thermal diffusivity

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Thermal Diffusivity Kushaji S. Parab Department of Physics Goa University Taleigao Plateau Goa 403206

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Page 1: Thermal diffusivity

Thermal Diffusivity

Kushaji S. ParabDepartment of Physics

Goa UniversityTaleigao Plateau

Goa 403206

Page 2: Thermal diffusivity

Diffusivity or Diffusion

• Diffusion, process resulting from random motion of molecules

by which there is a net flow of matter from a region of high

concentration to a region of low concentration.

• It describes the spread of particles through random motion

usually (but not always) from regions of higher

concentration to regions of lower concentration.

• Diffusion is one of several transport phenomena that occur in

nature.

• Diffusion processes may be divided into two types: (a) steady

state and (b) nonsteady state.

• Steady state diffusion takes place at a constant rate.

• This means that throughout the system dc/dx = constant and

dc/dt = 0.

Page 3: Thermal diffusivity

• Non-steady state diffusion is a time dependent process in

which the rate of diffusion is a function of time.

• Both types of diffusion are described quantitatively by Fick’s

laws of diffusion.

• Fick's first law relates the diffusive flux to the concentration

under the assumption of steady state. It postulates that the flux

goes from regions of high concentration to regions of low

concentration, with a magnitude that is proportional to the

concentration gradient. In 1-D the law is,

• J is the diffusion flux

• D is the diffusion coefficient or diffusivity

Page 4: Thermal diffusivity

• The negative sign in this relationship indicates that particle

flow occurs in a “down” gradient direction, i.e. from regions

of higher to regions of lower concentration.

• Fick’s 2nd law (non-steady state diffusion)

• Consider a volume element (between x and x+dx of unit cross

sectional area) of a membrane separating two finite volume

involved in a diffusion system.

• The flux of a given material into the volume element minus the

flux out of the volume element equals the rate of accumulation

of the material into this volume element:

(C is the average concentration in the volume element and Cdx is the total amount

of the diffusing material in the element at time (t).)

Page 5: Thermal diffusivity

• Using a Taylor series we can expand Jx+dx about x and obtain:

As dx→ 0

• If D does not vary with x (which is normally the case) we have

the formulation of Fick’s Second Law,

• In physical terms this relationship states that the rate of

compositional change is proportional to the “rate of change” of

the concentration gradient rather than to the concentration

gradient itself.

Page 6: Thermal diffusivity

Thermal Diffusivity:

• The thermal diffusivity of a material is k/rc

where k - thermal conductivity, r - density and c - specificheat of a material.

• This ratio expresses the speed with which the heatgenerated diffuses out.

• In steady state measurements of heat transport only theconductivity plays a role. The diffusivity has no role to play.But if the heat supplied varies as a function of time then thediffusivity determines how the temperature varies withspace and time in the medium.

Page 7: Thermal diffusivity

• Consider

• Let -κAT/x be the heat flowing out of ‘A’

• Heat flowing out of B is -κA[T/x + 2T/x2 dx]

• Therefore the heat flowing out of a rod of length dx in one second can be written as

{ -κA[T/x + 2T/x2 dx]} - κ AT/x = - κ A2T/x2 dx

Page 8: Thermal diffusivity

• The mass of material between the two sections is ρAdx. If thetemperature varies with time then the amount of heat usedup in one second in heating this element is

ρAdx c T/t

• In addition there could be heat loss due to radiation and thiscan be expressed by Newton’s Law of cooling

-εPdx(T-T0)

• If λAdx is the heat supplied then by law of conservation ofheat

λAdx = ρAdx c T/t - κ A2T/x2 dx -εPdx(T-T0)

Or

κ 2T/x2 - ρ c T/t +ε (P/A) (T-T0) = λ

Page 9: Thermal diffusivity

Solution of differential Equation

• Consider – heater is at the centre of the rod (x = 0)

– no distributed heat source (λ = 0)

• Then the equation satisfied by T becomes

κ 2T/x2 - ρ c T/t + ε (P/A) (T-T0) = 0 (0 < x < L)

or

2T/x2 – (ρ c/κ) T/t + ε (P/Aκ) (T-T0) = 0

Page 10: Thermal diffusivity

If the heat flowing per unit time varies as

Q = Q0 exp(-iωt)

