- Email: [email protected]
Stage concept in particle size analysis
Shorter Commumcations
1070
dimensionless parameter C,,/p,ar radial position radial position of reaction zone radius of particle time stoichiometnc coefficient dimensionless concentration, C/C, density of solid dimensionless radial position, dR dimensionless position of reaction zone. rJR mass transfer rate, defined by eqn (15) dmiensionless reaction parameter, ~(kIRz/D) dimensionless tmie, Dt/R*
The PSS solution and the higher order approximations are in close agreement for A = 0 001 For A = 10 the mass transfer rates predicted by the three solutions are identical It may be pointed out that for both values of N and for large values of A. the mass transfer rate tends to (A - 1) in the limit 6 + 1 CONCLUtRON Tbe PSS approximation to the reacted shell thickness and the mass transfer rate for the shrinking core model with a slow reaction IS valid for gas-solid systems for all values of the reaction parameter A It IS also valid for liquid-solid systems for large values of A, but could lead to erroneous results for A 4 1 Acknowledgement-The authors wish to acknowledge the partial support recetved from the Pittsburgh Energy Technology Center under DOE contract No EY-77-S-024163 S KRISHNAMURTHY YTSHAH Department of Chemical and Petroleum Enguteenng Unrverslty of Attsbugh
httsbugh,
PA 15261. USA
RRBXRRNCES
HI Yagi S and Kunu D, Ckem Engng Scr 196116 364 0, Chemical Reactton E~ueenng Wtiey. New _ [21 Levenspiel York 1962
[31 Bischoff K B , Chem Engng Scr 1%3 18 711 [41 Theofanous T G and Lim H C , Chem Engng SCI 1971 26 1297
I
[51 Friedman
S , LaCount R B and Warzmsh. Oxidative Desulfurixation of Coal Paper presented to Division of Fuel Chemistry, Amencan Chemrcal Sonety Natwnaf Meetutg, New Orleans. LA, March 1977 D,AIChEI 19772 1 679 WI KmgW E andPerlmutterD I73Scarborough J B , Numerical Alathematrcal Analysts. 6th Edn The Johns Hopkins Press, 1%6
NOTATION
C C, D ki
concentration of fluid reactant reference concentration drffusivity rate constant for slow reaction
stagecomept
ill particle
SiRe
anaiysis
(Received 18 Apni 1978. accepted 7 atcember Comminution and classification are mrportant steps m many chemical and metaRurgical mdustnes mvolvmg materials in solid state Many attempts have heen made in the past to study these aspects and the works of Gaudm{l], Schumann[2] and Kapur[3] are important land marks Thus article presents one such approach, which has lead to new results that are more useful for desrgn calculatrons
Development of the model Each sieve in a standard set of S-sieves IS considered as a stage These test sieves are stacked in such a manner that 40 IS the opening of the 0th sieve on which no material IS retained (feed) and 4 IS the opemng of the Sth sieve immediately ahove the pan Since there exists a detImte relationship between openmgs among the successive test sreves m a standard sieve set (McCabe and Srmth[4(a)]), the vanous openings D,, 4, D*_, and D, are related to D,, as follows 4
where r IS the ratio of the openings of the successive sieves (I e r = (0,-r/D,)) The top most sieve on which no material IS retained may he called as 0th stage and the last sieve (fictitious) through which no material IS passed through may be dtslgnated as Ntb stage Tben the fractron of material passed through the 0th stage (+_) wrll he one and that passed tbrougb Nth stage will be xero So, any equation proposed for representing the particle sixe distribution of a commmution should satisfy the ahove boundary conditions A number of such equations were tested of which the followmg equation IS found to be more satisfactory
1978) t$- = (N + l)CN
To test foflows
the
validity.
