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Fouriers Law and the Heat Equation Chapter Two Lecture 3

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

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

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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

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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: Net transfer of thermal energy into the control volume (inflow-outflow) (2.19) Thermal energy generation Change in thermal energy storage

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

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

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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: Constant Heat Flux: Applied FluxInsulated Surface Convection: T(0, t) = T s

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Properties Thermophysical Properties Thermal Conductivity: A measure of a materials ability to transfer thermal energy by conduction. Thermal Diffusivity: A measure of a materials 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

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Properties (Micro- and Nanoscale Effects) Micro- and Nanoscale Effects Conduction may be viewed as a consequence of energy carrier (electron or phonon) motion. For the solid state: Energy carriers also collide with physical boundaries, affecting their propagation. External boundaries of a film of material. thick film (left) and thin film (right). 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)

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

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Conduction Analysis Typical Methodology of a Conduction Analysis Solve appropriate form of heat equation to obtain the temperature distribution. Knowing the temperature distribution, apply Fouriers 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 Consider possible microscale or nanoscale effects in problems involving very small physical dimensions or very rapid changes in heat or cooling rates.

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

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Problem: Thermal Response (cont). < <

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

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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 due to Radiation Absorption

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Problem : Non-uniform Generation (cont.) <

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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/m 2. A uniform heat generation, =1000 W/m 3, is present in the wall of area 10 m 2 having the property of =1000 kg/m 3, k=40W/mK, and c P =4 kJ/kgK. 1.Determine the rate of heat transfer entering the wall (x=0) and leaving the wall (x=1m). 2.Determine the rate of change of energy storage in the wall. 3.Determine the time of temperature change at x =0, 0.25 and 0.5m.

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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 :

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Example 2.2 (pages 75-76) Schematic :

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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/m 3 ) Analysis: 1. For steady state 1-D conduction, Fouriers can be applied to calculate q in and q out. (q in = 120 kW ; q out = 160 kW)

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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,

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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:

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

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