Ind. Eng. Chem. Res. 1987,26, 2449-2454
2449
Acknowledgment
Literature Cited
We express our gratitude to Kuwait Institute for Scientific Research for the financial support during the course of this project.
ASTM D341, The American Society for Testing and Materials, 1983a. ASTM D445/IP71, The American Society for Testing and Materials, 193b. British Petroleum, Sunbury Report 3282, UK, 1947. Dean, D. E.; Stiel, L. I. AZChE J. 1965, 11(3), 526. Letsou, A.; Stiel, L. I. AIChE J. 1973, 19, 409. Lobe, V. M. M.S. Thesis, University of Rochester, New York, 1973. Lydersen, A. L.; Greenkorn, R. A.; Hougen, 0. A. Engineering Experimental Station Report 4,1955; College of Engineering, University of Wisconsin, Madison, WI. Orrick, C.; Erbar, J. A. Reported in Properties of Gases and Liquids, 3rd ed.; Ried, R. C., Prausnitz, J. M., Sherwood, T. K., Eds.; McGraw Hill New York, 1972. Ratcliff, G. A.; Khan, M. A. Can. J. Chem. Eng. 1971, 49, 125. Reid, R. C.; Sherwood, T. T. The Properties of Gases and Liquids, 2nd ed.; McGraw-Hill: New York, 1966; pp 431-440. Stiel, L. I.; Thodos, G. AZChE J. 1961, 7, 611. Twu, C. H.; Bulls, J. W. Hydrocarbon Process. 1981, 5, 217. Walther, C. Proc. 2nd World Pet. Congr. 1937, 2, 133. Wedlake, G. D.; Ratcliff, G. A. Can. J. Chem. Eng. 1973, 51, 511.
Nomenclature aij = interaction parameters ASTM = American Society for Testing and Materials GC = group contribution parameter I = Refutas index or blending index la= Refutas index of blend M = molecular weight n = number of carbon atoms P, = critical pressure T,= critical temperature x = mole fraction V = liquid molar volume W = weight fraction w o = Pitzer acentric factor Greek Symbols c$~ = liquid molar volume fraction v = kinematic viscosity 7 = absolute viscosity p = density
Received for review July 28, 1986 Revised manuscript received August 17, 1987 Accepted September 4, 1987
Analytical Design Equations for Multisolute Reverse Osmosis Systems Srinivas Palanki and Sharad K. Gupta* Chemical Engineering Department, Indian Institute of Technology, Hauz Khas, New Delhi 110016, India
Analytical equations are developed for tubular, spiral-wound, and plate-and-frame reverse osmosis (RO) modules to predict the membrane channel length for achieving a given fractional solvent recovery for a dilute feed with two highly rejected solutes. Extension to a large number of noninteracting solutes is straightforward. Analytical equations to predict the permeate solute concentrations are also developed. These design equations are simple to use and can be applied for both laminar and turbulent flow conditions. On the basis of these design equations, some approximate design equations are also derived. The predicted dimensionless length of the module and the average permeate concentrations are compared with the results obtained by numerical integration and found to be in excellent agreement. The above results are also compared with literature values. These design equations can be applied for systems where the solution-diffusion model is applicable. Reverse osmosis (RO) has rapidly emerged as a commercially attractive microsolute separation process. Applications of RO range from desalination, the dominant commercial application, to separation of products in bioreactions. Such widening activity suggests the need for development of simple and accurate design equations for making rapid process design calculations. Numerical design procedure and results for single-solute RO are available in the work of Ohya and Sourirajan (1971). Sirkar et al. (1982)developed a simple analytical design equation for single-solute RO. The flux equations were simplified by using a truncated series for the term exp(Nl/kliC) in the expression for concentration polarization, and then explicit expressions for solute and solvent fluxes were obtained. This approach was extended to multicomponent solute processes by Prasad and Sirkar (1984,1985).Prasad and Sirkar used the linear approximation of exp(Nl/kliC) for N,/kliC S 0.2 and quadratic approximation of exp(Nl/kEC)for Nl/k& 6 1. The design equations for N,/k,C > 1 may also be derived by using better approximations for exp(N1/ k&). However, the 0888-5885/87/2626-2449$01.50/0
resulting equations become very complex to integrate, and it may be easier to use numerical procedures. Recently Gupta (1985)developed an analytical procedure for a single-solute RO system where the use of a truncated series for the exponential term is not required. Thus, the resulting equations are valid for all values of Nl / k1, c. In this study, the design equations for plate-and-frame, spiral-wound, and tubular modules are developed by an analytical technique similar to that used by Gupta (1985). It is shown that the resulting design equations are accurate and simple to use. On the basis of these design equations, further approximate solutions are also obtained and compared with those obtained by Prasad and Sirkar (1985).
