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Exact uniqueness and multiplicity criteria for a positive-order Arrhenius reaction in a lumped system
Shorter Commumcatlons
135
temperature has been shown to @ve good resolution m determmmg the eff ecuve dtiuslvlty for automobile catalyst supports
CL 5r
2De. ,-
’ :
4-
Acknowfedgements-The author thanks J C Hegedus and J C Schlatter of the General Laboratones for helpful dIscussions
. ’
General Motors Research Laboratones
Cavendlsh, L L Motors Research
TAI-SHENG
CHOU
Wamen,MI4809Q,USA NOTATION
2CL CLln,
l-
D,
t 4
8
10
12
14
V XL
16
normahzed outlet concentration maximum normahzed outlet concentration effectwe ddfustvlty, cm% time, s tamer gas velocity, cm/s column length, cm
[l] Weisz P B , Z HaysdE?ue Folge 1957 11 1 [2] Meyer W W , Hegedus L L and Ans R , I Cat 1976 42 135 [3] Waidram S P , A I Ch E Meeting, Houston, Texas, March 1977 [4] Scott D S , Lee W and Papa J , Chem Engng Set 1974 29 2155 [5] Hslang T C S and Haynes H W Jr, Chem Engng Scl
F@ 3 Effective drffuslvity data for a typical catalyst support measured at 13 25 cm/s carrier gas velocity The curves labeled 20, and De/2 show the sensltrvlty of the technique Solid curve = expenmental data Dotted curves = model slmulatlons
This difference can be attrtbuted to a skm effect III the catalyst support, that IS, the tracer senses deeper into the pellet as a result of decreasing carrier gas flow rate This mflence of earner gas flow rates on the measured effective dtiuslvity has been treated m detad elsewhere [9]
1977 32 678 [6] Haynes H W Jr, Chem Engng Scr I975 30 955 [7] Gangwal S K , Hudgms R R , Bryson A W and Sdveston P L, Can I Chem Engng 1971 49 113 [8] Adelman A and Stevens W F , A I Ch E J 1972 18 20 [9] Chou T S and Hegedus L L , A I Ch E J 1978 24 255 [lo] Dang N D P and Gibdaro L G , Chem Engng J 1974 8
CONCLUSIONS The sensltlvlty of pulse dtiusivity measurements can be unproved by the appropriate selection of the tamer and tracer gases A mtrogen m helium system at 40°C or shghtly lower
157 [II] OmataH
andBrownL
F,AIChEI
197218967
ChemmlEnbvneem.g Scmee Vol 34 pp 135-137 0 Pcrmmon Press Ltd 1979 Pnnted m Great Bnlam
Exact uniqueness and muMiplicity criteria for a positive-order Arrhenius reaction in a lumped system (Recmed
10 May 1978)
Recent publications by Van Den Bosch and Luss [l, 21 have investigated the multiplicity and umqueness of steady states m reactlon systems for wluch the reaction order was different from unity We refer specfically to Ref [l] m which the authors estabhshed su&lent condlhons for uniqueness and suflicient con&tlons for multlphc~ty-both cnteria being m terms of system constants and parameters-through some algebrmc sunphficatrons In the course of analyzmg steady-state charactenstlcs for more compIex rate expressions, we succeeded m estabhshmg exact necessary and sufficient conditions for the lumped problem considered m Ref [l] We feel that tis result IS worth commumcatmg, because it IS not only exact (and therefore an Improvement. albed only a shght one, over the strong cntena sven m Ref [I]) but it ts also much sunpler m form and hence more hkely to be useful m apphcahons Furthermore, m other work, currently III progress, the same techmques as outhned m the analysis here have been apphed with slmdar success to problems mvolvmg Langmmr-Hmshelwood and Eley-&deal kmetlc models In order to save space, we adopt the notation of Ref [l] throughout Our attention 1s confined to the ntb order (n 20) exothermlc lumped problem (the analogous endothermic problem
always has umque states It]), the steady-state which IS defined by the following dunenslonless 1s the lirst equation m Ref [l]
temperature of equation which
