Chemical Reactors - American Chemical Society

7 Modeling of Fluidized Bed Combustion of Coal Char Containing Sulfur A.

1

REHMAT ,

2

S. C. SAXENA , and R. H . L A N D

Downloaded by CORNELL UNIV on October 15, 2016 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch007

Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL 60439

A mathematical model is developed f o r c o a l char combustion w i t h s u l f u r r e t e n t i o n by limestone or dolomite in a gas fluidized bed employing n o n c a t a l y t i c s i n g l e p e l l e t g a s - s o l i d r e a c t i o n s . The s h r i n k i n g core model is employed to describe the k i n e t i c s of chemical r e a c t i o n s taking place on a s i n g l e p e l l e t whose changes in s i z e as the r e a c t i o n proceeds are considered. The s o l i d s are assumed to be in back-mix c o n d i t i o n whereas the gas flow is regarded t o be in plug flow. The model is strictly v a l i d f o r the turbulent regime where the gas flow is q u i t e high and classical bubbles do not e x i s t . Formulation of the model i n c l u d e s s e t t i n g up heat and mass balance equations petaining t o a s i n g l e p a r t i c l e exposed to a v a r y i n g reactant concentration along the height of the bed with accompanying changes in its s i z e during the course of r e a c t i o n . These equations are then solved numerically to account f o r p a r t i c l e s of all s i z e s in the bed to obtain the o v e r a l l carbon conversion e f f i c i e n c y and r e s u l t a n t s u l f u r r e t e n t i o n . In p a r t i c u l a r , the i n f l u e n c e of s e v e r a l fluid-bed v a r i a b l e s such as oxygen concentration p r o f i l e , a d d i t i v e p a r t i c l e s i z e , r e a c t i o n rate f o r s u l f a t i o n r e a c t i o n , s u l f u r absorption e f f i c i e n c y are examined on a d d i t i v e requirement.

1

2

C u r r e n t Address: C u r r e n t Address:

I n s t i t u t e of Gas Technology, Chicago, I L U n i v e r s i t y o f I L a t Chicago C i r c l e , Chicago, I L

0097-6156/81/0168-0117$09.75/0 © 1981 American Chemical Society Fogler; Chemical Reactors ACS Symposium Series; American Chemical Society: Washington, DC, 1981.


118

CHEMICAL REACTORS

Many models have been used to describe fluidized-bed opera t i o n (1-7). Several a d d i t i o n a l models have been proposed during the l a s t three years and these w i l l be r e f e r r e d to l a t e r in this report. It is commonly assumed that the bed is composed of two d i s t i n c t phases, v i z . , a dense phase (emulsion) c o n s i s t i n g of s o l i d p a r t i c l e s and i n t e r s t i t i a l gas, and a bubble phase cons i s t i n g of r i s i n g voids with almost no s o l i d s . The most advanced models (1, 2^, 3) a l s o consider a d d i t i o n a l phases, v i z . , a cloud and wake a s s o c i a t e d with each bubble. Further v a r i a t i o n s appear in the c h a r a c t e r i z a t i o n of gas flow w i t h i n each phase, and mode of exchange among the phases, and the bubble shapes, v e l o c i t i e s , and growth r a t e s . I t is g e n e r a l l y assumed that in the two-phase theory of fluidization ( 8 ) , the flow r a t e of bubble voids through the fluidized bed is equal to the excess gas flow r a t e above that r e q u i r e d f o r minimum fluidization. Chemical r e a c t i o n s in the bed are assumed to occur e n t i r e l y in the emulsion phase. In the present a n a l y s i s , we s h a l l develop a b a s i c model f o r fluidized bed operation by extending our e a r l i e r a n a l y s i s (9_> 10, 11) f o r a s i n g l e p e l l e t r e a c t i o n to model the n o n c a t a l y t i c g a s - s o l i d r e a c t i o n s taking place in a fluidized bed. The e a r l i e r r e s u l t s have been derived with the assumption of constant gaseous reactant concentration surrounding the p e l l e t . However, in a fluidized bed, the p e l l e t encounters a considerable v a r i a t i o n in the gaseous reactant concentration due to i t s movement. A l s o , the fluidized bed is composed of p a r t i c l e s of d i f f e r e n t s i z e s , each of which w i l l behave d i f f e r e n t l y . The s o l i d m a t e r i a l in the bed is c o n s t a n t l y being consumed due to chemical r e a c t i o n s and is being depleted by entrainment and overflow. This solid material should be replenished continuously by feeding f r e s h reactant particles. In order to develop a r e a l i s t i c model, the p a r t i c l e s i z e d i s t r i b u t i o n of the feed and the bed must be taken i n t o account as a l s o the f a c t that the fluidized bed operates in a continuous mode with s o l i d s a d d i t i o n to the bed by feed and removal by overflow and e l u t r i a t i o n . The model presented here takes i n t o account the changes in the s i z e of a p a r t i c l e as a r e s u l t of chemical r e a c t i o n s in a fluidized bed. A number of modeling s t u d i e s r e l a t e d to the n o n c a t a l y t i c r e a c t i o n s and to c o a l combustion in p a r t i c u l a r , taking place in a fluidized bed have been reported (2-25). A review of these s t u d i e s i n d i c a t e that the c o a l combustion process is p r i m a r i l y d i f f u s i o n c o n t r o l l e d . The amount of gaseous reactant d i f f u s i n g through the gas f i l m surrounding the p a r t i c l e , w i l l depend on i t s s i z e . In most of the models r e f e r r e d to above, the p a r t i c l e s i z e is assumed to be constant throughout the r e a c t i o n i n s o f a r as the mass t r a n s f e r process is concerned. The shrinkage of p a r t i c l e s in those cases where e i t h e r no s o l i d product is formed or ash is f l a k e d o f f from the surface is used only in c a l c u l a t i n g the p a r t i c l e size d i s t r i b u t i o n in the bed, carryover, and overflow streams. To account f o r p a r t i c l e growth or shrinkage as the r e a c t i o n progresses in the r e a c t o r , a parameter, Z, is introduced. The

Fogler; Chemical Reactors ACS Symposium Series; American Chemical Society: Washington, DC, 1981.

7.

REHMAT E T A L .

Combustion of Coal Char

119

theory developed by K u n i i and L e v e n s p i e l ( 1 ) is f o r a r e a c t i n g system in which the p a r t i c l e s maintain a constant s i z e (Z = 1 ) . I t ( 1 ) deals with the d e r i v a t i o n of r e l a t i o n s g i v i n g the p a r t i c l e s i z e d i s t r i b u t i o n in the bed, overflow, and carryover streams and t h e i r r e s p e c t i v e weights. T h i s theory w i l l be extended to i n c l u d e the e f f e c t s of p a r t i c l e growth or shrinkage (Z>1 or Z < 1 ) . For t y p i c a l combustion of char c o n t a i n i n g s u l f u r followed by s u l f u r dioxide absorption by limestone, r e l a t i o n s w i l l be derived to determine the extent of s u l f u r r e t e n t i o n . The r e a c t i o n , carryover, and overflow r a t e s w i l l be evaluated with p a r t i c u l a r a t t e n t i o n to t h e i r dependence on Z.


