Conversion and reactor sizing

Chemical Reaction Engineering

Is this a reactor

Review - general mole balance

  • General form: Fj0−Fj+Gj=dNjdtF_{j0} - F_j + G_j = \frac{dN_j}{dt}Fj0​−Fj​+Gj​=dtdNj​​

  • Uniform generation Fj0−Fj+rjV=dNjdt F_{j0} - F_j + r_j V = \frac{dN_j}{dt} Fj0​−Fj​+rj​V=dtdNj​​

  • Non-uniform generation Fj0−Fj+∫VrjdV=dNjdt F_{j0} - F_j + \int^V r_j dV = \frac{dN_j}{dt} Fj0​−Fj​+∫Vrj​dV=dtdNj​​

Reactor mole balance

Reactor Differential form Algebraic form Integral form
Batch dNAdt=rAV \frac{dN_A}{dt} = r_AV dtdNA​​=rA​V t=∫NA0NAdNArAV t = \int_{N_{A0}}^{N_A} \frac{dN_A}{r_A V}t=∫NA0​NA​​rA​VdNA​​
CSTR V=FA0−FA−rAV = \frac{F_{A0} - F_A}{-r_A} V=−rA​FA0​−FA​​
PFR dFAdV=rA \frac{dF_A}{dV} = r_A dVdFA​​=rA​ V=∫FA0FAdFArA V = \int_{F_{A0}}^{F_A} \frac{dF_A}{r_A}V=∫FA0​FA​​rA​dFA​​
PBR dFAdW=rA′ \frac{dF_A}{dW} = r'_A dWdFA​​=rA′​ W=∫FA0FAdNArA′ W = \int_{F_{A0}}^{F_A} \frac{dN_A}{r'_A}W=∫FA0​FA​​rA′​dNA​​

Conversion, XXX

  • Conversion is convenient for relating: rj,V,υ,Nj,Fj,and Cjr_j, V, \upsilon, N_j, F_j, \text{and } C_jrj​,V,υ,Nj​,Fj​,and Cj​

  • Consider the generic reaction

    a A+b B→c C+d D \ce{a A + b B -> c C + d D}aA+bB​cC+dD

  • Choose limiting reactant A as basis of calculation and normalize

    A+ba B→ca C+da D \ce{ A + \frac{b}{a} B -> \frac{c}{a} C + \frac{d}{a} D}A+ab​ B​ac​ C+ad​ D

  • Conversion (X): The fraction or percentage of a reactant that is consumed during a chemical reaction.

    X=moles of A reactedmoles of A fed X = \frac{\text{moles of A reacted}}{\text{moles of A fed}} X=moles of A fedmoles of A reacted​

  • Batch system: “Moles A fed” is the amount of A at the start of the reactor (t=0) Flow system: “Moles A fed” is the amount of A entering the reactor

Conversion, XXX

Consider A+2 B→2 C\ce{A + 2B -> 2C}A+2B​2C, Start with 1 mole of A, and 1 mole of B

A is the basis:

At the end we have

  • 1 mole A, 1 mole B

    XA=0/1=0X_A = 0/1 = 0XA​=0/1=0 ⇒no reaction\Rightarrow \text{no reaction}⇒no reaction

  • 12\frac{1}{2}21​ mole A, 0 mole B

    XA=0.5/1=1/2(0.5)X_A = 0.5/1 = 1/2 (0.5)XA​=0.5/1=1/2(0.5)

  • 0 mole A, -1 mole B

    XA=1/1=1X_A = 1/1 = 1XA​=1/1=1 ⇒complete reaction, but not possible\Rightarrow \text{complete reaction, but not possible}⇒complete reaction, but not possible

B is the basis:

At the end we have

  • 1 mole A, 1 mole B

    XB=0/1=0X_B = 0/1 = 0XB​=0/1=0 ⇒no reaction\Rightarrow \text{no reaction}⇒no reaction

