isentropic expansion of condensing steam

8/11/2019 Isentropic Expansion of Condensing Steam

1/9

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 158754413

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 transferof latent heat toward the vapor portion of the two-phase mixture, rises its stagnation temperature. A non-dimensional

function , 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 engineer-ing importance.

Applications include condensing flows of most air

or combustion product, aerosol formation in mixing

processes, aerodynamic testing in cryogenic wind tun-

nels and wetness problems in steam turbines and ex-

pansion 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 prop-

erties [1]. Therefore, knowing the stagnation prop-erties such as total temperature and pressure at any

point is a vital issue. Following single-phase mea-

suring 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 be-

tween the two phases are maintained, and the pitot

tube would measure the equilibrium stagnation pres-

sure [5]. Hence, all interphase transfer processes re-

main essentially frozen. Although, imposing the as-

sumption of equilibrium thermodynamic model in the

wet flow studies is restrictive, but the development of

non-equilibrium multi-phase models begins with theknowledge 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 con-

densing 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 analyti-

cal solution, and reports a progress in our on-line de-

velopment. Here, we provide a new chart and table toconveniently 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 con-

ditions 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

1Assistant Professor, Corresponding Author, E-mail: [email protected] Student3

Professor


2/9



of state Lee-Kesler is suitable for higher values of

pressure too. Accuracy assessment tests show excel-

lent agreement between the predictions of numerical

results and analytical solutions.

Process Evaluation

Consider a dry steam flow entering a converging-

diverging nozzle that isentropically expands along the

duct, as illustrated in Fig. 1. According to the equi-

librium 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:

T0,res.T =

1 +

1

2 M2

dry(1)

where is the ratio of specific heats of the vapor,T0,res. is the stagnation temperature at inflow, andM andT represent the local Mach number and statictemperature, 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 stagna-

tion temperature for the vapor portion of the two-

phase mixture can be defined [1]:

T0,localT

=

1 +

1

2 M2

wet

(2)

T0,localis the local stagnation temperature of the va-por portion of the two-phase mixture, which is larger

than that of the inflow (T0,local > To,res.) due to thetransfer of latent heat from the condensate toward the

vapor.

We define a non-dimensional function repre-senting the rise in the stagnation temperature of the

vapor portion of the two-phase mixture, as:

T0,local T0,res.T

(3)

Using Eqn. 2, becomes:

=

1 +

1

2 M2

wet

(T0,res.

T ) . (4)

Applying the first law of thermodynamics for a

control volume between the nozzle inlet, and an arbi-

trary point in two-phase region along the nozzle, one

can write:

mtot. h0,res. = mg(hg +

V2g

2 ) + mf(hf+

V2f

2 ) , (5)

whereh0,res. is the stagnation enthalpy at the nozzleinlet, mtot. is the mass flow through the nozzle, mgand mfare the vapor and liquid mass flow, hg and hfare the enthalpy of the vapor and liquid, and Vgand Vfare the vapor and liquid velocities at the arbitrary sec-

tion. From the mass balance around our control vol-ume (with no mass accumulation within the nozzle),

we can write, mtot. = mg+ mf. If the slip velocitybetween 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 writtenas:

T0,res.T

= 1 (1 ) hfgCp T

+ V2

2 Cp T , (6)

where = mg/mtot. is the quality at any arbitrarysection. On the other hand, for a two-phase mix-

ture = (S0,res. Sf)/Sfg , where S representsentropy. Using the concept of frozen Mach number

(M2 =V2/RT), Eqn. 6 results:

Sg S0,res.CP

=

1 +

1

2 M2

wet

T0,res.

T (7)

Comparing the Eqns. 7 and 4:

= Sg S0,res.

CP(8)

The Eqn. 8 is an interesting and conceptual equation,

describing that is proportional to the entropy rise ofthe gas phase (vapor) from inlet. This entropy rise is

due to reversible heat flow from the condensate toward

the vapor phase. It is noted thattakes a zero valuefrom inlet to the c.o. point. However, beyond this

pointaccepts positive values, and it is an increasingfunction along the process. That is:

= 0.0 for T Tc.o. (9)

>0.0 for T < Tc.o. . (10)

The locus of isentropic process (along S = S0,res.=constant) onT Sdiagram is optional, and dependson the inlet conditionsT0,res. and P0,res.. The lowestvalue that the inflow entropy can take corresponds to

the value at the critical point, (Tcr., Pcr.) = (647.29K, 220.09 bar), and is Scr. = S0,res. = 4.4298kJ/kg.K, [8]. On the other hand, the highest valuethat the inflow entropy can possess corresponds to

the value of the triple point, (Ttr., Ptr.) = (273.16K, 0.00611 bar), which is Str. = S0,res. = 9.1562kJ/kg.K. Therefore, S0,res. can accepts any valuebetween the entropy of the critical point and that of

the triple point, i.e. Scr.< S0,res. < Str..


