r&k assigment 2

7/23/2019 R&K Assigment 2

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CH 3681 - Reactors and Kinetics

Assignment 2

Mohan Kumar Prakash (44212!

Pedro Henri"ue Magacho dePau#a (4$%4$34!

Part A: Analytical Modelling

1. Concentration profile of O2, inside and outside the tissue:

2. Transient Mass balance for the diffusion – reaction process:

Inside the tissue –

Mass &a#ance 'or 2 (A! )-

Accumu#ation * +n#et , ut#et Production

A

A A r x

j

t

C +

∂

∂−=

∂

∂

(1)

.A , di''usion '#u/ o' 2 through the tissue0

Hence e"uation 1 &ecomes & su&stituting icks 1st #a5 o' di''usion -



A

A A r x

C D

t

C +

∂

∂=

∂

∂2

2

he consum7tion o' o/gen & the tissues is considered as a 'irst order irreersi&#e reaction 5ith rate

constant

V k

0

AV A C k r −=

9o the 'ina# transient mass &a#ance e"uation 'or 2 is as 'o##o5s)

AV

A A C k

x

C D

t

C −

∂

∂=

∂

∂2

2

(2)

Boundary Conditions:

(1!

At x=0,

%=dx

dC A

:ue to smmetr at the midd#e CA 5i## &e at its minimum at this 7oint0 9mmetr is

considered &ecause o' the 'act that o/gen di''uses through the tissue 'rom &oth sides o' the tissue0

(2!

At x=L,

!!(( LC C k dx

dC D bext

A −=

(3)

: , :i''usion coe''icient inside the tissue

k ext – Mass trans'er coe''icient outside the tissue

bC

- Concentration o' 2 in the &u#k 7hase outside the tissue

C(L) - Concentration o' 2 at the e/terna# &oundar o' the tissue / * ;

3. Dimensionless form of the diffusion – reaction equations :

+nde7endent aria&#es) / t

:e7endent <aria&#es) CA



he 'o##o5ing are the sca#ed e"uations 'or these aria&#es each containing their res7ectie sca#ed

'actors

/ * =/> ?

t * =t> @

CA * =CA> Cmin

Cmin is considered ero &ased on the statement in "uestion that there 5as no 2 initia## in the tissue0

And moreoer considering Cmin as ero a#so sim7#i'ies the resu#tant di''erentia# e"uation0

A good guess 'or the sca#ed 'actors o' CA and / 5ou#d &e C & (&u#k concentration! and ; (ha#'-5idth!

res7ectie#0

9u&stituting the a&oe e"uations in the transient mass &a#ance e"uation resu#ts in the 'o##o5ing ,

θ η θ

τ θ

bV bb C k

LC D

t C −

∂∂=

∂∂

22

2

>=

B7on rearranging

>=>=

22

2

t k L

t D V θ

η

θ

τ

θ −

∂

∂=

∂

∂

Considering time sca#ed 'actor =t> * ;2:

θ η

θ

τ

θ

−

∂

∂=

∂

∂

D

Lk V 2

2

2

+n the a&oe e"uation k ;2: * 2 5here is ca##ed the hie#e modu#us0ϕ ϕ

θ φ η

θ

τ

θ 2

2

2

−∂

∂=

∂

∂

(4)

:imension#ess &oundar conditions ,

10 At / * %D that is ? * %

%==η

θ

d

d

dx

dC A



20 At / * ;D that is ? * 1

!>=(>=

>=θ

Abext

A A C C k dx

dC

x

C D −=

9u&stituting =CA> * C& =/> * ; and rearranging

>1= θ η

θ −

=

D

Lk

d

d ext

Here k e/t;: * Ei 5here Ei (Eiots num&er! is a dimension#ess num&er0

%=−+ Bi Bid

d θ η

θ

(5)

. !tead" state condition :

θ φ η

θ 2

2

2

% −=d

d

here'ore

θ φ η

θ 2

2

2

=d

d

Upon solving, the above equation yields the folloing solution,

nneC eC

φ φ

θ

−+= 21

he t5o &oundar conditions 5ere used to o&tain the a#ue o' &oth the constants C1 and C2 as

'o##o5s)

+

==

φ φ φ BiCoshSinh

BiC C

1

221



+=

φ φ φ

φ θ

BiCoshSinh

nCosh Bi

9u&stituting the a#ues o' C1 and C2 ie#ds the 'ina# so#ution

o' F

(!)

Part B: Analytical Modelling – MATLAB (Steady State)

1. #lot of dimensionless concentration $s tissue depth for $ar"in% and &i ϕ

2. Discussion of the abo$e plots

Figure 2 sho5s concentration 7ro'i#e 'or arious a#ues o' hie#e modu#us and Eiot num&er0 Ghen

the hie#e modu#us is #arge interna# di''usion rate is higher com7ared to the corres7onding rate o' ϕ

reaction and hence di''usion 5i## dominate the 7rocess0 9ma##er a#ues o' hie#e modu#us indicateϕ

the o77osite and hence di''usion 5i## dominate the 7rocess0 here'ore 5e can in'er 'rom the a&oe



gra7hs that the concentration dro7s to its minimum 'aster (c#oser to the &oundar ?*1! as increasesϕ

due to increasing dominance o' reaction oer di''usion0 he 7hsica# meaning is that 5ith increasing

2 gets consumed 'aster and 'aster com7ared to the rate at 5hich it di''uses through the tissue0ϕ

or #arge Eiot num&ers the characteristic time 'or interna# di''usion is #arge com7ared to e/terna# mass

trans'er and hence interna# di''usion dominates0 igure 1 re'#ects on this conce7t as the

concentrations &are# change oer the tissue 'or #o5 Eiot num&ers meaning that the interna# di''usion

is not im7ortant com7ared to the e/terna# mass trans'er0 B7on increasing the Eiot num&er it is

7ossi&#e to see that the interna# di''usion &ecomes more and more im7ortant as in genera# the

concentration gradient inside the tissue &ecomes #arger com7ared to #o5er Eiot num&ers0

3. 'o e(ternal mass transfer

or this case e"uation 6 can &e re-5ritten as

+=

φ φ φ

φ θ

Cosh

Bi

Sinh

nCosh

=

φ

φ θ

Cosh

nCosh

Ghen 5e consider there is no e/terna# mass trans'er resistance (Ei " ! the

a&oe e"uation sim7#i'ies to the 'o##o5ing 'orm

(I!

