14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
DEVELOPMENT OF A NEW THERMODYNAMIC CHART FOR
ISENTROPIC EXPANSION OF CONDENSING STEAM FLOW
M. J. Kermani 1
M. Zayernouri 2
M. Saffar-Avval 3
Department of Mechanical Engineering
Amirkabir University of Technology (Tehran Polytechnic)
Tehran, Iran 15875–4413
Abstract
A new thermodynamic chart for isentropic expansion of compressible steam flow is developed. The steam is assumed
to obey local equilibrium thermodynamic model, where condensation onsets as soon as the saturation line is crossed
at “c.o.”. Above the “c.o.”, the stagnation properties reflect those at inflow. However, beyond the “c.o.”, the transfer
of latent heat toward the vapor portion of the two-phase mixture, rises its stagnation temperature. A non-dimensional
function “ζ”, is defined, which represents the increase in vapor stagnation temperature. The vapor is assumed to be a
real gas obeying the “Lee-Kesler” EOS.
Keywords: Analytical Solution of Steam – Equilibrium Thermodynamics – Compressible Steam Flow
Introduction
Correct prediction of moisture levels in wet steam
flows is both scientifically interesting and of engineering importance.
Applications include condensing flows of most air
or combustion product, aerosol formation in mixing
processes, aerodynamic testing in cryogenic wind tunnels and wetness problems in steam turbines and expansion in nozzles. In many industrial equipments
such as vapor nozzles, it has been shown that focusing
on the gas phase and constructing a correct relation
between its static and stagnation conditions at each
point on the process line, it is possible to predict the
flow field characteristics and its thermodynamic properties [1]. Therefore, knowing the stagnation properties such as total temperature and pressure at any
point is a vital issue. Following single-phase measuring techniques, stagnation probes are often used in
two phase flow situation [2, 3, 4]. In practice if the
size of liquid droplets is small (less than one micron)
the momentum (inertia) and thermal equilibrium between the two phases are maintained, and the pitot
tube would measure the equilibrium stagnation pressure [5]. Hence, all interphase transfer processes re1 Assistant
2 Graduate
3 Professor
main essentially frozen. Although, imposing the assumption of equilibrium thermodynamic model in the
wet flow studies is restrictive, but the development of
non-equilibrium multi-phase models begins with the
knowledge of equilibrium state.
In our earlier work we developed an algorithm to
numerically compute the flow characteristics along a
converging-diverging duct, [6], and we modelled condensing steam flow under equilibrium thermodynamic
model. Later an analytical solution was provided for
an identical problem, [1], and an excellent agreement
between the results were obtained. In [1, 6] we used
the ideal gas equation of state for vapor.
The present paper is a continuation to our analytical solution, and reports a progress in our on-line development. Here, we provide a new chart and table to
conveniently determine the local stagnation states of
the vapor portion of a two-phase mixture through an
isentropic expansion of the mixture. These conditions
are used to fix the thermodynamic states and flow conditions along the duct. Here we use the “Lee-Kesler”
equation of state for the vapor. Although it is observed
that the equation of states “Lee-Kesler” and ideal gas
provide same results in low pressure application (less
than 30 kPa used in this study), however, the equation
Professor, Corresponding Author, E-mail: mkermani@aut.ac.ir
Student
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
of state “Lee-Kesler” is suitable for higher values of
pressure too. Accuracy assessment tests show excellent agreement between the predictions of numerical
results and analytical solutions.
Process Evaluation
Consider a dry steam flow entering a convergingdiverging nozzle that isentropically expands along the
duct, as illustrated in Fig. 1. According to the equilibrium thermodynamic model, the flow remains dry
up to the “condensation onset” point (“c.o.”), beyond
which a second phase (liquid water) in generated. The
stagnation conditions attributed to the dry flow (the
flow between the inlet and “c.o.”) stay constant, and
can be obtained from:
¶
µ
γ−1 2
T0,res.
