[HumanistEngineer]

Hi Guys,

Can any of you help me how to solve these three 1D heat balance equations simultaneously via finite difference method? The physical description is that this is an insulated pipe underground so the superscripts w, i, and g refer to the water, insulation, and ground while h is the heat transfer coefficient between the mediums (i.e. wi refers to water to insulation). T^u is a scalar value for the undisturbed ground temperature and delta_x is the pipe length.

The aim here is to obtain the temperature change of water medium through the pipe length over time (as well as the insulation, and the ground).



Ref: Pálsson, Hálldór - Methods for planning and operating decentralized combined heat and power plants - link

Thanks in advance.

[feride]

hi everybody,
I'm working on thermal hydrualic analysis of a nuclear reactor we can think fuel as heated cylindirical tubes which has bc with heat flux. There are 69 fuel rods ,heat flux is given as udf. working fluid temp is 300 K, The cooling mechanism is natural convection so ı try both boussienesq approx. tempereture dependent features with polynomial eq. in material box. The Ra number order is 10^13, I use k-e model for turbulent modeling, there are inlet pipe which is velocity inlet, outlet pipe which is out flow bc Btw fluid is water. I have tried many ways to converge continuity. but they didnt work. It would be much appreciated if you could share with me your opinions regards to this problem.

All the best

[FMDenaro]

Originally Posted by HumanistEngineer

Hi Guys,

Can any of you help me how to solve these three 1D heat balance equations simultaneously via finite difference method? The physical description is that this is an insulated pipe underground so the superscripts w, i, and g refer to the water, insulation, and ground while h is the heat transfer coefficient between the mediums (i.e. wi refers to water to insulation). T^u is a scalar value for the undisturbed ground temperature and delta_x is the pipe length.

The aim here is to obtain the temperature change of water medium through the pipe length over time (as well as the insulation, and the ground).



Ref: Pálsson, Hálldór - Methods for planning and operating decentralized combined heat and power plants - link

Thanks in advance.

This appears a system of hyperbolic-like equations for the temperature with source terms. You can find a lot of methods for solving numerically these equations using FD. I suggest start using an explicit time-marching method.

Here is a textbook http://sgpwe.izt.uam.mx/files/users/...tos/Morton.pdf

[HumanistEngineer]

Originally Posted by FMDenaro

This appears a system of hyperbolic-like equations for the temperature with source terms. You can find a lot of methods for solving numerically these equations using FD. I suggest start using an explicit time-marching method.

Here is a textbook http://sgpwe.izt.uam.mx/files/users/...tos/Morton.pdf

Thank you a lot again FMDenaro.

I kindly ask you to check if these FD approximations seem to be correct:

[FMDenaro]

Originally Posted by HumanistEngineer

Thank you a lot again FMDenaro.

I kindly ask you to check if these FD approximations seem to be correct:

No, FTCS scheme for hyperbolic equation is a very bad choice. Have a look to the chapter dedicated and check the consequences in terms of numerical stability properties.

[HumanistEngineer]

Originally Posted by FMDenaro

No, FTCS scheme for hyperbolic equation is a very bad choice. Have a look to the chapter dedicated and check the consequences in terms of numerical stability properties.

Thank you FMDenaro.

[CFD_10]

What about MacCormack scheme?

[HumanistEngineer]

Originally Posted by CFD_10

What about MacCormack scheme?

I used the implicit finite difference approximations given in the rereference:

[FMDenaro]

Originally Posted by HumanistEngineer

I used the implicit finite difference approximations given in the rereference:

Be careful, implicit schemes for hyperbolic equations are not generally used for some issue in the BCs. First of all, I suggest starting with the FTUS scheme, it is only first order accurate but provide you the first indication. Then the second order accurate scheme in time and space I suggest to try is the Lax-Wendroff scheme.

Be also careful that, according to the Godunov's theorem, using high order scheme you can have presence of spurious numerical oscillations

[CFD_10]

Both Lax-Wendroff and MacCormack schemes are second order accurate. But MacCormack scheme is very user friendly. it uses only two steps (predictor-corrector). The Lax-Wendroff is very tedious it requires you to express second order derivatives.