I n d . E n g . Chem. R e s . 1987, 26, 1365-1372
than the presently available prediction methods and is applicable to a significantly larger number of compounds. It also offers the advantage of a single method applicable to all families of compounds. The proposed method cannot be used, however, for compounds whose molecules contain a single polyvalent atom. This should not be considered as a drawback of the method, as such compounds are usually common compounds for which extensive data compilations are available. Within a particular family or homologous series, the errors for the lowest members are usually higher than the average errors for the rest of the members. For these compounds also there are usually ample data available, and it is not necessary to use any prediction method. Conclusions A second-order group contribution method has been proposed for the prediction of liquid thermal conductivity of organic compounds. The groups utilized are consistent with previously determined methods for ideal gas heat capacity, ideal gas entropy, heat of formation and combustion (Benson, 19761, second virial coefficient (McCann and Danner, 1984), and critical temperature and pressure (Jalowka and Daubert, 1986). The method compares favorably with the most accurate prediction method available (Baroncini et al., 1981), which is not applicable to all families of compounds. It is considerably more accurate than the best available method applicable to all families (Missenard, 1965). The application of the method is fairly simple, depending only upon the availability of group contribution values. The only other input parameter required is the critical temperature, which if unknown can be estimated to a fair degree of accuracy (Danner and Daubert, 1983). Acknowledgment
1365
Physical Property Data of the American Institute of Chemical Engineers. Nomenclature A , B = constants in eq 2, W/(m K) AA, AB = contribution of a group to the constants A and B, w/(m K) k = liquid thermal conductivity, W/(m K) T , = critical temperature T, = reduced temperature Literature Cited Baroncini, C.; Di Fillipo, P.; Latini, G.; Pacetti, M. Int. J . Thermophys. 1981,2,21. Benson, S.W. Thermochemical Kinetics, 2nd ed.; Wiley: New York, 1976. Benson, S. W.; Buss, J. H. J. Chem. Phys. 1958,29,546. Danner, R. P.; Daubert, T. E. Manual for Predicting Chemical Process Design Data: Data Prediction Manual; Design Institute for Physical Property Data, American Institute of Chemcial Engineers: New York, 1983; Procedures 2A-2C. Jalowka, J. W.: Daubert, T. E. Ind. Enp. - Chem. Process Des. Deu. 1986,25,139. McCann, D. W.; Danner, R. P. Ind. Eng. Chem. Process Des. Deu. 1984,23,529. Missenard, F.A. Conductivite Thermique des Solides, Liquides, gaz et de Leurs Melanges; Editions Eyrolles: Paris, 1965. Missenard, F. A. C. R. Acad. Sci., Ser. 3 1965,260,5521. More, J. J.; Garbow, B. S.; Hillstrom, K. E. User Guide for Y I N PACK-1; Argonne National Laboratory: Argonne, IL 1980. Nagvekar, M., “A Group Contribution Method for Liquid Thermal Conductivity”, M.S. Thesis, The Pennsylvania State University, University Park, 1984. Nagvekar, M.; Daubert, T. E.; Danner, R. P. Documentation of the Basis for Selection of the Contents of Chapter 9 Thermal Conductivity i n Manual for Predicting Chemical Process Design Data; Design Institute for Physical Property Data, American Institute of Chemical Engineers: New York.’ 1984: Table 7. Reidel, L. Chem. Ing. Tech. 1951a,23,59. Reidel, L. Chem. Ing. Tech. 1951b,23,321. Reidel, L. Chem. Ing. Tech. 1951c,23,465.
During the course of this work, M. Nagvekar was supported by funds provided by the Design Institute for
Received f o r review March 25, 1986 Accepted April 14, 1987
Newtonian and Inelastic Non-Newtonian Flow across Tube Banks Om Prakash, S.N. Gupta,* and P. Mishrat D e p a r t m e n t of Mechanical Engineering, I n s t i t u t e o f Technology, Banaras H i n d u University, Varanasi 221 005, I n d i a
Newtonian and inelastic non-Newtonian fluid flows across tube banks are analyzed by using two parallel-plate channel models. A simple parallel-plate channel model correlates the present and previous literature data on non-Newtonian fluids successfully, giving a single-valued correlation independent of tube spacings and the fluid rheology. Separate correlations are developed for staggered and inline arrangements. A comparison of a second model, the periodically converging-diverging channel flow, with experimental data shows that the flow does not expand from a minimum clearance t o the maximum possible available space between tubes due t o interaction of flow between them. It is further observed that the flow behavior index, n,has a marked effect on expansion ratio, D,/D,; the higher the pseudo-plasticity of the fluid, the higher is the expansion. 1. Introduction Most of the earlier attempts to correlate the friction factor and Reynolds number of Newtonian fluids flowing across tube banks are based on the conventional model
* Author t o whom correspondence should be addressed. Department of Chemical Engineering, Institute of Technology, Banaras Hindu University.