D e s c r i p t i o n of Char P a r t i c l e Combustion The combustion of s u l f u r - r i c h char is accompanied by the production of an undesirable r e a c t i o n product, v i z . , s u l f u r d i o x i d e . However, most of the s u l f u r dioxide should be removed from the combustion gases before they leave the combustor. T h i s may be accomplished by the i n t r o d u c t i o n i n t o the combustor of s u i t a b l e a d d i t i v e s which can absorb s u l f u r d i o x i d e . Limestone is such an a d d i t i v e . The limestone r e a c t s with s u l f u r dioxide in the presence of oxygen to form calcium s u l f a t e , which is a s o l i d product and can be e a s i l y removed from the r e a c t o r . In this work, a model is proposed f o r the p r e d i c t i o n of s u l f u r dioxide removal from the combustion gases, based on knowledge of g a s - s o l i d r e a c t i o n s taking place on a s i n g l e p e l l e t . The k i n e t i c s of g a s - s o l i d r e a c t i o n s obtained from s i n g l e p a r t i c l e s t u d i e s are u t i l i z e d to c a l c u l a t e the generation and u t i l i z a t o n o f s u l f u r dioxide f o r many p a r t i c l e s present in a fluidized-bed r e a c t o r . For s i m p l i c i t y , char ( i . e . , c o a l with almost all v o l a t i l e s removed) w i l l be the b a s i c feed to the r e a c t o r and it is assumed to contain carbon, ash, and s u l f u r . Carbon and s u l f u r r e a c t with oxygen to form t h e i r r e s p e c t i v e oxides. S u l f u r dioxide subsequently r e a c t s with limestone and excess oxygen to form calcium s u l f a t e . Char and limestone p a r t i c l e s undergo change in size as they r e a c t , and this w i l l be i n c l u d e d in o b t a i n i n g average conversions. U l t i m a t e l y , this model p r e d i c t s the average concentration of s u l f u r dioxide in the combustion gas stream, s o l i d flow r a t e s , and the p a r t i c l e s i z e d i s t r i b u t i o n s in the r e a c t o r and in the streams l e a v i n g the reactor. The f o l l o w i n g assumptions are made in the mathematical formulation of the process: 1. The p a r t i c l e s are completely mixed in the r e a c t o r . 2. Gases do not mix v e r t i c a l l y , i . e . , the gas flow through the bed is in plug flow. F u r t h e r , no gas concentration gradients e x i s t transverse to the d i r e c t i o n of flow. 3. The gas flow is s t a t i s t i c a l l y uniform over the bed cross s e c t i o n at a given bed height and is equal to a c e r t a i n mean value.


120

CHEMICAL REACTORS

4. 5. 6.

The temperature is uniform throughout the bed. The r e a c t o r is operated in the steady state mode. S u l f u r is uniformly d i s t r i b u t e d in the c o a l char particles. 7. Combustion and consequent generation and absorption of s u l f u r d i o x i d e occur throughout the bed. 8. The r e a c t i o n r a t e s are independent of r e a c t i o n product concentrations. 9. The g a s - s o l i d r e a c t i o n f o l l o w s a s h r i n k i n g core model. Char and limestone (calcium carbonate) p a r t i c l e s are fed to the r e a c t o r continuously at r a t e s , F χ and r e s p e c t i v e l y , and t h e i r p a r t i c l e s i z e d i s t r i b u t i o n s in the feed are given by Ρ χ ( ξ ) and Ρ^(ξβ), r e s p e c t i v e l y . The mass of these s o l i d components in the bed, the overflow and the carryover r a t e s , and t h e i r r e s p e c t i v e s i z e d i s t r i b u t i o n s are shown in F i g u r e 1. The c a r r y over and the overflow p a r t i c l e s are not r e c y c l e d . The f o l l o w i n g r e a c t i o n s are considered to take place in the fluidized bed. The s u b s c r i p t s by which the r e a c t a n t s and products are r e f e r r e d to throughout in this work are given in parentheses.


8

carbon

(J) + oxygen (A) -v carbon d i o x i d e (D)

(1)

s u l f u r (S) + oxygen (A) ->- s u l f u r d i o x i d e (B)

(2)

calcium carbonate

(N) + s u l f u r d i o x i d e (B) + oxygen (A) ->

calcium s u l f a t e (E) + carbon d i o x i d e (D)

(3)

When char r e a c t s with oxygen, s o l i d product ash is formed and it adheres to the p a r t i c l e . S i m i l a r l y , in the case of limestone, the s o l i d product calcium s u l f a t e adheres to the limestone p a r t i c l e . The change in the o v e r a l l s i z e of the p a r t i c l e depends on the amount of s o l i d formed and is r e l a t e d to the amount of s o l i d reactant consumed in the f o l l o w i n g manner: Ζ =

v

u m e

°l°f s o l i d product formed volume of s o l i d reactant consumed

The average r a d i u s , Γ, is d e f i n e d as f o l l o w s :

and is employed to normalize the d i s t a n c e s from the center of the pellet. In the f o l l o w i n g s e c t i o n s , the equations f o r a s i n g l e p e l l e t i n v o l v i n g one and two independent r e a c t i o n s are presented. F i r s t , we s h a l l derive equations p e r t a i n i n g to a s i n g l e r e a c t i o n


7.

REHMAT E T A L .

121


and then extend the d e r i v a t i o n to the two g a s - s o l i d r e a c t i o n s taking place with one gas and two s o l i d s . Gas-solid reactions given by Eqs. 1, 2, and 3 w i l l be used. Single P e l l e t : One Reaction. The s u l f a t i o n r e a c t i o n which is considered here f o r calcium carbonate is given by Eq. 3, and the temperature and concentration p r o f i l e s of a t y p i c a l growing limestone p a r t i c l e are shown in Figure 2. The r a t e of d i s appearance of s u l f u r dioxide is assumed to be the f i r s t order and is given by r

~ B = k3(T )


c

C

N 0

C

(6)

B C

The d i f f e r e n t i a l equation f o r the mass balance of gaseous reactant Β r e a c t i n g with a p e l l e t of r a d i u s R under the pseudosteady-state assumption is Q

_ Γγ.2

(7)

Ο

with the f o l l o w i n g boundary c o n d i t i o n s : at r

T

1

R , i . e . , a t ξ' = ξ , 8

d03B

= sRf ° N

at r

( (

B H

"

ω

β

(8)

)

i . e . , a t ξ' = ξ
d(jQ;

exp

dξ

The

δ

1

{κχ^ί " υ · ) }

s o l u t i o n of the above Equation

(9)

7 gives

^BH TTI Ν _ r'2 c SHr SI U

exp

The

TTt

N

i t j j

1

υ·)}

(10)

heat balance I s

_d

(11)

άζ


CHEMICAL REACTORS

122

FLUE GAS Aet Cee. Coe c

FEED STREAM F,,

P,(e ) s

CHAR

F"!. P|(Cs) DOLOMITE

V

Ho

';.·.:.p'ι{ξ' ):.··*


9

CARRYOVER STREAM F .P (€s) 4

Figure 1.

A

GAS

I

Y

4

Feed and exit streams of thefluidized-bedcombustor

ASH LAYER

R'

R'

r'

tl

Figure 2. Gas-solid reaction of a growing limestone particle at height Ηinthe fluidized bed: concentration and temperature profiles


7.

