EFFECTS OF PACKED BED PROPERTIES ON LOCAL C,ONCENTRATION AND TEMPERATURE PATTERNS D A V I D E. L A M B ' A N D
R I C H A R D
H. W I L H E L M
Department o j Chemical Engmeering, Princeton C'nrcersztj. PrincPton.
.V.J .
When a packed bed, such as a random assembly of granular catalyst particles, is repacked multiple times, variations in ploint concentrations and temperatures occur between one act of packing and the next because of local variations in position of particles throughout the bed. It may be presumed that such local effects will, under some circumstances, affect integral bed properties, such as productivity and selectivity in the case
of chemical reactors and effluent composition and temperature in the case of other packed bed equipment. Present work is an initial theoretical analysis and experimental study of the local granule-induced phenomena that cause variances to occur in the continuum properties of packed beds. The analysis is restricted to packed beds in which no chemical reaction occurs.
to the design of packed bed equipment for effecting heat transfer. mass transfer, or chemical reaction is the ability to determine concentration. temperature. and fluid velocity patterns produced within the bed. Because of the presence of packing pari.icles, these patterns may assume very complicated forms. A more accurate description of the patterns than those now available may be called for in the design of packed bed devices in which large gradients are present. It is useful to classify concentration. temperature, and velocity patterns in packed beds according to three scales of description. The largeijt scale consists of variations in the patterns Lvhich take place over distances in order of magnitude of the vessel diameter. At this scale of observation local disruptions in the patterns produced by the packing particles are neglected. Second. is the intermediate scale in which variations in the patterns occur in distances having the same order of magnitude as the. packing particle dimension. Finally. the smallest scale of classification encompasses variations in the patterns beginning at the boundary layer level and extending through flow processes in the pores of packing particles. ultimately to the molecular level. There is obviously a gradual transition from one category to the next. This article deals wiih prediction for design purposes of stationary concentration and temperature patterns of the second type. ,4n objecti.ve is to provide a theoretical basis for determining effects of random variations in packing voids on concentration and temperature patterns. An initial theoretical analysis considers the effects of random variations in void fraction resulting from irregularities in the packing structures. The treatment is based upon stochastic considerations, and equations are developed which give the variance of the distribution of concentration or temperature a t a point resulting from multiple repacking of the bed. This variance is related to concentration or tem'perature gradients in the bed and to the size distribution of void spaces within the bed. Development of the equations involves Monte Carlo calculations performed lvith a digital computer. ASIC
Packed Bed Models
In the design of packed bed devices, it is visualized that a hierarchy of design models will be needed to describe properly Present address, Uniwrsity of Delaware, Newark, Del.
situations of increasing complexity. suggested in outline form below.
Such a hierarchy is
Deterministic A . Continuum. Formulated in terms of differential equations with one or more independent space variables. B. Discrete. Finite stage models formulated in terms of difference equations. 2. Stochastic A . Random perturbation of deterministic model. B. Purely stochastic model. 1.
The degree to which reactors or other packed bed equipment can properly be described by any one of these models depends upon the purpose of analysis and the magnitude of temperature and composition gradients produced within the bed in steady or unsteady operation. Studies in the recent past have dealt primarily with deterministic models. Present tvork relates to a stochastic model of type 2A. The authors are not aware of other stochastic models which have been formulated to determine variation in concentration and temperature patterns arising from the random properties of a packed bed. The simplest of models is the well known plug flow model in which radial gradients in concentration, temperature, and velocity are taken everywhere to be zero. Formulation for design involves differential equations Lvith only one independent space variable. The model can provide satisfactory predictions of bed performance for beds with large tube to particle ratios. zero flux of mass and energy through the tube walls. and moderate rates of reaction and heat release. In many cases, these requirements are not satisfied and more accurate models must be used. Nonzero fluxes through the tube walls and nonuniform velocity distributions in reactors can produce radial gradients in concentration and temperature. Radial gradients have been taken into account in mathematical models by using partial differential equations with radial as well as axial space coordinates as independent variables. Further model refinements are sometimes required. Separate heat, mass, and momentum equations can be lvritten for each fluid and solid phase. Coefficients involving properties such as density. diffusivity, heat capacity, and void fraction can be made functions of axial and radial position within the bed. VOL. 2
