DEVELOPMENT OF A NEW THERMODYNAMIC CHART FOR

14th Annual (International) Mechanical Engineering Conference - May 2006Isfahan University of Technology, Isfahan, Iran

DEVELOPMENT OF A NEW THERMODYNAMIC CHART FORISENTROPIC EXPANSION OF CONDENSING STEAM FLOW

M. J. Kermani1 M. Zayernouri2 M. Saffar-Avval3

Department of Mechanical EngineeringAmirkabir University of Technology (Tehran Polytechnic)

Tehran, Iran 15875–4413

AbstractA new thermodynamic chart for isentropic expansion of compressible steam flow is developed. The steam is assumedto obey local equilibrium thermodynamic model, where condensation onsets as soon as the saturation line is crossedat “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-dimensionalfunction “ζ”, is defined, which represents the increase in vapor stagnation temperature. The vapor is assumed to be areal gas obeying the “Lee-Kesler” EOS.

Keywords: Analytical Solution of Steam – Equilibrium Therm odynamics – Compressible Steam Flow

Introduction

Correct prediction of moisture levels in wet steamflows is both scientifically interesting and of engineer-ing importance.

Applications include condensing flows of most airor combustion product, aerosol formation in mixingprocesses, aerodynamic testing in cryogenic wind tun-nels and wetness problems in steam turbines and ex-pansion in nozzles. In many industrial equipmentssuch as vapor nozzles, it has been shown that focusingon the gas phase and constructing a correct relationbetween its static and stagnation conditions at eachpoint on the process line, it is possible to predict theflow field characteristics and its thermodynamic prop-erties [1]. Therefore, knowing the stagnation prop-erties such as total temperature and pressure at anypoint is a vital issue. Following single-phase mea-suring techniques, stagnation probes are often used intwo phase flow situation [2, 3, 4]. In practice if thesize of liquid droplets is small (less than one micron)the momentum (inertia) and thermal equilibrium be-tween the two phases are maintained, and the pitottube 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 thewet flow studies is restrictive, but the development ofnon-equilibrium multi-phase models begins with theknowledge of equilibrium state.

In our earlier work we developed an algorithm tonumerically compute the flow characteristics along aconverging-diverging duct, [6], and we modelled con-densing steam flow under equilibrium thermodynamicmodel. Later an analytical solution was provided foran identical problem, [1], and an excellent agreementbetween the results were obtained. In [1, 6] we usedthe 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 ofthe vapor portion of a two-phase mixture through anisentropic expansion of the mixture. These conditionsare 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 observedthat the equation of states “Lee-Kesler” and ideal gasprovide same results in low pressure application (lessthan 30 kPa used in this study), however, the equation

1Assistant Professor, Corresponding Author, E-mail: [email protected] Student3Professor


of state “Lee-Kesler” is suitable for higher values ofpressure too. Accuracy assessment tests show excel-lent agreement between the predictions of numericalresults and analytical solutions.

Process Evaluation

Consider a dry steam flow entering a converging-diverging nozzle that isentropically expands along theduct, as illustrated in Fig. 1. According to the equi-librium thermodynamic model, the flow remains dryup to the “condensation onset” point (“c.o.”), beyondwhich a second phase (liquid water) in generated. Thestagnation conditions attributed to the dry flow (theflow between the inlet and “c.o.”) stay constant, andcan be obtained from:

T0,res.

T=

(

1 +γ − 1

2M2

)

dry

(1)

whereγ 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 condensatetoward the vapor, rises the stagnation temperature ofthe vapor portion of the two-phase mixture, whereEqn. 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,local

T=

(

1 +γ − 1

2M2

)

wet

(2)

T0,local is the “local stagnation” temperature of the va-por portion of the two-phase mixture, which is largerthan that of the inflow (T0,local > To,res.) due to thetransfer of latent heat from the condensate toward thevapor.

We define a non-dimensional function “ζ” repre-senting the rise in the stagnation temperature of thevapor portion of the two-phase mixture, as:

ζ ≡T0,local − T0,res.

T(3)

Using Eqn. 2, “ζ” becomes:

ζ =

(

1 +γ − 1

2M2

)

wet

− (T0,res.

T) . (4)

Applying the first law of thermodynamics for acontrol volume between the nozzle inlet, and an arbi-trary point in two-phase region along the nozzle, onecan write:

mtot. h0,res. = mg(hg +V 2

g

2)+mf (hf +

V 2

f

2) , (5)

whereh0,res. is the stagnation enthalpy at the nozzleinlet, mtot. is the mass flow through the nozzle,mg

andmf are the vapor and liquid mass flow,hg andhf

are the enthalpy of the vapor and liquid, andVg andVf

are 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 valueCP for the gas (vapor) phase, Eqn. 5 can be writtenas:

T0,res.