(1)
= 1+
M
T
2
dry
where γ is the ratio of specific heats of the vapor,
T0,res. is the stagnation temperature at inflow, and
M and T represent the local Mach number and static
temperature, respectively. However, beyond the “c.o.”
point, the transfer of latent heat from the condensate
toward the vapor, rises the stagnation temperature of
the vapor portion of the two-phase mixture, where
Eqn. 1 cannot be used. As a result a “local stagnation” temperature for the vapor portion of the twophase mixture can be defined [1]:
µ
¶
T0,local
γ−1 2
= 1+
M
(2)
T
2
wet
T0,local is the “local stagnation” temperature of the vapor portion of the two-phase mixture, which is larger
than that of the inflow (T0,local > To,res. ) due to the
transfer of latent heat from the condensate toward the
vapor.
We define a non-dimensional function “ζ” representing the rise in the stagnation temperature of the
vapor portion of the two-phase mixture, as:
ζ≡
T0,local − T0,res.
T
Using Eqn. 2, “ζ” becomes:
¶
µ
T0,res.
γ−1 2
−(
M
).
ζ = 1+
2
T
wet
(3)
(4)
Applying the first law of thermodynamics for a
control volume between the nozzle inlet, and an arbitrary point in two-phase region along the nozzle, one
can write:
ṁtot. h0,res. = ṁg (hg +
Vf2
Vg2
) + ṁf (hf +
) , (5)
2
2
where h0,res. is the stagnation enthalpy at the nozzle
inlet, ṁtot. is the mass flow through the nozzle, ṁg
and ṁf are the vapor and liquid mass flow, hg and hf
are the enthalpy of the vapor and liquid, and Vg and Vf
are the vapor and liquid velocities at the arbitrary section. From the mass balance around our control volume (with no mass accumulation within the nozzle),
we can write, ṁtot. = ṁg + ṁf . If the slip velocity
between the phases is ignored (i.e., Vf = Vg = V ),
and assuming an average iso-bar specific heat value
CP for the gas (vapor) phase, Eqn. 5 can be written
as:
hf g
V2
T0,res.
= 1 − (1 − χ)
+
,
T
Cp T
2 Cp T
(6)
where χ = ṁg /ṁtot. is the quality at any arbitrary
section. On the other hand, for a two-phase mixture χ = (S0,res. − Sf )/Sf g , where S represents
entropy. Using the concept of frozen Mach number
(M 2 = V 2 /γRT ), Eqn. 6 results:
Sg − S0,res.
=
CP
µ
1+
γ−1 2
M
2
¶
−
wet
T0,res.
(7)
T
Comparing the Eqns. 7 and 4:
ζ=
Sg − S0,res.
CP
(8)
The Eqn. 8 is an interesting and conceptual equation,
describing that ζ is proportional to the entropy rise of
the gas phase (vapor) from inlet. This entropy rise is
due to reversible heat flow from the condensate toward
the vapor phase. It is noted that ζ takes a zero value
from inlet to the “c.o.” point. However, beyond this
point ζ accepts positive values, and it is an increasing
function along the process. That is:
ζ = 0.0
for
T ≥ Tc.o.
(9)
ζ > 0.0
for
T < Tc.o. .
(10)
The locus of isentropic process (along S = S0,res. =
constant) on T − S diagram is optional, and depends
on the inlet conditions T0,res. and P0,res. . The lowest
value that the inflow entropy can take corresponds to
the value at the critical point, (Tcr. , Pcr. ) = (647.29
K, 220.09 bar), and is Sc r. = S0,res. = 4.4298
kJ/kg.K, [8]. On the other hand, the highest value
that the inflow entropy can possess corresponds to
the value of the triple point, (Ttr. , Ptr. ) = (273.16
K, 0.00611 bar), which is Str. = S0,res. = 9.1562
kJ/kg.K. Therefore, S0,res. can accepts any value
between the entropy of the critical point and that of
the triple point, i.e. Scr. < S0,res. < Str. .
