H = enthalpy of one mole of a mixture of c components H" = enthalpy of one mole of a mixture of c components in the standard state of a perfect gas a t the pressure of 1atm and a t the temperature i f f = partial molar enthalpy of component i ; defined by eq 1 H , = virtual value of the partial molar enthalpy; defined by eq 4. AH," = standard heat of reaction; enthalpy of the products in their standard states of perfect gases at 1 atm and temperature T minus the enthalpy of the reactants in their standard states of perfect gases a t one atmosphere and temperature T ni = moles of component i or molar flow rate of component i n~ = total moles of mixture or total molar flow rate of the mixture
ni
Subscripts k = counting integer for components; ranges over all components i = 1 through i = c i = counting integer for components; ranges overall components i = 1through i = c Greek Letters C Y , y = constants in the Benedict-Webb-Rubin equation of state; see eq 13 p = density of the mixture
Q = enthalpy function; departure of the enthalpy of one mole
of mixture a t P and T from one mole of the mixture in its standard state of a perfect gas at the temperature T ;see eq 3,4, and 15
Literature Cited Benedict, M., Webb, G. 6.. Rubin, L. C.. J. Chem. Phys., 8,334(1940); IO, 747 (1942). Earner, H. E.. Adler, S. B., lnd. Eng. Chem., Fundam., 9, 521 (1970). Dowling. D. W., Eubank, P. T., "Report of Investigation on the Calculationof the Compressibility Factor and Thermodynamic Propertiesof Methane. API Project 44", Thermodynamic Research Center, College Station, Texas, 1966. Holland, C. D., Eubank. P. T., Hydrocarbon Process., 53, (1 1). 176 (1974). Holland, C. D.. "Fundamentals and Modeling of Separation Processes, Absorption, Distillation, Evaporation, and Extraction", Prentice-Hall, Englewood Cliffs, N.J., 1975. Holland, C. D., "Multicomponent Distillation", Prentice-Hall.Englewood Cliffs, N.J.. 1963. Mickley, H. S., Sherwood, T. K.,Reed, C. E., "Applied Mathematics in Chemical Engineering", McGraw-Hill,New York, N.Y., 1957. Orye, R. V., lnd. Eng. Chem. Process Des. Dev., 8, 579 (1969). Papadopoulos. A., Pigford, R. L., Friend, L., Chem. Eng. Prog. Symp. Ser., 49, No. 7, 119(1953). Reid, R. C., Sherwood, T. K., "The Properties of Gases and Liquids. McGraw-Hill, New York, N.Y., 1958.
Receiued for review September 15, 1976 Accepted November 4, 1976 T h e paper is based on work which was supported in p a r t by D o w Chemical Company and the Texas Engineering Experiment Station. T h i s support is gratefully acknowledged.
Prediction of Heat Transfer under an Axisymmetric Laminar Impinging Jet N. R. Saad, W. J. M. Douglas,' and A. S. Mujumdar Department of Chemical Engineering, McGill University, and Pulp and Paper Research lnstitute of Canada, Montreal, Canada
An upwind finite-difference representation of the vorticity-stream function formulation of the full Navier-Stokes and energy equations was used to predict the flow and local heat transfer characteristics of a laminar semi-confined round jet impinging normally on a stationary plane wall. Effects of Reynolds number, distance between jet nozzle and impingement surface, diameter of the impingement and confinement surfaces, shape of the velocity profile at the nozzle exit, and application of uniform suction at the impingement plate were studied numerically. Favorable agreement between predicted heat transfer rates and published data was obtained for those few cases for which experimental data were available. The velocity profile at the nozzle exit was found to be a particularly important variable while the application of suction was found to enhance the heat transfer rates significantly.