the above
equation
(1) IS rearranged
as
6=p N+l
log [.#_/(N + l)] = N log C A plot of log [&/(N + l)] vs N should yreld a straight Ime wnh a slope log C Depending upon the number of stages (berng smag or large). one or more number of segments of strarght lines may be obtamed Equatron (1) can be mod&d for cases of two segments as follows 41- = (N-I- l)CINJgN* .&- = K(N + I) CzNI!$,
(2)
Similar extensions can be given for cases involving more than two segments also The size distribution of the particles retained m the pan IS unknown The largest size will be JUSt smaller than 9 and the smallest will be lust larger than & where 4y is the ouemng of the Nth stage sieve through which no material passes through This could be set from practical experience to fix the Ntb stage The hmltmg Ntb stage may be taken as that which retams 0 05% of the total sample
Number of partdes The number of psrticles per unit mass of mixture can be calculated using the following equation (McCabe and Smut M&)1)
la71
Shorter Communicauons Table I Comparison Quantity Specdic area sq cmlg Number of parttcles pergram
-3
of calculated
valuest
Present work
Prevtous work
4836
43 5$
2.14.ooO
2.29.600
tThe total numher of stages was found to be 18 #The value has been corrected for the mlssmg shape factor m the origmal text Replacmg .$+ and & reduces to
In terms of &.
SubsWutmg for d& mtegratmg one gets
0
and d&_
4.
as was
r and N, the equation
done
for
N,
and
Fig 1 Log [qL/(N + l)] vs stages (3) where I$+ IS the posltrve #$_as follows
cumulatwe
fraction
and
IS
related to
&+= I-$$Equation
(3) m terms of &.
L+, and r transforms
to
When only one segment IS obtamed,
A, = when there are two segments wrttte.n as
as shown m Ftg 1. eqn (4) can be
sun&lies
to
CNi” - Nf?“#“)
+(ln&‘;nr)z(l-CNP)] For the verdicatton of the model, typical data from the literature (McCabe and Snuth[4(d)]) are used Fme 1 aves the plot of log [&/(N + l)] vs N The number of part~clcs per unit mass and the specific surface computed from the model developed IIIthe present work and those from previous work are compared m Table 1 The agreement IS seen to be excellent
Np gN d+,_ +I”
?N d&2NP
Substitutmg
-j$,[,”;+“l.r (1 -
the equauon
for dd,_ and d&_ and mtegratmg one gets
(1 - C,N+Np - N,C,N~?N.)
Summary and concluswn
Treatmg each successtve stove IIIa standard set of slews as a stage, a model correlatm8 the ne8attve cumulative we&t fracbon and number of stages has been developed and venfied satlsfactonly Based on this new model, relationships have been
3 In r + (ln C, + 3 ln r)9 (c*Np?%
developed to premct the number of pticles and specific surface of a conumnutxon Compared with other models, the agreement IS found to be sahsfactory
- CW]
When only one segment IS obtamed, then C, = C, = C and K = 1 Thus the above equation reduces to NW=&
ln;~~,nr(l-CN+NCN?N) C
P SUBRAMANIAN P SHANMUGASUNDARAM VR ARUNACHALAM
Dcpatiment of Chemtcal Engmeenng Regwnal Engmeenng CoUege Tiruch~rapallr 620 0111 NOCATION
SpeciJic surface The
specific
smith Mm
surface
can
be obtamed
from
(McCabe
and
speatic surface, (sq cm/g) shape factor constants duuneter of the 0th stage sieve. (cm) constant number of stages number of stages for the first segment of FW 1
1072
Shorter Commumcations
N, r
number of particles per g ratio between the successive
REFERENCES
Sieve opemngs
p 129 McGraw-Hill, [ll Gaudm, Pnncrples of Materal D~%mng, New York (1939) [2j Schumann, AIME Tech Pub 1189, July, 1940 [31 Kapur P C , Chem Engng Scr 1972 27 425 [41 McCabe W L and Snuth J C , Unrt Operations of Chemrcal Engmeermg, 2nd Edn (a) p 794. (b) p 798, (c) P 797. Cd) P 812 McGraw-Hill, New York 1976
Greek symbols p density of solids A shape factor &+ positive cumulative weight fraction f& negative cumutative weight fraction Chemmd Engvleemg Scmce Vol 34 pp tO72-1075 Pcmamon Press Ltd 1979 Pnnted I” Great Bntam
