(1983) Heat Transfer to Packed and Stirred Beds (NO)

Prévia do material em texto

31 
Heat Transfer to Packed and Stirred Beds 
from the Surface of Immersed Bodies 
Wkmeiibergang zwischen ruhenden und mechanisch durchmischten 
Schiittgiitern und darin eingebetteten Heizfkhen 
E. U. SCHLiiNDER 
Institut fiir Thermische Verfahrenstechnik der Universittit Karlsruhe (TH), Karlsruhe (F.R.G.) 
(Received November 10,1983) 
Abstract 
The heat transfer between packed and stirred beds and immersed surfaces is controlled by the contact resistance at 
the surface followed by the heat penetration resistance of the bulk. Both resistances can be predicted from model 
equations with sufficient accuracy. The contact resistance and the bulk penetration resistance for packed beds follow 
from physical properties, while the prediction of the bulk penetration resistance for stirred beds requires the introduc- 
tion of an empirical parameter, the so-called mixing number in order to describe the random particle motion. The 
mixing number was found to lie between 2 and 25, depending on the design of the stirrer. 
Synopse 
Die Wciimetibertragung von Heiz_fl&hen an Schiitt- 
gr5’ter verursacht Temperaturprofile, wie sie die Abb. 1 
ze$t. Danach fdllt die Temperatur an der HeizJZiche 
fast sprungartig von Tw auf T@ Im Innern des &h&t- 
gutes dagegen bildet sich ein stetig geknimmtes Profil 
aus. Der integrale Mittelwert tiber dieses Profil liefert 
die kalorische Mitteltemperatur des Schiittgutes TB. 
Dieser Sachverhalt gibt Antiss, zwei in Reihe liegende 
Wtirmetibergangswiderst&nde zu definieren, den sog. 
Wandwiderstand 1 /cu, , GI. (31, und den sog. Kemwider- 
stand llcxsb, Cl. (4). Der Gesamtwiderstand I/Q ist 
gleich der Summe dieser beiden, Gl. (6). 
Nach dem heutigen Stand der Kenntnis lassen sich die 
beiden Teilwiderstliizde Ijoc, und I /cx,~ mit einer fir 
technische Zwecke meist auseichenden Genauigkeit 
vorausberechnen. 
Der Widerstand llcu, folgt allein aus physikalischen 
Stoffwerten, der relativen Fldchenbedeckung und der 
dquivalenten Oberfl&henrauhlgkeit, wie dies im AnhangA 
dargelegt ist. 
Zur Berechnung des Widerstandes 1 /LY,~ bentitigt man 
zumichst die effektive Leitftihekeit des Schtittgutes 
h,,. Auch diese kisst sich allein aus physikalischen 
Stoffwerten sowie einem sog. Kontaktfl&herumteil 
vorausberechnen. Der Rechengang kann dem Anhang B 
entnommen werden. Ausserdem findet sich im Anhang C 
eine Aujlistung von HP 41 CV-Programmen, die es 
gestatten, cx,, und ?I,~ in wenigen Sekunden zu berech- 
nen. Die darin enthaltenen Berechnungsgleiehungen 
beruhen auf den Forschungsarbeiten, deren Ergebnisse in 
Lit. 14-16 fir 01, und fiir Xbed in Lit. 17 und 18 mit- 
geteilt wurden. 
Kennt man nun Abed, so benatigt man zur Berechnung 
von (Y.~ nach Gl. (5) das Temperaturgefdlle an der Wand 
(aTIaY)y=o, das seinerseits aus der Anwendung der 
Fourierschen Theo& der W&meleitung folgt. Man erhcilt 
aus dieser j?ir (Y,,, den Ausdruck nach Gl. (S), in der als 
weitere Variable neben der volumetrischen Wdrmekapa- 
zit& des Schiittgutes (PC),,& die Kontaktzeit des Schtitt- 
gutes mit der Hetifl&he t auftritt. 
Der Gesamtwcirmtibergangskoefftiient 01 folgt dann 
aus Gl. (9), in der die einzige Variable die bezogene 
Kontaktzeit r nach Gl. (10) ist. Bei ruhendem Schtittgut 
ist die Kontakizeit gleich der Verweilzeit des Gutes auf 
der Heizji’dche und somit durch Vorgabe bekannt. Somit 
kann der W&m&bergangskoeffuient fir ruhendes 
Schtittgut (packed beds) nach Gl. (9) zahlenmci’ssig 
ausgerechnet werden. Entsprechende Vergleich e 
zwischen berechneten Werten und gemessenen Werten 
finden sich in den Abbildungen 10 bis 21. 
Bei mechanisch durchmischten Schrittgiitern sind 
effektiwe Kontaktzeit und Verweilzeit des Schiittgutes 
an der Heizji’tiche verschieden. Die effektive Kontaktzeit 
tR ist weder vorgebbar noch messbar. Sie muss daher 
unter Zuhilfenahme eines geeigneten Modells als eine 
fiktive Grdsse eingefihrt werden, von der lediglich 
gefordert wird, dass sie eine eindeutige Funktion der 
vorgebbaren Zeitkonstanten des Mischorganes ist. 
Diesem Konzept folgend kann der Wriimtibergangs- 
koeffizient 01 fir mechanisch durchmischte Schrittgti’ter 
eberifals nach Gl. (9) berechnet werden, wenn man dort 
die Verweilzeit t durch die Kontaktzeit t, und diese 
wiederum durch die Zeitkonstante des Mischorganes 
t,ix gemdss Gl. (12) ersetzt. Die in Gl. (12) enthaltene 
sog. Mischzahl Nmix kann im allgemeinen eine Funktion 
aller geometrischen, kinematischen und mechanischen 
O255-2701/84/$3.00 Chem. Eng. Process., 18 (1984) 31-53 Q Elsevier Sequoia/Printed in The Netherlands 
32 
Grossen des aus Schtittgut und Mischorgan gebikieten 
Systems sein. Sie sollte jedoch auf keinen Fall von irgend- 
welchen thermischen Eigenschaften des Systems abhan- 
gen. Mit Hilfe dieser Kennzahl kann nun der Warmeuber- 
gangskoeffEient (Y _, nach GI. (15) berechnet werden. 
Der Index mbesagt. dass die Venveilzeit des Schuttgutes 
auf der Heizflache auf jeden Fall grosser sein muss als die 
fiktive Kontaktzeit, da anderenfalls noch kein stationriier 
Zustand erreicht ist, wie dies u.a. die Abb. 16 verdeut- 
licht. Gleichung (1.5) ist als Diagramm mit Nmix als 
Parameter in Abb. 4 dargestellt. 
Im Kapitel 3 sind nun die GI. (9) und (15) mit Ver- 
suchsergebnissen nach Lit. 6-8 verglichen. Variiert 
wurden in den Versuchen der Partikekiurchmesser von 
250 pm bis 3100 pm, das Partikelmateriol (Polystyrol, 
Glas, Bronze), der Gasdruck von IO-’ bis 760 Torr, die 
durch Ruhrer versch iedener Form mit un tersch iedlichen 
Drehzahlen eingeleitete Mischbewegung. Die Abbildun- 
gen 13 bis 15 zeigen die Elgebnisse im Vergleich mit 
der Rechnung. 
In Einzelfallen sind Abweichungen bis zu 35% zu 
beobachten, meist in der Weise, dass die berechneten 
Werte zu niedrig, also auf der ‘sicheren Seite’ liegen. 
Die Mischzahl Nmix lag zwischen 2 und 25 je nach 
Ruhrertyp. 
Angesich ts der weithiufigen Variation aller Parameter 
erscheint indessen die iibereinstimmung zwischen Rech- 
nung und Messung fir technische Zwecke ausreichend 
zu sein. Auch Vergleiche mit Messwerten anderer Autoren 
(2,3, 11, 131 die in den Abb. 19 his 21 dargestellt sind, 
zeigen eine recht gute aereinstimmung mit den voraus- 
berechneten Werten. 
1. Introduction 
Heat transfer to packed and stirred beds from the 
surface of immersed bodies causes temperature profiles 
as shown in Fig. 1. There is a sharp temperature drop 
between the surface of the immersed body, which is 
at Tw, and the surface of the adjacent bed, which is at 
To. Within the bed the temperature drops further, 
following a more or less smooth curve. Such profiles 
have been observed, for example, by Seidel [ I]. 
The sharp temperature drop at the interface led to 
the suspicion that some sort of ‘contact resistance’ 
_ H 
Packed Bed 
Fig. 1. Temperature profile in a packed bed heated from the 
surface of an immersed body. 
exists. Ernst [2], Sullivan and Sabersky [3] and others 
gave empirical correlations for this resistance, while 
Schliinder [4] presented an equation which is based on 
physical fundamentals. Such work shows why the 
existence of some sort of ‘contact resistance’ is generally 
accepted today. 
It is also accepted that the temperature profile 
within the bed may be considered as steady and differ- 
entiable, i.e. the packed bed may be assumed to be a 
continuum, to which Fourier’s law of heat conduction 
[51 
aT 
G= -Abed-_ 
ay 
(1) 
applies. Hence eqn. (1) defines the apparent heat con- 
ductivity of the packed bed, Abed. 
Following these conceptional ideas we may introduce 
two heat transfer coefficients, one associated with the 
temperature difference Tw - To and the other associated 
with the temperature difference To - T,, where T, is 
the bulk temperature of the packed bed, defined as the 
calorimetric mean temperature, 
TB= ’ 
H 
(2) 
s k)beddv 
0 
These two heat transfer coefficients are defined by 
the following equations: 
4 
CYm= 
Tw - To 
4 
OLsb = 
To - TB 
Equation (3) defines the ‘wall surface to bed surface’ 
heat transfer coefficient (Y,, and eqn. (4) defines the 
‘bed surface to bulk’ heat penetration coefficient CY,~. 
While Tn follows from eqn. (2), the heat flux at the 
interface is given by eqn. (1): 
Eliminating To from eqns. (3) and (4) we obtain the 
overall heat transfer coefficient a! : 
1 1 1 
_= _ + _~ 
(Y %Vs &sb 
(6) 
Eventually, we have to distinguish between the instanta- 
neous and the time averaged heat transfer coefficients, 
ot and OL respectively: 
1 t 
(Y=- crtdt’ 
s 
t0 
(7) 
For engineering practice we need CX, while the outcome 
of research projects is usually ot. 
