Distribution of residence time

The expression of distribution of residence time is used in Génie of the processes. It makes it possible to give an account of the type of flow of a liquid in an engine (tank, led…).

The distribution of residence time is a model which makes it possible to characterize the Hydrodynamique of a chemical engine and to determine which model of engine defines best the studied installation (continuous engine or tubular engine). This characteristic is important to be able to calculate the performance of a reaction with known kinetics.

Principle

The models of the ideal engines are constuits on certain numbers of assumptions: the mixture of an entering flow of a continuous engine is regarded as complete and instatané with the reactional medium and in a tubular engine, the flow is defined like piston (not retro-mixture). However in reality, it is impossible to obtain such conditions, in particular for industrial engines which in general have a size ranging between 1 and several tens of m³.

To characterize the flow, one consequently uses the distribution of residence time which is a statisque approach. Indeed, one considers an element of the fluid at his entry in the engine and one measures time that this last met to reach the exit. If the experiment is repeated or one considers several elements at the same time, one will note that the results are not identical.

One can consequently establish a distribution of the Residence time, generally represented by a frequency distribution called usually E. With this intention, three assumptions are posed:

the engine is with the stationary state
the fluid is incompressible
at the entry and on the outlet side of the engine, transport takes place only by Convection

Under these conditions, the function E brings back the age of the elements which leave the engine at a given time. Dimension E (T) corresponds thus to a fraction of total flow having a certain age per unit of time.
The fraction of the fluid which remains during an time interval given in the system is given by the value $E (T) \ cdot \ Delta t$ (cf Figure 1).

The following relations can be defined:
$\ int \ limits_ {0} ^ \ infty E (T) \, \ mathrm dt = 1$
$\ int \ limits_ {0} ^ {T1} E (T) \, \ mathrm dt$ fraction of the fluid which leaves the engine with an age younger than t₁ $\ int \ limits_ {T1} ^ \ infty E (T) \, \ mathrm dt = 1 \ int \ limits_ {0} ^ {T1} E (T) \, \ mathrm dt$ fraction of the fluid which leaves the engine with an age older than t₁
The average of the distribution, which corresponds to the average of the residence times, is given by the first moment of the distribution
$\ overline {T} = \ int \ limits_ {0} ^ \ infty T \ cdot E (T) \, \ mathrm dt$
The average of the residence times is equal to the time of passge if the density of the system is constant
$\ overline {T} = \ tau = \ frac {V} {\ dowry V}$
The central moments of a nature higher than 1 provide significant information on the behavior of the function E (T). For example, the central moment of order 2 indicates dispersion around the average, C. - with-D. the variance.
$\ sigma^2 = \ int \ limits_ {0} ^ \ infty (T \ overline {T}) ^2 \ cdot E (T) \, \ mathrm dt$
The central moment of order 3 indicates the asymmetry function E (T) and that of a nature 4 the flatness of the latter.

One can also define an internal distribution of age, named I (T), for the contents of the engine. This value has the same definition as E (T): it is about the fraction of total flow within the engine having a certin age by unitéde time. The relation which connects E (T) and I (T) is given by

$E (T) = - \ tau \ cdot \ frac {dI (T)}{dt}$

Experimental determination of the SDR

To determine the distribution of residence time, a tracer (dyes (Colorimétrie) is introduced, salts (Conductimétrie) even of the radioactive elements (Radioactivité)) in the system according to a known function and one measures at the exit the changes of the function of injection. The selected tracer should not modify the physical properties of the medium (even density, even viscosity) and the introduction of the tracer should not modify the hydrodynamic conditions.

2 functions of introduction in general are used: the “level” (step) and the “impulse” (pulsates). Other functions are possible, but require more calculations to obtain E (T).

Injection by level

The concentration of the tracer at the entry of the engine passes abruptly from 0 to C₀. The concentration of the tracer at exit of the engine is measured and can be divided by the initial concentration C₀ to obtain the adimensional curve F (nomenclature of Danckwerts) which lies between 0 and 1: $F (T) = \ frac {C (T)}{C_ {0}}$ .

