Matakuliah
Tahun
: S0262-Analisis Numerik
: 2010
Engineering Aplication-3
Pertemuan 12
Material Outline
• Engineering Application
– 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 oC
22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m)
0
4.9
9.1
13.7
18.3
22.9
27.2
2.3
Locate the thermocline!
Heat Transfer
T oC
22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m)
0
4.9
9.1
13.7
18.3
22.9
27.2
2.3
The location of thermocline at inflection point
d2T/dz2=0. Several steps needs to be performed
1. Smoothing the data: use 3rd 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
d2T/dz2=0. Several steps needs to be performed.
3. The inflection point will be the maximum of dT/dz
T oC
22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m)
0
4.9
9.1
13.7
18.3
22.9
27.2
2.3
f3(z)
f3 ( z) 
Li ( z ) 
f3(z)
3
 L ( z) f ( z )
i 1
i
i
3
z  zj
j 0, j i
zi  z j

Heat Transfer
T oC
22.8 22.8
22.8
20.6
13.9
11.7
11.1
11.1
Z (m)
0
4.9
9.1
13.7
18.3
22.9
27.2
2.3
Find the the lagrange interpolation f3(z) using the first
4 data and the last 4 data and use them to find
dT/dz, etc……………………..