Control Control Valve Sizing for Complex Liquids Draft . . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . . .. . . . . . . . . . . .
By J.G. MacKinnon
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Where Y is an expansion factor which accounts for the density
Control Valve Sizing for Complex Liquids Draft
change through the valve. Squaring equation 2 and taking the limit as P2 approaches P1
By J. Gary MacKinnon, Manager of Engineering Standards
n
results in equation 4. 4
Abstract
A
Allowing γ to vary with P and integrating leads to eqution 5.
method is derived for determining the required control 5
valve capacity useful for complex uids including gas-liquid
mixtures and multiconstituent ashing liquids. The method agrees well enough with existing ISA sizing standards for the simple
Equating equations 3 and 5 yield equation 6.
uids such as non-ashing liquids, ashing liquids and gases to
6
provide condence in its use with the more complex uids. Introduction
Generally, γ is not available as an analytical function of pressure so this integral format is not very useful. Of more use is a
The Instrument Society of America publishes standard S75.01 entitled
conversion of this expression to a summation based on knowing
“Control Valve Sizing Equations”. This standard outlines the method
several (n) discrete values of density and pressure. Equation 7 is
of sizing control valves for ashing liquids, non-ashing liquids and
the solution of equation 6 assuming the pressure-density
gases which account for the vast majority of uids encountered in
relationship is linear between selected points.
industrial applications. Occasionally, valve sizing is required for uids that differ from those covered by the standard such as a 7
gas liquid mixture and multiconstituent liquids whose constituents ash at various pressures. The Standard is not applicable to these uids because the behavior of the uid density as the pressure drops through the valve does not t the uid models which form the basis
Comparison To Standard S75.01
of the standard. For liquids, the density is assumed constant and for
For example, if only the starting and ending density are known, it
gases the density is assumed to vary according to the laws governing
is reasonable to assume without any other information that the
ideal gases. But for complex uids the relation between pressure and
change in density is linear with pressure from inlet to outlet. In
density forms no pattern and can vary widely from uid to uid. In
that case, equation 7 reduces to equation 8.
addition, the problem is complicated by slip between the phases and thermal non-equilibrium. It is not surprising that a general solution
8
has not been found. This paper outlines a method of sizing control valves handling
If the uid is a liquid, then the density does not change and
complex uids using a set of apparently overly restrictive assumptions.
equation 8 reduces to the expression
Yet, when the results are compared to the present ISA equations where
9
these assumptions are not present, good agreement is found. As would be expected. The ISA equation for Liquid Sizing is For ideal gases, the density is proportional to pressure which 1
agrees with the linearity assumption in deriving equation 8. In
The liquid is assumed to be non-choking and that Reynolds
addition, this proportionality allows the following alternate of
number effects and piping geometry effects are neglibible.
equation 8 which applies to ideal gases.
Finally for the sake of later comparisons to the ISA gas and liquid
10
equations, the specic gravity term is converted to density which results in equation 2. Introducing the pressure drop ratio, x, dened as 2
The sizing equation we seek is of the form of equation 3.
11 3
2 Control Valve Sizing for Complex Liquids | 228
And substituting this into equaiton 10 yields
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12
This equation can be directly compared with the ISA equation for gas expansion factor, repeated here.
one with inlet pressure chosen at 30% of the critical pressure and one at 3%. These are illustrated in Figures 2 and 3 respectively. Note that there is fairly good agreement between the curves except under the conditions that the outlet pressure is close
13
The two methods of calculating Y are compared in Figure 1 for value of X t of .75 which is typcial for many plug valves and a
to the vapor pressure. However, this discrepancy is perhaps due more to the ISA sizing rules which do not recognize a flow reduction due to incipient choking and instead identify a discrete choking point at F f P v .