As pomted out m Ref [l], eqn (1) describes the general nomsothermal lumped situation mcludmg the famtiar gradtentless stured reactor and catalyst particle problems As obtamed in Ref [l], the necessary and sufficient condition (expressed m terms of y) that a steady-state soluuon from eqn (1) be umque for all values of a! 1s that the foilowmg mequahty be satisfied (n-l)y’+(B+y+l-n)y’--(2+/3)y+y(l+@)sO
(2)
From physical arguments y is restncted to the range (1. 1+ /3) If the above mequahty IS violated. mulhple steady states are assured for some value of a If it IS satisfied, steady states are umque for all values of ~1 (with a >O) Van Den Bosch and Luss[l] approxunated this cntenon m parameter space by first reducmg the cubic equation m (2) to a quadrauc one Our
Shorter Commumca~ons
136
analysis, wblcb follows, deals with the cubic equation dvectly and proceeds through tbe estabhsbment of necessary and sufficrent condltlons for multlphclty It IS convement for the moment to resmct n to one of the ranges 0 < n c 1 or n > 1 The special cases of n = 0 and n = 1 are consrdered later Representmg tbe funchon on tbe left side of (2) by r(y), we first pomt out that both r(l) and lY1 + @) are poslttve Thus d T(y) IS to vamsb on the Interval 1 c y < Ii #3, It must do so twice It follows that there IS always at least one real root of r(y) whlcb IS not on tb~s Interval Further exammation of T(y) for large negatlve and large poslfive values of y reveals that d (2) IS to be vlolated, then tbe graphIcal representatton of T(y) must be of the form shown m the sketches of F1.g 1 Clearly two necessary condrtlons for the multlpbclty of steady states are that r(y) have multiple roots and that a local mmunum m r(y) exist m the range 1~ y c I+ @ Furthermore, taken together these two condltlons establish necessary and sufficient condltlons for multlplcclty m the sense that tf eltber or both are violated, unique steady states are assured for alf values of a and d both are satisfied muluple states exist for some values of u To express the first of these condltlons m terms of tbe system parameters we employed a necessary and sufficient condlbon for the existence of multiple real roots of a cubic equatron t For T(y) tb~s condltron may be expressed as follows b@
where
x a) -z 0
(3)
range of values of n They were excluded earlier, not because they present unusual da&ties, but because some care must be taken m bandbng them and some earher comments do not apply For example, W&I n = 1, the leadutg coefficient m the expresslon for r(y) vanrsbes leavmg a quadratic equation Obviously the &cusslon regardmg tbe form of such curves as shown m Fs 1 must be changed Wltb n = 0, one root of r(y) occurs exactly at y = I+ & and agam some changes m earher comments must be made It turns out, however, that tbe above condlhons extend to these specml cases In fact, for these special cases, the function &#I. y, n) reduces to a rather sunple expresslon wblcb leads easdy to the necessary and sulliclent multlphclty condttion aven for n = 0 and n = I m Ref [l] Apparently exact multipbclty cntena of the type denved here have not been obtamed prev~ously for other reaction orders, that IS, for nf 0 and n# 1 Comhhons (3), (6) and (7) are tbe prmclpal results of our analysts To emphasize tbev possible utdlty we pomt out that for gven values of y.fl and n, the condltlons m (3) and (6), for OS; n < 1, or m (3) and (7), for n 2 1, may be easily tested If a condition IS vlolated, eqn (1) has a umque steady state for all values of (I On tbe other hand, d the condltlons are satisfied, eqn (1) has multiple solutions for some values of IX In addltron to bemg exact, these condttlons are much sampler to apply than
09
08
&%r,n)=-
Y3B2+2Y2f%1u%~~ -Yg*m~)+w+
1 -n)f(B
+
(4)
1)
and
07 06
1
g@*n)=(1-2n)BZ+2(2-n)B+1+n g~(~,n)=~4+4(3-5n)~3-2(4n2+lln-11)~Z +4/3(1 -n)(3+2n)+(l
-n)’
(9
