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 December 2005
library  >  Application Notes  >  Cathy Biber

## Pressure Drop Calculation for Rectangular Channels - Part I: Laminar Flow

yes, this does relate to cooling! i'm going to spend a little time on pressure drop because it affects the output of your fan or pump - and that is the volume of flow that is going do your cooling. over the next couple of months, we will address several elements of a fan or pump-cooled system by looking at how to do some analytical calculations.

these analytical calculations are just that - they're not perfect. but they're better than rules of thumb, and they're better than waiting for a test to tell you that you're in big trouble. and when you code the equations into a spreadsheet, they're really valuable for design work.

there are many examples of flow in rectangular channels in the cooling world. you have your basic heat exchangers, heat sinks (a misnomer if there ever was one - really they are heat exchangers too), stacks of two or more circuit boards, convection belt furnaces, even your car radiator. the analysis that follows might underestimate the pressure drop for circuit boards - they tend to be kind of rough topologically.

the most commonly applied correlations are for fully developed or developing laminar flow in parallel-plate or rectangular channels. fully developed implies no change in the width-wise velocity profiles along the flow length (see figure).

typically these 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. table 1 summarizes the correlations.

a quick note here: if you look at your basic fluid mechanics text, you might see the friction factor labeled as either cf or f (or maybe even both). also, the friction factor can be named after either "fanning" or "darcy." the darcy version is four times the fanning version, so the pressure drop equation either has a 4 in it or not. i'm using the fanning version here, which puts the 4 in the pressure drop equation instead of in the friction factor equation.

fully developed flow, parallel plates

there is an exact solution to the navier-stokes equations for fully developed laminar flow between parallel plates. the resulting velocity profile is parabolic. don't worry, i'm not going to make you go through the calculus. for now, just think of this case as a rectangular channel with infinite aspect ratio.

fully developed flow, rectangular channels

the pressure drop depends on the channel aspect ratio. the parallel plate value is modified by a function of the aspect ratio (see table 1) to obtain the values of the pressure drop parameter,fre, commonly tabulated in textbooks, for example white (1991). this tabulation also appears in the classic reference, shah and london (1979). the polynomial equation in guyer (1989) is more useful for spreadsheet work.

developing flow, parallel plates

the correlations for developing flow between parallel plates assume a uniform velocity profile at the channel inlet. the velocity profile gradually changes from uniform to having a parabolic ("fully developed," see figure) profile. the transition region is the hydrodynamic entry length, usually measured in terms of x+; x+ is the actual flow length scaled by the hydraulic diameter and the reynolds number (see table).

to obtain the pressure drop across the entire channel, an apparent (vs. local) friction factor is defined in guyer (1989), based on the effective channel length. i find this to be a lot of trouble to go to, and not that accurate if i'm looking at thick plates (like extruded heat sink fins), where the area contraction accelerates the flow development.

to simplify, stick with fully developed flow, and treat the result as a lower limit if you think you have a long region of developing flow. of course, you can go to the extra trouble and make yourself a spreadsheet that has all the detail in it. the beauty of writing this stuff into a spreadsheet is that you only have to do the coding once.

after that, the computer does the repetitive, boring calculations for you.

entrance and exit effects

the typical correlations given in the literature for pressure drops across changes in channel area are in the form of a pressure loss coefficient, k=dp/(½rv2). the coefficient usually depends on the free area ratio. see the table. the velocity to use is the one in the small cross-section (i.e. inside the channel). the pressure drop is the above equation rearranged (one of my college profs called it "turned inside out").

 fully developed, parallel plates, spacing s white (1991) fully developed,rectangular channels, area a and perimeter p aspect ratio a < 1 guyer (1989) shah and london (1979) developing, parallel plates x+= l / (dh re)guyer (1989) entrance loss coefficient adapted from kays & london (1964) exit loss coefficient adapted from kays & london (1964)

total pressure drop

calculate the pressure drop for all the flow lengths and area changes that an air particle would see as it flows through your system. add 'em all up, and you're there! at least, you've got the pressure drop for a specific value of the channel velocity, v. therein lies the rub - you usually don't know v. it's what you're after in the first place, because that's the fluid speed that is going to do your cooling.

the whole reason you are going through this little rigmarole is to find the temperature of some surface or other. so let's sketch out a sequence of calculations (we'll cover these in the next several columns).

calculation sequence:

1. assume a channel velocity.
2. calculate volume flow rate and pressure drop (making sure you have correct - laminar or 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.
4. revise velocity and repeat until you land on the curve (iterate).
5. then do the thermal analysis using the last velocity assumption (and associated volume flow rate).

we will cover how to do that in future columns, as well as how to get a spreadsheet to do step 4 without crashing. in the next column, we will look at pressure drop for turbulent flow.

references:
biber, c. r., and c. l. belady, "pressure drop prediction for heat sinks: what is the best method?" asme interpack '97 conference, hawaii, june 1997.
kays, w. m, and a. l. london, 1964, compact heat exchangers,2nd ed., mcgraw-hill.
guyer, e., 1989, ed., handbook of applied thermal design, mcgraw-hill, p.1-54, referencing r. k. shah and a. l. london, "laminar flow forced convection in ducts," in advances in heat transfer, r.f. irvine, jr., and j.p.hartnett (eds.), academic press, new york, 1978.
shah, r. k., and a. l. london, 1979 laminar flow forced convection in ducts, academic press.
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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