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December 2005
library  >  Application Notes  >  Cathy Biber

Heat Transfer in Rectangular Chancels - PART III


in the last couple of columns, we looked at how to find the volume flow rate through a rectangular channel based on the pressure drop and an air mover. now we are finally ready to tackle the thermal calculations, which was the whole point to begin with.

 

as part of the pressure drop calculation, we found a reynolds number based on the average speed in the channel, which is easy to get from the volume flow rate, and which your spreadsheet (you are doing this in a spreadsheet or other calculation tool, aren't you?) will have remembered for you. we'll use that as the starting point. let's sketch out the overall procedure and then go into the details.

we will start by looking for a heat transfer coefficient, which is a measure of how easily heat moves from a surface to the moving fluid adjacent to it. it's defined as the heat flux per unit area divided by the temperature difference between the bulk fluid and the surface. but what is the bulk fluid temperature?

 

that's where it can get tricky.

 

the trick is that you have to use the same definition for temperature (or temperature difference) as the author of the correlation for heat transfer coefficient. now for relatively constant fluid temperature, it's pretty straightforward; the heat flow q=ha(ts-tf), where h is the heat transfer coefficient, a is the wetted surface area, ts is the surface temperature and tf is the fluid temperature.

 

but if the fluid heats up significantly from one end of the channel to the other, then you need to use heat exchanger correlations and take fluid temperature rise into account. more on that in a bit.

so let's start with the heat transfer coefficient. just as the pressure drop varies with laminar or turbulent flow, and fully developed or developing flow, so does the heat transfer coefficient. the reynolds number lets you decide whether it's laminar or turbulent, and the developing-flow parameter x+ lets you decide whether it's fully developed or developing. for thermal calculations, x+ is modified by the prandtl number: x* = l/(d re pr).

 

using this parameter, we have correlations for the dimensionless form of the heat transfer coefficient, the nusselt number nu = hl/k where l is the characteristic length and k is the conductivity of the fluid. the characteristic length l is flow length for developing flow and hydraulic diameter for fully developed flow.

 

be sure to use the correct characteristic length! the table below gives the constant-wall-temperature correlations for parallel plates (infinite aspect ratio) and references.


flow character
flow development
correlation
reference
laminar
fully developed
7.54
white (1991)
laminar
developing x*>0.05
nut = 1.545455x*-0.4
curve fit to guyer (1989)
turbulent
fully developed
nut = [f/2(re-1000)pr]/[1+12.7(pr1/3-1)v(f/2)]
guyer (1989)
turbulent
developing
use fully developed
.

 

to keep things simple, i correct the nusselt number for aspect ratio after choosing the correlation and applying smoothing functions. the smoothing functions, as you may recall from the last column, let your computer do the iterations on a continuous function. for example, for nulam = (nufdm + nudevm)(1/m) i use m = 4; for nu = (nulamm + nuturbm) (1/m) i use m = 20. then the aspect ratio correction (based on a curve fit to data in guyer (1989)) is nua = [2.97 + 0.45(a-1)]nu with aspect ratio a >1.


now, after going to all that trouble to find the nusselt number, and then the heat transfer coefficient, you can finally get what you're after: the surface temperature ts. start with q=ha(ts-tf) and do the algebra, provided that the fluid temperature change from one end of the channel to the other changes little compared to (ts-tf). 

 

but what if the fluid heats up significantly as it moves down the channel? then you have a heat exchanger (or a heat sink, which is kind of the same thing) and you need to use log-mean temperature difference.

 

since you already know the definition, i'll go right to how to use it. simply put, the log-mean temperature difference dtlm = q/ha where q is the dissipated heat, h is the heat transfer coefficient found from the nusselt number above, and a is the wetted surface area over which fluid is moving. (for very thin heat sink fins you might want to modify the heat transfer coefficient by the fin efficiency, but we will get to that in another column.)

 

next, you want to calculate a dimensionless heat transfer effectiveness e = exp(dtair/ dtlm) where dtair = q/(mcp), m being the mass flow rate and cp being the specific heat of the fluid. then the surface temperature rise above ambient dtsa = dtair /(1- e). note that an effectiveness value close to 1 means that the fluid temperature rise is dominating the surface temperature rise.

 

improvement, meaning surface temperature reduction, requires more fluid flow and thus a tendency toward lower effectiveness. yes, it's somewhat counter-intuitive, but it just goes to show that you should not use the effectiveness as a design target; it's just a calculation convenience. a low effectiveness value means that either heat transfer coefficient or wetted surface area is the limiting factor.


in the next column, we will look at applying these calculations to heat sink design. we'll cover modifications for fin efficiency and heat spreading as well.

references:
guyer, e., 1989, ed., handbook of applied thermal design, mcgraw-hill
white, f. m., 1991, heat and mass transfer, addison-wesley.

 


about cathy biber

dr. catharina biber is senior thermal engineer at infocus corporation where she works with product design teams to solve optical and electronic cooling issues in advanced digital data/video projection systems. she particularly enjoys collaborating with cross-functional team members to address all the aesthetic, manufacturability and regulatory aspects of design needed for a successful product.

 

previously, she was a technical staff member at wakefield engineering, inc., where she was involved in the design, analysis, and optimization of high performance heat sinks. she has taught seminars on electronics cooling and basic thermal analysis throughout the u.s. and in europe.


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