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siw September 8, 2008 07:19

Help with unstanding Energy Equation

I'm working through J D Andersons book CFD The Basics with Applications and I'm a bit confused with how he's putting together the energy equation.

The book states the energy equation (eq 2.59) is:

A = B + C where,

A - rate of change of energy inside fluid element. B - net flux of heat into element. C - rate of work done on element due to body and surface forces.

How does A = B + C equate? Because the units of A must be Joules per second as it's just dE/dt. The units of C must be the same as it's d(F * x)/dt with units [N m / s] = [J / s].

But with B the heat flux is [J / s m^2]. Which is not the same as [J / s] in A and C. If it was heat flow rate rather than heat flux then the units would match.

Am I missing something really simple. I'm working through the book and to me it does not work out. So I'd appreciate your help clarrifying this.


ag September 8, 2008 08:55

Re: Help with unstanding Energy Equation
Net energy change = heat transfer + work done on element

First law of thermodynamics

siw September 8, 2008 09:36

Re: Help with unstanding Energy Equation
So the book stating heat flux (heat energy per unit time per unit area) is incorret and it should be heat transfer (heat energy per unit time) then?

andy2O September 8, 2008 10:30

Re: Help with unstanding Energy Equation
You've just forgotten that the heat flux in question is the heat flux **over the surface of the element**. Thus:

B = net heat flux into the element

= Area integral of (heat flux at element surface) over the surface of element

Now performing an area integral introduces a term like [m^2]. So whilst the heat flux has units of [J/s m^2], by the time it's been integrated over the element's surface area you end up with [J/s] as required.

Regards, andy2o

Ananda Himansu September 8, 2008 11:03

Re: Help with unstanding Energy Equation
"flux" carries the meaning of "flow" or "current". Some authors use the term flux to signify time-rate of flow per unit area, and then may refer to the integral of this flux with respect to the area as the "total flux" or the "total flow rate". Whereas other authors, such as Anderson, use "flux" to signify the time-rate of flow through the entire area, and they may use "flux density" or "flux current density" to signify the time-rate of flow per unit area.

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