33 
Equation (6) gives rise to the question whether in 
some special cases oL,u may be much larger than olW, 
which definitely simplifies the problem. In general, 
OL.,, is infinitely large as the bed is isothermal. There are 
three such cases: 
(1) during transient heat transfer, as time r goes to 
zero, then the bed is at the initial temperature; 
(2) when the heat capacity of the bed approaches 
infinity, which means that there is latent heat absorption 
owing to a strong heat sink in the bed as a result of a 
chemical reaction, evaporation, etc.; 
(3) for perfect mixing of the bed by stirring. 
These examples make clear that (Y, is identical with 
the maximum overall heat transfer coefficient max~. 
2. Present knowledge 
As pointed out above, the overall heat transfer coef- 
ficient ar (wall surface to bulk) can be predicted, provided 
that there are formulae available for the heat transfer 
coefficient cr, (wall surface to bed surface) and the heat 
conductivity of the bed hued and, last but not least, 
there is a method by which the temperature gradient at 
the interface, (dT/dy), = c, may be calculated. At present, 
we have reliable formulae for the prediction of both 
OL, and hued, at least for monodispersed beds of spheri- 
cal particles. 
As to the prediction of Abed, polydispersed beds of 
non-spherical particles are also no longer a serious 
problem. (Y, for polydispersed beds of non-spherical 
particles is still subject to current research. The state of 
the art for the prediction of both 1y and Xbed is given in 
Appendixes A and B. Consequently, the remaining 
problem is how to predict the temperature gradient at 
the interface, (dT/dy), = e . In the case of a packed bed, 
application of Fourier’s theory provides the solution. 
Still unsolved today from the theoretical point of view 
is @T/~&J), = e for stirred beds. There is still no theory 
for random particle motion in a stirred bed. At present 
we try to overcome this lack by introducing some sort 
of lumped parameter model, such as the ‘penetration 
model’, which seems to be more successful than other 
models. Fortunately we can predict at least an upper and 
a lower limit for the overall heat transfer coefficients for 
stirred beds. Figure 2 shows the overall heat transfer 
coefficient (Y versus the residence time r when the heat 
consumption is relatively large. The upper limit is formed 
by the contact heat transfer coefficient OL, which is 
a 
t upper limit 
I \ perfect stirring 
r:ig. 2. Upper and lower limits of heat transfer to a stirred bed. 
independent of time. The lower limit follows from the 
application of Fourier’s law to a packed bed, which 
yields (~,u usually decreasing with time t such that 
ash - I{fi While the lower limit represents the case of 
no stirring at all, the upper limit indicates perfect stirring, 
which keeps the bed permanently isothermal. The heat 
transfer coefficients for stirred beds are expected to lie 
within these limits. After a transition period they should 
become independent of time. Furthermore, there is a 
critical time r, at which the two limiting laws coincide. 
If the residence time of the bed at the heated wall is less 
than r, it makes no difference whether the bed is stirred 
or not, since cr is at its maximum value anyway. The 
critical tune r, can be predicted by the conditionolPb = 
CY,. It may be of the order of milliseconds up to the 
order of hours, depending on the external conditions 
(mainly on the gas pressure). 
For engineering applications ‘standard formulae’ for 
the prediction of heat transfer coefficients are usually 
recommended. In general, they are established for the 
standard boundary condition of constant surface temper- 
ature TO. In this case Fourier’s theory yields, for the 
time averaged heat penetration coefficient (Y,~ for a 
packed bed, 
(8) 
Elimination of (Y,~ in eqn. (6) then gives the standard 
formula for the time averaged overall heat transfer 
coefficient which can be written in a normalized form as 
(10) 
kx)bed 
being the dimensionless residence time of the packed 
bed at the heated wall. Putting (Y&, = cr, yields the 
standard critical residence time t,” 
(11) 
The starting point for the development of the standard 
formula for the stirred bed is eqn. (8). The stirred bed is 
assumed to be a packed bed for some fictitious period tR. 
During this period heat is transferred to the bed accord- 
ing to eqn. (9). Thereafter perfect mixing of the bed is 
assumed. This yields oscillating instantaneous heat 
transfer coefficients, as shown in Fig. 3. The time aver- 
age gives the overall heat transfer coefficient Q, for the 
stirred bed. The shorter the fictitious contact time rp, 
the closer (Y, is to ~1,. The final step is now to correlate 
the fictitious contact time fn with the time constant of 
the stirrer, tmix, which may be taken as the time required 
for one revolution. By introducing 
t 3 t, =lVmixtmir 
where Nmrx shall be called a ‘mixing number’, supposed 
to be merely a mechanical property of the system, r in 
eqn. (9) may be replaced by a product: 
0.2-a,-- 
O.l.a,,- 
Fig. 3. Instantaneous and average heat transfer coefficients for 
a stirred bed according to the ‘penetration model’. 
1.0 
a, 
0 
4 -\\u -3$ \\ / % I’0 
lc? lo-* loo lo2 
N therm 
Fig. 4. Reduced overall heat transfer coefficient olm/olws for a 
stirred bed versus Naerm with NW as parameter, according to 
the standard formula, eqn. (14). 
(13) 
N 
%a 2 *mix 
therm = (PCX)b& 
(14) 
contains the thermal properties of the system. Thus eqn. 
(9) takes the form 
fi 
(15) 
Gs 
Equation (15) is the standard formula for the prediction 
of the overall heat transfer coefficient a! for a stirred 
bed. It is depicted in Fig. 4. 
Nmix must be obtained from experimental data by a 
curve fitting procedure. It is the only empirical parameter 
in this model, One expects Nmix to be always greater 
than unity, since perfect mixing is not achieved after 
only one revolution of the stirrer. Most important, how- 
ever, is that Nmix represents solely a mechanical property 
of the system and does not depend on any thermal 
property; otherwise the ‘penetration model’ must be 
abandoned. In order to check this concept and also to 
find values for Nmix, suitable experiments have been 
carried out by Wunschmann [6 - 81. They are reported 
in the following sections. 
3. Wunschmann’s experiments 
3.1. Test equipment 
Figure 5 shows schematically the test equipment 
which consisted of a horizontal electrically heated 
copper plate (l), 240 mm in diameter and 5 mm thick, 
resting on Teflon fins supported by a plastic base (7). 
The packed (stirred) bed (2) rested on the copper plate. 
The bed height was in most cases 50 mm. The confining 
cylinder was insulated by 25 mm of polystyrene (3). 
The bed was agitated by a stirrer (4). The autoclave was 
formed by a glass cylinder (6) sealed with steel covers 
at the top and bottom (5). The autoclave couldbe 
evacuated down to 0.001 Torr. The gas in the autoclave 
was always air. The pressure was measured with a mem- 
brane gauge as well as by measuring the heat conductivity 
of the air. Both instruments were calibrated against a 
McLeod mercury gauge. Heat was supplied to the copper 
plate electrically. The maximum heat load was 400 W 
and was measured with a precision wattmeter (0.1% of 
full scale). The thermocouples were made from 0.1 mm 
Ni/NiCr with 1 mV per 24.0 K. 
-6 
-3 
-? 
,1,11//111,x,11,, 
Fig. 5. Schema of the autoclave: 1, copper plate; 2, packed 
(stirred) bed; 3, polystyrene; 4, stirrer; 5, steel covers; 6, glass 
cylinder; 7, plastic base; 8, vacuum seal. 
Figure 6 shows stirrer Nos. 1, 2 and 3. Stirrer No. 1 
consisted of three cylindrical rods, 5 mm in diameter, 
at a vertical distance of 10 mm from each other. Stirrer 
No. 2 had metal blades 15 mm wide, 2 mm thick and 
5 mm vertically apart. Stirrer No.3 had brushes on the 
lowest blade which continuously wiped the copper 
plate. Each stirrer had a radius of 115 mm (5 mm gap 
to the polystyrene insulation). 
3.2. Test material 
Table 1 shows the properties of the test material such 
as mean particle diameter d, particle density p, void frac- 
tion of the bed $, bed density pbed, particle heat 
conductivity As, particle heat capacity cs; all data are at 
room temperature. The particle size distribution was 
rather narrow. Figure 7 shows a typical distribution 
function for 1 mm glass beads. All the other size distribu- 
tion functions were similar. 
The physical properties used for the calculation of 
(Y,, and hbea are listed below: 
Particle roughness 6 = 0 for all experiments 
Temperature T = 3 15 K 
Gas viscosity vG = 1 .X8 X lops Pa s 
n 
L,=B I 
2 
3 
* 
L, 
2L - 
Stirrer No. 1 Stirrer No. 2 
Fig. 6. Stirrers used in the various runs. 
TABLE 1. Material properties 
Stirrer No, 3 
then kept at a constant value for t > 0. The temperature 
of the copper plate T,, which was always uniformly 
.cs _ . distributed, was measured and automatically plotted Material d ps J, Pbed AS 
(mm) (kg (kg (W m-’ (J kg-’ K-’ ) 
me3) mp3) K-l) 
versus time t. 
Figure 8 shows an original plot. The maximum 
temperature of the plate when the heat supply was 
interrupted was around 60 “C. The mean bulk tempera- 
ture of the bed was raised to between 10 and 20 “C, the 
total heating period did not exceed 300 s and the heat 
input was between 80 and 230 W. The total heat supply 
Qel was split into three parts: 
Glass 3.1/2.1 3000 0.4 1800 0.93 633 
l.O/OS 
0.25 
Polystyrene 1 .OS 1050 0.4 630 0.174 1255 
0.60 
Bronze 0.94 8600 0.4 5150 46.1 377 
0.50 
0.9 1.0 I 1.1 1.2 
I d. mm 
1.0205 
E%g. 7. Particle size distribution for 1 mm glass beads. 
Gas conductivity XG = 0.0275 W m-l K-’ 
Molecular mass of gas II& = 28.9 kg kmol-’ 
Accommodation coefficient 7 = 0.9 
Radiation emissivity e = 0.8 
Surface coverage factor (I~ = 0.85 
3.3. Test procedure 
At the beginning of each run the copper plate and 
the bed were at the same temperature, about 20 “C. The 
electrical heat input Qel was started at time t = 0 and 
&el = &ate * + dbed + Qloss (16) 
The first term covers the heat absorption of the plate, 
d plate = W)plate z 
the second that of the bed, 
dTB 
Qbed = (cM)bed dt 
40. 
