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 December 2005
library  >  Application Notes  >  Cathy Biber

## Pressure Drop Calculation for Rectangular Channels - PART II: Turbulent Flow

in last month's column, we looked at pressure drop calculation for laminar flow in rectangular channels. here, we'll extend the calculation for turbulent flow, and discuss how to implement the calculation in a spreadsheet.

to start, let's go over the correlations. then we'll look at when to use the turbulent calculation.

correlations

recall that any correlations are for a dimensionless form of the pressure drop, the friction factor f. the pressure drop is dp=4f(l/dh)(½rv2) where l is the flow length, dh is the hydraulic diameter, r is the fluid density, and v is the average fluid velocity in the channel. a quick look in the textbooks yields a bewildering array of nasty implicit formulas - meaning the variable you're after is on both sides of the equation.

let's avoid those!

an easier form is the following: 4f = 0.079 re-0.25 for smooth circular tubes. for rectangular channels, kraus et al (1983) recommend applying an aspect-ratio correction to the hydraulic diameter: 4f = 0.079 (re(.156+a)/a)-0.25 for aspect ratio a>1. (note, a = 1/a!)

when to use turbulent correlations

okay, so you have a pressure drop for turbulent flow through smooth rectangular channels. (for rough channel walls, you'll want to use a formula that includes roughness; i have never validated one like that.) now, when do you use this formula, and when do you use the laminar formula? if you remember one thing from undergraduate fluid mechanics class, it is probably the transition reynolds number being something like 2300.

that's the transition from laminar to turbulent character for pipe flow. i am going to argue that it doesn't really matter to you what value you assign to the transition number.

for one thing, most real equipment has surface roughness, intentional or not, that is going to "trip" turbulence at lower reynolds numbers than a careful laboratory experiment would predict. a perfect example of this is a circuit board. you've got all those little leads, packages, connectors, capacitors, etc.

the other reason that you won't need to worry about the exact transition reynolds number is the nature of the calculation, and the fact that you will probably be using a computer and a spreadsheet to do these calculations. recall the way we are using this calculation: we are trying to find the volume flow rate that a fan is delivering to a system whose main resistance is rectangular channels in series with area changes.

that's going to require iteration, and the spreadsheet iteration tool will gag if the function it's trying to iterate is discontinuous near the solution. to see what i mean, look up "moody chart" in a fluid mechanics or heat transfer textbook. in mine, there is a cross-hatched region to indicate that value of the friction factor is pretty much anybody's guess.

so here is what i suggest: a smoothing function as a means of guessing. this same smoothing function will let your spreadsheet work without gagging.

smoothing function

a generic smoothing function looks like this: yest = (y1m + y2m)(1/m). i found that if i used y = fre (product of friction factor and reynolds number) and smoothing exponent m = 6, the turbulent contribution to the value of the friction factor started to become significant at around re = 2000. that seems to fit well with the commonly accepted transition value of 2100--2300.

in any case you now have a smooth function without jumps to confuse the spreadsheet iteration tool. (in excel, this tool is called "goal seek;" in 1-2-3, it is called "backsolve," if i remember correctly.) let's review the calculation sequence from last month's column:

calculation sequence:

1. assume a channel velocity.
2. calculate volume flow rate and pressure drop (making sure you havethe smoothed laminar and turbulent correlations for the flow length; don't forget area changes).
3. compare to fan/pump curve, which is pressure drop as a function of volume flow rate. i usually approximate this as a line to avoid interpolation: dpfan = p0 - (p0/q0)q, where p0 and q0 are the shutoff pressure and free-delivery flow rate, respectively. if the curve doesn't look very linear, fit a line to the section of the curve in which you want the system to operate, or curve-fit another more suitable function.
4. revise velocity and repeat until you land on the curve (iterate). one way to look at this in spreadsheet-coding terms is that the difference between the fan pressure dpfan and the system pressure drop is zero. if you set up a cell that looks at this difference, the iteration tool has a target value of zero in that cell.
5. then do the thermal analysis using the last velocity assumption (and associated volume flow rate).

up to this point, we've covered calculation of pressure drop for both laminar and turbulent flow regimes in rectangular channels; pressure drop for area changes; the recipe for finding the volume flow rate and how to use a smoothing function. in the next column, we'll look at the thermal calculations that depend on the volume flow rate.

references:
guyer, e., 1989, ed., handbook of applied thermal design, mcgraw-hill.
kraus, a., and a. bar-cohen, 1983, thermal analysis and control of electronic equipment, mcgraw-hill.
white, f. m., 1991, heat and mass transfer, addison-wesley.

 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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