Derivation of Energy Equation

alibaig1991 · February 25, 2018, 08:45

Hello everyone,

I hope this finds you well.

In the research paper titled "Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems", the author has given an expression for stabilized form of 1 dimensional energy equation for a compressible fluid.

I am trying to derive this result. I have tried a few times but I am making mistakes, specifically in the sign of thermal conduction term and the velocity u in the first derivative on the RHS of Eq. 76. In my result, the sign of thermal conduction is minus, and the velocity u comes inside the derivative.

Can someone help me? Where am I making mistake?

Thank you.

FMDenaro · February 26, 2018, 05:31

If I assume h=0 and Q=0, I see the correct form of the total energy conservation, no error in the heat flux term.

LuckyTran · March 25, 2022, 03:09

Check that you have appropriately applied Fick's law. Heat/enegy flows down temperature gradients from high temperature to low temperature and there's a minus sign for the conductive heat flux.

To put the u inside (or take it out of the derivative), apply the product rule and use the continuity equation.

February 25, 2018, 08:45	Derivation of Energy Equation	#1
alibaig1991 New Member abcd efgh ijkl Join Date: Oct 2015 Posts: 26 Rep Power: 11	Hello everyone, I hope this finds you well. In the research paper titled "Derivation of stabilized equations for numerical solution of advective-diffusive transport and fluid flow problems", the author has given an expression for stabilized form of 1 dimensional energy equation for a compressible fluid. I am trying to derive this result. I have tried a few times but I am making mistakes, specifically in the sign of thermal conduction term and the velocity u in the first derivative on the RHS of Eq. 76. In my result, the sign of thermal conduction is minus, and the velocity u comes inside the derivative. Can someone help me? Where am I making mistake? Thank you.

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February 26, 2018, 05:31		#2
FMDenaro Senior Member Filippo Maria Denaro Join Date: Jul 2010 Posts: 6,896 Rep Power: 73	If I assume h=0 and Q=0, I see the correct form of the total energy conservation, no error in the heat flux term.

March 25, 2022, 03:09		#3
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,761 Rep Power: 66	Check that you have appropriately applied Fick's law. Heat/enegy flows down temperature gradients from high temperature to low temperature and there's a minus sign for the conductive heat flux. To put the u inside (or take it out of the derivative), apply the product rule and use the continuity equation.