Efficiency of Mass and Momentum Transport in Homogeneous and Two-Phase Flow Jean-Marc Engasser and Csaba Horvath* Chemical Engineering Group, Department of Engineering and Applied Science. Yale University, New Haven, Connecticut 06520
Plots of ~ / ( S C ) against ’ / ~ t h e viscous Nusselt number, Nu,, and ~ ( S C )against ~ ’ ~ t h e friction factor are used to compare the efficiency of radial mass and momentum transport in slug flow to that in homogeneous laminar and turbulent flow. A dimensionless group, t h e Le Goff n u m b e r , is suggested for t h e ratio of mass transferred and energy dissipated. With this n u m b e r it is shown quantitatively that t h e mass to momentum transfer efficiency is higher in open tubular t h a n in packed bed contactors.
In a previous study we have shown (Horvath et al., 1973) that the radial mixing in pipes is more efficient with slug flow than with homogeneous liquid flow and the enhancement of radial transport in the liquid slugs is accompanied by an increase in the pressure drop, that is, by a higher efficiency of momentum transfer. In order to assess the effect of various flow conditions on mass, heat, and momentum transport in contactors, whose efficiency is solely determined by transport in the bulk fluid, in this paper we examine the relationship between the efficiencies of these transport phenomena. It is shown that mass and momentum transport in slug flow can be conveniently compared to those in homogeneous laminar and turbulent flow by plotting data for mass transport against the corresponding momentum transport data. It has been found that such plots of the corresponding mixing and transfer efficiencies are more illustrative for the interrelation between mass and momentum transport than the conventional plots. The mass to momentum transport efficiency, that is the amount of mass transferred per unit of energy dissipated, is expressed by a new dimensionless number and used to compare contractors of different configurations.
Efficiency of Mass and Momentum Transport in Open Tubular Contactors According to Le Goff (1970a,b) most dimensionless groups used to measure the radial transport of momentum, mass, and heat are expressions of one of the following efficiencies: (a) the mixing efficiency in the fluid, (b) the overall transfer efficiency of an exchanger or contactor, and (c) the transfer efficiency per unit surface area of the contactor. This classification includes the Euler number and the viscous Nusselt number in addition to the more commonly used dimensionless groups as shown in Table I. Customarily, each of these dimensionless numbers is plotted against the Reynolds or Graetz number in order to illustrate the efficiency of mass or momentum transport under different flow conditions. We found, however, that it was more elucidative to plot the mass transport efficiencies against the corresponding momentum transfer efficiencies, because this graphical representation facilitates the comparison of the two transport efficiencies in a wide range of flow configurations. In order to illustrate the mixing efficiencies, which are also indicative of the rate of exchange a t the wall, m / S c l is plotted against Nu, for homogeneous laminar and turbulent flow, as well as for slug flow, in Figure 1. The average Nusselt number, Nu, for homogeneous flow was taken from the literature and for mass transport in slug flow from a previous study (Horvath et al., 1973). The viscous Nusselt number, Nuv, is the product of the
Reynolds number and the friction factor and given by
Nu, = Re(f/2)
(1)
For homogeneous flow Nu, was obtained with f / 2 values from the literature, whereas for slug flow Nu, was calculated from previous experimental pressure drop data by the procedure described in Appendix I. Corresponding to the conditions used in the slug flow experiments, the dimensionless tube length, Lld, and the Schmidt number are taken as 260 and 1700, respectively. Slug flow data are shown a t two different Reynolds numbers, Re = 220 and Re = 100, and a t different dimensionless slug lengths, p , in the range of 17 to 1. was divided by (SC)’’~,as first suggested by Colburn (1933), in order to make the numerical values of the dimensionless quantities describing momentum and mass transport comparable. Figure l shows that in slug flow, despite the low Reynolds number, the values of the mixing efficiencies for both mass and momentum transport are intermediate between those obtained in homogeneous laminar and turbulent flow. The efficiencies increase as the slug length decreases so that with short slugs turbulent mixing is approached. In Figure 2 another aspect of the relative efficiencies is illustrated by the plot of the transfer efficiencies, which are indicative of the “conversion” in the exchanger. For the same flow conditions as before, .%(SC)~3, often referred to as the j factor, is plotted against the friction factor, f / 2 . The average Stanton number was obtained by the relationship -
St = Ku/(ReSc)
and the friction factor was evaluated by eq 1. As seen in Figure 2 laminar flow at low Re yields the highest transfer efficiencies. Unlike the mixing efficiencies, both the momentum and mass transfer efficiencies decrease with increasing flow rate in homogeneous flow except in the transition domain. In slug flow, however, both the mixing and transfer efficiencies show the same trend since radial mixing increases a t a fixed liquid flow rate. The plots of the transfer efficiencies in Figure 2 also illustrate the applicability of the two most frequently used analogies: the Chilton-Colburn and the Prandtl analogy (Knudsen and Katz, 1958).