value of 1.0 which is typical for controlled velocity valves. Note that the present formula agrees very well with a controlled
1.0
velocity type valve. This is to be expected since ow through a .9
controlled velocity valve is nearly isenthalpic. PROPOSE
1.0 ISA : VELOCITY CONTROL VALVE
.9
) Y ( R O T C A F N O I S N A P X E
.8 .7 ISA : PLUG VALVE
.6
) Y ( R O T C A F N O I S N A P X E
.8 .7
O D T H
E G M
I N S I Z E D S O P P R O O D T H M E A I S
.6 .5 .4 .3
.5
.2
.4
.1
.3
0 0
.1
.2
.2
.3
.4
.5
.6
.7
.8
.9
1.0
PRESSURE RATIO (P 2/P1)
.1
Figure 2—Proposed Sizing Method vs. ISA Method for Flashing Water (PV/PC = 0.3)
0 0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
PRESSURE DROP RATIO (X) 1.0
Figure 1—Proposed Sizing Method vs. ISA Methods for Gases
.9
Finally, equation 6 is compared to the ISA equations for sizing of ashing liquids. Flashing liquids, especially those of low vapor pressures represent the opposite end of the spectrum from liquids. These uids have the greateset variation in density from the inlet to outlet of the valve. To make the comparison, an expression for the expansion factor Y must be derived for the ashing liquids using the ISA equations. Recognizing that Y is simply the ratio of the ow rate of an expanding uid to that of a non-ashing liquid under the same pressure and inlet density conditions equation, 14 is derived. 14
Where:
) Y ( R O T C A F N O I S N A P X E
.8 ISA METHOD
.7 .6 .5 .4 .3
PROPOSED SIZING METHOD
.2 .1
PO = F f P V if P2 < Ff P V PO = P 2
if P 2 ≥ F f P V
Values for equation 14 were calculated for water assuming a saturated inlet condition. Two sets of comparisons were made,
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0 0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
PRESSURE RATIO (P2/P1) Figure 3—Proposed Sizing Method vs. ISA Method for Flashing Water (PV/PC = 0.03)
228
| Control Valve Sizing for Complex Liquids 3
helps indentify the number of discrete points along the pressure-
LIQUID
1.0
density curve to use equation 7 with minimal error. The error is
.9
related to the proportion of the area that is added or subtracted from the exact pressure-density integral by connecting the chosen
.8
points with the straight lines. See Figure 5.
) 1 T .7 / T ( O I T A R Y T I S N E D
Conclusion
.6
A simple method of sizing control valves, owing complex
S G A
.5
liquids, utilizing the basic format of the ISA standard equations
R T E A W 3 N G C = . I R P H / T E A S P V A L F W 3 N G C = . 0 I H / P A S P V F L
.4 .3 .2 .1
has been proposed. The method uses the expansion factor Y to account for the variation in density from inlet to outlet. Y can be calculated using a simple summation formula once a relationship between pressure and density is determined. Comparisons between the proposed sizing method and existing sizing methods for non-ashing and ashing liquids and for
0 0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
gases indicates that the assumptions used in driving the sizing method are not overly restrictive. These comparisions give
PRESSURE RATIO (P/P 1)
reasonable assurance that the method will work for complex
Figure 4—Density vs. Pressure For Fluid Curves
uids with arbitrary density-pressure relationships.
Figure 4 illustrates the normalized pressure density
References
relationship for the four uids which were used in the comparisons. Note the wide range of curves in which fairly
(1) Control Valve Sizing Equations, ANSI/ISA –S75.01, 1977.
good agreement between the proposed sizing method and the ISA method is achieved. Complex uids involving gas-liquid
Nonmenclature
mixtures, two phase inlet conditions and multiconsituent ashing uids will in nearly all cases fall within the extremes
C V
Valve sizing coefcient
of these uids used in the comparisons.
F
Liquid critical pressure ratio factor
Fp
Piping geometry factor
F y
Liquid choked ow factor
G
Specic gravity
k
Ratio of specic heats
N
Numerical constant
N10
Numerical constant
n
Number of discrete points selected on the pressure-density curve
P1
Inlet pressure
P2
Outlet pressure
PO
Effective outlet pressure
P V
Vapor pressure
w
Mass ow rate
x
Pressure drop ratio
x t
Rated pressure drop ratio
Y
Expansion factor
f
1
1.0 .9
2
f
.8 ) 1 T .7 / T ( O I T A R Y T I S N E D
3
.6 .5 .4 .3 .2 .1 0 0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
PRESSURE RATIO (P/P 1) Figure 4—Error Caused by Using Discrete Pressure vs. Density Points
An intuitive feel for the value of the expansion factor Y can be achieved by recognizing that its value is equal to the square root of the area under the curves of Figure 4. This understanding
4 Control Valve Sizing for Complex Liquids | 228
γ
Density
γ 1
Inlet density
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