adiabatic two-phase flow of natural gas through...

ADIABATIC TWO-PHASE FLOW OF NATURAL GAS THROUGH VARIABLE AREA DUCTS

T. I. Sabry*, J. Huhn**, N. H. Mahmoud*, Mofreh H. Hamed* and A. A. El-Batawy* * Mechanical Power Engineering Department, Faculty of Engineering, Menoufiya University,

Shebin El-Kom, Egypt. ** Institute for Thermodynamics, Dresden University, 01062 Dresden, Germany.

Abstract: - The gas-dynamic equations are used to simulate a two phase flow of natural gas and condensed droplets through a convergent-divergent nozzle. A modified nucleation theory is used to estimate the nucleation coefficient and the nucleation rate. A numerical technique is used to solve the governing equations. The results obtained here show that increasing the correction coefficient as well as increasing the low initial pressure (Po from 1 - 5 bar) delayed the zone of condensation towards the nozzle exit, while at moderate initial pressure (Po from 10 - 30 bar) the observed shock condensation moves towards the nozzle throat. The increase of both initial pressure and correction coefficient leads to a rapid appearance of subcooling, the process becomes more non-equilibrium and the zone of spontaneous condensation is shifted downstream towards a region with larger subcooling. The increase of both initial pressure and the correction coefficient besides the decrease of condensation coefficient cause a rapid decay of shock condensation, rapid growth of droplets in the initial stage of condensation and slower further downstream.

Keywords: Natural gas, two-phase flow, nucleation, spontaneous condensation, supersonic nozzle.

1 Introduction The rapid development of gas dynamics of non-equilibrium supersaturated vapours is observed in recent years. The presence of two phase fluids may be primarily attributed to practically important problems in new branches of science and technology. These problems concerned with the condensation, nucleation and growth of condensed phase. These problems are accompanying the natural gas transportation through expansion turbine from the field to the users in industries and houses. When the gas is transported through a pipeline, it is transmitted at higher pressures from (15 to 105 bar) to reduce the volume of the gas and provide a pushing force to propel the gas through the pipe. When the gas flows over great distances, the pressure drops due to the friction of the gas molecules in the fluid as well as the friction between the fluid and the pipe walls. For continued transportation, the pressure must therefore be boosted at compressor stations spaced about 100–200 km along the transmission pipelines as noticed in [1]. The maximum transmitted pressure is 84 bar, while regional or local distribution mains operate at far lower pressures. Low-pressure lines take the gas to residential users at a pressure of about 20 mbar. The expansion turbine is fed with saturated or wet natural gas at pressures in the range from 15 to 200 bar. This turbine is used also to control the gas pressure. The low pressures lead to the presence of wetness in the expansion turbine blades. In the early days, the first model of expansion turbine is called expansion power module [EPM]. It was used in the natural gas transportation. The EPM turbine and the steam turbine have the same design without any modification. Knauf et

al. [2] modified the EPM turbine. The modified model of EPM turbine is called KKK turbine which is suitable for small and medium sized natural gas expansion stations. The KKK turbine’s simple layout and the easy way it can be incorporated into the gas regulating station have allowed the prime cost to be very low. The condensation and nucleation phenomena in the natural gas flow through expansion turbine decreases the thermal efficiency of the whole unit and causes blade erosion damage. Therefore, the study of condensation characteristics of natural gas has recently received a considerable attention due to its utilization in cloud physics, chemical industries and other areas. This paper is directed to study the factors influencing condensation in high speed expanding flows of natural gas. Theoretical treatment for the condensation problems of natural gas in supersaturated conditions was based on the modified nucleation theory. A kinetic theory approach for determining the rate of droplet growth has been developed. The governing flow equations such as continuity, momentum and energy equations are given here. On the other hand, the nonequilibrium characteristics during the spontaneous condensation (i.e. homogeneous condensation) within supersaturated natural gas are studied. Therefore, the objectives of the theoretical analysis presented here are as follows: 1- Designing a mathematical model using gasdynamic equations to simulate a two -phase flow of natural gas and condensed droplets through a supersonic nozzle. 2- A numerical technique is used to solve the governing flow equations released by the model using Rung-Kutta method.

