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CURVE FITTING OF AMMONIA-WATER MIXTURE PROPERTIES
by
David Urnes Johnson1, William E. Lear
1, and S.A. Sherif
1
1Department of Mechanical and Aerospace Engineering, University of Florida, P.O. Box 116300,
Gainesville, FL 32611-6300, USA
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ABSTRACT
In this paper two equations relating vapor-liquid equilibrium properties (T-P-x-y) of
ammonia-water mixtures are presented. Ryu et. al. [1] showed that the polynomial expressions
presented by Ptek and Klomfar [2] and El-Sayed and Tribus [3] show oscillatory behavior at
high ammonia concentrations and that the bubble and dew lines fail to meet at the pure substance
concentrations. This is thermodynamically impossible, and can cause iterative models to diverge.
A curve fitting procedure inspired by Lagrange polynomials that forces the bubble and dew line
to meet at pure components is developed. Numerical techniques are employed to reduce the
oscillatory behavior close to the pure substance values, and an improved data set selection is
chosen by reviewing the survey of Tiller Roth and Friend [4]. The equations presented in this
paper are an improvement of Ptek and Klomfars T(P,x) and T(P,x) equations, and are meant
for industrial calculations in absorption refrigeration systems.
INTRODUCTION
Thorin et. al [5] presented a review of all available correlations for thermodynamic
properties of ammonia-water mixtures. The correlations can be divided into seven groups
according to the way they are derived: cubic equations of state, virial equations of state, Gibbs
excess energy, the law of corresponding states, perturbation theory, group contribution method,
and polynomial functions. All of these correlations are semi-empirical except the correlations
based upon polynomial functions. The advantage of polynomial functions is their convenience of
use. The T(P,x) and T(P,x) equations presented by Ptek and Klomfar [2] do a good job in
general, but that fail to be thermodynamically consistent at high ammonia concentrations and the
bubble and the dew lines fail to meet at the pure substance values. These inconsistencies cancause dynamic models to diverge as iteration failure occurs. The aim of this paper is to improve
upon the T(P,x) and T(P,x) equations of Ptek and Klomfar by constraining the pure substance
values, by employing numerical techniques to reduce oscillation, and by selecting more accurate
experimental data as the input to the optimization process.
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DATA SELECTION
Experimental ammonia-water measurements of varying quality have been reported since
the mid-19th
century. Tillner-Roth and Friend [4] presented a comprehensive survey of the
available ammonia-water measurements, which has been used as the foundation for the data
selection in this study. This comprehensive survey was not available to Ptek and Klomfar and
hence there is a considerable potential of improvement in the data selection. In order to cover the
entire thermodynamic plane in interest, data of varying quality had to be chosen. To account for
these differences, the data was divided into three different groups, which was given different
weights in the subsequent least squares optimization. Weights of 2, 1, and 0.5 were assigned; a
weight of 2 to the most accurate data sets, and a weight of 0.5 to the less accurate. The T-P-x
data collected [7-16] is shown in a P-x plot below in Fig. 1-2. The T-P-y data collected [13-18] is
shown in Fig. 3. The weights assigned to each data set are shown in parenthesis.
Fig. 1. Distribution of selected (T, P, x) data points used to create the T(P,x) function.
The weights given to each data set is shown in parenthesis.
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All experimental data was converted to the ITS-90 temperature scale [20]. The data publishedbefore 1927 was assumed to follow the ITS-27 standard as there was no recognized international
temperature scale before 1927. The uncertainties associated with this assumption are believed tobe well below the uncertainties in the measurements themselves.
ANALYTICAL METHOD
Lagrange polynomials force a polynomial through a point (x0, y0) by letting all the
polynomial terms except one approach zero as x approaches x0, and by choosing the constant
term wisely. In this study, the interest is in constraining the end points of the polynomial which
first will be illustrated for a cubic polynomial. Consider a cubic polynomial with end points of
(x0, y0) and (xn, yn). One possible form of this cubic polynomial is shown in Eq. 1.