Introduction Impinging jets of various configurations are used in industrial equipment requiring high rates of heat and/or mass transfer. Drying of textiles and paper, cooling of turbine blades, and heat treatment of glass and metal sheets are some important practical applications (Black (1976), Hardisty (1973), and Keeple (1969)). Gauntner et al. (1969) and Mujumdar and Douglas (1972) among others have reviewed the flow and convective transport characteristics of the various types of impinging jets. Laminar, partially confined impinging jets have received little attention in the literature. Marple et al. (1974) predicted flow patterns for both a round and rectangular laminar jet impactor and verified their predictions experimentally, but were not concerned with transport phenomena at the im148
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
pingement plate. Van Heiningen et al. (1976a) predicted numerically the flow and heat transfer under a laminar slot jet with an upper confining plate, including the effect of suction at the impingement surface. Application of suction was demonstrated to increase heat transfer rates at the impingement surface, which is a particularly desirable feature when impinging jets are used for drying permeable materials like newsprint. Indeed, a proposed new process of newsprint drying utilizes this combination of impingement and percolation flow to achieve significantly increased drying rates (Burgess et al. (1972a,b)). The objective of the present study is to predict numerically the heat transfer characteristics of a semi-confined laminar round jet impinging normally on a stationary, permeable plate to which moderate suction may be applied. The full NavierStokes equations and the energy equations in their vortic-
ity-stream function formulation are solved numerically. The significance and effects on impingement heat transfer distribution of the jet Reynolds number, nozzle-to-plate spacing, nozzle-exit velocity profile, presence of a confining plate, and application of suction are presented.
CONFINING PLATE
I I
Mathematical Description of the Problem The impinging jet system studied is shown in Figure 1.The jet issues from a round nozzle of diameter Do, with an average velocity VJet, and impinges perpendicularly on a plate a t a distance H from the nozzle. Following Gosman e t al. (1969), the equations of motion in their velocity-stream function form and the energy equation can be represented by a general differential equation which, after nondimensionalization with respect to nozzle diameter Do, the average jet velocity Vjet and the inlet fluid properties may be written in as
I
ICFREE
~
STAGNATION
STREAMLINE
PERMEABLE
Tpli,T?P-LE
POINT
IMPINGEMENT PLATE
Figure 1. Semi-confined impinging jet system.
Table I. Coefficients in Eq 1
in the axisymmetric coordinate system. The general nondimensional dependent variable 4, with corresponding coefficients us, b,, c , ~ ,and d, is listed in Table I, where 4 stands alternately for the dimensionless vorticity w/r, dimensionless stream function $, and dimensionless temperature T , which are defined as (2) (3)
(4)
The assumptions pertaining to these equations are the following: laminar, incompressible, constant fluid property flow, no buoyancy effects, and neglect of viscous dissipation and temperature change due to compression. The boundary conditions are taken as follows. Boundary I. Nozzle Exit. For fully developed parabolic velocity profile
For flat velocity profile
Boundary 11. Plate of Confinement. Impermeability of the plate of confinement and the no-slip condition, Gosman et al. (1969) give $ = 0.125; T = 1
(7)
6 wlr
+T
a4
b,
r2
r3Re 1/ r rIRePr
0 1
b bT $ = 0; - ( w l r ) = 0; - = 0
br
br
Boundary IV. Outflow Boundary
a$
-=-
br
br
dT ( w l r ) = -= 0 br
(9)
d dl 0
1 1
0
-w
Boundary V. Permeable Impingement Plate $ = b2r 2 ; T = 0
(10)
and w/r as in eq 7. Boundary VI. Impermeable Part of the Impingement Plate, Same as boundary V except for the stream function $ = -VS",,
(11)
r2suct
where rsUctis the radius of the permeable part of the plate. Complete details of the above equations are given in the thesis of Saad (1976).
Numerical Procedure All results were obtained using a non-uniform, 21 X 24 grid, 21 lines parallel to the plates, 24 lines parallel to the jet centerline. Use of a grid spacing which was nonuniform in both directions permitted a finer grid spacing in regions of steeper gradients of variables, i.e., near the impingement and confining plate and, in the other direction, adjacent to the jet centerline. The grid spacing was monotonically increased away from these regions, with the ratio of neighboring grid spacings near the centerline and solid boundaries never exceeding 1.5 as recommended by Runchal e t al. (1969). A successive iterative procedure was followed until the convergence criterion h e w
where the subscript NP means the value of the variable a t the gridline next to the plate, and Az is the dimensionless spacing between this gridline and the plate. Boundary 111. Axis of Symmetry
cs 1
- 4old
< 10-4
(12)
@new
was satisfied. On an IBM 360-75 computer it required an average of 1.5 min to achieve a converged solution.
Results and Discussion Although Saad (1976) has made an extensive study of the entire flow field for this case of a semi-confined impinging jet, the following presentation focuses on the transport phenomena, principally heat transfer, occurring a t the impingement surface. However, because of the close interrelation Ind. Eng. Chem., Fundam., Vol. 16, No.
