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# Heat Transfer Across a Solid From Two Flowing Pipes

## Heat Transfer Across a Solid From Two Flowing Pipes

(OP)
I have two stainless tubes which are a distance apart, separated by aluminum (i.e tubes fitted inside aluminum block). One tube has hot fluid, the other has cool fluid to remove heat from the hot tube. I know the mass flow rate, inlet and outlet temperatures for the hot side, and the amount of heat to remove.

Questions:
- How do I calculate the flow rate and temperature inlet on the cold side to remove known heat?
- What impact does running in parallel vs. counterflow have on this heat removal calculation.

My thoughts:
I can calculate the heat transfer coefficient (W/m²K) for both flows using Re, Nu, Pr numbers, etc. However, as this appears to be 2x convection + conduction, unsure on what the dT to plug into the Q=h*A*dT equation should be. My thoughts are that: convective heat on the hot side to the aluminum = conduction in aluminum = convective heat to cold side from aluminum.
Replies continue below

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

This will only work in steady state with those calculations as you're not including any mass of aluminium to heat up or other heat losses from the aluminium.

What's the purpose of the AL? A heat sink / regualtor?

Counter flow is easier to calculate as you won't have radically different temperature gradients.

A drawing would help to chow the orientation and plan / section of the tubes relative to each other.

I think you're going to need a CFD analysis here to really see what's going on as there are too many unknowns and differences to do it by hand other than a simple heat rise calc in your cold water and then multiply by 2 to give you some spare to allow some sort of control.

Remember - More details = better answers
Also: If you get a response it's polite to respond to it.

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

506818,

I agree with LI here - CFD is likely needed to get an accurate result. Though calculating the two film coefficients is easy, you will run into geometric challenges pretty quickly if trying to solve this by hand. Even if the block is insulated (i.e. heat loss through the bock walls to atmosphere is negligible), you will still have to consider the average distance between each point on the radius of one tube to the heat sink tube surface area on the other.

Trying to solve this by hand would require several simplifying assumptions; in this case the assumption might be to reduce a 2D model by assuming an "average" thickness of metal between the two tubes, and treating the tubes as flat walls. The second dimension would be related the dT between the two bulk fluid temperatures that would vary with position along the tube walls. Heat loss to surroundings would be ignored. That might get you in the ballpark for initial estimation/sizing.

I believe solutions to 2D/3D heat transfer modeling are typically iterated numerical solutions to partial differential equations - something that is best handled with software that models heat flow.

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

(OP)
Understand CFD is the route to go. I'll see if I can get hold of the resources and update in due course.

In the meantime, it would be much appreciated if someone could direct me to perhaps a similar relevant example or set of equations. I've made some convection calcs based on assuming a constant heat flux (i.e. dT between cold-side fluid and wall is constant). Then varying the mass flow rate and temperature to achieve varying heat transfer coefficients. Is this at all a correct assumption?

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

I don't know of a similar real-life example.

Calculating Convective Film Coefficients: Perry's Chemical Engineering Handbook (5th Edition), Section 10-14, Turbulent Flow, Circular Tubes.

This should get you the film coefficient for the liquids inside your tubes. Everything else is conductive heat transfer.

Calculating Conductive Heat Transfer: Perry's Chemical Engineering Handbook (5th Edition), Section 10-6 to 10-8.

The equations found herein should provide you with the basis for a hand-calculation using Fourier's Law with a 1-dimensional analysis. You can apply this one-dimensional analysis to a 2D estimation in Excel by subdividing the tubes into small sections and calculating a heat transfer for each one, with the output of one section being used as the input for the other. You can then iterate and solve in Excel to get a 2-D analysis. This still relies upon you assuming an average thickness of metal between the tubes as well as assuming the tubes are flat walls.

That's the best I think you can do without CFD. Below I've quoted a note from Perry's under equation 10-18 (Section 10-8) for two-dimensional conduction, which supports my initial post above:

"Two-dimensional Conduction. If the temperature of a material is a function of two space variables, the two-dimensional conduction equation is [Eq 10-18]" "The analytical solution...is possible only for a few boundary conditions and geometric shapes."

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

(OP)
Thank you for the references - I now have a copy of the book.

Excuse my ignorance here. No trouble in calculating convective heat transfer coefficients (h, W/m²K). But how do I then relate this to conductive heat transfer? Is this a case of combined convection and conduction? Perhaps there is an example of a flat wall with flow on both sides that I'm struggling to find.

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

Give section 10-7 a read. For a flat wall, Q = A*dT/R.

A = assumed average surface area for heat transfer. Basically, it will be the length of one of your pipe sections multiplied by the circumference of the pipe.

dT = difference in the average bulk temperature between the two fluids in that particular section.

R = sum of the resistances to heat transfer (sum of all "R"). For convective heat transfer, R = 1/h (the reciprocal of the film coefficient). For conductive heat transfer, R = L/k, where L = your assumed "average" length between one boundary and another, and k is the thermal conductivity of that material.

For your example, you should have 5 terms that make up R - two film coefficients, two L/k that relate to heat transfer through the pipe walls, and one L/k that relates to heat transfer through the aluminum.

Use the calculated Q term for the first section, relate that to Q= m*Cp*dT to get the outlet temp of each fluid for that section. These temperature values will be the bulk fluid temperature for the next section calculation. Co-current flow, geometrically, would be easier to set up and solve in Excel.

This approach ignores any heat loss to the surroundings.

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

(OP)
TiCl4 - That's very clear, thank you. I've split the pipe into a number of sections and have a ballpark figure now. I can vary the inputs into my Excel calculator and get an idea of best and/or worst case.

IRstuff - I believe what you're suggesting is what TiCl4 has eluded to, and what I started off with (but without the knowledge on performing the calls). If not that's not correct, happy to hear your opinion.

### RE: Heat Transfer Across a Solid From Two Flowing Pipes

If you require temperatures at cold side, it is easy. You know the heat to be dissipated and using first law you can calculate the temperature rise in the cold side (assuming Al block is insulates on outside)

When you want to calculate the heat transfer surfaces it is complicated due to geometry. But it may be possible to develop a differential equation considering varying resistances of Al partition at different points.

An estimate of the surfaces can be made considering Al(as well tube wall) as highly conductive and neglecting the resistances. So effectively only two heat transfer coefficients need to be considered. Using analogy from HEX design, LMTD etc. can be calculated, thus calculating approximate heat transfer area.

But as indicated in earlier comments, CFD analysis with heat transfer (CHT analysis) can give accurate results after a few iterations.

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