3/9



In Eqn. 8,Sg is a function of only temperature,T.Therefore,, becomes a function of two independentvariablesT andS0,res.:

=(T, S0,res.) (11)

Noting that Sc.o.= S0,res., therefore, =(T, Sc.o.),and:

= Sg Sc.o.

CP. (12)

Therefore, the range of variation ofSc.o. [Scr., Str.],andT Ttr..

In the present study we assume the vapor as a real

gas, and a compressibility factor is employed to in-

clude deviations from ideal gases.

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 general-

ized equation of state has been used in this study. This

equation has twelve constants and is written as, [7]:

Z= P v

RT or P v= Z RT (13)

whereZis the compressibility factor that shows de-

viations from the ideal gas equation of state, v is thespecific volume of the gas, andR is the gas constant.The non-dimensional virial form of Eqn. 13 can be

written as:

Z = Prvr

Tr= 1 +

B(T)

vr+

C(T)

(vr)2

+D(T)

(vr)5

+ c4

(T3r)(v

r)2

(+

(vr)2

)exp(

(vr)2

)

(14)

where

B(T) = b1 (b2/Tr) (b3/T2

r) (b4/T3

r)C(T) = c1 (c2/Tr) (c3/T

3

r)

D(T) = d1+ (d2/Tr)

in which the non-dimensional variablesvr ,Tr and Prare:

vr = v

RTcr./Pcr., Tr =

T

Tcr.and Pr =

P

Pcr.,

(15)

where Tcr. and Pcr. are the critical temperature andpressure of steam, respectively. Empirical constants

for pure substances like water are given in Ap-

pendix A.

TheFunction

To develop an equation describing the variation of,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 points1and2in superheated re-gion (including the saturated vapor line) is obtained

using, [7]:

S2 S1 =

2

1

CpdT

T

2

1

(v

T)

PdP (16)

Differentiating Eqn. 13 along an iso-bar line:

(v

T)P =

R

P

Z+ T(Z

T)P

(17)

The compressibility factor along the saturated vapor

line, Zg, can be obtained from the Lee-Kesler equa-tion of state. Using the data provided for Zg andPralong the saturated vapor line, [7], we fit a polynomial

of degreen forZg:

Zg =AnPnr + An1P

n1r + ... + A1Pr+ A0 (18)

where n = 6 represents enough accuracy for the

curve-fit, and the coefficients A1 to A6 are givenin Appendix A. In the case of an ideal gas again,

Zg = 1 and in Eqn. 18, A0=1 and Ak = 0.0 fork {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):

Sg Sc.o. =

gc.o.

CpdT

T

gc.o.

R

Zg+ T(

ZgT

)P

dP

P . (19)

Again, in the case of ideal gases,Zg is a constant, =1,and Eqn. 19 is simplified to:

Sideal=Cpln( T

Tc.o.) R ln(

P

Pc.o.), (20)

whereSideal is the entropy rise in the case of idealgas. In the case of real gases, the entropy change is

obtained from:

Sg Sc.o. = Sideal+ Sdeviation (21)


4/9



in which Sdeviation represents the deviation fromthe ideal gas predictions, where,

Sdeviation = R[ (1 A0)ln( P

Pc.o.)

nk=1

Ak

k (Pkr,g Pkr,c.o.) ]

gc.o.

RT(ZgT

)PdP

P .(22)

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 forisdeveloped as:

= ln( T

Tc.o.)

1

ln(

P

Pc.o.) +

1

[ (1 A0)ln( P

Pc.o. )

nk=1

Akk (P

kr,g

Pkr,c.o.)

gc.o.

RT

ZgT

P

dP

P ] (23)

where= 1.32 for vapor.Equation 23 reiterates our earlier claim that is

a function of two variables T, and Sc.o. (or S0,res.).This is explained below. Along the saturated vapor

lineP = Psat, and it is a function of only tempera-ture. On the other hand the term(Zg/T)P in theintegrand can be written as:

ZgT

P

=

dZgdPr

dPrdP

dPdT

(24)

The first term in the right hand side of Eqn. 24 is ob-

tained 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 ofthe salutation line, and is obtained from Eqn. 15 in

Appendix A. On the other hand the Tc.o. andPc.o. inEqn. 24 are fixed based on the value Sg(Tc.o.) = Sc.o..Therefore, as stated in Eqn. 15,=(T, S0,res.) .

