heat and mass transfer - seoul national universityocw.snu.ac.kr/sites/default/files/note/545.pdf ·...
TRANSCRIPT
이 윤 우
서울대학교 화학생물공학부
Heat and Mass Transfer
19UNSTEADY-STATE
HEAT CONDUCTION
Supercritical Fluid Process Lab
Unsteady-State Heat Conduction
is of importance to engineers in many circumstances
It may control the rate at which process equipment is brought to stable operating conditions
It is also important in determining the processing time of many solid articles. For example, the curing time of objects made of molded plastic or rubber is often dependent on the time required to bring the center to some specifiedtemperature without causing thermal damage to the material at the surface.
There are also many applications of unsteady-state-conduction theory in the heat treating and casting of the metals.
A hot metal billet that is removed from a furnace
Re-entry of Columbia into the earth’s atmosphere
200oC 40oC
θ=20min
q1
200oC 40oC
θ=60min
q2
q1 = q2
200oC 40oC
θ=20min
q1
100oC 50oC
θ=60min
q2
q1 ≠ q2
(a) Steady (b) Unsteady
Steady vs. Unsteady State Heat Transfer
Supercritical Fluid Process Lab
Unsteady problem typically arise when the boundary conditions of a system are changed.If the surface temperature of a system is altered, the temperature at each point in the systemwill also begin to change. The changes will continue to occur until a new steady-state temperature distribution is reached.
The first law of thermodynamics
Ein Eout
Control volumeEgenEacc
Ein – Eout + Egen = Eacc
Ein : Energy transport by the fluid into the control volume surface phenomenonEout : Energy transport by the fluid out of the control volume surface phenomenonEgen : Energy generation in the control volume volumetric phenomenonEacc : Energy accumulation in the control volume volumetric phenomenon
re temperatu:T time,:
heat Specific:C volume,:V density,:
)(
volume:
eunit volumper rate generationheat :
p
t
VTCE
V
g
VgE
ptacc
gen
ρ
ρ=
=
∂∂
&
&
Supercritical Fluid Process Lab
qx
qy
qz
qx+Δx
qz+Δz qy+Δy
dz
dy
dx
General heat conduction equation
Energy Balance: In - Out + Generation = AccumulationEin - Eout + Egen =Eacc
Egen = gV (g: heat generation rate/volume)
Eacc = )( VTCpρθ∂∂
)(
)(
)(
)(
dxdydzgEztdxdykq
ytdxdzkq
xtdydzkq
gen
z
y
x
&=∂∂
−=
∂∂
−=
∂∂
−=
θρ
∂
⋅⋅∂=
∂∂
+=
∂
∂+=
∂∂
+=
Δ+
Δ+
Δ+
)( tdxdydzCE
dzz
qqq
dyy
qqq
dxxqqq
pacc
zzzz
yyyy
xxxx
Supercritical Fluid Process Lab
Supercritical Fluid Process Lab
General heat conduction equation
pp
p
CCk
tkg
zt
yt
xt
tCgztk
zytk
yxtk
x
ρρ
α
θα
ρθ
1
)(
2
2
2
2
2
2
=
∂∂
=+∂∂
+∂∂
+∂∂
∂∂
=+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
&
&
For constant properties (k, ρ, Cp)
: thermal diffusivity : thermal capacitance
Materials of large α will respond quickly to changes in their environment, taking shorter to reach a new equilibrium condition.
Supercritical Fluid Process Lab
Fundamental Equations
2
2
2
2
2
2
2
2
2
2
2
2
2
2
xt
Ckt
zt
yt
xt
Ckt
zt
yt
xt
Ckt
ztu
ytu
xtu
p
p
pzyx
∂∂
=∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
+∂∂
ρθ
ρθ
ρθ(8-11)
(8-12)
(19-1)
The basic differential equation for unsteady-state conduction in a solid(For constant thermal conductivity and no heat generation)
Unsteady-state conduction in only x direction
A differential energy balance
Supercritical Fluid Process Lab
0
0
2
2
2
2
2
2
2
2
=∂∂
=∂∂
+∂∂
+∂∂
xt
zt
yt
xt
(19-2)
Steady-state conduction in only x direction
Laplace’s equation
Fundamental Equations
The basic differential equation for steady-state conduction in a solid(For constant thermal conductivity and no heat generation)
Which is readily integrated to give a linear relation between temperature and distance. This was shown earlier in Eq. (18-3), obtained by integrating the Fourier conduction equation.