Proceedings of the 2006 WSEAS/IASME International Conference on Heat and Mass Transfer, Miami, Florida, USA, January 18-20, 2006 (pp24-32)

3- A modified nucleation theory is used to estimate the nucleation work and the nucleation rate. 4- The flow model equations are developed to study numerically the flow and condensation characteristics. 2 Governing Flow Equations and Mathematical Model The present study investigates the nucleation mechanisms, generation and growth of condensed phase which taking place during flow through a convergent–divergent nozzle. The general flow equations of adiabatic spontaneous condensation of natural gas flow and also supplementary equations are described below based on the following assumptions: a- The flow is one-dimensional, steady and velocities of the phases are equal. b-The natural gas is an ideal gas and consists of 100% methane and is not viscous. c- The influences of the walls under the form of mass deposition, viscous forces and heat transfer are neglected. d- Homogeneous spontaneous condensation and no foreign admixtures (solids, droplets, etc.) are encountered in the flow at the nozzle inlet. Following Danelen [3], the fundamental equations of motion for the two-phase steady state flow take the form: 2.1 Gas phase - Conservation of mass:

The continuity equation for gas phase in two-phase flow can be written as reported in [3 and 4] by, 1 A C .A

X∂ρ

= −χ∂

(1)

- Momentum equation:

The momentum change across the small element ∆x for two-phase flow in the flow direction, considering the pressure forces takes the form:

2

1( C P).A AC. .A PX X

∂ ρ + ∂= − χ +

∂ ∂ (2)

- Energy equation:

The conservation of energy of gas phase contained in the flow element is expressed by: ( ) ( )1 o1 1 1 o1C A i .A T S i

X∂

ρ = χ −∂

(3)

2.2 Liquid phase The following system of equations describe the density distribution of condensed phase (droplet) with respect to its size in one–dimensional flows as reported in [3 and 5] is as follows: 0A. .C J.A

X∂ ω

=∂

(4)

( )1* 2 0

A. .C A J r aX

∂ ω= + ω

∂ (5)

( )22* 2 1

A. .C A J r 2.aX

∂ ω= + ω

∂ (6)

where, 0ω , 1ω , 2ω are the density distribution, non-normalizing radius of droplet and non-normalizing square radius of droplet respectively and defined as,

0 Nω = (7)

1 dN rω = (8)

22 dN rω = (9)

where, N N(r)dr= ∫ (10)

where, N(r) is the non-normalizing density distribution of droplets. 2.3 Supplementary equations - The thermal equations simulate a multiphase flow of vapour phase and condensed phase are based upon the equation of state. This equation of state may be derived from a combination of Boyle’s, Charles, Gay Lussac’s and Avogadro’s laws as [5] and takes the form,

1 1P B R T= ρ - The enthalpy of the vapour phase.i1 is defined as in [5],

11

Pi . const1

γ= +γ − ρ

- The vapour phase entropy S1 is given as reported in [5],

in1 in

in 1

B.R PS S n ( ) ( )( 1) P

γ⎡ ⎤ρ= + ⎢ ⎥γ − ρ⎣ ⎦

l

- The nucleation rate J, is calculated similar to that of Frenkel-Zeldovioh’s formula [5],

2

o *

1 2 1

2 m w (r )P 1J Expk T k T

⎛ ⎞ ⎡ ⎤δ ∆= −β⎜ ⎟ ⎢ ⎥ρ π⎝ ⎠ ⎣ ⎦

Where; k is Boltzmann’s constant to be determined from theoretical consideration and following to Volmer [6]. While ∆W(r*) is the nucleation work is obtained according to [7] as, 2

* *4W(r ) . .r .3

∆ = π δ

The critical radius of a droplet r* in the case of equilibrium with its vapour can be determined according to the Kelvin-Helmhalz relation [4, 5, 7 and 8] takes the form. s

*2

2 Tr. . Tδ

=ρ ζ ∆

The mass flux of condensed phase (phase conversion rate, χ) may be defined based upon the equations of density distribution of droplet as reported in [3 and 9] as, dm(r)N(r) . .dr

dtχ = ∫ (11)

where, 2d 1

21

4. .P.r . Tdm(r) (1 )dt T2. .R.T

π α= −

π (12)

