Rate of gas absorption into a slurry accompanied by instantaneous reaction

RATE OF GAS ABSORPTION INTO A SLURRY ACCOMPANIED INSTANTANEOUS REACTION Department

of Chemical

BY

S UCHIDAt and C Y WEN Engmeermg, West Vugmla Umverslty. Morgantown, WV 26506, U S A

(Recerved 30 November

1976, accepted 27 January 1977)

Ah&a&-The problem of gas absorption mto a slurry IS theoretically treated by classlfymg it into SIX cases accordmg to the concentration profiles of the absorbed gas and the dissolved sohd m the hqmd phase For each

case, a mathemaucal model for the rate of gas absorption with the necessary conddlon IS gwen based on the film concept A numerlcal example IS given to illustrate the treatment of a practical problem INTRODUCTION

The absorption of gas into a liquid slurry contammg finely suspended solid pmcles IS theorehcally treated under dtierent operating conditions m this paper This problem IS frequently encountered m various industrial absorption processes For example, carbon dioxide IS absorbed m an aqueous slurry of lune m the manufacture of precipitated calcium carbonate which 1s required m large quantrues as a rubber filler, pigment, etc Au understandmg of the mechanism by which gas 1s absorbed mto a hme or hmestone slurry has also become nnportant m the development of hme or hmestone wet scrubbing processes which remove sulfur dioxide from flue gases Several experimental and theoretical studies on gas absorption mto a slurry have been reported[2-6] BJerle et al [3] performed an experiment on the absorption of SO2 mto a CaCO, slurry m a hquld let absorber The liquid m this type of absorber has a very short contact time urlth the gas so that the dissolution of CaCO, appears to have no effect on the rate of the absorption The effect of the sohd dissolution would become slgmlicant d the contact time were of longer duration Takeda[6] reported a series of experiments m which he determined the amount of SO* absorbed by hmestone slurries of different concentrations in a mlxmg tank He observed slgmficant dfierences m the rate of SO, absorption by the dtierent hmestone slurry concentrations In tlus case, limestone dlssolutlon probably plays an unportant role m the gas absorption process Ramachandran and Sharma[4] were the first to discuss thus problem and proposed the film models for two special cases (see Figs 1 and 2, case l-a and case 2-a) In their model, the rate constant for the solid dlssolutlon was assumed to be constant Uchlda et al [S] modified the Ramachandran-Sharma’s model based on the fact that the rate of hmestone dlssolutlon mto an acid solution LS accelerated by the concentration of hydromum Ions or the pH value of the solution The data obtained by Takeda [6] was explamed by this model tPresent

address

Shlzuoka Uruverslty,

Engneenng, Department of Chemical 3-5-l Johoku, Hamamatsu 432, Japan

The obJective of this study is to classdy the various mechanisms encountered m the three-phase gas absorption system and to develop a mathematical model for each case based on the fXm concept Apphcablhty of the models developed IS also discussed To fully understand the absorption mechanism of a process is important for the development and the e&lent operation of that process For example, under upset scrubber conditions, which promotes low pH, the concentration profiles of the absorbed gas and the dissolved sohd 1s either Case l-b or Case 2-c m Fig 1 or 2 and the blmdmg of hmestone m the wet scrubbmg process reported by Potts ef al [7]

Case

I

:

I-a

Case l-c -

Fig 1 Sohd dissolution m the hqmd film next to the gas-hqmd mterface IS not unportant Case l-a, Transfer of B IS faster than that of A so that the reactlon plane IS m the hqmd film near the gas-hqmd mterface Case l-b, Transfer of A is faster than that of B so that the reacuon plane IS m the hquld film around the sohd partlcles m the bulk hqmd phase Case l-c, Transfer of E IS much faster than that of A so that the reactlon plane is m the

hquld film near the gas-lrquld interface and the concentration of B m the bulk hqmd phase 1s saturation concentration of B

1277

1278

S UCHIDAand C Y WEN

may occur If the rate of SOZ absorption IS much faster than the rate of hmestone dlssolutlon, the reactlon plane wdl shift to the liquid film near the solid-hquld interface and the reaction product, CaSO, or CaSO.,, will be deposited on the surface of the limestone preventmg further dlssolutlon Therefore, recovery of the pH value of the slurry can not be expected and limestone utlhzatlon efficiency will be lowered Operation under such condltlon should always be avolded TaEoRETlcAL

TRE4TMEN-r

Pnor to the theoretical development, the followmg assumptions are made to slmpldy the treatment of the problem (1) The reactions occurring m the hquld-phase are represented by the followmg scheme

B(sohdW A(aq ) + B(aq +

B(aq )

(2)

Product

(3)

