1 lesson 1&2- general law of conduction i

7/31/2019 1 Lesson 1&2- General Law of Conduction I

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ME 306 HEAT TRANSFER

Dr. Y. Mukkamala


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1) Boiling water

(fotosearch.com)

2) Condensing steam

(Purdue University)


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1. Heat Transfer : is the rate of energytransfer (i.e., J/s or Watts) from a hightemperature source to a low temperature

sink.

2. Heat transfer deals with the rate of

energy transfer, while thermodynamics :deals with magnitude of energy and itsconversion.

3. An insulated object at a certaintemperature possesses a magnitude of

internal energy, with no attendant heattransfer.


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1. Conduction : Heat transfer mode thatoccurs mainly in solids, liquids andstationary gases due to molecular

vibration and electron transport.

2. Convection : Heat transfer mechanismobserved in liquids and vapors due tobulk movement of a fluid.

3. Radiation : Heat transfer phenomenain which heat can be transferred in the

form of photons and electromagneticrays, across a temperature differentialwithout any participating media.


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Tgas

Twall

Tbead qconv

qrad

Hot FlueGases

Thermocouple


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Cartesian Coordinates.

xdTq kAc dX

(1)

.

xq : Heat Transfer Rate (W/m)

k : Thermal conductivity (W/ m K)

Ac : Cross Section Area (m2)

dT

dx: Temperature Gradient (C/m)

Radial Coordinates (Radial Conduction)

. dTq kAr r dr (2)

Ar = 2rL (Cylindrical Area)

dT

dr: Radial Temperature Gradient

Spherical Coordinates. dTq kAr r dr

. dTq kA d

(3)

. dTq kAd

dT

dr: Radial Temperature Gradient

dT

d,

dT

d: Circumferential Temperature Gradients

Ar = A = A = 4r2

(Spherical Surface Area : Solid Sphere)


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Energy Balance (Cartesian Coordinates)

. . .( )dq q q c VT x x dx gen pdt (4)

Using Taylors expansion theorem (neglecting higher order terms)

. '''. . .( ) ( )

q dxq q dx q dV c VT x x px dt

(5)

. '''.. ( . )

q dx dx q dx A c dx AT px dt

(6)

'''.( ) . ( . )

T dkA dx q dx A c dx AT pdtx x

(7)

If variation in physical properties and cross-section are assumed to be negligible

2

2

'''.. . ( )

p

T dkA dx q dx A A dx c T dtx

2

2

.2 '''

2

'''.

p

p

T dT

k q c dtx

cT q dT

k k dt x

(8)

.2 '''

21T q dT

k dtx

(9)


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nergy Balance (Cylindrical Coordinates)

. . . . . . '''( ) ( ) ( ) [ ]dq q q q q q q dV c VT r r dr z z dz pt

(14)

Where

. . .

. . .

( ) ; ( ) ; ( )

( . ); ( . ); ( . )r zr z

T T T

q kA q kA q kAr r r

q k rd dz q k dr dz q k rd dr

Hence Eqn. 14 can be re-written as

.. . '''[ . . . ] ( . . ) [ ( . . ). ]qq qr zdr d dz q dr dz rd c dr dz rd T pr z t

'''[ ( ) ( ) ( ) ] ( . . . ) [ . . ]k rd dzT dr k rdrdzT d k rd drT dz q dr dz r d T c dr dz rd zr pr z t

'''[ ( ) ( ) ( )] (. . ) [ ]k rT k rT k rT q r T c rT zr pr z t

(15)

The above equation simplifies as (for isotropic solid)

'''2 2 21 1 12 2 2 2 qT T T T T r r tkr r z

(16)


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STEADY STATE HEAT DIFFUSION EQUATIONS (No Source Term)

Cartesian Coordinates (1-D):

20

2d T

dx

2-D :

2

2

20

2

TT

yx

Cylindrical Coordinates (Ignoring Axial Diffusion)

2 10

2

dTd T

r drdr

Spherical Coordinates

'''2 21 1 1 1( ) ( ) ( )2 2 2 2 2

qT T TrT Sinr tkr r Sin r Sin

(17)

Equation 17 is the general heat diffusion equation in spherical coordinates with a

volumetric source term.


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k : Constant of proportionality fromFourier law.

Solids : Conduction in solids is due to

lattice vibrations (phonon transport),and electron transport. Phonon vibration contributes

insignificantly. Good electrical conductors are hence,

good heat conductors as well. Exs:Copper, Aluminum, and most other

metals, and some alloys. Similarly insulators are also poor heat

conductors.

Exceptions : Semi conductors; somealloys.


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Combined Heat Transfer Eqn :

1 lesson 1&2- general law of conduction i

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