Mathematical Modeling of Chemical Processes

1
Mathematical Modeling of Chemical Processes
2
Mathematical Model

• a representation of the essential aspects of an
  existing system (or a system to be constructed)
  which represents knowledge of that system in a
  usable form
• Everything should be made as simple as possible,
  but no simpler.

3
Uses of Mathematical Modeling

• to improve understanding of the process
• to optimize process design/operating conditions
• to design a control strategy for the process
• to train operating personnel

4
General Modeling Principles

• The model equations are at best an approximation
  to the real process.
• Adage All models are wrong, but some are
  useful.
• Modeling inherently involves a compromise between
  model accuracy and complexity on one hand, and
  the cost and effort required to develop the
  model, on the other hand.
• Process modeling is both an art and a science.
  Creativity is required to make simplifying
  assumptions that result in an appropriate model.
• Dynamic models of chemical processes consist of
  ordinary differential equations (ODE) and/or
  partial differential equations (PDE), plus
  related algebraic equations.

5
A Systematic Approach for Developing Dynamic
Models

• State the modeling objectives and the end use of
  the model. They determine the required levels of
  model detail and model accuracy.
• Draw a schematic diagram of the process and label
  all process variables.
• List all of the assumptions that are involved in
  developing the model. Try for parsimony the
  model should be no more complicated than
  necessary to meet the modeling objectives.
• Determine whether spatial variations of process
  variables are important. If so, a partial
  differential equation model will be required.
• Write appropriate conservation equations (mass,
  component, energy, and so forth).

6
A Systematic Approach for Developing Dynamic
Models

• Introduce equilibrium relations and other
  algebraic equations (from thermodynamics,
  transport phenomena, chemical kinetics, equipment
  geometry, etc.).
• Perform a degrees of freedom analysis to ensure
  that the model equations can be solved.
• Simplify the model. It is often possible to
  arrange the equations so that the dependent
  variables (outputs) appear on the left side and
  the independent variables (inputs) appear on the
  right side. This model form is convenient for
  computer simulation and subsequent analysis.
• Classify inputs as disturbance variables or as
  manipulated variables.

7

• Conservation Laws

Theoretical models of chemical processes are
based on conservation laws.
Conservation of Mass
Conservation of Component i
8
Conservation of Energy
The general law of energy conservation is also
called the First Law of Thermodynamics. It can be
expressed as
9
Example

• Simple tank Problem

10
Degrees of Freedom Analysis

• List all quantities in the model that are known
  constants (or parameters that can be specified)
  on the basis of equipment dimensions, known
  physical properties, etc.
• Determine the number of equations NE and the
  number of process variables, NV. Note that time
  t is not considered to be a process variable
  because it is neither a process input nor a
  process output.
• Calculate the number of degrees of freedom, NF
  NV - NE.
• Identify the NE output variables that will be
  obtained by solving the process model.
• Identify the NF input variables that must be
  specified as either disturbance variables or
  manipulated variables, in order to utilize the NF
  degrees of freedom.

Chapter 2
11
Stirred-Tank Heating Process
Chapter 2
Stirred-tank heating process with constant
holdup, V.
12
Stirred-Tank Heating Process (contd.)

• Assumptions
• Perfect mixing thus, the exit temperature T is
  also the temperature of the tank contents.
• The liquid holdup V is constant because the inlet
  and outlet flow rates are equal.
• The density r and heat capacity C of the liquid
  are assumed to be constant. Thus, their
  temperature dependence is neglected.
• Heat losses are negligible.

Chapter 2
13
Degrees of Freedom Analysis for the Stirred-Tank
Model
3 parameters 4 variables 1 equation
Thus the degrees of freedom are NF 4 1 3.
The process variables are classified as
Chapter 2
1 output variable T 3 input variables Ti, w, Q
For temperature control purposes, it is
reasonable to classify the three inputs as
2 disturbance variables Ti, w 1 manipulated
variable Q
14
Degrees of Freedom Analysis
15
Degrees of Freedom Analysis

• System comprises of only 2 chemical species A and
  B
• Can write only 2 independent mass balances
• write for species A and species B
• write overall balance one component balance
  (either for species A or B)

16
Degrees of Freedom Analysis
17
Degrees of Freedom Analysis
Focus on the control volume (A ?z) over the time
interval t to t ?t
18
Degrees of Freedom Analysis
19
Dimensional Analysis

• A conceptual tool often applied to understand
  physical situations involving a mix of different
  kinds of physical quantities.
• It is routinely used by physical scientists and
  engineers to check the plausibility of derived
  equations.
• Only like dimensioned quantities may be added,
  subtracted, compared, or equated.
• When unlike dimensioned quantities appear
  opposite of the "" or "-" or "" sign, that
  physical equation is not plausible, which might
  prompt one to correct errors before proceeding to
  use it.
• When like dimensioned quantities or unlike
  dimensioned quantities are multiplied or divided,
  their dimensions are likewise multiplied or
  divided.

20
Dimensional Analysis

• Dimensions of a physical quantity is associated
  with symbols, such as M, L, T which represent
  mass, length and time
• Assume to determine the power required to drive a
  house fan. Torque is chosen as the dependent
  variable and the following are known physical
  variables
• Fan diameter (d)
• Fan design (R)
• Air density (r)
• Rotative speed (n)

21
(No Transcript)
22

• Dividing torque by density gives
• t/r divided by D5n2 gives

23

• Final analysis
• The torque for a given design R is proportional
  to the dimensionless product

24
Buckingham p theorem

• every physically meaningful equation involving n
  variables can be equivalently rewritten as an
  equation of n m dimensionless parameters, where
  m is the number of fundamental dimensions used
• it provides a method for computing these
  dimensionless parameters from the given
  variables, even if the form of the equation is
  still unknown

25
Buckingham p theorem

• In mathematical terms, if we have a physically
  meaningful equation such as
• where the qi  are the n  physical variables, and
  they are expressed in terms of k  independent
  physical units, then the above equation can be
  restated as
• where the pi are dimensionless parameters
  constructed from the qi  by p n - k  equations
  of the form
• where the exponents mi  are constants.

26
Example

• If a moving fluid meets an object, it exerts a
  force on the object, according to a complicated
  (and not completely understood) law. We might
  suppose that the variables involved under some
  conditions to be the speed, density and viscosity
  of the fluid, the size of the body (expressed in
  terms of its frontal area A), and the drag force.

27
Example

• Buckingham p theorem states that there will be
  two such groups

28

• Development of Dynamic Models
• Illustrative Example A Blending Process

An unsteady-state mass balance for the blending
system
29
or where w1, w2, and w are mass flow rates.

• The unsteady-state component balance is

The corresponding steady-state model was derived
in Ch. 1 (cf. Eqs. 1-1 and 1-2).
30
The Blending Process Revisited
For constant , Eqs. 2-2 and 2-3 become
31
Equation 2-13 can be simplified by expanding the
accumulation term using the chain rule for
differentiation of a product
Substitution of (2-14) into (2-13) gives
Substitution of the mass balance in (2-12) for
in (2-15) gives
After canceling common terms and rearranging
(2-12) and (2-16), a more convenient model form
is obtained
32
(No Transcript)