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
In Eqn. 8, Sg is a function of only temperature, T .
Therefore, ζ, becomes a function of two independent
variables T and S0,res. :
ζ = ζ(T, S0,res. )
(11)
Noting that Sc.o. = S0,res. , therefore, ζ = ζ(T, Sc.o. ),
and:
Sg − Sc.o.
.
(12)
ζ=
CP
Therefore, the range of variation of Sc.o. ∈ [Scr. , Str. ],
and T ≥ Ttr. .
In the present study we assume the vapor as a real
gas, and a compressibility factor is employed to include deviations from ideal gases.
The ζ Function
To develop an equation describing the variation of ζ,
Eqn. 12 will be used. To do so, we concentrate on
the entropy rise of the vapor portion of the two-phase
mixture (the numerator in Eqn. 12).
It can be shown that the entropy rise between two
arbitrary and distinct points 1 and 2 in superheated region (including the saturated vapor line) is obtained
using, [7]:
S2 − S1 =
Z
2
Cp
1
dT
−
T
Z
1
2
(
∂v
) dP
∂T P
(16)
Differentiating Eqn. 13 along an iso-bar line:
Equation of State
Deviations between the real and ideal gases at low
pressure and high temperature conditions (i.e. large
values of specific volume) is negligible, as shown in
Fig. 2 (the gray region), [7]. These deviations become
significant as the specific volume reduces. To take into
account the real gas effects, the “Lee-Kesler” generalized equation of state has been used in this study. This
equation has twelve constants and is written as, [7]:
Z=
Pv
RT
or
P v = ZRT
(13)
where Z is the compressibility factor that shows deviations from the ideal gas equation of state, v is the
specific volume of the gas, and R is the gas constant.
The non-dimensional virial form of Eqn. 13 can be
written as:
Z
=
Pr vr0
B(T ) C(T ) D(T )
= 1+
+ 0 2 + 0 5
Tr
vr0
(vr )
(vr )
c4
γ
γ
+ 3 0 2 (β + 0 2 ) exp(− 0 2 )
(Tr )(vr )
(vr )
(vr )
(14)
where
B(T ) = b1 − (b2 /Tr ) − (b3 /Tr2 ) − (b4 /Tr3 )
C(T ) = c1 − (c2 /Tr ) − (c3 /Tr3 )
D(T ) = d1 + (d2 /Tr )
in which the non-dimensional variables vr0 , Tr and Pr
are:
v
T
P
=
, Tr =
and Pr =
,
RTcr. /Pcr.
Tcr.
Pcr.
(15)
where Tcr. and Pcr. are the critical temperature and
pressure of steam, respectively. Empirical constants
for pure substances like water are given in Appendix A.
·
¸
R
∂v
∂Z
)P =
)P
(
Z + T(
∂T
P
∂T
(17)
The compressibility factor along the saturated vapor
line, Zg , can be obtained from the Lee-Kesler equation of state. Using the data provided for Zg and Pr
along the saturated vapor line, [7], we fit a polynomial
of degree n for Zg :
Zg = An Prn + An−1 Prn−1 + ... + A1 Pr + A0 (18)
where n = 6 represents enough accuracy for the
curve-fit, and the coefficients A1 to A6 are given
in Appendix A. In the case of an ideal gas again,
Zg = 1 and in Eqn. 18, A0 =1 and Ak = 0.0 for
k ∈ {1, 2, . . . , n}.
Applying Eqn. 16, along the saturated-vapor line
between the “c.o.” point and an arbitrary point along
the saturation vapor line (g):
Z g
dT
Sg − Sc.o. =
Cp
T
c.o.
¸
·
Z g
∂Zg
dP
−
)P
.
R Zg + T (
∂T
P
c.o.
(19)
Again, in the case of ideal gases, Zg is a constant, =1,
and Eqn. 19 is simplified to:
∆Sideal = Cp ln(
P
T
) − R ln(
),
Tc.o.
Pc.o.
