν u0 ν y0 u0 d , t= 2 , u= 2 ,  y= =   d d d gβ(Tw0 − Td0 ) ν Gr (4)  gβ(Tw0 − Td0 )d3 T 0 − Td0 µCp   Gr = , θ = , Pr = , ν2 Tw0 − Td0 k where Gr and Pr denote the Grashof and Prandtl numbers, respectively, and θ is the (dimensionless) temperature. Then, in view of (4), Eqs. (1)–(3) reduce to the following non-dimensional forms: ∂2u ∂u =θ+ 2 , ∂t ∂y Pr

∂θ ∂2θ = . ∂t ∂y 2

(5) (6)

The initial and boundary conditions in terms of the non-dimensional variables are given as follows:  t≤0 : u=0, θ=0 for 0 ≤ y ≤ 1 ,  u=0, θ=t at y = 0 , (7)  t>0 : u=0, θ=0 at y = 1 . The solutions to Eqs. (5) and (6) satisfying the initial and boundary conditions (7) are obtained by the Laplace transform technique: " ∞ X 4 1 a 2 u(y, t) = a + 12t (a + t) erfc √ − 24(Pr −1) n=0 2 t p 2 t/π e−a /4t − − 2a a2 + 10t b − b4 + 12t (b2 + t) erfc √ + 2 t p 2 + 2b b2 + 10t t/π e−b /4t − √ ! 4 2 a Pr 2 √ + − a Pr + 12t (a Pr +t) erfc 2 t p 2 + 2a a2 Pr +10t Pr t/π e−a Pr/4t + √ ! 4 2 b Pr 2 √ + b Pr + 12t (b Pr +t) erfc − 2 t # p 2 −b2 Pr/4t − 2b b Pr +10t Pr t/π e , (8)


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where a = 2n + y, b = 2n + 2 − y. The solution for the non-dimensional temperature distribution in the fluid can be found to be " √ ! ∞ p 2 1X a Pr 2 √ θ(y, t) = − 2a Pr t/π e−a Pr/4t − a Pr +2t erfc 2 n=0 2 t # √ ! p 2 b Pr √ − b2 Pr +2t erfc + 2b Pr t/π e−b Pr/4t . (9) 2 t

3. Solution for the case of Pr = 1 We observe that the solution for the velocity distribution given by Eq. (8) is not defined for a fluid of Prandtl number unity. Since the Prandtl number is a measure of the relative importance of the viscosity and thermal conductivity of the fluid, the case Pr = 1 corresponds to those fluids whose momentum and thermal boundary layer thicknesses are of the same order of magnitude. The exact solution of the present problem when Pr = 1 is given below. It may be noted that the solution for the temperature variable θ(y, t) follows from equation (9) by putting Pr = 1 in that equation. However, in the case of u(y, t), the solution has to be rederived starting from Eqs. (5) and (6): " ! ∞ X p 2 y a u(y, t) = t/π e−a /4t − a 6t + a2 erfc √ 2 4t + a2 − 12 2 t n=0 ! p 2 b 1 2b 4t + b2 t/π e−b /4t − b2 6t + b2 erfc √ + − 12 2 t !# p 2 (n + 1) c + 2 4t + c2 t/π e−c /4t − c 6t + c2 erfc √ , 6 2 t (10) where c = 2n + 2 + y.

4. Results and discussion Solution of the present problem by the Laplace transform method has yielded expressions for velocity and temperature fields in series form. The series in Eqs. (8), (9) and (10) can be shown to be absolutely convergent because of the presence of well known standard mathematical functions. In order to gain a physical insight into the problem, we have computed the numerical values of the velocity and temperature for different values of time t and for Prandtl number Pr = 0.71 (air), 7.0 (water) and 100 (oil).

