Short Version : 16. Temperature & Heat

16.1. Heat , Temperature & Thermodynamic Equilibrium

Thermodynamic equilibrium:State at which macroscopic properties of system remains unchanged over time.

Examples of macroscopic properties:

L, V, P, , , …

0th law of thermodynamics:2 systems in thermodynamic equilibrium with a 3rd system are themselves in equilibrium.

2 systems are in thermal contact if heating one of them changes the other.

Otherwise, they are thermally insulated.

Two systems have the same temperature

they are in thermodynamic equilibrium

A,B in eqm

B,C in eqm A,C in eqm

Gas Thermometers & the Kelvin Scale

Constant volume gas thermometer T P

Kelvin scale:

P = 0 0 K = absolute zero

Triple point of water 273.16 K

Triple point: T at which solid, liquid & gas phases co-exist in equilibrium

All gases behave similarly as P 0.

Mercury fixed at this level by adjusting h P T.

Temperature Scales

Celsius scale ( C ) :

Melting point of ice at P = 1 atm TC = 0 C.

Boiling point of water at P = 1 atm TC = 100 C.

Triple point of water = 0.01C

273.15CT T CT T

Fahrenheit scale ( F ) :

Melting point of ice at P = 1 atm TF = 32 F.

Boiling point of water at P = 1 atm TF = 212 F.

18032

100F CT T 9

5F CT T

Rankine scale ( R ) :

0 0R K

R FT T

16.2. Heat Capacity & Specific Heat

Q C T Heat capacity C of a body :

Q = heat transferred to body. /C J K

Specific heat c = heat capacity per unit mass Q m c T

/c J kg K

1 calorie (15C cal) = heat needed to raise 1 g of water from 14.5C to 15.5C.

1 BTU (59F) = heat needed to raise 1 lb of water from 58.5F to 59.5F.

1 4.184

1 1055

cal thermochemical J

BTU J

1 4kcal BTU

c = c(P,V) for gases cP , cV .

The Equilibrium Temperature

Heat flows from hot to cold objects until a common equilibrium temperature is reached.

For 2 objects insulated from their surroundings:

1 2 0Q Q 1 1 1 2 2 2m c T m c T

When the equilibrium temperature T is reached:

1 1 1 2 2 2 0m c T T m c T T

1 1 1 2 2 2

1 1 2 2

m c T m c TT

m c m c

16.3. Heat Transfer

Common heat-transfer mechanisms:

• Conduction

• Convection

• Radiation

Conduction

Conduction: heat transfer through direct physical contact.

Mechanism: molecular collision.

Thermal conductivity k ,

[ k ] = W / mK

dQH

dt

Tk A

x

Heat flow H , [ H ] = watt :

conductor

insulator

Specific Heat vs Thermal Conductivity

c ( J/kgK ) k (W/mK )

Al 900 237

Cu 386 401

Fe 447 80.4

Steel 502 46

Concrete 880 1

Glass 753 0.8

Water 4184 0.61

Wood 1400 0.11

TH k A

x

applies only when T = const over each (planar) surface

For complicated surface, use d T

H k Ad x

Prob. 72 & 78.

Composite slab:

H must be the same in both slabs to prevent

accumulated heat at interface

3 22 11 2

1 2

T TT TH k A k A

x x

Thermal resistance :x

Rk A

[ R ] = K / W

2 1

1

T T

R

1 3

1 2

T TH

R R

1 2 1T T H R

2 3 2T T H R Resistance in series

TH

R

3 2

2

T T

R

Insulating properties of building materials are described by the -factor ( -value ) .

xR A

k

R = thermal resistance of a slab of unit area

2 /m K WR

2 /ft F h BTU RU.S.

TA

H

2 21 / 0.176 /ft F h BTU m K W

d TH k A

d x

T

R

AT

R

Example 16.4. Cost of Oil

The walls of a house consist of plaster ( = 0.17 ), -11 fiberglass

insulation, plywood ( = 0.65 ), and cedar shingles ( = 0.55 ).

The roof is the same except it uses -30 fiberglass insulation.

In winter, average T outdoor is 20 F, while the house is at 70 F.

The house’s furnace produces 100,000 BTU for every gallon of oil,

which costs $2.20 per gallon. How much is the monthly cost?

