# Observation and Analysis of Affinity Law Deviations through Tested Performance of Liquefied Gas Reaction Turbines

Authors: Sarah D. Alison-Youel
Journal: International Journal of Rotating Machinery
 (2008)

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## Abstract

Liquefied gas reaction turbines are subject to the hydraulic affinity laws. Particularly for liquefied hydrocarbon gas-driven turbines, deviations from the affinity laws are encountered. In the case of reaction turbines, where the geometry is fixed, the affinity law relationships between flow, head, and rotational speed are relevant. Field experience confirms that the affinity law relationships are adequate, but that the predictions made also tend to deviate from real turbine performance. Part of the deviations seen may be attributed to the nonideal fluid; however, further examination is warranted. This paper presents an investigation into the affinity law relationships between head, flow, and rotational speed in conjunction with actual turbine performance. The three basic affinity law relationships are combined to form the most general performance equation. This equation subsequently incorporates both the affinity law relationships and the conservation of energy principal. Application of real turbine test data shows that this general performance equation presents a more accurate representation of turbine performance than the affinity law relationships alone.


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## Body

## 1. Introduction

The
liquefied  gas reaction turbine industry began in the mid 1990s. The design and
performance  of liquefied hydrocarbon gas reaction turbines
are
extensively described in US
patent   5659205 [1] and by
Habets and  Kimmel [2].  Lobanoff and  Robert
[3] stated that reaction turbines are pumps running in reverse; therefore, the
affinity laws are applicable to the reaction-type turbines. In 1996,
Kimmel [4]
presented a turbine performance equation that incorporated the affinity laws
and the energy of conservation. The intent of the current investigation is to
find the most general polynomial that satisfies both the affinity law
relationships for fixed geometry reaction turbines and the conservation of
energy.Relationships
provided by the affinity laws and the conservation of energy are formed into a
general polynomial resulting in a general performance equation.  This equation is the most universal equation
for constant geometry that incorporates both the affinity law relationships and
the conservation of energy law.The
developed general performance equation is evaluated and analyzed using tested
performance data from an Ebara International Corporation (EIC) (NV, USA) liquefied gas reaction turbine. The
results show an improvement in performance representation by the developed equation,
as well as a probable relationship to pump performance.


## 2. General Performance Equation

The
affinity laws  following relationship between the flow and rotational
speeds
at two operating points,a and b, (1)QaQb=NaNb=X, where
X is some constant. Again, at two operating points, a and b, the affinity law
relates the head, flow, and speed as follows: (2)HaHb=Qa2Qb2=Na2Nb2=Y, 	where
Y is some constant [1].  If (1) is squared,
then Y must be equal to X2, yielding the following relationships
between the two operating points: (3)Qa=X⋅Qb,Na=X⋅Nb,Ha=X2⋅Hb.     In
fluid-driven machinery, the head is a function of the flow and rotational
speed, H=f(Q,N). This function of flow and rotational speed is continuous;
therefore, it can be developed into a Taylor
series of two variables [5]: (4)H=∑m,ncmnQmNn, where
m and n are nonnegative integers. When (4) is expanded, it takes the form (5)H=c00+c10Q+c01N+c20Q2+c02N2+c11QN+c30Q3+c03N3+c21Q2N+c12QN2+⋯+cmnQmNn. Using (4), the head at one operating
point, a, is then (6)Ha=∑m,ncmnQamNan. Substituting the relationships shown in (3) into (6), a relationship is found for the operating point b: (7)X2Hb=∑m,ncmn(XQb)m(XNb)n. Collecting terms and factoring out constants, the
following results: (8)X2Hb=Xm+n∑m,ncmnQbmNbn. To relate this general function to the affinity laws, X2 must equal Xm+n, which leads to the constraint relationship m+n=2.
Since m and n must be positive integers, there are only three cases for values
of m and n that make a true statement. The values for the three cases are shown
in Table 1.Table 1
Values ofm and n.
mnm+n=2202+0=2111+1=2020+2=2To take (8) and make it specific to the affinity law,
the values of m and n in Table 1 are substituted into (5), yielding (9)H=c20Q2+c02N2+c11QN. Written more simply, substituting α, β, and γ for c20,c02, and c11,
respectively, a general performance equation is found (10)H=αQ2+βN2+γQN. The above equation is the most complete form of the
energy equation that also follows the affinity laws. The head is the total
static energy and it must equal the total kinetic energy. The traditional performance
equation proposed by Kimmel in 1996 [4] (see (21) below), does not include
all the terms for kinetic energy, whereas the general performance equation does.
The constants, α, β, and γ are specific to an individual turbine.