A changing grain size model for gas-so lid reactions (Received for publrcatwn 10 January 1979) In recent years there has been consIderable interest m the use of calcitic or dolormtic stones as the flmdizmg medium in fluid bed coal combustors m order to reduce sulfur &oxide emissions At the same time a great deal of work has been reported on the development of a gram model[l-31 as an alternative to the shnnkmg core model[4] for non-catalytic gas solid reactions While the gram model has advanced our understanding of gassold reactrons it has a major drawback III that structural changes occurring In the course of the reactIon are neglected Recently, Ramachandran and Srn&h[5] have presented a model to account for the combined effects of smtermg and density changes Qmte independently, Georgakis et al 161 have developed a similar model, the changing-gram-size, model wluch accounts for density changes during the reaction In the present commumcatron a brief description IS aven of this model and some analytical results not presented in [S] are detailed It IS shown that a linear relationship between porosity and conversmn exists not only m the average sense as it has been indicated before[7] but that such relationship also occurs between local values at each radml position in the pellet When the molar volume of product solid is larger than that for the reactant solid a cntlcal value for the mitral porosity IS established which determines whether complete conversion of the solid IS possible A cnt~cal time. pore plugging time, IS analytically calculated at which the pore at the pellet surface IS blocked if the initial porosity IS not larger than the critical value Previous models are shown to be hrnitmg cases of the present one and the predictive ability of the model IS compared with experimental data on the adsorption of SO2 in calcmes THE MODEL Consider a spherical pellet of a sobd reactant, made up of a large number of spherical grams of uniform imtml radms. r0 Assume that the solid material forming the pellet reacts ureversibly with a gas according to the followmg general scheme
A(g) + Ws)+
pP(s) + gQ(g)
(1)
If the pellet IS isothermal, the dtiusion within the pellet eqmmohv counter-drffuslon. or at low concentrations and the pseudo-steady state approximation for the d&usional phenomena v&d, the dimensionless form of the mathematical model IS gven as follows 161 Pellet
(2)
and
(6)
While all symbols are defined in the Notation, it IS worth mentiomg here that R IS a dimensionless r&al distance in the pellet. and gl and g2 are the dunensionless railu of the grams and their unreacted core respectively Parameter & plays the role of a Thiele modulus especially at mitral times and expresses the ratio of &ffusional to kinetic limitations Parameter /? represents the ratio of kinetic to d&usional resistances in the product layer Equation (4) expresses the fact that for each volume of reactant solid there appears (I volumes of solid product The local conversion at time t and at position R IS lpven by x(R, t) = 1 - gz3(R, t) from wluch follows
the overall conversion,
X(f) = 3
dgz
PC
dt=-#9+&(l-&t) g,3= a +(linitial and boundary
a)gz’
(3) (4)
con&ions
gz(R, 0) = gi(R. 0) = 1
(5)
(7)
X(r), may be calculated
as
1 I0
R*x(R, t) dR
(8)
The local porosity, l (R. t). which generally depends on tune, t. and on the radius, R, IS related to the grain size. g,, by Bi’ = (1 Where EOIS the mltial porosity to the local porosity by
l)/(l - 3
(9)
The local converSion x
l;~R~t)=~+(I-aH1-x(R.~))
IS
related
(10)
BY mtegratmg over the pellet volume a similar relationslup is obtained between the average stone porosity and average conversion ANALYTICAL REguLTg If external mass transfer lmutations are neglected then the reactant concentration at R = 1 IS equal to unity (eqn 6a). and eqn (3) can be integrated to obtain =2[B + g2(1II which, after some manipulations,
dgdl
&z = -fit
gives
1 t=(l-g,)+~(l-g,2)-28((r_,)(g,2-l)
Gram
w&h
C(l. t)= 1
1
(11)
