# Could you help me please with a mathematical model of the soil freezing process first studied by Harlan?

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Hello, dear colleague. Now I'm dealing with issues of modeling heat and mass transfer in frozen and thawed soils. I am solving this problem numerically using the finite volume method. Below I give an equation from Harlan for mass transfer,

∂/∂x(K(∂ψ/∂x)) = (∂θu/∂t) + (ρiw)(∂θi/∂t),

where K is the hydraulic conductivity of soil, [m/s]; θi is the volumetric ice fraction, i.e., the volume of the ice in per unit volume of frozen soil - dimensionless quantity; ψ is the soil suction potential, which controls the flow of the soil water [m]; T is the temperature, [K]; x is the position coordinate, [m]; t is the time, [s], θu is the volumetric unfrozen water fraction - dimensionless quantity, ρi - ice density [kg/m3], ρw - water density [kg/m3].

Give me please the most thorough explanation for these four questions:

What is the physical meaning of this equation?
What is the physical meaning of its left and right side?
In the right side of this equation, why has the second term a factor (ρiw)?
In which program can I put this equation to see its physical meaning?

P.S. Do you know quality and intuitive (with detailed explanations) articles, books, theses (in English language) on the subject of modeling of heat and mass transfer processes in frozen and thawed soils by the control (finite) volume method. This subject is very interesting. I am looking for the treatment one, two and three-dimensional problems, as well as software environments where you can realize the solution of these problems by “my” formulas (instead of formulas 'built into' these systems).

Please add a reference to the context from which you obtained the equation.

@ArnoldNeumaier

Article:

The equation you cite is a continuity equation describing mass transport. See Harlan's paper

Analysis of coupled heat-fluid transport in partially frozen soil

where you can find details. There is plenty of software for solving partial differential equations, see, e.g. here, but you need to find out for yourself which package is suitable for your problem.

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