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Direct-contact heat transfer with change of phase: a population balance model
Chemical Engineering Science 54 (1999) 3861}3871
Direct-contact heat transfer with change of phase: a population balance model M. Song*, A. Stei!, P.-M. Weinspach Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany Received 1 July 1998; accepted 19 January 1999
Abstract This paper presents a population balance model to predict the volumetric heat transfer coe$cient for direct-contact evaporation in a bubble column. The model is based mainly on the energy balance and the population balance. Growth and breakage of droplets are taken into account in the population balance model. Experimental data for the volumetric heat transfer coe$cient obtained in a column of 0.114 m diameter (Steiner, 1993) are used to validate the model. The predicted results are in reasonable agreement with experimental data at di!erent temperature di!erences for various #ow rates of the dispersed phase. In#uences of the size distribution (mean value and standard deviation) of initial drops on the volumetric heat transfer coe$cient are discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Direct contact heat transfer; Direct contact evaporation; Population balance; Bubble column; Volumetric heat transfer coe$cient
1. Introduction Direct-contact heat transfer between two immiscible liquids with change of phase has a lot of advantages over traditional heat exchange methods with metallic heat transfer surfaces: lower driving temperature di!erence, higher heat transfer rate, simple design and scale-up procedure, no surface corrosion and fouling, and so on. Practical applications have been found in a number of engineering processes such as spray cooling, water desalination, solar, geothermal and ocean-thermal energy conversion and thermal storage systems. A recent increase in publications dealing with the direct-contact heat transfer between two immiscible liquids with change of phase evidences a growing interest in this area. A critical review of early studies was given by Sideman (1966). Steiner (1993) reviewed more recent investigations in the course of studying the direct-contact heat transfer with a dispersed population of evaporating droplets in an another immiscible liquid as a measure of emergent cooling in
*Corresponding author. Present address: R.A.S. Industries Ltd., 8020-128th Street, Surrey, BC, Canada V3W 4E9.
chemical reactors in which strong exothermic chemical reactions exist. Considerable work has been devoted to studying the phenomena of single droplet evaporation for recent three decades. There are only a limited number of experimental, theoretical and numerical investigations dealing with a swarm of droplets undergoing evaporation in an immiscible liquid in the literature. Obviously, the heat transfer e$ciency is strongly related to the interfacial area between two phases, which can be determined in terms of the holdup and droplet-size distribution of the dispersed phase. Despite the same holdup the total interfacial area available for heat exchange can be very di!erent for di!erent size distributions of droplets. To describe the heat transfer characteristics properly, it is necessary to consider the size distribution of dispersed droplets in modelling. The objective of this work is to establish a population balance model to predict the volumetric heat transfer coe$cient for direct-contact evaporation in a bubble column, in which breakage and coalescence of evaporating droplets can be taken into account. The model mainly includes two parts: (1) the energy balance, and (2) the population balance. Coalescence of droplets is assumed negligible for the present system.
0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 0 2 9 - 9
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2. Previous studies on direct-contact heat transfer
method. Many researchers presented their theoretical and experimental Nusselt number for evaporation of a single droplet. These results are summarized in Table 1. In the literature we can "nd a few papers in which theoretical and/or numerical models were established to predict the volumetric heat transfer coe$cient for directcontact evaporation of a swarm of droplets in an another immiscible liquid. Because complicated particle phenomena such as coalescence and breakage are di$cult to precisely describe in mathematics and physics, it is impossible to experimentally determine the quantity of the interfacial area available for heat transfer between
The interest in the single droplet evaporation is concentrated on the con"guration and shape form of a single two-phase droplet and the Nusselt number representing heat transfer characteristic between the environmental #uid and the dispersed droplet. Mori's investigation (1978) is excellent in revealing the possible con"gurations of a two-phase droplet and the conditions of all those con"gurations. Sideman and Taitel (1964) and Tochitani et al. (1977a) determined the shape form and variation pattern of a rising two-phase droplet with a photograph
Table 1 Nusselt number for a single droplet evaporation Author(s)
Nusselt number
Year
Note
Klipstein
Nu"2#0.096 Re Pr
1963
Experimental
Sideman and Taitel
Nu"Pe
1964
Theoretical
Prakash and Pinder
Nu"0.0505
1967
Experimental
1967
Analytical
1970
Experimental
1972
Experimental
1974
Experimental
3 cos b!cos b#2 n
Pe o ! 1#g /g o " ! "
Nu"1.128 Pe Sideman and Isenberg
m!x (m!1)x
o D where m" "J , x" o D "E
0.0575 #10\ (o /o !1) ! "
Pe 1#g /g " !
Filatkin and Dolotov
Nu"1.45;10\
Adams and Pinder
Nu"7.55;10 Pr\
Simpson et al.
Nu"1.27 Pe
Tochitani et al.
sin 2b Nu"0.463 Pe n!b# 2
1977b
Analytical
Smith et al.
Nu"c ReV Pr where 0.00087(c(0.0405, 0.7(x(1
1982
Experimental
Battya et al. Battya et al.
Nu"0.64 Pe Ja\ Nu"0.68 Pe Ja\
1984 1985
Analytical Analytical
Raina and Grover
Nu"0.46929 Pe
1985
Theoretical
Raina and Wanchoo
Nu"1.069 Pe
1986
Theoretical
Shimizu and Mori
Nu"0.169 Pe for n}Pentane Nu"0.121 Pe for R113
1988
Experimental
Shimaoka and Mori Song et al.
Nu"e #M\ Ja\ #M> Pe Nu"(C #0.688 =eL)PeK
1990 1996
Experimental Analytical
g ! Bo g #g ! "
b!sin b 2
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two phases. The volumetric heat transfer coe$cient, de"ned as QQ k" T AHD¹
(1)
is used to represent the e$cient of heat exchange in direct-contact heat transfer systems both with and without change of phase by researchers (Smith et al., 1982; Coban and Boehm, 1989; Seetharamu and Battya, 1989; Core and Mulligan, 1990; Mori, 1991; Summers and Crowe, 1991; Shahidi and Ozbelge, 1995). For the directcontact heat transfer with change of phase, QQ in Eq. (1) is the heat removed from the continuous phase per unit time due to evaporation of the dispersed phase, H the column height, which is required for complete evaporation of all droplets, A the cross-section area of column, and D¹ the mean driving temperature di!erence between two phases. The logarithmic mean temperature di!erence is usually taken for this de"nition, that is (¹ !¹ )!(¹ !¹ ) "M !G "G , D¹ " !M (2) ln [(¹ !¹ )/(¹ !¹ )] !M "M !G "G where ¹ and ¹ are the temperature of the continuous ! " and the dispersed phase, and indices i and o stand for inlet and outlet, respectively. In a theoretical discussion ¹ "¹ "¹ is usually "G "M "@ assumed for direct-contact heat transfer with change of phase, where ¹ is the boiling point of the dispersed "@ phase. The axial temperature distribution may be described by using Fig. 1. Smith et al. (1982) developed an analytical model to calculate the volumetric heat transfer coe$cient for direct-contact evaporation. In this model heat transfer was calculated using single droplet correlations for the Nusselt number, and the #uid dynamics was described by the so-called drift-#ux model (Wallis, 1969). The analysis was divided into a preagglomeration and a post-agglomeration stage on the basis of an assumed maximum value for the dispersed phase volume fraction. Shiina and Sakaguchi (1984) incorporated the model of Smith et al. into their scheme to analyze the performance of a parallel-#ow spray column consisting of preheater, evaporator and superheater sections. The model of Smith et al. was also modi"ed by Seetharamu and Battya (1989) to predict the theoretical volumetric heat transfer coe$cient for the direct-contact evaporation of R113 and n-pentane in a stagnant column of distilled water. Mori (1991) established a model, assuming that the heat transferred to each of dispersed droplets of simultaneous evaporation could be approximated by an empirical correlation for heat transfer to an isolated droplet evaporating in a quiescent, su$ciently extended medium. A numerical model for a three-phase, direct-contact, spray-column heat exchanger was developed by Coban and Boehm (1989) to calculate the performance characteristic
Fig. 1. Axial temperature pro"le of two phases.