1, 1977
149
PARABOLIC VELOCITY PROFILE
0 130 0 142
-0040 -0 136
2:5
2:o
1.5
1.0
c
1 0
.+eo
0:5
1.0 /
1.5 2:O q. 0.000 q = 1.000
125 q.0
000
2:O
DIMENSIONLESS DISTANCE FROM STAGNATION POINT
Figure 2. Flow fields for semi-confinedjets: two cases.
Figure 3. Effect of plate diameter on Stanton number distribution.
between heat transfer and the flow field, the latter is considered here briefly. The pattern of streamlines and velocity profiles a t two Reynolds numbers, 950 and 1960, are shown in Figure 2. The dimensionless streamlines are designated by $ and q , where q is the flow between the streamline of interest and the free streamline, expressed as a fraction of the jet flow. Figure 2 indicates that an important feature of the flow field for a semi-confined impinging jet is the circulation of fluid from the environment through the region bounded by the two plates. Although the rate of inflow of entrained fluid, $ > 0.125 or q < 0, is always considerably less than the inlet jet flow, the inflow nonetheless fills considerably more than half of the gap between the plates for the case illustrated. The outflow, consisting of entrained fluid, $ > 0.125, and jet fluid, 0 < $ < 0.125, moves closer to the impingement plate for higher Re due to higher momentum of the inlet jet. A t smaller Reynolds number and spacing, Le., Re 6 200 and H 6 0.3, it was found that the free streamline attaches to the confining plate at r < 150
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
2.5. In such cases the region a t boundary IV designated in Figure 2 as inflow disappears, and boundary IV then comprises only outflow, Le., 0 < $ < 0.125. For conditions of larger Re and H , such as those represented in Figure 2, the free streamline, $ = 0.125, attaches to the confining plate only for plates considerably larger than r = 2.5. However, flow fields such as those of Figure 2 are typical for the conditions of most of the heat transfer results reported in the present study. Because of the elliptic character of eq 1, the flow a t boundary IV will influence the flow field and hence the heat transfer a t the impingement plate. Therefore a check was made of the effect of size of the plates, rm,on heat transfer at the impingement plate. These results, expressed as Stanton number distribution on the plate of impingement, are shown in Figure 3. The inflow, $ > 0.125, a t boundary IV is reduced as r , increases. Thus for Re = 100, H = 1.5, there is only outflow at boundary IV for r , b 8. The results shown in Figure 3 establish the fact that, for the range of Re and H tested, the influence on heat transfer a t the impingement plate of an inflow a t boundary IV does not extend closer to the stagnation point than about r = 1. Thus, convective transport in the central region of the impingement region, r 6 1, is essentially independent of whether or not inflow occurs a t boundary IV. Marple et al. (1974) and van Heiningen et al. (1976a) made similar observations concerning flow a t the impingement region. Since the objective of the present study was to investigate the impingement region flow and heat transfer, the size of the impingement plate was for the remainder of the study restricted to about r , = 2.5. Also, the convergence and stability characteristics become economically unfavorable for high values of r,. The local impingement heat transfer coefficient was determined by equating the convective flux with the conductive flux across the fluid layer adjacent to the impingement plate. Thus the location of the gridline adjacent to the impingement plate assumes special importance. The criterion used for this
--
-
0.10
a,
PARABOLIC VELOCITY PROFILE
0.08-
EXPERIMENTAL : SHOLTZ a TRASS H=2 A Re=960 R = 0.71 a Re = 1960
-
I
2.5
2 .o
PARABOLIC VELOCITY PROFILE
tj
h W
EXPERIMENTAL: SHOLTZ a TRASS Re = 960 0 Re =I960 Pr = 0.71
Re = 960
0 I 1 0 0.5 1.0 DIMENSIONLESS DISTANCE M STAQNATION POINT, r
I
I
I
1.5
1.0
0.5
I
1
2 .o
I .s
2.
Figure 4. Stanton number distributions for H = 2 and H = 4.
I
I
2.5
I
PARABOLIC VELOCITY PROFILE
PARABOLIC VELOCITY PROFILE
I
2 .o
I
1.5
1
1
I
0.5 0 0.5 I.o MMENSIONES3 DISTANCE FROM STAQNATION POINT
I
I .o
I
1.5
I
2.0
2.
