Thermal System Model?

I have been justly criticized for having an oversimplified thermal model in http://www.wescottdesign.com/articles/pidwophd.html . 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.
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Tim Wescott
Wescott Design Services
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remove ecks & kyoo to email.
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bruce varley wrote:

By "lag" do you mean a lowpass? I.e. something like
1 1 H(s) = ----------- * ------- ? (ts + 1)^20 e^
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.
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Tim Wescott
Wescott Design Services
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H(s):=G*exp(-sT)/((t0*s+1)(t1*s+1))
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snipped-for-privacy@deltacompsys.com wrote:

Dang misplaced signs.
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Tim Wescott
Wescott Design Services
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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
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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.
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