quantities of this type of device, such as temperature, holdup and volumetric heat transfer coe$cient. In general, these are sound models, which can be utilized to predict heat transfer coe$cients agreeable with experimental data to a certain extent. Expecially, the model of Smith et al. has been more widely used (Shiina and Sakaguchi, 1984; Seetharamu and Battya, 1989). However, all aforementioned models disregarded the population balance characteristic, an important nature in direct-contact heat transfer systems, and hence limited themselves in applications. For example, in the model of Smith et al. (1982) a relationship between the rate of evaporation of droplets and their displacement from the column bottom was formed so that the volumetric heat transfer coe$cient could be calculated as function of this displacement with the following equation: 1 X A (z )n (z )h (z ) dz , (3) k (z)" @ @ @ T z where z is the axial coordinate, A the droplet area, n the @ @ droplet number density, and h the heat transfer coe$@ cient per unit surface area for a single droplet. A , n and @ @ h could be expressed in terms of the instantaneous dia@ meter D of dispersed droplets. Because D was treated as a constant across the column cross section, this model is not able to handle variations of the size distribution of dispersed droplets. Core and Mulligan (1990) presented a population balance model to discuss the heat transfer characteristics of direct contact evaporation in a batch reactor. The population balance equation was solved using the Method of Classes, which was originally proposed by Marchal et al. (1988) in crystallization and precipitation engineering. However, they neglected both breakage and coalescence of dispersed droplets.
3. Model 3.1. Physical system Our research was carried out in bubble columns. Consider a bubble column, schematically shown in Fig. 2.
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The shape of actual droplets is very di$cult to describe in mathematics, and therefore Eq. (4) is used by nearly all researchers in establishing their heat transfer models. 3. The rate of direct-contact evaporation of a swarm of droplets can be described with correlations of the area-speci"c heat transfer rate for single droplets. This implies that in#uences of neighboring droplets on the heat transfer coe$cient from the continuous phase to a dispersed droplet are negligible. 4. Coalescence of dispersed droplets is negligible in the present system. In calculating the evaporation height we found that the values from numerical simulations are much larger than those from experiments (Song and Stei!, 1995). It can be assumed that breakage of droplets dominates in the present system. 3.3. Mathematical description
Fig. 2. Illustration of a bubble column.
One liquid, called dispersed phase, enters the column through a distributor at the bottom as liquid drops with a certain size distribution while another liquid, called continuous phase, #ow continuously from the bottom to the top. Because the continuous phase has a temperature higher than the boiling point of the dispersed phase, the drops will evaporate and hence extract heat from the continuous phase. The continuous phase is immiscible (or, in some cases, only poorly miscible) with either of both phases, vapor and unevaporated liquid, of the dispersed phase. As the dispersed droplets complete their evaporation, the vapor exists through the free surface of the continuous liquid and is subsequently withdrawn from the head part of column. 3.2. Fundamental assumptions
The model in this contribution is a one-dimensional model along the axis of column. The column height is equally discretized in numerical calculations. Fig. 3 schematically shows this discretization. Within height grids, called cells, the swarm of droplets are divided into groups, called classes, according to the droplet diameter. The discretization of the droplet diameter can be either equal or unequal. By selecting one class, we can observe complicated particle phenomena, including convection, growth, coalescence and breakage, as shown in Fig. 4. The population balance equation (PBE) o!ers a fundamental mathematical framework to describe such phenomena (Ramkrishna, 1985) and it will be used here to account for convection, growth and breakage of droplets. Such an equation can be written as *(vn) *(wn) # "BQ !DQ , *h *D
(5)
where n is the droplet number density. Birth rate BQ and death rate DQ are given by
L " l(y)b(y)p 6 D, y n(y, t) dy
BQ "
n
(6)
and
To establish the model, the following assumptions are employed to simplify the actual physical phenomena:
DQ "b
1. The continuous and the dispersed phase are uniform across the column cross section. Therefore, the model is one dimensional. 2. Evaporating two-phase droplets are spherical, with their diameter D being the equivalent diameter of actual droplets:
where p(y, x) is the breakage density function of a droplet of volume y from a mother droplet of volume x, b(x) is the breakage frequency of a droplet of volume x, and v(x) is the number of daughter droplets per breakage. In Eq. (5)
D"
6< " . n
(4)
dh v" dt
n n D n D, t , 6 6
(7)
(8)
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
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where C is the coe$cient of resistance. Applying " Levish's formula (Levish, 1949),
2 o gD ! C " . (11) " 3 1.82p ! Raina et al. (1984) obtained the following relationship v"1.91
1.91
o D p ! 1! " . (12) o D o D ! ! To account for several factors that are responsible for obvious deviation of theoretical results from experimental values, Raine et al. (1984) eventually gave
v" Fig. 3. Diagram of the discretization.
o D p ! 1! " DM \"M L o D o D ! ! , M ¹ #¹ " C k " " ! " N! ! 2¹ ¹ k ! " !
(13)
where DM is a dimensionless averaged diameter, de"ned as D#D . DM " (14) 2DD Eq. (13) was utilized by Coban and Boehm (1989) in their model for a three-phase, spray-column, direct-contact heat exchanger. Marrucci (1965) showed that the e!ect of the dispersed phase holdup could be expressed by 1!e v "v . 1!e
This correction was employed by Core and Mulligan (1990) and Song and Stei! (1995) in their population balance models. In the present analysis Eq. (13), with Marrucci's correction (15), is used. A single two-phase droplet is shown in Fig. 5. Let the volume of liquid evaporated from t to t#dt be d< , J which causes an increase of the volume of vapor within the droplet denoted by d< . From mass and energy E balance, we have
Fig. 4. Particle phenomena.
is the rising velocity, and dD w" dt
(9)
is the growth rate of droplets. Both v and w need to be modelled before numerical simulations are carried out by using Eq. (5). There are several empirical and semi-empirical formulas to represent the rising velocity of droplets. From the force balance for a single droplet Raina et al. (1984) derived the following expression:
v"
4 o D Dg 1! " 3 o D C ! "
(15)
(10)
o d< "o d< , !J J !E E n d< !d< "d D , E J 6
(16) (17)
nDh (¹ !¹ ) dt"Dh o d< . (18) D ! " T !J J From these equations we can obtain the growth rate of dispersed droplets due to evaporation as 2h (¹ !¹ )(o !o ) " !J !E . w" D ! (19) o o Dh !J !E T According to the assumptions given above, we use the correlations of the area-speci"c heat transfer rate for
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(1966), Coulaloglou and Tavlarides (1977) and Chatzi et al. (1989). b(x) is the breakage frequency of droplets, which represents when a droplet will break up from the viewpoint of statistics. Existing information from both theory and experiment is still very imprecise. Instead, we use the maximum diameter of stable droplets in dispersion systems given by Mersma nn (1977) to model it:
cp ! D " , (24)
(o !o )g ! " where c is an empirical constant whose value is around 9.0. In the present problem o is the averaged density of " a two-phase droplet, that is, m o " ", (25) " < " where m and < are the total mass and volume of " " a droplet, respectively. Apparently, Fig. 5. A two-phase droplet.