Tw .. - OC 
‘G.x -----------,q.,,, _.+ ; ,>.f .r 
35~------ ..+? -- .Q’ 
‘i,- 
,,,.%. 
.q* 
30 
*.: 
,.; 
;-- 
,s‘ 
25. .;; 
,:’ 
...--...~: 
20- 
t.0 200 300 100 
t. 5 
Fig. 8. Original plot of plate temperature Twversus time t. 
(17) 
(If-31 
36 
and the third the heat losses to the surroundings due to expression for the instantaneous overall heat transfer 
radiation and convection, coefficient at : 
Qlos, = WTw - T- 1 (19) 
Figures 9(a)-(e) show the absolute and relative 
magnitudes of these three energy fluxes. The flux 
Qloss is always rather low compared with the two 
others. At time zero no heat could be supplied to the 
bed. Therefore the first data were evaluated at t> 10 s. 
The instantaneous overall heat transfer coefficient at is 
defined as 
QbedW 
CYt = 
41adTw - TB) 
where the bulk temperature follows from 
1 t. 
TB = cca,,o !i?bed dt’ 
s 
(21) 
Q,, -was obtained from Q,r after subtraction of Q,,, 
and Qloss. 
An error analysis, taking into account the tolerances 
of the thermocouples, wattmeters and pressure gauges, 
yielded the following ranges of confidence for each data 
point: 
t=300s: (Yt f 4.3% 
t= 40s: (Yt f 13% 
t= 8 s: fft Zt 43% 
3.4. Test evaluation 
3.4.1. Packed beds 
The standard equation (9) is recommended for 
practical applications, where the thermal boundary con- 
ditions often cannot be stated precisely. For scientific 
purposes, however, a more rigorous analysis of the 
respective heat transfer process is required. The process 
used by Wunschmann is characterized by “transient heat 
transfer between a perfect conductor with heat generated 
in it and a semi-infinite solid (= packed bed) including a 
contact resistance”. This process has been analysed 
recently by Muchowski [9]. He obtained the following 
tl 
W 
QLs = {v[a- bW(afi)-aW(b&)] taW(a&)+ 
-1 
- bW(a&) 
with a t b = 1 and ab = l/p as well as W(x) = exp(x’) 
erfc(x). The parameters are the relative heat capacity of 
the perfect conductor, 
W'b) 
and the ratio-of heat generation Q and initial heat flux 
into the bed Qbed,e, 
h 
lJ= 
%JQ’o, o - Tbed,o) 
The variable is the dimensionless contact time 
(22c) 
(224 
In Wunschmann’s experiments the initial temperatures 
Tw. O. TO. o and G.,+ were all equal and therefore the 
parameter Y was infinite. In this case eqn. (22a) reduces 
to 
-%/%v = {a - b + bW(a&)-aW(b&)} (a-b) 
1 ( 
$fi+ 
-f +l 
1 
+%W(b\/;)-$W(a& 
i 
-1 
(22e) 
with 
a = (1 +4-X5)/2, b = (1 -d-)/2 (22f) 
The data for the copper plate were pPlate = 8900 kg 
m-‘, cprate = 390 J kg-’ K-r and L,r, = 5 mm. The 
al b) 
Fig. 9. Magnitude of del, tiplate, &,d and dloss for 2.1 mm glass beads, 50 mm bed height and various pressures as well as various 
stirrer speeds: (a) p = 760 Torr, 0 rev min-I; (b) p = 760 Torr, 102.2 rev An-*; (c) p = 0.10 Torr, 0 rev min-‘; Cd) p = 0.10 Torr, 
102.2 rev min-‘; (e) p = 0.001 Torr, 0 and 102.2 rev min-‘. 
37 
heat conductivity of the bed was around 0.18 W m-l 
K-’ at normal pressure and 0.010 at vacuum, while (Y, 
varied from about 1500 W rn-’ K-’ at normal pressure 
down to 5 W me2 K-’ at vacuum. This gives /J values 
between 100 and 10. Table 2 shows that eqn. (22e) gives 
almost the same results for p = 100 as for p= 10. In 
addition, the time averaged heat transfer coefficient 
according to eqn. (9) is given in this Table. Obviously 
eqn. (9) can be used as quite a good approximation for 
eqn. (22e) if the time averaged coefficient (Y is replaced 
by the instantaneous one, C.Q. 
TABLE 2. Q~/(Y~ as a function of 7 with was parameter accord- 
ing to eqn. (22e); a/a, according to eqn. (9) is given in the last 
column 
7 %l%vs 1 
- 
p= 10 jl= 100 J;; 
1+--J; 
2 
1ov 0.92048 0.92955 0.91859 
10-I 0.80390 0.80402 0.78110 
100 0.55438 0.55580 0.53016 
10 0.27679 0.27322 0.26299 
102 0.09517 0.10227 0.10140 
103 0.0295 1 0.03348 0.03445 
104 0.00905 0.01021 0.01116 
For the sake of simplicity, Wunschmann’s data will be 
compared with the results from the equation 
Figures lo-12 show the experimental data of (Y~ 
versus time r with pressure p ranging from 760 down to 
0.001 Torr as the parameter for glass, polystyrene and 
bronze spheres of various diameters ranging from 3.1 to 
0.25 mm. The lines on the left-hand side indicate (Y, 
according to eqn. (A-l), while the lines on the right-hand 
side show CY* according to 
(23a) 
” d/n x/t >I 
which follows from eqn. (23) for large r. For short 
times t, at approachesar,, while for long timesar, accord- 
ing to eqn. (23a) becomes dominant. In particular, for 
pressures around 0.10 Torr the experimental data for 
at slightly exceed the calculated ones (up to 25%). This 
can also be seen from Figs. 13-15, where the upper 
data points were obtained from packed bed experiments. 
These deviations, however, though lying rather system- 
atically to one side, are within the experimental errors. 
At very low pressures, heat transfer to glass spheres 
is solely radiation controlled (at = arad z 5 W rn-’ K-’ 
according to eqn. (A-6)). There is no contribution by 
direct solid to solid contact heat transfer. The same is 
true for polystyrene spheres, as can be seen from Figs. 
1 l(a) and (b). For bronze spheres, however, CQ at p = 
0.001 Torr is much higher than (Y,~, which indicates 
that there might be a contribution from direct contact 
heat conduction. Assuming the relative contact diameter 
a/d to be about 3 X 10e4, an additional ACQ of about 
30 W me2 K-’ is obtained, which is in agreement with 
the experimental findings. 
3.4.2. Stirred beds 
Figure 16 shows the effect of stirring on the instanta- 
neous heat transfer coefficient at for 1 mm glass spheres 
at various pressures p. The parameter in each plot is the 
stirrer speed 2 in rev min-‘. After about 300 s, CY~ 
approaches a terminal value CY, which is independent of 
time t but strongly dependent on the stirrer speed Z = 
1 /t,ix. It can also be seen that the critical time t8 accord- 
ing to eqn. (11) becomes larger as the pressure is lowered. 
Under normal pressure tz is of the order of milliseconds, 
while at vacuum tz may reach the order of hours. If the 
residence time of the stirred bed on the heated plate is 
less than t:, stirring has no effect on the heat transfer; 
i,t is always at = a,. In practice this may be important, 
especially for the design of vacuum equipment. Diagrams 
as presented in Fig. 16 for 1 mm glass spheres could be 
given for all the other experiments. They all look similar, 
which encouraged us to plot all experimental data in the 
reduced form according to eqn. (23) for packed beds 
and according to eqn. (15) for stirred beds. The results 
are shown in Fig. 13 for glass spheres, Fig. 14 for poly- 
styrene spheres and, Fig. 15 for bronze spheres. The data 
for the stirred beds, LY~ = (Y,, have been taken after a 
residence time t of 300 s. (Since CY, is constant, instan- 
taneous and time averaged values are the same.) 
The data for stirred beds have been fitted by selecting 
a suitable value for the mixing number Nmi,. For stirrer 
No. 2 the mixing number was found to be about 23, 
while for stirrer No. 3 (with brushes) the mixing number 
was around 3. Obviously stirrer No. 3 performed much 
better than stirrer No. 2. This is to be expected since 
wiping the particle boundary layer is a most effective 
way to enhance the heat transfer. 
In general it can be seen from Wunschmann’s experi- 
ments that eqn. (23) holds for packed beds, while eqn. 
(15), which follows from eqn. (23), gives a fairly good 
description of the heat transfer for stirred beds with one 
empirical parameter, the so-called mixing number Nmix, 
to be fitted to experimental data. Wunschmann’s experi- 
ments confirm that the mixing number depends only on 
geometrical and mechanical properties of the system. 
Another possibility to check eqns. (23) or (9) is to 
compare them with the data contained in refs. 2 and 10, 
respectively, by Ernst, who investigated the heat transfer 
to a ‘moving bed’, provided that the bed moved in plug 
flow. 
4. Ernst’s Experiments 
4.1. Test equipment 
The moving bed was realized by sliding a packed bed 
down a vertical tube, 50 mm in diameter. The length of 
38 
I I I I 
100 2 5 W’ 2 5 W’ 2 5 
-fS 
I 
W' 
30.. F c , ” 
a 
I 
100 2 5 10’ 2 5 wo’ 2 5 WJ 
-1s 
10’ 
(b) O.ZJ_ 
“.s*mm 
f, P E 5 1 IQ- 
0 I ” 
a’ 2 
I 
IO’ 
5 
2 
IO’ 
5 
2 
100 2 5 W ’ 2 5 a’ 2 5 w’ 
Fig. 10. Instantaneous heat transfer coefficients LY~ versus 
residence time 1 for packed beds of glass spheres of diameters 
(a) 3.1 mm, (b) 2.1 mm, Cc) 1.0 mm, (d) 0.5 mm, (e) 0.25 mm, 
and at various pressures. (From ref. 6.) 
aJo 2 5 rn’ 2 5 la2 2 5 10’ 
-fs 
I I I I I 
100 2 5 10’ 3 5 102 2 5 WJ 
-ts 
Fig. 11. Instantaneous heat transfer coefficients CY~ versus residence time t for packed beds of polystyrene spheres of diameters (a) 1.05 
mm, (b) 0.6 mm, and at various pressures. (From ref. 6.) 
100 I 
ro4 2 5 10’ 2 5 10’ 2 5 10’ 
-t+ 
ld_ ! I I ! I I. ! ! 
