heat transfer of stuff

7232019 Heat Transfer of stuff

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56 Chapter 2

Energy and the First Law of Thermodynamics

Conduction

Energy transfer by conduction can take place in solids liquids and gases Conductioncan be thought of as the transfer of energy from the more energetic particles of asubstance to adjacent particles that are less energetic due to interactions betweenparticles The time rate of energy transfer by conduction is quantified macroscopicallyby Fourierrsquos law As an elementary application consider Fig 212 showing a planewall of thickness L at steady state where the temperature T ( x) varies linearly with

position x By Fourierrsquos law the rate of heat transfer across any plane normal to the x direction Q

x is proportional to the wall area A and the temperature gradient in the x direction dTdx

Q

x 5 2kAdT

dx (231)

where the proportionality constant k is a property called the thermal conductivity

The minus sign is a consequence of energy transfer in the direction of decreasing temperature

in the case of Fig 212 the temperature varies linearly thus thetemperature gradient is

dT dx 5 T 2

2T 1

L 1 02and the rate of heat transfer in the x direction is then

Q

x 5 2kA c T 2 2 T 1

Ld b b b b b

Values of thermal conductivity are given in Table A-19 for common materials Sub-stances with large values of thermal conductivity such as copper are good conductorsand those with small conductivities (cork and polystyrene foam) are good insulators

Radiation

Thermal radiation is emitted by matter as a result of changes in the electronic configu-rations of the atoms or molecules within it The energy is transported by electromag-netic waves (or photons) Unlike conduction thermal radiation requires no interveningmedium to propagate and can even take place in a vacuum Solid surfaces gases andliquids all emit absorb and transmit thermal radiation to varying degrees The rateat which energy is emitted Q

e from a surface of area A is quantified macroscopicallyby a modified form of the StefanndashBoltzmann law

Q

e 5 esAT 4

b (232)

which shows that thermal radiation is associated with the fourth power of the abso-lute temperature of the surface T b The emissivity e is a property of the surface thatindicates how effectively the surface radiates 10 e 102 and s is the StefanndashBoltzmann constant

s 5 567 3 1028 W m2 K4 5 01714 3 10

28 Btu h ft2 8R4

In general the net rate of energy transfer by thermal radiation between two surfacesinvolves relationships among the properties of the surfaces their orientations withrespect to each other the extent to which the intervening medium scatters emits andabsorbs thermal radiation and other factors A special case that occurs frequently isradiation exchange between a surface at temperature T b and a much larger surround-ing surface at T s as shown in Fig 213 The net rate of radiant exchange between thesmaller surface whose area is A and emissivity is and the larger surroundings is

Q

e 5 esA3T 4b 2 T 4

s4 (233)

T 2

T 1

L

Area A

x

Q x

Fig 212 Illustration of

Fourierrsquos conduction law

Fourierrsquos law

StefanndashBoltzmann law

AAHT_Modes

A7 ndash Tab d

AAHT_Modes

A7 ndash Tab b



Convection

Energy transfer between a solid surface at a temperature T b and an adjacent gas orliquid at another temperature T f plays a prominent role in the performance of manydevices of practical interest This is commonly referred to as convection As an illus-tration consider Fig 214 where T b T f In this case energy is transferred in the

direction indicated by the arrow due to the combined effects of conduction within the

air and the bulk motion of the air The rate of energy transfer from the surface to the air can be quantified by the following empirical expression

Q

c 5 hA1T b 2 T f 2 (234)

known as Newtonrsquos law of cooling In Eq 234 A is the surface area and the proportion-ality factor h is called the heat transfer coefficient In subsequent applications of Eq 234a minus sign may be introduced on the right side to conform to the sign conventionfor heat transfer introduced in Sec 241

The heat transfer coefficient is not a thermodynamic property It is an empiricalparameter that incorporates into the heat transfer relationship the nature of the flowpattern near the surface the fluid properties and the geometry When fans or pumpscause the fluid to move the value of the heat transfer coefficient is generally greaterthan when relatively slow buoyancy-induced motions occur These two general categoriesare called forced and free (or natural) convection respectively Table 21 provides typicalvalues of the convection heat transfer coefficient for forced and free convection

983090983092983091 Closing Comments

The first step in a thermodynamic analysis is to define the system It is only afterthe system boundary has been specified that possible heat interactions with thesurroundings are considered for these are always evaluated at the system boundary

Surface of emissivity area A

and temperature T b

ε

Surroundingsurface at T s

Qe

Fig 213 Net radiation exchange

AT b

Qc

Cooling air flow

T f lt T b

Wire leads

Transistor

Circuit board

Fig 214 Illustration of Newtonrsquos law of cooling

Newtonrsquos law of cooling

Typical Values of the Convection Heat Transfer Coefficient

Applications h (Wm2 K) h (Btuh f t2 8R)

Free convection

Gases 2ndash25 035ndash44

Liquids 50ndash1000 88ndash180

Forced convection



TABLE 21

24 Energy Transfer by Heat 5

AAHT_Modes

A7 ndash Tab c



Convection

Energy transfer between a solid surface at a temperature T b and an adjacent gas orliquid at another temperature T f plays a prominent role in the performance of manydevices of practical interest This is commonly referred to as convection As an illus-tration consider Fig 214 where T b T f In this case energy is transferred in the

direction indicated by the arrow due to the combined effects of conduction within the

air and the bulk motion of the air The rate of energy transfer from the surface to the air can be quantified by the following empirical expression

Q

c 5 hA1T b 2 T f 2 (234)

known as Newtonrsquos law of cooling In Eq 234 A is the surface area and the proportion-ality factor h is called the heat transfer coefficient In subsequent applications of Eq 234a minus sign may be introduced on the right side to conform to the sign conventionfor heat transfer introduced in Sec 241

The heat transfer coefficient is not a thermodynamic property It is an empiricalparameter that incorporates into the heat transfer relationship the nature of the flowpattern near the surface the fluid properties and the geometry When fans or pumpscause the fluid to move the value of the heat transfer coefficient is generally greaterthan when relatively slow buoyancy-induced motions occur These two general categoriesare called forced and free (or natural) convection respectively Table 21 provides typicalvalues of the convection heat transfer coefficient for forced and free convection

983090983092983091 Closing Comments

The first step in a thermodynamic analysis is to define the system It is only afterthe system boundary has been specified that possible heat interactions with thesurroundings are considered for these are always evaluated at the system boundary

Surface of emissivity area A

and temperature T b

ε

Surroundingsurface at T s

Qe

Fig 213 Net radiation exchange

AT b

Qc

Cooling air flow

T f lt T b

Wire leads

Transistor

Circuit board

Fig 214 Illustration of Newtonrsquos law of cooling

Newtonrsquos law of cooling

Typical Values of the Convection Heat Transfer Coefficient

Applications h (Wm2 K) h (Btuh f t2 8R)

Free convection



Forced convection



TABLE 21

24 Energy Transfer by Heat 5

AAHT_Modes

A7 ndash Tab c

heat transfer of stuff

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