a single two-phase droplet. Sideman and Taitel (1964) derived a well-known analytical formula for the Nusselt number
Nu 3 cos b!cos b#2 , " Pe n
(20)
where b is the half-evaporation open angle, shown in Fig. 5. This formula has been cited widely in the literature. Assuming that the distorted droplet is an ellipsoid, the diameter ratio of which varies in terms of the Peclet number, Pe (Van Bekkum, 1978) we derived a correlation as follows (Song et al., 1996):
b)453, b+603, b+753, b+903, b*1053,
0.688 =e , 0.0817 =e , Nu "C # 2.40 =e\ , PeK 0.567 =e\ , 0.190 =e\ ,
(21)
where C is equal to the right side of Eq. (20), =e is the Weber number, and m is a "tting index, the value of which is around 0.5, given in Table 2. To model the breakage terms, l(x), p(y, x) and b(x) should be given in advance. In the present model the binary breakage with a normal distribution of daughter droplet diameter is assumed, that is l(x)"2
(22)
1 W\V p(y, x)" Exp \ NW (23) (2np W where p is the standard deviation of distribution, taken W equal to (x/2)/c (c "3}3.5). This distribution was ad opted in this study as used by Valentas and Amundson
o )o )o . (26) "E " "J If the droplet is a pure liquid drop, then Eq. (26) yields o "o , and if it is a pure gas bubble, o "o . " "J " "E 3.4. Numerical method Core and Mulligan (1990) used the Method of Classes to solve the population balance equation with the size growth of particles (Marchal et al., 1988). In the discussion of Core and Mulligan, both breakage and coalescence of droplets were not included so that the PBE was homogeneous. From our numerical example we found that if breakage and/or coalescence are taken into account, small non-smooth discretized values at some iteration step may lead to an unstable calculation. With the help of the principle of operator splitting, the authors proposed a new method, termed Splitting#Analogy, to solve the PBE with the size growth of particles. In this method the homogeneous PBE is simply discretized following an analogy with the one-dimensional continuity equation in #uid mechanics. A detailed analysis about this procedure can be found in Song et al. (1997). From the population balance model we can determine the droplet size distribution and hence the number of droplets in all classes. The overall heat transfer rate QQ in H cell j is then calculated by QQ " QQ , HG H G where
(27)
QQ "nD N h (¹ !¹ ) (28) HG HG HG D HG " ! H is the heat transfer rate in group i. The area-speci"c heat transfer coe$cient, h , is given by D k Nu h " ! , (29) D D
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Table 2 Values of m from Song et al. (1996) b)753
753(b(1053
b*1053
Pe"10 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.513 0.526 0.533
0.491 0.487 0.484
0.485 0.469 0.461
Pe"20 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.528 0.552 0.563
0.481 0.475 0.472
0.468 0.436 0.425
Pe"30 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.537 0.564 0.575
0.474 0.469 0.466
0.450 0.426 0.408
Pe*40 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.546 0.579 0.583
0.470 0.464 0.462
0.443 0.414 0.389
The two parameters, p and k, are determined by
Heat transferred in cell j is obtained from Dh Q " QQ . H HG v HG G
(30)
According to the energy conservation law we can calculate the temperature variation of the continuous phase Q H ¹ !¹ " . ! H> ! H (A dh! n D N )o C G HG HG ! N!
(31)
The total heat transfer rate is obtained from QQ " QQ . H H
(32)
From Eq. (31) the temperature of the continuous phase at the outlet, ¹ , can be determined. Eventually, the vol!M umetric heat transfer coe$cient for the whole column can be calculated by using Eqs. (1) and (2).
4. Results and discussions Numerical simulations are run with values of parameters that correspond to the experimental system studied by Steiner (1993): Materials system: R114/water, Column diameter: 0.114 m, Distributor: 35 holes of 2.0 mm diameter. The initial size distribution of drops is assumed to be a logarithmic-normal distribution:
(ln D !k) 1 f (D )" exp ! . 2p (2npD
(33)
DM , ((p /DM )#1 "
(34)
(35)
k"ln
p" ln
p " #1 , DM
where DM and p are the mathematical expectation " (mean value) and the standard deviation, respectively. The mean diameter is a very important parameter, which is found to depend mainly on the ori"ce velocity, the ori"ce diameter and the physical properties of materials (Hayworth and Treybal, 1950; Scheele and Meister, 1968; De Chazal and Ryan, 1971). Despite the fact that a considerable amount of work can be found in the literature on predicting the size of drops formed at ori"ces of a given diameter, we are still unable to predict the mean drop diameter or volume for a given system with high con"dence. Because of the uncertainty in this value, Smith et al. (1982) performed the calculations over a range of D (0.5, 1.0 and 2.0 mm) relative to their experimental system. Seetharamu and Battya (1989) predicted the volumetric heat transfer coe$cient using D "0.5, 1.0, 1.5 and 2.0 mm for the experimental system of R113 and water, with 19, distributor holes of 0.35 mm diameter and using D "1.0, 1.5 and 2.0 mm for the same materials system with 7 distributor holes of 1.00 mm diameter. For the former system they got the calculation results with D "1.0 mm in agreement with the experimental data, and for the latter the calculation results with D "1.5 mm matched the experimental data better. Mori (1991) estimated the initial drop diameter as about 1.4 mm for n-Pentane evaporating in water with the injection velocity of 0.22}0.44 m/s and the 19 and 36
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Fig. 6. k vs. D¹ for di!erent #ow rates of the dispersed phase. T G
distributor holes of 0.5 mm diameter in terms of the empirical formulas of Scheele and Meister (1968) and De Chazal and Ryan (1971). In the discussion below D "1.0, 1.5 and 2.0 mm are taken except that the value for this parameter is especially given. By taking DM "1.5 mm, we calculate the volumetric heat transfer coe$cient for di!erent #ow rates of the dispersed phase at di!erent values of the initial driving temperature di!erence. The results are shown in Fig. 6. This "gure shows that there is a very high heat transfer e$ciency at a very low initial temperature di!erence. If the temperature di!erence is higher than 6 K, a bigger value of this parameter cannot improve the heat transfer e$ciency. Note that a longer column is needed for a lower temperature di!erence (Song and Stei!, 1995). In engineering design we should "nd a balance between the heat transfer e$ciency and the evaporation height. It can be observed from Fig. 6 that the heat transfer coe$cient increases if the #ow rate of the dispersed phase goes up. However, the degree of increasing becomes lower and lower. The direct relationship of the heat transfer coe$cient vs. the #ow rate of the dispersed phase is drawn in Fig. 7. The curves are increasing, but gradually sloping down. Fig. 8 demonstrates the comparison of theoretical and experimental values of the volumetric heat transfer coef"cient. Experimental results used in this "gure were obtained by Steiner (1993). The agreement between theory and experiment is fairly good for the lower #ow rate of the dispersed phase (140 kg/h), and reasonably good for the higher #ow rate (200 kg/h). For the latter the value from our model are some lower. Trend of the theoretical curves is very similar to that of the experimental points for both cases. For two cases: D¹ "10 K and D¹ "5 K, we draw G G the temperature pro"les of both the continuous and the dispersed phase along the column height in Fig. 9. The temperature of the dispersed phase has been assumed to be always constant. The temperature of the continuous
Fig. 7. k vs. the #ow rate of the dispersed phase. T
Fig. 8. Comparison of theoretical and experimental values of k . T
Fig. 9. Temperature pro"les along the column height for two cases: D¹ "10 K and D¹ "5 K. G G
phase decreases more rapidly as the dispersed drops begin their evaporation. At a higher position of the column, the heat transfer rate becomes lower because the driving temperature di!erence is smaller and the areaspeci"c heat transfer coe$cient for a single droplet is
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
lower (Song et al., 1996). At a lower temperature di!erence, a bigger column height is needed for complete evaporation of all droplets, as mentioned before. Because of the imprecision in DM , we carry out calcu lations taking di!erent values of DM as most researchers such as Smith et al. (1982), Seetharamu and Battya (1989) and Mori (1991) have done previously. Taking DM as 1.0, 1.5 and 2.0 mm, the results of the volumetric heat transfer coe$cient are shown in Fig. 10. The calculations are performed assuming the log-normal distribution with p "0.5 mm for D . The "gure shows that this para" meter is very sensitive to the calculations. Therefore, precise determination of this parameter from experiment or from theory is a key to achieve a good agreement of the heat transfer coe$cient between theory and experiment to the present model and to all other models. In order to examine the in#uence of the distribution of D on numerical results, we employ two distributions used frequently in the past: log-normal and normal with the same mean value (1.5 mm) and standard deviation (0.2 mm). The results are shown in Fig. 11. From this
3869
"gure we cannot observe a big di!erence existing between curves for these two conditions. If we put the curve from the normal distribution onto Fig. 8, equal agreement with experimental data could be seen between these two distributions of D . At present DM cannot be determined with high con" dence. It is more di$cult to precisely determine the standard deviation of D from theory and experiment. Taking the standard deviation of D equal to 0.2 mm for the log-normal distribution, we calculate the volumetric heat transfer coe$cient, results of which are shown in Fig. 11 as well. From this "gure we "nd that the heat transfer coe$cient is lower for the smaller standard deviation. To draw an eventual conclusion this needs to check with values obtained in experiments. However, a reasonable explanation can be given. For a bigger standard deviation, more drops of larger diameter exist, and, therefore, they may break up earlier. This can be assumed to be one reason to contribute to a higher heat transfer coe$cient.
