Mws Che Nle Txt Newton Examples

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Chapter 03.04Newton-Raphson Method of Solving a NonlinearEquation More ExamplesChemical Engineering

Example 1

You have a spherical storage tank containing oil. The tank has a diameter of . You are

asked to calculate the height h to which a dipstick 8 ft long would be wet with oil when

immersed in the tank when it contains of oil.

ft6

3ft6

h

r

Spherical Storage Tank

Dipstick

Figure 1 Spherical storage tank problem.

The equation that gives the height h of the liquid in the spherical tank for the given volume

and radius is given by 08197.39 23 hhhf

Use the Newton-Raphson method of finding roots of equations to find the height to which

the dipstick is wet with oil. Conduct three iterations to estimate the root of the aboveequation. Find the absolute relative approximate error at the end of each iteration and the

number of significant digits at least correct at the end of each iteration.

h

03.04.1


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03.04.2 Chapter 03.04

Solution

8197.39 23 hhhf hhhf 183 2

Let us take the initial guess of the root of 0hf as 10 h .

Iteration 1

The estimate of the root is

0

001

hf

hfhh

11813

8197.31911

2

23

15

1803.41

2786901 .

721310.

The absolute relative approximate error a at the end of Iteration 1 is

1001

01

h

hha

100721310

1721310

.

.

%636.38

The number of significant digits at least correct is 0, as you need an absolute relative

approximate error of or less for one significant digit to be correct in your result.%5

Iteration 2The estimate of the root is

1

112

hf

hfhh

721310187213103

8197.37213109721310721310

2

23

..

...

423.11

48764.0721310

.

042690.0721310 .678620.

The absolute relative approximate error a at the end of Iteration 2 is

1002

12

h

hha


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Newton-Raphson Method More Equations: Chemical Engineering 03.04.3

100678620

72131067862.0

.

.

%2907.6

The number of significant digits at least correct is 0.

Iteration 3The estimate of the root is

2

223

hf

hfhh

6786201867862.03

8197.3678620967862067862.0

2

23

.

..

834.10

012536.0678620

.

0011572.0678620 .

677470.The absolute relative approximate error a at the end of Iteration 3 is

1003

23

h

hha

100677470

678620677470

.

..

%17081.0

Hence the number of significant digits at least correct is given by the largest value of for

which

m

m

a

2

105.0 m

2105.017081.0

m

21034162.0

m 234162.0log 4665.234162.0log2 mSo

2m

The number of significant digits at least correct in the estimated root 0.67747 is 2.

NONLINEAR EQUATIONSTopic Newton-Raphson Method-More ExamplesSummary Examples of Newton-Raphson Method

Major Chemical Engineering

Authors Autar KawDate August 7, 2009

Web Site http://numericalmethods.eng.usf.edu
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