Development of Design Equations We consider the case of two noninteracting solute species in the feed solution as in Prasad and Sirkar (1984). We assume that the solution diffusion model is applicable for transport across the membrane. In addition, we assume that the total molar density, C, of the solution is constant, 0 1987 American Chemical Society
2450
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
the osmotic pressures of dilute solutions are proportional to the salt concentrations in the solution, and the membranes are high salt rejection membranes. We define dimensionless parameters as
Yz = bCXZlf/AP =
(1)
bCX31f/hP
(2)
712
= k12C/A@
(3)
713
= k13C/AAP
(4)
73
62
= (D~~C)/(KZGA~~)
(5)
63
= (D,mC)/(K3GAW
(6)
Also, we define the dimensionless variables as
x;1 = X21/X2lf xi3
=
xi2
= X22/X21f
= X23/X21f = X32/X31f
X31/X31f
= X33/X31f
x3 ;
The differential equation (20) is to be solved subject to initial conditions
x=o
V = l
(23)
Note that (21) and (22) imply that qz In Z2 =
v3 In Z3
(24)
Analytical Solution
As in previous studies such as Prasad and Sirkar (1985) and Gupta (19851, we assume that the dimensionless parameters yz, y3, qz, q3, 02, and 63 are constant. Integrating (20) subject to (23), we have
(7)
(8)
v = UX/UXf
(9)
x = ax
(10)
Using (21) and (22) and following Gupta (1985), it can be easily shown that the above integral can be broken down into four separate integrals:
where CY
= AAF'/Chu,f
(11)
In terms of above dimensionless variables and parameters, the flux expressions for solvent and two solutes are N1 = A W 1 - Y~X:Z- Y3Xl2)
(12)
NZ= CX21d&m/K2S)X,+,
(13)
N3
= cx31r(D3m/K3s)x&
(14)
The concentration polarization for the two solutes using "film" theory may be written as
Furthermore, the solvent and solutes balances a t any location x for high rejection membranes are given by dV Ni cav,f = -dX h
vx,: vx;,
=1
(18)
=1
(19)
Substituting Nl from (12) in (17) and using (18)and (19), we have
Also (15) and (16) reduce to
Similarly the last two integrals in (26) can also be calculated, and we get
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2451
-
(ii) y2 and y3 0. This case corresponds to a solution of negligible osmotic pressure L+ = A (38) (iii) Large Values of q2 and q3. Following Gupta (1985), expanding each term of (34) for large q2 and q3, and neglecting second-order and higher order terms, we obtain
Now, it can be shown that
where A is the fractional recovery. Finally, by use of exponential integrals El(u) and Ei(w), the expression for L+ reduces to
Also,
( z 3 = 1 +(
z z = l + 1---Y2zz v 1---Y2z2
v
(2') (2')
Y3z3)/ 72+o
(40)
Y3z3)/
(41)
v
v
7 3 + 0
In (40) and (41), neglecting second-order and higher order terms and solving for Z2 and Z3, we obtain 2 2
=
Z3 =
+ v93 + 7 2 7 3 - 73773 7 2 9 3 + v'?2q3 + 9273 vq293 + v72 + 7 2 9 3 - 7 2 9 2 vq213
Y273
(42)
+ vq243 + 73%
Now using (35) and (42), it can be shown that (39) reduces to where the additional subscript 1 indicates the values at V = 1and the additional subscript 2 indicates the value a t any V or the outlet. Generalizing for n solutes gives
The method for solving (34) is as follows. (i) Give values of y2, y3, qZ, v3, and A. (ii) Calculate the values of 2, and Z3 at the inlet and outlet conditions by using any numerical method such as the Newton-Raphson's method. That is, calculate ZZl,Z31, Z z z p d 232. (111) Calculate uZl, u22,u3lt u32, wzl, w22, w31, and w32 by uqing (35). (iv) The exponential integrals in (34) can then be calculated since the tabulated values for these integrals are available in many references [for example, Abramowitz and Stegun (1968)l. Hence, L+ can be calculated from (34).
