INTRODUCTION

This subject has already been discussed in the Paper, Driskell. We will give a more fundamental derivation OF THE FUNDAMENTAL EQUATION for compressible fluid flow through venturis based on that of McCabe and Smith, Unit Operations of Chemical Engineering, Second Edition. We will compare any derived equation with that given in Perry, Chemical Engineers' Handbook, Sixth Edition.

What we will discussing here is the Y factor, which originally was a correction to the incompressible fluid formula for flow through venturis. This correction factor makes the incompressible fluid formula suitable for use in compressible fluid estimations. The Y factor concept has since been expanded to be used for flow through fittings of any nature and through valves.

INCOMPRESSIBLE FLUIDS (LIQUIDS)

McCabe and Smith give an equation (8-39) for incompressible fluid flow through venturis as found here in equation [1].

[1]

In equation [1], the mass flow rate is given as a function of the venturi correction for irreversibilities, c_v, the area at the throat, S_b, what is called the 'velocity of approach' (who knows?), 1/[1 - Ò⁴]^1/2, and the square root of twice the dimensional coefficient times the difference between the upstream and the throat pressures times the density of the incompressible fluid. The units are conventional American units – feet, lb_f/ft² and lb_m/ft³. Beta is the ratio of the throat to the upstream diameters. The dimensional coefficient, g_c, has units of ft-lb_m/lb_f-s².

Equation [1] is usually used with fixed upstream conditions, which fix P_a, fixed dimensions and specifications for the venturi, which fix c_v, S_b and Ò and with known fluid characteristics, which fix the density. The principle variable in equation [1] becomes the throat pressure P_b. A legitimate question is, what are the limits on equation [1]?. Under the constraints just mentioned, if the downstream pressure is equal to upstream pressure, there is no flow. This gives the upper pressure limit on the equation. Is there a limit on the reduction of downstream pressure? If the down stream pressure is lowered below the vapour pressure of the fluid, the liquid flashes to vapour and equation [1] no longer applies. Thus the lower pressure limit on the equation is the vapour pressure of the liquid at its temperature at the throat.

COMPRESSIBLE FLUIDS (GASES AND VAPOURS)

Since equation [1] is well proven for incompressible fluids, it was modified to include compressible fluids by the addition of the dimensionless Y factor (McCabe and Smith, 8-47).

[2]

The factor, Y, permits mass flow to be defined in terms of the upstream density. The factor, Y, is a correction of upstream density to the density at the section at which the downstream pressure is measured, the throat. This being the case, Y = (á_b/á_a)^1/2.

The upper pressure limit is the same as for incompressible fluids – when there is no pressure difference across the taps. The lower pressure limit is found when flow 'chokes' (not when it reaches sonic velocity as is commonly stated). Outside these limits, equation [2] does not hold.

MASS FLUX VERSUS MASS FLOW

The mass flux concept is useful when dealing with compressible fluids, in particular when the section of the conduit does not change. We introduce it in equation [3]. The units of mass flux are lb_m/s-ft² in the customary American system. The subscript b on G is kept because this is a constrained equation – mass flux is a function of area.

[3]

Equation [3] gives us a means of measuring the value of Y in a venturi. It is seen that the expansion factor can be measured if one knows the mass flux at the throat, the beta ratio, the venturi coefficient, the pressure difference across the venturi taps and the density at the upstream tap. The expansion factor so measured is normally correlated with upstream fluid conditions, with beta and with pressure drop to arrive at useful graphs and tables. The graphs and tables are so useful that frequently many of the constraints are forgotten.

[4]

Equation [4] is constrained to changes between the upstream measurement section, subscript a, and the throat of a venturi, subscript b.

DERIVATION FOR FLOW OF IDEAL GASES THROUGH IDEAL VENTURIS

We have used the word 'ideal' twice in the above heading to remind the reader of the constraints placed upon the equations that follow. These constraints are often glossed over, but should not be.

In order to approach ideality, it is necessary to take several precautions with the experimental, test setup. The venturi has to be carefully fabricated to recognised specifications on dimensions and finish. Any flow control valve has to be sufficiently far downstream (about 50 pipe diameters) so as not to disturb the flow profiles at the venturi. Only after all these precautions are taken do the following equations become applicable.