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Despite such refinements there are cases in which these models fail to provide satisfactory descriptions of packed bed performance. -4reason for this failure lies in the use of differential equations to describe packed bed behavior. Such use implies that the bed is a continuum whereas it is actually a heterogeneous mixture of solid packing particles and fluid. If concentration, temperature, and velocity patterns bvere viewed with a resolution comparable to the diameter of a single packing particle, it would be apparent that local patterns differ considerably from those obtained by solution of differential equations. The void volume of a packed bed can be regarded as an assembly of individual void spaces, connected to one another by small channels. Description of packed bed equipment in terms of differential equations is valid only when the effect of individual void spaces on the concentration and temperature profiles is small. However, the effect of an individual void space in disrupting concentration and temperature profiles becomes greater as the size of the void space increases and as the magnitude of concentration and temperature gradients in the vicinity of the void space increases. Use of finite stage models Lvhich result in difference equations instead of differential equations represents an improvement in the description of packed bed performance. Models involving difference equations have been used by Hill and \Yilhelm (5),Deans and Lapidus ( 3 ) and Beek ( 7 ) . In its simplest form: the finite stage model of a packed bed considers the interstices betxveen packing particles as a series of interconnected perfectly mixed stages positioned in an orderly manner throughout the bed. Many refinements in finite stage models are possible. These include more realistic description of diffusional effects and inclusion of the effects of variation in average void volume and fluid velocity with radial position ( 3 ) . Despite such refir.ements finite stage models as well as models based upon differential equations suffer from a common defect. They are deterministic models whereas packed beds exhibit stochastic behavior. In beds of dumped packing the orientation of packing particles and geometry of the void spaces can be described only in stochastic terms. This stochastic nature is manifested in changes in concentration, temperature, and velocity patterns which result when a packed bed is repacked? and changes in reactor conversion and yield when a packed bed reactor is repacked. Possibly. in some cases, the difficulty associated with scale-up and reproduction of results obtained with packed bed reactors in which large concentration or temperature gradients exist may be the result of random variations in the packing void spaces. Short packed bed reactors have a reputation for difficulties associated with reproducibility and scale-up. It is in these reactors that very large concentration and temperature gradients are often obtained, and in which the effects of variations in the packing void spaces would therefore be most strongly felt. As noted in the outline above. two types of stochastic models are suggested to describe packed bed performance. A purely stochastic model could be employed if sufficient statistical information were available concerning the structure of packed bed void spaces. Such a model could be developed by making the stage size variable in a finite stage model, and choosing stage sizes to conform with the size distribution of void spaces in the packed bed. Patterns of concentration, temperature, and fluid velocity through the bed could be calculated from the resulting stochastic model. However, the value of such a scheme is doubtful because of the long computation required for each new problem. 174
I&EC FUNDAMENTALS
Alternatively, a deterniinistic model based upon differential or difference equations can be used to calculate concentration and temperature patterns while considering void spaces in the packing as sources of perturbations in these patteriis. Void spaces can be positioned randomly throughout the bed, and statistical properties of the patterns can be calculated. A model of this type is developed in the next section of this paper. Results predicted by the model are compared with statistics derived from experimental measurements of concentration patterns in a packed bed. In the above discussion. the scale of concentration. temperature. and velocity patterns has been classified as large when only those aspects of the patterns larger than packing particles are of interest; intermediate when those parts of the patterns having the same order of magnitude as the packing particles are important; and small when those aspects smaller than the particles are needed to describe packed bed behavior. Models involving only differential equations are appropriate when consideration of large scale patterns only are sufficient. When intermediate scale patterns must be described. finite stage models are used. As a supplement to these deterministic models a new class of stochastic models is proposed here n.hich considers random variations in packed bed structure.
