Problem:
A pipe of diameter D is to be replaced by n pipes, each of diameter d, laid in parallel. Determine the value of d.

Solution:
To solve this problem, we can use the concept of flow rate conservation. The flow rate through the original pipe and the parallel pipes should be the same.

Flow Rate Conservation:
The flow rate (Q) through a pipe is given by the equation:
Q = A * V,
where A is the cross-sectional area of the pipe and V is the velocity of the fluid.

Flow Rate in the Original Pipe:
The cross-sectional area (A_1) of the original pipe is given by:
A_1 = π * (D/2)^2,
where D is the diameter of the original pipe.

The flow rate (Q_1) through the original pipe is:
Q_1 = A_1 * V_1.

Flow Rate in the Parallel Pipes:
The cross-sectional area (A_2) of each parallel pipe is given by:
A_2 = π * (d/2)^2,
where d is the diameter of each parallel pipe.

The total cross-sectional area of all the parallel pipes is:
A_total = n * A_2,
where n is the number of parallel pipes.

The flow rate (Q_2) through each parallel pipe is:
Q_2 = A_2 * V_2.

Flow Rate Conservation Equation:
Since the flow rate through the original pipe and the parallel pipes should be the same, we can equate Q_1 and Q_2:
Q_1 = Q_2.

Substituting the equations for Q_1 and Q_2, we have:
A_1 * V_1 = n * A_2 * V_2.

Simplifying the Equation:
Substituting the equations for A_1 and A_2, we have:
π * (D/2)^2 * V_1 = n * π * (d/2)^2 * V_2.
Simplifying further:
(D/2)^2 * V_1 = n * (d/2)^2 * V_2.
Taking the square root of both sides, we get:
(D/2) * √(V_1) = (d/2) * √(n * V_2).
Cancelling out the common factors of 2, we have:
D * √(V_1) = d * √(n * V_2).

Assumptions:
To further simplify the equation, we make the following assumptions:
1. The velocity of the fluid (V_1 and V_2) is the same in both the original pipe and the parallel pipes.
2. The flow rate is the same in both the original pipe and the parallel pipes.

Final Calculation:
With these assumptions, the equation becomes:
D * √(V) = d * √(n * V),
where V is the velocity of the fluid.

Since the square root of n * V is the same as the square root of n * (V/n), we can simplify further:
D * √(V) = d * √(V/n).

Squaring both sides of the equation