The concepts that play roles in this problem are similar to those in Multiple-Concept Example 15, except the fluid here moves upward rather than remaining horizontal. A liquid is flowing through a horizontal pipe whose radius is 0.0215 m. The pipe bends straight upward through a height of 8.00 m and joins another horizontal pipe whose radius is 0.0473 m. What volume flow rate will keep the pressures in the two horizontal pipes the same?

To determine the volume flow rate that will keep the pressures in the two horizontal pipes the same, we can use the principle of continuity. According to this principle, the mass flow rate of an incompressible fluid remains constant as it flows through a pipe of varying cross-sectional area.

To solve this problem, we'll use the following steps:

Step 1: Calculate the cross-sectional area of the first pipe.
Given the radius of the first pipe, we can use the formula for the area of a circle (A = πr^2) to find the cross-sectional area (A1) of the first pipe.

A1 = πr1^2

Step 2: Calculate the cross-sectional area of the second pipe.
Similar to step 1, we'll use the radius of the second pipe to find its cross-sectional area (A2).

A2 = πr2^2

Step 3: Set up the continuity equation.
According to the principle of continuity, the volume flow rate (Q) is equal in both pipes. We can express this relationship using the continuity equation:

A1V1 = A2V2

Where V1 is the velocity of the fluid in the first pipe, and V2 is the velocity of the fluid in the second pipe.

Step 4: Rearrange the equation to solve for the volume flow rate (Q).
Using the continuity equation from step 3, we can rearrange the equation to solve for the volume flow rate (Q):

Q = A1V1 / A2

Step 5: Calculate the volume flow rate (Q).
Substitute the values for A1, A2, and known variables into the equation from step 4 to calculate the volume flow rate (Q) that will keep the pressures in the two horizontal pipes the same.

Q = (πr1^2 * V1) / (πr2^2)

Simplify the equation by canceling out the π terms and substituting the given values for r1 and r2 to find the volume flow rate (Q).

Q = (0.0215 m^2 * V1) / (0.0473 m^2)

Finally, you would need to substitute the respective velocity in the first pipe (V1) to get the value of the volume flow rate (Q).