pressure loss in presence of conjugate heat transfer in internal flows

nima103 · September 2, 2019, 09:32

Hi Eeryone,

I am trying to calculate presssure loss of a flow in a pipe while the flow is being heated. I would not use the Darcy-Weisbach equation in presence of heat transfer because the quantity of heat being introduced into the system is not negligble.

Can anybody give me an idea where can I begin to make an analytical/numerical model?

LuckyTran · September 5, 2019, 14:03

There aren't analytical solutions for arbitrary heat transfer quantities unless you can constrain the process in some way to exploit some characteristics.

Consider the Darcy-Weisbach not in the discrete sense, but in differential sense.

Discrete form:
$\frac{\Delta P}{L}=f\frac{\rho}{2}\frac{U^2}{D}$

Differential form
$\frac{dP}{dx}=f\frac{\rho}{2}\frac{U^2}{D}$

This is now an ordinary differential equation that you can integrate numerically provided you have addition equations of state to get density and velocity along x and the friction factor f being a known property of the flow (dependent only on Reynolds number). Keep in mind of course that the Darcy-Weisbach relation is an empirical relation and doesn't work for all flow scenarios. At the end of the day, it is a model.

You can also integrate the discrete case as well (it artificially looks like you would treat the pipe as a bunch of successive segments with constant densities). This seems theoretically not sound to people not familiar with discrete calculus, so I like to show people this way via an ode how it would work.

September 2, 2019, 09:32	pressure loss in presence of conjugate heat transfer in internal flows	#1
nima103 New Member Join Date: Aug 2019 Posts: 14 Rep Power: 6	Hi Eeryone, I am trying to calculate presssure loss of a flow in a pipe while the flow is being heated. I would not use the Darcy-Weisbach equation in presence of heat transfer because the quantity of heat being introduced into the system is not negligble. Can anybody give me an idea where can I begin to make an analytical/numerical model? Last edited by nima103; September 2, 2019 at 10:56.

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September 5, 2019, 14:03		#2
LuckyTran Senior Member Lucky Join Date: Apr 2011 Location: Orlando, FL USA Posts: 5,672 Rep Power: 65	There aren't analytical solutions for arbitrary heat transfer quantities unless you can constrain the process in some way to exploit some characteristics. Consider the Darcy-Weisbach not in the discrete sense, but in differential sense. Discrete form: $\frac{\Delta P}{L}=f\frac{\rho}{2}\frac{U^2}{D}$ Differential form $\frac{dP}{dx}=f\frac{\rho}{2}\frac{U^2}{D}$ This is now an ordinary differential equation that you can integrate numerically provided you have addition equations of state to get density and velocity along x and the friction factor f being a known property of the flow (dependent only on Reynolds number). Keep in mind of course that the Darcy-Weisbach relation is an empirical relation and doesn't work for all flow scenarios. At the end of the day, it is a model. You can also integrate the discrete case as well (it artificially looks like you would treat the pipe as a bunch of successive segments with constant densities). This seems theoretically not sound to people not familiar with discrete calculus, so I like to show people this way via an ode how it would work.