Lecture Week 3 1

8/12/2019 Lecture Week 3 1

1/18

ME 3345 Heat Transfer

Week_3_1

Conduction with heat generation

Extended Surfaces


2/18

1 2 [K/W]tT T

Rq

Review of Thermal Resistance

2 1ln( / )

2t

r r

R Lk

1 21/ 1/

4t

r rR

k

One-dimensional, steady state, and constant k Without heat generation

t

LR

kA

1 2( ) lnT r C r C

1 2( )T x C x C

12( )

CT r C

r

0d dT

kdx dx

0

d dT

krdr dr

2 0d dT

krdr dr

Plane Wall

Cylindrical

Spherical

Definition:


3/18


4/18

21 2

10 ( ) ln

4

d dT q qr T r r C r C

r dr dr k k

Cylindrical:

Spherical:

2 2 122

10 ( )

6

Cd dT q qr T r r C

dr dr k k r r

22

1 220 ( )

2

d T q qT x x C x C

k kdx

Plane Wall:

With Heat Generation


5/18

21 2( )

2qT x x C x C k

Temperature Distribution

Plane Wall with Heat Generation

Symmetric

vs.

Adiabatic


6/18

Example 3.6.A plane wall is a composite of two materials, A and B. The

wall of material A has uniform heat generation 1.5 x 106W/m3, kA= 75W/m K, and thicknessLA= 50 mm. The wall material B has no generation

with kB= 150 W/m K and thicknessLB= 20 mm. The inner surface of

material A is well insulated, while the outer surface of material B is cooled

by a water stream with T = 30C and h= 1000 W/m2 K.

Determine the temperature distribution and T0, T1, T2.

Example 3.6


7/18

(a) parabolic in A(b) zero slope at wall

(c) linear in B

(d) slope change

(e) large gradient near surface/ 2B Ak k

How will qxchange?

It linearly increases in A

and remains the same in B.


8/18

Can we use thermal resistance in A?

"g out AE E q qL

/ 1/B BL k h

2 2" ( )AqLq h T T T T

h

1

1BA

B

L

T T qL k h 2 22

( ) 12

sqL xT x T

k L

2

0 1(0)2

A

A

qLT T T

k


9/18

The influence of h

Maximum temperature is often important for design consideration.


10/18

21 2

1 0 ( ) ln4

d dT q qr T r r C r C r dr dr k k

Cylindrical:

For a solid cylinder, there is only one boundary.

T(r)

Apply symmetric B.C.

10

20

20

0

0 0

( ) 4

4

r

s

dTC

dr

qT r T r k

qrT T

k

r


11/18

Example: An insulated wire of diameter D = 2mm

and uniform temperature T has an electrical resistance of

0.01 /m and a current of 20A. The insulation has an outerdiameter of 3mm and thermal conductivity of K=0.01 W/mK.

A) If heat is loss through convection, what is the surface

temperature of the rod and the insulation?

B) What is the rate of heat transfer per unit length at r =0, 0.5mm,

and 1.5 mm if the power density in the rod is 3x105W/m3?

D = 2mm

D = 3mm

h = 5 W/m2KT = 20 C


12/18

( )conv surface sq A h T T

How to enhance heat transfer

(without increasing the temperature difference) ??

Fins - Extended Surfaces


13/18

( )conv surface sq A h T T

How to enhance heat transfer

(without increasing the temperature difference) ??

(1) Increase hby strong forced convection (use fan, usewater instead of air, spray or inject water, etc.

(2) Increase the surface areaA. The second is often

achieved by using fins.

Fins - Extended Surfaces


14/18

Mobile PentiumProcessors

Extruded Heat Sink

Automobile Radiator

Examples of Extended Surfaces

Radiator (household heating)


15/18

Simple Structures:

We will perform the analysis for simple cases and discuss

engineering methods to deal with complicated geometry.


16/18

How much performance increase

Space

Weight/ Material

Manufacturing processCost


17/18

(a) Rectangular fin. (b) Pin fin.

Fins of Uniform Cross Section


18/18

Analysis of Heat Transfer Enhancement

The application of extended surfaces for heat transferenhancement must be carefully considered. This processes

induces additional manufacturing costs and complexity.

Thus, we must find a way to quantify the added benefits

of using extended surfaces to justify their application.

A) Determine the rate of heat transfer from an extended

surface. Involves finding the temperature distribution

in the fin structure.

B) Define some measure of efficiencyfor extended

surfaces. Use this as a basis for determining when to use them.

Lecture Week 3 1

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