Download Additional comments on Process modeling material

Additional comments on Process modeling material
Ville Alopaeus, 26.11.2015
Formulation of balances
Mathematical modeling should typically start by formulating appropriate balances. The basic form for
balances is always ACCUMULATION = IN – OUT + GENERATION. The most typical balances are material,
energy, and linear momentum balances. Material balance could be written for mass or moles. In chemical
engineering, it is typically formulated separately for each chemical component and each phase.
Energy balance is often separated in two parts: thermal energy balance (enthalpy) and mechanical energy
balance. The former is typically needed in heat transfer problems, and the latter in fluid flow problems.
Note that pressure drop correlations can be considered as mechanical energy balances (with some
assumptions, such as time independent flow etc.). Linear momentum balances are not written very often,
they are mainly used in Computational Fluid Dynamics (CFD). Very often mechanics related problems are
solved by setting forces equal. This is actually a special case of linear momentum balance, since forces
appear as source terms there. Linear momentum is mass times velocity, and time rate of change of linear
momentum is mass times acceleration for constant mass systems, which has a unit of force (N).
Other balances in chemical engineering are e.g. volume balance, entropy balance (e.g. for compressors),
and population balances for size distributions of bubbles, drops and particles.
Further important things that need to be considered already when the modeling work is started, is the time
allowed for the modeling project, and the desired accuracy. If time is limited, the number of variables and
model complexity must be kept at minimum. The model structure should also be simple if it needs to be
solved rapidly, e.g. nonlinear partial differential equations need to be solved in such ways, where lots of
additional variables (resulting from discretization discussed later) result in. If accuracy is of importance,
then the model typically will be more complicated, and time required for model solution (numerical
method implemented in a computer) is also longer.
Control volume
When writing balances, it is important to define control volume (boundaries) properly. The control volume
could be e.g. a slice of a tubular reactor, single process unit (assumed homogeneous, i.e. fully mixed), a
process plant or whatever part of space. The most important thing is to carefully consider inlet and outlet
streams, whether the system is time dependent or assumed steady state, and what production of the
modeled variable occurs within the control volume (e.g. reaction producing or consuming the component
considered). Particular care should be taken to avoid mistakes in the sign conventions.
Constitutive equations (rate laws)
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Typically the most difficult part is finding the correct constitutive equations, or rate laws that are needed
for the balances. These may include heat conduction, heat convection (advection), and radiation as the
three possible heat transfer mechanisms. For mass transfer, the possible mechanisms are convection and
diffusion. There are also lots of expressions for reaction rates, including detailed molecular level analysis of
catalyst surfaces etc.
Although the balances themselves are basically always correct (if all the terms are included), the
constitutive equations often contain lots of parameters that need to be estimated or measured. For
example diffusion coefficients, heat conductivity and other physical properties, or reaction kinetic model
parameters. Some of them can be estimated or are tabulated in the literature, such as chemical component
properties, but some of them must be experimentally determined, such as reaction kinetic parameters.
If the resulting model is a differential equation (ordinary or partial), boundary conditions are needed to
solve them. The number of boundary conditions needed depend on the order of the differential equation.
Boundary conditions typically can be obtained from geometric or other reasoning. One of the most typical
example of such is the film theory for mass transfer, where two boundary conditions are obtained for the
diffusion equation, one at the boundary (for example phase interface), and the other for bulk phase. The
model predicts that there is a finite distance (typically on the order of millimeter) between these two. This
distance is called the film, and its thickness depends on the fluid flow far away from the interface. After
integration, the model predicts mass transfer fluxes. Here is a comparison of differential form (one possible
way of writing Fick's law), and its integrated form, where l is the film thickness, and c1 and c2 are the
concentrations at the two ends of the film. Note that the notations may vary, so it is better to try to
understand the phenomenon first, which would result in remembering, understanding, and being able to
apply the equations. If you start by trying to remember, you will get lost due to different formulations and a
huge number of equations telling basically the same thing.
J  D
dc
dx
J
Differential form (Fick's law)
D
c1  c 2   k c1  c 2 
l
Integrated form
Inter-phase mass transfer is then a source term for material balances, "IN" for one phase, and "OUT" for
the other. In addition to the diffusion flux, the convective part is needed if that is important (IN or OUT
term due to fluid flow carrying the modeled component)
Dimensionless numbers and correlations
In practice, the film thickness for mass transfer, and many other parameters are not known. Mass transfer
(as well as other rate laws) is typically written so that there is a response (diffusion flux) that depends on a
proportionality coefficient (mass transfer coefficient) and driving force (composition difference) as shown
earlier. Typically there are parameters that cannot be directly measured, but there are correlations from
where these can be estimated. The correlations are typically formulated with dimensionless numbers.
Perhaps the most well-known is the Reynolds number, which appears also in fluid dynamics. Correlations
for mass transfer could be e.g. of the form
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Sh  2  a  Re m Sc n
Where Sh is the Sherwood number (containing the unknown mass transfer coefficient), Re is the Reynolds
number, and Sc is the Schmidt number. a, m and n are system (geometry) dependent parameters that are
fitted so that the correlation represents measured mass transfer rates as well as possible.
Solution of the model equations
The solution method must be selected according to the final model type. One often encountered model
type results in a set of algebraic equations. Examples of such result from time independent stirred tank
reactor or stage-wise separation process model, such as distillation column. Another typical category is
ordinary differential equations. These result e.g. from time independent tubular reactor or homogeneous
batch reactor, or other plug flow models, such as mass transfer on a distillation tray where bubbles could
be assumed to flow as plug in the liquid phase. Mass transfer occurs between these phases. If there is
enough time, the phases approach equilibrium; otherwise the model can be used to predict tray
efficiencies. Third model type is partial differential equations, which result in when there are significant
variations in many directions, or simultaneous time and space dependent variations.
The models are often solved in such numerical tools (e.g. finite differences), where changes as function of
position are discretized and changes at these discrete points are followed. In this way ordinary differential
equations can be transformed into a set of algebraic equations, or partial differential equations
transformed into a set of ordinary differential equations. If the ordinary differential equation is of first
order, such as tubular or batch reactor model, it is typically solved from a given initial state by marching
short steps in time (batch case) or space (tubular case), until the end. Efficient algorithms are available for
this, as well as other numerical solutions generally leading to some sort of matrix equations.
If the model is algebraic set of equations, that happen to be linear, the system can be written in a matrix
form and solved by using suitable matrix system solution techniques. Perhaps the most classical
formulation of a matrix equation could be the following, where (x) is a vector (column matrix) of known
values, [B] is a known matrix (in this example it could be anything, but typically a square matrix), and (y) is a
vector of unknowns. The equation on the left can be solved by simply multiplying left and right hand sides
by inverted [B] matrix, resulting in the equation on the right.
x   By
y   B1 x 
If the system is not linear, it can be linearized. Linearization generally means that you assume a linear
response, and in some way calculate the linear equation parameters (linearize). Linearization is typically
valid only near the linearization point. The validity region depends on how nonlinear the system really is.
Basically linearization is the same as drawing a tangent to a function at a given point.
y  Ax   b
In some cases, the constant term (b) is neglected during linearization.
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When computer programs, such as Matlab, are used to solve the model equations, the typical "flow chart"
is following:
Here the person using these programs typically specifies the problem statement in the uppermost box
(initialization), and calls for an appropriate solver. This solver could be e.g. ODE integrator (e.g. "ode15s" in
Matlab), solution of nonlinear set of algebraic equations (e.g. "fsolve" in Matlab), or numerical integration
of a given function (e.g. "integral" in Matlab). User model is the function to be evaluated (physical model),
which gives the necessary values for the solver, e.g. derivatives (ode15s), discrepancy functions (balance
errors, fsolve), or integrated function values at solver proposed points (integral). When the solver algorithm
is solved, e.g. tubular reactor solution reaches the outlet or errors in the algebraic equations are small
enough, or the full integral is calculated, the solver returns results to the main program. There the results
can be analyzed, graphs can be plotted etc.
Distributions and population balances
Distributions with respect to independent variables (so called internal coordinates) appear often in
Chemical engineering. If these distributions are important for the process operation or product quality,
they should be modeled in order to be able to predict process performance. One common modeled
distribution is related to particle, bubble or droplet sizes in multiphase systems. The reason could be that
we want to calculate mass transfer area between the phases, by summing up the surface areas of each
particle in our control volume, or predict the product size distribution, e.g. in crystallization or
polymerization.
One class of important parameter that can be calculated from a distribution is its moments. The
distribution moment is defined as

