A Finite Difference Scheme - Plume Entrainment

I am working on modelling of a heat storage tank. More specifically, I need to find out the transient temperature variation through the tank height.

My question is about the plume entrainment considering the case that hot water is injected from the bottom part of a heat storage tank. This bottom part is initially with cold water mass. So I need to model the plume entrainment (the hot water naturally rises to a higher level above the cold region due to its low density).

I could find a solution procedure (print-screen below) but I don't know how to replace these derivatives (Eq. 6, 7 and 8) with the finite difference quotients/approximations. Would you please help me solving these equations or any other approach is welcome?

The reference (of the print-screen) can be reached from here.

2018-09-19_14_29_26-_Khorasanizadeh_et_al_-_The_effect_of_an_inco.png
 

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Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
 
Are you asking how to discretize the pdes 6,7,8 so you can solve them numerically with a finite difference method? What is your background in numerical methods?
Did you follow a course that treats this stuff:
http://indico.ictp.it/event/a06220/session/18/contribution/10/material/0/2.pdf
I am not so experienced in numerical methods. But I could develop a model for the solution of convection diffusion equation at one dimension, in order to derive temperature profile for a single inlet/outlet storage tank through time. Here is the working Matlab code given: https://www.cfd-online.com/Forums/main/204476-inlet-mixing-1d-numerical-model-stratified-storage.html

My question at this forum is that, as you indicated, I don't know how to simultaneously solve (and discretize) pdes 6, 7, and 8.
 

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