PURPOSE OF PAPER

During an attempt to make use of some concepts developed in the book ISA HANDBOOK OF CONTROL VALVES, especially those regarding turning flows through angle valves, the author noticed that Driskell and the other contributors had made extensive use of American mixed units in their developments. These notes will be an analysis, in terms of non hydraulic (thermodynamic) equations of Driskell's methods for estimating flow through valves. The analysis will attempt to use modern customary American terminology.

The Y factor, which allows the equations developed for the flow of incompressible fluids to be used for the flow of compressible ones, will be discussed.

Entropy increase across an obstruction, in this case an angle valve, also will be discussed.

INCOMPRESSIBLE FLUIDS

Driskell begins by stating that a fluid passing through a control valve obeys the usual laws of conservation of mass and energy as expressed by the equations of fluid mechanics. He starts by considering only liquids. When a liquid passes through a pipe to a narrower section of a valve, or a vena contracta of any kind, it must accelerate. He says the energy for this acceleration must come from the pressure or static head. In fact, the energy comes from differential static energy, (P₁ – P₂)/á. It is transformed into differential kinetic energy, (U₂² – U₁²)/2g_c, and thermal energy, following the conservation of energy principle.

Of particular interest for our purpose is Driskell's Figure 1 which shows an angle valve and an orifice plate together with a representative pressure-distance profile. The profile shows the large pressure drop at the vena contracta, the recovered pressure (difference between final pressure and vena contracta pressure) and overall pressure loss (overall pressure drop at point of complete recovery of the flow profile).

If the overall pressure loss were divided by density, the quotient would represent energy per pound-mass converted to thermal energy (internal energy plus energy as heat flow). The distance required for full recovery of the flow profile is indicated. This distance shows that pressure loss between upstream and downstream sections close to the obstruction generally is not representative of mechanical energy loss and that conversion to thermal energy (irreversibilities) continues some distance downstream of the valve.

The conversion of mechanical energy to thermal energy is in conformity with Joule's 'mechanical equivalent of heat'. The same law is seen when a body falls through a height and is suddenly stopped – the temperature rises due to the increase in thermal energy. The figure also helps explain the term 'losses' used so loosely in the field of fluid mechanics. If we concentrate on energy conversions instead of pressure differences, perhaps the phenomena will be more understandable. The 'losses' are nothing more than conversions of one form of energy to another.

Driskell starts his analysis by assuming steady-state flow, an incompressible fluid and ideality, with the ideal Bernoulli equation (no conversion of mechanical energy to thermal energy) and horizontal flow (no elevation difference).

[1]

Equation [1] is an ideal energy balance. Dividing both sides by twice the dimensional constant reveals that the change in kinetic energy is equal to the change in static (pressure) energy. The subscripts show that a gain in kinetic energy is equal to a loss in pressure energy. The kinetic energy correction factor, alpha, applied to the two velocity terms, is assumed equal to one, in each case, for simplicity.

Driskell describes the variable U as being the mean axial speed (velocity) across areas 1 and 2 respectively. The theoretical steady-state flow, assuming an incompressible fluid and a flat velocity profile, is given by Equation 2.

[2]

The last equation of set [2] is a steady-state equation. The units are q in feet cubed per second, U in feet per second, a in feet squared, P in pounds-force per foot-squared and rho in pounds-mass per foot-cubed. The term g_c is the dimensional constant with units of pounds-mass-feet per pound-force-second-squared.

The actual flow through a restriction is always less than the theoretical flow due to irreversibilities so an experimentally obtained coefficient, C, is introduced. This coefficient is called the discharge coefficient. The discharge coefficient, C, is normally combined with the denominator of the last equation of set [2] to result in the following equation:

[3]

The variable, q, is no longer theoretical, it is empirical, so the subscript, t, has been removed.

Another correction factor is needed for the fact that the downstream section for pressure measurement no longer coincides with the vena contracta. This correction factor conventionally is included in the denominator. The use of the factor, F_L, should drive home the fact that reference sections are sections 1 and 2, defined as one pipe diameter upstream and 40 diameters downstream.

[4]

Now equation [3] can be modified as follows:

[5]

The factor, 38.0, was derived by converting pressure from psfa to psia (1/144), using 32.17 for the value of g_c, 62.371 as the pounds-mass per foot cubed of 60 F water, extracting the square root, using 7.48 as the conversion factor for q in feet cubed per second to US gallons per second and, lastly, 60 seconds per minute then rounding off. G is the so-called specific gravity or the ratio of mass per unit volume of the fluid to that of water, both normally at 60 F.

The traditional equation is established by replacing the factors of the term before the term containing the square root by the valve coefficient, C_V.

[6]

The units of the traditional valve coefficient, C_V, are USgpm/psi^1/2.