Theory .4n aim of the following stochastic theory is to provide a means whereby disturbances in concentration and temperature patterns resulting from random variations in the size and position of voids within the bed can be determined. By expressing disturbances in concentration and temperature patterns in statistical terms, the expected change in packed bed performance resulting from modification of bed void structure during repacking can be established. The theoretical analysis is divided into several parts. T h e structure of void spaces or interstitial holes is first considered and a simple model is developed which shows the effect of a single hole in producing a perturbation in the concentration or temperature pattern at the hole location. The effects on concentration and temperature patterns produced by diffusion of the perturbation away from the hole are next considered. Solution of a diffusion equation permits calculation of the effects of a single hole on concentration and temperature patterns a t any point in the bed. The magnitude of disruption in concentration and temperature patterns depends on distance from the hole. size of the hole, and magnitude of concentration or temperature gradients in the vicinity of the hole. A Monte Carlo procedure is used next to estimate the variance in the distribution of concentration or temperature disturbances produced at a point in a packed bed b>-random positioning of a single hole. Results are generalized to include the effect of hole size on the variance. Having calculated a n estimate of the variance produced by random positioning of a single hole, an estimate is obtained of the variance produced by an ensemble of void spaces randomly positioned throughout the bed and having a distribution of sizes. This variance applies. to the distribution of concentration or temperature perturbations at a point in a packed bed resulting from multiple repacking of the bed. Structure of Void Spaces. Void spaces in beds of,spheres which have been systematically packed can be represented in terms of the void space associated with unit cells of packing. These unit cell void spaces have been thoroughly studied ( 4 ) for each of the six systematic packing configurations obtainable with spheres. Examination of the structure of these void spaces sho\vs that they comist of a large void bounded by several
Blending in Holes
produces
Distorted Prof i I e
B
A
rminus
Distorted Profile
C
Original Profile
0
equals
Perturbation
E
Figure 1. Generation of temperature or composition perturbations
convex surfaces and joined to adjacent void spaces by a number of connecting channels. These channels are of variable cross section but have a minimum cross sectional area, small relative to the mean cross sectional area of the void. In beds of dumped packing, void space geometry is more complicated but the general structure consisting of a system of void spaces connected to one anothler by smaller channels remains unchanged. Bridging or arching of packing particles in beds of dumped packing produces some void spaces much larger than those found in systematically packed beds (2). More bridging and more exceptionally large void spaces would be expected in beds of nonspherical .packing because of the larger average void fraction usually associated with such packing. ‘ro develop a model through which the effects of void spaces on concentration and temperature profiles can be established, i t is useful to choose a simple geometry to represent the void spaces, but one which still retains the essential character of the void spaces. ,4n ensemble of randomly positioned spherical void spaces having a distribution of sizes, and connected to one another by a series of channels satisfies these requirements for many types of packing and will be used here. Generation of Pertwbations. The effect of a spherical hole on concentration or temperature profiles is illustrated Fluid enters the hole as a series of jets when in Figure I J . fluid velocity in the connecting channels is sufficiently high. Each jet entering the h’de may have a different composition and temperature as a result of having originated in a different location in the bed. However. because of rapid mixing resulting from jet interaction in the hole, the concentration and temperature of fluid leaving the hole through the various connecting channels is assumed to be uniform. The effect that mixing \vithin a hole has on a radial time average concentration or temperature profile is shown in Figure l , B to result in a reduction ‘of the radial concentration or temperature gradient to near zero over the space occupied by the hole. The difference hetween profiles with and without the presence of a hole in the packing \vi11 be termed the perturbation in the profile prmoduced by a hole. ‘4 perturbation in a radial profile is shown in Figure 1,E. Mixing within spherical holes affects both radial and axial concentration and temperature profiles. Ho\vever, because axial gradients are generally small relative to radial gradients. only perturbations of radial profiles will be considered.
Perturbations as Diffusion Sources. Mixing in holes within packed beds has an effect on concentration and temperature profiles not only a t the location of the hole but also at points downstream. T h e perturbation in the profile persists but with diminished amplitude as the fluid moves downstream through the packed bed. Ultimately the turbulent diffusion processes within the packed bed cause the magnitude of the perturbation to approach zero. The effect of a hole on concentration and temperature profiles downstream from the hole can therefore be treated as a diffusion problem. Mixing in a hole produces a perturbation in the radial profile which acts as a diffusion source. Because the perturbation has both positive and negative parts Lvhich act as diffusion sources of equal and opposite strength. as shoivn in Figure 1.E, the magnitude of the perturbation must decrease toward zero downstream of the source. In the course of diminishing in amplitude, however. the perturbation diffuses radially. Determination of the magnitude of the perturbation as a function both of distance downstream from the source and of radial distance from the source requires solution of the diffusion equation for the boundary conditions imposed by the geometry of the source and the geometry of the walls of the packed bed. It can be shoivn that unless a void space is very near the wall of the vessel the walls will have negligible effect on diffusion of the perturbation. Neglecting wall effects, the diffusion problem becomes one of diffusion into an infinite space. The perturbation source is not geometrically simple and cannot be represented by a simple boundary condition. The perturbation source showm in cross section in Figure 2.B, is three dimensional, the plan view of lvhich is circular. To simplify the mathematical description of the diffusion process it will be assumed that the perturbation source can be represented by two point sources, one positive and the other negative. The positive source will be located at a point in the positive section of the three dimensional perturbation source such that half of the volume of the positive part of this source lies to the left of the point and the other half to the right. The negstive source will be located a t the Corresponding position in the negative part of the three dimensional perturbation
source.
