Ind. Eng. Chem. Res. 1992,31, 736-743
736
Transport Phenomena Analysis of Through Drying Paper Osman Polat,* Reinhold H. Crotogino, and W. J. Murray Douglas Chemical Engineering Department and Pulp and Paper Research Centre, McGill University, Montreal, Canada
The rate of through drying paper in the constant rate drying period was measured for 210 combinations of temperature and throughflow rate of air, basis weight, and initial moisture content of paper. The volumetric heat and mass transfer was represented by use of a characteristic dimension, determined simultaneously from a new momentum transport treatment for drying air flow through a moist sheet. For the Sherwood-Reynolds number correlation, the complex Reynolds number dependency is documented. For paper heavier than that which is through dried industrially, the Sherwood number at high Reynolds numbers approaches independence from the paper thickness. For thin paper, end effects predominate. The results suggest that, during constant rate through drying of paper, the local drying rate may be increasing in some regions while it is already in the falling rate period in other regions.
Introduction The rate of drying sheets by throughflow may be expressed relative to either the sheet external area or, as a volumetric transport process, to the sheet volume or mass. As drying rates normalized to sheet area relate directly to dryer area and dryer cost, this is the most common choice. However as through drying takes place throughout the wet material, for a fundamental analysis the drying rate must be expressed relative to the amount of material. This standard practice for the rate of mass transfer for flow through packed beds and other porous media is used here for drying within a sheet of wet paper. Of the three periods of through drying paper identified by Polat et al. (1991a)-the increasing rate, constant rate, and falling rate periods-only in the constant rate period does the simplification of steady-state conditions apply. During the increasing rate period there is no quantitative treatment available for the rapid increase in the interfacial area of heat and mass transfer. In the increasing rate and falling rate periods there is the complication of large changes in sheet temperature. The vapor pressure of water changes correspondingly, while it changes with paper moisture content, as well during the falling rate period. Paper shrinkage during the falling rate period affects interfacial area. Thus there are formidable obstacles to obtaining a transport phenomena treatment of the increasing rate and falling rate periods of through drying. The focus here is then on the constant rate period, for which obtaining an integrated treatment of the fluid mechanics and heat- and mass-transfer aspects has not previously been accomplished. Mass Transport Parameters for the Constant Rate Drying Period Mass-Transfer Coefficient. The mass-transfer coefficient for through drying is defined by N = kGpa(Y*- Y) (1) where all terms represent local values within the sheet. Although the preferred driving force is the difference in water vapor partial pressure, at the highest transfer rates of the present study the two driving forces differ, according to the relation of Keey (1972), but only about 1%.Thus absolute humidity difference is used in eq 1. The local *To whom ail correspondence should be addressed. Present address: Winton Hill Technical Center, Procter and Gamble Company, 6105 Center Hill Rd, Cincinnati, OH 45224.
differential mass balance at position z within a sheet being dried by throughflow is G ( d Y / d z )= u ~ N = k@pp,(Y* - Y) (2)
For Y*, up, pa, and kG independent of position within the sheet, and for plug flow, integration of eq 2 across the sheet gives kGuppJ/G = In [(Y* - Y i ) / ( Y *- Yo)]= (Yo - Yi)/AYM (3) The nondimensional mass-transfer coefficient,Sherwood number, may be calculated directly from the number of transfer units for the constant rate period, NG,with the relation NG = kGappaL/G = (k,dp/D)(r/Gdp)(paD/r)upL = [Sh/ ((Re)(Sc))lapL (4) N, Yo,and Y* vary during both the increasing rate and falling rate periods, apvaries during the former, and p a and L vary during the latter because of sheet shrinkage. Thus eq 3 does not apply during the increasing rate or falling rate but may be applied to the period of constant rate drying during which N, p a , L, Yo, and Y* are at steady values and Y* is essentially constant across the sheet. If up for moist paper as a function of paper moisture content were not known, interfacial area could be combined with k G , as is standard practice, to give a volumetric masstransfer coefficient, kGuPpa. For the integration to eq 3 to obtain NG,plug flow was assumed here, as is normally applied, including for through drying by Schliinder et al. (1977). Moreover, as gas-phase axial dispersion for flow through paper has not been measured, there is no alternative to this standard assumption. For the correction to kG for a high mass-transfer rate, the conditions of maximum transfer rate used here give a correction factor (Bird et al., 1960) of 0.98, so this was neglected. Because sheet shrinkage takes place during the falling rate period, L in eq 4 is essentially constant in the paper moisture content range from Xa to XCf,the constant rate period limits. Moist never-dried fibers have a hollow cylindrical shape. As moisture leaves the wall and lumen of such fibers, they collapse, giving the ribbon-like shape of dry fibers (Figure 1). Once collapsed, a fiber never returns to its original cylindrical shape upon rewetting. Laboratory drying studies, the present one and almost all previous investigations, use handsheets formed by rewetting dry pulp. The industrial equivalent is paper made from market pulp. Sheet thickness change during the