This way of defining the equation for flow through a valve puts the onus on the valve manufacturer to perform tests on his product and to establish reliable correlations. Equation [6] is used as a starting point for the development of equations for compressible flow and for other factors that influence flow through a valve (reducers and expanders, etc.). Correction factors are introduced for deviations from the incompressible state and for reducers, expanders, etc.

COMPRESSIBLE FLUIDS AND THE Y FACTOR

Please note that in what follows, the Y factor is applied to control valves but is just as easily applied to venturis, orifices and fittings. Also note that the downstream pressure is measured at a location where it is thought that the velocity profile has re-established itself. The upstream pressure is not that at the face of the component, either, but is approximately one pipe diameter upstream. The approximate ratio 1:40 of upstream distance to downstream distance (the distances are specified for each tap location) is due to the fact that the smooth contraction upstream is almost isentropic, which accounts for the number 1, and the downstream section is not isentropic, which explains the number 40. Distance down the pipe is measured in pipe diameters.

The Y factor is a multiplying factor applied to the general equation for incompressible fluids usually transformed to a mass flow basis of kg/h or lb_m/h. When this transformation is made, the density appears in the numerator under the radical.

[7]

The units of w depend on the coefficient, N₆. Customary US units for mass flow are lb_m/h and N₆ is 63.3. Metric units usually give mass flow in kg/h and N₆ as 2.73. The coefficient, C_V, is the normal, liquid valve coefficient. Y is the expansion factor. The variable, x, is the ratio of the pressure drop to the upstream pressure, P₁. A subscript, T, on this variable indicates the terminal, limiting or choked value. The variable, r₁, is the upstream density.

Regarding the definition of x, it must be remembered that F_L corrects the pressure drop, P₂ – P₁, from the points of measurement (one diameter upstream and forty diameters downstream) to the vena contracta and that this factor is now included in the valve coefficient, C_v.

Experimentally, the Y factor, although used as a correction to the upstream density, is correlated to x and the ratio of specific heats.

[8]

The slope of Y versus x is set by the values of F_k, which depend on the fluid, and those of x_T, which depend on the valves and are the limiting values of x when Y reaches 0.667, according to Driskell, but there is some disagreement here (see the curves in Crane). We believe that we have resolved the disagreement in favour of Crane (see the paper Compressible Flow through Venturis by Mulley). In the meanwhile, for the purpose of this paper, consider the value of x_T to be that value of x that corresponds to the section at which Y reaches 0.667. This value serves to set the slope of the Y versus x curve . The limiting value of 0.667 is close, but not necessarily true. It was established theoretically using the calculus but was not corrected from experiment. Values for x_T, are given in Driskell's table of representative valve factors. They range from 0.8 for a flow-to-open restricted contour angle valve to 0.15 for a 2:1 tapered orifice flow-to-close angle valve. Under Driskell's hypothesis, this means that the variation of Y with the two types of angle valves operating on air under choked conditions at the valve outlet would be the same because the terminal value of x cancels but, if dry saturated steam between an inlet pressure of 1250 to 1400 psia were compared to air, the ratio of specific heats factor would be 0.88 and Y would be 0.621, a change of minus seven percent. This difference is sufficient to warrant the effort expended in the computations.

Mulley established a formula for Y as a function of K, g and the ratio DP/P₁ that reproduced fairly faithfully the values given by Crane.

[9]

Formula [9] has been used in computer simulations of compressible flow to represent expansion of gases across components of piping. The coefficient, K, is that established by Crane (incompressible fluids). It gives the starting point of the estimation.

ANALYSIS OF THE ROLES OF THE Y FACTOR AND F_L IN [7]

The various manipulations to simplify the steady-state flow equation [7] also serve to obscure certain facts.

1. Y = [á_vc/á₁]^1/2, therefore, the density under the radical, in the case of a valve, is the density at the vena contracta.

2. Because C_v contains F_L = [(P₁ – P₂)/( P₁ - P _vc)]^1/2 in the denominator, the driving pressure for flow through the valve is actually the delta P to the throat. This delta P cannot be measured, which is why the C_v is established by the vendor, by experiment.

3. x = (P₁ – P₂)/ P₁ so that the upstream pressure is cancelled under the radical, actually before F_L is applied.

THE HERSHEL VENTURI AND THE Y FACTOR

The Y factor grew out of studies on Hershel Venturis, flow nozzles and orifice plates. It is a coefficient that attempts to make the incompressible flow equations useable for compressible flows in flow metering. It has also proven useful in control valve sizing.