Supercritical Fluid Process Lab
General Heat Transfer Equation
Supercritical Fluid Process Lab
Cylindrical coordinates
zzryrx
===
θθ
sincos
zrrV ΔΔΔ=Δ θφrd
rΔzΔ
θΔr
θΔθ
Supercritical Fluid Process Lab
Cylindrical coordinates
θθρ
θ
θθ
θθ
θθ
θθθθθθ
′∂∂
ΔΔΔ
ΔΔΔ∂∂
ΔΔ−−∂∂
ΔΔ−=−
∂∂
ΔΔ−−∂∂
ΔΔ−=−
∂∂
ΔΔ−−∂∂
ΔΔ−=−
Δ+Δ+
Δ+Δ+
Δ+Δ+
trzrC
rzrSztzkr
ztrkrqq
rtzrk
rtzrkqq
rtzkr
rtzkrqq
p
zzzzzz
rrrrrr
)(
)(
)()(
)()(
)()(
Input-Output
r-direction:
θ-direction:
z-direction:
Generation:
Accumulation:
Supercritical Fluid Process Lab
Cylindrical coordinates
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=′∂
∂
′∂∂
=⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
′∂∂
=+Δ
∂∂
−∂∂
+Δ
∂∂
−∂∂
+Δ
∂∂
−∂∂
→ΔΔΔ
Δ+Δ+Δ+
2
2
2
2
2
20,,
11
11lim
ztt
rrtr
rrCkt
tCztk
ztk
rrtkr
rr
tCSz
ztk
ztk
rr
tktk
rrrtkr
rtkr
p
pzr
pzzzrrr
θρθ
θρ
θθ
θρ
θθθ
θ
θθθ
If k=constant
)( rzr ΔΔΔ÷ θ
0
Supercritical Fluid Process Lab
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=′∂
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=′∂
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂
=′∂
∂
rt
rrt
Ckt
trr
trr
tCkt
ztt
rrt
rrt
Ckt
p
p
p
1
11
11
2
2
2
2
22
2
2
2
2
2
22
2
ρθ
θρθ
θρθ(19-3)
No axial conduction
Fundamental Equations
The basic differential equation for unsteady-state conduction in a solid(For constant thermal conductivity and no heat generation)
Cylindrical coordinates
No axial conduction & angular conduction
RadialConduction
AxialConduction
θφθφθ
cossinsincossin
===
zryrxθΔr
φθ Δ⋅sinr
Δφ)Δθ, φΔr, θ(r +++
),,( φθr
rΔrrV Δ⋅Δ⋅Δ⋅=Δ θφθsin2
Spherical coordinates
Supercritical Fluid Process Lab
θφθθρ
φθθ
φθθ
φθθ
θφθ
θφθ
φθθφθθ
φφφ
θθθθθθ
′∂∂
ΔΔΔ
ΔΔΔ
∂∂
ΔΔ−−∂∂
ΔΔ−=−
∂∂
ΔΔ−−∂∂
ΔΔ−=−
∂∂
ΔΔ−−∂∂
ΔΔ−=−
Δ+Δ+
Δ+Δ+
Δ+Δ+
trrC
rrS
tr
zkrtr
rkrqq
rtrkr
rtrkrqq
rtkr
rtkrqq
p
zzz
rrrrrr
)sin(
)sin(
sin1)(
sin1)(
)sin()sin(
)sin()sin(
2
2
22
Input-Output
r-direction:
θ-direction:
z-direction:
Generation:
Accumulation:
Spherical coordinates
Supercritical Fluid Process Lab
Spherical coordinates
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=′∂
∂
′∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
′∂∂
=Δ
∂∂
−∂∂
+Δ
∂∂
−∂∂
+Δ
∂∂
−∂∂
→ΔΔΔ
Δ+Δ+Δ+
2
2
2222
2
2222
20,,
2
22
sin1sin
sin11
sin1sin
sin11lim
sinsinsin
sinsin
φθθθ
θθρθ
θρ
φφθθθ
θθ
θρ
θφθφφ
θθθ
θθ
θ
θ
φφφθθθ
tr
trr
trrrC
kt
tCtkr
tkrr
tkrrr
tCrr
tktk
rr
tktk
rrrtkr
rtkr
p
pzr
prrr
If k=constant
)sin( 2 φθθ ΔΔΔ÷ rr
Supercritical Fluid Process Lab
Differential Energy Balance
gtr
trr
trrrC
k
tr
utr
urtut
p
r
&+⎥⎦
⎤⎢⎣
⎡∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=
∂∂
+∂∂
+∂∂
+′∂
∂
2
2
2222
2 sin1sin
sin11
sin
φθθθ
θθρ
φθθθφθ
Supercritical Fluid Process Lab
gztt
rrtr
rrCk
ztut
ru
rtut
pzr &+⎥
⎦
⎤⎢⎣
⎡∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=∂∂
+∂∂
+∂∂
+′∂
∂2
2
2
2
2
11θρθθ
θ
gzt
yt
xt
Ck
ztu
ytu
xtut
pzyx &+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
+∂∂
2
2
2
2
2
2
ρθ
Rectangular coordinates
Spherical coordinates
Cylindrical coordinates
Energy Equations by ConductionConstant Properties
gtr
trr
trrrC
kt
p
&+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=′∂
∂2
2
2222
2 sin1sin
sin11
φθθθ
θθρθ
Supercritical Fluid Process Lab
gztt
rrtr
rrCkt
p
&+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=′∂
∂2
2
2
2
2
11θρθ
gzt
yt
xt
Ckt
p
