Abstract
The steady mixed convection flow in a vertical channel is investigated for laminar and fully developed flow regime. In the modelling of the heat transfer the viscous dissipation term was also considered. Temperature on the right wall is assumed constant while a mixed boundary condition (Robin boundary condition) is considered on the left wall. The governing equations are expressed in non-dimensional form and then solved both analytically and numerically. It was found that there is a decrease in reversal flow with an increase in the mixed convection parameter.
Authors
Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Teodor Groşan
(Babeş-Bolyai University Faculty of Mathematics and Computer Sciences)
Adrian Vasile Lar
(Școala Cojocna, Romania)
Keywords
viscous fluid; forced convection; heat transfer; fully developed flow
Cite this paper as:
F. Pătrulescu, T. Groşan, A.V. Lar, Mixed convection in a vertical channel subject to Robin boundary condition, Studia Universitatis Babeş-Bolyai, seria Mathematica, vol. 55, no. 2 (2010), pp. 167-176
About this paper
Publisher Name
Universitatea Babeş-Bolyai, Cluj-Napoca; Cluj University Press, Cluj-Napoca
Paper on the journal website
Print ISSN
0252-1938
Online ISSN
2065-961X
MR
2661883
ZBL
1224.76037
Google scholar
References
[1] Aung, Fully developed laminar free convection between vertical plates heated asymmetrically, Int. J. Heat Mass Transfer 15 (1972), 1577-1580.
[2] Aung, L.S. Fletcher, V. Sernas, Developing laminar free convection between vertical flat plates with asymmetric heating, Int. J. Heat Mass Transfer, 15 (1972), 2293-2308.
[3] Aung, G. Worku, Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures, J. Heat Transfer, 108 (1986), 299-304.
[4] Aung, G. Worku, Theory of fully developed, combined convection including flow reversal, J. Heat Transfer, 108 (1986), 485-488.
[5] Barletta, Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions, J. Heat Transfer, 121 (1999), 349-356.
[6] Barletta, Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux, Int. J. Heat Mass Transfer, 45 (2002), 641-654.
[7] Boulama, N. Galanis, Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer, J. Heat Transfer, 126 (2004), 381-388.
[8] Bejan, Convection Heat Transfer (2nd edition), Wiley, New York (1995).
[9] Barletta, Laminar mixed convection with viscous dissipation in a vertical channel, Int. J. Heat Mass Transfer, 41 (1988), 3501-3513.
[10] Pop, D. B. Ingham, Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford (2001).
[11] Kohr, I. Pop, Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press, Southampton (2004).
Paper (preprint) in HTML form
Mixed convection in a vertical channel subject to Robin boundary condition
Abstract
The steady mixed convection flow in a vertical channel is investigated for laminar and fully developed flow regime. In the modelling of the heat transfer the viscous dissipation term was also considered. Temperature on the right wall is assumed constant while a mixed boundary condition (Robin boundary condition) is considered on the left wall. The governing equations are expressed in non-dimensional form and then solved both analytically and numerically. It was found that there is a decrease in reversal flow with an increase in the mixed convection parameter.