What is frequently glossed over is the fact, stated admirably by McCabe and Smith, Unit Operations, in their derivations, that the Y factor applies to ideal gases (those that can be described by the ideal gas 'law') flowing isentropically (ideally) and adiabatically through venturis. The Y factor, under these circumstances is almost a linear function of the pressure ratio (vena contracta to inlet). Based on the theoretical equation developed for venturis, an empirical equation was developed for sharp-edged orifice plates and another for valves and fittings.

Before giving the equations of the ASME Research Committee on Fluid Meters involving the Y factor, we will explore the approach taken by McCabe and Smith. The numbering of the equations will be that of second edition of their book, Unit Operations of Chemical Engineering. We will change the symbols to be those used by Mulley in Flow of Industrial Fluids.

The basic theoretical equation for velocity of an incompressible fluid through a venturi is given in [8-37].

[8-37]

A venturi coefficient is included to account for the small irreversibilities between section 1 at the entrance and section 2 at the throat. The alpha coefficients are the corrections for the velocity profiles at sections 1 and 2. It is to be noted that these sections no longer correspond to those used for valves. Section 1 is in the straight pipe at the entrance; section 2 is at the throat.

Equation [8-38] results when the empirical coefficient, C, is introduced in order to eliminate the alpha coefficients in the denominator of the venturi coefficient.

[8-38]

Practically, mass flow is of more interest than velocity, so equation [8-38] is transformed by multiplying it by the density of the incompressible fluid and by the throat area.

[8-39]

There is no subscript on density because, for incompressible fluids, it is constant.

For compressible fluids, a factor, Y, is introduced that, although outside the radical, acts upon the density under the radical. A subscript is introduced on upstream density because downstream density is now variable.

[8-47]

Note that the mass rate of flow divided by the area at the throat is equivalent to the mass flux at the throat, G_T. The mass flux, generally, is simply G, obtained by dividing the mass flow rate by the area at a specific location. Note also that the Y factor changes the upstream density of the last equation to the density at the section of measurement, which could be the throat or it could be any section upstream of the throat in a venturi.

McCabe and Smith give an equation (consult the original) for mass flux of an ideal gas through an ideal nozzle modified as [6-29].

[6-29]

The variables G and P are referenced to the same section of the nozzle. Equation [6-29] gives mass flux at a section in terms of the reservoir conditions, the ratios of specific heats and the ratio of pressure at the same section to the reservoir pressure. If we substitute the mass flux from [6-29] into [8-47], we obtain equation [8-49].

[8-49]

If flow through the venturi is considered to have normal turbulent, flattened profiles at the upstream and at the throat locations, C equals one and the equation can be further simplified.

The ASME Research Committee on Fluid Meters established the basic equations for liquids or gases flowing through a restriction as follows:

[10]

The constants and variables of equation [8-49 and 10] are:

A₂ = Cross sectional area, ft², at throat or vena contracta.

C = Dimensionless coefficient of discharge. Ratio of actual flow to theoretical flow that allows for irreversibilities. Hershel, C = 0.984.

K = C/[1-b⁴].

P₁, P₂ = Upstream and downstream pressure, psfa, measured at tap locations.

W, q₁ = mass and volumetric flow rates, lb_m or kg per hour or scfh or Nm³/h. The bases for scfh and Nm³/h are 14.73 psig and 60 F and 1.013 bar and 15 C.

Y = Expansion factor, Y = 1, incompressible flow, Y_lim = 0.667, compressible flow (Driskell). The limiting value changes slightly. The Crane method is correct.

b = ratio of smaller to larger diameter.

R = ratio of downstream to upstream pressures

g = ratio of specific heats of gas as ideal gas.

For a Hershel venturi, the ASME equation is given as follows:

[11]

This rather complicated equation should be compared to the simpler equation [8] which was derived from empirical observations. Equation [11] results in a series of straight lines of negative slope, when Y is plotted versus r, each line starting at Y = 1. A new series of lines can be generated for each different primary element. The closer to ideal the element, the steeper the series. For instance, the slopes of the lines for venturis and flow nozzles are steepest; orifices are represented by equation [11] and they also have there origin at Y = 1, but are less steep.

These observations permit equation [11] to be replaced by equations [8] and [9] for control valves. Note that the title in Crane refers to expansion to a larger flow area and that in [9] the area ratio no longer appears, but the mechanical energy loss coefficient, K, does. This is an empirical equation. The coefficient K is a substitute for the effects of b on the density through a restriction that is no longer an orifice bore but is the effective vena contracta of a control valve.

There is a fundamental difference in the use of the Y factor for primary elements (measuring elements) and its use for fittings or valves. In the first case, it is recommended to keep the factor constant or as close to one as possible because it varies with the flow rate. In the second case, measurement is not involved and the factor can vary between one and its minimum value. At the minimum value of Y, in the second case, choked flow is established, The maximum flow rate no longer changes even when the downstream pressure is reduced further, only if the upstream pressure is increased or decreased.

