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4.4. Separated Flow Model for Critical Flow

One of the key assumptions utilized in the past two models was that the velocity of each phase was equal. This assumption is known to be in error particularly under conditions where the pressure is low and the density ratio can be large. The concept of separated flow which was introduced for pressure drop can also be used here to develop a critical flow model. Let us consider the case where the phases are in thermodynamic equilibrium although there is slip between the phases. As stated previously, phase equilibrium is a good assumption when the phases are initially well-mixed or dispersed, or when the channel is long to allow equilibrium to develop.

Once again under isentropic flow conditions we can express the pressure drop as consisting of only acceleration effects (i.e. equ 4.9). In this case one must remember the relationship between density and specific volume contains the velocity ratio


where S is the ratio of tex2html_wrap_inline4813 . The expression for acceleration pressure drop (equ. 4.9) is in general


If we eliminate the void fraction, tex2html_wrap_inline4259 , from this expression using equ. 4.22 we arrive at the general expression for the critical mass velocity


One should note that for equal phase velocities, (S = 1) we simplify to the HEM expression. The flow quality at this "choking" condition can be found from the energy balance and equ. 4.22 as


Now if one can specify the critical slip ratio at the "choking" condition then one can use equ's 24 and 25 to find X and tex2html_wrap_inline4791 (2 equs and 2 unknowns).

Based on some theoretical arguments and empirical matching of experimental data there are two widely utilized models [8-10] for the critical velocity slip ratio, tex2html_wrap_inline4823 . Both models assume that the slip ratio at "choking" conditions is equal to the inverse density ratio to an exponent, m


where Fauske [8,9] assumes tex2html_wrap_inline4827 and Moody uses tex2html_wrap_inline4829 . The only real justification for either exponent is comparison to data and matching of the exponent and the critical pressure ratio. Over a wide range of data Fauske's empirical value shows somewhat better agreement. In addition the Fauske model has been solved parametrically and a convenient graphical solution is available (Figure 4.6 and 4.7) with the effect of geometry also empirically included. Therefore, we discuss this model results below, primarily as an example of this class of semi-empirical models.

Fauske Model

One interesting note should be made. Because in Fauske's model the exponent is empirically "fit" to data the isentropic flow assumption becomes somewhat of a moot point. One might consider that frictional effects, if important are indirectly accounted for in the empirical part of the model. However, this is only correct for similar L/D ratios. If one uses the model outside of its range of data then additional frictional effects may have to be included. As one will see in the graphical results this would be in the range of pipe length to diameter ratio of 15 or greater. Let us discuss the model results in general and then examine the effect of geometry.

Figure 4.6 is a plot of the choked mass velocity as a function of stagnation enthalpy at various critical pressures for steam-water mixtures. For any enthalpy and choked pressure one can immediately locate the choked mass velocity.

In order to calculate the critical pressure at the choked plane, it is necessary to refer to Figure 4.6 Here the choked pressure ratio is shown as a function of the L/D of the test section [9]. The choked flow is calculated using several different equations depending on the length to diameter ratio of the outlet pipe section.

  1. L/D - O - Sharp edged orifice. There is no choked flow. The following equation should be used just as it is shown below. This is essentially the orifice equation for single phase incompressible flow as an estimate for an orifice.


    where tex2html_wrap_inline4467 is the density of the initial mixture.

  2. O < L/D < 3 - Nozzles and short tubes. Use Equation 4.27 with tex2html_wrap_inline4841 from 4.6 in Region I replacing tex2html_wrap_inline4755 . There are substantial departures from equilibrium for these conditions. Perhaps the jet breaks free of the tube but the space between the tube and the jet is filled with vapor at a pressure higher than the back pressure.
  3. 3 < L/D < 12 - Middle length tubes. For this geometry no convenient prediction scheme exists as departures from equilibrium, which are neither negligible nor governing, exist. for these conditions replacing tex2html_wrap_inline4755 by tex2html_wrap_inline4841 in Region II from 4.6 overestimates the flow rate to some extent. Apparently, the jet is breaking up as equilibrium is being established.
  4. 12 < L/D < 40. The flow rate is almost independent of the tube length so the tex2html_wrap_inline4841 from 4.6 in Region III can be used to determine the critical flow rate from 4.7. Almost the whole pressure drop is due to momentum changes and wall friction is negligible.
  5. L/D < 40. Wall friction forces become increasingly important, but very gradually so, and the critical flow rate drops slowly. Here it is suggested that one can begin by guessing a critical mass velocity and going to Figure 4.7 to obtain the corresponding critical pressure. Figure 4.6 can be used to estimate the pressure 40 L/D's upstream from the discharge. From that point to the entrance, the appropriate single or two phase pressure drop can be estimated with existing correlations. If the calculated entrance pressure is the right one, the initial mass velocity assumption is correct. If not, try another assumption. Small changes in the assumed mass velocity will change the pressure drop appreciably and the calculation will coverage rapidly.

The experiments justifying the calculation method just described are performed on an apparatus such as illustrated in Figure 4.11. A typical pressure-length curve is shown. Data comparisons reported in References [8] and [12], among others, are shown in Figures 4.12 and 4.13 for low and high quality cases. The comparison between the Fauske prediction methods and the data in this quality range is good. As can be seen, the homogeneous model predicts poorly in the moderately low quality region but gives better predictions as the quality increases.

Similar experiments have been run on a long capillary with Freon 12 in and 18-foot tube .042 inches in diameter. The L/D is over 1000 in the two phase region II. The pressure and temperature trace is shown in Figure 4.14. The steeper pressure gradient in the two phase region, beyond 12.5 feet, is evident on this figure.

As mentioned in Reference [9], a number of other choked flow theories exist for separated flow, which do about as well with the data of these Figures (4.12-13) and as that of Fauske (e.g., Ref. 10). The details of the assumptions somewhat differ but the result is about the same. In fact one should realize that the agreement may be the result of compensating errors.

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Next: Fluids Other Than Water Up: SOUND SPEED AND CRITICAL Previous: Homogeneous Frozen Flow Model

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Last Modified: Tue Sep 2 15:06:55 CDT 1997