# NEW EQUATION FOR FRICTION-FACTOR APPROXIMATION DEVELOPED

April 4, 1994
Brian F. Towler, Timothy L. Pope University of Wyoming Laramie A new equation for approximating the Moody friction factor in gas flow calculations includes gas viscosity and gives a better approximation for a wide range of Reynolds numbers than four existing, more widely known equations. All of these latter four discount effects of gas viscosity. Only the IGT equation (Institute of Gas Technology, Chicago) applies to a wide range of flow velocities (Reynolds number = Nre).
Brian F. Towler, Timothy L. Pope
University of Wyoming
Laramie

A new equation for approximating the Moody friction factor in gas flow calculations includes gas viscosity and gives a better approximation for a wide range of Reynolds numbers than four existing, more widely known equations.

All of these latter four discount effects of gas viscosity. Only the IGT equation (Institute of Gas Technology, Chicago) applies to a wide range of flow velocities (Reynolds number = Nre).

In comparisons of the four equations among themselves as well as with the newly developed equation, the Weymouth approximation was shown to be a poor estimator of friction for most flow conditions and should not be used.

Viscosity is a laminar phenomenon, and it is often stated that in turbulent flow, the flow rate and pressure drop are independent of viscosity.

This new equation shows that the viscosity dependence is indeed small and that if the practicing engineer assumes a constant viscosity for turbulent gas pipeline flow, the error involved would be small.

The new equation shows, however, the specific dependence of viscosity in turbulent flow.

### MOODY FRICTION FACTOR

The relationship between pressure and flowrate in pipelines is given by the well known Moody Equations Equation I (equations box) is the most general form of the equation when applied to gas flow.

The Moody friction factor which is an integral part of this equation, is a highly nonlinear function which must be read from a chart or determined iteratively from a nonlinear equation.

Approximations to the Moody friction factor have been widely used because they allow the gas-pipe flow equation to be solved directly instead of iteratively. The most widely published friction-factor approximations are the Weymouth, Panhandle A, Panhandle B, and IGT equations.12

The Weymouth equation approximates the Moody friction factor by Equation 2, and the remaining three equations approximate the friction factor by Equation 3 in which e and g are constants.

These constants are the following: for Weymouth, e = 0.032, g = 1/3; for Panhandle A, e = 0.085, g = 0.147; for Panhandle B, e = 0.015, g = 0.0392; and for IGT, e = 0.187, g = 0.20.

In the approximations involving the Reynolds number, the Reynolds number can be given by Equation 4.

These approximations are substituted into the flow equation for f, and the equation is then solved for the flow rate to yield a unique gas-flow equation in Equation 5.

In Equation 5, the values al through a5 are constants that are functions of the friction-factor approximations and the gas-flow equation. These constants are given in Table 1 for which q = cfd (measured at Pb and Tb); T = R.; P psia; L = miles; and d in.

The efficiency factor (E) generally varies between 0.7 and 0.92, depending on the pipe condition.

### COMPARISONS

The research reported on here evaluated the accuracy of the previously mentioned friction-factor approximations and their respective ranges of applicability. It also investigated the validity of the derivation of the specific pipe-flow equation (Equation 5).

The studies yielded an improved friction-factor approximation (that is, the new equation) that will be discussed presently.

The range of applicability for the friction-factor approximations will be defined for this discussion as an error of +-0.002 from the Moody smooth-pipe friction factor (f).

The friction-factor approximations were evaluated by comparing them with the Moody friction factor for smooth pipe flow. The Reynolds number was varied from 10 4 to 10 8.

An equation was used to generate the Moody smooth-pipe friction factor, which is the benchmark for comparing the approximations. This equation, which was presented by Seghides in 1984, has an average deviation factor of 0.0002% and a maximum deviation of 0.0023%.3

For the Weymouth equation, a pipe diameter of 18 in. was used. The four friction-factor approximations and the Moody smooth-pipe friction factor are shown in Fig. 1.

Because the Weymouth equation is a function of the pipe diameter only, it produced a straight line on the Moody diagram (Fig. I).