5. Concluding remarks
Fig. 10. The calculated volumetric heat transfer coe$cients for di!erent values of DM .
Fig. 11. In#uence of the distribution of D on the volumetric heat transfer coe$cient.
A population balance model is presented to calculate the volumetric heat transfer coe$cient for a bubble column with direct-contact evaporation of one liquid in an another immiscible liquid. An important feature of this model is that breakage of dispersed droplets is included, and the size distribution of dispersed droplets can be accounted for. Without considering breakage of droplets in simulations we obtain much lower values in the volumetric heat transfer coe$cient than those found from experiments. The reason is that a larger evaporation height of column would be needed to complete the evaporation of all droplets if there were no breakage of droplets (Song and Stei!, 1995). The model has a good ability to predict the volumetric heat transfer coe$cient. The predicted results are in good agreement with experimental data not only qualitatively but also quantitatively. The mean diameter of dispersed drops, just leaving the bottom distributor, is a very sensitive parameter to our model calculations and, we believe that the same type of sensitivity applies also to other models existing in the literature. At present it is impossible to even roughly predict its value from theory and experiment. On the other hand, the function form of the diameter distribution of the initial drops was found to show low sensitivity to the calculations if the mean value and standard deviation are kept unchanged. Both the log-normal and the normal distribution seem to be a good approximation for D . The standard deviation of the initial size of drops has some in#uence on the results presented in this analysis. For example, a bigger value of the standard deviation may lead to a higher heat transfer coe$cient.
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Notation A BQ b(x) C N C " D D DM DM DQ D
g H h h D Dh T k k T m N Nu n Pe p(y, x) Q QQ ¹ D¹ < v =e w
Subscripts column cross-sectional area, m birth rate of droplets, m\ s\ breakage frequency function, s\ speci"c heat, J kg\ K\ resistance coe$cient diameter of droplets, m initial diameter of drops, m dimensionless mean diameter of droplets initial mean diameter of drops, m death rate of droplets, m\ s\ maximum diameter of stable droplets, m\ s\ acceleration due to gravity, m s\ column height for complete evaporation, m axial coordinate of column, m heat transfer coe$cient for a single droplet, W m\ K\ evaporation enthalpy of the dispersed phase, J kg\ thermal conductivity, W m\ K\ volumetric heat transfer coe$cient, W m\ K\ mass, kg number of droplets in one group Nusselt number number density of droplets, m\ Peclet number density distribution function of daughter droplets, m\ transferred heat, J heat transfer rate, W temperature, K temperature di!erence between two phases, K volume, m rising velocity of droplets, m s\ Weber number growth rate of droplets, m s\
Greek symbols b e l(x) k o p p, k p " p W
half evaporation open angle dispersed phase holdup number of daughter droplets viscosity, kg m\ s\ density, kg m\ surface tension, N m\ two parameters in the logarithmic-normal distribution standard deviation of a distribution for D ,m standard deviation of distribution function p(y, x), m
C D g i l o
continuous phase dispersed phase gas phase inlet liquid phase outlet
References Adams, A.E.S., & Pinder, K.L. (1972). Average heat transfer coe$cient during the evaporation of a liquid drop. Canadian Journal of Chemical Engineering, 50, 707}713. Battya, P., Ragharan, V.R., & Seetharamu, K.N. (1984). Parametric studies on direct contact evaporation of a drop in an immiscible liquid. International Journal of Heat Mass ¹ransfer, 27, 263}272. Battya, P., Ragharan, V.R., & Seetharamu, K.N. (1985). A theoretical correlation for the Nusselt number in direct evaporation of a moving drop in an immiscible liquid. =arme- und Sto+uK bertragung, 19, 61}66. Chatzi, E.G., Gavrielides, A.D., & Kiparissides, C. (1989). Generalized model for prediction of the steady-state drop size distributions in batch stirred vessels. Industrial and Engineering Chemistry } Research, 28, 1704}1711. Coban, T., & Boehm, R. (1989). Performance of a three-phase, spraycolumn, direct-contact heat exchanger. ¹ransactions of the ASME, Journal of Heat ¹ransfer, 111, 166}172. Core, K.L., & Mulligan, J.C. (1990). Heat transfer and population characteristics of dispersed evaporating droplets. ¹he American Institute of Chemical Engineers Journal, 36, 1137}1144. Coulaloglou, C.A., & Tavlarides, L.L. (1977). Description of interaction processes in agitated liquid}liquid dispersions. Chemical Engineering Sciencees, 32, 1289}1297. De Chazal, L.E.M., & Ryan, J.T. (1971). Formation of organic drops in water. ¹he American Institute of Chemical Engineers Journal, 17, 1226}1229. Filatkin, V.R., & Dolotov, A.G. (1970). Heat transfer during the direct contact of a mixture of boiling hydrocarbons with water. Cholodil1naja ¹echnika, 47, 26}29. Hayworth, C.B., & Treybal, R.E. (1950). Drop formation in two liquidphase systems. Industrial and Engineering Chemistry, 42, 1174}1181. Klipstein, D.H. (1963). Heat transfer to a vaporizing immiscible drop. Ph.D. thesis, MIT, USA. Levish, V.G. (1949). The motion of gas bubbles (in Russian), Zhurnal Eksperimental1noi i ¹eoreticheskoi Fiziki, 19, 18. Marchal, P., David, R., Klein, J.P., & Villermaux, J. (1988). Crystallization and precipitation engineering, I. An e$cient method for solving population balance in crystallization with agglomeration. Chemical Engineering Science, 43, 59}67. Marrucci, G. (1965). Rising velocity of a swarm of spherical bubbles. Industrial and Engineering Chemistry, Fundamentals, 4, 224}225. Mersmann, A. (1977). Auslegung und MapstabsvergroK perung von Blasen-und TropfensaK ulen. Chemie}Ingieur}¹echnik, 49, 679}691. Mori, Y.H. (1978). Con"guration of gas}liquid two-phase bubbles in immiscible liquid media. Inernational Journal of Multiphase Flow, 4, 383}396. Mori, Y.H. (1991). An analytic model of direct-contact heat transfer in spray-column evaporators. ¹he American Institute of Chemical Engineers Journal, 37, 539}546. Prakash, C.B., & Pinder, K.L. (1967). Direct contact heat transfer between two immiscible liquids during vaporization, Part I: Measurement of heat transfer coe$cient. Canadian Journal of Chemical Engineering, 45, 210}214.
