- Email: [email protected]
The design of shallow fluidised bed heat exchange equipment
Powder TechnoZogy-Ekvier
The Design
Sequoia SA., Lamanne-Primed
of Shallow
159
in tic Netherlands
Fiuidised Red Heat Exchange
Equipment
D. R. McGAW
(Received August 23. 1971; in revised form Januxy
25. 1972)
Sunimary Tlze shallow jhddked bed is now a standard industrial techniquefor heating or cooling particulate solids. In spite of this. satirjactory design procedures for this type of equipment hare not Jet been formulated_ Analyses are presented in this paper which can be used to design both sikgIe and double stage equipment. The analyses corer both possibiliries of thermal equilibrium or non-thermal equilibrium in the bed, and also may be usedfor various soIi&flo~r conditions.
ImODU~ON
The method of cooiing or heating particulate solids in a shallow. rectangular section fluidised bed is now a standard industhal technique. Its use. for instance, has been reported for process heating of granular coker and for cooling _m_ranuIar fertilizersZe3, sulphur5 and powdered milk’_ This kind of equipment is generally operated either as a single-stage or as a two-stage unit with fluidised bed depths as low as can satisfactorily be operated, e.g. an inch or two. The use. of low fluidised beds reduces the pressure drop in the system. In the two-stage unit, the solid particles are fed on to the top stage and leave from the bottom stage, the gas entering the unit under the bottom stage and then being re-used in the top stage. The use of the two-stage device results in more efficient use of the gas. In spite of the wide use of the technique, only limited analytical work has been reported as yet, and this only for a single-stage device3-6-7. Thus a satisfactory design method has not yet been established This paper proposes methods which can be used to enable a rigorous design of either a single or two-stage device to be made for a given heat load. Powder Technol_ 6 (1972)
GEXER4L
APPR0.4CH
It has been established* that thermai equilibrium in a fhridised bed system can be reached only a short distance above the distributor plate. The magnitude of this distance depends on the size of the particles and conditions existing within the bed_ This means that for small particle systems such as powders. shallow bed operation will most likely be under \onditions of thermal equilibrium_ Hovvev-er_ for larger particles the possibility of thermal gradients within the particles exists. and the depth of bed may not bc enough for thermal equilibrium to be achrev-ed_ Thus two types of analyses are possible one for the case where thermal equilibrium has been achieved_ this being the more important. and the other for the use where thermal equilibrium has not been achieved. One important factor which previous investigators have neglected is that of the effect of solids flow conditions within the bed on the process of heat transfer. If the bed was operating without bubbies being formed lateral mixing of solids in the bed would be minimised and plug flow conditions would be approximated_ But because bubbles do cause lateral mixing, plug flow conditions will only be approached in longer beds. The deeper the bd up to a certain limit, the larger the bubbles. hence the greaier the mixing and the tendency to move towards the condition of perfect mixing- This has been verified experimentally by Verloop et aL9_ The effect of changing the residence time distribution in the system is to change the system heat transfer characteristics Thus it is important to inclnde the effect of residence time distribution on the process of heat transfer in the system under consideration_ It should also be noted that a residence time distribution can be introduced by other factors such as non-uniform solids inlet distribution. In order to accommodate ‘rhesetwo phenomena. the approach adopted has been to !irst consider a
160
D_ R McGAW
single-stage device operating under conditions of plug flow solids. The analysis assuming thermal equilibrium was developed and then modified to allow for the residence time distribution of solids_ This was then extended to a two-stage device. The same approach was extended to account for nonthermal equilibrium in both a single and two-stage device_ This is given in Appendix I since it is of less practical
importance.
THEORETICqL
AN.4LYSES
The following assumptions are made in each analysis : (i) The particles are of uniform temperature at any time. (ii) The temperature of operation is small enough to neglect heat losses from the unit by convection and radiation_ (iii) Uniform distribution of gas velocity-no chanelling. The analyses are based on a case where solids entering the unit at temperature rSiare to be cooled by gas entering at t,, I_ Single stage
1. Single stage tluidisedbed.
Fig
1.2 General casefir
any residence time distribution
If there is a distribution of residence times, then the relationship for determining particle temperature having a dimensionless residence time 0 may be given by
The following general relation may then be used to determine the mean outlet temperature: zc
I _1 Plug flow of solids.
Applying a heat balance over the differential element of length da as shown in Fig. 1 gives M,C,dr,
= G,C,(t,,-
t&da
t,E(O)dO
f_= 0
For the case under consideration stituted
(1)
This is the basic relationship for the single-stage unit from which all cases are derived. When thermal equilibrium is achieved in the bed, fro = ts_ Hence :
It is convenient to take all temperatures with reference to f,, thus (t,-ff,J may be taken as T,__ Integrating between the limits T=T,,ata=A
T=T,,ata=O
by the relation
derived
td may be sub
for T,, at dimension-
less residence time 0. giving (I,),
r Ti exp
=
I0
(-
w)
E(O)dfl
In order to integrate this, it is necessary to know the relationship between E(B) and 8. If this relationship is in numeric form then the inte_graI would have to be. solved numerically but if an algebraic relation is available it may be integrated directly. If, for instance, flow through the bed can be described by a perfect mixers in series model then the relationship is given by”
E(O)=
PeJ-’ (J
_,#
e
Inserting this and integrating gives
the analysis for the simplest case, has already been reported3.