REHMAT E T A L .


123

with the f o l l o w i n g boundary c o n d i t i o n s :

dU

= R', i . e . , V

f

at r

= ξ»,

T

N

" Nur

1

and a t r

=r ,

^

i.e., ξ

c

1

- ξ£ c

_ /RT \ /dU'\ 0

= Φ Γ*3 3

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(12)

- D

exp

1

( ^") {^ i ~ ϋ,τ)}

(13)

U£ is then given by the f o l l o w i n g expression:

1

« "

1

« - f o i

- i s ) } (14)

I f we assume that the gas in the r e a c t o r is i d e a l and the gas pressure is constant, the f o l l o w i n g r e l a t i o n holds true throughout the r e a c t o r : CT

constant

(15)

Thus, we can w r i t e the expression f o r the r a t e of conversion f o r the p e l l e t in terms of i t s core r a d i u s as f o l l o w s :

de

exp U

3

c

{lio"

1

i u )}

(16)

c

where τ

3 " M k (T )C N

3

and θ ο =

o

N 0

C

(17) A 0

—

(18)

Since the behavior of the fluidized bed depends upon the o v e r a l l p a r t i c l e s i z e , it is necessary to derive an expression f o r dÇs/ 3' l I shorn (9) that f o r s p h e r i c a l p a r t i c l e s , d 9

de

f

(

3

1

}

c

s

ξ 2d e 3 έ


(19)

124

CHEMICAL REACTORS

S u b s t i t u t i n g the expression f o r c^ /d03 from Eq. 16 i n t o Eq. we get the r e q u i r e d r e l a t i o n , v i z . , c

ει

dEl

19,

2

{«> £ (> - fe)}

(20)


Single P e l l e t : Two Independent Reactions. The two inde pendent r e a c t i o n s considered f o r char combustion are given by Eqs. 1 and 2, and t h e i r r e s p e c t i v e r a t e s of r e a c t i o n are r

D

r

B

=

k

c

c

a

l J0 AC> k

c

n

(21)

d

(22)

c

= 2 S0 AC

Reactions 1 and 2 take place independently w i t h i n a s i n g l e p e l l e t which c o n t a i n s both of the s o l i d r e a c t a n t s , J and S. The m a t e r i a l balance f o r the gaseous reactant A in the ash l a y e r of the p e l l e t under the pseudo-steady-state assumption is represented by: d2q)A

+

2 do) ξ dξ

(23)

The boundary c o n d i t i o n at ξ = ξ da)

= N

dC

and at ζ = ξ

S H

r(o

A H

-

ω

Α δ

8

is (24)

)

is

ε

dCOjj

(25)

S o l u t i o n of the above equations g i v e s the f o l l o w i n g r e s u l t f o r ω

Α0

^ U

:

=ι

+

1_

AC

U

Φΐ

Γ

c

N h r ξ2

+ ?c

S

exp

Η)

·+

ex

(26)

The heat balance equation is d U dç2 2

2 du ξ

.


(27)

7.

REHMAT E T A L .

125


with the f o l l o w i n g boundary c o n d i t i o n s : At ξ = C c

dU dξ

(28)

N

- NuF ( U - D s

'ζ - ^s and at ξ = ξ

ε


exp . U

c

U

c

is obtained from the f o l l o w i n g expression:

Φΐϊ3ι I I .

- 1

RT

e

x

0

p

( j - (ι -

"(RT \ 0

L-)\ U /J c

(30) The conversion of the s o l i d p e l l e t expressed in terms of core r a d i u s is given by d^ dt

M 1

ki(To>CjoCAO Τ***

(31) Let τ. = ?JI 1 Mjk^T^CjoCAo' 2

M k (T )C C > s

2

o

S 0

(32) (33)

A 0

and (34) S u b s t i t u t i n g Eqs. 32, 33, and 34 i n t o 31, we get: _ d£c - ω 0 Up d9i Α

(35)

On the b a s i s of Eqs. 19 and 35, the r a t e of change of o v e r a l l p a r t i c l e s i z e is given by

(36)


126

CHEMICAL REACTORS

The above equations w i l l be employed in the mathematical modeling of fluidized bed presented below. When two s o l i d r e a c t a n t s (char and limestone) are present, we s h a l l use primed (limestone) and unprimed (char) symbols to d i s t i n g u i s h between them.


Mathematical Model f o r F l u i d i z e d - B e d Combustion Process The development of mathematical models to describe the thermochemical process o c c u r r i n g in a fluidized bed i n v o l v e s s e t t i n g up the m a t e r i a l and energy balance equations. The t o t a l process is represented in terms of a set of independent equations which are solved simultaneously to obtain such q u a n t i t i e s as combustion e f f i c i e n c y , s u l f u r r e t e n t i o n , oxygen u t i l i z a t i o n , oxygen and s u l f u r dioxide concentration p r o f i l e s in the bed, e t c . The r e l a t i o n s h i p between v a r i o u s streams, flow r a t e s and p a r t i c l e s i z e s w i l l be d e r i v e d f o l l o w i n g the method of K u n i i and L e v e n s p i e l (1). F i r s t , the r e l a t i o n s are d e r i v e d f o r char, whose p a r t i c l e s g e n e r a l l y shrink as they r e a c t with oxygen. The o v e r a l l mass balance f o r char p a r t i c l e s of the system is given by: F

— F 1 3

F 4

.~v

ί mass of carbon and s u l f u r / jconsumed by chemical r e a c t i o n f

7

In order to evaluate the r i g h t side of Eq. 37, we w i l l c a l c u l a t e the mass l o s s f o r a s i n g l e p e l l e t due to r e a c t i o n and sum up such l o s s e s f o r all p a r t i c l e s present in the fluidized bed. Upon combustion, char leaves behind a l a y e r of ash having a d i f f e r e n t d e n s i t y than that of coke. Thus, the mass of a s i n g l e char p a r t i c l e , w^, of size ξ in the bed is given by: 8

w

3p

?

i - 7 ** Q

+

38

(*2 - c ) 52]

Therefore, the r a t e of change of mass of a s i n g l e p a r t i c l e dwj dt

4π3 Ρ

PQ

size

[ cJ « a + (i - ) 4 g*] ai

(39)

ai

where

(40) The volume, dV, of the fluidized bed of c r o s s - s e c t i o n a l area A and elemental height dH is A dH. Let f q be the f r a c t i o n of char p a r t i c l e s in the bed voidage ε· The volume of char p a r t i c l e s of s i z e ξ g in the elemental volume dV is f g ( l - ε ) Α Ρ 2 ( ξ ^ ξ ^ Η . The number of p a r t i c l e s of s i z e ξ in this elemental volume is 0

Q

0

8

8

* . f ( l - ε)Α Ρ (ξ ) Q ( 1 - ε)Α

P3(?s)

dH

+ Η κ (ξ ) 0

ρ

8

dt

0

c

/ 1

+

Ms dt

dH

dt

/ 3(α -1)ξ3 \ Ί

Ζ

+

3 α ^

1§£

3(1-O.^2

+

dH

dH


dt

7

d£, dt

dH

dξ

(59)

s

Ρ 3 ( ξ ) is the s i z e d i s t r i b u t i o n of the p a r t i c l e s in the bed and overflow stream f o r a feed of f i x e d p a r t i c l e s i z e ξ . The r e l a t i o n connecting ξ , ξ , ξ and Ζ is, 8

0

8

α

0

(60) Therefore (61) (1-Ζ)ξ2 Substituting

Eqs. 49, 60, and 61 i n t o Eq. 59 we f i n a l l y g e t ,

dH

dt 8

H ,

Ρ3(ξ )

0

d

«ο

?