Flow does not increase indefinitely with falling downstream pressure. With liquids, if the downstream pressure is maintained below the vapour pressure, some of the liquid flashes to vapour. With vapour and gases, the fluid accelerates and the energy for acceleration is taken from the static energy. At some point, there is no energy left to overcome additional irreversibilities caused by increasing velocity and the flow rate remains constant - it 'chokes'. For gases, the velocity of the fluid at the choked flow point corresponds closely with sonic velocity. For two phase flow, there is no single sonic velocity. The choked flow concepts developed in Flow of Industrial Fluids must be used.

SIZING EQUATION FOR GASES AND VAPOURS

For gases and vapours, it is common to use volumetric units in the valve sizing equation, although mass units can be used. An equation given by Driskell is

[12]

The units have to be watched carefully. F_P is a factor to correct for pipe reducers and N₇ has the value of 1360 in customary American units or 4.17 in metric units. The American units are scfh and the metric units are m³/h with the standard foot cubed being at 14.73 psia and 60 F and the standard meter cubed being at 101.3 kPa(a) and 15 C. Pressure is measured in psia or kPa(a). Temperature is in R or K. The compressibility, Z, is necessary because the density is a computed term based on inlet conditions. The mass flow formula will give better results.

COMPARISON OF DRISKELL'S VALVE SIZING FORMULAE WITH MILLER'S LOSS COEFFICIENTS

Miller's coefficients are applied to the obstruction between the faces of a fitting (assumed flanged). Driskell's formulae apply to a length of pipe approximately one diameter upstream to approximately 40 diameters downstream of the valve (also assumed flanged), corrected empirically by the C_V. The difference between the two methods for establishing coefficients will have to be resolved.

Driskell's formulae are concerned with actual (measured) pressure drops at locations removed from the faces of the component, a valve, versus flow. Miller's are concerned with differences in total pressure, (P₁ + á₁U₁²/2g_c) – (P₂ + á₂U₂²/2g_c) with the values of the variables being projected to the faces of the component, a fitting, as a ratio to the kinetic energy at a section.

We will examine the difference in total pressure for a known gas, air, in a known valve, a venturi angle valve to see if the data can help our understanding of mechanical energy losses due to a component.

DATA USED IN THE ANALYSIS

The justification for using Driskell's formulae is that they have been used by many different engineers and technicians over many years. There has been ample time and opportunity to highlight errors or discrepancies. These equations have been proven by correlations with experimental data. Therefore, they can be used with a certain confidence that only minor corrections will be made in the future.

The fundamental data will come from Driskell's table of representative values. The sizing equation is equation [7] using customary US units. The data follows from assuming incipient choking so that the extreme case could be analysed. The table entitled ANGLE VALVE DATA summarises the sizing data used for estimations.

It must be remembered that Driskell used ideal gas concepts for the most part and that, for the sake of this exploratory analysis, we have done the same thing. A more stringent anlysis would be to perform a computer simulation using the R-K equation.

We have used Driskell's data for the angle valve and have developed other data using the constant stagnation enthalpy assumption. This is justified in most industrial cases of short, well insulated pipes.

ANGLE VALVE DATA USED FOR ESTIMATIONS
Line size 2 inch, Sch. 40, same for both US and SI
US Customary Units SI units
Cd (1) 22.0, same for both US and SI
CV = Cdd2, gpm/P1/2	(2) 94.0	same	94
Pipe i.d. inches	(3) 2.067	same	2.067
P1, psia	(4) 114.7	P1, kPa(a)	790.8
P2, psia	(5) 91.76	P2, kPa(a)	632.7
TF1 (F), (R)	(6) 60.0, 519.7	TF1, (C), (K)	15.6, 288.7
TF2 (F), (R)	(7) 56.1, 515.8	TF2, (C), (K)	13.4, 286.6
Incipient choked Y	(8) 0.667	dimensionless	0.667
xT	(9) 0.2	dimensionless	0.2
Density, á1, lbm/ft3	(10) 0.6	kg/m3	9.55
	(11) 0.481	kg/m3	7.69
Density, á2, lbm/ft3
	(12) 14 724	kg/h	6679
W, lbm/h
	(13) 294.3	m/s	89.75
U1, ft/s
	(14) 365.1	m/s	111.4
U2, ft/s
		kJ/kg-k
	(15) 0.241		1.0035
CP, Btu/lbm-R
		n/a
	(16) 778.16		n/a
J, ft-lbf/Btu
		kJ/kg
Enthalpy, hST, Btu/lbm	(17) 127.0		295.5
		kJ/kg
Enthalpy, hF1, Btu/lbm	(18) 125.3		291.5
		kJ/kg
Enthalpy, hF2, Btu/lbm	(19) 124.3		289.2

DISCUSSION OF DATA DERIVED FROM DRISKELL'S FORMULAE

This is a survey-type investigation. Therefore, several shortcuts have been taken.

