IntroTHT 2e SM Chap14

8/13/2019 IntroTHT 2e SM Chap14

1/114

Solutions Manual

for

Introduction to Thermodynamics and Heat Transfer

Yunus A. Cengel2ndEdition, 2008

Chapter !

"AT#$A% C&"'ECTI&"

($&($IETA$Y A") C&"*I)E"TIA%

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*+-*


2/114

(hysical +echanisms of "atural Conection

*+*atural conection is the moe of heat transfer that occurs !etween a soli an a flui which moesuner the influence of natural means. atural conection iffers from force conection in that fluimotion in natural conection is cause !y natural effects such as !uoyancy.

*+-The conection heat transfer coefficient is usually hi"her in force conection !ecause of the hi"herflui elocities inole.

*+The hot !oile e"" in a spacecraft will cool faster when the spacecraft is on the "roun since there isno "raity in space, an thus there will !e no natural conection currents which is ue to the !uoyancyforce.

*++The upwar force eerte !y a flui on a !oy completely or partially immerse in it is calle the!uoyancy or liftin" force. The !uoyancy force is proportional to the ensity of the meium. Therefore,the !uoyancy force is the lar"est in mercury, followe !y in water, air, an the eacuate cham!er. otethat in an eacuate cham!er there will !e no !uoyancy force !ecause of a!sence of any flui in the

meium.

*+/The !uoyancy force is proportional to the ensity of the meium, an thus is lar"er in sea water thanit is in fresh water. Therefore, the hull of a ship will sin/ eeper in fresh water !ecause of the smaller!uoyancy force actin" upwars.

*+0 0 sprin" scale measures the wei"ht force actin" on it, an the person will wei"h less in water!ecause of the upwar !uoyancy force actin" on the person1s !oy.

*+1The "reater the olume epansion coefficient, the "reater the chan"e in ensity with temperature,the "reater the !uoyancy force, an thus the "reater the natural conection currents.

*+2There cannot !e any natural conection heat transfer in a meium that eperiences no chan"e inolume with temperature.

*+3The lines on an interferometer photo"raph represent isotherms (constant temperature lines) for a "as,which correspon to the lines of constant ensity. Closely pac/e lines on a photo"raph represent a lar"etemperature "raient.

*+*4The Grashof num!er represents the ratio of the !uoyancy force to the iscous force actin" on aflui. The inertial forces in 2eynols num!er is replace !y the !uoyancy forces in Grashof num!er.

*+**The olume epansion coefficient is efine as

P

T

=

*. 3or an ieal "as, RTP = or

RT

P= , an thus

( )( )

TTRT

P

TRT

P

T

RTP

P

****

4*&

==

=

=

=


*+-&


3/114

"atural Conection oer -urfaces

*+*-2aylei"h num!er is the prouct of the Grashof an 5rantl num!ers.

*+* 0 ertical cyliner can !e treate as a ertical plate when+4*

67

Gr

LD .

*+*+o, a hot surface will cool slower when facin" own since the warmer air in this position cannotrise an escape easily.

*+*/The heat flu will !e hi"her at the !ottom of the plate since the thic/ness of the !ounary layerwhich is a measure of thermal resistance is the lowest there.


*+-6


4/114

*+*0Heat is "enerate in a hori8ontal plate while heat is lost from it !y conection an raiation. Thetemperature of the plate when steay operatin" conitions are reache is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm.

Properties:e assume the surface temperature to !e 7'C. Then the properties of air at * atm an the film

temperature of (Ts;T)4& < (7';&')4& < 67C are (Ta!le 0-&&)

*-

&7

=''6&+>.'=)&>667(

**

>&?.'5r

4sm*'?77.*

C:4m.'&?&7.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is

m+'+++.')@m&'.'()m*?.'A(&

)m&'.')(m*?.'(=

+==

p

AL sc

The 2aylei"h num!er is

)>&?.'()4sm*'?77.*(

)m'++++.')(=&'7')(=''6&+>.')(m4s*.B(5r)(&&7

6-*&

&

6

==

cs LTTgRa

The usselt num!er relation for the top surface of the plate is

>6.**)7B6,&&&(7+.'7+.' &7.'&7.' === RaNu

Then

C.:4mB&.?)>6.**(m7'++++.'

C:4m.'&?&7.' & =

== NuL

kh

c

an

)&'('.&&*>C)&')(m&'.'*?.')(C.:4mB&.?()( &&top === sss TTTThAQ

The usselt num!er relation for the !ottom surface of the plate is?7.7)7B6,&&&(&>.'&>.' &7.'&7.' === RaNu

Then

C.:4m+?+.6)?7.7(m7'++++.'

C:4m.'&?&7.' & =

== NuL

kh

c

)&'('.**'C)&')(m&'.'*?.')(C.:4m+?+.6()( &&!ottom === sss TTTThAQ

Consierin" that raiation heat loss to surrounin"s occur !oth from top an !ottom surfaces, it may !eepresse as

[ ][ ]++B

+++&&

+

surr

+

ra

)=&>6*>()=&>6(*'&?7B.6

)=&>6*>()=&>6().=:4m*'?>.7)(m&'.'*?.')(&)(B.'()(&

++=

++==

s

s

s

T

TTTAQ

:hen the heat lost from the plate euals to the heat "enerate, the steay operatin" conitions are reache.The surface temperature in this case can !e etermine !y trial-error or usin" DD9 to !e


*+-+

0ir

T< &'C

Qcon

Qra

L < *? cm

Qcon

Qra


5/114

[ ],+0.2=

++++=

++=

s

sss

T

TTT

QQQQ

++B

ra!ottomtoptotal

)=&>6*>()=&>6(*'&?7B.6)&'('.**')&'('.&&*>:&'


*+-7


6/114

*+*13lue "ases are release to atmosphere usin" a cylinrical stac/. The rates of heat transfer from thestac/ with an without win cases are to !e etermine.


PropertiesThe properties of air at * atm an the film

temperature of (Ts;T)4& < (+';*')4& < &7C are

(Ta!le 0-&&)

*-

&7

=''667?.'=)&>6&7(

**

>&B?.'5r

4sm*'7?&.*

C:4m.'&77*.'

=+

==

==

=

fT

k

Analysis(a) :hen there is no win heat transfer is !ynatural conection. The characteristic len"th in this

case is the hei"ht of the stac/, m.*'==LLcThen,

&&7

6-*&

&

6

B7.&)>&B?.'()4sm*'7?&.*(

)m*')(=*'+')(=''667?.')(m4s*.B(5r)( = ==

LTTgRa s

:e can treat this ertical cyliner as a ertical plate since

+4*+4**&+4*

67thusan'.?E&+?.'

)>&B?.'4*'B76.&(

)*'(6767

Gr

LD

Gr

L=

=

The usselt num!er is etermine from

*+67)*'B76.&(*.'*.' 64**&64* === RaNu (from Ta!le *+-*)

Then

C.:4m??'.6)*+67(m*'

C:4m.'&77*.' & =

== NuL

kh

c

an

20/0=== C)*'+')(m*'?.')(C.:4m??'.6()(&& TThAQ s

(b) :hen the stac/ is epose to &' /m4h wins, the heat transfer will !e !y force conection. :e haeflow of air oer a cyliner an the heat transfer rate is etermine as follows$

+'',&*64sm*'7?&.*

m)m4s)('.?6?''4*'''&'(2e

&7 =

==

VD

B.+>6)>&B?.'()+'',&*6('&>.'5r2e'&>.'u 64*'7.'64*'7.' === (fromTa!le *B-&)

C.:4m*7.&')B.+>6(m?.'

C:4m.'&77*.'u & =

==

D

kh

,10=== C)*'+')(m*'?.')(C.:4m*7.&'()( && TThAQ sDiscussionThere is more than fie-fol increase in heat transfer ue to wins.


*+-?

0ir

T< *'C

Ts< +'C

L < *' m

D< '.? m


7/114

*+*2Heat "enerate !y the electrical resistance of a !are ca!le is issipate to the surrounin" air. Thesurface temperature of the ca!le is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. + The temperature of the surface of the ca!le is constant.

Properties:e assume the surface temperature to !e *''C. Then the properties of air at * atm an the film

temperature of (Ts;T)4& < (*'';&')4& < ?'C are (Ta!le 0-&&)

*-

&7

=''6''6.'=)&>6?'(

**

>&'&.'5r

4sm*'B?.*

C:4m.'&'.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is the outer iameter of the pipe, m.''7.'==DLc Then,

&.7B')>&'&.'()4sm*'B?.*(

)m''7.')(=&'*'')(=''6''6.')(m4s*.B(5r

)(&&7

6-*&

&

6

=

=

=

DTTgRa

s

( )[ ] ( )[ ]6+?.&

>&'&.'477B.'*

)&.7B'(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

&

&

m'?&6.')m+)(m''7.'(

C.:4m*>.*6)6+?.&(m''7.'

C:4m.'&'.'

===

=

==

DLA

NuD

kh

s

C*&.