(20)
vr0
where ∆Sideal is the entropy rise in the case of ideal
gas. In the case of real gases, the entropy change is
obtained from:
Sg − Sc.o. = ∆Sideal + ∆Sdeviation
(21)
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
in which ∆Sdeviation represents the deviation from
the ideal gas predictions, where,
∆Sdeviation
= R [ (1 − A0 ) ln(
−
−
n
X
Ak
k=1
Z g
c.o.
k
P
)
Pc.o.
k
k
(Pr,g
− Pr,c.o.
)]
RT (
∂Zg dP
)P
.(22)
∂T
P
It is noted that in case of ideal gases, ∆Sdeviation = 0,
so, Eqn. 21 is converted to Eqn. 20.
Now, using Eqns. 12, 21 and 20, a formula for ζ is
developed as:
γ−1
P
γ−1
T
)−
ln(
)+
×
Tc.o.
γ
Pc.o.
γ
n
X
Ak k
P
[ (1 − A0 ) ln(
)−
(Pr,g −
Pc.o.
k
k=1
¶
µ
Z g
dP
∂Zg
k
]
(23)
Pr,c.o.
)−
RT
∂T P P
c.o.
ζ = ln(
where γ = 1.32 for vapor.
Equation 23 reiterates our earlier claim that ζ is
a function of two variables T , and Sc.o. (or S0,res. ).
This is explained below. Along the saturated vapor
line P = Psat , and it is a function of only temperature. On the other hand the term (∂Zg /∂T )P in the
integrand can be written as:
¶ µ
¶
¶ µ
¶
µ
µ
dPr
dP
dZg
∂Zg
×
×
(24)
=
∂T P
dPr
dP
dT
The first term in the right hand side of Eqn. 24 is obtained from the polynomial curve fit of Eqn. 15 being
a function of pressure and consequently temperature
only along the saturated vapor line, the second term
is equal to 1/Pcr. , and the last term is the slope of
the salutation line, and is obtained from Eqn. 15 in
Appendix A. On the other hand the Tc.o. and Pc.o. in
Eqn. 24 are fixed based on the value Sg (Tc.o. ) = Sc.o. .
Therefore, as stated in Eqn. 15, ζ = ζ(T, S0,res. ) .
The ζ Chart
In this section, we derive a relationship between
T0,res. and T0,local for an isentropic process.
As shown in Fig. 3, for any point along an isentropic expansion process and within the two-phase region, there exists a point on the saturated vapor line
that if an imaginary stagnant condition (called “local
stagnation”) adiabatically and reversibly expands, it
will arrive to the same point on saturated-vapor line.
Similarly, as the flow marches along the nozzle, a set
of “local stagnation” points are sought, as shown in
Fig. 3. The locus of these “local stagnation” points
form a curve as shown in Fig. 4.
Replacing Eqn. 2 in Eqn. 4, we obtain:
T0,local = T0,res. + T ζ(T, S0,res. ),
(25)
and substituting ζ from Eqn. 23 into Eqn. 25, one can
obtain:
T0,local
T
γ−1
)−
×
Tc.o.
γ
P
P
( ln(
) + (1 − A0 ) ln(
)
Pc.o.
Pc.o.
n
X
Ak k
k
)
−
(Pr,g − Pr,c.o.
k
k=1
Z g
∂Zg dP
−
RT (
)P
)]
∂T
P
c.o.
(26)
= T0,res. + T [ ln(
It is noted that for T ≥ Tc.o. the flow is dry,
and T0,local is equal to T0,res. . Otherwise, T0,local >
T0,res. (see Figs. 3 and 4), as the flow is in wet region.
Assessing Eqns. 11 and 25, it is noticed that
T0,local is a function of three variables, including two
reservoir properties T0,res. , S0,res. and the local temperature T . Since the inflow stagnation properties, i.e.
T0,res. and S0,res. , are optional, and given we can suppose them as two constants C1 and C2 , respectively,
T0,res. = C1
S0,res. = C2 .
