F4. Fluid Flow Formulas

Flow Measurement: There are a number of devices used in industry. Those include pitot tubes, venturi meters, nozzles, orifice meters, and rotameters. Manometers give a measure of the pressure drop as follows:

where subscript m represents the manometer fluid and h represents the manometer reading. Numerical values of g and g_c are equal to 32.174.

Pitot Tube: This device measures the local velocity. For liquids the density of the fluid at these two locations remains constant. So velocity at the center of the pipe can found as

where C is a constant that is used to compensate for frictional losses. The value of C depends on its design. For a standard pitot tube C lies between 0.98 and 1.0. For S-type pitot tube its value is 0.84.

For gases at velocities greater than 200 ft/s, pressure drop is found from the following equation.

where subscripts 1 and 2 represent inlet and outlet conditions, respectively.

In general flow coefficient, K, is given as

where C_c º A_{vena contracta}/A_throat and C_v = V_{2 actual}/V_{2 ideal}

For nozzles and venturi meters, the section of minimum flow area is located at the throat. There is no vena contracta and C_c = 1. For these cases

where β is given as

The factor 1/(1 - β⁴)^1/2 is called the "velocity of approach" factor.

Some of the pressure is recovered in the diverging section. An expansion factor Y is introduced to give a better representation of the velocity.

For liquids, expansion factor (Y) is 1.

For gases, expansion factor (Y), can be obtained using Figure 4.1 (Crane Fig. A-20). Figure 4.1 presents Y for various values of (1 - r)/k and β², where r is defined as

This equation can be rearranged to the following form:

Nozzles: If the flow of a gas is occurring under critical conditions, then pressure ratio (r) values for compressible flow through nozzles and venturi tubes is given in Figure 4.2 (Crane A-21). Pressure ratio r is plotted as a function of k and b. Now values of r and β can be utilized in Figure 4.1 (Crane A-20) to obtain expansion factor (Y).

Orifice Meters: The velocity through an orifice can be computed knowing the value of orifice coefficient C_V, which is a function or Reynolds number. The expansion factor for liquids is 1.0, whereas for gases it is found from the following equation.

Assume velocity of the fluid in the pipe (V). Calculate Reynolds number (Re) Read friction factor (f) from Figure 4.3 Calculate frictional loss (h_f) using Equation 4.12Determine pressure drop (ΔP) using Bernoulli equation. Compare calculated pressure drop with the given value. Repeat the steps till an agreement is achieved. We recommend that you use the following steps: Use Bernoulli equation to obtain frictional head (h_f) Calculate the value of a group ReÖf given as

Read friction factor (f) from Figure 4.4. Calculate Reynolds number (Re). Calculate velocity of the fluid (V) using Equation 4.12.

Case 2: Pipe diameter unknown for given flow rate and pressure drop:

Assume a value of friction factor (say 0.005).

Estimate the diameter of the pipe (d) using the following equation.

Calculate Reynolds number (Re).

Read friction factor using Figure 4.3

Compare the revised and assumed friction factor values

Repeat the steps till convergence for d is achieved.

Case 3: Relative roughness of the pipe unknown

Frictional Losses through Fittings and Valves: These losses are reported as resistance coefficient K, where K is the number of the velocity heads that is lost because of the fittings or an obstruction. These values are independent of friction factor. Fitting friction loss h_ff is given as

And sum of form friction loss and skin friction loss is designated as friction loss h_f given as

Bernoulli equation for incompressible fluid is:

which can be written as

The equivalent length of a fitting or an obstruction is the length of the pipe that offers the same pressure drop due to friction. It is usually expressed in terms of the equivalent pipe diameters L_e/d.

There are minor losses that include contraction, entrance, enlargement and exit losses. Sudden contraction loss coefficient for turbulent flow conditions is given as a function of β² where β is the ratio of d₂ to d₁.

The frictional loss for a sudden enlargement with turbulent flow, is given by Borda-Carnot Equation as

where V₁ is the velocity in smaller duct. The expansion loss coefficient, K_e, is shown as follows

Ideal pump work, called water horsepower, can be calculated as follows:

If the overall efficiency of the pump is η, then the actual power requirement, brake horsepower (BHP), can be calculated as follows:

For calculating P_b, we can utilize Newton's method. The nonlinear function for a circular cross section is

And the derivative is given by the following function

When pressure drop is small, then density of the gas can be calculated at an average conditions and the pressure drop can be found from the equation used for liquids.

For compressible fluids, Reynolds number is calculated by utilizing mass velocity, G. So mass velocity is

The Reynolds number is found to be