wakao-1979

Prévia do material em texto

EFFECT OF 
000P-2509/791030I432SIS02 0010 
FLUID DISPERSION COEFFICIENTS ON 
PARTICLE-TO-FLUID HEAT TRANSFER COEFFICIENTS 
IN PACKED BEDS 
CORRELATION OF NUSSELT NUMBERS 
N WAKAO,t S KAGUEI and T FUNAZKRI 
School of Engmeermg, Yokohama National Umverslty, Hodgaya-ku, Yokohama, Japan 240 
(Recerued 24 February 1978. accepted 17 May 1978) 
Abstract-The pubhshed heat transfer data obtamed from steady and nonsteady measurements are corrected for 
the axial thud thermal dispersion coefficient values proposed by Wakao[ I] 
The corrected data m the range of Reynolds number from 15 to 8500 are correlated by the analogous form of 
the mass correlation proposed by Wakao and Funazkn[Z] 
Nu=2+ 1 1 Pr”)Reo6 
This work for particle-to-flmd heat transfer 1s an ex- 
tenston of that reported by Wakao and Funazkn[2] m 
which the pubhshed particle-to-flmd mass transfer data 
were corrected for axial fluld dlsperslon coefficients 
Slmdarly to the data collectlon for mass transfer, we 
confine ourselves to the heat transfer measurements 
which satisfy the followmg condltlons 
1 Particles m bed bemg all actlve 
2 Number of particle layers m heat transfer bed being 
greater than two 
Table 1 summarizes the expenmentai condltlons of the 
collected measurements The data are shown as Nu 
vs Re m Fig 1 ConsIderable scattering and 
anomalous decrease m Nusselt number are seen at low 
Reynolds numbers 
In fact, the anomaly of Nusselt numbers has been a 
subject of dlscusslon for long However, not all the 
experimental mvestigators have supported the anomaly 
of Nusselt numbers From frequency response 
measurements, Llttman and Shva[3], and Gunn and De 
Souza[4] have estimated, respectively, from their own 
experiments the llmltmg Nusselt numbers of about 0 4 
and 10 
Some theoretical studies have also been made on this 
subject There IS, of course, no exact theory which 
describes the transport phenomena m packed beds 
Different theoretlcal estimates have given different hml- 
tmg Nusselt numbers Just hke the expenmental data, the 
two completely dIvergent conclusions have been drawn 
from the estimates 
Cormsh[S] indicates, assummg an analogy between 
electrostatics and heat, that the heat transfer coefficient 
m dense system of particles IS considerably smaller than 
that for a single sphere m an mfirute medium Kunu and 
Suzukl[6] pomt out that flow channeling m the bed IS the 
reason for the anomaly Nelson and Galloway [7] indicate 
that the anomaly IS explamed by a renewal of fluld 
tTo whom correspondence should be made 
element surrounding each particle Schlunder [8] shows, 
on the assumption that packed bed IS a bundle of parallel 
capillaries, that transfer coefficients decrease with a 
decrease of flow rate at lower Reynolds numbers 
On the other hand, Pfeffer and Happel[9] obtam, by 
applymg a free surface model, a lrmitmg Nusselt number 
of about 13 at the bed void fraction 0 4 From an analysis 
of steady transfer m a stagnant fluid m a concentrlc 
hollow sphere, Mlyauchl[lO] shows a hmltmg Nusselt 
number of about 18 at the bed void fraction 0 4 Stiren- 
sen and Stewart[ll] study a creepmg flow through a 
cubic array of spheres to find a hmltmg Nusselt number 
of about 3 9 
Wakao et al [l, 12-141, however, find the defect of the 
fundamental equations responsible for the anomalous 
decrease m Nusselt number at lower Reynolds numbers 
REVIEW AND CORRECTI ON OF THE DATA OBTAINED FROM 
STEADY MEASUREMENTS 
Sdmultaneous heat and mass transfer study evaporatron 
of water and drfiusron-controlled chemrcal reactton on 
part&e surface 
The mass transfer data obtained from the stmultaneous 
heat and mass transfer studies have been used for the 