TheChart

In this section, we derive a relationship between

T0,res. and T0,localfor an isentropic process.As shown in Fig. 3, for any point along an isen-

tropic expansion process and within the two-phase re-

gion, 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 (T, S0,res.), (25)

and substitutingfrom Eqn. 23 into Eqn. 25, one canobtain:

T0,local = T0,res.+ T[ ln( T

Tc.o.)

1

(ln( P

Pc.o.) + (1 A0)ln(

P

Pc.o.)

n

k=1

Akk

(Pkr,g Pkr,c.o.)

gc.o.

RT(ZgT

)PdP

P ) ]

(26)

It is noted that for T Tc.o. the flow is dry,andT0,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 tworeservoir propertiesT0,res.,S0,res. and the local tem-peratureT. Since the inflow stagnation properties, i.e.T0,res.and S0,res., are optional, and given we can sup-pose them as two constants C1and C2, respectively,

T0,res. = C1 S0,res. = C2 . (27)

This makes Eqn. 11 a general formula to depict a fam-ily of curves describing the locus of local stagnation

conditions in a TSdiagram. These family of curvesfor given C1 and C2 are obtained in the followingform:

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 con-ditions of the vapor portion of a two-phase mixture?

Figure 6 shows a superheated inflow vapor that ex-

pands 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 va-

por line. Then this point is vertically extended until to

meet a curve that the inflow stagnation point belongs


5/9



to that curve at T = T0,local and S = S0,local, seeFig. 6.

The limiting case, when the inflow stagnation

point is located right on the saturated vapor line, is

illustrated in Fig. 7.

TheTable

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 mix-

ture. 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 fo-

cuss on the variation offunction.Using the Eqns. 8, 9, 10 and 23, a 3-D surface

is obtained in S-T- space, which is describing thebehavior of function in any isentropic expansion ofcondensing steam flow. Figure 8 shows the -surface.

Now, it is possible to develop a table using the

data of surface. This table can simplify the calcu-lation procedure of the local stagnation conditions.

As shown in Fig. 9, the first left column is represent-

ing the temperature and along the another columns

which has an special entropy, the function is vary-ing 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 stagnationstate 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 hass=s0,res.. So, at each cell on thecolumn one can read a value for corresponding toa static temperature T in the same row on the firsleft column. Now, havingT0,res,s0,res and extractedT and from the table, the local stagnation corre-sponding to Talong the process line, is derived as :T0,local = T0,res.+ T and s0,local =s0,res.+ CP.It is clear that, before c.o. point ( the cell which is

specified by light green on the column), function isequal to zero, hence the local stagnation properties are

same with those of initial values of upstream imagi-

nary reservoir. But beyond the c.o. point on the col-

umn,accepts the positive values and therefore localstagnation state is varied.

Verification of the Computation

For the compressions of analytical solutions (devel-

oped in the present paper) with numerical computation

(developed in Ref. [6]), several test cases with vari-

ous nozzle geometries and different expansion rates

are tested. Excellent agreement in all the cases were

achieved. A sample of compressions (between the nu-

merical results and analytical solutions) are given in

this section. To do so, a nozzle geometry with a rela-

tively high value of expansion rate (i.e. a large exit

to throat area ratio) has been chosen. These com-parisons 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 TNum. and Mach num-berMNum., and analytical solutions (obtained in thepresent study), forTAnal. andMAnal. are given. Asshown in Table 1, excellent agreement between the re-

sults are achieved.

Summary and Conclusions

The highlights of the present study are given here. A

new non-dimensional function is introduced in thepresent study, which represents the deviation of local

stagnation condition from that of the inflow. A ther-

modynamic chart and table for this function has been

provided. 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 re-

sults were compared with numerical computations [6].

Excellent agreement in all cases were obtained. The

vapor, in the present study, has been taken as a realgas 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 = +0.1181193b2 = +0.265728

b3 = +0.154790b4 = +0.030323c1 = +0.0236744c2 = +0.0186984c3 = 0.0c4 = +0.042724d1 10

4 =+0.155488d2 104 =+0.623689= +0.65392= +0.060167

(29)

Using the above equation of state, compressibility

factor of saturated vapor, Zg, can be determined by


6/9



sixth order as a function of reduced pressure of vapor:

Zg = A6P6

r + A5P5

r + A4P4

r + A3P3

r

+A2P2

r + A1P1

r + A0 (30)

where the coefficientsA0to A6are:

A6 = 14.7523A5 = -45.2802A4 = +52.6399A3 = -29.7745A2 = +8.6910A1 = -1.7379A0 = +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 =273.15K.