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871 Raina, G.K., Wanchoo, R.K., & Gover, P.D. (1984). Direct contact heat transfer with phase change: Motion of evaporating droplets. ¹he American Institute of Chemical Engineers Journal, 30, 835}837. Raina, G.K., & Gover, P.D. (1985). Direct contact heat transfer with change of phase: Theoretical model incorporating slashing e!ects. ¹he American Institute of Chemical Engineers Journal, 31, 507}510. Raina, G.K., & Wanchoo, R.K. (1986). Direct contact heat transfer with phase change: Bubble growth and collapse. Canadian Journal of Chemical Engineering, 64, 393}398. Ramkrishna, D. (1985). The status of population balances. Review of Chemical Engineering, 3, 49}95. Seetharamu, K.N., & Battya, P. (1989). Direct-contact evaporation between two immiscible liquids in a spray column. ¹ransactions of the ASME, Journal of Heat ¹ransfer, 111, 780}785. Scheele, G.F., & Meister, B.J. (1968). Drop formation at low velocities in liquid}liquid systems. ¹he American Institute of Chemical Engineers Journal, 14, 9}19. Shahidi, M.K., & Ozbelge, T.A. (1995). Direct contact heat transfer between two immiscible liquids #owing in a horizontal concentric annulus. International Journal of Multiphase Flow, 21, 1025}1036. Shiina, K., & Sakaguchi, S. (1984). Investigation of the heat transfer evaluation method in a direct liquid}liquid contact type evaporator (in Japanese). Proceedings of the 21st National Heat ¹ransfer Symposium of Japan (p. 526). Shimaoka, H., & Mori, Y.H. (1990). Evaporation of single liquid drops in an immiscible liquid: Experiments with n-pentane drops in water and preparation of new heat transfer correlations. Experimental Heat ¹ransfer, 3, 159}172. Shimizu, Y., & Mori, Y.H. (1988). Evaporation of single drops in an immiscible liquid at elevated pressure: Experimental study with n-pentane and R113 drops in water. International Journal of Heat Mass ¹ransfer, 31, 1843}1851. Sideman, S., & Taitel, Y. (1964). Direct-contact heat transfer with change of phase: Evaporation of drops in an immiscible liquid medium. International Journal of Heat Mass ¹ransfer, 7, 1273}1289. Sideman, S. (1966). Direct contact heat transfer between immiscible liquids. Advances in Chemical Engineering, 6, 207}286.
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Sideman, S., & Isenberg, J. (1967). Direct contact heat transfer with change of phase: Bubble growth in three-phase systems. Desalination, 2, 207}214. Simpson, H.C., Beggs, G.C., & Nazir, M. (1974). Evaporation of a droplet rising through a second liquid: A new theory of the heat transfer process. Heat ¹ransfer, 5, 59}63. Smith, R.C., Rohsenow, W.M., & Kazimi, M.S. (1982). Volumetric heat-transfer coe$cient for direct-contact evaporation. ¹ransactions of the ASME, Journal of Heat ¹ransfer, 104, 264}270. Song, M., & Stei!, A. (1995). Simulation of vaporization height by considering the size distribution of drobbles in drobble columns. Chemical Engineering Processing, 34, 87}95. Song, M., Stei!, A., & Weinspach, P.-M. (1996). Nusselt number of direct-contact evaporation of a moving ellipsoidal drop in an immiscible liquid. Chemical Engineering Communications, 145, 73}87. Song, M., Stei!, A., & Weinspach, P.-M. (1997). A very e!ective new method to solve the population balance equation with particle-size growth. Chemical Engineering Science, 52, 3493}3498. Steiner, J. (1993). Zur direkien =a( rmeuK bertragung bei der
Direct-contact heat transfer with change of phase: a population balance model M. Song*, A. Stei!, P.-M. Weinspach Department of Chemical Engineering, University of Dortmund, 44221 Dortmund, Germany Received 1 July 1998; accepted 19 January 1999
Abstract This paper presents a population balance model to predict the volumetric heat transfer coe$cient for direct-contact evaporation in a bubble column. The model is based mainly on the energy balance and the population balance. Growth and breakage of droplets are taken into account in the population balance model. Experimental data for the volumetric heat transfer coe$cient obtained in a column of 0.114 m diameter (Steiner, 1993) are used to validate the model. The predicted results are in reasonable agreement with experimental data at di!erent temperature di!erences for various #ow rates of the dispersed phase. In#uences of the size distribution (mean value and standard deviation) of initial drops on the volumetric heat transfer coe$cient are discussed. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Direct contact heat transfer; Direct contact evaporation; Population balance; Bubble column; Volumetric heat transfer coe$cient
1. Introduction Direct-contact heat transfer between two immiscible liquids with change of phase has a lot of advantages over traditional heat exchange methods with metallic heat transfer surfaces: lower driving temperature di!erence, higher heat transfer rate, simple design and scale-up procedure, no surface corrosion and fouling, and so on. Practical applications have been found in a number of engineering processes such as spray cooling, water desalination, solar, geothermal and ocean-thermal energy conversion and thermal storage systems. A recent increase in publications dealing with the direct-contact heat transfer between two immiscible liquids with change of phase evidences a growing interest in this area. A critical review of early studies was given by Sideman (1966). Steiner (1993) reviewed more recent investigations in the course of studying the direct-contact heat transfer with a dispersed population of evaporating droplets in an another immiscible liquid as a measure of emergent cooling in
*Corresponding author. Present address: R.A.S. Industries Ltd., 8020-128th Street, Surrey, BC, Canada V3W 4E9.
chemical reactors in which strong exothermic chemical reactions exist. Considerable work has been devoted to studying the phenomena of single droplet evaporation for recent three decades. There are only a limited number of experimental, theoretical and numerical investigations dealing with a swarm of droplets undergoing evaporation in an immiscible liquid in the literature. Obviously, the heat transfer e$ciency is strongly related to the interfacial area between two phases, which can be determined in terms of the holdup and droplet-size distribution of the dispersed phase. Despite the same holdup the total interfacial area available for heat exchange can be very di!erent for di!erent size distributions of droplets. To describe the heat transfer characteristics properly, it is necessary to consider the size distribution of dispersed droplets in modelling. The objective of this work is to establish a population balance model to predict the volumetric heat transfer coe$cient for direct-contact evaporation in a bubble column, in which breakage and coalescence of evaporating droplets can be taken into account. The model mainly includes two parts: (1) the energy balance, and (2) the population balance. Coalescence of droplets is assumed negligible for the present system.
0009-2509/99/$ } see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 9 - 2 5 0 9 ( 9 9 ) 0 0 0 2 9 - 9
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2. Previous studies on direct-contact heat transfer
method. Many researchers presented their theoretical and experimental Nusselt number for evaporation of a single droplet. These results are summarized in Table 1. In the literature we can "nd a few papers in which theoretical and/or numerical models were established to predict the volumetric heat transfer coe$cient for directcontact evaporation of a swarm of droplets in an another immiscible liquid. Because complicated particle phenomena such as coalescence and breakage are di$cult to precisely describe in mathematics and physics, it is impossible to experimentally determine the quantity of the interfacial area available for heat transfer between
The interest in the single droplet evaporation is concentrated on the con"guration and shape form of a single two-phase droplet and the Nusselt number representing heat transfer characteristic between the environmental #uid and the dispersed droplet. Mori's investigation (1978) is excellent in revealing the possible con"gurations of a two-phase droplet and the conditions of all those con"gurations. Sideman and Taitel (1964) and Tochitani et al. (1977a) determined the shape form and variation pattern of a rising two-phase droplet with a photograph
Table 1 Nusselt number for a single droplet evaporation Author(s)
Nusselt number
Year
Note
Klipstein
Nu"2#0.096 Re Pr
1963
Experimental
Sideman and Taitel
Nu"Pe
1964
Theoretical
Prakash and Pinder
Nu"0.0505
1967
Experimental
1967
Analytical
1970
Experimental
1972
Experimental
1974
Experimental
3 cos b!cos b#2 n
Pe o ! 1#g /g o " ! "
Nu"1.128 Pe Sideman and Isenberg
m!x (m!1)x
o D where m" "J , x" o D "E
0.0575 #10\ (o /o !1) ! "
Pe 1#g /g " !
