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8/12/2019 Heat Chap04 026
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Chapter 4 Transient Heat Conduction
Transient Heat Conduction in Large Plane Walls, Long Cylinders, and Spheres
426C A cylinder whose diameter is small relative to its length can be treated as an infinitely longcylinder. When the diameter and length of the cylinder are comparable, it is not proper to treat thecylinder as being infinitely long. It is also not proper to use this model when finding the temperatures nearthe bottom or top surfaces of a cylinder since heat transfer at those locations can be twodimensional.
427C Yes. A plane wall whose one side is insulated is equivalent to a plane wall that is twice as thickand is eposed to convection from both sides. !he midplane in the latter case will behave like an insulatedsurface because of thermal symmetry.
428C !he solution for determination of the onedimensional transient temperature distribution involvesmany variables that make the graphical representation of the results impractical. In order to reduce thenumber of parameters, some variables are grouped into dimensionless quantities.
429C !he "ourier number is a measure of heat conducted through a body relative to the heat stored. !husa large value of "ourier number indicates faster propagation of heat through body. #ince "ourier number
is proportional to time, doubling the time will also double the "ourier number.
430C !his case can be handled by setting the heat transfer coefficient h to infinity ∞ since thetemperature of the surrounding medium in this case becomes equivalent to the surface temperature.
431C !he maimum possible amount of heat transfer will occur when the temperature of the body
reaches the temperature of the medium, and can be determined from Q mC T T p ima $ %= −∞ .
432C When the &iot number is less than '.(, the temperature of the sphere will be nearly uniform at alltimes. !herefore, it is more convenient to use the lumped system analysis in this case.
433 A student calculates the total heat transfer from a spherical copper ball. It is to be determinedwhether his)her result is reasonable.
Assumptions !he thermal properties of the copper ball are constant at room temperature.
Properties !he density and specific heat of the copper ball are ρ * + kg)m, and C p * '.+ k/)kg.°0
$!able A%.
Analysis !he mass of the copper ball and the maimum amount of heat transfer from the copper ball are
k/('120%33''%$0k/)kg.+.'%$kg4,.($56
kg4,.(1
m%(.'$%kg)m+,$
1
ma


=°−°=−=
=
=
==
∞T T mC Q
DV m
i p
π π ρ ρ
Discussion !he student7s result of 23' k/ is not reasonable since it isgreater than the maimum possible amount of heat transfer.
2(4
Copperball, 200 C
Q
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Chapter 4 Transient Heat Conduction
434 An egg is dropped into boiling water. !he cooking time of the egg is to be determined. √
Assumptions 1 !he egg is spherical in shape with a radius of r ' * 3.4 cm. 2 8eat conduction in the egg isonedimensional because of symmetry about the midpoint. 3 !he thermal properties of the egg areconstant. 4 !he heat transfer coefficient is constant and uniform over the entire surface. 4 !he "ourier
number is τ 9 '.3 so that the oneterm approimate solutions $or the transient temperature charts% are
applicable $this assumption will be verified%.
Properties !he thermal conductivity and diffusivity of the eggs are given to be k * '.1 W)m.°0 and α *'.(2×('1 m3)s.
Analysis !he &iot number for this process is
Bihr
k
o= = °
° =
$ %$ . %
$ . %.