Expressmg the condttlon regardmg a local mmunum m T(y) m terms of system parameters requves a relattvely strmgbforward analysis of the roots of tbe first denvahve of r(y)-& roots of a quadratic equation--and an exammation of tbe sign of the second denvahvea lmear expresslon Tbls analysts yields the followmg necessary and sufliclent condmons that a local mnnmum m extsts In tbe range 1 c y c 1 + /3
rfy)
I '0
I
2
4
6
8
10
12
14
16
18
Y
ForQ
Fu 2 Grapb~cal representation of multlplrclty condmons (3) and (7) for n = 2 Condmons (3) IS sattsfied throughout tbe shaded regons and condition (7) throughout the cross-batched resons
1,(~+~+1-n)-t/([~+~+1-n~-3(1-n~~+2)~~),I 3(1-n) +B
(6)
For n 2 I Y' Notice that now tbe values
Ftg I Sketches
2/3+n-1 B of 0 and 1 have been Included m
of the functton r(y) when mtdttple steady states extst
tTbere appear to be varrous ways See, for example, Refs [3, 41
of dertvmg
tbrs condltton
Fu 3 Bifurcation curves IIIthe y, &plane Umque steady states extst below the curves, multiple states exist (for some values of a) above the curves
137
Shorter Commumcations those presented m Ref [l] The range of values of 4 over which muIt& solutions exist, ti Indeed they do exist, may be located by first computmg the two real roots of T(y) on the Interval I < y C I+ j? and subsequently usmg those roots m eqn (1) to compute duectly the endpomt values of cx The reeons m the 7, &plane throughout which conditions (3) and (7) are met are shown for n = 2 m Fig 2 It turns out, as suggested by the reaons m that figure, that the condition (3) IS stronger than conditions (6) and (7) except for small values of y and B Exact b,rfurcotlon curves, obtamed from the above condlhons. whEh separate the y, p-plane rnto regrons of multiplsty and remans of uniqueness are shown for several values of nmF= 3
*Present address verslty of Southern
Department of Chemical Engmeermg. UmCahforma, Los Angeles, CA 90000, U S A
Acknowledgement-The work descnhed m tlus paper IS part of research project supported by the Natronal Scrence Foundation (Grant ENG 74-21094) T T TSOTSISt R A SCHMITZ
Department of Chemrcal Engmeenng Unruersrty of Illrnocs Urbana. II 61801, US
A REFERENCES
[ll
Van Den Bosch B and Luss D Cfiem Engng SKI 1977 32 203 [2] Van Den Bosch B and Luss D Chem Engng Scl 1977 32 560 [3] Neumark S Solu~on of Cubrc and Quartz Equatrons Pergamon Press, New York 1%5 [41 Gellert W , Kustner H , Hellwtch M and Kastner H (Eds), The VNR Concrse Encyclopedra of Mathematxs Van Nostrand Remhoid 1977
Charcn~Eit~~rug Scmxr Vol 34 up 137-143 0 PcrmnonPressLtd 1979 PrmtcdmGrsat Entam
Particle
segregation
in
liquhkolid
tluidised beds
(Received 7 March 1978. accepted 11 May 1978) Several worlcers[l-51 have reported segregation m hqmd-sold Ruidaed beds composed of particles of more than one size but there has been little attempt to study the tranSItIon regon m the Rmdlsatlon of a mixture of two sizes of particles Where segregation IS mcomplete the voidage. and hence bed density, wdl vary with he&t and thus measurements of pressures w&n the bed will enable an estimate to be made of the degree of segregation In the present work a pressure transducer has been used to measure the pressure gradient throughout the depth of a bed of partrcles of two sues of glass ballotml flmdlsed by water, and hence to obtam the concentration gradlent
The flmdlsatlon coIumn[6] conslsted of a length of Q V F glass prpe of dmmeter 104’ 1 mm The fiquld dlstrtbutor was formed from three pieces of 42 mesh stamless steel woven wtre cloth arranged with consecutive meshes at 45 deg to one another The free area was 37% of the cross-section of the tube This dlstrtbutor gave d reasonably umform flow dlstnbutron with a pressure drop conslderably greater than that across the bed Itself Llqtnd cuculatlon (see Rg 1) was effected by a WorthmgtonSimpson smgle stage centnfugal pump gvmg a maxnnum flowrate of 2 27 l/s (267 mm/s m the bed) wth a pressure Qfferentml of 8 X I@ N/m2 Flowrates were measured by means of a “Gapmeter” with