be,
,
If 7 I S the unperturbed concentration or tem-
bY
perature gradient through the hole center, H. shown in Figure 2,B. and if R is the radius of the hole producing the perturbation, then the volume of the positive or negative half of the 2 bCh perturbation is- R3 - The radial position measured from
3
the hole center corresponding to a volume one half of this value is 0.6 R. In addition to locating point sources to represent the more complicated perturbation produced by a spherical hole, it is necessary to establish the strength of these sources which will be equivalent to the total distributed strength of the original perturbation. Strength of the positive source is found by multiplying volume of the positive half of the perturbation by the point fluid velocity. C:: through the per2 bC turbation. This product is: - C R3 ’ ’br” The negative source 3 has the same magnitude but opposite sign. Strength of the t\\o point sources is more conveniently expressed by defining k = 2R ID, and r = r’,’Dp. The absolute value of the strength of each source. GI. is then given by
VOL. 2
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175
I
I
I
Figure 3. Coordinate systems locating H and P relative to 0
f
0
Figure 2.
Diffusion from perturbation source
Diffusion of Perturbation from a Single Hole. To begin quantitative study of the effects of void spaces or holes in the packing, a single hole will be considered. A rectangular coordinate system is established in the packed bed with origin a t point 0 as shown in Figure 3. The coordinate axes are made dimensionless by dividing by the particle diameter, D,: and are labeled x , y , z ; the absence of primes denoting nondimensionality. The spherical hole \\ill be located at some arbitrarily chosen point, H. which has nondimensional coordinates xh, .>h, z h . It is of interest to determinr the effect which this hole produces in the concentration or temperature profile at any point P in the packed bed located by nondimensional coordinates x p . y p j z p . This will be done by solving the diffusion equation for heat or mass diffusing from the point source pair into a n infinite space. To do this. the distance measured from the point sources to P must be found. The location of point P relative to that of the hole, H , is (x, - x h ) , ( y p - , y h ) . ( z , - z h ) . Distance from the center or the hole to each of' the point sources is 0.6 R. Separation distance benveen the tlvo point sources is therefore 1.2 R = 0.6 k D , or in nondimensiorial form 0.6 k. The tww point sources must lie on either side of the center of the hole along a line drawn in the direction of maximum radial Concentration or temperature gradient. In most cases. this line will be a radius of the packed bed because of cylindrical s:mmrtry of 176
l&EC FUNDAMENTALS
'hl
'h2
'h
Figure 4. Location of point sources relative to hole center
the unperturbed profiles. I n Figure 4, the two point sources are shown located a t points 1 and 2. The coordinates of these t\yo points in terms of the coordinates of H a r e : Xh
- -/
yh, = J h
-6
Xh,
X h , = Xh yh,
= yh
-/
+6
(2) (3) (4)
(5)
Distance between source 1 and the point P i s given by: X, Yp
Zp
- X h + -/ - Yh, = YP - Y h + 6 - Zh, = Z p - Z h -
Xh,
=
Xp
Similarly for source 2 : Xp
- Xh3
=
Xp
-
Y P - Yiia = Y P Zp
-
Zh2
=
Zp
-
Xh
Yh
-
y
-6
(7)
zh
Now that the distance from each of the point sources to the point P has been established, and since the strength of the sources is known. the concentration perturbation produced at P by the holi at H can be found. Magnitude of the per-
turbation a t P is determined by the sum of the separate contributions of the t u 0 point sources. The concentration perturbation, c p . a t point P produced by source 1 is obtained from solution of the point source diffusion problem (70) :
Magnitude of the concentration perturbation produced by two sources of equal and opposite strength-i.e.. G? = - GI is given by :
turbation. Because positive and negative parts of the perturbation are mirror images of one another, their mutual cancellation as t -+ 03 reduces the magnitude of the perturbation toward zero. Equation 9 shokvs that for all finite values of x P , y,. z,, the magnitude of the perturbation is greater than zero. It is of interest to know over what range of x p , y p , t p the magnitude of the perturbation is significant. In Figure 2.B: c , ~is the maximum value of the perturbation produced by a hole centered at point H. The curves shown in Figure 5 are the values o f t and Y for lvhich the magnitude of the perturbation has been reduced to 10% and 1% of c,. These curves are sections of surfaces which extend in the plus and minus y direction. However. the maximum extent of the surfaces in the y direction is no greater than in the r direction. The curves were obtained by setting
aci,
sional by dividing both sides of the equation by ar ' Figure 2 . A .
