J m
= 6~y~l.r~~
ps
= gas density, g./cc.
= mass of bubble, g. = ( P L - P ~ 8/m )
p,
= liquid density, g./cc.
Q
M N Q
= 6~yk/m = volumetric flow rate of gas, cc./sec.
R
= radius of “force balance bubble,” cm. = capillary radius, cm.
t t,
= time variable used to define stage of detachment, sec. = time of detachment after bubble formation due to force
T
= V,’
u
=
u, u,
V VF V,’
= = = = = =
x
=
y
= =
r rs
uh
p
literature Cited
= radius of bubble, cm.
balance is complete, sec. Qt velocity of center of free bubble away from tip after force balance bubble formation i s complete, cm./sec. velocity of gas through capillary, cm./sec. velocity of bubble center during expansion, cm./sec. velocity of residual bubble, cm./sec. bubble volume, cc. final bubble volume, cc. force balance bubble volume, cc. distance covered by base of bubble, cm. surface tension of liquid, dynes/cm. viscosity of liquid, poises
+
Benzing, R. J., Myers, J. E., Ind. Eng. Chem. 47,2087 (1955). Bryan, J . C., Garber, H. J., Symposium on Mechanics of Bubbles and Drops, A . I. Ch. E. Annual Meeting, November 1955. Datta, R. L., Napier, D. H., Newitt, D. M., Trans. Inst. Chem. Engrs. (London)28, 14 (1950). Davidson, J. F., Schuler, B. 0. G., Trans. Inst. Chem. Engrs. (London) 38, 144 (1960). Davidson, L., .L\mick, E. H., A.I.Ch.E. J . 2 , 337 (1956). Hayes, LY. B., Hardy, B. \$’., Holland, C. D., A.I.Ch.E. J . 5, 319 11959). Jackson, R. i V . , Chem. Eng. (London) 178, CE107 (1964). Jackson, R. W., Ind. Chemist 28, 350 (1952). Krishnamurthi, S., Kumar, R., Datta, R. L., Kuloor, N. R., J . Sa.Ind. Res. 21B,No. 11, 554-5 (1962). Siemes, LV,, Kauffmann, J. F., Chem. Eng. Sci. 5 , 127 (1956). RECEIVED for review March 21, 1966 RESUBMITTED June 13, 1968 ACCEPTEDJune 20, 1968
GAS-PARTICLE HEAT TRANSFER COEFFICIENTS IN PACKED BEDS A T LOW REYNOLDS NUMBERS HOWARD L I T T M A N Department of Chemical Engineering, Rensselaer Polytechnic Institute, Troy, N . Y . RONALD G. BARILE Department of Chemical Engineering, Purdue Uniuersity, Lafayette, Ind.
47907
ALLEN H. PULSIFER Department of Chemical Engineering, Iowa State University, Ames, Iowa
50010
12181
Experimental data for the gas-particle heat transfer coefficient in packed beds a t Reynolds numbers between 2 and 100 were determined using a model which takes into account axial dispersion of heat in the gas phase and axial conduction in the solid phase. Beds of copper, lead, and glass were used with particle sizes ranging from 0.0198 to 0.080 inch. Dynamic thermal conductivities of the solid phase in the direction of flow were measured simultaneously.