a &unless steel float m a glass tube, It could measure flows m the range 0 25 to 2 5 I/s (2 5 x lob4 to 2 5 x IO-’m3/s) The pressure drop across the bed was measured usmg a pressure transducer, (first developed by Fand[7f), mcorporatmg a stamless steel probe (5 mm dla X 3 m long) whose position could be vmed w&un the bed The bed pressure was transmitted through four holes of 1 mm dla at a distance of 5 mm from the sealed end of the probe In the pressure transducer, shown m l+g 2, a thm stainless steel plate dlvldes a small chamber mto two compartments, one of wtuch IS connected to the probe which transmits the pressure signal At the centre of the other compartment was a small brass fixed electrode and the vmable capacitance was formed between this and the flexible stamless steel disc The thickness of the stamless steel chsc determines the sensltlvlty of the transducer and the distance between the electrodes can be adjusted to ave the desued pressure range The
Fly
IO
l-l
l’li.’$c._
Fig 1 Flow diagram 1, Water tank, 2, Centiugal pump, 3, Gapmeter, 4. Fhndlsation column, 5. Return hne, 6. Basket, 7, Water manometer, 8, Pressure transducer, 9, Chart recorder, 10, Stainless steel probe whole of the pressure transmlttmg system mcludmg the compartment of the transducer must be completely full of hqmd The second compartment of the transducer 1s open to atmosphere as reference pressure The vartation m the capacitance as a result of a pressure different& unbalances the voltages on two tuned cucmts and operates a ddferentml valve voltmeter, the output of winch @ves a reading proportional to pressure difference The transducer was calibrated by applymg known hydrostatic pressures, up to 100 mm of water, and the lmaty of the Instrument was checked over thus range
Experunents were camed out first by flu&smg 2 mm, 3 mm and 4 mm lead glass spheres of density 2960 kg/m’ with water at 20°C m order to obtam plots of log UC (superiicuil velocity) vs loge (voidage) from which the slope g was measured The
135
temperature has been shown to @ve good resolution m determmmg the eff ecuve dtiuslvlty for automobile catalyst supports
CL 5r
2De. ,-
’ :
4-
Acknowfedgements-The author thanks J C Hegedus and J C Schlatter of the General Laboratones for helpful dIscussions
. ’
General Motors Research Laboratones
Cavendlsh, L L Motors Research
TAI-SHENG
CHOU
Wamen,MI4809Q,USA NOTATION
2CL CLln,
l-
D,
t 4
8
10
12
14
V XL
16
normahzed outlet concentration maximum normahzed outlet concentration effectwe ddfustvlty, cm% time, s tamer gas velocity, cm/s column length, cm
[l] Weisz P B , Z HaysdE?ue Folge 1957 11 1 [2] Meyer W W , Hegedus L L and Ans R , I Cat 1976 42 135 [3] Waidram S P , A I Ch E Meeting, Houston, Texas, March 1977 [4] Scott D S , Lee W and Papa J , Chem Engng Set 1974 29 2155 [5] Hslang T C S and Haynes H W Jr, Chem Engng Scl
F@ 3 Effective drffuslvity data for a typical catalyst support measured at 13 25 cm/s carrier gas velocity The curves labeled 20, and De/2 show the sensltrvlty of the technique Solid curve = expenmental data Dotted curves = model slmulatlons
This difference can be attrtbuted to a skm effect III the catalyst support, that IS, the tracer senses deeper into the pellet as a result of decreasing carrier gas flow rate This mflence of earner gas flow rates on the measured effective dtiuslvity has been treated m detad elsewhere [9]
1977 32 678 [6] Haynes H W Jr, Chem Engng Scr I975 30 955 [7] Gangwal S K , Hudgms R R , Bryson A W and Sdveston P L, Can I Chem Engng 1971 49 113 [8] Adelman A and Stevens W F , A I Ch E J 1972 18 20 [9] Chou T S and Hegedus L L , A I Ch E J 1978 24 255 [lo] Dang N D P and Gibdaro L G , Chem Engng J 1974 8
CONCLUSIONS The sensltlvlty of pulse dtiusivity measurements can be unproved by the appropriate selection of the tamer and tracer gases A mtrogen m helium system at 40°C or shghtly lower