0888-58851921263I-0736$03.O0/0 0 1992 American Chemical Society
Ind. Eng. Chem. Res., Vol. 31,No. 3, 1992 737 Table I. Chmteristic Dimension of Paper during Constant Rate Period of Through Drying d p for given Ti("C), lun G,kg/(mzs) 2 3 % 42'C 64OC 88OC dh$m
Ms = 25 g/m2 0.09 0.16 0.29 0.52
39.5 27.9 29.1 21.4
44.0 34.3 39.4 35.9
39.4 41.6 31.1 26.1
39.3 37.4 39.7
40.5 35.3 32.3 30.8
dp
29.5
38.4
34.5
36.5
34.7
0.52
11.4
12.5
13.5
12.0
12.4
dp
11.7
12.6
13.5
12.6
12.6
10.3
9.5
12.2
11.4
10.8
14.0 11.7 9.8 9.9
9.3 9.4 9.3 10.0
-
Figure 1. Scanning electron micrograph of 100 g/mZ paper cross section. (The scale is indicated at the bottom of the photograph by the line which follows the two dashed lines. The length of the line corresponds to 10 pm.)
drying of paper made from the collapsed fibers of dried pulp is much smaller than from never-dried fibers. Thus taking moist paper thickness, L,in eq 4 as that of dry paper, as done here, does not introduce appreciable error. Selection of Characteristic Dimension for Nondimensional Numbers. For a nondimensional analysis of transport phenomena occurring in through drying of a porous medium of the complexity of paper (Figure l),a key aspect is selection of the Characteristic dimension for use in the Reynolds, S h e r w d , Nuaselt, or Peclet number. In the absence of reliable measurements with moist paper, those previous investigators who have used a d, for momentum, heat, or mass transport analysis of flow through paper have based such a dimension on measurements for dry paper. Polat et al. (1989)established this to be an inappropriate approximation. Empirical definitions for a characteristic dimension for flow through paper are reciprocal of specific surface (Robertson and Mason, 1949), square root of permeability (Rajand Emmons, 1975),fiber diameter (Wedel and Chance, 1977),and equivalent diameter from the Hagen-Poiseuille equation (Gummel, 1977). Moreover, all previous work has used the Darcy law approximation which applies for purely viscous flow, ignoring the inertial contribution that Polat (1989)has shown to be a dominant component of a pressure drop for air throughflow at industrially relevant rates for drying. But permeability and specific surface, determined without this error, give k1I2 and l/apvalues in the order of a micrometer whereas Polat et d. (1989)show that the pore size distribution of paper extends to about 50 pm. Thus the empirical definitions of d,,which have been used for Re, Sh, Nu, and Pe are inconsistent, even for dry paper, with the actual pore structure. A new characteristicdimension for flow through paper, dp = @/a,was determined by Polat et al. (1989)from the application of a momentum transport analysis in which the parameters a and @ represent complex integrals determined from theoretical solutions. This theoretically based dpgives values consistent with their pore structure observations by 8cBnninp electron microscopy (SEM). This dp = @/a was successfully applied to a nondimensional friction factorReynolds number treatment of flow through moist paper. They documented the relationship dp = f ( X M B for ) 25 and 50 g of fiber/(m* of paper) (shortened henceforth to g/m2), from wet to dry. Their dp determinations and the mean pore radius measurements of
4
29.7
MB= 150 g/mz 0.09 0.16 0.29 0.52 -
5.8 6.7 9.0 9.2
6.4 9.5 8.8 9.6
11.0 9.9 9.5 11.3
4 7.7 8.6 10.4 11.4 9.5 Bliesner (1964)by the gas-drive technique show that dp for dry paper is independent of basis weight for MB2 50 g/m2. The pore size shown by SEM cross sections is likewise similar for 50,100,and 150 g/m2 sheets. As the best estimate currently possible, therefore, the dp-X r e lationship for 100 and 150 g/m2 kraft paper can he approximated as that determined here for the 50 g/m2 basis weight. This new dp = j(X&fB)relation is now applied in a transport phenomena analysis of through drying paper. Use of these dp-X-MB relations to determine dpfor Re and Sh in the constant rate period of drying evidently requires the paper moistme content. The initial and final critical moisture contents, Xci and X ,which mark the beginning and end of constant rate through drying, were defmed and related to the deterpining variables by Polat et al. (1991a)as Xci = j(Ti,GMB,Xo)and Xc,= j(Ti,Xo). For the dp-X relation, X = (Xci+ Xcr)/2 is used. Thus the dpused in Re and Sh depends directly on MBand X and indirectly, through Xci and X C ,on the conditions of the throughflow air, Ti and G. Each dp value of Table I, for one of the 64 combinations of TrG-MB, was obtained by averaging the dpvalues from 2-4 experiments at various
x,.
