Distribution of residence time

Chemical Reaction Engineering

General Considerations

  • Models developed so far are for perfectly mixed batch reactor, the plug flow tubular reactor, packed bed reactor, and perfectly mixed continuous tank reactor.

  • Real world behavior is often very different from the ideal behavior.

  • Use residence time distribution to analyze and characterize non-ideal reactors.

    • diagnose problems of reactor operations

    • predict conversion in existing reactor when new chemical reaction is used in the reactor.

Examples of non-ideality

  • Describing deviation from ideal reactor mixing pattern

    • Residence time distribution (RTD)
    • Quality of mixing
    • Model used to describe the system

  • Residence time distribution (RTD) function

    Popularized by Prof. P. V. Dankwerts

Residence Time Distribution (RTD) function

Residence time: The time atoms have spent in the reactor.

  • Plug flow reactor: Atoms spend exactly the same time in these two reactors.

  • Ideal batch reactor: Atoms spend exactly the same time in these two reactors.

  • CSTR: Feed introduced into a CSTR becomes completely mixed with the material already in the reactor.

    • Some atoms entering the CSTR leave almost immediately.
    • Other atoms remain in the reactor almost forever as all the material recirculates within the reactor and is virtually never removed from the reactor at one time.

  • Distribution of residence times can significantly affect reactor performance.

    • The RTD is a characteristic of the mixing that occurs in the chemical reactor.

    • RTD yields distinctive clues to the type of mixing occurring within it and is one of the most informative characteristics of the reactor.

Measurement of the RTD

  • Determined experimentally
    • Injecting tracer into the reactor at some time t=0t=0t=0 and then measuring the tracer concentration CCC in the effluent stream as a function of time.

  • Properties of Tracer
    • Inert (non-reactive)
    • Easily detectable
    • Similar physical properties to the reacting mixture
    • Completely soluble in the reacting mixture
    • Does not adsorb on reactor walls
    • Tracer behavior should mimic the behavior of material flowing in the reactor.
  • Common tracers: colored dye, radioactive material, inert gases

Pulse input experiment

  • An amount of tracer N0N_0N0​ is suddenly injected in one shot into the feed stream.

  • Outlet concentration is measured with time.

  • Consider single-input and single-output system:

    • Only flow carries the tracer material.
    • No dispersion.
    • Increment of time Δt\Delta tΔt is sufficiently small that the concentration of tracer C(t)C(t)C(t) exiting between ttt and t+Δtt + \Delta tt+Δt is essentially the same.

  • Amount of tracer material leaving the reactor between ttt and (t+Δt)(t + \Delta t)(t+Δt): ΔN=C(t)⋅υ⋅Δtwhere υ is the volumetric flow rate(1) \Delta N = C(t) \cdot \upsilon \cdot \Delta t \quad \text{where } \upsilon \text{ is the volumetric flow rate} \tag{1}ΔN=C(t)⋅υ⋅Δtwhere υ is the volumetric flow rate(1)

Pulse input experiment

  • Dividing by the Total Amount of Material that was Injected

ΔNN0=υC(t)N0Δt(2) \frac{\Delta N}{N_0} = \frac{\upsilon C(t)}{N_0} \Delta t \tag{2}N0​ΔN​=N0​υC(t)​Δt(2)

⇒\Rightarrow⇒ Fraction of material that has residence time in the reactor between ttt and t+Δtt + \Delta tt+Δt

  • For pulse injection let

E(t)=υC(t)N0... Residence time function E(t) = \frac{\upsilon C(t)}{N_0} \quad \text{... Residence time function} E(t)=N0​υC(t)​... Residence time function

  • Residence time function

    Function that describes in a quantitative manner how much time different fluid elements have spent in the reactor

∴ΔNN0=E(t)Δt \therefore \frac{\Delta N}{N_0} = E(t) \Delta t ∴N0​ΔN​=E(t)Δt

Pulse input experiment

  • E(t)dtE(t) dtE(t)dt is the fraction of fluid exiting the reactor that has spent between time ttt and t+Δtt + \Delta tt+Δt inside the reactor.

