Does anybobody have (or a link to) 1

[zekeman]

Does anybobody have (or a link to) the transient solution to the heatup of a hot water tank?
For simplicity, assume the system adiabatic with the tank water stagnant and it's initial temperature T1.
The heat exchanger area is A and the overall conductance is U. The heat source is hot water coming from the boiler at a constant temperature T0.
I need the time history of the tank water temperature,T
I found this equation:

UA/wc=ln[1-Mc/wct*ln{(T0-T1)/(T0-T)}]
But I need proof.
Anybody??
Thank you for your help.

M weight of water in tank
c specific heat
w flow rate
T1 initial tank water temperature
T0 incoming boiler temperature(constant)
t time

 

[trashcanman]

I am working on just such a program. The equation you found does not account for ambient air temp or heat loss from convection and radiation. Mine does.
Provide the tank diameter, height (assuming a vertical type cylindrical tank), wall thickness, insulation type, insulation thickness, type of outside surface, ambient air temp. and heat input please.

 

[Latexman]

Look in Kern's Process Heat Transfer. It is derived there.

Good luck,
Latexman

 

[trashcanman]

Latexman: That is where I got the equations for convective heat transfer hc formulas.

 

[chicopee]

In your OP, w is a flow rate so the water is not "stagnant."

 

[prex]

As noted by Latexman it is the same equation (after rearranging a bit) as eq.18.9 , page 627, of Kern. I think however that a minus sign should precede the external logarithm.

prex
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[Latexman]

Kern references Bowman, R. A., A. C. Mueller, and W. M. Nagle, Trans. ASME, 62, 283-294 (1940) for the documented derivation if that's easier for you to fine.

Good luck,
Latexman

 

[zekeman]

Latexman,

Thanks for the references, but cannot get either.

So I guess I'll derive it myself when I get time.

 

[Latexman]

The derivation can be found on the web with some persistant searching.

Good luck,
Latexman

 

[zekeman]

Thanks,

I was easier to derive it.