,r
Figure 5. Stanton number distributions for H = 8 and H = 12.
gridline was that it be located within the region for which temperature varies linearly with distance from the plate for all Reynolds numbers considered. This spacing was thus set at Az = 0.0156 from the plate. A check performed by varying this value confirmed that at less than this distance the results were independent of the spacing. Most of the heat transfer results are expressed in the form of Stanton number as this dimensionless heat transfer coefficient is independent of any of the characteristic lengths of this system. For the case of a parabolic velocity profile a t the nozzle, local Stanton number distributions were obtained for a range of Reynolds numbers from 100 to 2500 and for values of nozzle-to-plate distance from 0.2 to 12. Marple et al. (1974), using a flow visualization technique to study the flow field of water impinging on a flat plate, observed that the flow is laminar for Reynolds number up to 2300. A number of Stanton number distributions are presented in Figures 4 and 5. The radial distributions are smooth and bell-shaped. The maxima are always a t the stagnation point where the momentum of the jet is highest. The data show a decrease of St with increasing Re because the heat transfer coefficient increases with inlet jet velocity to a power less than 1. There are no experimental heat transfer data with which to make a direct comparison with these predicted results. However, good agreement was obtained between the present work and experimental values of Sholtz and Trass (1970), who measured mass transfer rates from a naphthalene plate into an unconfined impinging round air jet with a parabolic velocity profile at the nozzle exit. It should be noted that this comparison is only indirect in that it was necessary to convert the experimental mass transfer data of Sholtz and Trass (1970) to the heat transfer equivalent by using the heatmass-transfer analogy relation Nu = (P~/SC)O.~ Sh. As nozzle-to-plate spacing is increased over a sufficiently wide range the rate of heat transfer a t the impingement plate must of course decrease. However, the presentation of results given in Figure 6 shows that, over the range 1.5 < H < 12,
t
0.05
PAMBOLIC MLOClrY PROFILE
0 Ra = 450 OR*.-
z 0.03
0.al 0
I 1
I I 1 I I I ' I 1 2 3 4 5 6 7 8 9 1 0 1 1 DIMENSIONLESS NOZZLE TO PLATE DISTANCE , H
1
,
1
2
Figure 6. Effect of nozzle-to-plate spacing on Stanton number a t the stagnation point.
stagnation point heat transfer is quite insensitive to nozzle spacing. At Re = 450 the decrease in stagnation point heat transfer is only about 15%,while a t Re = 950 there is no perceptible decrease in heat transfer as nozzle spacing is increased over this range. As the effect of H on St a t r = 2.5 was similar to that at the stagnation point, the value of mean heat transfer coefficient in this range of variables is relatively independent of the nozzle-to-plate spacing. The heat transfer rate will decrease at still larger nozzle spacings, but the range of nozzle spacing shown covers the region of principal industrial interest. To investigate whether conditions at the impingement plate are sensitive to velocity profile a t the nozzle, a series of comparative runs was made with a flat velocity profile at the nozzle exit. The slight increase of the transfer coefficient some distance from the stagnation point for Re = 950 and 1960 that may be seen in Figure 7 is the result of an inherent inaccuracy Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
151
0.08
2.0
412
Dd2
FLAT VELOCITY PROFILE
-
1.0 FLAT VELOCITY PROFILE
on
H=8 Pr = 0.71
3 060 0
z 04f
w
0.0202
9 p 3 D
0
0.5 DIMENSIONLESS
1.0 1.5 2.0 DISTANCE FROM STAGNATION POINT
2.5
,r
Figure 7. Stanton number distributions for flat velocity profile at the nozzle exit.
0 D0/2
L
14.0
.P
PI. 0.71 EXPERIMENTAL SHOLTZ a TRASS
P
nn 8 Pr
- COMWTEU
i
PARABOLIC VELOCITY PROFILE
l$2 . 0
0.1s
= 0.71
H.2
0
5
3.0 n
$
0.05
A 0.031 = I
f
,I 0
,
,
:
i ;
1 6 i b S l d o o REYNOLDS NUMBER
2ooo I
I 3
I
4
1
1
56000
Re
Figure 10. Effect of Reynolds number on stagnation point Stanton
,
y
+
o
z
0.5 1.0 1,s 2.0 DIMENSIONLESS DISTANCE FROM STAGNATION POINT
number. ,r
2.5
Figure 8. Nondimensional heat transfer and skin friction distributions for H = 8 and parabolic velocity profile a t the nozzle exit.
