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Mathematical Modeling of Chemical Processes

Mathematical Model (Eykhoff, 1974)“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.

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.

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Table 2.1. A Systematic Approach for Developing Dynamic Models

1. State the modeling objectives and the end use of the model. They determine the required levels of model detail and model accuracy.

2. Draw a schematic diagram of the process and label all process variables.

3. 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.

4. Determine whether spatial variations of process variables are important. If so, a partial differential equation model will be required.

5. Write appropriate conservation equations (mass, component, energy, and so forth).

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6. Introduce equilibrium relations and other algebraic equations (from thermodynamics, transport phenomena, chemical kinetics, equipment geometry, etc.).

7. Perform a degrees of freedom analysis (Section 2.3) to ensure that the model equations can be solved.

8. 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.

9. Classify inputs as disturbance variables or as manipulated variables.

Table 2.1. (continued)C

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Modeling Approaches Physical/chemical (fundamental, global)

• Model structure by theoretical analysis Material/energy balances Heat, mass, and momentum transfer Thermodynamics, chemical kinetics Physical property relationships

• Model complexity must be determined (assumptions)

• Can be computationally expensive (not real-time)

• May be expensive/time-consuming to obtain• Good for extrapolation, scale-up• Does not require experimental data to obtain

(data required for validation and fitting)

• Conservation Laws

Theoretical models of chemical processes are based on conservation laws.

Conservation of Mass

rate of mass rate of mass rate of mass(2-6)

accumulation in out

Conservation of Component i

rate of component i rate of component i

accumulation in

rate of component i rate of component i(2-7)

out produced

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Conservation of Energy

The general law of energy conservation is also called the First Law of Thermodynamics. It can be expressed as:

rate of energy rate of energy in rate of energy out

accumulation by convection by convection

net rate of heat addition net rate of work

to the system from performed on the sys

the surroundings

tem (2-8)

by the surroundings

The total energy of a thermodynamic system, Utot, is the sum of its internal energy, kinetic energy, and potential energy:

int (2-9)tot KE PEU U U U

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Black box (empirical)

•Large number of unknown parameters

•Can be obtained quickly (e.g., linear regression)

•Model structure is subjective

•Dangerous to extrapolate

Semi-empirical

•Compromise of first two approaches

•Model structure may be simpler

•Typically 2 to 10 physical parameters estimated (nonlinear regression)

•Good versatility, can be extrapolated

•Can be run in real-time

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• linear regression

• nonlinear regression

• number of parameters affects accuracy of model, but confidence limits on the parameters fitted must be evaluated

• objective function for data fitting – minimize sum of squares of errors between data points and model predictions (use optimization code to fit parameters)

• nonlinear models such as neural nets are becoming popular (automatic modeling)

2210 xcxccy

/1 teKy

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Number of sightings of storks

Number of births (West Germany)

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

•Development of Dynamic Models•Illustrative Example: A Blending Process

An unsteady-state mass balance for the blending system:

rate of accumulation rate of rate of(2-1)

of mass in the tank mass in mass out

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• The unsteady-state component balance is:

1 1 2 2

ρ(2-3)

d V xw x w x wx

dt

The corresponding steady-state model was derived in Ch. 1 (cf. Eqs. 1-1 and 1-2).

1 2

1 1 2 2

0 (2-4)

0 (2-5)

w w w

w x w x wx

or

where w1, w2, and w are mass flow rates.

1 2

ρ(2-2)

d Vw w w

dt

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The Blending Process Revisited

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

1 2 (2-12)dV

w w wdt

1 1 2 2 (2-13)

d Vxw x w x wx

dt

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Equation 2-13 can be simplified by expanding the accumulation term using the “chain rule” for differentiation of a product:

(2-14)

d Vx dx dVV x

dt dt dt

Substitution of (2-14) into (2-13) gives:

1 1 2 2 (2-15)dx dV

V x w x w x wxdt dt

Substitution of the mass balance in (2-12) for in (2-15) gives:

/dV dt

1 2 1 1 2 2 (2-16)dx

V x w w w w x w x wxdt

After canceling common terms and rearranging (2-12) and (2-16), a more convenient model form is obtained:

1 2

1 21 2

1(2-17)

(2-18)

dVw w w

dt

w wdxx x x x

dt V V

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Stirred-Tank Heating Process

Figure 2.3 Stirred-tank heating process with constant holdup, V.