By a matter assessment, one can define the following relation

$F (T) + \ tau \ cdot I (T) = 1$ what gives $I (T) = \ frac {(1-F (T))}{\ tau}$

By the preceding relation binding E (T) and I (T), one finds
$F (T) = \ int \ limits_ {0} ^t E (T) \, \ mathrm dt$ what gives $E (T) = \ frac {dF (T)}{dt}$
The value of the average and the variance can also be deduced from the function F (T)

$\ overline {T} = \ int \ limits_ {0} ^ \ infty T \ cdot E (T) \, \ mathrm dt = \ int \ limits_ {0} ^1 T \, \ mathrm dF (T) = - \ int \ limits_ {0} ^1 T \, \ mathrm D (1-F (T)) = \ int \ limits_ {0} ^ \ infty (1-F (T))\, \ mathrm dt$ $\ sigma^2 = \ int \ limits_ {0} ^ \ infty (T \ overline {T}) ^2 \ cdot E (T) \, \ mathrm dt = \ int \ limits_ {0} ^1 (T \ overline {T}) ^2 \, \ mathrm dF (T) = \ int \ limits_ {0} ^1 t^2 \, \ mathrm dF (T) - \ overline {T} ^2 = 2 \ int \ limits_ {0} ^ \ infty (1-F (T)) \, \ mathrm dt - \ overline {T} ^2$

Injection by impulse

This work grouting consists in introducing all the tracer on an interval very court at the entry of the engine so as to approach the Fonction of Dirac. In practice, the duration of the injection must be small compared to average residence time. The concentration of the tracer at exit of the engine is measured and divided by the virtual concentration $\ bar C_ {0}$ (defined by the number of mole of the tracer n₀ injected divided by the volume of the engine) to obtain the adimensional curve C (nomenclature of Danckwerts) which lies between 0 and 1: $\ mathbf {C} = \ frac {C (T)}{\ bar C_ {0}}$ .

By a matter assessment, one obtains:

$E (T) = \ frac {\ dowry V} {n_ {0}} \ cdot C (T) = \ frac {1} {\ tau} \ frac {C (T)}{\ bar C_ {0}} = \ frac {1} {\ tau} \ mathbf {C}$

SDR of the ideal engines

Ideal tubular engine

The tubular engine ideal has only one retarding function and does not change the entry signal. For an injection by implulsion, one obtains the same signal at the exit after a certain time which corresponds to the average residence time.

$E (T) = \ delta (T \ bar T)$

The same reasoning can be applied to the injection level.

Perfectly mixed engine

If one introduces into a engine perfectly mixed an impulse of n₀ moles of tracer, the system instantaneously will reach the maximum average concentration

$C_0 = \ frac {n_0} {V} T = 0$

The evolution of the concentration according to time can then be déduie by integration of the matter assessment:

$V \ cdot \ frac {cd. (T)}{dt} = - Q \ cdot C (T)$

what gives

$\ frac {C (T)}{C_0} = exp (- \ frac {T} {\ tau})$

Models of the real engines

Tubular engine with laminar flow

The tubular engine with laminar flow does not form part of the category of the ideal engines, it has a hydrodynamic behavior known and defined well, which makes it possible to predict the distribution of stay of it. The difference in residence time of the elements of volume in •engine is the consequence of the profile of vitesseparabolic. In a net of fluid of constant radial position, each particle crosses the engine without being influenced by the others.

If one defines $y = R/R_ {0}$ as being the radial coordinate brought back to the ray of the tube, the profile speed gives:

$U = u_ {max} \ cdot (1 \ frac {R^ {2}} {R_ {0} ^ {2}}) = 2 \ cdot \ bar U \ cdot (1 there ^ {2})$

where $u_ {max}$ is speed in the center of the tube and $\ bar u$ the median value on the section ( $\ bar U = \ frac {Q} {S})$ .

The residence time of an element of volume which is in a net of fluid given is worth as follows:

$t = \ frac {L} {U} = L \ cdot (u_ {max} \ cdot (1 there ^ {2}) ^ {- 1}) = \ frac {t_ {min}} {(1 there ^ {2})}$

from where $t_ {min} = \ frac {L} {u_ {max}} = \ frac {L} {2 \ cdot \ bar U}$ .

The fraction of the total flow which is in the position there, which thus has a residence time of T, is calculated starting from the surface of the corresponding annular section:

$\ frac {dQ} {Q} = \ frac {U \ cdot 2 \ cdot \ pi \ cdot R Dr.} {\ pi \ cdot R_ {0} ^ {2} \ cdot \ bar U} = \ frac {2 \ cdot U \ cdot RdR} {\ bar U \ cdot R_ {0} ^ {2}} = E (T) dt$

And by using the relation of the residence time of an element of volume (cf above):

$E (T) dt = \ frac {2 \ cdot t^ {2} _ {min}} {t^ {3}} dt = \ frac {\ tau^ {2}} {2 \ cdot t^ {3}} dt$

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