(P ~~
dC, dr
is shou n as a function of
Y
and
t
In
for a ho!e of
diameter 4 D,-i e.. i,= 4 This figure sho\\s that the concentration perturbation spreads radially but decreases in amplitude Lvith increasing axial distance from the hole. The decrease in amplitude rrsults both from the spreading and from cancellation of the positive and negative parts of the per-
CP
and
J,
Zp
a value of concentration can be calculated from a d i l h i o n model. The difference between that value and the coricentration measured experimentally is the concentration perixrbation at the point. For each of several radial positions, the concentration perturbation was calculated as a function of angular position. The variance of each function was then determined and used in Equation 24 to find M I . The concentration gradient required in Equation 24 was calculated from the di-ffusion equation. Similar calculation Lvere performed using a second contour map prepared from data obtained in R u n 2 at a bed depth of 3 inches. \.slues of MI calculated. from the contour maps are summarized in Table I . Also shown in the table are values of M1 calculated from measurements of concentration patterns obtained a t the top of a bed packed to a depth of 7 inches. The variance a t
Table 1. r'
7
Summary of Experimental Results b(ClC m 1 br
Run 2
Run 7 U
~
E
~
M i
cP2
MI
1.14 0.67 0.71 037 019 0 03
1.61 0.82 1.25 138 215 1 43
z' = 3 inches, S = 37
0.2 0.3 0.4 0 5 0 6 0 7
1.33 2.00 2.67 333 400 467
2.04 2.20 1.84 126 073 036
0 2 0 3 0 4 0 5 0.6 0.7
133 200 267 333 4.00 4.67
044 057 062 060 0.53
0.86 1.03 0.60 038 021
1.12 1.26 1.08 139 235
z' = 7 inches, A' = 9
0.42
VOL. 2
008 005 012 012 0.12 0.06
NO. 3
248 099 188 298 2.48 1.84
AUGUST
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181
a given radial position was calculated from a series of concentration measurements. each made after fluidizing the bed. a CIC- 7 1 used to calculate M I in Table The values of u p z 2and -
1 vary by more than a factor of 20. When the values of M1 in Table I are weighted by *Y. the mean value of M I is 1.5 with a variance of 0.3. The fact that M I remains relatively con-
(z)
stant despite large changes inuPh2and ___ 3‘c~rcm)is in accord with Equation 24 which shokvs upEzto be proportional to
’.
The fact that M I does not vary greatly with radial and axial position in the bed is in accord with the assumption that holes are distributed uniformly throughout the bed. Hoivever. it is expected that different values ofM1 would be obtained near the wall where void fraction variations caused by the wall occur. T o examine further the validity of M I , it is of interest to calculate the average hole size corresponding to M I If all holes in the bed were of uniform size, then
Since u =
77k3
7it
follows that
dimensional distance coordinate normal to
Conclusions
For beds of dumped spheres in which no reaction takes place. Equation 24 with M I = 1.5 can be used to obtain estimates of the variance of concentration and temperature a t each point in the bed. These estimates \vi11 probably be slightly high and hence conservative because M I was established for beds with large void fraction. The present analysis is a n initial attempt to develop a stochastic model for fixed bed equipment. The analysis does not include the coupling effects of chemical reaction and flow disturbances on concentration and temperature profiles. In its present state of development, the model should therefore be used only in estimating variance for packed bed equipment in which no reaction occurs. Modification of the stochastic model to include chemical reaction may be possible.