The maximum and minimum values of dp during the increasing rate and falling rate periods, respectively, are 44 and 16.7 pm for 25 g/m2 sheets and 14 and 5 pm for 50-150 g/m2 sheets (Polat et al., 1989). Comparison with the Table I values shows that d, for the constant rate drying period is closer to the maximum dpof the inueaaing rate period than to the minimum dp reached during the falling rate period. Previous through drying studies that used dpbased on one of the empirical definitions, i.e. k'I2, l/ap,etc., have therefore reported values of Re, Sh, or Pe between 1 and 2 orders of magnitude too low because of the use of definitions of d too low to that extent. The error from the definition o!d, has been further increased because in all eases the value determined for dry paper was used while
738 Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 Table 11. Transfer Area of Moist Pawr. during the Constant Rate Period. and of Dry P a w r moist paper constant rate period dry paper
-
MBt g/m2 25 50 100 150
dp, pm 34.7 12.6 10.8 9.5
4/&, lo6 m2/m3 0.12 0.32 0.37 0.42
4/&, 106 m2/m3 0.25 0.76 0.80 0.85
up,
10s m2/m3
~
~
~
~
_
_
_
_
L,w
= ap/(4/dp) 3.0 1.4 1.5 1.6
7
0.76 1.04 1.19 1.34
_
58 114 227 350
the present work shows that d, for moist paper is higher by 250-300% than that for the dry sheet. Determination of Transfer Area. Measurement or prediction of interfacial mass-transfer area is a difficult aspect of the determination of the mass-transfer coefficient, kc. Polat (1989) determined the specific surface, up, of dry paper by using the widely accepted Carman-Kozeny equation. The accuracy of such up values is very sensitive With no method to the porosity function, e3/(l available for accurate determination of the e-X relation, no work has been published on up-e-X relations of paper. Thus wet paper up cannot be determined from wet paper
k. If the pores were straight cylinders, i.e. of tortuosity 7 = 1,with pore length equal to the paper thickness, L, then up would be inversely proportional to dp. For straight cylindrical pores, up = (n&&)/(nndp2L/4) = 4/&. Although individual dp and 4/dp values were calculated for each sheet dried, for illustration purposes Table I1 shows only the estimates with the average value, &, from Table I. Dry paper comparative values are shown, with up as correctly determined by Polat (1989) and as estimated from the relation 4/dp. For dry paper, the true up is substantially larger than that given by the 4/dp approximation. Tortuosities, 7 = up/(4/dp),are in the range 1.4-3 for dry paper (Brown, 1950). G o i i from wet to dry paper, 7 will increase because the smallest pores have the highest tortuosity. In the absence of any quantitative basis for the 7-X relation, 7 = 1was assumed for moist paper during the constant rate period of drying, and up correspondingly approximated as up = 4/dP Gummel (19771, in the only previous treatment of through drying paper involving specific surface, used dry paper AP to estimate up, assuming only pin holes are open to flow. Although throughflow is mainly governed by the larger pores or pin holes during the constant rate period, the above up estimate using dry paper AP is erroneous because of an inaccurate surface porosity determination. Moreover, the data in Table I1 indicate that use of up for dry instead of moist paper gives an up value too high by a factor of about 3-6.