()= ( ) +( )( ) +( )( ) +( )
where a0to a3are arbitrary constants. Note that when x0is entered into Equation 1, every term on
the right hand side becomes zero except the last term. Similarly, when xnis entered into Equation
1, every term except the first term becomes zero. Hence, by plugging in the end points (x0, y0)
and (xn, yn), Eq. 2 and Eq. 3 are deduced.
= ( ) (2)
= ( ) (3)
Equation 2 and 3 can be solved for a3and a0, respectively. Plugging a0and a3back into Equation
1 yields Eq. 4.
()= ( )
( )+( )( ) +( )( ) +
( )
( ) (4)
It is trivial to show that if the exponents of the first and last term are removed, the polynomial
will still have the desired characteristics. Hence, a cubic polynomial going through the two
points (x0, y0) and (x3, y3) can be expressed as in Eq. 5.
()= ( )( )
+( )( ) +( )( ) +( )( )
(5)
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It should be noted that by constraining the end points of the cubic polynomial the degrees
of freedom is reduced from four to two. The result of a cubic polynomial can easily be
generalized to polynomials of degree n going through the points (x0, y0) and (xn, yn).
y(x)= y(x x)(x x)
+a(x x)(x x) ++a(x x)(x x) +y(x x)(x x)
(6)
The polynomial of degree n shown in Eq. 6 has n-1 degrees of freedom.
The result derived for polynomials above can be extended to functions of any form as
long as the following requirement is met.
Every term in the equation except the last has to vanish if the function is evaluated at x0,
and every term except the first has to vanish if the function is evaluated at xn.
Consequently, Eq. 7, shown below, will also go through the points (x0, y0) and (xn, yn).
y(x)= y(x x)(x x)
+a1 e()
1 e()+a(1 cos(x x))(1
cos(3(x x))) ++a(x x) sin((1+x x)) +y (x x)(x x) (7)
3-D Curve Fitting of TPx and TPy data
The equations sought in this paper, T(P,x) and T(P,y), are three dimensional, and hence
the goal is not to constrain a line to two points, but to constrain a surface to two lines. The two
constrained lines are the lines of temperature as a function of pressure for pure water and pure
ammonia. The equations for the two lines were derived by using the very accurate polynomials
presented by Reynolds [19] converted to the ITS-90 temperature scale [20]. The polynomials
presented by Reynolds were saturation pressures as a function of temperature. Since the inverse
of this was desired, the polynomials were sampled at small intervals and fitted by an appropriate
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power function. The resulting functions for pure water and pure ammonia are shown below in Eq.
9 and Eq. 10, respectively.
()= =269.8. +52.79. +130.4 (9)
()= =177.9. +40.28. +79.83 (10)
Both Eq. 9 and Eq. 10 were found to agree within 0.006 K of the sampled data for the
entire pressure range of interest. The procedure for 3-D fitting is identical to that of 2-D fitting
expect for that the coefficients a0and anin the 3-D case will be functions of pressure. The
surface tool in MATLAB was used for the 3-D linear least square optimization.
Since the polynomial expressions by Ptek and Klomfar have be shown to represent the
thermodynamic plane relatively well for most ammonia concentrations, the polynomial form of
the equations presented in this paper was chosen to resemble that of Ptek and Klomfar. The
functional form of the T(P,x) and T(P,y) equations of Ptek and Klomfar is shown below in Eq.
11 and Eq. 12, respectively.
00( , ) (1 ) ln (11)
i
i
n
m
i
i
pT P x T a x
p
/ 4 00( , ) (1 ) ln (12)
i
i
n
m
i
i
pT P y T a y
p
Because of the condition discussed above, the functional form of the proposed equations could
not resemble those of Ptek and Klomfar exactly. The functional form of the proposed T(P,x)
and T(P,y) equations are shown in Eq. 13 and Eq. 14, respectively.