## 3. General Performance Equation and The No-Load Condition

Equation (10) governs the performance of the turbine;
therefore, it should also describe the no-load condition. In 1997,
Kimmel [6]
stated that under the no-load condition the inlet and outlet momentums must be
equal, leading to the following relationship: (11)Q=λN. Inserting this relationship into (10) and then
simplifying, the following relationship  is produced: (12)H=δQ2, where, (13)δ=α+βλ2+γλ. Equation (10) coupled with the offshoot equation for
no-load operation, (12), provides general performance equations that follow the
affinity law, conservation of energy, and momentum allowing for complete and superior
representation of turbine performance.


## 4. Turbine Data Analysis

Test data from EIC turbine model LX14-07 was used in
the analysis and application of the general performance equation. Model LX14-07
is a single-phase liquefied gas reaction turbine designed for a nominal flow of1400m3/h and a nominal head of 700 m.During performance testing of the turbine,
performance parameters relating to flow and head were recorded in addition to
the rotational speed. The turbine flow parameter is recorded by measuring the
pressure drop across a Venturi flowmeter. Flow is proportional to the square of
the pressure resulting in flow
parameter tested data in units ofkPa1/2.
The turbine head parameter is recorded by measuring the pressure drop across
the turbine. Head is directly proportional to the differential pressure
resulting in head parameter tested data in units of kPa. Typically turbine flow
and head are then calculated using the measured flow and head parameter data;
however, in the analysis of the general performance equation, all calculations
were made using the flow and head parameter tested data points and not the
typically calculated head and flow. This was done in order to eliminate
calculation error and to empirically evaluate the general performance equation.
The turbine speed is measured by accelerometers and is recorded in revolutions
per minute which is easily converted to revolutions per second for use in the
analysis of the equation. The flow and head parameter data was recorded during
the no-load condition and for three rotational speeds, 2400 RPM, 3110 RPM, and
4000 RPM.  For this particular test, at
the speed of 3110 RPM the data was only recorded for a range approximately
between flow parameters 3.71 and 4.25. This does not greatly affect the results
as a surface fit is applied to all the speed data.A three-dimensional surface fit using the general performance equation
and the tested data was performed to determine the values ofα, β, and δ in (10) specific to the tested data. It is assumed
from the theory explained above that (10) was the best fit for the data. It
follows, if a certain test point, i, is plugged into this equation, that there
is an error associated, Ei, if the test point does not lie exactly
on the surface generated by (10): (14)Ei=αQi2+βNi2+γQiNi−Hi. Some of these Ei values are positive and
some are negative.  To eliminate the
possibility of positive and negative error values canceling out, each Ei is squared; this is called the square error. The square error for each data
point is then summed as follows: (15)∑iEi2=∑i(αQi2+βNi2+γQiNi−Hi)2. To make a best fit, this square error sum should be a
minimum. The minimum condition is found when the partial derivatives relative
to α, β, and γ are found and set equal to zero. This  provides
three linear equations that are solved  for the constants α, β, and γ for the data set. The following three linear
equations are the result of the partial derivatives set to zero to find the
constants α, β, and γ: (16)α=s22s5−s3s5s7+s1s72+s3s6s8−s2(s4s5+s7s8)s23+s32s6+s4s72−s2(s4s6+2s3s7),(17)β=s32s5+s1(s22−s3s7)+s4s7s8−s2(s4s5+s3s8)s23+s32s6+s4s72−s2(s4s6+2s3s7),(18)γ=s1s3s6+s4s5s7−s2(s3s5+s1s7)+s22s8−s4s6s8s23+s32s6+s4s72−s2(s4s6+2s3s7). Application of the test points to
(16)–(18) yielded α=8.088724,β=0.0886066, and γ = −0.2323394, resulting in a general performance
equation specific to the tested data: (19)H=8.088724Q2+0.0886066N2−0.2323394QN. The negative sign associated with the third term was
expected. The negative sign of third term is explained by the leakage flow and
recirculation [7]. The results of this surface fit are plotted two-dimensionally
by holding the rotational speed constant and are shown with corresponding tested
flow and head parameter data points in Figure 1.Figure 1
Head parameter versus flow parameter
tested data with general performance equation surface fit.Application of the same square error method to (11),
for the no-load condition, yields a relationship between λ and the tested data as follows: (20)λ=∑iQiNi∑iNi2. The
value of λ was found to be 0.051374. Substituted this into (13), along with α,β, and γ,
yielded the value, δ  =  37.13874. This generated a surface fit for the no-load
condition.  Using the now  known values
for α,β,γ,λ, and δ, the
surface fit for the  no-load curve was to be added to the constant speed curves from
Figure 1 and included the tested
flow and head parameter data to show complete  turbine performance envelope (see
Figure 2).Figure 2
Head parameter versus flow parameter tested data with general
performance equation surface fit including no load.The results from the three speeds and the  no-load curve clearly
show the applicability of the general performance equation. Furthermore, it can
be shown that this equation is also applicable to pumps.  Pumps also follow the affinity law
relationships and conservation of energy principal; therefore, the general
performance equation must also be applicable to pumps. To model this, the
principal that a pump is a turbine running in reverse is employed.  Essentially, a turbine running in reverse is
just a turbine with negative flow.  To
show this, a range of flow and head parameter values for the same three speeds
using (19) was plotted in Figure 3. As the flow parameter becomes negative, the
curves in Figure 3 show a similarity to pump performance curves, supporting the
applicability of (10) to pumps as well as turbines. These curves show a
striking similarity to those presented by Stepanoff [7].Figure 3
General performance equation for pumps and turbines.


