Thermal entrance region heat transfer for MHD laminar flow in parallel-plate channels with unequal wall temperatures |
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Authors: | Ray -Shing Wu K C Cheng |
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Institution: | (1) Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, Canada |
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Abstract: | The problem of thermal entry heat transfer for Hartmann flow in parallel-plate channels with uniform but unequal wall temperatures considering viscous dissipation, Joule heating and axial conduction effects is approached by the eigenfunction expansion method. The series expansion coefficients for the nonorthogonal eigenfunctions are obtained by using a method for nonorthogonal series described by Kantorovich and Krylov 21]. Numerical results are obtained for the case with entrance condition parameter o=1 and open circuit condition K=1. The parametric values of Ha=0, 2, 6, 10 and Br=0, –1 are considered for Hartmann and Brinkman numbers, respectively.
Zusammenfassung Das Problem der Wärmeübertragung im thermischen Einlauf einer Hartmannströmung im ebenen Spalt mit einheitlichen, aber ungleichen Wandtemperaturen wurde unter Berücksichtigung viskoser Dissipation, Joulescher Heizung und axialer Wärmeleitung mit Hilfe einer Entwicklung nach Eigenfunktionen behandelt. Die Koeffizienten der Entwicklung für nichtorthogonale Eigenfunctionen wurde nach einer Methode für nichtorthogonale Reihen nach Kantorovicz und Krylow 21] berechnet. Numerische Ergebnisse werden für den Eintrittsparameter o=1 und die Bedingung für den offenen Stromkreis K=1 erhalten. Die Parameterwerte Ha=0, 2, 6, 10 und Br=0, –1 werden für die jeweiligen Werte der Hartmann- und der Brinckman-Zahl betrachtet. Nomenclature a
one-half of channel height
- ¯B,B0
magnetic field Induction vector and magnitude of applied magnetic field
- Br
Brinkman number, f Um
2/(kc)
- Cn,Dn
coefficients in the series expansion of e, see eq. 16
- cp
specific heat at constant pressure
- ,E0
electric field intensity vector and component
- En,On
even and odd eigenfunctions
- Ha
Hartmann number, (/f)1/2 Bo a
- h1,h2
local heat transfer coefficients at lower and upper plates
- ¯J,Jy
electric current density vector and component
- K
external loading parameter, Eo/(Bo Um)
- k
thermal conductivity
- Nu1, Nu2
local Nusselt numbers, h1,a/k and h2a/k, respectively
- P
fluid pressure
- Pe
Peclet number, PrRe
- Pr
Prandtl number, Cp f/k
- q1,q2
rates of heat transfer per unit area,–k(T/Z)Z=–a'–k(T/Z) Z=a respectively
- Re
Reynolds number, Uma/uf
- T,T0,T1,T2
fluid temperature, uniform entrance temperature, uniform but different lower and upper plate temperatures, respectively
- Tb,Tm
bulk temperature and (T1+T2)/2
- U,Um,u
axial, mean and dimensionless velocities, respectively
- ¯V
velocity vector
- X,Z
axial and transverse coordinates
- x,z
dimensionless coordinates
- n,n
even and odd eigenvalues
- ,0,b
dimensionless fluid, entrance and bulk temperatures, respectively
- c,e,f
characteristic temperature difference (T2-Tm), and dimensionless fluid temperatures, defined by eq. (10)
- e,f
magnetic permeability and viscosity of fluid
-
fluid density
-
electric conductivity
-
viscous dissipation function
-
(1-)/2 |
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