Heat transfer/heat exchanger question

[KevinH673]

I'm analysing a heat transfer problem, but I am a bit rusty as it's been a while since I've taken the course. I have two concentric tubes (annulus), and the outside tube has water flowing through it, to cool the solid rod in the middle. I am interested in the temperature gradiants of both the rod and the water. I have the power of the rod (q), the flow rate (Q), the initial temperature of the water, and the area of both the rods. Can this be solved from this? It seems as though it would be an easy problem, but I have not found the correct equation yet.

[CaptainPanic]

Step 1:

Find out the Reynolds number (Re) for the water flow.

 

Step 2:

From the Reynolds number, find out whether it's turbulent or laminar flow.

 

In case it is laminar flow, you'll need to choose a model for calculating the temperature profile... there exist several... and all are a pain in the ass on maths. I never work with these because a good heat exchanger has a turbulent flow.

 

In the case of turbulent flow, you can either: look in detail at the temperature profile at the wall (can't really help you with that) , or assume constant temperature through the liquid, and calculate the Nusselt number (Nu), which in turn is a function of other dimensionless numbers. That number can then be used to determine the heat transfer on the liquid side.

 

Step 3:

Find the heat transfer of the metal. The only mechanism is conduction. You'll be needing Fourier's Law for that (look up heat conduction). And in addition, I'm not sure whether any heat is also generated in the metal to complicate things. I also don't know the dimensions of the metal. I'd simply assume a constant temperature because compared to a liquid interface, metal is a pretty good conductor.

 

Please note: I'm trying to work towards a solution, and I am not working towards a complete set of equations that you might want... that set of equations is a lot of work

[KevinH673]

Thanks for the help. Would this be a convection problem, though?

 

What about if this was simplified as a flat plate with external flow over it. Taking it as a convection problem, I haven't found helpful equations that would deal with a temperature gradient in the fluid and the plate. My heat transfer books only deal with heat exchangers that have two moving fluids.

 

To model the fluid, could I use an equation such as:

 

Q=m*Cp*(To-Ti)

 

Where "Q" is the power of the Yag rod? I'm not sure if I'm using Q correctly.

[CaptainPanic]

Well... the water is flowing - that is convection. So, yes, the problem is very much related to convection.

 

Q = heat, in J.

 

There are 2 formulas that are almost the same:

 

Q=m*Cp*(T(start)-T(end))

where Q = heat (energy), in J, this is for a certain amount of mass (m, in kg)

and:

 

P=F*Cp*(To-Ti)

where P = power, in J/s, or Watt (W)... this is for a mass flow (F, in kg/s)

 

 

Since you have a flow, I'd go with the 2nd formula. But those formulas will not give you a temperature profile. It'll give the average temperature...

 

I still don't completely understand the problem, but you probably need time-dependent formulas (differential equations).

 

If you're really interested in the temperature profile, then you'll simply have to study the steps I mentioned. There are loads of wikipedia websites. Search on "Reynolds number", "energy balance" and "heat transfer" and then click a bunch of other links on those websites.

 

Sorry for not explaining the entire theory of heat transfer. It's a university study which takes a bit of time to understand (so also: don't despair when you don't understand it in 1 day! That's normal!)

[KevinH673]

Thank you for the quick reply.

 

I actually have taken courses on heat transfer as well as thermofluids, and have done quite a bit of calculation on this already; however with all the variables that need to be accounted for (convection coefficients for the YAG rod, surface area, flow rate, area of the rod, beginning and end temperatures of both the rod and the water) and it has just gotten a bit overwhelming. I have the Reynolds number, as well as the Nusseult number, but am not sure I'm always using the correct equations (or variables in those equations) to get it.

 

Ontop of this, dealing with power levels that oscillate makes it more complicated. Power can spike up to 12 kW or higher for a millisecond, then be off for another 99 milliseconds. At a constant power of 12 kW, the surface temperature is much too high.

 

The second equation you wrote is helpful, but I worry about it not factoring in surface area. Any suggestions?

[CaptainPanic]

You calculate the P (power) from the heat transfer coefficient. You use the Nusselt number to get the heat transfer coefficient for the liquid side of the rod... What is the Reynolds' number? If it's Re > 1000, you can assume a turbulent flow, and I certainly hope you can assume that. Laminar problems are so much more nasty.

 

Then you use:

 

P = U * A * dT

 

P = power (W = J/s)

U = (W/m2K) overall heat transfer coefficient (note: assume a constant temperature inside the rod, then you can say that the heat transfer coefficient of the liquid side equals the overall heat transfer coefficient - the real equation will be much more nasty, because you'll have a heat conductivity inside the rod, and the center of the rod is warmer than the outside, and you'll probably end up with a bunch of nasty differential equations describing the temperature profile inside the rod as a function of the width).

A = area (m2)

dT = temperature difference between the two sides across which you calculate the power (heat transfer) (Kelvins or deg C).

 

Hope this helps.

 

I am still clueless what kind of temperature profile you need: along the length of the system, or the width? (In reality, you get both! Which is why modeling this is so hard.)

 

Also, I re-read the original question. Please note that I use Q for the heat (Joules), not for flow. Always specify the units and name of any symbol you use.

[KevinH673]

Thanks for all the help, CaptainPanic. I believe I have solved the problem.

[CaptainPanic]

That's great news

 

Heat and mass transfer problems can be a real pain

 

I remember how happy I was when I passed the exam. I swore never to get a job where I need that stuff... of course, I failed, and I need it a lot.