2010 Mathematics Subject Classification : 76D05,80A20
Keywords: viscous fluid, forced convection, heat transfer, fully developed flow
Keywords: viscous fluid, forced convection, heat transfer, fully developed flow
1 Introduction
Heat transfer in channels occurs in many industrial processes and natural phenomena. It has been, therefore, the subject of many detailed, mostly numerical studies for different flow configurations. Most of the interest in this subject is due to its practical applications, for example, in the design of cooling systems for electronic devices and in the field of solar energy collection. Some of the published papers, such as by Aung [1], Aung et al. [2], Aung and Worku [3, 4], Barletta [5, 6], and Boulama and Galanis [7], are concerned with the evaluation of the temperature and velocity profiles for the vertical parallel-flow fully developed regime. As is well known, heat exchangers technology involves convective flows in vertical channels. In most cases, these flows imply conditions of uniform heating of a channel, which can be modelled either by uniform wall temperature (UWT) or uniform wall heat flux (UHF) thermal boundary conditions. In the present paper, new types of boundary conditions are considered. The right wall is kept at constant temperature while a convective heat flux is considered on the left wall (see, Bejan[8]):
{:(1)(k(del T)/(del y))_(y=0)+h_(a)(T_(a)-T)_(y=0)=0:}\begin{equation*}
\left(k \frac{\partial T}{\partial y}\right)_{y=0}+h_{a}\left(T_{a}-T\right)_{y=0}=0 \tag{1}
\end{equation*}
where kk is the thermal conductivity, h_(a)h_{a} is the external heat transfer coefficient and T_(a)T_{a} is the external temperature (see Figure 1). This kind of boundary condition is appropriate to express matematically heat loosing in insullation problems.In addition we have taken in account in this paper the effect of viscous dissipation, see Barletta[9].
Figure 1: Geometry of the problem and the co-ordinate system
2 Basic Equations
Consider a viscous and incompressible fluid, which steadily flows between two infinite vertical and parallel plane walls. At the entrance of the channel the fluid has an entrance velocity U_(0)U_{0} parallel to the vertical axis of the channel. The geometry of the problem, the boundary conditions, and the coordinate system are shown in Fig. 1. The variation of density with temperature is given by the Boussinesq approximation and the fluid rises in the duct driven by buoyancy forces and initial velocity. Hence, the flow is due to difference in temperature and in the pressure gradient. The flow being fully developed the following relations apply here v=0,del v//del y=0,del p//del y=0v=0, \partial v / \partial y=0, \partial p / \partial y=0, where vv is the velocity in the transversal direction and pp is the pressure. Thus, from the continuity equation, we get del u//del x=0\partial u / \partial x=0 so that the velocity component along xx-axis depends only by y,u=u(y)y, u=u(y). Based on the fact that the flow is fully developed we can assume that the temperature T=T(y)T=T(y). Under these assumptions the momentum and energy equations for the flow and heat transfer have the following form:
