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Buoyancy effects on heat transfer and liquid solidification-free zone in a convectively-cooled horizontal square channel

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The effects of free convection on thermal entrance region problem and liquid solidification-free zone in a convectively-cooled horizontal square channel are studied by numerical method for steady-state fully developed laminar conditions using the large Prandtl number assumption. The Rayleigh number effects on the heat transfer results and the length of liquid solidification-free zone are investigated for Biot number of 2 and 20. It is concluded that the free convection effects should be included in the analysis of ice formation in a convectively-cooled pipe or channel when the Rayleigh number is greater than say 104.

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A :

cross-sectional area of a channel

Bi :

Biot number,hD e/k

C :

a constant, (D e 2 /μW f)∂P f/∂Z

c p :

specific heat

D e :

hydraulic diameter, 4A/S

Gr :

Grashof number, [(T 0T )D e 3 ]/ν 2

f :

dummy variable

g :

gravitational acceleration

h :

average heat transfer coefficient

h :

heat transfer coefficient between inner wall and ambient defined by −k(∂T/∂N)w=h(T wT )

k :

thermal conductivity

M, N :

number of divisions inX andY directions, respectively, or outward normal at wall forN

Nu :

local Nusselt number,hD e/k

n :

dimensionless outward normal distance to the wall,N/D e, ornth iteration

P f :

pressure for fully developed laminar flow

Pr :

Prandtl number,ν/β

Ra :

Rayleigh number,Pr Gr

Re :

Reynolds number,u f D e/ν

S :

circumference of cross-section

T :

liquid temperature

T 0,T :

uniform entrance temperature and ambient temperature, respectively

T f :

freezing temperature of liquid

U, V :

velocity components of secondary flow inX andY directions, respectively

u, v :

dimensionless quantities foru, v, respectively

W f :

fully developed axial velocity

w f :

dimensionless quantity forW f

X, Y, Z :

rectangular coordinates

x, y, z :

dimensionless quantities forX, Y, Z respectively

z f :

liquid solidification-free length

β :

coefficient of thermal expansion

ε :

superheat ratio, (T 0T f)/(T fT )

θ :

dimensionless temperature difference, (TT )/(T 0T )

κ :

thermal diffusivity,k/ρc p

μ :


ν :

kinematic viscosity

ξ :

vorticity function, 2 ψ

ρ :


ψ :

dimensionless stream function

b :

bulk average value

w :

value at wall


average value


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Ou, J., Cheng, K.C. Buoyancy effects on heat transfer and liquid solidification-free zone in a convectively-cooled horizontal square channel. Appl. Sci. Res. 30, 355–366 (1975).

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  • Heat Transfer
  • Biot
  • Prandtl Number
  • Rayleigh Number
  • Free Convection