  • 12\frac{1}{2}21​ mole A, 0 mole B

    XB=1/1=1X_B = 1/1 = 1XB​=1/1=1 ⇒complete reaction\Rightarrow \text{complete reaction}⇒complete reaction

Pick the limiting reagent as the basis to calculate conversion

Moles and molar flow rate in terms of XAX_AXA​

A+ba B→ca C+da DX=moles of A reactedmoles of A fed \ce{ A + \frac{b}{a} B -> \frac{c}{a} C + \frac{d}{a} D} \qquad \qquad X = \frac{\text{moles of A reacted}}{\text{moles of A fed}} A+ab​ B​ac​ C+ad​ DX=moles of A fedmoles of A reacted​

  • Batch systems: Longer reactant is in reactor, more reactant is converted to product (until reactant is consumed or the reaction reaches equilibrium)

    NA=NA0−NA0XAN_A = N_{A0} - N_{A0} X_ANA​=NA0​−NA0​XA​

    NA=NA0(1−XA)N_A = N_{A0}( 1 - X_A)NA​=NA0​(1−XA​)

  • Flow systems: For a given flow rate, the larger the reactor, the more time it takes the reactant to pass through the reactor, the more time to react

    FA=FA0−FA0XAF_A = F_{A0} - F_{A0} X_AFA​=FA0​−FA0​XA​

    FA=FA0(1−XA)F_A = F_{A0}( 1 - X_A)FA​=FA0​(1−XA​)

Design equation in terms of XXX: Batch reactor

  • Ideal batch reactor design equation

    dNAdt=rAV \frac{dN_A}{dt}=r_A V dtdNA​​=rA​V

  • NA=NA0(1−XA)N_A = N_{A0}( 1 - X_A)NA​=NA0​(1−XA​)

  • Taking the derivative of NAN_ANA​ equation

    ddtNA=ddt(NA0(1−XA)) \frac{d}{dt}N_A = \frac{d}{dt}\left( N_{A0}(1 - X_A)\right) dtd​NA​=dtd​(NA0​(1−XA​))

    dNAdt=0−NA0dXAdt \frac{dN_A}{dt} = 0 - N_{A0}\frac{dX_A}{dt} dtdNA​​=0−NA0​dtdXA​​

  • Substituting

    NA0dXAdt=−rAV N_{A0} \frac{dX_A}{dt} = -r_A V NA0​dtdXA​​=−rA​V

t=NA0∫0XAdXA−rAV t = N_{A0} \int_0^{X_A}\frac{dX_A}{-r_A V} t=NA0​∫0XA​​−rA​VdXA​​

Design equation in terms of XXX: CSTR

  • Ideal steady state CSTR design equation

    V=FA0−FA−rA V = \frac{F_{A0}-F_A}{-r_A} V=−rA​FA0​−FA​​

  • Substitute for FAF_AFA​

    FA=FA0(1−XA)F_A = F_{A0}( 1 - X_A)FA​=FA0​(1−XA​)

    V=FA0−[FA0−FA0XA]−rA V = \frac{\cancel{F_{A0}} - \left[ \cancel{F_{A0}} - F_{A0} X_A \right]}{-r_A} V=−rA​FA0​​−[FA0​​−FA0​XA​]​

  • V=FA0XA−rA V = \frac{F_{A0} X_A}{-r_A} V=−rA​FA0​XA​​

VVV is the CSTR volume required to achieve a specified conversion. XAX_AXA​ and –rA–r_A–rA​ are evaluated at the exit of the CSTR

Design equation in terms of XXX: PFR

  • Ideal steady state PFR design equation

    dFAdV=rA \frac{dF_A}{dV} =r_A dVdFA​​=rA​

  • FA=FA0(1−XA)F_A = F_{A0}( 1 - X_A)FA​=FA0​(1−XA​)

