Particle Diameter Ratio

20 Heat Transfer in Packed Beds of Low Tube/Particle Diameter Ratio

Downloaded by KTH ROYAL INST OF TECHNOLOGY on February 13, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch020

A. G . D I X O N and W . R. P A T E R S O N Department of Chemical Engineering, University of Edinburgh, Edinburgh, Scotland D. L. C R E S S W E L L Systems Engineering Group, E . T . H . Zentrum, CH-8092 Zurich, Switzerland

1.

Introduction

In spite of much research (1-7), identification of the rele vant heat transfer parameters in packed beds and their subsequent estimation continue to provide challenging problems, especially so for beds having a small tube to particle diameter r a t i o , where so few experimental data are reported. The aims of this paper are as follows: (1) to rigorously evaluate homogeneous continuum models as applied to heat transfer in beds of low tube to particle diameter ratio (typically d / d - 5 - 1 2 ) . (2) to examine the effect of gas flow rate, particle size and conductivity on the estimated heat transfer parameters. (3) to progress towards a priori prediction of the important para meters by modelling the underlying heat transfer mechanisms. t

2.

p

Experimental Equipment and Procedure

The experimental apparatus is similar to that used by Gunn and Khalid (7), although temperature measurement is different. The bed consisted of two sections of internal diameter 70.8 mms, insulated at the plane z=0 by a sandwich of rubber and PVC gaskets (see fig. 1). Twelve thermocouples were inserted r a d i a l l y through the central PVC gasket to provide duplicate temperature measurements at six radial positions, distant 12,16,23,28,31 and 34 mms. from the central axis. Both sections were packed with similar s o l i d particles to provide a continuous length of packing. The experiments consisted of measuring the temperature distribution in the a i r stream leav ing the top of the packing in the heated section (b) for a range of gas velocities by means of 32 Cr/Al thermocouples (30 SWG) supported at 8 radial positions (r=9,11,13,18,24,29,31 and 33 mms) and at 90° intervals by a brass cross, similar to that used by ©

0-8412-0401-2/78/47-065-238$05.00/0

In Chemical Reaction Engineering—Houston; Weekman, Vern W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

20.

DIXON ET AL.

Heat Transfer in Packed Beds

239


deWasch and Froment (6J. The procedure was repeated for a range of bed heights. The temperature uniformity of the a i r stream entering the calming section (a) and of the heated wall (b) was checked by additional thermocouples. Altogether, 107 separate experiments were performed (plus several repeats) covering the range of variables shown in. Table 1.

Table 1 : Ranges of Experimental Variables

d

p

(mms)

12.,7 9..5 6..4 9.5

d /d t

p

^

(

i

B

f

t

e

d

H e i

9

h t

L

( )

(

c m s

)

ceramic beads k0 by z

]

c

f

= ^ = 0

/

at r=0

-k |i=h r

/

w

( t - t ) , r=R, w

z>0 (2)

3t

/ -K-ww · ' Boundary conditions in the axial direction follow readily for the long calming section and the adjoining plane (z=0): h

t - t„ ·,

ζ» -

t -t,

%

c

c

r = R

z < 0

(3)

= ||

=0

;

(4,

Z

The boundary condition at the bed exit (z=L) is much less certain. If the bed were long we might use the condition t-t ;

z + +~

w

(IBC)

(5)

as was done by Gunn and Khalid (7). However, in many of our experiments the beds were necessarily quite short ( L « 5 0 dp), otherwise i t would have been impossible to observe significant radial temperature gradients. In these cases, an alternative condition would be §~ = 0

z=L

5

(FBC)

(6)

which assumes plug flow in the space above the packing. We have analyzed our data using both boundary conditions. For the i n f i n i t e bed boundary condition (IBC) the equations can be integrated to give for the test section (7) Vt

-

(__

) = Σ

\

0

n

2

0

where A

Bi(UA ) J ( ) 2

V

n=l A ( B i + a J . n

η

= (1+16

l

/ 6 Pe^ P e J r a 2

a

π

-Pe (A exp { — ι J (a ) °p a

n

1)2

}

ά

J


(7)

CHEMICAL REACTION ENGINEERING—HOUSTC

242 Gc dp Gc d Pe , Pe = —rf-— , ι » Ρ a r K κ a

Bi

(axial and radial Peel et numbers)

Γ

= h.R/k (wall Biot number), w r

y = r/R, 3 =

and a are the roots of c* ϋ ( α ) = Β ΐ n

n


For the t -t

Ί

η

η

(FBC), t h e s o l u t i o n becomes - Bi (1+A ) J ( a y )

A-l

-Pe exp i - - { A ρ

-

p e

a(

A

1

n

- )

z

a

Ë

+

? a

η

(Β) The Plug Flow Model

n

(2L-z)-z}}}

(Pe ==•>)

A special case which has received much attention (1-2, 4-6) is the plug flow model, resulting from Eqn (7) in the limit Pe ~*»: a

09

Λ"*

5.