&+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
2
2
2
2
2
2
ρθ
Spherical coordinates
Cylindrical coordinates
Rectangular coordinates
One-Dimensional Conduction in a Planar Medium with Constant Properties and No Generation
gzt
yt
xt
Ckt
p
&+⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
2
2
2
2
2
2
ρθ
2
2
xt
Ckt
p ∂∂
=∂∂
ρθ
gzt
yt
xt
Ck
ztu
ytu
xtut
pzyx &+⎟⎟
⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
+∂∂
+∂∂
+∂∂
2
2
2
2
2
2
ρθ
Boundary and Initial Conditions
Constant Surface Temperature
Constant Heat Flux
Applied Flux
Insulated Surface Convection
2
2
xt
Ckt
p ∂∂
=∂∂
ρθ
For transient conduction, heat equation is first order in time, requiring specification of an initial temperature distribution:
Since heat equation is second order in space, two boundary conditions must be specified. Some common cases:
st
),( θxt ),( θxt),( θxt
),( θxt
),0( θt
Boundary Conditions at the surface (x=0)
1. Constant Surface Temperature
stt =),0( θst
),( θxt
2. Constant Surface Heat Flux(a) Finite heat flux
Boundary Conditions at the surface (x=0)
Applied Flux
),( θxt
sx
qxtk ′′=∂∂
−=0
2. Constant Surface Heat Flux(b) Adiabatic or insulated surface
Boundary Conditions at the surface (x=0)
00
=∂∂
=xxt
),( θxt
3. Convection Surface Condition
Boundary Conditions at the surface (x=0)
),( θxt
),0( θt
ht ,∞
( )[ ]θ,00
tthxtk
x
−=∂∂
− ∞=
Supercritical Fluid Process Lab
Ex. 19-1Problems of heating or cooling spheres are best solved using the differential energy balance written in spherical coordinates. Derive the equation for the case where there is no variation of temperature with angular position.
rt
r+dr
t+dt
Control volume
Rate of heat flow into control volume
Rate of heat flow out of control volume
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=
⎟⎠⎞
⎜⎝⎛
∂∂
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
+∂∂
+∂∂
+∂∂
−=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
++−=
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛∂∂
+∂∂
+−=
∂∂
−=
drrtrdr
rtrk
rtrk
drrtdrdr
rtrdrdr
rtr
rtdr
rtrdr
rtrk
drrt
rtdrrdrrk
rtd
rtdrrk
rtrk
24
)4(
224
)2(4
])(4[
)4(
2
22
2
2
22
2
2
2
2222
2
222
2
2
π
π
π
π
π
π
Net heat flow000
spherical coordinates
Ex. 19-1
rt
r+dr
t+dt
Control volume
The rate of accumulation energy in the control volume
Net heat flow = rate of accumulation
⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
ρ=
θ∂∂
θ∂∂
ρπ=⎟⎟⎠
⎞⎜⎜⎝
⎛
∂∂
+∂∂
π
θ∂∂
ρπ
rt
rrt
Ckt
tCdrrdrrtrdr
rtrk
tCdrr
p
p
p
2
)4(24
)4(
2
2
22
22
2
spherical coordinates
Supercritical Fluid Process Lab
The equation for unsteady conduction in a sphere
Modeling in Heat Transfer
Potato
100oC
PhysicalPhenomena
H2O
100oC
Actual Ideal(simplification)
Modeling
PhysicalPrinciples & laws
2
2
xtt
∂∂
=∂∂ αθ
ODE, PDE
Mathematicalformulation
Solution
Supercritical Fluid Process Lab
Physical PhenomenaPhysical Phenomena
Solution of the Fundamental Equations
ODE, PDEODE, PDE
Numerical SolutionNumerical Solution
Modeling
Analytical SolutionAnalytical Solution
(1) Separation of variables(2) Similarity variable(3) Laplace, Fourier transform
- FDM, FEM
Supercritical Fluid Process Lab
Similarity Variable
Conduction into a plate of infinite thickness
Analytical Solution
αθ
αθ
4
t t,at x :2 B.C.t t0,at x :1 B.C.t t0,θat : I.C.