(2'' ' '34
' 112 3
W ) m . 0 m
W ) m. 0
3
!he constants λ ( (and A corresponding to this &iot number
are, from !able 2(,
1.( and '+44. (( == Aλ
!hen the "ourier number becomes
3.'(+.'%1.($4+
44' 33( %'+44.$
('
,' ≈= → =−
− → =
−
−=
−−
∞
∞τ θ
τ τ λ ee AT T
T T
i sph
!herefore, the oneterm approimate solution $or the transient temperature charts% is applicable. !hen the
time required for the temperature of the center of the egg to reach 4'°0 is determined to be
min17.8==×
=α
τ=
−s('1+
)s%m('(2.'$
m%'34.'%$(+.'$31
33or t
2(+
Water
97 C
Egg
T i = 8 C
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Chapter 4 Transient Heat Conduction
43
"!PROBLEM 435""GIVEN"D=0.055 "[m]"T_i=8 "[C]""T_o=0 [C] #$#m%&%$ &o '% (#$i%)"T_i*+i*i&,= "[C]"
=/400 "[1m2 C]"
"PROPERTIE"=0.6 "[1mC]"#7#=0./4E6 "[m21]"
"9N9L:I"Bi=;<$_o1$_o=D1">$om T#'7% 4/ ?o$$%o*)i*@ &o &i Bi *Am'%$ % $%#)"7#m')#_/=/.6
9_/=3. 0863;T_oT_i*+i*i&,1;T_iT_i*+i*i&, =9_/<%; 7#m')#_/ 2 <& #A
&im%=;&#A <$_o21#7#<Co*(%$&; mi*
!o "C# time "min#
' .+1 23.2
1' 2.31
1 2+.2
4' 3.+
4 4
+' 13.+3
+ 4'.1+
' +3.+
(((.(
50 55 60 65 0 5 80 85 0 5
30
40
50
60
0
80
0
/00
//0
/0
To [C]
t i m e
[ m i n ]
2(
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Chapter 4 Transient Heat Conduction
436 :arge brass plates are heated in an oven. !he surface temperature of the plates leaving the oven is to be determined.
Assumptions 1 8eat conduction in the plate is onedimensional since the plate is large relative to itsthickness and there is thermal symmetry about the center plane. 3 !he thermal properties of the plate areconstant. 4 !he heat transfer coefficient is constant and uniform over the entire surface. !he "ourier
number is τ 9 '.3 so that the oneterm approimate solutions $or the transient temperature charts% are
applicable $this assumption will be verified%.
Properties !he properties of brass at room temperature are given to be k * ((' W)m.°0, α * .×('1
m3)s
Analysis !he &iot number for this process is
BihL
k = =
°
° =
$+' %$ . %
$ %.
W ) m . 0 m
W ) m. 0
3''(
(('''('
!he constants λ ( (and A corresponding to this &iot
number are, from !able 2(,
λ ( ('(' (''(+= =. . and A
!he "ourier number is
τ α
= = × ×
= >
−t
L3
1 (' ('
''(' 2 ' 3
$ .
$ .. .
m ) s%$ min 1' s ) min%
m%
3
3
!herefore, the oneterm approimate solution $or the transient temperature charts% is applicable. !hen thetemperature at the surface of the plates becomes
C44°= → =−
−
==λ=−
−=θ
−τλ−
∞
∞
%,$4+.'4''3
4''%,$
4+.'%('.'cos$%''(+.($%)cos$%,$
%,$ %2.'$%('.'$((
33(
t LT t LT
e L Le AT T
T t xT t L
iwall
Discussion !his problem can be solved easily using the lumped system analysis since &i ; '.(, and thusthe lumped system analysis is applicable. It gives
°+°=−+=→=−−
=°⋅⋅×
°⋅=====
°⋅⋅×=×
°⋅==→=
−−
∞∞
−
∞
∞ s1''%$s''(122.'$
1
3
1
31
(
0%4''$304''%$%$ %$
''.'0%s)mW('32.m%$'(.'$
0W)m+'
%)$%$
0s)mW('32.)m('.
0W)m(('
eeT T T t T eT T
T t T
k L
h
LC
h
C LA
hA
VC
hAb
s
k C
C
k
bt i
bt
i
p p p
p p
α ρ ρ ρ
α ρ
ρ α
which is almost identical to the result obtained above.