1. The ideal gas assumption with constant heat capacity has been used. This assumption is justified by the fact that we are using low-pressure air as the test fluid, but a real-gas analysis should be ultimately made. A polynomial giving the relationship between temperature and C_P0. The R-K equation, as discussed in Flow of Industrial Fluids, probably will be acceptable for this case.

2. The main data is gathered in the above table for convenience. The notes that follow briefly discuss each element of the table.

3. One of the often-overlooked problems that must be cited is the fact that locations 1 and 2 associated with Driskell's methods are one pipe diameter upstream and 40 diameters downstream of the valve; these locations do not coincide with the faces of the valve. This problem is most apparent when the velocities associated with the choked flow condition are estimated. The downstream velocity is not the choked velocity at the exit from the valve. The choked velocity at the valve will be substantially greater than the one estimated as U₂.

4. Arbitrarily chosen values upstream of the valve can result in the temperature close to the vena contracta dipping below the freezing point of water. This problem would have to be addressed in the real world because of potential ice formation or sweating on the downstream piping.

NOTES (Notes are indexed with numbers found in previous table)

1. From Driskell's Tables of Representative Values (Venturi, flow to close)

2. From Driskell's Tables of Representative Values

3. ID of standard Sch. 40 pipe

4. Arbitrarily chosen value (100 psig equals 114.7 psia or 790.8 kPa(a))

5. Estimated using P₂ = P₁ - x_TP₁

6. Arbitrarily chosen value (can result in T_vc dipping below 32 F or 0 C)

7. Estimated using constant stagnation enthalpy, ideal gas.

8. Driskell's standard figure, used for convenience

9. Driskell's standard figure, used for convenience

10. Estimated from ideal gas 'law', á₁ = P₁/RT₁

11. Estimated using from ideal gas assumption using T₂ from constant stagnation enthalpy

12. Estimated using equation [7]

13. Estimated using U₁ = G/á₁ = 14 724/[(à[d/12]²/4)á₁3600] in US customary units

14. Estimated using U₂ = G/á₂ = 14 724/[(à[d/12]²/4)á₂3600] in US customary units

15. Average value of specific heat at constant pressure using customary US units

16. Standard value of Joule's “mechanical equivalent of heat” using customary US units

17. Estimated using h_F1 = C_PT₁ and h_ST = h_F1 + U₁²/(2g_cJ)

18. Estimated using h_F1 = C_PT₁

19. Estimated using h_F2 = C_PT₂

The subscripts on enthalpy indicate F for flowing, ST for stagnation and 1 and 2 for location.

Note that although we refer to the figure to indicate a possible calculation path, given the input data, we were able to perform the estimations directly from section 1 to section 2, because of the adiabatic assumption.

Given the choice of a line sized angle valve and a line size of two inches, schedule 40, from the relationship C_V = C_dd², the C_V will be about 94. If P₁ is 100 psig and T₁ is 60 F, the choked value of Y will be 0.677, assuming that Driskell's derivation is correct, x will equal x_T which is 0.2 and r₁ will be 0.596 pounds mass per foot cubed as an ideal gas. The mass flow rate through the wide open valve when the flow is choked would be 14 724 lb_m/h. The pressure drop from one pipe diameter upstream to about 40 diameters downstream would be given by x_TP₁ = 0.2(114.7) = 22.94 psi; so, P₂ will be 91.76 psia. Assuming constant stagnation enthalpy, the temperatures upstream and downstream would be T₁ = 519.67 R and T₂ = 515.8 R. This temperature was estimated as follows:

The last equation has the form of ax² + bx + c = 0, therefore we can use the quadratic equation to solve for the downstream temperature.

In the above, the coefficients must first be estimated. We will give the details using customary US units. The equivalent SI units will be tabulated subsequently.

Only the positive sign before the radical was taken into consideration. The negative sign has no meaning physically.

The average velocity in a section depends of the mass flow rate, the diameter and the density. Taken as an ideal gas , U₁ = 292.5 fps, U₂ = 364.9 fps. If the enthalpy reference point is taken as zero R for an ideal gas and R is considered a constant h_ST = 126.95 Btu/lbm, h₁ = 125.25 Btu/lbm, and h₂ would be 124.31 Btu/lbm.m The author believes that the constant heat capacity assumption is a major source of error but, for the purpose of exploration, it can be ignored.

TABULATION OF VARIABLE VALUES ACROSS THE ANGLE VALVE BETWEEN SECTIONS 1 AND 2

We will first make a sketch to demonstrate the path of the estimations.