In a previous paper (Littman and Barile, 1966), gas-particle heat transfer coefficients in packed and fluidized beds were investigated using frequency response techniques to obtain suitable models for prediction of the frequency response and measurements of the interphase heat transfer coefficient and axial conductivity of the solid phase of the bed. I n this paper, data are presented for the gas-particle Nusselt number in the particle Reynolds number range 1.92 to 99.2, which take into account the effects of both axial dispersion of heat ip the gas phase and axial conduction in the solid phase. Beds of copper, lead, and glass were used with particle sizes ranging from 0.0198 to 0.080 inch in diameter (Table I). I n addition to the Nusselt numbers, dynamic thermal conductivities of the solid phase in the axial direction were measured simultaneously. Reliable gas-particle heat transfer coefficients in packed beds are available at Reynolds numbers from about 50 to 10,000. At lower Reynolds numbers, however, data are not only scarce but their reliability is in question for two reasons: (1) Nusselt numbers well below 2 are encountered with no ap554
I&EC FUNDAMENTALS
parent limit as the Reynolds number decreases, and (2) there is an effect of the Reynolds number on the Nusselt number, which is difficult to justify in terms of a convective heat transfer process between a fluid and a solid surface. The scarcity of data at low Reynolds numbers exists to a large extent because the method of Gamson, Thodos, and Hougen (1943) is inapplicable in that range. Their method involves the vaporization of water from porous catalyst carriers in an air stream during the constant rate drying period; it fails at low Reynolds numbers because the temperature difference between the gas and particle at the outlet to the bed becomes so small that the mean temperature difference between gas and particle for the bed cannot be determined accurately. Sinusoidal input techniques have been used successfully a t higher Reynolds numbers by several investigators (Dayton et al., 1952; Meek, 1961 ; Shearer, 1962) employing Schumann’s equations (Schumann, 1929) as the model to evaluate the heat transfer coefficient. [Equations do not account for solid phase conduction or gas phase dispersion (Equations 3 and 4 with y o and ys+L equal to zero).] I n that range, the model gives
Table 1.
S.C.F.M.
Run
D,, Inch
11 25 26 27 28 29 30 31 37 38 49 50
0.0225 0,0282 0.0282 0.0282 0.0282 0.0282 0.0282 0.0282 0,0427 0,0427 0.0225 0.0198
3.069 0.260 1.230 0.570 0,260 0.570 1.230 2.340 0,260 3.100 0.981 0.260
25.8 2.76 13.0 6.02 2.74 6.02 13.2 24.9 4.16 49.6 8.31 1.93
34 35
0.0198 0.0198
0.461 0.260
3.38 1.92
u,
Bed and Flow Data
x
SL, Lb.
L , Inches
E
LID,
COPPER PARTICLES 1 .ooo 0.133 0.133 0.133 0.265 0.265 0.265 0.530 0,200 0.500 0,500 0,170
1.87 0.30 0.30 0.30 0.52 0.52 0.52 0,99 0.42 0.94 0.94 0.38
0,433 0.532 0,532 0.532 0,460 0.460 0.460 0.432 0.495 0,436 0.436 0.526
83.1 10.9 10.9 10.9 18.5 18 i 18.5 35.2 9.74 22.0 41.8 19.2
0,352 0.298 0.408 0.326 0.319 0.487
0.39 0.39
0.490 0.490
19.7 19.7
0.455 0.455
2.00 1.09
0.404 0.431
25.0 13.5
0.603 0.683
.VRe
(ffQ/$L)
10
0,317 0.503 0.499 0.499 0.375
n- z m
GLASS PARTICLES 0.056 0.056
LEADPARTICLES 40 41
0.080 0.080
3.310 1.660
SEPARATOR
99.2 49.7
1.540 0.800
PRESSURE GAGE
Figure 1.