  • If N0N_0N0​ is not known directly, it can be obtained from the outlet concentration measurements by summing up all the amounts from 0 to ∞0 \text{ to } \infty0 to ∞

  • Writing Equation 1 in Differential Form

    dN=υC(t)dt dN = \upsilon C(t) dt dN=υC(t)dt

  • Integrating

    N0=∫0∞υC(t)dt N_0 = \int_{0}^{\infty} \upsilon C(t) dt N0​=∫0∞​υC(t)dt

    υ\upsilonυ is usually constant, therefore

    E(t)=C(t)∫0∞C(t)dt(3) E(t) = \frac{C(t)}{\int_{0}^{\infty} C(t) dt} \tag{3}E(t)=∫0∞​C(t)dtC(t)​(3)

    • E(t)E(t)E(t) is also called exit age distribution function

Pulse input experiment

  • The E curve is just the C curve divided by the area under the C curve.

Pulse input experiment

  • Fraction of material leaving the reactor that has resided in the reactor between t1t_1t1​ and t2t_2t2​

    =∫t1t2E(t) dt = \int_{t_1}^{t_2} E(t) \, dt =∫t1​t2​​E(t)dt

  • Fraction of all material that has resided for a time t in the reactor between t=0t = 0t=0 and t=∞t = \inftyt=∞ is 1

    ∫0∞E(t) dt=1 \int_{0}^{\infty} E(t) \, dt = 1 ∫0∞​E(t)dt=1

Difficulties with pulse technique

  • Obtaining a reasonable pulse at the reactor entrance
    • Injection time should be very short compared to residence times in various segments of the reactor.
    • There must be negligible dispersion between the point of injection and the entrance to the reactor.
  • If these conditions are achieved, pulse technique is a simple and direct way to obtain RTD.

Step tracer experiment

  • Inlet is either:

    • Perfect pulse input (Dirac delta function)
    • Imperfect pulse, determine E(t)E(t)E(t)

  • Cumulative distribution F(t)F(t)F(t) can be determined from step input.

  • Cumulative distribution gives the fraction of material F(t)F(t)F(t) that has been in the reactor at time ttt or less.

Step tracer experiment

  • Consider constant tracer addition to a feed that is initiated at t=0t = 0t=0

Cin(t)={0t<0C0,constantt≥0 C_{\text{in}}(t) = \begin{cases} 0 & t < 0 \\ C_0, \text{constant} & t \ge 0 \end{cases} Cin​(t)={0C0​,constant​t<0t≥0​

  • The concentration of tracer in feed is kept at this level until the concentration in the effluent is almost the same as the feed.

  • As inlet concentration is constant with time, C0C_0C0​, we can take it out of the integral sign:

Cout(t)=C0∫0tE(t′)dt′ C_{\text{out}}(t) = C_0 \int_{0}^{t} E(t') dt' Cout​(t)=C0​∫0t​E(t′)dt′

  • Dividing by t0t_0t0​:

[Cout(t)C0]step=∫0tE(t′)dt′=F(t) \left[ \frac{C_{\text{out}}(t)}{C_0} \right]_{\text{step}} = \int_{0}^{t} E(t') dt' = F(t) [C0​Cout​(t)​]step​=∫0t​E(t′)dt′=F(t)

Step tracer experiment

F(t)=[Cout(t)C0]step(4) F(t) = \left[ \frac{C_{\text{out}}(t)}{C_0} \right]_{\text{step}} \tag{4}F(t)=[C0​Cout​(t)​]step​(4)

  • We differentiate Equation 4 to obtain RTD function:

E(t)=dFdt=ddt[Cout(t)C0]step E(t) = \frac{dF}{dt} = \frac{d}{dt} \left[ \frac{C_{\text{out}}(t)}{C_0} \right]_{\text{step}} E(t)=dtdF​=dtd​[C0​Cout​(t)​]step​

  • Positive step is usually easier to carry out experimentally than the pulse test.