0
s
0
2
2
xn
xtt
=
=∞=====
∂∂
=∂∂
Let
(19-1)
(19-4)
Supercritical Fluid Process Lab
t0
tS
x
θ
∞0
tS
x
t0
Similarity VariableAnalytical Solution
2
2
2
2
2
2
23
41
41
41
41
2
41;
221
421
4
nt
xn
nt
xn
nt
nt
xt
nt
xn
nt
xt
ntnn
ntt
xnnxxn
∂∂
⋅=∂∂⋅
∂∂
⋅=∂∂⋅⎟⎠
⎞⎜⎝
⎛∂∂
⋅∂∂
=∂∂
∂∂
⋅=∂∂⋅
∂∂
=∂∂
∂∂
⋅−=∂∂
⋅∂∂
=∂∂
=∂∂
−=⋅−=⎟⎟⎠
⎞⎜⎜⎝
⎛−=
∂∂ −
αθαθαθ
αθ
θθθ
αθθθαθθ
αθ
Supercritical Fluid Process Lab
Similarity VariableAnalytical Solution
02
41
2
41
2
2
2
2
2
2
2
2
2
=+
∂∂
⋅⋅=∂∂
⋅−
∂∂
⋅=∂∂
∂∂
⋅−=∂∂
dndtn
dntd
nt
ntn
nt
xt
ntnt
αθα
θ
αθ
θθ (19-5)
(19-6)
(19-7)
Supercritical Fluid Process Lab
Similarity VariableAnalytical Solution
201
1
2
2
02
: variablenew
CdneCt
eCp
npdndp
dndtp
n n
n
+=
=
=+
=
∫ −
−
(19-8)
(19-9)
(19-10)
(19-7)
Two boundary conditions are needed. Supercritical Fluid Process Lab
022
2
=+dndtn
dntd
Similarity VariableAnalytical Solution
αθπ
π
π
θ
αθ
42
)(22
,n 0, , x:2 B.C.C ,t t0,n 0, x:1 B.C.
40
0
01
1
010
0
2s
2
2
xerfdnetttt
ttC
tC
tdneCt
ttt
xn
s
s
s
s
sn
s
==−−
−=
+=
+=
=∞→=∞→====
∫
∫
−
∞ −(19-11)
(19-12)
(19-13)
Gauss error function or probability functionSupercritical Fluid Process Lab
The error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. One of its main purposes is for graphing a normal distribution. It is defined as:
By expanding the right-hand side in a Taylor series and integrating, one can express it in the form
for every real number x, or any complex number z = x. The error function is evidently odd.
Plot of the error function
Analytical Solution
αθπαθ
42 4
00
2 xerfdnetttt x
n
s
s ==−−
∫ −
Gauss error function (probability function)
erf(z)
z
zπ2erf(z) z, smallfor =
Supercritical Fluid Process Lab
Analytical Solution
αθαθπ
αθπαθ
αθπαθ
018.0~42
01.0
42
401.0
42 4
00
2
=
≈=
==−−
∫ −
T
TT
xn
s
s
x
xxerf
xerfdnetttt
Thermal penetration depth, xT
x @ 1% of driving force
Supercritical Fluid Process Lab
zπ2erf(z) z, smallfor =
Analytical Solution
)(
)(1
@4
24
0
0
0
0
ttCk
ttk
xtk
Aq
smallisxxxerftttt
sp
s
x
s
s
s
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛=
−⎟⎠
⎞⎜⎝
⎛ −−=
∂∂
−=
≈=−−
=
πθ
ρ
παθ
αθπαθ
Thermal penetration depth, xT
x @ 1% of driving force
pCkρ : Thermal effusivity
Supercritical Fluid Process Lab
Analytical Solution
Thermal penetration depth, xT
과도기 상태에서는 가 중요하다.pCkρ
겨울에 나무 손잡이와 쇠 손잡이를 잡을 때더 빨리 차가움을 느끼는 것
뜨거운 공기와 뜨거운 물에 손을 담글 때화상을 입는 정도
100oC 스팀 사우나에서는 오랫동안 견디면서45oC 목욕탕 물 속에서는 잠깐도 견디기 어려운 것
Supercritical Fluid Process Lab
Separation of Variables
Unsteady state heat conduction in a rubber sheet
Analytical Solution
hft
xtt
/0028.0Ck
F290 tand 0 x@ ?θ
0dt/dx 0,at x:2 B.C.
F292t t,1/4"xat x:1 B.C.
F70t t0,θat : I.C.
2
p
s0
0
2
2
=ρ
=α
===
==
====
===
∂∂
α=θ∂∂
o
o
o
(19-1)
x
x0=1/4”
292oF
Supercritical Fluid Process Lab
292oF
Curing at 292oFfor 50min
Analytical Solution
2
2
20
02
2
002
20
0
20
0
2000
2
2
1)(1)(
1)(
)(
;;
nY
xtt
xn
nY
xtt
xt
nY
xtt
xn
nY
Yt
xt
Yx
ttYYtt
xxxn
ttttY
xtt
ss
s
s
s
s
∂∂
−=∂∂
∂∂
−=∂∂
∂∂
−=∂∂
∂∂
∂∂
=∂∂
∂∂
−=∂∂
∂∂
∂∂
=∂∂
==−−
=
∂∂
=∂∂
τα
θτ
τθ
αθτ
αθ (19-1)
Dimensionless variables
(1)
(1-1)
(1-2)
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
0,1,:2..
0,00,0:1..
1,0,0:..
1)()(
0
0
2
2
2
2
20
020
0
2
2
==→==
=∂∂
=→=∂∂
=
==→==∂∂
=∂∂
∂∂
−=∂∂
−
∂∂
=∂∂
YnttxxCBnYn
xtxCB
YttCInYY
nY
xttY
xtt
xtt
s
ss
τθτ
ατ
α
αθ (19-1)
(1-1) & (1-2) (19-1)
(1)
(2)
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
Supercritical Fluid Process Lab
0,0@:B.C.2
0,1@:B.C.11,0@:I.C.