Layout of apparatus
a n adequate description of frequency response data and the coefficients calculated give results which agree with those of Gamson, Thodos, and Hougen. At low Reynolds numbers, conduction in the solid phase and dispersion in the gas phase affect the frequency response to such an extent that the simple model does not fit (Littman and Barile, 1966): and the data presented in this work had to be interpreted using a model which accounts for both of these effects (see Equations 3 and 4). Both Kim and Brodkey (Brodkey, 1964; Kim, 1965) and Kunii and Smith (1961) have used the same model to describe packed bed heat transfer. Existing data (De Acetis and Thodos, 1960; Eichorn and White, 1952; Glaser and Thodos, 1958; Kim, 1965; Pulsifer, 1965; \t’ilke and Hougen, 1945) at Reynolds numbers below 100 are given in Figures 6 and 7 in terms of lYSu a n d j , us. N x , . St’ilke and Hougen (1945) and De Acetis and Thodos (1960) used the method of Gamson, Thodos, and Hougen to obtain their results. Glaser and Thodos (1958) maintained steady-state conditions in the bed by passing an electrical current through a bed of metallic particles and then measured the gas and solid temperature profiles. Dielectric heating was used by Eichorn and White (1952) to generate heat continuously in a bed of ion exchange particles. In all these investi-
gations? the heat transfer coefficient was evaluated from the steady-state heat balance dq = vBp3C,dT, = - h A p ( T g - T,)dt (1) or its integrated form
q =
uBPgc3(Tg,r,
-
Tgaut)
= hApL(Tg
-
TS)a\
(2)
from which
Pulsifer (1965) employed the frequency response method. H e used beds of copper particles and the work reported here was performed using a modified version of his equipment. Data at Reynolds numbers below 1 are given by Kunii and Smith (1960), who obtained them as a by-product of packed bed thermal conductivity measurements. The conductivity was related to the heat transfer coefficient using the steady-state version of Equations 3 and 4. A discussion of their coefficients and the reason they are much too low is given in the Results and Discussion section. Equipment and Procedure
The apparatus consists of a 2-inch i.d. Plexiglas column, a 250-mesh screen heater which impresses a periodic temVOL. 7
NO. 4
NOVEMBER 1968
555
perature variation on the gas entering the bed, quick response thermocouples, and auxiliary regulating, measuring, and recording equipment. Beds are made up of spherical particles of copper, glass, or lead in the size range 0.0198 to 0.08 inch supported on a layer of fine Saran filter cloth and heated by up\vard flowing gas until steady-state periodic conditions are reached. Temperatures are measured below the screen and just above the bed a t radial points which are representative of the circular planes perpendicular to the column axis. Details of the equipment and procedure are given elsewhere (Barile, 1966). An apparatus layout is given in Figure 1.
(5)
bU - = 0
atx=1
bX
-ys$L
dV
D, =
$ L ( U - V ) at x
=
0
Experimental Frequency Response
L7 and V are dimensionless gas and solid temperatures, reys@L,and spectively; the dimensionless parameters, a@r, yg represent the dimensionless frequency, heat transfer coefficient, solid phase conductivity, and gas phase dispersion of heat, respectively. Equations 3, 4, 5, and 6 are solved simultaneously to obtain the amplitude ratios and phase lags predicted by this model for given values of the parameters. yQ was generally calculated from dispersion data in the literature and @L and ySOLwere obtained from the measured amplitude ratios and phase lags. Because of the steady-state periodic boundary condition, the form of the solution of Equations 3 and 4 subject to Conditions 5 and 6 is U = A(x)eiqeand V = B(x)e'qoB. These relationships Models for Interpretation substituted into Equations 3 and 4 give: Energy Conservation Equations. CONDCCTION-DISPERSION MODEL. T h e conduction-dispersion model (C-D model) which (7) is used to interpret the heat transfer measurements has three parameters. These are needed to describe the heat fluxes d2B from gas dispersion, interphase transfer, and solid phase conyS - - (1 iq)B = - A dx2 duction. A pictorial representation of the model is as follows: The periodic inlet and outlet waves are expanded in a full Fourier series to obtain the amplitude and phase of the fundamental and the harmonics. The method is simple and is given elsewhere (Barile, 1966; Littman and Stone, 1966). At low velocities and high frequencies correction of the amplitude for heat interaction with the bed support is necessary. The correction can be as high as 307, and is one of the largest sources of error in this work. Correction of the phase lag for support interaction is minor. A run is composed of several frequency response measurements made a t constant gas velocity, each point taken at a different temperature-wave period.
oL,
+
Gas Out
i
Dbpr ion
14
Gas
The dependent variable, B , is eliminated by substituting into Equation 8 expressions for B and d2B/d,Y2as obtained from Equation 7, giving a fourth-order linear differential equation with constant but complex coefficients.
d4A
1 d3A dxd - y gdx3 -
Conduction
-
('y + ); +? ( i
+ );
d2A
+
Interphase Transf0.P
I S o w , the complete solution is:
In
A
The energy balance for the gas and solid phases involves the following assumptions: Fluid velocity independent of position No radial temperature gradients in the bulk fluid Adiabatic system Uniform void spacing in the packing Physical properties of the bed and gas independent of temperature Solid and fluid phases separately having the properties of a continuum Negligible radiation No radial temperature gradients inside a particle Segligible free convection The energy conservation equations and boundary conditions for the gas and solid phases in dimensionless form are :
(4)
556
l&EC FUNDAMENTALS
A =
ajemP j=1
and similarly 4
B =
bjemP j=1
since exactly the same differential equation is obtained if A is eliminated instead of B. The solution A = em', which is substituted into Equation 9, gives the characteristic polynomial from which the four complex roots, mj, are determined by Muller's method (Lapidus, 1962).