£

8

dH

3(ατ1)ζ3ζ

xi(i-z)?| where Χ

χ

= α

χ

3(1-α Ζ) Ί

8

(62)

xid-z)?s

+ (1-α )(ξ2-Ζξ§)/(1-Ζ)ξ| 1

I n t e g r a t i o n with respect to ξ

8

leads to the f o l l o w i n g :


(63)

7.

in 4 ^ = ?3(?o)

133


REHMAT ET AL.

/

U ^ J "Ο

lis dt

dH

dt

s -ln dH

J

dH =

^s ?o

if

+ ln

(64)

The above expression gives Ρ3(ξ ) i f we could evaluate Ρβίξο) which comes out to be Downloaded by CORNELL UNIV on October 15, 2016 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch007

8

V

Ρ3(ξο)

f (l-e)A Q

l

) Ho. Γ I d ^ dH J dt

0

(65)

S u b s t i t u t i n g Eq. 65 i n t o Eq. 64 we get, p* ζ (

. ν (1-ε)ζ3ΐ(ξ.,ξ )

8 )

1

(66)

0

ϋ (1-ε)Α ζ3 0

"ο

ο

^

dH

where ξ

_ J £

I( s>?o) "

e x

S

/ν^(Ι-ε,) \ίη(1-ε)Α

+Η κ (ζ )\ J 0

η

η

Β

(67)

P J

dH

dt

The above r e s u l t s apply to a s i n g l e size feed. wide d i s t r i b u t i o n

For a

ξο,max ρ

3(ξ8)Δξ

-

8

Σ ο,ιη1η

ρ

ς

3(ξ8)Δξ ΐ(ξο)Δξ

(68)

ρ

8

E

0

where ξ and K i are the l a r g e s t and the smallest size p a r t i c l e s in the feed. The output d i s t r i b u t i o n f u n c t i o n Ρ;3(ξ ) f o r constant input size ξ is already derived and is given by Eq. 66, d i v i d i n g Eq. 68 by Δ ξ and taking l i m i t s as Δ ξ -> 0, we get 0 m

a

x

Q

m

n

8

0

8

8


134

CHEMICAL REACTORS

? o,max p

3(?s) =

/

69

Ρ3(ξ ) Ρΐ(ξ )

0

? o,min Substituting

Eq. 66 i n t o Eq. 69 we get ô,max (70)


ô.min

ξ

3 / | ^ μ

Η

Equation 72 d e f i n e s Ρ 3 ( ξ ) in which there are two unknowns v i z . , V3 and f q . One of them can be e l i m i n a t e d by u t i l i z i n g the normal i z a t i o n condition f o r Ρ3(ξ ). Integration of Eq. 70 f o r all s i z e s of p a r t i c l e s in the bed ( ζ ι to C ) yields 8

8

8 > π ι

f (l-c)A f t

0

m

η

s > m a x

^s.max Γ

?o,max f ξ^Ρ (ζ )Ι(ζ ζ )

?s,min

? ο,min

Ί

η

Η >

J

ξ3

0

,

d

(

7

1

)

dt

I t may be noted that ξ = ξ f o r shrinking p a r t i c l e s . I f Η is nondimensionalized such that: η = (H/H ). The r e s u l t i n g equations are then u t i l i z e d to compute F 3 from Eq. 45 as a f u n c t i o n of f q . Next we proceed to e s t a b l i s h a mathematical framework in a somewhat analogous fashion f o r c a l c u l a t i n g F 4 as a f u n c t i o n of f q . Combining Eqs. 54 and 56 and s u b s t i t u t i n g f o r Ρ 3 ( ξ ) from Eq. 70 we get, 8m

a

x

0m

a

x

0

8

ô,max **«•>

= Κ0«.)

3

/ ?

o,min

1

' Κ ψ ç3 J

* ^

d.

(72)

?0

Id^L o '

I

The n o r m a l i z a t i o n of the s i z e d i s t r i b u t i o n f u n c t i o n finally yields

Ρ4(ξ ) 8


7.

£s,max ν

*α- Λ) ε

Vi(l-ei)

135


REHMAT ET AL.

?o,max

Γ

Γ

I S,min

I ô,min

ύ^α«)^(^ηα«Λη) 1 f

3

(

7

3

)

d

£«

dn

ο Equations 72, 73 and 71 can be solved simultaneously to obtain Ρ 4 ( ξ ) d 4 as a f u n c t i o n of f q . These r e l a t i o n s are then u t i l i z e d in Eq. 46 to obtain F 4 as a f u n c t i o n of f q only. The mathematical development presented so f a r enables us to employ the mass balance of Eq. 43 to determine uniquely the f r a c t i o n of char present in the bed f o r a given set of feed and operating c o n d i t i o n s . A s i m i l a r set of equations can be derived f o r the v a r i o u s streams of limestone. For the general case when the size of the p a r t i c l e changes as it r e a c t s with oxygen and s u l f u r d i o x i d e , the f o l l o w i n g equations apply. The o v e r a l l mass balance is given by the f o l l o w i n g r e l a t i o n which is analogous to Eq. 43 developed above f o r char p a r t i c l e s : a n

v


8

5s,max F

F

l

=

F

" 3 "4

/

1

/ ξ s,min

3

%σ- ) οΗ Ρ^(ξέ)ΡΝ ε

Α

0

0

where the feed rate of limestone F { , the overflow r a t e F ^ , and the c a r r y o v e r r a t e F ' may be expressed in terms of the c o r r e sponding volumetric flow r a t e s . Adopting the approach developed above f o r the char p a r t i c l e s combustion, the s i z e d i s t r i b u t i o n f u n c t i o n of limestone p a r t i c l e s as a r e s u l t of s u l f a t i o n r e a c t i o n in the overflow stream which is the same as in the bed is given by, .

.

£o,max

.

