Chapter 2: Introduction to Conduction Heat Conduction: Heat transfer in which energy exchange takes place in solid or fluids in rest from the region of high temperature to the region of low temperature due to the presence of a temperature gradient in the body – due to atomic and molecular activities. General assumptions: The medium is assumed to be a continuum (macroscopic). The medium is assumed to be homogeneous (conductivity does no vary from point to point at the same temperature) and isotropic (conductivity is the same in all directions).
Fourier’s law of conduction Consider a solid flat plate of thickness x . For sufficiently small values of the temperature difference between the surfaces of the plate, experimental observations of different solids lead to where k is a constant, the so-called thermal conductivity of the material of the plate.
Fourier’s law of conduction Since heat flux is direction dependent (a vector), similar relations can be written for y and z directions under a Cartesian coordinate system: A heat flux vector can thus be constructed: = - kT The vectorial form of Fourier’s law for isotropic continua where is the three dimensional del operator. The magnitude of the heat flux at any point P in the direction of m is given by the following relation in terms of the differentiation in the direction of m:
Thermal conductivity (physical interpretations) 1. Metallic solids: 10 400 W/m-K Heat conduction occurs by virtue of diffusion or motion of free electrons from a high temperature region to low temperature region 2. Nonmetallic solids: 0.1 10 W/m-K Heat conduction occurs in insulators, which lack “conduction Electrons”, by virtue of interatomic lattice vibrations – molecules behave as mass-spring-mass interconnected dynamic system. 3. Fluids: 0.01 10 W/m-K Heat conduction occurs in liquid via elastic collisions of fluid molecules; gases: may be explained in terms of kinetic theory of gases (random molecular motion) 4. Insulation systems: 0.01 0.1 A combination of nonmetallic solid and gases. Conduction Through the solid materials, conduction or convection through the air in the void spaces, radiation exchange between the surfaces of solid matrix
The Differential Equation of Heat Conduction (The Heat Diffusion Equation) Objective: to find a relation that can describe the temperature distribution in a solid. Energy balance equation for a control volume: Consider the energy-balance for the small control volume V: The net rate of heat The rate of energy The rate of change entering through the + generation in V = of energy stored bounding surfaces of V in V
The Differential Equation of Heat Conduction Taylor series: Let The Differential Equation of Heat Conduction Taylor series: Let The net heat entering V in the x direction: The net heat entering V in the y and z directions, respectively:
The Differential Equation of Heat Conduction The Differential Equation of Heat Conduction Therefore, The net rate of heat entering through the bounding surfaces of V Heat generation per unit time, per unit volume, may be due to nuclear, electrical, chemical, gamma-ray, or other sources that may be a function of time and/or position Storage: Substituting all three terms into the energy balance equation, the differential equation of heat conduction under a Cartesian system is obtained:
Conduction (continued) Thermal conductivity k: Metallic solids: 10 – 400 W/m-K Nonmetallic solids: 0.1 – 10 W/m-K Fluids: 0.01 – 10 W/m-K Insulation systems (a combination of nonmetallic solids/gases): 0.05 – 0.1 W/m-K Conduction is the most important heat transfer mode for solids
The Differential Equation of Heat Conduction The Differential Equation of Heat Conduction When k is a constant, the above equation can be written as thermal diffusivity In the cylindrical coordinate system (r, , z), the differential equation is
The Differential Equation of Heat Conduction The Differential Equation of Heat Conduction In the spherical coordinate system (r, , ):
Boundary and initial conditions the energy storage rate per unit volume Substituting these terms into the energy balance equation to obtain (exception: NOT applicable to a brake system)
Boundary and initial conditions According to Fourier’s law of heat conduction Constant surface heat flux, B.C. of the second kind: Constant, or A special case: adiabatic or insulated surface, Convection surface condition, B.C. of the third kind: Constant surface temperature, B.C. of the first kind: Constant surface temperature: constant If the situation is time dependent, the initial condition specifies the temperature distribution in the medium at the origin of the time coordinate.