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Asymmetric solidification of flowing liquid in a convectively cooled parallel-plate channel

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Abstract

An analysis is made of steady-state unsymmetric liquid solidification of forced laminar flow in a parallel-plate channel with different uniform external convection coolings at the upper and lower plates, respectively. The classical Graetz type analysis is made by using the confluent hypergoemetric function. The case of one plate with perfect insulation and the other plate with a uniform external convection cooling is studied in detail. Numerical results are obtained for liquid solidification-free length, ice layer thickness, pressure drop, bulk temperature, heat transfer rate and local Nusselt number for Bi 1=0, Bi 2=1, 2, 10, 100, ε=1 ∼ 10, Pr=13.2 and k 1/k s=0.25 using 10 eigenvalues. The configuration and thermal conditions may be encountered, for example, in heat exchangers using cryogenics and freezing of ice sheets on northern rivers and lakes.

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Abbreviations

a, b :

parameters in Kummer's equation

Bi 1, Bi 2 :

Biot numbers at lower and upper plates, h 1 L/k, h 2 L/k

Bi s :

(k 1/k s)Bi2

C 1j , C 2j :

constants in eq. (8)

C j, Dj :

constants

c p :

specific heat at constant pressure

E j, Kj :

eigenconstants, eqs. (4) and (33)

H :

function of x only, equation (31)

h :

local heat transfer coefficient, eqs. (54) and (55)

h 1, h 2 :

overall heat transfer coefficients defined by k∂T(X, 0)/∂Y= h 1[T(X, 0) − T ∞], −k∂T(X,L)/∂Y=h 2[T(X,L)−T ∞]

k :

thermal conductivity of liquid

L :

distance between parallel plates

M(a, b, z):

confluent hypergeometric function

N j, Yj :

eigenfunctions

Nu :

Nusselt number, hL/k

P, P 0 :

Pressures at X and X=0

Pe :

Peclet number, 4 U m L/α

Pr :

Prandtl number, ν/α

p :

dimensionless pressure drop, (P 0−P)/(ρU 2m /2)

Q, Q :

total heat transfer rate and dimensionless Q, eq. (51)

q :

local heat transfer rate, eqs. (54) and (55)

T, T b, T f :

liquid temperature, bulk temperature and freezing temperature

T 0, T ∞ :

uniform entrance temperature and ambient temperature

U, V :

axial and transverse velocities

U m :

average axial velocity

v :

solution of Kummer's equation

X, Y :

rectangular coordinates

X′ :

axial coordinate with origin at solid-phase entrance

x, y :

(X/LPe), (Y/L)

x y :

(X′/LPe), (Y/L) in freezing zone

x f :

dimensionless solidifcation-free length

z :

transformed variable

α :

thermal diffusivity

α j, βj :

eigenvalues

δ, \(\bar \delta \) :

transverse coordinate of liquid-solid interface and δ/L

ε :

superheat ratio, (T 0−T f)/(T f T ∞)

η :

dimensionless transverse coordinate, \(\bar y/\bar \delta = Y/\delta \)

θ :

dimensionless temperature difference, (T−T ∞)/(T 0−T ∞)

θ b :

dimensionless bulk temperatures, eqs. (48) and (50)

θ f :

(T f−T ∞)/(T 0−T ∞)

μ, ν :

dynamic viscosity and kinematic viscosity

ρ :

density

φ :

dimensionless temperature difference, (T−T f)/(T 0−T f)

1, s:

liquid and solid-phase

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Authors and Affiliations

  1. Dept. of Mech. Eng., Univ. of Alberta, Edmonton, Alberta, Canada

    K. C. Cheng & Sum-Lok Wong

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  1. K. C. Cheng
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  2. Sum-Lok Wong
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Cheng, K.C., Wong, SL. Asymmetric solidification of flowing liquid in a convectively cooled parallel-plate channel. Appl. Sci. Res. 33, 309–335 (1977). https://doi.org/10.1007/BF00383958

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