## NATURAL- cooling by convection

General discussions about how to formulate a script for FlexPDE.

### NATURAL- cooling by convection

Dear Sirs,
I would like to clarify the disagreement that exists between flexpde 6.20 manual for convective cooling and the book "Fields of Physics by Finite Element Analysis - An Introduction". For example see the manual on (a) page 289,"Sample Problems : applications" where convective cooling on the outer arc is specified as "natural(Temp)=BI*(TW-Temp)" and (b) page 339 "Sample Problems : applications" where natural(Temp)=Tzero-Temp { "Temperature-difference" flow boundary. Negative value means negative K*grad(Temp) or POSITIVE heat flow INTO coolant hole } and compare it with the book where (a) on page 167 it defines Convection on the boundary as "natural(temp)=50*(temp- 273)" and (b) on page 174 where cooling by forced convection is " natural(temp)=1e3*(temp- temp2)".
My trials show that the manual is wrong. Is it so?
George Florides
gflo

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### Re: NATURAL- cooling by convection

As we explain in the documentation (see "Natural" in the Help Index), the NATURAL boundary condition defines the value of the outward normal component of the boundary flux term implied by the Divergence Theorem (or equivalent integration by parts). As such, the sign of the Natural BC carries the sign of the flux term in the PDE. Our examples include terms like Div(K*grad(temp)), whereas Backstrom's examples use terms like Div(flux) or Div(-k*grad(temp)). That is, the sign of the flux terms differ between the two documents. The sign of the Natural must therefore also differ between the various problems. I think you will see that this fact both explains the apparent discrepancy between the statements in the various examples, and also highlights the fact that the meaning of the Natural Boundary Condition depends on the statement of the PDE.
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