Calculate the kinetic energy change in the water that occurs when 10,000lb/hr flow in a pipe that is reduced from a diameter of 2 in. to a diameter to 1 in.
To calculate the kinetic energy change in the water flow, we need to determine the velocity of the water at each diameter, and then use the kinetic energy equation.
Step 1: Calculate the velocity at each diameter
To calculate the velocity, we can use the equation of continuity, which states that the product of the cross-sectional area and velocity remains constant for an incompressible fluid like water.
The cross-sectional area (A) of a pipe can be determined using the formula for area of a circle: A = π * (d/2)^2, where d is the diameter.
For the initial diameter of 2 inches:
A1 = π * (2/2)^2 = π square inches
For the reduced diameter of 1 inch:
A2 = π * (1/2)^2 = π/4 square inches
Now, let's calculate the velocity at each diameter. We can use the volumetric flow rate (Q), given in lb/hr, and the density of water (ρ), which is approximately 62.4 lb/ft^3. The formula to calculate velocity (V) is V = Q / (A * ρ).
For the initial diameter:
V1 = (10,000 lb/hr) / (A1 * ρ)
For the reduced diameter:
V2 = (10,000 lb/hr) / (A2 * ρ)
Step 2: Calculate the kinetic energy change
The kinetic energy (KE) of a fluid can be calculated using the formula KE = 0.5 * m * V^2, where m is the mass of the fluid.
To convert the volumetric flow rate (Q) from lb/hr to mass flow rate (m) in lb/s, divide by 3600 (since there are 3600 seconds in an hour): m = Q / 3600.
For the initial diameter:
KE1 = 0.5 * (m1) * (V1)^2
For the reduced diameter:
KE2 = 0.5 * (m2) * (V2)^2
Finally, to calculate the kinetic energy change, subtract KE2 from KE1:
ΔKE = KE2 - KE1
Now you can plug in the known values and calculate the kinetic energy change in the water flow when 10,000 lb/hr of water flows through a pipe with diameters 2 in. and 1 in.