1. Momentum Heat Mass Transfer
MHMT8
RTD
Residence time distribution.
Rudolf Žitný, Ústav procesní a zpracovatelské techniky ČVUT FS 2010

2. Residence Time Distribution
MHMT8
Solution of Navier Stokes equation enables to calculate residence times of particles flowing through a continuous system. Residence time is in fact exposition time (exposition to temperature, radiation, catalyser,…) determining yield of chemical/biochemical reactions. First of all, experimentally determined residence time distribution is a diagnostics tool of continuous apparatuses.

3. Residence Time Distribution
MHMT8
Residence time  = time spend by a particle inside a continuous system (time between entering and leaving the system).
Let us assume that at a time interval (0,dt) we analyze all fluid particles passing through outlet and that we evaluate their residence times by backtracking streamlines using solution of Navier Stokes equations, or experimentally. Histogram of residence times of the collected particles is an important characteristic of continuous system, RTD-residence time distribution:
INLET
OUTLET
Fast particle (short residence time)
RTD Residence time distribution E() = histogram of residence times  of all particles passing through outlet. E()d is relative amount of particles having residence times within the interval (,+d).

4. RTD of circular pipe MHMT8
Let us consider as a system a straight pipe with fully developed velocity profile u(r). Streamlines are horizontal lines, and residence times =L/u(r) depend only upon the radial coordinate of fluid particles. Distribution of residence times is determined by the shape of velocity profile u(r): r dr R L
u(r)
inlet
outlet
Radius r must be expressed as a function of , according to velocity profile
Mean residence time is the ratio of active internal volume and flowrate
Remark: Active volume cannot be evaluated just only from geometry of apparatus, because it is necessary to eliminate “dead” or stagnant regions, problems are with active volume of fixed porous beds etc. Measurement of mean residence time is usually the only way how to determine the active internal volume of continuous apparatuses.
Example: Laminar velocity profile for Newtonian liquid

5. RTD of circular pipe MHMT8
The result holds only for the residence time >min where min is the shortest residence time of the fastest particle moving at axis (min is sometimes called the time of the first appearance). For the parabolic velocity profile when the maximum velocity is twice the mean velocity
because E() is distribution
mean residence time

6. RTD=E(t) Impulse response of a system
MHMT8
Let us assume that during an infinitely short time interval (0,dt) are all particles passing through inlet labeled (for example exposed to a short flash of light). Detector located at outlet counts at a time t the labeled particles (these particles have residence time within the interval (t,t+dt)) and their concentration is proportional to E(t). Normalised time response recorded by the detector is therefore identical with the residence time distribution E(t) (which is frequently called impulse response). L
Experiments: Instantaneous labeling of fluid particle is usually realised by a very short injection of tracer (radionuclide, solution of salt,…) into the inlet stream, and concentration of tracer at outlet (impulse response) is recorded by a suitable transducer (Geiger Muller counter, conductivity probe,…).
Remark: Later on we shall analyze the diffusion of tracer in more details (Axial Dispersion Model in the last lecture on mass transport)

7. RTD=E(t) Impulse response of a system
MHMT8
Mathematically: The impulse response E(t) is the response of a general system to infinitely short pulse (→0) having unit area (Dirac delta function).
Delta function (t) is zero for t≠0 and
In this way the residence time distribution E(t) can be evaluated theoretically by solving transient transport equations for a “tracer” concentration with boundary condition at inlet prescribed in form of a short pulse of concentration.
1/
E(t) 
t [s]

8. ? Example response of system to a pulse (1/3) MHMT8
Derive response cout(t) of a perfectly mixed tank of inner volume V to a short pulse of tracer concentration cin(t)
Constant flowrate at inlet
Constant flowrate at outlet
 t[s]
Initial condition: zero concentration c in the tank at time t=0. ?
Concentration in tank is the same as at outlet (perfect mixing, concentration is uniform inside)

9. Example response of system to a pulse (2/3)
MHMT8
First step – describe the system by differential equation for concentration c(t) in the mixed tank
 t[s]
Mass balance of tracer (concentration c in the perfectly mixed tank)
Mean residence time

10. Example response of system to a pulse (3/3)
MHMT8
Response to a unit step stimulus function cin(t)=1 (up to the time )
Response to zero concentration at inlet for t> (initial concentration )

11. RESPONSE of a system to arbitrary stimulus
MHMT8
An arbitrary function cin() can be substituted by a series of infinitely short rectangular pulses. Response to a pulse acting at time  is the shifted impulse response E(t-) multiplied by the area of pulse. Response of system to all pulses is the integral:
cin()
t [s]
E(t-)
 d
E(t-)cin()d
To understand the second integral use substitution t-=*

12. F(t)-integral distribution
MHMT8
Inlet concentration in form of unit step (Heaviside function-integral of Dirac delta function) results to the concentration response expressed as the convolution integral
Relationship between the impulse response and F-distribution L
Inlet is switched to „red“ tracer since the time 0.
Experiments: Inlet stream is switched to the tracer stream since the time 0. Concentration of tracer at outlet is proportional to the integral distribution F(t). The value F(t) is the ratio of flowrate of the particles having residence time less then t to the overall flowrate.

13. F(t) of circular pipe MHMT8
Example: Stabilised laminar parabolic profile in a circular tube results to the integral distribution of residence times (derived from the known impulse response E(t))
Application: Let us consider an extruder producing coloured fibers (spinning). For how long the quality of product will be decreased by ingredients of old colour after switching to a new colour? The F(t) graph indicates that as soon as 25% of “old” colour is tolerated (F=0.75) this time is 1 s (or more precisely ).

14. F(t) of circular pipe MHMT8
Example: Integral residence time distribution F(t) for stabilised laminar power-law velocity profile (n-flow behaviour index) in a circular tube
inlet
outlet R r
n=1 L
n=0.5
Relationship between radius and residence time t=L/u(r)

15. EXAM
MHMT8
Residence time distribution

16. What is important (at least for exam)
MHMT8
Definition of RTD and impulse response
Distribution, mean residence time
Integral distribution

17. What is important (at least for exam)
MHMT8
Stimulus response relationship. Convolution