T= 1 − (1 − χ)

hfg

Cp T+

V 2

2Cp T, (6)

whereχ = mg/mtot. is the quality at any arbitrarysection. On the other hand, for a two-phase mix-ture χ = (S0,res. − Sf )/Sfg, whereS representsentropy. Using the concept of frozen Mach number(M2 = V 2/γRT ), Eqn. 6 results:

Sg − S0,res.

CP

=

(

1 +γ − 1

2M2

)

wet

−T0,res.

T(7)

Comparing the Eqns. 7 and 4:

ζ =Sg − S0,res.

CP

(8)

The Eqn. 8 is an interesting and conceptual equation,describing thatζ is proportional to the entropy rise ofthe gas phase (vapor) from inlet. This entropy rise isdue to reversible heat flow from the condensate towardthe vapor phase. It is noted thatζ takes a zero valuefrom inlet to the “c.o.” point. However, beyond thispoint ζ accepts positive values, and it is an increasingfunction 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) onT − S diagram is optional, and dependson the inlet conditionsT0,res. andP0,res.. The lowestvalue that the inflow entropy can take corresponds tothe value at the critical point,(Tcr., Pcr.) = (647.29K, 220.09 bar), and isScr. = S0,res. = 4.4298kJ/kg.K, [8]. On the other hand, the highest valuethat the inflow entropy can possess corresponds tothe value of the triple point,(Ttr., Ptr.) = (273.16K, 0.00611 bar), which isStr. = S0,res. = 9.1562kJ/kg.K. Therefore,S0,res. can accepts any valuebetween the entropy of the critical point and that ofthe triple point, i.e.Scr. < S0,res. < Str..


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

ζ = ζ(T, S0,res.) (11)

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

ζ =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 realgas, and a compressibility factor is employed to in-clude deviations from ideal gases.

Equation of State

Deviations between the real and ideal gases at lowpressure and high temperature conditions (i.e. largevalues of specific volume) is negligible, as shown inFig. 2 (the gray region), [7]. These deviations becomesignificant as the specific volume reduces. To take intoaccount the real gas effects, the “Lee-Kesler” general-ized equation of state has been used in this study. Thisequation has twelve constants and is written as, [7]:

Z =Pv

RTor Pv = ZRT (13)

whereZ is 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 bewritten as:

Z =Prv

′

r

Tr

= 1 +B(T )

v′

r

+C(T )

(v′

r)2

+D(T )

(v′

r)5

+c4

(T 3r )(v′

r)2(β +

γ

(v′

r)2) exp(−

γ

(v′

r)2)

(14)

where

B(T ) = b1 − (b2/Tr) − (b3/T 2

r ) − (b4/T 3

r )

C(T ) = c1 − (c2/Tr) − (c3/T 3

r )

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

in which the non-dimensional variablesv′

r, Tr andPr

are:

v′

r =v

RTcr./Pcr.

, Tr =T

Tcr.

and Pr =P

Pcr.

,

(15)whereTcr. andPcr. are the critical temperature andpressure of steam, respectively. Empirical constantsfor pure substances like water are given in Ap-pendix A.

The ζ Function

To develop an equation describing the variation ofζ,Eqn. 12 will be used. To do so, we concentrate onthe entropy rise of the vapor portion of the two-phasemixture (the numerator in Eqn. 12).

It can be shown that the entropy rise between twoarbitrary and distinct points1 and2 in superheated re-gion (including the saturated vapor line) is obtainedusing, [7]:

S2 − S1 =

∫

2

1

Cp

dT

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 vaporline, Zg, can be obtained from the Lee-Kesler equa-tion of state. Using the data provided forZg andPr

along the saturated vapor line, [7], we fit a polynomialof degreen for Zg:

Zg = AnPnr + An−1P

n−1

r + ... + A1Pr + A0 (18)

where n = 6 represents enough accuracy for thecurve-fit, and the coefficientsA1 to A6 are givenin Appendix A. In the case of an ideal gas again,Zg = 1 and in Eqn. 18,A0=1 andAk = 0.0 fork ∈ {1, 2, . . . , n}.

Applying Eqn. 16, along the saturated-vapor linebetween the “c.o.” point and an arbitrary point alongthe saturation vapor line (g):

Sg − Sc.o. =

∫ g

c.o.

Cp

dT

T

−

∫ g

c.o.

R

[

Zg + T (∂Zg

∂T)P

]

dP

P. (19)

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

∆Sideal = Cp ln(T

Tc.o.

) − R ln(P

Pc.o.