Filatkin and Dolotov
Nu"1.45;10\
Adams and Pinder
Nu"7.55;10 Pr\
Simpson et al.
Nu"1.27 Pe
Tochitani et al.
sin 2b Nu"0.463 Pe n!b# 2
1977b
Analytical
Smith et al.
Nu"c ReV Pr where 0.00087(c(0.0405, 0.7(x(1
1982
Experimental
Battya et al. Battya et al.
Nu"0.64 Pe Ja\ Nu"0.68 Pe Ja\
1984 1985
Analytical Analytical
Raina and Grover
Nu"0.46929 Pe
1985
Theoretical
Raina and Wanchoo
Nu"1.069 Pe
1986
Theoretical
Shimizu and Mori
Nu"0.169 Pe for n}Pentane Nu"0.121 Pe for R113
1988
Experimental
Shimaoka and Mori Song et al.
Nu"e #M\ Ja\ #M> Pe Nu"(C #0.688 =eL)PeK
1990 1996
Experimental Analytical
g ! Bo g #g ! "
b!sin b 2
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two phases. The volumetric heat transfer coe$cient, de"ned as QQ k" T AHD¹
(1)
is used to represent the e$cient of heat exchange in direct-contact heat transfer systems both with and without change of phase by researchers (Smith et al., 1982; Coban and Boehm, 1989; Seetharamu and Battya, 1989; Core and Mulligan, 1990; Mori, 1991; Summers and Crowe, 1991; Shahidi and Ozbelge, 1995). For the directcontact heat transfer with change of phase, QQ in Eq. (1) is the heat removed from the continuous phase per unit time due to evaporation of the dispersed phase, H the column height, which is required for complete evaporation of all droplets, A the cross-section area of column, and D¹ the mean driving temperature di!erence between two phases. The logarithmic mean temperature di!erence is usually taken for this de"nition, that is (¹ !¹ )!(¹ !¹ ) "M !G "G , D¹ " !M (2) ln [(¹ !¹ )/(¹ !¹ )] !M "M !G "G where ¹ and ¹ are the temperature of the continuous ! " and the dispersed phase, and indices i and o stand for inlet and outlet, respectively. In a theoretical discussion ¹ "¹ "¹ is usually "G "M "@ assumed for direct-contact heat transfer with change of phase, where ¹ is the boiling point of the dispersed "@ phase. The axial temperature distribution may be described by using Fig. 1. Smith et al. (1982) developed an analytical model to calculate the volumetric heat transfer coe$cient for direct-contact evaporation. In this model heat transfer was calculated using single droplet correlations for the Nusselt number, and the #uid dynamics was described by the so-called drift-#ux model (Wallis, 1969). The analysis was divided into a preagglomeration and a post-agglomeration stage on the basis of an assumed maximum value for the dispersed phase volume fraction. Shiina and Sakaguchi (1984) incorporated the model of Smith et al. into their scheme to analyze the performance of a parallel-#ow spray column consisting of preheater, evaporator and superheater sections. The model of Smith et al. was also modi"ed by Seetharamu and Battya (1989) to predict the theoretical volumetric heat transfer coe$cient for the direct-contact evaporation of R113 and n-pentane in a stagnant column of distilled water. Mori (1991) established a model, assuming that the heat transferred to each of dispersed droplets of simultaneous evaporation could be approximated by an empirical correlation for heat transfer to an isolated droplet evaporating in a quiescent, su$ciently extended medium. A numerical model for a three-phase, direct-contact, spray-column heat exchanger was developed by Coban and Boehm (1989) to calculate the performance characteristic
Fig. 1. Axial temperature pro"le of two phases.
quantities of this type of device, such as temperature, holdup and volumetric heat transfer coe$cient. In general, these are sound models, which can be utilized to predict heat transfer coe$cients agreeable with experimental data to a certain extent. Expecially, the model of Smith et al. has been more widely used (Shiina and Sakaguchi, 1984; Seetharamu and Battya, 1989). However, all aforementioned models disregarded the population balance characteristic, an important nature in direct-contact heat transfer systems, and hence limited themselves in applications. For example, in the model of Smith et al. (1982) a relationship between the rate of evaporation of droplets and their displacement from the column bottom was formed so that the volumetric heat transfer coe$cient could be calculated as function of this displacement with the following equation: 1 X A (z )n (z )h (z ) dz , (3) k (z)" @ @ @ T z where z is the axial coordinate, A the droplet area, n the @ @ droplet number density, and h the heat transfer coe$@ cient per unit surface area for a single droplet. A , n and @ @ h could be expressed in terms of the instantaneous dia@ meter D of dispersed droplets. Because D was treated as a constant across the column cross section, this model is not able to handle variations of the size distribution of dispersed droplets. Core and Mulligan (1990) presented a population balance model to discuss the heat transfer characteristics of direct contact evaporation in a batch reactor. The population balance equation was solved using the Method of Classes, which was originally proposed by Marchal et al. (1988) in crystallization and precipitation engineering. However, they neglected both breakage and coalescence of dispersed droplets.
3. Model 3.1. Physical system Our research was carried out in bubble columns. Consider a bubble column, schematically shown in Fig. 2.
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The shape of actual droplets is very di$cult to describe in mathematics, and therefore Eq. (4) is used by nearly all researchers in establishing their heat transfer models. 3. The rate of direct-contact evaporation of a swarm of droplets can be described with correlations of the area-speci"c heat transfer rate for single droplets. This implies that in#uences of neighboring droplets on the heat transfer coe$cient from the continuous phase to a dispersed droplet are negligible. 4. Coalescence of dispersed droplets is negligible in the present system. In calculating the evaporation height we found that the values from numerical simulations are much larger than those from experiments (Song and Stei!, 1995). It can be assumed that breakage of droplets dominates in the present system. 3.3. Mathematical description
Fig. 2. Illustration of a bubble column.
One liquid, called dispersed phase, enters the column through a distributor at the bottom as liquid drops with a certain size distribution while another liquid, called continuous phase, #ow continuously from the bottom to the top. Because the continuous phase has a temperature higher than the boiling point of the dispersed phase, the drops will evaporate and hence extract heat from the continuous phase. The continuous phase is immiscible (or, in some cases, only poorly miscible) with either of both phases, vapor and unevaporated liquid, of the dispersed phase. As the dispersed droplets complete their evaporation, the vapor exists through the free surface of the continuous liquid and is subsequently withdrawn from the head part of column. 3.2. Fundamental assumptions
The model in this contribution is a one-dimensional model along the axis of column. The column height is equally discretized in numerical calculations. Fig. 3 schematically shows this discretization. Within height grids, called cells, the swarm of droplets are divided into groups, called classes, according to the droplet diameter. The discretization of the droplet diameter can be either equal or unequal. By selecting one class, we can observe complicated particle phenomena, including convection, growth, coalescence and breakage, as shown in Fig. 4. The population balance equation (PBE) o!ers a fundamental mathematical framework to describe such phenomena (Ramkrishna, 1985) and it will be used here to account for convection, growth and breakage of droplets. Such an equation can be written as *(vn) *(wn) # "BQ !DQ , *h *D
(5)
where n is the droplet number density. Birth rate BQ and death rate DQ are given by
L " l(y)b(y)p 6 D, y n(y, t) dy
BQ "
n
(6)
and
To establish the model, the following assumptions are employed to simplify the actual physical phenomena:
DQ "b
1. The continuous and the dispersed phase are uniform across the column cross section. Therefore, the model is one dimensional. 2. Evaporating two-phase droplets are spherical, with their diameter D being the equivalent diameter of actual droplets:
where p(y, x) is the breakage density function of a droplet of volume y from a mother droplet of volume x, b(x) is the breakage frequency of a droplet of volume x, and v(x) is the number of daughter droplets per breakage. In Eq. (5)
D"
6< " . n
(4)
dh v" dt
n n D n D, t , 6 6
(7)
(8)
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
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where C is the coe$cient of resistance. Applying " Levish's formula (Levish, 1949),
2 o gD ! C " . (11) " 3 1.82p ! Raina et al. (1984) obtained the following relationship v"1.91
1.91
o D p ! 1! " . (12) o D o D ! ! To account for several factors that are responsible for obvious deviation of theoretical results from experimental values, Raine et al. (1984) eventually gave
v" Fig. 3. Diagram of the discretization.
o D p ! 1! " DM \"M L o D o D ! ! , M ¹ #¹ " C k " " ! " N! ! 2¹ ¹ k ! " !