The reactlons are consldered to be ureverslble and instantaneous Although addltlonal St&es are needed to verify d such a simple scheme can be apphed to the SO&aC03 system, the mechanism shown IS probably applicable m the case of the CO&a0 system The absorbed CO*(=A(aq)) ~11 react wth the dissolved species CatOH), ( = B(aq )) to produce CaCO, (u) There IS no gas-side resistance to mass transfer (m) The rate of dissolution of the solid 1s enhanced by the reaction between the absorbed gas and the dissolved solid species m the hqmd film around the solid particle When a gas IS absorbed mto a slurry contammg fine solid particles, there are two categories to be encountered from the film concept point of view Category I Solid dissolution m the liquid film next to the gas-liquid interface 1s ummportant This category 1s encountered when the concentration of the solid 1s relatively small or when the particle size of the solid IS relatively large as compared to the thickness of the hquld film Under these condltlons, the extent of solid dlssolutlon m the hquld film next to the gas-liquid interface can be neglected and the solid dlssolutlon and chemical reaction can be assumed to take place m series The condltlon under which this assumption would be valid can be shown to be the followmg[4] k,A,D,’ 4kL2D, Q ’

(4)

There are three cases m tlus category according to the concentration profiles of the species A and B as shown m Fig 1 Category II Solid dissolution m the liquid film next to the gas-liquid interface 1s unportant When the average diameter of the particle 1s consrderably less than the thickness of the liquid film and the solid concentration 1s high, the solid dlssolutlon m the hqmd film next to the gas-llquld interface becomes

important (reverse of condltlon 4) In this case, the solid dissolution and the chemical reaction become parallel steps The effect of solid dissolution in the film IS to increase the local concentration of the reactive species m the film, thereby enhancmg the rate of absorption This category 1s further classfied into 3 cases according to the concentration profiles as shown m Fig 2(a) The concentration profiles of A and B around the solid particle present m the hquld film next to the gas-hquld interface are also mven m Fig 2(b) In the followings, each of these SIX cases is discussed The rate equation governing mass transfer across the interface for each case IS developed by the film concept Category I Solid drssolutronm the lrqurdfilm next to the gas-hqutd mterface IS not mportant Case l-a Reaction plane IS m the liquid film near the gas hquld interface (Fig 1) This case was first treated by Ramachandran and Sharma[4] When the rate of gas absorption IS relatively

I

I

Case 2-a

Case

2-b

Fig 2(a) Solid drssolution m the liquid film next to the gas-hqmd mterface IS uaportant Case 2-a, Transfer of B is much faster than that of A so that the mam reactlou plane 1s m the hquld film near the gas-hquld Interface and the concentration of B m the bulk hqmd phase IS saturation concentration of B Case 2-b, Transfer of B IS faster than that of B so that the mam reactlon plane 1s m the hqmd film near tbe gas&quid Interface Case 2-c, Transfer of A ISfaster than that of B so that the mam reachon

plane IS m the liquid film around the solid particles m the bulk hquld phase

i/ Fig

2(b)

-“‘,1”

Concentration profiles of A and B around the sobd particle present m the hquld tilm next to the gas-hquld Interface

1279

Rate of gas absorption Into a slurry accompamed by instantaneous reaction slow as compared with the rate of sohd dlssolutlon, the concentration of the dissolved sohd m the bulk hquld phase IS mamtamed at a certam value under the steady state condltlon The reaction plane between species A and B IS then m the liquid film near the gas-hqmd interface Two steps are mvolved m this case, the dissolution of the solid species B m the bulk liquid phase and the ddFuslon and simultaneous chemical reactions of the dissolved gas m the hqmd film near the gas&quid interface These steps are m senes and the rate equation for gas absorption can be given by[4] A*+g RA =

B,

(5)

L+B DA* k= D,ksA,

bulk hqmd phase IS mamtamed at the saturation concentration B, (Fig 2a) This case corresponds to Case l-a The solid dlssolution rate 1s much faster than the rate of gas absorption and therefore the concentration of the dissolved sohd B 1s mamtamed at Its saturation concentration Ramachandran and Sharma[4] first theoretically treated et al [S] thrs case and obtamed the rate equation U&da modified their model according to the expenmental fact that the rate of solid dlssolutlon IS enhanced by the reaction between species A and B and that the enhancement factor IS Bven by eqn (7) They obtamed the equation for concentration profiles of A and 3 as follows A*+D,Bs

A= (

The condition lows

under

which

this case

prevads

k,aA* C %A,&

DA2 >

1s as fol(6)

Case l-b Reaction plane 1s m the liquid film around the solid particles m the bulk hqmd phase (Fig 1) When the rate of gas absorption 1s relatively fast as compared with that of solid dlssolutlon, the dissolved solid species B m the bulk hqmd phase 1s consumed and the reactlon plane shifts to the hqmd film around the solid If the rate of solid dlssolutlon 1s enhanced by the reaction and the enhancement factor IS gven by[2,6]

