Hold-up correlation

In addition to flow regime, hold-up and pressure drop are two other important parameters in two-phase gas-liquid flows. Hold-up is defined as the relative portion of space occupied by a phase in the pipe. It can be expressed on a time or space average basis, with the actual method chosen depending on the intended use of the hold-up value, and the measurement method employed. There are numerous correlations in the literature for hold-up, but most are based upon a pressure drop-hold-up correlation. The following expression is a widely recognized empirical relationship between hold-up and pressure drop ... 

A second hold-up correlation reported by T. Otake and K, Okada [55] represents a survey of considerable literature, and is applicable to aqueous and non-aqueous systems for Reynolds numbers from 10, - 20,000 [40]. 

M.J. Elmann, N. Midoux, G. Wild, A. Laurent and J.C. Charpentier, A new, improved liquid hold-up correlation for trickle-bed reactors, Chem. Engng. Science, 45 (1990) 1677-1684. 

Early SBCR models were reviewed by Ramachandran and Chaudhari (5) and by Deckwer (9). They require hold-up correlations as an input and do not compute flow patterns. The most complete and useful of these models applied to the Fischer-Tropsch (F-T) conversion of synthesis gas in a SBCR is that of Prakash and Bendale (79). They sized commercial SBCR for DOE. They gave syngas conversion and production as a function of temperature, pressure and space velocity. Input parameters with considerable uncertainty that influenced production rates were the gas hold-up, the mass transfer coefficient and the dispersion coefficient. Krishna s group (77) extended such a model to compute product distribution using a product selectivity model. Air Products working with Dudukovic measured dispersion coefficients needed as an input into such model. The problem with this approach is that the dispersion coefficients are not constant. They are a function of the local hydrodynamics. 

Tjatjopoulos, "Hold-up Correlations in Slurry-Solid Fluidized Beds". AIChE Journal 28(1982) 346-348. 

Lockhart and Martinelli used pipes of one inch or less in diameter in their test work, achieving an accuracy of about -l-/-50%. Predictions are on the high side for certain two-phase flow regimes and low for others. The same -l-/-50% accuracy will hold up to about four inches in diameter. Other investigators have studied pipes to ten inches in diameter and specific systems however, no better, generalized correlation has been found.The way... 

The precise nucleation mechanism occurring in any particular case is often a subject of debate, however, and in practice the data are normally correlated empirically by an expression including a dependence on solids hold up of the form ... 

Figure 9-44 presents water hold-up data that are correlated by [40] ... 

Monsanto and other companies are working independently on design methods to size vents more rigorously using two-phase flow calculations in complex computer programs. Several assumptions have been made in an effort to allow a wide range of application. Most notable is the use of the correlations of Martinelli and co-workers for pressure drop H) and hold-up( ). 

Correlations are needed to predict whether two-phase flow will occur after vapor venting is initiated by rupture disc failure or relief valve opening. Research is needed in this area, but for the present we recommend the following correlations to predict batch swell. For systems with low viscosity (less than 500 cp) an equation based on bubble column hold-up is used to obtain a swell ratio ... 

The presence of a gas in the suspension results in an increase of the stirrer speed required to establish the state of complete suspension. The propeller usually requires a higher speed than the turbine. Furthermore, a critical volume gas flow exists above which drastic sedimentation of particles occurs. Hence, homogenisation of the suspension requires an increase of the rotational speed and/or a decrease of the gas flow rate. The hydrodynamics of suspensions with a solid fraction exceeding 0.25-0.3 becomes very complex because such suspensions behave like non-Newtonian liquids. This produces problems in the scale-up of operations. Hydrodynamics, gas hold-up, mass-transfer coefficients, etc. have been widely studied and many correlations can be found in literature (see e.g. Shah, 1991). 

A well-substantiated correlation for air-water systems taken from the trickle bed literature (Morsi and Charpentier, 1981) was used for the volumetric mass transfer coefficients in the / , and (Rewap)i terms in the model. The hi term was taken from a correlation of Kirillov et al. (1983), while the liquid hold-up term a, in Eqs. (70), (71), (74), (77), and (79) were estimated from a hold-up model of Specchia and Baldi (1977). All of these correlations require the pressure drop per unit bed length. The correlation of Rao and Drinkenburg (1985) was employed for this purpose. Liquid static hold-up was assumed invariate and a literature value was used. Gas hold-up was obtained by difference using the bed porosity. 

The operating parameter for the CSTR reactor is the liquid flow rate Q, which sets the residence time of the liquid through the ratio Q/VL and finally the conversion. From a production viewpoint, the (residence) time required to achieve a given conversion of S (or outlet concentration of S) is obtained by solving the set of Eqs. (33) and (34). The characteristics of the reactor kLa and VL must be known. In general, whereas VL is easily determined in a batch reactor, it is not in a CSTR. Rather, VL=fiLVR will be used, which requires knowledge of the liquid hold-up L. Correlations provide kLo (see below) and L characteristics for the different reactor types [3]. 

The effects of gas hold-up and bubble diameter have also been studied by Sridhar and Potter and, again, the correlations obtained by Calderbank are recommended. 

Below the upper transition point, hold-up data may be correlated by an equation analogous to that used for spray towers, except as Gayler et al. 21 and Gayler and Pratt 28 point... 

Only a limited number of attempts have been made to correlate the liquid hold-up or the equivalent in the pipe. Again it seems clear that the liquid hold-up will be significally influenced by the flow pattern, and that either separate correlations will be required for each flow pattern, or a master correlation incorporating those variables which influence flow pattern will need to be developed. With one or two exceptions the few existing correlations suffer from the same kind of disadvantage as those for the prediction of pressure drop. ... 

The influence of the liquid velocity and of the liquid viscosity are taken into account via the liquid saturation fi. Unfortunately, very few reliable experimental investigations on the influence of viscosity on the liquid hold-up are to be found in the literature. The correlation proposed by Wijffels et al. [21] is based mainly on data obtained with water. 

The residual (or capillary) liquid hold-up was correlated also by Saez and Carbonell [26] as ... 

This correlation gives, for perfectly wettable solids, fairly good estimates of the static holdup for different particle-geometries and sizes. Saez and Carbonnel [26] used the hydraulic diameter, instead of the nominal particle diameter, as the characteristic length in the Eotvos number, to include the influence of the particle geometry on the static hold-up. However, no improvement could be obtained in correlating the data with this new representation. 

The available correlations in the literature are not able to represent the experimental results derived from 1500 high pressure data on the liquid hold-up. To correlate all the data, the effects of fluid inertia, surface forces, and liquid shear stress have again been accounted for, by using the corresponding dimensionless groups in the following empirical correlation [37] ... 

With the same approach as described for the two-phase pressure drop, Iliuta et al. [47] also proposed a new correlation for the total external liquid hold-up in the high interaction regime, with an average absolute relative error of 13.6 %. 

Figure 5.2-32. Parity plot of the correlation (5.2-28) to estimate the total liquid hold-up (after Larachi et al. [37]).

I. Iliuta, F. Larachi and B.P.A. Grandjean, Pressure drop and liquid hold-up in trickle flow reactors improved Ergun constants and slip correlations for the slit model, Ind. Engng. Chem. Res., 37 (1998) 4542-4550. 

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