I
Figure 4. culate y o
0.60 -
L 1.0
0.40
A
74
= 20, i n = 0.03,
1
= 0.06,
I
! I / I I
1
0 AMPLITUDE RATIO DATA A PHASE LAG DATA
I
- 80 - 60 - 40
-
-
v)
202 0
a
a a
W
n
10
0.10-
2 0.08-
Effect of solid phase conduction in conduction-dispersion model $L
I
RUN 26
2 !z 0.20-
Figure 2. Amplitude ratio vs. dimensionless frequency 07$J D
I
I
10.0
4.0
I
Peclet numbers from literature data used to cal-
0.80-
O.O1
I l l l l l
4
-8 w
z a 0.06-
au = 0.0005
$
-6 2
3
$L
V*,,$L= ~ . O , O : . $ L = O . ~ , A ~ ~ $ L = O . ~$,~A=- 0, . 0 4 = 20, a/,, = 0,y $ L = O] - _ - - AB [$L [ $ L = 20, "/" = 0,^iA$L = m l
----
(n10.0 P
4 K "9,
Figure 5. Amplitude ratio and phase lag vs. dimensionless frequency ( v $ ~ )
W v)
U I
Comparison of experimental data (run 261 with conduction dispersion model with $L = 10.9, y o = 0.047,-/,$L = 0.084
n / I
4.0
1
I
,
10.0
7+L
Figure 3.
Phase lag vs. dimensionless frequency ( q + L )
Effect of solid phase conduction in conduction-disDersion model D +L = 20, y o = 0.03, I ' = 0.06, ?!? = 0.0005
1 +L 0 -,&L = 1 . 0 , 0 ' / , $ ~ = 0 . 4 , A - / & ~= 0.1 A Y&L = 0.04 - - _ _ A [ $ L = 20, -i= o 0, y . 6 ~= 01 - _ _ _B [ $ L = 20, Ya = 0,Y , $ L = m l
The resulting equations constitute a system of eight linear, complex-coefficient, algebraic equations in eight unknowns, u3 and b,, j = 1.2,3,4. Tliey were solved by numerical matrix inversion techniques: consisting of an adaptation to complex matrices of the Gauss-Jordan method with pivotal selection (Hamming. 1962). A typical model plot with y o = 0.03, 4L = 20, and y S o L taking values from 0 to m is shown in Figures 2 and 3. OTHERRELATED MODELS. These models are identified as: C M . Conduction mo(cie1, y o = 0 , no dispersion
DAM. Dispersion model, y s $ ~= 0, no conduction SM. Simple model, y u = 0; y s $ ~= 0, no dispersion or conduction IM. Infinite conduction model, yo = 0; ys$L = m , no dispersion, infinite conduction They have been described fully and typical model plots are available (Littman and Barile, 1966). CALCULATION OF PARAMETERS. The C-D model has three parameters (qL,y S o L , and yo) which can all be evaluated simultaneously from the frequency response data. I n this work, however, mass dispersion data in the literature were used to calculate y o . This simplifies the calculation of $t and ys$L considerably. The use of mass dispersion data for heat dispersion is justified by the analogous mechanism of transport for each phenomenon. I n addition, such data are as accurate for our purposes as are any results obtained by the more difficult procedure of evaluating three parameters simultaneously. The Peclet-Reynolds number relation used in this work is shown in Figure 4. I t was established from mas? dispersion measurements by McHenry and Wilhelm (1957) and from the interpretation of existing data by Wilhelm (1962). VOL. 7
NO. 4
NOVEMBER 1 9 6 8
557
Table II.