-
Approximate Design Equations (i) y3 0. This case reduces to an RO process with a single solute
(43) which is the same as (22) of Prasad and Sirkar (1985). Permeate Solute Concentration The mole fractions of both solute species in the permeate stream can be determined even though their concentrations in the permeate were assumed to be negligible in the calculation of L+. Following Gupta (1985),it can be shown that
Similarly, for the second solute,
Adding (44) and (45) and integrating, we obtain
2452 Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987
Equation 49 can also be written as
Table I. Dimensionless Parameters from Prasad and Sirkar (1985) for Turbulent Flow Conditions feed 3000 ppm NaC1, 2000 ppm MgSO,; B,(NaCl) = 0.02, 83(MgS04) = 0.0007 3000 ppm NaC1, 2000 ppm MgClZ; O,(NaCl) = 0.01, B,(MgClZ) = 0.0056 25 000 ppm NaC1, 1000 ppm MgSO,; 02(NaC1)= 0.01, 03(MgS0), = 0.007
parameter set
92
93
Y2
Y3
5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0 5.0
4.0 4.0 4.0 4.0 1.3 1.6 3.8 1.0 4.3 4.1 4.7 4.0
0.1 0.1 0.1 0.3 0.1
0.0325 0.0325 0.0325 0.0975 0.0411 0.0411 0.0411 0.0821 0.0195 0.0390 0.0390 0.0585
0.1 0.1 0.2 0.1 0.2 0.2 0.3
T1 T2
0.2 0.3 0.5 0.3 0.2 0.3 0.5 0.3 0.3 0.2 0.3 0.3
T3 T4 T5 T6 T7 T8 T9 T10 T11 T12
JV+ [ 2 2
-23
In 23
Table 11. Dimensionless Parameters from Prasad and Sirkar (1985) for Laminar Flow Conditions feed 3000 DDm NaC1. 2060' MgSO,; ' O,(NaCl) = 0.01, B3(MgS0,) = 0.0007 3000 ppm NaC1, 2000 ppm MgC12; MNaCl) = 0.01, Oj(MgC1,) = 0.0056 25 000 DDm NaC1. ~. 10060'ppm MgSO, B,(NaCl) = 0.01, 03(MgS04)= 0.0007
1.0 0.276 1.0 0.277 1.0 0.365 1.0 0.385 1.0 0.285 1.0 0.275 1.0 0.255
Yz 0.1 0.1 0.2 0.3 0.1 0.1 0.2
0.0325 0.0325 0.0650 0.0975 0.0411 0.0411 0.0821
0.3 0.5 0.3 0.3 0.3 0.5 0.3
parameter set L1 L2 L3 L4 L5 L6 L7
0.205 0.215 0.215 0.175
0.1 0.2 0.2 0.3
0.0195 0.0389 0.0389 0.0585
0.3 0.2 0.3 0.2
L8 L9 L10 L11
7lz
7l3
1.0 1.0 1.0 1.0
Y3
Equation 46 is similar to the expression for a single-solute RO process. Equations 44 and 45 are not integrable analytically. However, we can integrate them by making some approximations.