£ô,min ..3

iia

f

dn

The n o r m a l i z a t i o n property of P^(£ ) l e e d s to the f o l l o w i n g t i o n s h i p between V£ and V^: s

f (l-e)A H VJd-ε,·) N

n

n

_

£s,max Γ J £s,min

ô,max -, r gjfol(ξΑ)Ι'(ξ&,ξη)άζ άξ J 1 £o,min ? 3 f \άζΙ Co / | ^ | d n 0

Γ

Β

3

Ο

rela-

1


V

'

CHEMICAL REACTORS

136

Here

I

|f (l-e)A H N

0

0

(77)

and (78) N

U

s

;

Α Η (1-ε)ί Ρ:(ξέ)


0

0

Ν

The size d i s t r i b u t i o n of limestone in the e l u t r i a t e d stream, 4(£s)> obtained from Eq. 78 in which PjCÇs) replaced with 3 fζ s) (backmix approximation) as d e f i n e d by Eqs. 75 and 77. In the present formulation the size of the fluidized bed is kept constant by the presence of an overflow pipe, Figure 1, and consequently A and H are constant. The f r a c t i o n of dolomite in the bed, f , is r e l a t e d to the f r a c t i o n of char in the bed such that p

l

s

is

P

0

Q

N

(79) fq has already been determined and hence it may be assumed that fjj is known. In case the s i z e and d e n s i t y of the limestone p a r t i c l e s remain the same as a r e s u l t of chemical r e a c t i o n , the above r e l a t i o n s are s i m p l i f i e d . We next develop the mass balance equations f o r the gaseous reactant (oxygen) and the product ( s u l f u r d i o x i d e ) . The gas flow in the r e a c t o r is assumed to be in plug flow and hence the concentration of these gases w i l l depend only on the height H, in the bed above the d i s t r i b u t o r p l a t e . The rate of consumption of oxygen by r e a c t i o n s 1, 2 and 3 can be obtained from Eqs. 43 and 80 and the stoichiometry of these r e a c t i o n s . We w i l l f i r s t examine Eq. 43 which may be r e w r i t t e n as f o l l o w s a f t e r appropriate substitutions. £s,max RHS of Eq. 43 = 3ν ( l - y p ^ i - ^ z 1 Ί

£o,max

J

J r ô,min

r ^s,min

1 J

^ c

°

ρ

Κ ξ > ξ ο ) ΐ ( ξ ο ) exp 8

ld£ dÇs 0

J

dt

dn



7.

REHMAT

The f i r s t term of Eq. 80 represents the mass l o s s p a r t i c l e s due to carbon combustion and the second term the mass l o s s of char p a r t i c l e s due to s u l f u r r e a c t i o n oxygen. Using the stoichiometry of r e a c t i o n s 1 and 2, obtain the moles of oxygen used up in these r e s p e c t i v e f o r any a r b i t r a r y height η as,

fMoles of oxygen used upto bed height η due to = ^ n ·. r e a c t i o n 1]

I(Cs»?o) l(?o)

of char represents with we can reactions

n

£s,max £o,max 3Vi(1 -ε )ρρ(1-α Ζ) — Μ τ τι J ± 1

J

p

137


ET AL.

e x

1

0

r j Ε ^ ^s,min ^»o,min

u

P

/ lis

c

(81) dn

dt

[Moles of oxygen used upto bed 3V (l-e )Pq(l-a Z) height n due to = M T2 r e a c t i o n 2] 1

1

^s,max ô,max

η

^s,min ô,min

°

1

c

S

(82)

/


CHEMICAL REACTORS

138

[Moles of consumed height η reaction

oxygen upto bed due to = 31

ξβ,πιβχ £o,max η 3V^(1 -εχ)Pfl(l-ct2 ) r r f J J J r' r' . ° ξs,min ξο,παη f

f

z1

5

S

i

F

b

J

Ι'(ξέ»ξο)Ρί(ξο) exp j t


c

-

!RTO

/

BC

3

J

3

W

(83)

dndC dCè 0

dn

dt

Since there is only a small change in the t o t a l moles of gases in the fluidized bed combustor as a r e s u l t of chemical r e a c t i o n s , we can assume that the gas flow remains unchanged. L e t this flow be Y. Also the bulk temperature of the fluidized bed remains constant, t o t a l gas concentration remains constant throughout the r e a c t o r and hence Ω

Α Η

-

CAH/ AO

(

C

The r a t e of change of oxygen concentration with height in fluidized bed is given by the f o l l o w i n g r e l a t i o n .

the

^s,max ô,max YC A

A n 0

i!5AE = - 3 V ( l - c ) p ( l - a Z )

exp£RT

?o,max f

f

mi U

J

c

so,min

ω Γ Α

ς

ο,πιιη

r

JL

/

US dt

dn


8 4

>

7.

REHMAT

ET


AL.

£s,max

_

f

τ

3νΐ(1-ε )ρ (1-αοΖ') Ί

2M

Ν

139

ξο,τηβχ

f

a)

B C

τ3

S

ζ s,min

'(ξ ,ξο)Ρΐ(ξο) e x p j f ^ 8

£o,min

( l- \ A \ (85)

u

o t,s

1

/ Downloaded by CORNELL UNIV on October 15, 2016 | http://pubs.acs.org Publication Date: September 21, 1981 | doi: 10.1021/bk-1981-0168.ch007

The boundary c o n d i t i o n f o r Eq. 85 is ω

Α

Η

m

l at n - 0

(86)

The s o l u t i o n of Eqs. 85 with 86 w i l l y i e l d the oxygen concentration p r o f i l e along the fluidized bed combustor. S i m i l a r l y the mass balance equation f o r s u l f u r dioxide is £s,max YCΑΠ

= 3V (l-e )p (l-a Z) 1

1

Q

1

p

e x

(

[

£s,min

I(£s>£o) l(£o)

5o,max

f

£o,min

°

j=t

P

dn

dt

T

3Vi(l-£i)P (l-q Z ) N

2

τ3 S M

£s,max f

ξ ο,max Γ

\ £o,max

\ £o,min

u

c

(87)

/S

4

">


140

CHEMICAL REACTORS The boundary c o n d i t i o n f o r Eq. 87 is given by

ω

= 0 at η = 0

Β Η

(88)

The s o l u t i o n of Eq. 87 with Eq. 88 w i l l e s t a b l i s h the s u l f u r dioxide concentration p r o f i l e along the fluidized bed combustor. There are two more q u a n t i t i e s that must be defined to complete the d e s c r i p t i o n of the fluidized bed combustor v i z . , the carbon combustion e f f i c i e n c y , nçcE» * s u l f u r absorption e f f i c i e n c y , n$AE are: a n c

e


F

l

f

c

n

c

C

E

T

n

e

s

e

£s,max £o,max = 3ν (1-ε )Ρ (1-α ) / / Ί

}

0

^ /

Ί Ζ

£s,min £o,min

I«..ç )Pl«o) 0

Up

expj|-(l-j-)j

1 dt

dn

ο and

" «

-

ι

-

c

^

g

The s u l f u r adsorption e f f i c i e n c y , sAE> is defined as the r a t i o of moles of s u l f u r d i o x i d e consumed by the s u l f a t i o n r e a c t i o n to the moles of s u l f u r dioxide produced due to char combustion. The above equations w i l l be u t i l i z e d to analyze the parametric s e n s i t i v i t y of the fluidized-bed combustion operation. Numerical C a l c u l a t i o n s :

Parametric I n v e s t i g a t i o n s

The mathematical model f o r char combustion described in the previous two s e c t i o n s is a p p l i c a b l e to a bed of constant volume, i . e . , to a fluidized bed of f i x e d height, H , and having a constant c r o s s - s e c t i o n a l area, AQ. The constant bed height is maintained by an overflow pipe. For this type of combustor operating f o r a given feed rate of char and limestone p a r t i c l e s of known s i z e d i s t r i b u t i o n s , the model presented here can p r e d i c t the f o l l o w i n g : (1) the f r a c t i o n of char p a r t i c l e s in the bed, f q ; (2) the f r a c t i o n of limestone p a r t i c l e s in the bed, (3) the size d i s t r i b u t i o n of char p a r t i c l e s in the bed or in the overflow, Ρ 3 ( ξ ) ; 0

8


7.

REHMAT ET AL.

