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# David Steinberg's formula

## David Steinberg's formula

(OP)
Guys:
I'm probably showing my age, but I have this book:
Steinberg, Dave S., Cooling Techniques for Electronic Equipment 2nd Ed.; John Wiley & Sons, Inc., 1991.

Any of you out there have ANY OTHER edition of this book?
In my edition his equation for the hydraulic diameter D of a W X L horizontal rectangular surfaces is D=0.5LW/(L+W). What does your edition say?

For years in my career I used this formula in spreadsheet calculations. The natural convection coefficient for that surface is inversely proportional to the fourth root of the hydraulic diameter.

I looked many other places and found that it SHOULD read D=2LW/(L+W) which is four times bigger. So I have been UNDERESTIMATING the natural convection coefficients by a factor of (4)^(-0.25) = .707 for years in my spreadsheet calculations? Do you agree?

I actually used the definition of hydraulic diameter to prove the correction:
D=4X(cross-sectional area)/(wetted perimeter)=4(LW)/[2L+2W]=2LW/(L+W).

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: David Steinberg's formula

Page 6-12 in Perry Chem Engg Handbook (7th ed) has the same formula as the DS book, while my first year BSc text on Chem Engg by Coulson and Richardson Vol 1 equation 3.62 matches your derivation from first principles.

### RE: David Steinberg's formula

I have the same 1991 edition of Steinberg's book "Cooling Techniques for Electronic Equipment."
In the index it lists three places where hydraulic diameter is mentioned. In all three places it is correctly printed:

Dh = 4x area/perimeter (page 170)

for a rectangular duct Dh = 2ad/(a+d)

Can you go back and look at page 170 in your copy of the book? My best guess is that it got copied into the spreadsheet wrong (I do it all the time).

### RE: David Steinberg's formula

(OP)
Tony you are right. I was in the middle of moving and couldn't find my Steinberg book and Ellison book. I just found them and Steinberg's book is OK. It is Gordon Ellison's Thermal Computations for Electronic Equipment that has the wrong formula.
Thanks all!

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: David Steinberg's formula

Can't let that go! I have the 1989 reprint of "Thermal Computations for Electronic Equipment" by Ellison. Page 56 has the definition of hydraulic diameter:
D = 4 Ac/Pw where Ac is the cross-section area and Pw is the wetted perimeter.

Obviously I can't check every instance in the book, but I think this source also is correct. Maybe it was one of my books that steered you wrong?

### RE: David Steinberg's formula

(OP)
OK Tony I'm not imagining this. I looked for my copy of Ellison and couldn't find it. Are there other places in the book where he goes on to show D=0.25LW/(L+W) for horizontal planes?

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: David Steinberg's formula

Now I'm getting curious why you say it is the formula for a horizontal plane. Hydraulic diameter is for a duct. It is a construct for creating fudge factors so that pressure drop and convection coefficient correlations for circular pipes can be applied to ducts of other cross sections, like rectangular ducts. The definition of hydraulic diameter does not change if the duct is vertical or horizontal or running at an angle to gravity.
Even if you are thinking about natural convection, the concept of hydraulic diameter does not change. Maybe you are thinking of another concept?

### RE: David Steinberg's formula

(OP)
Tony you are right that I was using the wrong terminology. The right term is "characteristic length".

Now I have searched high and low and cannot find my copy of Ellison's "Thermal Computations for Electronic Equipment" but I DID find a copy of his short course notes where he uses the D=0.5WL/(L+W) formula. I have attached the notes. If you search for the word "characteristic" you'll find it. I can only assume that if he used the wrong formula in his course notes then he must have the wrong formula in his book.

Thanks,
Bruce

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

### RE: David Steinberg's formula

(OP)
Steinberg's book uses the term "hydraulic diameter" which I now see is incorrect terminology for describing laminar natural convection flow over and around plane surfaces. The correct terminology is "characteristic length".

H. Bruce Jackson
ElectroMechanical Product Development
UMD 1984
UCF 1993

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