(13)
where DM is a dimensionless averaged diameter, de"ned as D#D . DM " (14) 2DD Eq. (13) was utilized by Coban and Boehm (1989) in their model for a three-phase, spray-column, direct-contact heat exchanger. Marrucci (1965) showed that the e!ect of the dispersed phase holdup could be expressed by 1!e v "v . 1!e
This correction was employed by Core and Mulligan (1990) and Song and Stei! (1995) in their population balance models. In the present analysis Eq. (13), with Marrucci's correction (15), is used. A single two-phase droplet is shown in Fig. 5. Let the volume of liquid evaporated from t to t#dt be d< , J which causes an increase of the volume of vapor within the droplet denoted by d< . From mass and energy E balance, we have
Fig. 4. Particle phenomena.
is the rising velocity, and dD w" dt
(9)
is the growth rate of droplets. Both v and w need to be modelled before numerical simulations are carried out by using Eq. (5). There are several empirical and semi-empirical formulas to represent the rising velocity of droplets. From the force balance for a single droplet Raina et al. (1984) derived the following expression:
v"
4 o D Dg 1! " 3 o D C ! "
(15)
(10)
o d< "o d< , !J J !E E n d< !d< "d D , E J 6
(16) (17)
nDh (¹ !¹ ) dt"Dh o d< . (18) D ! " T !J J From these equations we can obtain the growth rate of dispersed droplets due to evaporation as 2h (¹ !¹ )(o !o ) " !J !E . w" D ! (19) o o Dh !J !E T According to the assumptions given above, we use the correlations of the area-speci"c heat transfer rate for
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(1966), Coulaloglou and Tavlarides (1977) and Chatzi et al. (1989). b(x) is the breakage frequency of droplets, which represents when a droplet will break up from the viewpoint of statistics. Existing information from both theory and experiment is still very imprecise. Instead, we use the maximum diameter of stable droplets in dispersion systems given by Mersma nn (1977) to model it:
cp ! D " , (24)
(o !o )g ! " where c is an empirical constant whose value is around 9.0. In the present problem o is the averaged density of " a two-phase droplet, that is, m o " ", (25) " < " where m and < are the total mass and volume of " " a droplet, respectively. Apparently, Fig. 5. A two-phase droplet.
a single two-phase droplet. Sideman and Taitel (1964) derived a well-known analytical formula for the Nusselt number
Nu 3 cos b!cos b#2 , " Pe n
(20)
where b is the half-evaporation open angle, shown in Fig. 5. This formula has been cited widely in the literature. Assuming that the distorted droplet is an ellipsoid, the diameter ratio of which varies in terms of the Peclet number, Pe (Van Bekkum, 1978) we derived a correlation as follows (Song et al., 1996):
b)453, b+603, b+753, b+903, b*1053,
0.688 =e , 0.0817 =e , Nu "C # 2.40 =e\ , PeK 0.567 =e\ , 0.190 =e\ ,
(21)
where C is equal to the right side of Eq. (20), =e is the Weber number, and m is a "tting index, the value of which is around 0.5, given in Table 2. To model the breakage terms, l(x), p(y, x) and b(x) should be given in advance. In the present model the binary breakage with a normal distribution of daughter droplet diameter is assumed, that is l(x)"2
(22)
1 W\V p(y, x)" Exp \ NW (23) (2np W where p is the standard deviation of distribution, taken W equal to (x/2)/c (c "3}3.5). This distribution was ad opted in this study as used by Valentas and Amundson
o )o )o . (26) "E " "J If the droplet is a pure liquid drop, then Eq. (26) yields o "o , and if it is a pure gas bubble, o "o . " "J " "E 3.4. Numerical method Core and Mulligan (1990) used the Method of Classes to solve the population balance equation with the size growth of particles (Marchal et al., 1988). In the discussion of Core and Mulligan, both breakage and coalescence of droplets were not included so that the PBE was homogeneous. From our numerical example we found that if breakage and/or coalescence are taken into account, small non-smooth discretized values at some iteration step may lead to an unstable calculation. With the help of the principle of operator splitting, the authors proposed a new method, termed Splitting#Analogy, to solve the PBE with the size growth of particles. In this method the homogeneous PBE is simply discretized following an analogy with the one-dimensional continuity equation in #uid mechanics. A detailed analysis about this procedure can be found in Song et al. (1997). From the population balance model we can determine the droplet size distribution and hence the number of droplets in all classes. The overall heat transfer rate QQ in H cell j is then calculated by QQ " QQ , HG H G where
(27)
QQ "nD N h (¹ !¹ ) (28) HG HG HG D HG " ! H is the heat transfer rate in group i. The area-speci"c heat transfer coe$cient, h , is given by D k Nu h " ! , (29) D D
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
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Table 2 Values of m from Song et al. (1996) b)753
753(b(1053
b*1053
Pe"10 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.513 0.526 0.533
0.491 0.487 0.484
0.485 0.469 0.461
Pe"20 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.528 0.552 0.563
0.481 0.475 0.472
0.468 0.436 0.425
Pe"30 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.537 0.564 0.575
0.474 0.469 0.466
0.450 0.426 0.408
Pe*40 000
0.9)k(1.0 0.8)k(0.9 k)0.8
0.546 0.579 0.583
0.470 0.464 0.462
0.443 0.414 0.389
The two parameters, p and k, are determined by
Heat transferred in cell j is obtained from Dh Q " QQ . H HG v HG G
(30)
According to the energy conservation law we can calculate the temperature variation of the continuous phase Q H ¹ !¹ " . ! H> ! H (A dh! n D N )o C G HG HG ! N!
(31)
The total heat transfer rate is obtained from QQ " QQ . H H
(32)
From Eq. (31) the temperature of the continuous phase at the outlet, ¹ , can be determined. Eventually, the vol!M umetric heat transfer coe$cient for the whole column can be calculated by using Eqs. (1) and (2).