TABULATION OF RESULTS OF ESTIMATION ACROSS TRANSITION

Notes:

1. The entropy for US units is in Btu/lb_mole-R but for SI units is kJ/kg-K. The US units are based on the mole; the SI units are mass-based.

2. The temperature and the pressure increases with enthalpy on the plot, but the pressure curves slope upwards.

3. The values of P and T at section 1 were picked arbitrarily. The density at section 1 was estimated from the ideal gas relationship. The specific volume is the inverse of the density. The enthalpy, h₁, was obtained from the constant specific heat assumption, h₁ = C_P0T₁ with C_P0 equal to 0.241 Btu/lb_m-R in customary US units and 1.0035 kJ/kg-K in SI units. The fixed mass flow rate was a constant at 14 724 lb_m/h, estimated from equation [7]. The mass flow rate is 4.09 lb_m/s and the mass flux, G, is 4.09/(àD²/4) or 175.5 lb_m/s-ft².

4. The velocity at section 1 was estimated from U₁ = Gv₁ = 175.5(1.677) = 294.3 fps.

5. The stagnation enthalpy at section 01 was estimated from h_ST = h₁ + U²₁/2g_c = 127.0 Btu/lb_m.

6. The stagnation temperature at the same section was obtained by dividing the stagnation enthalpy by the heat capacity at constant pressure, 127.0/0.241 = 526.9 R.

7. Given the adiabatic constraint, the stagnation enthalpy and the stagnation temperature were both constant and were inserted in the column representing conditions at section 02. Similarly, the velocities at these two sections were zero given the definition of the stagnation condition.

The sketch is an enthalpy-entropy plot showing changes across a transition. The two curves marked P₁ and P₂ represent the initial and final flowing pressures at locations 1 and 2. The two curves marked P₀₁ and P₀₂ represent stagnation pressures. These are theoretical pressures that would be obtained if the fluid were brought to rest adiabatically and isentropically.

The end points of the conditions caused by the presence of the valve are those sections one pipe diameter upstream and forty pipe diameters downstream.

It is possible to estimate changes from sections 1 to 01 using the isentropic constraint, from sections 01 to 02 using the constant stagnation enthalpy constraint, and from sections 02 to 2 again using the isentropic constraint. The reader is invited to consult any good textbook on thermodynamics such as Van Wylen and Sontag for more information.

ENTROPY CHANGES OF IDEAL GASES

The entropy changes for ideal gases are given by Equation [13].

[13]

The units of entropy are those of R which are same as those of c_P0. We will use 1.987 Btu/lb_m-mole-R converted to mass units by multiplying by the inverse of the molar mass of air. The heat capacity at constant pressure, C_p0, has units of Btu/lb_m-R. The molar mass of air is 28.97 lb_m/lb_m-mole. The entropy change depends on a starting temperature and pressure. The starting temperature and pressure at section 1 were 519.67 R and 114.7 psia and the same variables at section 2 are 515.8 R and 91.76 psia. The change in entropy between the two sections is

Entropy seems to be such a hard-to-grasp concept that we should spend a little time studying the fundamentals of its development. Carnot, French, laid the ground rules when he developed his ideas on the ideal Carnot 'heat' engine that could convert some heat energy into mechanical energy but always transfered some heat energy to an environmental sink. People such as Kelvin and Joule, both English, helped emphasise the concept of the 'mechanical equivalent of heat' that was nothing more than a first law statement equating the two recognised forms of energy transfer to a body. Rudolf Clausius, German, took Carnot's ideas a step further when he formalized the concept of entropy, gave it its name, Greek, which means 'that which gives direction'. Many other people of all nationalities saw connections to other concepts and realized that entropy was a property of all matter related to its state of order or, more specifically, disorder.

Being a property of matter, entropy has a given value for a given state. This state can be specified by two other properties such as pressure and temperature in the case of pure substances. If a mixture of pure substances exists, more than two properties must be specified if we are to fix the state. Gibbs, American, quantified the solution to this problem with his phase rule.

The peculiarity of the property, entropy, is that its total value always increases in any spontaneous (natural) process. This is the reason Clausius coined the name 'entropy'. Entropy has been used by scientists to give them information on the direction of spontaneous processes.

Benedict et al wrote a paper linking what is commonly termed 'form friction' to increase in entropy as fluids traverse obstructions. The present author happens to think that the methods of Benedict et al will prove to be a breakthrough in the calculation of irreversibilities in piping systems. These methods will ultimately replace the commonly used methods involving K factors to estimate the irreversibilities created by fittings. This subject has been discussed by R. Mulley in a companion paper.