## 5. Improvement

Figures1 and 2 show that the general performance
equation is an excellent fit to the tested data. To show actual improvement by
(10), the same method and the tested data were applied to the traditional performance
equation proposed by Kimmel in 1996 (see (21)) and then compared to results
obtained by the general performance equation: (21)H=AQ2+BN2. The sum of the square error for (21) is then as
follows: (22)∑iei2=∑i(AQi2+BNi2−Hi)2. In this  case, there are only two constants to  solve for, A and B;
therefore, there are only two partial derivatives of the sum of the square
error to set to zero resulting in two linear equations as follows: (23)A=s2s5−s1s6s22−s4s6,(24)B=s1s6−s4s5s22−s4s6. Application of the tested data points to
(23) and (24)
yielded A = 7.120981, and B = 0.0768359, resulting in (25)H=7.120981Q2+0.0768359N2. Now   with (19) and
(25), again the error was employed to
check how well
each equation fits the tested data. The square error was calculated for each
method and then a ratio was taken to indicate any improvement. The ratio of the
square error is then as follows: (26)Ratio=∑iei2∑iEi2. The ratio of the square error was calculated to be
1.10 and the root mean-square error is then the square root of this ratio. This
gives a root mean-square error of 1.05, indicating that the general performance
equation is 1.05 times better than the traditional equation. These results show
considerable improvement without a significantly more complicated equation.


## 6. Conclusions

A new general performance equation relating flow,
head, and rotational speed for liquefied gas reaction turbines was developed
and presented with performance test data. The general performance equation is
the most general and complete equation for the performance of liquefied gas
reaction turbines with constant geometry. In addition, from the analysis
presented, there are several strong findings supporting the claim that the general
performance equation is an improvement over the traditional performance
equation presented by Kimmel [4]. The general performance employs the affinity law
relationships between flow, head, and rotational speed with more accuracy. The
third term in the general performance equation incorporates the experienced
negative slope in a turbine performance curve. Furthermore, the general performance
equation is also applicable to pumps. The strongest evidence supporting the
improvement over the known affinity law is found in the error analysis. The error
generated by the general performance equation is considerably less, ultimately
providing a better engineering tool for the representation of  both turbine and pump
performances.



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*Source: 737285-2008-11-24.xml*