Examining the curvature dependency of surface tension in a nucleating steam flow

Abstract

Many researches have been performed on the phenomenon of vapor condensation at the hypersonic stages of the steam turbine. This phenomenon consists of nucleation and droplet growth consequently. One of the most important parameters affecting the nucleation and droplet growth is surface tension, which is often assumed to be equal to the surface tension of a planar interface. The nucleated droplets have the nano-scale size at the beginning of the process and they will not be bigger than macro-scale after the growth. On the other hand, there are many reports about the dependency of surface tension on the curvature of the surface in such a small size. In order to evaluate the effects of curvature dependency on the surface tension, three famous models, proposed by Tolman, Benson and Rasmussen, were incorporated with nucleation theory and their effects on the wet steam flow were investigated. The results revealed that the dependency of the surface tension on the droplet radius significantly improved the accuracy of the predicted droplet radius. Also, it was shown that Tolman’s theory have offered the best results.

Appendix

1.1 Governing equations of wet steam

The wet steam flow was also analyzed in a Eulerian-Lagrangian coordinate. In this model, gas dynamic equations (conversation of mass, energy, and momentum of the flow) was solved in a Eulerian coordinate, while nucleation and droplet growth equations were performed in a Lagrangian frame. In order to combine Eulerian and Lagrangian coordinates, the flow path was divided into very small steps. At each step, at first, the flow variables such as temperature, velocity, density, and pressure were determined in the Eulerian coordinate. Then, considering the pressure and temperature, the Lagrangian equations were solved and wetness parameters consisting of droplet size distribution and wetness fraction were obtained.

Regarding an infinite-small control volume of a one-dimensional condensing steam flow with a length of dx (Fig. 10), the main governing equations of flow were written as follows.

Fig. 10

Control volume of flow in Eulerian coordinate

1.2 Mass balance

Mass conservation equation along the axis of control volume was written as follows:

$$ M={M}_L+{M}_G= const $$

(12)

where subscripts L and G refer to liquid and gas phases, respectively. Also, M is the mass flow rate.

Differentiating Eq. (12) and performing a few mathematical operations, a computational form of the law of mass conservation for the one-dimensional condensing flow was obtained as:

$$ \frac{d{\rho}_G}{\rho_G}+\frac{dA}{A}+\frac{d{U}_G}{U_G}+\frac{d{M}_L}{M_G}=0 $$

(13)

1.3 Momentum balance

Neglecting interphase slipping due to the microscopic size of the liquid droplets and the drag force between droplets and surrounding vapor, the momentum balance was obtained as follows:

$$ \frac{dP}{P}+\frac{f{\rho}_G{U}_G^2}{2P{D}_e} dx+\frac{\left({M}_G+{M}_L\right){U}_G}{AP}\frac{d{U}_G}{U_G}=0 $$

(14)

where D_e and f are the hydraulic diameter and wall friction factor, respectively.

1.4 Energy balance

By imposing the conservation of energy on the one dimensional, adiabatic, compressible, and steady state nucleating flow, the energy equation was written as follows:

$$ \frac{U_G^2}{C_P{T}_G}\frac{d{U}_G}{U_G}+\frac{d{h}_G}{C_P{T}_G}-\frac{d\left(L{M}_L\right)}{C_P{T}_G\left({M}_G+{M}_L\right)}=0 $$

(15)

where C_pand L_hare the specific heat capacity of vapor in constant pressure and the latent heat, respectively.

1.5 Equation of state

A density based virial state equation was also used for super-saturated vapor [25]:

$$ \frac{P}{\rho_GR{T}_G}=1+{B}_1{\rho}_G+{B}_2{\rho}_G^2+{B}_3{\rho}_G^3+{B}_4{\rho}_G^4+{B}_5{\rho}_G^5 $$

(16)

Whereβ₁, β₂, β₃, β₄ and β₅ are known as density virial coefficients, and are functions of the vapor temperature [18].

The differential form of the state equation was given by the following equation:

$$ \frac{dP}{P}-X\frac{d{\rho}_G}{\rho_G}-Y\frac{d{T}_G}{T_G}=0 $$

(17)

where Y and X are defined as follows:

$$ {\displaystyle \begin{array}{l}Y=\frac{T_G}{P}{\left(\frac{\partial P}{\partial {T}_G}\right)}_{\rho_G}=1+\frac{\rho_G{T}_G}{1+{B}_1{\rho}_G+{B}_2{\rho}_G^2+{B}_3{\rho}_G^3+{B}_4{\rho}_G^4+{B}_5{\rho}_G^5}\times \\ {}\left[\frac{d{B}_1}{d{T}_G}+{\rho}_G\frac{d{B}_2}{d{T}_G}+{\rho}_G^2\frac{d{B}_3}{d{T}_G}+{\rho}_G^3\frac{d{B}_4}{d{T}_G}+{\rho}_G^4\frac{d{B}_5}{d{T}_G}\right]\kern0.24em \end{array}} $$

(18)

$$ X=\frac{\rho_G}{P}{\left(\frac{\partial P}{\partial {\rho}_G}\right)}_{T_G}=\frac{1+2{B}_1{\rho}_G^2+3{B}_2{\rho}_G^2+4{B}_3{\rho}_G^3+5{B}_4{\rho}_G^4+6{B}_5{\rho}_G^5}{1+{B}_1{\rho}_G+{B}_2{\rho}_G^2+{B}_3{\rho}_G^3+{B}_4{\rho}_G^4+{B}_5{\rho}_G^5} $$

(19)

1.6 Mach Number

The Mach number was defined as follows:

$$ Z=M{a}^2=\frac{u^2}{\left(\frac{\gamma P}{\rho_G}\right)} $$

(20)

Taking the differentiation of Eq. (20) led to:

$$ \frac{dZ}{Z}=2\frac{d{U}_G}{U_G}+\frac{d{\rho}_G}{\rho_G}-\frac{dP}{P} $$

(21)

The Eqs. (13), (14), (15), (17) and (21) were solved and integrated into a Eulerian framework.