This,
T o= T,i
JJ
or if there are P perfect mixers per unit length of bed:
THE DESIGN
T
2
=
Ki
OF SHALLOW
FLUIDISED
BED HEAT EXCHANGE
Hence
+ 1]- pA
& [
d&s = xTsB da
2. Two smges 21 Plug flow of soiids The physical situation of this case where the two stages have the same dimensions is depicted in Fig 2 The length term a is increasing in the direction of flow on the top stage and decreasing in the direction of flow on the bottom stage Subscript B refers to the bottom stage and subscript T to the top stage. On the bottom stage eqn. (1) holds as for the single stage. If we simplify this by putting z= G,C,/MsC., then this reduces to
dT,, =
IT
da
0
If the two stages are relatively close together there will be little or no lateral mixing of gas between stages. Hence for the top stage the differential heat balance gives - M~C,dr,, or -
d7;, da
= G,C,(r,-
= ~x(T~r- TEB)
161
EQUIPMENT
r,,)dq
integrating eqn. (2) for the bottom stage between 0 and CIwhere 0 c a -C A results in KB = ?;O exp (zca)
d7r da
The sign is negative because on the top stage rs is decreasing as a is increasing These two relationships (4) and (5) form the basis from which the analyses for the two-stage unit are derived. Under conditions of thermal equilibrium
i
x7& = rl,
exp(ya)
This is a linear first-order differential equation which can be solved directly using terminal condition (a = 0. Tq~= Ti) to give T,r exp(xn) = +
exp(Zxn) -
+
f
Tsi
(S)
But at n= A. 7&= 7;r and from eqn. (7) it is seen that at this point Li
(3
(7)
Inserting this general relation for xB into eqn. (6) gi\-es
(9)
= T, exp Wl)
Inserting eqn. (9) into (8) and simplifying gia-es
T,i_ T - +[exp (22~4) + I] This analysis is based on a personal communieation’ ‘_ 37 General case for any residence time distribution
Equations (4) and (5) still hold, but in this case applying the residence time distribution function on the bottom stage gives
~sg=
T,, =
JI IIT_exp (w)]
E(B)dB
Hence
Fig 2 Two stas fltidised beds
In order to evaluate this. it will be necessary to know the relationship between E(6) and 0 so that the integral can be solved before the differential equation. This equation only takes into account the residence time distribution on the lower stage, however_ Thus, after a relationship has been obtained for Kr, the residence time distribution of solid on
162
D. R MffiAW
the top stage must be accounted for in a similar manner. If solid flow conditions are such that flow can be described by a simple perfect mixers in series model, eqn. (10) can be modified to give
The distinction is made between zg and xr because the residence time distribution for solids flow has been accounted for on the bottom stage but not yet on the top stage_ They will be numerically the same. Solving this in a similar manner to that described in Section 2 1 gives for TcToat n = A [esp (- r,A)) x~cj;~ exp il [P In( 1 + z#) [P In (1 ta&P)tz,] t
TsTo =
UT Ts,
+ G
-
P ln(1 +rr,/P)+aJ
+ xT]
\
We must now take into account the residence time distribution on the top stage_ which would be based on solids flow on the top stage only, ie_ by using x&I instead of zT and then integrating as follows:
Go =
J‘; :exp(-z+L4): are T,, exp A [IP In(It z&P) i aTO] [P In (I + ag/P) + a#] 1
.
+ T,i -
[P In (1 TfBTi) + aTo] I E(e)de
(11) but at a=A TSTo= &
= T,(l
-~rr,/P)~
(12)
The integral would have to be determined numerically as indicated in the worked example in Appendix Il.