2
2
=∂∂
=
====
∂∂
=∂∂
nYn
YnY
nYY
τ
τ
0dt/dx 0, x@:2 B.C.t t,x x@:1 B.C.
tt0,θ @: I.C.
s0
0
2
2
====
==
∂∂
=∂∂
xtt α
θ
2000
;;xx
xnttttY
s
s αθτ ==−−
=
Dimensionless variables
Separation of Variables
Analytical Solution
2
2
2
2
2
2
2
2
11
)()(
dnNd
NddT
T
dnNdT
ddTN
dnNdT
nY
ddTNY
nNTY
=
=
=∂∂
=∂∂
⋅=
τ
τ
ττ
τSolution: (3)
(4)
(5)Supercritical Fluid Process Lab
Separation of variables:
Separation of VariablesT is a function of only τN is a function of only n
2
2
nYY
∂∂
=∂∂τ
Left is a function of only τRight is a function of only n
Analytical Solution
( )anCanCeCNTY
anCanCNNadn
Nd
eCTTaddT
adn
NdNd
dTT
a
a
cossin
cossin0
0
11
321
322
2
2
12
22
2
2
2
+=⋅=
+=→=+
=→=+
−==
−
−
τ
τ
τ
τ(5)
(10)
(6) (8)
(7) (9)
4 unknowns
Since time and distance are independent of each other. The left side of this equation is independent of distance and the right side is independent of time. Hence each side must be constant.
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
( )
ππππ
τθ
τ
τ
ττ
212,2/5,2/3,2/
cos0:2..
)A (wherecos
0
0sincos:1..
0,1,:2..
0,00,0:1..
1,0,0:..
2
2
22
31
2
21321
0
0
−=
=
==
=
==−=∂∂
==→==
=∂∂
=→=∂∂
=
==→==
−
−
−−
ia
aAeCB
CCanAeY
C
eCaCanaCanaCeCnYCB
YnttxxCBnYn
xtxCB
YttCI
a
a
aa
s
L
(11)
Supercritical Fluid Process Lab
This equation contains four constants C1, C2, C3, and a: however C1 can be combined with C2 and C3, so that only three IC & BCs are needed to complete the solution.
n=0, sinan=0, cosan=1
(symmetrical system)
Separation of Variables
Analytical Solution
L
L
+⎟⎠⎞
⎜⎝⎛ −
+
+⎟⎠⎞
⎜⎝⎛+
⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛=
−−
−
−−
nieA
neA
neAneAY
ii 2
)12(cos
25cos
23cos
2cos
2
2
22
)2/)12[(
)2/5(3
)2/3(2
)2/(1
π
π
ππ
τπ
τπ
τπτπ
GeneralSolution
(12)
The substitution of one of these values for a in Eq. (11) would meet the requirement of the second boundary condition, but it is not possible to represent an arbitrary temperature distribution in the slab by a cosine curve.
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
LL +⎟⎠⎞
⎜⎝⎛ −
++⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= niAnAnA i 2
)12(cos2
3cos2
cos1 21πππ
(13)
Both side of this equation are multiplied byAnd integrated over the range of 0 to 1.
I.C.
ni π⎟⎠⎞
⎜⎝⎛ −
2)12(cos
i
ini
idnni )1(1
122
212sin1
122
212cos
1
0
1
0−
−−=⎥
⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
−=⎟
⎠⎞
⎜⎝⎛ −
∫ ππ
ππ
( )
( ) 02
sin)22(1sin
2sin
)22(1sin
212cos
2cos
1
1
01
1
0 1
=⎥⎦
⎤⎢⎣
⎡+
−−
=
⎥⎦
⎤⎢⎣
⎡+
−−
=⎟⎠⎞
⎜⎝⎛ −
⎟⎠⎞
⎜⎝⎛
∫
ππ
ππ
ππ
ππππ
ii
iiA
ini
iniAdnninA
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
LL +⎟⎠⎞
⎜⎝⎛ −
++⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= niAnAnA i 2
)12(cos2
3cos2
cos1 21πππ
(13)
Both side of this equation are multiplied byAnd integrated over the range of 0 to 1.