  • Total amount of tracer in feed over the period of test does not have to be known

  • Drawbacks

    • It may be difficult to maintain constant tracer concentration in the feed.
    • Obtaining RTD involves differentiation of the data. ( differentiation can lead to large errors. )
    • A large amount of tracer is required.

  • Other Tracer Techniques

    • Negative step (elution), Frequency response method, Methods that use inputs other than pulse or step

    • These techniques are much more difficult to carry out and are not encountered often.

Integral relationships

Sometimes FFF curve is used in the same manner as the RTD in modeling chemical reactors.

  • Fraction of effluent that has been in the reactor for less than ttt

    F(t)=∫0tE(t) dt F(t) = \int_{0}^{t} E(t) \, dt F(t)=∫0t​E(t)dt

  • Fraction of effluent that has been in the reactor for longer than ttt

    1−F(t)=∫t∞E(t) dt 1 - F(t) = \int_{t}^{\infty} E(t) \, dt 1−F(t)=∫t∞​E(t)dt

Mean residence time

  • First moment of the RTD function

tm=∫0∞tE(t)dt∫0∞E(t)dt=∫0∞tE(t)dt t_m = \frac{\int_0^\infty t E(t) dt}{\int_0^\infty E(t) dt} = \int_0^\infty t E(t) dt tm​=∫0∞​E(t)dt∫0∞​tE(t)dt​=∫0∞​tE(t)dt

  • In absence of dispersion, and for constant volumetric flow rate,

    tm=τ⇒V=υtm only for closed systems  t_m = \tau \qquad \Rightarrow \qquad V = \upsilon t_m \qquad \text{ only for closed systems } tm​=τ⇒V=υtm​ only for closed systems 

Other moments of RTD

  • Variance (σ2\sigma^2σ2): Magnitude indicates spread of distribution

    σ2=∫0∞(t−tm)2E(t)dt \sigma^2 = \int_0^\infty (t - t_m)^2 E(t) dt σ2=∫0∞​(t−tm​)2E(t)dt

  • Skewness (s3s^3s3): Magnitude measures extent that distribution is skewed in one direction in reference to mean

    s3=1σ3/2∫0∞(t−tm)3E(t)dt s^3 = \frac{1}{\sigma^{3/2}} \int_0^\infty (t - t_m)^3 E(t) dt s3=σ3/21​∫0∞​(t−tm​)3E(t)dt

Normalized RTD function

  • Frequently a normalized function is used instead of E(t)E(t)E(t).

  • Let θ≡tτ\theta \equiv \frac{t}{\tau}θ≡τt​ : Number of reactor volumes of fluid based on entrance conditions that have flowed through the reactor in time ttt

    E(θ)=τE(t);∫0∞E(θ)dθ=1 E(\theta) = \tau E(t) \qquad; \qquad \int_{0}^{\infty} E(\theta) d\theta = 1 E(θ)=τE(t);∫0∞​E(θ)dθ=1

  • The flow performance inside reactors of different sizes can be compared directly.

  • If normalized function E(Θ)E(\Theta)E(Θ) is used, all perfectly mixed CSTRs have numerically the same RTD. If the simple function E(t)E(t)E(t) is used, numerical values of E(t)E(t)E(t) can differ substantially.

Internal-age distribution I(α)I(\alpha)I(α)

  • A function such that I(α)ΔαI(\alpha) \Delta \alphaI(α)Δα is the fraction of material inside the reactor that has been inside for a period of time between α\alphaα and α+Δα\alpha + \Delta \alphaα+Δα.

  • In catalytic reaction using catalyst whose activity decays with time, I(α)I(\alpha)I(α) is of importance and can be used to model the reactor.