Then the temperature will also vary as

T (x,t) = θ(x) exp(-iωt) + T0

Therefore ,

d2θ/dx2 + i(ωρ c/κ) θ + ε(P/Aκ) θ = 0

Writing, ε(P/Aκ) = a and (ωρ c/κ) = b, we get

d2θ/dx2 + (a +ib) θ = 0

Page 11: Thermal diffusivity

Boundary conditions

• Condition I: - κA (T/x)x = 0 = Q

• Condition II: The end of the rod is at a fixed temperature T0

Writing a + ib = h2 and putting u = hx we get

d2 q/du2 + q = 0

The solution of this equation is

q (x) = C exp(hx ) + B exp(-hx)

The temperature T (x, t) varies as

T (x,t) = [C exp(hx ) + B exp(-hx )]exp(-iwt)+ T0

Page 12: Thermal diffusivity

h = (α+iβ) = (a + ib)1/2 = (a2 + b2)1/4 exp(iφ/2)

Therefore,

T (x,t) = [C exp(αx) exp(i βx) + B exp(-αx) exp(-i βx)] exp(-iwt) + T0

α = (a2 + b2)1/4 cos(φ/2)

β = (a2 + b2)1/4 sin(φ/2)

• At x = L, T = T0 at all time. So,

C exp (αL) exp(iβL) + B exp(-αL) exp(-iβL) = 0

C = -B exp(-2aL) exp(-2iβL)

Page 13: Thermal diffusivity

Now a and b have the dimension of the inverse of length. If wechoose rod of appropriate length (aL > 5) such that T = T0 at x =L, then C = 0

T (x, t) = B exp (-(α+iβx ) exp(-iωt)+ T0

The boundary condition at x = 0 is - k A[dT/dx] = Q0 exp(-i ωt)

B = [Q0 /(k A)](α2 + β2)1/2

= [Q0/(kA)] (a2+ b2)1/4

So the temperature distribution is given by

T(x,t) = [(Q0/κA) ((a2+ b2)1/4 )]exp(- αx) exp (-i(ωt-βx))amplitude phase

Page 14: Thermal diffusivity

The temperature is therefore not in phase with the heatsupplied.

The amplitude of temperature oscillation is a function of x andvaries as

Amp (T(x)) = [Q0/ (kA (a2+b2)1/4)]exp(-ax)

Therefore the ratio of amplitudes at say x1 and x2 is given by

Amp (x1)/Amp(x2) = exp [-α(x1-x2)]

The difference in phase is given by

f(x1) - f(x2) = b(x1-x2)

Page 15: Thermal diffusivity

Therefore one can find a and β and hence thermal diffusivity ofthe material by

– measuring the ratio of amplitudes of temperature variation at twopoints separated by known distance

– by measuring the phase difference of the temperature wave at two

points separated by known distance.

1,2 - brass tube3 - heater4 & 5- thermocouples6- the insulating box/container

Page 16: Thermal diffusivity

For a heat pulse of the type

Q = I2R nτ < t < (n+f) τ= 0 (n+f) τ < t < (n+1) τ

With B-n = B*n , Bn = [H-n/ (κAn η)] exp(-iπ/4). same periodic variation as heat pulse

Page 17: Thermal diffusivity

We measure temperature time graph at two points and Fouriertransform the to get Fourier components at w0.

