Chapter 05.02
Direct Method of Interpolation – More Examples
Chemical Engineering
Example 1
To find how much heat is required to bring a kettle of water to its boiling point, you are
asked to calculate the specific heat of water at 61C . The specific heat of water is given as a
function of time in Table 1.
Table 1 Specific heat of water as a function of temperature.
Temperature, T Specific heat, C p
C
 J 


 kg  C 
22
4181
42
4179
52
4186
82
4199
100
4217
Determine the value of the specific heat at T  61C using the direct method of interpolation
and a first order polynomial.
05.02.1
05.02.2
Chapter 05.02
Figure 1 Specific heat of water vs. temperature.
Solution
For first order polynomial interpolation (also called linear interpolation), we choose the
specific heat given by
C p T   a 0  a1T
y
x1 , y1 
f1 x 
x0 , y0 
x
Figure 2 Linear interpolation.
Direct Method of Interpolation – More Examples: Chemical Engineering
05.02.3
Since we want to find the specific heat at T  61C , and we are using a first order
polynomial, we need to choose the two data points that are closest to T  61C that also
bracket T  61C to evaluate it. The two points are T0  52 and T1  82 .
Then
T0  52, C p T0   4186
T1  82, C p T1   4199
gives
C p 52  a0  a1 52  4186
C p 82  a0  a1 82  4199
Writing the equations in matrix form, we have
1 52 a0  4186
1 82  a   4199

 1  

Solving the above two equations gives
a0  4163.5
a1  0.43333
Hence
C p T   a0  a1T
 4163.5  0.43333T , 52  T  82
At T  61 ,
C p 61  4163.5  0.4333361
 4189.9
J
kg  C
Example 2
To find how much heat is required to bring a kettle of water to its boiling point, you are
asked to calculate the specific heat of water at 61C . The specific heat of water is given as a
function of time in Table 2.
Table 2 Specific heat of water as a function of temperature.
Temperature, T Specific heat, C p
C
 J 


 kg  C 
22
4181
42
4179
52
4186
82
4199
100
4217
05.02.4
Chapter 05.02
Determine the value of the specific heat at T  61C using the direct method of interpolation
and a second order polynomial. Find the absolute relative approximate error for the second
order polynomial approximation.
Solution
For second order polynomial interpolation (also called quadratic interpolation), we choose
the specific heat given by
C p T   a0  a1T  a2T 2
y
x1 , y1 
x2 , y2 
f 2 x 
 x0 , y 0 
x
Figure 3 Quadratic interpolation.
Since we want to find the specific heat at T  61C , and we are using a second order
polynomial, we need to choose the three data points that are closest to T  61C that also
bracket T  61C to evaluate it. The three points are T0  42, T1  52, and T2  82.
Then
T0  42, C p T0   4179
T1  52, C p T1   4186
T2  82, C p T2   4199
gives
C p 42  a0  a1 42  a2 42  4179
2
C p 52  a0  a1 52  a2 52  4186
2
C p 82  a0  a1 82  a2 82  4199
Writing the three equations in matrix form, we have
1 42 1764  a0  4179
1 52 2704  a   4186

 1  

1 82 6724 a 2  4199
2
Solving the above three equations gives
Direct Method of Interpolation – More Examples: Chemical Engineering
05.02.5
a0  4135.0
a1  1.3267
a2  6.6667  10 3
Hence
C p T   4135.0  1.3267T  6.6667 10 3 T 2 , 42  T  82
At T  61 ,
2
C p 61  4135.0  1.326761  6.6667  10 3 61
J
kg  C
The absolute relative approximate error a obtained between the results from the first and
second order polynomial is
4191.2  4189.9
a 
 100
4191.2
 0.030063%
 4191.2
Example 3
To find how much heat is required to bring a kettle of water to its boiling point, you are
asked to calculate the specific heat of water at 61C . The specific heat of water is given as a
function of time in Table 3.
Table 3 Specific heat of water as a function of temperature.
Temperature, T Specific heat, C p
C
 J 


 kg  C 
22
4181
42
4179
52
4186
82
4199
100
4217
Determine the value of the specific heat at T  61C using the direct method of interpolation
and a third order polynomial. Find the absolute relative approximate error for the third order
polynomial approximation.
Solution
For third order polynomial interpolation (also called cubic interpolation), we choose the
specific heat given by
C p T   a0  a1T  a2T 2  a3T 3
05.02.6
Chapter 05.02
y
 x3 , y 3 
f 3 x 
x1 , y1 
 x0 , y 0 
x2 , y2 
x
Figure 4 Cubic interpolation.
Since we want to find the specific heat at T  61C , and we are using a third order
polynomial, we need to choose the four data points closest to T  61C that also bracket
T  61C to evaluate it. The four points are T0  42, T1  52, T2  82 and T3  100.
(Choosing the four points as T0  22 , T1  42 , T2  52 and T3  82 is equally valid.)
Then
T0  42, C p T0   4179
T1  52,
T2  82,
C p T1   4186
C p T2   4199
T3  100, C p T3   4217
gives
C p 42  a0  a1 42  a2 42  a3 42  4179
2
3
C p 52  a0  a1 52  a2 52  a3 52  4186
2
3
C p 82  a0  a1 82  a2 82  a3 82  4199
2
3
C p 100  a0  a1 100  a2 100  a3 100  4217
Writing the four equations in matrix form, we have
1 42 1764 7.4088  10 4   a 0   4179

 

5 
1 52 2704 1.4061  10   a1    4186
1 82 6724 5.5137  10 5  a 2   4199

  

10 6
1 100 10000
  a3  4217
Solving the above four equations gives
2
3
Direct Method of Interpolation – More Examples: Chemical Engineering
05.02.7
a0  4078.0
a1  4.4771
a2  0.062720
a3  3.1849  10 4
Hence
C p T   a0  a1T  a2T 2  a3T 3
 4078.0  4.4771T  0.062720T 2  3.1849  10 4 T 3 , 42  T  100
T 61  4078.0  4.477161  0.06272061  3.1849  10 4 61
J
 4190.0
kg  C
The absolute relative approximate error a obtained between the results from the second
2
and third order polynomial is
4190.0  4191.2
a 
 100
4190.0
 0.027295%
INTERPOLATION
Topic
Direct Method of Interpolation
Summary Examples of direct method of interpolation.
Major
Chemical Engineering
Authors
Autar Kaw
February 12, 2016
Date
Web Site http://numericalmethods.eng.usf.edu
3