I.C.
ni π⎟⎠⎞
⎜⎝⎛ −
2)12(cos
2000
212
21
122
212)2sin(
41
212
21
122
212cos
1
0
1
0
21
ii
i
Aii
A
ninii
A
dnniA
=⎥⎦⎤
⎢⎣⎡ −−+
−−
=
⎥⎦⎤
⎢⎣⎡ −
+−
−=
⎟⎠⎞
⎜⎝⎛ −
∫
ππ
πππ
π
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
LL +⎟⎠⎞
⎜⎝⎛ −
++⎟⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛= niAnAnA i 2
)12(cos2
3cos2
cos1 21πππ
(13)I.C.
i
i)1(1
122
−−
−π
= 0 + 0 + 0 + … + 2iA
L,54,
34,4
)12()1(4
2)1(1
122
321 πππ
π
π
=−==
−−−
=
=−−
−
AAA
iA
Ai
i
i
ii
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
[ ] niei
Y
ne
neneY
i
i
i
i
i
⎟⎠⎞
⎜⎝⎛ −
−−−
=
−⎟⎠⎞
⎜⎝⎛+
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛=
−−∞=
=
−
−−
∑ 2)12(cos
2/)12()1(2
25cos
54
23cos
34
2cos4
2
2
22
]2/)12[(
1
)2/5(
)2/3()2/(
ππ
ππ
ππ
ππ
τπ
τπ
τπτπ
L
(15)
The general solution is
(14)
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
L−π
+π
−π
=
=
−−
=
τπ−τπ−τπ− 222 )2/5()2/3()2/(
54
344009.0
0@70292
290292
eee
n
Y
The specific example:
The solution for τ must be obtained by trial and error.As a first approximation, only the first term on the right hand side will be considered. This gives
min)7.18(313.0)01.2(121
41
0028.011
01.24
)009.0)((ln2
220
20
2
hrxxk
C p =⎟⎠⎞
⎜⎝⎛=τ
α=τ
ρ=θ
=π
⎟⎠⎞
⎜⎝⎛π
−=τ
Supercritical Fluid Process Lab
(18.7min)
Separation of Variables
Analytical Solution
L
L
)1040.1)(254.0()1050.3)(424.0()00694.0)(27.1(
54
344
01.2@
5420
1248.4497.4
−−
−−−
×+×−=
−π
+π
−π
=
=τ
eeeY
Check the relative magnitude of the terms in the series solution
The validity of the approximation which employed only the first term of the series is apparent.
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
Temperature distribution in rubber sheet
0 1 70 70 70 70
1 9.76x10-1 75 92 139 209 292
5 3.33x10-1 217 223 239 263 292
10 9.02x10-2 272 273 278 284.3 292
20 6.37x10-3 290.6 290.7 291 291.5 292
30 4.52x10-4 291.9 291.9 292 292 292
40 3.19x10-5 292 292 292 292 292
TimeElapsed(min)
τπ
π2)2/(4 −e
Temperature in sheet, oF
12
cos
0
=
⎟⎠⎞
⎜⎝⎛
=
πnn
924.02
cos
4/1
=
⎟⎠⎞
⎜⎝⎛
=
πnn
707.02
cos
2/1
=
⎟⎠⎞
⎜⎝⎛
=
πnn
383.02
cos
4/3
=
⎟⎠⎞
⎜⎝⎛
=
πnn
02
cos
1
=
⎟⎠⎞
⎜⎝⎛
=
πnn
Supercritical Fluid Process Lab
Separation of Variables
Analytical Solution
Temperature profiles in rubber sheet
300
200
100
00 1/4 1/2 3/4 1surfacecenter
10min
5min
1min
Distance, n(=x/x0)Supercritical Fluid Process Lab
Tem
pera
tur e
, oF
292oF
Separation of Variables
Analytical Solution Conduction in Cylinder
sttRrrtrat
ttat
ztt
rrtr
rrt
==
=∂∂
=
==
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+⎟⎠⎞
⎜⎝⎛
∂∂
∂∂
=′∂
∂
,
0,0
,0
11
0
2
2
2
2
2
θ
θα
θ
0 0 (L>>R)
I.C.:
B.C.1:
B.C.2:
ts
0=∂∂rt
Supercritical Fluid Process Lab
Analytical Solution Conduction in Cylinder
Supercritical Fluid Process Lab
0,0 =∂∂
=rtr
sttRr == ,0,0 tt ==θ
),( θ= rft
Conduction in CylinderAnalytical Solution
22
2
2
2
2
20
2
2
111
)()(
11
1
addT
TdrdN
rNdrNd
N
ddTN
drdN
rT
drNdT
rNTY
YrY
rrY
ttttY
rt
rrtt
s
s
−==+
=+
⋅=
∂∂
=∂∂
+∂∂
−−
=
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
=′∂
∂
θα
θα
θ
θα
αθ
Solution:
GE:
Supercritical Fluid Process Lab
Conduction in CylinderAnalytical Solution
0
)exp(0
111
222
22
2
21
2
22
2
2
2
=++
×
−=→=+
−==+
=+
NardrdNr
drNdr
Nr
aCTTaddT
addT
TdrdN
rNdrNd
N
ddTN
drdN
rT
drNdT
θααθ
θα
θα
Supercritical Fluid Process Lab
Conduction in CylinderAnalytical Solution
∫
∑
∞ −−
∞
=
+
=Γ
++Γ
⎟⎠⎞
⎜⎝⎛−
=
+=
=−++
0
1
0
2
222
22
)(
)1(!21)1(
)(
)()(
0)(
dtetz
kmm
xxJ
xBYxAJy
ykxdxdyx
dxydx
tz
m
kmm
k
kk
Bessel’s equation of order k
Solution:
Supercritical Fluid Process Lab
Conduction in CylinderAnalytical Solution
constant sEuler' :5772.0ln131
211lim
)!(21)1(
2)(21ln2)(
)1(!21)1(
)(
)()(
00
12
2
00
0
2
0
00
222
22
LL =⎟⎠⎞
⎜⎝⎛ −++++=
⎟⎠⎞
⎜⎝⎛−
−⎥⎦
⎤⎢⎣
⎡+⎟
⎠⎞
⎜⎝⎛=
+Γ
⎟⎠⎞
⎜⎝⎛−
=
+=
==→=++
∞→
∞
=
∞
=
∑
∑
mm
m
ararJararY
mm
ararJ
arBYarAJN
arxandkNardrdNr
drNdr
m
m
mm
m
mm
γ
πγ
π
Bessel function of the 1st kind of order zero
Bessel function of the 2nd kind of order zero
Supercritical Fluid Process Lab
( )
0,0:2..