Figure 2. Mass-transfer coefficients for through drying of paper: experimental results, MB = 25 g/mZ.
10-1
ea"
-
0-
0 59
:
3 10-8-
&
a
8 0
t t
j
Figure 3. Mass-transfer coefficients for through drying of paper: experimental results, MB = 50 g/m2.
.
1
ll.ll..ll.llll.,
1
I
.1111.1..1....
., .
,
,..
B ~ i aweight 100 g/m' t
10-1
Experimental Section Handsheets made from standard laboratory kraft pulp (unbeaten, unbleached, 100% black spruce, 18-19 kappa number, CSF 685 mL) were through dried in a laboratory dryer at constant mass flow rate of air. Experiments were made for 210 combinations of four parameters, initial moisture content and basis weight of the paper, temperature, and mass flow rate of the throughflow air. Monitoring of througMow exit air humidity, Yo,with a sampling frequency of 5-10 Hz, enabled determination of the drying history of each sheet. The drying facility was described by Polat et al. (1987). The experimental program, and a list of all values of MB,G, Ti,NG,k ~Re, , and Sh,is given by Polat (1989). Because valid Sh-Re data for through drying paper have not been available, these results are shown in Figures 2-5 for all 210 sheets dried in the present study, with the
15
026 di 41
AT,.
'C o 15 11
A
n
26 41
0 58
\
a"
10-4
10-8
lo-' 100 Re, Gdp/p
10'
Figure 4. Mass-transfer coefficients for through drying of paper: experimental results, MB = 100 g/m2.
exclusion of only those few cases where the exiting air was within 1% of saturation, for which the mass-transfer driving force becomes too small for reliable determination
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 739
ATb 'C o 15
-
10-1
15 26 41 59 15 26 41 59 15 26 41 59 15 26 41 59
41
A
b 59
.
a*
25
02!3
n
\
Table 111. Parameters for Sherwood Number-Reynolds Number Correlation a MB,g/mZ ATi, O C n 'Jn OSh
.
3 10-8;
50
100 0
104
''".l....ll.l.d
10-8
'
"
10-1
'
1
-
'
100
150
" '
10'
Re, Gdp/p Figure 5. Mass-transfer Coefficients for through drying of paper: experimental resulta, MB = 150 g/m2.
of the mass-transfer coefficient. The same Re and S h scales are used for all data sets to provide uniform perspective. At the exit of the sheet AY, varied from -1 to -48% of that entering the sheet, AY? Correspondingly, AYM varied from 1 X to 16 X kg of H20/(kg of dry air) (shortened to kg/kg hereafter). For example, for the heaviest paper (150 g/m2) dried with air of the lowest flow rate (G = 0.09 kg (m2s) and lowest ATi (15 "C), AY, is less than 0.1 X 10- kg/kg and AYM is about 1 X kg/kg. Even for thew extreme conditions the experimental scatter in S h is surprisingly small, thus c o n f i i the high precision of the humidity measurements and, thereby, of the reported values of NG,kG, and Sh. Only two investigations (Raj and Emmons, 1975; Gummel, 1977) report mass-transfer coefficients for through drying paper. Both studies are incomplete as to the effect of two important variables, MB and AT? Ftaj and Emmom varied the basis weight from 15 to 65 g/m2, but only at a single, low value of ATi, 15 "C. Gummel varied ATi from -15 to -60 OC, but only for a single basis weight of paper, 20 g/m2. Other shortcomings of these studies are noted subsequently. The results for the 210 combinations of MB-ATi-G-Xo thus cover a wide range of both MB, 25-150 g/m2, and ATi, 14-64 "C, with air throughflow rate, G , varied from 0.09 to 0.52 kg/(m2 s), producing wide variation in AY, and AYM,i.e. (0.05-11) X kg/kg and (1-16) X kg/kg. Thus mass-transfer coefficients for throughflow drying of paper can now be reported for the first time over a wide range for all parameters, Xo-MB-ATi-G-Re-AYi-AYM
/
-
Relations for Sh-Re-MB-ATi General Correlation. The Reynolds number dependence of the nondimensional mass-transfer coefficient is expressed as S h = a(Re") (5) Nondimensional representation as Peclet number, P e = (Re)(Sc),facilitates comparison of systems of different Schmidt number. Conversion to the (Re)(&) form may be made using Sc = 0.61 for air-water vapor. Table I11 gives the eq 5 regression resulta. The relative experimental error, uah/Sh,typically about 25%, reflects the difficulty of such rate measurements. This relative error increases with MB and decreases with ATi because of the associated closer approach to saturation for the throughflow leaving the sheet. Effect of MB and ATi on Reynolds Number Dependency. The strong dependency of n on both basis weight and throughflow inlet temperature driving force,