data correlation m the mass transfer paper[2] The cor- 
responding heat transfer data are, therefore, used m the 
present paper The data are those obtatned from the 
measurements of evaporation of water by Hougen et 
al [15, 161, Hurt[l7], Galloway et al [18], Thodos et 
al [19-231 and Bradshaw and Myers[24], and those 
determmed from the catalytic decomposltlon of 
hydrogen peroxide on metal spheres by Satterfield and 
Resmck[25] 
The measurements were made with sohd particles 
havmg codstant surface temperatures throughout the 
beds In the studies the system was described as 
dTF h,a U~+CbC.FU-~- T,.)=O 
325 
326 N WAKAO et al 
Kunu and Smith [271 
C Cybulskl e/a/ C281 
K Bradshow et al C461 
L Hacdley and Hsggs C453 
(Hougen eta/ Cl5. l-63 
D Elchharn and White C301 
E Llttman et o/ 1291 M 
F Gum and De Sauza 141 
G Balokrlshnan and Pet C323 i 
Hurt Cl71 
Galloway et a/ Cl81 
Thadas et a/ 119-231 
Brodshaw and Myers 1241 
H Bhattacharyya and Pel m31 N Llndouer C471 
I Glosw and Thodas 1351 0 Eaumslster and Bennett C3ll 
J Satterfleld and Resruck C251 P Turner et 01 C39. 401 
A Liquid-Sdld B-P Gas-S&d 
FIN I Heat transfer data pubhshed In the literature 
and the heat transfer coefficients. h,, have been deter- 
mmed 
If, however, axial fluld thermal dlsperslon IS CO~SI- 
dered, the system IS 
~TF h,a dZTF 
Udx+tEbCFPF(T~-Tps)=aox~ (2) 
The axial flmd thermal dlsperslon coefficient a,, has 
been consldered as a_ = (0 6- 0 8)aF m lammar flow 
range and aur = 0 SD,U m turbulent flow range, or ap- 
proximately over the range from lammar to turbulent 
(3) 
Wakao [ 11 and coworkers [ 12,141, however, recently 
pomted out that the heat transfer coefficients should be 
determmed from eqn (2) with the followmg ore values 
(as described later, this ts called the moddied D-C model) 
Instead of a._ of eqn (3) 
a&= k tax 
GCFQF (4) 
The axial fluid dlsperslon IS expressed by eqn (3), but 
the right hand side of eqn (2) should have sohd-phase 
conductlon contrlbutlon, as well as the dlsperslon m fluld 
phase Therefore, a Zz should rather be called the 
effective dlsperslon coefficient 
The heat transfer measurements revlewed m this 
sectlon have been made m the Reynolds number up to 
8500 As far as the authors know, no measurements 
have been made on k,, at such high flow rates At htgh 
flow rates, however, the axial fluld dlsperslon IS, as 
described before, expressed as a Peclet number of two 
Equation (4) IS, therefore, rewritten as 
a%- 
k,” , 
GCFPF P (5) 
where k,” 1s the effective thermal conductlvlty of a 
stagnant packed bed Note that Wakao and Kato[26] 
have given a chart for esttmatmg k,” dues 
The heat transfer coefficients are recalculated from all 
of the steady heat transfer studies revlewed m thts 
section except the papers[l5, 17,241 m which no detalled 
data on bed height and/or void fraction are Gven The 
Effect of Ruld dlsperslon coefficients on particle-to-fluld heat transfer coeficlents m packed beds 327 
Pr 
Walke and Hougen 1161 07 
Sotterfteld and Resnick t251 IO 
Galloway el 01 Cl81 07 
De Acetls and Thodos [I91 07 
McConnoch~e and Thodos [211 07 
Sen Gupta and Thodos C22, 231 07 
Mollkng and Thodos I201 07 
Nu-2+ I I Pr”3Rew, for Pr=07 
Ag 2 Heat transfer data of steady state measurements corrected for azx 
correctlon procedure IS ldentmal wrth that employed m 
the mass transfer study[Z] except for the difference 
between heat and mass This IS briefly described m 
Appendix A 
The corrected data are plotted as Nu vs Re in Ftg 2 
The solid lme shows the analogous form of the mass 
transfer correlation proposed m the work[2] 
Nu = 2 + I 1 Pr”3 Re”” (6) 
At lower Reynolds numbers the data are slightly higher 
than the solid line In fact, Sattertield and Resmck[25] 