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Fig. 2. Velocity profiles at various t and Pr

Fig. 3. Temperature profiles at various t and Pr

The velocity profiles for different values of the Prandtl number and time are shown in Fig. 2. It is observed that an increase in the Prandtl number has a diminishing effect on the velocity. However, increasing time has the reverse effect on the local velocity in the channel. The velocity profile is expectedly asymmetric, the maximum of it leaning towards the plate having an increasing


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temperature and correspondingly lower density of the fluid in its vicinity. Also the velocity distribution at a constant dimensionless time becomes steeper with decreasing Prandtl number that corresponds to decreasing momentum diffusivity of the fluid as compared to its thermal diffusivity. The dimensionless temperature, which is numerically equal to the dimensionless time in accordance with the definition of the dimensionless quantities used in this work, for different time and Prandtl number is plotted in Fig. 3. The fall in temperature with increasing y is sharper for higher Prandtl numbers because of the lesser contribution of transport of energy as compared to that of momentum. The other two practically important measures of transport of momentum and energy are the skin friction (or shear stress) and the heat flux at the plates. The quantities in dimensionless form may be easily derived from the solutions for the distribution of velocity Eq. (8) and temperature Eq. (9) . Let τ10 denote the skin friction at the plate y 0 = 0. When Pr 6= 1, the non-dimensional skin frictions τ1 and τ2 at plate-1 and plate-2, respectively, are as follows: • plate-1: " ∞ X τ10 du n 2 2 τ1 = = n 2n + 3t erfc √ − = 0 0 dgβ(Tw − Td ) dy y=0 3(Pr −1) n=0 t p 2 n+1 − 2 t/π n2 + t e−n /t + (n + 1) 2(n + 1)2 + 3t erfc √ − t √ ! p −(n+1)2/t 2 n Pr 2 − 2 t/π (n + 1) + t e − n Pr 2n Pr +3t erfc √ + t p 2 + 2 Pr t/π n2 Pr +t e−n Pr /t − √ ! (n + 1) Pr 2 √ + − (n + 1) Pr 2(n + 1) Pr +3t erfc t # p −(n+1)2 Pr /t 2 + 2 Pr t/π (n + 1) Pr +t e (11) and • plate-2: " ∞ X du −1 2n + 1 2 √ τ2 = − = (2n + 1) (2n + 1) + 6t erfc − dy y=1 3(Pr −1) n=0 2 t p 2 − 2 t/π (2n + 1)2 + 4t e−(2n+1) /4t − √ ! (2n + 1) Pr 2 √ − (2n + 1) Pr (2n + 1) Pr +6t erfc + 2 t p 2 + 2 Pr t/π (2n + 1)2 Pr +4t e−(2n+1) Pr /4t . (12)

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The dimensionless heat flux, which is nothing but the Nusselt numbers Nu1 and Nu2 at the two plates, can be derived as • plate-1: dθ q10 d = − = k(Tw0 − Td0 ) dy y=0 " √ ! ∞ p √ √ X n Pr −n2 Pr /t t/π e − n Pr erfc √ + = 2 Pr t n=0 √ !# p √ (n + 1) Pr −(n+1)2 Pr /t √ + t/π e − (n + 1) Pr erfc t

Nu1 =

(13)

and • plate-2: dθ q20 d =− Nu2 = = k(Tw0 − Td0 ) dy y=1 " √ !# ∞ p √ √ X (2n + 1) Pr −(2n+1)2 Pr /4t √ − (2n + 1) Pr erfc 2 t/π e , = 2 Pr 2 t n=0 (14) where q10 and q20 denote the rate of heat transfer at plate-1 and plate-2, respectively. Computed values of the dimensionless shear stress and Nusselt number at typical values of the parameters are presented in Table 1 against the time t. For all times, plate-1 is subjected to a significantly higher shear stress than plate-2 because of the steeper velocity profile in the vicinity. The relative increase in shear stress is found to be larger for a fluid of higher Prandtl number as the dimensionless time increases from 0.2 to 1.0. This is commensurate with the dampening effect of the fluid viscosity and hence of the Prandtl number on the velocity distribution between the plates. The Nusselt number at plate-1 is much higher than that at plate-2 indicating that much of the energy released from plate-1 because of its increasing temperature with time is convected out by the fluid before it reaches the other plate. This observation is practically important for a solar air heating device similar in construction and operation. The volume flow rate, i.e., volumetric rate of up-flow of the fluid because of free convection, can be calculated from the integral of the axial velocity distribution [6], [11]. The volume flow rate M in non-dimensional form is given by Z 1