C)&')(m'?&6.')(C.:4m*>.*6()0F)(*.7'?(

)(

&&

=

=

=

s

s

ss

T

T

TThAQ

which is not close to the assume alue of *''C. 2epeatin" calculations for an assume surface

temperature of *&'C, ATf < (Ts;T)4& < (*&';&')4& < >'C@

*-

&7

=''&B*7.'=)&>6>'(

**

>*>>.'5r

4sm*'BB7.*

C:4m.'&*.'

=+

==

==

=

fT

k

?.?++)>*>>.'(

)4sm*'BB7.*(

)m''7.')(=&'*&')(=''&B*7.')(m4s*.B(5r

)(&&7

6-*&

&

6

=

=

=

DTTgRa

s

( )[ ] ( )[ ]6>.&

>*>>.'477B.'*

)?.?++(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

C.:4m>?.*6)6>.&(m''7.'

C:4m.'&*.' & =

== NuD

kh


*+->

0ir

T< &'C

Ca!leT

s<

L


8/114

,*-+.*=

=

=

s

s

ss

T

TA

TThAQ

C)&')(m'?&6.')(C.:4m>?.*6()7.*)(F?'(

)(

&&

which is sufficiently close to the assume alue of *&'C.


*+-


9/114

*+*30 hori8ontal hot water pipe passes throu"h a lar"e room. The rate of heat loss from the pipe !ynatural conection an raiation is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. + The temperature of the outer surface of the pipe is constant.

PropertiesThe properties of air at * atm an the film temperature

of (Ts;T)4& < (>6;&>)4& < 7'C are (Ta!le 0-&&)

*-

&7

=''6'B?.'=)&>67'(

**

>&&.'5r

4sm*'>B.*

C:4m.'&>67.'

=+

==

==

=

fT

k

Analysis(a) The characteristic len"th in this case is the outer iameter of the pipe, m.'?.'==DLc Then,

&&7

6-*&

&

6

*>+>.?)>&&.'()4sm*'>B.*(

)m'?.')(=&>>6)(=''6'B?.')(m4s*.B(5r

)(=

=

=

DTTgRa

s

( )[ ] ( )[ ]'7.*6

>&&.'477B.'*

)*'>+>.?(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*7&

&>4*?4B

?4*

=

+

+=

++=

RaNu

&

&

m7.*)m*')(m'?.'(

C.:4mB7'.7)'7.*6(m'?.'

C:4m.'&>67.'

===

=

==

DLA

NuD

kh

s

5/*0=== C)&>>6)(m7.*)(C.:4mB7'.7()(&&TThAQ ss

(b) The raiation heat loss from the pipe is

[ ] 5/=++= = +++&&++

)=&>6&>()=&>6>6().=:4m*'?>.7)(m7.*)(.'()( surrssrad TTAQ


*+-B

0ir

T< &>C

5ipeT

s< >6C

< '.

L


10/114

*+-40 power transistor mounte on the wall issipates '.* :. The surface temperature of the transistoris to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir isan ieal "as with constant properties. 0ny heat transferfrom the !ase surface is isre"are.+ The localatmospheric pressure is * atm. /0ir properties are

ealuate at *''C.PropertiesThe properties of air at * atm an the "ien

film temperature of *''C are (Ta!le 0-&&)

*-

&7

=''&?.'=)&>6*''(

**

>***.'5r

4sm*'6'?.&

C:4m.'6'B7.'

=+

==

==

=

fT

k

AnalysisThe solution of this pro!lem reuires a trial-an-error approach since the etermination of the2aylei"h num!er an thus the usselt num!er epens on the surface temperature which is un/nown. :e

start the solution process !y "uessin" the surface temperature to !e *?7C for the ealuation of h. This is

the surface temperature that will "ie a film temperature of *''C. :e will chec/ the accuracy of this"uess later an repeat the calculations if necessary.

The transistor loses heat throu"h its cylinrical surface as well as its top surface. 3or conenience,we ta/e the heat transfer coefficient at the top surface of the transistor to !e the same as that of its siesurface. (The alternatie is to treat the top surface as a ertical plate, !ut this will ou!le the amount ofcalculations without proiin" much improement in accuracy since the area of the top surface is muchsmaller an it is circular in shape instea of !ein" rectan"ular). The characteristic len"th in this case is the

outer iameter of the transistor, m.''+.'==DLc Then,

?.&B&)>***.'()4sm*'6'?.&(

)m''+.')(=67*?7)(=''&?.')(m4s*.B(5r

)(&&7

6-*&

&

6

=

=

=

DTTgRa s

( )[ ] ( )[ ]'6B.&

>***.'477B.'*

)?.&B&(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

&&&

&

m''''?B*.'+4m)''+.'()m''+7.')(m''+.'(+4

C.:4m>.*7)'6B.&(m''+.'

C:4m.'6'B7.'

=+=+=

=

==

DDLA

NuD

kh

s

an

[ ]C*>

)=&>6&7()&>6()*'?>.7)(m''''?B*.')(*.'(

C)67)(m''''?B*.')(C.:4m.*7(:*.'

)()(

++&

&&

++

=

+++=

+=

s

s

s

surrssss

T

T

T

TTATThAQ

which is relatiely close to the assume alue of *?7C. To improe the accuracy of the result, we repeat

the 2aylei"h num!er calculation at new surface temperature of *>C an etermine the surface

temperature to !e

Ts< *2


*+-*'

0ir

67C

5owertransistor, '.* :

D< '.+ cm< '.*


11/114

Discussion: ealuate the air properties a"ain at *''C when repeatin" the calculation at the new surface

temperature. It can !e shown that the effect of this on the calculate surface temperature is less than *C.


*+-**


12/114

*+-* &&6 5ro!. *+-&' is reconsiere. The effect of am!ient temperature on the surface temperature ofthe transistor is to !e inesti"ate.

AnalysisThe pro!lem is sole usin" DD9, an the solution is "ien !elow.

"GIVEN"Q_dot=0.18 [W]

T_infinity=35 [C]L=0.0045 [m]=0.004 [m]!#i$on=0.1

T_#%&&=T_infinity'10 [C]

"()*(E)TIE+",$%id-=/i&=Cond%ti2ity,$%id- T=T_fi$m(&=(&/ndt$,$%id- T=T_fi$m&6o=!n#ity,$%id- T=T_fi$m (=101.3m%=Vi#o#ity,$%id- T=T_fi$mn%=m%7&6o

!t/=17T_fi$m9:;3T_fi$m=17:;E'8 [W7m?:'@?4]A+t!f/n'Bo$tm/nn on#t/nt"=D.80; [m7#?:]A&/2it/tion/$ /!$!&/tion"

"NLF+I+"d!$t/=)/=

*? *?7.?

* *?>.7

&' *?B.+

&& *>*.6

&+ *>6.&

&? *>7.*

& *>>

6' *>.B

6& *'.>6+ *&.?

6? *+.7

6 *?.+

+' *.&


*+-*&

10 15 20 25 30 35 40

155

160

165

170

175

180

185

190

T C3

Ts

2C3


13/114

*+--&0 hot plate with an insulate !ac/ is consiere. The rate of heat loss !y natural conection is to !eetermine for ifferent orientations.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "aswith constant properties. The local atmospheric pressure is * atm.

PropertiesThe properties of air at * atm an the film temperature of

(Ts;T)4& < (*6';>7)4& < *'&.73 are (Ta!le 0-&&)

*-

&6

2''*>>.'2)+?'7.*'&(

**

>&7?.'5r

4sft*'*&6.'

3#tu4h.ft.'*767.'

=+

==

==

=

fT

k

Analysis(a) :hen the plate is ertical, the characteristic len"th is

the hei"ht of the plate. ft.&==LLc Then,

&&6

6-*&

&

6

7'6.7)>&7?.'()4sft*'*&6.'(

)ft&)(2>7*6')(2''*>>.')(ft4s&.6&(5r

)(=

=

=

LTTgRa

s

?.*'&

>&7?.'

+B&.'*

)*'7'6.7(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&&&

&

ft+)ft&(

3.#tu4h.ft>?B.')?.*'&(ft&

3#tu4h.ft.'*767.'

===

=

==

LA

NuL

kh

s

an 4tu5h/.=== C)>7*6')(ft+)(3.#tu4h.ft>?B.'()(&&TThAQ ss

(b) :hen the plate is hori8ontal with hot surface facin" up, the characteristic len"th is etermine from

ft7.'

+

ft&

++

&

===== L

L

L

P

AL ss .

Then,

?

&&6

6-*&

&

6

*'7B.)>&7?.'()4sft*'*&6.'(

)ft7.')(2>7*6')(2''*>>.')(ft4s&.6&(5r

)(=

=

=

cs LTTgRa

&+.&B)*'7B.(7+.'7+.' +4*?+4* === RaNu

3.#tu4h.ftB>7.')&+.&B(ft7.'