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00( , ) (1 ) ln (13)
i
i i
s
q r
i
i
PT P x T a x x
P
/4 /400( , ) (1 ) ln (14)
i
i i
s
q r
i
i
PT P y T a y y
P
Since the functional form was chosen to be similar to that of Ptek and Klomfar, the behavior of
the resulting equations also showed a similar behavior. To reduce the oscillation near the pure
substance values, a numerical technique was used. For the ammonia side, the average slope
between y = 0.999 and y =1 was found and used to sample imaginary data points in this region.
These virtual data points punish oscillatory behavior during the least square optimization. Asimilar method was used on the water side, but here different slopes were used depending on the
pressure range and the behavior of the function within this range. An example of this numerical
technique at high ammonia concentrations is shown in Fig. 4 and Fig. 5.
Fig. 4. Virtual points in regions of oscillation punish thermodynamic inconsistent behavior in
the least square optimization and improve the behavior of the curve. The improvements can be
seen in Fig. 5.
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Fig. 5.The dew line in Fig. 4 after the least square optimization was run with the virtual data
points. Even though the oscillation is not eliminated, a clear improvement is shown.
Numerical techniques was only used to improve the behavior of the dew line equation, T(P,y).
No numerical tools were needed for the bubble line equation, T(P,x).
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RESULTS AND DISCUSSION
The coefficients to Eq. 13 and Eq. 14 are shown in Table I and Table II.
Table I: Coefficients for Proposed T(P,x) Equation
i qi ri si ai0 1.11 0 0 (177.9*P0.09397+40.28*P0.3898+79.83)/100
1 0.53 0.66 0 0.6862
2 0.8 1.02 1 -0.1223
3 1.08 1.4 2 0.03215
4 0.55 1.1 3 -0.001288
5 0.35 2 4 -3.355E-05
6 0.6 1 0 -1.181
7 0.1 1 1 -0.005287
8 0.1 1 2 0.001947
9 0.85 2 3 0.0008481
10 0.9 4 0 -2.88811 1 5 0 13.89
12 0.76 5 1 -0.07483
13 1.1 6 0 -12.01
14 1.5 13 1 -0.1027
15 0 1.12 0 (269.8*P . +52.79*P . +130.4)/100
T0= 100 K, P0= 2 MPa
Table II: Coefficients for Proposed T(P,y) Equation
i qi ri si ai0 4.3 0 0 (177.9*P
0.09397+40.28*P
0.3898+79.83)/100
1 6 1.1 0 2.8222 3.03 2.2 1 -17.05
3 3.05 2.2 2 -1.592
4 37 2.7 3 0.04388
5 2.7 1 0 -6.151
6 5 1 1 0.2648
7 4 1 2 0.2028
8 3.1 2 0 -7.961
9 3.01 2 1 14.27
10 3 3 0 46.96
11 3 3 1 2.699
12 3 4 0 -91.913 3 4 2 5.545
14 3 5 0 81.52
15 3 5 2 -5.084
16 3.1 6 0 -24.18
17 3 7 2 0.9451
18 0 1.05 0 (269.8*P.
+52.79*P.
+130.4)/100
T0= 100 K, P0= 2 MPa
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The constants qi, ri, and siare exponents which were determined by a brute force
optimization process. The coefficients in bold are the same as the exponent of the corresponding
term in Ptek and Klomfars equations. The aicoefficients are the constant coefficients
determined by a least square optimization. The first and the last constant coefficient are used to
constrain the equation to the pure substance values. The intended pressure range for Eq. 13 is
between 0.002 MPa and 2 Mpa. Eq. 14 is intended to be used in pressures between 0.05 MPa and
2 MPa.
In Fig. 6 the developed equations are compared with those of Ptek and Klomfar. Both
the bubble line and the dew line are found to agree well over the thermodynamic plane in general.
The differences arise close to the pure substance values as shown in Fig. 7-9.