M=

u dy . 0

(15)

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Table 1. Typical computed values of τ1 , τ2 , Nu1 , Nu2 , M and Q t

Pr

τ1

τ2

Nu1

Nu2

M

Q

0.2 0.4 0.6 0.8 1.0

0.71 0.71 0.71 0.71 0.71

0.036102 0.085755 0.119625 0.171303 0.295337

0.007703 0.034646 0.066931 0.100109 0.133420

0.427742 0.636113 0.836632 1.036665 1.236667

0.090590 0.282220 0.481701 0.681669 0.881667

0.002753 0.009791 0.017912 0.026214 0.034542

0.000226 0.001806 0.005194 0.010384 0.017358

0.2 0.4 0.6 0.8 1.0

7.0 7.0 7.0 7.0 7.0

0.018462 0.052522 0.096584 0.146053 0.196532

0.000543 0.005359 0.016038 0.031938 0.052013

1.335116 1.888139 2.312491 2.670274 2.985737

0.000021 0.004192 0.030121 0.088606 0.178410

0.000537 0.002652 0.006246 0.010998 0.016629

0.000020 0.000229 0.000917 0.002392 0.004927

0.2 0.4 0.6 0.8 1.0

100 100 100 100 100

0.006117 0.017320 0.031825 0.048837 0.067792

0.000033 0.000327 0.001004 0.002082 0.003563

5.046265 7.136496 8.740387 10.092530 11.283792

2.0611×10−56 8.0178×10−29 1.6260×10−19 8.2730×10−15 5.9254×10−12

0.000047 0.000246 0.000613 0.001147 0.001842

4.7747×10−7 0.000006 0.000027 0.000074 0.000161

The total heat rate added to the fluid can be calculated from the integral of the product of the axial velocity and the temperature [6], [11]. The nondimensional total heat rate added to the fluid Q is given by Z Q=

1

uθ dy .

(16)

0

Computed time-varying values of M and Q obtained by numerical integration with Simpson’s 1/3 rule are also presented in Table 1 for different values of the Prandtl number. The flow induced by the monotonically rising wall temperature predictably increases the volume flow rate and the total heat rate added to the fluid because of the enhanced buoyancy. The Prandtl number, however, has a diminishing influence on the volume flow rate and the total heat rate added to the fluid. This is because a higher Prandtl number is associated with a larger kinematic viscosity and correspondingly higher viscous drag. A comparison with data reported by Singh et al. [12] for the case of uniform constant wall temperature shows the dimensionless shear and Nusselt number to be lower in the case under consideration. This appears to be because of a step increase in the wall temperature at time t = 0+ for the case considered by Singh et al. [12] compared to the ramp increase in the temperature of one of the plates in the present situation while the other plate is maintained at the initial fluid temperature.

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5. Conclusions We have presented an exact analytical solution to the problem of unsteady free convective flow and heat transfer of a viscous incompressible fluid between two long vertical parallel plates with the plate temperature linearly varying temperature with time at one boundary, that at the other boundary being held constant. The dimensionless governing equations are solved by the Lapace transform technique. The solution for the velocity field at Pr = 1 is derived separately because of a singularity. The solutions which are in the form of infinite series converge very rapidly. The effects of the Prandtl number and time on the flow and temperature fields are discussed. It is found that both the velocity and temperature profiles are asymmetric and are strongly influenced by the Prandtl number. The velocity profile is steeper at a lower Prandtl number while the temperature profile is so at a higher value of the parameter. Computed values are presented to show that time variation of dimensionless skin friction and Nusselt number. The skin friction or the dimensionless shear increases more with the low values of the Prandtl number. The Nusselt number is consistently much higher at the plate with time-varying temperature. A major part of the heat transferred from this plate is removed by the free convection flow before it reaches the other plate. The solutions for the velocity and temperature fields and the associated transport rates are likely to be useful for design calculation of parallel plate solar heaters.

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Received May 14, 2010