157 [II] OmataH
andBrownL
F,AIChEI
197218967
ChemmlEnbvneem.g Scmee Vol 34 pp 135-137 0 Pcrmmon Press Ltd 1979 Pnnted m Great Bnlam
Exact uniqueness and muMiplicity criteria for a positive-order Arrhenius reaction in a lumped system (Recmed
10 May 1978)
Recent publications by Van Den Bosch and Luss [l, 21 have investigated the multiplicity and umqueness of steady states m reactlon systems for wluch the reaction order was different from unity We refer specfically to Ref [l] m which the authors estabhshed su&lent condlhons for uniqueness and suflicient con&tlons for multlphc~ty-both cnteria being m terms of system constants and parameters-through some algebrmc sunphficatrons In the course of analyzmg steady-state charactenstlcs for more compIex rate expressions, we succeeded m estabhshmg exact necessary and sufficient conditions for the lumped problem considered m Ref [l] We feel that tis result IS worth commumcatmg, because it IS not only exact (and therefore an Improvement. albed only a shght one, over the strong cntena sven m Ref [I]) but it ts also much sunpler m form and hence more hkely to be useful m apphcahons Furthermore, m other work, currently III progress, the same techmques as outhned m the analysis here have been apphed with slmdar success to problems mvolvmg Langmmr-Hmshelwood and Eley-&deal kmetlc models In order to save space, we adopt the notation of Ref [l] throughout Our attention 1s confined to the ntb order (n 20) exothermlc lumped problem (the analogous endothermic problem
always has umque states It]), the steady-state which IS defined by the following dunenslonless 1s the lirst equation m Ref [l]
temperature of equation which
As pomted out m Ref [l], eqn (1) describes the general nomsothermal lumped situation mcludmg the famtiar gradtentless stured reactor and catalyst particle problems As obtamed in Ref [l], the necessary and sufficient condition (expressed m terms of y) that a steady-state soluuon from eqn (1) be umque for all values of a! 1s that the foilowmg mequahty be satisfied (n-l)y’+(B+y+l-n)y’--(2+/3)y+y(l+@)sO
(2)
From physical arguments y is restncted to the range (1. 1+ /3) If the above mequahty IS violated. mulhple steady states are assured for some value of a If it IS satisfied, steady states are umque for all values of ~1 (with a >O) Van Den Bosch and Luss[l] approxunated this cntenon m parameter space by first reducmg the cubic equation m (2) to a quadrauc one Our
Shorter Commumca~ons
136
analysis, wblcb follows, deals with the cubic equation dvectly and proceeds through tbe estabhsbment of necessary and sufficrent condltlons for multlphclty It IS convement for the moment to resmct n to one of the ranges 0 < n c 1 or n > 1 The special cases of n = 0 and n = 1 are consrdered later Representmg tbe funchon on tbe left side of (2) by r(y), we first pomt out that both r(l) and lY1 + @) are poslttve Thus d T(y) IS to vamsb on the Interval 1 c y < Ii #3, It must do so twice It follows that there IS always at least one real root of r(y) whlcb IS not on tb~s Interval Further exammation of T(y) for large negatlve and large poslfive values of y reveals that d (2) IS to be vlolated, then tbe graphIcal representatton of T(y) must be of the form shown m the sketches of F1.g 1 Clearly two necessary condrtlons for the multlpbclty of steady states are that r(y) have multiple roots and that a local mmunum m r(y) exist m the range 1~ y c I+ @ Furthermore, taken together these two condltlons establish necessary and sufficient condltlons for multlplcclty m the sense that tf eltber or both are violated, unique steady states are assured for alf values of a and d both are satisfied muluple states exist for some values of u To express the first of these condltlons m terms of tbe system parameters we employed a necessary and sufficient condlbon for the existence of multiple real roots of a cubic equatron t For T(y) tb~s condltron may be expressed as follows b@