(27)
This makes Eqn. 11 a general formula to depict a family of curves describing the locus of “local stagnation”
conditions in a T −S diagram. These family of curves
for given C1 and C2 are obtained in the following
form:
T0,local = T0,local (C1 , C2 , T )
(28)
Figure 5 shows a family of curves which describe the
“local stagnation” conditions of the vapor portion of
the two-phase mixture. These curves start from the
saturated-vapor line, where the first curve onsets from
the critical point.
At this point the question is how can this figure
(Fig. 5) help us to obtain the “local stagnation” conditions of the vapor portion of a two-phase mixture?
Figure 6 shows a superheated inflow vapor that expands through an isentropic process from the reservoir
stagnation conditions. When the process crosses the
saturated vapor line, the gas phase starts to move the
saturated-vapor line (the blue line shown in Fig. 6).
The “local stagnation” conditions of an arbitrary
point on saturated vapor line is obtained as follows.
A horizontal line, extended from an arbitrary point on
the process line (see Fig. 6), crosses the saturated vapor line. Then this point is vertically extended until to
meet a curve that the inflow stagnation point belongs
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
to that curve at T = T0,local and S = S0,local , see
Fig. 6.
The limiting case, when the inflow stagnation
point is located right on the saturated vapor line, is
illustrated in Fig. 7.
The ζ Table
In the previous section, we developed a new chart that
contains a family of curves, which its significance was
to give a profile for the behavior of “local stagnation”
conditions of the vapor portion of the two-phase mixture. In this section, we aim to reword our discussions
in previous section in order to give a user friendly way
to extract the local stagnation states. To do so, we focuss on the variation of ζ function.
Using the Eqns. 8, 9, 10 and 23, a 3-D surface
is obtained in S-T -ζ space, which is describing the
behavior of ζ function in any isentropic expansion of
condensing steam flow. Figure 8 shows the ζ-surface.
Now, it is possible to develop a table using the
data of ζ surface. This table can simplify the calculation procedure of the “local stagnation” conditions.
As shown in Fig. 9, the first left column is representing the temperature and along the another columns
which has an special entropy, the ζ function is varying corresponding to temperature. In fact each column
in this table is representing a isentropic process with
special amount of entropy which is specified at the top
of each column. Now, to extract the local stagnation
state along an isentropic expansion process, knowing
the initial stagnation properties, (T0,res. and s=s0,res.
), it is enough that one marches downward along the
column which has s=s0,res. . So, at each cell on the
column one can read a value for ζ corresponding to
a static temperature T in the same row on the firs
left column. Now, having T0,res , s0,res and extracted
T and ζ from the table, the local stagnation corresponding to T along the process line, is derived as :
T0,local = T0,res. + T ζ and s0,local = s0,res. + CP ζ.
It is clear that, before “c.o.” point ( the cell which is
specified by light green on the column), ζ function is
equal to zero, hence the local stagnation properties are
same with those of initial values of upstream imaginary reservoir. But beyond the “c.o.” point on the column, ζ accepts the positive values and therefore local
stagnation state is varied.
Verification of the Computation
For the compressions of analytical solutions (developed in the present paper) with numerical computation
(developed in Ref. [6]), several test cases with various nozzle geometries and different expansion rates
are tested. Excellent agreement in all the cases were
achieved. A sample of compressions (between the numerical results and analytical solutions) are given in
this section. To do so, a nozzle geometry with a relatively high value of expansion rate (i.e. a large exit
to throat area ratio) has been chosen. These comparisons are performed in Table 1. Table 1 shows
the nozzle cross sectional area (A) along the nozzle
axis (X), and compares the numerical values (obtained
in Ref. [6]) for temperature TN um. and Mach number MN um. , and analytical solutions (obtained in the
present study), for TAnal. and MAnal. are given. As
shown in Table 1, excellent agreement between the results are achieved.