With these preliminaries out of the way, we will concentrate on trying to verify equation [13] for entropy change. As in all thermodynamic equations, there is no abolute value. What appears to be absolute is frequently simply a convenient reference value that has become accepted by convention. The thermodynamic equations are mostly derived from integral calculus, so the difference in values between two equilibrium states is what counts. Their values are related to certain standard states used as starting points in integrations. The multiplicity of units used in engineering and science adds to the confusion.

Looking at the integrated differential equation [13], we see that the independent variables are pressure and temperature. The dependent variable is entropy. The direction in which each independent variable drives the dependent variable may be reasoned by the fact that the entropy of a pure crystal is close to zero at zero Rankine or Kelvin. This state reflects perfect order. As we reduce the absolute temperature, all gaseous substances become liquid, then solid. They may form crystaline substances. The conclusion is that a reduction in temperature causes a reduction in entropy and we see that this is the case with our example.

In the case of pressure, if we start with a gas and compress it so as to increase the pressure, the molecules are forced to be closer in proximity to one another. They become more ordered, thus also decreasing the value of the property, entropy. Lowering the pressure has the opposite effect. The conclusion is that lowering the pressure increases the disorder and increases the value of the property known as entropy. In the example the combination of the natural log of the fraction and the minus sign before the term containing the pressures gives us the positive increment that we were looking for.

The example equation gave us the direction of change expected of both influences. The sum of the two was also positive, which is what we were hoping for. It remains to obtain a comfort level for the numerical result. We will try to do this with SI units.

The change in entropy is positive as predicted by theory. The change is within 0.5% of that predicted using customary US units. Note that estimations of entropy change require both temperature and pressure values upstream and downstream as well as heat capacity information. Entropy at section 1 can be arbitrarily set at zero. Delta is the final less the initial values of the variable. The formula for delta s may be checked against the differential entropy of the steam tables. It will be found that for low pressures and high temperatures, the values are reasonably close. The largest error comes from the assumption of a constant heat capacity. The steam tables use a polynomial to calculate specific heat and they use a more complicated equation-of-state to calculated entropy. The reference point is usually liquid water at the triple point (this can be seen from the steam tables). Entropy and internal energy at this point are taken as zero.

PROJECTION OF VARIABLE VALUES TO FACES OF COMPONENT

The presence of the component, in this case an angle valve, caused a disturbance to the velocity profile that resulted in a reduction in mechanical energy (and an equal gain in thermal energy) that was not substantially complete for about 40 diameters downstream of the downstream face of the component. Similarly, the upstream flow profile was also disturbed, but for about one diameter upstream of the upstream face (also assumed flanged). Without these disturbances, the profile would have been normal, fairly flat. In order to quantify the affect of the presence of the component, we should compute the upstream and downstream affects separately; then we can say which irreversibilities were due to the valve alone and which were due to the piping.

MECHANICAL ENERGY IMMEDIATELY UPSTREAM OF ANGLE VALVE

To estimate the theoretical mechanical energy immediately upstream of the valve with completely turbulent flow and no valve in place, we have to subtract the piping losses over a length of one pipe diameter from the mechanical energy at section 1. The temperature at the upstream section 1 is 519.67 R.

The Reynolds number is

From the Churchill-Usagi relation (discussed in detail around [15]), the upstream friction factor with completely turbulent flow is 0.0190. This number is close to the downstream friction factor and also checks closely with the charts in Crane.

Assuming the conditions are constant over one pipe diameter of length, the mechanical energy losses to the upstream face are:

The mechanical energy at the upstream face of the component is estimated from:

MECHANICAL ENERGY IMMEDIATELY DOWNSTREAM OF ANGLE VALVE

We will start by computing the mechanical energy changes downstream of the valve with an ideal profile (as if the valve were not present). Subscript 1 refers to the beginning of the downstream pipe; subscript 2 refers to the end. The normal downstream change is given by equation [14].

[14]

For longer lengths, more accuracy may be had if the computation is performed stepwise. The subscript 2 is the starting point of a backwards calculation and the subscript 1 is the terminal, upstream section. The elevation is considered constant. For the sake of this exercise, we will consider a single set of calculations to suffice. The velocity, U₂, will be considered constant for simplicity.

The Moody friction factor, f_M, can be found with the help of the Churchill-Usagi relationships or from Crane. The Churchill-Usagi relationships were given in Flow of Industrial Fluids, AII-10.

[15]

From the above set, the input variables are the Reynolds number, the absolute roughness and the pipe internal diameter in feet. The Reynolds number was given in Flow of Industrial Fluids, I-1, as N_re =6.31W/(dm_cP) with the variables on the right being in mixed units of lb_m/h, inches and centipoise. The Reynolds number is dimensionless because the units of the coefficient cancel those of the mixed units.

We will use the viscosity at the downstream temperature calculated from Sutherland's approximation given by Crane.