This work was supported by generous financial aid in the form of a research assistantship by the Shell Development Co. and a fellowship by the Celanese Corporation of .4merica. Computer calculations were performed at the University of Delaware Computing Center. Nomenclature = concentration a t H = mixed average concentration a t any bed cross section
c,
= maximum value of concentration perturbation pro-
cp
= concentration perturbation at any point,
duced by hole
182
P
= average value of distribution of concentration perturbal&EC FUNDAMENTALS
’ and
t’,
dimensionless distance coordinate normal to ,r and 2 , .?’!LIP dimensional distance coordinate normal to Y ’ and z’,
(L)
dimensionless distance coordinate normal to \ and y > Z’,’D, dimensional distance coordinate normal to 2 ’ and J ’%
(L) Greek letters = t component of distance from hole center to point y source used to represent effect of hole = 1. component of distance from hole center to point 6 source used to represent effect of hole = fraction of bed void volume occupied by holes eh = kinematic viscosity, ( L ) z (T)-l v u , ~ = variance of distribution of concentration perturbations at P. resulting from positioning a single hole within the bounds given by Equation 10 ulE2 = estimate of u> u p 2 = defined exactly as u I 2 except that distribution results from the effects of many holes upE2 = estimate of up%
Subscripts 1. 2 = refers to point sources used to represent perturbation produced by hole = refers to hole, H h = refers to point, P p literature Cited
Acknowledgment
Ci
J
(L)
For eh = 0.4 and M I = 1.5, this equation gives k = 1.9. ,41though this value of k is of the right order of magnitude and some holes of this size probably occur in beds of spheres, the estimate is probably too large. I n developing the stochastic model, the interaction of perturbations from different holes was neglected. Had this assumption not been made the coefficient in Equation 24 would be larger and M I calculated from this equation lvould be smaller.
C, C,
+
x ‘/OD
(2) 1 3
k =
tions at a point. P, resulting from positioning a single hole within the bounds given by Equation 10 estimate of Z i diameter of packing particle, ( L ) eddy diffusivity, ( L ) 2 T)-l ( hole void volume distribution strength of point source (M) ( L )-I( T ) hole size distribution location of hole in packed bed ratio of average hole diameter to particle diameter zeroth moment off(&#) first moment ofJ(z’) number of data points used in calculation of?,, and uiL number of holes in size range k to k dk per unit volume of bed location of arbitrarily chosen point in packed bed probability density function Peclet number, D,L7/E radius of hole, ( L ) Reynolds number, D,Co/ Y dimensionless radial position = r ’ / D p radial position in packed bed measured from axis, ( L ) mean point velocity in t direction, ( L )( T ) superficial velocity, ( L ) (T)-1 volume of region defined by Equation 10, dimensionless volume of a single hole, 7rk3/6 dimensionless distance coordinate normal to J and z ,
(1 ) Beek, J., “Advances in Chemical Engineering,” Vol. 3, p. 203, Academic Press, 1962. (2) Brown, R. L., Hawksley, P. G. W., “Particle Mechanics,” Tzmes Sci. Rez’.p. 6, Summer, 1954. (3) Deans, H. A,, Lapidus, L., A.Z.Ch.E. Journal 6, 656, 663. (1960). (4) Graton, L. C., Fraser, H. J., J . Geol. 43, 785 (1935). (5) Hill, F. B., Wilhelm, R. H., A.I.Ch.E. Journal 5 , 486 (1959). (6) Lamb, D. E., Ph.D. dissertation, Princeton University, 1962. ( 7 ) Lamb, D. E., Manning, F. S., Wilhelm, R. H., A.I.Ch.E. Journal 6, 682 (1960). (8) Roblee, L. H. S., Baird, R. M., Tierney, J. W., Zbid., 4, 460 (1958). (9) Rotenberg, A . , J . Asroc. Corn$. .Mach. 7, 7 5 (1960). (10) Wilson, H. A , : Proc. Cambridge Phil. SOC.12,406 (1904). RECEIVED for review February 11, 1962 ACCEPTEDMay 29, 1962