Here g, and g2 represent g,(l, I) and gz(l, 1). respectively The dependence of the local converSIon on time. at the surface techniques, with of the pellet, may be calculated usmg analytmal the aid of eqns (4). (7) and (11) If P IS smaller than one, it follows that the porosity increases with time and at the time the local conversion at the surface of the pellet (R = 1) IS equal to 100% the value of g2(1, r.) IS equal to
1070
dimensionless parameter C,,/p,ar radial position radial position of reaction zone radius of particle time stoichiometnc coefficient dimensionless concentration, C/C, density of solid dimensionless radial position, dR dimensionless position of reaction zone. rJR mass transfer rate, defined by eqn (15) dmiensionless reaction parameter, ~(kIRz/D) dimensionless tmie, Dt/R*
The PSS solution and the higher order approximations are in close agreement for A = 0 001 For A = 10 the mass transfer rates predicted by the three solutions are identical It may be pointed out that for both values of N and for large values of A. the mass transfer rate tends to (A - 1) in the limit 6 + 1 CONCLUtRON Tbe PSS approximation to the reacted shell thickness and the mass transfer rate for the shrinking core model with a slow reaction IS valid for gas-solid systems for all values of the reaction parameter A It IS also valid for liquid-solid systems for large values of A, but could lead to erroneous results for A 4 1 Acknowledgement-The authors wish to acknowledge the partial support recetved from the Pittsburgh Energy Technology Center under DOE contract No EY-77-S-024163 S KRISHNAMURTHY YTSHAH Department of Chemical and Petroleum Enguteenng Unrverslty of Attsbugh
httsbugh,
PA 15261. USA
RRBXRRNCES
HI Yagi S and Kunu D, Ckem Engng Scr 196116 364 0, Chemical Reactton E~ueenng Wtiey. New _ [21 Levenspiel York 1962
[31 Bischoff K B , Chem Engng Scr 1%3 18 711 [41 Theofanous T G and Lim H C , Chem Engng SCI 1971 26 1297
I
[51 Friedman
S , LaCount R B and Warzmsh. Oxidative Desulfurixation of Coal Paper presented to Division of Fuel Chemistry, Amencan Chemrcal Sonety Natwnaf Meetutg, New Orleans. LA, March 1977 D,AIChEI 19772 1 679 WI KmgW E andPerlmutterD I73Scarborough J B , Numerical Alathematrcal Analysts. 6th Edn The Johns Hopkins Press, 1%6
NOTATION
C C, D ki
concentration of fluid reactant reference concentration drffusivity rate constant for slow reaction
stagecomept
ill particle
SiRe
anaiysis
(Received 18 Apni 1978. accepted 7 atcember Comminution and classification are mrportant steps m many chemical and metaRurgical mdustnes mvolvmg materials in solid state Many attempts have heen made in the past to study these aspects and the works of Gaudm{l], Schumann[2] and Kapur[3] are important land marks Thus article presents one such approach, which has lead to new results that are more useful for desrgn calculatrons
Development of the model Each sieve in a standard set of S-sieves IS considered as a stage These test sieves are stacked in such a manner that 40 IS the opening of the 0th sieve on which no material IS retained (feed) and 4 IS the opemng of the Sth sieve immediately ahove the pan Since there exists a detImte relationship between openmgs among the successive test sreves m a standard sieve set (McCabe and Srmth[4(a)]), the vanous openings D,, 4, D*_, and D, are related to D,, as follows 4
where r IS the ratio of the openings of the successive sieves (I e r = (0,-r/D,)) The top most sieve on which no material IS retained may he called as 0th stage and the last sieve (fictitious) through which no material IS passed through may be dtslgnated as Ntb stage Tben the fractron of material passed through the 0th stage (+_) wrll he one and that passed tbrougb Nth stage will be xero So, any equation proposed for representing the particle sixe distribution of a commmution should satisfy the ahove boundary conditions A number of such equations were tested of which the followmg equation IS found to be more satisfactory
1978) t$- = (N + l)CN
To test foflows
the
validity.
