hAs (T −T )dt =mC ( ) =− ρ ln ) ∞ =− ρ · 2009-11-11 · 6 Copyright © The McGraw-Hill...
41
1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display 1 FIGURE 4-1 A small copper ball can be modeled as a lumped system, but a roast beef cannot Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display 2 FIGURE 4-2 The geometry and parameters involved in the lumped system analysis. ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ dt during body the of energy the in increase The dt during body the o transfer Heat int ( ) dT mC dt T T hA p s = − ∞ ( ) dt VC hA T T T T d p s ρ − = − − ∞ ∞ t VC hA T T T t T p s i ρ − = − − ∞ ∞ ) ( ln bt i e T T T t T − ∞ ∞ = − − ) ( p s VC hA b ρ =
Transcript of hAs (T −T )dt =mC ( ) =− ρ ln ) ∞ =− ρ · 2009-11-11 · 6 Copyright © The McGraw-Hill...
1
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display1
FIGURE 4-1
A small copper ball can be modeled as a lumped system, but a roast beef cannot
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display2
FIGURE 4-2
The geometry and parameters involved in the lumped system analysis.
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
dtduringbodytheofenergy
theinincreaseThe
dtduringbodytheotransferHeat int
( ) dTmCdtTThA ps =−∞
( )dt
VChA
TTTTd
p
s
ρ−=
−−
∞
∞
tVChA
TTTtT
p
s
i ρ−=
−−
∞
∞)(ln
bt
i
eTTTtT −
∞
∞ =−−)(
p
s
VChA
bρ
=
2
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display3
FIGURE 4-3
The temperature of a lumped system approaches the environment temperature as time gets larger.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display4
FIGURE 4-4
Heat transfer to or from a body reaches its maximum value when the body reaches the environment temperature.
[ ]∞−= TtThAtQ s )()(&
[ ]ip TtTmCQ −= )(
( )ip TTmCQ −= ∞max
sc A
VL =
khLBi c=
3
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display5
FIGURE 4-5
bodythewithinConductionbodytheofsurfacetheatConvection
TT
LkhBi
c
=ΔΔ
=/
bodytheofsurfacetheatceresisConvectionbodythewithinceresisConduction
hkLBi c
tantan
1==
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display6
FIGURE 4-6
Small bodies with high thermal conductivities and low convection coefficients are most likely to satisfy the criterion lumped system analysis.
4
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display7
FIGURE 4-7
Analogy between heat transfer to a solid and passenger traffic to an island.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display8
FIGURE 4-8
When the convection coefficient h is high and k is low, large temperature difference occur between the inner and outer regions of a large solid.
5
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display9
FIGURE 4-9
Schematic for Example 4-1.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display10
Assumption•The junction is spherical in shape with a diameter of D = 0.001 m.•Thermal properties of the junction and the heat transfer coefficient are constant.•Radiation effects are negligible
6
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display11
( ) mmDD
D
AVL
sc
42
3
1067.1001.061
616
1−×=====
π
π
( )( ) 1.0001.0/35
1067.1/210 42
<=•
ו==
−
CmWmCmW
khLBi O
Oc
Analysis
In order to read 99 percent of the initial temperature difference
between the junction and the gas, we must have
01.0)(=
−−
∞
∞
TTTtT
i
∞− TTi
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display12
Analysis
The value of the exponent b is
( )( )( )1
43
2
462.01067.1/320/8500
/210 −− =
ו•
=== smCkgJmkg
CmWLC
hVC
hAb O
O
cpp
s
ρρ
tsbt
i
eeTTTtT )462.0( 1
01.0)( −−−
∞
∞ =→=−−
st 10=
7
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display13
FIGURE 4-10
Schematic for Example 4-2.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display14
Assumption
.The body can be modeled as a 30-cm-diameter, 1.70-m-long cylinder..The thermal properties of the body and the heat transfer coefficient are constant..The radiation effects are negligible..The person was healthy (!) when he or she died with a body temperature of 37 0C.
8
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display15
Analysis
( ) ( )( )( ) ( )
mmmm
mmrLr
LrAVL
sc 0689.0
15.027.115.027.115.0
22 2
2
200
20 =
+=
+==
πππ
πππ
( )( ) 1.089.0/617.0
0689.0/8 2
>=•
•==
CmWmCmW
khLBi O
Oc
( )( )( )15
3
2
1079.20689.0/4178/996
/8 −−×=•
•=== s
mCkgJmkgCmW
LCh
VChAb O
O
cpp
s
ρρ
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display16
Analysis
tsbt
i
eeTTTtT )1079.2( 15
20372025)( −−×−−
∞
∞ =−−
→=−−
hst 2.12860,43 ==
9
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display17
FIGURE 4-11
Schematic of the simple geometries in which heat transfer is one-dimensional.