where
x a) -z 0
(3)
range of values of n They were excluded earlier, not because they present unusual da&ties, but because some care must be taken m bandbng them and some earher comments do not apply For example, W&I n = 1, the leadutg coefficient m the expresslon for r(y) vanrsbes leavmg a quadratic equation Obviously the &cusslon regardmg tbe form of such curves as shown m Fs 1 must be changed Wltb n = 0, one root of r(y) occurs exactly at y = I+ & and agam some changes m earher comments must be made It turns out, however, that tbe above condlhons extend to these specml cases In fact, for these special cases, the function &#I. y, n) reduces to a rather sunple expresslon wblcb leads easdy to the necessary and sulliclent multlphclty condttion aven for n = 0 and n = I m Ref [l] Apparently exact multipbclty cntena of the type denved here have not been obtamed prev~ously for other reaction orders, that IS, for nf 0 and n# 1 Comhhons (3), (6) and (7) are tbe prmclpal results of our analysts To emphasize tbev possible utdlty we pomt out that for gven values of y.fl and n, the condltlons m (3) and (6), for OS; n < 1, or m (3) and (7), for n 2 1, may be easily tested If a condition IS vlolated, eqn (1) has a umque steady state for all values of (I On tbe other hand, d the condltlons are satisfied, eqn (1) has multiple solutions for some values of IX In addltron to bemg exact, these condttlons are much sampler to apply than
09
08
&%r,n)=-
Y3B2+2Y2f%1u%~~ -Yg*m~)+w+
1 -n)f(B
+
(4)
1)
and
07 06
1
g@*n)=(1-2n)BZ+2(2-n)B+1+n g~(~,n)=~4+4(3-5n)~3-2(4n2+lln-11)~Z +4/3(1 -n)(3+2n)+(l
-n)’
(9
Expressmg the condttlon regardmg a local mmunum m T(y) m terms of system parameters requves a relattvely strmgbforward analysis of the roots of tbe first denvahve of r(y)-& roots of a quadratic equation--and an exammation of tbe sign of the second denvahvea lmear expresslon Tbls analysts yields the followmg necessary and sufliclent condmons that a local mnnmum m extsts In tbe range 1 c y c 1 + /3
rfy)
I '0
I
2
4
6
8
10
12
14
16
18
Y
ForQ
Fu 2 Grapb~cal representation of multlplrclty condmons (3) and (7) for n = 2 Condmons (3) IS sattsfied throughout tbe shaded regons and condition (7) throughout the cross-batched resons
1,(~+~+1-n)-t/([~+~+1-n~-3(1-n~~+2)~~),I 3(1-n) +B
(6)
For n 2 I Y' Notice that now tbe values
Ftg I Sketches
2/3+n-1 B of 0 and 1 have been Included m
of the functton r(y) when mtdttple steady states extst
tTbere appear to be varrous ways See, for example, Refs [3, 41
of dertvmg
tbrs condltton
Fu 3 Bifurcation curves IIIthe y, &plane Umque steady states extst below the curves, multiple states exist (for some values of a) above the curves
137
Shorter Commumcations those presented m Ref [l] The range of values of 4 over which muIt& solutions exist, ti Indeed they do exist, may be located by first computmg the two real roots of T(y) on the Interval I < y C I+ j? and subsequently usmg those roots m eqn (1) to compute duectly the endpomt values of cx The reeons m the 7, &plane throughout which conditions (3) and (7) are met are shown for n = 2 m Fig 2 It turns out, as suggested by the reaons m that figure, that the condition (3) IS stronger than conditions (6) and (7) except for small values of y and B Exact b,rfurcotlon curves, obtamed from the above condlhons. whEh separate the y, p-plane rnto regrons of multiplsty and remans of uniqueness are shown for several values of nmF= 3
*Present address verslty of Southern
Department of Chemical Engmeermg. UmCahforma, Los Angeles, CA 90000, U S A
Acknowledgement-The work descnhed m tlus paper IS part of research project supported by the Natronal Scrence Foundation (Grant ENG 74-21094) T T TSOTSISt R A SCHMITZ
Department of Chemrcal Engmeenng Unruersrty of Illrnocs Urbana. II 61801, US