), (20)

where∆Sideal is the entropy rise in the case of idealgas. In the case of real gases, the entropy change isobtained from:

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


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

∆Sdeviation = R [ (1 − A0) ln(P

Pc.o.

)

−

n∑

k=1

Ak

k(P k

r,g − P kr,c.o.) ]

−

∫ g

c.o.

RT (∂Zg

∂T)P

dP

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 forζ isdeveloped as:

ζ = ln(T

Tc.o.

) −γ − 1

γln(

P

Pc.o.

) +γ − 1

γ×

[ (1 − A0) ln(P

Pc.o.

) −

n∑

k=1

Ak

k(P k

r,g −

P kr,c.o.) −

∫ g

c.o.

RT

(

∂Zg

∂T

)

P

dP

P] (23)

whereγ = 1.32 for vapor.Equation 23 reiterates our earlier claim thatζ is

a function of two variablesT , andSc.o. (or S0,res.).This is explained below. Along the saturated vaporline P = Psat, and it is a function of only tempera-ture. On the other hand the term(∂Zg/∂T )

Pin the

integrand can be written as:(

∂Zg

∂T

)

P

=

(

dZg

dPr

)

×

(

dPr

dP

)

×

(

dP

dT

)

(24)

The first term in the right hand side of Eqn. 24 is ob-tained from the polynomial curve fit of Eqn. 15 beinga function of pressure and consequently temperatureonly along the saturated vapor line, the second termis equal to1/Pcr., and the last term is the slope ofthe salutation line, and is obtained from Eqn. 15 inAppendix A. On the other hand theTc.o. andPc.o. inEqn. 24 are fixed based on the valueSg(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 betweenT0,res. andT0,local for 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 linethat if an imaginary stagnant condition (called “localstagnation”) adiabatically and reversibly expands, itwill arrive to the same point on saturated-vapor line.Similarly, as the flow marches along the nozzle, a setof “local stagnation” points are sought, as shown in

Fig. 3. The locus of these “local stagnation” pointsform 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 substitutingζ from Eqn. 23 into Eqn. 25, one canobtain:

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

Tc.o.

) −γ − 1

γ×

( ln(P

Pc.o.

) + (1 − A0) ln(P

Pc.o.

)

−

n∑

k=1

Ak

k(P k

r,g − P kr,c.o.)

−

∫ g

c.o.

RT (∂Zg

∂T)P

dP

P) ]

(26)

It is noted that forT ≥ Tc.o. the flow is dry,andT0,local is equal toT0,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 thatT0,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. andS0,res., are optional, and given we can sup-pose them as two constantsC1 andC2, 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 aT −S diagram. 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 ofthe two-phase mixture. These curves start from thesaturated-vapor line, where the first curve onsets fromthe critical point.

At this point the question ishow 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 reservoirstagnation conditions. When the process crosses thesaturated vapor line, the gas phase starts to move thesaturated-vapor line (the blue line shown in Fig. 6).

The “local stagnation” conditions of an arbitrarypoint on saturated vapor line is obtained as follows.A horizontal line, extended from an arbitrary point onthe process line (see Fig. 6), crosses the saturated va-por line. Then this point is vertically extended until tomeet a curve that the inflow stagnation point belongs


to that curve atT = T0,local andS = S0,local, seeFig. 6.

The limiting case, when the inflow stagnationpoint is located right on the saturated vapor line, isillustrated in Fig. 7.

The ζ Table

In the previous section, we developed a new chart thatcontains a family of curves, which its significance wasto 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 discussionsin previous section in order to give a user friendly wayto extract the local stagnation states. To do so, we fo-cuss on the variation ofζ function.

Using the Eqns. 8, 9, 10 and 23, a 3-D surfaceis obtained inS-T -ζ space, which is describing thebehavior ofζ function in any isentropic expansion ofcondensing steam flow. Figure 8 shows theζ-surface.

Now, it is possible to develop a table using thedata ofζ 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 columnswhich has an special entropy, theζ function is vary-ing corresponding to temperature. In fact each columnin this table is representing a isentropic process withspecial amount of entropy which is specified at the topof each column. Now, to extract the local stagnationstate along an isentropic expansion process, knowingthe initial stagnation properties, (T0,res. ands=s0,res.

