Kuliah 12 Konduksi Metoda Numerik
-
Upload
perpindahanpanas -
Category
Documents
-
view
418 -
download
3
Transcript of Kuliah 12 Konduksi Metoda Numerik
![Page 1: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/1.jpg)
1
NUMERICAL HEAT NUMERICAL HEAT CONDUCTIONCONDUCTION
DR. Ir. Nazarudin Sinaga, MS
Laboratorium Efisiensi dan Konservasi EnergiUniversitas Diponegoro
![Page 2: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/2.jpg)
IntroductionIntroduction
2
Analytical solution methods such as those presented in Chapter 2 are based on solving the governing differential equation together with the boundary conditions.
They result in solution functions for the temperature at every point in the medium.
Numerical methods, on the other hand, are based on replacing the differential equation by a set of n algebraic equations for the unknown temperatures at n selected points in the medium, and the simultaneous solution of these equations results in the temperature values at those discrete points.
![Page 3: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/3.jpg)
3
There are several ways of obtaining the numerical formulation of a heat conduction problem, such as the finite difference method, the finite element method, the boundary element method, and the energy balance (or control volume) method.
Each method has its own advantages and disadvantages, and each is used in practice.
In this chapter we will use primarily the energy balance approach since it is based on the familiar energy balances on control volumes instead of heavy mathematical formulations, and thus it gives a better physical feel for the problem.
Besides, it results in the same set of algebraic equations as the finite difference method.
![Page 4: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/4.jpg)
4
FIGURE 5–2 Analytical solution methods are limited to simplified problems in simple geometries.
![Page 5: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/5.jpg)
5
FIGURE 5–3The approximate numerical solution of a real-world problem may be more accurate than the exact analytical) solution of an oversimplified model of that problem.
![Page 6: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/6.jpg)
6
FIGURE 5–4Some analytical solutions are very complex and difficult to us
![Page 7: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/7.jpg)
Finite Difference FormulationFinite Difference FormulationOf Differential EquationsOf Differential Equations
7
The numerical methods for solving differential equations are based on replacing the differential equations by algebraic equations.
In the case of the popular finite difference method, this is done by replacing the derivatives by differences.
![Page 8: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/8.jpg)
8
Derivatives are the building blocks of differential equations, and thus we first give a brief review of derivatives.
Consider a function f that depends on x, as shown in Figure 5–6.
The first derivative of f(x) at a point is equivalent to the slope of a line tangent to the curve at that point and is defined as
![Page 9: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/9.jpg)
9
FIGURE 5–6 The derivative of a function at a point represents the slope of the function at that point.
![Page 10: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/10.jpg)
10
If we don’t take the indicated limit, we will have the following approximate relation for the derivative:
This approximate expression of the derivative in terms of differences is the finite difference form of the first derivative.
The equation above can also be obtained by writing the Taylor series expansion of the function f about the point x,
![Page 11: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/11.jpg)
11
and neglecting all the terms in the expansion except the first two.
The first term neglected is proportional to x2, and thus the error involved in each step of this approximation is also proportional to x2.
However, the commutative error involved after M steps in the direction of length L is proportional to x since M x2 = (L/x)x2 = L x.
Therefore, the smaller the x, the smaller the error, and thus the more accurate the approximation.
![Page 12: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/12.jpg)
12
FIGURE 5–7Schematic of the nodes and the nodal temperatures used in the development of the finite difference formulation of heat transfer in a plane wall.
![Page 13: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/13.jpg)
13
Using Eq. 5–6, the first derivative of temperature dT/dx at the midpoints m-1/2 and m+1/2 of the sections surrounding the node m can be expressed as
The second derivative of temperature at node m can be expressed as
![Page 14: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/14.jpg)
14
The the differential equation
which is the governing equation for steady one- dimensional heat transfer in a plane wall with heat generation and constant thermal conductivity, can be expressed in the finite difference form as (Fig. 5–8)
![Page 15: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/15.jpg)
15
FIGURE 5–8The differential equation is valid at every point of a medium, whereas thefinite difference equation is valid at discrete points (the nodes) only.
![Page 16: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/16.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
16
Metode Relaksasi
Untuk menyelesaikan soal konduksi panas secara numerik, kita membagi sistem menjadi sejumlah subvolume yang kecil dan memberi nomor acuan pada tiap subvolume dan asumsikan bahwa tiap subvolume bersuhu yang sama dengan suhu titik pusatnya dan kita mengganti sistem fisik dengan jaringan batang-batang khayal.
![Page 17: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/17.jpg)
17
![Page 18: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/18.jpg)
18
![Page 19: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/19.jpg)
19
Untuk mendapatkan persamaan temperattur untuk nodal-m nodal maka berlakukan hukum kelestarian energi pada nodal-m, yang dikelilingi oleh nodal m-1 dan nodal m+1
![Page 20: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/20.jpg)
20
![Page 21: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/21.jpg)
21
![Page 22: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/22.jpg)
22
![Page 23: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/23.jpg)
23
Dengan asumsi distribusi suhu linier, maka konduktansinya adalah :
atau
![Page 24: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/24.jpg)
24
Kalau disederhanakan:
Pada dasar sirip (gb. 3-11b) suhu T1 sama dengan suhu dinding dan tetap konstan:
![Page 25: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/25.jpg)
25
Pada ujung sirip panas berpindah secara konveksi:
![Page 26: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/26.jpg)
26
![Page 27: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/27.jpg)
27
![Page 28: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/28.jpg)
28
![Page 29: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/29.jpg)
29
![Page 30: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/30.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
30
Sistem Dua Dimensi:
Metode numerik dapat dengan mudah diperluas bagi sistem dua maupun tiga dimensi seperti terlihat pada gambar 3-13:
![Page 31: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/31.jpg)
31
![Page 32: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/32.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
32
Keseimbangan panas keadaan Stedi pada suatu titik dibagian dalam adalah:
Dengan mensubstitusi persamaan konduksi untuk masing-masing titik/nodal maka diperoleh:
![Page 33: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/33.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
33
Atau:
![Page 34: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/34.jpg)
34
![Page 35: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/35.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
35
![Page 36: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/36.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
36
![Page 37: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/37.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
37
![Page 38: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/38.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
38
![Page 39: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/39.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
39
![Page 40: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/40.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
40
![Page 41: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/41.jpg)
Konduksi 2D dan 3D StediKonduksi 2D dan 3D Stedi
41
![Page 42: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/42.jpg)
42
![Page 43: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/43.jpg)
43
![Page 44: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/44.jpg)
44
![Page 45: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/45.jpg)
45
![Page 46: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/46.jpg)
46
FIGURE 5–26 Schematic for Example 5–3 and the nodal network (the boundaries of volume elements of the nodes are indicated by dashed lines).
![Page 47: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/47.jpg)
47
![Page 48: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/48.jpg)
48
![Page 49: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/49.jpg)
49
![Page 50: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/50.jpg)
50
![Page 51: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/51.jpg)
51
![Page 52: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/52.jpg)
52
![Page 53: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/53.jpg)
53
![Page 54: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/54.jpg)
54
![Page 55: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/55.jpg)
55
![Page 56: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/56.jpg)
56
![Page 57: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/57.jpg)
57
![Page 58: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/58.jpg)
58
![Page 59: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/59.jpg)
59
![Page 60: Kuliah 12 Konduksi Metoda Numerik](https://reader033.fdocuments.us/reader033/viewer/2022050808/547730335806b55f068b4617/html5/thumbnails/60.jpg)
Powerpoint TemplatesPage 60
The EndThe EndTerima Terima kasihkasih