∞
∞
−−
=TT
TtxTtxi
),(),(θ
LxX =
khLBi =
2Ltατ =
Dimensionless temperature
Dimensionless distance from the center
Dimensionless heat transfer coefficient
Dimensionless time
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display18
FIGURE 4-12
Transient temperature profiles in a plane wall exposed to convection from its surfaces for Ti > T∞.
2.0,)cos(),(),( 11
21 >=
−−
= −
∞
∞ τλθ τλ LxeATT
TtxTtxi
wall
2.0,)(),(),( 0101
21 >=
−−
= −
∞
∞ τλθ τλ rrJeATT
TtrTtri
cyl
2.0,)sin(),(),(01
011
21 >=
−−
= −
∞
∞ τλλθ τλ
rrrreA
TTTtrTtr
isph
Plane wall
Cylinder
Sphere
τλθ21
10
,0−
∞
∞ =−−
= eATTTT
iwall
τλθ21
10
,0−
∞
∞ =−−
= eATTTT
icyl
τλθ21
10
,0−
∞
∞ =−−
= eATTTT
isph
Center of plane wall (x=0)
Center of cylinder (r=0)Center of sphere (r=0)
10
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display19
FIGURE 4-13
Transient temperature and heat transfer charts for a plane wall of thickness 2L initially at a uniform temperature Ti subjected to convection from both sides to an environment at temperature T∞with a convection coefficient of h.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display20
FIGURE 4-14
Transient temperature and heat transfer charts for a long cylinder of radius r0 initially at a uniform temperature Ti subjected to convection from all sides to an environment at temperature T∞ with a convection coefficient of h.
11
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display21
FIGURE 4-15
Transient temperature and heat transfer charts for a sphere of radius r0 initially at a uniform temperature Ti subjected to convection from all sides to an environment at temperature T∞ with a convection coefficient of h.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display22
FIGURE 4-16
The specified surface temperature corresponds to the case of convection to an environment at T∞ with a convection coefficient h that is infinite.
( )( )ip
ip
TTVC
TTmCQ
−=
−=
∞
∞
ρmax
12
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display23
FIGURE 4-17
The fraction of total heat transfer Q/Qmax up to a specified time t is determined using the Gröber charts.
1
1,0
max
sin1λλθ wall
wallQ
Q−=⎟⎟
⎠
⎞⎜⎜⎝
⎛
( )1
11,0
max
21λλθ J
cylcyl
−=⎟⎟⎠
⎞⎜⎜⎝
⎛
31
111,0
max
cossin31λ
λλλθ −−=⎟⎟
⎠
⎞⎜⎜⎝
⎛sph
sphQ
Q
Plane wall
Cylinder
Sphere
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display24
FIGURE 4-18
Fourier number at time t can be viewed as the ratio of the rate of heat conducted to the rate of heat stored at that time.
( )TT
tLCLkL
Lt
p ΔΔ
== 3
2
2
1ρ
ατ .
3
3
LvulumeofbodyainstoredisheatwhichatrateThe
LvolumeofbodyaofLacrossconductedisheatwhichatrateThe
=
13
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display25
FIGURE 4-19
Schematic for Example 4-3.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display26
Assumption
•.The egg is spherical in shape with a radius of r0=2.5cm.•.Heat conduction in the egg is one-dimension because of thermal symmetry about the midpoint.•.The thermal properties of egg and the heat transfer coefficient are constant.•The Fourier number is τ>0.2 so that the one term approximate solutions are applicable.
14
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display27
Analysis
( )( ) 1.08.47/627.0
025.0/1200 20 >=
••
==CmW
mCmWk
hrBi O
O
The coefficients λ1 and A1 for a sphere corresponding to this Bi are, from table 4-1,λ1=3.0753A1=1.9958
2.0209.09958.19559570 22
1 )0753.3(1
0 >=→=−−
→=−− −−
∞
∞ τττλ eeATTTT
i
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display28
Analysis
( )( ) min4.14865/10151.0
025.0209.026
220 ≈=
×== − s
smmrt
ατ
15
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display29
FIGURE 4-20
Schematic for Example 4-4.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display30
Assumption
•. Heat conduction in the plate is one-dimensional since the plate is large relative to its thickness and there is thermal symmetryabout the center plane.•The thermal properties of the plate and the heat transfer coefficient are constant.•The Fourier number is τ>0.2 so that the one-term approximate solutions are applicable.
