Void Fraction Correlation Coefficients

Simulation Review

Void fraction, the volume taken by each vapor and liquid phase in vapor liquid mixture, is an important parameter for accurate refrigerant charge inventory calculation. The void fraction is given in terms of vapor quality (x), vapor and liquid density (ρ_v and ρ_l), and slip ratio (S) as used by Thome (2004).

α=11+(1-x)/xρvρlS, where S=vvaporvliquid.

Slip ratio is the ratio of vapor velocity, V_vapor and liquid velocity, V_liquid. Slip ratio of unity means both the vapor and liquid have same velocity and mixture is homogeneous.

However, the vapor velocity is higher than liquid velocity in general and slip ratio is between 0 and 1.

In 1940, the need to mathematically model the complex two-phase flows first came to prominence for nuclear, pipeline, and process industry. The initial void fraction models were based on no-slip assumption. However, later researchers such as Hughmark (1962), Baroczy (1965), Taitel and Duckler (1976), and many others developed the void fraction correlations for different flow regimes considering non-unity slip.

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The three general two-phase flow regimes are; separated flow, intermittent flow, and distributed flow with each regime containing different types of sub flow regimes. However, for typical air conditioning and heat pump conditions, three important flow regimes are: annular flow (separated flow), intermittent flow, and wavy flow (separated flow). Most of the studies assume separated flow for void fraction model development while few studies analyze specific flow regimes. Due to the complex nature of the two-phase flow, only simple two-phase flows are modeled using an analytical approach.

Most of the developed void fraction correlations are empirical models containing dimensionless parameters obtained from the experimental data.

Rice (1987) performed a comprehensive overview of ten available void fraction models. He found void fraction correlations developed by as Hughmark (1962), Baroczy (1965), Premoli et al. (1971), and Tandon et al. (1985) gave best overall agreement with the experimental data for the refrigeration and air conditioning applications. Void fraction correlation developed by Hughmark (1962) yielded highest refrigerant charge predictions in comparison to the correlations developed by other researchers. Rice further found that Hughmark correlation over predicted the charge in the condenser.

Two system models, HPSIM developed by Domanski and Didion (1983) and PUREZ developed by Rice and Jackson (1994) under predicted the total refrigerant charge in the system. This is due to the charge inventory model error, unaccounted internal volume, and void fraction modeling assumptions. Void fraction correlations for micro-finned tubes were examined by Yashar et al. (2001). They proposed few modifications to the existing correlations to incorporate the microchannel heat exchanger internal enhancements.

Harms (2002) found unaccounted system volume and the improper choice of void fraction correlation to be the main cause of inaccurate refrigerant charge inventory modeling. Some of the commonly used correlations found to be inconsistent with the flow regime in the two-phase region of the condenser. The homogeneous model under predicted the charge by 11 to 23%. He found unaccounted refrigerant charge dissolved in the compressor oil as another important issue affecting accurate charge inventory modeling. He recommended Yashar’s model for separated flow and the Taitel’s and Barnea’s model for the slug flow for use in system modeling. For simple void fraction model, he recommended Zivi (1964) model.

Bo Shen (2006) suggested not using system simulations at the system level to validate a void fraction model because many factors such as accurate liquid line calculations, unknown inside cross-sectional area etc. affect the refrigerant charge inventory predictions. He developed two-point tuning method to take into account the unaccounted system volume and the inaccuracies related to the liquid line length. The void fraction choice is not important while using his two-point tuning method. However, two-point tuning method requires experimental data and at the design phase, refrigerant charge model still requires an accurate void fraction correlation for accurate refrigerant charge inventory calculation.

Void Fraction Models Overview

Different void fraction correlations developed by researchers have different values of the coefficients (B_B,n_1,n_2,n_3) used in the equation 2. Table 2 (Carey, 1992) lists the value of different coefficients for equation 2.

α=[1+B_B x_r^n1 ρ_r^(-n2) μ_r^n3 ]^(-1), (2)

where ρ_r is the ratio of saturated vapor to saturated liquid density i.e. ρ_r=ρ_l/ρ_v , x_r=(1-x)/x, and μ_r is the ratio of saturated vapor to saturated liquid dynamic viscosity i.e. μ=μ_l/μ_v .

Table 2: Void fraction correlation coefficients for equation 2 

Homogeneous void fraction model yields an overestimated value of void fraction and as a result, tends to underestimate the system charge. Rigot’s (1973) model recommended a slip ratio of 2. Smith’s (1969) model used equation 3 for the slip ratio. The equation is derived based on the assumption of annular liquid phase flow with the droplets entrainment in the vapor region in the center.

S=K+(1-K) [(ρ_r+Kx_r)/(1+Kx_r )]^0.5,K=0.4 (3)

Analytical void fraction model developed by Zivi uses minimum entropy generation principle. His model assumed no wall friction and no liquid entrainment for the annular configuration. Equation 4 gives the slip ratio used by Zivi.

