**2. Literature review**

#### **2.1 Early Investigations**

Among the pioneers in the investigation of thermodynamic condensation was (Aitken, 1880), who in his experiments in 1880 observed that any dust or salt particles present in the expansion of saturated air will act as centres for condensation. A similar investigation was also carried out by (Von Helmholtz, 1886) but it was (Wilson, 1897) who made a detailed study of spontaneous condensation where it was found that in the absence of ions or foreign nuclei during the expansion of saturated air, condensation was delayed. The formulation of the ratio of vapour pressure, *P* to the saturation pressure corresponding to local vapour temperature *Ps*(*TG*) was later formulated and called supersaturation ratio, *S.* This parameter is used as a measure of supersaturation and is given by:

$$\mathcal{S} = \frac{p}{\mathcal{P}\_{\mathcal{S}}(\mathcal{T}\_{\mathcal{G}})} \tag{1}$$

Supersaturation ratio, *S* measures the departure of fluid from thermodynamic equilibrium. Subsequently, (Henderson, 1913) reported that the discharge of steam in nozzles, expanding in the wet region in Mollier chart was approximately 5% greater than the value that would be expected from equilibrium calculations. Following this observation, (Stodola, 1915) published his own results of nozzle expansions. A thorough discussion of supersaturation effects in nozzles was first given by (Callender, 1915). His discussion includes a prediction of droplet size based on the Kelvin–Helmholtz equation. Subsequently, by assuming that condensation will always produce droplets of similar size, (Martin, 1918) calculated the limiting supersaturation at other pressures and plotted them on the Mollier chart. This limiting supersaturation line is called Wilson line. For the next 10 years, much work on spontaneous condensation was carried out by Stodola and most of his work is summarized in his book, (Stodola, 1927).

#### **2.2 Nucleation theory**

The development of nucleation theory started almost at the same time as the study of condensation but significant results were obtained only after 10 years. Among the earlier investigators on limiting supersaturation were (Yellot, 1934), Yellot & Holland, 1937; Rettaliata, 1938). In their works, attempts were made to define the position of Wilson line more precisely. It was found that the limiting supersaturation was dependent on the nozzle shape and experimental condition and they suggested the replacement of the Wilson line by Wilson zone. Following his work, (Binnie & Woods, 1938) and (Binnie & Green, 1943) performed further accurate measurement of axial pressure distribution in nucleating flows in convergent-divergent nozzles.

The nucleation theory was first combined with the gas dynamics equations by (Oswatitsch, 1942). He applied the treatment to condensing flows of water vapour in nozzles both as pure vapour and part of atmospheric air and reported good agreement with experimental observations. In parallel with the investigations mention, the development of steam turbines was progressing with remarkable speed and the problems associated with the presence of liquid were being experienced. One of the particular serious consequences resulting from the presence of liquid in steam turbines is blade erosion. However, with the introduction of reheat cycles after the Second World War, the problem was temporarily alleviated. Following new developments and designs of larger steam turbine, the steam velocity could reach much higher values. This led to considerable impact velocities which brought renewed interest in wetness problems. Among the many researchers who have reported their investigations into the field are (Gyarmathy, 1962, Pouring, 1965, Hill, 1966, Puzyrewski, 1969, Wegener, 1969, Campbell & Bakhtar, 1970, Barschdorff, 1970, Filippov & Povarov, 1980). In addition to these investigations, a number of studies aimed at measuring the size of droplets formed by spontaneous nucleation were reported by (Gyarmathy & Meyer, 1965, Krol, 1971, Deich et al., 1972). The measurements have provided further data for comparison with the theoretical solutions.

#### **2.3 Condensation in nozzle**

444 Mechanical Engineering

method such as quadrature method of moments (Gerber & Mousavi, 2007). Attempt was also made to use commercial CFD package to calculate the three dimensional steam

Among the pioneers in the investigation of thermodynamic condensation was (Aitken, 1880), who in his experiments in 1880 observed that any dust or salt particles present in the expansion of saturated air will act as centres for condensation. A similar investigation was also carried out by (Von Helmholtz, 1886) but it was (Wilson, 1897) who made a detailed study of spontaneous condensation where it was found that in the absence of ions or foreign nuclei during the expansion of saturated air, condensation was delayed. The formulation of the ratio of vapour pressure, *P* to the saturation pressure corresponding to local vapour temperature *Ps*(*TG*) was later formulated and called supersaturation ratio, *S.* This parameter

