Question

A sugar solution undergoes laminar flow from a reservoir to a product preparation vessel through a(n) 3.0-in. diameter pipe. The pipe needs replacing and a vendor suggests replacing the 3.0-in. pipe with a number of 1.5-in. diameter pipes. How many 1.5-in. pipes will it take to provide the same flow rate as one 3.0-in. pipe

Answer 1

To determine the number of 1.5-in. pipes needed to provide the same flow rate as one 3.0-in. pipe, we can use the equation Q = A * v, where Q is the flow rate, A is the cross-sectional area, and v is the velocity of the fluid. By calculating the cross-sectional areas for both pipes, we find that it would take 4 1.5-in. pipes to provide the same flow rate as one 3.0-in. pipe.

Step-by-step explanation:

The flow rate of a fluid through a pipe is determined by the cross-sectional area of the pipe. To determine the number of 1.5-in. pipes that will provide the same flow rate as one 3.0-in. pipe, we can use the equation:

Q = A * v

where Q is the flow rate, A is the cross-sectional area, and v is the velocity of the fluid.

Since the flow rate is the same for both pipes, we can set up the equation:

A1 * v1 = A2 * v2

where A1 and v1 are the cross-sectional area and velocity of the 3.0-in. pipe, and A2 and v2 are the cross-sectional area and velocity of the 1.5-in. pipe.

Given that the diameter of the 3.0-in. pipe is 3.0 inches, the radius is 1.5 inches, and the diameter of the 1.5-in. pipe is 1.5 inches, the radius is 0.75 inches.

Using the formula for the cross-sectional area of a pipe, A = π * r2, we can calculate the values for A1 and A2 as follows:

A1 = π * (1.5)2 = 2.25π

A2 = π * (0.75)2 = 0.5625π

To find the ratio of A1 to A2, we divide A1 by A2:

A1 / A2 = (2.25π) / (0.5625π) = 4

This means that it would take 4 1.5-in. pipes to provide the same flow rate as one 3.0-in. pipe.