Void fraction model

Rhodes, and Scott Can. j. Chem. Eng., 47,445 53 [1969]) and Aka-gawa, Sakaguchi, and Ueda Bull JSME, 14, 564-571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles Can. ]. Chem. Eng., 52, 25-35 [1974]) and Barnea, Shoham, and Taitel Chem. Eng. Sci, 37, 741-744 [1982]). Use of drift flux theoiy for void fraction modeling in downflow is presented by Clark anci Flemmer Chem. Eng. Set., 39, 170-173 [1984]). Downward inclined two-phase flow data and modeling are given by Barnea, Shoham, and Taitel Chem. Eng. Set., 37, 735-740 [1982]). Data for downflow in helically coiled tubes are presented by Casper Chem. Ins. Tech., 42, 349-354 [1970]). 

Chexal, G., J. Horowitz, and G. Lellouche, 1987, An Assessment of Eight Void Fraction Models, in Heat Transfer—Pittsburgh, AIChE Symp. Ser. 83 (257) 249-254 Also 1986, EPRI, Nuclear Safety Analysis Center, Rep. NSAC-107. (3)... 

Akagawa, Sakaguchi, and Ueda (Bull JSME, 14, 564-571 [1971]). Correlations for flow patterns in downflow in vertical pipe are given by Oshinowo and Charles (Can. J. Chem. Eng., 52,25-35 [1974]) and Barnea, Shoham, and Taitel (Chem. Eng. Sci., 37, 741—744 [1982]). Use of drift flux theory for void fraction modeling in downflow is... 

The above equation uses the interface diameter, Di. This same expression can be represented in terms of die more convenient tube diameter, D, through the use of a void fraction model [30] as follows ... 

Despite the fact Chat there are no analogs of void fraction or pore size in the model, by varying the proportion of dust particles dispersed among the gas molecules it is possible to move from a situation where most momentum transfer occurs in collisions between pairs of gas molecules, Co one where the principal momentum transfer is between gas molecules and the dust. Thus one might hope to obtain at least a physically reasonable form for the flux relations, over the whole range from bulk diffusion to Knudsen streaming. 

The relation between the dusty gas model and the physical structure of a real porous medium is rather obscure. Since the dusty gas model does not even contain any explicit representation of the void fraction, it certainly cannot be adjusted to reflect features of the pore size distributions of different porous media. For example, porous catalysts often show a strongly bimodal pore size distribution, and their flux relations might be expected to reflect this, but the dusty gas model can respond only to changes in the... 

Since the void fraction distribution is independently measurable, the only remaining adjustable parameters are the A, so when surface diffusion is negligible equations (8.23) provide a completely predictive flux model. Unfortunately the assumption that (a) is independent of a is unlikely to be realistic, since the proportion of dead end pores will usually increase rapidly with decreasing pore radius. 

Besides the void fraction a very important parameter is the size of the particles. Using a simple cubic model for packing spherical balls, which represent the particles, we get the following expression for the void fraction... 

Fig. 5.22a,b Comparison of measured void fractions by Triplett et al. (1999b) for circular test section with predictions of various correlations (a) homogeneous flow model (b) Lockhart-Martinelli-Butterworth (Butterworth 1975). Reprinted from Triplett et al. (1999b) with permission... 

For a micro-channel connected to a 100 pm T-junction the Lockhart-Martinelli model correlated well with the data, however, different C-values were needed to correlate well with all the data for the conventional size channels. In contrast, when the 100 pm micro-channel was connected to a reducing inlet section, the data could be fit by a single value of C = 0.24, and no mass velocity effect could be observed. When the T-junction diameter was increased to 500 pm, the best-fit C-value for the 100 pm micro-channel again dropped to a value of 0.24. Thus, as in the void fraction data, the friction pressure drop data in micro-channels and conventional size channels are similar, but for micro-channels, significantly different data can be obtained depending on the inlet geometry. 

Similar void fraction data can be obtained in micro-channels and conventional size channels, but the micro-channel void fraction can be sensitive to the inlet geometry and deviate significantly from the homogeneous flow model. 

Taitel Y, Bamea D, Dukler AE (1980) Modeling flow pattern transitions for steady upward gas-liquid flow in vertical tubes. AlChE J 26 345-354 Triplett KA, Ghiaasiaan SM, Adbel-Khalik SI, Sadowski DL (1999a) Gas-liquid two-phase flow in microchannels. Part 1 two-phase flow patterns. Int J Multiphase Flow 25 377-394 Triplett KA, Ghiaasiaan SM, Abdel-Khalik SI, LeMouel A, McCord BN (1999b) Gas-liquid two-phase flow in microchannels. Part 11 void fraction and pressure drop. Int J Multiphase Flow 25 395 10... 

Kawahara et al. (2002) presented void fraction data obtained in a 100 pm micro-channel connected to a reducing inlet section and T-junction section. The superficial velocities are Uqs = 0.1-60m/s for gas, and fAs = 0.02-4 m/s for liquid. The void fraction data obtained with a T-junction inlet showed a linear relationship between the void fraction and volumetric quality, in agreement with the homogeneous model predictions. On the contrary, the void fraction data from the reducing section inlet experiments showed a non-linear void fraction-to-volumetric quality relationship ... 

The Martinelli correlations for void fraction and pressure drop are used because of their simplicity and wide range of applicability. France and Stein (6 ) discuss the method by which the Martinelli gradient for two-phase flow can be incorporated into a choked flow model. Because the Martinelli equation balances frictional shear stresses cuid pressure drop, it is important to provide a good viscosity model, especially for high viscosity and non-Newtonian fluids. 