DISCUSSION
The relatiorr derived can be used to calculate the base area requirement of either a single or twostage unit for a given heat load Alternatively_ they may be used to determine the amount of heat transferred in a given unit operating under a particular set of conditions_ in order to apply the relations, however, information must be available on the heat transfer coefficient and flow properties ofthe Powder TechnoL 6 (1972)
system. Information on the heat transfer coefficient would first be used to de?ermine whether thermal equilibrium isapproximated in the bed or not. lfthis approach temperature is low [(r,- fsO)< 0.1” C] at the hot end of the bed, then the assumption of thermal equilibrium may be considered valid. The information on the solid flow properties of the system would be used to determine whether the bed is operating under plug flow conditions or whether a residence time distribution would have to be taken into account. A typical design example for the cooling of urea from 100°C to 40°C using_ air at 30°C in both single and two-stage units is gwen in Appendix Il. Examination of eqn. (3) shows that the longer the bed, the closer is the approach to plug flow conditions_ Hence the method suggested is to choose as long a bed as is practical. consistent with its physical location and then calculate the required width for the given heat load. using the appropriate desi_F air velocity_ It is worthwhile noting that for the smgle-stage case, twenty-five perfect mixers in series gave a cooling air requirement only 4% greater than that for plug flow conditions_ For the two-stage case, twenty perfect mixers in series gav-e a cooling air or area requirement 22% greater than that for the plug flow condition. Hence if the bed length is greater than the equivalent of about 40 perfect mixers for the single-stage unit and about 50 perfect mixers for the two-stage unit, the plug flow equation can be used with acceptable accuracy (2-3 %)_ A residence time distribution can. however, be set up in other ways than just by the fhtidised solid mixing mechanism implied in the example_ The most likely way is by there being an uneven solids inlet distribution_ This can be minimised by having a narrow bed. Thus it is seen that, in order to minimise the air flow requirement for a given load, the equipment should be operated under plug flow conditions wherever possible_ These conditions are associated with relatively long narrow rectangular shaped beds and small bubbles, i.e. low bed depths. The effect of increasing the bed depth may not be significant when operating large beds where plug flow conditions are already approached, but for small beds it can be signi&ant The effect of increasing the outlet solid temperature by increasing the bed depth has been demonstrated by the author in a small laboratory scale unit, the results being tabled in Appendix III. It is unlikely that a large scale unit would be able to operate at a bed depth below 3 cm, so that under normal conditions, it would be
THE DESIGN OF SHALLOW
FLUIDISED BED HEAT MCHANGE
advisable to operate at the lowest operable bed depth. In order to maintain steady operation, a stable downflow of solids between stages is necessary in the two-stage unit. The most satisfactory method of achieving this is by using a rotary valve as air seal between the two stages, as indicated in Fig. 2. The heat transfer coefficient may be obtained using the appropriate correlation from the reviewI on gas-particle heat transfer in fluidised beds It is important to note, however, that not every investigator has used the same definition of heat transfer coefficient. So it must bestressed that when attempting to use one of the correlations, the designer must check that the definition of heat transfer coefficient in the correlation coincides with that used in the analysis. All published work on residence time distributions in fluid&xi beds has also been reviewed13_ Most work was in circular section fluid&d bed systems, but one set of investigators9 carried out an investigation on a rectangular section fluidiscd bed. Their work in a small scale unit showed that solids flow in the system was somewhere in between the extremes of plug flow and perfect mixing. More work is, however, necessary in this field. The analyses derived can be used to optimise operating conditions in a unit and also to compare single-stage with two-stage operation_ A comparison has been made in the sample design calculations in Appendix II. This showed the actual cooling air requirement for a two-stage unit to bc somewhat more than half that for a sing!e-stage unit. Hence, since the pressure drop is doubled in a twostage unit, the single-stage unit seems to offer cheaper running costs. Normally, however, a cyclone is n ecessary in the system and this could tend to bring the running costs in favour of the twostage system because of the reduction in the a~ flow rate. It is unlikely that a three-stage system will have lower running costs than a two-stage system. More efficient use of the air is made in a two-stage system than the single-stage sys:em. If u-e define a gas phase cooling efficiency by the equation 9
8
=
EQUIPMENT
163
cheaper to operate than other equipment capable of doing the same job_
CONCLUSIONS The analyses derived can be used zo design and oplimise operating conditions in a fluidised bed heat exchanger_ A comparison has been made between single and two-stage units and this shows that, taking into account economics and eflicienties, a single-stage unit appears to offer cheaper running costs, provided that the pressure drop in the rest of The gas circuit is small. APPENDIX
I
NO THERMAL
EQUILIBRIUM
IN BED
Single stage
In this case, eqn. (1) still holds but it is necessary to introduce a heat transfer coeffLient by carrying out a differential heat balance in a vertical element of unit cross-sectional area Referring to Fig. 3 this is seen to be puCrdrs Integrating
= k(t,-
r.Jd=
this between
(AlI the limits
te = roi at z=(? tp=t,,at==z gives
Eak in gas temp. = LX,fi max. possible gain tpi - t,,
then for the example quoted in Appendix II this eficiency is increased from 33.7 oAto 42 o/Oby adding the second stage. The analyses also enable a comparative study to be made to decide if this kind of equipment will be
Fig 3. Vertical difTerentiaI heat balanaz
164
D- R
(A2) Plug jZow of solids Rearranging eqn. (A2) to put it in terms of
McGAW
General case for any residence time distribution The differential equation for the solid tempem-
ture on the top stage in this case is given by
The solution is obtained by a similar method to that obtained in Section 2.2.
