Chapter 2

1. Chapter 2 Conduction

2. Conduction Heat Transfer • Conduction refers to the transport of energy in a medium (solid, liquid or gas) due to a temperature gradient. • The physical mechanism is random atomic or molecular activity • Governed by Fourier’s law • In this chapter we will learn • The definition of important transport properties and what governs thermal conductivity in solids, liquids and gases • The general formulation of Fourier’s law, applicable to any geometry and multiple dimensions • How to obtain temperature distributions by using the heat diffusion equation. • How to apply boundary and initial conditions

3. Thermal Properties of Matter • Recall from Chapter 1, equation for heat conduction: • The proportionality constant is a transport property, known as thermal conductivity k (units W/m.K) • Usually assumed to be isotropic (independent of the direction of transfer): kx=ky=kz=k • Is thermal conductivity different between gases, liquids and solids?

4. Thermal Conductivity: Solids • Solid comprised of free electrons and atoms bound in lattice • Thermal energy transported through • Migration of free electrons, ke • Lattice vibrational waves, kl where • What is the relative magnitude in pure metals, alloys and non-metallic solids? See Figure 2.5, Appendix Tables A.1, A.2 and A.3 text

5. Thermal Conductivity: Fluids • Intermolecular spacing is much larger • Molecular motion is random • Thermal energy transport less effective than in solids; thermal conductivity is lower • Kinetic theory of gases: • where n the number of particles per unit volume, the mean molecular speed and l the mean free path (average distance travelled before a collision) • What are the effects of temperature, molecular weight and pressure?

6. Thermal Conductivity: Fluids • Physical mechanisms controlling thermal conductivity not well understood in the liquid state • Generally k decreases with increasing temperature (exceptions glycerine and water) • k decreases with increasing molecular weight. • Values tabulated as function of temperature. See Tables A.5 and A.6, text.

7. Thermal Conductivity: Insulators • How can we design a solid material with low thermal conductivity? • Can disperse solid material throughout an air space – fiber, powder and flake type insulations • Cellular insulation – Foamed systems • Several modes of heat transfer involved (conduction, convection, radiation) • Effective thermal conductivity: depends on the thermal conductivity and radiative properties of solid material, volumetric fraction of the air space, structure/morphology (open vs. closed pores, pore volume, pore size etc.) Bulk density (solid mass/total volume) depends strongly on the manner in which the solid material is interconnected. See Table A.3.

8. Thermal Diffusivity Thermophysical properties of matter: • Transport properties: k (thermal conductivity/heat transfer), n (kinematic viscosity/momentum transfer), D (diffusion coefficient/mass transfer) • Thermodynamic properties, relating to equilibrium state of a system, such as density,r and specific heat cp. • the volumetric heat capacity r cp (J/m3.K) measures the ability of a material to store thermal energy. • Thermal diffusivitya is the ratio of the thermal conductivity to the heat capacity:

9. T1(high) qx” A T2 (low) x1 x2 The Conduction Rate Equation Recall from Chapter 1: • Heat rate in the x-direction • Heat flux in the x-direction We assumed that T varies only in the x-direction, T=T(x) Direction of heat flux is normal to cross sectional area A, where A is isothermal surface (plane normal to x-direction) x

10. The Conduction Rate Equation In reality we must account for heat transfer in three dimensions • Temperature is a scalar field T(x,y,z) • Heat flux is a vector quantity. In Cartesian coordinates: for isotropic medium Where three dimensional del operator in cartesian coordinates:

11. Summary: Fourier’s Law • It is phenomenological, ie. based on experimental evidence • Is a vector expression indicating that the heat flux is normal to an isotherm, in the direction of decreasing temperature • Applies to all states of matter • Defines the thermal conductivity, ie.

12. Element of volume: dx dy dz CV T(x,y,z) The Heat Diffusion Equation • Objective to determine the temperature field, ie. temperature distribution within the medium. • Based on knowledge of temperature distribution we can compute the conduction heat flux. • Reminder from fluid mechanics: Differential control volume. We will apply the energy conservation equation to the differential control volume

13. z x y z y x Heat Diffusion Equation (2.1) Energy Conservation Equation where from Fourier’s law

14. Heat Diffusion Equation • Thermal energy generation due to an energy source: • Manifestation of energy conversion process (between thermal energy and chemical/electrical/nuclear energy) • Positive (source) if thermal energy is generated • Negative (sink) if thermal energy is consumed is the rate at which energy is generated per unit volume of the medium (W/m3) • Energy storage term • Represents the rate of change of thermal energy stored in the matter in the absence of phase change. is the time rate of change of the sensible (thermal) energy of the medium per unit volume (W/m3)

15. Heat Diffusion Equation Substituting into Eq. (2.1): Heat Equation (2.2) time rate of change of thermal energy per unit volume rate of energy generation per unit volume Net conduction of heat into the CV • At any point in the medium the rate of energy transfer by conduction into a unit volume plus the volumetric rate of thermal energy generation must equal the rate of change of thermal energy stored within the volume

16. Heat Diffusion Equation- Other forms • If k=constant is the thermal diffusivity (2.3) • For steady state conditions (2.4) • For steady state conditions, one-dimensional transfer in x-direction and no energy generation • Heat flux is constant in the direction of transfer

17. Heat Diffusion Equation • In cylindrical coordinates: (2.5) • In spherical coordinates: (2.6)

18. Example (Problem 2.23 textbook) The steady-state temperature distribution in a one-dimensional wall of thermal conductivity 50 W/m.K and thickness 50 mm is observed to be T(°C)=a+bx2, where a=200°C, b=-2000°C/m2, and x is in meters. • What is the heat generation rate in the wall? • Determine the heat fluxes at the two wall faces. In what manner are these heat fluxes related to the heat generation rate?

19. Ts T(x,t) x Boundary and Initial Conditions • Heat equation is a differential equation: • Second order in spatial coordinates: Need 2 boundary conditions • First order in time: Need 1 initial condition • Boundary Conditions Example: a surface is in contact with a melting solid or a boiling liquid • B.C. of first kind (Dirichlet condition):

20. qx” T(x,t) x T(x,t) x Boundary and Initial Conditions Example: What happens when an electric heater is attached to a surface? What if the surface is perfectly insulated? • B.C. of second kind (Neumann condition): Constant heat flux at the surface

21. T(0,t) T(x,t) x Boundary and Initial Conditions • B.C. of third kind: When convective heat transfer occurs at the surface

22. Example 2.3, textbook A long copper bar of rectangular cross section, whose width w is much greater than its thickness L, is maintained in contact with a heat sink (for example an ice bath) at a uniform initial temperature To. Suddenly an electric current is passed through the bar, and an airstream of temperature is passed over the top surface, while the bottom surface is maintained at To. Obtain the differential equation, the boundary and initial conditions that can be used to determine the temperature as a function of position and time inside the bar.

23. Example (Problem 2.39, textbook) Passage of an electric current through a long conducting rod of radius ri and thermal conductivity kr results in uniform volumetric heating at a rate of . The conducting rod is wrapped in an electrically nonconducting cladding material of outer radius ro and thermal conductivity kc, and convection cooling is provided by an adjoining fluid. For steady-state conditions, write appropriate forms of the heat equations for the rod and cladding. Express appropriate boundary conditions for the solution of these equations.