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Stirred-Tank Heating Process (cont’d.)

Assumptions:

1. Perfect mixing; thus, the exit temperature T is also the temperature of the tank contents.

2. The liquid holdup V is constant because the inlet and outlet flow rates are equal.

3. The density and heat capacity C of the liquid are assumed to be constant. Thus, their temperature dependence is neglected.

4. Heat losses are negligible.

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For the processes and examples considered in this book, itis appropriate to make two assumptions:

1. Changes in potential energy and kinetic energy can be neglected because they are small in comparison with changes in internal energy.

2. The net rate of work can be neglected because it is small compared to the rates of heat transfer and convection.

For these reasonable assumptions, the energy balance inEq. 2-8 can be written as

int (2-10)dU

wH Qdt

int the internal energy of

the system

enthalpy per unit mass

mass flow rate

rate of heat transfer to the system

U

H

w

Q

denotes the differencebetween outlet and inletconditions of the flowingstreams; therefore

-Δ wH = rate of enthalpy of the inlet

stream(s) - the enthalpyof the outlet stream(s)

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For a pure liquid at low or moderate pressures, the internal energy is approximately equal to the enthalpy, Uint , and H depends only on temperature. Consequently, in the subsequent development, we assume that Uint = H and where the caret (^) means per unit mass. As shown in Appendix B, a differential change in temperature, dT, produces a corresponding change in the internal energy per unit mass,

H

ˆ ˆintU H

ˆ ,intdU

intˆ ˆ (2-29)dU dH CdT

where C is the constant pressure heat capacity (assumed to be constant). The total internal energy of the liquid in the tank is:

int intˆ (2-30)U VU

Model Development - IC

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An expression for the rate of internal energy accumulation can be derived from Eqs. (2-29) and (2-30):

int (2-31)dU dT

VCdt dt

Note that this term appears in the general energy balance of Eq. 2-10.

Suppose that the liquid in the tank is at a temperature T and has an enthalpy, . Integrating Eq. 2-29 from a reference temperature Tref to T gives,

H

ˆ ˆ (2-32)ref refH H C T T

where is the value of at Tref. Without loss of generality, we assume that (see Appendix B). Thus, (2-32) can be written as:

ˆrefH H

ˆ 0refH

ˆ (2-33)refH C T T

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ˆ (2-34)i i refH C T T

Substituting (2-33) and (2-34) into the convection term of (2-10) gives:

ˆ (2-35)i ref refwH w C T T w C T T

Finally, substitution of (2-31) and (2-35) into (2-10)

(2-36)idT

V C wC T T Qdt

Model Development - IIIFor the inlet stream

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steam-heating: s vQ w H

( ) (1) i s v

dTV C wC T T w H

dt

0 ( ) (2) si vwC T T w H

subtract (2) from (1)

( ) ( ) ss v

dTV C wC T T w w H

dt

divide by wC

( )

vss

HV dTT T w w

w dt wC

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Define deviation variables (from set point)

1

1

is desired operating point

( ) from steady state

Vnote that and

wdy

note when 0dt

General linear ordinary differential equation solution:

s ss

v vp

p

p

y T T T

u w w w T

H HV dyy u K

w dt wC wC

y K u

dyy K u

dt

1

sum of exponential(s)

Suppose u 1 (unit step response)

( ) 1t

py t K e

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Example 1:Example 1:C0=T C,90=T C,40=T o

ioo

i

wC(T - T ) = w Hi s v6

s

v

o

4

3 3

3

4

w =0.83 10 g hr

H =600 cal g

C=l cal g C

w=10 kg hr

=10 kg m

V=20 m

2 10 kgV

4

4

V 2 10 kg2hr

w 10 kg hr

dynamic model2dy

dt= -y + 6 10 u-5

y T Tu w ws s

s.s. balance:

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Step 1: t=0 double ws

final

T(0) = T y(0) = 0

hrg10+0.83=u 6

2dy

dt= -y + 50

y = 50 l - e-0.5t

T = y + T = 50 + 90 = 140 Csso

Step 2: maintain

Step 3: then set

hr 24 / C140 = T o

u = 0, w = 833 kg hrs

2dy

dt= -y + 6 10 u, y(0) = 50-5

Solve for u = 0y = 50e t y 0

-0.5t (self-regulating, but slow)

how long to reach y = 0.5 ?