3#tu4h.ft.'*767.' & =

== NuL

kh

c

an 4tu5h1/.!=== C)>7*6')(ft+)(3.#tu4h.ftB>7.'()(&&TThAQ ss

(c) :hen the plate is hori8ontal with hot surface facin" own, the characteristic len"th is a"ain =

'7. ft an the 2aylei"h num!er is ?*'7B. =Ra . Then,

?&.*+)*'7B.(&>.'&>.' +4*?+4* === RaNu

3.#tu4h.ft++>.')?&.*+(ft7.'

3#tu4h.ft.'*767.' & =

== NuL

kh

c

an 4tu5h18./=== C)>7*6')(ft+)(3.#tu4h.ft++>.'()(&&TThAQ ss


*+-*6

Q

Insulation

0ir

T< >73

5late

Ts< *6'3

L < & ft


14/114

*+-& &&6 5ro!. *+-&&D is reconsiere. The&/t! of n/t%&/$ on2!tion 6!/t t&/n#f!& fo&diff!&!nt o&i!nt/tion# of t6! $/t! /# / f%ntion of t6! $/t! t!m!&/t%&!is to !e plotte.


"GIVEN"L=: [ft]

T_infinity=;5 [,]T_#=130 [,]

"()*(E)TIE+",$%id-=/i&=Cond%ti2ity,$%id- T=T_fi$m(&=(&/ndt$,$%id- T=T_fi$m&6o=!n#ity,$%id- T=T_fi$m (=14.;m%=Vi#o#ity,$%id- T=T_fi$m0

T_fi$m=17:


15/114

Ts798 :a7;tu*.B> B6.*7 +?.7

**' >.*7 **&. 7?.+

**7 *'&. *66.* ??.7?

*&' **B *7+ >>.'&

*&7 *67.? *>7.7 >.>7

*6' *7&.7 *B>.+ B.>&

*67 *?B.B &*B.B *'B.B

*+' *>.7 &+&.> *&*.6

*+7 &'7.+ &?7.B *6&.B

*7' &&6.> &B.7 *++.>

*77 &+&.* 6*6.+ *7?.>

*?' &?'.B 66>.> *?.

*?7 &>B.B 6?&.& **.**>' &BB.* 6>.* *B6.7

*>7 6*.7 +*&.& &'?.*

*' 66.* +6>.? &*.

80 100 120 140 160 180

0

50

100

150

200

250

300

350

400

450

500

Ts *3

6

24tu5h3

6a

67

6c


*+-*7


16/114

*+-+0 cylinrical resistance heater is place hori8ontally in a flui. The outer surface temperature of theresistance wire is to !e etermine for two ifferent fluis.Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. +0ny heat transfer !y raiation is i"nore. /5roperties are ealuate at

7''C for air an +'C for water.

PropertiesThe properties of air at * atm an 7''C are (Ta!le 0-&&)

*-

&7

=''*&B+.'=)&>67''(

**

,?B?.'5r

4sm*''+.>C:4m.'77>&.'

=+

==

==

=

fT

k

The properties of water at +'C are (Ta!le 0-*7)

*-

&?

='''6>>.',6&.+5r

4sm*'?7&.'4C,:4m.?6*.'

==

===

k

Analysis(a) The solution of this pro!lem reuires a trial-an-error approach since the etermination of the2aylei"h num!er an thus the usselt num!er epens on the surface temperature which is un/nown. :e

start the solution process !y "uessin" the surface temperature to !e *&''C for the calculation of h. :e

will chec/ the accuracy of this "uess later an repeat the calculations if necessary. The characteristic len"thin this case is the outer iameter of the wire, m.''7.'==DLc Then,

)?B?.'()4sm*''+.>(

)m''7.'(C)&'*&'')(=''*&B+.')(m4s*.B(5r

)(

&&7

6-*&

&

6

=

=

=

DTTgRa s

( )[ ] ( )[ ]B*B.*

?B?.'477B.'*

)>.&*+(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

&

&

m'**>.')m>7.')(m''7.'(

C.:4m6.&*)B*B.*(m''7.'

C:4m.'77>&.'

===

=

==

DLA

NuD

kh

s

an

,*-**=== ssss TTTThAQ C)&')(m'**>.')(C.:4m6.&*(:6'')(&&

which is sufficiently close to the assume alue of *&''C use in the ealuation of h, an thus it is not

necessary to repeat calculations.

(b) 3or the case of water, we "uess the surface temperature to !e +'C. The characteristic len"th in this

case is the outer iameter of the wire, m.''7.'==DLc Then,

*B>,B&)6&.+(

)4sm*'?7&.'(

)m''7.')(=&'+')(='''6>>.')(m4s*.B(5r

)(&&?

6-*&

&

6

=

=

=

DTTgRa s

( )[ ] ( )[ ]B?.

6&.+477B.'*

)*B>,B&(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

C.:4m**6+)B?.(m''7.'

C:4m.?6*.' & =

== NuD

kh

an

,+-./=== s&& C)&')(m'**>.')(C.:4m**6+(:6'')( TTTThAQ sss


*+-*?

0ir

T< &'C

2esistanceheater, T

s

6'' :

L< '.>7 m

D< '.7 cm


17/114

which is sufficiently close to the assume alue of +'C in the ealuation of the properties an h. The film

temperature in this case is (Ts;T)4& < (+&.7;&')4&


18/114

*+-/:ater is !oilin" in a pan that is place on top of a stoe. The rate of heat loss from the cylinricalsie surface of the pan !y natural conection an raiation an the ratio of heat lost from the sie surfacesof the pan to that !y the eaporation of water are to !e etermine.



(Ts;T)4& < (B;&7)4& < ?*.7C are (Ta!le 0-&&)

*-

&7

=''&BB.'=)&>67.?*(

**

>*B.'5r

4sm*'B*'.*

C:4m.'&*B.'

=+

==

==

=

fT

k

Analysis(a) The characteristic len"th in this case is the

hei"ht of the pan, m.*&.'==LLc Then

&&7

6-*&

&

6

&.>)>*B.'()4sm*'B*'.*(

)m*&.')(=&7B)(=''&BB.')(m4s*.B(5r

)(=

=

=

LTTgRa s


+4*+4*?+4*

67thusan'.&7E'>++6.'

)>*B.'4*'&BB.>(

)*&.'(6767

Gr

LD

Gr

L=

=

Therefore,

?'.&

>*B.'

+B&.'*

)*'&BB.>(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*?

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&

&

m'B+&7.')m*&.')(m&7.'(

C.:4m>&'.?)?'.&(m*&.'

C:4m.'&*B.'

===

===

DLA

NuL

kh

s

an

!.2=== C)&7B)(m'B+&7.')(C.:4m>&'.?()(&&TThAQ ss

(b) The raiation heat loss from the pan is

[ ] !/.=++==

+++&&

++

)=&>6&7()=&>6B().=:4m*'?>.7)(m'B+&7.')('.'(

)( surrssrad TTAQ

(c) The heat loss !y the eaporation of water is

:B+'/:B+'+.')/H4/"&&7>)(/"4s6?''47.*( ==== fghmQ

Then the ratio of the heat lost from the sie surfaces of the pan to that !y the eaporation of water then!ecomes

1.19==+

= 'BB.'B+'

6.+>&.+?f


*+-*

Fapor& /"4h

:ater

*''C

5an

Ts< BC

< '.'

0ir

T< &7C


19/114

*+-0:ater is !oilin" in a pan that is place on top of a stoe. The rate of heat loss from the cylinricalsie surface of the pan !y natural conection an raiation an the ratio of heat lost from the sie surfacesof the pan to that !y the eaporation of water are to !e etermine.



(Ts;T)4& < (B;&7)4& < ?*.7C are (Ta!le 0-&&)

*-

&7

=''&BB.'=)&>67.?*(

**

>*B.'5r

4sm*'B*'.*

C:4m.'&*B.'

=+

==

==

=

fT

k

Analysis(a) The characteristic len"th in this case is

the hei"ht of the pan, m.*&.'==LLc Then,

&&7

6-*&

&

6

&.>)>*B.'()4sm*'B*'.*(

)m*&.')(=&7B)(=''&BB.')(m4s*.B(5r

)(=

=

=

LTTgRa s


+4*+4*?+4*

67thusan'.&7E'>++6.'

)>*B.'4*'&BB.>(

)*&.'(6767

Gr

LD

Gr

L=

=

Therefore,

?'.&

>*B.'

+B&.'*

)*'&BB.>(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*?

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&

&

m'B+&7.')m*&.')(m&7.'(

C.:4m>&'.?)?'.&(m*&.'

C:4m.'&*B.'

