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26/08/2013 1 School of Chemical Engineering Sunday, August 25, 2013 Life Impact | The University of Adelaide Introduction to Process Modeling - Introduction to Excel - Statistical analysis - Regression analysis - Solving a nonlinear algebraic equation School of Chemical Engineering Life Impact | The University of Adelaide Slide 1 finding v for a given p and T Poling, B. E., J. M. Prausnitz, et al. (2001). The properties of gases and liquids. New York, McGraw-Hill. Volume of a real gas Slide 1 v v b v RT p 2 2 2 1 2 2 2 1 2699 . 0 5422 . 1 37464 . 0 1 ; 2 0778 . 0 ; 45724 . 0 c c c c c c c c c T T b b RT bP RT v P Z RT ap is critical pressure is critical temperature is critical volume is the acentric factor c p c T c v

Nonlinear Equations

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Page 1: Nonlinear Equations

26/08/2013

1

School of Chemical Engineering

Sunday, August 25, 2013

Life Impact | The University of Adelaide

Introduction to Process Modeling - Introduction to Excel - Statistical analysis - Regression analysis - Solving a nonlinear algebraic equation

School of Chemical Engineering

Life Impact | The University of Adelaide Slide 1

• finding v for a given p and T

Poling, B. E., J. M. Prausnitz, et al. (2001). The properties of gases and liquids. New York,

McGraw-Hill.

Volume of a real gas

Slide 1

vvbv

RTp

2

2

21

2

2

2

12699.05422.137464.01

;2

0778.0 ;

45724.0

c

c

c

c

cc

c

c

c

TT

bb

RT

bP

RT

vPZ

RT

ap

is critical pressure is critical temperature is critical volume is the acentric factor

cp

cT

cv

Page 2: Nonlinear Equations

26/08/2013

2

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations

Slide 2

• graphical method

• bisection

• secant

• false position

• Newton-Raphson

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: graphical method

Slide 3

• to calculate when

axf x

axfxg

Page 3: Nonlinear Equations

26/08/2013

3

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: graphical method

Slide 4

• Find the molar volume of propane at 300 K and 10 bar by using the following EoS.

where

vvbv

RTp

2

2212

22

2

c

13

126992.054226.137464.01

45724.0 0778.0

2 0.152

2762.0 Z 0.200 48.42 83.369

r

c

c

c

c

ccc

Ta

p

TRa

p

RTb

bb

molcmvbarpKT

School of Chemical Engineering

Life Impact | The University of Adelaide

-20.0000

-15.0000

-10.0000

-5.0000

0.0000

5.0000

10.0000

15.0000

20.0000

0 500 1000 1500 2000 2500 3000

pa

nd

p-

10

(b

ar)

v (cm3/mol)

p(v) 10 bar p(v) - 10 bar

Numerical methods for solving nonlinear equations: graphical method

Slide 5

• Find the molar volume of propane at 300 K and 10 bar by using the following EoS.

vvbv

RTp

2

Vg

Vl

Page 4: Nonlinear Equations

26/08/2013

4

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 6

The bisection method is based on the following theorem:

An equation f(x) = 0, where f(x) is a real continuous function, has at least one root between xl and xu only if f(xl)×f(xu) < 0.

xl

xu

f(x)

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 7

What about if f(x) is a real continuous function and f(xl)×f(xu) > 0.

xl xu

f(x)

xl xu

f(x)

Page 5: Nonlinear Equations

26/08/2013

5

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 8

Algorithm

1. Select two points xl and xu such that f(xl)×f(xu) < 0.

xl

xu

f(x)

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 9

Algorithm

2. Calculate a new point xm as xm = (xl + xu) / 2.

xl

xu

f(x)

Page 6: Nonlinear Equations

26/08/2013

6

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 10

Algorithm

3. Change the xl or xu according to the following criteria:

a) if f(xl)×f(xm) < 0, then the root lies between xl and xm; then xl = xl, xu = xm;

b) if f(xm)×f(xu) < 0, then the root lies between xm and xu; then xl = xm, xu = xu;

c) if f(xm) < tolerance value, then xm is the root; stop the algorithm.

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 11

Algorithm

2. Calculate the next point xm as xm = (xl + xu) / 2.

xl

xu

f(x)

xm

Page 7: Nonlinear Equations

26/08/2013

7

School of Chemical Engineering

Life Impact | The University of Adelaide Slide 12

• Excel example

Find the molar volume of propane at 300 K and 10 bar by using the following EoS.

where

vvbv

RTp

2

2212

22

2

c

13

126992.054226.137464.01

45724.0 0778.0

2 0.152

2762.0 Z 0.200 48.42 83.369

r

c

c

c

c

ccc

Ta

P

TRa

P

RTb

bb

molcmvbarpKT

Numerical methods for solving nonlinear equations: bisection method

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Slide 13

Excel example

• First we set up the information:

Page 8: Nonlinear Equations

26/08/2013

8

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Excel example

We calculate values in the table according to the bisection algorithm.

first two input values vm = (vl + vu) / 2

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Excel example

We calculate values in the table according to the bisection algorithm.

Use the IF function to set up vl and vu. vm = (vl + vu) / 2

Page 9: Nonlinear Equations

26/08/2013

9

School of Chemical Engineering

Life Impact | The University of Adelaide

Numerical methods for solving nonlinear equations: bisection method

Excel example

We calculate values in the table according to the bisection algorithm.

solution

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

Excel has a built in function that allows calculating the root of a function. Example in Excel 2007

Page 10: Nonlinear Equations

26/08/2013

10

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

Excel has a built in function that allows calculating the root of a function. Example in Excel 2007

This is the cell where we have f(x).

This is the value that we need, so it is 0 if f(x) – a = 0.

This is the cell where x is stored.

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

In our example

pcal - p is defined in $G$27.

Page 11: Nonlinear Equations

26/08/2013

11

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

In our example

We need: pcal - p = 0.

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

In our example

We change the volume in $G$25 in order to find the value for which pcal - p = 0.

vvbv

RTp

2

Page 12: Nonlinear Equations

26/08/2013

12

School of Chemical Engineering

Life Impact | The University of Adelaide

Goal Seek

In our example

The solution depends on the initial value selected.

If the first value is v = 85 cm3/mol:

If the first value is v = 2600 cm3/mol:

vvbv

RTp

2

Goal Seek

Goal Seek

School of Chemical Engineering

Life Impact | The University of Adelaide

-20.0000

-15.0000

-10.0000

-5.0000

0.0000

5.0000

10.0000

15.0000

20.0000

0 500 1000 1500 2000 2500 3000

pa

nd

p-

10

(b

ar)

v (cm3/mol)

p(v) 10 bar p(v) - 10 bar

Goal Seek

The solution depends on the initial value selected.

initial condition for the liquid phase

solution for the liquid phase

initial condition for the gas phase

solution for the gas phase

Goal Seek