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Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above

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Abstract

When porous media saturated with initially stagnant cold water around the density maximum temperature are cooled from above, convection may be induced in an unstable lower layer. In this study, the onset of buoyancy-driven convection during time-dependent cooling is investigated using the propagation theory, which transforms disturbance equations similarly, and also considering the density inversion effect. The critical Darcy–Rayleigh number Ra D,c is found as a function of the dimensionless density maximum temperature θ max. For Ra D > Ra D,c the dimensionless critical time τ c to mark the onset of instability is presented as a function of Ra D and θ max. These critical conditions are compared with previous theoretical results.

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Abbreviations

a :

Dimensionless wavenumber, \(\sqrt{a_x^2 +a_y^2 }\)

a*:

Modified dimensionless wave number, \(a\sqrt{\tau}\)

c :

Specific heat (J/(kg K))

d :

Porous layer depth (m)

Da :

Darcy number, K/d 2

g :

Gravitational acceleration vector (m/s2)

K :

Permeability (m2)

k :

Thermal conductivity (W/(m K))

P :

Pressure (Pa)

Ra D :

Darcy–Rayleigh number \(\varepsilon ((\rho c)_{\rm f}/(\rho c)_{\rm s})\,{\bf g}2\gamma\,(\Delta T)^{2}\,Kd/(\alpha\nu)\)

\(Ra_{\rm D}^\ast\) :

Modified Darcy–Rayleigh number, \(Ra_{\rm D}\sqrt{\tau}\)

T :

Temperature (K)

t :

Time (s)

U :

Velocity vector (m/s)

w :

Dimensionless vertical velocity component

(x, y, z):

Dimensionless Cartesian coordinates

α :

Thermal diffusivity (m2/s)

γ :

Constant in Eq.2.4

Δ T :

Thermal penetration depth (m)

ε :

Porosity

θ :

Dimensionless temperature (T i −T)/(T i −T u)

μ :

Viscosity (Pa s)

ν :

Kinematic viscosity (m2/s)

ρ :

Density (kg/m3)

τ :

Dimensionless time (αt/d 2)

ζ:

Similarity variable, \(z/\sqrt{\tau}\)

c:

Critical state

e:

Effective value

f:

Fluid phase

i:

Initial state

max:

Density maximum condition

s:

Solid matrix

0:

Basic quantity

1:

Perturbed quantity

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Authors and Affiliations

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

    Min Chan Kim

  2. Department of Chemical Engineering, Kwangwoon University, Seoul, 139-701, Korea

    Do-Young Yoon

  3. School of Chemical and Biological Engineering, Seoul National University, Seoul, 151-744, Korea

    Joo Hyung Moon & Chang Kyun Choi

Authors
  1. Min Chan Kim
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  2. Do-Young Yoon
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  3. Joo Hyung Moon
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  4. Chang Kyun Choi
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Correspondence to Min Chan Kim.

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Kim, M.C., Yoon, DY., Moon, J.H. et al. Onset of Buoyancy-Driven Convection in Porous Media Saturated with Cold Water Cooled from Above. Transp Porous Med 74, 369–380 (2008). https://doi.org/10.1007/s11242-008-9205-0

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