By inserting equations (9), (10) and (12) in equation (11) then,

2a .χ = ω (13)

where, 1

21

T4 Pa 1T2 R T

⎛ ⎞πα= −⎜ ⎟⎜ ⎟π ⎝ ⎠

, 2

2

aa4

=πρ


2.3 Solution procedure The nozzle was divided into a number of elements of a step (∆x). The element thickness is obtained from the relation, x L / n∆ = where; n is the number of steps and L is the length of the nozzle measured from the throat. Under the assumptions (a) and (b), the natural gas expands isentropically and without change in its composition along the convergent section of the tested nozzle. The flow properties along the convergent section are obtained from the isentropic relations, which were reported in many textbooks [9-10]. For every step (∆x) in the expansion, the pressure distribution P(x), the amount of gas subcooling ∆T(x), nucleation rate (J), critical radius of nucleated droplets (r*), mean radius of droplets (rm), and density distribution (ω0) along the nozzle axis are calculated using the system of equations from (1) to (13). This system of equations is integrated using Rung–Kutta method. 3 Results The predicted results presented below indicate the effects of changing the condensation coefficient (α), the correction coefficient of nucleation work (β) and the initial pressure (Po) on the flow and condensation characteristics during the expansion of natural gas through a convergent-divergent nozzle. Geometry of utilized nozzle is shown in Fig. (1.a) by the dotted line curve. The presented flow characteristics are described here by the nondimensional static pressure distribution (P/Po). While the predicted condensation characteristics are illustrated here as the amount of gas subcooling (∆T), nucleation rate (J), critical radius of nucleated droplets (r*), mean radius of droplets (rm), and density distribution (ω0) along the nozzle axis. All the predictions discussed here indicate the changes in flow and condensation characteristics of the natural gas along the divergent section of the utilized nozzle. Also, it is of great importance to note that the initial conditions are the parameters of natural gas at the saturation condition. The results are divided into two groups. The first group concerning with the effects of condensation and correction coefficients of nucleation work on the two-phase flow of natural gas in the case of low initial conditions of pressure (up to 5 bar) as shown in Figures (1 to 4). While the second one deals also with the effect condensation and correction coefficients in the case of moderate pressure (up to 30 bar) as shown in Figures (5 to 8). The condensation coefficient α is taken here from (0.03 up to 1.0) while correction coefficients of nucleation work is changed from (β = 1, 2 and 3) taken as advised by Saltanov et al [5], Hill [11], Littlewood et al [12], Burrows et al [13] and Campell et al [14].

3.1 Low pressures Plots of Fig. (1) illustrate the effect of changing

the condensation coefficient (α) on the flow and condensation characteristics. The results in Fig. (1.a) to (1.e) are obtained for different values of condensation coefficient (α = 0.03, 0.5 and 1.0), a constant value equals

