Surface Temperature of Insulated Duct

[bgtengineer]

I'm attempting to calculate the insulation required for condensation control on an insulated duct (material is duct board) in a ventilated attic space (essentially outdoors). I'm using the ASHRAE formula for One-Dimensional Steady-State Conduction from the Fundamentals Handbook (Chapter 4), assuming a constant cross-sectional area slab:

q = k * A * (t1 - t2) / L

I may be overthinking this but the issue I'm running into is that I have two unknowns, q (heat transfer, btu/h) and L (insulation thickness, in.). I'm ultimately trying to figure out the L required to maintain the dT (t1 - t2). Otherwise my inputs are

k = 0.23 (Btu * in) / (hr * ft^2 * °F) - thermal conductivity of the insulation
t1 = 80 °F - design dew-point temperature
t2 = 55 °F - airstream temperature
A = (this is the area, I guess this is technically unknown too, or at least abstracted, I think I may need this to cancel out somewhere)

I feel like I'm overlooking something very obvious but I've been staring at this for a while with no conclusion, I'd appreciate any help to point me in the right direction, thanks. (By the way this isn't a homework problem or anything, it's a real-world design application -- if there's another method altogether that I should be using, then I am listening.)

 

[MintJulep]

Your equation deals with heat transfer though the duct board by conduction.

But the heat doesn't arrive at the outside surface by magic. It arrives by the combination of convection and radiation.

And the heat doesn't magically stop at the inside surface. It goes into the air stream by convection.

A can be 1 square foot. Because you can (probably) safely assume that every square foot behaves the same.

 

[bgtengineer]

Thanks, I did know that there is also some convection and radiation involved, but I thought I'd try a simpler check first, as I assumed (possibly incorrectly) that much of the heat transfer will occur through conduction. But even if I include convection and radiation, I believe I'll still have q and L as unknowns. There is an example in the Fundamentals Handbook, Chapter 4 Example 1, which goes through this except for an insulated chilled water pipe (which is essentially identical to a supply air duct for demonstration purposes), but it just assumes a (seemingly arbitrary?) value for t1 to solve for q. Chapter 23 also covers this under the Condensation Control section, but that only really demonstrates how to select a design dew-point temperature (as well as some other helpful general information), and doesn't go into selecting insulation thickness or determining the surface temperature for condensation control. I believe I'm still missing something obvious for how I should be handling the variable q if I'm only concerned about the surface temperature t1 as it corresponds to the insulation thickness L.

 

[bgtengineer]

Aha! I believe I've figured this out. I did oversimplify it in my initial attempt, and MintJulep was trying to point me in the right direction even if it wasn't clear to me how that impacted my missing variables.

So I must first solve for q based on the airstream temperature and the ambient temperature, by totaling the convective heat transfer between the airstream and the duct board inner wall, conductive heat transfer across the inner and outer surfaces of the duct board, then the convective and radiative heat transfer between the outer surface of the duct board and its surroundings. That will result in q, which I can then take to work out the intermediate surface temperatures step by step. I believe this full process eluded me because I thought I remembered it being much simpler.

As for my previous comment about the ASHRAE example making an arbitrary assumption for the surface temperature -- this is correct, because you need the surface temperature to calculate the heat transfer coefficient of radiation, and then this assumption must be checked when you later use it to solve for the surface temperature. If these numbers don't match within an acceptable tolerance, then you'll need to make a new assumption and iterate until the result matches the assumption.

One note about this process for a real-world application, is that there are assumptions in the heat transfer coefficient for convection, at both the inside and outside surfaces, which seem to have a large possible range (the ASHRAE data for my conditions range by a factor of 10). This could have a large impact on the result if you don't have a precise value to use here, so this calculation may not be as useful to me as I had hoped if I can't determine good values to use here.

 

[IRstuff]

As for my previous comment about the ASHRAE example making an arbitrary assumption for the surface temperature -- this is correct, because you need the surface temperature to calculate the heat transfer coefficient of radiation, and then this assumption must be checked when you later use it to solve for the surface temperature. If these numbers don't match within an acceptable tolerance, then you'll need to make a new assumption and iterate until the result matches the assumption.

That's only partially correct, and only applies if you aren't using software that can solve systems of equations. You have a number of simultaneous equations
> heat flow from duct to duct wall
> heat flow through the duct wall
> heat flow from duct wall to ambient

All 3 are related by the same Q; a equation solver simultaneously solves for Q and the intermediate temperatures

TTFN (ta ta for now)
I can do absolutely anything. I'm an expert!

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