Problem 4.4.1:

Water flows through a pipe that has inside diameter of 0.5 inch. Pipe is 100-foot long. If water is flowing at 7.25 feet per second, then estimate pressure drop for different types of pipes.

Density of water is given as 62.37 pounds per cubic foot and viscosity is 1.2 centipoise.

Let’s write Bernoulli’s equation between two points that are located within a pipe of constant diameter. There are three energy terms in this equation.

1. Pressure energy, P₁ over rho
2. Kinetic energy, V₁²/2g_c
3. Potential energy, Z g/g_c

With pressure units in pound-force per square foot and density in pound-mass per cubic foot, energy term has units of foot pound-force per pound-mass. g_c has a value of 32.174 pound-mass foot per pound-force second squared.

For given situation, Z₁ = Z₂ = 0. (Points are at same elevation).

V₁ = V₂ (Points are within the pipe having constant diameter).

Pump work, w_p, = 0

The Bernoulli’s equation reduces to P₁ over rho is equal to P₂ over rho plus h_f, where h_f is form friction loss. Form friction loss has a major contribution from skin friction.

Friction Factor: Skin friction loss is equal to 4 f_F L/D V²/2g_c or f_M L/D V²/2g_c. Note that f_F is Fanning friction factor. However, Moody’s friction factor, f_M, is widely used in civil engineering discipline. Friction factor can directly be read from the chart that has a log-log scale. For Reynolds number smaller than 2100, friction factor can be also be calculated using a simple formula: 64 over Reynolds number. Flow is considered turbulent when Reynolds number is greater than 4000. One can see that friction factor is also dependent upon the pipe roughness. A parameter relative roughness is used to incorporate pipe roughness.

For the given situation, there are no fittings installed between these two points.

Reynolds number is given as V d rho over mu. In this equation, V is velocity of the fluid, d is inside diameter of the pipe, rho is density, and mu is viscosity of the fluid. [One centipoise is equal to 2.42 pounds per foot per hour]. With these values, Reynolds number can be found to be 2.5 times 10⁴.

Let’s estimate the pressure drop for a pipe that is made of commercial steel. Roughness (epsilon or k) is 0.00015 foot. That gives a relative roughness (epsilon over d) as 0.004. Friction factor can be read as 0.0079. Skin friction losses, h_fs, are calculated as 62 foot pound-force per pound-mass. Bernoulli’s equation for this situation, gives, pressure drop of 26.8 pounds per inch squared. Table summarizes calculation for a number of materials. One can see that pressure drop increases with roughness.