� � �

Supersaturation ratio, *S* measures the departure of fluid from thermodynamic equilibrium. Subsequently, (Henderson, 1913) reported that the discharge of steam in nozzles, expanding in the wet region in Mollier chart was approximately 5% greater than the value that would be expected from equilibrium calculations. Following this observation, (Stodola, 1915) published his own results of nozzle expansions. A thorough discussion of supersaturation effects in nozzles was first given by (Callender, 1915). His discussion includes a prediction of droplet size based on the Kelvin–Helmholtz equation. Subsequently, by assuming that condensation will always produce droplets of similar size, (Martin, 1918) calculated the limiting supersaturation at other pressures and plotted them on the Mollier chart. This limiting supersaturation line is called Wilson line. For the next 10 years, much work on spontaneous condensation was carried out by Stodola and most of his work is summarized

The development of nucleation theory started almost at the same time as the study of condensation but significant results were obtained only after 10 years. Among the earlier investigators on limiting supersaturation were (Yellot, 1934), Yellot & Holland, 1937; Rettaliata, 1938). In their works, attempts were made to define the position of Wilson line more precisely. It was found that the limiting supersaturation was dependent on the nozzle shape and experimental condition and they suggested the replacement of the Wilson line by Wilson zone. Following his work, (Binnie & Woods, 1938) and (Binnie & Green, 1943) performed further accurate measurement of axial pressure distribution in nucleating flows

The nucleation theory was first combined with the gas dynamics equations by (Oswatitsch, 1942). He applied the treatment to condensing flows of water vapour in nozzles both as pure vapour and part of atmospheric air and reported good agreement with experimental observations. In parallel with the investigations mention, the development of steam turbines

�����) (1)

properties (Dykas et al., 2007, Nikkhahi et al. 2009, Wroblewski et al., 2009).

is used as a measure of supersaturation and is given by:

**2. Literature review 2.1 Early Investigations** 

in his book, (Stodola, 1927).

in convergent-divergent nozzles.

**2.2 Nucleation theory** 

An expansion of steam from superheated to wet condition in a typical convergent-divergent nozzle is illustrated in Fig. 1. The process can be also illustrated on an *h − s* diagram as shown in Fig. 2. Steam enters the nozzle as dry superheated vapour at point (1) and during its passage through the nozzle, it expands to the sonic condition represented by point (2). At point (3) in Fig. 2, the saturation line is crossed which may occur before or after the throat and droplet embryos begin to form and grow in the vapour. The nucleation rates associated with these early embryos are so low that the steam continues to expand as a dry singlephase vapour in a metastable, supercooled or supersaturated state. Depending on the local conditions and the rate of expansion, the nucleation rate increases dramatically and reaches

Fig. 1. Axial pressure distribution in nozzle with spontaneous condensation.

$$
\Delta T = T\_\ $(P) - T\_\$ \tag{2}
$$

$$m = m\_{\mathcal{G}} + m\_{\mathcal{L}} = \text{constant} \tag{3}$$

$$
\sigma m\_G = \, \rho\_G \, A \, u \tag{4}
$$

$$\frac{d\rho\_G}{\rho\_G} + \frac{dA}{A} + \frac{du}{u} + \frac{dm\_L}{m - m\_L} = 0\tag{5}$$

$$\frac{d\frac{dP}{dt} + \frac{\rho\_G u^2}{\rho} \frac{f}{2} \frac{dx}{d\_\theta} + \frac{\rho\_G u^2}{(1 - \omega)\rho} \frac{du}{u} = 0 \tag{6}$$

$$\frac{u^2}{c\_{P\_L}\tau\_G}\frac{du}{u} + \frac{dT\_G}{\tau\_G} + \frac{d(\ell \, m\_L)}{c\_{P\_L}\tau\_G \, m} = 0\tag{7}$$

$$
\frac{d\rho}{d\rho} - \frac{d\rho\_G}{\rho\_G} - \frac{d\tau\_G}{\tau\_G} = 0\tag{8}
$$

$$r^\* = \frac{^{2\sigma}}{\rho\_L \, ^{\text{R } T\_G} \ln \left\{ ^{p\_s} / \_{p\_s \, ^\*T\_G} + ^{2B } [\rho - \rho\_s] \right\}} \tag{9}$$