W” 2 5 W’ 2 5 10’ 2 5 W’ 
-tS 
Fig. 12. Instantaneous heat transfer coefficients ~lt versus residence time f for packed beds of bronze spheres of diameters (a) 0.94 mm, 
(b) 0.5 mm, and at various pressures. (From ref. 6.) 
the heated section was varied between 5 and 100 mm 
(see Fig. 17(a)). By measuring the average velocity and 
the particle velocity at the wall, Ernst verified that 
almost perfect plug flow was established (vw> 0.95v,). 
Heat was supplied electrically. Figure 17(b) shows the 
heated section in detail. The wall temperature was 
measured at both ends of this section, which was made 
from copper. 
4.2. Test material 
The test material was quartz sand of particle diameter 
ranging from 100 m up to 800 pm. A size distribution 
is not specified, only the minimum and maximum 
diameters: 
lOO~m<d<200~m dS 150pm 
300~m<d<500~m d~400/.ml 
500~m<d<700~m ~Z?OO~rn 
For comparison with the theory, the arithmetic mean 
diameters d will be taken. The material properties were: 
ps = 2300 kg m-‘, 9 = 0.42, &,d = 1335 kg m-‘, 
cs = 730 J kg-’ K-‘, hs = 1.4 W m -’ K~’ 
-El- 
=,, 
Packed 
Bed 
t 
10-3 lUZ 103 100 IO’ 102 ‘3 10 
Packed Bed 
(a) 
-N q L L Stirred Bed 
therm ipch)bed 2 
Packed Stirred 
Cc) 
103 1u* 10-l 102 103 
- for Packed Bed 
- Nmerm = af -!- for Stirred Bed 
fPcxhm, z 
Packed Stirred 
W 
la2 10-l 100 10’ 
&t 
102 103 104 
- 
t = @cX)bed 
Packed Bed 
--) Ntbrm” (pcAjbd z L L Stirred Bed 
For the calculation of CY, and hued the following param- 
eters were chosen: 
T = 350 K, no = 20.7 X lo-’ Pa s, 
ho = 0.030 W m-r K-l, y = 0.85, e = 0.80, 
GA = 0.80, I@, = 28.9 kg kmol-‘, p = lOsPa, 
With these data, cr, and hued can be calculated accord- 
Packed Stirred 
Bed Bed 
IO- ! ! \ 
(b) 
-I 
10-3 lcs 10-l 100 10’ 102 
- &t 
103 
T= @CA),, 
Pocked Bed 
- N,,,,= & + Stirred Bed 
Packed St irrd 
0ed Bed 
103 104 
Packed Bed 
Cd) 
- Ntherm’ ad’ -!- Stirred Bed 
bcA)bed z 
Fig. 13. Terminal heat transfer coefficients CI _ for stirred beds 
and instantaneous heat transfer coefficients at for packed beds of 
glass spheres of various diameters at various pressuresp and various 
stirrer speeds Z rev min-‘. Pressure and speed of stirrer No. 2 are 
indicated by the symbols given above. (Data from ref. 6.) 
ing to Appendixes A and B. Table 3 shows the results. 
CY, was calculated for surface roughness 6 = 0, 0.5 and 
1 pm. 
4.3. Test evaluation 
It is assumed that within the heated copper section 
there was always a uniform temperature because of the 
high heat conductivity. This means that the thermal 
Packed 
Bed 
103 104 
Packed Bed 
t Stirred Bed 
GO 
5 aws 
Packed 
Bed 
t 
lo-2 lo-1 100 
- &5g2 
103 104 
Packed Bed 
(b) 
--C N = a’s 
therm IpcX Ibed 
+ Stirred Bed 
Fig. 14. Terminal heat transfer coefficients no. for stirred beds Fig. 15. Terminal heat transfer coefficients cxco for stirred beds 
and instantaneous heat transfer coefficients at for packed beds 
of polystyrene spheres of various diameters at various pressures 
and instantaneous heat transfercoefficients tit for packed beds 
p and various stirrer speeds Z rev min-‘. Pressure and speed of 
of bronze spheres of various diameters at various pressures p 
stirrer No. 2 are indicated by the symbols given opposite. (Data 
and various stirrer speeds Z rev min-‘. Pressure and speed of 
from ref. 6 .) 
stirrer No. 2 are indicated by the symbols given opposite. (Data 
from ref. 6 .) 
2L 
4vs 
Pocked 
Bed 
t 
4 
100 10’ 102 103 104 
&?AL 
(PcX)bed 
Pocked Bed 
- Ntherm= (pcX Ibed z 
L L Stirred Bed 
(a) 
a, 
aws 
Packed 
Bed 
t 
lo-’ 1DD 10’ 10” 103 104 
- T_ a 
hA)b,d 
Packed Bed 
(b) 
= L L Stirred Bed - Nth=m (pcXlbed z 
Zlrpm 0 12,7125,6150,6176,31102 1157 
P/mm Hg a, la,, a, laws 
760 l •1*1.1.-1~1+ 
TABLE 3. Abea and olws for surface roughness 6 = 0.5 and 1 I.rm boundary condition was that of constant wall tempera- 
ture even though there might have been a constant rate 
d (m) hbed am (W me2 K-‘) of heat generation per unit length of the section. This 
(W rn-’ K-l) 
6=0 6=lMm S = 0.5 flrn 
me’ans that the heat flux from the wall into the moving 
bed was not constant downstream. We have to admit 
150 0.193 
400 0.197 
600 0.198 
3169 2122 2515 that it is very difficult to find out which boundary con- 
1427 1026 1214 dition really holds for these experiments. Fortunately, 
1018 749 880 as we have seen in 93.4, the boundary conditions do not 
effect the heat transfer coefficients very much. There- 
42 
102 
5 
2 
10’ 
I a, 
I I I I I I I 
10’ 2 5 102 2 5 103 
t. a 
10; 
5 
at 
W 
mZK 
2 
10’ 
5 
2 
loo 
(e) 
p = 0.1, Torr 
103 
m2K 
2 
5 
2 
10’ 
IO’ 
5 
at 
W - 
m2K 
2 
10’ 
5 
2 
10’ 
10: 
5 
at 
W 
iPi? 
2 
10’ 
5 
2 
100 
-., 
--t I I I I I I I 
10' 2 5 102 * 5 
t. s 
(0 
t-r b = 0.01 u. 0.001 Torr _ I I I a, act. l q. 23 a 
Fig. 16. Instantaneous heat transfer coefficients (it versus residence time t for a stirred bed of 1 mm glass spheres at various pressures 
and various stirrer speeds Z rev min-t. (Data from ref. 6.) 
r :: * 
t 
moving bed 
(a) (b) 
Fig. 17. Test equipnent for moving bed heat transfer according 
to Ernst 12, IO]. Dimensions are in mm. 
fore, Ernst’s data may be compared with the standard 
equation (9) which applies to time averaged heat transfer 
coefficients. Figure 18 shows the experimental results 
according to ref. 10 for quartz sand of three different 
particle diameters in the form OL versus contact time t. 
Figure 19 shows the same data plotted in the reduced 
form CY/oI, versus 1. The best fit is achieved assuming 
the surface roughness to be 0.5 /.ur~. Nevertheless, the 
deviations seem to be systematic such that (Y is over- 
predicted for fine particles and underpredicted for the 
coarse ones. The deviations, however, are within the 
experimental errors. In general, Ernst’s data confirm the 
applicability of eqn. (9) under normal pressure. 
Another investigation under normal pressure with 
very fine powders ranging from 400 to 4 pm was 
published by Harakas and Beatty [l l]. Their results 
will be compared with eqn. (9) in the next section. 
5. Experiments by Ha&as and Beatty 
Harakas and Beatty [l l] used a moving bed which 
flowed past a submerged flat plate. Thus the contact 
time can be calculated from the bed velocity and the 
plate length. Since these authors measured time averaged 
heat transfer coefficients at nearly constant wall temper- 
ature, their results may be compared with eqn. (9). 
5. I. Test equipment 
The basic apparatus consisted of an electrically heated 
surface immersed in a bed of solid particles contained in 
a rotating trough. The entire apparatus could be enclosed 
in a steel bell jar (dome), of diameter 30 in. and height 
30 in., thus permitting control of the interstitial gas 
around the particles. The trough was driven by a shaft 
equipped with a vacuum seal. The heat transfer surface, 
immersed vertically in the moving bed, was supported 
from a frame mounted rigidly on the base plate of the 
bell jar system. The length was set tangential to this 
43 
circle of rotation. This provided essentially zero angle of 
incidence, that is parallel flow, of the solid particles past 
the heat transfer surface. A plough-like mixer was placed 
in the moving bed 18O’around the trough circumference 
from the heat transfer surface. This mixer virtually 
eliminated radial temperature gradients so that a single 
thermocouple, 3 in. upstream from the heat transfer 
surface, could be used to measure the bed temperature. 
5.2. Test material 
The solid materials used included glass beads, 
powdered alumina, powdered mica, and celite. The two 
nearly uniform sizes of glass beads were 147 pm and 
a 
W 
mZK 
Fia. 
t. s 
18. Wall to bed heat transfer coefficients OL versus residence 
time f obtained by Ernst [2, lo]. 
100 
a 
t 
rd 
1oZ 
102 10-l IO0 10 102 103 104 
a* t 
t:WS 
(@)b,, 
Fig. 19. Ernst’s data in the reduced form ol/~l~= f(r). The full 
curve is from eqn. (9). 
381 pm in diameter. The alumina powder had a narrow 
particle size distribution with an average particle size of 
43 h. The mica had a nominal particle size of 14 m 
and the celite 3 -5 pm. The pertinent physical properties 
of these materials are listed in Table 4. The interstitial 
gases used were helium, air and dichlorodifluoromethane. 
For the physical properties under normal pressure 
(10’ Pa) at 350 K see Table 5. 