{:[(2)nu(d^(2)u)/(dy^(2))-(1)/(rho)(dp)/(dx)+g beta(T-T_(0))=0],[(3)alpha(d^(2)T)/(dy^(2))+(nu)/(c_(p))((du)/(dy))^(2)=0]:}\begin{gather*}
\nu \frac{d^{2} u}{d y^{2}}-\frac{1}{\rho} \frac{d p}{d x}+g \beta\left(T-T_{0}\right)=0 \tag{2}\\
\alpha \frac{d^{2} T}{d y^{2}}+\frac{\nu}{c_{p}}\left(\frac{d u}{d y}\right)^{2}=0 \tag{3}
\end{gather*}
subject to the boundary condition given by Eq. (1), noslip condition for for velocity at the walls and constant temperature at the left wall:
{:(4)u(0)=0","u(L)=0","T(L)=T_(w):}\begin{equation*}
u(0)=0, u(L)=0, T(L)=T_{w} \tag{4}
\end{equation*}
where alpha\alpha is the thermal diffusivity of the viscous fluid, rho\rho is the fluid density and c_(p)c_{p} is the specific heat at constant pressure. In the sistem (2) and (3) there is an additional unknown, the gradient of pressure, dp//dxd p / d x. In order to close the above system subject to the boundary conditions (1) and (4) it is necessary to consider the equation of the mass flux conservation:
{:(5)U_(0)=(1)/(L)int_(0)^(L)u(y)dy:}\begin{equation*}
U_{0}=\frac{1}{L} \int_{0}^{L} u(y) d y \tag{5}
\end{equation*}
where LL is the channel width. Further, we introduce the following dimensionless variables (see Pop and Ingham[10] or Kohr and Pop[11]):
{:(6)U=(u)/(U_(0))","X=(xRe)/(L)","Y=(y)/(L)","theta=(T-T_(0))/(T_(w)-T_(0))","P=(L^(2))/(rhonu^(2))p:}\begin{equation*}
U=\frac{u}{U_{0}}, X=\frac{x R e}{L}, Y=\frac{y}{L}, \theta=\frac{T-T_{0}}{T_{w}-T_{0}}, P=\frac{L^{2}}{\rho \nu^{2}} p \tag{6}
\end{equation*}
where Re=U_(0)L//nu\operatorname{Re}=U_{0} L / \nu is the Reynolds number and T_(0)=(T_(a)+T_(w))//2T_{0}=\left(T_{a}+T_{w}\right) / 2 is a characteristic temperature. Using (6) in the equations (2)-(3), in the boundary conditions (1) and (4) and in the mass flux conservation (5) we obtain:
{:[(7)(d^(2)U)/(dY^(2))+lambda theta-gamma=0],[(8)(d^(2)theta)/(dY^(2))+Br((dU)/(dY))^(2)=0],[(9)U(0)=0","U(1)=0","((d theta)/(dY))_(Y=0)=kappa(1+theta)_(Y=0)","theta(1)=1],[(10)int_(0)^(1)U(Y)dY=1]:}\begin{gather*}
\frac{d^{2} U}{d Y^{2}}+\lambda \theta-\gamma=0 \tag{7}\\
\frac{d^{2} \theta}{d Y^{2}}+B r\left(\frac{d U}{d Y}\right)^{2}=0 \tag{8}\\
U(0)=0, U(1)=0,\left(\frac{d \theta}{d Y}\right)_{Y=0}=\kappa(1+\theta)_{Y=0}, \theta(1)=1 \tag{9}\\
\int_{0}^{1} U(Y) d Y=1 \tag{10}
\end{gather*}
In Eqs(7)-(10)gamma\operatorname{Eqs}(7)-(10) \gamma is the pressure gradient in XX direction, BrB r is the Brinkman number, lambda\lambda is the mixed convection parameter and kappa\kappa is the convection heat transfer parameter given by
{:(11)gamma=(dP)/(dX)","Br=Pr Ec=(muU_(0)^(2))/(k(T_(w)-T_(0)))","lambda=(Gr)/(Re)=(g beta(T_(w)-T_(0))L^(2))/(U_(0)nu)","kappa=(h_(a)L)/(k):}\begin{equation*}
\gamma=\frac{d P}{d X}, B r=\operatorname{Pr} E c=\frac{\mu U_{0}^{2}}{k\left(T_{w}-T_{0}\right)}, \lambda=\frac{G r}{R e}=\frac{g \beta\left(T_{w}-T_{0}\right) L^{2}}{U_{0} \nu}, \kappa=\frac{h_{a} L}{k} \tag{11}
