Bioprocess Assignment Derive Equations

QUESTION

Develop the design equation and mass balance for batch reactor, fed batch reactor

(FBR), Continuous Stirred Tank Reactor (CSTR), Plug Flow Reactor (PFR) with respect

cell or substrate.

Solution:

Introduction to Bioreactor

From kinetic studies, we arrive at rate equation which tells us how fast a

reaction proceeds by converting the reactants to produce the products, and the

dependence of the rate of reaction on the concentration of the reactants. A differential

equation for rate expression can be expressed as,

ri=1V ( dN i

dt )Where, Ni is number of moles of component i, dN i/dt indicates the rate of change of N i

with time, in a reactor of volume V. Therefore, r i will have units of mol/(unit vol.)/(unit

time).

The general mass balance equation for reactor design equation is as shown below,

(Componentinlet flow )=(componentoutlet flow )+( disappearanceofcomponent by reaction)+(accumulationof component¿ the reactor volume )

If any component is forming in the reactor, it is taken place of the second term

in the RHS with a negative sign. Let us consider an ideal design of bioreactor for figure

below,

Bioreactor

Product leaves

Unreacted reactant leaves

Reactant enters

Reactant accumulates Reactant disappears by reaction

Therefore, the general material balance in bioreactor will be,

(reactant flowrate∈inlet )=(reactant flowrate∈outlet )+(rate of loss of reactantdue ¿chemical reaction¿)+(rate of accumulationof reactant )

(a) Design Equation For Batch Reactor

For a batch reactor, there is no in flow and out flow which means the reactant flow for

inlet and outlet will be zero. Whatever is produced is accumulated inside the reactor at

constant volume.


−(rate of disappear anceofAby reaction )=(rate of accumulation ofA∈reactor )(1)NA0 = Initial number of moles of A or substract

NA = Number of moles of A present at time, t

V = Volume of the reactor

CA0 = Initial concentration of A or substract

CA = Concentration of A at time, t

rA = Reaction rate of disappearance of A

XA = Conversion factor

Page | 1

Reactant accumulates Reactant disappears by reaction

Batch Bioreactor

Reactant accumulates of A Reactant disappears of A by reaction

Therefore,

X A=CA 0−C A

C A0

C A=C A0(1−X A) (2)

N A=N A0(1−X A) (3)

r A=1V ( dN A

dt )(4)By rearrange equation (4), equation (1) become,

(−r¿¿ A)V=−( dN A

dt )(5)¿By substituting equation (3) to equation (5),

−N A 0¿¿

N A 0( dX A

dt )=(−r¿¿A)V (6)¿

Where, (- rA) indicates the rate of conversion or depletion of A. Equation (6) can

be called as the design equation in differential form for a batch reactor. However, we

can write the design equation in the form of integral form,

t=N A 0∫0

X A dX A

(−r A )V(7)

Equation (7) had shown the time that required achieving the conversion XA. If

the density of the fluid remains constant, or the volume is constant, it can be taken

out of the integral sign, therefore, equation (7) will become equation (8)

t=C A0∫0

X A dX A

(−r A )V=−∫

C A0

CA dC A

(−rA )(8)

Equation (8) had shown the time required for a certain amount of conversion to

take place. Normally, in a batch reactor, the reaction proceeds as time increases, or

until the equilibrium had reached.

Page | 2

(b) Design Equation For Fed Batch Reactor

For a batch reactor, there is only an inflow but no outflow which means the reactant

flow for outlet will be zero. Whatever is produced is accumulated inside the reactor at

changing of volume, this leading to a change in reaction volume (V) with time. The

feed flow is intermittent leading to fluctuating concentration inside the vessel. The

mass balance equation is given as


(Componentinlet flow )=( disappearance ofcomponent by reaction)+(accumulationof component¿ the reactor volume )

(Rate of accumulationofcomponent )=(Rateof Componentinlet flow )−(Rate of disappearance ofcomponent byreaction )(1)Since, volume is changing with time, therefore we include it in the rate of

accumulation of component.

( dSVdt )=S ( dVdt )+V ( dSdt )❑(2)

V = Volume of the reactor

Si = Inlet concentration

Page | 3

Fed Batch Bioreactor

Reactant accumulates Reactant disappears of subtract by reaction

Substrate Inlet, Fi , Si

Volume, VSubstrate, S

Product, PCells, X

S = Concentration of A at time, t

rs = Reaction rate of disappearance of A

Fi = Inlet Flowrate

Therefore,

r s=1V ( dN s

dt )(3)By rearrange equation (3),

(−r¿¿ A)=−( dSdt )(4)¿Multiply V on both side since reaction volume,

(−r¿¿ s)V=−V ( dSdt )(4)¿Next, for the inlet flow,

( dVdt )=Fi

(7)

Multiply S on both side of the equation,

S( dVdt )=Si Fi

(8)

Substitute equation (8) and equation (4) into equation (2),

( dSVdt )=S ( dVdt )+V ( dSdt )=FiS i−r SV (9)

Equation (9) is the general substrate balance for the fed batch reactor. Consider

the rate of cell growth with a single limiting substrate given by (Bailey and Ollis, 1986;

Fogler,2006),

r g=μX (10)

Where rg is the rate of cell growth per unit volume, X is the cell mass, and µ is

the specific growth rate, which is well described by Monod equation,

μ=μmaxS

K s+S(11)

Page | 4

The rate of product formation per unit volume rp can be expressed as,

r p=Y PX

rg (11)

Where the product Yield coefficient YP/X is defined as,

Y PX

= massof product formedmassof newcells formed

(12 )

Therefore, the individual component balance will be,

Cells :d (XV )dt

=Vrg

(13)

Product :d (PV )dt

=VrP

(14)

Substrate :d (SV )dt

=F iSi

− 1Y X /S

Vrg(15)

Therefore, equation (15) is the design equation for fed batch reactor in the differential

form.

