TIME-DEPENDENT HEAT TRANSFER

Edward Tefft

ME 5180

12/2/2016

Outline of Presentation

I. Refresher on heat transfer concepts

II. Finite element application

III. Numerical time integration

IV. Example: MATLAB 1-D

V. Example: ANSYS 2-D

Concepts

Types of Heat Transfer

• What is heat flux again?𝑞 (W/m2) – heat transfer rate per perpendicular area

• Conduction:

𝑞" = −𝐾𝑑𝑇𝑑𝑥

• Convection:𝑞) = ℎ(𝑇, − 𝑇∞ )Heat transfer illustration [1]

𝜎" = 𝐸𝑑𝑢𝑑𝑥

analogous to:

1-D Heat Transfer Equation

• For the amount of energy in a volume:𝐸in + 𝐸generated = Δ𝑈 + 𝐸out

• For a system with: Heat IN: conduction, generationHeat OUT: convection

𝜕𝜕𝑥 𝐾

𝜕𝑇𝜕𝑥 + 𝑄 = 𝜌𝑐

𝜕𝑇𝜕𝑡 +

ℎ𝑃𝐴 (𝑇, − 𝑇∞ )

Change of energy stored in volume (time dependent!!)

Δ𝑈 = specificheat×mass×changeintemperature

Outline of Presentation

I. Refresher on heat transfer concepts

II. Finite element application

III. Numerical time integration

IV. Example: MATLAB 1-D

V. Example: ANSYS 2-D

1-D Finite Element Equation

𝐾 𝑇 + 𝐶 �̇� = 𝐹

Thermal conductivity

matrix

Heat capacity matrix

Thermal loads

Considered for an axial bar element undergoing conduction and convection.

𝑇 𝑥 = 𝑁R 𝑥 𝑇R + 𝑁S 𝑥 𝑇S

𝑁R = 1 −𝑥𝐿

𝑁S =𝑥𝐿

movingfluid

T1

x, T(x)

T2

h, TA, k, ρ

8

L

Thermal Conductivity Matrix

𝐾 𝑇 + 𝐶 �̇� = 𝐹

Thermal conductivity

matrix

𝐾 =𝐴𝑘𝐿

1 −1−1 1 +

ℎ𝑃𝐿6

2 11 2

Conduction Part of the convection equation

𝑞" = −𝐾𝑑𝑇𝑑𝑥

𝑞) = ℎ(𝑇, − 𝑇∞ )

Heat Capacity Matrix

𝐾 𝑇 + 𝐶 �̇� = 𝐹

Heat capacity matrix 𝐾 =

𝐴𝑘𝐿

1 −1−1 1 +

ℎ𝑃𝐿6

2 11 2

So far…

𝐶 =𝑐𝜌𝐴𝐿6

2 11 2

𝐶 =𝑐𝜌𝐴𝐿2

1 00 1

This is analogous to the mass matrix!

Consistent-heat capacity matrix• virtual work principle• uses shape functions

Lumped-heat capacity matrix• assumes all of the mass is

lumped at the nodes• generally not as accurate [2]

Thermal Loads

𝐾 𝑇 + 𝐶 �̇� = 𝐹

Thermal loads𝐾 =

𝐴𝑘𝐿

1 −1−1 1 +

ℎ𝑃𝐿6

2 11 2

So far…

𝐶 =𝑐𝜌𝐴𝐿6

2 11 2

𝐹 = 𝑄𝐴𝐿2

11 + 𝑞

𝑃𝐿2

11 + ℎ𝑇∞

𝑃𝐿2

11

Internal heat

generation

Heat flow into lateral

surface

Part of the convection equation

𝑞) = ℎ(𝑇, − 𝑇∞ )

Outline of Presentation

I. Refresher on heat transfer concepts

II. Finite element application

III. Numerical time integration

IV. Example: MATLAB 1-D

V. Example: ANSYS 2-D

Numerical Time Integration

Galerkin method:

1. Rewrite the equation to this form:

2. Givenaknown 𝑇_ at 𝑡 = 0 and a time step of Δ𝑡, solve for 𝑇Rat 𝑡 = Δ𝑡

3. Using 𝑇R , find 𝑇S at 𝑡 = 2(Δ𝑡) and so on

=1Δ𝑡

𝑀 −13𝐾 𝑇b +

13𝐹b +

23𝐹bcR

1Δ𝑡

𝑀 +23𝐾 𝑇bcR

𝐴 𝑇bcR = 𝐵 𝑇b + 𝐶

Outline of Presentation

I. Refresher on heat transfer concepts

II. Finite element application

III. Numerical time integration

IV. Example: MATLAB 1-D

V. Example: ANSYS 2-D

Example: MATLAB 1-D

Example 16.7from A First Course in the Finite Element Method by D.L. Logan

1. Find 2-element solution

2. Find 4-element solution

1 2 3 Insulated tip𝑇∞ = 25℃85℃

Results

Outline of Presentation

I. Refresher on heat transfer concepts

II. Finite element application

III. Numerical time integration

IV. Example: MATLAB 1-D

V. Example: ANSYS 2-D

Example: ANSYS 2-D

Example 13.6from A First Course in the Finite Element Method by D.L. Logan

Extended to include transients – assumed similar to lead

𝑇 = 100℉

4

2 ft

2 ft

ℎ = 20Btu

hftS℉

𝑇∞ = 50℉

1 2

3

5

x

y𝐾 = 25

Btuhft℉

𝑐j = 0.03Btulbm℉

𝜌 = 709lbmfto

Results

ANSYS results:𝑇S = 69.23℉𝑇o = 69.23℉𝑇p = 84.62℉

Book results:𝑇S = 69.33℉𝑇o = 69.33℉𝑇p = 84.62℉

References

[1] http://www.tutorvista.com/physics/example-of-convection-heat-transfer

[2] Archer, J.S., “Consistent Matrix Formulations for Structural Analysis Using Finite Element Techniques,” Journal of the American Institute of Aeronautics and Astronautics, Vol. 3, No. 10, pp. 1910-1918, 1965.