TABLE 4. Physical properties of the solid material 
Particle diameter d (pm) 
Bulk density pbeu (kg me3 1 
Parttcle therm. conduct. hi&l 
Particle heat cap. cg (.I kg- ) 
-1 K-1 
1 
Void fraction $ 
Radiation emissivity e 
Relative flattened particle-surface contact 
area @g, eqn. (B-l) 
Surface coverage factor @A, eqn. (A-l) 
Surface roughness S (Mm) 
Celite Mica powder Alumina powder 
~~ 
4 
160 
830 
10.2 
0.90 
0.80 
0.005 
14 43 147 381 
834 957 1500 1500 
418 836 753 753 
15.8 210 1.05 1.05 
0.14 0.71 0.40 0.40 
0.80 0.80 0.80 0.80 
0.0005 0.0004 0.0077 0.0077 
0.65 0.65 0.65 0.65 
0 0 0 0 
0.40 
0 
__~ 
Glass beads Glass beads 
TABLE 5. Physical properties of the interstitial gases 
Air He CF,Cl2 
Accommodation y 0.85 0.3 0.90 
Heat conduct. 0.028 0.160 0.111 
hG (W m-t K-t) 
Viscosity nG (Pa s) 1 .o x 10-s 2.1 x10-s 1.35x10-5 
Molar mass 28.9 4 120.0 
(kg kmol-‘1 
_ 
5.3. Test evaluation 
Botterill et al. [12] reported on moving bed heat 
transfer at normal pressure. Their data will be compared 
with eqn. (9) in the next section. 
The experimental data are plotted in the reduced 
form o/o, versus r =cr,‘r/(p~)l)r,~ in Fig. 20. The 
essential parameters such as (Y, and Xr,, were calculated 
according to Appendixes A and B using the physical 
properties listed in Tables 4 and 5. Figure 20 reveals 
fairly good agreement between experiments and theory. 
Around r = 1 the two heat transfer resistances l/au, and 
l/c~~r, are of the same order of magnitude, while for 
r 2: 10’ only I/Q, becomes rate controlling. At r < 10 
eqn. (9) underpredicts a! by about 25% compared with 
the experimental data. This can also be seen from 
Wunschmann’s data presented in Figs. 13 and 14. It may 
be that the assumption of surface roughness 6 = 0 does 
not precisely reflect the true situation. However, the 
o-data of Harakas and Beatty in no case exceed the upper 
limit, which is given by crws according to Appendix A. 
6. Experiments by Botterill et al. 
Botterill and his collaborators mainly investigated 
fluidized bed heat transfer.. In ref. 12, however, they 
report on experiments with moving beds. Their test 
equipment was similar to that used by Ernst [2, lo]. 
The bed slid down insidea vertical tube, a narrow ring 
section of which was heated electrically. Thus residence 
times as short as 200 ms were achieved. They measured 
time averaged heat transfer coefficients for beds of glass 
and copper spheres respectively, ranging from 110 to 
745 pm in diameter. The interstitial gas was air at 
normal pressure and temperature. 
With the appropriate physical property data, the heat 
transfer coefficient o, and the heat conductivity hbed 
could be calculated from Appendixes A and B, respec- 
tively . 
lo2 10-l loo 10’ lo2 lo3 loL 
a&t 
- ‘=o,,, 
Fig. 20. Wall to bed heat transfer coefficients for moving beds 
according to Harakas and Beatty [ 111, presented in the reduced 
form o/a, = f(r). The full curve is from eqn. (9). 
In Fig. 21 the data of Botterill et al. are compared 
with eqn. (9). Agreement is found within the limits of 
Sullivan and Sabersky [3] also reported on experi- 
ments with moving beds. The results will be discussed 
experimental error. 
in the next section. 
7. Sullivan and Sabersky’s experiments 
Sullivan and Sabersky [3] investigated a vertically 
downward moving bed within a rectangular channel. 
A pair of heated copper plates was imbedded on opposite 
sides of the channel. They measured time averaged heat 
transfer coefficients for beds of glass beads, finegrained 
45 
1oc 
(1 
a,, 
t 
10-l 
0 293 2495 0.171 
Copper beads 
10” 
10-2 10-l IO0 10’ 102 
a: 
_T=A 
[PC 
TABLE 6. Physical properties and test evaluation 
Glass beads Mustard Fine- 
seed grained 
sand 
d (crm) 
f’bed resting 
(kg me3) moving 
a* resting 
moving 
&bed resting 
(W m-t K-t) moving 
Y* 
ows moving bed 
(W n?K-‘) 
ows, theor packed bed 
(W mm2 K-l) 
Gaseous gap due to bed 
expansion (Mm) 
330 1346 2160 203 
1600 1800 800 1700 
1500 1700 700 1500 
0.60 0.59 0.59 0.63 
0.73 0.71 0.74 0.91 
0.211 0.225 0.156 0.519 
0.208 0.190 0.138 0.348 
0.022 0.058 0.062 0.042 
622 216 146 545 
\ 
A 
104 
bed 
Fig. 21. Wall to bed heat transfer coefficients for moving beds 
according to Botterill er al. [ 121 in the reduced form @/LX, = 
f(7). The full curve is from eon. (9). 
1493 457 305 2225 
7 10 12 18 
sand and mustard seed at normal pressure and tempera- 
ture. The particle diameters ranged from 203 to 2 160 m. 
The moving bed velocity was varied from 0.5 to 5 cm 
S -l, thus realizing residence times from 0.5 to 5 s. 
In one respect the test section used by these authors 
differed from those used by Ernst or Botterill in that its 
cross-sectional area was reduced by inserts. Consequently, 
the moving bed was accelerated when entering the test 
section. This may explain why these authors report a 
considerably higher void fraction $* of the moving bed 
in the test section than that of the packed bed at rest 
(see Table 6). The void fraction I&* was defined as the 
ratio of interstitial volume to solid volume. It corre- 
sponds to the commonly used void fraction $, which is 
defined as the ratio of interstitial volume to total volume 
by the equation 
$ = jl*/(l + JI*) (24) 
The authors correlated their experimental results by 
On the other hand, (Y, may be calculated from 
Appendix A for the packed bed at rest. With zero surface 
roughness S and a surface coverage factor of GA = 0.75 
one finds considerably higher values than have been 
found in these experiments (see Table 6). This gives rise 
to the presumption that, owing to the bed expansion in 
the test section, the particles did not always slide down 
in direct contact with the heating wall, thus allowing for 
a certain time averaged gaseous gap between particles 
and wall. Calculation of the gap width (which formally 
would be identical with the surface roughness a-see 
Appendix A) from the experimental data yields some- 
thing between 7 and 18 p. 
An orthorhombic packed bed of spheres has a void 
fraction +* = 0.654. Expansion of the lattice by 2% 
gives J/*= 0.755, while at 5% expansion we get IL*= 
0.914; 2% of 330 p is just 7 m (glass beads), while 
5% of 203 p yields 10 pm (fine-grained sand). It can 
be seen that this is at least the correct order ofmagnitude. 
However, because of these imponderabilities any further 
analysis seems inappropriate. 
(25) 
1 
N”=- \/;; 1 
y*+L-- 
2* 
where 
Nu=$ and Pe = z (pc)bed 
8. Experiments by Gloski et al. 
Gloski et al. [ 131 measured heat transfer coefficients 
(Y between a narrow flat plane and a packed bed of glass 
spheres around 1 mm in diameter at normal pressure 
and temperature with air as the interstitial gas. The 
contact times ranged from 20 to 200 ms. They found a! 
to be almost independent of the contact time. Evalua- 
tion of the experiments yielded the value of 12 for the 
Nusselt number Nu = cud/X,. They also report that the 
surface coverage factor GA was around 0.5 in their 
experiments. Since neither (Y nor Nu was found to 
depend on the time, the experimental values may be 
identified with those for (Y, according to Appendix A. 
With $n =0.5 and d = 1 mm, eqn. (A-l) yields Nu = 
Abed Abed 
L being the length of the heated copper plates. The 
empirical parameter y* was fitted to the experimental 
results and is listed in Table 6. With &,& m,,vrng a$ given 
by the authors* and plate length L = 15.2 mm, eqn. (25) 
can be evaluated so as to extract OLD, which is the upper 
limit as Pe goes to infinity: 
1 Xbed,movin.s 
%vs, exp = - 
YY L 
(26) 
The results are listed in Table 6. 
*The ABd as measured by the authors and listed in Table 6 
seem to be rather high. They are only used, however, to 
recalculate 01 from Nu and y* . 
46 
cu,d/X, = 13.7 for S = 0 (zero roughness) and Nu = 11.2 
for 6 = 0.5 pm. The experimental value lies in between. 
Step 2. Calculation of hubed 
Print-out of HP 41 CV: 
9. Conclusions 
Heat transfer between packed beds, moving beds, 
stirred beds and immersed surfaces can be predicted 
from the physical properties of the solids and gases, 
provided that the actual surface area, the equivalent 
surface roughness and the actual contact time are 
known. In most applications the actual surface area is 
around 80% of the geometrical area. The equivalent 
surface roughness can be assumed to be zero in most 
cases; if at all necessary, 1 pm would be quite a good 
guess. The actual contact time is identical with the 
residence time for packed beds and for moving beds as 
well, provided plug flow exists. For stirred beds the 
actual contact time may be correlated with the stirrer 
speed by an empirical function, the so-called mixing 
number. This mixing number was found to be of the 
order of 20 for stirred beds in a bench-scale autoclave. 
It is a mechanical property of the system and depends 
on the stirrer design as well as on the particle frictional 
characteristics. 
Predicted and experimental heat transfer coefficients 
from various sources agree fairly well, to about *25%, 
within the following parameter variations: 
Particle diameter 4pm<d<3100pm 
Pressure 1 0m3 Ton < p < 760 Torr 
Particle material polystyrene, glass, sand, alumin- 
ium, bronze, copper, celite 
Interstitial gas air, helium, Freon 
The measured heat transfer coefficients were within 
a range of 5 W me2 K-’ at low vacuum (radiation heat 
transfer only) and 1500 W me2 K-’ for dense packed 
fine aluminium powder with helium gas at norinal 
pressure and short contact times. 
10. Examples 
(a) Calculation of the time averaged heat transfer 
coefficient (Y for an air filled packed bed of glass spheres 
of 1 mm particle diameter in contact with a plate for 
60 s. Reference temperature 50 “C, pressure 10’ Pa, 
surface roughness zero, surface coverage 85%. Particle 
size uniform. 
Step I. Calculation of crws, Appendix A 
Print-out of HP 41 CV: 
ALFA WS 
PARTRAD/M ? 
ROUGH/M ? 
PRESSURE/PA ? 
TEMPIK ? 
GASVISCOIPAS 
GASCONDlWlMK 
MOLMAS GAS ? 
ACCOMMODATION 
EMISSION ? 