have found Jr+/& = I 37, and De Acetls and Thodos[l9] 
JJJD = 1 5 1 McConnachle and Thodos [2l], Sen Gupta 
and Thodos[ZZ, 231, and Malhng and Thodos[ZO], 
however, have found JwlJD = I 0 In general, heat trans- 
fer measurements and determmatlon of heat transfer 
coefficients are more difficult than mass transfer studies 
Consldermg this, we may say that the heat transfer data 
shown m Fig 2 are well represented by eqn (6) 
Measurement of tetnperature m bed of no heat generating 
partrcles 
Kunu and Smlth[27], and Cybulskl et of [28], respec- 
tively,determined the heat transfer coefficients on the 
Continuous Solid Phase (C-S) model from the axml and 
radial heat transfer measurements 
Assuming that the gas temperatures computed on the 
C-S model are equal to the measured temperature 
profiles, Cybulskl et al obtam the coefficients Wakao et 
al [l3], however, pomt out that the C-S model can not be 
extended to the determmatlon of heat transfer 
coefficients at steady state 
In fact, under steady condlbon, the gas temperatures 
calculated with low heat transfer coefficrents on the C-S 
model agree with the temperatures computed on a smgle- 
phase model (m which heat transfer coefficient 1s not 
Involved, the temperature IS considered to be almost 
equal to the measured temperature), but the solid 
temperatures evaluated on the C-S model are far 
different from the temperatures on the single-phase 
model It IS considered that If Cybulskr et al had noticed 
the large difference m calculated temperature between 
the gas and solid, they have hesitated to determme the 
heat transfer coefficients 
Kunn and Smith also employ the C-S model In ad- 
dmon, they determine the anomalously low heat transfer 
coefficients by incorrect mterpretatlon of the algebraic 
relatIonshIp between the heat transfer coefficient and the 
effective thermal conductlvlty of the bed This has been 
cntlclzed by Llttman et al [29], and Gunn and De 
Souza[4] Gunn and De Souza point out that If Kunu et 
al had interpreted the algebraic relatlonshlp correctly, 
mfimtely large heat transfer coefficients would have been 
obtained 
As far as no heat source or smk exists m solid partl- 
cles, fluid- and sohd-phase temperatures under steady 
state condition are considered to be substantially the 
same Heat transfer coefficients cannot, therefore, be 
determined from the steady heat transfer measurements 
unless heat 1s generated or removed m solid particles 
Heat transfer between heat generatrng partrcles and fluid 
E~chhorn and Whlte[30], Baumelster and Bennett1311 
and Pel et al [32,33] have employed high-frequency 
heating to generate heat m sohd particles 
Elchhorn and White find that the sohd temperature 
profiles are linear function of axial distance (see then 
Fig 3) The solid temperature at bed exit IS then estl- 
mated by the straight hne-extrapolation of the solrd 
temperatures to the exit They assume that the tempera- 
ture difference between the solid and gas throughout the 
bed IS the same as the difference between the solid 
temperature extrapolated to the bed exit and the 
328 N WAKAO et QI 
Fig 3 Comparison 
Calculotm model 
Schumann with Nu - IO - 
Modlfled D-C Nu- I6 
Modlfed D-C IW- 9 005 
Modified D-C Mr= 12 002 
Modlfed D-C Nu=23 002 
Modbed D-C Nu=44 005 
0 
f. set 
of step response curves, lead-afr system, Re = 100, cb = 0 36, Dp = 0 3 cm, L = 3 3 cm, 
3 1 cm2/sec 
a:= = 
temperature of the gas leaving the bed However, the 
temperature difference 1s small (0 9-3 8°F) and yet the 
solid temperature profiles seem to have relatively large 
errors (about 1°F m then Fig 3) compared to the 
temperature difference The obtained heat transfer 
coefficients are, therefore, considered to be less r&able 