    I(α)=(1−F(α))τ I(\alpha) = \frac{(1 - F(\alpha))}{\tau} I(α)=τ(1−F(α))​

    E(α)=−ddα[τI(α)] E(\alpha) = -\frac{d}{d\alpha} \left[ \tau I(\alpha) \right] E(α)=−dαd​[τI(α)]

  • For CSTR

    I(α)=−1τe−α/τ I(\alpha) = -\frac{1}{\tau} e^{-\alpha / \tau} I(α)=−τ1​e−α/τ

RTD in ideal reactors: batch and PFR

  • E(t)=δ(t−τ)E(t) = \delta(t - \tau)E(t)=δ(t−τ) (Dirac delta function)

    δ(x)={0when x≠0∞when x=0 \delta(x) = \begin{cases} 0 & \text{when } x \neq 0 \\ \infty & \text{when } x = 0 \end{cases} δ(x)={0∞​when x=0when x=0​

  • Properties of Dirac delta function

∫−∞∞δ(x)dx=1;∫−∞∞g(x)δ(x−τ)dx=g(τ) \int_{-\infty}^{\infty} \delta(x) dx = 1 \qquad ; \qquad \int_{-\infty}^{\infty} g(x) \delta(x - \tau) dx = g(\tau) ∫−∞∞​δ(x)dx=1;∫−∞∞​g(x)δ(x−τ)dx=g(τ)

  • Mean Residence Time

tm=∫0∞tE(t)dt=∫0∞tδ(t−τ)dt=τ t_m = \int_{0}^{\infty} t E(t) dt = \int_{0}^{\infty} t \delta(t - \tau) dt = \tau tm​=∫0∞​tE(t)dt=∫0∞​tδ(t−τ)dt=τ

  • Variance σ2=∫0∞(t−τ)2δ(t−τ)dt=0 \sigma^2 = \int_{0}^{\infty} (t - \tau)^2 \delta(t - \tau) dt = 0 σ2=∫0∞​(t−τ)2δ(t−τ)dt=0
  • Variance Calculation

    • The variance σ2\sigma^2σ2 measures the spread of residence times around the mean residence time τ\tauτ.

    • For an ideal plug flow reactor, there is no spread because every particle has the same residence time.

    • Therefore, when calculating the variance using the delta function, the integrand (t−τ)2(t - \tau)^2(t−τ)2 is zero at t=τt = \taut=τ, and since δ(t−τ)\delta(t - \tau)δ(t−τ) is zero everywhere else, the integral evaluates to zero:

      σ2=∫0∞(t−τ)2δ(t−τ) dt=0 \sigma^2 = \int_{0}^{\infty} (t - \tau)^2 \delta(t - \tau) \, dt = 0 σ2=∫0∞​(t−τ)2δ(t−τ)dt=0

Single-CSTR RTD

  • Concentration in effluent stream is identical to the concentration throughout the reactor.

  • Material balance on an inert tracer injected as a pulse at t=0t = 0t=0

In−Out=Accumulation⇒0−υC=VdCdt \text{In} - \text{Out} = \text{Accumulation} \Rightarrow 0 - \upsilon C = V \frac{dC}{dt} In−Out=Accumulation⇒0−υC=VdtdC​

  • Initial Condition: at t=0C=C0\text{at } t = 0 \quad C = C_0at t=0C=C0​

C(t)=C0e−t/τ C(t) = C_0 e^{-t/\tau} C(t)=C0​e−t/τ

  • RTD Function

E(t)=C(t)∫0∞C(t)dt=C0e−t/τ∫0∞C0e−t/τdt=e−t/ττ E(t) = \frac{C(t)}{\int_{0}^{\infty} C(t) dt} = \frac{C_0 e^{-t/\tau}}{\int_{0}^{\infty} C_0 e^{-t/\tau} dt} = \frac{e^{-t/\tau}}{\tau} E(t)=∫0∞​C(t)dtC(t)​=∫0∞​C0​e−t/τdtC0​e−t/τ​=τe−t/τ​

Single-CSTR RTD

  • Normalized RTD Function

E(θ)=e−θwhereθ=tτ;E(θ)=τE(t) E(\theta) = e^{-\theta} \quad \text{where} \quad \theta = \frac{t}{\tau} ; \quad E(\theta) = \tau E(t) E(θ)=e−θwhereθ=τt​;E(θ)=τE(t)

