Finite Difference Scheme - Plume Entrainment

In summary: What is your background in numerical methods? If you are not familiar with the code, could you please provide a brief summary of it?In summary, Hossein Khorasanizadeh is working on modelling a heat storage tank. He is asking for help with solving equations 6, 7, and 8 which are related to the transient temperature variation through the tank height. Khorasanizadeh does not have experience in numerical methods, but he is willing to develop a model for the solution of the convection diffusion equation at one dimension in order to derive the temperature profile for a single inlet/outlet storage tank through time.
  • #1
HumanistEngineer
18
2
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 https://www.researchgate.net/profile/Hossein_Khorasanizadeh2/publication/258446557_Effect_of_an_Incoming_Jet_on_Thermal_Stratification_of_Hot_Water_Tanks/links/547df0570cf285d6caa99543/Effect-of-an-Incoming-Jet-on-Thermal-Stratification-of-Hot-Water-Tanks.pdf?_sg%5B0%5D=hrAxHaPXcYNiXYkj__x1IRAO7It9HuvSQOeprFhP9HMIMF6I4eEuZLvZtxO7urTn-266pLke9XoMC8HEAzyUew.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_sg%5B1%5D=zOlVzI6fVeWfpofXWMIhT1GkYWS8bwV8OE8DoWc1X8TlD7MniMslQ5-vBJRxC9rwIb1YmMGEo6LUnWbef7qfTvVZLTraCIOXWO4k4VzoFq9X.rymjI4Md_XKMYh0NtvMmDYvN2XC71vY32wxogZL727v6TR31DGT3A9uAFHRyRW8KN3jxRsRlrQLf9hOkRtPoKA&_iepl=.

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

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  • #3
bigfooted said:
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/m...ng-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.
 

1. What is a finite difference scheme?

A finite difference scheme is a numerical method used to solve differential equations. It involves dividing a continuous domain into a discrete grid and approximating the derivatives of the equation at each grid point using finite differences. This allows for the equation to be solved using algebraic equations rather than analytical solutions.

2. How does a finite difference scheme work?

A finite difference scheme works by breaking down a differential equation into a discrete set of points on a grid. The derivatives of the equation at each point are approximated using finite differences, which are then solved using algebraic equations. This process is repeated for each point on the grid until a solution is obtained.

3. What is the purpose of using a finite difference scheme for plume entrainment?

The purpose of using a finite difference scheme for plume entrainment is to accurately model the dispersion of a plume in a fluid, taking into account the effects of entrainment. This allows for the prediction of the plume's behavior and impact on the surrounding environment.

4. What are the advantages of using a finite difference scheme for plume entrainment?

Some advantages of using a finite difference scheme for plume entrainment include its ability to handle complex geometries and boundary conditions, its computational efficiency, and its flexibility in handling different types of equations. It also allows for the incorporation of additional physical phenomena, such as turbulence, into the model.

5. Are there any limitations to using a finite difference scheme for plume entrainment?

While finite difference schemes are a powerful tool for solving differential equations, they do have some limitations. They may not accurately capture the behavior of highly non-linear systems or those with rapidly changing conditions. Additionally, the accuracy of the solution depends on the grid size and the stability of the scheme, which can be challenging to determine in some cases.

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