Santanu Barua, Yugdutt Sharma, Mark G. Brosius

Simulation Sciences Inc.

Brea, Calif.

A method for calculating accelerational pressure drop in two-phase compressible gas-liquid flow has been developed.

Knowing this pressure drop is important in the design and analysis of high-velocity compressible flow systems such as flare networks where critical two-phase flow may occur.

This article sets out a simple relationship between critical mass flux and accelerational pressure drop for single phase and two-phase compressible flow in pipes.

Based on this relationship and on the Moody model for critical flow, the new method for calculating acceleration pressure drop was developed.

Previous pressure-drop methods were developed for relatively low acceleration flows. Their applicability to high acceleration flows has never been thoroughly investigated.

Indeed, the connection between critical mass flux and acceleration pressure drop has never been established. Therefore, results for high acceleration flows were inadequately interpreted.

The relationship proposed here enables the calculation of acceleration pressure drop consistent with physical phenomena observed at critical flow. This makes the new approach applicable to the entire range of velocities encountered in pipe flow (low velocity to critical velocity).

Also, current and future acceleration pressure-drop methods will be more amenable to physical interpretation and detailed analysis.

Acceleration pressure drops generated by the method proposed here were compared with those obtained with the widely used Beggs and Brill correlation. The proposed method predicted lower acceleration pressure drops than the Beggs and Brill method.

### GRADIENT ANALYSIS

Essential in the design of two-phase flow is the calculation of total pressure gradient.

This consists of terms which evaluate the gravitational, frictional, and accelerational components and is expressed in Equation 1 (equations box).

ManN, applications in two-phase gas-liquid transport occur at conditions at which the kinetic-energy effect (acceleration term) is negligible.

In these cases, the frictional and gravitational components are the terms computed to evaluate the total pressure gradient, consequently the pressure drop.

There are situations, however, in which the acceleration term is significant. In these cases, it is necessary accurately to determine what it contributes to the total pressure drop. An example situation is offered by the design of low-pressure, high-velocity systems such as flare networks.

In such systems, more than 50% of the pressure drop resulting from acceleration is not uncommon.

Critical two-phase flow may occur in parts of the flare network. Proper sizing of the piping can occur only when all the terms of Equation 1 are rigorously determined.

The method presented here evaluates accelerational pressure drop in two-phase flow. The derivation is based on fundamental principles.

Brief reviews of critical flow phenomena and acceleration pressure-drop calculation methods precede presentation of the method.

### CRITICAL FLOW

Consider a compressible fluid flowing in a horizontal pipe for a fixed inlet pressure.

As the outlet pressure decreases, the flow rate increases. If the pressure falls below a certain level, however, called the "critical pressure," the flow rate does not change.

This is called "choked flow" and this flow rate, the "critical flow rate."

Further decrease of the outlet pressure will only introduce a pressure discontinuity at the point where the critical flow occurs, resulting in an infinite pressure gradient. This can occur only at the outlet for a uniform pipe. Fig. 1 shows a plot of pressure vs. distance.

At outlet pressures below the critical value, pressure change in downstream conditions will not affect the flow rate or the upstream pressure.

For single-phase isentropic gas flow, the critical flow velocity and sonic velocity are the same. The critical-flow velocity is given by Equation 2; the critical mass flux, by Equation 3.

For two-phase flow, critical-flow phenomena are more complex.

Critical-flow velocity and sonic velocity are not the same. In addition, critical two-phase flow occurs at velocities slower than the critical velocity of either phase for a given pressure and temperature (Fig. 2).

The minimum occurs at a liquid holdup of about 0.5.

The accelerational pressure gradient is commonly defined by the relationship shown in Equation 4. The total pressure gradient can then be expressed as in Equation 5. Several investigators have proposed relationships for evaluating the factor Ek as defined by Equation 4. Eaton1 suggested the relationship shown in Equation 6. Brill and Beggs' use Equation 7.

Dukler1 proposed the relationship given by Equation 9; Guzhov1 suggested Equation 10.

The term pn (Equations 10-12) is the no-slip density evaluated by a relationship similar to Equation 8 with HL replaced by lambda-L.

It should be noted that all the expressions presented in this section to evaluate the factor Ek show a dependence on the liquid holdup.

### PROPOSED METHOD

Lahey and Moody2 presented Equation 13 for calculating the total pressure gradient.

Assuming a constant cross sectional area for flow (dA/dZ = 0), and noting the relationships evident in Equations 14 and 15, allows Equation 13 to be simplified to yield Equation 16.

At critical flow, (dP/dZ)t is infinite resulting in a pressure discontinuity at the point where critical flow begins. The denominator in Equation 16 goes to zero which yields the condition shown in either Equation 17 or Equation 18.

Substituting Equation 18 into Equation 16 yields Equation 19.

Comparing term-for-term Equation 19 with Equation 5 shows that Ek may be expressed as in Equation 20.

The accelerational pressure drop is evaluated with Equation 21.

Evaluating the total pressure gradient by this method requires that the critical mass flux of the two-phase mixture be determined. In this work, the Moody pulse model was used to calculate the critical mass flux.

Moody3 examined the movement of a pressure pulse in an adiabatic uniform pipe.

The temperature of the vapor phase was assumed to be equal to the temperature of the liquid phase. Also, the phase entropy change was assumed to be zero.

The changes in properties across the pressure pulse were determined to be isentropic with these assumptions. The relationship expressed in Equation 22 or Equations 23 and 24 was derived to describe critical flow rate.