141


(4) the s i z e d i s t r i b u t i o n of char p a r t i c l e s in the bed or in the overflow, Ρβίξ^); ( 5 ) the overflow r a t e of char p a r t i c l e s , F 3 ; ( 6 ) the overflow r a t e of limestone p a r t i c l e s , F 3 ; ( 7 ) the flow r a t e of char p a r t i c l e s in the carryover stream, F 4 ; (8) the p a r t i c l e s i z e d i s t r i b u t i o n in the carryover stream, (9) the flow r a t e of limestone p a r t i c l e s in the carryover stream, F 4 ; (10) the p a r t i c l e s i z e d i s t r i b u t i o n of limestone p a r t i c l e s in the carryover stream, Ρ4(ξ ); (11) the concentration p r o f i l e of oxygen along the fluidized bed, ω ; (12) the concentration p r o f i l e of s u l f u r d i o x i d e along the fluidized bed, ω ^ ; (13) the carbon conversion e f f i c i e n c y , ncCE» * (14) the s u l f u r absorption e f f i c i e n c y , nsAE' The above c a l c u l a t i o n is q u i t e tedious and gets complicated by the f a c t that the p r o p e r t i e s which u l t i m a t e l y c o n t r o l the magnitude of these fourteen unknown q u a n t i t i e s f u r t h e r depend on the p h y s i c a l and chemical parameters of the system such as r e a c t i o n r a t e constants, initial s i z e d i s t r i b u t i o n of the feed, bed temperature, e l u t r i a t i o n constants, heat and mass t r a n s f e r c o e f f i c i e n t s , p a r t i c l e growth f a c t o r s f o r char and limestone p a r t i c l e s , flow r a t e s of s o l i d and gaseous r e a c t a n t s . In a complete a n a l y s i s of a fluidized bed combustor with s u l f u r absorption by limestone, the i n f l u e n c e of all the above parameters must be evaluated to enable us to optimize the system. In the present r e p o r t we have l i m i t e d the scope of our c a l c u l a t i o n s by c o n s i d e r i n g only the initial s i z e of the limestone p a r t i c l e s and the r e a c t i o n r a t e constant f o r the s u l f a t i o n r e a c t i o n . F u r t h e r , it is not necessary to c a r r y out excessive c a l c u l a t i o n s to i n v e s t i g a t e the parametric s e n s i t i v i t y o f the combustor o p e r a t i o n . The same goal can be accomplished by assuming some of the fourteen unknowns and determining the remaining by the s o l u t i o n of the above mentioned equations. T h i s procedure is adopted here. We assume a form f o r the oxygen p r o f i l e , values of the carbon combustion and s u l f u r absorption e f f i c i e n c i e s , char feed r a t e , and the v a r i o u s constants of the system and then the framework of mathematical model is employed to evaluate the amount of dolomite, Fj_, needed to o b t a i n such an operation. In g e n e r a l , the f u n c t i o n a l form f o r oxygen p r o f i l e in a combustor is as f o l l o w s :


8

Α Η

a n c

ω

ΑΗ

- a* + (1 - a * ) e - * ^ b

(91)

Here the constants a* and b* are t o be s p e c i f i e d . The constant a* is g e n e r a l l y r e l a t e d to the excess oxygen in the f l u e gas whereas b* e s t a b l i s h e s the slope of the p r o f i l e . A l a r g e r value


142

CHEMICAL REACTORS

of b* s i g n i f i e s that a major degree of combustion r e a c t i o n s take place near the bottom of the r e a c t o r . The fluidized-bed c o a l combustion c a l c u l a t i o n s described below according to the mathematical model developed here w i l l employ the parameter values given in Table 1. The a n a l y s i s is confined to a f i x e d temperature of 1225 Κ f o r the bed. Figure 4 g i v e s the normalized p a r t i c l e s i z e d i s t r i b u t i o n f o r char and dolomite feeds f o r a p a r t i c u l a r case when r = r = O.04 cm. The p a r t i c l e s i z e d i s t r i b u t i o n f o r char in the feed is held constant, whereas f o r dolomite it is changed such that r v a r i e s from O.02 to O.08 cm. The r e a c t i o n r a t e between s u l f u r d i o x i d e and dolomite is changed by v a r y i n g the r e a c t i o n r a t e constant, k 3 ( T ) , from 380 to 960 cmVmol's. The changes in the dolomite s i z e and r e a c t i o n r a t e constant r e f l e c t in 3 and Φ351 as seen from Table 1. The v a r i a t i o n in the heat and mass t r a n s f e r c o e f f i c i e n t s due to change in p a r t i c l e s i z e w i t h i n the range considered here is found to be n e g l i g i b l e (±5%) and, t h e r e f o r e , is ignored in the present c a l c u l a t i o n s . In one of our t e s t runs the s u l f u r absorption e f f i c i e n c y is kept constant at O.99 while the carbon combustion e f f i c i e n c y is v a r i e d from O.7 to O.995. In all of the remaining runs, the carbon combustion e f f i c i e n c y is held constant at O.995 while the s u l f u r absorption e f f i c i e n c y is v a r i e d from O.7 to O.99. I t may be pointed out that the dolomite feed r a t e is d i r e c t l y p r o p o r t i o n a l to the rate of change of s i z e of the dolomite p a r t i c l e , Eq. 73, and consequently the dolomite r e q u i r e ment is s t r o n g l y dependent upon the f a c t o r s i n f l u e n c i n g the r a t e of change of the dolomite p a r t i c l e . Let us examine the e f f e c t of changing oxygen p r o f i l e on the dolomite requirement. f

f

Q


τ

a

n

d

a

r

e

x e

t

n

e

I f the values of ncCE ^SAE fi B

3

3

C

C

A0 AC> AS

3

C

3

c

C

c

Ae> Be AH> BH C

3

C

C

BC» BS


3

c

J0> S0» N0

c

c

D

A> B

D

eA> eB

D

D

E ,E ,E 1

2

3

F ,F3 3

F ,F4 4

f

c> *

f

f

Q> N

f

h,h

1

2

H «ο ΛΗ ,ΔΗ ,ΔΗ 1

2

k k

k'

^nA>^nB k

k

k

l» 2> 3

Mj,M ,M N

3

s

s

N*


7. N

REHMAT

Nu>^


ET AL.