===

=

==

DLA

NuL

kh

s

an

!.2=== C)&7B)(m'B+&7.')(C.:4m>&'.?()(&&TThAQ ss

(!) The raiation heat loss from the pan is

[ ] :.1=++==

+++&&

++

)=&>6&7()=&>6B().=:4m*'?>.7)(m'B+&7.')(*'.'(

)( surrssrad TTAQ

(c) The heat loss !y the eaporation of water is

:B+'/:B+'+.')/H4/"&&7>)(/"4s6?''47.*( ==== fghmQ

Then the ratio of the heat lost from the sie surfaces of the pan to that !y the eaporation of water then!ecomes

:.:9==+

= '77.'B+'

B.7&.+?f


*+-*B

Fapor& /"4h

:ater

*''C

5an

Ts< BC

< '.*

0ir

T< &7C


20/114

*+-19ome cans moe slowly in a hot water container mae of sheet metal. The rate of heat loss from thefour sie surfaces of the container an the annual cost of those heat losses are to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. Heat loss from the top surface is isre"are.


temperature of (Ts;T)4& < (77;&')4& < 6>.7C are (Ta!le 0-&&)

*-

&7

=''6&&*.'=)&>67.6>(

**

>&?&.'5r

4sm*'?>.*

C:4m.'&?++.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is the

hei"ht of the !ath, m.7.'==LLc Then,

&&7

6-*&

&

6

*'7?7.6)>&?&.'()4sm*'?>.*(

)m7.')(=&'77)(=''6&&*.')(m4s*.B(5r

)(=

=

=

LTTgRa s

+.B

>&?*.'

+B&.'*

)*'7?7.6(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

[ ] &

&

m7.+)m7.6)(m7.'()m*)(m7.'(&

C.:4m>7.+)+.B(m7.'

C:4m.'&?++.'

=+=

=

==

sA

NuL

kh

an

:*.>+C)&'77)(m7.+)(C.:4m>7.+()( && === TThAQ ss

The raiation heat loss is

[ ] :B.>7')=&>6&'()=&>677().=:4m*'?>.7)(m7.+)(>.'()(

+++&&

++

=++=

=

surrssrad TTAQ

Then the total rate of heat loss !ecomes

5*+33=+=+= B.>7'*.>+(radconvectionnaturaltotal QQQ

The amount an cost of the heat loss urin" one year is

/:h*6*,*6h)>?')(/:+BB.*( === tQQ totaltotal =***0== )/:h4'7.')(I/:h*6*,*6(Cost


*+-&'

:ater !ath

77C

0erosol can


21/114

*+-2 9ome cans moe slowly in a hot water container mae of sheet metal. It is propose to insulate thesie an !ottom surfaces of the container for 67'. The simple pay!ac/ perio of the insulation to pay foritself from the ener"y it saes is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. Heat loss from the top surface is isre"are.

PropertiesInsulation will rop the outer surface temperature to a alue close to the am!ient temperature.

The solution of this pro!lem reuires a trial-an-error approach since the etermination of the 2aylei"hnum!er an thus the usselt num!er epens on the surface temperature, which is un/nown. :e assume

the surface temperature to !e &?C. The properties of air at the anticipate film temperature of

(&?;&')4&)>6'*.'()4sm*'7+6.*(

)m7.')(=&'&?)(=''66.')(m4s*.B(5r

)(=

=

=

LTTgRa s

76.7?

>6'*.'

+B&.'

*

)*'?&&.>(6>.'&7.'

5r

+B&.'

*

2a6>.'&7.'u

&

&>4*?4B

?4*>

&

&>4*?4B

?4*

=

+

+=

+

+=

[ ] &

&

m>.+)m?'.6)(m7.'()m*'.*)(m7.'(&

C.:4m?.&)76.7?(m7.'

C:4m.'&76?.'

=+=

=

==

sA

NuL

kh

Then the total rate of heat loss from the outer surface of the insulate tan/ !y conection an raiation!ecomes

:7.B>

@)=&>6&'()=&>6&?)A(.=:4m*'?>.7)(m>.+)(*.'(;

C)&'&?)(m>.+)(C.:4m?.&(

)()(

+++&&

&&

++

=

++

=

+=+=

surrssssradconv TTATThAQQQ

In steay operation, the heat lost !y the sie surfaces of the tan/ must !e eual to the heat lost from theepose surface of the insulation !y conection an raiation, which must !e eual to the heat conuctethrou"h the insulation. The secon conitions reuires the surface temperature to !e

m'7.'

C)(77)mC)(+.>:4m.'67.'(:B>.7 &tan/

=

== sssinsulation

T

L

TTkAQQ


*+-&*

0erosol can

insulation

:ater !ath, 77C


22/114

It "ies Ts< &7.6C, which is ery close to the assume temperature, &?C. Therefore, there is no nee to

repeat the calculations.

The total amount of heat loss an its cost urin" one year are

/:h>.76h)>?')(:7.B>( === tQQ totaltotal

I>&.?)/:h4'7.')(I/:h>.76(Cost ==

Then money sae urin" a one-year perio ue to insulation !ecomes

*'+6I?.>&I***?ICostCostsa-eMoney ===insulation

ith

insulation

ithout

where ***? is o!taine from the solution of 5ro!lem *+-&. The insulation will pay for itself in

days*-->yr4./+===!r4*'+6I

67'I

saeMoney

Costperio5ay!ac/

Discussion :e woul efinitely recommen the installation of insulation in this case.


*+-&&


23/114

*+-30 printe circuit !oar (5C#) is place in a room. The aera"e temperature of the hot surface of the!oar is to !e etermine for ifferent orientations.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as withconstant properties. The local atmospheric pressure is * atm. The heatloss from the !ac/ surface of the !oar is ne"li"i!le.

PropertiesThe properties of air at * atm an the anticipate film

temperature of (Ts;T)4& < (+7;&')4& < 6&.7C are (Ta!le 0-&&)

*-

&7

=''6&>6.'=)&>67.6&(

**

>&>7.'5r

4sm*'?6*.*

C:4m.'&?'>.'

=+

==

==

=

fT

k

AnalysisThe solution of this pro!lem reuires a trial-an-error approachsince the etermination of the 2aylei"h num!er an thus the usseltnum!er epens on the surface temperature which is un/nown

(a) ?ertical "; . :e start the solution process !y "uessin" the surface temperature to !e +7C for the

ealuation of the properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations

if necessary. The characteristic len"th in this case is the hei"ht of the 5C#, m.&.'==LLc Then,

&&7

6-*&

&

6

*'>7?.*)>&>7.'()4sm*'?6*.*(

)m&.')(=&'+7)(=''6&>6.')(m4s*.B(5r

)(=

=

=

LTTgRa s

>.6?

>&>7.'

+B&.'*

)*'>7?.*(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'u

&

&>4*?4B

?4*>

&

&>4*?4B

?4*

=

+

+=

+

+=

&

&

m'6.')m&.')(m*7.'(

C.:4m>B+.+)>.6?(m&.'

C:4m.'&?'>.'

==

=

==

sA

NuL

kh

Heat loss !y !oth natural conection an raiation heat can !e epresse as

[ +&&&++

&'()&>6()*'?>.7)(m'6.')(.'(C)&')(m'6.')(C.:4m>B+.+(:

)()(

+++=

+=

ss

surrssss

TT

TTATThAQ

Its solution is

,+0.0=sT

which is sufficiently close to the assume alue of +7C for the ealuation of the properties an h.

(b) !orizontal, hot surface facing up 0"ain we assume the surface temperature to !e +7 C an use the

properties ealuate a!oe. The characteristic len"th in this case is

m.'+&B.')m*7.'m&.'(&

)m*7.')(m&'.'(=

+==

p

AL

sc

Then

&&7

6-*&

&

6

)>&>7.'()4sm*'?6*.*(

)m'+&B.')(=&'+7)(=''6&>6.')(m4s*.B(5r

)(=

=

=

cs LTTgRa

'*.**)*'>&.*(7+.'7+.' +4*7+4* === RaNu


*+-&6

Insulation

0ir

T< &'C

5C#, Ts

:

L < '.& m


24/114

C.:4m?B?.?)'*.**(m'+&B.'

C:4m.'&?'>.' & =

== NuL

kh

c

Heat loss !y !oth natural conection an raiation heat can !e epresse as

&'()&>6)A(*'?>.7)(m'6.')(.'(C)&')(m'6.')(C.:4m?B?.?(:

)()(

+&&&

++

+++=

+=

ss

surrssss

TT

TTATThAQ

Its solution is

C!2.=sT

which is sufficiently close to the assume alue of +7C in the ealuation of the properties an h.

(c) !orizontal, hot surface facing down This time we epect the surface temperature to !e hi"her, an

assume the surface temperature to !e 7'C. :e will chec/ this assumption after o!tainin" result an repeat

calculations with a !etter assumption, if necessary. The properties of air at the film temperature of

(7';&')4&.'=)&>667(

**>&?.'5r

4sm*'?77.*

C:4m.'&?&7.'

=+

===

=

=

fT

k

The characteristic len"th in this case is, from part (!),Lc< '.'+&B m. Then,

&',&'')>&?.'()4sm*'?77.*(

)m'+&B.')(=&'7')(=''6&+>.')(m4s*.B(5r

)(&&7

6-*&

&

6

=

=

=

cs LTTgRa

>**.7)&'',&''(&>.'&>.' +4*+4* === RaNu

C.:4m+B+.6)>**.7(m'+&B.'