0.17 11 0.65 0.55wall R 12.370.17 30 0.65 0.55roof R 31.37

2 36 28 10rectA ft ft ft 21280 ft21164 ft 14

2 36cos30roof

ftA ft 1

2 28 14 tan 302gableA ft ft 2226 ft

21506wall rect gableA A A ft

2 21/ / / 1506 70 20

12.37wallH BTU h ft F ft F F

2 21/ / / 1164 70 20

31.37roofH BTU h ft F ft F F

6073 /BTU h

1853 /BTU h

6073 1853 / 24 / 30 /Q BTU h h d d month 5.7 MBTU

5.7 10 / $ 2.20 /Cost MBTU gal MBTU gal $126

Convection

Convection = heat transfer by fluid motion

T rises

Convection cells in liquid film between glass plates(Rayleigh-Bénard convection, Benard cells)

Radiation

Glow of a stove burner it loses energy by radiation

4Pe T

AStefan-Boltzmann law for radiated power:

= Stefan-Boltzmann constant = 5.67108 W / m2 K4.

A = area of emitting surface.

0 < e < 1 is the emissivity ( effectiveness in emitting radiation ).

e = 1 perfect emitter & absorber ( black body ).

Black objects are good emitters & absorbers.

Shiny objects are poor emitters & absorbers.

Wien‘s displacement law : max = b / T

P T4 Radiation dominates at high T.

Wavelength of peak radiation becomes shorter as T increases.

Sun ~ visible light.

Near room T ~ infrared.

4Pe T

AStefan-Boltzmann law :

sun sunRT

RT

T

T

32.898 10b mK

.502 5778

300

m K

K

9.66 m

Example 16.5. Sun’s Temperature

The sun radiates energy at the rate P = 3.91026 W, & its radius is 7.0 108

m.

Treating it as a blackbody ( e = 1 ), find its surface temperature.4P e AT

= 5.67108 W / m2 K4

1/4

26

28 2 4 8

3.9 10

5.67 10 / 4 7.0 10

W

W m K m

1/4

24

PT

e R

35.8 10 K

Conceptual Example 15.1. Energy-Saving Windows

Why do double-pane windows reduce heat loss greatly compared with single-paned windows?

Why is a window’s -factor higher if the spacing between panes is small?

And why do the best windows have “low-E” coatings?

Thermal conductivity (see Table 16.2):Glass k ~ 0.8 W/mK Air k ~ 0.026 W/mK

Layer of air reduces heat loss greatly & increases the -factor .

This is so unless air layer is so thick that convection current develops.

“low-E” means low emissivity, which reduces energy loss by radiation.

Making the Connection

Compare the for a single pane window made from 3.0-mm-thick glass with that of a double-pane window make from the same glass with a 5.0-mm air gap between panes.

x

k

R Glass k ~ 0.8 W/mK

Air k ~ 0.026 W/mK

3

single

3.0 10

0.8 /

m

W m K

R

2double 0.2 /m K W R

20.004 /m K W

3 33.0 10 5.0 10

20.8 / 0.026 /double

m mR

W m K A W m K A

double single50 R R

xR

A k A

R

20.2/m K W

A

16.4. Thermal Energy Balance

A house in thermal-energy balance.

System with fixed rate of energy input

tends toward an energy- balanced state

due to negative feedback.

Heat from furnace balances

losses thru roofs & walls

Example 16.7. Solar Greenhouse

A solar greenhouse has 300 ft2 of opaque -30 walls,

& 250 ft2 of -1.8 double-pane glass that admits solar energy at the rate of 40 BTU / h / ft2.

Find the greenhouse temperature on a day when outdoor temperature is 15 F.

T A TH

R

R

67 F

2

2

300

30 /wall

ft TH

ft F h BTU

10 / /BTU h F T

2

2

250

1.8 /glass

ft TH

ft F h BTU

139 / /BTU h F T

2 240 / / 250sunH BTU h ft ft 410 /BTU h

410 /

149 / /

BTU hT

BTU h F

15 67T F F 82 F

wall glassH H

Application: Greenhouse Effect & Global Warming

Average power from sun :

2960 /S W m

Total power from sun : 2S EH S R

Power radiated (peak at IR) from Earth :

2 44E EH e R T

S EH H

18 C

1/42

8 2 4

960 /

5.67 10 / 4

W mT

W m K

255 K1e

C.f. T 15 C natural greenhouse effect

Greenhouse gases: H2O, CO2 , CH4 , …

passes incoming sunlight, absorbs outgoing IR.

Mars: none

Venus: huge

CO2 increased by 36%

0.6 C increase during 20th century.

1.5 C – 6 C increase by 2100.