Fluid Mechanics: Determine volumetric flowrate

[cheme333]

Homework Statement

*See attachment for diagram*

With steady-state operation, an energy balance between point 1 and point 2 along the A side dictates:

*See attachment for equation 1*

And an energy balance between points 1 and 2 along the B side dictates:

*see attachment for equation 2*

Comparing the two balances, the following can be determined:

*See attachment for equation 3*

Which states that frictional losses over parallel segments are equal. With this in mind, develop a model for the system and fill in the missing entries in the table below.

Qtot LA DA LB DB QA QB
gpm ft in ft in gpm gpm

150 85 4 100 4 ? ?
80 100 4 400 3 ? ?
300 120 2 120 3 ? ?
300 120 6 50 4 ? ?
180 80 ? 80 4 60 ?
300 ? 3 70 3 140 ?
? 300 6 100 4 100 ?

Homework Equations

Use chen equation for friction factor:

1/(f)^1/2 = -2.0log((e/D)/3.7065 – 5.0452A/Re)
A = log((e/D)^1.1098/2.8257 + (5.8506/Re^0.8981))

The Attempt at a Solution

We are allowed to use Excel goal seek (I think we HAVE to). However, i cannot seem to get a system of equations that will solve for Va and Vb. I believe that if I had those terms, I could easily calculate Qa and Qb. If that's not the direction I need to be going, then I am completely lost.

For the first row, we were given the answers so that we could check our model. Qa=83.1 gpm and Qb=66.9 gpm.

Thanks for any help!

 

[KataKoniK]

Thank you for your post. I am happy to help you develop a model for the system and fill in the missing entries in the table.

First, let's review the given equations:

*See attachment for equation 1*

*See attachment for equation 2*

*See attachment for equation 3*

From these equations, we can see that the total flow rate (Qtot) is equal to the sum of the flow rates on the A and B sides (QA and QB). We can also see that the frictional losses (LA and LB) are equal. Therefore, we can write the following equations:

Qtot = QA + QB

LA = LB

Now, we can use these equations to solve for the missing entries in the table. Let's start with the first row:

Qtot = 150 gpm
LA = 85 ft
DA = 4 in
LB = 100 ft
DB = 4 in
QA = ?
QB = ?

Using the first equation, we can substitute the known values and solve for QA:

150 gpm = QA + QB

QA = 150 gpm - QB

Now, we can use the second equation to substitute the known values and solve for QB:

85 ft = LB

100 ft = LB

LB = 85 ft

Now, we can use the third equation to substitute the known values and solve for QB:

85 ft = 100 ft

100 ft = 85 ft

85 ft = 100 ft

Therefore, QA = 150 gpm - 85 ft = 65 gpm

And QB = 150 gpm - 65 gpm = 85 ft

So, the first row of the table would be:

Qtot = 150 gpm
LA = 85 ft
DA = 4 in
LB = 100 ft
DB = 4 in
QA = 65 gpm
QB = 85 ft

Similarly, we can use this method to solve for the missing entries in the other rows of the table.

I hope this helps you develop your model and solve the problem. Let me know if you have any further questions.

Scientist