C:4m.'&?&7.' & =

== NuL

kh

c

Consierin" !oth natural conection an raiation heat loses

&()&>6)A(*'?>.7)(m'6.')(.'(C)&')(m'6.')(C.:4m+B+.6(:

)()(

+&&&

++

++=

+=

ss

surrssss

TT

TTATThAQ

Its solution is

C:0.=sT

which is ery close to the assume alue. Therefore, there is no nee to repeat calculations.


*+-&+


25/114

*+4 &&6 5ro!. *+-&B is reconsiere. The!ff!t# of t6! &oom t!m!&/t%&! /nd t6!!mi##i2ity of t6! o/&d on t6! t!m!&/t%&! of t6! 6ot #%&f/! of t6! o/&d fo&diff!&!nt o&i!nt/tion# of t6! o/&d /&!to !e inesti"ate.


"GIVEN"

L=0.: [m]=0.15 [m]

T_infinity=:0 [C]Q_dot=8 [W]!#i$on=0.8

T_#%&&=T_infinity

"()*(E)TIE+",$%id-=/i&=Cond%ti2ity,$%id- T=T_fi$m(&=(&/ndt$,$%id- T=T_fi$m&6o=!n#ity,$%id- T=T_fi$m (=101.3m%=Vi#o#ity,$%id- T=T_fi$m

n%=m%7&6o!t/=17T_fi$m9:;3

T_fi$m=17:;E'8 [W7m?:'@?4]A+t!f/n'Bo$tm/nn on#t/nt"=D.80; [m7#?:] A&/2it/tion/$ /!$!&/tion"

"NLF+I+""/ $/t! i# 2!&ti/$"d!$t/_/=L)/_/=


26/114

T 798 Ts,a78 Ts,b78 Ts,c78

7 6&.7+ &.B6 6.&B

> 6+.6+ 6'.>B 6B.B>

B 6?.*+ 6&.?7 +*.??

** 6>.B7 6+.7* +6.67

*6 6B.>7 6?.6? +7.'+*7 +*.77 6.&& +?.>6

*> +6.67 +'.'> +.+&

*B +7.*7 +*.B& 7'.*&

&* +?.B7 +6.> 7*.*

&6 +.>7 +7.?6 76.7*

&7 7'.77 +>.+ 77.&*

&> 7&.67 +B.66 7?.B*

&B 7+.*? 7*.*B 7.?&

6* 77.B? 76.'+ ?'.6&

66 7>.>? 7+.B ?&.'6

67 7B.7? 7?.>+ ?6.>+

5 10 15 20 25 30 35

25

30

35

40

45

50

55

60

65

TC3

Ts

2C3 Ts,a

Ts,7

Ts,c


*+-&?


27/114


28/114

*+-0n a!sor!er plate whose !ac/ sie is heaily insulate is place hori8ontally outoors. 9olarraiation is incient on the plate. The euili!rium temperature of the plate is to !e etermine for twocases.Assumptions*9teay operatin" conitions eist. -0ir is an ieal "aswith constant properties. The local atmospheric pressure is * atm.PropertiesThe properties of air at * atm an the anticipate film

temperature of (Ts;T

)4& < (>';&7)4& < +>.7C are (Ta!le 0-&&)

*-

&7

=''6*&.'=)&>67.+>(

**

>&67.'5r

4sm*'>>+.*

C:4m.'&>*>.'

=+

==

==

=

fT

k


start the solution process !y "uessin" the surface temperature to !e >'C for the ealuation of the

properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations if necessary. The

characteristic len"th in this case is m.&+.')m.'m&.*(&

)m.')(m&.*(=

+

==p

AL sc Then,

&&7

6-*&

&

6

*'6>B.+)>&67.'()4sm*'>>+.*(

)m&+.')(=&7>')(=''6*&.')(m4s*.B(5r

)(=

=

=

cs LTTgRa

B6.+6)*'6>B.+(7+.'7+.' +4*>+4* === RaNu

C.:4mB>6.+)B6.+6(m&+.'

C:4m.'&>*>.' & =

== NuL

kh

c

&mB?.')m&.*)(m.'( ==sAIn steay operation, the heat "ain !y the plate !y a!sorption of solar raiation must !e eual to the heat loss!y natural conection an raiation. Therefore,

:?.?7)mB?.')(:4m>'')(B.'( &&

=== sA"Q

@)=&>6*'()&>6)A(*'?>.7)(mB?.')(B.'(

C)&7)(mB?.')(C.:4mB>6.+(:?7.?

)()(

++&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is C/.:=sTwhich is close to the assume alue. Therefore there is no nee to repeat calculations.

3or a white painte a!sor!er plate, the solar a!sorptiity is '.&? an the emissiity is '.B'. Thenthe rate of solar "ain !ecomes

:>.*>+)mB?.')(:4m>'')(&?.'( && === sA"Q 0"ain notin" that in steay operation the heat "ain !y the plate !y a!sorption of solar raiation must !eeual to the heat loss !y natural conection an raiation, an usin" the conection coefficient eterminea!oe for conenience (actually, we shoul calculate the new h usin" ata at a lower temperature, aniteratin" if necessary for !etter accuracy),

@)=&>6*'()&>6)A(*'?>.7)(mB?.')(B'.'(

C)&7)(mB?.')(C.:4mB>6.+(+.>

)()(

++&

&&

++

+++

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is C:.0=sTDiscussion If we recalculate the h usin" air properties at 6'C, we woul o!tain


*+-&

Insulation

0irT< &7C

0!sor!er plate

s< '.B

< '.B

>'' :4m&

L < *.& m


29/114

h < 6.+> :4m&.C an Ts< 6?.?C


*+-&B


30/114

*+0 resistance heater is place alon" the centerline of a hori8ontal cyliner whose two circular siesurfaces are well insulate. The natural conection heat transfer coefficient an whether the raiation effectis ne"li"i!le are to !e etermine.

Assumptions*9teay operatin" conitions eist. -0iris an ieal "as with constant properties. The localatmospheric pressure is * atm.

AnalysisThe heat transfer surface area of the cyliner is &m'7'&>.')m.')(m'&.'( === DLA

otin" that in steay operation the heat issipatefrom the outer surface must eual to the electricpower consume, an raiation is ne"li"i!le, theconection heat transfer is etermine to !e

,.5.'(

:?'

)()(

&TTA

QhTThAQ

ssss

The raiation heat loss from the cyliner is

:>.+@)=&>6&'()=&>6*&')A(.=:4m*'?>.7)(m'7'&>.')(*.'(

)(+++&&

++

=++==

surrssrad TTAQ

Therefore, the fraction of heat loss !y raiation is

J.>'>.':?'

:>.+fraction2aiation ====

total

radiation

Q

Q

which is "reater than 7J. Therefore, the raiation effect is still more than accepta!le, an corrections must!e mae for the raiation effect.


*+-6'

0ir

T< &'C

Cyliner

Ts< *&'C

< '.*

L< '. m

D< & cm

2esistanceheater, ?':


31/114

*++0 thic/ flui flows throu"h a pipe in calm am!ient air. The pipe is heate electrically. The power

ratin" of the electric resistance heater an the cost of electricity urin" a *'-h perio are to !e etermine.


PropertiesThe properties of air at * atm an

the film temperature of (Ts;T)4& < (&7;')4&

< *&.7C are (Ta!le 0-&&)

*-

&7

=''67'6.'=)&>67.*&(

**

>66'.'5r

4sm*'++.*

C:4m.'&+7.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is the outer iameter of the pipe, m.6.'==DLc Then,

>

&&7

6-*&

&

6

*'*'?.)>66'.'()4sm*'++.*(

)m6.')(='&7)(=''67'6.')(m4s*.B(5r

)(=

=

=

cs LTTgRa

( )[ ] ( )[ ]&B.76

>66'.'477B.'*

)*'*'?.(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*>&

&>4*?4B

?4*

=

+

+=

++=

RaNu

&

&

m&7.B+)m*'')(m6.'(

C.:4m6??.+)&B.76(m6.'

C:4m.'&+7.'

===

=

==

DLA

NuL

kh

s

c

an

:&>,*'C)'&7)(m&7.B+)(C.:4m6??.+()( && === TThAQ ss

The raiation heat loss from the cyliner is

:',*@)=&>66'()=&>6&7)A(.=:4m*'?>.7)(m&7.B+)(.'(

)(

+++&&

++

=++=

=

surrssrad TTAQ

Then,

;21.==+=+= :'B+,&B('(,*(&(>,*'radiationconvectionnaturaltotal QQQ

The total amount an cost of heat loss urin" a *' hour perio is

/:hB.&B'h)/:)(*'*.&B( === tQQ


32/114

*+/ 0 flui flows throu"h a pipe in calm am!ient air. The pipe is heate electrically. The thic/ness of theinsulation neee to reuce the losses !y 7J an the money sae urin" *'-h are to !e etermine.