T, = (t,, - trzi) gtves
(A3 Combining eqns. (AI) and (A3) and integrating between the limits a=0
T=T,, T=
a=A
Ti
APPENDIX SAMPLE
II DESIGN
CALCU-LATION
It is required to calculate the size of both a singlestage and a two-stage fluidised bed to cool 5000 kg/h of urea from 1WC to WC using air at 3BC.
grres
case for anv residence time distribution Modifying the equation in the usual manner gives
Generai
T,=
T,i
exp
-
w-7 1 MsC, [1 -eelip
(-
%)I}
E(B)dfl
Two stages Plug flow OJ solids
In this case the relationship for the solids temperature on the bottom stage is given by the equation
-dT,, = da
q9TsB
0.0015 m (1500 pm) 1.00 0.5 0.05 m 1.165 kg/m’ 2 m/set ODDOO18kg/m set 1-W J/kg% 1.33 J/kg =K _. It may be assumed that solid flow conditions can be given by a perfect mixers in series model with 10 perfect mixers per metre length of bed. It is lirst necessary to calculate if thermal equilibrium is achieved in the bed. This is done by use of eqn. (-42) &=p”d=
P
195
From Fig 1 in the review” on heat transfer coeflicients in fluidised beds, we get j” = 0.2
where
and St = 0.256
For the top stage, eqn. (5)may be modified to give
Using a similar procedure to that given previously in Section 2 results in T,i= T, Powder
+ [exp (2zflA) + l]
Tech&
6 (1972)
fs--f,, _
--exp r, - tgi
C ---
h
WC=
6(1 -e)z sD.dP
1=
exp (-
25.6)
Thus t, - toDis so small that thermal equilibrium can be assumed. Single stage
Assume that the largest value of A that can be
THE
DESIGN
OF SHALLOW
accommodated eqn. (3) gives
is 25
FLUIDISED
m. Then,
BED
HEAT
substitution
EQUIPMENT
in
gas particle heat transfer coefficient number of perfect mixers mass flow rate surface area of solid per unit bed volume surface area of solid per unit base crosssectional arca temperature temperature with refercnoe to inlet gas temperature (= f - fFi) superficial gas velocity in bed depth of bed dimensionless residence time. defined as residence time divided by mean residence time bed porosity shape factor viscosity density efiiciency G,C,/Xlr,C, [l-exp(--AS/G,C,)]
G,A = 13,050 kg/h with bed dimensions 25 m x 0.62 m. Note that if plug flow conditions had been assumed, the air requirement would have been 12.550 kg/h and bed dimensions 25 m x 0.6 m. Two stages
A bed length of 2 m is assumed in this case. The relationship to be used to calculate G, is a combination of eqns (11) and (12). A trial and error solution is neozssary7 since a(= G= Cs/&Zs C,) appears as a complex function within the integral. The relationship between 0 and E(0) was obtained from the perfect mixers in series equation using J= 20. The trial and error calculation of G, using an IBM 1620 computer gave an air requirement of IO.200 kg/h which corresponds to a bed 2 m x 0.61 m. Note that if plug tlow conditions had prevailed, the air requirement would have been 8320 kzJh corresponding to bed dimensions 2 m x O-5 m.
APPENDIX
III
RESULTS OF BED COOLER
TESTS
ON
LABORATORY
FLUIDISED
(cm)
Air flow. kg/h Solid flow, k@h Inlet solid temp. t,, OC Outlet solid temp. t_ OC Inlet air temp. tn. OC
LIST
A C
4
E(e) G
Ponder
2.5
5
7J
90 81.9 110.0 54.9 36.7
90 81.0 108.8 555 372
90 81.0 1092
90 Sl.0 109.1 613 3s.5
38.4
OF SYMBOLS
length of bed specific heat particle diameter fraction of solids having residence time bctween8andetdQ mass flow rate per unit length of bed
TechnoZ..
6 (19JZ)
B g i m 0
bottom stage gas phase bed inlet mean value bed outlet solid phase top stage at residence time 6
REFERENCES
13
59.7
Subscriprs
; 0
30 cm long x 5 cm wide Size of cooler I sand 14-18 mesh Solid material : CooIing medium : air Beddepth
165
EXCHANGE
1 Yv_ P. Ryazantser. intern Chcm Eng..S (1)(196X) 131. 2 H. Schrcibcr. Brir. Chem Eng_ 7 (8) (1962) SSJ. 3 EA. Kazakova et nL. Khim Prom . (5) (1961) 330. 4 J. Cibrowski and J. Roaak. Pcem Chem. 3 (195s) 103. 5 G. A_ Dormadieu. Genie Chim, 85 (1) (1962) 1. 6 N. I. Gelpcrin and V. G. Ainshtein. Intern Chem &ng_. 3 (2) ( 1963) 259. 7 V_ABorodu1~aInrenrChemEng..4(1)~1964)110. 8 J. F. FrantL. Chem Eng. Progr., 57 (7) (1961) 35. 9 J. Vcrloop. I_ H. Dc Nit and P. M. Hcertjes. Pror 1nre-m. SJmp. on Fluidiscztion, June 6-9, 1967, Netherlands Uni%ersity Press, p_ 476. IO 0. Levenspiel. Chemical Reoc:ion Engineering. Wilq. New York. 1965 11 J. A McWilliam, Fmns Ltb, Felixstowz, U.K. personal communication 1965. 12 J. J_ Barker. Ind. Ens. Chem. 57 (3 (1965) 33. 13 J. Verloop. I_ H De Nieand P.M. Heertjes_ Powder TechnoL. 1 (196s) 32
The Design
Sequoia SA., Lamanne-Primed
of Shallow
159
in tic Netherlands
Fiuidised Red Heat Exchange
Equipment
D. R. McGAW
(Received August 23. 1971; in revised form Januxy
25. 1972)
Sunimary Tlze shallow jhddked bed is now a standard industrial techniquefor heating or cooling particulate solids. In spite of this. satirjactory design procedures for this type of equipment hare not Jet been formulated_ Analyses are presented in this paper which can be used to design both sikgIe and double stage equipment. The analyses corer both possibiliries of thermal equilibrium or non-thermal equilibrium in the bed, and also may be usedfor various soIi&flo~r conditions.