Another reason for discarding their data 1s that the 
sohd temperatures may not be extrapolated to the bed 
exit Equations (Bl-3) solved with the Danckwerts’ 
boundary conditions show that both the sohd and gas 
temperatures should level off at the exit If this 1s the 
case, the temperature difference evaluated by Elchhom 
et al IS larger than the real temperature difference m the 
bed 
Note that the solid temperatures m the bed bemg hnear 
function of axial distance may ave an impression that 
the axial fluid dispersion coefficient IS small However, 
the substitution of cu& of eqn (5) mto eqn (B3), m fact, 
gives almost hnear increase of sohd temperature m the 
axial direction except m the vicuuty of the bed exit 
The data of Baumelster er al have been criticized by 
Jeff reson [34] because large temperature differences 
existed m radial direction of the bed Pel et al found the 
particle temperatures uniform throughout the beds and 
have calculated the heat transfer coefficients under the 
condition of uniform parMe temperature However, 
there may be a question whether a uniform temperature 
IS attamed by the unrform heat generatlon m particles In 
fact, as mentloned above, Elchhorn ef al found sohd 
temperature increase m the axial direction of the bed 
Baumelster et al have also observed considerable 
difference m sohd temperature between the bed inlet and 
outlet 
Glaser and Thodos[35] heated the particles by passing 
elecmc current directly through the bed of metaI 
spheres We are afraid, however, that heat must have 
generated at the points of partlcle contact and most of 
the heat transfer would have taken place near the contact 
pornts 
We, therefore, conclude that all of the data revlewed 
m this section should not be Included m the data cor- 
relation 
REVIEW AND CORRECTION OF THE DATA OBTAINED FROM 
NONSI’EADY MFASUREMENTS 
From step, frequency and shot response measure- 
ments, the heat transfer coefficients have been obtamed 
on any of the heat transfer models shown in Table 2 
Schumann model, Contmuous Sohd Phase (C-S) model, 
and Dlsperslon Concentric (D-C) model 
The Schumann model [36] incorporates the following 
assumptions 
(I) Fluid IS m plug flow with no dlsperslon 
(II) No temperature gradient m particle 
The C-S model[29], when applied for axial heat trans- 
fer, has the assumptions 
(I) Fluid 1s m dispersed plug flow 
(11) Solid IS In axially contmuous phase through which 
heat conduction takes place m the axial direction 
The ortgmal D-C model and the modified D-C 
mode1[12, 141, both, have the assumptions 
(I) Fluid IS m dispersed plug flow 
(ii) Concentric temperature profile m particle 
The difference between the two models IS that the 
axial fluid drsperslon coefficient, a, of eqn (3), 1s used 
for the ongmal D-C model, while the large axial effective 
dispersion coefficient, aL of eqn (4), IS employed for the 
modified D-C model 
Gunn and De Souza[4] were the first to find, from the 
frequency response measurements, that the axial flmd 
thermal dispersion coefficients on the D-C model were 
considerably larger than those for extrapartlcle mass 
dispersion Vortmeyer(371 has discussed the meaning of 
the large dispersion coefficient values Wakao et al [38] 
have also obtamed, from the shot response measure- 
ments, the large thermal dlsperslon coefficients 
The C-S model has been cntlclzed by Kaguel et 
al [12], and the orlgmal D-C model by Wakao et 
al [ 1,141 They have shown that the moddied D-C model 
has advantage over the C-S and ongmal D-C models 
One of the important thmgs pomted out by them IS that 
the heat transfer coefficients determined on the C-S or 
original D-C model are different from those on the 
modified D-C model, particularly m the range of low 
