Lecture Notes for CO2 (Part 3) 1-D STEADY STATE HEAT

CONDUCTION

Wan Azmi bin Wan Hamzah Universiti Malaysia Pahang

Week – 5

Course Outcome 2 (CO2)

Students should be able to understand and

evaluate one-dimensional heat flow and in

different geometries

2

Lesson Outcomes from CO2 (Part 3)

To derive the equation for temperature

distribution in various geometries

Thermal Resistance concept – to derive

expression for various geometries

To evaluate the heat transfer using thermal

resistance in various geometries

To evaluate the critical radius of insulation

To evaluate heat transfer from the rectangular fins

3

CRITICAL RADIUS OF INSULATION

Adding more insulation to a wall or

to the attic results:

a) decrease heat transfer and,

b) increase the thermal resistance

of the wall.

In a cylindrical pipe or a spherical

shell, the additional insulation

results :

a) increase the conduction

resistance but,

b) decreases the convection

resistance because of the surface

for convection become larger .

An insulated cylindrical pipe exposed to

convection from the outer surface and

the thermal resistance network

associated with it.

• Cylindrical pipe of outer radius r1 whose outer

surface temperature T1 is maintained constant.

• The pipe is covered with an insulator (k and r2).

• Convection heat transfer at T∞ and h.

• The rate of heat transfer from the insulated

pipe to the surrounding air can be expressed

as

1 1

2 1

2

ln / 1

2 2

ins conv

T T T TQ

r rR R

Lk h r L

5

The critical radius of insulation

for a cylindrical body:

The critical radius of insulation

for a spherical shell:

The variation of heat transfer

rate with the outer radius of the

insulation r2 when r1 < rcr.

• The rate of heat transfer from the cylinder increases with the addition of insulation for r2 < rcr

• Reaches maximum, when r2 = rcr

• Decreases when r2 > rcr

• Thus, insulating the pipe may actually increase the rate of heat transfer instead of decreasing it when r2 < rcr.

A 3 mm diameter and 5 m long electrical wire is tightly wrapped with a 2 mm

thick plastic cover whose thermal conductivity is k = 0.15 W/m°C. Electrical

measurements indicate that a current of 10A passes through the wire and there

is a voltage drop of 8V along the wire.

If the insulating wire is exposed to a medium at T∞ = 30°C with a heat transfer

coefficient of h = 24 W/m2 °C, find the temperature at the interface of the wire

and the plastic cover in steady state operation. Also, determine the effect of

doubling the thickness of the plastic cover on the interface temperature.

6

Problem

1. Heat transfer is steady state.

2. Heat transfer is 1-D,

3. Thermal properties are constant.

4. The thermal contact resistance at the interface is negligible.

5. Heat transfer coefficient accounts for the radiation effects, if any.

6. Heat generation is uniform

Properties:

k= 0.15 W/m0C, h= 12 W/m2 0C

Analysis:

,

7

W80108 VIWQ e

Solution

• Outer surface Area

• Thermal resistances

• The interface temperature can be determined

• Critical radius

• Doubling thickness?

8

2

2 m 11.050035.022 LrA

C/W 0.940180.76R

C/W 18.0515.02

5.15.3ln

2

ln

C/W 76.011.012

11

0

plasticconvtotal

012plastic

0

2

conv

RR

kL

rrR

hAR

C 103T94.0

03-T80

T-T 0

11

total

1

RQ

mm 12.5m 0012512

15.0rcr

h

k

cr

crnew2,

r reaches radiusouter theuntilfer heat trans theincreasing Thus

r than less is which mm, 6r

A 5-mm-diameter spherical ball at 50 0C is covered by a 1-mm-thick

plastic insulation (k=0.13 W/m ·0C). The ball is exposed to a medium

at 15 0C, with a combined convection and radiation heat transfer

coefficient of 20 W/m2·0C.

Determine if the plastic insulation on the ball will help or hurt heat

transfer from the ball.

9

Problem

Assumptions : 1. steady state, 2. one-dimensional , 3. Thermal properties are constant.

4 The thermal contact resistance at the interface is negligible.

Properties: k = 0.13 W/m⋅°C.