16
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display31
Analysis
( )( ) 8.4502.0120
10012 =•
•==
mCmWCmW
hLk
Bi o
o
( )( )( )
6.3502.0
607/109.332
26
2 =××
==−
mssm
Ltατ
46.00 =−−
∞
∞
TTTT
i
8.451==
hLk
Bi
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display32
Analysis
1==LL
Lx
99.00
=−−
∞
∞
TTTT
455.099.046.00
0
=×=−−
−−
=−−
∞
∞
∞
∞
∞
∞
TTTT
TTTT
TTTT
ii
( )( ) ( )( )( )
13
2
00185.002.0/380/8530
/1202
2 −=•
•==== s
mCkgJmkgCmW
LCh
LACAh
VChAb o
o
ppp
s
ρρρ
17
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display33
Analysis
( )ssbt
i
etTeTTTtT 420)00185.0( 1
50020500)()( −−−
∞
∞ =−−
→=−−
CtT o279)( =
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display34
FIGURE 4-21
Schematic for Example 4-5.
18
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display35
Assumption
•Heat conduction in the shaft is one-dimensional since it is long and it has thermal symmetry about the centerline.•The thermal properties of the shaft and the heat transfer coefficient are constant.•The Fourier number is τ>0.2 so that the one-term approximate solutions are applicable.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display36
Analysis
( )( ) 86.11.080
9.1412
0
=•
•==
mCmWCmW
hrk
Bi o
o
( )( )( )
07.11.0
6045/1095.32
26
20
=××
==−
mssm
rtατ
40.00 =−−
∞
∞
TTTT
i
( ) ( ) CTTTT oi 3602006004.02004.00 =−+=−+= ∞∞
19
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display37
Analysis
( ) ( ) ( ) kgmmmkgLrVm 2.24811.0/7900 2320 ==== πρπρ
( ) ( )( )( ) kJCCkgkJkgTTmCQ ooip 354,47200600/477.02.248max =−•=−= ∞
537.086.11
11
===Bi
Bi
( ) ( ) 309.007.1537.0 222
2
=== τα Bik
th
62.0max
=Q
Q
( ) kJkJQQ 360,29354,4762.062.0 max =×==
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display38
Analysis Alternative solution
( )( ) 537.0/9.14
1.0/80 20 =
••
==CmW
mCmWk
hrBi O
O
970.01 =λ 122.11 =A
( ) ( ) 41.0122.1 07.1970.01
00
221 ===
−−
= −−
∞
∞ eeATTTT
i
τλθ
( ) ( ) CTTTT oi 36420060041.020041.00 =−+=−+= ∞∞
20
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display39
Analysis Alternative solution
( )( ) 636.0
970.0430.041.02121
1
110
max
=×−=−=λλθ J
( ) kJkJQQ 120,30354,47636.0636.0 max =×==
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display40
FIGURE 4-22
Schematic for a semi-infinite body.
i
i
i TTTtxT
TTTtxTtx
−−
=−−
−=−∞∞
∞ ),(),(1),(1 θ
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−
∞ kth
txerfc
kth
khx
txerfc
TTTtxT
i
i αα
αα 2
exp2
),(2
2
dueerfc u∫ −−=ξ
πξ
0
221)(
⎟⎟⎠
⎞⎜⎜⎝
⎛=
−−
txerfc
TTTtxT
is
i
α2),(
21
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display41
FIGURE 4-23
Variation of temperature with position and time in a semi-infinite solid initially at Ti subjected to convection to an environment at T∞with a convection heat transfer coefficient of h (from P.J. Schneider, Ref. 10).
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display42
FIGURE 4-24
Schematic for Example 4-6.
22
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display43
Assumption
•The temperature in the soil is affected by the thermal conditions at one surface only, and thus the soil can be considered to be asemi-infinite medium with a specified surface temperature of –100C.•The thermal properties of the soil are constant.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display44
Analysis
∞=k
th α
( ) ( )( ) 6.0
10151001,1 =−−−−
−=−−
−∞
∞
TTTtxT
i
36.02
==t
xα
ξ
( )( )( ) shsdayhdayst 61078.7/3600/2490 ×==
( )( ) mssmtx 77.01078.7/1015.036.022 626 =×××== −αξ
23
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display45
Analysis alternative solution
( ) 60.02
,=⎟⎟
⎠
⎞⎜⎜⎝
⎛=
−−
txerfc
TTTtxT
is
i
α
( )( ) mssmtx 80.01078.7/1015.037.022 626 =×××== −αξ
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display46
FIGURE 4-25
The temperature in a short cylinder exposed to convection from all surfaces varies in both the radial and axial directions, and thus heat is transferred in both directions.
cylinderinitei
wallplanei
cylindershorti TT
TtrTTT
TtxTTT
TtxrTinf
),(),(),,(⎟⎟⎠
⎞⎜⎜⎝
⎛−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
∞
∞
∞
∞
∞
∞
24
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display47
FIGURE 4-26
A short cylinder of radius r0 and height a is the intersection of along cylinder of radius r0 and plane wall of thickness a.
