Transient Heat Conduction through an Insulated Pipe - Biot Number?

[palebluedot]

Hi everyone,

I am attempting to determine the amount of time it might take for a process fluid to heat up from 46.4 deg F to 50 deg F. The buffer fluid can be modeled with the same properties as water sitting stagnant in a 3/4" ID SS pipe (with 0.65" wall thickness). I am assuming a horizontal cylinder for the geometry.

The purpose of the calculation is to understand the extra time gained to increase the water 3.6 deg F through insulating the pipe. This will help me determine whether the insulation would be effective enough to keep the buffer cool, and whether it would be necessary at all.

I have been able to understand my heat transfer equations well enough to determine the time required to heat up the water without insulation surrounding the pipe, but once I add the insulation I run into problems.

In my methodology for figuring out the time to heat the water (without insulation), I used the lumped capacitance method. In order to do so I needed to calculate a Biot number <0.1, so the approach was valid.

To find the Biot number one is required to observe the properties of the convection fluid and the conduction solid to solve Bi# = h*Lc/k

To find the h value I used the grashof and prandtl numbers, along with the operating temperatures and system geometry.
h=(0.27)*(((Ts-Tinf)/d)^.25)

Using properties of steel (conductivity and characteristic length) along with the calculated h value, I found the Biot number was in fact <0.1
I then calculated the Fourier number in terms of time: Fo = k*t/p*(Cp)*(Lc)^2

Plugging into the non-dimensionalized equation to find the temperature at time t: Tt = Tinf + (Ti-Tinf)e^(-Bi*Fo)

This approach seemed to work well, stating it would take 29.1 min to heat the water up 3.6 deg F through the steel pipe once I solved for t.

HOWEVER! I am unsure of how to approach the problem once I add the insulation.
There are two main problems I seem to be facing:

1) How should the Biot number be calculated with two different conductivities? The insulation has a much lower conductivity than steel obviously, and would drastically alter the Biot number.

2) I assume that if I could correctly calculate the Biot number, it will prove the lumped capacitance method invalid, with a value over 0.1 (due to the insulation). If this is the case I am left to solve for the time using the exact equation (which I am unsure of how to approach) or use the Heisler charts. Unfortunately, Heisler charts require the use of the Fourier number, which requires a conductivity value for the steel and the insulation. So still, I am left with the question of which parameter properties to use.
Does it make sense that I scale and combine the conductivities, densities, and specific heats of the steel and the insulation?

I wouldd suggest to more simply scale only the diffusivity (equivalent to the k/p*Cp term in Fo) but those values are not supplied for the insulation. Which brings me to my last and final question! Does anyone know where I might be able to find information on the diffusivity or specific heat of t-tubes?

http://www.t-tubes.com/technical_data.php

<- this is all the information I can find/am supplied with and it seems essential that I know either the diffusivity or specific heat.
For now, I am using an insulation specific heat of 0.4 BTU/lb-F, does that seem reasonable?

Thank you so much for your help!

 

[ione]

You can find useful info at the link below

http://books.google.it/books?id=b22...age&q=transient conduction biot > 0.1&f=false

Also google for “Analytical Solution of Heat Conduction in a Multi-Layer cylinder”.

 

[corus]

The biot number allows you to make a judgemnet of whether or not there will be a significant temperature gradient through the object. If there is a low temperatuer gradient then the heat transfer equation reduces down to an easily solveable problem of temperature against time. The purpose of insulation is to provide a significant temperature gradient, and hence there is no point in trying to calculate a Biot number, as you know the answer before you start. You could use an FE model to calculate the temperatures, or a Finite Difference (FD) model if any are available.

Tara

http://tinyurl.com/4ydjg7m

 

[davefitz]

I would think the addition of the insulation would greatly improve the accuracy of the lumped parameter model. A comparison of the relative resistance to heat transfer would show that you could likely ignore the resistance to heat transfer of everything except the insulation and the outer fouling factor + outer heat transfer coeficient , and treat the pipe metal + contained fluid as a single composite.

 

[ione]

I’m with corus on this one. Lumped capacitance approach gives acceptable results when conduction heat transfer prevails over convective mechanism. Lumped capacitance method is more suitable for those scenarios involving high conductivity materials, possibly in still air. In this case a graphical method or finite difference model have to be preferred.