0,:1..1,0:..
)()()()( 001
2
=∂∂
=
====
+=⋅= −
rYrCB
YRrCBYCI
arBYarAJecrNTY a
θ
θ θα
Conduction in CylinderAnalytical Solution
Supercritical Fluid Process Lab
differentiation
)()()(
)()()(
1
1
xxYxkYxYdydx
xxJxkJxJdydx
kkk
kkk
αααα
αααα
+
+
−=
−=
)()()()(
)()()()(
1010
1010
araYarYdrdararYarY
drdr
araJarJdrdararJarJ
drdr
−=→−=
−=→−=
Conduction in CylinderAnalytical Solution
Supercritical Fluid Process Lab
,
0,0@ =∂∂
=rYr
))()()(exp( 112
1 arBaYarAaJacrY
+−−=∂∂ θα
))0()0()(exp( 112
10 BaYAaJacrY
r +−−=∂∂
= θα
001 =→=∴ BBc
Conduction in CylinderAnalytical Solution
0
Supercritical Fluid Process Lab( ))()()()( 001
2
arBYarAJecrNTY a +=⋅= − θαθ
0,@ == YRr
,
)()exp(0 02
1 aRJaAc θα−=
0)( ofroot th theis a 0 =RaJ iιι
∑∞
=
−=0
02 )()exp(
iiii raJacY θα
Conduction in CylinderAnalytical Solution
Supercritical Fluid Process Lab( ))()()()( 001
2
arBYarAJecrNTY a +=⋅= − θαθ
1,0@ == Yθ
)(1 00
raJc ii
i∑∞
=
=
integralthen),(0 raxJa jj×
)()( 1,kwhen
function :)()1()(
)()(
11
1
xJxJ
xJxJ
xJxdxxJx
kk
k
kk
kk
αα
αα
ααα
−==
Γ−=
=
−
−
−∫
Conduction in CylinderAnalytical Solution
Left: )()( 10 1 RaRJdxxJx j
R
kk =∫ − αα
)()( 10 0 RaRJdrraxJa j
R
jj =∫
Supercritical Fluid Process Lab∑∞
=
−=0
02 )()exp(
iiii raJacY θα
right: ji ≠
∫ =−−
=R
jijijiji
ijii RaJRaJaRaJRaJa
aaRCdxxaJxaxJC
0 10102200 0)]()()()([)()(
ji =
∫ −−=R
iiiii
ii RaJRaJRaJRaCdxxaxJC0 11
20
20 )]()()([
2)(
)]()([2
)( 111 RaJRaJRaCRaRJ iiii
j −−=
)(2
)(2
11 RaJRaRaJRaC
iiiii =
−=∴
−
Conduction in CylinderAnalytical Solution
0
0
0
Supercritical Fluid Process Lab
)()(
)/exp(2
)()exp()(
2
01 1
2
02
1 10
raJRaJa
CkaR
raJaRaJRatt
ttY
ii ii
pi
iii iis
s
∑
∑
∞
=
∞
=
−=
−=−−
=
ρθ
θα
Conduction in CylinderAnalytical Solution
Final Solution
Supercritical Fluid Process Lab
)()(
)/exp(20
1 1
2
0
raJRaJa
CkaRtt
tti
i ii
pi
s
s ∑∞
=
−=
−− ρθ
Conduction in CylinderAnalytical Solution
Final Solution
Supercritical Fluid Process Lab
0)( ofroot th theis a 0 =RaJ iιι
J0 and J1: Bessel function of the 1st kind (zero and first order)
(19-15)
Supercritical Fluid Process Lab
The Fourier equation has been solved for many geometries and sets of conditions. A set of general solutions has been plotted for use in obtaining reasonably good solutions with less work. These are the Gurney-Lurie or Heisler Charts. Figures 19-3, 19-4, and 19-5 are simple versions of such a chart.
