Matakuliah : S0262-Analisis Numerik
Tahun : 2010

– Heat Transfer

 Heat Transfer
Lakes in temperature zone can be thermally stratified especially in tropical area such as Indonesia. Such stratification can divide lake vertically in to 2 layers: epilimnion and hypolimnion seperated by thermocline layer. The hard task is to locate the thermocline. The location of thermocline is at deplection point.

 Heat Transfer
A set of data of temperature as a function of depth (z) was collected and given below.
T o C 22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m) 0 2.3
4.9
9.1
13.7
18.3
22.9
27.2
Locate the thermocline!

 Heat Transfer
T o C 22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m) 0 2.3
4.9
9.1
13.7
18.3
22.9
27.2
The location of thermocline at inflection point
 d 2 T/dz 2 =0. Several steps needs to be performed
1. Smoothing the data: use 3 rd order lagrange interpolation to interpolate the data such that the values of T are available for z: 0 - 28 for any values of Z
2. Use the function found from step-1 to find dT/dz for z:0 - 28

 Heat Transfer
The location of thermocline at inflection point
 d 2 T/dz 2 =0. Several steps needs to be performed.
3. The inflection point will be the maximum of dT/dz
T o C 22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m) 0 2.3
4.9
9.1
13.7
18.3
22.9
27.2
f
3
(z) f
3
( z )

L i
( z )
 j i
3 

1
L i
( z )



0
3
, j i z z i

 f ( z i
) z j z j f
3
(z)

 Heat Transfer
T o C 22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m) 0 2.3
4.9
9.1
13.7
18.3
22.9
27.2
Find the the lagrange interpolation f
3
(z) using the first
4 data and the last 4 data and use them to find dT/dz, etc……………………..