For isentropic, adiabatic processes of ideal gases, there are three relationships given in [5].

[5]

The terms to the left of the first two equations of the set [5] are the result of isentropic, adiabatic processes starting from the conditions subscripted with 1 on the right. The third equation is a differential equation representing changes through an horizontal venturi acting isentropically.

The third equation comes from an overall energy balance, including irreversibilities, constrained to eliminate elevation changes, profile changes and the irreversibilities. We give the source equation as equation [6]. Equation [6] is the Bernoulli equation, including irreversibilities and corrections for differences in flow profiles.

[6]

In equation [6], each term except the last represents energy per unit mass at sections 1 or 2. The last term represents the conversion of mechanical energy per unit mass flowing between sections 1 and 2 to thermal energy. The negative sign reflects the fact that this conversion is a 'loss' of mechanical energy (it actually is a conversion of mechanical energy to thermal energy). The alpha terms are corrections for the fact that the velocity profiles at sections 1 and 2 are not flat and that U is an average velocity across a supposedly flat profile.

The author cannot emphasise enough that equation [6] is an overall energy balance consisting of mechanical energy terms and one thermal energy term. Total energy is conserved; only mechanical energy is 'lost' or converted to thermal energy.

If the venturi is horizontal, the Z terms cancel. If the flow is ideal, the h_f is zero. If the profile is flat, the alpha terms equal one. (These are many of the assumptions that are usually made.)
If we differentiate, the density is constant over the differential length and we can state all of the preceding assumptions as in [7]. The final equation of [7] is identical to the final one of [5].

[7]

Using the first relationship in [5], we can develop an equation for isentropic, adiabatic flow through an ideal venturi. The constraints of adiabatic and isentropic flow apply to the first two equations of the set [5]. The assumption of an ideal (perfect) venturi permits us to treat the venturi coefficient, c_v, as one.

We can manipulate the first equation of [5] as in [8].

[8]

Substituting the last equation of [8] into the differential equation of [5] and integrating between limits, we obtain the development in [9].

[9]

We can multiply and divide the first term by P₁^1-1/Ó and change the signs as in [10]. The ratio of the two pressures is then replaced by the symbol, r.

[10]

The variable r refers to the ratio of P₂ to P₁. We can evaluate U₂² – U₁² as follows in [11].

[11]

In [11], we have replaced average velocity squared (at section 2) with the mass flow rate divided by the area and density at the same section (all squared). Mass flow rate divided by area equals mass flux.

Equation [10] can now be written as in [12].

[12]

We can now substitute in equation [4] the relationship just developed and we can further manipulate it to replace the downstream density with downstream pressure. The pressure ratio has been replaced by the symbol r.

[13]

[14]

The expansion factor, Y, which is a correction applied to the upstream density, is seen to be a function of the physical dimensions of the venturi through beta, of the beta coefficient, of the fluid properties, through gamma, and of the pressure ratio at the section of interest (section pressure to reservoir pressure, r).

Equation [14], which we have just derived, is identical to equation (5-15) in the Sixth Edition of Perry with the exception of the venturi coefficient, c_v. Our derivation has given us the opportunity to examine the constraints and assumptions that are normally made. The most obvious ones are those involving ideal gases flowing through ideal venturis undergoing ideal processes (adiabatic, isentropic).

GRAPHING THE STRAIGHT LINE RELATIONSHIPS OF THE EXPANSION FACTOR

Note that neither Perry nor McCabe and Smith use the venturi coefficient when plotting the Y factor. This means that they assume no irreversibilities. Their curves are theoretical, not experimental. One means of establishing the Y versus r or ÇP/P₁ relationship for real gases would be to make use of the projection technique described in Flow of Industrial Fluids by R. Mulley to estimate upstream and downstream densities and pressures and then perform correlations on the (corrected) data. The nozzle used would be the upper half of a venturi and the downstream pipe would be the same diameter as the throat. The factor Y would simply be the square root of the ratio of the two densities.

Alternatively, if the upstream pressure were kept low, so that the gas could be considered ideal, equation [14] can be used with c_v estimated for the venturi. In this case, the low downstream pressure associated with the choking phenomenon would have to be estimated independently.

There are several methods of graphing the relationships of the Y factor on the ordinate with whatever is used on the abscissa. These methods will now be described.