Analysis: The critical radius of plastic insulation is

10

2

cr

ran greater th iswhich

mm 13m 013.020

13.022

h

kr

Solution

11

HEAT TRANSFER FROM FINNED SURFACES

When Ts and T are fixed, there are two

ways to increase the rate of heat transfer:

• Increase the convection heat transfer

coefficient h. This may require the

installation of a pump or fan or add larger

size of fan. Not adequate.

• Increase the surface area As by attaching

to the surface extended surfaces called

fins made of highly conductive materials

such as aluminum.

Newton’s law of cooling: The rate of heat transfer

from a surface to the surrounding medium

12

The thin plate fins of a car

radiator greatly increase the

rate of heat transfer to the air.

13

Fin Equation

Volume element of a fin at location x

having a length of x, cross-sectional

area of Ac, and perimeter of p.

14

Fin Equation

Volume element of a fin at location x

having a length of x, cross-sectional

area of Ac, and perimeter of p.

where

Temperature

excess

and

15

The general solution of the

differential equation

Boundary condition at fin base

Boundary conditions at the fin

base and the fin tip.

16

Under steady conditions, heat

transfer from the exposed surfaces

of the fin is equal to heat conduction

to the fin at the base.

1 Infinitely Long Fin

(Tfin tip = T)

Boundary condition at fin tip

The variation of temperature along the fin

The steady rate of heat transfer from the entire fin

17

2 Negligible Heat Loss from the Fin Tip

(Adiabatic fin tip, Qfin tip = 0)

Boundary condition at fin tip

The variation of temperature along the fin

Heat transfer from the entire fin

Fins are not likely to be so long that their temperature approaches the

surrounding temperature at the tip. A more realistic assumption is for

heat transfer from the fin tip to be negligible since the surface area of

the fin tip is usually a negligible fraction of the total fin area.

18

3 Specified Temperature (Tfin,tip = TL)

In this case the temperature at the end of the fin (the fin tip) is

fixed at a specified temperature TL.

This case could be considered as a generalization of the case of

Infinitely Long Fin where the fin tip temperature was fixed at T.

19

4 Convection from Fin Tip

The fin tips, in practice, are exposed to the surroundings, and thus the proper

boundary condition for the fin tip is convection that may also include the effects

of radiation. Consider the case of convection only at the tip. The condition

at the fin tip can be obtained from an energy balance at the fin tip.

20

Replace the fin length L in the relation for

the insulated tip (slide 49) by a corrected

length defined as

Corrected fin length Lc is defined such

that heat transfer from a fin of length Lc

with insulated tip is equal to heat transfer

from the actual fin of length L with

convection at the fin tip.

t is the thickness of the rectangular fins.

D is the diameter of the cylindrical fins.

4 Convection from Fin Tip - cont

21

Fin Efficiency

Zero thermal resistance or infinite

thermal conductivity (Tfin = Tb)

22

mL

23

Efficiency of straight fins of rectangular, triangular, and parabolic profiles.

Triangular and parabolic are more efficient than rectangular, contain less

material and more suitable for applications requiring less weight and less

space

24

25

• The fin efficiency decreases with increasing fin length because

of decrease in fin temperature with length.

• Fin lengths that cause the fin efficiency to drop below 60 percent

usually cannot be justified economically.

• The efficiency of most fins used in practice is above 90 percent.

26

Fin

Effectiveness

Effectiveness

of a fin

• The thermal conductivity k of the fin

material should be as high as possible.

Use aluminum, copper, iron.

• The ratio of the perimeter to the cross-

sectional area of the fin p/Ac should be

as high as possible. Use slender or thin

pin fins.

• Low convection heat transfer coefficient

h. Place fins on the gas (air) side.

27

Proper Length of a Fin

Because of the gradual temperature drop

along the fin, the region near the fin tip makes

little or no contribution to heat transfer.

mL = 5 an infinitely long fin

mL = 1 offers a good compromise

between heat transfer

performance and the fin size.

28

• Heat sinks: Specially

designed finned surfaces

which are commonly used in

the cooling of electronic

equipment, and involve one-

of-a-kind complex

geometries.

• The heat transfer

performance of heat sinks is

usually expressed in terms of

their thermal resistances R.

• A small value of thermal

resistance indicates a small

temperature drop across the

heat sink, and thus a high fin

efficiency.