Summary and Conclusions
The highlights of the present study are given here. A
new non-dimensional function ζ is introduced in the
present study, which represents the deviation of “local
stagnation” condition from that of the inflow. A thermodynamic chart and table for this function has been
provided. ζ takes values equal to zero in dry regions,
and positive in wet regions. The developed table is
general and can be used for any geometry of nozzle,
with arbitrarily selected inflow stagnation properties.
The method is applied to several test cases and the results were compared with numerical computations [6].
Excellent agreement in all cases were obtained. The
vapor, in the present study, has been taken as a real
gas obeying the “Lee-Kesler” equation of state.
Appendix A
Constants of Lee-Kesler Equation of state. The
sets of constants of Lee-Kesler Equation of state is as
follows:
b1 = +0.1181193
b2 = +0.265728
b3 = +0.154790
b4 = +0.030323
c1 = +0.0236744
c2 = +0.0186984
c3 = 0.0
c4 = +0.042724
d1 × 104 = +0.155488
d2 × 104 = +0.623689
β = +0.65392
γ = +0.060167
(29)
Using the above equation of state, compressibility
factor of saturated vapor, Zg , can be determined by
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
sixth order as a function of reduced pressure of vapor:
Zg
= A6 Pr6 + A5 Pr5 + A4 Pr4 + A3 Pr3
+A2 Pr2 + A1 Pr1 + A0
(30)
where the coefficients A0 to A6 are:
A6
A5
A4
A3
A2
A1
A0
= 14.7523
= -45.2802
= +52.6399
= -29.7745
= +8.6910
= -1.7379
= +0.9995
(31)
Saturated Pressure Value. The saturation pressure
for steam is determined by a fifth order polynomial
least square curve fit to the steam data taken from [6]
and [8]. given by:
psat
= B5 (T − t0 )5 + B4 (T − t0 )4
+B3 (T − t0 )3 + B2 (T − t0 )2
+B1 (T − t0 ) + B0 ,
(32)
where p and T are in terms of P a and K, t0 =
273.15 K.
Entropy. The entropy of the mixture is determined
from s = sf + χsf g , where sf g = hf g /T and sg
is obtained from sg = Cp ln T − R ln p, and sf =
sg − sf g .
References
[1] Zayernouri, M. and Kermani, M. J. (2006) “Development of an Analytical Solution for Com-
pressible Two-Phase Steam Flow”, Transctions–
Canadian Society for Mechanical Engineers, Accepted.
[2] Petr, V. & Kolovrant’k, M. 1994 “Laboratory
and field measurements of droplet nucleation in
expansion steam ”. 12th Int. Conf. on Properties
of Water and Steam, Sept. 11-16. FL, ASME.
[3] Stastny, M. & Sejna, M. 1994 “Condensation
effects in transonic flow through turbine cascade”. 12th Int. Conf. on Properties of Water and
Steam, Sept. 11-16. FL, ASME.
[4] White, A. J., Young, J. B. & Walters, P. T. 1996 “
Experimental validation of condensing flow theory for a stationary cascade of steam turbine
blade”. Phi. Trans. R. Soc. Lond. A354, 59-88
[5] Guha, A., “A unified theory for the interpretation
of total pressure and temperature in two-phase
flows at subsonic and supersonic speads,” Proc.
R. Soc. Lond. A (1998) 454, 671-695.
[6] Kermani, M. J., Gerber, A. G., and Stockie, J.
M., Thermodynamically based Moisture Prediction using Roe’s Scheme, The 4th Conference of
Iranian AeroSpace Society, Amir Kabir University of Technology, Tehran, Iran, January 27–29,
2003.
[7] Van Wylen, Borgnakke, Sonntag, “Fundamentals of Thermodynamics,” 6th Edition, John Wiley & Sons, 2002.
[8] Moran, M. J. and Shapiro, H. N., “Fundamentals
of Engineering Thermodynamics,” 4th Edition,
John Wiley & Sons, 1998.
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
Figure 1: Schematic of an isentropic expansion of
steam flow through a nozzle.
