Fourier’s Law and the Heat Equation Chapter Two Lecture 3

Fourier’s Law Fourier’s Law A rate equation that allows determination of the conduction heat flux from knowledge of the temperature distribution in a medium Its most general (vector) form for multidimensional conduction is: Implications: Heat transfer is in the direction of decreasing temperature (basis for minus sign). Fourier’s Law serves to define the thermal conductivity of the medium Direction of heat transfer is perpendicular to lines of constant temperature (isotherms). Heat flux vector may be resolved into orthogonal components.

Cartesian Coordinates: Heat Flux Components Cartesian Coordinates: (2.3) Cylindrical Coordinates: (2.24) Spherical Coordinates: (2.27)

Heat Flux Components (cont.) In angular coordinates , the temperature gradient is still based on temperature change over a length scale and hence has units of C/m and not C/deg. Heat rate for one-dimensional, radial conduction in a cylinder or sphere: Cylinder or, Sphere

Heat Equation The Heat Equation A differential equation whose solution provides the temperature distribution in a stationary medium. Based on applying conservation of energy to a differential control volume through which energy transfer is exclusively by conduction. Cartesian Coordinates: (2.19) Change in thermal energy storage Thermal energy generation Net transfer of thermal energy into the control volume (inflow-outflow)

Heat Equation (Radial Systems) Cylindrical Coordinates: (2.26) Spherical Coordinates: (2.29)

Heat Equation (Special Case) One-Dimensional Conduction in a Planar Medium with Constant Properties and No Generation becomes

Boundary and Initial Conditions Boundary Conditions Boundary and Initial Conditions For transient conduction, heat equation is first order in time, requiring specification of an initial temperature distribution: Since heat equation is second order in space, two boundary conditions must be specified. Some common cases: Constant Surface Temperature: T(0, t) = Ts Constant Heat Flux: Applied Flux Insulated Surface Convection:

Thermophysical Properties Thermal Conductivity: A measure of a material’s ability to transfer thermal energy by conduction. Thermal Diffusivity: A measure of a material’s ability to respond to changes in its thermal environment. Property Tables: Solids: Tables A.1 – A.3 Gases: Table A.4 Liquids: Tables A.5 – A.7

Properties (Micro- and Nanoscale Effects) Conduction may be viewed as a consequence of energy carrier (electron or phonon) motion. For the solid state: average energy carrier velocity, energy carrier specific heat per unit volume. mean free path → average distance traveled by an energy carrier before a collision. (2.7) Energy carriers also collide with physical boundaries, affecting their propagation. External boundaries of a film of material. thick film (left) and thin film (right).

Properties (Micro- and Nanoscale Effects) (2.9b) Grain boundaries within a solid Measured thermal conductivity of a ceramic material vs. grain size, L. Fourier’s law does not accurately describe the finite energy carrier propagation velocity. This limitation is not important except in problems involving extremely small time scales.

Typical Methodology of a Conduction Analysis Consider possible microscale or nanoscale effects in problems involving very small physical dimensions or very rapid changes in heat or cooling rates. Solve appropriate form of heat equation to obtain the temperature distribution. Knowing the temperature distribution, apply Fourier’s Law to obtain the heat flux at any time, location and direction of interest. Applications: Chapter 3: One-Dimensional, Steady-State Conduction Chapter 4: Two-Dimensional, Steady-State Conduction Chapter 5: Transient Conduction

Problem: Thermal Response of Plane Wall Problem 2.57 Thermal response of a plane wall to convection heat transfer.

Problem: Thermal Response (cont). < <

Problem: Thermal Response (Cont). < d) The total energy transferred to the wall may be expressed as Dividing both sides by AsL, the energy transferred per unit volume is <

Problem: Non-uniform Generation due to Radiation Absorption Problem 2.37 Surface heat fluxes, heat generation and total rate of radiation absorption in an irradiated semi-transparent material with a prescribed temperature distribution.

Problem : Non-uniform Generation (cont.) <

Problem : Non-uniform Generation (cont.)

Example 2.3 (pages 75-76) The temperature distribution across a wall 1m thick at a certain instant of time is given as: where T is in degree Celsius and x is in meters, while a=900C, b=-300C/m, c=-50C/m2. A uniform heat generation, =1000 W/m3, is present in the wall of area 10 m2 having the property of =1000 kg/m3, k=40W/mK, and cP=4 kJ/kgK. Determine the rate of heat transfer entering the wall (x=0) and leaving the wall (x=1m). Determine the rate of change of energy storage in the wall. Determine the time of temperature change at x =0, 0.25 and 0.5m.

Example 2.2 (pages 75-76) Solution Known: Temperature distribution T(x) at an instant of time t in a 1-D wall with uniform generation Find: 1. Heat rates entering and leaving; 2. Rate of change of energy storage in the wall, 3. Time rate of temperature change at several locations Schematic:

Example 2.2 (pages 75-76) Schematic:

Example 2.2 (pages 75-76) Assumptions: 1. 1-D conduction in x-direction 2. Isotropic medium with constant properties 3. Uniform internal heat generation, (W/m3) Analysis: 1. For steady state 1-D conduction, Fourier’s can be applied to calculate qin and qout. (qin= 120 kW ; qout = 160 kW)

Example 2.2 (pages 75-76) Analysis: 2. The rate of change of energy storage in the wall ( ) can be calculated by applying an overall energy balance to the wall. Using Equation 1.1 for control volume about the wall,

Example 2.2 (pages 75-76) Analysis: 3. The time rate of change of the temperature at any point in the medium may be determined from the heat Equation 2.19, rewritten as: From the prescribed temperature distribution, it follows:

Example 2.2 (pages 75-76) Analysis: Lecture 3