Conduction with Generation

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1 Conduction with Generation
Thermal energy may be generated or consumed due to conversion from some other energy form. If thermal energy is generated in the material at the expense of some other energy form, we have a source: is +ve Deceleration and absorption of neutrons in a nuclear reactor Exothermic reactions Conversion of electrical to thermal energy: where I is the current, Re the electrical resistance, V the volume of the medium If thermal energy is consumed we have a sink: is -ve Endothermic reactions Chapter 3 Chee 318 24

2 The Plane Wall Heat diffusion equation (eq. 2.3) : General Solution:
Consider one-dimensional, steady-state conduction in a plane wall of constant k, with uniform generation, and asymmetric surface conditions: Heat diffusion equation (eq. 2.3) : General Solution: Boundary Conditions: Chapter 3 Chee 318 25

3 Temperature Profile (3.3) Profile is parabolic.
Heat flux not independent of x What happens when: Chapter 3 Chee 318 26

4 Symmetrical Distribution
When both surfaces are maintained at a common temperature, Ts,1= Ts,2 = Ts (3.4a) What is the location of the maximum temperature? (3.4b) Chapter 3 Chee 318 27

5 Symmetrical Distribution
Note that at the plane of symmetry: Equivalent to adiabatic surface Chapter 3 Chee 318 28

6 Calculation of surface temperature Ts
In equations (3.4a) and (3.4b) the surface temperature, Ts is needed. Boundary condition at the wall: Substituting (dT/dx)x=L from equation (3.4a): (3.5) Chapter 3 Chee 318 29

7 Example (Problem 3.71 textbook)
The steady-state temperature distribution in a composite plane wall of three different materials, each of constant thermal conductivity, is shown in the schematic below. Does heat generation occur in any of sections A,B, or C? Based on the schematic, what is the boundary condition at location (4)? Comment on the relative magnitudes of q2” and q3”. Comment on the relative magnitudes of kA and kB. Chapter 3 Chee 318 30

8 Example (Problem 3.72 textbook)
A plane wall of thickness 0.1 m and thermal conductivity 25 W/m.K having uniform volumetric heat generation of 0.3 MW/m3 is insulated on one side, while the other side is exposed to a fluid at 92°C. The convection heat transfer coefficient between the wall and the fluid is 500 W/m2.K. Determine the maximum temperature in the wall. Chapter 3 Chee 318 31

9 Radial Systems Spherical Wall (Shell) Solid Cylinder (Circular Rod)
Cylindrical (Tube) Wall Spherical Wall (Shell) Solid Cylinder (Circular Rod) Solid Sphere Chapter 3 Chee 318 32

10 Radial Systems Heat diffusion equation in the r-direction for steady-state conditions: General Solution: Boundary Conditions: L Temperature profile: (3.6) Calculation of surface temperature: (3.7) and Chapter 3 Chee 318 33

11 Example (Problem 3.84 textbook)
A cylindrical shell of inner and outer radii ri and ro, respectively, is filled with a heat-generating material that provides a uniform volumetric generation rate. The inner surface is insulated, while the outer surface of the shell is exposed to a fluid with a convection coefficient h. Obtain an expression for the steady-state temperature distribution T(r) in the shell. Determine an expression for the heat rate q’(ro) at the outer radius of the shell in terms of the heat generation rate and the shell dimensions Chapter 3 Chee 318 34

12 Extended Surfaces (Fins)
An extended surface (also know as a combined conduction-convection system or a fin) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction Chapter 3 Chee 318 35

13 Extended Surfaces (Fins)
Extended surfaces may exist in many situations but are commonly used as fins to enhance heat transfer by increasing the surface area available for convection (and/or radiation). They are particularly beneficial when h is small, as for a gas and natural convection. Solutions for various fin geometries can be found in the literature (see for example Table 3.4 textbook). Chapter 3 Chee 318 36

14 PERSAMAAN KELESTARIAN ENERGI PADA SIRIP
Chapter 3 Chee 318

15 Sirip dengan luas tampang seragam
Konstanta Integrasi C1 dan C2 tergantung syarat batas Syarat batas pertama pada x = 0  Chapter 3 Chee 318

16 Syarat batas kedua pada x = L ada 4 KASUS A. Konveksi di Ujung Sirip
Chapter 3 Chee 318

17 B.Perpindahan Panas Konveksi Ujung Sirip diabaikan
Karena perpan di ujung sirip diabaikan  C.Suhu di Ujung Sirip Tertentu (TL) D. Sirip Panjang  TL ≈ T∞ Chapter 3 Chee 318

18 Chapter 3 Chee 318

19 Chapter 3 Chee 318

20 Unjuk Kerja Sirip Efektivitas sirip (εf) :Nisbah antara Laju perpan melalui sirip dgn laju perpan dipangkal sirip bila tanpa sirip. Harus diusahakan sebesar mungkin Untuk kasus D  εf = M/(h Ac,b θb) Implikasinya : bahan sirip k besar, koefisien konveksi kecil  natural, gas sirip tipis melebar, memanjang. Untuk kasus B  εf = M tanh mL/(h Ac,b θb)≥ 2 Harga tanh mL maksimum ≈ 1, untuk tanh mL= 0.99  mL ≈ 2.50 Dengan demikian kasus B dapat didekati manakala L ≈ 2.50/m Chapter 3 Chee 318

21 3.6.3. Unjuk Kerja Sirip Tahanan sirip (Rt,f) didefinisikan sebagai
Bila tahanan termal di pangkal sirip Maka efektivitas sirip dinyatakan sebagai Efisiensi sirip (ηf) “Nisbah antara laju perpan sirip dengan laju perpan maksimum/ideal (seluruh sirip pada Tb)” Untuk kasus B  Chapter 3 Chee 318

22 Unjuk Kerja Sirip Panjang sirip dikoreksi  Lc = L + t/2 ; Lc = L + D/4 Kesalahan karena pendekatan dapat diabaikan bila (ht/k) atau (hD/2k) ≤ Bila w >> t  P ≈ 2w dan Luas fin terkoreksi Ap = Lc x t  mLc = (2h/kAp)0.5 Lc1.5 Dengan demikian η = C.η [(h/k.Ap )0.5 Lc1.5] qf = ηf qmax = ηf h Af θb Perhatikan baik-baik harga Af Chapter 3 Chee 318

23 Unjuk Kerja Sirip Chapter 3 Chee 318

24 Unjuk Kerja Sirip Chapter 3 Chee 318

25 Unjuk Kerja Sirip Efisiensi Permukaan Menyeluruh (ηo) menggambarkan keadaan permukaan yang dilekati sejumlah sirip dan permukaan yang tidak dilekati sirip. Dengan qt = laju perpan total = qf + qb Chapter 3 Chee 318

26 Unjuk Kerja Sirip Chapter 3 Chee 318