Heat Transfer and mass trtransfer lecture

Heat

1. Heat Transfer

2. Introduction Heat is energy in transition under the motive force of a temperature difference, and the study of heat transfer deals with the rate at which such energy is transferred. Three modes of heat transfer may be distinguished. Conduction Convection Radiation 2

3. Conduction Heat Transfer In conduction, energy is transferred on a molecular scale with no movement of macroscopic portions of matter relative to one another. In solids the conduction of heat is partly due to the impact of adjacent molecules vibrating about their mean positions, and partly due to internal radiation. When the solid is a metal, there are also large numbers of mobile electrons which can easily move through the matter, passing from one atom to another, and they contribute to the redistribution of energy in the metal. 3

4. Convection Heat Transfer Convection heat transfer occurs when temperature differences exist between a fluid and a solid boundary. The redistribution of energy is partly due to conduction and partly due to transport of enthalpy by the motion of the fluid itself. Such motion can be generated by a pump, and this mode of heat transfer is called forced convection. On the other hand, motion may be entirely due to density gradients in the fluid, caused by temperature gradients and this is called free or natural convection. 4

5. Radiation Heat Transfer Radiation heat transfer does not depend on the existence of an intervening medium. Earth receives energy from the sun through radiation heat transfer. All matters at temperatures above absolute zero emit heat in the form of electromagnetic waves. The calculation of radiation heat transfer is based mainly on the Stefan-Boltzmann, Kirchhoff and Lambert Laws. 5

6. Fourier’s Law of Heat Conduction 6 Fourier’s Law states that the rate of flow of heat through a single homogeneous solid is directly proportional to the area (A) of the section at right angles to the direction of heat flow, and to the temperature gradient in the direction of heat flow. T 1 T 2 x dx Y Y

7. Fourier’s Law of Heat Conduction contd.. 7 Hence k – Constant of proportionality k is known as the thermal conductivity of the material (W/ mK ). Let us consider the transfer of heat through section Y-Y.

8. Fourier’s Law of Heat Conduction contd.. 8 Hence or By integration or

9. Fourier’s Law of Heat Conduction contd.. 9 For most solids, the value of the thermal conductivity is approximately constant over a wide range of temperatures and therefore k will be taken as constant. or

10. 1-D Steady Conduction through a Composite Wall 10 A wall built up of three different materials is shown in the figure. Surface temperatures of the wall are T 1 and T 4 and the temperatures of the interfaces are T 2 and T 3. x 1 x 2 x 3 T 1 T 2 T 3 T 4 k 1 k 2 k 3

11. 1-D Steady Conduction through a Composite Wall contd.. 11 For steady flow through the wall, the heat flow rate through successive slabs must be the same for the reasons of continuity, and hence,

12. 1-D Steady Conduction through a Composite Wall contd.. 12 Hence By adding the equations together,

13. 1-D Steady Conduction through a Composite Wall contd.. 13 Hence If U is known as the Overall heat transfer coefficient for the composite wall

14. 1-D Steady Conduction through a Composite Wall contd.. 14 In general for n number of layers

15. Heat Transfer in Solid-Liquid Interfaces Consider the transfer of heat from fluid A to fluid B through a dividing wall of thickness x and thermal conductivity k. 15 k h A h B x T 1 T 2 T A T B Fluid A Fluid B Solid Wall

16. 16 Convective heat transfer coefficients between the wall and fluids A and B are h A and h B respectively. Heat transfer from fluid A to wall per unit area is expressed as: Heat Transfer in Solid-Liquid Interfaces contd.. Heat transfer from wall to fluid B per unit area is expressed as:

17. 17 Heat flow through the wall per unit area is expressed as: Heat Transfer in Solid-Liquid Interfaces contd.. For steady state heat transfer, the heat flowing from fluid A to the wall is equal to the heat flowing through the wall, which is also equal to the heat flowing from the wall to fluid B.

