curve fitting of ammonia-water mixture properties

8/11/2019 Curve Fitting of Ammonia-Water Mixture Properties

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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.


low ammonia concentration and P = 1 MPa.


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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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REFERENCES

[1]. Ryu, C. J., Lear, W. E., Sherif, S. A., Crisalle, O. D., Thermodynamic Properties of

Ammonia-Water Mixtures, Int. J. of Hear and Mass Transfer, Submitted, 2010

[2]. Patek, J., Klomfar, J., Simple functions for fast calculations of selected thermodynamic

properties of the ammonia-water system, Int. J. of Refrigeration, 1995, Vol. 18, No. 4, pp

228-234.

[3]. El-Sayed, Y.M., Tribus, M., Thermodynamic properties of water-ammonia mixtures

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[4].

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[20]. International Atomic Energy Agency, Thermophysical properties database of materials

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92-0-104706-1

curve fitting of ammonia-water mixture properties

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