1 Teaching Innovation - Entrepreneurial - Global The Centre for Technology enabled Teaching & Learning D M I E T R, Wardha DTEL DTEL (Department for Technology Enhanced Learning)

DEPARTMENT OF MECHANICAL ENGINEERING V -SEMESTER HEAT TRANSFER 2 UNIT NO. 02 CONDUCTION WITH INTERNAL HEAT GENERATION

CHAPTER 1:- SYLLABUSDTEL 1)Conduction with internal heat generation for plane wall, Cylinder and sphere, 2)Extended Surfaces, Types of Fins, 3)Fins of uniform cross section area, temperature distribution and their heat transfer rate, 4)Fin efficiency and effectiveness, 5)Error in temperature measurement, Steady state Heat transfer, 6)Lumped Heat Capacity analysis, 7)Heister charts, Biot number, 8)Fourier Number and their Significance 3

DTEL CARTESIAN COORDINATE 4 LECTURE 01:- CONDUCTION WITH INTERNAL HEAT GENERATION General conduction equation in Cartesian Coordinate System Rate of energy generation

DTEL CARTESIAN COORDINATE 5 LECTURE 01:- CONDUCTION WITH INTERNAL HEAT GENERATION For an isotropic and homogeneous material:

DTEL CYLINDRICAL COORDINATE 6 LECTURE 02:- CONDUCTION WITH INTERNAL HEAT GENERATION General conduction equation based on Polar Cylindrical Coordinates

DTEL SPHERICAL COORDINATE 7 LECTURE 03:- CONDUCTION WITH INTERNAL HEAT GENERATION General conduction equation based on Polar Spherical Coordinates X Y

DTEL SPHERICAL COORDINATE 8 LECTURE 03:- CONDUCTION WITH INTERNAL HEAT GENERATION Thermally Heterogeneous Materials

DTEL EXTENDED SURFACES / FINS 9 LECTURE 04:- CONDUCTION WITH INTERNAL HEAT GENERATION EXTENDED SURFACES / FINS Convection: Heat transfer between a solid surface and a moving fluid is governed by the Newton’s cooling law: q = hA(T s -T  ). Therefore, to increase the convective heat transfer, one can  Increase the temperature difference (T s -T  ) between the surface and the fluid.  Increase the convection coefficient h. This can be accomplished by increasing the fluid flow over the surface since h is a function of the flow velocity and the higher the velocity, the higher the h. Example: a cooling fan.  Increase the contact surface area A. Example: a heat sink with fins.

DTEL EXTENDED SURFACES / FINS 10 LECTURE 04:- CONDUCTION WITH INTERNAL HEAT GENERATION Extended Surface Analysis x TbTb A C is the cross-sectional area P: the fin perimeter A c : the fin cross-sectional area

DTEL EXTENDED SURFACES / FINS 11 LECTURE 04:- CONDUCTION WITH INTERNAL HEAT GENERATION Extended Surface Analysis (contd….)

DTEL EXTENDED SURFACES / FINS 12 LECTURE 04:- CONDUCTION WITH INTERNAL HEAT GENERATION Extended Surface Analysis (contd...) For example: assume the tip is insulated and no heat transfer d  /dx(x=L)=0 The temperature distribution is given by The fin heat transfer rate is given by

DTEL EXTENDED SURFACES / FINS 13 LECTURE 05:- CONDUCTION WITH INTERNAL HEAT GENERATION Temperature distribution for fins of different configurations

DTEL EXTENDED SURFACES / FINS 14 LECTURE 05:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Design Total heat loss: q f =Mtanh(mL) for an adiabatic fin, or q f =Mtanh(mL C ) if there is convective heat transfer at the tip TbTb TT

DTEL EXTENDED SURFACES / FINS 15 LECTURE 06:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Effectiveness How effective a fin can enhance heat transfer is characterized by the fin effectiveness  f : Ratio of fin heat transfer and the heat transfer without the fin. For an adiabatic fin:

DTEL EXTENDED SURFACES / FINS 16 LECTURE 06:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Effectiveness (contd...)  To increase  f, the fin’s material should have higher thermal conductivity, k.  It seems to be counterintuitive that the lower convection coefficient, h, the higher  f. But it is not because if h is very high, it is not necessary to enhance heat transfer by adding heat fins. Therefore, heat fins are more effective if h is low. Observation: If fins are to be used on surfaces separating gas and liquid. Fins are usually placed on the gas side. (Why?)

DTEL EXTENDED SURFACES / FINS 17 LECTURE 07:- CONDUCTION WITH INTERNAL HEAT GENERATION  P/AC should be as high as possible. Use a square fin with a dimension of W by W as an example: P=4W, AC=W2, P/AC=(4/W). The smaller W, the higher the P/AC, and the higher  f.  Conclusion: It is preferred to use thin and closely spaced (to increase the total number) fins. Fin Effectiveness (contd...)

DTEL EXTENDED SURFACES / FINS 18 LECTURE 07:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Effectiveness (contd...)

DTEL EXTENDED SURFACES / FINS 19 LECTURE 07:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Efficiency

DTEL EXTENDED SURFACES / FINS 20 LECTURE 07:- CONDUCTION WITH INTERNAL HEAT GENERATION T(x)<T b for heat transfer to take place Total fin heat transfer q f Real situation Ideal situation For infinite k T(x)=T b, the heat transfer is maximum Ideal heat transfer q max TbTb x x Fin Efficiency (contd…)

DTEL EXTENDED SURFACES / FINS 21 LECTURE 07:- CONDUCTION WITH INTERNAL HEAT GENERATION Fin Efficiency (cont.) Use an adiabatic rectangular fin as an example:

DTEL EXTENDED SURFACES / FINS 22 LECTURE 08:- CONDUCTION WITH INTERNAL HEAT GENERATION Overall Fin Efficiency Overall fin efficiency for an array of fins: Define terms: A b : base area exposed to coolant A f : surface area of a single fin A t : total area including base area and total finned surface, A t =A b +NA f N: total number of fins qbqb qfqf

DTEL EXTENDED SURFACES / FINS 23 LECTURE 08:- CONDUCTION WITH INTERNAL HEAT GENERATION Overall Fin Efficiency (contd…)

DTEL MODES OF HEAT TRANSFER 24 LECTURE 01:- CONDUCTION WITH INTERNAL HEAT GENERATION