Page | 5

(c) Design Equation For Continuous Stirred Tank Reactor (CSTR)

CSTR is one of the flow reactors. There is continuous inflow of reactants and

continuous outflow of products after reaction is over. The flow reaction system calls for

the need for defining term of space time and space velocity. Reaction time, τ in CSTR

is the time required to process one reactor volume of feed measured at specific

conditions.

τ=Vv= volumeof reactorvolumetric feed rate

τ=S iV

F i

Where, Fi is the molar feed rate of substrate. Space velocity can be defined as

the number of reactor volumes of feed at specified condition which can be treated in a

unit time.

SpaceVelocity=s=1τ

For a CSTR, we assumed that inlet and outlet flow rates are equal (F). Hence the

reactant volume is constant. The concentration of the outgoing fluid (S) is assumed to

be that of the concentration of the fluid inside the vessel. There will not be any

accumulation of reactants or products in the continuous stirred tank reactor.

Therefore, the fourth term in the general mass balance equation will be zero.

Page | 6Continuous Stirred

Tank Reactor Bioreactor

Substrate Inlet, Fi , Si Volume, V

Substrate, SProduct, P

Cells, X

Outlet, F, S

Where, V = Volume of the reactor


S = Outlet Concentration

Fi = Inlet Flowrate

F = Outlet Flowrate

Xs = Conversion for the substrate



(Compone ntinlet flow )=(componentoutlet flow )+( disappearance ofcomponent by reaction)(1)

Therefore, equation (1) become,

F i=F+F i X s(2)

F=F i (1−X s )(3)

Conversion of substrate by reaction (in mol/time)

F=(−rA )V (4 )

Thus,

F iX s=(−r A )V (5)

or

VF i

=X s

(−r¿¿ s)¿

τ=Vv=SiV

Fi=S i

X s

(−r¿¿ s )=tm(6)¿

Where, tm is average residence time in a fully mixed flow reactor. In the above

equation, generally, rs stands for reaction rate at exit conditions, since the exit

Page | 7

Reactant disappears of subtract by reaction

composition from the CSTR is the same as that in the reactor. (S i – S) is also called

conversion. The reaction residence time is given as,

τCSTR=VF

=Si−S

(−r¿¿ s)(7)¿

or

V=Fi X s

(−r¿¿ s )exit(8)¿

The flow rate F can be estimated from the annual production, and the reactor

volume can be determined from above equation for a proposed conversion. Equation

(7) or (8) is considered as the design equation for the CSTR.

(d) Design Equation For Plug-Flow Reactor (PFR)

PFR is also one of the flow reactors. There is continuous inflow of reactants and

continuous outflow of products after reaction is over. The flow reaction system calls for

the need for defining term of space time and space velocity. Reaction time, τ in PFR is

the time required to process one reactor volume of feed measured at specific

conditions.

τ=Vv= volumeof reactorvolumetric feed rate

τ=S iV

F i

Where, Fi is the molar feed rate of substrate. Space velocity can be defined as

the number of reactor volumes of feed at specified condition which can be treated in a

unit time.

SpaceVelocity=s=1τ

The plug flow reactor or tubular flow reactor as in figure below with the notation

of flow parameters into and out of a small differential element of PFR.

Page | 8

PFR Bioreactor

Substrate Inlet, Si, F , Xs

Volume, VSubstrate, S

Product, PCells, X

Outlet Flow, Si + dSi, F + dF , Xs + dXs

Where, V = Volume of the reactor


S = Outlet Concentration

Fi = Inlet Flowrate

F = Outlet Flowrate

Xs = Conversion for the substrate


(Componentinlet flow )=(componentoutlet flow )+( disappearanceofcomponent by reaction)+(accumulationof component¿ the reactor volume )(1)

Let us consider first three terms in above equation (1) for the element volume of

reactor, dV.

Molar flow rate of reactant into the element = F

Molar flow rate of reactant out of the element = F + dF

Disappearance of reactant in the element by reaction = (-rs)(dV)

Therefore, equation (1) become,


(Compone ntinlet flow )=(componentoutlet flow )+( disappearance ofcomponent by reaction)(2)

F=(F+dF )+(−r¿¿ s)(dV )(3)¿

On simplification, we get,

−dF=(−r¿¿ s)(dV )(4)¿

dV= −dF(−r¿¿ s)(5)¿

τ=S iV

F i

(6)

From equation (6), we have,

Page | 9

VF i

= τS i

∨VF

= τS

(7)VF

=Vv1S(8)

Or

vS=F (9)

From equation (5),

dV=−v dS(−r¿¿ s)(10)¿

On integration, it yields,

V=−v∫Si

SdS

(−r¿¿ s)(11)¿

Or

Vv

=τ=−∫Si

SdS

(−r¿¿ s)(12)¿

Conversion,

X s=S i−SSi

=1− SSi

(13)

Or

1−X s=SSi

(14)

Which on differentiation will gives,

dS=Si(−d X s)

Hence, the above equation (12), on replacement, become,

τS i

=∫0

X s d X s

(−r¿¿ s)(15)¿

Or

τ=S i∫0

X s d X s

(−r¿¿ s )(16)¿

We can write the residence time in the form of concentration,

Page | 10

τ PFR=VF

=∫Si

SdS−rs

(17)

By combining equation (7) and equation (15), yields,

V=F i∫0

X s d X s

(−r¿¿ s )(18)¿

Equation (18) and equation (16) or equation (17) is the design equation for plug

flow reactor (PFR).

Page | 11

Bioprocess Assignment Derive Equations

Documents

Transcript of Bioprocess Assignment Derive Equations