u

= Nusselt Number f o r char and limestone par t i c l e s , 2Rh/k, 2 R h 7 k , dimensionless = N (r/2R)(k/k ), N^ (r/2R')(k/k »), dimensionless = Sherwood number f o r char and limestone par t i c l e s , 2 R k / D , 2 R k / D , dimensionless = N (f/2R)(D /D ), N» (f/2R')(D /D ), dimensionless = size frequency d i s t r i b u t i o n of char and l i m e stone feed, 1/m = size frequency d i s t r i b u t i o n of char and l i m e stone in the fluidized-bed, 1/m = size frequency d i s t r i b u t i o n of char and l i m e stone in the overflow, 1/m = value of Ρ 3 ( ξ ) corresponding to p a r t i c l e of size ξ in the feed, 1/m = size frequency d i s t r i b u t i o n of char and l i m e stone in the c a r r y o v e r , 1/m = char,— = radial p o s i t i o n in the limestone p a r t i c l e , m = radial p o s i t i o n in the limestone p a r t i c l e , m = average r a d i u s g e n e r a l , limestone defined by Eq. 5, m = r a d i u s of the i t h s i z e f r a c t i o n of feed, m = rate of formation of gas product B, mol/m^-s mol of s o l i d reactant = r a d i u s of the unreated core of the char and limestone p a r t i c l e s , m = rate of formation of gas product D, mol/m^-s mol of s o l i d reactant = instantaneous r a d i u s of char and limestone particles, m = r a d i u s of the l a r g e s t ( s m a l l e s t ) char p a r t i c l e in the feed, m = r a d i u s of the l a r g e s t ( s m a l l e s t ) char p a r t i c l e a f t e r complete r e a c t i o n , m = r a d i u s of the l a r g e s t ( s m a l l e s t ) limestone par t i c l e in the feed, m = r a d i u s of the l a r g e s t ( s m a l l e s t ) limestone par t i c l e a f t e r complete r e a c t i o n , m = gas constant, J/mol Κ = initial r a d i u s of the char and limestone par t i c l e s in the feed, m = time, s = temperature of the char p a r t i c l e a t r a d i u s r , limestone p a r t i c l e at r , Κ = temperature of the fluidized bed, Κ = temperature of the unreacted core surface of char and limestone, Κ = temperature of the outer surface of the char and limestone p a r t i c l e s , Κ f

N u

N

Sh>N£

h

e

u

e

,

mA

Nshr>^ ρ

h f

p

ΐ ( ξ ) , { ( ξ s) 8

p


?2(ξ s ) > 2 ^ s )

ρ

3(ζ )

s h

A

A

e A

m B

h

B

e

eB

8

8

0

4(ξ )» 4(ζ8)

ρ

ρ

8

Q r r f,r f

r

1

i B

r

r

r

c> c

f

R,R R

o , max ( o , min)

R

R

s,max( s,min)

R

o,max( o,min)

R

s,max ( s,min)

R

R

R

R R

R

o> ô

t T,T T

T

c» c

Τ 1

1

1

f

τ s» s 1


152

CHEMICAL REACTORS f

U,U

= reduced temperature of the char and limestone p a r t i c l e s , T / T , dimensionless = reduced core temperature of the char and l i m e stone p a r t i c l e s , T / T , dimensionless = reduced surface temperature of the char and l i m e stone p a r t i c l e s , T / T , T / T , dimensionless = s u p e r f i c i a l v e l o c i t y , m/s = volume of the fluidized bed, m volumetric feed, overflow and carryover r a t e s of char, m /s volumetric feed, overflow and carryover r a t e s of limestone, m /s = weight f r a c t i o n of p a r t i c l e s in the feed of r a d i u s , r ^ , dimensionless = weight of char and limestone p a r t i c l e s in the fluidized-bed, kg mole f r a c t i o n of components A and B, dimen sionless value of x a t the unreacted core surface, bottom of the bed and outer surface of the char p a r t i c l e , dimensionless = value of X A a t a height Η in the fluidized bed, dimensionless value of xg a t the unreacted core surface, outer surface of limestone p a r t i c l e and at height H in the bed, dimensionless = average r a t e of gas flow through the r e a c t o r bed, m /s = parameter d e f i n i n g p a r t i c l e growth or shrinkage of char and limestone defined by Eq. 4, dimen sionless 0

U

c

U

s

u V 1> 3> A

c

0

S

Q

s

0

3

V

V

V

=

V

, V

, V

=

3

1

3

4

3


wj[ WQ,W

N

X

A> B

X

x

AC> AO> AS

x

=

x

=

A

ΧΑΗ X

X

X

BC> BS> BH

=

Y

3

Ζ,Ζ

1

Greek L e t t e r s ε

ε

ε

> 1> 3>ε4

= average v o i d f r a c t i o n of the bed, char feed, overflow and c a r r y o v e r , dimensionless = average v o i d f r a c t i o n of limestone feed, dimensionless ξ,ξ = any reduced distance f o r char and limestone p a r t i c l e s , r / r and r / r , dimensionless ξ ,ξ = reduced unreacted core r a d i u s of char and l i m e stone p a r t i c l e s , τ/7 and r ' / F , dimensionless ξ >ξέ = reduced r a d i u s of the char and limestone par t i c l e s , R/r and R /F, dimensionless £o,max(£o,min) reduced r a d i u s of the l a r g e s t ( s m a l l e s t ) par t i c l e in the char feed, dimensionless £s,max(£s,min) " reduced r a d i u s of the l a r g e s t ( s m a l l e s t ) char p a r t i c l e a f t e r complete r e a c t i o n , dimensionless ξο,π^χίξο,πιΐη) reduced r a d i u s of the l a r g e s t (smallest) l i m e stone p a r t i c l e in the feed, dimensionless 1

Q

α

Q

α

8

f

=

=


7.

ξ

REHMAT

f

f

max(ξ min)

=

ξο»ξο

153


ET AL.

reduced r a d i u s o f the l a r g e s t ( s m a l l e s t ) l i m e stone p a r t i c l e a f t e r complete r e a c t i o n , dimen sionless = reduced value of R and R , R / r and R / f , dimensionless = d e n s i t y o f the s o l i d product ash and calcium s u l f a t e , kg/m = d e n s i t y o f the s o l i d r e a c t a n t J , and limestone N, kg/m = d e n s i t y o f char and s o l i d r e a c t a n t S , kg/m = a parameter to c h a r a c t e r i z e the r a t e s of i n t r a p a r t i c l e d i f f u s i o n r e s i s t a n c e to the r e a c t i o n residence f o r r e a c t i o n 1 = r * k i ( T ) C j / D ( T ) , dimensionless a parameter to c h a r a c t e r i z e the r a t e s of i n t r a p a r t i c l e d i f f u s i o n r e s i s t a n c e to the r e a c t i o n residence f o r r e a c t i o n 2 = r k 2 ( T ) C s o / D A ( o ) > dimensionless = a parameter to c h a r a c t e r i z e the r a t e s of i n t r a p a r t i c l e d i f f u s i o n r e s i s t a n c e to the r e a c t i o n residence f o r r e a c t i o n 3 = ï k 3 ( T ) C / D ( T ) , dimensionless reduced values of x , x , x ^ , X A / A O > A C / A0> A H / A O > dimensionless reduced value of x and x , X A S / A O > Bc/ A0> dimensionless reduced value of x , x , and x , Χ β θ / Α Ο > Β Η / Α Ο > B s / A O t dimensionless = e l u t r i a t i o n constant f o r char p a r t i c l e s of size ξ , 1/s = e l u t r i a t i o n constant f o r limestone p a r t i c l e s of s i z e ξ , 1/s = reduced bed h e i g h t , H / H , dimensionless = carbon conversion e f f i c i e n c y d e f i n e d by Eq. 89, dimensionless s u l f u r a b s o r p t i o n e f f i c i e n c y d e f i n e d by Eq. 90, dimensionless Q

p ,pg a

0

Q

Q

3

pj,pjj

3

3

pq,Pg φ If


0

β

2f

0

e A

0

T

0

φ3f

e

o

ω

ω

AÂC> ΑΗ

=

X

A

X

ω

ω

o

X

A S

X

x

x

B

=

χ

B C

χ

8

e A

X

A S

U B C ^ B H ^ B S

κρ(ξ )

x

=

Α8» Β

N 0

χ

x

B H

B S

x

8

κ^(ξ ) 8

8

η ncCE η sAE

0

=

Literature Cited 1. 2. 3.