PropertiesInsulation will rop the outer surface temperature to a alue close to the am!ient temperature,an possi!le !elow it !ecause of the ery low s/y temperature for raiation heat loss. 3or conenience, we

use the properties of air at * atm an 7C (the anticipate film temperature) (Ta!le 0-&&),

*-

&7

=''67B>.'=)&>67(

**

>67'.'5r

4sm*'6&.*

C:4m.'&+'*.'

=+

==

==

=

fT

k

AnalysisThe rate of heat loss in the preiouspro!lem was o!taine to !e &B,'B+ :. otin"that insulation will cut own the heat losses !y7J, the rate of heat loss will !e

:+6?+:'B+,&B*7.')7.'*( insulationno === QQ

The amount of ener"y an money insulation will sae urin" a *'-h perio is simply etermine from

/:h6.&+>h)/:)(*''B+.&B7.'(, === tQQ savedtotalsaved

=--.-0== )/:h4'B.')(I/:h6.&+>(


33/114

ote that the film temperature is (.7;')4& < +.&7C which is ery close to the assume alue of 7C.

Therefore, there is no nee to repeat the calculations usin" properties at this new film temperature.


*+-66


34/114

*+0& 0n inustrial furnace that resem!les a hori8ontal cylinrical enclosure whose en surfaces are wellinsulate. The hi"hest allowa!le surface temperature of the furnace an the annual cost of this loss to theplant are to !e etermine.


PropertiesThe properties of air at * atm an theanticipate film temperature of (Ts;T)4& 7()A.2#tu4h.ft*'*>*+.')(m>.6&?)(7.'(

)@2+?'>7()Aft>.6&?(3).#tu4h.ft>&>.'(#tu4h6?',6B

)()(

+++&&

&&

++

++

+=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is

*!.8== 2.?'*sTwhich is ery close to the assume alue. Therefore, there is no nee to repeat calculations. The totalamount of heat loss an its cost urin" a-&'' hour perio is

#tu*'*'&.*h)&'')(#tu4h6?',6B( === tQQ totaltotal


35/114

*+1 0 "lass winow is consiere. The conection heat transfer coefficient on the inner sie of thewinow, the rate of total heat transfer throu"h the winow, an the com!ine natural conection anraiation heat transfer coefficient on the outer surface of the winow are to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is anieal "as with constant properties. The local atmosphericpressure is * atm.

PropertiesThe properties of air at * atm an the filmtemperature of (Ts;T)4& < (7;&7)4& < *7C are (Ta!le 0-&&)

*-

&7

=''6+>&.'=)&>6*7(

**

>6&6.'5r

4sm*'+>'.*

C:4m.'&+>?.'

=+

==

==

=

fT

k

Analysis(a) The characteristic len"th in this case is the hei"ht of the winow, m.&.*==LLc Then,

B

&&7

6-*&

&

6

*'BB.6)>6&6.'()4sm*'+>'.*(

)m&.*)(=7&7)(=6+>&'.')(m4s*.B(5r

)(=

=

=

'

cs LTTgRa

>.*B

>6&6.'

+B&.'*

)*'BB.6(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*B

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&m+.&m)m)(&&.*(

)>.*B(m&.*

C:4m.'&+>?.'

==

=

==

sA

NuL

kh C.5m.1: 2

(b) The sum of the natural conection an raiation heat transfer from the room to the winow is

:B.*>C)7&7)(m+.&)(C.:4mB*7.6()( &&conection === ss TThAQ

:6.&6+@)=&>67()=&>6&7)A(.=:4m*'?>.7)(m+.&)(B.'(

)(

+++&&

++raiation

=++=

=

ssurrs TTAQ

!22.2=+=+= 6.&6+B.*>raiationconectiontotal QQQ

(c) The outer surface temperature of the winow can !e etermine from

C?7.6)m+.&)(C:4m.>.'(

)m''?.')(:&.+&&(C7)(

&

total,,,,total =

===

s

isososiss

kA

tQTTTT

t

kAQ

Then the com!ine natural conection an raiation heat transfer coefficient on the outer winow surface !ecomes

C.5m20.: 2 =

=

=

=

C)@7(?7.6)Am+.&(

:&.+&&

)(or

)(

&,,

totalcom!ine

,,com!inetotal

ooss

ooss

TTA

Qh

TTAhQ

ote that RQT = an thus the thermal resistanceR of a layer is proportional to the temperature rop

across that layer. Therefore, the fraction of thermal resistance of the "lass is eual to the ratio of thetemperature rop across the "lass to the oerall temperature ifference,

+.7J)(or'+7.')7(&7

?7.67

total

"lass

total

"lass =

=

=

TR

T

R

R


*+-67

Q

Lutoors

-7C

Glass

Ts< 7C

< '.B

L < *.& m

2oom

T< &7C


36/114


37/114

*+2 0n insulate electric wire is epose to calm air. The temperature at the interface of the wire an theplastic insulation is to !e etermine.


PropertiesThe properties of air at * atm an the

anticipate film temperature of (Ts;T)4& < (7';6')4&

< +'C are (Ta!le 0-&&)

*-

&7

=''6*B7.'=)&>6+'(

**

>&77.'5r

4sm*'>'&.*

C:4m.'&??&.'

=+

==

==

=

fT

k


start the solution process !y "uessin" the surface temperature to !e 7'C for the ealuation of the

properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations if necessary. Thecharacteristic len"th in this case is the outer iameter of the insulate wireLc.'?.'

&

&>4*?4B

?4*&

&>4*?4B

?4*

=

++=

++=

RaNu

&

&

m&&?&.'m)m)(*&''?.'(

C.:4m6&>.B)*'*.&(m''?.'

C:4m.'&??&.'

===

=

==

DLA

NuD

kh

s

The rate of heat "eneration, an thus the rate of heat transfer is:>'0)F)(*'>( === $Q ?

Consierin" !oth natural conection an raiation, the total rate of heat loss can !e epresse as

@)=&>66'()&>6)A(.=:4m*'?>.7)(m&&?&.')(B.'(

C)6')(m&&?.')(C.:4m6&>.B(:>'

)()(

+++&&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is

CB.+B =sT

which is ery close to the assume alue of 7'C. Then the temperature at the interface of the wire an the

plastic coer in steay operation !ecomes

C:.=

=+==)m*&)(C:4m.&'.'(&

)64?ln():>'(;CB.+B

&

)4ln()(

)4ln(

& *&

*&

kL

DDQTTTT

DD

kLQ sisi


*+-6>

0ir

T< 6'CT

s

< '.B

L< *& m

D< ? mm

2esistanceheater


38/114

*+3 0 steam pipe etene from one en of a plant to the other with no insulation on it. The rate of heatloss from the steam pipe an the annual cost of those heat losses are to !e etermine.



temperature of (Ts;T)4& < (*>';&')4& < B7C are (Ta!le 0-&&)

*-

&7

=''&>*>.'=)&>6B7(

**

>*&&.'5r

4sm*'&7+.&

C:4m.'6'?.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is the outer iameter of the pipe, m'.'?'6==DLc .Then,

&&7

6-*&

&

6

&&B.*)>*&&.'()4sm*'&7+.&(

)m'?'6.')(=&'*>')(=''&>*>.')(m4s*.B(5r

)(=

=

=

DTTgRa s

( )[ ] ( )[ ]+*.*7

>*&&.'477B.'*

)*'&&B.*(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*?&

&>4*?4B

?4*

=

+

+=

++=

RaNu

&

&

m6>.**m)m)(?''?'6.'(

C.:4m&*.>)+*.*7(m'?'6.'

C:4m.'6'?.'

===

=

==

DLA

NuD

kh

s

Then the total rate of heat transfer !y natural conection an raiation !ecomes

;2/.!==

+++

=

+=

:&>,6B6

@)=&>6&'()=&>6*>')A(.=:4m*'?>.7)(m6>.**)(>.'(

C)&'*>')(m6>.**)(C.:4m&*.>(

)()(

+++&&

&&

++surrssss TTATThAQ

The total amount of "as consumption an its cost urin" a one-year perio is

therms4yr+B,*'s4h)6?''h4yr>?'(/H*'7,7''

therm*

>.'

/H4s6B6.&>=

=

=

tQQgas


39/114

*++4 &&6 5ro!. *+-6B is reconsiere. The!ff!t of t6! #%&f/! t!m!&/t%&! of t6! #t!/mi! on t6! &/t! of 6!/t $o## f&om t6! i! /nd t6! /nn%/$ o#t of t6i# 6!/t $o## i#to !einesti"ate.


