Khalid Aziz, Liang-Biao OuyangStanford University

Stanford, Calif.

A newly developed form of the gas flow equation for predicting flow rate or pressure drop along pipes employs fewer assumptions and applies to a wider range of conditions than previous methods.

Both a general integrated and a simplified form of the equation are presented here.

### NEED FOP ACCURACY

Natural gas continues to assume a greater role in worldwide energy supply. In the U.S., the value of natural gas produced in 1993 exceeded that of all crude oil produced (OGJ, Feb. 28,1994, p. 26).

Major projects are being planned to move massive amounts of gas over long distances through pipelines. Most of the gas used in the world is in fact transported through pipes (well bores, gathering systems, transmission lines, and distribution networks).

Optimum pipeline design requires accurate methods for predicting pressure loss for a given flow rate or predicting flow rate for a specified pressure loss. Steady-state pressure loss or flowrate calculation methods for single pipes are the most widely used and the most basic relationships in the engineering of gas-delivery systems.

They also form the basis of other more complex transient calculations and network design.

From the classical work of Weymouth 1 to the paper of Towler and Pope 2 engineers have presented practical methods for relating gas flow rate to pressure drop in pipes. Unfortunately, many of the publications on this subject, including that of Towler and Pope 2 are confusing and often contain serious errors.

Hence the practicing engineer, faced with selecting among dozens of available methods, has a difficult task. Presented here is a summary of a more detailed paper on the state of knowledge on single-phase, steady-state gas flow in pipes, and guidelines for the practical design of gas pipelines.

Fortunately, the problem of gas flow in pipes is well understood, and there is no reason for many of the assumptions or approximations often made by practicing engineers.

### MOMENTUM BALANCE

Development of gas-flow equations starts with momentum balance on a differential volume of gas.

The frictional term in this balance is expressed in terms of a friction factor. This factor is related to fluid-flow rate, fluid properties, and pipe characteristics through two dimensionless groups: Reynolds number and relative roughness.

This relationship is well established in graphical form as the Fanning Friction Factor Chart or in mathematical form as the Colebrook-White equation.

In principle, this relationship completely defines the problem, and equations can be numerically integrated to any desired degree of accuracy without assumptions.

While this procedure requires many computations, it is easy to implement on a personal computer and is the basis of many commercial programs.

This summarizes the solution of all pipeline gas-flow problems.

But because engineers are always looking for simple design methods that can be applied even without computers, many simplified equations continue to be applied.

All of them are derived from integrating the momentum-balance relation by employing one or more of the following simplifications:

- Assume that the pipe is smooth and then use one of many approximate equations for estimating smooth-pipe friction factor.

The effect of pipe roughness is then accounted for by introducing an efficiency factor. Often this factor is assumed to be constant but is in fact a strong function of both Reynolds number and relative roughness.

- Assume friction factor and fluid properties to be constant over the length of the pipe.

Because fluid properties depend on pressure and temperature, an iterative procedure is needed when pressure drop is to be calculated.

- Fix standard conditions and/or fluid properties. For example, in previously published simplified flow equations, such as Panhandle, IGT, Weymouth, the gas viscosity is fixed at about 0.0105 CP.
- Ignore or simplify kinetic energy and/or hydrostatic head terms in the equation.

### TWO EQUATIONS

Presented here are only two equations (accompanying box).

Equation 1,(37812 bytes) the most general form of the integrated flow equation, accounts for hydrostatic, frictional, and kinetic energy effects.

This equation can be solved iteratively with the friction factor calculated from the Colebrook-White equation that applies to both smooth and rough pipes over the full range of the Reynolds number.

If a simplified friction-factor relation is desired, it should be selected with care for the specific range of the Reynolds number and pipe roughness of interest.

Equation 2 (37812 bytes) is the new simplified form of flow equation from which one can derive most of the simplified equations discussed in the literature.

Equation 2 (37812 bytes) employs approximate explicit representations of the friction factor for smooth pipes and introduces the efficiency factor (E) to account for pipe roughness.

The constants (a1 to a5) depend on the approximation used for the friction factor; their values for commonly used relationships are given in Table 1.(25448 bytes)

(These values have been carefully checked and are correct, although they differ in some cases from the values reported in previous publications.)

Unfortunately none of the simple explicit relationships for friction factor is valid for all ranges of the Reynolds number.

The Blasius equation can be used for the Reynolds number range of 3,000 to 10 5, the modified 1/9th power law for 2 X 10 4 to 10 6, and the Panhandle equation for 2 x 10 6 to 10 8 (the latter being the range of Reynolds number encountered in most commercial gas pipelines).

The efficiency factor (E) corresponding to the Panhandle friction-factor relationship is shown in Fig. 1.(42561 bytes) Note that this efficiency factor is sensitive to both Reynolds number and pipe roughness and that the commonly used value of 0.92 can lead to large errors.

Two main recommendations are evident:

- The most general integrated form of the flow equation is Equation 1.(37812 bytes)

It should be used whenever numerical integration of the momentum balance along the pipe is not the option. We have tested this equation with field data with excellent results.

- If a simplified equation is to be used, the best solution is Equation 2,(37812 bytes) with an appropriate set of constants from Table 2 for the range of Reynolds numbers of interest and the correct value of efficiency factor E.

If Panhandle equation is applied, then the efficiency factor can be read directly from Fig. 1.(42561 bytes) Similar figures can be generated easily for other simplified equations.

### REFERENCES

- Weymouth, T. R., "Problems in Natural Gas Engineering," Transactions ASME, Vol. 34, 1913, pp. 185-231.
- Towler, B. F., and Pope, 'I'. L. "New Equation for Friction-Factor Approximation Developed," OGJ, Apr. 4,1994, pp. 55-58.
- Ouyang, L. B., and Aziz, K., "Steady-State Gas Flow in Pipes," 1995, accepted by J. Petroleum Science & Engineering.
- Gregory, G. A., Aziz, K., and Moore, R. G., "Computer Design of Dense Phase Pipelines," J. Petroleum Technology, Vol. 31 (I 979), No. 1, pp. 40-50.
- User Manual, ASA Systems (1995), Mississauga, Ont.

*Copyright 1995 Oil & Gas Journal. All Rights Reserved.*