\end{equation*}
and Pr,Ec,Gr\operatorname{Pr}, E c, G r and Re are the Prandtl number, Eckert number, Grashoff number and Reynolds number,respectively, defined as:
{:(12)Pr=(nu )/(alpha)","Ec=(U_(0)^(2))/(c_(p)(T_(w)-T_(0)))","Gr=(g beta(T_(w)-T_(0))L^(3))/(nu^(2))","Re=(U_(0)L)/(nu):}\begin{equation*}
\operatorname{Pr}=\frac{\nu}{\alpha}, E c=\frac{U_{0}^{2}}{c_{p}\left(T_{w}-T_{0}\right)}, G r=\frac{g \beta\left(T_{w}-T_{0}\right) L^{3}}{\nu^{2}}, \operatorname{Re}=\frac{U_{0} L}{\nu} \tag{12}
\end{equation*}
The physical quantity of interest in this problem are the skin friction coefficient C_(f)C_{f} and the Nusselt number NuN u, which are defined as:
{:(13)C_(f)=(mu)/(rhoU_(0)^(2))((du)/(dy))_(y=0,L)","Nu=((h_(f)L)/(k))_(y=0,L):}\begin{equation*}
C_{f}=\frac{\mu}{\rho U_{0}^{2}}\left(\frac{d u}{d y}\right)_{y=0, L}, N u=\left(\frac{h_{f} L}{k}\right)_{y=0, L} \tag{13}
\end{equation*}
In Eq. (13) h_(f)h_{f} is the internal heat transfer coefficient which can be calculated from the heat transfer balance at the wall:
{:(k(del T)/(deln))_("wall ")=h_(f)(T_("wall ")-T_("fluid ")))\left.\left(k \frac{\partial T}{\partial \mathbf{n}}\right)_{\text {wall }}=h_{f}\left(T_{\text {wall }}-T_{\text {fluid }}\right)\right)
where n\mathbf{n} is the normal to the wall. Using dimensionless varibles (6) we obtain:
{:(14)C_(f)Re=((dU)/(dY))_(Y=0,1)","Nu|_(Y=0)=kappa((theta(0)+1)/(theta(0)-1))","Nu|_(Y=1)=-(((d theta)/(dY))_(Y=1))/(theta(0)-1):}\begin{equation*}
C_{f} R e=\left(\frac{d U}{d Y}\right)_{Y=0,1},\left.N u\right|_{Y=0}=\kappa\left(\frac{\theta(0)+1}{\theta(0)-1}\right),\left.N u\right|_{Y=1}=-\frac{\left(\frac{d \theta}{d Y}\right)_{Y=1}}{\theta(0)-1} \tag{14}
\end{equation*}
3 Results and Discussions
Equations (7) to (10) admit an analytical solution in two particular cases:
i) Case Br =0=0
i) Case Br =0=0
In this case the system (7) and (8) becomes:
{:(15)(d^(2)U)/(dY^(2))-(dP)/(dX)+lambda theta=0:}\begin{equation*}
\frac{d^{2} U}{d Y^{2}}-\frac{d P}{d X}+\lambda \theta=0 \tag{15}
\end{equation*}
{:(16)(d^(2)theta)/(dY^(2))=0:}\begin{equation*}
\frac{d^{2} \theta}{d Y^{2}}=0 \tag{16}
\end{equation*}
subject to the boundary conditions (9). Further, from Eq.(15), (16) and condition (10) we obtain
{:[theta(Y)=(2kappa)/(1+kappa)Y+(1-kappa)/(1+kappa)],[(17)U(Y)=-(kappa lambda)/(1+kappa)(Y^(3))/(3)+(gamma+(1-kappa)/(1+kappa)lambda)(Y^(2))/(2)+((kappa lambda)/(3(1+kappa))-(1)/(2)(gamma+(1-kappa)/(1+kappa)))Y],[gamma=-12+(lambda)/(1+kappa)]:}\begin{gather*}
\theta(Y)=\frac{2 \kappa}{1+\kappa} Y+\frac{1-\kappa}{1+\kappa} \\
U(Y)=-\frac{\kappa \lambda}{1+\kappa} \frac{Y^{3}}{3}+\left(\gamma+\frac{1-\kappa}{1+\kappa} \lambda\right) \frac{Y^{2}}{2}+\left(\frac{\kappa \lambda}{3(1+\kappa)}-\frac{1}{2}\left(\gamma+\frac{1-\kappa}{1+\kappa}\right)\right) Y \tag{17}\\
\gamma=-12+\frac{\lambda}{1+\kappa}
\end{gather*}
ii) Case lambda=0\lambda=0
For lambda=0\lambda=0 the forced convection only is considered. The system (7) and (8) takes the following form:
{:[(18)(d^(2)U)/(dY^(2))-gamma=0],[(19)(d^(2)theta)/(dY^(2))+Br((dU)/(dY))^(2)=0]:}\begin{gather*}
\frac{d^{2} U}{d Y^{2}}-\gamma=0 \tag{18}\\
\frac{d^{2} \theta}{d Y^{2}}+B r\left(\frac{d U}{d Y}\right)^{2}=0 \tag{19}
\end{gather*}
Taking in account that gamma\gamma is constant, using the boundary conditions (9) and mass flux conservation (10) we have:
{:[U(Y)=-6Y^(2)+6Y],[(20)theta(Y)=-12 BrY^(4)+24 BrY^(3)-18 BrY^(2)+(2kappa)/(1+kappa)(1+3Br)Y+(1+6Br-kappa)/(1+kappa)],[gamma=-12]:}\begin{gather*}
U(Y)=-6 Y^{2}+6 Y \\
\theta(Y)=-12 B r Y^{4}+24 B r Y^{3}-18 B r Y^{2}+\frac{2 \kappa}{1+\kappa}(1+3 B r) Y+\frac{1+6 B r-\kappa}{1+\kappa} \tag{20}\\
\gamma=-12
\end{gather*}
Equations (7) and (8) subject to (9) and (10) were solved numerically for different values of the parameters, lambda,kappa\lambda, \kappa and Br(lambda=0,100,250,500;kappa=0.01,0.1,1,10;Br=0,0.001,0.01,0.025)\operatorname{Br}(\lambda=0,100,250,500 ; \kappa=0.01,0.1,1,10 ; \mathrm{Br}=0,0.001,0.01,0.025) using an implicit finite-difference method for velocity and a Gauss-Seidel iteration for temperature. Dimensionless velocity profiles, U(Y)\mathrm{U}(\mathrm{Y}), and temperature profiles, theta(Y)\theta(Y), are presented in Figs. 2 to 7 for different values of the above parameters. Analytical solutions ( lambda=0,Br=0\lambda=0, \mathrm{Br}=0 ) are also presented on figures with a circle marker.
The variation of the velocity U(Y)U(Y) and temperature theta(Y)\theta(Y) with the mixed convection parameter lambda\lambda is presented in Figs. 2 and 5. We notice
Acknowledgements. The work of the second author was supported from UEFISCU Grant PN-II-ID-PCE-2007-1/525 (Romanian Ministry of Education and Research).
| lambda\lambda | kappa\kappa | Br=0B r=0 | Br=0.001B r=0.001 | Br=0.01B r=0.01 |
| 0 | 0.1 | 5.940594 | 5.940594 | 5.940594 |
| 1 | 5.940594 | 5.940594 | 5.940594 | |
| 10 | 5.940594 | 5.940594 | 5.940594 | |
| 100 | 0.1 | 4.470594 | 4.501963 | 4.799567 |
| 1 | -2.144405 | -2.134825 | -2.047795 | |
| 10 | -8.759405 | -8.784130 | -9.001301 | |
| 500 | 0.1 | -1.409405 | -1.27279 | -0.125966 |
| 1 | -34.484405 | -32.905781 | -25.265381 | |
| 10 | -67.559405 | -68.912398 | -68.791864 |
Table 1: Friction coefficient C_(f)Re|_(Y=0)\left.C_{f} R e\right|_{Y=0}
| lambda\lambda | kappa\kappa | Br=0B r=0 | Br=0.001B r=0.001 | Br=0.01B r=0.01 |
| 0 | 0.1 | -5.940594 | -5.940594 | -5.940594 |
| 1 | -5.940594 | -5.940594 | -5.940594 | |
| 10 | -5.940594 | -5.940594 | -5.940594 | |
| 100 | 0.1 | -7.410594 | -7.370828 | -6.999151 |
| 1 | -14.025594 | -13.992628 | -13.700667 | |
| 10 | -20.640594 | -20.606004 | -20.288425 | |
| 500 | 0.1 | -13.290594 | -13.053063 | -11.199059 |
| 1 | -46.365594 | -43.025955 | -27.986534 | |
| 10 | -79.440594 | -74.606586 | -42.694129 |
Table 2: Friction coefficient C_(f)Re|_(Y=1)\left.C_{f} R e\right|_{Y=1}
| lambda\lambda | kappa\kappa | Br=0B r=0 | Br=0.001B r=0.001 | Br=0.01B r=0.01 |
| 0 | 0.1 | -0.999999 | -1.032983 | -1.451739 |
| 1 | -1.000000 | -1.005839 | -1.059969 | |