employing an equivalent diameter and the maximum velocity at the minimum cross section. Chilton and Genereaux (1933) presented pressure drop vs. Reynolds number data for inline and staggered tube arrangements and correlated the transition and turbulent region data using minimum clearange between two adjacent tubes as the characteristic length. In the turbulent region, the exponent on Reynolds number was found to
oasa-~ss~/s~/~~~~-0 i ~1987 ~ ~American $ o 1 . ~Chemical o / o Society
1366 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
be -0.2 for both the staggered and inline tube arrangements, whereas laminar flow data through inline arrangement gave -1.0. Pierson’s (1937) data showed a significant effect of the tube spacing on the pressure drop, and Huge’s (1936) results were consistent with those of Pierson (1937). Grimison (1937) presented a graphical correlation of friction factor, while Guntur and Shaw (1945) proposed a single correlation for all the tube layout configurations and spacing. Boucher and Lapple (1948) recommended the correlation presented by Grimison (1937) and Chilton and Genereaux (1933) and commented that the use of equivalent diameter as proposed by Guntur and Shaw (1945) should not correlate the data from geometrically different tube banks into a single curve. Bergelin and his co-workers (1949, 1950a,b, 1952), in an experimental study of pressure drop and heat transfer in tube banks of varying geometrical configurations, used volumetric mean diameter and also equivalent diameter to correlate their data. Laminar to turbulent region was found to be smooth for staggered arrangements, but a discontinuity was observed for inline arrangements. Hughmark (1972) and Whitaker (1972, 1976) have used capillary flow model to correlate the pressure drop data. Zukauskas (1972) pointed out that in staggered banks, the flow is comparable with the flow in curved channels of periodically converging and diverging cross sections and the velocity distribution in different rows in a staggered bank has a similar character, whereas the flow in an inline tube bank is sometimes comparable with that in a straight channel. Vossoughi and Seyer (1974) suggested that the flow situation in a staggered tube arrangement will be closely identical with that in a packed bed. Cruzan (1964) and Adams and Bell (1968) presented the correlations using a modified Reed-Metzner Reynolds number for tube banks of different geometries. The use of Reed-Metzner Reynolds number for correlating cross-flow pressure drop is unrealistic as the fluid streams flowing through tube banks can be seen to be more similar to the flow through a large number of parallel plate channels rather than flowing through a bundle of capillary tubes. The capillary flow model has been successfully used to predict pressure drop across pipes or beds of particles in packed or expanded state by assuming the bed consists of a large number of capillary tubes having diameters equal to the hydraulic diameter of the bed. Similarly, the flow across the tube bank can also be assumed to consist of the flow through a number of parallel-plate channels having a hydraulic diameter equal to that of the tube bundle. The flow across a tube bank can be assumed to be divided into different parallel streams (similar to flow through parallel-plate channel) flowing around the surface of the tube and mixing with one another in the interstitial space downstream of the tubes. This process will be repeated for all the tubes encountered by the fluid stream. Therefore, the whole situation can be visualized as a number of parallel streams having tortuous paths. This situation is more realistic during laminar flow where contraction and expansion losses are negligible. The length of the tortuous path will be different for different tube arrangements and tube diameters. The present analysis is, therefore, based upon the parallel-plate channel model as described above, and later an attempt is made to analyze the flow by applying a converging and diverging parallel-plate channel model. 2. Theoretical Development 2.1. Parallel-Plate Channel Model. For a power law fluid obeying
r = K ( g ) flowing laminarly through a parallel-plate channel having gap b between the plates, the shear stress-shear rate relation is
For the tube bundle as a whole, the mean shear stress at the wall is rw
= rH(
z) -AP
(3)
where rH =
DOC 4(1 - E )
(4)
~
where t is the void fraction and ,6 is the tortousity factor. Based on rn and the realtionship U = U s / € the , friction factor is
and Reynolds number is
Applying the parallel-plate channel flow relationship, using the above Reynolds number, and defining the friction factor as
(7) it can be shown that 1
24
afmrr = Rem”
1’
For laminar flow in the parallel-plate channel
r w = K r r [12U*(1- t ) Dot2
Therefore,
Rem” =
DOnUs2-np
( p ).-lo
K”(12)n-l -
(10) -
E)
2.2. Converging-Diverging Parallel-PlateChannel Model. The flow situation accounted for in the parallel-plate channel model is closer to that existing in an inline tube bank. In the case of a staggered tube bank, the flow path is tortuous and cannot be described by such an over-simplified model. For this situation, a convergingdiverging parallel-plate channel model will be more appropriate. It is difficult to analyze the periodically contracting and expanding plate channel because of the variation of velocity profile and the wall shear stress in the direction of flow (Batra et al., 1970). Assuming that the channel top and bottom curved edges, as looking from the front in the
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1367 where n is the flow behavior index and K" is the consistency constant.