SURF COVER ‘l 
GTO 01 
ALFAWSlWlSQM. K = 
5 .ooo - 04 
0.000 + 00 
1.000 + 06 
3.230 + 02 
1.900 - 06 
2.800 - 02 
2.890 + 01 
8.600 - 01 
8.000 - 01 
8.500 - 01 
6.609 + 02 
LAMBDA BED 
SHAPEFACTORS 
C SPHERE = 1.250 + 00 
C CYLINDER = 2.500 + 00 
C CHOPPED = 1.400 + 00 
DIRECT CONTACT AREAS 
CERAMIC = 1.100 - 03 
STEEL = 1.300 - 03 
SAND = 1 .ooo - 03 
PART RADIM ? 5.000 - 04 
C SHAPE ? 1.250 + 00 
VOIDFRAC ? 4.000 - 01 
ZETA ONE ? 0.000 + 00 
PRESSURE/PA ? 1.000 + 05 
TEMPERATURE/K 3.230 + 02 
GASVISCOIPAS 1.900 - 05 
GASCOND/W/MK 2.800 - 02 
MOLMAS GAS 7 2.890 + 01 
ACCOMMODATION 8.500 - 01 
EMISSION ? 8.000 - 01 
SOLIDCONDIWIMK 9.300 - 01 
DIRECT CONTACT I .700 - 03 
GTO 01 
LAMBEDILAMGAS = 6.380 + 00 
LAMBEDIWIMK = 1.786 -01 
Step 3. Calculation of 01, eqn. (9) 
Further input: bulk density pbed = 1800 kg rn-’ ; 
solid heat capacity cs = 653 .I kg-’ K-r; t = 60 s. This 
gives 
T = c~,~t/(pc&_~ = 124.6 
o/Gs = 0.092 
(Y = 60.1 W me2 K-r 
The heat transfer coefficient (Y reaches only 9.2% of 
its maximum value. Reduction of the residence time 
from 60 s to 1 s results in a considerable increase in 0~: 
r = 2.08 
&&S = 0.439 
CY = 290.3 W me2 K-’ 
(b) Calculation of (Y for the same conditions as in (a), 
but at reduced pressure,p = 100 Pa. 
Step 1 
ALFA WS 
PART RAD/M ? 
ROUGH/M ? 
PRESSURE/PA ? 
TEMPlK ? 
GASVISCOIPAS 
GASCONDIWIMK 
MOLMAS GAS ? 
ACCOMMODATION 
EMISSION ? 
SURF COVER ? 
GTO 01 
ALFAWSIWISQM. K = 
5.000 - 04 
0.000 + 00 
1 .ooo + 02 
3.230 + 02 
1.900 - 05 
2.800 - 02 
2.890 + 01 
3.500 -Of 
8.000 - 01 
8.500 - 01 
8.139 + 01 
47 
Step 2 
LAMBDA BED 
SHAPEFACTORS 
C SPHERE = 1.250 + 00 
C CYLINDER = 2.500 + 00 
C CHOPPED = 1.400 + 00 
DIRECT CONTACT AREAS 
CERAMIC = 7.700 - 03 
STEEL = 1.300 - 03 
SAND = 1.000 - 03 
PART RADIM ? 5.000 - 04 
C SHAPE ? 
VOIDFRAC ? 
ZETA ONE 7 
PRESSURE/PA ? 
TEMPERATURE/K 
GASVISCO/PAS 
GASCONDIWIMK 
MOLMAS GAS ? 
ACCOMMODATION 
EMISSION ? 
SOLIDCONDfW/MK 
DIRECT CONTACT 
GTO 01 
LAMBEDILAMGAS = 
LAMBEDIWIMK = 
1.250 + 00 
4.000 - 01 
0.000 + 00 
1 .ooo + 02 
3.230 + 02 
1.900 - 06 
2.800 - 02 
2.890 + 01 
8.500 ~ 01 
8.000 - 01 
9.300 - 01 
7.700 - 03 
2.474 + 00 
6.927 - 02 
Step 3 
t=60s: 
7 = 4.0 
~/%m = 0.338 
a = 27.5 W m-* K-’ 
t=ls: 
r=O.O82 
h, = 0.80 
~1~64.9 Wm~2K-1 
In this case reduction of the residence time brings Q 
quite close to its maximum value cr,. 
(c) Calculation of cr for stirred beds. The physical 
properties are the same as in (a). Nmrx may be assumed 
to be 23. 
The equivalent contact time is 
Gws 
2 
T = NrhermNmir = ___- 
@CA&Z Nmix 
where Z is the number of revolutions per unit time. 
For Z = 60 rev min-’ the equivalent contact time is 
N . 
t=-- mlx X60=23s 
60 
This gives, fOrp = 10’ h, 
r = 29.5 
o/oI, = 0.172 
(~=113.7Wm-~K-~ 
and, for p = 10’ Pa, 
r= 1.88 
4%va = 0.452 
(Y = 36.8 W me2 K-’ 
(d) Calculation of rx for extreme conditions. Ceramic 
particles, 100 pm in diameter, 40% void fraction, 
hydrogen gas, 10 bar, 1000 “C, residence time 1 s. 
Step I. Calculation of Q, 
ALFA WS 
FART RADIM ? 
ROUGH/M ? 
PRESSURE/PA ? 
TEMPIK ? 
GASVISCOIPAS 
GASCONDlWlMK 
MOLMAS GAS ? 
ACCOMMODATION 
EMISSION ? 
SURF COVER ? 
GTO 01 
ALFAWSIWISQM. K = 
5.000 - 05 
0.000 + 00 
1 .OOO + 06 
1.273 + 03 
2.330 - 05 
6.100 - 01 
2.000 + 00 
1.000 - 01 
6 .ooo - 01 
8.000 - 01 
3.660 + 04 
Step 2. Calculation of Abed 
Note that &,, is below h, and &rid! This is because 
of the Knudsen effect near the contact point. 
LAMBDA BED 
SHAPEFACTORS 
C SPHERE = 1.250 + 00 
C CYLINDER = 2.600 + 00 
C CHOPPED = 1.400 + 00 
DIRECT CONTACT AREAS 
CERAMIC = 7.700 - 03 
STEEL = 1.300 - 03 
SAND = 1 .ooo - 03 
PART RADIM 7 6.000 - 05 
C SHAPE 7 
VOIDFRAC 7 
ZETA ONE 1 
PRESSURE/PA ? 
TEMPERATURE/K 
GASVISCOIPAS 
GASCONDIWIMK 
MOLMAS GAS ? 
ACCOMMODATION 
EMISSION ? 
SOLIDCONDlW/MK 
DIRECT CONTACT 
GTO 01 
LAMBEDILAMGAS = 
LAMBEDlW/MK = 
1.250 + 00 
4.000 - 01 
0.000 + 00 
1 .OOO + 06 
1.273 + 03 
2.380 - 05 
5.100 - 01 
2.000 + 00 
1 .ooo - 01 
6.000 - 01 
5.000 - 01 
7.700 - 03 
9.551- 01 
4.871 - 01 
Step 3. Calculation of r = am2t/(pch),,, 
pbd = 1200 kg mm3 
cs = 600 J kg-’ K-r 
r = 3925 
+wS = 0.01 
CI = 659 W me2 K-’ 
The heat transfer is entirely bulk conduction control- 
led. Stirring could improve OL considerably. 
48 
Nomenclature 
A 
(I 
Cl2 
z 
I 
ti 
Nmix 
N therm 
p. 
e 
1 
s 
T 
t 
tmix 
Y 
z 
CY 
; 
E 
:: 
A 
P 
u 
A 
@K 
J/ 
wall surface, m2 
radius of direct contact area, m 
radiation coefficient 
heat capacity at constant pressure, J kg-’ K-r 
particle diameter, m 
modified mean free path of gas molecules, 
eqn. (A-3) 
molar mass, kmol kg-’ 
eqn. (12) 
*eqn. (14) 
pressure, Pa 
heat flux, W 
heat flux density, W rn-’ 
gas constant, J kmol-’ K-r 
gap width, m 
temperature, K 
time, s 
= 1 /Z, time constant of mixing device, s 
coordinate, m 
stirrer speed, s-’ 
heat transfer coefficient, W mm2 K-t 
accommodation coefficient 
surface roughness, m 
emissivity 
viscosity, Pa s 
mean free path, m 
heat conductivity, W m-’ K-’ 
density, kg me3 
radiation constant, W m-’ KA4 
eqns. (10) and (22d) 
surface coverage factor 
direct contact area 
void fraction 
Su bscri@ ts 
B bulk 
bed bed 
dir direct 
G gas 
rad radiation 
S solid 
sb surface to bulk 
W wall 
wP wall to particle 
ws wall to surface 
0 surface 
References 
1 H. P. Seidel, Untersuchungen zum Warmetransport 
in Ftillkijrpersliulen, C/rem.-Zng.-Tech., 37 (1965) 
1125-1132. 
2 R. Ernst, Warmeiibergang an Warmetauschern im 
Moving Bed, Chem.-Zng.-Tech., 32 (1960) 17- 
32. 
3 W. N. Sullivan and R. M. Sabersky, Heat transfer to 
flowing granular material, Znt. J. Heat Mass Transfer, 
18 (1975) 97-107. 
4 E. U. Schliinder, Warmetibergang an bewegte Kugel- 
schiittungen bei kurzfristigem Kontakt, Chem.-Ing.- 
Tech., 43 (1971) 651-654. 
5 J. P. Fourier, Zheorie Analytique de la Chaleur, 
1821. 
6 J. Wunschmann, W&metibertragung von beheizten 
Flachen an bewegte Schtittungen bei Normaldruck 
und im Vakuum, Dissertation, Univ. Karlsruhe, 
1974. 
7 J. Wunschmann and E. U. Schltinder, Heat transfer 
from heated plates to stagnant and agitated beds of 
spherical shaped granules under normal pressure and 
vacuum, Proc. 5th Int. Heat Transfer Conf, Tokyo, 
1974, Vol. V, CT 2.1, p. 49-53. 
8 J. Wunschmann and E.U. Schliinder, Warmetibergang 
von beheizten FILchen an Kugelschiittungen, VT- 
Verfahrenstechnik, 9 (1975) 501-505; Znt. Ci’zem. 
Eng., 43 (1980) 555-563 (English trans.). 
9 E. Muchowski, Transient heat transfer between a 
perfect conductor with heat generated in it and a 
semi-infinite solid including a contact resistance, 
W&me- Stoffiibertrag., I2 (1979) 161-164. 
