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The onset of buoyancy-driven convecion in a horizontal fluid layer subjected to evaporative cooling

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Abstract

The onset of buoyancy-driven convection in an initially isothermal, quiescent fluid layer experiencing evaporative cooling from above is analyzed under linear theory. Based on propagation theory, the self-similar stability equations are forced. The dimensionless critical time τ c to mark the onset of convective instability is presented as a function of the Rayleigh number Raq and the Prandtl number Pr. The present model predicts that τ c decreases with increasing Pr for a given Raq. When the temporal growth rate of disturbances is considered, the present predictions compare reasonably well with existing experimental data of water. It seems that for large Pr manifest convection is detected at a certain time τ ≅ 4τ c .

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Abbreviations

a :

dimensionless horizontal wave number \( = {\sqrt {a^{2}_{x} + a^{2}_{y} } }\)

a*:

modified wave number \( = a{\sqrt \tau } \)

D :

d/dζ

d :

depth of layer, m

g :

gravitational acceleration, m/s2

k :

thermal conductivity, W/mK

P :

pressure, Pa

Pr:

Prandtl number = ν/α

q w :

upper-wall heat flux, W/m2

Raq :

Rayleigh number based on the heat flux = gβq w d 4/(kαν)

Ra*:

modified Rayleigh number = Raqτ2

T :

temperature, K

t :

time, s

U :

velocity vector, m/s

W :

vertical velocity, m/s

w :

dimensionless vertical velocity

x,y,z :

dimensionless Cartesian coordinates = (X,Y,Z)/d

α:

thermal diffusivity, m2/s

β:

volumetric thermal expansion coefficient, 1/K

Δ T :

thermal penetration depth, m

δ T :

dimensionless thermal penetration depth = Δ T /d

θ1 :

dimensionless perturbed temperature = T 1 gβd 3/(αν)

θ0 :

dimensionless basic temperature = k (T 0 – T i )/ (q w d)

ν:

kinematic viscosity, m2/s

ρ:

density, kg/m3

τ:

dimensionless time = αt/d 2

ζ:

similarity variable = \(z/{\sqrt \tau }\)

c :

critical state

i :

initial state

o :

observable

s :

upper surface

0:

basic state

1:

perturbed state

*:

amplitude functions

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Authors and Affiliations

  1. School of Chemical Engineering, Seoul National University, Seoul 151-744, Korea

    Chang Kyun Choi &  Joung Hwan Park

  2. Department of Chemical Engineering, Cheju National University, Cheju, 690-756, Korea

    Min Chan Kim

Authors
  1. Chang Kyun Choi
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  2. Joung Hwan Park
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  3. Min Chan Kim
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Corresponding author

Correspondence to Chang Kyun Choi.

Additional information

This work was supported by LG Chemical, Seoul under the Brain Korea 21 Project of the Ministry of Education.

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Chang Kyun Choi, ., Joung Hwan Park, . & Min Chan Kim, . The onset of buoyancy-driven convecion in a horizontal fluid layer subjected to evaporative cooling. Heat Mass Transfer 41, 155–162 (2004). https://doi.org/10.1007/s00231-004-0503-y

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  • DOI: https://doi.org/10.1007/s00231-004-0503-y

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