( )wallplanei
wall TTTtxTtx ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=
∞
∞),(,θ
cylinderinitei
cyl TTTtrTtr
inf
),(),( ⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=∞
∞θ
( )solid
initesemiisemi TT
TtxTtxinf
inf),(,
−∞
∞− ⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=θ
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display48
FIGURE 4-27
A long solid bar of rectangular profile a X b is intersection of two plane wall of thicknesses a and b.
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
1max2max1max2,max
1Q
Q
Dtotal
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟⎟
⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
2max1max3max1max2max1max3,max
111Q
Q
Dtotal
),(),(),,(tan
tytxTT
TtyxTwallwall
bargularreci
θθ=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−
∞
∞
rectangular bar
25
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display49
FIGURE 4-28
Schematic for Example 4-7.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display50
FIGURE 4-28
Assumption
•Heat conduction in the short cylinder is two-dimensional, and thus the temperature varies in both the axial x- and the radial r-directions. • The thermal properties of the cylinder and the heat transfer coefficient are constant.•The Fourier number is τ > 0.2 so that the one-term approximate solutions are applicable.
26
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display51
Analysis (a)
( )( )( )
48.806.0
900/1039.32
25
2 =×
==−
mssm
Ltατ
( ) ( ) 8.0,0,0 =⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=∞
∞
TTTtTt
iwallθ
( )( ) 63006060
11012 .
m.Cm/WCm/W
hLk
Bi O
O
=⋅
•==
( )( )( )
2.1205.0
900/1039.32
25
2 =×
==−
mssm
rtατ
( ) ( ) 5.0,0,0 =⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=∞
∞
TTTtTt
icylθ
( )( ) 73605060
11012 .
m.Cm/WCm/W
hrk
Bi O
O
o
=⋅
•==
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display52
Analysis (a)
( ) ( ) ( ) 4.05.08.0,0,0,0,0=×=×=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
∞
∞ ttTT
TtTcylwall
cylindershorti
θθ
( ) ( )∞∞ −+= TTTtT i4.0,0,0
( ) C°=−+= 63251204.025
27
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display53
Analysis (b)
106.006.0
==mm
Lx
( ) 98.0,=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
∞
∞
TTTtLT
o
( )( ) 6.3006.0/60
/11012 =⋅
•==
mCmWCmW
hLk
Bi O
O
( ) ( ) ( ) 784.08.098.0,,, =×=⎟⎟⎠
⎞⎜⎜⎝
⎛−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−−
=−−
=∞
∞
∞
∞
∞
∞
TTTT
TTTtLT
TTTtLTtL
i
o
oiwallθ
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display54
Analysis (b)
( ) ( ) ( ) 392.05.0784.0,0,,0,=×=×=⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−
∞
∞ ttLTT
TtLTcylwall
cylindershorti
θθ
( ) ( )∞∞ −+= TTTtLT i392.0,0,
( ) C°=−+= 2.6225120392.025
28
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display55
FIGURE 4-29
Schematic for Example 4-9.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display56
FIGURE 4-28
Assumption
•Heat conduction in the semi-infinite cylinder is two dimensional, and thus the temperature varies in both the axial x- and the radial r-directions. • The thermal properties of the cylinder and the heat transfer coefficient are constant.•The Fourier number is τ > 0.2 so that the one-term approximate solutions are applicable.