As an example, a diameter of 18 in. was used. This corresponds to a Moody friction factor of approximately 0.012 because the Weymouth equation is a straight line on the Moody diagram and the range of applicability would be 3.5 x 10 5 - 10 6 Nre for smooth pipe of this diameter.

It could also be applied to Reynolds numbers of 10 7 to 10 8 for pipe with a relative roughness (E/d) of approximately 0.00005. The range of applicability, therefore, is very narrow and depends mainly on the choice of pipe diameter.

There seems to be little physical basis for the approximation, and it is usually inaccurate. Whether smooth or rough pipes are used, there is no correlation to the Moody friction factor except in the small range where the two curves meet, which is at a Reynolds number of approximately 8 x 10 5 for smooth pipe of this pipe diameter.

The residuals (differences between the friction factor determined by the Moody equation and that determined by the four major equations) are shown in Fig. 2. Those for the Weymouth equation are shown in Fig. 2a.

The Panhandle A equation's range of applicability is for 1.1 x 101-108 Nre. Over the total range investigated, the maximum deviation was 0.0089.

The plot of residuals in Fig. 2b shows that this is a good approximation at high Reynolds numbers.

The range of applicability for the Panhandle B equation is 3 X 10 6 - 10 8 Nre. The maximum deviation over the experimental limits was 0.02.

A plot of residuals (Fig. 2c) shows these results.

The IGT equation is by far the most accurate of the previously published friction-factor approximations. The applicability range for the IGT is 7.5 x 10 3 - 10 8 NRe. The maximum deviation over the entire tested range was 0.0015.

The plot of residuals for IGT is shown in Fig. 2d; Fig. 3 shows a comparison of the residuals for all four published equations.

### MISSING EFFECT

Combining these friction-factor approximations with Equation 1, the general flow equation, yields specific flow equations.

The previously published specific equations are of the form of Equation 5. This derivation seems to have left out the effect of the gas viscosity.

The discussion will present a new form of Equation 5 which includes the viscosity as well as other minor changes. As will be seen, the rate does not strongly depend on the viscosity at high Reynolds numbers, but we include it to complete the functional form.

The new equation is Equation 6.

As can be seen by Equation 6, the exponent on the base temperature (Tb) and base pressure (Pb) term should be 1.0. Also, the average gas viscosity (@g) in centipoise (cp) is included with an exponent (a5).

All of the units used in Equation 6 are identical to those in Equation 5. The constants al through a5 for the proposed equation are shown in Table 2.

As can be seen, the exponent for viscosity (a5) is quite small which is why it was previously neglected. This is because viscosity is primarily a laminar phenomenon and gas pipelines are usually operated in the turbulent region.

It should be included, however, so that its effect can be examined.

### NEW EQUATION

Although the IGT approximation has a wide range of applicability, there is room for improvement. Consequently, a new friction-factor approximation was investigated.

The new friction factor was of the form of Equation 3. A regression analysis was carried out on the Moody smooth pipe friction-factor data for 10 4 - 10 8 Nre and Equation 7 was obtained.

Fig. 4a shows the residuals between Moody and the new equation, and Fig. 4 shows a comparison with the residuals of these residuals and the IGT equation.

As can be seen, the difference between the two equations is quite small. The IGT equation is slightly closer to the Moody friction factor for 10 4 - 2.2 @ 10 4 Nre and also 2 x 10 5 - 1.8 x 10 6 Nre.

For all other Reynolds numbers investigated, the new equation was superior.

When the new equation is substituted into the general flow equation, a new specific equation results.

The new constants for Equation 6 are shown in Table 3 along with the IGT constants.

### REFERENCES

1. Beggs, H. Dale, Gas production Operations, Tulsa, OGCI, 1984, pp. 100-10.

2. Kennedy, John L., Oil and Gas Pipeline Fundamentals, Tulsa, PennWell Publishing Co., 1994, pp. 76-79.

Serghides, C. K., "Estimate Friction Factors Accurately," Chemical Engineering, Mar. 5, 1984, pp. 63-64.