B i

2

-4a z

Data Analysis The models were subjected to two stages of analysis:

(A) Overall.Analysis A stringent test of the models i s provided by f i t t i n g them simultaneously to data measured at several bed depths. In the axial dispersion model, the parameters Pe , Pe and Bi were est ated by minimising the sum of squares of residuals on the 32 be exit temperatures: a

Ν F

=;

32 ;

r

9

(texp.0 - W o )

o°

where Ν is the number of bed depths; t ] p is calculated from either Eqn. (7) or (8), depending upon which boundary condition is adopted at the bed exit, at the appropriate radial measuring points for z=L. c a

Ç j

(B) Depth_bY_Degth_AnalYsis The a b i l i t y of the models to f i t the data at individual bed depths (N=l) was next examined in order to detect any trend in t parameters with bed depth. The non-linear function minimisations of Eqn.(10) and i t s simpler cases were carried out by the Marquardt search algorithm


20.

DIXON

ETAL.


243

(Fortran sub-routine E04 FBF NAG library, NAG Ltd., Oxford). A preliminary grid search was made to check for irregularities in the sum of squares surface and provide a starting point for the search.


Previous analysis of experimental errors (8) substantiated the validity of the unweighted least squares criterion (]0) for estimation of the model parameters. 6.

Evaluation of Models Results of Depth by Depth Analysis

Neither model showed significant lack of f i t to the data at the 95% confidence level. However, the plug flow model parameters were found to decrease systematically with increasing bed depth. Fig. (3) shows this effect quite clearly in the case of the effective radial conductivity. No such effect was observed with the axial dispersion model, as i s apparent from Fig. (4). DeWasch and Froment (6) also noted the dependence of the plug flow model parameters with bed depth. They therefore only correlated their data obtained on the longest beds hoping to minimize axial dispersion effects. However, i f Figs. (3) and (4) are superimposed, the estimates of k obtained from the axial dispersion model are significantly greater than those obtained on the longest bed using the plug flow model, even at the quite large Reynolds numbers of industrial practice. The two sets of estimates ultimately merge at large Reynolds numbers. No doubt the differences would have been even greater had a larger bed been used. r

This behaviour of the plug flow model may be a significant factor in explaining some of the scatter in literature correlations obtained on beds of different length. Results of Overall Analysis When a l l the bed depths were analysed simultaneously, the plug flow model was clearly rejected for a]J[ the different particles and Reynolds numbers considered. The ratio Fcalc/Fo.05 found to be between 1.5 and 8, where F ] is the estimated F ratio from analysis of variance and FQ.05 is the appropriate s t a t i s t i c at the 5% significance level. w

c a

a

s

c

For a l l the beads the axial dispersion model (Eqn. 7) showed no significant lack of f i t at any Reynolds number, F Ç ^ C / F Q lying between 0.4 and 0.9. Fig. (5) shows a typical t i t of this model. Fig. (5), however, i s unusual in that the calculated and experimental entrance profiles (z=0) agree well. In the majority of cases this was not so, which we attribute in part to unsatisfactory measurements at the entrance.



244

CHEMICAL

SL5mmcERAMic

à

too

Figure 3.

ENGINEERING—HOUSTON*

SPHERES

—ièo Ν

REACTION

RE

ώο -

Correlation of k with bed depth for the plug flow model r


1

DIXON

ET AL.



20.


245


Figure 5. Fit of axial dispersion model to angular smoothed radial temperature profiles


DIXON E T

20.


AL.

247

Incorporation of the f i n i t e bed boundary condition into the model (Eqn. 8) generally led to a poorer f i t , in many cases leading to a significant lack of f i t . Its main effect was to increase estimates of Pe by some 10-20%, the other parameters Pe and Bi remaining virtually unaffected.


a

r

It was observed that, in general, the model f i t is worst at low Reynolds number and improves progressively as the Reynolds number increases. This is probably due to the d i f f i c u l t y of measuring the development of the radial temperature profile, since at low Reynolds number the bed attains the wall temperature within a few particle diameters of the entrance (z=0). A typical cross-section of the parameter cross correlations is shown in Table 2. Table 2:

Typical Parameter Cross-Correlations: Ceramic Beads: Bed Depth 17.5 cms.

'Re

Vw -0.72 -0.77 -0.82 -0.91

535 430 290 140

r* a

Vw

-0.10 -0.05 +0.06 +0.34

-0.11 -0.07 -0.07 -0.26

k

h

12.7 ntn

h

k

Estimates of (k ,k ) and (k »h ) are virtually uncorrected except possibly at low Reynolds number; those for (k hw) strongly correlated at a l l Reynolds numbers. While k i s not a conductivity in the true sense, i t nevertheless has a sound theoretical basis, as proposed by Argo and Smith (9_); h on the other hand is perhaps no more than an empirical parameter needed in the model to account for a decreasing k near the wall. r

a

a

w

a r e

rJ

r

w

r

7.