Numerical Modeling of Wet Steam Flow in Steam Turbine Channel 449

4. The values of *mL* and *dmL* found from Eq. 3 and Eq. 5 are averaged and assumed to

5. Using the average values of *mL* and *dmL* found in Eq. 6, the flow equations are

6. To ensure that all properties are compatible, the droplet temperature at exit is adjusted

Then starting at inlet to the nozzle, the flow equations are integrated step by step until the

In this investigation, upon completion of the nucleation phase, the properties of fluids are calculated based on the two methods. The first method is carried out by combining all droplet groups into a single population and their properties are averaged when there is no new droplet embyo forms in the subsequent step length. This is called the "Average" method. The other method of calculation is when every single droplet groups and their properties are retained even after the completion of nucleation phase. This is called the "Non-Averaged" method. The droplet radius calculation using this method is based on r.m.s. values for each droplet group. In theory, this method should results in more accurate solution but it requires an extremely huge processing power to complete the whole

In order to test for the accuracy of the numerical code, the model was applied to a few test cases at dry condition in a convergent-divergent nozzle. Three cases were calculated namely the subsonic-supersonic flow, purely subsonic flow and the flow involving shock. In this test, the inlet temperature and total pressure for all cases are set to 421.3K and 72,700 Pa respectively while the back pressure is varied according to the desired cases. For purely subsonic case, the back pressure is set to 68,000 Pa (giving a pressure ratio of 1.069) while for the supersonic flow, linear interpolation is employed for the determination of the back

Fig. 4 illustrates the general Mach number distribution along x direction for all cases. The numerical results are compared with the exact solutions which are given by solid lines close to each case respectively. It can be clearly seen that all numerical results show a slight deviation from the exact solutions except for the supersonic case where both numerical and exact solutions agrees very well with each other. For the pure subsonic flow case, the Mach number increases with distance until it reaches a peak value of 0.57 at the throat. Downstream of the throat, the Mach number decreases in the divergent section due to the decrease in the flow velocity. Comparison with the exact solution shows a slightly higher value and this could be attributed to the numerical error presented in the model. For the subsonic-supersonic case, the Mach number increases along the nozzle. The numerical solution for this case shows good agreement with the exact solution. The Mach number increases starting from the nozzle inlet to the nozzle exit. In order to emulate the shock case, the exit pressure is reduced slightly below the pressure imposed on the subsonic case. A

pressure. On the other hand, the back pressure for the shock case is set to 60,400 Pa.

represent the variables over incremental step, *Δx*.

so that it correspond to the mean droplet radius.

end of nozzle is reached.

**4. Results, analysis and discussion** 

**4.1 Calculation on dry, steady flow** 

calculation.

integrated to obtain the final values of *Pout* and *TG,out*.

where *r\** is the critical droplet radius, is the surface tension of liquid droplet, *R* is the gas constant, *ps(TG)* is the saturation pressure corresponding to local vapour temperature *TG* and *B* is the virial coefficient. For given vapour conditions, droplets with radius *r\** will be in unstable equilibrium with the vapour condition. For equilibrium, larger droplets need a lower supersaturation and will grow. On the other hand, smaller droplets will find the surrounding supercooling state insufficient and therefore tend to evaporate. In order to condense, the molecules must form droplet of radius *r\** which is against their natural tendency. The only route to the formation of super-critical droplets is through collision within the body of the vapour. From eq. (7), it can be seen that the critical radius is inversely proportional to the supersaturation (� �⁄ �) and therefore, the lower the supersaturation, the larger the size of critical droplets and the smaller the chance that new droplet embryo will be formed.

Original investigation of the rate of formation of critical clusters within the supercooled vapour was carried out by (Volmer & Weber, 1996, Farkas, 1927, Becker & Doring, 1935, Frenkel, 1946, Zeldovich, 1942). This was later refined, modified and reviewed by numerous investigators. The expression for the nucleation rate as the number of droplets formed per unit volume and time as given by the classical nucleation theory is:

$$J\_{\parallel} = q \left(\frac{2\sigma}{\pi m^3}\right)^{1/2} \frac{\rho\_G^{\*2}}{\rho\_L} \exp\left[-\frac{4\pi}{3} \frac{r^\*\sigma}{T\_G}\right] \tag{10}$$

where *J* is the nucleation rate, *k* is the Boltzmann's constant and *q* is the condensation coefficient, which is defined as the fraction of molecular collisions which results in condensation. Substituting for critical radius, *r\** in terms of the supersaturation ratio, it will be seen that very small changes in the supersaturation of the fluid can influence the nucleation rate drastically.