Case l-c Reaction plane 1s m the hqmd film near the gas-liquid mterface and the concentration of B m the bulk hqmd phase 1s constant at the saturation concentration B, (Fig 1) When the solid IS abundant m the hqmd phase and the rate of gas absorption IS much slower than that of sohd dissolution, then the second term m the denommator of eqn (5) 1s negligible compared to the first term, and the concentration of dissolved solid species B m the bulk hqmd phase IS mamtamed at the saturation solub&ty B. The rate equation IS, then gven by R

A

=k

1 1

(OSXGA) jy=B

*

c

p’“hm(S-4 slnh m(6 - A) >

The rate equation

IS aven

(

Z

cothmh--

(A SXSS)

(coth

m&B, =-cothm(S-A)+m

(

(10) (11)

by

RA = mDAA* coth mA + T

x

the rate equation m this case can also be gven by the same equation as eqn (5) The condltton for this case to hold 1s k,A,B, < k,aA* (8)

smhmx +DsB, --DAz ( slnh mh

smhm(A -x) smh mh

mA - &)

D,A*fy

2DzJ3s

1

)

(12)

smh mh )

Case 2-b Reaction plane 1s m the liquid film m the gas-hqmd interface and the concentration of dissolved solid species B m the bulk hqmd phase 1s lower than the saturation concentratron B, (Fig 2a) This case corresponds to Case l-a When the solid concentratton 1s very low or the concentration of the dissolved gas at the gas-hqmd interface 1s high, the rate of gas absorption becomes relatively fast compared to that of solid dissolution, and the concentration of dlssolved species B m the bulk hqmd phase becomes lower than the saturation solublllty & The bulk concentration wdl vary with the operating condltlons When the concentration of the dissolved species B 111 the bulk liquid phase IS not equal to the saturation solubdtty of the solid, the rate equation 1s aven by eqn (12) with B, instead of B. The rate of solid dissolution m the bulk liquid phase, &%. 1s then aven by

L

R = k, (+) This equation 1s the famllmr expression sorption accompamed by an instantaneous reaction 111

for gas abn-reversible

Category II Solid dwolution zn the lrqurdfilm next to the gas-lrqutd mterface IS important Case 2-a Reaction plane 1s m the hqmd film near the gas-liquid interface and the concentration of B m the

(B, -B,)

(13)

Smce the concentration profile of the solid species m the hquld film for A d x G S IS gven by eqn (1 l), the rate of dtiuslon of the solid species from the bulk hqmd phase to the liquid film, RB, 1s obtamed by

x-6

= mDB

B, slnh m(S - A)>

(14)

S

1280

UCHIDA

and C Y WEN

These two rates must be the same under steady-state condltlons Equations (13) and (14) are combined to @ve the concentration of the solid species m the bulk hquld phase as follows t&

B, =

1 smhm(6-A)+;

(15)

m

gven

by

% = k,

(

(21)

And the rate of the dfiuston of the absorbed gas into the bulk liquid phase through the plane x = S IS gven by

The rate of gas absorption mto the slurry IS then calcalculated by replacing B, m eqn (12) by B, gven by eqn (15) When the sohd concentration 1s high or the mterfacml area m the absorber 1s small, the second term in the denominator of eqn (15) becomes dominant over the first B, If the concentration of the term and B, approaches absorbed gas at the gas-liquid interface IS high, the posltlon of the reaction plane, h, approaches S and the first term m the denommator becomes much larger than the second term Then, the value of B, approaches zero Case 2-c Mam reaction plane 1s m the lrqmd film around the solid particle m the bulk hqmd phase (Fig 2a) In contrast to the conditions of gas absorption 1s relatively solid dlssolutlon, the reaction film near the solid particle m case corresponds to Case l-b hquld lilm near the gas-hquld bulk liquid phase 1s Important For 0
D,+-$ The boundary

I+%$

condltlons

in Case 2-b, when the rate fast compared to that of plane shifts to the liquid the bulk liquid phase This The solid dlssolutlon in the interface as well as m the balance

B

‘>

for the dissolved

A,,B, -0

(16)

are

atx=O,

A=A*,

at x=S,

A=A,

(18)

A,+zB,Q = mD,

smh m8

smh m6

--

a B,

(19)

@-I

When the gas absorption process con&tlons, % = zR,,, that is,

A*+gB, A, =

1s under

A

cash ma + t smh m8

RA = mD,(A*

1s represented

123)

