Chapter 1: Fourier Equation and Thermal Conductivity 1.1 …………. Introduction of Heat Transfer 1.2 …………. Fourier’s law of heat conduction 1.3 …………. Thermal Conductivity of material 1.4 …………. General heat conduction equation (a) Cartesian co-ordinates (b) Cylindrical co-ordinates (c) Spherical co-ordinates (d) General one dimensional conduction equation

1.1… Introduction of Heat Transfer Heat transfer is a science that studies the energy transfer between two bodies due to temperature difference. There can be no net heat transfer between two mediums that are at the same temperature. Basic requirement for heat transfer : presence of temperature difference . Note: Heat flow occurs only in the direction of decreasing temperature The temperature difference is the driving force for heat transfer, just as the voltage difference is the driving force for electric current flow and pressure difference is the driving force for fluid flow.

Introduction of Heat Transfer (continue…) Modes of Heat Transfer Conduction Convection Radiation Conduction: An energy transfer across a system boundary due to a temperature difference by the mechanism of intermolecular interactions. Conduction needs matter and does not require any bulk motion of matter. Convection: An energy transfer across a system boundary due to a temperature difference by the combined mechanisms of intermolecular interactions and bulk transport. Convection needs fluid matter. Radiation: Radiation heat transfer involves the transfer of heat by electromagnetic radiation that arises due to the temperature of the body. Radiation does not need matter.

Introduction of Heat Transfer (continue…) Heat travels along the rod Conduction is the transfer of heat through a solid or from one solid to another. When you heat a metal strip at one end, the heat travels to the other end. HOT (lots of vibration) COLD (not much vibration) Heat travels along the rod As you heat the metal, the particles vibrate, these vibrations make the adjacent particles vibrate, and so on and so on, the vibrations are passed along the metal and so is the heat. We call this? http://education.jlab.org/jsat/ [Accessed 13 November 11] 4

Introduction of Heat Transfer (continue…) When the handle of a spoon stirring a cup of hot chocolate gets hot, it’s because of conduction. How ??????? When the particles of a solid are heated they gain energy and vibrate more quickly. They bump into neighbor particles and transfer the energy to them.

1.2… Fourier’s Law of Heat Transfer The heat flux is proportional to the temperature gradient: …… (1) Where k=thermal conductivity (W/m°C or Btu/h ft °F) -- a measure of how fast heat flows through a material -- k(T), but we usually use the value at the average temperature q can have x, y, and z components; it’s a vector quantity x hot wall cold wall temperature profile ….. (2) Fourier’s Law …… (3) Heat Flux In most practical situations conduction, convection, and radiation appear in combination

1.3… Thermal Conductivity of Material The heat transfer characteristics of a solid material are measured by a property called the thermal conductivity (k) measured in W/m.K. It is a measure of a substance’s ability to transfer heat through a solid by conduction. K = Q × L / (A × ΔT) Thermal conductivity is defined as the quantity of heat (Q) transmitted through a unit thickness (L) in a direction normal to a surface of unit area (A) due to a unit temperature gradient (ΔT) under steady state conditions and when the heat transfer is dependent only on the temperature gradient. Note: The thermal conductivity of most liquids and solids varies with temperature. For vapors, it depends upon pressure.

Thermal Conductivity of Material (continue…) Thermal conductivity values for various materials at 300 K Silver 410 W/m. °C 237 Btu/h.ft.°F Copper 385 W/m. °C 223 Btu/h.ft.°F Metals Window glass 0.780 W/m. °C 0.045 Btu/h.ft.°F Brick 0.720 W/m. °C 0.0461 Btu/h.ft.°F Glass wool 0.038 W/m. °C 0.022Btu/h.ft.°F Nonmetallic solids Ammonia 0.147 W/m. °C 0.085 Btu/h.ft.°F Water 0.556 W/m. °C 0.327 Btu/h.ft.°F Liquids Hydrogen 0.175 W/m. °C 0.101 Btu/h.ft.°F Steam 0.0206 W/m. °C 0.0119 Btu/h.ft.°F Air 0.024 W/m. °C 0.0138 Btu/h.ft.°F Gases Note: 1 W/(m.K) = 1W/(m.oC) = 0.85984 kcal/(hr.m.oC) = 0.5779 Btu/(ft.hr.oF)

Heat Quantities

1.4… General Heat Conduction Equation (a) Cartesian (Rectangular) Coordinates: Consider a medium within which there is no bulk motion (advection) and the temperature distribution T(x,y,z) is expressed in Cartesian coordinates. First define an infinitesimally small (differential or elemental) control volume, dx.dy.dz, as shown in Fig.

Cartesian Coordinates system (continue…) Conduction Heat Rates If there are temperature gradients, conduction heat transfer will occur across each of the control surfaces and the conduction heat rates perpendicular to each of the control surfaces at the x, y, and z coordinate locations are indicated by the terms qx , qy and qz respectively. The conduction heat rates at the opposite surfaces can then be expressed as a Taylor series expansion with neglecting higher order terms, ……(4) ……(5) ……(6) Above equations simply states that the x component of the heat transfer rate at x + dx is equal to the value of this component at x plus the amount by which it changes with respect to x times dx.

Cartesian Coordinates system (continue…) Thermal energy generation ……(7) ……(8) ……(9) Energy storage Conservation of energy ……(10) From equation (10), ……(11)

Cartesian Coordinates system (continue…) Where, ……(12) ……(13) ……(14) Net conduction heat flux into the controlled volume, ……(15)

Cartesian Coordinates system (continue…) Heat (Diffusion) Equation: at any point in the medium the rate of energy transfer by conduction in a unit volume plus the volumetric rate of thermal energy must equal to the rate of change of thermal energy stored within the volume. ……(16) Equation (16) is final form of heat conduction equation for rectangular co-ordinates system. If the thermal conductivity (k) is constant. ……(17) Where α= k/(ρCp) is the thermal diffusivity i.e. rate of heat diffuse from system Under steady-state condition, there can be no change in the amount of energy storage. ……(18) Poisson's equation

Cartesian Coordinates system (continue…) If the no heat generation in volume, ……(19) Fourier's equation If steady state heat conduction with no heat generation in volume, ……(21) Laplace's equation If the heat transfer is one-dimensional, steady state and there is no energy generation, the above equation reduces to ……(22)

(b) Cylindrical Coordinates: ……(23)

(c) Spherical Coordinates: ……(24)

(c) General one dimensional conduction equation: In compact form, ……(25) Coordinate system X value n value Cartesian X=x Cylindrical X=r 1 Spherical 2