Condensation occurs by the nucleation of new droplets and the growth of any existing droplets within specified incremental step. The incremental mass of liquid formed, *dmL* can be determined by using the nucleation and droplet growth equations. The equation for nucleation rate is derived from classical nucleation theory. Once the mass of liquid over an incremental step is known and regarding the terms ��� ��� , �� � and ��� as independent variables, Eqns. 5-8 can be solved for four unknowns, �� � , ��� �� , �� � and ��� �� . The resulting expression can be integrated using fourth order Runge-kutta technique to yield the changes in the flow properties over the step. But to increase the accuracy of the coupling between the main flow equation and the equations describing droplet behavior, the calculation is carried out in the following steps:


constant, *ps(TG)* is the saturation pressure corresponding to local vapour temperature *TG* and *B* is the virial coefficient. For given vapour conditions, droplets with radius *r\** will be in unstable equilibrium with the vapour condition. For equilibrium, larger droplets need a lower supersaturation and will grow. On the other hand, smaller droplets will find the surrounding supercooling state insufficient and therefore tend to evaporate. In order to condense, the molecules must form droplet of radius *r\** which is against their natural tendency. The only route to the formation of super-critical droplets is through collision within the body of the vapour. From eq. (7), it can be seen that the critical radius is inversely proportional to the supersaturation (� �⁄ �) and therefore, the lower the supersaturation, the larger the size of

is the surface tension of liquid droplet, *R* is the gas

critical droplets and the smaller the chance that new droplet embryo will be formed.

� ���

unit volume and time as given by the classical nucleation theory is:

incremental step is known and regarding the terms ���

variables, Eqns. 5-8 can be solved for four unknowns, ��

Original investigation of the rate of formation of critical clusters within the supercooled vapour was carried out by (Volmer & Weber, 1996, Farkas, 1927, Becker & Doring, 1935, Frenkel, 1946, Zeldovich, 1942). This was later refined, modified and reviewed by numerous investigators. The expression for the nucleation rate as the number of droplets formed per

> � �� ��� ��

where *J* is the nucleation rate, *k* is the Boltzmann's constant and *q* is the condensation coefficient, which is defined as the fraction of molecular collisions which results in condensation. Substituting for critical radius, *r\** in terms of the supersaturation ratio, it will be seen that very small changes in the supersaturation of the fluid can influence the

Condensation occurs by the nucleation of new droplets and the growth of any existing droplets within specified incremental step. The incremental mass of liquid formed, *dmL* can be determined by using the nucleation and droplet growth equations. The equation for nucleation rate is derived from classical nucleation theory. Once the mass of liquid over an

expression can be integrated using fourth order Runge-kutta technique to yield the changes in the flow properties over the step. But to increase the accuracy of the coupling between the main flow equation and the equations describing droplet behavior, the calculation is carried

1. The droplet growth equations are integrated at constant inlet pressure, *Pin* and temperature, *TG,in* to the element *Δx* to obtain the first approximation to the values of *mL*

2. The values of *mL and dmL* found in Eq. 3 are inserted into the flow equations which are then integrated to give a first approximation for the exit consition, *Pout* and *TG,out*. 3. The droplet growth equation are integrated for a second time over *Δx*, assuming constant vapour condition, *Pout* and *TG,out* and new values of *mL* and *dmL* are

��� �� �� �∗ � � � ��

> ��� , �� �

> > � , ��� �� , �� �

� (10)

and ��� as independent

. The resulting

 and ��� ��

where *r\** is the critical droplet radius,

� ��� ��

nucleation rate drastically.

out in the following steps:

and *dmL*.

calculated.


Then starting at inlet to the nozzle, the flow equations are integrated step by step until the end of nozzle is reached.

In this investigation, upon completion of the nucleation phase, the properties of fluids are calculated based on the two methods. The first method is carried out by combining all droplet groups into a single population and their properties are averaged when there is no new droplet embyo forms in the subsequent step length. This is called the "Average" method. The other method of calculation is when every single droplet groups and their properties are retained even after the completion of nucleation phase. This is called the "Non-Averaged" method. The droplet radius calculation using this method is based on r.m.s. values for each droplet group. In theory, this method should results in more accurate solution but it requires an extremely huge processing power to complete the whole calculation.