23'
<lates
3°0
"urnace,
4''°0
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Chapter 4 Transient Heat Conduction
437 "!PROBLEM 43"
"GIVEN"L=0.031 "[m]"T_i=5 "[C]"T_i*+i*i&,= 00 "[C] #$#m%&%$ &o '% (#$i%)"&im%=/0 "[mi*] #$#m%&%$ &o '% (#$i%)"
=80 "[1m2 C]"
"PROPERTIE"=//0 "[1mC]"#7#=33.E6 "[m21]"
"9N9L:I"Bi=;<L1">$om T#'7% 4/ ?o$$%o*)i*@ &o &i Bi *Am'%$ % $%#)"7#m')#_/=0./03
9_/=/. 00/8&#A=;#7#<&im%<Co*(%$&;mi* 1L2;T_LT_i*+i*i&,1;T_iT_i*+i*i&,=9_/<%; 7#m')#_/ 2 <& #A <Co;7#m')#_/<L1L
! "C# !$ "C#
'' 3(.1
3 4.3
' 3.
4 1+.
1'' +2.(
13 .41' 2(.
14 2'.
4'' 221.
43 213.(
4' 244.+
44 2.2+'' '
+3 32.1+' 2'.3
+4 .+
'' 4(.2
time "min# !$ "C#
3 (21.42 322.+
1 3.+ (.
(' 221.
(3 2(.(2 3+.
(1 +.
(+ +.
3' 1'2.
33 13(.232 1.2
31 121.+
3+ 11.3
' 112
23(
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Chapter 4 Transient Heat Conduction
439
"!PROBLEM 43"
"GIVEN"$_o=0.351 "[m]"T_i=400 "[C]"T_i*+i*i&,=/ 50 "[C]"
=60 "[1m2 C]""& im%= 0 [mi*] #$#m%&%$ &o '% (#$i%)"
"PROPERTIE"=/4. "[1mC]"$o=00 "[@1m23]"C_=4 "[1@C]"#7#=3.5E6 "[m21]"
"9N9L:I"Bi=;<$_o1">$om T#'7% 4/ ?o$$%o*)i*@ &o &i Bi *Am'%$ % $%#)"7#m')#_/=/.035
9_/=/. /558_/=0.40 ">$om T#'7% 4 ?o$$%o*)i*@ &o 7#m')#_/"&#A=;#7 #<&im %<Co*(%$&; mi* 1$_o2;T_oT_i*+i*i&,1;T_iT_i*+i*i&, =9_/<%; 7#m')#_/ 2 <& #A
L=/ "[m] / m 7%*@& o+ &% ?,7i*)%$ i ?o*i)%$%)"V=i<$_o2<Lm=$o<V_m#=m<C_<;T_iT_i*+i*i&,<Co*(%$&; 1_m#= / <;T_oT_i*+i*i&, 1;T_iT_i*+i*i& ,<_/17#m')#_/
time [min] To [C] Q [kJ]
5 45. 44/
/0 4/3.4 8386
/5 40/.5 //05
0 30./ /5656
5 3.3 /046
30 368. 83
35 35 534
40 34.6 835
45 340.5 3//4
50 33/. 3383
55 33. 3640/
60 3/5.8 38853
232
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Chapter 4 Transient Heat Conduction
0 /0 0 30 40 50 60
300
30
340
360
380
400
40
440
0
5000
/0000
/5000
0000
5000
30000
35000
40000
time [min]
T o
[ C ]
Q
[ k J ]
temperature
heat
23
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Chapter 4 Transient Heat Conduction
440E :ong cylindrical steel rods are heattreated in an oven. !heir centerline temperature when theyleave the oven is to be determined.
Assumptions 1 8eat conduction in the rods is onedimensional since the rods are long and they havethermal symmetry about the center line. 2 !he thermal properties of the rod are constant. 3 !he heat
transfer coefficient is constant and uniform over the entire surface. 4 !he "ourier number is τ 9 '.3 so that
the oneterm approimate solutions $or the transient temperature charts% are applicable $this assumptionwill be verified%.
Properties !he properties of AI#I stainless steel rods are given to be k * 4.42 &tu)h.ft.°", α * '.( ft3)h.
