Distributions of Residence Times (rtd) for Chemical Reactors-Part 1- By: Mdm. Noor Amirah Abdul Halim
Theory of RTD • Definition of Residence Time Distribution (RTD) - Probability distribution function that describes the amount of time that a fluid element could spend inside the reactor • Purpose of RTD Analysis - To characterize the mixing and flow within reactors - To compare the behavior of real reactors to their ideal models Why?.. - For reactors troubleshooting - Estimation of yield for given reaction • Future reactor design • The theory of RTD generally begins with 3 assumptions: 1. The reactor is at steady-state, 2. Transports at the inlet and the outlet takes place only by advection 3. The fluid is incompressible (v = constant).
Residence Time Distribution (RTD) Function E (t) • The distribution of residence times is represented by an external residence time distribution or an exit age distribution, E(t). The function E(t) has the units of time^-1 and is defined such that: • The fraction of the fluid that spends a given duration, (t) inside the reactor is given by the value of
The Cumulative Distribution Function F(t) • The fraction of the fluid that leaves the reactor with an age less than (t1) is; Where F (t) is called ‘cumulative distribution’ • Thus, The fraction of the fluid that leaves the reactor with an age greater than (t1) is
Mean Residence Time tm • The average residence time is given by the first moment of the age distribution: • For no dispersion/diffusion in the reactor: space time (τ) equal to mean residence time (tm)
Variance σ2 • The behavior of the function E(t) also could indicate the degree of dispersion around the mean through the variance (σ2),
Measurement of RTD • RTD can be determined experimentally by; 1. Injecting an inert chemical/molecule/atom called tracer ,into the reactor at t=0 2. Measure the tracer concentration ,C in the effluent stream as a function of time.
Properties of tracer : - Non reactive species - Easily detectable - Physical properties similar to reacting mixture - Completely soluble in the mixture - Not adsorb on the wall or other reactor surface • Common type tracer- Colored an radioactive materials along with inert gases • RTD can be determined by two experimental methods (based on injection method: pulse or step) 1. Pulse experiment 2. Step experiment
Pulse Experiment • This method required the introduction of a very small volume of concentrated tracer at the inlet of the reactor, .The outlet tracer concentration C (t) is then measured as a function of time.
Example: • From our experiment data of the exit tracer concentration from pulse tracer test • We can obtain;
Step Experiment • In a step experiment, the concentration of tracer at the reactor inlet changes abruptly from 0 to C0. The concentration of tracer at the outlet is measured and normalized to the concentration C0 to obtain the non-dimensional cumulative distribution curve F(t) which goes from 0 to 1:
Discussions • The RTD function E(t) can be determined directly from a pulse input, • The cumulative distribution F(t) can be determined directly from a step input. • Relation of the step- and pulse-responses of a reactor are given by ;
The value of themean residence time (tm)and the variance (σ2) can also be deduced from the cumulative distribution function F(t):
Exercise 1 The following is an E curve calculated for reactor Y 1. What is the maximum value of E shown on this curve? 2. What fraction of the molecules spend between 2 and 2.5 minutes in the reactor 3. What fraction spend between 3.5 and 4
Exercise 2 The F curves is shown below for a real reactor What is the mean residence time?
Find E (t) 1. Determine the area of C(t) curve
F(t) and, the fraction of material that spends 25 s or less in the reactor
Variance aArea= 15 s^2
RTD for Ideal Reactors • The RTD of a reactor can be used to compare its behavior to that of two ideal reactor models: the PFR and the CSTR (or mixed-flow reactor). • This characteristic is important in order to calculate the performance of a reaction with known kinetics.
Batch & Plug Flow Reactors (pfr) • In an ideal PFR, there is no mixing and the fluid elements leave in the same order they arrived. • Therefore, fluid entering the reactor at time t will exit the reactor at time t + τ, where τ is the residence time of the reactor. • The residence time distribution function,E(t) is therefore a dirac delta function (δ) at τ. • E (t) for PFR is given by: • Where the dirac delta function (δ) at is given as • Assume (x=t)
Thus: Mean residence time: : Variance: Cumulative distributions:
CSTR (Mixed Flow Reactor) • An ideal CSTR is based on the assumption that the flow at the inlet is completely and instantly mixed into the bulk of the reactor. • The reactor and the outlet fluid have identical, homogeneous compositions at all times. An ideal CSTR has an exponential residence time distribution: • Equation for CSTR
Predicting Conversion And Exit Concentration • The RTD tells us how long the various fluid elements have been in the reactor, but it does not tell us anything about the exchange of matter between the fluid elements. • The length of time each molecule spends in the reactor is all that is needed to predict conversion.
Type Of Mixing • Macromixing • Produces a distribution of residence times • Micromixing • Describes how molecules of different ages encounter one another in the reactor. 2 types: 1. Complete Segregation : All molecules of the same age group remain together as they travel through the reactor and are not mixed with any other age until they exit the reactor 2. Complete Micromixing Molecules of different age groups are completely mixed at the molecular level as soon as they enter the reactor
Conversion (X) • For batch reactor : • For PFR : • For CSTR tm = τ CA = CA0 (1-X)
Reactor Modeling Using RTD Segregation Model Maximum Mixedness Model
Segregation Model In the segregation model globules behave as batch reactors operated for different times Mean conversion for the segregation model The segregation model has mixing at the latest possible point.
The farther the molecules travel along the reactor before being removed, the longer their residence time. Each globule exiting the real reactor at different times will have a different conversion. (X1,X2,X3...)
MAXIMUM MIXEDNESS MODEL Maximum mixedness: Mixing occurs at the earliest possible point As soon as the fluid enters the reactor, it is completely mixed radially (but not longitudinally) with the other fluid already in the reactor.
Let 𝜆 be the time it takes for the fluid to move from a particular point to the end of the reactor. In other words, 𝜆 is the life expectancy of the fluid in the reactor at that point