Entropy. The entropy of the mixture is determined

from s = sf +sfg , where sfg = hfg/T andsgis obtained from sg = Cpln T R lnp, and sf =sg sfg.

References

[1] Zayernouri, M. and Kermani, M. J. (2006) De-

velopment of an Analytical Solution for Com-

pressible Two-Phase Steam Flow, Transctions

Canadian Society for Mechanical Engineers, Ac-

cepted.

[2] Petr, V. & Kolovrantk, 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 cas-

cade. 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 the-

ory for a stationary cascade of steam turbine

blade. Phi. Trans. R. Soc. Lond. A354, 59-88

[5] Guha, A., A unified theory for the interpretationof 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 Predic-

tion using Roes Scheme, The 4th Conference of

Iranian AeroSpace Society, Amir Kabir Univer-

sity of Technology, Tehran, Iran, January 2729,

2003.

[7] Van Wylen, Borgnakke, Sonntag, Fundamen-

tals of Thermodynamics,6th Edition, John Wi-ley & Sons, 2002.

[8] Moran, M. J. and Shapiro, H. N., Fundamentals

of Engineering Thermodynamics, 4th Edition,John Wiley & Sons, 1998.


7/9



Figure 1: Schematic of an isentropic expansion of

steam flow through a nozzle.

Figure 2: Deviation of superheated and saturated va-

por from ideal-gas equation of state.

Figure 3: Schematic of isentropic processes from the

saturated vapor line to their corresponding local stag-

nation conditions.

Figure 4: The locus of local stagnation conditions

of the vapor portion of the two phase mixture.

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 )

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

1400

T(K

)

Two Phase Region

critical point

Figure 5: Locus of the family of curves for the local

stagnation conditions.


8/9



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 )

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

T(K

) Stagnation state ofimaginary reservoir

Condensation onset

T = T0, local

Local stagnation state of the vapor,corresponding to specified point on

the saturated vapor line.

An arbitrary point onthe process line

Inlet point

( S0,local

, T0,local

)

S = S0,local

Figure 6: Schematic of the procedure to determine the

local stagnation.

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 )

300

350

400

450

500

550

600

650

700

750

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

T(K

)

Stagnation state ofimaginary reservoir

T = T0, local

Local stagnation state of the vapor,corresponding to specified point on

the saturated vapor line.

An arbitrary point onthe process line

Inlet point

( S 0,local, T 0,local)

S = S0,local

Figure 7: Schematic of the procedure to determine the

local stagnation (the limiting case in which inflow is

on the saturated vapor line).

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

Zeta

4.55

5.56

6.57

7.58

8.59 Entr

opy(kJ/k

g.K)

300

350

400

450

500

550

600

650

Temperature

(K)XY

Z

Zeta

2.48978

2.40419

2.3421

2.244312.13911

2.03391

1.9287

1.8235

1.7183

1.6131

1.5079

1.40269

1.29749

1.19229

1.08709

0.981886

0.876684

0.771482

0.66628

0.561078

0.455876

0.350674

0.245472

0.140269

0.0422393

0.0139029

0

Figure 8: Chart (surface view) of function.

X (m) -0.2 -0.1 0 0.1

A(m2) 0.0379 0.0353 0.0315 0.0366TNum.(K) 337 332.2 326.3 315.9TAnal.(K) 337.5 332.2 326.7 315.9

MNum. 0.562 0.650 0.928 1.203MAnal. 0.561 0.650 0.933 1.203

X (m) 0.2 0.3 0.4 0.50

A(m2) 0.0417 0.0468 0.0519 0.0570TNum.(K) 311.1 307.4 304.5 302TAnal.(K) 311.1 307.4 304.5 302

MNum. 1.440 1.546 1.630 1.696MAnal. 1.439 1.545 1.629 1.695

Table 1: Comparison between temperature and Mach

number along an arbitrary nozzle with a relatively

high value of expansion rate. The subscriptsNum.and Anal. refer to numerical values (obtained inRef. [6]) and analytical solutions (obtained in the

present study), respectively.


9/9



Figure 9: Table offunction.

isentropic expansion of condensing steam

Documents