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

logo1 Underlying Principles Derivation Visualization of the Derivation

The Heat Equation

Bernd Schr¨

• der

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

The Heat Equation is Another Manifestation of the Principle of Conservation of Energy

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 ✐ ✏ ✏ ✶

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, where u is the

temperature.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, where u is the

temperature. O H T thermal flux

✲

C O L D

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, where u is the

temperature.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, where u is the

temperature.

• 3. Consider the net heat transfer through the surface S (per

time unit).

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, 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

• S

−grad(u)·d S.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, 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

• S

−grad(u)·d 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),

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 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 −gradu, 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

• S

−grad(u)·d 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 − ∂ ∂t

• B u dV.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

−

• S

grad(u)·d S = −k ∂ ∂t

• B u dV

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

−

• S

grad(u)·d S = −k ∂ ∂t

• B u dV
• S

grad(u)·d S =

• B k∂u

∂t dV

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

−

• S

grad(u)·d S = −k ∂ ∂t

• B u dV
• S

grad(u)·d S =

• B k∂u

∂t dV

• B div
• grad(u)
• dV

=

• B k∂u

∂t dV

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

−

• S

grad(u)·d S = −k ∂ ∂t

• B u dV
• S

grad(u)·d S =

• B k∂u

∂t dV

• B div
• grad(u)
• dV

=

• B k∂u

∂t dV lim

a→0

1

4 3πa3

• B div
• grad(u)
• dV

= lim

a→0

1

4 3πa3

• B k∂u

∂t dV

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

Deriving a Partial Differential Equation.

−

• S

grad(u)·d S = −k ∂ ∂t

• B u dV
• S

grad(u)·d S =

• B k∂u

∂t dV

• B div
• grad(u)
• dV

=

• B k∂u

∂t dV lim

a→0

1

4 3πa3

• B div
• grad(u)
• dV

= lim

a→0

1

4 3πa3

• B k∂u

∂t dV div

• grad(u)
• (
• r,t)

= k∂u ∂t (

• r,t)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

q

P(

• r)

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ q

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ q

As the radius a shrinks, the approximations Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

• B div(grad(u)) dV ≈ 4

3 πa3div(grad(u)),

• B k ∂u

∂t dV ≈ 4 3 πa3k ∂u ∂t

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ q

As the radius a shrinks, the approximations Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

• B div(grad(u)) dV ≈ 4

3 πa3div(grad(u)),

• B k ∂u

∂t dV ≈ 4 3 πa3k ∂u ∂t

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ q

As the radius a shrinks, the approximations improve towards equality. Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

• B div(grad(u)) dV ≈ 4

3 πa3div(grad(u)),

• B k ∂u

∂t dV ≈ 4 3 πa3k ∂u ∂t

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✙ q q

As the radius a shrinks, the approximations improve towards equality. Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

• B div(grad(u)) dV ≈ 4

3 πa3div(grad(u)),

• B k ∂u

∂t dV ≈ 4 3 πa3k ∂u ∂t

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✙ ◗◗◗ ◗ s q q q

As the radius a shrinks, the approximations improve towards equality. Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation

logo1 Underlying Principles Derivation Visualization of the Derivation

• B div(grad(u)) dV =
• B k ∂u

∂t dV

• B div(grad(u)) dV ≈ 4

3 πa3div(grad(u)),

• B k ∂u

∂t dV ≈ 4 3 πa3k ∂u ∂t

q

P(

• r)

P(

• r)

❅ ❅ ❅ ❘ ✟ ✟ ✟ ✟ ✟ ✙ ◗◗◗ ◗ s q q q

div(grad(u))(

• r,t) = k ∂u

∂t (

• r,t)

As the radius a shrinks, the approximations improve towards equality. Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science The Heat Equation