of the finite difference approximation, discussed by Gosman et al. (1969), Saad (1976), and van Heiningen et al. (1976b). The small error involved does not affect the results significantly. Comparison of Stanton number profiles for a flat inlet velocity profile, Figure 7, with comparable profiles in Figure 5 for the parabolic case, shows that inlet velocity profile is indeed a very important variable. With a flat inlet profile it is not only that the impingement plate profiles are predictably flatter than when the inlet jet is parabolic, but, more important, for a flat inlet velocity profile the heat transfer rate is significantly less over the entire impingement plate. The reduction in heat transfer coefficient for a flat relative to a parabolic inlet profile varies from about 500h a t the stagnation point to about 15%a t r = 2.5. This effect is a consequence of much higher momentum a t the stagnation point when the inlet jet profile is parabolic rather than flat. Thus, for convective transfer processes a t an impingement surface, these results establish a clear advantage of a parabolic over a flat profile in the inlet jet, and that this advantage applies over the entire impingement surface. In examining the effect of Reynolds number for laminar heat and momentum transfer, results are frequently presented as the dimensionless groupings N u l a and 0 . 5 C f 6 because laminar boundary layer theory indicates a 0.5 power dependence for Re. In some cases such as representation has 152
Loo
Ind. Eng. Chem., Fundam., Vol. 16, No. 1, 1977
the effect of collapsing a family of curves into a single, more general correlation. This approach was checked in the present case for all values of nozzle-to-plate spacing studied, and representative results are shown in Figures 8 and 9 for a single spacing ( H = 8) and for both inlet velocity profiles investigated. For a parabolic inlet profile Figure 8 indicates that, over the range investigated, the 0.5 power on Re in fact overcorrects for the dependence of Nu on Re. The same overcorrection may be seen for the dimensionless skin friction, 0 . 5 C f 6 .For the case of a flat inlet velocity profile the dimensionless heat transfer curves do nearly collapse into a single curve. I t is interesting to note that the position of maximum skin friction a t the impingement plate occurs a t somewhat under r = 0.5 for a parabolic inlet profile but a t somewhat greater than r = 0.5 for a flat inlet profile. This shift in location of maximum skin friction is a natural consequence of the difference in radial distribution of axial momentum between the extremes of flat and parabolic inlet profiles. Although the results shown are for only one nozzle-to-plate spacing, the location of these maxima was essentially the same for the entire range of nozzle spacings studied. This observation further confirms that conditions a t the impingement surface are rather insensitive to nozzle-to-plate spacing over the fairly wide range investigated. Reference to Figures 8 and 9 further indicates that the maximum skin friction for a flat inlet profile is only about 'h that for the same conditions with a parabolic inlet profile, and that this relative difference prevails all the way out to the edge of the impingement plate. Thus the significant differences in flow fields between the cases of parabolic and flat inlet velocity
-I
PARABOLIC VELOCITY PROFILE
PARABOLIC VELOCITY PROFILE
Pr = 0 71 Re = 950
I2.5
V,"ff
I
2.0
I
1.5
I
I.o
I
0.5
0.01v.
I
I
0.5
0
1.0
DIMENSIONLESS DISTANCE FROM STAGNATION POINT
P r = 0.71 Re = 950
I
1.5
I
2.0
2.5
,r
Figure 11. Effect of uniform suction on Stanton number distributions.