4. Results and discussions Numerical simulations are run with values of parameters that correspond to the experimental system studied by Steiner (1993): Materials system: R114/water, Column diameter: 0.114 m, Distributor: 35 holes of 2.0 mm diameter. The initial size distribution of drops is assumed to be a logarithmic-normal distribution:
(ln D !k) 1 f (D )" exp ! . 2p (2npD
(33)
DM , ((p /DM )#1 "
(34)
(35)
k"ln
p" ln
p " #1 , DM
where DM and p are the mathematical expectation " (mean value) and the standard deviation, respectively. The mean diameter is a very important parameter, which is found to depend mainly on the ori"ce velocity, the ori"ce diameter and the physical properties of materials (Hayworth and Treybal, 1950; Scheele and Meister, 1968; De Chazal and Ryan, 1971). Despite the fact that a considerable amount of work can be found in the literature on predicting the size of drops formed at ori"ces of a given diameter, we are still unable to predict the mean drop diameter or volume for a given system with high con"dence. Because of the uncertainty in this value, Smith et al. (1982) performed the calculations over a range of D (0.5, 1.0 and 2.0 mm) relative to their experimental system. Seetharamu and Battya (1989) predicted the volumetric heat transfer coe$cient using D "0.5, 1.0, 1.5 and 2.0 mm for the experimental system of R113 and water, with 19, distributor holes of 0.35 mm diameter and using D "1.0, 1.5 and 2.0 mm for the same materials system with 7 distributor holes of 1.00 mm diameter. For the former system they got the calculation results with D "1.0 mm in agreement with the experimental data, and for the latter the calculation results with D "1.5 mm matched the experimental data better. Mori (1991) estimated the initial drop diameter as about 1.4 mm for n-Pentane evaporating in water with the injection velocity of 0.22}0.44 m/s and the 19 and 36
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M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
Fig. 6. k vs. D¹ for di!erent #ow rates of the dispersed phase. T G
distributor holes of 0.5 mm diameter in terms of the empirical formulas of Scheele and Meister (1968) and De Chazal and Ryan (1971). In the discussion below D "1.0, 1.5 and 2.0 mm are taken except that the value for this parameter is especially given. By taking DM "1.5 mm, we calculate the volumetric heat transfer coe$cient for di!erent #ow rates of the dispersed phase at di!erent values of the initial driving temperature di!erence. The results are shown in Fig. 6. This "gure shows that there is a very high heat transfer e$ciency at a very low initial temperature di!erence. If the temperature di!erence is higher than 6 K, a bigger value of this parameter cannot improve the heat transfer e$ciency. Note that a longer column is needed for a lower temperature di!erence (Song and Stei!, 1995). In engineering design we should "nd a balance between the heat transfer e$ciency and the evaporation height. It can be observed from Fig. 6 that the heat transfer coe$cient increases if the #ow rate of the dispersed phase goes up. However, the degree of increasing becomes lower and lower. The direct relationship of the heat transfer coe$cient vs. the #ow rate of the dispersed phase is drawn in Fig. 7. The curves are increasing, but gradually sloping down. Fig. 8 demonstrates the comparison of theoretical and experimental values of the volumetric heat transfer coef"cient. Experimental results used in this "gure were obtained by Steiner (1993). The agreement between theory and experiment is fairly good for the lower #ow rate of the dispersed phase (140 kg/h), and reasonably good for the higher #ow rate (200 kg/h). For the latter the value from our model are some lower. Trend of the theoretical curves is very similar to that of the experimental points for both cases. For two cases: D¹ "10 K and D¹ "5 K, we draw G G the temperature pro"les of both the continuous and the dispersed phase along the column height in Fig. 9. The temperature of the dispersed phase has been assumed to be always constant. The temperature of the continuous
Fig. 7. k vs. the #ow rate of the dispersed phase. T
Fig. 8. Comparison of theoretical and experimental values of k . T
Fig. 9. Temperature pro"les along the column height for two cases: D¹ "10 K and D¹ "5 K. G G
phase decreases more rapidly as the dispersed drops begin their evaporation. At a higher position of the column, the heat transfer rate becomes lower because the driving temperature di!erence is smaller and the areaspeci"c heat transfer coe$cient for a single droplet is
M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
lower (Song et al., 1996). At a lower temperature di!erence, a bigger column height is needed for complete evaporation of all droplets, as mentioned before. Because of the imprecision in DM , we carry out calcu lations taking di!erent values of DM as most researchers such as Smith et al. (1982), Seetharamu and Battya (1989) and Mori (1991) have done previously. Taking DM as 1.0, 1.5 and 2.0 mm, the results of the volumetric heat transfer coe$cient are shown in Fig. 10. The calculations are performed assuming the log-normal distribution with p "0.5 mm for D . The "gure shows that this para" meter is very sensitive to the calculations. Therefore, precise determination of this parameter from experiment or from theory is a key to achieve a good agreement of the heat transfer coe$cient between theory and experiment to the present model and to all other models. In order to examine the in#uence of the distribution of D on numerical results, we employ two distributions used frequently in the past: log-normal and normal with the same mean value (1.5 mm) and standard deviation (0.2 mm). The results are shown in Fig. 11. From this
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"gure we cannot observe a big di!erence existing between curves for these two conditions. If we put the curve from the normal distribution onto Fig. 8, equal agreement with experimental data could be seen between these two distributions of D . At present DM cannot be determined with high con" dence. It is more di$cult to precisely determine the standard deviation of D from theory and experiment. Taking the standard deviation of D equal to 0.2 mm for the log-normal distribution, we calculate the volumetric heat transfer coe$cient, results of which are shown in Fig. 11 as well. From this "gure we "nd that the heat transfer coe$cient is lower for the smaller standard deviation. To draw an eventual conclusion this needs to check with values obtained in experiments. However, a reasonable explanation can be given. For a bigger standard deviation, more drops of larger diameter exist, and, therefore, they may break up earlier. This can be assumed to be one reason to contribute to a higher heat transfer coe$cient.
5. Concluding remarks
Fig. 10. The calculated volumetric heat transfer coe$cients for di!erent values of DM .
Fig. 11. In#uence of the distribution of D on the volumetric heat transfer coe$cient.
A population balance model is presented to calculate the volumetric heat transfer coe$cient for a bubble column with direct-contact evaporation of one liquid in an another immiscible liquid. An important feature of this model is that breakage of dispersed droplets is included, and the size distribution of dispersed droplets can be accounted for. Without considering breakage of droplets in simulations we obtain much lower values in the volumetric heat transfer coe$cient than those found from experiments. The reason is that a larger evaporation height of column would be needed to complete the evaporation of all droplets if there were no breakage of droplets (Song and Stei!, 1995). The model has a good ability to predict the volumetric heat transfer coe$cient. The predicted results are in good agreement with experimental data not only qualitatively but also quantitatively. The mean diameter of dispersed drops, just leaving the bottom distributor, is a very sensitive parameter to our model calculations and, we believe that the same type of sensitivity applies also to other models existing in the literature. At present it is impossible to even roughly predict its value from theory and experiment. On the other hand, the function form of the diameter distribution of the initial drops was found to show low sensitivity to the calculations if the mean value and standard deviation are kept unchanged. Both the log-normal and the normal distribution seem to be a good approximation for D . The standard deviation of the initial size of drops has some in#uence on the results presented in this analysis. For example, a bigger value of the standard deviation may lead to a higher heat transfer coe$cient.
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M. Song et al. /Chemical Engineering Science 54 (1999) 3861}3871
Notation A BQ b(x) C N C " D D DM DM DQ D
g H h h D Dh T k k T m N Nu n Pe p(y, x) Q QQ ¹ D¹ < v =e w
Subscripts column cross-sectional area, m birth rate of droplets, m\ s\ breakage frequency function, s\ speci"c heat, J kg\ K\ resistance coe$cient diameter of droplets, m initial diameter of drops, m dimensionless mean diameter of droplets initial mean diameter of drops, m death rate of droplets, m\ s\ maximum diameter of stable droplets, m\ s\ acceleration due to gravity, m s\ column height for complete evaporation, m axial coordinate of column, m heat transfer coe$cient for a single droplet, W m\ K\ evaporation enthalpy of the dispersed phase, J kg\ thermal conductivity, W m\ K\ volumetric heat transfer coe$cient, W m\ K\ mass, kg number of droplets in one group Nusselt number number density of droplets, m\ Peclet number density distribution function of daughter droplets, m\ transferred heat, J heat transfer rate, W temperature, K temperature di!erence between two phases, K volume, m rising velocity of droplets, m s\ Weber number growth rate of droplets, m s\
Greek symbols b e l(x) k o p p, k p " p W
half evaporation open angle dispersed phase holdup number of daughter droplets viscosity, kg m\ s\ density, kg m\ surface tension, N m\ two parameters in the logarithmic-normal distribution standard deviation of a distribution for D ,m standard deviation of distribution function p(y, x), m
C D g i l o
continuous phase dispersed phase gas phase inlet liquid phase outlet
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