), it is enough that one marches downward along thecolumn which hass=s0,res.. So, at each cell on thecolumn one can read a value forζ corresponding toa static temperatureT in the same row on the firsleft column. Now, havingT0,res, s0,res and extractedT and ζ from the table, the local stagnation corre-sponding toT along the process line, is derived as :T0,local = T0,res. + Tζ ands0,local = s0,res. + CP ζ.It is clear that, before “c.o.” point ( the cell which isspecified by light green on the column),ζ function isequal to zero, hence the local stagnation properties aresame with those of initial values of upstream imagi-nary reservoir. But beyond the “c.o.” point on the col-umn,ζ 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 wereachieved. A sample of compressions (between the nu-merical results and analytical solutions) are given inthis section. To do so, a nozzle geometry with a rela-tively high value of expansion rate (i.e. a large exitto throat area ratio) has been chosen. These com-parisons are performed in Table 1. Table 1 showsthe nozzle cross sectional area (A) along the nozzleaxis (X), and compares the numerical values (obtainedin Ref. [6]) for temperatureTNum. 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. Anew non-dimensional functionζ is introduced in thepresent study, which represents the deviation of “localstagnation” condition from that of the inflow. A ther-modynamic chart and table for this function has beenprovided.ζ takes values equal to zero in dry regions,and positive in wet regions. The developed table isgeneral 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. Thevapor, 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. Thesets of constants of Lee-Kesler Equation of state is asfollows:

b1 = +0.1181193b2 = +0.265728b3 = +0.154790b4 = +0.030323c1 = +0.0236744c2 = +0.0186984c3 = 0.0c4 = +0.042724d1 × 104 = +0.155488d2 × 104 = +0.623689β = +0.65392γ = +0.060167

(29)

Using the above equation of state, compressibilityfactor of saturated vapor,Zg, can be determined by


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 coefficientsA0 to A6 are:

A6 = 14.7523A5 = -45.2802A4 = +52.6399A3 = -29.7745A2 = +8.6910A1 = -1.7379A0 = +0.9995

(31)

Saturated Pressure Value. The saturation pressurefor steam is determined by a fifth order polynomialleast 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 ofPa and K, t0 =273.15 K.

Entropy. The entropy of the mixture is determinedfrom s = sf + χsfg , wheresfg = hfg/T andsg

is obtained fromsg = Cp lnT − R ln p, andsf =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. & Kolovrant’k, M. 1994 “Laboratoryand field measurements of droplet nucleation inexpansion steam ”. 12th Int. Conf. on Propertiesof Water and Steam, Sept. 11-16. FL, ASME.

[3] Stastny, M. & Sejna, M. 1994 “Condensationeffects in transonic flow through turbine cas-cade”. 12th Int. Conf. on Properties of Water andSteam, 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 turbineblade”. Phi. Trans. R. Soc. Lond. A354, 59-88

[5] Guha, A., “A unified theory for the interpretationof total pressure and temperature in two-phaseflows 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 Roe’s Scheme, The 4th Conference ofIranian AeroSpace Society, Amir Kabir Univer-sity of Technology, Tehran, Iran, January 27–29,2003.

[7] Van Wylen, Borgnakke, Sonntag, “Fundamen-tals of Thermodynamics,”6th Edition, John Wi-ley & Sons, 2002.

[8] Moran, M. J. and Shapiro, H. N., “Fundamentalsof Engineering Thermodynamics,”4th Edition,John Wiley & Sons, 1998.


Figure 1: Schematic of an isentropic expansion ofsteam 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 thesaturated vapor line to their corresponding “local stag-nation” conditions.

Figure 4: The locus of “ local stagnation” conditionsof 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 localstagnation conditions.


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 of

imaginary 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 thelocal 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

( S0,local , T0,local )

S = S0,local

Figure 7: Schematic of the procedure to determine thelocal stagnation (the limiting case in which inflow ison the saturated vapor line).

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

2.25

2.5

Zet

a

4.555.566.577.588.59 Entropy (kJ/ kg.K)

300

350

400

450

500

550

600

650

Temperature (K)XY

Z

Zeta2.48978

2.40419

2.3421

2.24431

2.13911

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 “ζ” function.

X (m) -0.2 -0.1 0 0.1A (m2) 0.0379 0.0353 0.0315 0.0366

TNum. (K) 337 332.2 326.3 315.9TAnal. (K) 337.5 332.2 326.7 315.9MNum. 0.562 0.650 0.928 1.203MAnal. 0.561 0.650 0.933 1.203

X (m) 0.2 0.3 0.4 0.50A (m2) 0.0417 0.0468 0.0519 0.0570

TNum. (K) 311.1 307.4 304.5 302TAnal. (K) 311.1 307.4 304.5 302MNum. 1.440 1.546 1.630 1.696MAnal. 1.439 1.545 1.629 1.695

Table 1: Comparison between temperature and Machnumber along an arbitrary nozzle with a relativelyhigh value of expansion rate. The subscriptsNum.and Anal. refer to numerical values (obtained inRef. [6]) and analytical solutions (obtained in thepresent study), respectively.


Figure 9: Table ofζ function.

Documents

DEVELOPMENT OF A NEW THERMODYNAMIC CHART FOR