Analysis !he time the steel rods stays in the oven can be determined from
t = = =length
velocity
ft
ft ) min min * (+' s
'
('
!he &iot number is
2'4.'%"&tu)h.ft.42.4$
%ft(3)3%$".&tu)h.ft3'$ 3
=°
°==
k
hr Bi o
!he constants λ ( (and A corresponding to this &iot number are, from !able 2(,
λ ( (' +4+2 ('= =. . and A
!he "ourier number is
τ α = = =
t
r o3
'(
3 (3'32
$ .
$ ).
ft ) h%$ ) 1' h%
ft%
3
3
!hen the temperature at the center of the rods becomes
θ λ τ '
'(
' +4+2 ' 32(3 3
(' ' (3,$ . % $ . %
$ . % .cyl i
T T
T T A e e=
−
− = = =∞
∞
− −
)228°= → =−
−
''
(3.'(4''+
(4''T
T
231
*+en, 1700 )
'teel ro, 8 )
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Chapter 4 Transient Heat Conduction
441 #teaks are cooled by passing them through a refrigeration room. !he time of cooling is to bedetermined.
Assumptions 1 8eat conduction in the steaks is onedimensional since the steaks are large relative to theirthickness and there is thermal symmetry about the center plane. 3 !he thermal properties of the steaks areconstant. 4 !he heat transfer coefficient is constant and uniform over the entire surface. !he "ourier
number is τ 9 '.3 so that the oneterm approimate solutions $or the transient temperature charts% are
applicable $this assumption will be verified%.
Properties !he properties of steaks are given to be k * '.2 W)m.°0 and α * '.(×('4 m3)s
Analysis !he &iot number is
3''.'%0W)m.2.'$
%m'(.'%$0.W)m,$ 3
=°
°==
k
hL Bi
!he constants λ ( (and A corresponding to this
&iot number are, from !able 2(,
λ ( (' 23+ ('((= =. . and A
!he "ourier number is
3.'1'(.%23+.'cos$%'((.($%(($3
%(($3
%)cos$%,$
3
3(
%23+.'$
((
>= → =−−
−−
=−−
−
−
∞
∞
τ
λ
τ
τ λ
e
L Le AT T T t LT
i
!herefore, the oneterm approimate solution $or the transient temperature charts% is applicable. !hen thelength of time for the steaks to be kept in the refrigerator is determined to be
min102.6==×
==−
s1()s%m('(.'$
m%'(.'%$1'(.$34
33
α
τ Lt
234
#teaks
3°0
>efrigerated air
((°0
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Chapter 4 Transient Heat Conduction
442 A long cylindrical wood log is eposed to hot gases in a fireplace. !he time for the ignition of thewood is to be determined.
Assumptions 1 8eat conduction in the wood is onedimensional since it is long and it has thermalsymmetry about the center line. 2 !he thermal properties of the wood are constant. 3 !he heat transfer
coefficient is constant and uniform over the entire surface. 4 !he "ourier number is τ 9 '.3 so that the
oneterm approimate solutions $or the transient temperature charts% are applicable $this assumption will be verified%.
Properties !he properties of wood are given to be k * '.(4 W)m.°0, α * (.3+×('4 m3)s
Analysis !he &iot number is
''.2%0W)m.(4.'$
%m'.'%$0.W)m1.($ 3
=°
°==
k
hr Bi o
!he constants λ ( (and A corresponding to this
&iot number are, from !able 2(,
21+.( and '+(.( (( == Aλ
=nce the constant J ' is determined from !able 23
corresponding to the constant λ (*(.'+(, the "ourier number isdetermined to be
3(.'%344(.'$%21+.($''('
''23'
%)$%,$
3
3(
%'+(.($
('(
=τ → =−
−
λ=−
−
τ−
τλ−
∞
∞
e
r r J e AT T
T t r T oo
i
o
which is above the value of '.3. !herefore, the oneterm approimate solution $or the transienttemperature charts% can be used. !hen the length of time before the log ignites is
min81.7==×
=α
τ=
−s2'2
)s%m('3+.($
m%'.'%$3(.'$
34
33or t
23+
(' cm Woo log, 10 C
8ot gases
4''°0
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Chapter 4 Transient Heat Conduction
443 A rib is roasted in an oven. !he heat transfer coefficient at the surface of the rib, the temperature ofthe outer surface of the rib and the amount of heat transfer when it is rare done are to be determined. !hetime it will take to roast this rib to medium level is also to be determined.