S=1+aRe_l^(-0.19) (ρ_r )^0.22 [y/(1+b⋅y⋅We_l⋅Re_l^(-0.53) (ρ_r )^(-0.08) )-b⋅y⋅We_l⋅Re_l^(-0.53) (ρ_r )^(-0.08) ]^0.5,

where:

a=1.58,b=0.0273,y=β/(1-β),We_l=((G^2 D)/(σρ_l g)), (4)

G is mass flux, σ is surface tension, Re_l is liquid Reynolds number, and β=[1+x_r ρ_r ]^(-1).

Permoli (1971) void fraction model takes into account the liquid density error and as a result, more accurately predicts the apparent two-phase refrigerant density. Slip ratio used by Permoli for void fraction correlation is given by equation 5.

S=(ρ_r )^(1/3), (5)

Lockhart and Martinelli (1947) and Martinelli and Nelson (1948) introduced an important parameter, Lockhart-Martinelli parameter( X)_tt, which is the ratio between single phase frictional pressure gradient of vapor and liquid. Both the vapor and liquid phases assumed to have turbulent flow. (X)_tt is given by equation 6.

(X)_tt=((1-x)/x)^0.9 (ρ_v/ρ_l )^0.5 (μ_v/μ_l )^0.1, (6)

The void fraction correlation developed by Lockhart and Martinelli is given by equation 7.

α=(1+0.28⋅X_tt^0.71 )^(-1), (7)

Baroczy (1965), Wallis (1969), and Domanski and Didion (1983), and many others developed void fraction correlations as semi-empirical functions of Lockhart-Martinelli parameter.

Void fraction correlation in equation 8 is developed by Wallis while correlation in equation 9 is developed by Domanski and Didion.

α=(1+X_tt^0.8 )^(-0.378),'for' X_tt≤10 (8)

α=0.823-0.157 ln⁡(X_tt,) 'for' X_tt>10 (9)

Void fraction correlation developed by Tandon considers the mass flux by using Reynolds number (Re) as given by equation 10.

α=[1-(1.928⋅Re_l^(-0.315))/F(X_tt ) +(0.9293⋅Re_l^(-0.63))/(F(X_tt )^2 )]^ ,for 501125,

where:

F(X_tt )=0.15⋅(1/X_tt +2.85/(X_tt^0.476 )),Re_L=GD(1-x)/μ_l ,D ' is the tube diameter' (10)

Hughmark developed void fraction correlation by applying correction factor K_H to the homogeneous model as given by equation 11. Figure 1 gives the equation of K_H as a function of Z.

α=K_H β

Z=[DG/(μ_l+α(μ_v-μ_l ) )]^(1/6) [1/gD (Gx/(ρ_v β(1-β) ))^2 ]^(1/8) (11)

To account for volumetric flow rates of vapor and liquid phase with respect to a moving reference frame at mean velocity, drift flux void fraction model developed by Zuber and Findlay (1965) as given by the equation 12.

α=x/ρ_v [C_0 (x/( ρ_g )+(1-x)/ρ_l )+U ̅_GU/m ̇ ]^(-1), (12)

where m ̇ is the mass flow of refrigerant.

Parameters C_0 and U ̅_GU in equation 13 for drift flux void fraction correlation for vertical channels (equation 13) are given by Rouhani-Axelsson (1970) and for horizontal channels (equation 14) by Steiner (1993).

C_0=1+0.2(1-x) (gd_i ρ_l^2)/m ̇^2 ,U ̅_GU=1.18[gσ(ρ_l-ρ_v )/(ρ_l^2 )]^(1/4) (13)

C_0=1+0.2(1-x) ,U ̅_GU=1.18(1-x)[gσ(ρ_l-ρ_v )/(ρ_l^2 )]^(1/4) (14)

Harms et al. (2002) developed the explicit model for void fraction as given by equation 15. He took into account the momentum eddy diffusivity damping at the interface of the film. With the increase in the mass flux, the slip ratio decreases. In comparison to the Yashar et al. (2001) empirical model, this model had less dependence on mass flux and as a result under predicted the slip ratio at low mass flow rates in comparison to Yashar et al. model.

α=[1-10.06 Re_l^(-0.875) (1.74+0.104 Re_l^0.5 )^2 (1.376+7.242/(X_tt^1.655 ))^(-1/2) ]^2 (15)

In this project, heat pump model developed by graduate student in OSU, Iu (2007) will be modified for the comparison of various void fraction correlations for refrigerant charge calculation in two-phase region of condenser and evaporator.

Conclusion

Choosing an appropriate void fraction correlation is crucial for accurate refrigerant charge calculations. This literature review provides insights into various correlations and their applicability in heat exchanger models, highlighting the need for further research to refine existing models and improve charge inventory accuracy.