Fig. 6. Comparison between the T(P,x) and T(P,y) equations of Ptek and
Klomfar and those proposed in this study.
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Fig. 7. Comparison with the T(P,x) and T(P,y) equations of Ptek and Klomfar at
high ammonia concentration and P = 2Mpa. The developed bubble and dew line
do meet at x = 1 even though this cannot be seen from the graph.
Fig. 8. Comparison with the T(P,x) and T(P,y) equations of Ptek and Klomfar at
low ammonia concentration and P = 1 MPa.
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Fig. 9. Comparison with the T(P,x) and T(P,y) equations of Ptek and Klomfar at
low ammonia concentration and P = 0.05 MPa. The proposed dew line shows
non-monotonic behavior close to the pure water composistion.
From Fig. 7, one sees that the proposed equation reduces the numerical instabilities
shown by Ptek and Klomfars equations. To solve this problem at high ammonia concentration
was the main motivation for this work and Fig. 7 shows that this goal has been partially reached.
Paradoxically, Fig. 7 suggests that the bubble and the dew line fail to meet at pure ammonia; this
despite that this convergence was a fundamental condition embedded in the formulation of the
functional form in Eq. 14. The bubble and the dew line do actually meet at pure ammonia, but
because of the number polynomial terms approaching zero in this region, the sensitivity of the
function with respect to y is extremely high, and the dew line function exhibits what one in
practical terms could call an instantaneous jump just before y = 1. The same behavior can be
seen in Fig. 9 close to pure water concentrations. These are limitation embedded into the nature
of the polynomial expression in Eq. 14, and there is little that can be done to avoid it with the
current method. The numerical methods discussed in the previous section can reduce this non-
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physical behavior, but cannot eliminate it completely. The bubble line T(P,x) does not exhibit
this abrupt behavior.
The statistical accuracy of the equations by Ptek and Klomfar and the equations
developed in this work is compared in Table III. The bubble lines show a similar performance.
The dew line of Ptek and Klomfar shows a slightly more accurate behavior than the developed
curves. This difference can be attributed to the numerical technique that reduced oscillatory
behavior of the developed dew line. There is a tradeoff between a well behaved curve at the pure
components and the overall accuracy of the curve. The statistical comparisons in Table III are
made for all the data points used in this study, but a similar conclusion is reached if only a subset
of the experimental data (i.e. the most accurate) is compared.
Table III: Statistical Comparison between the Discussed Equations
Standard
Deviation (K)
Ave Abs
Error (K)
Systematic
Error (K)
Max Error
(K)
Johnson et. al.
Bubble Line0.74 0.51 0.04 3.69
Patek & Klomfar
Bubble Line
0.72
0.50
0.08
3.77
Johnson et. al
Dew Line2.04 1.40 -0.44 10.07
Patek & Klomfar
Dew Line1.67
1.15
0.08
8.28
A graph comparing the temperature difference between the dew line and the bubble line
at pure components for Ptek and Klomfars equations and for the equations developed in this
paper is shown in Fig.10.
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Fig.10. Temperature difference at pure components as a function of
pressure.
From Fig.10, one sees the main strength of the equations developed; they eliminate the
temperature discrepancies between the dew and bubble line at pure components.
CONCLUSIONS
Two equations, T(P,x) and T(P,y), were presented as an improvement to the
corresponding equations of Ptek and Klomfar. The proposed equations were successfully forced
to meet at the pure substance values, even though erupt changes close to the pure components
had to occur in some cases to meet this condition. Erupt changes close to the pure substance
values are embedded in the nature of the functional forms presented in Eq. 13 and Eq. 14 and
cannot be avoided completely. Despite this, Fig. 6 shows that the proposed equations are well-
behaved in general, and Fig. 7 shows that the numerical instabilities at high ammonia
concentrations are significantly reduced. Overall, the proposed equations are considered an
improvement of the T(P,x) and T(P,y) equations presented by Ptek and Klomfar.
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