+
2 2
1
dV (50)
Neglecting the second term, we have
Substituting this in (48), we obtain
Approximate Solute Concentration Consider (44). This equation can also be written as
-%r(
7222
Y2
v
+73
+ 7323
I---7222
v
dV
+
Similarly,
Y3Z9
v
which readily simplifies to
The second term in (48) cannot be integrated exactly. Here we show a simple method for estimating this approximately. Using (24), it can be easily shown that
(;
2 2
-
-23
;)vT12ln 2 2
dV=
sv=
2 2
Vln-
23
dV (49)
In the derivation of (52), the second term in (50) was assumed to be negligible. This was checked by integration (44)numerically to determine (X.$Javand comparing these values with the values of (Xi3)av obtained from (52). As shown in Tables I11 and IV, the two values are within 1% of each other. Therefore, neglecting the second term in (50) is justified. In addition, using (40) and (41) for Z2and 2, for large 0, it can be easily shown that (52) and (53) simplify to (28) and (29) of Prasad and Sirkar (1985),which are also valid for large 0 only. Results and Discussion Prasad and Sirkar (1985) have calculated the values of L+, (X;Jav,and (X&)av by using their design equations for three dilute feeds with the following compositions: (1)3000 ppm NaCl and 2000 ppm MgS04;(2) 3000 ppm NaC 1 and
Ind. Eng. Chem. Res., Vol. 26, No. 12, 1987 2453 Table 111. Comparison of Predictions of Dimensionless Length (L') and Averaged Permeate Solute Concentrations ((X& and (X&) from Analytical Design Equations" with the Values Obtained by (i) Numerical Integrationband (ii) Prasad and Sirkar (1985)' for Turbulent Flow parameter set L+" L+b L+' (Xi3),,. (Xi3)a: (XiJa; (Xi3),,. (X3aYb (X&)a$ T1 T2 T3 T4
Tk T6 T7
T8 T9 T10 T11 T12
0.2428 0.3698 0.6401 0.6351 0.2547 0.3842 0.6524 0.5351 0.3609 0.2891 0.4457 0.5758
0.2428 0.3693 0.6402 0.6351 0.2547 0.3842 0.6525 0.5351 0.3610 0.2892 0.4458 0.5758
0.2421 0.3681 0.6380 0.6320 0.2511 0.3803 0.6500 0.5187 0.3600 0.2880 0.4440 0.5731
0.01598 0.017 25 0.020 94 0.027 94 0.016 64 0.017 84 0.021 30 0.023 87 0.016 92 0.018 55 0.020 30 0.025 53
0.01598 0.017 25 0.020 95 0.027 94 0.016 64 0.017 84 0.021 30 0.023 87 0.016 92 0.018 55 0.020 30 0.025 53
0.015 75 0.017 00 0.020 65 0.027 70 0.016 25 0.017 49 0.021 01 0.023 07 0.016 67 0.018 32 0.020 06 0.025 29
0.001 17 0.001 26 0.001 52 0.002 00 0.014 56 0.013 91 0.012 51 0.020 88 0.012 17 0.013 39 0.014 33 0.018 34
0.001 17 0.001 26 0.001 52 0.002 00 0.014 56 0.01391 0.012 52 0.020 88 0.012 17 0.013 39 0.014 33 0.018 34
0.001 14 0.001 23 0.001 49 0.001 98 0.012 66 0.012 62 0.012 26 0.018 25 0.011 94 0.013 17 0.014 15 0.018 12
"Calculated from (34), (52), and (53). bObtained from numerical integration of (25), (44), and (45). 'From Prasad and Sirkar (1985).
Table IV. Comparison of Predictions of Dimensionless Length ( L') and Averaged Permeate Solute Concentrations and (A'&) from Analytical Design Equations" with the Values Obtained by (i) Numerical Integrationband (ii) Prasad and Sirkar (1985)' for Laminar Flow
L2 L3 L4 L5 L6 L7 L8 L9 L10 L11
1.0043 0.7393 1.0687 0.5887 1.0729 0.9298 0.5825 0.5061 0.7899 0.7546
1.0044 0.7394 1.0688 0.5888 1.0730 0.9300 0.5826 0.5061 0.7900 0.7546
0.7700 0.5310 0.7010 0.4530 0.8090 0.6690 0.4360 0.3700 0.5780 0.5330
0.046 06 0.044 17 0.056 63 0.038 89 0.047 69 0.051 11 0.038 60 0.041 95 0.045 88 0.054 98
0.046 30 0.044 19 0.056 65 0.038 94 0.047 97 0.051 19 0.038 72 0.041 99 0.045 98 0.05503
0.019 9 0.018 0.021 20 0.016 90 0.02040 0.020 60 0.017 30 0.017 30 0.018 90 0.021 0
0.011 80 0.006 26 0.006 20 0.078 15 0.091 13 0.073 47 0.019 95 0.012 44 0.012 87 0.013 44
0.011 75 0.006 25 0.006 20 0.078 08 0.090 75 0.073 37 0.019 91 0.012 43 0.012 84 0.013 43
0.00168 0.001 3 0.001 43 0.016 80 0.020 20 0.019 90 0.001 30 0.001 38 0.001 50 0.001 70
Calculated from (34), (52), and (53). *Obtained from numerical integration of (251, (44), and (45). C F:om ~ Prasad and Sirkar (1985).