Use of the charts is restricted to cases where: 1. there is no internal heat source (generation) 2. the thermal diffusivity of the object is constant 3. the problem can be approximated as one-dimensional 4. the initial temperature of the object is uniform 5. the system is forced by a step change in temperature
of the surroundings (or of the surface, when 1/h=0)
Supercritical Fluid Process Lab
Graphical Solution: Gurney-Lurie Chart
Graphical Solution: Gurney-Lurie ChartThe chart below is only an example of how a Gurney-Lurie chart might look and is not based on any actual data. The chart shows how four different dimensionless groups depend on each other.
Bihxkm
xxn
xCk
xX
ttttY
ps
s 1,,,00
20
200
=====−−
=ρ
θαθ
0ttttY
s
s
−−
=
20
20 xC
kx
Xpρθαθ
==
Supercritical Fluid Process Lab
Gurney-Lurie Chart
To use the charts, some variables need to be defined. Various versions of the charts are slightly different in how the variables are defined:
resistance relative :1
position relative :
time)(relativenumber Fourier :
0
0
20
20
max0
Bihxkm
xxn
FoxC
kx
X
tt
ttttY
p
s
s
==
=
===
ΔΔ
=−−
=
ρθαθ
change tempshedunaccompli fractional the:
Supercritical Fluid Process Lab
Fourier number (Fo)
Supercritical Fluid Process Labstored
conducted
LFo &
&== 2
αθ
Gurney-Lurie Chart for Plate
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Plate
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Plate
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Cylinder
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Cylinder
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Cylinder
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Sphere
Supercritical Fluid Process Lab
Gurney-LurieChart for Sphere
Supercritical Fluid Process Lab
Supercritical Fluid Process Lab
Gurney-Lurie Chart for Sphere
Newman’s Rule :the technique for solving many problems of systems with finite dimensions in all direction
″
21
″
21
″
21
Finite dimension in all direction
If a brick-shape object is heated or cooled, the general solution describing temperature as a function of time and three distance variables, x, y, and z.
Y=YxYyYz
Yx=f1(x, θ) : x-dir. unsteady state conductionYy=f2(y, θ) : y-dir. unsteady state conductionYz=f3(z, θ) : z-dir. unsteady state conduction
70oF
292oF
290oF
208.0
009.070292
29029270292
292
0
=
=−−
=
−−
=−−
=
x
s
s
Y
tttttY
Supercritical Fluid Process Lab
t = 290oF at center θ = ?
( )
min)7(113.044.673.0
48/10028.044.673.0
3-19 Fig. From.0,0
208.00090.0
220
00
3
h
x
hxkm
xxn
YY
YY
YYY
x
x
zyx
==θ
⎟⎟⎠
⎞⎜⎜⎝
⎛ θ=θ==
αθ
====
=→=
=
==
Newman’s Rule: Y=YxYyYz
Supercritical Fluid Process Lab
208.0
009.070292
29029270292
292
0
=
=−−
=
−−
=−−
=
x
s
s
Y
tttttY
h→∞
Three dimensional conductionThree dimensional conduction
2
2
2
2
2
2
2
2
2
2
2
2
zY
yY
xYY
zt
yt
xt
Ckt
p ∂∂
+∂∂
+∂∂
=∂∂
→⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
+∂∂
=∂∂
τρθ
2
2
2
2
2
2
2
2
2
2
2
2
1111zY
YyY
YxY
Y
zYYY
yY
YYxYYYYYY
z
z
y
y
x
x
zyx
yzx
xzyzyx
∂∂
+∂
∂+
∂∂
=∂∂
∂∂
+∂
∂+
∂∂
=∂∂
ττ
τ
τττττ
)()()( zYyYxYY zyx=τ
The basis for validity of Newman’s Rule
Solution:
(1)
(2)
Differentiating of Eq. (2)
(3)
Rearrangement yields
Supercritical Fluid Process Lab
)(1
1
1
1
23
22
21
232
2
222
2
212
2
aaa
azY
Y
ayY
Y
axY
Y
z
z
y
y
x
x
++−=∂∂
−=∂∂
−=∂
∂
−=∂∂
ττ
τ
Newman’s RuleEach part of Eq(3) is a function of one of the four independent variables and hence, it is reasoned, each of the four terms is a constant.
Supercritical Fluid Process Lab
( )
( ) ( )( )( )zaCzaCyaCyaC
xaCxaCeCY
zaCzaCY
yaCyaCYxaCxaCY
eC
aaa
z
y
x
aaa
37362524
13121
3736
2524
1312
1
cossincossincossin
cossin
cossincossin
23
22
21
23
22
21
+++=
+=
+=+=
=
++−
++−
τ
ττ
Newman’s Rule
These individual solutions are combined
These four expressions can be written as ordinary differential equationsfor which the solutions are
(4)
(5)
(6)
(7)
(8)
These general solution for three-dimensional conduction is consist of the products of each of the one-dimensional solutions.
Supercritical Fluid Process Lab
Homework #2
PROBLEMSPROBLEMS
1919--111919--22
Due date: October 19Due date: October 19
Supercritical Fluid Process Lab