. )alf *idth of the tissue



A77# one &oundar condition to get the a#ue o' at 5hich the minimum concentration o' C is 1%Jϕ

o' C&0

B.C:

At ? * % C * %01C& (to &e maintained! " F * %01

"uation I &ecomes

φ Cosh

Cosh !%(10% =

here'ore * Cosh-1(1%! * 20 L 30ϕ

As 5e kno5 k ;2: * 2 5e can 'ind the a#ue o' ; 'orϕ k * %0%3 s-1 and : * 1%-$ cm2s0

ck

D L

V

%$40%%30%

1%M $2

===−φ

At ha#' 5idth (;! * %0%$$cm a## the ce##s in the tissue 5i## hae the minimum 2 concentration

re"uired 5hen * 30ϕ

+. Dimensionless Concentration profile as a function of , at - ϕ



Figure 3 sho5s the dimension#ess concentration 7ro'i#e as 'unction o' at the center o' the tissueϕ

(?*%!0 rom the gra7h it is eident that the dimension#ess o/gen concentration at the center is %01 at

e"ua# toϕ three0 his matches 5ith the ca#cu#ated 'rom the ana#tica# so#ution0 Hence 5e can saϕ

that &oth ana#tica# and numerica# a77roach ie#d the same resu#t0

Part C: Nuerical Modelling – MATLAB (Tran!ient)

1. &oundar" conditions *ith no e(ternal mass transfer resistance

At / * %

%=dx

dC A

#his bounda$y %ondition is un%hanged as the sy&&et$y at the %ent$e is

una'e%ted by the va$iation in ete$nal &ass t$ansfe$ $esistan%e

*t + L

!!(( LC C k dx

dC D bext

A −=

No e/terna# mass trans'er resistance means et , substituting this value in above

equation and $ea$$anging,

%!!(( =−=∂

∂ LC C

x

C

k

Db

A

ext

here'ore C& * C (;! or F * 1 is the ne5 &oundar condition at / * ; 5hen there is no e/terna# mass

trans'er resistance0

2. Dimensionless concentration as function of / at 0 - and 2ϕ



Figure " sho5s the ariation in concentration 5ith time at t5o 'i/ed 7ositions in the tissue * % and

* ;20 rom the gra7h it is eident that the reaction o' 2 is accom7anied & corres7onding

di''usion o' 2 into the tissue0 At t*% the concentration is ero at &oth the #ocations 5hich suggest

that 2 has not di''used into these regions et0 As time 7rogresses 5e can see &ui#d-u7 o' 2 at these

t5o #ocations (due to di''usion! 5hich eentua## reach a stead state0 ne can a#so o&sere that the

concentration o' 2 at * ;2 is genera## higher than at * % 5hich is due to the concentration

gradient that arises due to di''usion0

3. Comparison of numerical and anal"tical solution

he ana#tica# so#ution o&tained in E-3 is 'or the stead state case0 Hence the dimension#ess

concentration F o&tained 'or ? * % ( * %! and ? * %0$ ( * ;2! 'rom e"uation (I! 5i## &e the steadstate concentrations at these t5o #ocations0

At ? * % O * 3 F * Cosh (%! Cosh (3! * %0% L %01

At ? * %0$ O * 3 F * Cosh (10$! Cosh (3! * %02336 L %024

+n 'igure 4 5e can o&sere that the concentration at ? * % ( * %! and ? * %0$ ( * ;2! reaches a

stead state at F * %01 and %024 a77ro/imate#0 rom this com7arison 5e can a''irm that the

numerica# so#ution 'or the transient state is consistent 5ith the ana#tica# so#ution 'or the stead state0



. ill all the cells sur$i$e4

rom 'igure 4 it is c#ear that 2 concentration at the centre ( * %! 5i## a#5as &e #o5er than at *

;20 Hence our 'ocus 5i## &e at the centre ( * %!0 ien the transient nature o' the 7rocess 5e need

to ca#cu#ate the time re"uired 'or the 2 concentration to rise to 1%J C & (min re"uired! at the centre (

* %!0 +' the time taken is #ess than $ minutes 5e can con'irm that a## the ce##s 5i## surie0

rom 'igure 4 5e can see that the concentration F gets c#oser to %01 as / 10 ;ets consider the

5orst case (@ * 1!0

Ge kno5

t * =t> @

=t> * ;2:0

here'ore t * (;2:! @

9u&stitute ; * %0%$4 cm : * 1%-$ cm2s and @ * 1 (5orst case! into the 7reious e"uation 5e get

t * 2106s * 4086 min0

rom this 5e can in'er that the time re"uired 'or 2 to &ui#d u7 to 1%J C & at the centre is #ess than $

min0 9o the ce##s at * % 5i## surie as it gets its minimum 2 concentration &e'ore $ min0

Since# t$e cell! at t$e centre can !ur%i%e& it can 'e !aely con!idered t$at all t$e cell! in t$e ti!!ue

ill !ur%i%e.

r&k assigment 2

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