unity for the correction coefficient of nucleation work (β) and a constant value of the initial pressure (Po) equals 3.0 bar. Figure (1.a) indicates axial dimensionless static pressure variation along the divergent section of the tested nozzle. Predicted pressure variation includes a notable pressure bumps due to shock condensation. This shock condensation vanishes rapidly. Generally, the condensation shock is observed during supersonic flow of condensing vapours due to the liberation of heat of phase transition with phase conversion of condensation. It can be noticed also in Fig.(1.a) that increasing the condensation coefficient (α) moves the point of condensation shock incipience (i.e., the location at which the amount of subcooling appears in the calculation) downstream towards nozzle exit. However increasing the condensation coefficient (α) from (0.03 to 1.0) decreases the amount of subcooling (∆T) as advised from Fig. (1.b). While Fig. (1.c) indicates that, the mean size of droplets differs from the critical one. The critical radius of droplets (r*) decreases to a minimum value at the location of condensation onset. Nucleated droplets grow up through the condensation process and reaches to higher sizes comparing to its initial size at the point of condensation onset. When the condensation process vanishes, different behaviours for critical and mean radius variations along the divergent section of the nozzle can be noticed as shown in Fig. (1.c). As the condensation stops, the critical radius of droplets (r*) decreases slightly and the growth of the mean radius of droplets (rm) continues along the divergent section of the nozzle. The calculations show that, the nucleated droplets with the condensation coefficient (α) equals unity starts from critical radius (r*) of 3.5x 10-5 µm and reaches to 5x 10-9 µm at the condensation onset which lies at (X = 0.275) downstream of the throat as depicted from Fig. (1.c). The critical radius reaches to a value of 1.4x10-8 µm, when the condensation vanishes at (X = 0.3) from the throat as shown in Fig. (1.c). It can be observed also in this figure that, increasing condensation coefficient leads to a rapid increase on the mean radius of droplets (rm). It is evident in Fig. (1.d) that increasing α, tends to delay the nucleation in the direction of nozzle throat and causes to decrease the nucleation rate. The tendency of the plots in Fig. (1) can be explained as the condensation coefficient (α) increases the rate of heat dissipation from droplets surface into the ambient vapour increases also and then decreases the amount of subcooling (∆T). Furthermore, increasing the vapour phase temperature leads to decrease the nucleation rate as predicated from the calculations. Also, decreasing the amount of subcooling (∆T) leads to decrease the condensation rate. On the other hand, as a result of (∆T) decreasing, the critical radius increases as expected from Kelvin-Helmhalz relation [4, 5, 7 and 8]. The normalized density (ω0) which is shown in Fig. (1-e) remains constant until the condensation starts. While in the zone of condensation, the normalized density increases to its maximum value. It can be noticed also in Fig. (1.e) that as the condensation coefficient increases, the normalized density decreases.


0.00 0.15 0.30 0.45 0.600.46

0.48

0.50

0.52

0.54

p/po

po=3bar β=1

x

32 1

1.00

1.01

1.02

1.03

A/A

*

p/p 0

A/A*

a- dimensionless static pressure

0.00 0.15 0.30 0.45 0.600

1

2

3

4

5

po= 3bar β= 1

321

∆T

x

b- vapour subcooling

0.00 0.15 0.30 0.45 0.600

4

8

12

16

20

po=3 bar β=1

1

2

3 2 1

r *x10

6

x

0

4

8

12

16

r*

....... rm

r mx1

05

3

c- droplet radius

0.00 0.15 0.30 0.45 0.600.0

0.5

1.0

1.5

2.0

1

po=3bar β=1

32

Jx10

-20

x

d- nucleation rate

0.00 0.15 0.30 0.45 0.600.0

0.2

0.4

0.6

0.8

po=3 bar β=1

2 , 3

1

ωo

x

e- droplet density distribution Fig. 1 Effect of condensation coefficient, α on the

flow and condensation characteristics. (1- α = 0.03 , 2- α = 0.5 , 3- α = 1.0)

Figure (2) illustrates the influence of changing the correction coefficient of nucleation work (β) on the two-phase flow parameters and condensation characteristics along nozzle axis. The plots of Fig. (2) show that increasing the coefficient β, causes to decrease, damp and decay the shock condensation within the nozzle divergent section. Furthermore, increasing the coefficient β leads to move the onset of shock condensation towards the nozzle exit. Also, as a result of increasing β, phase temperature decreases in turn, and causes to increase the amount of subcooling as clearly shown in Fig. (2.b). Besides, the critical radius is decreased as described in Fig. (2.c) from (r* =1.27x 10-4 µm) at nozzle throat to ( r* = 2.9474x 10-9 µm) at the point of condensation onset. It can be concluded that, the final size of droplet grow by condensation becomes about 2.2 times its initial size. Figure (2.c) indicates also that increasing the coefficient β tends to advance the condensation shock in the direction of nozzle exit. Furthermore, increasing the coefficient β yield a decrease in the mean radius of condensing droplets (rm). The calculations show that, increasing the correction coefficient of nucleation work (β) from 1.0 to 3.0 decreases the maximum value of nucleation rate as shown in Fig. (2.d). Furthermore, increasing β tends to advance the location of the maximum value of the nucleation rate (J) towards the nozzle exit. It can be observed also in Fig. (2.d) the existence of nucleated droplets with decreased sizes in the distance between nozzle throat and the point of condensation onset. The normalized density (ω0) which is shown in Fig. (2.e) remains constant until the condensation starts within the zone of condensation, then ω0 increases to its maximum value. Gyarmathy [15] proved that changing nucleation rate is reflected in changing the subcooling (∆T). Furthermore, increasing ∆T tends to decrease the size of nucleated droplets as expected from nucleation theory. Therefore, decrease of phase conversion rate, critical radius and density distribution are accompanied with the decrease of nucleation rate. Generally it can be concluded that, increasing the coefficient β, leads to delay the


