HMT Lec-2 of Week 05 of Mechanical Engineering.pptx

HMT Lec-2 of Week 05 of Mechanical Engineering.pptx

1. HEAT & MASS TRANSFER Dr. Muhammad Usman Lecture # 2

2. Introduction to Conduction Understand Fourier’s Law. Derive heat diffusion equation. Apply to various geometries & materials. How does heat move inside materials? How do material properties affect conduction? How do we model conduction mathematically Objectives Key questions

3. The Conduction Rate Equation Steady-state heat conduction experiment T 1 >T 2 Heat is always transferred in the direction of decreasing temperature. The direction of heat flow will always be normal to a surface of constant temperature, called an isothermal surface.

4. The Conduction Rate Equation Relationship between coordinate system, heat flow direction, and temperature gradient in one dimension Recognizing that the heat flux is a vector quantity, we can write a more general statement of the conduction rate equation ( Fouriers law ) as follows:

5. Thermal Conductivity Range of thermal conductivity for various states of matter Thermal conductivity, a transport property, provides an indication of the rate at which energy is transferred by the diffusion process. F or an isotropic material the thermal conductivity is independent of the direction of transfer, kx = ky = kz = k

6. Thermal Conductivity Transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibrational waves . When viewed as a particle-like phenomenon, the lattice vibration quanta are termed phonons. In pure metals, the electron contribution to conduction heat transfer dominates Whereas in nonconductors and semiconductors, the phonon contribution is dominant. Solid State

7. Thermal Conductivity Kinetic theory yields the following expression for the thermal conductivity: Solid State For conducting materials such as metals, C= Ce is the electron specific heat per unit volume, is the mean electron velocity, and λ mfp = λ e is the electron mean free path. In nonconducting solids, C = C ph is the phonon specific heat, is the average speed of sound, and λ mfp = λ ph is the phonon mean free path.  

8. Thermal Conductivity Solid State k e is inversely proportional to the electrical resistivity, ρ e . For pure metals, k e >> k ph . For alloys, k ph is not negligible. For nonmetallic solids, k is determined primarily by k ph F or crystalline, nonmetallic solids such as diamond and beryllium oxide, k ph can be quite large, exceeding values of k associated with good conductors , such as aluminum.

9. Thermal Conductivity Electron or phonon trajectories in ( a ) a relatively thick film and ( b ) a relatively thin film with boundary effects. Large L / λ mfp Small L / λ mfp k x < k y < k λ mfp / L is a dimensionless parameter known as the Knudsen number. Large Knudsen numbers ( small L / λ mfp ) suggest potentially significant nano- or microscale effects. Solid State-Micro- and Nanoscale Effects

10. Thermal Conductivity Solid State-Micro- and Nanoscale Effects Measured thermal conductivity of yttria-stabilized zirconia as a function of temperature and mean grain size, L

11. Thermal Conductivity The thermal conductivity of gases and liquids is generally smaller than that of solids. Why? Fluid State is the mean molecular speed, λ mfp is the mean free path, ρ is the density, kB is Boltzmann’s constant, d is the diameter of the gas Molecule, and p is the pressure.  

12. Thermal Conductivity Fluid State The temperature dependence of the thermal conductivity of selected gases at normal pressures. Molecular diameters ( d ) are in nm. Molecular weights ( M ) of the gases are also shown. Thermal conductivity can be expressed as

13. Thermal Conductivity Fluid State The temperature dependence of the thermal conductivity of selected nonmetallic liquids under saturated conditions.

14. Thermal Conductivity Liquid State-Micro- and Nanoscale Effects Bulk thermal conductivity of a fluid may be modified when the characteristic dimension of the system becomes small, in particular for small values of L / λ mfp . Nanofuids are base liquids that are seeded with nanometer-sized solid particles. Their very small size allows the solid particles to remain suspended within the base liquid for a long time. A nanofluid exhibits the high thermal conductivity.

15. Thermal Diffusivity In heat transfer analysis, the ratio of the thermal conductivity to the heat capacity is an important property termed the thermal diffusivity α , which has units of m 2 /s: It measures the ability of a material to conduct thermal energy relative to its ability to store thermal energy. Large α ………………… time required to reach equilibrium? Small α …………………time required to reach equilibrium?

16. The Heat Diffusion Equation In the absence of motion (or with uniform motion), there are no changes in mechanical energy and no work being done on the system . Only thermal forms of energy need be considered.

17. The Heat Diffusion Equation Rate of thermal energy generation The energy storage term may be expressed as sensible thermal energy Conservation of energy can be expressed as

18. The Heat Diffusion Equation S substituting values of q x+dx , q y+dy , and q z+dz From Fourier’s law

19. The Heat Diffusion Equation This is the general form, in Cartesian coordinates, of the heat diffusion equation . This equation, often referred to as the heat equation . Constant thermal conductivity Steady-state ……….?

20. The Heat Diffusion Equation Cylindrical Coordinates Differential control volume, dr . r d φ . dz , for conduction analysis in cylindrical coordinates ( r , φ , z ).

21. The Heat Diffusion Equation In cylinderical coordinates, the general form of the heat flux vector and Fourier’s law is Cylindrical Coordinates Derive general form of heat equation in cylindrical coordinates by yourselves.

22. The Heat Diffusion Equation Spherical Coordinates Differential control volume, dr . r sin ϴ d φ . r d ϴ , for conduction analysis in spherical coordinates ( r, φ , ϴ ) .

23. The Heat Diffusion Equation Spherical Coordinates In spherical coordinates, the general form of the heat flux vector and Fourier’s law is

24. Boundary and Initial Conditions

25. Thanks Any Questions…???