the above
equation
(1) IS rearranged
as
6=p N+l
log [.#_/(N + l)] = N log C A plot of log [&/(N + l)] vs N should yreld a straight Ime wnh a slope log C Depending upon the number of stages (berng smag or large). one or more number of segments of strarght lines may be obtamed Equatron (1) can be mod&d for cases of two segments as follows 41- = (N-I- l)CINJgN* .&- = K(N + I) CzNI!$,
(2)
Similar extensions can be given for cases involving more than two segments also The size distribution of the particles retained m the pan IS unknown The largest size will be JUSt smaller than 9 and the smallest will be lust larger than & where 4y is the ouemng of the Nth stage sieve through which no material passes through This could be set from practical experience to fix the Ntb stage The hmltmg Ntb stage may be taken as that which retams 0 05% of the total sample
Number of partdes The number of psrticles per unit mass of mixture can be calculated using the following equation (McCabe and Smut M&)1)
la71
Shorter Communicauons Table I Comparison Quantity Specdic area sq cmlg Number of parttcles pergram
-3
of calculated
valuest
Present work
Prevtous work
4836
43 5$
2.14.ooO
2.29.600
tThe total numher of stages was found to be 18 #The value has been corrected for the mlssmg shape factor m the origmal text Replacmg .$+ and & reduces to
In terms of &.
SubsWutmg for d& mtegratmg one gets
0
and d&_
4.
as was
r and N, the equation
done
for
N,
and
Fig 1 Log [qL/(N + l)] vs stages (3) where I$+ IS the posltrve #$_as follows
cumulatwe
fraction
and
IS
related to
&+= I-$$Equation
(3) m terms of &.
L+, and r transforms
to
When only one segment IS obtamed,
A, = when there are two segments wrttte.n as
as shown m Ftg 1. eqn (4) can be
sun&lies
to
CNi” - Nf?“#“)
+(ln&‘;nr)z(l-CNP)] For the verdicatton of the model, typical data from the literature (McCabe and Snuth[4(d)]) are used Fme 1 aves the plot of log [&/(N + l)] vs N The number of part~clcs per unit mass and the specific surface computed from the model developed IIIthe present work and those from previous work are compared m Table 1 The agreement IS seen to be excellent
Np gN d+,_ +I”
?N d&2NP
Substitutmg
-j$,[,”;+“l.r (1 -
the equauon
for dd,_ and d&_ and mtegratmg one gets
(1 - C,N+Np - N,C,N~?N.)
Summary and concluswn
Treatmg each successtve stove IIIa standard set of slews as a stage, a model correlatm8 the ne8attve cumulative we&t fracbon and number of stages has been developed and venfied satlsfactonly Based on this new model, relationships have been
3 In r + (ln C, + 3 ln r)9 (c*Np?%
developed to premct the number of pticles and specific surface of a conumnutxon Compared with other models, the agreement IS found to be sahsfactory
- CW]
When only one segment IS obtamed, then C, = C, = C and K = 1 Thus the above equation reduces to NW=&
ln;~~,nr(l-CN+NCN?N) C
P SUBRAMANIAN P SHANMUGASUNDARAM VR ARUNACHALAM
Dcpatiment of Chemtcal Engmeenng Regwnal Engmeenng CoUege Tiruch~rapallr 620 0111 NOCATION
SpeciJic surface The
specific
smith Mm
surface
can
be obtamed
from
(McCabe
and
speatic surface, (sq cm/g) shape factor constants duuneter of the 0th stage sieve. (cm) constant number of stages number of stages for the first segment of FW 1
1072
Shorter Commumcations
N, r
number of particles per g ratio between the successive
REFERENCES
Sieve opemngs
p 129 McGraw-Hill, [ll Gaudm, Pnncrples of Materal D~%mng, New York (1939) [2j Schumann, AIME Tech Pub 1189, July, 1940 [31 Kapur P C , Chem Engng Scr 1972 27 425 [41 McCabe W L and Snuth J C , Unrt Operations of Chemrcal Engmeermg, 2nd Edn (a) p 794. (b) p 798, (c) P 797. Cd) P 812 McGraw-Hill, New York 1976
Greek symbols p density of solids A shape factor &+ positive cumulative weight fraction f& negative cumutative weight fraction Chemmd Engvleemg Scmce Vol 34 pp tO72-1075 Pcmamon Press Ltd 1979 Pnnted I” Great Bntam