2000 ppm MgC1,; (3) 25 000 ppm NaCl and 10 000 ppm MgS04. Their ranges of dimensionless parameters yi,-y3, q2, q3, and A are shown in Tables I and 11. To compare the design equations developed here with those of Prasad and Sirkar (1985), we have calculated the values of L+, (Xi3)av, and from (341, (52), and (53) for the same range of dimensionless parameters as in Tables I and 11. To further verify our results, (251, (441, and (45) were investigated numerically by using Simpson's rule with a step size of the order of The values of L+, (X13)av, and (X13)avfrom the design equations developed here, from Prasad and Sirkar (1985), and from numerical integration are also shown in Tables 111and IV for comparison. Table I11 is for q, = 5 (turbulent flow conditions), while Table IV is for q2 = 1 (laminar flow conditions). In Table 111, we observe that the value of L+ calculated by (34) is in excellent agreement with the value obtained by Prasad and Sirkar (1985) and by numerical integration. This is not surprising,as it has been shown earlier that (34) reduces to the equation developed by Prasad and Sirkar (1985) for turbulent flow by neglecting second-order and higher order terms of the exponential term expansion. In Table IV, we observe that the value of L+ calculated by (34) is in excellent agreement with the value obtained by numerical integration. However, these results do not match with those obtained by Prasad and Sirkar (1985). First we thought the discrepansies were due to the following reason. The values of q3 were not listed by Prasad and Sirkar (1985). Due to this, the value of q3 had to be worked back using their results which may have resulted in some errors in the values of q3. To further check, we took arbitrary values of q,, q3, y,, y3,and A and compared the results obtained by (34), (52), and (53) with those obtained by the equations of Prasad and Sirkar. These results also do not match for low values of 0, and q3. After some examination, it became clear that the equations of
Prasad and Sirkar cannot be applied for low values of q, and q3. For the conditions listed in Table 11, the values of q3 are of the order of 0.2. Equations 52 and 53 developed for (Xi3)av and (Xi3), are approximate. The values of (Xi3), and (Xi3)av obtained by (52) and (53) are compared with the values obtained by numerical integration for various cases of turbulent and laminar flow in Tables I11 and IV. It is noted that the results obtained analytically are within 1% of those obtained numerically. So, we conclude that (52) and (53) are good approximations. We further observe from Tables I11 and IV that as in the case of L+,the values of (Xi3)av and (Xg3Iav calculated by (52) and (53) are in excellent agreement with the values obtained by Prasad and Sirkar (1985) for turbulent flow conditions; however, these values do not match for laminar flow conditions. The reason for these errors is already discussed above. A , AP, and kli are assumed to be constant in the development of design equations. Due to these assumptions there may be errors, especially in the case of turbulent flows. In such cases for better results, the procedure of Prasad and Sirkar (1984) of dividing the module in two or more sections is recommended. In each section, A , AP, and kL may be assumed constant and calculations carried out by using (34), (52), and (53). After each section, new values of A , AP,and kli can be found and used for the calculations of the succeeding sections.
Conclusion The design equations developed analytically predict more accurately than the equations of Prasad and Sirkar (1985). Unlike the equations of Prasad and Sirkar, these equations are valid for all values of N,/k,C. On the basis of these design equations, some approximate design equations are also derived.
Ind. Eng. Chem. R e s . 1987,26, 2454-2461
2454
Acknowledgment We are grateful to Baldeep Koundal for neatly typing the manuscript.