Figure 4: The locus of “ local stagnation” conditions
of the vapor portion of the two phase mixture.
Figure 2: Deviation of superheated and saturated vapor from ideal-gas equation of state.
1400
1350
1300
1250
1200
1150
1100
1050
1000
T(K)
950
900
850
800
750
700
650
600
critical point
550
500
450
400
Two Phase Region
350
300
4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 6.2 6.4 6.6 6.8 7 7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8 9
S ( kJ / kg.K )
Figure 5: Locus of the family of curves for the local
stagnation conditions.
Figure 3: Schematic of isentropic processes from the
saturated vapor line to their corresponding “local stagnation” conditions.
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
Zeta
2.48978
2.40419
2.3421
2.24431
1300
Local stagnation state of the vapor,
corresponding to specified point on
the saturated vapor line.
1250
1200
1150
1.8235
1.7183
( S 0,local , T0,local )
T = T0, local
1100
2.13911
2.03391
1.9287
1.6131
1.5079
1.40269
950
1.29749
1.19229
2.5
1.08709
0.981886
0.876684
2.25
0.771482
0.66628
2
0.561078
0.455876
0.350674
1.75
T(K)
900
850
Stagnation state of
imaginary reservoir
800
750
S = S 0,local
700
Inlet point
650
0.245472
0.140269
600
550
1.5
0.0422393
0.0139029
0
Condensation onset
500
1.25
1
450
400
350
Zeta
1000
1050
300
0.75
An arbitrary point on
the process line
350
300
Z
4.6 4.8
5
5.2 5.4 5.6 5.8
6
6.2 6.4 6.6 6.8
7
7.2 7.4 7.6 7.8
8
8.2 8.4 8.6 8.8
9
S ( kJ / kg.K )
X
Te
m
pe 450
ra
tu
re
Y
0.5
400
0.25
500
(K
)
550
5.5
600
Figure 6: Schematic of the procedure to determine the
local stagnation.
650
9
8.5
5
0
4.5
6
6.5
)
7
kg.K
(kJ/
opy
E ntr
8
7.5
Figure 8: Chart (surface view) of “ζ” function.
1300
Local stagnation state of the vapor,
corresponding to specified point on
the saturated vapor line.
1250
1200
1150
1100
( S 0,local , T0,local )
T = T0, local
1050
1000
950
T(K)
900
850
800
750
S = S 0,local
700
Stagnation state of
imaginary reservoir
650
600
550
500
Inlet point
450
An arbitrary point on
the process line
400
350
X (m)
A (m2 )
TN um. (K)
TAnal. (K)
MN um.
MAnal.
-0.2
0.0379
337
337.5
0.562
0.561
-0.1
0.0353
332.2
332.2
0.650
0.650
0
0.0315
326.3
326.7
0.928
0.933
0.1
0.0366
315.9
315.9
1.203
1.203
X (m)
A (m2 )
TN um. (K)
TAnal. (K)
MN um.
MAnal.
0.2
0.0417
311.1
311.1
1.440
1.439
0.3
0.0468
307.4
307.4
1.546
1.545
0.4
0.0519
304.5
304.5
1.630
1.629
0.50
0.0570
302
302
1.696
1.695
300
4.6 4.8
5
5.2 5.4 5.6 5.8
6
6.2 6.4 6.6 6.8
7
7.2 7.4 7.6 7.8
8
8.2 8.4 8.6 8.8
9
S ( kJ / kg.K )
Figure 7: Schematic of the procedure to determine the
local stagnation (the limiting case in which inflow is
on the saturated vapor line).
Table 1: Comparison between temperature and Mach
number along an arbitrary nozzle with a relatively
high value of expansion rate. The subscripts N um.
and Anal. refer to numerical values (obtained in
Ref. [6]) and analytical solutions (obtained in the
present study), respectively.
14th Annual (International) Mechanical Engineering Conference - May 2006
Isfahan University of Technology, Isfahan, Iran
Figure 9: Table of ζ function.