Where:

Ü_P = viscosity in centipoise at temperature, T

Ü₀ = viscosity in centipoise at reference temperature, T₀

C = constant for fluid (120 for air)

T = temperature in degrees Rankine at which viscosity is estimated

T₀ = reference temperature

The estimated Reynolds number will be

The relative pipe roughness for commercial steel is the absolute roughness, 0.00015 feet, divided by the diameter in feet. The pipe diameter, D, is the inside diameter in inches divided by 12 inches per foot. It is 2.067/12 = 0.172 feet. The relative roughness is 0.000871 .

The Moody friction factor by the Churchill-Usagi relationship is 0.0192 which checks fairly closely with the charts given in Crane.

The average velocity at section 2 is 365.1 fps. We will estimate the permanent mechanical energy losses (irreversibilities) between the valve and section 2 as 0.0192(40)365.1²/2g_c = 1591 ft-lb_f/lb_m. The equivalent in Btu/lb_m is about 2.0 Btu/lb_m.

To estimate the mechanical energy immediately downstream of the valve assuming completely turbulent flow, we have to add the irreversibilities to the mechanical energy at section 2 according to [14]. The units are ft-lb_f/lb_m. The following projection could be improved by a stepwise computer simulation while varying the velocity using the R-K equation. (M.E.)₁ = 91.76(144)/0.481 + 1591 = 29 062 ft-lbf/lb_m or 36.5 Btu/lb_{m .} The losses in mechanical energy for the 40 pipe diameters were about 5.5% of the mechanical energy immediately downstream of the valve.

ANALYSIS OF PROBLEMS CAUSED BY ACCEPTANCE OF HYDRAULICS TERMINOLOGY

Even McCabe and Smith fall into the trap of using the shortcuts of hydraulics terminology. For instance they give the pressure difference at the Hg manometer taps of a water venturi as follows:

[16]

If equation [16] is examined dimensionally, we find the terms on the left have units of ft-lb_f/ft² and the terms on the right have units of inches/(inches/foot) times lb_m/ft³ or lb_m/ft^2.. It looks as though they have equated the acceleration of gravity, g = 32.17 ft/s², with g_c =32.17 ft-lb_m/lb_f-s². Dividing g by g_c gives units of lb_f/lb_m. Multiplying the units on the right by lb_f/lb_m gives lb_f/ft², which is what we should have. So the equation should have been written as follows to maintain dimensional consistency.

[17]

Figure 1 will now be analysed.

FIGURE 1

ORIFICE PLATE MEASURING WATER FLOW

WITH MERCURY MANOMETER

Figure 1 represents an orifice plate with an attached mercury manometer. The fluid being measured is water. Sections A and B are the projections from the upstream and downstream taps to the pipe centreline (B is not shown). Since the meter run is horizontal, H_H and H_L are equal and represent the distance from the pipe centreline to the common point on the manometer. This common point is actually the section at which pressure is equal in each leg of the manometer. On the left, pressure at base section is due to the pressure at A plus the pressure due to a water column of height H_H plus the pressure due to the height of the mercury column on the left. On the right, the same pressure at the base is due to 50/12 feet of mercury plus a water column of H_L-50/12 feet of water plus an equivalent height of the mecury column on the left. The additional height of the mercury column on the right is 50 inches.

We can write equations starting at the common section.

[18]

Each term has units of pressure, in this case, lb_f/ft².

We can rearrange the last two equalities while leaving out the common term to find the difference between P_A and P_B.

[19]

The last equation of the set [19] is the same as [16] but the dimensional conversion g/g_c has been included as a multiplier.

SUMMARY

The paper analysed, in thermodynamic terms, some of the work of Les Driskell, ISA Handbook of Control Valves. It also developed the concept and utility of the Y factor. To support the analysis, the text by McCabe and Smith, Unit Operations, was also cited.

Driskell made use of hydraulic terminology as did McCabe and Smith. This terminology was transformed to thermodynamic terminology.

Driskell's contention that the limiting value of Y is a fixed number was shown to be not correct. Choked flow was cited as the principal reason.

The fundamental equation for liquid flow through orifices, its transformation to an equation for flow through valves and its further transformation to an equation for gaseous flow was examined. The origins of the various correction factors were explained. The Y factor, which corrects initial compressible fluid density as the fluid expands through a restriction was examined in some detail. The text by McCabe and Smith, cited above, was quoted.

The work of the ASME Research Committee on Flow Meters was used to support some of the analysis.

An angle valve was chosen to demonstrate changes that occur upstream and downstream of an obstruction. The projection method of computing physical properties was discussed.

The Churchill-Usagi equations were given. These equations permit the estimation of the friction factor without the use of graphs.