condensation onset towards nozzle exit and decreasing the nucleation rate.

0.00 0.15 0.30 0.45 0.60

0.44

0.48

0.52

0.56

p/p o

po=3bar α =1

32

1

x

a-dimensionless static pressure

0.00 0.15 0.30 0.45 0.600

2

4

6

8

∆T

po= 3 bar α = 1

3

2

1

x

b- vapour subooling

0.00 0.15 0.30 0.45 0.600

1

2

3

4

5

po=3bar α =1

r*.......... rm

3

2

321

r *x10

8

x

0

4

8

12

16

r mx1

05

1

c. droplet radius

0.00 0.15 0.30 0.45 0.600

2

4

6

8

10

12

Jx

10-2

0

po=3bar α=1

3

2

1

x

d- nucleation rate

0.00 0.15 0.30 0.45 0.600.00

0.02

0.04

0.06

3

2

1po=3bar α =1ω

o

x

e- droplet density distribution- Fig. 2 Effect of correction coefficient β, on the flow

and condensation characteristics (1- β = 1.0 , 2- β = 2.0 , 3 - β = 3.0)

Plots of Fig. (3) illustrate the influence of

changing the correction coefficient (β) on the location of condensation onset (xonset) at different values of condensation coefficient and constant value of initial pressure Po= 3 bar. A higher value of this coefficient β, causes the shock condensation to move downstream in the nozzle. A lower value of the coefficient α, causes the shock condensation to move downstream along the nozzle. However, increasing α tends to increase the rate of static pressure, decrease the natural gas subcooling, and decrease the nucleation rate. This behavior can be explained by the fact that as α increases the rate of heat dissipation from droplets surface into the ambient vapour and consequently the vapour phase temperature are increased, and then the vapour subcooling (∆T) is decreased. The rate of heat dissipation is accompanied by an adverse pressure gradient ( i.e., condensation shock). Therefore, increasing (α) is accompanied by the increase of the phase conversion and the increase of the mean radius of condensing droplets (rm).

0.5 1.0 1.5 2.0 2.5 3.0

0.2

0.3

0.4

0.5po= 3 bar

1 α = 0.032 = 0.53 = 1.0

32

1

x on

set

β

Fig. 3 Effect of correction coefficient , β on the location of condensation onset.

Figure (4) summarizes the effect of increasing the initial pressure from 1 to 5 bar on the condensation onset (xonset) at constant values of (β) and (α) during expansion of the natural gas through a nozzle. It can be noticed in Fig. (4) that, as the initial pressure Po increases the shock condensation displaces towards the nozzle exit.


1 2 3 4 50,24

0,27

0,30

0,33

α = 0,03 , β =1

x onse

t

po, bar

Fig. 4 Effect of initial pressure on the location of condensation onset

3.2 Moderate pressures

Plots of Figure (5) illustrate the influence of changing the condensation coefficient (α) on the flow and condensation characteristics at constant value of correction coefficient of nucleation work equal unity and constant value of initial pressure Po = 10 bar. These plots indicate the same tendency as was noticed and discussed in the case of low initial pressure Po = 3 bar as shown in Figures (1-3). A lower value of this coefficient of condensation causes the shock condensation to move downstream in the nozzle.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.48

