The Heat Equation Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . ❪ ❏ ✣ ✡ ❏ ✡ ❅ ■ � ✒ ❅ � P ✐ ✏ ✶ P ✏ ✏ ✏ ✮ � ✠ � ✡ ✡ ✢ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. C H O ✲ O thermal L T flux D logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. 3. Consider the net heat transfer through the surface S (per time unit). logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. 3. Consider the net heat transfer through the surface S (per time unit). It is proportional to the surface integral �� − grad ( u ) · d � � S . S logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. 3. Consider the net heat transfer through the surface S (per time unit). It is proportional to the surface integral �� − grad ( u ) · d � � S . S 4. The net heat transfer through S (per time unit) is the rate of change of the net heat content of B (per time unit), logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation The Heat Equation is Another Manifestation of the Principle of Conservation of Energy 1. Consider a small ball B centered at � r with radius a and surface S . 2. Heat flux is proportional to − grad u , where u is the temperature. 3. Consider the net heat transfer through the surface S (per time unit). It is proportional to the surface integral �� − grad ( u ) · d � � S . S 4. The net heat transfer through S (per time unit) is the rate of change of the net heat content of B (per time unit), which is proportional to − ∂ ��� B u dV . ∂ t logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. − k ∂ �� ��� grad ( u ) · d � − � S = B u dV ∂ t S logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. − k ∂ �� ��� grad ( u ) · d � − � S = B u dV ∂ t S B k ∂ u �� ��� grad ( u ) · d � = � S ∂ t dV S logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. − k ∂ �� ��� grad ( u ) · d � − � S = B u dV ∂ t S B k ∂ u �� ��� grad ( u ) · d � = � S ∂ t dV S B k ∂ u ��� ��� � � grad ( u ) = B div dV ∂ t dV logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. − k ∂ �� ��� grad ( u ) · d � − � S = B u dV ∂ t S B k ∂ u �� ��� grad ( u ) · d � = � S ∂ t dV S B k ∂ u ��� ��� � � grad ( u ) = B div dV ∂ t dV 1 1 B k ∂ u ��� ��� � � lim B div grad ( u ) dV = lim ∂ t dV 4 4 3 π a 3 3 π a 3 a → 0 a → 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation Deriving a Partial Differential Equation. − k ∂ �� ��� grad ( u ) · d � − � S = B u dV ∂ t S B k ∂ u �� ��� grad ( u ) · d � = � S ∂ t dV S B k ∂ u ��� ��� � � grad ( u ) = B div dV ∂ t dV 1 1 B k ∂ u ��� ��� � � lim B div grad ( u ) dV = lim ∂ t dV 4 4 3 π a 3 3 π a 3 a → 0 a → 0 k ∂ u � � grad ( u ) ( r , t ) = ∂ t ( r , t ) div � � logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation B k ∂ u ��� ��� B div ( grad ( u )) dV = ∂ t dV P ( � r ) q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation B k ∂ u ��� ��� B div ( grad ( u )) dV = ∂ t dV P ( � r ) q ❅ ❅ ❅ ❘ P ( � r ) q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation B k ∂ u ��� ��� B div ( grad ( u )) dV = ∂ t dV P ( � r ) q ❅ ❅ ❘ ❅ As the radius a shrinks, the approximations P ( � r ) q logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation

Underlying Principles Derivation Visualization of the Derivation B k ∂ u ��� ��� B div ( grad ( u )) dV = ∂ t dV P ( � r ) q ❅ ❅ ❅ ❘ As the radius a shrinks, the approximations P ( � r ) B k ∂ u 3 π a 3 k ∂ u ��� B div ( grad ( u )) dV ≈ 4 ��� ∂ t dV ≈ 4 3 π a 3 div ( grad ( u )) , q ∂ t logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Heat Equation