29
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display57
Analysis
( )( )( )
2.091.21.0
605/1071.92
25
2 >=××
==−
mssm
Ltατ
( )( ) 05.0/273
1.0/120 20 =
•⋅
==CmW
mCmWk
hrBi O
O
( )⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛+−⎟
⎠
⎞⎜⎝
⎛=− − kth
txerfc
kth
khx
txerfctxsemi
αα
αα
θ2
exp2
,1 2
2
inf
( ) ( ) ( ) 762.00124.1,0 91.23126.010
221 ==== −− eeAtcylτλθθ
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display58
Analysis
( ) 0074.0086.0 22
2
2
==⎟⎟⎠
⎞⎜⎜⎝
⎛=
kth
kth αα
( )( )44.0
605/1071.9215.0
2 25=
××==
− ssmm
txα
ξ
( ) ( )( )086.0
/237300/1071.9/120 252
=°⋅
×°⋅=
−
CmWssmCmW
kth α
( )( ) 0759.0/237
15.0/1202
2
=°⋅
°⋅=
CmWmCmW
khx
30
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display59
Analysis
( ) ( ) ( ) 73407620963000 ...t,t,xTT
Tt,,xTcyleintinfsemi
cylindereintinfsemii
=×=×=⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−
−∞
∞ θθ
( ) ( )∞∞ −+= TT.Tt,,xT i73400
( ) C. °=−+= 15115200734015
( ) 457008330533801 ..exp. ×+−=
( ) ( ) ( ) ( )086.044.00074.00759.0exp44.01,inf +++−=− erfcerfctxsemoθ
9630.=
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display60
FIGURE 4-30
Schematic for Example 4-10.
31
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display61
FIGURE 4-28
Assumption
•Heat conduction through the steaks is one-dimensionalsince the steaks from a large layer relative to their thickness and there is thermal symmetry about the center plane. • The thermal properties of the steaks and the heat transfer coefficient are constant.•The Fourier number is τ > 0.2 so that the one-term approximate solutions are applicable.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display62
Analysis
( ) ( )( ) 770
157152
0
.TT
Tt,LT=
−−−−
=−−
∞
∞
( ) Ccm/Wm/.
Ccm/W.Lk
.h O
O
⋅=⋅
== 22
30100151
45051
1
111
==cmcm
Lx
511 .hLk
Bi==
32
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display63
FIGURE 4-31
Typical growth curve of microorganisms.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display64
FIGURE 4-32
The factors that affect the rate of growth of microorganisms.
33
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display65
FIGURE 4-33
The rate of growth of microorganisms in a food product increases exponentially with increasing environmental temperature.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display66
FIGURE 4-34
Freezing may stop the growth of microorganisms, but it may not necessarily kill them.
34
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display67
FIGURE 4-35
Recommended refrigeration and freezing temperatures for most perishable foods.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display68
FIGURE 4-36
Typical freezing curve of food item.
35
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display69
FIGURE 4-37
Typical cooling curve of a beef carcass in the chilling and holding rooms at an average temperature of 0oC (from ASHRAE, Handbook: Refrigeration, Ref. 3, Chap. 11, Fig. 2).
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display70
FIGURE 4-38
Various cuts of beef (from National Livestock and Meat Board).
36
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display71
FIGURE 4-39
Variation of tenderness of meat stored at 2oC with time after slaughter.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display72
FIGURE 4-40
The freezing time of meat can be reduced considerably by using low temperature air at high velocity.
37
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display73
FIGURE 4-41
A refrigerated truck dock for loading frozen items to a refrigerated truck.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display74
FIGURE 4-42
Air chilling causes dehydration and thus weight loss for poultry, whereas immersion chilling causes a weight gain as a result of water absorption.
38
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display75
FIGURE 4-43
The storage life of fresh poultry decreases exponentially with increasing storage temperature.
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display76
FIGURE 4-44
The variation of freezing time of poultry with air temperature (from van der Berg and Lentz, Ref. 11).
39
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display77
FIGURE 4-45
The variation of temperature of the breast of 6.8-kg turkeys initially at 1oC with depth during immersion cooling at -29oC (from van der Berg and Lentz, Ref. 11).
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display78
FIGURE 4-46
Schematic for Example 4-5.
40
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display79
FIGURE 4-28
Assumption
•Water evaporates at a rate of 0.080 kg/s. •All the moisture in the air freezes in the evaporator.
Properties
( ) Ckg/kJ....contentwater..Cp °⋅=×+=×+= 143580512681512681
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display80
Analysis (a)
( )( ) ( ) s/kg.s/carcass/kgcarcasses 563360010285450 =×=
gainheatlightsfanbeefchillroom,total QQQQQ &&&&& +++=
( ) ( )timeCooling/cooledmassbeefTotalmbeef =
( ) ( )( )( ) kWCCkg/kJ.s/kg.TmCQ beefbeef 2351536143563 =°−°=Δ=&
kW27713326235 =+++=
41
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display81
Analysis (b)
s/m.m/kg.s/kgmV
air
airair
33 978
2921102
===ρ&&
( ) ( )( ) kWkg/kJs/kg.mhQwaterfgeevaporativ,beef 19924900800 ===&
( ) ( ) ( )[ ] s/kg.C.Ckg/kJ.
kWTC
Qmairp
airair 0102
2700061277
=°−−°⋅
=Δ
=&
&