| 10 | -1.000000 | -1.003203 | -1.032116 | |
| 100 | 0.1 | -0.999999 | -1.027827 | -1.377468 |
| 1 | -1.000000 | -1.006256 | -1.063425 | |
| 10 | -1.000000 | -1.009053 | -1.092552 | |
| 500 | 0.1 | -0.999999 | -1.031473 | -1.360956 |
| 1 | -1.000000 | -1.133834 | -2.040945 | |
| 10 | -1.000000 | -1.276971 | -3.552673 |
Table 3: Nusselt number on the left wall Nu|_(Y=0)\left.N u\right|_{Y=0}
| lambda\lambda | kappa\kappa | Br=0B r=0 | Br=0.001B r=0.001 | Br=0.01B r=0.01 |
| 0 | 0.1 | 1.000000 | 0.967016 | 0.548260 |
| 1 | 1.000000 | 0.994160 | 0.940030 | |
| 10 | 1.000000 | 0.996796 | 0.967883 | |
| 100 | 0.1 | 1.000000 | 0.959813 | 0.498805 |
| 1 | 1.000000 | 0.981394 | 0.813917 | |
| 10 | 0.999999 | 0.978624 | 0.787469 | |
| 500 | 0.1 | 1.000000 | 0.906917 | 0.033431 |
| 1 | 1.000000 | 0.810584 | -0.328300 | |
| 10 | 0.999999 | 0.683490 | -0.854004 |
Table 4: Nusselt number on the right wall Nu|_(Y=1)\left.N u\right|_{Y=1}
Figure 2: Velocity profiles for different values of lambda\lambda
Figure 3: Velocity profiles for different values of BrB r
Figure 4: Velocity profiles for different values of kappa\kappa
Figure 5: Temperature profiles for different values of lambda\lambda
Figure 6: Temperature profiles for different values of BrB r
Figure 7: Temperature profiles for different values of kappa\kappa
References
- Aung, W.: Fully developed laminar free convection between vertical plates heated asymmetrically. In:Int. J. Heat Mass Transfer, vol. 15, 1972, p. 1577-1580.
- Aung, W., Fletcher, L.S., Sernas, V.: Developing laminar free convection between vertical flat plates with asymmetric heating. In: Int. J. Heat Mass Transfer, vol.15, 1972), p. 2293-2308.
- Aung, W., Worku, G.: Developing flow and flow reversal in a vertical channel with asymmetric wall temperatures. In: J. Heat Transfer, vol. 108, 1986, p. 299-304.
- Aung, W., Worku, G.: Theory of fully developed, combined convection including flow reversal. In: J. Heat Transfer, vol. 108, 1986, p. 485-488.
- Barletta, A.: Analysis of combined forced and free flow in a vertical channel with viscous dissipation and isothermal-isoflux boundary conditions. In: J. Heat Transfer, vol. 121, 1999, p. 349-356.
- Barletta, A.: Fully developed mixed convection and flow reversal in a vertical rectangular duct with uniform wall heat flux. In: Int. J. Heat Mass Transfer, vol. 45, 2002, p. 641-654.
- Boulama, K., Galanis, N.: Analytical solution for fully developed mixed convection between parallel vertical plates with heat and mass transfer. In: J. Heat Transfer, vol. 126, 2004, p. 381-388.
- Bejan, A., 1995. Convection Heat Transfer (2nd edition), Wiley, New York.
- Barletta, A.: Laminar mixed convection with viscous dissipation in a vertical channel. In: Int. J. Heat Mass Transfer, vol. 41, 1988, p. 3501-3513.
- Pop, I., Ingham, D.B., 2001. Convective Heat Transfer: Mathematical and Computational Modeling of Viscous Fluids and Porous Media, Pergamon, Oxford.
- Kohr, M., Pop, I., 2004.Viscous Incompressible Flow for Low Reynolds Numbers, WIT Press, Southamton.