The mass of the liquid phase was unknown in the above-mentioned equations. In fact, the mass of the liquid phase was obtained by calculating the nucleation rate and droplet growth in a Lagrangian framework.

1.7 Nucleation model

When the single-phase vapor in the absence of any external particle was expanded and passed from the saturation line, it would become supersaturated. This was a quasi-equilibrium state. Then, the flow returned to equilibrium state through formation and growth of a large number of tiny liquid droplets. This process is called homogeneous nucleation. From a thermodynamic view, the changes in Gibb’s free energy for the formation of a tiny liquid droplet from the supersaturated vapor consisted of two bulk and surface terms as follows [26]:

$$ \varDelta G=\varDelta {G}_V+\varDelta {G}_S=-{m}_rR{T}_G\ln \left(\frac{P}{P_s\left({T}_G\right)}\right)+4\pi {r}^2{\sigma}_r $$

(22)

where $\frac{P}{P_s\left({T}_G\right)}$is the supersaturation ratio, ${m}_r=\frac{4}{3}\pi {r}^3{\rho}_L$is the mass of a droplet, and σ_ris the surface tension of the liquid droplet.

Based on the thermodynamic equilibrium, a stable nucleated droplet had a minimum radius which was called critical radius. The formation Gibb’s free energy of a critical clusterΔG^∗ and r^∗were obtained from differentiating Eq. (22) with respect to r [27]:

$$ {r}_{\infty}^{\ast }=\frac{2{\sigma}_{\infty }}{\rho_LR{T}_G Ln\left(\frac{P}{P_s\left({T}_G\right)}\right)} $$

(23)

$$ \varDelta {G_{\infty}}^{\ast }=\frac{4\pi \sigma {{r_{\infty}}^{\ast}}^2}{3}=\frac{16\pi \kern0em {\sigma}_{\infty}^3}{3{\left({\rho}_LR{T}_G Ln\left(\frac{P}{P_s\left({T}_G\right)}\right)\right)}^2} $$

(24)

where ∞ refers to the flat surface tension.

Steady state nucleation rate was also obtained as follows [28,29,30]:

$$ {J}_{classic}={q}_c\frac{\rho_G^2}{\rho_L}\sqrt{\frac{2{\sigma}_{\infty }}{\pi {m}^3}}\exp \left(-\frac{\varDelta {G}_{\infty}^{\ast }}{K{T}_G}\right)\kern2.04em $$

(25)

where Kis Boltzmann’s constant and q_c is the condensation coefficient.

Sometimes, the classic Nucleation equation (Eq. 25) is not compatible with available experimental results [31]. Hence, numerous researchers required to correct the equation [32,33,34,35].

Recently, Hale [36,37,38] developed a nucleation model based on scaling arguments, which had a temperature dependence. This model was matched with the experimental results quite closely.

$$ J={J}_0\exp \left(-\frac{W}{k{T}_G}\right) $$

(26)

where:

$$ {J}_0={10}^{26}c{m}^{-3}{s}^{-1} $$

(27)

$$ W=\frac{16\pi {\varOmega}^3{\left(\frac{T_c}{T}-1\right)}^3}{3{\left(\ln \left[\frac{P}{P_s\left({T}_G\right)}\right]\right)}^2} $$

(28)

T_c is vapor-droplet critical temperature, Wis the formation work of a critical cluster, and Ω is the dimensionless surface entropy. The value of Ω was derived from experimental nucleation rate data or estimated from the physical properties of the desired substance, which was mainly considered to be 1.47 for water [15]. This model was used by several researchers such as Strey [39], Sinha [40] and Amiri Rad et al. [15].

1.8 Droplet growth

The growth of the droplets was controlled by mass and energy transfer to the liquid and vapor phase, respectively. Applying the first law of thermodynamics to a small droplet, the conservation of energy was written as:

$$ \frac{d}{dt}\left(\frac{4}{3}\pi {r}^3{\rho}_LL\right)=4\pi {r}^2{\alpha}_r\left({T}_L-{T}_G\right) $$

(29)

where α_r is the heat transfer coefficient which can be expressed as [41]:

$$ {\displaystyle \begin{array}{l}{\alpha}_r=\frac{\lambda }{r\left(\frac{1}{1+2\beta kn}+3.78\frac{kn}{\Pr}\right)};\kern1.08em \beta =0.75\kern0.75em Kn=\frac{\overline{l}}{2r}\\ {}\Pr =\frac{C_p{\mu}_G}{\lambda },\end{array}} $$

(30)

where Pr is Prantel number and λ is the heat conduction coefficient of the vapor phase.

Bakhtar and Zidi [42] introduced a semi-empirical equation for droplet growth in all regimes as follows:

$$ \frac{dr}{dt}=\frac{kn}{kn+0.375{q}_c{S}_c}\frac{q_c}{\rho_L}{\left(\frac{R}{2\pi}\right)}^{0.5}\left[{\rho}_G\sqrt{T_G}-{\rho}_s\left({T}_L,r\right)\sqrt{T_L}\right] $$

(31)

where S_c is the Schmidt number and T_L is the temperature of the droplet.

The droplet radius and temperature were calculated by a simultaneous numerical solution of differential Eqs. (29) and (31) along the length of the element.