ImODU~ON
The method of cooiing or heating particulate solids in a shallow. rectangular section fluidised bed is now a standard industhal technique. Its use. for instance, has been reported for process heating of granular coker and for cooling _m_ranuIar fertilizersZe3, sulphur5 and powdered milk’_ This kind of equipment is generally operated either as a single-stage or as a two-stage unit with fluidised bed depths as low as can satisfactorily be operated, e.g. an inch or two. The use. of low fluidised beds reduces the pressure drop in the system. In the two-stage unit, the solid particles are fed on to the top stage and leave from the bottom stage, the gas entering the unit under the bottom stage and then being re-used in the top stage. The use of the two-stage device results in more efficient use of the gas. In spite of the wide use of the technique, only limited analytical work has been reported as yet, and this only for a single-stage device3-6-7. Thus a satisfactory design method has not yet been established This paper proposes methods which can be used to enable a rigorous design of either a single or two-stage device to be made for a given heat load. Powder Technol_ 6 (1972)
GEXER4L
APPR0.4CH
It has been established* that thermai equilibrium in a fhridised bed system can be reached only a short distance above the distributor plate. The magnitude of this distance depends on the size of the particles and conditions existing within the bed_ This means that for small particle systems such as powders. shallow bed operation will most likely be under \onditions of thermal equilibrium_ Hovvev-er_ for larger particles the possibility of thermal gradients within the particles exists. and the depth of bed may not bc enough for thermal equilibrium to be achrev-ed_ Thus two types of analyses are possible one for the case where thermal equilibrium has been achieved_ this being the more important. and the other for the use where thermal equilibrium has not been achieved. One important factor which previous investigators have neglected is that of the effect of solids flow conditions within the bed on the process of heat transfer. If the bed was operating without bubbies being formed lateral mixing of solids in the bed would be minimised and plug flow conditions would be approximated_ But because bubbles do cause lateral mixing, plug flow conditions will only be approached in longer beds. The deeper the bd up to a certain limit, the larger the bubbles. hence the greaier the mixing and the tendency to move towards the condition of perfect mixing- This has been verified experimentally by Verloop et aL9_ The effect of changing the residence time distribution in the system is to change the system heat transfer characteristics Thus it is important to inclnde the effect of residence time distribution on the process of heat transfer in the system under consideration_ It should also be noted that a residence time distribution can be introduced by other factors such as non-uniform solids inlet distribution. In order to accommodate ‘rhesetwo phenomena. the approach adopted has been to !irst consider a
160
D_ R McGAW
single-stage device operating under conditions of plug flow solids. The analysis assuming thermal equilibrium was developed and then modified to allow for the residence time distribution of solids_ This was then extended to a two-stage device. The same approach was extended to account for nonthermal equilibrium in both a single and two-stage device_ This is given in Appendix I since it is of less practical
importance.
THEORETICqL
AN.4LYSES
The following assumptions are made in each analysis : (i) The particles are of uniform temperature at any time. (ii) The temperature of operation is small enough to neglect heat losses from the unit by convection and radiation_ (iii) Uniform distribution of gas velocity-no chanelling. The analyses are based on a case where solids entering the unit at temperature rSiare to be cooled by gas entering at t,, I_ Single stage
1. Single stage tluidisedbed.
Fig
1.2 General casefir
any residence time distribution
If there is a distribution of residence times, then the relationship for determining particle temperature having a dimensionless residence time 0 may be given by
The following general relation may then be used to determine the mean outlet temperature: zc
I _1 Plug flow of solids.
Applying a heat balance over the differential element of length da as shown in Fig. 1 gives M,C,dr,
= G,C,(t,,-
t&da
t,E(O)dO
f_= 0
For the case under consideration stituted
(1)
This is the basic relationship for the single-stage unit from which all cases are derived. When thermal equilibrium is achieved in the bed, fro = ts_ Hence :
It is convenient to take all temperatures with reference to f,, thus (t,-ff,J may be taken as T,__ Integrating between the limits T=T,,ata=A
T=T,,ata=O
by the relation
derived
td may be sub
for T,, at dimension-
less residence time 0. giving (I,),
r Ti exp
=
I0
(-
w)
E(O)dfl
In order to integrate this, it is necessary to know the relationship between E(B) and 8. If this relationship is in numeric form then the inte_graI would have to be. solved numerically but if an algebraic relation is available it may be integrated directly. If, for instance, flow through the bed can be described by a perfect mixers in series model then the relationship is given by”
E(O)=
PeJ-’ (J
_,#
e
Inserting this and integrating gives
the analysis for the simplest case, has already been reported3.
