Pump curve problem

A recent exchange here on the boards inspired me to create a cool problem. Here goes:

An old copy of a pump curve from a manufacturer shows the curve corresponding to an impeller diameter of 12 inches (shown here in blue). Another curve is shown in the graph (shown here in green), but a coffee stain on the graph covered the impeller diameter corresponding to this curve.


The impeller diameter (inches) corresponding to the green curve is most nearly:

(A) 10.5

(B) 11.0

(C) 11.5

(D) 12.5

Post your solution and explain your reasoning.

Speed is held constant at 1750. I picked 380 gpm and looked at the head for each curve at that flow. P1=140, P2=90, D1=12, D2=?. I used 18.47 in the MERM, P2/P1=(D2/D1)^3 and solved for D2 to get 10.4 inches.

namod65 said:

Speed is held constant at 1750. I picked 380 gpm and looked at the head for each curve at that flow. P1=140, P2=90, D1=12, D2=?. I used 18.47 in the MERM, P2/P1=(D2/D1)^3 and solved for D2 to get 10.4 inches.

Click to expand...

Unfortunately, that is incorrect.

First of all, the "P" in equation 18.47 refers to power, not head. I believe you meant to use equation 18.46 instead.

However, applying 18.46 with h1 = 140 ft, h2 = 90 ft, D1 = 12 in and solving for D2 will still be incorrect. Since speed is held constant, both 18.45 and 18.46 apply. If you look at 18.45 you see that the flow rate will also change (not remain constant as you have assumed)

First, just by looking at the curves, you know that the diameter is going to be smaller.  So assuming a person had to guess, they could reduce their choices to 3 instead of 4.  Second, I note that the plot is head vs. flow, suggesting that the appropriate equation for the relationship (from MERM 13) is 18.46, which is h2/h1=(D2/D1)^2.  Using any particular flow (keeping with namod65's choice of 380 gpm) I can use h1=140 and h2=90, D1 =12 and get 14.9" or B.

Audi driver said:

First, just by looking at the curves, you know that the diameter is going to be smaller.  So assuming a person had to guess, they could reduce their choices to 3 instead of 4.  Second, I note that the plot is head vs. flow, suggesting that the appropriate equation for the relationship (from MERM 13) is 18.46, which is h2/h1=(D2/D1)^2.  Using any particular flow (keeping with namod65's choice of 380 gpm) I can use h1=140 and h2=90, D1 =12 and get 14.9" or B.

Click to expand...

That's the correct answer, but I disagree with the steps.

Why pick points with the same flow rate? and why not points with the same head added instead? I could pick  Q1 = 438 gpm, Q2 = 320 (both have h = 110 ft) use D1 = 12" and use equation 18.45 to solve for D1 which yields 8.8"

The key is that when diameter changes, 18.46 and 18.47 both apply. If we pick a point in the blue curve say (380, 140) and pick one of the choices as D2 (say, 10.5 as a first attempt) we get  Q2 = 319 qpm and  h2 = 99 ft. This point is not on the green curve, so (A) would be incorrect. Next we try  D2 = 11" and we get  Q2 = 334 qpm and  h2 = 108 ft. This point is on the green curve, so (B) is correct

I get 10.36" or 'A'...   Same reasoning as Audi.  

I am gonna have to go over this again..   Apparently I have been doing this all wrong.

Yeah I messed myself up by labeling head with P and then I used the wrong equation lol. So obviously the point of this is to prove the SMS HVAC #30 wrong since they use 18.46 as well. 

Are there any other affinity law cases where plugging and chugging through the formula doesnt work? Every single practice problem I've done in the neecs, PPI, engproguides with affinity laws always gives you 3 variables and you just plug and chug with the correct equation to get the 4th variable. I feel like that's the type of question I'll see tomorrow. 

Slay the P.E. said:

That's the correct answer, but I disagree with the steps.

Why pick points with the same flow rate? and why not points with the same head added instead? I could pick  Q1 = 438 gpm, Q2 = 320 (both have h = 110 ft) use D1 = 12" and use equation 18.45 to solve for D1 which yields 8.8"

The key is that when diameter changes, 18.46 and 18.47 both apply. If we pick a point in the blue curve say (380, 140) and pick one of the choices as D2 (say, 10.5 as a first attempt) we get  Q2 = 319 qpm and  h2 = 99 ft. This point is not on the green curve, so (A) would be incorrect. Next we try  D2 = 11" and we get  Q2 = 334 qpm and  h2 = 108 ft. This point is on the green curve, so (B) is correct

Click to expand...

I don't see how this makes any more sense than what I did.  If you pick a different point (say 438, 110) and use 11", you get Q2=401.5 and h2= 92.4.  That point is not on the green curve either.

Audi driver said:

I don't see how this makes any more sense than what I did.  If you pick a different point (say 438, 110) and use 11", you get Q2=401.5 and h2= 92.4.  That point is not on the green curve either.

Click to expand...

Curses!

I posted the wrong figure in the original problem. That green curve in the top post is drawn incorrectly. This is the right one:



So my response to you should have been (edited numbers in bold):

The key is that when diameter changes, 18.46 and 18.47 both apply. If we pick a point in the blue curve say (380, 140) and pick one of the choices as D2 (say, 10.5 as a first attempt) we get  Q2 = 333 qpm and  h2 = 107 ft. This point is not on the green curve, so (A) would be incorrect. Next we try  D2 = 11" and we get  Q2 = 348 qpm and  h2 = 118 ft. This point is on the green curve, so (B) is correct

..

..

Here's further illustration of why picking two points with the same flow rate and using only equation 18.45 is wrong. The answer you get depends on the flow rate you pick:

If  you pick Q2 = 410 qpm then set h1 = 131 ft and h2 = 84 ft, use eq. 18.45 to get D2 = 9.6"

If  you pick Q2 = 380 qpm then set h1 = 140 ft and h2 = 108 ft, use eq. 18.45 to get D2 = 10.5"

If  you pick Q2 = 300 qpm then set h1 = 148 ft and h2 = 123 ft, use eq. 18.45 to get D2 = 10.9"

namod65 said:

Every single practice problem I've done in the neecs, PPI, engproguides with affinity laws always gives you 3 variables and you just plug and chug with the correct equation to get the 4th variable. I feel like that's the type of question I'll see tomorrow. 

Click to expand...

I agree. Tomorrow you are more likely to encounter correctly designed problems. 

I'm still not sure this is the valid approach to use.  First, if you note, I went back and edited my comment because I was doing some math incorrectly using my original approach.  Second, it appears that your graph here was developed using your approach, so yes... it's going to work out according to your approach because that is how you generated your curves.  If I use your methodology and use the graph shown in figure 18.11b, I don't end up on a pump curve.  (I picked point 280, 60 and end up with 251, 49).  If I operate assuming your approach is correct, I'm not 100% certain why that that would be, maybe efficiency?  I don't know.

Yes, efficiency. The affinity laws are based on the assumption that efficiency remains constant. So, in that figure (assuming it is to scale) if you pick (240,75) which is on the 77% efficiency line you would get (216, 61) which is the point on the 4.5" curve with that same efficiency.  But... there are many other points one can pick with a known efficiency -- for example (130, 100) -- that don't fall on the corresponding point on the other curve; you in fact get (117, 81).  Could be that equation 18.48 would have to come into play.

Interestingly enough, if you pick (200,90) the point of peak efficiency for 5" you get (180,73)...roughly the point of peak efficiency for 4.5"

At least if you pick (280, 60) you can get something approximate like (251, 49).  Seems like the whole thing falls apart for lower efficiency.