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Mathematical Modeling of Chemical Processes

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.

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

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.

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).

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.

- Conservation Laws

Theoretical models of chemical processes are

based on conservation laws.

Conservation of Mass

Conservation of Component i

Conservation of Energy

The general law of energy conservation is also

called the First Law of Thermodynamics. It can be

expressed as

Example

- Simple tank Problem

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

Stirred-Tank Heating Process

Chapter 2

Stirred-tank heating process with constant

holdup, V.

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

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

Degrees of Freedom Analysis

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)

Degrees of Freedom Analysis

Degrees of Freedom Analysis

Focus on the control volume (A ?z) over the time

interval t to t ?t

Degrees of Freedom Analysis

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.

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)

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

- Final analysis
- The torque for a given design R is proportional

to the dimensionless product

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

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.

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.

Example

- Buckingham p theorem states that there will be

two such groups

- Development of Dynamic Models
- Illustrative Example A Blending Process

An unsteady-state mass balance for the blending

system

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).

The Blending Process Revisited

For constant , Eqs. 2-2 and 2-3 become

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

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