This,
T o= T,i
JJ
or if there are P perfect mixers per unit length of bed:
THE DESIGN
T
2
=
Ki
OF SHALLOW
FLUIDISED
BED HEAT EXCHANGE
Hence
+ 1]- pA
& [
d&s = xTsB da
2. Two smges 21 Plug flow of soiids The physical situation of this case where the two stages have the same dimensions is depicted in Fig 2 The length term a is increasing in the direction of flow on the top stage and decreasing in the direction of flow on the bottom stage Subscript B refers to the bottom stage and subscript T to the top stage. On the bottom stage eqn. (1) holds as for the single stage. If we simplify this by putting z= G,C,/MsC., then this reduces to
dT,, =
IT
da
0
If the two stages are relatively close together there will be little or no lateral mixing of gas between stages. Hence for the top stage the differential heat balance gives - M~C,dr,, or -
d7;, da
= G,C,(r,-
= ~x(T~r- TEB)
161
EQUIPMENT
r,,)dq
integrating eqn. (2) for the bottom stage between 0 and CIwhere 0 c a -C A results in KB = ?;O exp (zca)
d7r da
The sign is negative because on the top stage rs is decreasing as a is increasing These two relationships (4) and (5) form the basis from which the analyses for the two-stage unit are derived. Under conditions of thermal equilibrium
i
x7& = rl,
exp(ya)
This is a linear first-order differential equation which can be solved directly using terminal condition (a = 0. Tq~= Ti) to give T,r exp(xn) = +
exp(Zxn) -
+
f
Tsi
(S)
But at n= A. 7&= 7;r and from eqn. (7) it is seen that at this point Li
(3
(7)
Inserting this general relation for xB into eqn. (6) gi\-es
(9)
= T, exp Wl)
Inserting eqn. (9) into (8) and simplifying gia-es
T,i_ T - +[exp (22~4) + I] This analysis is based on a personal communieation’ ‘_ 37 General case for any residence time distribution
Equations (4) and (5) still hold, but in this case applying the residence time distribution function on the bottom stage gives
~sg=
T,, =
JI IIT_exp (w)]
E(B)dB
Hence
Fig 2 Two stas fltidised beds
In order to evaluate this. it will be necessary to know the relationship between E(6) and 0 so that the integral can be solved before the differential equation. This equation only takes into account the residence time distribution on the lower stage, however_ Thus, after a relationship has been obtained for Kr, the residence time distribution of solid on
162
D. R MffiAW
the top stage must be accounted for in a similar manner. If solid flow conditions are such that flow can be described by a simple perfect mixers in series model, eqn. (10) can be modified to give
The distinction is made between zg and xr because the residence time distribution for solids flow has been accounted for on the bottom stage but not yet on the top stage_ They will be numerically the same. Solving this in a similar manner to that described in Section 2 1 gives for TcToat n = A [esp (- r,A)) x~cj;~ exp il [P In( 1 + z#) [P In (1 ta&P)tz,] t
TsTo =
UT Ts,
+ G
-
P ln(1 +rr,/P)+aJ
+ xT]
\
We must now take into account the residence time distribution on the top stage_ which would be based on solids flow on the top stage only, ie_ by using x&I instead of zT and then integrating as follows:
Go =
J‘; :exp(-z+L4): are T,, exp A [IP In(It z&P) i aTO] [P In (I + ag/P) + a#] 1
.
+ T,i -
[P In (1 TfBTi) + aTo] I E(e)de
(11) but at a=A TSTo= &
= T,(l
-~rr,/P)~
(12)
The integral would have to be determined numerically as indicated in the worked example in Appendix Il.
