system curve and bernoulli equation problem
1. The problem statement, all variables and given/known data
I have a pump pumping 50 gallons per minute of water from a tank at atmospheric pressure to another tank at atmospheric pressure. The main discharge pipe (internal diameter is 4.03 inches) is divided onto 5 smaller pipes (internal diameter is 1.05 inches). The pressure on each on the smaller pipes is 26.3 psia. But as I said before, the discharge tank is at atmospheric pressure.
I’m trying to do 2 different things:
1. Apply the Bernoulli equation to this system, specifically between points A, which is the pump suction, and points B1, B2, B3, B4 and B5 which are the 5 discharge points in order to know the work that the pump must supply to achieve this pressure on points B1-B5.
2. Draw the system curve for this situation. The system curve is basically total head vs flow rate.
2. Relevant equations
From what I understand the total head of a pump can be expressed as follows:
Total head = Static head + Friction loss throughout the system + pressure head (if the system has it) + velocity head (negligible)
Static Head: Static head represents the net change in height, in feet, that the pump must overcome
Pressure Head: When liquid is pumped from a vessel at one pressure to a vessel at another pressure, pressure head exists. If there is no difference between the 2 vessels, then it does not.
When applying the Bernoulli equation between the points, I get:
ηWp = (5Pb/ρ) + (g*Zb/gc) + (5(Vb^2)/(2*gc)) - (Pa/ρ) + ((Va^2)/(2*gc))
I know there are friction losses in this system due to several factors, but I decided to ignore them on this post because I dont have doubts about that.
3. The attempt at a solution
Pb is 26.3 psia
Pa is 14.7 psia
Zb is 50 ft
Vb is 0.98 ft/s
Va is 0.33 ft/s
5Pb and 5Vb because the pressure and the velocity on all 5 pipes is the same. When I solve the equation with the values given above, I get:
ηWp = 319.75 ft*lbf /lbm = 1.77 hp
On the other hand when I assume Pb to be 14.7 psia, ηWp = 1.03 hp
Is my energy balance correct?
Or, should I assume Pb to be 14.7 psia since the tank is under atmospheric pressure?
I preferred to assume Pb = 26.30 psia because this problem is very much like a shower problem where the water needs to have a certain pressure (above 14.7 psia) when it exits the shower nozzle and one needs to assume Pb as the pressure of the water exiting the nozzle even though the bathroom is under atmospheric pressure. As a mater of fact, I’ve got nozzles on all 5 pipes. Am I right about my approach to this problem?
Second question: According to the pressure head definition above, this system does not have pressure head because the pressure on both tanks is the same. Or should I assume the pressure head to be 26.30-14.7 like I did on the Bernoulli equation when drawing the system curve?
Third question: can the static head for the system curve and Zb for the Bernoulli equation be different?