Equations 17 and 19 give
aP ax Therefore, the pressure drop for one unit length xo is
Figure 1. Periodically contracting and expanding plate channel.
direction of flow, as shown in Figure 1,are sinusoidal in nature and in the equation form, it can be written as Y = a sin (cx) (11)
Substituting the value of DH from eq 16 in eq 21, we get AP1=
4Kf'(
(1- b)2n+' dx ~
)
n
~
x
(2D1)2n+1[1 + b sin
where a and c are constants. If the boundary condition for one cycle length xo of the channel is applied, eq 11 can be reduced to 4
Let us assume that the condition at any section of converging and diverging plate channel is comparable to a parallel-plate channel of hydraulic diameter DH; that means the pressure gradient at a point on the central axis of the converging and diverging channel at axial position under consideration will be the same as the pressure gradient in the central axis of a parallel-plate channel whose hydraulic diameter is the same as that at the corresponding cross section a t the point of consideration in the converging-diverging channel. From Figure 1 it is obvious that D _ -- Y + a + ,D1 L
This relationship can readily be written as
"-[
D= l-b
1 + b sin
(E).]
(14)
Hydraulic diameter at position x = DH = 20. Substituting the value of D from eq 14, we get
3[ 1 + b sin
(22)
($)XI
4Kffx0(
(1 - b)2n+1 1 + n(2n - l ) b 2 / 2 (2D1)2n+1 (1- bZ)Zn+1/2
2)
Now the total pressure drop for the total length L of the parallel-plate channel is
L AP= -AF1 XO
or
Substituting the value of b from eq 15 in terms of D, and D2 and simplifying it, we get (D2 + D1)2n-2 (2 - n - 2n2) x (DzD1)2n+1/2 2 n - 2n2 D,2 + 2D1D2 + D?] (25) 2 - n + 2n2
+
where
DH = I - b
($)x]'~+'
The solution of the above equation gives AP1 =
Y = -(D2 1 - Dl) sin
L
a
(16)
Equation 25 can be expressed in terms of friction factor and Reynolds number relationship for three following positions: (i) at the minimum cross-section area U = Ul and DH = 2 0 , fl
Volumetric flow rate is
=
h(Dl/D2, n)
(26)
Re1
where where 2, is the depth of the parallel-plate channel. Therefore, the average velocity is
The relationship between the wall shear stress and the hydraulic diameter of a parallel-plate channel is given by
.l(3,n ) 0 2
3(2 - n =
+ 2n2)( Z ) ' I 2 (
1+
22(n-1)
(ii) at the maximum cross-sectional area U = U , and DH = 2D2
1368 Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987
where
(iii) a t a position where average velocity U is at the average value of D1 and D2 and hydraulic diameter, DH, is based on the mean value of D1 and D2, that is, DH = (01+ D2)
E - T u b e bank
W
F Monmrter G. ~ o o l ~ n g y-
D-Calmrng
d~(Di/Dz,n) Re
f =
- Reservoir - Pump C - Weighing b a l a n c e
A
(30)
-
section T-Thermometer
H-Hot woter
I-Immersion heaters attached with thermostot
Figure 2. Schematic diagram of the experimental rig.