Reynolds number 
The published nonsteady heat transfer data are, there- 
fore, converted mto those on the modified D-C model 
and compared with the steady heat transfer data recal- 
culated m the preceding section Note that the steady 
data have been reevaluated on the modtied D-C model 
Turner et al [39,40] and Gunn and De Souza[4] have 
obtamed, from the frequency response measurements, 
the heat transfer coefficients and the large axial dls- 
persion coefficients on the D-C model Their measure- 
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at r = R. aTS 
ks(--$ = hp(TF-Ts) 
ments obviously support the modified D-C model, but 
theu data are examined m sensitlvlty 
Step response 
Step response measurements were made by 
Furnas[41], Saunders and Ford [42]. Lof and 
Hawley[43]. Coppage and London[44], Handley and 
Heggs[45], and Bradshaw et al [441 Except Bradshaw et 
al who employed the D-C model. all of them have 
determmed the heat transfer coefficients on the Schu- 
mann model However, m the early stuches[414l] the 
heat transfer coefficients seem to have been determined 
by less reliable graphIcal method 
The data of the two relatively recent measurements, 
Handley ef al [453 and Bradshaw et al [&I, are reevalu- 
ated on the modified D-C model Handley et al have 
obtamed the heat transfer coef?icIents on the Schumann 
mode1 The exact solution to a step response IS shown m 
Appendix D Usmg the Schumann model solution, the 
response curves T,(t) of Handley et al are predicted with 
the data reported in their paper The response curves TZ( t) 
are also calculated on the mod&d D-C model wrth vaned 
heat transfer coefficient values Figure 3 illustrates the 
comparison of T,(f) and IQ(t) The error es IS calculated 
by 
where we choose the times. 7, and 7*, respectively. as 
T&T,) = 0 05 To and T=(T~) = 0 95 To 
We may probably say that the agreement of the two 
curves IS good when l sFig 9, the 
valley with the error 0 05 IS steep As far as the example 
is concerned, the determined Nusselt number IS consi- 
dered to be enough reliable 
I 
In fact, the experiments of Turner et al were conduc- 
ted at high Reynolds numbers 1200-4600 As mentioned 
already, reliable heat transfer coefficients are determined 
at high flow rates The data of Turner et al are, there- 
fore,-included in our data correlation 
Id I II, 
Fig 9 Error map for a simulation illustration of Goss and 
Turner[391 (+ shows the data obtamed by them), Re = 950, 
o = 0 5 rad/sec, other data shown in theu Table 1, a:, of eqn 
(5) = 250 cm2/sec 
Shot response 
From the analysis of shot response measurements, 
Wakao et al [l, 381 examined the heat transfer 
coefficients on the modified D-C model, pointing out the 
advantage of the model No definite Nusselt numbers 
were obtained, but they have found that Nusselt 
numbers were considered to be in the range from 0 1 to m 
at Reynolds numbers 0 24 
ex$ession of mass correlation, eqn (6) - 
I I 7 
CORRELATION OF NUSSELT NUMBERS 
From the review in the preceding sections the heat 
transfer data which have passed our criteria are [16, 
M-23, 25, 39. 40, 45, 463 Their heat transfer coefficients 
reevaluated on the modified D-C model are plotted in 
Fig 10 The data are well reoresented bv the analogous 
L 
e 
0 
- + 
D 
14 - I> 
V 
A 
NU 
10 - 
Steady state 
Wllke and Hougen Cl61 
Satterf@d and Resnrk C251 
Galloway eZ a/ Cl81 
De Acetls and Thodos Cl91 
McCo-he and Thodos C211 
Sen Gupta and Thodos C22. 231 
Mallmg and Thadas C201 
Nonsteady state 
l Hand!ey and Heggs L451 
N&,=2+1 lPr”sR&s 
I Bradshow ef o/ C461 
. Goss and Turner C391 
A Turner and Otten [401 
IO 10- ID- IO- Id 
(Pr”%M v 
Fig 10 Correlation of reevaluated Nusselt numbers 