  • Taking the derivative of FAF_AFA​ equation

    ddVFA=ddV(FA0(1−XA)) \frac{d}{dV}F_A = \frac{d}{dV}\left( F_{A0}(1 - X_A)\right) dVd​FA​=dVd​(FA0​(1−XA​))

    dFAdV=0−FA0dXAdV⇒dFAdV=−FA0dXAdV \frac{dF_A}{dV} = 0 - F_{A0}\frac{dX_A}{dV} \qquad \Rightarrow \qquad \frac{dF_A}{dV} = - F_{A0}\frac{dX_A}{dV} dVdFA​​=0−FA0​dVdXA​​⇒dVdFA​​=−FA0​dVdXA​​

  • Substituting

    FA0dXAdV=−rA⇒V=FA0∫0XAdXA−rA F_{A0} \frac{dX_A}{dV} = -r_A \qquad \Rightarrow \qquad V = F_{A0} \int_0^{X_A} \frac{dX_A}{-r_A} FA0​dVdXA​​=−rA​⇒V=FA0​∫0XA​​−rA​dXA​​

Design equation in terms of XXX: PBR

  • Ideal steady state PBR design equation

    dFAdW=rA′ \frac{dF_A}{dW} =r'_A dWdFA​​=rA′​

  • FA=FA0(1−XA)F_A = F_{A0}( 1 - X_A)FA​=FA0​(1−XA​)

  • Taking the derivative of FAF_AFA​ equation

    ddWFA=ddW(FA0(1−XA)) \frac{d}{dW}F_A = \frac{d}{dW}\left( F_{A0}(1 - X_A)\right) dWd​FA​=dWd​(FA0​(1−XA​))

    dFAdW=−FA0dXAdW \frac{dF_A}{dW} = - F_{A0}\frac{dX_A}{dW} dWdFA​​=−FA0​dWdXA​​

  • Substituting

    FA0dXAdW=−rA′⇒W=FA0∫0XAdXA−rA′ F_{A0} \frac{dX_A}{dW} = -r'_A \Rightarrow W = F_{A0} \int_0^{X_A} \frac{dX_A}{-r'_A} FA0​dWdXA​​=−rA′​⇒W=FA0​∫0XA​​−rA′​dXA​​

::::

Reactor mole balance in terms of XXX

Reactor Differential form Algebraic form Integral form
Batch NA0dXdt=rAV N_{A0}\frac{dX}{dt} = r_AV NA0​dtdX​=rA​V t=NA0∫0XdXrAV t = N_{A0}\int_0^{X} \frac{dX}{r_A V}t=NA0​∫0X​rA​VdX​
CSTR V=FA0X−rAV = \frac{F_{A0}X}{-r_A} V=−rA​FA0​X​
PFR FA0dXdV=−rA F_{A0}\frac{dX}{dV} = -r_A FA0​dVdX​=−rA​ V=FA0∫0XdX−rA V = F_{A0} \int_{0}^{X} \frac{dX}{-r_A}V=FA0​∫0X​−rA​dX​
PBR FA0dXdW=−rA′ F_{A0}\frac{dX}{dW} = -r'_A FA0​dWdX​=−rA′​ W=FA0∫0XdX−rA′ W = F_{A0} \int_{0}^{X} \frac{dX}{-r'_A}W=FA0​∫0X​−rA′​dX​

Sizing of continuous flow reactors

  • Sizing refers to either of

    • Determining reactor volume for specified conversion
    • Determining conversion for a specified volume

  • For All irreversible reactions of order greater than 0, As we approach complete conversion, the reciprocal rate approaches infinity

  • Irreversible reaction:

    As X→1;−rA→0X \rightarrow 1 ; -r_A \rightarrow 0X→1;−rA​→0

  • Reversible reaction:

    As X→Xe;−rA→0X \rightarrow X_e ; -r_A \rightarrow 0X→Xe​;−rA​→0

⇒1−rA→∞∴V→∞ \Rightarrow \frac{1}{-r_A} \rightarrow \infty \therefore V \rightarrow \infty ⇒−rA​1​→∞∴V→∞