where
'( ') ':
(1
+
+
= 3(2 -24n-1 n 2n2)( 1 + 1+
;)n-2[
2-n
ill)'/' -
X
+ 272' (31)
Equation 31 can be written in terms of superficial velocity, Us,of fluids flowing across tube banks. The number of channels in a tube bank, n,, will be equal to W / p , where W is the width of the tube bank and p is the pitch, and total volumetric flow rate of fluid (QJ flowing across the tube bank will be equal to n, times the volumetric flow rate through one channel ( Q ) , that is, Qt
= Qn, = Q W / P
or
Q
=
~ J P
(32)
As discussed in the previous section, the flow through a tube bank may be considered as the flow through a bundle of converging-diverging parallel-plate channel with a tortuous path factor, P. The general eq 31 relating friction factor and Reynolds number based on average hydraulic diameter of a convergent-divergent plate channel can be used to predict the friction factor-Reynolds number relationship for tube banks by replacing the friction factor and Reynolds number defined by eq 7 and 6, respectively. The actual fluid path length may be taken equal to 6 times the length of the tube bank in the direction of flow. Replacing f and Re in eq 30 by the modified friction factor (f,") and modified Reynolds number (Re,") from eq 7 and 6, respectively and taking the tortuous path factor, P, into consideration, we can write the generalized friction factor-Reynolds number relationship for laminar flow through a tube bank as (33) The applicability of the above equation will be discussed below. 3. Experimental Section The experimental rig used in the present study is shown in Figure 2. The test section was an ideal tube bank having 110 copper tubes of 0.64-cm i.d., 0.95-cm o.d., and 15.24-cm length arranged in staggered triangular fashion having p or SI = 1.43 cm, SL= 1.239 cm, and p / D o = 1.5. The width and length of the tube bank were 15 and 12.01 cm, respectively. The number of longitudinal and tran-
102
eu 0
10
10
Figure 3. Shear stress vs. shear rate for 3% CMC-B solution. Table I. Range of Parameters Covered parameter range fluid water, 1-270 CMC-A, 3-4% CMC-B 20-58 temperature, "C K , g/(cm@") 0.01-20.59 n 0.56-1.00 Re," 0.10-15000
sverse rows were 11and 10, respectively. The total number of exposed tubes was 105. The test fluid was allowed to flow crosswise outside the tubes. Mercury in glass and thermocouples were used to measure temperatures; the pressure drop was measured with the help of manometers, having least count of h0.5 mm. Both Newtonian (water) and non-Newtonian fluids (l%, 1.5%,and 2% CMC-A and 3% and 4% CMC-B) were used as test fluids. The rheological properties of the fluids were measured by a capillary tube viscometer. The viscometer data in terms of shear stress-shear rate diagrams were plotted on a log-log graph as shown in Figure 3 to determine the values of the consistency constant ( K ) and flow behavior index (n). There was no effect of the temperature on the flow behavior index, and the values obtained for 1%,1.5%, and 2% CMC-A and 3% and 4% CMC-B were obtained as 0.73, 0.64, 0.56,0.77, and 0.61, respectively. The flow behavior index of the fluid and the consistency constant decreased with an increase in the temperature and increased with an increase in concentration. The range of the parameters covered in the present work is shown in Table I.
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1369
1.5Y.CMC-A WXCMC-A 4.0% CMC - B
A
-do-
3OI.CMC-8
0.77
-do -do-
TAP WATER
1.00
-do-
I
Thick
0
Trranwlar
125
011
t
BERGELIN
A
* A *A
A
'
n
A
0
4
0
.. . . 0
.
0..
01
.
I
1
I l l 1 1 1 1 1
I
I
I 1 1 1 1 1 1 1
I
I 1 1 1 1 1 1
102
101
'
I
I I I I I
10 4
103
R;,
Figure 6. Plot of Reynolds number vs. friction factor for triangular tube arrangements ( p / D , = 1.25).
10.0;
:
/m
\mu;
Pfb,
SYMBOL FLUID 0 Thick otl 0 lXCMC
1.25 125
-do-
ADAMS
15Y.CMC
1.25
-do-
ADAMS
TUBE ARRANGEMENT AUTHOR
S t a g g r r r d rquor.
BERGELIN
A
0.1
01
101
102
103
R'e m
a1 0 ,
,
I 1 1 1 1 ' 1 1
I
I
I I 1 ' 3 1 1
I
t
I I ' " ' 1
I
102
IO la
I
l''lIll
103
I
I 1 1 1 ' 1 1 1
I
'
Figure 7. Plot of Reynolds number vs. friction factor for staggered square tube arrangements @/Do= 1.25).
IO'
R;rn
Figure 5. Plot of Reynolds number vs. friction factor for triangular tube arrangements @/Do= 1.5).