Effect of tluld drsperslon coetliclents on particle-to-flutd beat transfer coefficrents m packed beds 335 
One more thmg we have to pay attention to IS that Ftg 
1 shows the mixture of heat transfer data obtamed on the 
different models Some of them are, as mentioned al- 
ready, less rehable and some have been crltlclzed on the 
calculation procedure 
COIWLUSIONS 
1 The published heat transfer data are corrected for 
the ax&al fluid thermal dlsperslon coefficients proposed 
by Wakao[l 1 The reevaluated data on the mod&d D-C 
model are correlated by the analogous form of the mass 
transfer correlation, eqn (6) 
2 When the heat transfer correlation 1s used for the 
design and analysis of packed bed reactors, the axial 
effective thermal dlsperslon coefficients gven by eqn (4) 
should also be employed 
a 
CF 
CS 
DP 
G 
h, 
h;, 
JrhJw 
ke" 
k aax 
k ef 
k es 
k, 
ks 
L 
Nu 
Pr 
R 
Rd 
TF 
Tk 
TL 
Tt 
NCYTATION 
partlcle surface area per umt volume of packed 
bed 
specific heat of fluid 
specific heat of sohd 
particle diameter 
~,bupF, fluld mass velocrty 
heat transfer coefficient 
heat transfer coefficient determmed with aaX = 0 
J-factors, respectively, for mass and heat 
effective thermal conductlvlty of packed bed 
with stagnant fluld 
axial effective therma conductlvlty of packed 
bed 
effectlve fhud thermal conductlvlty 
effective sohd thermal conductlvlty 
fiuld thermal conductlvlty 
sohd thermal conductlvlty 
packed bed helgkt 
hPDJkF, Nusselt number 
C&kF, Prandtl number 
partrcle radms 
radial distance variable 
D+bUpJp, Reynolds number 
fhud temperature 
Inlet fluId temperature 
fluid temperature at bed exit 
fluld temperature at bed exrt calculated on 
modified D-C model 
fluld temperature at bed exit calculated on D-C 
model with varied crrx and Nu 
temperature change Imposed on tnlet fluid 
temperature on particle surface 
sohd temperature 
amplitude m complex 
ttme 
mterstttlal fluid velocity 
axial distance variable 
Greek symbols 
aar axial fluld thermal dlsperslon coefficrent 
a?; axml effective thermal dlsperslon coefficient, 
defined by eqn (4) 
aF fluld thermal drffuslvlty 
as sohd thermal dlffuslvlty 
Q bed void fraction 
4 error defined by eqn (8) 
eS error defined by eqn (7) 
pF flurd denstty 
PS sohd density 
7,, 72 trmes 
p fluld vlscoslty 
w frequency 
REFERENCES 
[l] Wakao N, Chem EngngScr 197631 1115 
[2] Wakao N and Funazti T, Chem Engng Scr 1978 33 1375 
131 Lntman H and Shva D E . Proc Int’l Heat Transfer Conf 
Versailles, Paper CT1 4 (1970) 
r41 Gunn D J and De Souza J F C Chem Engng Scd 1974 29 - 
1363 
[5] Cormsh A R H , Trans Instn Chem Engrs 1%5 43 T332 
161 Kunu D and Suzuki M . Inl J iYear Mass Truns 1%7 10 _ - 
845 
[7] Nelson P A and Galloway T R , Chem Engng Scr 1975 30 
1 
[8] Schlfinder E U , Chem Engng Scl 1977 32 845 
191 Pfeffer R and Happel J , A ICh E J 1964 10 605 
[lo] Mlyauchl T , J Chem Engng Japan 1971 4 238 
[I I] Serensen J P and Stewart W E , Chem Engng Scr 1974 29 
827 
[12] Kaguel S , Shrozawa B and Wakao N , Chem Engng SCI 
1977 32 507 
[ 131 Wakao N , Kaguel S and Nagal H , Chem Engng Scr 1977 
32 1261 
[14] Wakao N , Kaguel S and Shlozawa B , Chem Engng Scl 
1977 32 451 
[IS] Gamson B W , Thodos G and Hougen 0 A, Truns Am 
Inst Chem Engrs 1943 39 1 
[16] Wllke C R and Hougen 0 A. Trans Am Insl Chem 
Engrs 1945 41 445 
[17] Hurt D M , Ind Engng Chem 1943 35 522 
[18] Galloway L R , Komarmcky W and Epstem N , Con J 
Chem l&n= 1957 35 139 
[I91 
1201 
PII 
WI 
WI 
~241 
[251 
[261 
[271 
[=I 
r291 
r301 
[311 
~321 
r331 
[341 
f351 
K361 
[371 
r3g1 
[391 
(401 
De Acetlsj and Thodos G , Ind Engng Chem 1960 52 1003 
Malhng G F and Thodos G , Inf J Heat Mass Truns 1%7 
IO 489 
McConnachle J T L and Thodos G , A Z Ch E J 1963 9 60 