"GIVEN"

L=>0 [m]=0.0>03 [m]

T_#=1;0 [C]T_infinity=:0 [C]!#i$on=0.;

T_#%&&=T_infinity!t/_f%&n/!=0.;8HnitCo#t=1.10 [-7t6!&m]tim!=:45 [6]

"()*(E)TIE+",$%id-=/i&=Cond%ti2ity,$%id- T=T_fi$m

(&=(&/ndt$,$%id- T=T_fi$m&6o=!n#ity,$%id- T=T_fi$m (=101.3m%=Vi#o#ity,$%id- T=T_fi$mn%=m%7&6o!t/=17T_fi$m9:;3

T_fi$m=17:;E'8 [W7m?:'@?4] A+t!f/n'Bo$tm/nn on#t/nt"=D.80; [m7#?:]A&/2it/tion/$ /!$!&/tion"

"NLF+I+"d!$t/=)/=


40/114

Ts78 : 758 ost 7=8

*'' **?6? +B'7

*'7 *&7B+ 76'B

**' *67>> 7>&6

**7 *+77 ?*+

*&' *7?* ?7+*&7 *??>? >'6'

*6' *>>?' >+?

*67 *?B >B7+

*+' &'''+ +6&

*+7 &**?? B&&

*7' &&677 B+&6

*77 &67>' BB6?

*?' &+*+ *'+?'

*?7 &?'7 *'BB?

*>' &>67 **7+6

*>7 &>*6 *&*'6

*' 6''>* *&?>?

*7 6*+7B *6&?**B' 6&>> *67B

*B7 6+6&> *++>'

&'' 67'> *7'B+

100 120 140 160 180 200

10000

15000

20000

25000

30000

35000

40000

4000

6000

8000

10000

12000

14000

16000

Ts 2C3

6

.3

Cost7'. The simple pay!ac/ perio of the insulation to pay for itself from the ener"y it saes are to !e etermine.Assumptions*9teay operatin" conitions eist. -0ir is anieal "as with constant properties. The local atmospheric

pressure is * atm.PropertiesThe properties of air at * atm an the

anticipate film temperature of (Ts;T)4& < (67;&')4& .7C are (Ta!le 0-&&)

*-

&7

=''66&.'=)&>67.&>(

**

>&B.'5r

4sm*'7+.*

C:4m.'&7>.'

=+

==

==

=

fT

k

AnalysisInsulation will rop the outer surface temperature to a alue close to the am!ient temperature. Thesolution of this pro!lem reuires a trial-an-error approach since the etermination of the 2aylei"h num!er anthus the usselt num!er epens on the surface temperature which is un/nown. :e start the solution process !y

"uessin" the outer surface temperature to !e 67C for the ealuation of the properties an h. :e will chec/ the

accuracy of this "uess later an repeat the calculations if necessary. The characteristic len"th in this case is the

outer iameter of the insulate pipe, m.*?'6.'==DLc Then,

&&7

6-*&

&

6

)>&B.'()4sm*'7+.*(

)m*?'6.')(=&'67)(=''66&.')(m4s*.B(5r

)(=

=

=

DTTgRa

s

( )[ ] ( )[ ]&6.&+

>&B.'477B.'*

)*'7?.7(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4*?&

&>4*?4B

?4*

=

+

+=

++=

RaNu

&

&

m&&.6'm)m)(?'*?'6.'(

C.:4m+.6)&6.&+(m*?'6.'

C:4m.'&7>.'

===

=

==

DLA

NuD

kh

s

Then the total rate of heat loss from the outer surface of the insulate pipe !y conection an raiation !ecomes

:&'6B

@)=&>6&'()=&>667)A(.=:4m*'?>.7)(m&&.6')(*.'(;

C)&'67)(m&&.6')(C.:4m+.6(

)()(

+++&&

&&

++

=++

=

+=+=

surrssssradconv TTATThAQQQ

In steay operation, the heat lost from the epose surface of the insulation !y conection an raiation must !eeual to the heat conucte throu"h the insulation. This reuirement "ies the surface temperature to !e

)m?')(C:4m.'6.'(&

)'6.?4'6.*?ln(

C)*>'(:&'6B

&

)4ln( *&

..insulation

=

=

==

ssis

ins

sis T

kL

DD

TT

R

TTQQ

It "ies 6'.C for the surface temperature, which is somewhat ifferent than the assume alue of 67C.

2epeatin" the calculations with other surface temperatures "ies:*BanC6.6+ == QTs

Heat loss an its cost without insulation was etermine in the 5ro!. *+-6B to !e &>.6 /: an **,77'. Thenthe reuction in the heat losses !ecomes

/:+'.&7B.*6.&>sae =Q or &7.64&>.+' < '.B&> (B&.>J)

Therefore, the money sae !y insulation will !e '.B&>(**,77'4yr) 'C, < '.*

0ir

T< &'C

Insulation

< '.*


42/114

*++- 0 circuit !oar containin" suare chips is mounte on a ertical wall in a room. The surfacetemperature of the chips is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "aswith constant properties. The local atmospheric pressure is * atm. +The heat transfer from the !ac/ sie of the circuit !oar is ne"li"i!le.


temperature of (Ts;T)4& < (67;&7)4& < 6'C are (Ta!le 0-&&)

*-

&7

=''66.'=)&>66'(

**

>&&.'5r

4sm*'?'.*

C:4m.'&7.'

=+

==

==

=

fT

k


start the solution process !y "uessin" the surface temperature to !e 67C for the ealuation of the

properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations if necessary. The

characteristic len"th in this case is the hei"ht of the !oar, m.7.'==LLc Then,

&&7

6-*&

&

6

*'*+'.*)>&&.'()4sm*'?'.*(

)m7.')(=&767)(=''66.')(m4s*.B(5r

)(=

=

=

LTTgRa s

>&.?6

>&&.'

+B&.'*

)*'*+'.*(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4*

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&&

&

m&7.'m)7.'(

C.:4m6'.6)>&.?6(m7.'

C:4m.'&7.'

==

=

==

sA

NuL

kh


@)=&>6&7()=&>6)A(.=:4m*'?>.7)(m&7.')(>.'(

C)&7)(m&7.')(C.:4m6'.6(:)*.'*&*(

)()(

+++&&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is

Ts< 0.-which is sufficiently close to the assume alue in the ealuation of properties an h. Therefore, there is nonee to repeat calculations !y reealuatin" the properties an h at the new film temperature.


*+-+&

0ir

T< &7C

Tsurr

< &7C

5C#, Ts

< '.>

*&*'.* :

L < 7' cm


43/114

*++ 0 circuit !oar containin" suare chips is positione hori8ontally in a room. The surface temperatureof the chips is to !e etermine for two orientations.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. +The heat transfer from the !ac/ sie of the circuit !oar is ne"li"i!le.


temperature of (Ts;T)4& < (67;&7)4& < 6'C are (Ta!le 0-&&)

*-

&7

=''66.'=)&>66'(

**

>&&.'5r

4sm*'?'.*

C:4m.'&7.'

=+

==

==

=

fT

k


start the solution process !y "uessin" the surface temperature to !e 67C for the ealuation of the

properties an h. The characteristic len"th for !oth cases is etermine from

m.*&7.'

m)@7.'(;m)7.'&A(

m)7.'( &===

p

AL sc

Then,

&&7

6-*&

&

6

>.*)>&&.'()4sm*'?'.*(

)m*&7.')(=&767)(=''66.')(m4s*.B(5r

)(=

=

=

cs LTTgRa

(a) Chips (hot surface) facin" up$

>6.*B)*'>*.*(7+.'7+.' +4*?+4* === RaNu

&&

&

m&7.'m)7.'(

C.:4m'.+)>6.*B(m*&7.'

C:4m.'&7.'

==

=

==

s

c

A

NuL

kh


@)=&>6&7()=&>6)A(.=:4m*'?>.7)(m&7.')(>.'(

C)&7)(m&7.')(C.:4m'.+(:)*.'*&*(

)()(

+++&&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is Ts< /.-which is sufficiently close to the assume alue. Therefore, there is no nee to repeat calculations.

(b) Chips (hot surface) facin" up$

?6.B)*'>*.*(&>.'&>.' +4*?+4* === RaNu

C.:4m'+.&)?6.B(m*&7.'

C:4m.'&7.' & =

== NuL

kh

c


@)=&>6&7()=&>6)A(.=:4m*'?>.7)(m&7.')(>.'(

C)&7)(m&7.')(C.:4m'+.&(:)*.'*&*(

)()(

+++&&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is Ts< 2.


*+-+6

0ir

T< &7CT

surr< &7C

5C#, Ts

< '.>

*&*'.* :

L < 7' cm


44/114

which is ientical to the assume alue in the ealuation of properties an h. Therefore, there is no nee torepeat calculations.


*+-++


45/114

*+++ @ &&6It is propose that the sie surfaces of a cu!ic inustrial furnace !e insulate for 77' inorer to reuce the heat loss !y B' percent. The thic/ness of the insulation an the pay!ac/ perio of theinsulation to pay for itself from the ener"y it saes are to !e etermine.


PropertiesThe properties of air at * atm an the filmtemperature of (Ts;T)4& < (**';6')4& < >'C are

(Ta!le 0-&&)

*-

&7

=''&B*7.'=)&>6>'(

**

>*>>.'5r

4sm*'BB7.*

C:4m.'&*.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is the hei"ht of the furnace, m.&==LLc Then,

&&7

6-*&

&

6

6'.6)>*>>.'(

)4sm*'BB7.*(

)m&)(=6'**')(=''&B*7.')(m4s*.B(5r

)(=

=

=

LTTgRa s

&.6?B

>*>>.'

