How to find the partial derivatives of a composite function

  • #1
Sashatgu
Hello, dear colleague. Now I'm dealing with issues of modeling processes of heat and mass transfer in frozen and thawed soils. I am solving this problems numerically using the finite volume method (do not confuse this method with the finite element method). I found your article: "Numerical simulation of coupled heat-fluid transport in freezing soils using a finite volume method" by Yang Zhou and Guoqing Zhou. The original data of the article: Heat Mass Transfer (2010) 46:989–998, DOI 10.1007/s00231-010-0642-2. I started reading it and completely analysis it. And immediately ran into an obvious misprint. It is located on page 989 at the bottom right. There we are talking about the fact that: : ψ-soil suction potential, [pa], but according to the equation No. 3 located on page 990 on the bottom right, ψ must have the dimension [m], so that the dimensions of the right side and left side of the equation No. 3 is matched:


$$\frac{\partial }{\partial x}(K\frac{\partial \psi}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t}(3)$$


where K is the hydraulic conductivity of soil, [m/s]; θ_i 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; T is the temperature, [K]; x is the position coordinate, [m]; t is the time (s), θ_u is the volumetric unfrozen water fraction - dimensionless quantity, ρ_i - ice density [kg/m^3], ρ_w- water density[kg/m^3].


I also think that this article has an error, it is located on page 990 (bottom right, equation No. 3) and page 991 (top left, equations No. and 5). I do not understand how from the formula No. 3, substituting in it the formula No. 4, it is possible to receive the formula No. 5 (may be K depends on x, may be in formula No. 3 instead of K there should be D):



$$\frac{\partial }{\partial x}(K\frac{\partial \psi}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t}(3)$$



$$D = K\frac{\partial \psi}{\partial \theta_u}(4)$$


where D - The soil water diffusivity [m^2/s]


$$\frac{\partial }{\partial x}(D\frac{\partial \theta_u}{\partial x})=\frac{\partial \theta_u}{\partial t}+\frac{\rho_i}{\rho_w}\frac{\partial \theta_i}{\partial t} - \frac{\partial K}{\partial x} (5)$$

The second question: Do you know the quality and intuitive (with detailed explanations) articles, books, theses (on English language) on the subject: 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 searching of 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).

Here is the article:

https://www.researchgate.net/public..._in_freezing_soils_using_finite_volume_method
 

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