Optimal Design of Energy Systems

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Optimal Design of Energy Systems

Chapter 4 Equation Fitting

Min Soo KIM

Department of Mechanical and Aerospace Engineering

Seoul National University

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Chapter 4. Equation Fitting

4.1 Mathematical Modeling

performance characteristics of equipment behavior of processes

thermodynamic properties

• Equation development is required

Facilitate the process of system simulation

Develop a mathematical statement for optimization

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4.2 Matrices

11 12 1

1 2

n

m m mn

a a a

 

 

 



 



Order of matrix

m n 

  A

^T Transpose of a matrix [A]

ex>

 

1 4 2 5 3 6 A

 

 

  

 

 

 

¹ ² ³

4 5 6

A T  

  

 

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4.2 Matrices

• Simultaneous linear equations

1 2 3

1 2

1 2 3

2 3 6

3 1

4 2 0

x x x

x x

x x x

  

 

  

1 2 3

2 1 3 6

1 3 0 1

4 2 1 0

x x x

      

      

     

      

     

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4.2 Matrices

• Determinant

11 12 13

21 22 23

31 32 33

a a a

D a a a

a a a

 

^{11 22 33} ¹² ^{23 31} ^{13 21 32}

31 22 13 21 12 33 11 23 32

a a a a a a a a a a a a a a a a a a

  

  

11 11 21 21 13 31

a A a A a A

  

cofactor of

22 33 23 32

a a a a

 

a11 ^{[( 1)} ^]

i

[ ]

i j ij

submatrix formed by striking out A

th row and j th column of A

  

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4.3 Solution of Simultaneous Equations

11 12 13 1 1

21 22 23 2 2

31 32 33 3 3

[ ][ ] [ ]

a a a x b

A X a a a x b B

a a a x b

     

     

        

     

     

1 1 1 1 2 2 1 3 3 1

2 1 1 2 2 2 2 3 3 2

3 1 1 3 2 2 3 3 3 3

a x a x a x b a x a x a x b a x a x a x b

  

 Simultaneous linear equations

 Matrix form

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Chapter 4. Equation Fitting

1 2 3

2 1 1 3

1 2 2 9

1 0 3 0

x x x

      

      

     

     

     

[ ] [ ] i

i

A matrix with B matrix substituted in th column

x  A

ex>

1

3 1 1

9 2 2

0 0 3 45

2 1 1 15 3

1 2 2

1 0 3

x



   

 



2

2 3 1

1 9 2

1 0 3 2 1 1 2

1 2 2

1 0 3

x



   



3

2 1 3

1 2 9

1 0 0

2 1 1 1

1 2 2

1 0 3

x



  



• Crammer’s rule

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Coefficient matrix [A]

 

 

 

 

 

 

Triangular matrix

Back substitution

• Gaussian elimination

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4.4 Polynomial representation

2

0 1 2

n

y  a  a x  a x    a x

n

# of data point = n+1 → exact expression

> → best fit

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Chapter 4. Equation Fitting

4.6

Simplification when the independent variable is uniformly spaced

 Points are equally spaced

 Derive 4^th degree polynomial

 n=4

2

0 1 0 2 0

3 4

3 0 4 0

( ) ( )

n n

y y a x x a x x

R R

n n

a x x a x x

R R

   

        

   

       

1 0 n n 1

x x x x x _

     

2

0 1 2

n

y  a  a x  a x  a xn

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Eq. (4.16) 4

x

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4.6

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4.6

2 3 4

1 0 1 0 1 0 1 0

1 1 2 3 4

1 2 3 4

4(x x ) 4(x x ) 4(x x ) 4(x x )

y a a a a

R R R R

a a a a

         

          

   

0 0

1 1 1 2 3 4

2 2 1 2 3 4

3 3 1 2 3 4

4 4 1 2 3 4

, ,

, 2 4 8 1 6

, 3 9 2 7 6 4

, 4 1 6 6 4 2 5 6

x x y y

x x y y a a a a

x x y a a a a

 

      

     

 if substitute (x1,y1)

 Substitute all the points to Equation (4.16)

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2

0 1 2

y  a  a x  a x

4.7 Lagrange Interpolation

1( 2)( 3) 2( 1)( 3) 3( 1)( 2)

y  c x  x x  x  c x x x  x  c x  x x  x

1, 1 1( 1 2)( 1 3)

x  x y  c x  x x  x

2, 2 2( 2 1)( 2 3)

x  x y  c x  x x  x

3, 3 3( 3 1)( 3 2)

x  x y  c x  x x  x

1 1

( ) ( )

n n

j i

i

i j i j i i

x x ommiting x x

y y

x x ommiting x x

 

 



 

 

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4.8 Function of two variables

2

1 : 1 1 1 1

S  P a  b Q  c Q

2

2 : 2 2 2 2

S  P a  b Q  c Q

2

3 : 3 3 3 3

S  P a  b Q  c Q

2

0 1 2

2

0 1 2

2

0 1 2

( ) ( ) ( )

a S A A S A S b S B B S B S c S C C S C S

  

( ) ( ) ( ) 2

P a S b S Q c S Q

   

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Chapter 4. Equation Fitting

4.9 Exponential Forms

ln ln ln

y bxm

y b m x



 

y  b axm

If y approaches some value b, as orx   x  

Estimate b

Calculate m with log-log plot (y-b vs. x) Fitting (y vs. x^m)

Correct value of b



 Log-log plot (y=bx^m)

Curve y=b+ax^m

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4.10 Best fit : Method of Least Squares

The sum of the squares of the deviation is a minimum

Misuse of least square method Example of least square method

 Misuses of least square method

(a) Questionable correlation (b) Applying too low degree

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Chapter 4. Equation Fitting

• Method of least squares for ^y ^{ }^a ^bx

2( _i _i) 0

z a bx y

a

    





2 1

( ) min

m

i i

i

z a bx y





  

2( _i _i) _i 0

z a bx y x

b

    





i i

ma  b  x   y

2

i i i i

a  x  b  x   x y

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<Example> provide a best fit (least square)

0 1

y  a  a x

0 1

4a 14a  49.9

x y

1 4.9

3 11.2

4 13.7

6 20.1

0 1

14a  62a  213.9

1.908 3.019

y   x

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• Method of least squares for ^y ^{ }^a ^bx ^ ^cx²

2

2 3

2 3 4 2

1 _i _i _i

i i i i i

a b x c x y

a x b x c x y x

  

   

sin ln y  a x b x

(sin ) 2

(ln ) 2

(sin ) (ln ) sin

(sin ) (ln ) ln

i

x

i i i i

x

i i i i

a x b x y x

 

  

• For

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Chapter 4. Equation Fitting

4.11 Method of Least Squares Applied to Nonpolynomial Forms

• Method of least squares

→ apply to equation with constant coefficients cf)

y  sin 2 ax  bx

^c

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4.12 The are of equation fitting

• Choice of the form of the equation

Polynomials with negative exponent⁽¹⁾ Exponential eq.⁽²⁾

Gompertz eq⁽³⁾ combination

, 1

cx

y  ab where b c 

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