Nomenclature A = pure water permeability constant b = constant defined as b = T , / C ,
C = total molar density of solution, g-mol/cm3 C , = molar concentration of species i at location j , i = 1 for solvent, i = 2,3 for solute species;j = 1 for feed stream j = 2 for high-pressuremembrane-water interface, j = 3 for permeate stream D,,/K,6 = solute transport parameter of species E l ( u ) = exponential integral defined as E,(u) = S:[exp(-u)/ul du Ei(w) = exponential integral defined as Ei(w) = S-! [exp(w)/wI dw l / h = membrane surface area per unit volume of feed channel kl, = mass-transfer coefficient on high-pressure side, cm/s K , = ratio of solute i concentration in brine over solute i concentration in membrane with which the brine is in contact L = length of membrane from channel entrance, cm L+ = dimensionless length of membrane = aL N1 = solvent flux at location x , g-mol/cm2/s N , = flux of solute species i at location x , g-mol/cm2/s A€' = applied pressure difference across membrane at location x , atom v, = bulk velocity on the high-pressure side at any location x , cm/s V = dimensionless velocity = v,/vXf V1 = partial molar volume of solvent, cm2/g-mol x = axial distance from channel, cm X = dimensionless distance = a x X l l , X12,X L 3= mole fractions of solute i in feed stream, high-pressure side membrane-solution interface, and permeate solution at any x X = solvent mole fraction at any x X lil1 , X z , X: = normalized mole fraction of solute i of feed stream, high-pressure side membrane-solution interface,
and permeate stream at any x = Xij/Xilf u = dimensionless variable = yz/sV w = dimensionless variable = l / q - u uzl,~ 3 1 wZl, , ~ 3 =1 values of u and w at the channel entrance uZ2,~ 3 2 wZ2, , ~ 3 = 2 values of u and w at any location x Z, Z3 = concentration polarization as defined by (15) and (16) Greek Symbols a = parameter = A A P / C h u Z f cm-' , riJ= osmotic pressure of solute in the solution of concentration
C , , atm dimensionless osmotic pressure = bCXilf/AP,i = 2, 3 A = fractional solvent recovery = 1 - V 6 = membrane thickness, cm Oi = dimensionless solute permeability = ( D i , / K i 6 ) / ( A A F / C ) vi = dimensionless mass-transfer coefficient = k l i C / A A P ,i = 2, 3 yi =
Subscripts av = averaged quantity from module f = high-pressure side feed at channel entrance
i = 2 solute 2, 3 = solute 3 j = 1 = feed stream, 2 = high-pressure side membrane-solution interface, 3 = permeate stream 1, 2 = conditions at the entrance and at any location x
Literature Cited Abramowitz, M.; Stegun, I. A,, Ed. Handbook of Mathematical Functions; Applied Mathematics Series, 55; Bureau of Standards: 1968; p 238. Washington, D.C., Gupta, S. K. Znd. Eng. Chem. Process Des. Deu. 1985, 24, 1240. Ohya, H.; Sourirajan, S. Reverse Osmosis System Specification and Performance Data for Water Treatment; Thyer School of Engineering, Dartmouth College: Hanover, NH, 1971. Prasad, R.; Sirkar, K. K. Ind. Eng. Chem. Process Des. Deu. 1984, 23, 320. Prasad, R.; Sirkar, K. K. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 350. Sirkar, K. K.; Dang,P. T.; Rao, G. H. Znd. Eng. Chem. Process Des. Deu. 1982,21, 517.
Received f o r review September 2, 1986 Accepted July 21, 1987
Drying and Mixing of Solids and Particles Residence Time Distribution in Four Impinging Streams and Multistage Two Impinging Streams Reactors? Amir Kitron, Ron Buchmann, Kfir Luzzatto, and Abraham Tamir* Department of Chemical Engineering, Ben-Gurion University of the Negeu, Beer-Sheva, Israel
The novel four impinging streams (FIS) and multistage two impinging streams (TIS) reactors were investigated as efficient tools for effecting gas-solids operations. Some aspects of their hydrodynamics and heating and mixing performance are discussed. The FIS reador is shown superior to conventional and one-stage TIS reactors for solids mixing, and its solids drying performance is shown to improve for increased solids load. The parameters obtained for a stochastic model, employed to describe t h e particle residence time distribution (RTD) in multistage TIS reactors, are related t o reactor operation parameters. The single-stage two impinging streams (TIS) reactor for performing heat- and mass-transfer operations between solid particles and a gaseous phase was first suggested by Elperin (1961) and was subsequently modified by Elperin and Tamir (1982). The main feature of the reactor is the impingement of two gas-particle streams, flowing coaxially in opposing directions. The reactor has been thoroughly investigated by Tamir et ai. (1984, 1985), Luzzatto et al. 'This paper is dedicated to Professor I. Elperin, who passed away on March 13, 1983.
0888-5885/8~/2626-2454$01.50/0
(1984), and Tamir and Luzzatto (1985). A comprehensive review on applications of impinging streams in chemical engineering has been recently presented by Tamir and Kitron (1987). The reactor promises the following advantages over conventional reactors, such as fluidized-bed, spouted-bed, or cyclone-type reactors: (a) high relative velocities between the gas phase and the particles, enhancing considerably convective heat- and mass-transfer rates; (b) the specific energy for solids mixing is in the order of 0.5 kJ/kg, which is much lower than that consumed by batch fluidized beds, 0 1987 American Chemical Society
