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Is there an equation I can use to calculate the temperature (as a function of time) of an object which is gaining or losing heat by convection? Or equivalently, the rate of energy transfer from the object to the surrounding fluid (or vice-versa)? It can involve constants representing properties of the object and the surrounding fluid.

I know about the heat equation for conduction and the Stefan-Boltzmann law for radiation, but I can't think of one for convection.

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I guess you've seen this already, but here are wikipedia articles with some information: Convection vs. conduction, Newton's law of cooling. –  Marek Dec 29 '10 at 16:26
@Marek: hm, interesting. I glanced over those before, but taking a second look I notice that at the latter link, it says there are formulas to calculate the heat transfer coefficient $h$ for certain systems. I'd be very interested to see information about hose formulas posted here, if you (or anyone else) is inclined to do so. –  David Z Jan 1 '11 at 7:37
I am afraid that I don't know anything about these topics besides the general overview. But I too am very eager to learn more, so I hope someone else will elaborate. –  Marek Jan 1 '11 at 14:04
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5 Answers

up vote 1 down vote accepted

Nothing as easy from basic principles as for conduction or radiation.

You can multiple the mean heat carried by the convecting liquid by it's mass rate of flow

$$ \Delta T = c_p * (T_{in} - T_{out}) * \frac{\Delta V}{\Delta t} \rho \Delta t .$$

Where $T_{in}$ is the mean temperature of the liquid moving toward the sink (cool side) and $T_{out}$ is the temperature of the liquid moving the other way.

The problem is that you can't get the temperature(s) of the convecting liquid or the flow rate without detailed calculations or measurements.

For the generaly case you pretty much have to go to CFD.

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"conduction of radiation"? I'm pretty sure that makes no sense. Perhaps you meant something else? –  Noldorin Dec 29 '10 at 3:24
@Noldorin: You've uncovered by darkest secret: I'm a bad typist and rely on the spell-check squiggles to catch it, thus consistently missing wordos... –  dmckee Dec 29 '10 at 3:50
Heh, well if we could get by without squiggles we'd be on english.SE. (though: it missed "wordos"? :-P) Anyway I was hoping for a simple formula that wouldn't require the flow rate or temperatures but I'm not too surprised to hear that there isn't one. –  David Z Dec 29 '10 at 8:19
Haha; that makes a lot more sense now! Funnily enough the "or" didn't occur to me else I would have corrected it myself. –  Noldorin Dec 29 '10 at 15:47
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here you can find some formulas for calculating Nusselt, Prandtl, Reynolds, Rayleigh and Grasshoff numbers. Those are important for evaluating conditions in different systems. Numbers will tell you which state of convection is around your geometry (natural, forced, laminar, turbulent, external, internal). For each case there exist some correlation.

More info about Nusselt number and others you can find eg. on Wikipaedia.

After calculating convenient numbers, you can calculate heat transfer coefficient zumbeispiel from Nusselt nuber.

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For convection, you have a similar formula as conduction for heat transfer. http://en.wikipedia.org/wiki/Nusselt_number

$$\frac{dQ}{dt}=-\frac{N_uk_fA\Delta T}{L}$$

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There are some formulas (for example, for cylinders, such as pipes) in the following book: Bosworth R.C.L. Heat transfer phenomena. The flow of heat in physical systems. – Associated general publications, PTY. LTD. – Sydney. – 1952, 211 p. Both heat conduction (in the so called stagnant film near the pipe) and convection are important for pipe cooling in the air. It is interesting that the relevant coefficients depend on the orientation of the pipes (vertical/horizontal).

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If you were interested in the temperature distribution within a solid in contact with a fluid (e.g. gas or liquid), you would solve the heat equation, e.g.: $$ \frac{\partial T}{\partial t} = k\nabla^2T$$. The loss of heat via convection to the surrounding fluid would show up in the choice of boundary conditions and specifically you would use the Robin Condition (or mixed condition) that is a balance of conduction in the solid and convection in the fluid at the interface and can be derived from applying Newton's Law of Cooling at the interface. For example at the boundary $L$: $$ k\frac{\partial T(t,L)}{\partial n}+h T(t,L)=g(t)$$ where $k$ is the thermal conductivity in the solid and $h$ is the convective heat transfer coefficient for the fluid.

If you were interested in the temperature variation within a fluid with convective heat transfer, you want to solve a more general heat equation that contains a convective term that then couples its solution with a solution for the motion of the fluid (e.g. Navier Stokes), $$ \frac{\partial T}{\partial t} + \tilde{u} \cdot \nabla T= k\nabla^2T$$ where $\tilde{u}$ is the velocity vector (thus making this a vector equation).

Hope this helps some.


Paul Safier

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