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Answer :
To determine how many feet of head must be added by the pump, we need to consider the total head loss in the system. The total head loss includes both the frictional head loss and the elevation head difference. In this case, the elevation head difference is 65 feet.
The Darcy-Weisbach equation is an important empirical formula used in fluid mechanics to calculate the pressure drop or head loss in a pipe or channel due to friction. It provides a relationship between pressure loss, flow rate, pipe characteristics, and fluid properties. The equation is derived from the principles of fluid dynamics and is widely used for both laminar and turbulent flow conditions, as long as the flow is fully developed and the pipe is hydraulically smooth.
First, let's calculate the frictional head loss using the Darcy-Weisbach equation:
hL = (f * L * V^2) / (2 * g * D)
Where:
hL is the frictional head loss
f is the Darcy friction factor (which depends on the Reynolds number and the roughness of the pipe)
L is the length of the pipe
V is the velocity of the fluid
g is the acceleration due to gravity
D is the diameter of the pipe
Since the diameter of the pipe is given as 6 inches, we need to convert it to feet:
D = 6 inches = 6/12 feet = 0.5 feet
The velocity of the fluid can be calculated using the volumetric flow rate and the cross-sectional area of the pipe:
Q = A * V
Where:
Q is the volumetric flow rate (8000 lbm/hr)
A is the cross-sectional area of the pipe (π * (D/2)^2)
Substituting the values, we can solve for V:
8000 lbm/hr = (π * (0.5/2)^2) * V
V = (8000 lbm/hr) / (π * (0.5/2)^2)
Now that we have the velocity, we can calculate the frictional head loss using the Darcy-Weisbach equation. However, we need the Darcy friction factor (f) for the calculation. The Darcy friction factor depends on the Reynolds number and the roughness of the pipe.
Without information about the Reynolds number and the pipe roughness, it's not possible to calculate the exact value of the Darcy friction factor and, consequently, the frictional head loss. However, we know that the frictional head loss is given as 2 feet.
Therefore, the total head loss is the sum of the frictional head loss and the elevation head difference:
Total head loss = Frictional head loss + Elevation head difference
Total head loss = 2 feet + 65 feet
To find the feet of the head that must be added by the pump, we simply add the frictional head loss and the elevation head difference:
Feet of head added by the pump = 2 feet + 65 feet
Feet of head added by the pump = 67 feet
Therefore, the pump must add 67 feet of head to overcome the frictional head loss and lift the water 65 feet higher than the reservoir.
To learn more about the Darcy-Weisbach equation:
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