McCabe and Smith use the downstream to the upstream pressure ratio on the abscissa. The slope is negative, but the values on the abscissa decrease from 1.00 to 0.60. Beta is used as parameter. Four values of beta are shown for orifice plates with flange taps or vena contracta taps and three values of beta are shown for venturis – seven straight lines in all. Little is said about the limits. The slopes of the orifices, the more irreversible devices, are less than those of the venturis. Increasing values of beta increase the slopes.

Perry uses values of (1-r)/Ó for the values on the abscissa. The values increase from 0.00 to 0.16. Four values of beta serve as parameters for orifices and five values are used for venturis. Again, the slopes for the orifices are less than those of the venturis and increasing values of beta increase the slopes.

Crane, in Technical Paper No. 410, 17^th Printing, has four separate graphs for the expansion factor. There are two graphs of compressible flow through nozzles (and venturis) and orifices and two graphs for pipe expanders. Each of the pairs of graphs is for a different value of gamma, 1.3 and 1.4. The abscissa is the ratio of the pressure difference to the upstream pressure and the values increase from 0.0 to 0.6 for nozzles, venturis and orifices and from 0.0 to 1.0 in the case of the expanders. The slopes of the curves are negative. Crane does specify a limiting factor for what is called 'sonic' velocity (choked flow). It is interesting to see that Crane has enlarged the concept of the net expansion factor to include piping enlargements.

Les Driskell, Control-Valve Selection and Sizing, ISA, shows a series of curves showing correlations based on test data for different types of valves. The values on the abscissa are x, which is the ratio of the pressure drop to the upstream pressure (the same as Crane). He is rather adamant in his contention that the lower limit on Y is 0.667, in all cases, which is a value arrived at by calculus. He states that experimental data conform to straight lines within about 2.0%. He uses a different value of the limiting value of x, x_T, for each different type of valve, always corresponding to the limiting value of Y of 0.667.

In analysing the difference between Driskell's statement and the plotted results of Crane, we come down on the side of Crane. Driskell's results stem from theoretical analysis of ideal (perfect) gases not corrected against reality. Use of the real gas, choked flow equations shows that there is a different Y_c-r_c pair for each gas or vapour.

As we have pointed out in Flow of Industrial Fluids, it is not necessary to use the ideal gas law for other than exploration and the concept of 'sonic' velocity should be replaced by the concept of choked flow. In this paper, we limit ourselves to showing how close the last equation of the set [14] conforms to a straight line. We make use of the fact that air at low pressures conforms fairly closely to ideal gas behaviour in order to establish the 'critical' pressure ratio as a starting point. Equation [14] will then be used to establish Y_c.

The ratio of specific heats, gamma, we take as 1.4. We arbitrarily set beta at 0.5.

The results of starting the calculations at the choke point and calculating another nine intervals to the no-flow condition follow.

TABLE OF Y VERSUS r VALUES
P2	G	BETA
2.16E+03	9.06E+01	5.00E-01
RHO2	GAMMA	rc
8.43E-02	1.40E+00	5.28E-01
P1
4.09E+03
	r	Y
1	5.28E-01	6.91E-01
2	5.75E-01	7.26E-01
3	6.22E-01	7.60E-01
4	6.70E-01	7.93E-01
5	7.17E-01	8.25E-01
6	7.64E-01	8.56E-01
7	8.11E-01	8.86E-01
8	8.58E-01	9.15E-01
9	9.06E-01	9.44E-01
10	9.53E-01	9.72E-01

The correlation coefficient for the above two sets of numbers was 0.999 which demonstrates an almost perfect fit to a straight line.

CONCLUSIONS

1. The expansion factor, although based on ideal gases, is a useful concept because charts such as those of Crane are plentiful and have been proven by generations of engineers.

2. The Y factor does not stop at 0.667 as Driskell maintained, but varies around this number.

3. When deciding on whether or not to trust data derived from the use of the expansion factor, first decide on how closely the gas or vapour conforms to ideal gas behaviour. Gases or vapours such as air, nitrogen, oxygen and steam conform well. CO₂, an acid gas, does not. It all depends on the interaction between molecules.

4. The straight line relationship shown by the charts in Crane, etc., can be misleading. Computations beyond the choke point are meaningless.