0.50

0.52 p/p

o

po=10 bar β =1

2,3 1

p/p o

x

1.00

1.01

1.02

1.03

A/A

*

A/A*


0.0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4po=10 barβ =1

2,3 1

∆T

x

b- vapour subooling

0,0 0,1 0,2 0,3 0,4 0,5 0,60

1

2

3

4

5 po=10 barβ =1

3

2

1

3 21r *x

108

x

0

1

2

3

4

r*------- rm

r mx1

0 4

c. droplet radius

0.0 0.1 0.2 0.3 0.4 0.5 0.60

4

8

12

J x1

0-24

po =10 barβ =1

32

1

x

d- nucleation rate

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

0.4

0.8

1.2

1.6

po=10 bar β =1

32

1

ωo

x

e- droplet density distribution Fig. 5 Effect of condensation coefficient, α on the

flow and condensation characteristics. (1- α = 0.03 , 2- α = 0.5 , 3- α = 1.0)

Figure (6) illustrates the changes in the predicted flow and condensation characteristics with increasing the coefficient of nucleation work β from (1 up to 3) at constant value of condensation coefficient equals unity and constant initial pressure (Po =10 bar) during expansion of the natural gas through nozzle. These plots indicate the same tendency as was noticed with low initial pressure Po= 3 bar. One can noticed that increasing coefficient β, moves the shock condensation towards the nozzle exit, decreases the rate of static pressure, the vapour phase temperature and the normalized density while increasing the coefficient β yields to increase of the amount of subcooling, and the nucleation rate. Downstream the point of condensation onset, the critical radius of droplet


decreases while the mean radius of droplet increases with the increasing of coefficient. The reasons for these tendencies are discussed previously with the results of low pressures. Plots of Fig. (7) illustrate the influence of changing the correction coefficient (β) on the location of condensation onset (xonset) at different values of condensation coefficient and constant value of initial pressure Po= 10 bar.

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.48

0.50

0.52po=10 barα =1

2

3

1

p/p 0

x.


0.0 0.1 0.2 0.3 0.4 0.5 0.601234567

po=10 bar α =1

32

1

∆T

x

b- vapour subooling

0.0 0.1 0.2 0.3 0.4 0.5 0.60.0

1.5

3.0

4.5

r*------- rm

p =10 barα =1

32

1

x

0

2

4

6

8

r *x10

8

r mx1

04

3

2

1

c. droplet radius

0.0 0.1 0.2 0.3 0.4 0.5 0.60

1

2

3

4

po=10 bar α =1

32

1

J x1

0-24

x

d- nucleation rate

0.0 0.1 0.2 0.3 0.4 0.5 0.60.000

0.005

0.010

0.015

0.020

po=10 bar α =1

3

2

1ω

ο

x

e- droplet density distribution Fig. 6 Effect of correction coefficient β, on the flow

and condensation Characteristics (1- β = 1.0 , 2- β = 2.0 , 3 - β = 3.0)

1.0 1.5 2.0 2.5 3.0

0.18

0.20

0.22

0.24

0.26

0.28

0.30

3

21po =10 bar

1 α = 0,032 = 0,53 = 1x on

set

β

Fig. 7 Effect of correction coefficient , β on the location of condensation onset.

Figure (8) summarizes the changes in the investigated flow and condensation characteristics with increasing the initial pressure Po from 10 up to 30 bar at constant values of α and β. One can be observed that with the increasing of the initial pressure Po, the observed shock condensation moves towards the nozzle throat. as shown in Figure (8).


5 10 15 20 25 30

0.1

0.2

0.3

0.4

0.5

α = 0,03β = 1

x onse

t

po, bar

Fig. 8 Effect of initial pressure on the location of condensation onset

The reason for this behavior is that as a result of increasing the initial pressure, the rate of static pressure increases and consequently the vapour phase temperature increases. These tendencies cause the amount of subcooling to be decreased and the rate of phase conversion is increased. 4 Conclusion