The RP of the integral on the right is

And the imaginary part is

Page 18: Thermal diffusivity

Test Results:time s ThC1 mV T1Cos T1sin ThC2 mV T2Cos T2Sin

15 58.3 57.58218 9.120438 51.9 51.26098 8.119224

45 73.6 65.57754 33.41476 54.3 48.38126 24.65246

75 86.8 61.37522 61.37852 60.8 42.99094 42.99325

105 98.3 44.62398 87.58762 67.6 30.68749 60.23319

135 107.4 16.79594 106.0785 74.1 11.58826 73.18827

165 114.9 -17.981 113.4843 79.7 -12.4725 78.71802

195 121.6 -55.2128 108.3425 85 -38.5945 75.73286

225 127.4 -90.0927 90.07815 89.3 -63.1497 63.13955

255 132 -117.618 59.91601 93.1 -82.9566 42.25895

285 136.4 -134.723 21.32392 96.9 -95.7085 15.14874

315 136.5 -134.817 -21.3685 100.3 -99.0634 -15.7015

345 121.4 -108.161 -55.1278 98.1 -87.4022 -44.5473

375 107.8 -76.2159 -76.2363 91.8 -64.9037 -64.9211

405 97.3 -44.1607 -86.7013 84.9 -38.5328 -75.652

435 88.2 -13.784 -87.1163 78.1 -12.2055 -77.1404

465 81.1 12.70016 -80.0994 73 11.43171 -72.0993

495 74.7 33.92488 -66.5522 68 30.88209 -60.583

525 69.2 48.94098 -48.9226 63.3 44.76827 -44.7515

555 64.3 57.29752 -29.1802 59 52.5747 -26.775

585 60.1 59.36204 -9.38929 55.5 54.81852 -8.67064

615 60.6 59.85183 9.4931 52.1 51.45677 8.161559

-333.451 130.2172 -215.507 -6.63604

I1Cos -16.6726 I1sin 6.510862 I2cos -10.7754 I2sin -0.3318

Amp1 17.89877 Amp2 10.78047

Phase1 2.769398 Phase2 3.172483

Alpha 0.169

Beta 0.134

Diffusivity 0.231 cm^2/s

For brass the actual value of diffusivity is 0.3 /s.cm^2

Page 19: Thermal diffusivity

Using Simpson’s Rule

Page 20: Thermal diffusivity

Amp of q(x1,w0) = [ RP2 + IP2]1/2

Phase of q(x1,w0) = φ(x1) = tan-1 [IP/ RP]

From the ratio of amplitudes we can calculate α = ln(|θ(x1,ω0)/ θ(x2,ω0)|)/(x2-x1)

And β = (φ(x2) - φ(x1) )/(x2-x1)

αβ = (a2+b2)1/2 cos(φ/2) = ½(a2 + b2)1/2 sin φ = b/2 = ωρc/2κ

Therefore diffusivity

D = κ/ρc = ω/(2αβ)

Page 21: Thermal diffusivity

0 300 600 900 1200 1500 1800 2100 2400 2700 3000 3300

0

100

200

300

No Reads

He

ate

rPo

we

r

Time(sec)

No Readings TC1 TC2

1215 1815 2415 3015

Page 22: Thermal diffusivity

Test Results:time s ThC1 mV T1Cos T1sin ThC2 mV T2Cos T2Sin

15 58.3 57.58218 9.120438 51.9 51.26098 8.119224

45 73.6 65.57754 33.41476 54.3 48.38126 24.65246

75 86.8 61.37522 61.37852 60.8 42.99094 42.99325

105 98.3 44.62398 87.58762 67.6 30.68749 60.23319

135 107.4 16.79594 106.0785 74.1 11.58826 73.18827

165 114.9 -17.981 113.4843 79.7 -12.4725 78.71802

195 121.6 -55.2128 108.3425 85 -38.5945 75.73286

225 127.4 -90.0927 90.07815 89.3 -63.1497 63.13955

255 132 -117.618 59.91601 93.1 -82.9566 42.25895

285 136.4 -134.723 21.32392 96.9 -95.7085 15.14874

315 136.5 -134.817 -21.3685 100.3 -99.0634 -15.7015

345 121.4 -108.161 -55.1278 98.1 -87.4022 -44.5473

375 107.8 -76.2159 -76.2363 91.8 -64.9037 -64.9211

405 97.3 -44.1607 -86.7013 84.9 -38.5328 -75.652

435 88.2 -13.784 -87.1163 78.1 -12.2055 -77.1404

465 81.1 12.70016 -80.0994 73 11.43171 -72.0993

495 74.7 33.92488 -66.5522 68 30.88209 -60.583

525 69.2 48.94098 -48.9226 63.3 44.76827 -44.7515

555 64.3 57.29752 -29.1802 59 52.5747 -26.775

585 60.1 59.36204 -9.38929 55.5 54.81852 -8.67064

615 60.6 59.85183 9.4931 52.1 51.45677 8.161559

-333.451 130.2172 -215.507 -6.63604

I1Cos -16.6726 I1sin 6.510862 I2cos -10.7754 I2sin -0.3318

Amp1 17.89877 Amp2 10.78047

Phase1 2.769398 Phase2 3.172483

Alpha 0.169

Beta 0.134

Diffusivity 0.231 cm^2/s

For brass the actual value of diffusivity is 0.3 /s.cm^2