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Heat Conduction in a Body Subject to an Oscillating Heat Source

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Heat Conduction in a Body Subject to an Oscillating Heat Source

By Seth Flaxman

Evanston Township High School

Problem Statement

- Copper plate in contact with an oscillating heat source
- Tests at various velocities
- Measured temperature distribution
- Computer simulations
- Physical experiments

Purpose

- Heat conduction problems
- Novel problem: oscillating heat sources at various velocities
- Importance

Background

Heat flow is a part of everyday life.

Examples:

Radiation: Heat transfer due to the sun

Convection: Hot air currents in an oven

Conduction: Touching something that is hot

Heat Flux

Fourier’s Law of Conduction

q - heat flux

k - thermal conductivity

T -temperature distribution, dependent on time and space

Heat Conduction

Heat Equation

k - thermal conductivity

q - heat generation term

ρ - density of the material

c - heat capacity of the material

T - temperature distribution

Previous Work

- Work by Krishnamurthy on oscillating heat sources
- Computer model developed by Backstrom

A steel bar heated by a moving torch

Methods

Computer Simulation and Laboratory Experiment

- Copper plate
- Heat source
- Heat source velocity
- Boundary conditions
- Time period

Laboratory Experiment

Thermistor positions in the copper plate

Laboratory Experiment

Laboratory Experiment

Laboratory Experiment

Laboratory Experiment

- Each run lasted 120 seconds
- Temperature readings recorded every 5 milliseconds

Computer Model

- Heat equation
- Boundary conditions
- Insulated
- Initial temperature
- Heat flux

Contact area in the laboratory

Contact area in the computer model

Results

Temperature vs. Time at 0.40 m/sec

Computer Simulation

Laboratory Results

Temperature (ºC)

Temperature (ºC)

Time (sec.)

Time (sec.)

Results

Temperature vs. Time at 0.040 m/sec

Computer Simulation

Laboratory Results

Temperature (ºC)

Temperature (ºC)

Time (sec.)

Time (sec.)

Results

Temperature vs. Time at 0.0079 m/sec

Computer Simulation

Laboratory Results

Temperature (ºC)

Temperature (ºC)

Time (sec.)

Time (sec.)

Results

Temperature vs. Time at 0.0040 m/sec

Computer Simulation

Laboratory Results

Temperature (ºC)

Temperature (ºC)

Time (sec.)

Time (sec.)

Results

Laboratory Results

Temperature vs. Time at 0.40 m/sec

Temperature vs. Time at 0.0040 m/sec

Temperature (ºC)

Temperature (ºC)

Time (sec.)

Time (sec.)

Laboratory Results

Temperature Change by Probe (ºC)

Discussion

- Interesting behavior at low velocities
- Computer simulations and experimental results agree
- Average rise in temperature is independent of velocity
- New results, not in literature

Further Research

- Analytic solution
- Other types of oscillatory motion: simple harmonic motion
- Slower velocities
- Different materials

Acknowledgments

Special thanks to Dr. Mark Vondracek, my parents, my brothers, Dr. Russell Kohnken, and Ms. Lisa Oberman.

Also, thanks to David L. Vernier of Vernier Software and Technology and Marcus H. Mendenhall, author of the Python Laboratory Operations Toolkit.

Results

Heat Vector Field Plot

Heat Vector Field Plot (0.040 m/sec.)

Location of heat source

y (cm)

x (cm)

Methods

P = IV

P = 2.12 * 10 = 21.2 W

Heat flux = 21.2 / (.023 * .0275) = 33,517 W / m2

2.75 cm

Methods

Heat Source Velocities (m/sec.)

Thermistor Circuit

FlexPDE Descriptor File

TITLE 'Temperature vs. Time at 0.0040 m/sec' { intel.pde }

SELECTerrlim=1e-3 spectral_colors

VARIABLES

temp( range=400)

DEFINITIONS

Lx = .2286 { in meters }

Ly = .1294 { in meters }

RealHeight = .1524 { in meters }

d0 = .0275 { length of heat source in meters }

heat_flux = -33517 { W / m^2 }

time_interval = 120

vx = .0040 { Moving heat source velocity in m/sec }

period = (Lx - d0) / vx

oscillations = time_interval / period

length = Lx

heat = 0

tempi = 26.5 { degrees Celsius }

k = 386 {Thermal conductivity of copper }

rcp= 8954 * 383.1

{ Density of copper times specific heat }

{ Source: p. 657 of Heat Conduction Ed. 2 by Ozisik [7] }

fluxd_x = -k*dx(temp)

fluxd_y = -k*dy(temp)

fluxd = vector(fluxd_x, fluxd_y)

fluxdm = magnitude(fluxd)

step = abs(ustep( mod(t, period * 2) - period)

* period - mod(t, period) )

{ In FlexPDE, mod does accept floats. This means repeated subtraction rather than remainder. }

fluxd0 = heat_flux * [ustep(vx* step + d0 - x)

- ustep(vx * step - x)]

INITIAL VALUES

temp = tempi

EQUATIONS

div(fluxd) + rcp * dt(temp) = 0

BOUNDARIES

region 'domain'

start (0,0) natural(temp) = 0 line to (Lx,0)

natural(temp) = 0 line to (Lx,Ly)

natural(temp) = fluxd0 line to (0,Ly)

natural(temp)= 0 line to finish

TIME

from 0 to oscillations * period

PLOTS

for t=0 by period * .1 to

oscillations * period

contour( temp) painted

vector( fluxd) norm

elevation( fluxd0 ) from (0,0) to (Lx,0)

HISTORIES

HISTORY(temp) at

(Lx/5,2*RealHeight/3) (2*Lx/5, 2*RealHeight/3)

(3*Lx/5,2*RealHeight/3) (4*Lx/5, 2*RealHeight/3)

(Lx/5,RealHeight/3) (2*Lx/5, RealHeight/3)

(3*Lx/5,RealHeight/3) (4*Lx/5, RealHeight/3)

END

Boundary Conditions

Initial temperature (26.5 ºC):

Insulated on all sides:

Subject to oscillating heat flux on top:

fluxd0 = heat_flux * [ustep(vx * step + d0 - x) - ustep(vx * step - x)]

Presentation Outline

- Problem Statement
- Purpose
- Methods
- Results
- Further Research