18. 18 Hence Heat Transfer in Solid-Liquid Interfaces contd.. By adding the equations

19. 19 Heat Transfer in Solid-Liquid Interfaces contd.. Where U is called the Overall Heat Transfer Coefficient or

20. Composite Wall & Electrical Analogy 20 Consider the general case of a composite wall surrounded by fluids as shown in the figure. x 1 x 2 x 3 T T 1 T 2 T 3 k 1 k 2 k 3 T A Fluid A Fluid B T n-1 T B T n k n x n ……………. R A R 1 R 2 R 3 R n R B h A h B

21. Composite Wall & Electrical Analogy 21 The most convenient method of solving such a problem is by making use of an electrical analogy. The flow of heat can be thought of as analogous to an electric current. The heat flow is caused by a temperature difference whereas the current flow is caused by a potential difference, hence it is possible to postulate a thermal resistance analogous to an electrical resistance. Thermal Resistance

22. Composite Wall & Electrical Analogy 22 is analogous to I and (T 1 -T 2 ) is analogous to V. The composite wall is analogous to a series of resistances and resistances in series can be added to give the total resistance. Thermal Resistance of the fluid film

23. Composite Wall & Electrical Analogy 23 Total resistance to heat flow is then, For any number of layers of material, Total Resistance

24. Composite Wall & Electrical Analogy 24 Electrical analogy for the overall heat transfer For any number of layers It is proved that It can be seen that that the reciprocal of U is simply the thermal resistance for unit area

25. Composite Wall & Electrical Analogy 25 Hence where or

26. Heat Transfer through a Cylinder One of the most common heat transfer problems in practice is the case of heat being transferred through a pipe or cylinder. 26 r 2 r 1 r T 2 T 1 dr k h out h in

27. Heat Transfer through a Cylinder Apply Fourier’s Law for the small element of unit length in the axial direction. 27 Integrating between the inside and outside surfaces

28. Heat Transfer through a Cylinder 28 From electrical analogy, thermal resistance Thermal resistance through the film of fluid inside and outside surfaces can be expressed as follows.

29. Heat Transfer through a Cylinder 29 From electrical analogy, total thermal resistance and For any number of layers of material,

30. Heat Transfer through a Cylinder 30 Overall heat transfer through the cylinder

31. Heat Flow through a Sphere Consider a hollow sphere of internal radius r 1 and external radius r 2 as shown in the figure. r 2 r 1 r T 2 T 1 dr k 31 h out h in

32. Heat Flow through a Sphere Consider a small element of thickness dr at any radius r. Rate of heat transfer through the sphere is given by: By integrating 32

33. Heat Flow through a Sphere Therefore 33

34. Heat Flow through a Sphere Hence by applying the electrical analogy, we have: Thermal resistance through the film of fluid of inside and outside surfaces can be expressed as follows: and 34

35. Heat Transfer through a Sphere 35 From electrical analogy, total thermal resistance For any number of layers of material,

36. Heat Transfer through a Sphere 36 Overall heat transfer through the sphere

37. Convection This is concerned with the calculation of rates of heat exchange between fluids and solid boundaries. Main resistance to heat flow from a solid wall to a fluid is in a comparatively thin boundary layer adjacent to the wall. Within the boundary layer, viscous forces dominate and main mode of heat transfer is through conduction. Away from the solid wall, pressure forces dominate and heat transfer is due to convection. 37

38. Flow of fluid over a flat plate Fluid flows over a flat plate with a free stream velocity U s . Fluid velocity adjacent to the surface of plate is zero. There is no-slip between fluid and solid surface. In the direction perpendicular to the plate, the stream velocity increases and approaches the free stream velocity. The two regions in fluid, both boundary layer and free stream is shown in the figure below: 38

39. Flow of fluid over a flat plate 39 Free Stream

40. Flow of fluid over a flat plate If the flow approaching the leading edge of the plate is laminar, a laminar boundary layer of thickness δ builds up on the plate. At the edge of the plate δ is zero and it increases gradually with the distance x. The shear stress at any point in the fluid is given by: 40 where μ is the dynamic viscosity

