lecture heat diffusion (1)

8/10/2019 Lecture Heat Diffusion (1)

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Heat Diffusion

MSE 308

Jian-Ku Shang

University of Illinois at Urbana-Champaign


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Heat Diffusion

Governed by Diffusion Equations


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Heat Diffusion Equations

Fouriers Law

q(x; t) is the heat flux

k> 0 is the thermal conductivity

T(x; t) indicates the temperature

The heat equation

cis the heat capacity

D is the thermal diffusivity

D


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Solutions to Heat Diffusion Equation:

a= D: thermal diffusivity

TASK: Given IC and BCs, from T = function (t,x), determine a= D: thermal diffusivity


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Flash Method

for Measuring Thermal Diffusivity

Parker 1961


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Lab Version

Halogen Lamp

To

Outlet

Chopper

To Choppers

Freq. Generator

Lens

Spectru

mD

etector

Detector

To T-piece/Signal of Lock-in Amp

Signal Out of Freq. Generator

To T-piece/ Reference In of Lock-in Amp

Sample in detector


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Experimental Procedures

Equipment and samples Sample disks made of graphite and carbon-coated steel

Pyroelectric detector, lockin amplifier.

High intensity lamp; dc power supply; optical chopper;

Computer, plotting software, computerbased oscilloscope.

Procedure Calibrate the detector (without the sample)

Collect the data on frequency response from 2 samples

Measure the phase angles at a function of the frequency

Conduct data analysis to find the thermal diffusivity


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Calibration of Detector

Turn on light, detector and chopper

Set chopper to 4.5Hz

Minimize the light by shining in off direction as to not saturate the detector

Use the Easy Sense BNC Live Voltmeter in parallel (via T-piece) to ensurea less noisy signal (at x.01V) and the reference and signal from detector are

measurable. Set lock-in to appropriate Range:

High-pass filter: 1Hz

Low-pass filter 50Hz

Minimize the max voltage appropriately.

Change the time constant to 3 sec.

Record (type in excel) the Phase and Magnitude with FrequencyIncrease the freq. in regular intervals and wait for the Phase and magnitude to

stablize before changing the freq and recording.

Continue to 30-40Hz


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Adding sample

Sample

Detector Side View

Unscrew Lense

Apply

15/16diameter

Sample.

DO NOT

Touch Detector

Detector

Screw a cover

without a lens

back on to keep

sample in place


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Sample Measurements

Install sample

Turn on light, detector and chopper

Set chopper to 4.5Hz

Maximize the light hitting the sample by adjusting lens and distances.

Keep lock-in set to appropriate Range: High-pass filter: 1Hz

Low-pass filter 50Hz

Minimize the max voltage appropriately.

Change the time constant to 3 sec.

Record (type in excel) the Phase and Magnitude with Frequency

Increase the freq. in regular intervals and wait for the Phase and magnitude to stabilizebefore changing the frequency and recording.

Continue to ~30-40Hz or higher frequencies until you have less signal and singal maynot stabilize.


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Data Analysis

2

2

( , ) ( , )T x t T x t Dt x

[ ] [ ]( , ) i kx t i kx t T x t Ae Be

Heat Equation

General Solution

2 (1 )i i f f k iD D D

22

2

i fJ D A i fD A

D

JA

i fD

BC: x = 0

IC: t =0, at x = d, T(x,t)= 0

2cosh( )

2sinh([ )

i fid i t

DB Ai f

d i tD

2 2

( , ) sinh( 2 ) cosh( 2 )

i f i f

T x t A x i ft B x i ft D D


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Phase Analysis

( , ) 24

fd t d n

D

Im{ ( , )}

( , ) arctan( ) 2Re{ ( , )}

T d t

d t nT d t

21( , )2 2

sinh( 2 )

i ftJT x d t ei fD i f

d i ft D

At x = d,

graph (x,t) vs.f (1/2)where the slope, m, is dD

the thermal diffusivity is:2

2

dD

m


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Sample Data for 0.8mm Carbon Steel


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Angstroms method


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A simple Rod Experiment

(Based on Angstroms Classic Exp.)

Heater

Thermocouple

SiC or AlN rod.

L

T

t

t

T0

DT ~ 10oC

Steady-state:

In practice,

lecture heat diffusion (1)

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