mi 
i


n
L
L
dL

L 0
Here mi is i'th moment of the distribution, n(L) is the number density (number of particles in the control
volume), L is the size (moments could be calculated for other properties as well, but ony size is considered
here for simplicity). One useful property of the moments is that many of them have physical meanings.
Zeroth moment of the number density distribution would be the total number of particles, second moment
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is proportional to the total surface area (which is important for mass transfer), and third moment is
proportional to the total volume of the particles (related to dispersed phase volume fraction). Also some
other well-known properties, such as average or standard deviation of the distribution can be calculated
directly from the moments.
Integration of size distributions is one typical task where numerical integration could be needed. Numerical
integration (here for one variable only) formulas can be often written in the form of quadratures:
b
n
a
i 1
y   f x dx   w i f x i 
Where the integral of a continuous function is approximated by evaluating the function at given quadrature
points xi, multiplying these values by corresponding quadrature weights wi, and summing up each
contribution. Perhaps the simplest numerical integration method would be to divide the interval a-b into
equal distance rectangles, calculate rectangle height from the function value at mid-point of the rectangle,
and then summing up each rectangle areas to approximate the integral. This would lead to equally spaced
quadrature points and equal quadrature weights (b-a)/(number of points). There are, however, much more
efficient quadrature formulas available, e.g. in Matlab.
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