41. Flow of fluid over a flat plate Thickness of the boundary layer increases with the distance x from the leading edge of the plate according to: 41 where Re x is the Reynolds number based on the distance x:

42. Flow of fluid over a flat plate The shear stress at the wall is expressed as: The average friction factor f is found to be: Beyond a certain critical distance x cr , from the leading edge, the flow becomes fully turbulent as shown below: 42 where f is the dimensionless friction factor

43. Flow of fluid over a flat plate The critical distance, in terms of the Reynolds number is found from experiments as: 43 U s U s U s where Re cr = 500,000

44. Radiation Heat Transfer Any matter with temperature above absolute zero (0 K) emits electromagnetic radiation Intensity of radiant energy flux depends upon the temperature of the body and the nature of its surface Radiation heat transfer does not need a medium In order to understand radiation heat transfer, study of electro-magnetic spectrum is needed 44

45. Electromagnetic Spectrum 45

46. Black Body Radiation A black body absorbs all energy that reaches it completely It is also a perfect emitter 46

47. Radiative Properties When radiation strikes a surface, a portion of it is reflected, and the rest penetrates the surface Out of the radiation that enters the object, some proportion is absorbed by the material, and the rest is transmitted through 47

48. Radiative Properties contd.. The ratio of reflected energy to the incident energy is called Reflectivity ( ρ) Transmissivity ( τ ) is defined as the fraction of the incident energy that is transmitted through the object A bsorptivity ( α ) is defined as the fraction of the incident energy that is absorbed by the object Hence, All three radiative properties have typical values between 0 and 1 48

49. Stefan-Boltzmann Law The radiant energy emitted by a blackbody per unit time can be determined from the Stefan-Boltzmann Law: 49 where - Blackbody emissive power (W) A - Surface area (m 2 ) T - Absolute temperature of the surface (K) σ - Stefan-Boltzmann constant (5.67 x 10 -8 W/m 2 K 4 )

50. Emissive Power 50

51. Emissivity ( ε ) Emissivity is the ratio of a surface's ability to emit radiant energy compared with the ability of a perfect black body of the same area at the same temperature Emissivity is a dimensionless constant having values between 0 and 1 A material with high emissivity is efficient in both absorbing radiant energy as well as emitting it Therefore, a good absorber is also a good emitter 51 ε – Emissivity of surface

52. Grey Body A Grey body has a constant emissivity over all wavelengths and for all temperatures For a grey body at all temperatures, where α and ε are total absorptivity and the total emissivity over all wavelengths 52

53. Radiation Exchange Consider a body of emissivity ε at a temperature T 1 , completely surrounded by black surroundings at a lower temperature T 2 . Energy leaving the body is completely absorbed by the surroundings, where rate of radiant energy emission per unit area = Rate of energy emitted by the black surroundings per unit area = Fraction of this energy which is absorbed by the body depends on the absorptivity of the same 53

54. Radiation Exchange contd.. For a grey body at all temperatures and hence: Rate of energy absorption = Hence, rate of heat transferred from the body to its surroundings per unit area is: 54

55. Net Radiant Heat Transfer between Surfaces 55

56. Net Radiant Heat Transfer between Surfaces contd.. Figure shows two arbitrary surfaces exchanging radiant energy between them If surfaces are “black” then the net radiant energy exchanged: View factor is purely a geometrical parameter (ranges between 0 and 1) that accounts for the effect of orientation on radiant heat exchange between surfaces Furthermore or 56 where F 12 - View factor (Shape factor) between the surfaces

57. Radiant heat transfer between surfaces Radiant heat exchange between two surfaces is given by: View factor is purely a geometrical parameter that accounts for the effect of orientation on radiant heat exchange between surfaces. The view factor ranges between 0 and 1 57 where F - View factor (Shape factor) between surfaces