The value of the friction factor in a pipe flow depends on the Reynolds number and the relative roughness of the pipe. The Reynolds number is a dimensionless quantity that characterizes the flow regime, and the relative roughness is the ratio of the roughness height to the pipe diameter.

To calculate the friction factor, we first need to calculate the Reynolds number using the following formula:

Re = (ρVD) / μ

where Re is the Reynolds number, ρ is the fluid density, V is the average velocity of the fluid, D is the pipe diameter, and μ is the dynamic viscosity of the fluid.

Since the pipe diameter is given in centimeters, we need to convert it to meters by dividing it by 100.

Given that the diameter of the pipe is 100 cm, which is equivalent to 1 meter, and the roughness height is 0.25 cm, we can proceed with the calculations.

Let's assume a fluid with a density of 1000 kg/m^3 and a dynamic viscosity of 0.001 kg/(m·s). We can use these values for the calculations, but note that the actual values may vary depending on the fluid.

Calculating the Reynolds number:

Re = (ρVD) / μ
= (1000 kg/m^3)(V)(1 m) / (0.001 kg/(m·s))

Now, let's consider two extreme cases to find the range of the friction factor.

Case 1: Maximum friction factor
Assume the fluid velocity is very high, resulting in a turbulent flow regime. In this case, the friction factor can be approximated using the Colebrook equation:

1 / sqrt(f) = -2.0 * log10((ε/D) / 3.7 + (2.51 / (Re * sqrt(f))))

where ε is the roughness height and f is the friction factor.

Since we want to find the maximum friction factor, we can assume a very high value for the Reynolds number, such as 10^6. Substituting the values into the equation and solving for f will give us the maximum friction factor.

Case 2: Minimum friction factor
Assume the fluid velocity is very low, resulting in a laminar flow regime. In this case, the friction factor can be approximated using the Hagen-Poiseuille equation:

f = 16 / Re

where f is the friction factor and Re is the Reynolds number.

Since we want to find the minimum friction factor, we can assume a very low value for the Reynolds number, such as 10^3. Substituting the values into the equation will give us the minimum friction factor.

By considering both extreme cases, we can conclude that the value of the friction factor lies between the maximum and minimum values obtained in the calculations.