K u n i i , D.; L e v e n s p i e l , O. " F l u i d i z a t i o n Engineering"; John Wiley: New York, 1969. Chen, T. P . ; Saxena, S. C. AIChE Sym. S e r i e s 1968, No. 176, 74, 149-161. Chen, T. P.; Saxena, S. C. F u e l , 1977, 56, 401-413.


154

4.

5.

6.

7.

CHEMICAL REACTORS

Saxena, S. C.; Chen, T. P.; Jonke, A. A. "A Slug Flow Model for Coal Combustion with S u l f u r Emission C o n t r o l by Limestone or Dolomite"; Presented at the 70th Annual AIChE Meeting, New York C i t y , 1977, Paper No. 104d. Rengarajan, P.; Krishnan, R; Wen, C. Y. "Simulation of F l u i d i z e d Bed Coal Combustors"; Report No. NASA CR-159529, February 1979, 199 pp. Becker, Η. Α.; Beer, J. M.; Gibbs, Β. M. "A Model f o r F l u i d i z e d - B e d Combustion of Coal"; Inst. of F u e l Symposium S e r i e s No. 1, Proc. F l u i d i z e d Combustion Conference 1, Paper No. AI, 1975, A1•1-A1•10. Horio, M; Mori, S.; Muchi, I . "A Model Study f o r the Develop ment of Low NO F l u i d i z e d - B e d Coal Combustors"; Proc. 5th Intern. Conf. on F l u i d i z e d - B e d Combustion, V o l . II, 1977, 605-624. Davidson, J. F.; H a r r i s o n , D. " F l u i d i z e d P a r t i c l e s " ; Cambridge U n i v e r s i t y Press: New York, 1963. Rehmat, Α.; Saxena, S. C. "Single Nonisothermal N o n c a t a l y t i c Gas-Solid Reaction. E f f e c t of Changing P a r t i c l e S i z e " ; Ind. Eng. Chem. Process Des. Dev. 1975, 16, 343-350. Rehmat, Α.; Saxena, S. C. " M u l t i p l e Nonisothermal Noncata lytic Gas-Solid Reaction. E f f e c t of Changing P a r t i c l e S i z e " ; Ind. Eng. Chem. Process Des. Dev. 1977, 16, 502-510. Rehmat, Α.; Saxena, S. C.; Land, R. H.; Jonke, A. A. "Nonc a t a l y t i c Gas-Solid Reaction with Changing P a r t i c l e S i z e : Unsteady State Heat T r a n s f e r " ; Canadian J. Chem. Eng. 1978, 56, 316-322. Yagi, S.; K u n i i , D. "Studies on Combuston of Carbon P a r t i c l e s in Flames and F u l i d i z e d Beds"; 5th Symposium (Int'l) on Combustion, 1955, 231-244. Burovoi, I . Α.; E l i a s h b e r g , V. M.; D'yachko, A. G.; Bryukvin, V. A. "Mathematical Models f o r Thermochemical Processes Occurring in F l u i d i z e d Beds"; I n t . Chem. Eng. 1962, 2, 262-258. S t r e l ' t s o v , V. V. "Approximate R e l a t i o n s h i p s f o r C a l c u l a t i n g the K i n e t i c s of Reactions of a S o l i d Phase in a F l u i d i z e d Bed"; I n t . Chem. Eng. 1969, 9, 511-513. B e t h e l l , F. B.; Gill, D. W.; Morgan, Β. B. "Mathematical Modelling of the Limestone-Sulfur Dioxide Reaction in a F l u i d i z e d - B e d Combustor"; F u e l 1973, 52, 121-127. Koppel, L. "A Model f o r P r e d i c t i n g the Extent of Reaction of Limestone and S u l f u r Dioxide During Fluidized-Bed Combustion of Coal"; Appendix C, pp. 60-77, in Jonke, A. A. "Reduction of Atmospheric P o l l u t i o n by the A p p l i c a t i o n of F l u i d i z e d Bed Combustion"; Argonne National Laboratory Annual Report, ANL/ES-CEN-1002, J u l y 1969-June 1970. Avedesian, M. M.; Davidson, J. F. "Combustion of Carbon P a r t i c l e s in a F l u i d i z e d Bed"; Trans. I n s t . Chem. Engrs. 1973, 51, 121-131.


X

8. 9.

10.

11.

12.

13.

14.

15.

16.

17.


7.

18.

19.

20.


21. 22.

23.

24. 25.

26.

REHMAT

ET

AL.


155

Evans, J. W.; Song, S. " A p p l i c a t i o n of a Porous P e l l e t Model to F i x e d , Moving, and F l u i d i z e d Bed Gas-Solid Reactors"; Ind. Eng. Chem. Process Des. Dev. 1974, 13, 146-152. Campbell, Ε. K.; Davidson, J. F. "The Combustion of Coal in F l u i d i z e d Beds"; Inst. of F u e l Sym. S e r i e s No. 1, Proc. F l u i d i z e d Combustion Conference 1, Paper No. A2, 1975, Α 2 • 1 A2•9. Gibbs, Β. M.; "A Mechanistic Model f o r P r e d i c t i n g the P e r f o r mance of a F l u i d i z e d Bed Coal Combustor"; Inst. of Fuel Sym. S e r i e s No. 1, Proc. F l u i d i z e d Combustion Conference Paper No. A5, 1975, Α5•1-Α5•10. Beer, J. M.; "The F l u i d i z e d Combustion of Coal"; XVIth Sym. (Int'l) on Combustion, 1976, 439-460. Hovmand, S.; Davidson, J. F. Chapter 5 in " F l u i d i z a t i o n " ; E d i t o r s J. F. Davidson and D. H a r r i s o n ; Academic Press: London, 1971. Horio, M.; Wen, C. Y. "Simulation of F l u i d i z e d Bed Com bustors: Part I. Combustion E f f i c i e n c y and Temperature Profile"; AIChE Symp. S e r i e s 1978, No. 176, V o l . 74, 101-111. Mori, S.; Wen, C. Y. "Estimation of Bubble Diameter in Gaseous F l u i d i z e d Beds"; AIChE J 1975, 21, 109-115. Gibbs, B. M.; P e r e i r a , F. J.; Beer, J. M. "Coal Combustion and NO Formation in an Experimental F l u i d i z e d Bed"; I n s t i t u t e of F u e l Symp. S e r i e s No. 1, Proc. F l u i d i z e d Combustion Con ference 1, Paper No. D6, 1975, D6•1-D6•13. Rehmat, Α.; Saxena, S. C.; Land, R. H. " A p p l i c a t i o n of Nonc a t a l y t i c Gas-Solid Reactoins f o r a Single P e l l e t of Changing Size to the Modeling of F l u i d i z e d - B e d Combustion of Coal Char Containing S u l f u r " ; Argonne N a t i o n a l Laboratory Report, ANL/ CEN/FE-80-13, September 1980, 86 pp.

RECEIVED

July 15, 1981.


Chemical Reactors - American Chemical Society

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