profile produce comparable effects at the impingement surface for both momentum and heat transfer. The above comment does not imply that the simple Reynolds analogy between heat and momentum transfer applies for impingement flows. Indeed, Figures 8 and 9 show that heat transfer is at a maximum a t the stagnation point where skin friction is zero, and that over much of the impingement region skin friction is increasing while heat transfer is decreasing. These characteristics indicate just how wrong it is to assume, as some have, that the analogy between heat and momentum transfer can be applied in impingement flows. This confusion has been treated by van Heiningen et al. (1976~). The effect of Reynolds number on impingement heat transfer may also be represented by its effect at the stagnation point, as given in Figure 10. The results as shown in Figure 6 have demonstrated that, above about Re = 950, stagnation point heat transfer is effectively independent of H over the range 1.5 < H < 12. Therefore the solid line in Figure 10 in fact represents the predicted value of Sto for any nozzle-to-plate spacing over this range. For Re > 950 the computed values of Sto are proportional to For Re < 950 this single straight line would become a family of curves which, at Re = 450, would pass through three computed points shown for nozzle-to-plate spacings of 1.5, 4,and 8. The heat transfer equivalent of the experimental mass transfer results of Sholtz and Trass (1970),recalculated as described earlier, are again seen to agree quite closely with the predictions of the present study. It is of interest to note that, for the case of an impinging semi-confined laminar slot jet, van Heiningen et al. (1976a) found that S b was proportional to R ~ c O .Thus ~ . although the behavior of round and slot jets is similar there are significant quantitative differences. The effect of uniform suction at the impingement surface on the heat transfer distribution with a parabolic velocity profile jet is displayed in Figure 11.The plate was taken to be permeable up to r = 2.5 and impermeable from r = 2.5 to r , = 2.68, the small impermeable section at the outer edge of the plate being added in order to maintain the same boundary conditions a t the outflow as for a simple impermeable plate. For nozzle spacings of both 8 and 12 the absolute magnitude of the increase in heat transfer coefficient for V,,,, = 0.01 VJet was found to be approximately constant over the entire impingement plate. The approximately constant increase in Stanton number of about 0.006 due to this amount of suction corresponds to a relative increase in heat transfer which varies from about 10%at the stagnation point to about a 30% increase at r = 1.5. For a permeable plate of radius r = 2.5, a suction velocity equal to 0.01 Vjet corresponds to 25% of the jet flow being sucked through the impingement plate.
Conclusion The described numerical simulation provides a convenient means of investigating various laminar transport rates under impinging jets without the expense of physical experimentation and with only a modest computing time requirement.
All characteristics of impinging round jets were found to be a sensitive function of inlet jet velocity profile between the limits of parabolic and flat profile, while the application of suction was found to enhance the heat transfer rate significantly. Transport rates a t the impingement plate were found to be relatively insensitive to nozzle-to-plate spacing over the range of spacing 1.5 < H < 12. These numerical predictions of the heat transfer characteristics of semi-confined laminar round jets were found to compare favorably with the limited experimental data available. As compared to the results found by van Heiningen et al. (1976a) for a slot jet, the present study shows that the general trends for a round jet are similar but that there are significant differences quantitatively.
Acknowledgment The authors acknowledge with thanks the helpful comments made by Mr. Adriaan van Heiningen in the course of this research. Nomenclature C, = specific heat at constant pressure Cf = skin friction coefficient, 2r/pV2jet D o = nozzlediameter D A B = diffusivity h = heat transfer coefficient H = nozzle-to-plate spacing k = thermal conductivity kD = mass transfer coefficient Nu = Nusselt number, hDo/k P r = Prandtl number, pC,lk q = dimensionless streamline = 1 - $10.125 r = dimensionless radial coordinate r , = dimensionless size of confinement and impingement plate rsuct = dimensionless radius of permeable section of impingement plate Re = Reynolds number, D0Vjetp/jt Sc = Schmidt number, p / p D A B Sh = Sherwood number, kDDOIDAB St = Stanton number, hlpCpVjet Sto = Stanton number a t the stagnation point T = dimensionless temperature T' = temperature Tplate= temperature of the impingement plate Tjet = temperature of the air at the nozzle exit Vj,, = average velocity a t the nozzle exit Vsuct = uniform suction velocity a t the impingement plate V, = dimensionless radial velocity component V, = dimensionless axial velocity component z = dimensionless axial coordinate Az = dimensionless spacing between the plate of impingement and the adjacent gridline Greek Letters viscosity p = density w/r = dimensionless vorticity y5 = dimensionless stream function p =
Ind. Eng. Chern., Fundarn., Vol. 16, No. 1, 1977
153
-6 = shear dimensionless general variable stress at the impingement plate 7
=
Literature Cited Black, J., Hardisty, H., Sixth Thermodynamicsand Fluid Mechanics Convention, C . 73/76, 99, University of Durham, Apr 6, 1976. Burgess, B. W., Chapman, S.M., Seto. W., Pulp Paper Mag. Can., 7 3 (1l), 314 (1972a). Burgess, B. W.. et al., Pulp Paper Mag. Can., 73 (1l ) , 323 (1972b). Gauntner, J. W., Livingood, J. N. B., Hrycak, P., NASA TN D-5652 (1969). Gosman, A. D., Pun, W. M., Runchai, A. K.,Spaiding, D. B.,Wolfshtein, M., "Heat and Mass Transfer in RecirculatingFlows". Academic Press, New York, N.Y., 1969. Hardisty, H., Report No. 226 and 227, University of Bath, School of Engineering, 1973. Keebie, T. S., Aero. Res. Council, Department of Defence, Australia, ARL/ME308 Aug 1969.
