Transcription of GOVERNING EQUATION AND BOUNDARY CONDITIONS OF …
1 GOVERNING EQUATION AND BOUNDARY CONDITIONS OF heat transfer introduction1. Deriving GOVERNING equation2. BOUNDARY CONDITIONS 3. Deriving The bioheat transfer equation4. GOVERNING Equations for heat condition in Various coordinate systems5. Problem formulationGoverning EQUATION for heat transfer Derived from Energy Conservation and Fourier s lawFigure 1. Control volume showing energy inflow and outflow by conduction ( diffusion ) and EQUATION for heat transfer Derived from Energy Conservation and Fourier s law(1)(2)(3) GOVERNING EQUATION for heat transfer Derived from Energy Conservation and Fourier s lawGoverning EQUATION for heat transfer Derived from Energy Conservation and Fourier s law(4)(5) GOVERNING EQUATION for heat transfer Derived from Energy Conservation and Fourier s law(6)(8)(7)(9)Meaning of Each Term in the GOVERNING EQUATION ( )TermWhat does it representWhen can you ignore itStorageRate of change of stored energysteady state (no variation of temperaturewith time)
2 ConvectionRate of net energy transportdue to bulk flowTypically in a solid, with no bulk flowthrough itConductionRate of net energy transportdue to conductionSlow thermal conduction in relation togeneration or convection. For example,in short periods of microwave heatingGeneration Rate of generation of energyNo internal heat generation due tobiochemical reactions, of Thermal Source (Generation) Term in Biological SystemsA working muscle such as in the heart or limbs produce heatFermentation, composting and other biochemical reactions generate heatUtility of the Energy EQUATION It is very general 1. it is useful for any it is useful for any size of shape . similar equations can be derived for other coordinate it is easier to derive the more general EQUATION and it is safer as you drop terms you are aware of the reasons.
3 Can we make it general?1. To use with compressible To use when all properties vary with temperature. We neednumerical solutions to solve such To include mass Solution to Specific Situations: Need for BOUNDARY CONDITIONS (11)(12)(13)(14)(15)(16)Genera l BOUNDARY conditions1. Surface temperature is specifiedGeneral BOUNDARY conditions1. Surface temperature is specifiedFigure 2. A surface temperature specified BOUNDARY BOUNDARY conditions2. Surface heat flux is specifiedGeneral BOUNDARY conditions2. Surface heat flux is specifiedFigure 3. A heat flux specified BOUNDARY BOUNDARY conditions2a) special case: Insulated conditionFigure 4. An insulated (zero heat flux specified) BOUNDARY General BOUNDARY conditions2b) special case: Symmetry conditionFigure 5.
4 A symmetry (zero heat flux specified) BOUNDARY condition at the BOUNDARY conditions3. Convection at the surfaceGeneral BOUNDARY conditions3. Convection at the surfaceFigure 6. A convection BOUNDARY EQUATION 1. It is a mathematical statement of energy conservation. It is obtained bycombining conservation of energy with Fourier s law for heat Depending on the appropriate geometry of the physical problem choose agoverning EQUATION in a particular coordinate system from the equations3. Different terms in the GOVERNING EQUATION can be identified with conductionconvection , generation and storage. Depending on the physical situation some terms may be conditions1. BOUNDARY CONDITIONS are the CONDITIONS at the surfaces of a Initial CONDITIONS are the CONDITIONS at time t= BOUNDARY and initial CONDITIONS are needed to solve the GOVERNING equationfor a specific physical One of the following three types of heat transfer BOUNDARY CONDITIONS typically exists on a surface:(a) Temperature at the surface is specified (b) heat flux at the surface is specified(c) Convective heat transfer condition at the surface