A changing grain size model for gas-so lid reactions (Received for publrcatwn 10 January 1979) In recent years there has been consIderable interest m the use of calcitic or dolormtic stones as the flmdizmg medium in fluid bed coal combustors m order to reduce sulfur &oxide emissions At the same time a great deal of work has been reported on the development of a gram model[l-31 as an alternative to the shnnkmg core model[4] for non-catalytic gas solid reactions While the gram model has advanced our understanding of gassold reactrons it has a major drawback III that structural changes occurring In the course of the reactIon are neglected Recently, Ramachandran and Srn&h[5] have presented a model to account for the combined effects of smtermg and density changes Qmte independently, Georgakis et al 161 have developed a similar model, the changing-gram-size, model wluch accounts for density changes during the reaction In the present commumcatron a brief description IS aven of this model and some analytical results not presented in [S] are detailed It IS shown that a linear relationship between porosity and conversmn exists not only m the average sense as it has been indicated before[7] but that such relationship also occurs between local values at each radml position in the pellet When the molar volume of product solid is larger than that for the reactant solid a cntlcal value for the mitral porosity IS established which determines whether complete conversion of the solid IS possible A cnt~cal time. pore plugging time, IS analytically calculated at which the pore at the pellet surface IS blocked if the initial porosity IS not larger than the critical value Previous models are shown to be hrnitmg cases of the present one and the predictive ability of the model IS compared with experimental data on the adsorption of SO2 in calcmes THE MODEL Consider a spherical pellet of a sobd reactant, made up of a large number of spherical grams of uniform imtml radms. r0 Assume that the solid material forming the pellet reacts ureversibly with a gas according to the followmg general scheme
A(g) + Ws)+
pP(s) + gQ(g)
(1)
If the pellet IS isothermal, the dtiusion within the pellet eqmmohv counter-drffuslon. or at low concentrations and the pseudo-steady state approximation for the d&usional phenomena v&d, the dimensionless form of the mathematical model IS gven as follows 161 Pellet
(2)
and
(6)
While all symbols are defined in the Notation, it IS worth mentiomg here that R IS a dimensionless r&al distance in the pellet. and gl and g2 are the dunensionless railu of the grams and their unreacted core respectively Parameter & plays the role of a Thiele modulus especially at mitral times and expresses the ratio of &ffusional to kinetic limitations Parameter /? represents the ratio of kinetic to d&usional resistances in the product layer Equation (4) expresses the fact that for each volume of reactant solid there appears (I volumes of solid product The local conversion at time t and at position R IS lpven by x(R, t) = 1 - gz3(R, t) from wluch follows
the overall conversion,
X(f) = 3
dgz
PC
dt=-#9+&(l-&t) g,3= a +(linitial and boundary
a)gz’
(3) (4)
con&ions
gz(R, 0) = gi(R. 0) = 1
(5)
(7)
X(r), may be calculated
as
1 I0
R*x(R, t) dR
(8)
The local porosity, l (R. t). which generally depends on tune, t. and on the radius, R, IS related to the grain size. g,, by Bi’ = (1 Where EOIS the mltial porosity to the local porosity by
l)/(l - 3
(9)
The local converSion x
l;~R~t)=~+(I-aH1-x(R.~))
IS
related
(10)
BY mtegratmg over the pellet volume a similar relationslup is obtained between the average stone porosity and average conversion ANALYTICAL REguLTg If external mass transfer lmutations are neglected then the reactant concentration at R = 1 IS equal to unity (eqn 6a). and eqn (3) can be integrated to obtain =2[B + g2(1II which, after some manipulations,
dgdl
&z = -fit
gives
1 t=(l-g,)+~(l-g,2)-28((r_,)(g,2-l)
Gram
w&h
C(l. t)= 1
1
(11)
Here g, and g2 represent g,(l, I) and gz(l, 1). respectively The dependence of the local converSIon on time. at the surface techniques, with of the pellet, may be calculated usmg analytmal the aid of eqns (4). (7) and (11) If P IS smaller than one, it follows that the porosity increases with time and at the time the local conversion at the surface of the pellet (R = 1) IS equal to 100% the value of g2(1, r.) IS equal to