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.
Any ideas on solving this? The answer is 2.7 lbf*ft.
To calculate the kinetic energy change in the water flow, we can use the principle of conservation of energy. The kinetic energy change can be determined using the equation:
ΔKE = KE2 - KE1
Where:
ΔKE = Change in kinetic energy
KE2 = Final kinetic energy
KE1 = Initial kinetic energy
To find the initial and final kinetic energy, we need to calculate the velocities at the two points where the diameter of the pipe changes. We can use Bernoulli's equation, which states that the total energy of a fluid is conserved along a streamline.
Bernoulli's equation can be written as:
P1 + 0.5 * ρ * V1^2 = P2 + 0.5 * ρ * V2^2
Where:
P1 = Pressure at point 1
V1 = Velocity at point 1
P2 = Pressure at point 2
V2 = Velocity at point 2
ρ = Density of the fluid
Since we are dealing with water, let's assume a density of ρ = 62.4 lb/ft^3.
Now let's solve step-by-step to find the change in kinetic energy:
Step 1: Convert the flow rate from lb/hr to lb/s:
10,000 lb/hr = 10,000 / 3600 lb/s = 2.778 lb/s
Step 2: Calculate the initial velocity (V1) using the diameter of 2 in.:
Diameter1 = 2 in. = 2/12 ft = 0.167 ft
Radius1 = Diameter1/2 = 0.167/2 ft = 0.0835 ft
Area1 = π * Radius1^2 = 3.1415 * 0.0835^2 ft^2
Flow rate (Q) = Velocity1 * Area1
Velocity1 = Q / Area1
Substituting the values:
Velocity1 = 2.778 lb/s / (3.1415 * 0.0835^2 ft^2) = 16.01 ft/s (approx)
Step 3: Calculate the final velocity (V2) using the diameter of 1 in.:
Diameter2 = 1 in. = 1/12 ft = 0.0833 ft (approx)
Radius2 = Diameter2/2 = 0.0417 ft (approx)
Area2 = π * Radius2^2 = 3.1415 * 0.0417^2 ft^2
Flow rate (Q) = Velocity2 * Area2
Velocity2 = Q / Area2
Substituting the values:
Velocity2 = 2.778 lb/s / (3.1415 * 0.0417^2 ft^2) = 73.95 ft/s (approx)
Step 4: Calculate the initial kinetic energy (KE1):
KE1 = 0.5 * ρ * V1^2
Substituting the values:
KE1 = 0.5 * 62.4 lb/ft^3 * (16.01 ft/s)^2 ≈ 8,049 lb*ft/s^2 ≈ 8,049 lbf*ft
Step 5: Calculate the final kinetic energy (KE2):
KE2 = 0.5 * ρ * V2^2
Substituting the values:
KE2 = 0.5 * 62.4 lb/ft^3 * (73.95 ft/s)^2 ≈ 171,558 lb*ft/s^2 ≈ 171,558 lbf*ft
Step 6: Calculate the change in kinetic energy (ΔKE):
ΔKE = KE2 - KE1
ΔKE = 171,558 lbf*ft - 8,049 lbf*ft ≈ 163,509 lbf*ft
The change in kinetic energy in the water flow is approximately 163,509 lbf*ft. The value you provided (2.7 lbf*ft) might be incorrect or may have been rounded for simplicity.
To solve this problem, we need to apply the principle of conservation of energy for fluid flow, specifically the Bernoulli's equation. Bernoulli's equation states the conservation of total energy of a fluid as it flows along a streamline. It involves the sum of three components: pressure energy, kinetic energy, and potential energy.
In this case, we are interested in the change in kinetic energy of the water as it flows through the pipe. We know the initial and final conditions: the initial diameter of the pipe is 2 inches, and the final diameter is 1 inch.
To calculate the change in kinetic energy, we can use the equation:
ΔKE = (ρ * A1 * V1^2) / 2 - (ρ * A2 * V2^2) / 2
where:
ΔKE represents the change in kinetic energy
ρ represents the density of the fluid (water in this case)
A1 represents the cross-sectional area of the pipe at the initial diameter
V1 represents the velocity of the water at the initial diameter
A2 represents the cross-sectional area of the pipe at the final diameter
V2 represents the velocity of the water at the final diameter
Now, let's break down the problem and calculate the values step by step:
1. Initial diameter: 2 inches. This gives us the radius of the initial pipe, r1 = 1 inch = 0.0833 ft.
Initial cross-sectional area: A1 = π * r1^2 = π * (0.0833 ft)^2
2. Final diameter: 1 inch. This gives us the radius of the final pipe, r2 = 0.5 inch = 0.0417 ft.
Final cross-sectional area: A2 = π * r2^2 = π * (0.0417 ft)^2
3. Given the flow rate is 10,000 lb/hr, we can find the velocity using the equation:
Flow rate (Q) = A * V
V = Q / A
V1 = (10,000 lb/hr) / A1 : Convert flow rate to lb/sec if necessary
4. We also need to determine the velocity at the final diameter (V2). Since the mass flow rate (10,000 lb/hr) remains constant, the product of density (ρ) and cross-sectional area (A) remains constant as well. Therefore, V1 * A1 = V2 * A2.
5. Substitute the values into the equation for ΔKE:
ΔKE = (ρ * A1 * V1^2) / 2 - (ρ * A2 * V2^2) / 2
6. Solve the equation to obtain the change in kinetic energy, which is the answer to the problem.
Note: To find the value in lbf*ft, you would need to multiply the value obtained for ΔKE by the conversion factor from lb to lbf (pound to pound-force) and the conversion factor from ft-lb to lbf-ft (foot-pound to pound-force-foot).
Make sure to use accurate and consistent units throughout the calculations to attain the correct answer.