DISCUSSION
The relatiorr derived can be used to calculate the base area requirement of either a single or twostage unit for a given heat load Alternatively_ they may be used to determine the amount of heat transferred in a given unit operating under a particular set of conditions_ in order to apply the relations, however, information must be available on the heat transfer coefficient and flow properties ofthe Powder TechnoL 6 (1972)
system. Information on the heat transfer coefficient would first be used to de?ermine whether thermal equilibrium isapproximated in the bed or not. lfthis approach temperature is low [(r,- fsO)< 0.1” C] at the hot end of the bed, then the assumption of thermal equilibrium may be considered valid. The information on the solid flow properties of the system would be used to determine whether the bed is operating under plug flow conditions or whether a residence time distribution would have to be taken into account. A typical design example for the cooling of urea from 100°C to 40°C using_ air at 30°C in both single and two-stage units is gwen in Appendix Il. Examination of eqn. (3) shows that the longer the bed, the closer is the approach to plug flow conditions_ Hence the method suggested is to choose as long a bed as is practical. consistent with its physical location and then calculate the required width for the given heat load. using the appropriate desi_F air velocity_ It is worthwhile noting that for the smgle-stage case, twenty-five perfect mixers in series gave a cooling air requirement only 4% greater than that for plug flow conditions_ For the two-stage case, twenty perfect mixers in series gav-e a cooling air or area requirement 22% greater than that for the plug flow condition. Hence if the bed length is greater than the equivalent of about 40 perfect mixers for the single-stage unit and about 50 perfect mixers for the two-stage unit, the plug flow equation can be used with acceptable accuracy (2-3 %)_ A residence time distribution can. however, be set up in other ways than just by the fhtidised solid mixing mechanism implied in the example_ The most likely way is by there being an uneven solids inlet distribution_ This can be minimised by having a narrow bed. Thus it is seen that, in order to minimise the air flow requirement for a given load, the equipment should be operated under plug flow conditions wherever possible_ These conditions are associated with relatively long narrow rectangular shaped beds and small bubbles, i.e. low bed depths. The effect of increasing the bed depth may not be significant when operating large beds where plug flow conditions are already approached, but for small beds it can be signi&ant The effect of increasing the outlet solid temperature by increasing the bed depth has been demonstrated by the author in a small laboratory scale unit, the results being tabled in Appendix III. It is unlikely that a large scale unit would be able to operate at a bed depth below 3 cm, so that under normal conditions, it would be
THE DESIGN OF SHALLOW
FLUIDISED BED HEAT MCHANGE
advisable to operate at the lowest operable bed depth. In order to maintain steady operation, a stable downflow of solids between stages is necessary in the two-stage unit. The most satisfactory method of achieving this is by using a rotary valve as air seal between the two stages, as indicated in Fig. 2. The heat transfer coefficient may be obtained using the appropriate correlation from the reviewI on gas-particle heat transfer in fluidised beds It is important to note, however, that not every investigator has used the same definition of heat transfer coefficient. So it must bestressed that when attempting to use one of the correlations, the designer must check that the definition of heat transfer coefficient in the correlation coincides with that used in the analysis. All published work on residence time distributions in fluid&xi beds has also been reviewed13_ Most work was in circular section fluid&d bed systems, but one set of investigators9 carried out an investigation on a rectangular section fluidiscd bed. Their work in a small scale unit showed that solids flow in the system was somewhere in between the extremes of plug flow and perfect mixing. More work is, however, necessary in this field. The analyses derived can be used to optimise operating conditions in a unit and also to compare single-stage with two-stage operation_ A comparison has been made in the sample design calculations in Appendix II. This showed the actual cooling air requirement for a two-stage unit to bc somewhat more than half that for a sing!e-stage unit. Hence, since the pressure drop is doubled in a twostage unit, the single-stage unit seems to offer cheaper running costs. Normally, however, a cyclone is n ecessary in the system and this could tend to bring the running costs in favour of the twostage system because of the reduction in the a~ flow rate. It is unlikely that a three-stage system will have lower running costs than a two-stage system. More efficient use of the air is made in a two-stage system than the single-stage sys:em. If u-e define a gas phase cooling efficiency by the equation 9
8
=
EQUIPMENT
163
cheaper to operate than other equipment capable of doing the same job_
CONCLUSIONS The analyses derived can be used zo design and oplimise operating conditions in a fluidised bed heat exchanger_ A comparison has been made between single and two-stage units and this shows that, taking into account economics and eflicienties, a single-stage unit appears to offer cheaper running costs, provided that the pressure drop in the rest of The gas circuit is small. APPENDIX
I
NO THERMAL
EQUILIBRIUM
IN BED
Single stage
In this case, eqn. (1) still holds but it is necessary to introduce a heat transfer coeffLient by carrying out a differential heat balance in a vertical element of unit cross-sectional area Referring to Fig. 3 this is seen to be puCrdrs Integrating
= k(t,-
r.Jd=
this between
(AlI the limits
te = roi at z=(? tp=t,,at==z gives
Eak in gas temp. = LX,fi max. possible gain tpi - t,,
then for the example quoted in Appendix II this eficiency is increased from 33.7 oAto 42 o/Oby adding the second stage. The analyses also enable a comparative study to be made to decide if this kind of equipment will be
Fig 3. Vertical difTerentiaI heat balanaz
164
D- R
(A2) Plug jZow of solids Rearranging eqn. (A2) to put it in terms of
McGAW
General case for any residence time distribution The differential equation for the solid tempem-
ture on the top stage in this case is given by
The solution is obtained by a similar method to that obtained in Section 2.2.