Mass to Momentum Transport Efficiency in Open Tubular and Packed Bed Contactors The mixing or transfer efficiencies discussed in the previous section were singularly related to either mass or momentum transport. In many cases however, the efficiency of mass transport with respect to that of momentum transport would be of interest in order to evaluate and compare the efficacy of various types of exchangers a t different flow conditions. Ind. Eng. Chem.. Fundam., Vol. 14, No. 2, 1975
107
TURBULENT
10
/ SLUG
-
, , ' I
L
I 10
100
Nu, Figure 1 . Plot of t h e mixing efficiency for mass transport, % (Sc)l 3 , against t h a t for momentum transport, Nu,, for homogeneous laminar and turbulent as well as for slug flow. T h e increasing values of both Re and t h e dimensionless slug length, 8, are indicated by arrows. Sc = 1700 a n d for slug flow c = 0.5. L / d = 260.
Table I. Dimensionless Groups Used to Express the Different Efficiencies of Mass, Heat, and Momentum Transport Mass and heat transfer
Momentum transfer
Mixing efficiency
Nu Nusselt number
Over all transfer efficiency
NT U Number transfer units St Stanton number
Nu, Viscous nusselt number Eu Euler number
Efficiency
Transfer efficiency per unit area
f/2
Friction factor
We suggest that the efficiency of mass to momentum transport is expressed by the ratio of the j factor to the friction factor, and the resulting dimensionless group is called the Le Goff number, Lf. It is easy to show that Lf is equal to the ratio of any given pair of the corresponding mass and momentum transport efficiencies, that is
Thus, the physical meaning of Lf is evident; it expresses the amount of mass or heat transported to the contact surface per unit of amount of momentum transferred when mass or heat transport in the bulk fluid is the rate-determining step. A total Schmidt number, which takes into account both eddy and molecular diffusion and has a similar physical meaning as the reciprocal Le Goff number, has already been suggested in the literature (Opfell and Sage, 1956). Nevertheless, the practical usefulness of the total Schmidt number is greatly reduced by the limited availability of data for eddy diffusivity, whereas the Le Goff number can be evaluated from a variety of data frequently measured in chemical engineering. Thus, Lf can be calculated from the conversion, X, and the pressure drop, I P , by
where c is the void fraction in the two phase flow, p is the mass density of the liquid, and V is the average axial ve108
Ind. Eng. Chem., Fundam., Vol. 14, No. 2, 1975
B
,
16' I f/2 Figure 2. Graphical illustration of t h e analogy between mass and momentum transport by the plot of the mass transfer efficiency, ~ ( S C3 , )against ~ the momentum transfer efficiency, f / 2 , for homogeneous turbulent and laminar flow as well as for slug flow. The increasing values of both Re and the dimensionless slug length, 3, are indicated by arrows. Sc = 1700, L / d = 260 and for slug flow t = 0.5. The broken lines represent the Chilton-Colburn and the Prandtl analogies. I
16'
locity. For small conversions, X < 0.1, the Le Goff number is therefore proportional to the ratio of conversion and pressure drop. In view of the recent literature (Engasser and Horvath, 1974) eq 4 is also applicable to first-order heterogeneous chemical reactors which are not bulk diffusion controlled. A plot of Lf against a suitable parameter of the hydrodynamic conditions such as the Reynolds number allows the assessment of the mass transfer efficiency with respect to the frictional energy loss in open tubes with homogeneous liquid and slug flow as shown in Figure 3. In order to make the data for slug flow comparable with those for homogeneous flow, the Reynolds number Re!, was calculated with the volumetric liquid flow rate, F, by the equation 4FP Re, = ndu
(5)
where p and p are the viscosity and mass density of the liquid. For slug flow the physical meaning of Reb is different from that in homogeneous flow because the flow velocity depends not only on F but also on the void fraction. Figure 3 shows that the mass to momentum efficiency is much greater for turbulent than laminar homogeneous flow. At relatively low values of ReF, however, Lf can be significantly increased by generating slug flow and the improvement of Lf is the greatest with relatively long slugs. All the three graphs presented so far can conveniently be used to illustrate the efficiency of mass, heat, and momentum transport in open tubes of various configurations and with all kinds of single and two-phase flows. Similar plots can also be prepared for packed and fluidized beds. Nevertheless, the classical dimensionless groups are of limited value for the comparison of the mass and momentum transport efficiencies in open tubes and packed beds because of the different choices of the characteristic length in the two types of conduits. The Le Goff number, however, is independent of the characteristic distance of the system; thus it permits a direct comparison of the mass to momentum efficiencies in pipes and packed beds. In order to illustrate this point, in Figure 4 Lf is plotted against the Reynolds number for homogeneous liquid flow in beds packed with spherical particles a t the same Schmidt number. The appropriate Reynolds number, Re,, is defined by
1,2c'I
I
"
I
OPEN TUBES
( 1
0l o8 [
-
PACKED BEDS
TURBULENT
I
1
02
I
0 ' IO
IOP
IOS
10'
d
Re, Figure 3. Plot of the Le Goff number. which expresses t h e mass to momentum transfer efficiency, against t h e Reynolds number for open tubular contactors or exchangers. Slug flow d a t a for different values of the dimensionless slug length, ,d, are also indicated for Re, = 50 a n d 110. Sc = 1700 a n d for slug flow I = 0.5; L / d = 260.