The results obtained from the present paper appear to give a reasonable description for the condensation behavior of natural gas during supersonic flow in a convergent-divergent nozzle. The gasdynamic equations are used to simulate two phase flow of natural gas and condensed droplets through a nozzle. The flow equations are developed to compute the pressure distribution, droplets formation, nucleation rate and the subcooling along the nozzle axis. The results obtained here show that increasing the correction coefficient (ß) as well as increasing the initial pressure (Po from 1 - 5 bar) delayed the zone of condensation towards the nozzle exit. While at moderate initial pressure (Po from 10 - 30 bar) the observed shock condensation moves towards the nozzle throat. The increase of initial pressure and correction coefficient (β) causes rapid appearance of subcooling (∆T). This means that the process becomes more non-equilibrium and the zone of spontaneous condensation is shifted down stream into a region with larger subcooling. The increase of both initial pressure and the correction coefficient (β) beside the decrease of condensation coefficient (α) leads to an intensive decay of shock condensation, rapid growth of droplets in the initial stage of condensation and slower further downstream. NOMENCLATURE A nozzle area; m 2 B compressibility; - C velocity; m.s-1 i enthalpy; J.kg-1 J nucleation rate; s-1. m-3 k Boltzman constant; J.K-1 L nozzle length; m

m droplet mass; kg mo molecular weight; mol.kg-1 P pressure; N.m-2 rd droplet radius; m r

* critical dimension of nucleus; m

R gas constant; J.kg-1.K-1 S entropy; J.kg-1.K-1 X distance along nozzle axis; m

Greek Letters

α condensation coefficient; - β correction factor for nucleation work; - δ surface tension coefficient; N.m-1 ∆w nucleation work; N.m ς: latent heat of evaporation; J. kg-1 χ phase conversion rate; kg.m-3.s-1 ρ2 density of condensed phase; kg.m-3

Subscripts

1 vapour phase; 2 liquid phase; * critical; c condensed; d droplet; o stagnation, initial; s saturation; REFERENCES [1] Ruhrgas AG, Natural Gas in the energy market in

1998. Report on the Industry. [2] Knauf, H. Kühnle, AG., kopp, F. und Kausch. F;

Wirtschaftliche Erdgasenentspannung bis 200 KW mit dem Expansion Power Module (EPM), Gaswärme international 49 (2000) Heft 3. März.

[3] Danenlen, V. C, Tsiklauri, J.V. and Seleznev, L. I., Adiabatic two-phase flow. Moscow; Atom Publishers, (1973).

[4] Hamed, H M, Investigation of unsteady two-phase flow of wet steam through nozzles. Alexandria Engineering journal. Vol. 37. No.5, A237-A249. September (1998).

[5] Saltanov G. A, Seleznev L. I. and Tsiklauri, G. V., Generation and growth of condensed phase in high – velocity flows. Int. J Heat Mass Transfer Vol. 16, Pp. 1577-1587 Pergamon press (1973).

[6] Volmer, M Kinetic der Phasenbildung , Chap. 4. Dresden and leipzig, Steinkopf., (1939).

[7] Dohrmann U., Ein Numerisches Verfahren zur Berechnung stationärer Transsonischer Strömung mi Energiezufuhr durch homogene Kondensation .,Karles ruhe, (1989).

[8] Moore, M.J and Sieverding, C.H., Two –phase steam flow in turbines and separators, McGraw – Hill, (1976).

[9] Stiver, H., A condensation phenomenon in high- velocity flows. Chapter 3 in “ Fundamentals of Gas Dynamics”, Edited by Emmons, W, Princeton University Press, New Jersey, U.S.A, (1958)

[10] PatricK, H and William, E., Compressible Fluid Flow. The MCGRAW – Hill Companie.


[11] Hill, ph. G: Condensation of water vapor during supersonic expansion in nozzles, J. Fluid Mech. 25, pp. 593-620, (1966).

[12] Littlewood, R and Rideal, E, On the evaporation coefficient. Trans. Faraday Soc.52.1598 (1956).

[13] Burrows, G., Evaporation at low pressures. J. Appl. Chem. 7,375 (1957).

[14] Campell, P. A.,Bakhtar,F., Condensation phenomena in high speed flow of steam. Proc. Instn Mech. Engrs. Vol. 185 25/71 (1970).

[15] Gyarmathy, G. and Meyer, H., Spontane Kondensation ,VDI Forschungsheft 508 ,VDI– verlag, Düsseldorf , (1965).


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