4. Results a n d Discussion
It is observed from Figure 4 that pressure drop is more for fluids having a higher consistency index. None of these sets of data show an abrupt change in pressure drop or a sharp transition from laminar to turbulent flow. 4.1. Comparison w i t h Parallel-Plate Channel Model. It has been pointed out earlier that in defining the dimensionless parameters most of the previous workers have ignored the actual flow pattern prevailing in the tube bank. In an attempt to correlate the friction factor and Reynolds number assuming the flow to be similar to that through a parallel-plate channel having the spacing equal to the minimum clearance between the tubes in the tube bank, the characteristic diameter, De,, was taken as equal to 2(p -Do); the friction factor and Reynolds number were calculated for both Newtonian and non-Newtonian fluids and were plotted. These data did not fall on a single curve even for the same tube bank. In order to obtain a general correlation for both Newtonian and non-Newtonian fluids, ",f is plotted against Re," for triangular tube banks of p / D o = 1.5 and 1.25 in Figures 5 and 6, respectively, and for staggered square tube bank of p / D o = 1.25 in Figure 7. The data shown in Figure 5 cover the Reynolds number range from 0.16 to 1.5 X lo4 and flow behavior index range from 0.56 to 1.0. A t low Reynolds number, f," is seen to vary, as Re,"-l.
It is observed that the flow behavior index, n,has no effect on these plots. The slope of the probable curve gradually decreases as Reynolds number increases, and finally it seems to be approaching a constant value. If the total resistance is considered to be consisting of viscous and form resistances, we can write where a and b depend on the void fraction and geometry of the tube bank. The first term in the above equation represents the viscous part and second the form resistance. In terms of friction factor and Reynolds number, the above equation can be transformed as f,"
= A +E
Re,"
(35)
The data for both Newtonian and non-Newtonian fluids fall on a single line without showing any effect on flow behavior index. A regression analysis of all the data of p / D o = 1.5 gave the values of a = 65 and b = 0.35. Equation 35 thus takes the form fm 11 =
65 + 0.35
Re,"
(36)
The data of Bergelin et al. (1952) for thick oil (n = 1) and those of Adams (1968) for 0.5% and 1% CMC solutions flowing across triangular tube arrangements (p/ Do = 1.25) are compared with eq 36 in Figure 6. The data of Bergelin et al. (1952) and Adams (1968) for staggered
Table 11. Values of P and Of for Different Tube Arrangements tube arrangement 1. triangular 2. staggered square 3. inline 4. inline
100
I
I
-
SYMBOL FLUID
p/Oo
Thick oil
150
0
JUBEARRANGEMENT In line squore
I
AUJHOR BERGELIN
t 01
PlDo
P
Pf
1.25, 1.5 1.25 1.25 1.5
1.57 (7r/2) 1.57 1.00 1.00
1.72 1.72 1.46 1.25
(1979))and Tandon (1976)for triangular tube arrangement @/Do = 1.5) with a mean deviation of f14%, and the data of Bergelin et al. (1952) and Adams (1968) for staggered square tube arrangement ( p / D o= 1.25) with a mean deviation of k570. Equation 37 correlates the data of Bergelin et al. (1952) and Adams (1968) for inline tube arrangement ( p / D , = 1.25) with a mean deviation of f16%, and eq 38 correlates the data of Bergelin et al. (1952) for inline tube arrangement ( p / D o = 1.5) with a mean deviation of &7 70. Comparing eq 8 and 35, we find that A = 246, where p is the tortuous path factor. Equations 36-38 show that if parallel-plate plane model is applicable, 24p = 65 for all staggered (triangular or square) tube arrangements and 24p = 35 and 30 for inline tube arrangements of p / D o = 1.25 and 1.5, respectively. It is interesting to note that the tortuous path facotr, p, should be almost equal to unity for inline tube arrangements, and it should have a value aroung ?r/2 for staggered tube arrangements. But by applying the parallel-plate plane model, we find that p = 2.7 for staggered tube arrangements and p = 1.46 and 1.25 for inline tube arrangements of p / D o = 1.25 and 1.5, respectively. The increase in experimental value appears to be due to (i) the possibility of converging and diverging nature of the main stream of the fluid and (ii) form resistance caused by circulating eddies on the rear portion of the tubes. In this approach, if converging and diverging fluid streams are ignored and the increase in drag is considered to be only due to the form resistance, we can modify eq 8 as
I 101
102
103
104
R;rn
Figure 9. Plot of Reynolds number vs. friction factor for inline tube arrangements @/Do= 1.50).
