Presentation on theme: "Fourier’s Law and the Heat Equation"— Presentation transcript:
1 Fourier’s Law and the Heat Equation Chapter TwoLecture 3
2 Fourier’s LawFourier’s LawA rate equation that allows determination of the conduction heat fluxfrom knowledge of the temperature distribution in a mediumIts 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 themediumDirection of heat transfer is perpendicular to lines of constanttemperature (isotherms).Heat flux vector may be resolved into orthogonal components.
4 Heat Flux Components (cont.) In angular coordinates , the temperature gradient is stillbased on temperature change over a length scale and hence hasunits of C/m and not C/deg.Heat rate for one-dimensional, radial conduction in a cylinder or sphere:Cylinderor,Sphere
5 Heat EquationThe Heat EquationA differential equation whose solution provides the temperature distribution in astationary medium.Based on applying conservation of energy to a differential control volumethrough which energy transfer is exclusively by conduction.Cartesian Coordinates:(2.19)Change in thermalenergy storageThermal energygenerationNet transfer of thermal energy into thecontrol volume (inflow-outflow)
7 Heat Equation (Special Case) One-Dimensional Conduction in a Planar Medium with Constant Propertiesand No Generationbecomes
8 Boundary and Initial Conditions Boundary ConditionsBoundary and Initial ConditionsFor transient conduction, heat equation is first order in time, requiringspecification of an initial temperature distribution:Since heat equation is second order in space, two boundary conditionsmust be specified. Some common cases:Constant Surface Temperature:T(0, t) = TsConstant Heat Flux:Applied FluxInsulated SurfaceConvection:
9 Thermophysical Properties Thermal Conductivity: A measure of a material’s ability to transfer thermalenergy by conduction.Thermal Diffusivity: A measure of a material’s ability to respond to changesin its thermal environment.Property Tables:Solids: Tables A.1 – A.3Gases: Table A.4Liquids: Tables A.5 – A.7
10 Properties (Micro- and Nanoscale Effects) Conduction may be viewed as a consequence of energy carrier (electron orphonon) motion.For the solid state:average energy carrier velocity,energy carrierspecific heat perunit volume.mean free path → average distancetraveled by an energy carrier beforea collision.(2.7)Energy carriers also collide withphysical boundaries, affectingtheir propagation.External boundaries of a film of material.thick film (left) and thin film (right).
11 Properties (Micro- and Nanoscale Effects) (2.9b)Grain boundaries within a solidMeasured thermal conductivity of a ceramic material vs. grain size, L.Fourier’s law does not accurately describe the finite energy carrier propagationvelocity. This limitation is not important except in problems involving extremelysmall time scales.
12 Typical Methodology of a Conduction Analysis Consider possible microscale or nanoscale effects in problems involving verysmall physical dimensions or very rapid changes in heat or cooling rates.Solve appropriate form of heat equation to obtain the temperaturedistribution.Knowing the temperature distribution, apply Fourier’s Law to obtain theheat flux at any time, location and direction of interest.Applications:Chapter 3: One-Dimensional, Steady-State ConductionChapter 4: Two-Dimensional, Steady-State ConductionChapter 5: Transient Conduction
13 Problem: Thermal Response of Plane Wall Problem Thermal response of a plane wall to convection heat transfer.
15 Problem: Thermal Response (Cont). <d) The total energy transferred to the wall may be expressed asDividing both sides by AsL, the energy transferred per unit volume is<
16 Problem: Non-uniform Generation due to Radiation Absorption Problem Surface heat fluxes, heat generation and total rate of radiationabsorption in an irradiated semi-transparent material with aprescribed temperature distribution.
19 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=900C, b=-300C/m, c=-50C/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.
20 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 generationFind: 1. Heat rates entering and leaving;2. Rate of change of energy storage in the wall,3. Time rate of temperature change at several locationsSchematic:
22 Example 2.2 (pages 75-76)Assumptions: 1. 1-D conduction in x-direction2. Isotropic medium with constant properties3. 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)
23 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,
24 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: