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 :
- D e :
hydraulic diameter, 4A/S
- Gr :
Grashof number, [gβ(T 0−T ∞)D e 3 ]/ν 2
- f :
- g :
- h :
average heat transfer coefficient
- h :
heat transfer coefficient between inner wall and ambient defined by −k(∂T/∂N)w=h(T w−T ∞)
- k :
- 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 :
- Ra :
Rayleigh number,Pr Gr
- Re :
Reynolds number,u f D e/ν
- S :
circumference of cross-section
- T :
- 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 :
- x, y, z :
dimensionless quantities forX, Y, Z respectively
- z f :
liquid solidification-free length
- β :
coefficient of thermal expansion
- ε :
superheat ratio, (T 0−T f)/(T f−T ∞)
- θ :
dimensionless temperature difference, (T−T ∞)/(T 0−T ∞)
- κ :
thermal diffusivity,k/ρc p
- μ :
- ν :
- ξ :
vorticity function,▽ 2 ψ
- ρ :
- ψ :
dimensionless stream function
- b :
bulk average value
- w :
value at wall
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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). https://doi.org/10.1007/BF00705610
- Heat Transfer
- Prandtl Number
- Rayleigh Number
- Free Convection