The Relationship between Channel Width, Discharge, and Head

In fluid mechanics, the relationship between channel width, discharge, and head is governed by the principle of continuity, also known as the continuity equation. According to this principle, the product of cross-sectional area and velocity of a fluid must remain constant along a streamline. This means that if the channel width changes, the velocity of the fluid must change accordingly to maintain the same discharge.

Given Data
- Initial channel width: 1 cm
- Initial head: 8 cm
- Final channel width: 8 cm
- Discharge remains the same

Understanding the Problem
We are given the initial conditions of a channel with a certain width and head, and we are asked to determine how the head changes when the channel width increases while the discharge remains constant.

Applying the Principle of Continuity
To solve this problem, we can apply the principle of continuity. According to this principle, the product of the cross-sectional area and velocity of the fluid must remain constant. Mathematically, this can be expressed as:

A1 × V1 = A2 × V2

Where:
- A1 and A2 are the cross-sectional areas of the channel at the initial and final states, respectively.
- V1 and V2 are the velocities of the fluid at the initial and final states, respectively.

Calculating the Cross-Sectional Areas
The cross-sectional area of a channel can be calculated using the formula:

A = W × H

Where:
- A is the cross-sectional area
- W is the width of the channel
- H is the height (head) of the fluid in the channel

Let's calculate the cross-sectional areas for the initial and final states.

Initial state:
A1 = 1 cm × 8 cm = 8 cm²

Final state:
A2 = 8 cm × ? = ?

Calculating the Velocity of the Fluid
Since the discharge remains the same, we can use the formula for discharge to calculate the velocity of the fluid. The formula for discharge is:

Q = A × V

Where:
- Q is the discharge
- A is the cross-sectional area
- V is the velocity of the fluid

Since the discharge remains constant, we have:

A1 × V1 = A2 × V2

Let's calculate the velocity of the fluid for the initial and final states.

Initial state:
Q = A1 × V1
V1 = Q / A1

Final state:
Q = A2 × V2
V2 = Q / A2

Substituting the Values
Now, let's substitute the values into the equations to find the velocities of the fluid at the initial and final states.

Initial state:
V1 = Q / A1

Final state:
V2 = Q / A2

Conclusion
To summarize, when the channel width increases from 1 cm to 8 cm while maintaining the same discharge, the head will decrease. The exact amount of decrease in the head can be calculated by comparing the cross-sectional areas and velocities at the initial and final states. By applying the principle of continuity, we can determine the relationship between channel width, discharge, and head in fluid mechanics.