square tube arrangement ( p / D o = 1.25) are shown in Figure 7. It is interesting to note that for both staggered square and triangular tube arrangements, the correlation of modified friction factor c f , ” ) with modified Reynolds number (Re,”) is the same. The data of Bergelin et al. (1952) on the flow of thick oil through inline arrangement ( p / D o = 1.25) are shown in Figure 8, and Adams’ (1968) data for 0.5% and 1% CMC and Bergelin et al.’s (1952) data for thick oil flowing through inline tube arrangements ( p / D o = 1.5) are in Figure 9. Attempts to correlate all the inline data by a single-valued correlation were not successful. The regression analysis of these data gave the correlation for p / D o = 1.25 f,”
=35,, + 0.42
(37)
f’ = 30 + 0.25
(38)
Rem
for p / D , = 1.5 f m
Re,”
Equations 37 and 38 differ from eq 36 only in the values of the constants. It is further observed that constants for higher p / D o (1.5)are lower than those of lower p / D o (1.25). Equation 36 correlates the experimental data of Adams (1968) and Bergelin et al. (1952) with a mean deviation of f7% for triangular tube arrangement @ / D o = 1.25), the present data and the data of Bergelin et al. (19521, Chand
(39) where Pf is the form factor and is determined experimentally. The values of p and Pf are given in Table 11. Comparing the values of Pf for various inline arrangements, it is found that the form factor or form resistance is much more for p / D o = 1.25 than that for p / D o = 1.5. It is further noted that Pf for staggered tube arrangements remains the same for p / D o = 1.25-1.5. In the case of staggered tube arrangements, the value of should not have such a higher value compared to that of inline tube arrangements. The position of impact points and pattein of flow past a tube in a longitudinal row in an inline tube bank was investigated by Zukauskas (1968) and Kostic (1969). They presented a qualitative picture of the pressure distribution along the perimeter of the subsequent tubes. The general flow pattern and the intensity of the wake formation change with the decrease in distance between the tubes in the flow direction. As the longitudinal pitch decreases, the velocity profiles are more and more straightened and the concave shapes in the curve diminish accordingly. At the higher longitudinal pitch, convergence and divergence of the main stream are much more than that at lower longitudinal pitches. This results in a decrease in Pr values with increasing p / D o values. 4.2. Comparison with Converging-Diverging Plate Channel Model. The laminar flow of Newtonian and non-Newtonian fluids flowing through a periodically con-
Ind. Eng. Chem. Res., Vol. 26, No. 7, 1987 1371
’
O
0
m
90-
-
80
-
70
-
. -s C
$
60-
50-
8
40
-
30-
- 0 2 0
0.2
0.4
D2
0.6
0.8
1.0
the tubes in the succeeding rows. It is further observed that the flow behavior index has a marked effect on expansion ratio ( D z / D l ) . The higher the pseudo-plasticity of the fluid, the higher is the expansion. Turbulent analysis for converging and diverging plate channel shows that the friction factor should vary as Re-lf4 but in a tube bank it is observed that the modified friction factor approaches a constant value a t high Reynolds number. In turbulent flow region, the tube bank appears to behave similar to the converging-divergingplate channel with a high degree of roughness. There is marked transition in inline arrangements (Re,” = 100-300), whereas in staggered arrangements such a transition region cannot be easily delineated. The value of constant b is obtained (b = 0.42 for p / D o = 1.25 and 0.25 for p / D o = 1.5). This shows that inline arrangements with lower p / D o ratios behave analogous to flow through rough channels with higher relative roughness.
Dl
Figure 10. Variation of d(D1f Dz, n) with Dl/Dz and n. Table 111. Experimental Values of $ ( D 1 / D zn, ) eor Various Tube Arrangements tube arrangement plDo Pd(Dl/D2,n) P $(D1/Dz,n) inline 1.25 35 1.00 35.0 inline 1.50 30 1.00 30.0 65 1.57 41.4 staggered square 1.25 triangular 1.25, 1.50 65 1.57 41.4