When the concentration of the solid approaches zero, that is, m +O, the concentration of A 111the bulk hqud phase, A,, approaches A*, and m eqn (22), RA approaches zero If the value of ma or (m/a) increases, A, grven by eqn (23) becomes zero at a certam value and the absorption mechanism shifts from Case 2-c to Case 2-b

Numerical

example

Consider the following example which may be encountered and which \Klll serve as a practical application A gas, A, IS absorbed mto a slurry contanung fine sohd partxles of B It 1s assumed that the reactions mvolved are mstantaneous The values used in this calculation are gven m Table 1 The film thickness 1s gven by 6 = DA/k, = 5 x lo-’ cm To know which category this case belong, eqn (4) IS used

krApD.az6k, VW**

(6)(2x lo-3(0

2)(10-3’

= (4)(2 x 10-2)*(1)( lo-‘)( 10-5) = 3 ’

Therefore, the sohd dlssolutlon m the hquld film next to the gas-hqmd interface 1s not neghmble and tis case 1s in Category II Equation (23) 1s used to see which case m Category II this case belongs Table 1 Values used m numerical examule A* = 3 x lo-’ mole/cm3 0, = Ds = IO- cm*/sec

by

coth m8 - A,) + y

steady-state

-- a B, =DA

B. = 3 x lo-’ mole/cm”

The rate equation

1

film next to the

smhmx A=

)

(22)

4kL2D, - 4k,fd,& The concentratron profile m the hquld gas-hqutd interface 1s then grven as

coshms

A

(coth ms - 1) (20)

The value of A, m eqn (20) can be obtamed as follows The rate of sohd dlssolutlon m the bulk hqmd phase 1s

/c, = 2 X lo-* cmlsec p=lOg/cm’ d, = 1O-4cm W=02g/cm3 Ap = 6Wlpd, a = 10 cm’1cm k, = 1 x lo-* cmlsec

Rate of gas absorphon

mto a slurry accompanied

by mstantaneous

1281

reaction

liquid-flm mass transfer coefficient, cmlsec dissolution rate constant, cm/set m 5

&g$jqj=l

= ~/(k,A,JD,)

R.4 rate of gas absorption,

mole/cm’ see

RB rate of dtiuslon of B to or from bulk liquid phase,

3 x lo-5 + 3 x 10-5 A’ = cash (3464)(5 x lo-‘) + 3464 smh (3464)(5 x lo-*) -3x10-5=-3x10-5<0 Assuming that the concentration of B, IS equal to B,, the rate of gas absorption IS calculated from eqn (12) by trml and error procedure as . RA = 2 2 x 10m6mole/cm’ set The value of I3, gven by eqn (15) IS also nearly equal to B, The absorption scheme IS, therefore, m Case 2-a Acknowledgement-This work IS supported by a grant from EPA Grant No R800781-03-3

%

’ :

mole/cm’ set rate of dlssolutlon of solid m bulk hquld phase, mole/cm’ set distance from gas-hqmd interface, cm stolchlometrlc factor

Greek symbols B enhancement factor S thickness of the hqmd film next to interface, cm S’ thickness of the hqmd film around sohd the bulk liquid phase, cm A reaction plane m hqmd film next to Interface, cm A’ reaction plane m liquid film around solid the bulk liquid phase, cm

gas-liquid particle In gas-hqmd particle m

RFNERENCES NOTATION

gas-liquid mterfaclal area, cm2/cm3 species to be absorbed concentration of A m the bulk hqmd phase, mole/cm3 solid-liquid contact area, cm2/cm3 concentration of A at the gas-liquid interface, mole/cm” sohd species to be dissolved concentration of B m the bulk hqmd phase, mole/cm3 solubrllty of solid B, mole/cm’ dlffuslvlty of A, cm*/sec dlffuslvlty of B, cm2/sec

[l] Danckwerts P V , Gas-Llquld Reactions, p 112 McGraw Hdl, New York 1970 [21 Ucluda S , Wen C Y and McMxzhael W J , Prog Rept Envvonmental Protection Agency, U S A, 1973, Contract No EHS-D-71-20, No 21, also Ch ICh E J 1974 5(2) 111 [3] Bjerle I , Bengtsson S and Farnkvlst K , Chem Engng Sa 1972 27 1853 [41 Ramachandran P A and Sharma M M , Ckem Engng Scr 1%9 24 1681 [5] U&Ida S , Koide K and Shmdo M , Chem Engng Scr 1975 30644 161 Takeda T , Moriguclu H , Ucluda S and Kolde K , a paper submitted to 12th fall meetmg of the Society of Chem Engrs, Japan, Nagoya, 1976 [71 Potts J M , Slack A V and Hatfield J D, Proc 2nd Int Llme/Lonestone Wet-Scrubbmg Symp New Orleans, LOIUSIana 1971 1 198