A REFERENCES
[ll
Van Den Bosch B and Luss D Cfiem Engng SKI 1977 32 203 [2] Van Den Bosch B and Luss D Chem Engng Scl 1977 32 560 [3] Neumark S Solu~on of Cubrc and Quartz Equatrons Pergamon Press, New York 1%5 [41 Gellert W , Kustner H , Hellwtch M and Kastner H (Eds), The VNR Concrse Encyclopedra of Mathematxs Van Nostrand Remhoid 1977
Charcn~Eit~~rug Scmxr Vol 34 up 137-143 0 PcrmnonPressLtd 1979 PrmtcdmGrsat Entam
Particle
segregation
in
liquhkolid
tluidised beds
(Received 7 March 1978. accepted 11 May 1978) Several worlcers[l-51 have reported segregation m hqmd-sold Ruidaed beds composed of particles of more than one size but there has been little attempt to study the tranSItIon regon m the Rmdlsatlon of a mixture of two sizes of particles Where segregation IS mcomplete the voidage. and hence bed density, wdl vary with he&t and thus measurements of pressures w&n the bed will enable an estimate to be made of the degree of segregation In the present work a pressure transducer has been used to measure the pressure gradient throughout the depth of a bed of partrcles of two sues of glass ballotml flmdlsed by water, and hence to obtam the concentration gradlent
The flmdlsatlon coIumn[6] conslsted of a length of Q V F glass prpe of dmmeter 104’ 1 mm The fiquld dlstrtbutor was formed from three pieces of 42 mesh stamless steel woven wtre cloth arranged with consecutive meshes at 45 deg to one another The free area was 37% of the cross-section of the tube This dlstrtbutor gave d reasonably umform flow dlstnbutron with a pressure drop conslderably greater than that across the bed Itself Llqtnd cuculatlon (see Rg 1) was effected by a WorthmgtonSimpson smgle stage centnfugal pump gvmg a maxnnum flowrate of 2 27 l/s (267 mm/s m the bed) wth a pressure Qfferentml of 8 X I@ N/m2 Flowrates were measured by means of a “Gapmeter” with a &unless steel float m a glass tube, It could measure flows m the range 0 25 to 2 5 I/s (2 5 x lob4 to 2 5 x IO-’m3/s) The pressure drop across the bed was measured usmg a pressure transducer, (first developed by Fand[7f), mcorporatmg a stamless steel probe (5 mm dla X 3 m long) whose position could be vmed w&un the bed The bed pressure was transmitted through four holes of 1 mm dla at a distance of 5 mm from the sealed end of the probe In the pressure transducer, shown m l+g 2, a thm stainless steel plate dlvldes a small chamber mto two compartments, one of wtuch IS connected to the probe which transmits the pressure signal At the centre of the other compartment was a small brass fixed electrode and the vmable capacitance was formed between this and the flexible stamless steel disc The thickness of the stamless steel chsc determines the sensltlvlty of the transducer and the distance between the electrodes can be adjusted to ave the desued pressure range The
Fly
IO
l-l
l’li.’$c._
Fig 1 Flow diagram 1, Water tank, 2, Centiugal pump, 3, Gapmeter, 4. Fhndlsation column, 5. Return hne, 6. Basket, 7, Water manometer, 8, Pressure transducer, 9, Chart recorder, 10, Stainless steel probe whole of the pressure transmlttmg system mcludmg the compartment of the transducer must be completely full of hqmd The second compartment of the transducer 1s open to atmosphere as reference pressure The vartation m the capacitance as a result of a pressure different& unbalances the voltages on two tuned cucmts and operates a ddferentml valve voltmeter, the output of winch @ves a reading proportional to pressure difference The transducer was calibrated by applymg known hydrostatic pressures, up to 100 mm of water, and the lmaty of the Instrument was checked over thus range
Experunents were camed out first by flu&smg 2 mm, 3 mm and 4 mm lead glass spheres of density 2960 kg/m’ with water at 20°C m order to obtam plots of log UC (superiicuil velocity) vs loge (voidage) from which the slope g was measured The