Assumptions 1 !he rib is a homogeneous spherical ob?ect. 2 8eat conduction in the rib is onedimensional because of symmetry about the midpoint. 3 !he thermal properties of the rib are constant. 4 !he heat
transfer coefficient is constant and uniform over the entire surface. !he "ourier number is τ 9 '.3 so that
the oneterm approimate solutions $or the transient temperature charts% are applicable $this assumption
will be verified%.
Properties !he properties of the rib are given to be k * '.2 W)m.°0, ρ * (3'' kg)m, C p * 2.( k/)kg.°0,
and α * '.(×('4 m3)s.
Analysis $a% !he radius of the roast is determined to be
m'+1'.'2
%m''3114.'$
2


2
m''3114.'kg)m(3''
kg3.





=== → =
=== → =
π π π
ρ ρ
V r r V
mV V m
oo
!he "ourier number is
(3(4.'m%'+1'.'$
1'%s2@1'')s%$3m('(.'$3
34
3 =
×××==
−
or
t α τ
which is somewhat below the value of '.3. !herefore, the oneterm approimate solution $or the transienttemperature charts% can still be used, with the understanding that the error involved will be a little morethan 3 percent. !hen the oneterm solution can be written in the form
%(3(4.'$((
','
3(
3( 1.'
(1.2
(11' λ τ λ θ −−
∞
∞==
−
− → =
−
−= e Ae A
T T
T T
i sph
It is determined from !able 2( by trial and error that this equation is satisfied when Bi * ', which
corresponds to λ ( ('43 (++= =. . and A . !hen the heat transfer coefficient can be determined from
C.Wm16.92°=
°== → =
%m'+1'.'$
%'%$0W)m.2.'$
o
o
r
kBih
k
hr Bi
!his value seems to be larger than epected for problems of this kind. !his is probably due to the "ouriernumber being less than '.3.
$b% !he temperature at the surface of the rib is
C19. °= → =
−
−
==−
−=
−−
∞
∞
%,$'333.'
(1.2
(1%,$
'43.
%rad'43.sin$%++.($
)
%)sin$%,$%,$
%(3(4.'$%'43.$
(
((
33(
t r T t r T
er r
r r e A
T T
T t r T t r
oo
oo
oo
i
o spho
λ
λ θ
τ λ
23
=ven
(1°0
ib,
4. C
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Chapter 4 Transient Heat Conduction
$c% !he maimum possible heat transfer is
k/3'+'0%.2(1%$0k/)kg.(.2%$kg3.$%$ma =°−°=−= ∞ i p T T mC Q
!hen the actual amount of heat transfer becomes
Q
Q
Q Q
o sphma
,
ma
sin$ % cos$ %$ . %
sin$ . % $ . % cos$ . %
$ . %.
. $ . %$
= − −
= − −
=
= = =
( ( ' 1'43 '43 '43
'43'4+
' 4+ ' 4+ 3'+'
( ( (
(
θ λ λ λ
λ
k/% 1629 %&
$d % !he cooking time for mediumdone rib is determined to be
hr3≅==×
==
= → =−
− → =
−
−=
−
−−
∞
∞
min(+(s+11,(')s%m('(.'$
m%'+1'.'%$(1.'$
(1.'%++.($(1.2
(14(
34
33
%'43.$(
','
33(
α
τ
τ θ τ τ λ
o
i sph
r t
ee AT T
T T
!his result is close to the listed value of hours and 3' minutes. !he difference between the two results isdue to the "ourier number being less than '.3 and thus the error in the oneterm approimation.
