How to find the partial derivatives of a composite function

[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/publication/226696141_Numerical_simulation_of_coupled_heat-fluid_transport_in_freezing_soils_using_finite_volume_method

 

[jvicens]

Dear colleague,

Thank you for bringing this misprint to my attention. I have also noticed the discrepancy between the dimension of ψ in equation 3 and the stated unit in the text. It seems like a simple error that can be easily corrected, and it does not affect the overall validity of the study.

Regarding your second question, I am not familiar with any specific articles or books on this topic, but I can provide some general advice for finding reliable resources. First, I recommend checking out reputable journals in the field of soil science, such as the Journal of Geotechnical and Geoenvironmental Engineering or the Vadose Zone Journal. These journals often publish high-quality articles on numerical modeling of heat and mass transfer in soils.

Additionally, I suggest looking into the works of other researchers in this field, such as Yang Zhou and Guoqing Zhou, who wrote the article you mentioned. They may have published other articles or books that could be useful for your research.

As for software environments, I am not familiar with any specific ones that cater to the finite volume method for modeling heat and mass transfer in soils. However, there are many general finite volume software packages available, such as OpenFOAM or Fluent, that can be used for this purpose. It may require some additional coding and customization, but it is possible to implement your own formulas in these software environments.

I hope this helps. Best of luck with your research.