Calculated Results h,
Run 40 41 38 11 31 30 26 49 27 29 37 34 25 28 50 35
Material Pb Pb cu cu cu cu cu cu cu cu cu G1 cu cu cu G1 ysqi~too small ,for accurate
‘\IRe
99.2 49.7 49.6 25.8 24.9 13.2 13.0 8.31 6.02 6.02 4.16 3.38 2.76 2.74 1.93 1.92 eoaluation.
YO
+L
Y d L
0,020 0.037 0.022 0.006 0.014 0.027 0.047 0.014 0.027 0.027 0.051 0.025 0.047 0.027 0.034 0,033
25 18 20 100 60 30 11 60 11 30 9.2 20 12 31 40 40
0.005
0.010 0.014 0,005 0.030
0.050 0,084 0,020 0.20 0.14 0.43 0.15 1 .oo 0.47 1.10 0.35
;t”“ 15.9 13.5 9.3 7.2 9.8 4.6 3.3 2.3 1.6 2.1 0.91 0.79 0.78 1 .o 1 .o 0.90
IH
0.17 0.31 0.21 0.31 0.43 0.40 0.29 0.29 0.29 0.40 0.24 0.26 0,32 0.41 0.58 0.52
B.t.u./ Hr. Sq. Ft.”F. 36 32 41 58 62 30 22 18 10 14 4.0 7.7 5.4 6.7 9.4 8.7
kea, B.t.u./ Hr . Ft . F.
...
(I
0.13 0.29 0.23 0.54 0.24 0.27 0.15 0.30 0.32 0.38 0.22 0.69 0.48 0.93 0.29
An analysis for a complete run is obtained from graphs such as Figure 5, in which the amplitude ratio and phase s!iift data of run 26 are plotted against ?qL and compared to calculated answers for the C-D model. To obtain an individual point analysis, or and y S o Lfor each frequency, cross plots of solutions in the form of amplitude ratio us. ys are used. T h e analysis involves mapping several possible parametric curves on both amplitude and phase plots for one frequency, followed by a simultaneous graphical solution. In this manner, any pronounced frequency dependence is apparent and validity judgments are possible. Some frequency dependence of ysq$L was observed at loiv Reynolds number, in which case the parameters reported were averaged for a given run.
oL
Results and Discussion
I
0. I
!
I
1
I
I
I
I
I
I
I
1.0 IO REYNOLDS NUMBER
1
100
Figure 6. Nusselt-Reynolds number plot for gas-particle heat transfer data Comparison of experimental results with literature data 0 This work A. Kunii and Smith ( 1 961 E. Eichorn and White (1952); D , = 0.0109 inch C. Eichorn and White (1952); D , = 0.0206 inch D. Eichorn and White (1952); D, = 0.0259 inch E . Pulsifer (1 965) F. Wilke and Hougen (1 9 4 5 ) G. Glaser and Thodos (1 958) H. De Acetir and Thodos (1 960)
For each run, the value of the parameters #L, and yg, the gas-particle heat transfer coefficient, and the solid phase conductivity are given in Table 11. Plots of Nusselt us. Reynolds number and lx us. Reynolds number are shown in Figures 6 and 7. 558
l&EC FUNDAMENTALS
Heat Transfer Coefficients. T h e Nusselt-Reynolds number plot (Figure 6) shows that the data are in general agreement with existing gas phase heat transfer measurements in packed beds for Reynolds numbers bet\veen 25 and 100. These include the data of Pulsifer (1965), Wilke and Hougen (1945), De Acetis and Thodos (1960), and Glaser and Thodos (1958). The data of McConnachie and Thodos (1963) were not included because they scatter too much in this range. Pulsifer (1965) worked with beds of copper particles similar to those used in this work and the data reported here were obtained using a modified version of his equipment. Preliminary results using his original equipment indicated that some channeling was present and modifications ivere made to prevent it. His Q L values agree with those obtained in this work down to a Reynolds number of about 25. Despite the fact that his phase lag data were too high, @ L (and so N x u ) could be evaluated from the amplitude ratio data using the SAM, D M , and, with modification, the C M model. Glaser and Thodos (1958) have only a few data points below N,, of 100 and none belo\\ 70. Conduction and dispersion effects are minor for their beds and the agreehent between their work and ours is good. Wilke and Hougen (1945) and De Acetis and Thodos (1960) used the vaporization method (Gamson et al., 1943) which, as previously mentioned, fails at low Reynolds numbers because the mean temperature difference in the bed cannot be measured accurately. Solid phase conduction is negligible in both investigations and the effect of gas phase dispersion is difficult to assess because the beds used were 1 to 5 particles deep. In-
