My quest for information led me to this weblog post. Essentially, the author used the same approach as me to improve his draught balance calculations. Unfortunately, he assumed that vinyl tubing was perfectly smooth and that beer at serving temperature had the same viscosity as water at room temperature. No wonder his calculated resistance for 3/16" ID vinyl tubing was only a third of the Draught Quality Manual value. The author of this post equated the viscosity of beer with the viscosity of water at 38-39 degf and assigned a non-zero roughness value to vinyl tubing, which are both steps in the right direction, but he didn't explain the rationale behind his chosen roughness value. In addition, he marginalized the impact of a keg spear (for a corny keg, it's like adding two feet of 1/4" OD stainless tubing to your tap line) and faucet shank (not a big deal for most home draught systems, but it's often a major source of restriction for commercial setups). Although I question their assumptions, both authors explain the math well and I'll spare you the drudgery of repeating it.

In my calculations, I assume the dynamic viscosity of beer at a given temperature equals the dynamic viscosity of water at the same temperature x 1.7 / 1.002 lb-s/ft2 (the value for beer is between the lager and stout values in Malting and Brewing Science, Volume 2 by Hough, Briggs, Stevens and Young, and the value for water is from the same paragraph). To correlate viscosity and temperature, I used the values from this website. At typical serving temperatures, the dependence of viscosity on temperature had a smaller impact than I expected. It wasn't zero, though, so it was worth investigating. On the other hand, researching the surface roughnesses of vinyl tubing, barrier tubing and stainless tubing was a total dead end. Defeated, I fudged the values by (1) driving the calculations for serving beer on my home setup at 14 psig and 44 degf to require 6" more 3/16" ID vinyl tubing than the Draught Quality Manual and (2) making my calculations match the performance of Ale Asylum's downstairs draught system, which uses all three tubing materials. If you're wondering why I don't just use the same resistance values as draught technicians, the taps at Ale Asylum were designed to pour beer at 2 fluid oz/sec but actually operate around 2.4 fluid oz/sec. Like my home tap lines, the resistance assumptions were too high and I suspect the same is true of many commercial draught systems.

My assumptions will probably change as I acquire more data points, but here's how my current resistance values (in psi/foot) compare with the Draught Quality Manual for a flowrate of 2 fluid oz/sec and a temperature of 38 degf:

By design, my value for 3/16" ID vinyl tubing is slightly lower than the manual's. From there, the two sets of values diverge as diameter increases. For barrier tubing, my values are higher than the Draught Quality Manual. In fact, if my viscosity assumptions are reasonable, the manual's numbers for barrier tubing aren't physically possible. Stainless was hard to compare because OD is a terrible way to describe draught tubing. A given OD can have a wide range of IDs that depend on wall thickness, and the differences can profoundly influence fluid flow. For what it's worth, I assumed a wall thickness of 0.020". If you'd like to examine the math behind these numbers, you can download it in spreadsheet form at the usual place. The file name is Draught_Balance.xlsx.

Really interesting post. One quick correction -- I think you mean 2 fluid oz/sec instead of 2 fluid oz/min?

ReplyDeleteYup, I meant fluid oz/sec and it's now fixed (all three times... ugh). Thanks!

ReplyDeleteThanks for the link, and nice catch regarding my assumptions of kinematic viscosity. I wasn't able to find a reference for beer. I did want to point out that I used a roughness of 0.0025 mm for vinyl tubing, though at the flow rates involved it hardly matters.

ReplyDeleteIncorporating your assumed values for roughness and viscosity, the theoretical resistance for 3/16" ID vinyl is still only about 0.9 psi/ft. Any thoughts on why there's such a discrepancy? In my experience (a few kegerators and one long-draw beer gas system) assuming a value of 2-3 psi/ft would be far from a balanced pour.

I figured you assumed zero roughness because of where Re = 11,000 and f = 0.031 intersect on the Moody Chart. Sorry about that! Anyway, I think the biggest difference between our calculations is that mine are based on 2 fl-oz/sec instead of ~1.35 fl-oz sec. If I change my flowrate to match yours, I calculate ~1.3 psi/ft for 3/16" ID vinyl. When you used my roughness assumption, did you convert it to relative roughness before you determined the friction factor? If you did, I'd guess that most of the remaining 0.4 psi/ft difference between our calculations is because I used the Swamee-Jain equation (http://en.wikipedia.org/wiki/Darcy_friction_factor_formulae) to calculate friction factor instead of looking it up in a Moody Chart.

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