Thermal System Model?

I have been justly criticized for having an oversimplified thermal model in

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In that article I just wanted something that demonstrated the need for an integrator to hit a setpoint, and I didn't care that the model may be wildly inadequate otherwise.

Now I'd like to fix that, but I'm no thermodynamicist. Is it possible to generate, and does anyone have, or have links to, a good example thermal model for something like a solid bar heated at one end and measured at the other, or a stirred flask. Something that has a closed-form, linear differential equation would be nice (but discrete parameters aren't necessary -- I'll probably survive partial differential equations). I need something that I can generate frequency responses and impulse responses from yet is still "nasty" enough to be considered a proper thermal model.

If you guys let me down I'll probably just use a 2nd-order heavily damped lowpass (like in the article) with lots of pure delay added.

Thanks.

Reply to
Tim Wescott
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Tim, Your pure delay with 2O lags is likely to be a good - if uncalibrated - representation of a heated bar, which in fact is a cascaded series of lags. A (well) stirred reactor is also often representable this way, although TD/DT ratio may be higher. Not sure there's much point in going beyond this.

If you really want to struggle with the maths, then most 'advanced' engineering maths texts cover PDEs and the associated stuff. My tattered and battered copy of Greenberg (Advanced Engineering Mathematics, Prentice Hall) covers the heated bar rigorously. Look for 'diffusion equation', 'separation of variables'.

If you're in Perth you're welcome to borrow the book but technical libraries are certain to have an equivalent.. Cheers

snipped-for-privacy@weqstnet.com.au remove ecks & kyoo to email.

Reply to
bruce varley

Check this out. I didn't add dead time. You can just change the u(n-1) to be a u(n-deadtime) in the iteration loop. ftp://ftp.deltacompsys.com/public/PDF/Mathcad%20-%20TempPID.pdf

This jumps right in to the model in th s domain with out much explaination, but I am sure you can understand this. I am also using Jerry's favorite form of PID. I-PD is perfect for this model.

Reply to
peter

By "lag" do you mean a lowpass? I.e. something like

1 1 H(s) = ----------- * ------- ? (ts + 1)^20 e^{-sT}

where t is the lowpass (or lag) time constant and T is the pure delay?

I would stop by and borrow the book but I'm almost one earth diameter away from you as the Tom Swift earth burrowing machine travels -- I'm at about 122 degrees lat by 46 degrees long.

Reply to
Tim Wescott

H(s):=G*exp(-sT)/((t0*s+1)(t1*s+1))

Reply to
peter

Dang misplaced signs.

Reply to
Tim Wescott

uncalibrated -

For large n, the (1/(1+sT))^n tends to a form rather like what you wrote above, but you generally get a reasonable fit with only about 3 time constants. I'm sure there are numerous references, but the only one that springs to mind is in Mr Shinskeys immortal classic Process Control Systems. It shows response graphs for n around 20 iirc. It wouldn't be too hard to simulate the problem and do a curve fit using excel or any number of other methods.

Not sure why I thought you were in Perth, obviously you're not. Cheers

Reply to
bruce varley

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