10 R. Ernst, Der Mechanismus des Warmeiiberganges an 
Warmeaustauschern in Fliessbetten,Chem.-Zng.-Tech., 
31 (1959) 166-173. 
11 N. K. Ha&as and K. 0 Beatty, Jr., Moving bed heat 
transfer: I. Effect of interstitial gas with fine particles, 
Chem. Eng. Symp. Ser. 59 (41) (1963) 122- 
128. 
12 J. S. M. Botterill, M. H. D. Butt, G. L. Clain and 
K. A. Redish, The effect of gas and solids thermal 
properties on the rate of heat transfer to gas-fluidizedbeds, Proc. Int. Symp. on FZuidization, Eindhoven, 
June 1967, p. 450. 
13 D. Gloski, L. Glicksman and N. Decker, Thermal 
resistance at a surface immersed in a fluidized bed. 
14 S. Giines, E. U. Schliinder and V. Gnielinski, 
Kontakttrocknung von grobkomigem Granulat 
im Vakuum, VT- Verfahrenstechnik, 14 (1980) 
31-39. 
15 E. U. Schliinder, ijber den Stand der wissenschaftli- 
then Grundlagen zur Auslegung von Kontakttrock- 
nem fur grobkbmiges, rieselfahiges Trocknungsgut, 
Chem.-Zng.-Tech., 53 (1981) 925-941. 
16 E. U. Schltinder, Particle heat transfer, Proc. 7th 
Znt. Heat Transfer Conf, Munich, 1982, Vol. 1, RK 
10,~~. 195-212. 
17 P. Zehner, Experimentelle und theoretische Bestim- 
mung der effektiven Wdrmeleitftiigkeit durchstrom- 
ter Kugelschtittungen bei massigen und hohen 
Temperaturen, VDI-Forschungsh., (5.58) (1973). 
18 R. Bauer, Effektive radiale W%rmeleitf%higkeit gas- 
durchstrijmter Schtittungen aus Partikeln unter- 
schiedlicher Form und Griissenverteilung, VDI- 
Forschungsh., (582) (1977); see also Znt. Chem. 
Eng.. 18 (1978) 181-203. 
19 N. Mollekopf and H. Martin, Zur Theorie des WCrme- 
tiberganges an bewegte Kugelschtittungen bei kurz- 
fristigem Kontakt, VT-Verfahrenstechnik, 16 (1982) 
70 l-706. 
20 S. Imura and E. Takegoshi, Nippon Kikai Gakkai 
Rombunshu, 40 (1974) 489. 
Appendix A 
Prediction of the contact heat transfer coefficient h 
(wall surface to bed surface) 
The main reason for the contact heat transfer resis- 
tance is that the heat conductivity of the gas in the 
wedge between particle and wall goes to zero when 
approaching the contact point, as set forth in ref. 4. 
Later slight modifications of the original equation have 
been published [14-l 91. The equation as recommended 
in refs. 14 and 15 is used in this paper: 
2hdd 
%?3 = @Aa, + (l - @A) fi f (2Z+ 26)/d 
cyIvp is the heat transfer coefficient for a 
and is to be calculated by 
+ arad + adir 
(A-1 1 
single particle 
o-=?[(l+ y)ln(l+ A)-1] (A-2) 
ho is the continuum heat conductivity of the gas. $A is a 
plate surface coverage factor and is of the order of 0.80. 
d is the particle diameter and 6 the roughness of the 
particle surface. I is the modified mean free path of the 
gas molecules and follows from 
2-Y 
1=2hp (A-3) 
T 
where A is the mean free path and y is the accommoda- 
tion coefficient, which is around 0.8-l for normal gases 
at moderate temperatures. A may be calculated from 
(A-4) 
where 17 is the dynamic viscosity, p the pressure, T the 
temperature, &? the molecular mass and i? the universal 
gas constant. 
(Y,,d accounts for heat transfer by radiation and can 
be calculated from the linearized Stefan-Boltzman law: 
arad = 4C12 T,” (A-5) 
with 
1 
c,*=a- 
1 1 
(A-6) 
-+--1 
%dl %ed 
being the overall radiation exchange coefficient, which 
follows from the black body radiation coefficient u = 
5.67 X lo-’ W m-* KM4 and the emissivities of the wall 
surface ew and the bed surface futi as well. 
Actually, ‘ydir accounts fur direct solid to solid heat 
conduction. This contribution may be estimated from 
the formula / 
a h3 
(Y&=2- - 
dd 
(A-7) 
where a is the diameter of that surface area in direct 
solid to solid contact with the plate when the particles 
49 
TABLE A-l (a). uwIl for 1 mm glass beads at various air pressures 
p and various surface roughnesses 6 (hS = 0.93 W me1 K-‘;n/d = 
3 X10-4) 
P (mbar) 6=0 s=lpm 6=10pm &=lOOnm 
10s 627.5680 485.2854 295.0458 119.5723 
102 511.2243 383.4911 275.7123 118.0232 
10 209.5610 206.8179 186.085 1 104.8636 
100 65.6040 65.4585 64.1828 54.0039 
10-r 14.1510 14.1488 14.1221 13.8642 
10-2 6.0291 6.0290 6.0287 6.0257 
1 o-3 5.1515 5.1515 5.1515 5.1515 
TABLE A-l(b). oM for bronze spheres at various air pressures p 
and various surface roughnesses 6 (AS= 47.1 W m-l K-l; a/d = 
3 X10-4) 
~~~~~ 
p (mbar) 6=0 6=lpm 6=10pm 6 =lOOfim 
10s 656.3820 514.0994 323.8598 148.3863 
102 440.0383 412.3051 304.5263 146.8372 
10 238.3758 235.6319 214.8991 133.6776 
loo 94.4180 94.2725 92.9968 82.8179 
10-r 42.9657 42.9628 42.9361 42.6782 
10-2 34.843 1 34.8430 34.8427 34.8397 
10-s 33.9655 33.9655 33.9655 33.9655 
are not perfectly spherical all over and, moreover, when 
this plane part of the particle surface is as close to the 
wall surface as the crystal lattice spacing (otherwise no 
direct solid to solid heat conduction could be established; 
when the metal particles contact a metal plate, both 
must be welded together over the surface area n(a/2)*). 
In general, the direct solid to solid contribution is 
negligibly small except for high vacuum and metal 
particles (good heat conductors). 
Table A-l and Fig. A-l show (lr, as a function of 
pressure p for 1 mm glass beads with h, = 0.93 W m-l 
10s 
cwr 
W 
mZK 
10' 
1 
P mber 
Pig. A-l. Contact heat transfer coefficient ow versus pressure p 
for glass (full curves) and bronze (broken curves) spheres. Sphere 
diameter = 1 mm; interstitial gas is air at room temperature. 
50 
K-r and 1 mm bronze spheres with Xa =47.1 W m-l 
K-r. The gas is air and the parameter is the surface 
roughness 6. For both materials the same diameter ratio 
a/d=3XlO --4 for the direct solid to solid heat conduc- 
tion has been chosen. The other physical properties are : 
temperature 320 K, no = 1.9 X lo-’ Pa s, Xo = 0.027 W 
m -1 K-1, co = 1010 J kg-’ K-r, iI& = 28.9 kg kmol-‘, 
y=0.90;ew=E~~=0_90;C#l*=O.85. 
The Tables and diagrams reveal the following: 
(1) At normal pressure and zero roughness both Xs 
and a/d have very little effect on o,. 
(2) At normal pressure and large roughness both As 
and a/d still have little, but not negligible, effect on (Y,. 
(3) At vacuum, Xs may have a strong effect on (Y, 
provided a/d is not zero. 
(4) Assuming a/d = 3 X 10e4, direct solid to solid 
conduction has a significant effect on (Y, at vacuum for 
good conducting particles (copper), however, it has no 
effect on 01, for poor conducting particles (glass). This 
conclusion follows from CY, = 5.15 15 W rn-’ K-’ for 
glass beads under vacuum (see Table A-l(a)), which is 
solely due to radiation, according to eqn. (A-6). For 
copper spheres, (Y, = 33.9655 W m-* K-’ at vacuum 
(see Table A-l(b)) indicates that in addition to radiation 
there is a significant contribution by direct solid to solid 
conduction (33.9655 - 5.1515 = 28.814 W rnp2 K-l). 
(The contribution of gas conduction is negligibly small 
for both glass and bronze spheres, at this low pressure, 
anyway.) 
(5) a/d = 3 X lop4 means that the gap widths between 
a perfectly smooth sphere and a perfectly smooth plate 
at distance a/2 from the contact point is very small. 
From 
2 
(A-8) 
it follows that, for a sphere of diameter d = 1 mm and 
a/d =3 X 10p4, the gap width s is 0.225 X lo-” rns 
0.225 A. Since the crystal lattice spacing is of the order 
of some angstroms, a/d = 3 X 10m4 is within the direct 
solid to solid region. Nevertheless, one should bear in 
mind that perfectly smooth surfaces are an oversimplifi- 
cation of the physical reality. Therefore, s - 0.1 A, 
which is much less than the diameter of a single atom, 
is a model parameter and not a physical one! Since the 
real physical situation at the contact point is unknown, 
the parameter a/d must be fitted to experimental data, 
particularly those obtained at high vacuum. a/d may 
depend on the particle material and the particle diameter. 
Appendix B 
Predictions of the packed bed heat conductivi@ Abed 
Various correlations for the prediction of &,bed have 
been published over the last decades. The mosfelaborated 
and reliable one seems to be that developed by Zehner 
[ 171 and Bauer [ 181. It applies to monodispersed as well 
as polydispersed packed beds of spherical and non- 
sphericalparticles of poor and good conductors within a 
wide temperature and pressure range (100 < T < I 500 K, 
10m3 < p < 100 bar). The calculation procedure has 
been summarized in ref. 15 and is repeated here. Accord- 
ing to refs. 17 and 18 the overall heat conductivity of 
packed beds hbed depends on the following parameters: 
Abed =f(&, AG. ~Rv hD,d, ‘h, @K. CFormf(~r)) 
These parameters are: 
AS heat conductivity of particles 
hG heat conductivity of gas 
hrl equivalent heat conductivity due to radiation 
AD equivalent heat conductivity due to molecular 
flow 
d particle diameter 
!! 
void fraction 
relative flattened particle-surface contact area 
Form particle shape factor 
f(S;) particle size distribution function 
As set forth in refs. 17 and 18 the following correla- 
tions may be recommended for the prediction of Abed: 
1 (B-1) 
X’bed 2 
-=_ 
%&G + XR/~G - ~XXG/~D)@G/M x 
hG K K2 
X In 
(&/ho + hR/kG)hG/kD 
B 11 + @G/ID - ~&/XG + XR/XG)I 
+;I$? -B[l+(% -l)#++ 
B-l A, 
--- 
K An 
where 
K= 2 [I +(2 -&);f]+ 
-B(z -,)(I+ 2 2) 
and 
Further 
AR 4cs _=- 
AG 2/e - 1 
Tm3F 
and 
(B-2) 
(B-3) 
(B-4) 
(B-5) 
(B-6) 
51 
with 
XR = R~ormd 
and 
(B-7) 
XD = &or& (B-8) 
d is the equivalent particle diameter 
d=3m (B-9) 
V being the particle volume. RForm and DForm are shape 
factors for the interstitial energy transport by radiation 
and molecular flow, respectively. 
