Trying to derive a transfer function for a cross flow tube and fin HX (liquid to air, tube is air side & hot side). Temperature control feedback loop, temp sensor for feedback and valve for flow rate control to regulate temperature.
derive the formula where delta flow rate in is proportional to delta T out.
transfer function Tout as a function of flow rate in.
Generally the heat transfer coefficient (htc) on the tube out-side (1) is proportional to Re1^.65 (as an approximation), and the inside coefficient (2) is proportional to Re2^.8.
For changes in flow where thermal properties are not widely varying it is possible to use inside film coefficient = K1. Flow1^.8 and outside coefficient = K2.Flow2^.65. The overall coefficient is then calculated as normal.
When the inside fluid is condensing or boiling (or liquid metal), and the inside htc is very much higher than the outside htc, it is possible to use only the variation of the outside coefficient and the overall htc becomes simply U = K1.Flow1^.8. Then T1out = T2+(T1in-T2).EXP(-U.A/M/Cp), so that a simple transfer function can be used. Where U is the overall htc = external htc; A = outside surface area; M= flowrate; Cp= external fluid Specific Heat.
Dear @Julius
( mdot*CP*(Tin-Tout))|liquid * Hx effeciency= ( mdot *Cp*( Tout-Tin))|air
Regards.
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