  • Cumulative Distribution Function F(t)F(t)F(t)

    F(t)=∫0tE(t)dt=∫0te−t/ττdt F(t) = \int_{0}^{t} E(t)dt = \int_{0}^{t} \frac{e^{-t/\tau}}{\tau} dt F(t)=∫0t​E(t)dt=∫0t​τe−t/τ​dt

    F(t)=1−e−t/τ;F(θ)=1−e−θ F(t) = 1 - e^{-t/\tau}; F(\theta) = 1 - e^{-\theta} F(t)=1−e−t/τ;F(θ)=1−e−θ

  • Mean residence time tm=∫0∞tE(t)dt=∫0∞tτe−t/τdt=τ t_m = \int_{0}^{\infty} t E(t) dt = \int_{0}^{\infty} \frac{t}{\tau} e^{-t/\tau} dt = \tau tm​=∫0∞​tE(t)dt=∫0∞​τt​e−t/τdt=τ

    σ2=∫0∞(t−τ)2τe−t/τdt=τ2∫0∞(x−1)2e−xdx=τ2 \sigma^2 = \int_{0}^{\infty} \frac{(t - \tau)^2}{\tau} e^{-t/\tau} dt = \tau^2 \int_{0}^{\infty} (x - 1)^2 e^{-x} dx = \tau^2 σ2=∫0∞​τ(t−τ)2​e−t/τdt=τ2∫0∞​(x−1)2e−xdx=τ2

  • σ=τ\sigma = \tauσ=τ: Standard deviation is as large as the mean

PFR/CSTR series RTD

  • In some stirred tanks, there is a highly agitated zone in the vicinity of the impeller →\rightarrow→ CSTR

  • Depending on the location of the inlet and outlet, the reacting mixture may follow a tortuous path either before entering or after leaving the perfectly mixed zone →\rightarrow→ PFR

PFR/CSTR series RTD

  • Early mixing: C=C0e−t/τsC = C_0 e^{-t/\tau_s}C=C0​e−t/τs​; τs\tau_sτs​: CSTR mean RT; τp\tau_pτp​: PFR mean RT

    This conc. output will be delayed by τp\tau_pτp​ at the outlet plug flow section

  • RTD E(t)={0t<τpe−(t−τp)/τsτst≥τp E(t) = \begin{cases} 0 & t < \tau_p \\ \frac{e^{-(t - \tau_p) / \tau_s}}{\tau_s} & t \ge \tau_p \end{cases} E(t)={0τs​e−(t−τp​)/τs​​​t<τp​t≥τp​​

  • Late Mixing

E(t)={0t<τpe−(t−τp)/τsτst≥τp E(t) = \begin{cases} 0 & t < \tau_p \\ \frac{e^{-(t - \tau_p) / \tau_s}}{\tau_s} & t \ge \tau_p \end{cases} E(t)={0τs​e−(t−τp​)/τs​​​t<τp​t≥τp​​

  • Exactly same as early mixing
  • Even though RTD will be the same for both these cases, conversion can be very different
  • RTD is not a complete description of the structure for a particular reactor / reactor systems

Diagnostics and troubleshooting

Diagnostics and troubleshooting

Chemical Reaction Engineering

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Distribution of residence time Chemical Reaction Engineering

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  • Distribution of residence time
  • General Considerations
  • Examples of non-ideality
  • Residence Time Distribution (RTD) function
  • Measurement of the RTD
  • Pulse input experiment
  • Pulse input experiment
  • Pulse input experiment
  • Pulse input experiment
  • Pulse input experiment
  • Difficulties with pulse technique
  • Step tracer experiment
  • Step tracer experiment
  • Step tracer experiment
  • Integral relationships
  • Mean residence time
  • Normalized RTD function
  • Internal-age distribution I(\alpha)
  • RTD in ideal reactors: batch and PFR
  • Single-CSTR RTD
  • Single-CSTR RTD
  • PFR/CSTR series RTD
  • PFR/CSTR series RTD
  • Diagnostics and troubleshooting
  • Diagnostics and troubleshooting
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