Biological process simulation and optimization

생물공정모사 및 최적화

Biological process simulation and optimization

Major: Interdisciplinary program of the integrated biotechnology

Graduate school of bio- & information technology Youngil Lim (N110), Lab. FACS

Youngil Lim (N110), Lab. FACS

phone: +82 31 670 5200 (secretary), +82 31 670 5207 (direct) phone: +82 31 670 5200 (secretary), +82 31 670 5207 (direct)

Fax: +82 31 670 5445, mobile phone: +82 10 7665 5207 Fax: +82 31 670 5445, mobile phone: +82 10 7665 5207

Email:

Email: [email protected][email protected], homepage: , homepage:  http://http://

hknu.ac.kr/~limyi/index.htm hknu.ac.kr/~limyi/index.htm

Part I. Problem Formulation

Mathematical form :

• Objective function (economic criteria): profit,

variables.

• Process model (constrains): interrelationship

equations).

Part I. Problem Formulation

• Ch. 1: Examples in chemical engineering

• Ch. 2: Process models: material/energy

balances, equilibrium equations, empirical equations.

• Ch. 3 Objective functions: capital

Ch 3. Formulation of the objective function

3.1 Economic objective functions - cost functions, profit functions

3.2 The time value of money in objective functions 3.3 Measures of profitability

Example 3.1 Operating profits as the objective function

m k

x h

n j

x g

x t

s

x c x

c x

f Min

k j

i i i i i i

x

,...

2 , 1 ,

0 )

(

,...

2 , 1 ,

0 )

( , 0

. .

)

(

1 8

7











 

 



 



  





Cost function Profit

function Given data:

1. raw material cost 2. processing cost 3. selling prices

4. raw material availability 5. reactant mass balance

Mass balance

Raw material constraints

See p. 86

- the model can be classified as LP, NLP, MILP, or MINLP

Example 3.2 Capital costs as the objective function

D t

s

D D V

t D S D

f

Min

D



 

 



 

 0 . .

4 ) 2

(

2

 

 

Cost function Given data for capital cost minimization of cylindrical pressure vessel:

1. best dimensions (L/D) ?

2. vessel volume, V_vessel, is fixed

3. compare with the rule-of-thumb, L/D=3.0 4. constant thickness (t) with density () 5. vessel fabrication cost (S): $/kg

2 2

2 2

2 2

4 4

2 2

2 2 4

D L V

D L V

D DL t

S Cost

D DL t

W

D DL t

V

D DL Area

vessel vessel

material





 



 



 

 









 



 





 



 





 



 









Example 3.3 Optimum thickness of insulation

x t

s

C C

x f

Max

x





 0 .

.

)

(

Given data for cost minimization of insulation thickness:

1. heat loss (conduction and convection): Q [kJ/hr]

2. cost per area: C_area [$/m²]=C₀ + C₁x 3. interest rate (r) for 5 years

4. cost per heat loss: H_t [$/kJ]

5. working hour per year: Y[hr/yr]

c c

h k

x

T Q A

T A

h Q

where

Q Q

saving Energy

/ 1 /

,

0





 













hc

k x

T Q A

/ 1 / 

 

A x

C C

r C

H Y Q Q

C where

C C

saving energy

f

t saving

energy

saving energy





















) (

) (

,

) (

1 0

int

0 _

int _

3.2 The time value of money in objective functions

A. Present value and future worth

+ annual cash flow (compound interest, simple interest)

B. Various present and future worth/value

+ present value of a series of payment, F

+ present value of a series of uniform future payment, 1 + future value of a series of payment, P

+ future value of a series of uniform future payment, 1 + capital recovery factor

+ repayment multiplier, r

n n

n n

i P F

i P

F

) 1 (

) 1 (

 





Example 3-4 Paying off a loan

Given data for a loan:

1. interest per year: i= 0.105 2. loan, P=35,000$

3. payment: $325 per month

4. how many months (n) are required to pay off the loan.

 





 





 

 

 

i i i

F

P ( 1 )

... 1 )

1 (

1 )

1 (

1

2





 



 









k

k

F i P

1

1 0 1

) 0 1

(

1 )

1

( 

 





 









 

i i

F i

P

Example 3-5 Selection of the cheapest anodes

Given data for a loan:

1. option 1: - normal anode replacement period = 2 years - its replacement price, P

= $20,000

2. option 2: - impregnated anode replacement period = 6 years - its replacement price, P

= $56,000

3. interest rate = 0.06 per year.

4. prices do not change in 6 years

 





 





 

 



1

( 1 )

1 )

1 ( 1 1

i P i

P

P

 P

 $ 56 , 000

Option 1 Option 2

3.3 Measures of profitability

A. Deterministic approach (probabilistic approach) for profitability + numerous measurement of economic performance (=profit) + Min. of payback period (PBP):

+ Max. of Return on investment (ROI):

most common performance index in economics + Max. of Net present value (NPV)

+ Max. of Internal rate of return (IRR)

B. Comparison of various methods (see p102)

Example 3.3 Optimum thickness of insulation

x t

s

C C

x f

Max

x





 0 .

.

)

(

Given data for cost minimization of insulation thickness:

1. heat loss (conduction and convection): Q [kJ/hr]

2. cost per area: C_area [$/m²]=C₀ + C₁x, C₁=34$/cm /m² 3. interest rate (r) for 5 years

4. cost per heat loss: H_t [$/kJ] = 3.8$/10⁶kJ, 80% thermal efficiency (boiler) 5. working hour per year: Y[hr/yr] = 8000

6. conductivity of insulator: k=0.8 kJ/h/m/K, h_c=32.7kJ/h/m²/K 7. heat exchanger length/diameter, L=100m, D=0.2 m; A=?

8. Temperature difference, T=(260-27)K

hc

k x

T Q A

/ 1 / 

 

c c

h k

x

T Q A

T A

h Q

where

Q Q

saving Energy

/ 1 /

,

0





 













A x

C C

r C

H Y Q Q

C where

C C

saving energy

f

t saving

energy

saving energy





















) (

) (

,

) (

1 0

int

0 _

int _

Optimum value depends on

the criterion for selection

1.6 General procedure for solving optimization problems

1. Analyze the process itself so that the process variables and specific characteristics of interest are defined  make a list of the variables/parameters

2. Determine the criterion for optimization, and specify the objective function w.r.t variables and parameters  performance model

3. Using mathematical expressions, develop a valid process or equipment model that relates the input/output variables. Include both equality and inequality constraints. Use first-principle models (mass/energy balances, equilibrium equations), empirical equations, implicit concepts and external restrictions. Identify the number of degree of freedom.  equality/inequality constraints

4. If the problem formulation is too large in scope,  reduced model development

1. Break it up into manageable parts or

2. Simplify the objective function and model

5. Apply a suitable optimization technique (SQP, GA, GCMC, etc. or Matlab, GAMS, etc.) to the mathematical statement of the problem.

6. Check the answers, and examine the sensitivity of the result to change in the parameters  parameter sensitivity analysis.

Rep eat

Exercise and homework 3

• Select one problem of ch. 3 and solve it using Matlab or others.

• Each student should select a different problem each other.

• If there is no specific value to be needed, please set the

values yourself.