Marple, V. A., Lui, B. V. H.. Whitby, K. T., J. Fluids Eng., 96, 394 (1974). Mujumdar, A. S.,Douglas, W. J. M., presented at TAPPI Meeting, New Orleans, La., Oct 3, 1972. Runchal, A. K., Spaiding, 0. B., Wolfshtein, M., High Speed Comp. Fluid Dyn., Phys. FluidSuppl., 2, 21 (1969). Scholtz, H. T., Trass, 0.. AlChEJ., 16, 90 (1970). Saad, N. R., Master of Engineering Thesis, McGili University, Montreal, 1976. van Heiningen, A. R. P., Mujumdar, A. S.,Douglas, W. J. M., J. Heat Transfer (1976a). van Heiningen, A. R. P., Mujumdar, A. S., Douglas, W. J. M., A.S.M.E. Winter Annual Meeting, New York, N.Y., Dec 5 , (1976b). van Heiningen, A. R. P., Mujumdar, A. S.,Douglas, W. J. M., Lett Heat Mass Transfer, (1976~).
Received f o r reuieu; November 15, 1976 Accepted November 15,1976
Numerical Solution of Surface Controlled Fixed-Bed Adsorption D. U. von Rosenberg;'
R. P. Chambers, and G. A. Swan
Chemical Engineering Depadrnent, Tulane University, New Orleans, Louisiana
A simple numerical method has been developed for the solution of fixed bed adsorption which can be described by equations of the form -B1 [u(dc/dx) 4- (dc/dt)] = B2(dw/dt) = f(c,w), where the function f may take various forms depending upon the mechanism controlling the adsorption process. Breakthrough curves calculated by this method show very good agreement with experimental curves for two different systems. The computation time can be decreased by a modification which allows the time increment to be increased for a fixed length increment.
Introduction The adsorption of one component from a moving fluid stream by a fixed solid in a packed bed is a common operation. In many cases transfer in the direction of flow by diffusion or dispersion is negligible with respect to convective transfer, and composition does not vary normal to the flow. Furthrmore, the adsorption process can often be described by a mass transfer or chemical reaction relationship, and variation of loading within the solid particles can be neglected. I t is this case which is treated. The method of solution developed was applied to two distinct physical problems. The first problem was the adsorption of carbon dioxide from a stream of nitrogen by a molecular sieve. The second problem treated was the adsorption of heavy metal compounds from waste water by a selective adsorbent. Governing Equations The governing equations for this model consist of two partial differential equations obtained from material balances on the fluid and solid phases. In addition, an equation describing the adsorption equilibrium or the reaction rate is needed. This equation is usually nonlinear. For the general case, the governing equations can be written as
(
-B1 u - + ax ac
dW
at
=Bzz
aw B:! - = f(c,w) at
where c is the concentration of sorbate in the fluid, w is the 1 Chemical Engineering Department, U n i v e r s i t y of Tulsa, Tulsa, Okla. 74104.
154
Ind. Eng. Chern., Fundam., Vol. 16, No. 1, 1977
concentration of sorbate on the solid, x is the length dimension in the direction of flow, t is time, u is the fluid velocity, B 1 is the capacity of the fluid for sorbate, Bz is the capacity of the solid for sorbate, and f(c,w) is a function which describes the rate of adsorption. It is this set of equations for which a simple numerical solution was developed.
Previous Methods A number of methods for the design of fixed-bed adsorbers have been developed. Vermeulen et al. (1973) developed a graphical procedure which has been used extensively for design purposes but which was not particularly adaptable to the analysis of the experimental data from the heavy metal removal. Acrivos (1956) developed a method of numerically solving the equations along the characteristics. One limitation of this method is that the sizes of the space increment and of the time increment cannot be changed once the solution has been started. The nature of adsorption problems is that small space increments must be used throughout the solution, but, as the solution proceeds, the time derivatives decrease in magnitude. Thus, it is desirable to use a method of solution in which the size of the time increments can be increased as the solution progresses. Method of Solution The numerical method to be described makes use of two types of finite difference equations. The equation used in the beginning is based on the centered difference method which has been shown to be unconditionally stable by Wendroff (1960) and which is second order correct in both length and time. The method is based on the characteristics of the system in that the ratio of the space increment to the time increment
