RTD in Ideal Reactors
RTDs in Batch and Plug-Flow Reactors
• The RTDs in plug-flow reactors and ideal batch reactors are the
simplest to consider.
• All the atoms leaving such reactors have spent precisely the same
amount of time within the reactors.
• The distribution function in such a case is a spike of infinite height
and zero width, whose area is equal to 1.
RTDs in Batch and Plug-Flow Reactors
• The spike occurs at
• E(t) (a)
• F(t) (b)
RTDs in Batch and Plug-Flow Reactors
• Mathematically, this spike is represented by the Dirac delta function:
RTDs in Batch and Plug-Flow Reactors
• Properties of the Dirac delta function:
RTDs in Batch and Plug-Flow Reactors
• To find the mean residence time set g(x) = t .
• To find the variance set
• As all the material spends the same amount of time in the reactor,
RTDs in Batch and Plug-Flow Reactors
• Cumulative distribution function:
Ideal Single-CSTR RTD
• All concentrations in the effluent are the same as in the reactor.
• It is therefore possible to obtain the RTD from conceptual
considerations only.
• A material balance on an inert tracer that has been injected as a pulse
at time t = 0 into a CSTR yields for t > 0:
Ideal Single-CSTR RTD
• Separating the variables and integrating with C = C0 at t = 0:
• Consequently:
Ideal Single-CSTR RTD
• Where
Ideal Single-CSTR RTD
• (F(t) curve
Ideal Single-CSTR RTD
• First and second moments:
• The standard deviation of the residence time distribution is as large as
the mean itself!!
The Segregation Model
Modeling Nonideal Reactors
Using the RTD
Modeling Overview
• Presently, we can characterise a reactor by using RTD analysis and we
can determine the reaction kinetics from experimental data.
• We need to choose a model to predict conversion in a real reactor.
Modeling Overview
• There are five models that use the RTD data to predict conversion in
nonideal reactors:
Modeling Overview
• Each model is classified according to the number of adjustable
parameters.
• Chapter 17 covers the zero adjustable parameter models.
• Chapter 18 covers one and two adjustable parameter models.
• No intermediate calculations are made when using the zero adjustable
parameter models.
• i.e. E- and F-Curves are used directly to predict the conversion given the
kinetic parameters.
Modeling Overview
• One adjustable parameter models:
• Use RTD to calculate mean residence time, tm, and variance,
• The calculated values are used to:
(1) to find the number of tanks in series necessary to accurately model a
nonideal CSTR.
(2) calculate the Peclet number, Pe, to find the conversion in a tubular flow
reactor using the dispersion model.
Modeling Overview
• Peclet Numbers
Advection is the transport of a substance by bulk
• Defined as: motion of a fluid.
Advected substances are normally also fluids.
• Mass Transfer:
• Heat Transfer:
Modeling and Mixing Overview
• Two adjustable parameter models:
• We create combinations of ideal reactors to model the nonideal reactor.
• We then use the RTD to calculate the model parameters such as fraction
bypassed, fraction dead volume, exchange volume, and ratios of reactor
volumes that then can be used along with the reaction kinetics to predict
conversion.
Modeling Overview
• In this subject we will address only the Segregation Model.
Mixing
• RTD tells us how long the various fluid elements have been in the
reactor.
• RTD does not tell us anything about the exchange of matter between
the fluid elements (i.e., the mixing).
• For first-order reactions, the conversion is independent of
concentration and residence time is all that is needed to predict
conversion (mixing is not important).
Mixing
• For reactions other than first order, the degree of mixing of molecules
must be known in addition to how long each molecule spends in the
reactor.
• Models that account for the mixing of molecules must be developed.
• More complex models of nonideal reactors must contain information
about micromixing and macromixing.
Mixing
• Macromixing produces a distribution of residence times without,
specifying how molecules of different ages encounter one another in the
reactor (RTD).
• Micromixing describes how molecules of different ages encounter one
another in the reactor. Extremes of micromixing:
(1) 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 (complete segregation).
(2) Molecules of different age groups are completely mixed at the molecular
level as soon as they enter the reactor (complete micromixing).
Mixing
• For a given RTD these two extremes of micromixing will give the
upper and lower limits on conversion in a nonideal reactor.
• For single reactions with orders greater than one or less than zero, the
segregation model will predict the highest conversion.
• For reaction orders between zero and one, the maximum mixedness model
will predict the highest conversion.
Mixing
• A globule is a fluid particle containing millions of molecules of the
same age.
• A fluid in which the globules of a given age do not mix with other globules is
called a macrofluid. The globules do not coalesce.
• A fluid in which molecules are not constrained to remain in the globule and
are free to move everywhere is called a microfluid.
Mixing
• Extremes of mixing of the macrofluid globules:
• late mixing (a)
• early mixing (b)
• The extremes of late and early mixing are referred to as complete
segregation and maximum mixedness, respectively.
Zero-Adjustable-Parameter
Models (p810)
Segregation Model
• In a “perfectly mixed” CSTR, the the entering fluid is assumed to be
distributed immediately and evenly.
• Furthermore, elements of different ages mix together thoroughly and a
completely micromixed fluid is formed
Segregation Model
• If fluid elements of different ages do not mix together at all, the
elements remain segregated.
Segregation Model (CSTR)
• Globules do not interchange material with other globules during their
period of residence (segregated).
• Each globule spends a different amount of time in the reactor.
• All molecules with the same residence time are assumed to be in the
same globule.
Segregation Model (PFR)
• Each exit stream corresponds to a specific residence time in the
reactor.
If each globule is
regarded as a
batch reactor, E(t)
matches the
removal of these
batch reactors.
Segregation Model (PFR)
• Batches of molecules are removed from the reactor so as to duplicate
the RTD function, E(t).
• The farther the molecules travel along the reactor the longer their
residence time.
• The points at which the various batches of molecules are removed
correspond to the RTD function for the reactor.
• As there is no molecular interchange between globules, each acts as its
own batch reactor with its own residence time.
Segregation Model
• Conversion will be predict as shown in the box:
Segregation Model
• To determine the mean conversion in the effluent stream, we must
average the conversions of all of the various globules in the exit
stream:
Segregation Model
• From the previous slide:
• Summing all the globules:
Segregation Model (1st order reaction)
• Reaction:
• Each globule is a batch reactor, therefore
• For constant volume:
Segregation Model (1st order reaction)
• Mean conversion for a first-order reaction:
Segregation Model
• Example 17.1
• For a first-order reaction whether you assume complete micromixing
[Equation (E17-1.6)] or complete segregation [Equation (E17-1.5)] in a
CSTR, the same conversion results.
Reference