It should be noted that Equations 18 and 24 are identical for isentropic flow and arrived at independently.

The critical mass flux of the two-phase flow was derived as a function of flow pattern. For a separated flow structure, it is calculated by Equation 25.

The expression for an homogeneous two-phase flow was evaluated to be as shown in Equation 26. And the adiabatic compressibility of the two-phase mixture is defined by Equation 27.

The adiabatic compressibility of the vapor phase is defined by Equation 28; the adiabatic compressibility of the liquid phase, by Equation 29.

The adiabatic compressibilities of the vapor and liquid phases can be calculated from the thermodynamic relationship given by Equations 30 and 31.

It should be noted that Cp, Cv, and kT are measurable quantities or can be easily calculated.

### COMPARATIVE METHODS

The method to calculate accelerational pressure gradients presented here was used to compute the values for the factor Ek for different values of no-slip liquid holdup (HLNS).

Because the Beggs and Brill method (Equation 7) is one of the most common methods used to evaluate the accelerational pressure-drop component in two-phase flows, it was also used to compute the values of Ek for the same conditions to afford a comparison.

The Taitel and Dukler flow-pattern map4 was used to predict the flow structure, while the Beggs and Brill holdup correlation was used to calculate the slip holdup.

The two-phase mixture was modeled compositionally; the composition appears in Table 1. The phase no-slip holdup was controlled by varying the temperature of the fluid.

Plots of Ek vs. G (mixture mass flux) were generated for the two methods for the liquid holdup range of 0.19 to 0.86. The plots show that the proposed method predicts a lower Ek factor and hence a lower acceleration pressure drop for the same flow conditions.

Note that at choked flow, Ek = 1.0. This implies that the proposed method predicts choked flow at higher mass fluxes than the Beggs and Brill method.

It is interesting to investigate the two methods' predictions for single-phase gas flow.

In this case a = 1.0, Vm VSG = VG, and ptp = pG. These conditions and the fact that at choked flow Ek = 1 transform Equations 7 and 20, respectively, to Beggs and Brill (Equation 32) and the Moody model (Equation 33).

The expression from the Moody model is applicable for adiabatic flow and polytropic flow (heating or cooling).5 The Beggs and Brill method is applicable for isothermal flow only.

The critical velocity calculated from the Beggs and Brill method is lower by a factor of K when compared to the Moody pulse model.

Note that for gas flow at high velocities where fluid temperature changes, the Moody pulse model should be more applicable.

The trends shown in Fig. 3 are consistent with the simpler case of single-phase gas flow. The proposed method predicts a lower acceleration pressure drop compared to the Beggs and Brill method for both single and multiphase flow.

The proposed method is based on physical and mechanical properties of the phases such as isentropic compressibility and ratio of specific-heat capacities (Cp/Cv).

Because these are functions of pressure, temperature, and composition of the fluid, this method should be consistent over a large range of pressure, temperature, and flow conditions.

### IMPLICATIONS

Some of the major conclusions from the development of this method and its comparison with the Beggs and Brill method are the following:

- A relationship between acceleration pressure drop and critical mass flux has been established for single and two-phase compressible flow. The form of the equation is consistent with what is experimentally observed at critical flow.
- The Moody pulse model was used to predict the critical mass flux for a given pressure and temperature. An acceleration factor was then calculated with this critical mass flux and used to calculate the two-phase acceleration pressure drop.
When compared to the Beggs and Brill method, the proposed method predicted lower acceleration pressure drops for the range of variables studied.

- A rigorous analysis for single-phase gas flow indicated that the proposed method behaved consistently with observed and expected critical-flow phenomena. The form of the equation suggests that the engineer can try other two-phase critical-flow models to calculate acceleration pressure drop.
Conversely, existing acceleration pressure-drop methods can be linked to predict critical mass flux. This will probably allow a more thorough and meaningful analysis of their behavior at velocities tending towards critical flow.

- The Beggs and Brill method reduces to the case of isothermal choked flow for single-phase gas. If adiabatic flow occurs or flow with cooling or heating, the Beggs and Brill method will overpredict the pressure drop.
In these cases the proposed method should be more applicable.

- The Moody pulse critical-flow model predicts the general trends we see in two-phase critical flow. This model in conjunction with the proposed acceleration pressure-drop equation at the very least should correctly model the phenomenological trends.

### ACKNOWLEDGMENTS

The authors would like to thank the management of Simulation Sciences Inc. for permission to present this work.

### REFERENCES

- Brill, J.P., and Beggs, H.D., Two Phase Flow in Pipes, University of Tulsa, Tulsa, Okla., 1979.
- Lahey Jr., R.T., and Moody, F.J., The Thermal-Hydraulics of a Boiling Water Nuclear Reactor, American Nuclear Society, La Grange Park, 1984, pp. 356-358.
- Moody, F.J., "A Pressure Pulse Model for Two-Phase Critical Flow and Sonic Velocity," Journal of Heat Transfer, Trans. ASME, Vol. 91, 1969, pp. 371-384.
- Taitel, Y., and Dukler, A.E., "A Model for Predicting Flow Regime Transitions in Horizontal and Near Horizontal Gas-Liquid Flow," AlChE journal, Vol. 22, No. 1, January 1976, pp. 47-55.
- Hughes, W.F., and Brighton, J.A., Fluid Dynamics-Schaum's Outline Series, New York: McGraw-Hill Inc., 1967, pp. 138-144.

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