**Introduction**

In the case of sewage flowing in old cast iron pipes, the value of head loss does not depend on the length of the pipe, velocity of flow, or friction factor. This can be explained by understanding the factors that contribute to head loss in a fluid flow system.

**Head Loss in Fluid Flow**

Head loss refers to the loss of energy or pressure in a fluid flow system due to various factors such as friction, elevation changes, and fittings. It is an important parameter to consider in the design and analysis of piping systems as it affects the efficiency and performance of the system.

**Factors Affecting Head Loss**

There are several factors that contribute to head loss in a fluid flow system, including:

1. **Length of Pipe:** The longer the pipe, the greater the head loss. This is because as the fluid flows through a longer length of pipe, there is more surface area for frictional losses to occur.

2. **Velocity of Flow:** The higher the velocity of flow, the greater the head loss. This is because higher velocities result in increased turbulence and shear stress, leading to greater frictional losses.

3. **Friction Factor:** The friction factor, also known as the Darcy-Weisbach friction factor, represents the resistance to flow caused by the roughness of the pipe wall. It depends on factors such as the pipe material, roughness, and Reynolds number. A higher friction factor corresponds to a higher head loss.

**Explanation for Sewage Flow in Old Cast Iron Pipes**

In the case of sewage flowing in old cast iron pipes, the value of head loss does not depend on the factors mentioned above. This is because of the following reasons:

1. **Length of Pipe:** Old cast iron pipes are typically fixed in place and do not vary in length. Therefore, the length of the pipe remains constant, resulting in a consistent head loss.

2. **Velocity of Flow:** Sewage flow in old cast iron pipes is typically slow and steady, as it is gravity-driven. The velocity of flow is relatively low, and turbulence is minimal. As a result, the impact of velocity on head loss is negligible.

3. **Friction Factor:** Old cast iron pipes have a relatively high roughness compared to modern pipes. This roughness can lead to increased frictional losses. However, since we are considering sewage flow, which is typically a viscous and non-turbulent flow, the impact of the friction factor on head loss is minimal.

**Conclusion**

In conclusion, the value of head loss for sewage flowing in old cast iron pipes does not depend on the length of the pipe, velocity of flow, or friction factor. This is due to the nature of sewage flow, which is slow, steady, and non-turbulent. The fixed length of the pipe and the high roughness of old cast iron pipes contribute to a consistent head loss.