Correlation of Heat Transfer Data

Analysis of the data revealed that the radial conductivity (k ) is of particular importance. It would be desirable, therefore, to develop a model which gives a priori prediction of this parameter in terms of flow rate, particle diameter and conductivity and compare the predictions with our experimental data. r

7.1

A Model for Prediction of the Radial Conductivities

Starting along the lines of Argo and Smith (9J, the radial conductivity is given by k

r

= k

q

+ k, , + k . td series American Chemical Society Library

In Chemical Reaction Engineering—Houston; 1155 16th St.» M.W. Weekman, Vern W., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978. Washington, O.C. 20036

(11)

CHEMICAL REACTION ENGINEERING—HOUSTON

248

where k , k j and k · represent the molecular conductivity of the f l u i d , the turbulent conductivity and the effective conductivi ty of the solid, a l l based on unit of void + non-void area. g

tc

$

s

The turbulent conductivity k^d i s given in terms of the Rey nolds, Prandtl and Peel et numbers by td m = Re Pr rm < ) where k is the molecular conductivity of the f l u i d . From turbuleni mixing data (10), Pe = 10 for N R > 4 0 , and for a i r Np =0.72. Thus, Eqn. (12) simplifies to / k


k

N

N

/ P e

12

m

rm

td'

k

k

=

m

g

°-

0 7 2

N

f

N

Re

( Re

> 4

13

°)

Heat transfer between contacting particles i s assumed to occur by a static process controlled by conduction across stagnant gas f i l l e t s at the point of contact, and a dynamic process involv ing a series mechanism of solid conduction, film convection and turbulent mixing,as in Fig. (6). The static and dynamic processes occur in p a r a l l e l , thus k

series

=

k

st

+

k

( 1 4 )

dyn

The static contribution can be measured experimentally (Y\J or estimated from the model of Kunii and Smith ( 1 2 ) . The dynamic term is obtained by f i r s t integrating the heat flux over the hemispher ical surface between e=0 and θ=90° in Fig. (6). After some algebra, the total heat flow is given by 0 Q

P td - φ )

2 π

-

T

"

R

k

β

{

_J_ Vl

1 η β

.

/dL ' dH

1 }

1 Π β

, }

(

Π5) f l u i d

( , b )

where B=hk / k V j (h+k /R ) and (dT/dr) -j 4 is the temperature gradient in tne f l u i d in the direction of heat flow (assumed linear). Eqn. (15) enables an effective conductivity k^yn to be defined, based on solid projected area i T R p , f

U

d

2

'dyn

•

k

T^frtFr"*-"

(,6

»

For a packed bed, Eqn. (16) must be modified to account for the bed voidage and for the number of contacts (n) a pellet makes with i t s neighbours, corrected for the cross-sectional areas norma to the direction of heat flow and for the frequency of the orient ations. Assuming an actual bed is a composite of loose and close packings then, according to Kunii and Smith (12), n-2 for beds of voidage ε=0.44 to 0.46, as measured in our studies. Thus,

dyn = ^ t d T F i j i F T ^ -

k

1

1

}


( 1 7 )

20.

DIXON E T A L .

Heat

Transfer

in

Packed

249

Beds

Eqns. (11), (13), (14) and (17) permit a priori prediction of k in terms of the underlying heat transfer processes. No adjustable parameters are involved. A comparison of this model with our data is shown in Fig. (7). Static conductivities were measured separate ly using Sehr's electrical heating method (Vl_) and the correlation of DeAcetis and Thodos (13J was used to estimate h. Downloaded by KTH ROYAL INST OF TECHNOLOGY on February 13, 2016 | http://pubs.acs.org Publication Date: June 1, 1978 | doi: 10.1021/bk-1978-0065.ch020

r

The results show an encouraging agreement over a wide range of flow rate,particle size and conductivity. In particular, i t is found that (a) k increases linearly with N for N >40 but does not extra polate linearly to the static results. (b) k is virtually independent of pellet diameter. (c) k„ is only weakly dependent on pellet conductivity (k ) - a 100-fold increase in k produces a 50% increase in k r

Re

Re

r

s

s

r

Also shown in Fig. (7), for comparison, is the contribution to k due to turbulent conduction (k^d). It is apparent that heat transfer through the solid forms a significant, i f not dominant, fraction of the total radial heat transfer within the Reynolds number range of interest. r

7.2

The_Wall_Biot_Number

If the data are plotted as (Bi) χ (d /dL)^vs. N then the results for different particle sizes and conductivity are brought together on a single curve, at least to within the scatter of the data. The results show that the Biot number decreases with Reynolds number, according to (see Fig. 8). ι r Q κι -0.262 ,„ (Bi) (dp/d )J = · * (18) Re

ΛΧ

5

3

t

R e

which correlates the data to within 15% in the range 100

Particle Diameter Ratio

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