Discussion !he temperature of the outer parts of the rib is greater than that of the inner parts of the ribafter it is taken out of the oven. !herefore, there will be a heat transfer from outer parts of the rib to theinner parts as a result of this temperature difference. !he recommendation is logical.
444 A rib is roasted in an oven. !he heat transfer coefficient at the surface of the rib, the temperature ofthe outer surface of the rib and the amount of heat transfer when it is welldone are to be determined. !hetime it will take to roast this rib to medium level is also to be determined.
Assumptions 1 !he rib is a homogeneous spherical ob?ect. 2 8eat conduction in the rib is onedimensional
because of symmetry about the midpoint. 3 !he thermal properties of the rib are constant. 4 !he heattransfer coefficient is constant and uniform over the entire surface. !he "ourier number is τ 9 '.3 so that
the oneterm approimate solutions $or the transient temperature charts% are applicable $this assumptionwill be verified%.
Properties !he properties of the rib are given to be k * '.2 W)m.°0, ρ * (3'' kg)m, C p * 2.( k/)kg.°0,
and α * '.(×('4 m3)s
Analysis $a% !he radius of the rib is determined to be
m'+1'.'
2
%m''314.'$
2


2
m''314.'kg)m(3''
kg3.





=== → =
=== → =
π π
π
ρ ρ
V r r V
mV V m
oo
!he "ourier number is
(++(.'m%'+1'.'$
1'%s(@1'')s%$2m('(.'$3
34
3 =
×××==
−
or
t α τ
which is somewhat below the value of '.3. !herefore, the oneterm approimate solution $or the transienttemperature charts% can still be used, with the understanding that the error involved will be a little morethan 3 percent. !hen the oneterm solution formulation can be written in the form
2'
=ven
(1°0
ib,
4. C
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Chapter 4 Transient Heat Conduction
%(++(.'$((
','
3(
3( 2.'
(1.2
(144 λ τ λ θ −−
∞
∞==
−
− → =
−
−= e Ae A
T T
T T
i
sph
It is determined from !able 2( by trial and error that this equation is satisfied when Bi * 2., which
corresponds to 42'3.(and2''.3 (( == Aλ . !hen the heat transfer coefficient can be
determined from.
C.Wm22. 2 °=°== → =%m'+1'.'$
%.2%$0W)m.2.'$o
o
r kBih
k hr Bi
$b% !he temperature at the surface of the rib is
C142.1 °= → =−
−
==−
−=
−−
∞
∞
%,$(3.'(1.2
(1%,$
2.3
%2.3sin$%42'3.($
)
%)sin$%,$%,$
%(++(.'$%2.3$
(
((
33(
t r T t r T
er r
r r e A
T T
T t r T t r
oo
oo
oo
i
o spho
λ
λ θ
τ λ
$c% !he maimum possible heat transfer is
k/3'+'0%.2(1%$0k/)kg.(.2%$kg3.$%$ma =°−°=−=
∞ i p T T mC Q
!hen the actual amount of heat transfer becomes
%&112===
=−
−=−
−=
k/%3'+'%$434.'$434.'
434.'%2.3$
%2.3cos$%2.3$%2.3sin$%2.'$(
%cos$%sin$(
ma
(
(((,
ma
Q
Q spho
λ
λ λ λ θ
$d % !he cooking time for mediumdone rib is determined to be
hr4===×
=α
τ=
=τ → =−
− → =
−
−=θ
−
τ−τλ−
∞
∞
min'.32's2',(2)s%m('(.'$
m%'+1'.'%$(44.'$
(44.'%42'3.($(1.2
(14(
34
33
%2.3$(
','
33(
o
i
sph
r t
ee AT T
T T
!his result is close to the listed value of 2 hours and ( minutes. !he difference between the two results is probably due to the "ourier number being less than '.3 and thus the error in the oneterm approimation.