Infinite reactor volume is necessary to reach complete conversion

Sizing of continuous flow reactors

CSTR

V=(FA0(−rA)exit)⋅X V = \left( \frac{F_{A0}}{(-r_A)_{exit}} \right) \cdot X V=((−rA​)exit​FA0​​)⋅X

CSTR volume

area of rectangle bound by XAX_AXA​ and FA0−rA,exit\frac{F_{A0}}{-r_{A, exit}}−rA,exit​FA0​​

PFR

V=∫0X(FA0(−rA))dX V = \int_0^X \left( \frac{F_{A0}}{(-r_A)}\right) dX V=∫0X​((−rA​)FA0​​)dX

PFR volume

area under the curve FA0−rA=f(XA)\frac{F_{A0}}{-r_{A}} = f(X_A)−rA​FA0​​=f(XA​)

Levenspiel plots - reactor sizing

  • Given –rA–r_A–rA​ as a function of conversion, −rA=f(X)-r_A= f(X)−rA​=f(X), one can size any type of reactor.

  • We do this by constructing a Levenspiel plot.

  • Here we plot either FA0−rA\frac{F_{A0}}{-r_A}−rA​FA0​​ or 1−rA\frac{1}{-r_A}−rA​1​ as a function of XXX.

CSTR sizing

Using following data: Calculate VCSTRV_{CSTR}VCSTR​ for X=0.4X = 0.4X=0.4, and X=0.8X = 0.8X=0.8

X 0.00 0.10 0.20 0.40 0.60 0.70 0.80
FA0−rA\frac{F_{A0}}{-r_A}−rA​FA0​​ 0.89 1.08 1.33 2.05 3.56 5.06 8.00

V=(FA0(−rA)exit)⋅X V = \left( \frac{F_{A0}}{(-r_A)_{exit}} \right) \cdot X V=((−rA​)exit​FA0​​)⋅X

  • For XXX = 0.4; FA0(−rA)exit\frac{F_{A0}}{(-r_A)_{exit}}(−rA​)exit​FA0​​ = 2.05 m3m^3m3

    VVV = 0.82 m3m^3m3

  • For XXX = 0.8; FA0(−rA)exit\frac{F_{A0}}{(-r_A)_{exit}}(−rA​)exit​FA0​​ = 8.00 m3m^3m3

    VVV = 6.4 m3m^3m3

PFR sizing

Using following data: Calculate VPFRV_{PFR}VPFR​ for X=0.4X = 0.4X=0.4, and X=0.8X = 0.8X=0.8

X 0.00 0.10 0.20 0.40 0.60 0.70 0.80
FA0−rA\frac{F_{A0}}{-r_A}−rA​FA0​​ 0.89 1.08 1.33 2.05 3.56 5.06 8.00

V=∫0X(FA0(−rA))dX V = \int_0^X \left( \frac{F_{A0}}{(-r_A)}\right) dX V=∫0X​((−rA​)FA0​​)dX

  • For XXX = 0.4; Numerically integrate the data

    VVV = 0.55 m3m^3m3

  • For XXX = 0.8; FA0(−rA)exit\frac{F_{A0}}{(-r_A)_{exit}}(−rA​)exit​FA0​​ = 8.00 m3m^3m3

    VVV = 2.15 m3m^3m3

Reactors in series

  • Reactors are arranged sequentially, with the output of one reactor feeding directly into the next.

  • Advantages:

    • Achieve higher overall conversion rates
    • Handle reactions requiring different conditions at different stages.
    • Different reaction conditions can be optimized in each reactor.
    • Staging: Allows for the introduction of intermediates or the removal of by-products between stages.

  • Both Continuous Stirred Tank Reactors (CSTRs) and Plug Flow Reactors (PFRs) can be configured in series, either separately or in a mixed arrangement, to optimize reaction conditions and efficiencies.

  • Key Considerations

    • Reactor volume and design optimization
    • Maintenance and operational complexity

Levenspiel plots can be used to visualize and sequence reactors in series.