verging and diverging plate channel was analyzed in the previous section. Friction factor-Reynolds number relationships, eq 26,28, and 30, were derived for the positions a t the minimum, maximum, and average cross-sectional areas, respectively. Equations 26, 28, and 30 reduce for Newtonian fluids, substituting only n = 1. D1/Dzis plotted against h ( D l / D Z , 4,$ J Z ( D , / D Z4, , and q W / D z , n) in Figure 10 for various values of n. From this figure we find that values of $l(D1/Dz,n) decrease steeply with decreasing D 1 / D zand marginally with n,whereas those of &(Dl/Dz, n) increase sharply with decreasing D l / D z and decrease sharply with decreasing n. These are contrary to the trend shown by the actual data as well requirements of the model. This indicates that both of these functions do not correspond to the effective flow mechanism in the tube bank. The values of $(D1/Dz,n),on the other hand, show a gradual increase with decreasing D l / D z and with decreasing n. This agrees with the assumption of the model and trend by the data. The values of $(D1/D2,n) and eq 30 are thus of prime importance in this context. Because of this fact, various dimensionless groups are expressed in terms of average quantities or a t a position of mean hydraulic radius. If eq 33 is compared with eq 36-38, values of j3q5(D1/D2,n) can be evaluated. These are listed in Table 111. As discussed in the previous section, if p is taken to be equal to 1 for inline arrangements and 1.57 for staggered arrangements, the values of 4(D1/Dz,n) were recalculated and are presented in Table 111. From the experimental values of $(D,/Dz, n) as presented in Table I11 and by use of eq 30, experimental values of D1/Dzwere calculated. In a tube bank the minimum cross-sectional area equals p Do, and the maximum possible cross-sectional area equal p per unit depth of the tube bank. The experimental value of D 2 / D 1is compared with the maximum possible value p / ( p - Do). It is observed that the maximum possible values of ratio D 2 / D 1are much higher than the experimental value of Dz/D1. This means that the fluid does not expand from a minimum dimension (p - Do)to a maximum possible available space, i.e., p , due to interaction of
5. Conclusions The use of maximum velocity a t the minimum flow passage in friction factor and Reynolds number as prevalent in the available literature does not give a single-valued correlation for tube banks of different geometries. The “parallel-plate channel model” correlates the present and previous literature data on non-Newtonian fluids successfully, giving a single-valued correlation. The converging-diverging parallel-plate channel model is more successful in analyzing and correlating the pressure drop data for the flow of both Newtonian and inelastic nonNewtonian fluids across tube banks. According to this model, friction factor is not only a function of Reynolds number but is also a function of minimum to maximum diameter ratio and flow behavior index, n. The expanded fluid stream dimension, D z , is found to increase with a decrease in flow behavior index, n.
Nomenclature b = width of the parallel-plate channel, m D, = equivalent hydraulic diameter based on minimum flow passage in the tube bank, m DH = hydraulic diameter of the tube bank, m Do = outside tube diameter, m D, = minimum spacing of the converging-diverging channel, m D2 = maximum spacing of converging-diverging channel, m du/dy = shear rate, s-l f, = friction factor based on parallel-plate channel models and minimum cross-section areas defined as D,,APg,p/ 2LGm2 f,” = modified friction factor based on parallel-plate channel model and average hydraulic diameter defined as DohPg,p~~/2LG,2( 1 - C) G = mean mass velocity, kg/(h.m2) G, = maximum mass velocity, kg/(h.m2) g, = conversion factor K = power law consistency constant, kg/ (m@) K” = generalized power law consistency constant based on parallel-plate channel model, kg/ (m.s2-n) L = length of the tube bank, m n = power law fluid flow behavior index P = absolute pressure, kg f/m2 AP = differential pressure drop, kg f / m 2 p = transverse pitch, m Q = volumetric flow rate, CU m/s rH = hydraulic radius, m Reo = Reynolds number based on parallel-plate channel model and minimum flow passage defined as DeoUmp/K”(~w/ K!q(n-l)/n --
Re,” = modified Reynolds number based on parallel-plate channel model and average flow passage defined as D ~ u K~”( ~T w // K ” ) ( n - l ) / n (1 -
Ind. Eng. Chem. Res. 1987, 26, 1372-1381
1372
U = average velocity, m/s Urn= maximum velocity, m/s U , = superficial velocity, m/s W = width of the tube bank, m Y = distance of the elementary strip from the entrance, m 2 = height of the tube bank, m Greek Symbols (3 = tortuous path factor
= viscosity of the fluid, CP = density, kg/m3 7 = shear stress, kg f/m2 T~ = shear stress at wall, kg f/m2 A = difference t = void fraction
p p
Literature Cited Adams, D. Ph.D. Dissertation, Oklahoma State University, Stillwater, 1968. Adams, D.; Bell, K. J. Chern. Eng. Prog. Syrnp. Ser. 1968, 64(82), 133. Batra, V. K.: Fulford, G. D.: Dullien, F. A. L. Can. J . Chem. Eng. 1970, 48, 622. Bergelin, 0. P.; Brown, G. A.; Hull, H. L.; Sullivan, F. W. Trans. ASME 1950a. 72. 881. Bergelin, 0. P.; Brown, G. A.; Doberstein, S. C. Trans. ASME 1952, 74, 953.