Discussion !he temperature of the outer parts of the rib is greater than that of the inner parts of the ribafter it is taken out of the oven. !herefore, there will be a heat transfer from outer parts of the rib to theinner parts as a result of this temperature difference. !he recommendation is logical.
2(
8/12/2019 Heat Chap04 026
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Chapter 4 Transient Heat Conduction
44 An egg is dropped into boiling water. !he cooking time of the egg is to be determined.
Assumptions 1 !he egg is spherical in shape with a radius of r ' * 3.4 cm. 2 8eat conduction in the egg isonedimensional because of symmetry about the midpoint. 3 !he thermal properties of the egg areconstant. 4 !he heat transfer coefficient is constant and uniform over the entire surface. !he "ourier
number is τ 9 '.3 so that the oneterm approimate solutions $or the transient temperature charts% are
applicable $this assumption will be verified%.
Properties !he thermal conductivity and diffusivity of the eggs can be approimated by those of water at
room temperature to be k * '.1'4 W)m.°0, α = k C p) ρ * '.(21×('1 m3)s $!able A%.
Analysis !he &iot number is
3.1%0W)m.1'4.'$
%m'34.'%$0.W)m+''$ 3
=°
°==
k
hr Bi o
!he constants λ ( (and A corresponding to this
&iot number are, from !able 2(,
3.(and'. (( == Aλ
!hen the "ourier number and the time period become
(1.'%3.($(''+
(''1' 33( %'.$
('
,' = → =−
− → =
−
−=
−−
∞
∞τ θ
τ τ λ ee AT T
T T
i sph
which is somewhat below the value of '.3. !herefore, the oneterm approimate solution $or the transienttemperature charts% can still be used, with the understanding that the error involved will be a little morethan 3 percent. !hen the length of time for the egg to be kept in boiling water is determined to be
min14.1==×
==−
s+21)s%m('(21.'$
m%'34.'%$(1.'$
31
33
α
τ or t
23
Water
100 C
Egg
T i = 8 C
8/12/2019 Heat Chap04 026
http://slidepdf.com/reader/full/heatchap04026 17/17
Chapter 4 Transient Heat Conduction
446 An egg is cooked in boiling water. !he cooking time of the egg is to be determined for a location at(1('m elevation.
Assumptions 1 !he egg is spherical in shape with a radius of r ' * 3.4 cm. 2 8eat conduction in the egg isonedimensional because of symmetry about the midpoint. 3 !he thermal properties of the egg and heattransfer coefficient are constant. 4 !he heat transfer coefficient is constant and uniform over the entire
surface. !he "ourier number is τ 9 '.3 so that the oneterm approimate solutions $or the transient
temperature charts% are applicable $this assumption will be verified%.
Properties !he thermal conductivity and diffusivity of the eggs can be approimated by those of water at
room temperature to be k * '.1'4 W)m.°0, α = k C p) ρ * '.(21×('1 m3)s $!able A%.
Analysis !he &iot number is
3.1%0W)m.1'4.'$
%m'34.'%$0.W)m+''$ 3
=°
°==
k
hr Bi o
!he constants λ ( (and A corresponding to this
&iot number are, from !able 2(,
3.(and'. (( == Aλ
!hen the "ourier number and the time period become
(434.'%3.($2.2+
2.21' 33( %'.$
('
,' = → =−
− → =
−
−=
−−
∞
∞τ θ
τ τ λ ee AT T
T T
i sph
which is somewhat below the value of '.3. !herefore, the oneterm approimate solution $or the transienttemperature charts% can still be used, with the understanding that the error involved will be a little morethan 3 percent. !hen the length of time for the egg to be kept in boiling water is determined to be
min14.9==×
==−
s+)s%m('(21.'$
m%'34.'%$(434.'$31
33
α
τ or t
2
Water
94.4 C
Egg
T i = 8 C