If the packed bed consists of mass fractions AZj with 
various particle diameters di, xR and xD must be calcu- 
lated as follows: 
1 i=n AZ 
-= 
xD ’ FormI idi i=ID 
1 i=n AZi 
_= 
XD ’ D~orm, idi i=l 
(B-10) 
(B-l 1) 
The particle size distribution function f({,) was found 
to be [18] 
f(L)= 1 +3<1 
where the distribution parameter cr is given by 
(B-l 2) 
(B-13) 
The set of eqns. (B-l)-(B-13) contains three shape 
factors, such as CForm, RForm and DForm, as well as the 
relative particle to particle contact surface area $K, 
which must be evaluated from experiments. Some of the 
parameters have been evaluated in refs. 16 and 17 and 
are listed in Table B-l. 
TABLE B-l. Shape factors and particle to particle contact areas 
@K according to refs. 16 and 17 
Particle 
shape 
CForm R~orrn D~orm OK Remarks 
Spheres 1.25 1 1 0.0077 Ceramic 
0.0013 Steel 
0.0253 Copper 
Cylinders 2.50 1 ? ? 
Hollow di2 1 ? ? 
cylinders 2.5 [ 1+ 01 - 
& 
Arbitrary 1.4 1 ? 0.001 Sand 
shaped ? Clay 
Figure B-l shows predicted and experimental [20] 
data for &&I, versus the gas pressure p for a packed 
bed of ceramic spheres. 
The parameter GK becomes important only at low 
pressure. Further comparisons between predicted and 
3 
P. mm Hg 
Fig. B-l. Packed bed heat conductivity Abed for monodispersed 
beds. Data according to ref. 20. Full curves from eqns. (B-l)- 
(B-13). 
TABLE B-2. Histograms of particle size distribution, taken from 
ref. 18. 
6 
51 
Fig. B-2. Packed bed heat conductivity h&d for polydispersed 
beds versus distribution parameter cl. Data according to ref. Lg. 
Full curves from eqns. (B-l)-(B-13). Normal pressure and tem- 
perature. Particle size distributions are given in Tables B-2 and B-3. 
52 
experimental data such as in Fig. B-l for other solid experimental data are depicted in Fig. B-2. It seems 
materials and other gases may be found in ref. 18. AI1 to be obvious that the distribution parameter {r, as 
predictions are within the experimental error. given by eqn. (B-13) is sufficient to represent rather 
Arbitrary shaped material has also been investigated different size distributions, such as, for example, cases 
in ref. 18. The particle size distribution functions are (g) and (i) in Table B-2. For further details see ref. 
given in Tables B-2 and B-3, and the predicted and 18. 
TABLE B-3. Evaluation of Table B-2. Numerical values, A,zi (%) 
d; (mm) 
- 
. . 
2 3 4 5 6 7 8 9 10 12.5 14 16 18 20 
ii (mm) Sl &=XR hxd 
2.9 ______ ~ 3.9 4.74 6.15 1.24 8.41 9.65 10.15 11.9 13.75 16.2 17.75 19.36 =xD 
h 
(a) 
(b) 
(c) 5.94 
(d) 
(e) 0.80 
(f) 1 .oo 
0.98 
(g) 13.71 
(h) 17.33 
(0 2.25 
100.00 
17.26 16.82 40.75 25.16 
4.82 5.60 6 .OO 5.78 5.26 5.97 5.67 
0.13 1.83 18.55 29.88 38.80 4.76 5.26 
2.70 6.86 15.91 24.08 24.65 18.12 5.76 
1.12 3.54 3.80 6.75 7.68 8.27 11.67 
1.17 1.51 1.85 3.82 7.00 8.51 11.35 
11.91 8.57 6.53 4.42 1.53 1.36 1.57 
12.33 10.25 8.00 6.51 6.00 5.10 4.25 
2.11 2.62 2.73 2.77 2.58 3.33 4.49 
13.33 8.21 11.65 1 
0.80 
1.10 
28.44 10.77 12.10 
30.82 15.58 10.97 
2.14 4.08 6.57 1, 
10.3s 5.55 5.00 4.95 4.30 0.585 5.80 23.50 
10.49 8.31 11.27 17.64 29.33 0.694 11.77 26.07 
0 8.47 14.45 
0.129 11.49 17.53 
1.61 10.15 0.677 9.05 25.36 
0.171 7.56 17.39 
0.292 1.26 19.37 
3.91 0.93 0.447 9.91 22.32 
4.96 1.48 0.435 
4.12 23.48 0.727 6.82 25.60 
Appendix C 
Input Data from ST0 01 to ST0 10 
Ol*LBL“ALFAWS",a, 
79aLBLOl 
80 RCL04 
81 1323 
82 * 
83 RCLO7 
84 I 
85 SQRT 
86 3.2 
87 * 
68 RCLOS 
89 * 
90 RCL03 
91 I 
92 ST011 
93 2 
94 * 
95 2 
96 RCLOB 
97 - 
98 + 
99 RCL08 
100 I 
101 ST012 123 ST014 144 * 
102 RCL02 124 RCL04 145 RCLl2 
103 + 125 3 146 + 
104 RCLOl 126 YtX 147 RCLOZ 
105 I 127 22.28E-8 148 + 
106 ST013 128 * 149 / 
107 1 129 2 150 RCL15 
108 + 130 RCLOS 151 + 
109 RCL13 131 I 152 RCL14 
110 l/X 132 1 153 RCLlO 
111 1 133 - 154 * 
112 + 134 / 155 + 
113 LN 135 ST015 156 ST0 16 
114 * 136 1 157 “ALFAWS/W/SQM,K =I' 
115 1 137 RCLlO 158 RCL16 
116 - 138 - 159 ACA 
117 2 139 RCLOG 160 FMT 
118 * 140 * 161 ACX 
119 RCLOl 141 RCLOl 162 PRBUF 
120 I 142 2- 163 RTN 
121 RCL06 143 SQRT 164 .END. 
122 * 
Input Data from ST0 01 to ST0 13 
141eLBLOl 148 RCL03 155 Ytx 162 SQRT 
142 RCL04 149 1 156 RCLOP 163 RCL07 
143 3 150 - 157 * 164 * 
144 * 151 CHS 158 ST021 165 RCL05 
145 1 152 RCL03 159 RCL06 166 I 
146 + 153 I 160 RCL09 167 116.4 
147 ST020 154 1.11 161 I 168 * 
53 
169 RCLlO 
110 2 
111 / 
112 l/X 
113 1 
114 - 
115 * 
116 ST022 
111 RCLOl 
118 / 
119 1 
180 + 
181 ST023 
182 RCLOG 
183 3 
184 YtX 
185 RCLOl 
186 11.34 
167 + 
168 RCLll 
189 2 
190 1 
191 l/X 
192 1 
193 - 
194 / 
195 RCLOS 
196 / 
191 ST024 
198 RCL04 
199 RCLl2 
200 I 
201 ST025 
202 RCL21 
203 RCL23 
204 1 
205 RCL24 
206 - 
201 CHS 
208 RCL25 
209 * 
210 1 
211 + 
212 RCL23 
213 * 
214 RCL23 
215 1 
216 - 
211 RCL21 
218 * 
219 RCL25 
220 RCL24 
221 * 
222 1 
223 + 
224 * 
225 - 
226 ST026 
227 RCL25 
228 l/X 
229 RCL24 
230 + 
231 1 
232 - 
233 RCL23 
234 * 
235 RCL25 
236 * 
231 RCL21 
238 * 
239 RCL26 
240 Xf2 
241 1 
242 ST027 
243 RCL25 
244 l/X 
245 RCL24 
246 + 
247 RCL23 
2413 * 
249 RCL25 
250 l/X 
251 RCL24 
252 + 
253 RCL23 
254 1 
255 - 
256 * 
257 1 
258 + 
259 RCL21 
260 * 
261 I 
262 LN 
263 RCLZI 
264 * 
265 RCL23 
266 1 
261 - 
268 RCL24 
269 * 
270 1 
211 + 
212 RCL21 
213 * 
214 RCL24 
215 RCL23 
216 * 
277 - 
278 CHS 
219 RCL21 
280 1 
281 + 
282 * 
283 2 
284 / 
285 RCL21 
286 I 
281 + 
288 RCL21 
289 1 
290 - 
291 RCL23 
292 * 
293 RCL26 
294 1 
295 - 
296 2 
291 + 
298 RCL26 
299 j 
300 ST028 
301 RCL13 
302 1 
303 - 
304 CHS 
305 * 
306 RCL13 
307 RCL25 
308 1 
a09 + 
.310 RCLOS 
311 1 
312 - 
313 CHS 
314 SQRT 
315 * 
816 RCL03 
317 RCL24 
318 * 
319 RCL03 
320 1 
321 - 
322 RCL23 
323 + 
324 RCL03 
325 / 
326 I/X 
321 + 
328 RCLOS 
329 1 
330 - 
331 CHS 
332 SQRT 
333 1 
334 - 
335 CHS 
336 * 
331 + 
338 ST029 
339 “LAMBEDILAMGAS =" 
340 RCL29 
341 ACA 
342 FMT 
343 ACX 
344 PRBUF 
345 RCL29 
346 RCLOS 
347 * 
348 ST030 
349 “LAMBED/W/MK=" 
350 RCL30 
351 ACA 
352 FMT 
353 ACX 
354 PRBUF 
355 RTN 
356 END