T, = (t,, - trzi) gtves
(A3 Combining eqns. (AI) and (A3) and integrating between the limits a=0
T=T,, T=
a=A
Ti
APPENDIX SAMPLE
II DESIGN
CALCU-LATION
It is required to calculate the size of both a singlestage and a two-stage fluidised bed to cool 5000 kg/h of urea from 1WC to WC using air at 3BC.
grres
case for anv residence time distribution Modifying the equation in the usual manner gives
Generai
T,=
T,i
exp
-
w-7 1 MsC, [1 -eelip
(-
%)I}
E(B)dfl
Two stages Plug flow OJ solids
In this case the relationship for the solids temperature on the bottom stage is given by the equation
-dT,, = da
q9TsB
0.0015 m (1500 pm) 1.00 0.5 0.05 m 1.165 kg/m’ 2 m/set ODDOO18kg/m set 1-W J/kg% 1.33 J/kg =K _. It may be assumed that solid flow conditions can be given by a perfect mixers in series model with 10 perfect mixers per metre length of bed. It is lirst necessary to calculate if thermal equilibrium is achieved in the bed. This is done by use of eqn. (-42) &=p”d=
P
195
From Fig 1 in the review” on heat transfer coeflicients in fluidised beds, we get j” = 0.2
where
and St = 0.256
For the top stage, eqn. (5)may be modified to give
Using a similar procedure to that given previously in Section 2 results in T,i= T, Powder
+ [exp (2zflA) + l]
Tech&
6 (1972)
fs--f,, _
--exp r, - tgi
C ---
h
WC=
6(1 -e)z sD.dP
1=
exp (-
25.6)
Thus t, - toDis so small that thermal equilibrium can be assumed. Single stage
Assume that the largest value of A that can be
THE
DESIGN
OF SHALLOW
accommodated eqn. (3) gives
is 25
FLUIDISED
m. Then,
BED
HEAT
substitution
EQUIPMENT
in
gas particle heat transfer coefficient number of perfect mixers mass flow rate surface area of solid per unit bed volume surface area of solid per unit base crosssectional arca temperature temperature with refercnoe to inlet gas temperature (= f - fFi) superficial gas velocity in bed depth of bed dimensionless residence time. defined as residence time divided by mean residence time bed porosity shape factor viscosity density efiiciency G,C,/Xlr,C, [l-exp(--AS/G,C,)]
G,A = 13,050 kg/h with bed dimensions 25 m x 0.62 m. Note that if plug flow conditions had been assumed, the air requirement would have been 12.550 kg/h and bed dimensions 25 m x 0.6 m. Two stages
A bed length of 2 m is assumed in this case. The relationship to be used to calculate G, is a combination of eqns (11) and (12). A trial and error solution is neozssary7 since a(= G= Cs/&Zs C,) appears as a complex function within the integral. The relationship between 0 and E(0) was obtained from the perfect mixers in series equation using J= 20. The trial and error calculation of G, using an IBM 1620 computer gave an air requirement of IO.200 kg/h which corresponds to a bed 2 m x 0.61 m. Note that if plug tlow conditions had prevailed, the air requirement would have been 8320 kzJh corresponding to bed dimensions 2 m x O-5 m.
APPENDIX
III
RESULTS OF BED COOLER
TESTS
ON
LABORATORY
FLUIDISED
(cm)
Air flow. kg/h Solid flow, k@h Inlet solid temp. t,, OC Outlet solid temp. t_ OC Inlet air temp. tn. OC
LIST
A C
4
E(e) G
Ponder
2.5
5
7J
90 81.9 110.0 54.9 36.7
90 81.0 108.8 555 372
90 81.0 1092
90 Sl.0 109.1 613 3s.5
38.4
OF SYMBOLS
length of bed specific heat particle diameter fraction of solids having residence time bctween8andetdQ mass flow rate per unit length of bed
TechnoZ..
6 (19JZ)
B g i m 0
bottom stage gas phase bed inlet mean value bed outlet solid phase top stage at residence time 6
REFERENCES
13
59.7
Subscriprs
; 0
30 cm long x 5 cm wide Size of cooler I sand 14-18 mesh Solid material : CooIing medium : air Beddepth
165
EXCHANGE
1 Yv_ P. Ryazantser. intern Chcm Eng..S (1)(196X) 131. 2 H. Schrcibcr. Brir. Chem Eng_ 7 (8) (1962) SSJ. 3 EA. Kazakova et nL. Khim Prom . (5) (1961) 330. 4 J. Cibrowski and J. Roaak. Pcem Chem. 3 (195s) 103. 5 G. A_ Dormadieu. Genie Chim, 85 (1) (1962) 1. 6 N. I. Gelpcrin and V. G. Ainshtein. Intern Chem &ng_. 3 (2) ( 1963) 259. 7 V_ABorodu1~aInrenrChemEng..4(1)~1964)110. 8 J. F. FrantL. Chem Eng. Progr., 57 (7) (1961) 35. 9 J. Vcrloop. I_ H. Dc Nit and P. M. Hcertjes. Pror 1nre-m. SJmp. on Fluidiscztion, June 6-9, 1967, Netherlands Uni%ersity Press, p_ 476. IO 0. Levenspiel. Chemical Reoc:ion Engineering. Wilq. New York. 1965 11 J. A McWilliam, Fmns Ltb, Felixstowz, U.K. personal communication 1965. 12 J. J_ Barker. Ind. Ens. Chem. 57 (3 (1965) 33. 13 J. Verloop. I_ H De Nieand P.M. Heertjes_ Powder TechnoL. 1 (196s) 32