Influence of the thermal boundary layer on the contact layer between a liquid and a cold plate in a solidification process

Abstract

In this paper, the influence of both the hydrodynamic and the thermal boundary layer on the solidification process of the flowing liquid on a cold plate is theoretically analyzed. Heat transfer between a frozen layer which is created and a laminar flowing liquid over that layer is considered. The development of the boundary layers and the relation between them on the solidification process are studied. An integral method for the solution of the boundary layer equations was used to obtain approximative solutions. The influence of the Prandtl and Reynolds number on the formation of the solid crust is shown and discussed for time dependent and steady-state solutions.

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Abbreviations

λ:

Thermal conductivity [W/(mK)]

c p :

Specific heat [J/(kgK)]

ρ:

Density [kg/m3]

\( \nu \) :

Kinematic viscosity [m2/s]

a :

Heat diffusivity [m2/s]

L :

Latent heat of liquid fusion [J/kg]

l :

Length of the cold plate [m]

b :

Width of the cold plate [m]

T :

Temperature of liquid [K]

T F :

Liquid fusing temperature [K]

T :

Temperature of the liquid in the free-stream [K]

T W :

Temperature of the wall [K]

u :

Velocity of the liquid [m/s]

u :

Velocity in the free-stream [m/s]

\( \dot{q} \) :

Heat flux [W/m2]

α:

Heat-transfer coefficient \( = {{\dot{q}} \mathord{\left/ {\vphantom {{\dot{q}} {\left( {T_{\infty } - T_{F} } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {T_{\infty } - T_{F} } \right)}},\left[{{\text{W}}/({\text{m}}^{2} {\text{K}})} \right] \)

δ:

Frozen layer thickness [m]

δ s :

Steady-state thickness of the frozen layer [m]

\( \bar{\delta } \) :

Averaged axial thickness of the frozen layer [m]

δ l :

Hydrodynamic boundary layer thickness [m]

δ t :

Thermal boundary layer thickness [m]

t :

Time [s]

x, y :

Coordinates [m]

Ste :

Stefan number = c p (T  − T F )/L

Fo :

Fourier number = at/l 2

Nu :

Nusselt number = α l

Re:

Reynolds number = u l

Pr:

Prandtl number = ν/a

τ :

Dimensionless time = FoSte

\( \tilde{\delta },\tilde{\delta }_{l} ,\tilde{\delta }_{t} \) :

Dimensionless quantities = δ/l; δ l /l; δ t /l

\( \tilde{x} \) :

Dimensionless axial coordinate \( = {{\tilde{x}} \mathord{\left/ {\vphantom {{\tilde{x}} l}} \right. \kern-\nulldelimiterspace} l} \)

r :

Ratio of thermal to hydrodynamic boundary layer thicknesses = δ t l

\( \vartheta \) :

Dimensionless temperature ratio = (T F  − T W )/(T  − T F )

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Lipnicki, Z., Weigand, B. Influence of the thermal boundary layer on the contact layer between a liquid and a cold plate in a solidification process. Heat Mass Transfer 47, 1629 (2011). https://doi.org/10.1007/s00231-011-0823-7

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Keywords

  • Boundary Layer
  • Nusselt Number
  • Prandtl Number
  • Flowing Liquid
  • Solidification Process