How can we decide which geometry would have faster heat transfer rate while working in negative temperature if we are using copper and the mode of heat transfer is pure conduction?
Also if the geometry is made up of two parts, then how can we ensure proper contact of the two surfaces for efficient heat transfer.
Conductive heat transfer in a body in contact with an infinite medium with temperature T1 can be approximated in dimensionless variables (which are independent if temperature is positive or negative) as,
Y=exp(-bFo) (1)
where Y=(T-T1)/(T0-T1), Fo=kt/l^2 (Fourier number), b first eigenvalue of a Sturm-Liouville problem, T is body temperature,T0 is body initial temperature, k body heat conductivity (W/m-K), l body characteristic length (m) and t time.
In the three basic geometries:
Flat slab (1 side at T1 the other side isolated): l = total thickness, b=(3.14159/2)^2
Flat slab (2 sides at T1): l = half thickness, b=(3.14159/2)^2
Large cylinder: l = radius, b=(2.72)^2
Sphere: l = radius, b=(3.14159)^2
Eq. (1) is only an approximation but is a good approximation at Y
Thanks for adding a detailed answer.Can You provide me the reference material which I can study for better understanding?
Hi Nauman Khan,
I don't understand if your problem is transient or steady stade. If your doubt is regarding the mathematical model for the geometries described by Prof. Miguel, he is right. But the better geometry is a tree shaped, described by Adrian Bejan. You can model a tree shaped and simulate in ansys cfx or other cfd software.
Best regards from Brazil,
Prof. Strobel, PhD.
In my opinion the better references are the classical books,
Carslaw, H.S., (1921). Introduction to the Mathematical Theory of the Conduction of
Heat in Solids. MacMillan and Co., London.
Carslaw, H.S. and Jaeger J.C. (1959). Conduction of Heat in Solids. Editorial Clarendon Press, Oxford University Press, Great Britain.
I gave you the asymptotic analytical solutions of non-steady heat transfer by pure conduction in the three basic geometries (you can find the exact solution in the above references). As Prof. Strobel wrote, may be faster geometries. You must remember that the geometries with faster heat transfer are the geometries with greater S/ V ratio. Where “S” is the heat transfer surface and “V” is the body volume. For basic geometrics,
Flat slab: S/ V=1/l
Large cylinder: S/ V=2/l
Sphere: S/ V=3/l with the same nomenclature of my first answer.
Dear Mr. Khan,
under steady state conditions, and for one-dimensional heat transfer by conduction, we try to lower the thermal resistance in order to increase the heat transfer rate. thus, if your geometry is a simple one (square/rectangle, circular..) you can compare their thermal resistances and choose the lowest one. if your problem is a transient one you better follow the indication of Mr. Garcia-Alvarado. in all cases see the attached file for pure conduction under steady-state and transient conditions.(reference: J.P. Holman, Heat Transfer, 10th edition)
in the case of multidimensional heat transfer, you can use analytical solutions (if they exist) or any CFD software to simulate you geometry and solve the PDE equations by numerical methods. For 2-D heat transfer you can use the PDEtool command of Matlab to plot different geometries and compare their behavior in pure conduction.
Regards,
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