Re P Figure 1. Plot of t h e Le Goff number, which expresses t h e mass t o momentum transfer efficiency, against the Reynolds number for beds packed with spherical particles. Sc = 1700 a n d the void volume of the bed is 0.4.
EU = D F Re, = 2ii
where Drl is the particle diameter. The values of Lf have been calculated as outlined in Appendix 11. It is seen from Figure 4 that the ratio of efficiencies for mass and momentum transport in packed beds is optimal in a certain intermediate range of Re,. The maximum value of Lf is about 0.02 a t Sc = 1700. On the other hand, open tubes yield much higher efficiencies even with laminar flow, as shown in Figure 3 for the same value of Sc. Consequently, the efficiency of mass transfer relative to that of momentum transfer is significantly lower in packed beds than in open tubes. In other words, the energy input required to transport a given amount of mass from the fluid stream to the surface is much greater in packed beds than in pipes. Figures 3 and 4 give insight into the analogy between mass and momentum transport under various conditions and show the deviation from the Chilton-Colburn analogy which predicts that Lf is unity. The results also demonstrate that the capillary bundle model, which is frequently employed in the treatment of packed beds, fails to give a useful approximation for the relative magnitude of mass and momentum transfer. In terms of the amount of mass transported per unit amount of energy dissipated, open tubes a t fully developed turbulent flow and packed beds appear to be the best and poorest contactors, respectively. The importance of the relationship between the mass transfer efficiency and the required pressure drop in contactors notwithstanding, packed beds are frequently used because of their relatively high contact surface area per unit volume. Under some circumstances, however, the superiority of open tubes to packed beds in terms of the mass to momentum transfer efficiency has been exploited in practical applications. Such considerations, for instance, prompted Golay to introduce capillary columns in gas chromatography (Ettre, 1965) and Pretorius and Smuts (1966) to suggest the use of open tubes with turbulent flow in liquid chromatography. Appendix I . Efficiency of Momentum Transport in Slug Flow The overall efficiency of momentum transfer in an open tube is conveniently expressed by the Euler number Eu, calculated from the pressure drop, 1P,as
AP
(1
-
(AI- 1)
4pv2
where c is the void fraction, p the mass density, and V the average axial velocity of the liquid. The transfer efficiency per unit area of the tube wall is given by the friction factor, f / 2 , defined by S
. f / 2 = -Eu
(AI- 2 )
A
where S is cross-sectional area and A is the internal surface area of the tube. The mixing efficiency that indicates the relative importance of convective momentum transport, is expressed by the viscous Nusselt number, Nuv, and is obtained from the E u number by the relation L mte
EU = 4Nu,-
(AI-3)
where Re is the Reynolds number and L and d are the tube length and diameter, respectively. Combination of eq AI-1 and AI-3 yields the following relationship between Nub and the pressure drop across the tube Nu,
1 d APRe 4 L (1 - 4 p V 2
(AI- 4)
--
Appendix 11. Calculation of the Le Goff Number for Packed Beds In order to compare the J factor to the friction factor ratios for open tubes and packed beds, both factors have to express the transfer efficiency per unit contact surface. The j factor for packed beds was calculated by using the correlation of Williamson et al. (1963). The values of the Schmidt number and the void fraction were 1700 and 0.4, respectively. Thus, the J factor was calculated as a function Re, by the equations j = 2.5(Re,)-'1'C'