REYNOLDS NUMBER Figure 7. j factor vs. Reynolds number plot for gas-particle heat and mass transfer data Comparison of experimental results with literature data 0 This work A. Kunii and Smith ( 1 961 ) B. Eicharn and White ( 1 9 5 2 ) ; D, = 0 . 0 1 0 9 inch C. Eicharn and White (1952); D, = 0.0206 inch D. Eichorn and White ( 1 9 5 2 ) ; D, = 0.0259 inch E. Pulsifer (1 9 6 5 ) F. Wilke and Hougen ( 1 9 4 5 ) G. Glaser and Thodos ( 1 9 5 8 ) H. D e Acetis and Thodos ( 1 5'60) 1. Bar-llan and Resnick ( 1 9 5 7 ) ; gas phase mass transfer, D, = 0.0146 inch J. Bar-llan and Resnick ( 1 957); gas phase mass tronsfer, D, = 0.01 8 9 in.ch K Wilson and Geankoplis (1966); liquid phase mass transfer, E = 0.40
clusion of the gas phase dispersion would make the difference bet\veen their results ancl those presented in this work larger than shown in Figure 6. I n our opinion the data obtained using the frequency response measurements are more reliable than similar data obtained using the vaporization method a t Reynolds numbers below about 100. Only the data of Eichorn and White (1952) are available for comparison purposes at Reynolds numbers between 2 and 25. Their data cover the range from 2.77 to 17.4 using ion exchange resin particles 0.0109 to 0.0259 inch in diameter. For the 0.0109-inch particles a t a Reynolds number of 2.77, their Nusselt number is a factor of 10 below that obtained in this work (Figure 6). Such a discrepancy cannot be accounted for by conduction or dispersion effects, since y S + L is about 0.01 and ys about 0.005. From their heat balance,
At these low Reynolds numbers, their method should fail for the same reason that the method of Gamson, Thodos, and Hougen fails. If Nsu E 1 as our data indicate, $ L for run E-6 of Eichorn and White (D, = 0.0109 inch, N R e = 2.77) would be 187. Using this result, tht: temperature difference between gas and particle a t the outlet would be 0.185' F . Eichorn and White's figure is 2.2' F., which, of course, accounts
for their low Nusselt number. Naturally it is not possible to prove that their outlet temperature differences are in error, but it seems so for the reasons given above and because a Nusselt number of 0.1 seems too low from a theoretical point of view. Furthermore, at higher Reynolds numbers and higher values of D J L , better agreement between the two investigations does result, primarily, in OUT opinion, because ( T o - Tp)outis larger and the error is smaller. Finally, our results do not show any large particle size effect as was found by Eichorn and White. Kunii and Smith's Susselt numbers at Reynolds numbers from 0.1 to 1 are about t\vo orders of' magnitude below those obtained in this \vork at a Reynolds number of 2. I n adtlition, their coefficients appear to be dropping off \vith the square of the Reynolds number, \vhile those obtained in this ivork appear to be becoming independent of Reynolds number. Kunii and Smith obtain their heat transfer coefficients indirectly from measurements of the axial gas temperaturt: profiles. These profiles are first used to evaluate the effective thermal conductivity of the bed, assuming that the gas and particle temperatures are the same a t any point in the bed. This single-phase model demands an exponential gas temperature profile (xvhich the measured profiles satisfy) dependent only on the effective thermal conductivity of the bed. They then develop an algebraic relationship between this conductivity and the heat transfer coefficient using the steady-state version of the C-D model (Equations 3 and 4) and their measured exponential gas temperature profile. The heat transfer coefficients are finally calculated from this relationship. Their procedure is not mathematically correct, because the C-D model leads to a fourth-order ordinary differential
