Proportional Depth of a Circular Sewer

The proportional depth of a circular sewer refers to the ratio of the depth of the sewer to its diameter. It is commonly used in hydraulic calculations to determine the hydraulic characteristics of the sewer.

To find the proportional depth of a circular sewer running partially half, we can use the Manning's equation. The Manning's equation is commonly used to calculate the flow rate and flow velocity in open channels, including circular sewers.

The Manning's equation is given as:
Q = (1 / n) * A * R^(2/3) * S^(1/2)

Where:
Q is the flow rate,
n is the Manning's roughness coefficient,
A is the cross-sectional area of flow,
R is the hydraulic radius,
S is the slope of the channel.

Derivation of the Proportional Depth

To derive the proportional depth, we need to consider the cross-sectional area of the flow in a circular sewer. The cross-sectional area of a circular sewer is given by:

A = (π / 4) * D^2

Where D is the diameter of the sewer.

The hydraulic radius can be calculated as:

R = A / P

Where P is the wetted perimeter of the sewer, which is given by:

P = π * D

Substituting the values of A and P, we get:

R = (π / 4) * D^2 / (π * D)
R = D / 4

Calculation of the Proportional Depth

Now, let's consider the slope of the sewer, denoted by K. The slope is defined as the change in water surface elevation per unit horizontal distance. In this case, since the sewer is running partially half, the slope can be defined as the change in water surface elevation divided by half the diameter.

The change in water surface elevation can be calculated using the trigonometric relationship:

Change in elevation = D * sin(K/2)

Substituting the value of the hydraulic radius (R) and the change in elevation into the Manning's equation, we get:

Q = (1 / n) * (π / 4) * D^2 * (D / 4)^(2/3) * (D * sin(K/2))^(1/2)

Simplifying the equation, we get:

Q = (1 / n) * (π / 4) * D^(5/3) * (sin(K/2))^(1/2)

Now, the flow rate (Q) can be expressed as:

Q = V * A

Where V is the flow velocity. Assuming a constant flow velocity, we can write:

Q = V * (π / 4) * D^2

Equating the two expressions for Q, we get:

V * (π / 4) * D^2 = (1 / n) * (π / 4) * D^(5/3) * (sin(K/2))^(1/2)

Cancelling out the common terms, we get:

V = (1 / n) * D^(1/3) * (sin(K/2))^(1/2)

The proportional depth is defined as the ratio of the depth of the sewer (H) to its diameter (D):

Proportional depth = H / D

Since