Owing to the high computational load, it is tempting to assume rotational symmetry to reduce to 2D simulations. However, the symmetrical axis is a wall in the simulations that allows slip but no transport across it. The flow in bubble columns or bubbling fluidized beds is never steady, but instead oscillates everywhere, including across the center of the reactor. Consequently, a 2D rotational symmetry representation is never accurate for these reactors. A second problem with axis symmetry is that the bubbles formed in a bubbling fluidized bed are simulated as toroids and the mass balance for the bubble will be problematic when the bubble moves in a radial direction. It is also problematic to calculate the void fraction with these models. 

In the model equations, A represents the cross sectional area of reactor, a is the mole fraction of combustor fuel gas, C is the molar concentration of component gas, Cp the heat capacity of insulation and F is the molar flow rate of feed. The AH denotes the heat of reaction, L is the reactor length, P is the reactor pressure, R is the gas constant, T represents the temperature of gas, U is the overall heat transfer coefficient, v represents velocity of gas, W is the reactor width, and z denotes the reactor distance from the inlet. The Greek letters, e is the void fraction of catalyst bed, p the molar density of gas, and rj is the stoichiometric coefficient of reaction. The subscript, c, cat, r, b and a represent the combustor, catalyst, reformer, the insulation, and ambient, respectively. The obtained PDE model is solved using Finite Difference Method (FDM). 

In the following sections, the flow patterns, void fraction and slip ratio, and local phase, velocity, and shear distributions in various flow patterns, along with measuring instruments and available flow models, will be discussed. They will be followed by the pressure drop of two-phase flow in tubes, in rod bundles, and in flow restrictions. The final section deals with the critical flow and unsteady two-phase flow that are essential in reactor loss-of-coolant accident analyses. 

The Wilson bubble rise model (a void fraction of steam bubbling through stagnant water)... 

Interfacial area measurement. Knowledge of the interfacial area is indispensable in modeling two-phase flow (Dejesus and Kawaji, 1990), which determines the interphase transfer of mass, momentum, and energy in steady and transient flow. Ultrasonic techniques are used for such measurements. Since there is no direct relationship between the measurement of ultrasonic transmission and the volumetric interfacial area in bubbly flow, some estimate of the average bubble size is necessary to permit access to the volumetric interfacial area (Delhaye, 1986). In bubbly flows with bubbles several millimeters in diameter and with high void fractions, Stravs and von Stocker (1985) were apparently the first, in 1981, to propose the use of pulsed, 1- to 10-MHz ultrasound for measuring interfacial area. Independently, Amblard et al. (1983) used the same technique but at frequencies lower than 1 MHz. The volumetric interfacial area, T, is defined by (Delhaye, 1986)... 

As both phases occupy the full flow field concurrently, two sets of conservation equations correspond to these two phases and must be complemented by the set of interfacial jump conditions (discontinuities). A further topological law, relating the void fraction, a, to the phase variables, was needed to compensate for the loss of information due to model simplification (Boure, 1976). One assumption that is often used is the equality of the mean pressures of the two phases, ... 

Using a homogeneous model proposed by Owens (1961) for low void fractions (a < 0.30) and high mass flux, as is usually encountered in a water-cooled reactor, the momentum change (or acceleration) pressure gradient term is obtained from... 

Hsieh and Plesset assumed that the two-phase homogeneous mixture can be represented as a uniform medium with physical properties synthesized from the constituent phases and weighted according to void fraction, a, and quality, X. Using such a model, they were able to show that the gas compression is essentially isothermal and the acoustic velocity can be approximated as... 

Weisman and Pei Model Weisman and Pei (1983) assumed that CHF occurs when the steam void fraction in the bubble layer just exceeds the critical void fraction (they estimated it to be 0.82) at which an array of ellipsodal bubbles can be maintained without significant contact between the bubbles. The steam void fraction in the bubble layer is determined by a balance between the outward flow of vapor and the inward flow of liquid at the bubble layer-core interface. 

Xr the true quality, and a, the void fraction, have to be predicted under the conditions of subcooled or low-quality flow boiling. Again, with the homogeneous two-phase flow model,... 

The uncertainty in the predicted CHF of rod bundles depends on the combined performance of the subchannel code and the CHF correlation. Their sensitivities to various physical parameters or models, such as void fraction, turbulent mixing, etc., are complementary to each other. Therefore, in a comparison of the accuracy of the predictions from various rod bundle CHF correlations, they should be calculated by using their respective, accompanied computer codes.The word accompanied here means the particular code used in developing the particular CHF correlation of the rod bundle. To determine the individual uncertainties of the code or the correlation, both the subchannel code and the CHF correlation should be validated separately by experiments. For example, the subchannel code THINC II was validated in rod bundles (Weismanet al., 1968), while the W-3 CHF correlation was validated in round tubes (Tong, 1967a). 

Smith, S. L., 1970, Void Fraction in Two-Phase Flow A Correlation Based upon an Equal Velocity Head Model, Proc. Inst. Mech. Engrs., JS4(Part I), (36) 657. (3)... 

Staub, F. W., 1967, The Void Fraction in Subcooled Boiling—Prediction of the Initial Point of Net Vapor Generation, ASME Paper 67-HT-36, National Heat Transfer Conf., ASME, New York. (3) Staub, F. W., 1969, Two Phase Fluid Modeling, The Critical Heat Flux, Nuclear Sci. Eng. 3J.T 90-199. (5)... 

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