# Transient heat transfer

Hi,

I am studying the heat transfer in a stirling engine. The basic situation is a pipe blocked off at both ends, with air inside, and a displacer also inside, that moves backwards and forwards inside. It is not tight against the walls of the pipe, and air can pass around it.

One end of the pipe is heated, and so I have modelled the pipe hear transfer as steady state, after a period of time. So the insde wall temperature of the pipe varies along it but at any point the temperature is constant.

I am now interested in modelling the air temeprature inside, which changes because the displacer is moving the air back and forth between the heated end, and the cooler end.

I am reading Adrian Bejan's book on heat transfer, but I struggled to find a concluding equation for transient heat transfer, if any body could help me with that.

For now I am saying the displacer is 100% insualted.

Using nodal analysis, please could anybody give me tips on how to go about modelling this with the displacer moving over time back and forth?

Thanks alot

Alex

## Answers and Replies

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First let me say I do not have the answer to your question. But I think to solve this you need to proceed in small steps, beginning with a simplified model. First, can you solve the heat transfer problem where at time zero you have a cylinder with hot walls and a cool gas inside? Assume that the wall temp & gas temp are initially uniform (at two temperatures Tgas & Twall). How does the gas temp change with time? You will need to estimate a heat transfer coefficient at the wall surface. This simple model can then be complicated to address the fact that the gas is not all at one temp, and that the cylinder wall is hotter at the 'bottom' than at the top. Finally you can look at the dynamics of the gas being 'squirted' into the hot end as the displacement piston rises towards the cold end.

I'd say it is a pretty complicated problem to solve in detail, and alot of the result is going to depend on your heat transfer coefficient at the wall.

Or maybe a more global approach might be interesting - take the device as a kind of black box: heat is added at the hot end and removed at the cold end, the difference is the work done by the shaft. Might be hard to figure out the temperatures, though.

I would also continue to look in books & the 'net, see how others have solved this before you. When I was in school I built a small Stirling engine; it was as much about learning how to run a lathe & a mill as thermo. My little toy engine still works, 30 years on.

I'm not sure how much heat transfer you know, so maybe start here:

$$q = hA(Twall - Tgas)$$

$$q= MCp\frac{dTgas}{dt}$$

q is the heat transfer rate (Btu/sec)
h is the heat transfer coefficient (Btu/sec-ft2-F)
A is the wall area (ft2)
Twall & Tgas are the temps (F)
M is the mass of the working fluid (air?) (lb)
Cp is the specific heat of the air (Btu/lb-F)
and dTgas/dt is the time derivative of the air temp

the two q's are equal, the equations just say 'adding heat to the gas heats it up' If you don't like 'english' units use the corresponding SI units.

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my steady state heat transfer I would say was reasonable. Its the transient heat transfer that I am not so good at, but the major problem for me is that its not the temperatures that are changing of the walls of the pipe, it is the fact that the displacer is moving the air, between the hotter and cooler ends of the pipe, how should this be accounted for in the transient equations

but those two equations are definately what I am after, but I think it is the gas temperature that is changing??? How to model it over time, WITH a moveing displacer?

my steady state heat transfer I would say was reasonable. Its the transient heat transfer that I am not so good at, but the major problem for me is that its not the temperatures that are changing of the walls of the pipe, it is the fact that the displacer is moving the air, between the hotter and cooler ends of the pipe, how should this be accounted for in the transient equations

but those two equations are definately what I am after, but I think it is the gas temperature that is changing??? How to model it over time, WITH a moveing displacer?
Right - the gas temp rises when it is in the hot end. Then it gets pushed bac into the cold end and it gives the heat back to that wall. Same equation, but the q heat rate has a minus sign (or, Tgas > Twall). So, as an approximation, you have two states: Gas in the hot end heating up, or gas in the cold end cooling down.

Now look at the overall device - how long does the gas stay in each end? Depends on the speed of the engine, the rpm. The faster it is turning, the shorter the time. And if the time gets very short, the gas will not be at a uniform temp - the gas in the center of the cylinder won't heat all the way up before it is pushed back to the cold end. It is a transient problem with a periodic boundary condition (that is, think of the gas as being stationary and the wall temp alternates hot, cold, hot, cold....). The displacement piston is just a mechanical way to make that happen, right?

Hi I'm studying nuclear energy engineering and there is a project about heat transfer. In the project it is asked us to write the heat conduction equation for the rod with boundary conditions. ( In a given figure which is a PWR rod coolent system) put this equation in to finite difference form. and obtain a matrix equation in AX=b form. I will code my own program and solve the finite difference equation. (MATLAB)
My problem is :
if the user inputs the nodes number how can I calculate this number of node in sequence and put tempreatures coefficient in to matrix [coefficients][T]= ? If the user inputs 150-150 our nodes will be the number of intersection of the inputs (like (150-1)*(150-1)) and I can't give number to each nodes for large inputs. Can you give me an advice ?

Thank you

i think you are over complicating it, I am doing exactly the same thing but for a pipe rather than a rod, using the Gauss-seidal finite method, with the user being able to enter the number of nodes.

I havent finished yet, but I will try to remember to post my solution here,

Please I beg you because I have been trying it for a week and there is not a solution yet I can go crayz. I will post my project in 2 days (first I have to scan the paper if you are interested I can mail that to you)

thank you

Hopefully By the weekend I can get something up that will show you the finite method for heat transfer. In the meantime, see if you can get your hands on a book called Heat transfer by Adrian Bejan. Around page 120 is an excellent description of the matrix method you talk of (for steady state), and the (better) Gauss-Seidal method I was talking of.

I have learnt heat transfer with the view to using software such as matlab to use the finite method to solve large heat transfer problems. If you also do a google search for finite method heat transfer alot of googlebooks come up with much more information on the topic. Have a read, its actually quite straight forward

I found something like that if you're interested. It may help you (ıf you are still struggling with your project) It contains also heat transfer

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im about starting my project of designing heat pipe for my indoor solar cooker, because heat is not available in my place even if available i may not afford its cost, pls advice me more on the design and probably the working fliud to use if im to use copper pipe which readily available and cheap in my place.
Best Regards
Bubuwa