Bergelin, 0. P.; Colburn, T.; Hull, H. L. Engineering Experimental Station Bulletin 2, 1950b; University of Delaware, Newark. Bergelin, 0. P.; Davis, E. S.; Hull, H. L. Trans. ASMe 1949, 71,369. Boucher, D. F.; Lapple, G. F. Chem. Eng. Prog. 1948,44, 117. Chand, P. M. Tech. Dissertation, Banaras Hindu University, Varanasi, India, 1979. Chilton, T. H.; Genereaux, R. P. J. AIChE 1933, 29, 161. Cruzen, C. G. M.S. Thesis, Oklahoma State University, Stillwater, 1964. Grimison, E. D. Trans. ASME 1937, 59, 583. Guntur, A. Y.; Shaw, W. A. Trans. ASME 1945, 67, 643. Huge, E. S. Trans. ASME 1936, 59, 573. Hughmark, G. A. J . AIChE 1972, 18, 1020. Kostic, Z. Presented at the International Seminar, Hercerg-Novi, Yugoslavia, 1969. Pierson, 0. L. Trans. ASME 1937, 50, 563. Tandon, S. K. M. Tech. Dissertation. Banaras Hindu University, Varanasi, India, 1976. Vossoughi, S.; Seyer, F. A. Can. J . Chem. Eng. 1974, 52, 666. Whitaker, S. J . AlChE 1972, 18, 361. Whitaker, S. Elementary Heat Transfer Analysis; Pergamon: New York, 1976. Zukauskas, A. Advances in Heat Transfer; Academic: New York, London, 1972; Vol. 8, p 93. Zukauskas, A.; Makarvicius, V.; Shalncianskas, Mintis, Vilnius, Lithuania, 1968. Received for review March 27, 1986 Accepted February 24, 1987
A Modified UNIFAC Model. 1. Prediction of VLE, hE, and y m U l r i c h Weidlicht and Jiirgen Gmehling" Lehrstuhl f u r Technische Chemie B, Universitat Dortmund, 0-4600 Dortmund 50, FRG
A modified UNIFAC model (mod. UNIFAC) has been developed which differs from the original UNIFAC method (orig. UNIFAC) in that it has a different combinatorial part and that temperature-dependent group interaction parameters and different van der Waals quantities have been introduced. The parameters were fitted simultaneously t o a large number of VLE, ym,and hE data by using the Dortmund Data Bank. Compared to orig. UNIFAC, the modified model gave a relative improvement of 73% for the calculation of activity coefficients a t infinite dilution, 23% for the calculation of consistent binary vapor-liquid equilibria, and 70% for the calculation of binary excess enthalpies. T h e pure prediction of ternary VLE data gave an improvement of 11% , that of ternary hE data a n increase in exactness of ca. 78% (in each case compared to orig. UNIFAC). 1. Introduction The most successful methods presently used for calculation of activity coefficients in the liquid phase are the group contribution methods, in which the liquid phase is considered to be a mixture of structural groups. This has the great advantage that any system of technical interest can be calculated by using a relatively small number of parameters which describe the interactions between the structural groups. The best-known and most successful of the group contribution methods so far proposed is the UNIFAC (UNIQUAC functional group activity coefficients) model (Fredenslund et al., 1977). It has already been used successfully in many areas (Gmehling, 1982), e.g., (1)for calculating vapor-liquid equilibria (Fredenslund et al., 1977), (2) for calculating liquid-liquid equilibria (Magnussen et al., 1981), (3) for calculating solid-liquid equilibria (Gmehling et al., 1978), (4) for determining activities in polymer solutions (Oishi and Prausnitz, 1978; Present address: Huls AG, ZB FE-Zentrale/Verfahrenstechnik, D-4370 Marl, FRG. 0888-5885/87/2626-1372$01.50/0
Gottlieb and Herskowitz, 1981), (5) for determining vapor pressures of pure components (Jensen et al., 1981), (6) for determining the influence of solvent on reaction rate (Gmehling and Fellensiek, 1980; Lo and Paulaitis, 1981), ( 7 ) for determining flash points of solvent mixtures (Gmehling and Rasmussen, 1982), (8) for determining solubilities of gases (Nocon et al., 1983; Sander et al., 1983). Many comparisons made using experimental data for various combinations of substances have shown that the UNIFAC method is particularly well suited for calculating vapor-liquid equilibria. However, various publications (Kikic et al., 1980;Thomas and Eckert, 1984) have made clear that the results obtained for the calculation of activity coefficients a t infinite dilution are in most cases unsatisfactory especially when systems with molecules very different in size are considered. However, the exact knowledge of this quantity is particularly important for separation techniques, since the number of plates can be particularly high for very dilute systems. The orig. UNIFAC model is also not able to calculate enthalpies of mixing and thus the temperature dependence of the Gibbs excess energy to the required degree of exactness. It was 0 1987 American Chemical Society