Reactors in series

  • In absence of any side streams (inlets or outlets)

    Conversion up to point iii:

    Xi=total moles of A reacted up to point iMoles A fed into 1st reactor X_i = \frac{\text{total moles of A reacted up to point } i }{\text{Moles A fed into } 1^{st} \text{ reactor}} Xi​=Moles A fed into 1st reactortotal moles of A reacted up to point i​

  • Molar Flow rate of species A at point iii:

    FAi=FA0−FA0Xi F_{Ai} = F_{A0} - F_{A0} X_i FAi​=FA0​−FA0​Xi​

Reactors in series

Example: An adiabatic liquid-phase isomerization

The isomerization of butane

n-CX4HX10⇌i-CX4HX10 \ce{n-C4H10 <=> i-C4H10} n-CX4​HX10​​i-CX4​HX10​

was carried out adiabatically in the liquid phase. The data for this reversible reaction are given below. Calculate the volume of each of the reactors for an entering molar flow rate of n-butane of 50 kmol/hr.

X 0.00 0.20 0.40 0.60 0.65
−rA,kmolm3⋅h-r_A, \frac{kmol}{m^3 \cdot h}−rA​,m3⋅hkmol​ 39.00 53.00 59.00 38.00 25.00

Reactors in series

Example: An adiabatic liquid-phase isomerization

Reactors in series

  • If FA0−rA\frac{F_{A0}}{-r_A}−rA​FA0​​ monotonically increases with XXX

Vone PFR≤∑iVPFR(i)+∑jVCSTR(j)≤Vone CSTR V_{\text{one PFR}} \leq \sum_i V_{\text{PFR(i)}} + \sum_j V_{\text{CSTR(j)}} \leq V_{\text{one CSTR}} Vone PFR​≤i∑​VPFR(i)​+j∑​VCSTR(j)​≤Vone CSTR​

for any combination of PFRs and CSTRs in series

  • In general

    1 PFR = any number of PFRs in series

    1 PFR = ∞\infty∞ number of CSTRs in series

  • For large number of CSTRs in series, the total volume is ‘roughly’ same as volume of PFR

The concept of using CSTRs in series to model PFR is used in larger number of situations such as modeling catalyst decay in packed bed reactors, or studying transient heat effects in PFR.

Space time (τ\tauτ)

  • Time necessary to process one reactor volume, also called mean residence time or holding time

    τ=Vυ0 \tau = \frac{V}{\upsilon_0} τ=υ0​V​

  • Space velocity (SV): inverse of space time, but vo may be measured under different conditions than the space time

    SV=υ0V=1τ SV = \frac{\upsilon_0}{V} = \frac{1}{\tau} SV=Vυ0​​=τ1​

    • LHSV: Liquid hourly space velocity: LHSV=υ0∣liquidV\mathrm{LHSV} =\frac{\upsilon_{0 \mid \mathrm{liquid}}}{V}LHSV=Vυ0∣liquid​​
    • GHSV: Gas hourly space velocity: GHSV=υ0∣STPV\mathrm{GHSV}=\frac{\upsilon_{0 \mid \mathrm{STP}}}{V}GHSV=Vυ0∣STP​​
    • υ0∣\upsilon_{0 \mid}υ0∣​ is the volumetric flow rate measured at specified condition

Chemical Reaction Engineering

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Conversion and reactor sizing Chemical Reaction Engineering

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  • Conversion and reactor sizing
  • Is this a reactor
  • Review - general mole balance
  • Reactor mole balance
  • Conversion, X
  • Conversion, X
  • Moles and molar flow rate in terms of X_A
  • Design equation in terms of X: Batch reactor
  • Design equation in terms of X: CSTR
  • Design equation in terms of X: PFR
  • Design equation in terms of X: PBR
  • Reactor mole balance in terms of X
  • Sizing of continuous flow reactors
  • Sizing of continuous flow reactors
  • Levenspiel plots - reactor sizing
  • CSTR sizing
  • PFR sizing
  • Reactors in series
  • Reactors in series
  • Reactors in series
  • Reactors in series
  • Reactors in series
  • Space time (\tau)
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