Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Heat Transfer Problems Introductory Course on Multiphysics Modelling T OMASZ G. Z IELI ´ NSKI bluebox.ippt.pan.pl/˜tzielins/ Institute of Fundamental Technological Research of the Polish Academy of Sciences Warsaw • Poland
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle 2 Heat transfer equation Balance of thermal energy Specific thermal energy Fourier’s law Heat equation
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle 2 Heat transfer equation Balance of thermal energy Specific thermal energy Fourier’s law Heat equation Boundary and initial conditions 3 Mathematical point of view Physical interpretations Initial-Boundary-Value Problem
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle 2 Heat transfer equation Balance of thermal energy Specific thermal energy Fourier’s law Heat equation Boundary and initial conditions 3 Mathematical point of view Physical interpretations Initial-Boundary-Value Problem 4 Convective heat transfer Heat transfer by convection (and conduction) Non-conservative convective heat transfer
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle 2 Heat transfer equation Balance of thermal energy Specific thermal energy Fourier’s law Heat equation Boundary and initial conditions 3 Mathematical point of view Physical interpretations Initial-Boundary-Value Problem 4 Convective heat transfer Heat transfer by convection (and conduction) Non-conservative convective heat transfer
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Three mechanisms of heat transfer Heat transfer: a movement of energy due to a temperature difference.
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Three mechanisms of heat transfer Heat transfer: a movement of energy due to a temperature difference. Thermal energy is transferred according to the following three mechanisms : Conduction – heat transfer by diffusion in a stationary medium due to a temperature gradient. The medium can be a solid or a liquid.
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Three mechanisms of heat transfer Heat transfer: a movement of energy due to a temperature difference. Thermal energy is transferred according to the following three mechanisms : Conduction – heat transfer by diffusion in a stationary medium due to a temperature gradient. The medium can be a solid or a liquid. Convection – heat transfer between either a hot surface and a cold moving fluid or a hot moving fluid and a cold surface. Convection occurs in fluids (liquids and gases).
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Three mechanisms of heat transfer Heat transfer: a movement of energy due to a temperature difference. Thermal energy is transferred according to the following three mechanisms : Conduction – heat transfer by diffusion in a stationary medium due to a temperature gradient. The medium can be a solid or a liquid. Convection – heat transfer between either a hot surface and a cold moving fluid or a hot moving fluid and a cold surface. Convection occurs in fluids (liquids and gases). Radiation – heat transfer via electromagnetic waves between two surfaces with different temperatures.
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Three mechanisms of heat transfer Heat transfer: a movement of energy due to a temperature difference. Thermal energy is transferred according to the following three mechanisms : Conduction – heat transfer by diffusion in a stationary medium due to a temperature gradient. The medium can be a solid or a liquid. Convection – heat transfer between either a hot surface and a cold moving fluid or a hot moving fluid and a cold surface. Convection occurs in fluids (liquids and gases). Radiation – heat transfer via electromagnetic waves between two surfaces with different temperatures. Motivation for dealing with heat transfer problems: In many engineering systems and devices there is often a need for optimal thermal performance . Most material properties are temperature-dependent so the effects of heat transfer enter many other disciplines and drive the requirement for multiphysics modeling .
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Heat conduction and the energy conservation law Problem : to find the temperature ∂ T ∂ x · n > 0 in a solid, T = T ( x , t ) =? � � . K (warm) ∂ T ∂ x · n < 0 n Temperature is related to heat (cold) which is a form of energy . ∂ B The principle of conservation of energy should be used to B heat sink determine the temperature. ( ̺, c , k ) f < 0 Thermal energy can be: stored , heat source generated (or absorbed), and f > 0 supplied (transferred).
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Heat conduction and the energy conservation law Problem : to find the temperature ∂ T ∂ x · n > 0 in a solid, T = T ( x , t ) =? � � . K (warm) ∂ T ∂ x · n < 0 n Temperature is related to heat (cold) which is a form of energy . ∂ B The principle of conservation of energy should be used to B heat sink determine the temperature. ( ̺, c , k ) f < 0 Thermal energy can be: stored , heat source generated (or absorbed), and f > 0 supplied (transferred). The law of conservation of thermal energy The rate of change of internal thermal energy with respect to time in B is equal to the net flow of energy across the surface of B plus the rate at which the heat is generated within B .
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Outline Introduction 1 Mechanisms of heat transfer Heat conduction and the energy conservation principle 2 Heat transfer equation Balance of thermal energy Specific thermal energy Fourier’s law Heat equation Boundary and initial conditions 3 Mathematical point of view Physical interpretations Initial-Boundary-Value Problem 4 Convective heat transfer Heat transfer by convection (and conduction) Non-conservative convective heat transfer
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Balance of thermal energy The internal thermal energy , E � � : J � ̺ e d V � kg � ̺ = ̺ ( x ) – the mass density m 3 B e = e ( x , t ) – the specific � J � internal energy kg
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Balance of thermal energy • E = d E The rate of change of thermal energy , � � : W d t � � d ̺ ∂ e ̺ e d V = ∂ t d V � kg � ̺ = ̺ ( x ) – the mass density d t m 3 B B e = e ( x , t ) – the specific � J � internal energy kg
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Balance of thermal energy • E = d E The rate of change of thermal energy , � � : W d t � � d ̺ ∂ e ̺ e d V = ∂ t d V � kg � ̺ = ̺ ( x ) – the mass density d t m 3 B B e = e ( x , t ) – the specific � J � internal energy kg � (the amount of heat per unit time � The flow of heat , Q W flowing-in across the boundary ∂ B ): � − q · n d S � W � q = q ( x , t ) – the heat flux vector m 2 ∂ B n – the outward normal vector
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Balance of thermal energy • E = d E The rate of change of thermal energy , � � : W d t � � d ̺ ∂ e ̺ e d V = ∂ t d V � kg � ̺ = ̺ ( x ) – the mass density d t m 3 B B e = e ( x , t ) – the specific � J � internal energy kg � (the amount of heat per unit time � The flow of heat , Q W flowing-in across the boundary ∂ B ): � − q · n d S � W � q = q ( x , t ) – the heat flux vector m 2 ∂ B n – the outward normal vector � (the amount of heat per The total rate of heat production , F � W unit time produced in B by the volumetric heat sources): � f d V f = f ( x , t ) – the rate of heat production � W � per unit volume B m 3
Introduction Heat transfer equation Boundary and initial conditions Convective heat transfer Balance of thermal energy • The thermal energy conservation law, E = Q + F , leads to the following balance equation. The global form of thermal energy balance � ̺ ∂ e � � ∂ t d V = − q · n d S + f d V B ∂ B B
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