0.6
-
0.4
-
v
a
2 a a
0.86 0.84 0.47 0.46 0.75 0.54 0.38 0.29 0.89 0.60 0.52 0.46 0.97 0.86 0.95 0.65
0.12 0.09 0.12 0.09 0.09 0.10 0.09 0.09 0.10 0.12 0.07 0.07 0.08 0.1% 0.08 0.13
0.021 0.013 0.017 0.015 0.0020 0.0018 0.0012 0.0010 0.0010 0.0010 0.0006 0.0004 O.ooOo5 0.0010 0.0003 0.0004
i
P
n
2
0.143 0.137 0.101 0.092 0.314 0.0165 0.0134 0.0068 0.0161 0.0070 0.0086 0.0068 0.0081 0.0070 0.0106 0.0068
-
o.2
o.o/
i
A
41
0 59
_____ 100
50
150
Eqn. 6
200
Basis Weight, Mg, g/m2 Figurq6. Effect of basis weight on the Re exponent of the Sh-Re relation.
0 150
__-__Epn. o.oo
(I
20
40
60
80
Temperature Driving Force, AT,, 'C Figure 7. Effect of throughflow inlet temperature driving force on the Re exponent of the Sh-Re relation.
apparent in Table I11 and displayed in Figures 6 and 7, is a central feature of the results. Figure 6 shows that over the 50-150 g/m2 range, as M B increases, n approaches unity regardless of the ATi level. Tissue paper of basis weight 25 g/m2 forms a separate case because paper this thin cannot be treated a0 a packed bed, as elaborated subsequently. Correspondingly, Figure 7 shows that n approaches unity as ATi becomes small. Thus aSh/aRe approaches unity as the packed bed becomes thick and as the mass-transfer driving force becomes small. When the results of various researchers for mass transfer at low Reynolds number in packed beds of small particulates are expressed in the Sh-Re form of eq 5, the Re exponent is
100
r
. . . ..
, ,
1.11......,
.
'
>-""l
' ' ''
AT, = 59%
YS, a b g 26
--_50
10-1
..... c1
\
100 150
__.*
,.---
__I-
//--
.-*-
a^
g 10-8 t
- 4 - 01
10-8
..,'
.I.-
10-1
100
101
Re, Gd,/p Figure 8. Mass-transfer coefficients for through drying of paper: regression results, ATi = 59 O C .
found to be unity (Martin, 1978). For the present results the best-fit correlation for n = f(MB,ATi),consistent with Figures 6 and 7, is n = 0.58 + 0.0037MB - 0.0084ATi
(6)
The expectation is that, for high MB (Figure 6) and low ATi (Figure 7), these lines would curve to become asymptotic to a value of n of approximately unity. Effect of MBon Sherwood Number. For Figure 8, only the eq 5 regression lines are shown, as the data appear on Figures 2-5. The regression lines for the thinnest paper, 25 g/m2, stand clearly separate from those for all heavier grades at the Figure 8 value of ATi and for all ATi measured. When dry,this 25 g/m2 kraft paper is about 58 pm thick. From the Table I values of dp, the 25 g/m2 sheet is thus seen to be only 1.7& deep. Transport processes for a porous media this thin are dominated by entrance and exit effects. For L = 1.7&, the mass transfer is effectively for a screen, not a bed. The acceleration and deceleration of flow with the corresponding sudden contraction and expansion at the entrance and exit of this "screen" account for the extra frictional losses and the enhanced heat- and mars-transfer rates measured here for 25 g/m2 paper. Higher transfer rates in very short sections of whatever geometry are well-documented phenomena. For 50,100, and 150 g/m2 paper, the sheet becomes 9dp, 21dp, and 37dpthick and the end effect contribution would decrease in relative importance. Figure 8 (and similar figures at other values of ATi) indicates some basis weight effect on Sh over the range 50-150 g/m2, presumably becoming minimal for MB > 150 g/m2. Interestingly, for all ATi tested, the Sh-Re regression lines for 50-150 g/m2 paper converge as Re increases. This behavior is consistent with the expectation that, above some higher value of Re, the Sh-Re relation would become independent of MB. The approach of the Reynolds number exponent of Sherwood number to a common value of about one at higher values of MB, Figure 6, further supports this expectation. As mass-transfer coefficientswere determined here using the standard assumption of plug flow through the bed, the possible contribution of axial dispersion to the Sh-ReMB-AT~relations was evaluated. Axial dispersion changes the temperature and concentration profiles in the flow direction so as to decrease the driving force for heat or mass transfer. To the extent that axial dispersion is present, values of Sh calculated with the assumption of plug flow would be correspondingly lower than the "true" values, i.e. those determined with the true driving force for the actual degree of axial dispersion. The effect of axial
dispersion on mass transfer in a packed bed decreases sharply with increasing bed depth (Ercan, 1989). As in Figure 8, the trend is always for Sh to be higher for thinner paper for which this effect should be more important; axial disperson is not the source of the segregation of results with respect to paper thickness observed here. In any case, no measurements of gas-phase axial dispersion for flow through paper have been reported. Axial dispersion in a fiber bed is determined by the web structure, the wide pore size distribution, and the high porosity (Sherman, 1964). As there is no geometrical similarity between the structure of paper, a consolidated bed of ribbonlike fibers, and conventional beds of particles of simple shape, axial dispersion for paper cannot be predicted reliably from information available for such packed beds. Martin (1978) also found that the effect of axial dispersion is not sufficient to explain the low transfer coefficients reported by various researchers. A significant effect, ignored in previous work, is the convective heat and mass transfer which occur before the throughflow enters the paper. The standard way transfer coefficients are calculated attributes all the mass transfer to transfer within the packed bed, i.e including whatever convective transfer occurs prior to the throughflow entering the bed. The relative contribution of this effect increases as bed thickness decreases. The increase in Sh with decreasing MB observed here would in part be due to this effect. For the thickest paper used, 150 g/m2, this effect is probably negligible. The trend of Sh decreasing with increasing bed thickness is consistently observed. Kat0 et al. (1970) analyzed the bed thickness effect for their own data as well as that of others for naphthalene sublimation, decompositon of H202, and evaporation of water and organic solvents; with throughflow fluids as varied as air, N2, C02, H2, and a mixture of H20-H202vapor; with particle shapes of pellets, granules, spheres, and cylinders; with particle size 0.2 < dp < 32 mm; and for a wide range of L/dp, 0.3-25. Their analysis indicated that Sh a (L/dp)-0.6at constant Reynolds number. They argued that, according to theory, the fluid boundary layer thickness is inversely proportional to some power of Re and that the ratio of mass boundary layer thickness to that of momentum is inversely proportional to some power of Sc. Thus at low Re for gassolid systems where Sc is rather small, these boundary layers overlap, reducing the effective mass-transfer area. With a particle surface area (used to define the masstransfer coefficient) much larger than the effective surface area, Kat0 et al. reasoned that this phenomenon becomes accentuated with increasing bed depth, with higher bulk concentration of the transferring component in the gas, or with increasing L/dp. Other researchers attribute the decrease in Sh with increasing bed thickness to nonuniform flow in the bed. With a channeling model, Kunii and Suzuki (1967)showed that as the ratio of average channel length to particle diameter increases, Sh (or Nu) should decrease. Schliinder (1977) modeled the flow maldistribution resulting from pore size distribution using an array of tubes consisting of one large diameter capillary embedded in a matrix of smaller pores. Assuming plug flow through each capillary and using the combined Graetz and Leveque solutions to predict the transfer coefficients, he showed that the Sherwood number for the combined flow is less than that of individual capillaries. For packed beds of uniform sized spheres the section adjacent to the wall has a lower porosity. Martin (1978) predicted the reported Nusselt number decrease with in-
Ind. Eng. Chem. Res., Vol. 31, No. 3, 1992 741 creasing L / d p ratio by assuming plug flow in each section and basing the transfer coefficients on a Ranz-Marshall type equation. Martin also evaluated the influence of a small bypass, which he found to be negligible for large values of (Re)(&), >200, but which reduces Sh greatly for small (Re)(&),