Sen Gunta A and Thodos G , A I Ch E3 1963 9 751 
Sen Gupta A and Thodos G , Ind Engng Chem Fundls 
1964 3 218 
Bradshaw R D and Myers J E t A ICh EJ 1963 9 590 
SattertIeld C N and Resmck H , Chem Engng Progr 1954 
50 504 
Wakao N and Kato K , J Chem Engng Japan 1969 2 24 
Kunu D and Smith J M , A I Ch E J 1961 7 29 
Cybulskl A, Van Dale” M J . Verkerk J W and Van Den 
Berg P J Chem Engng Scr 1975 30 I015 
Lrttman H , Barde R G and Pulsifer A H , Ind Engng 
Chem Fundls 1%8 7 554 
Erchhorn J and White R R , Chem Engng Progr Symp 
Ser 195248(4) 11 
Baumelster E B and Bennett C 0 , A ICh E J 1958 4 69 
Balakrlshnan A R and Pel D C T, Ind Engng Chem 
Proc Des Da, 197413441 
Bhattacharyya D and Pel D C T , Chem Engng Scr 1975 
30 293 
Jeffreson C P A ICh El 1972 18 409 
Glaser M B and Thodos G , A I Ch E J 1958 4 63 
Schumann T E W , / Frankfm hsl 1929 2@ii 405 
Vortmeyer D , Chem Engng Scl 1975 30 999 
Wakao N , Tamsho S and Shlozawa B , Kogaku Kogaku 
Ronbunshu (Japan) 1976 2 422 
GossM J andTumerG A AfChEJ 1971 175% 
Turner G A and Otten L , Ind Engng Chem Proc Des 
Dev 1973 12 417 
336 
N WAKAO et al 
I411 Furnas C C , Ind Engng Chem 1930 22 721 
[421 Saunders 0 A and Ford H , J Iron SteelInsr 1940 141 291 
[43] Lof G 0 G and Hawley R W , Ind Engng Chem 1948 40 
1061 
[441 Coppage J E and London A L Chem Engng Progr 1956 
52 57-F 
[451 Handley D and Heggs P J Trans Instn Chem Engn 1968 
46 TZSI 
r461 Bradshaw A V , Johnson A McLachlan N H and Chm 
Y-T Trans Instn Chem Engrs 1970 48 T77 
I471 Lmdauer G C , A ICh El I%7 13 I181 
APPENDIX A STEADY HEAT TRANSFER B ETWRRN FLUID AND 
PARTlCLE SURFACE AT CONSTANT TEMPERATURE 
For heat transfer between thud and particles havmg constant 
surface temperature T,, throughout the bed of finite length, the 
change m thud temperature IS given by (refer to eqn (7) m 121) 
Tm - TL = 4Aev 2rrw L-1 T,, - Ttn (t +A)‘exp[A&]-(l:)‘exp[-A$$] 
where 
When n:, = 0 eqn (Al) reduces to 
T’s - TL _ 
Tps - T,, - exp 
h;aL 
- CCZJCFQF 1 
ConversIon of h; into h, IS, therefore made 
(Al) and (A2) 
(Al) 
(A3 
by equating eqns 
APPENDIX B STEADY HEAT TRANSFER FROM HEAT GENERATING 
PARTKLES 
When heat 1s generated at constant rate m particles the flmd 
and sohd temperatures m the bed of finite length are 
TF = T,,++(&)(+) 
and 
x[f+$(l-‘=P[-~(l-;)])] (RI) 
T,= T+@+*(R’-9) 
3h, 6ks 
where q = heat generatlon rate per umt volume of particleThe 
sohd temperatures Elchhorn and WhrteI301 claim_ to have 
measured are the particle-volume-mean-temperatures Ts 
(B3) 
APPENDIX C FREQUENCY RESPONSE 
For the fundamental equations hsted m Table 2, the boundary 
condltlon on the Schumann model IS 
atx=O TF = Re [fOeelp)#] (Cl) 
and those on D-C model are 
at x = 0. U(TF - Re f?Oel”‘]) = a., % 
aTF=O 
(C2) 
atx=L 
dX 
When the stationary solution of TF at x = L 1s expressed as 
TL = Re [ fL cur] (C3) 
on Schumann model f,+ IS 
%=exp[-+(I+-$-)] (C4) 
where 
k,= h*a 
(I- 4CsPs 
and on D-C model, 
UL . 
TL= 
exp - 
[ 1 
TO cash 2,~1+8)]+~slnh,~,lts)l C 0.X 
where 
1 
I 
t$ coth r$ - 1 
In both Schumann and D-C model, the error Ed of eqn (8) IS 
W4 
Using eqn (Cl), Gunn et al [41 have obtained the semi-mtimte 
solution of the fundamental equations 
f 
f=exp 
TO C, 
e (I- ~‘(1 + B))] (C7) 
APPENDIX D Sl%P RESPONSE 
The mltial and boundary condlttons for Schumann model are 
at f=O, TF = 0 
at x =0, TF = T,, I WI) 
and those for D-C model are 
at f=O, TF = 0 
at x = 0. U(T,z - To) = a_ % (D2) 
at x = L, aT,=O 
l%X 
The solution IS 
TL=l+22 ’ ---Im [ (2) eUf],=, 
TO 2 ?r,,=,2n-I 
(D3) 
where w. = (2n - I)F/$and lo IS a time sufficiently long enough 
to attam Tr = To TJTo IS given for Schumann model by eqn 
((24) and for D-C model by eqn (C5) 
The error of eqn (7) IS approximately expressed as