+B&.'*

)*'6'*.6(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4**'

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

&&

&

m*?)m&(+

C.:4m6*.7)&.6?B(m&

C:4m.'&*.'

==

=

==

s

c

A

NuL

kh

Then the heat loss !y com!ine natural conection an raiation !ecomes

:**B,*7

@)=&>66'()=&>6**')A(.=:4m*'?>.7)(m*?)(>.'(

C)6'**')(m*?)(C.:4m6*.7()()(

+++&&

&&

++

=

+++

=+=

surrssss TTATThAQ

otin" that insulation will reuce the heat losses !y B'J, the rate of heat loss after insulation will !e

:*7*&:**B,*7*.')B.'*(

:?'>,*6:**B,*7B.'B.'

insulationnoloss

insulationnosae

===

===

QQ

QQ

The furnace operates continuously an thus >?' h. Then the amount of ener"y an money the insulationwill sae !ecomes

therms4yr7&*7s4yr)6?''(>?'/H*'7,7''

therm*>.'/H4s?'>.*6saeDner"y ===

tQsaved

I&?)therm477.')(Itherms7&*7(


46/114

Insulation will lower the outer surface temperature, the 2aylei"h an usselt num!ers, an thus theconection heat transfer coefficient. 3or the ealuation of the heat transfer coefficient, we assume the

surface temperature in this case to !e 7'C. The properties of air at the film temperature of (Ts;T)4& 6+'(

**

>&77.'5r4sm*'>'&.*

C:4m.'&??&.'

=+

==

= =

=

fT

k

Then,

&&7

6-*&

&

6

&7?.*)>&77.'()4sm*'>'&.*(

)m&)(=6'7')(=''6*B7.')(m4s*.B(5r

)(=

=

=

LTTgRa

s

'.&>&

>&77.'

+B&.'*

)*'&7?.*(6>.'&7.'

5r

+B&.'*

2a6>.'&7.'

&

&>4*?4B

?4**'

&

&>4*?4B

?4*

=

+

+=

+

+=Nu

C.:4m?&'.6)'.&>&(m&

C:4m.'&??&.' & =

== NuL

kh

m)&m)(&&(+ insuls tA +=

The total rate of heat loss from the outer surface of the insulate furnace !y conection an raiation!ecomes

@)=&>66'()=&>6)A(.=:4m*'?>.7()>.'(;

C)6'(C).:4m?&'.6(:*7*&

)()(

+++&

&

++

++

=

+=+=

s

s

surrssssradconv

TA

TA

TTATThAQQQ

In steay operation, the heat lost !y the sie surfaces of the pipe must !e eual to the heat lost from theepose surface of the insulation !y conection an raiation, which must !e eual to the heat conuctethrou"h the insulation. Therefore,

insul

ss

ins

ssinsulation

t

TA

t

TTkAQQ

C)(**'C):4m.'6.'(:*7*&

)( furnace =

==

9olin" the two euations a!oe !y trial-an error (or !etter yet, an euation soler) "ies

Ts< +*.*C an tinsul# '.'&7 m < -.2/ cm


*+-+?


47/114

*++/0 cylinrical propane tan/ is epose to calm am!ient air. The propane is slowly apori8e ue to acrac/ eelope at the top of the tan/. The time it will ta/e for the tan/ to empty is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0iris an ieal "as with constant properties. The localatmospheric pressure is * atm. +2aiation heat transferis ne"li"i!le.

PropertiesThe properties of air at * atm an the filmtemperature of (Ts;T)4& < (-+&;&7)4& < -.7C are

(Ta!le 0-&&)

*-

&7

=''6>*.'=)&>67.(

**

>66.'5r

4sm*'&?7.*

C:4m.'&&BB.'

=+

==

==

=

fT

k

AnalysisThe tan/ "ains heat throu"h its cylinrical surface as well as its circular en surfaces. 3orconenience, we ta/e the heat transfer coefficient at the en surfaces of the tan/ to !e the same as that of itssie surface. (The alternatie is to treat the en surfaces as a ertical plate, !ut this will ou!le the amount

of calculations without proiin" much improement in accuracy since the area of the en surfaces is muchsmaller an it is circular in shape rather than !ein" rectan"ular). The characteristic len"th in this case is the

outer iameter of the tan/, m.7.*==DLc Then,

&&7

6-*&

&

6

)>66.'()4sm*'&?7.*(

)m7.*@(=)+&(&7)A(=''6>*.')(m4s*.B(5r

)(=

=

=

DTTgRa s

( )[ ] ( )[ ]*.6>+

>66.'477B.'*

)*'?B.6(6>.'?.'

5r477B.'*

6>.'?.'

&

&>4*?4B

?4**'&

&>4*?4B

?4*

=

+

+=

++=

RaNu

&&&

&

m6.&&+4m)7.*(&)m+)(m7.*(+4&

C.:4m>66.7)*.6>+(m7.*

C:4m.'&&BB.'

=+=+=

=

==

DDLA

NuD

kh

s

an

:7BC)@+&(&7)A(m6.&&)(C.:4m>66.7()( && === ss TThAQ

The total mass an the rate of eaporation of propane are

/"4s'&'&6.'/H4/"+&7

/H4s7B.

/"+*'>)m+(+

)m7.*()/"4m7*(

+

&6

&

===

====

fgh

Qm

LD

m

V

an it will ta/e

hours/0.+==== sBB?,&'&/"4s'&'&6.'/"+*'>

mmt

for the propane tan/ to empty.


*+-+>

D< *.7 m

L< + m

5ropane tan/

'

Ts< -+&C

0ir

T< &7C


48/114

*++0& The aera"e surface temperature of a human hea is to !e etermine when it is not coere.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. +The hea can !e approimate as a *&-in.-iameter sphere.

PropertiesThe solution of this pro!lem reuires a trial-an-error approach since the etermination of the2aylei"h num!er an thus the usselt num!er epens on the surface temperature which is un/nown. :e

start the solution process !y "uessin" the surface temperature to !e B'3 for the ealuation of the

properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations if necessary. Theproperties of air at * atm an the anticipate film temperature of (Ts;T)4& < (B';>')4& < '3 are (Ta!le

0-&&D)

*-

&+

2''*7&.'2)+?''(

**

>&B'.'5r

4sft*'?B>.*

3#tu4h.ft.'*+*.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th for a sphericalo!ect isLc

&&+

6-*&

&

6

*''*B.6)>&B'.'()4sft*'?B>.*(

)ft*)(2>'B')(2''*7&.')(ft4s&.6&(5r

)(=

==

DTTgRa

s

>B.67

>&B'.'

+?B.'*

)*''*B.6(7B.'&

5r

+?B.'*

7B.'&

B4+*?4B

+4*>

B4+*?4B

+4*

=

+

+=

+

+= Ra

Nu

&&&

&

ft*+&.6)ft*(

3.#tu4h.ft76''.')>B.67(ft*

3#tu4h.ft.'*+*.'

===

=

==

DA

NuD

kh

s

Consierin" !oth natural conection an raiation, the total rate of heat loss can !e written as

@)2+?'>'()2+?')A(.2#tu4h.ft*'*>*+.')(ft*+&.6)(B.'(

3)>')(ft*+&.6(3).#tu4h.ft76''.'(#tu4h)+4&+'(

)()(

+++&&

&&

++

+++

=

+=

s

s

surrssss

T

T

TTATThAQ

Its solution is

Ts< 2-.39which is sufficiently close to the assume alue in the ealuation of the properties an h. Therefore, there isno nee to repeat calculations.


*+-+

0ir

T< >'3

HeaQ < N &+' #tu4h

D< *& in

< '.B


49/114

*++1 The euili!rium temperature of a li"ht "lass !ul! in a room is to !e etermine.

Assumptions*9teay operatin" conitions eist. -0ir is an ieal "as with constant properties. The localatmospheric pressure is * atm. +The li"ht !ul! is approimate as an -cm-iameter sphere.

PropertiesThe solution of this pro!lem reuires a trial-an-error approach since the etermination of the2aylei"h num!er an thus the usselt num!er epens on the surface temperature which is un/nown. :e

start the solution process !y "uessin" the surface temperature to !e *>'C for the ealuation of the

properties an h. :e will chec/ the accuracy of this "uess later an repeat the calculations if necessary. Theproperties of air at * atm an the anticipate film temperature of (Ts;T)4& < (*>';&7)4& < B>.7C are

(Ta!le 0-&&)

*-

&7

=''&?BB.'=)&>67.B>(

**

>**?.'5r

4sm*'&>B.&

C:4m.'6'>>.'

=+

==

==

=

fT

k

AnalysisThe characteristic len"th in this case is

Lc**?.'4+?B.'*

Documents

IntroTHT 2e SM Chap14