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Step 4: How can we speed up the return from 140°C to 90°C?

ws = 0 vs. ws = 0.83106 g/hr

at s.s ws =0

y -50°C T 40°C

Process DynamicsProcess control is inherently concerned with unsteady state behavior (i.e., "transient response", "process dynamics")

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Stirred tank heater: assume a "lag" between heating element temperature Te, and process fluid temp, T.

heat transfer limitation = heA(Te – T)

Energy balances

Tank:

Chest:

At s.s.

Specify Q calc. T, Te

2 first order equations 1 second order equation in T

Relate T to Q (Te is an intermediate variable)

i e e

dTwCT +h A(T -T)-wCT=mC

dt

dt

dTCm=T)-A(Th-Q e

eeee

0dt

dT 0,

dt

dT e

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fixed T Q-Q=u T-T=y i

Note Ce 0 yields 1st order ODE (simpler model)

Fig. 2.2

h R

1=q

v

Rv: line resistance

Rv

uwC

1y

dt

dy

w

m

wC

Cm

Ah

Cm

dt

yd

Awh

Cmm ee

ee

ee2

2

ee

ee

linear ODE

If

nonlinear ODE

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57)-(2 1

hR

qdt

dhA

vi

* (2-61)i v i v

dhA q C ρgh q C h

dt

ghpP

pressureambient : P P-P=q aa*

vC

ghP p

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Table 2.2. Degrees of Freedom Analysis

1. 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.

2. 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.

3. Calculate the number of degrees of freedom, NF = NV - NE.

4. Identify the NE output variables that will be obtained by solving the process model.

5. 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.

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Thus the degrees of freedom are NF = 4 – 1 = 3. The process variables are classified as:

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 for the Stirred-Tank Model:

3 parameters:

4 variables:

1 equation: Eq. 2-36

, ,V C

, , ,iT T w Q

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Biological Reactions

• Biological reactions that involve micro-organisms and enzyme catalysts are pervasive and play a crucial role in the natural world.

• Without such bioreactions, plant and animal life, as we know it, simply could not exist.

• Bioreactions also provide the basis for production of a wide variety of pharmaceuticals and healthcare and food products.

• Important industrial processes that involve bioreactions include fermentation and wastewater treatment.

• Chemical engineers are heavily involved with biochemical and biomedical processes.

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Bioreactions• Are typically performed in a batch or fed-batch reactor.

• Fed-batch is a synonym for semi-batch.

• Fed-batch reactors are widely used in the pharmaceutical and other process industries.

• Bioreactions:

• Yield Coefficients:

substrate more cells + products (2-90)cells

/ (2-92)P Smass of product formed

Ymass of substrate consumed to form product

/ (2-91)X Smass of new cells formed

Ymass of substrate consumed to form new cells

Fed-Batch Bioreactor

Figure 2.11. Fed-batch reactor for a bioreaction.

Monod Equation

Specific Growth Rate

(2-93)gr X

max (2-94)s

S

K S

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• Modeling Assumptions

1. The exponential cell growth stage is of interest.2. The fed-batch reactor is perfectly mixed.3. Heat effects are small so that isothermal reactor operation can

be assumed.4. The liquid density is constant.5. The broth in the bioreactor consists of liquid plus solid

material, the mass of cells. This heterogenous mixture can be approximated as a homogenous liquid.

6. The rate of cell growth rg is given by the Monod equation in (2-93) and (2-94).

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• General Form of Each Balance

/ (2-96)P Xmass of product formed

Ymass of new cells formed

• Modeling Assumptions (continued)

7. The rate of product formation per unit volume rp can be expressed as

/ (2-95)p P X gr Y r

where the product yield coefficient YP/X is defined as:

8. The feed stream is sterile and thus contains no cells.

(2-97)Rate of accumulation rate in rate of formation

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• Individual Component Balances

• Cells:

• Product:

• Substrate:

• Overall Mass Balance

• Mass:

( )(2-98)g

d XVV r

dt

1 1(2-100)f g P

X / S P / S

d( SV )F S V r V r

dt Y Y

( )(2-101)

d VF

dt

(2-99)p

d PVVr

dt

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