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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-1
Problem Formulation
in Conduction Heat
Transfer
Formulation of general laws
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-2
Overview
• Determined andundetermined problems
• Lumped formulation
•
Integral formulation• Differential formulation
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-3
Determined and
undetermined
problems
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-4
Example 1: Steady one-dimensional
flow of an inviscid fluid in an adiabatic
diffuser
• Enthalpy is conserved.
• Flow is isentropic.
• Apply massconservation.
• Apply the linearmomentum equation(Newton’s 2nd Law ofMotion).
2
A
u
V
p
ρ
1
A
u
V
p
ρ
CV
V δ
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-5
Example 1 (continued)
2
A
u
V
p
ρ
1
A
u
V
p
ρ
CV
V δ
( )
+=
⋅
+−= ∫∫
ρ
Ω ρ ρ
pu
dx
d
d nV p
E CV
CVS
0
0
Energy conservation
( ) ( )
( )
( )
( ) ( )
( )[ ]V AV pAdx
d
d nV V n p
d nV V d n
AV dx
d
AV AV
d nV
CV
CVS
CV
CVS
CV
CVS
CV
CVS
x x x
cv
CVS
ρ
Ω ρ Ω
Ω ρ Ω
ρ
ρ ρ
Ω ρ
∆
+=
⋅−−=
⋅−⋅ℑ=
=
+−=
⋅−=
∫∫ ∫∫
∫∫ ∫∫
∫∫ +
0
0
0
0
0
0
Mass conservation
Linear momentum
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-6
Example 1 (continued)
2
A
u
V
p
ρ
1
A
u
V
p
ρ
CV
V δ
+=
ρ
10 pd du
Incompressible fluid – Density is constant, and du = 0.The problem is completely determined and theequations can be solved.
Compressible fluid – The problem is undeterminedand cannot be solved. An equation of state is theadditional condition that is needed. Pressure is afunction of density, p =p(ρρρρ,u), e.g., p = ρρρρRT, and theproblem can be solved.
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-7
Example 2. First law of
thermodynamics
x
z
y
( )t E
For a lumped formulation, the FirstLaw of Thermodynamics is written,
Γ &&& +−= W Qdt
dE
( )t E
t
( )it E
Problem is undetermined unlessrelations between heat transfer,
work, and energy generation withtime or another variable are known.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-8
Lumped formulation
of general laws
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-9
Lumped system configuration
with control volume
Control volume
System
Initial
State,B1,B’
Final
State,
B2, B’’.
iii ,m ,b ϑ ∆∆ Note: “b” is the specific value of the quantityof interest (“property/mass”).
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-10
Control volume, σσσσ
System
InitialState,B1,B’
FinalState,B2, B’’.
iii ,m ,b ϑ ∆∆
B Bmb B B
B B B
ii
system
′′=+′=
−=
2
1
12
∆
∆
∑
∑
∑
−=
−=
∆−∆=∆
∆=∆−∆
σ
σσ
σ
σ
i
ii
i
ii
system
i
iisystem
systemii
wbdt
dBdt
dmb
dt
dB
dt
dB
mbBB
BmbB
Combine system and controlvolume relations to get the
transformation between the two.
This expression is the general transformation
from system to control volume formulation.
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-11
Conservation of mass
∑−==
===
=
i
iwdt
dm
dt
dm
m
Bb ,m B
dt
dm
σ 0
1
0Control volume, σσσσ
System
InitialState,B1,B’
FinalState,B2, B’’.
iii ,m ,b ϑ ∆∆
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-12
Conservation of energy – First
law of thermodynamics
nuclchem
by
by
cycle cycle
U U mgzmV
U E
PqW Q E
W Q E
W Q
++++=
−≡−=
−=
=∫ ∫
2
2
&&&
δ δ ∆
δ δ
S1
S2
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-13
Conservation of energy
esd
i
ii
by
PPPP
Pqwedt
dE
eb , E B
PqW Q E
++=
−=−
==
−≡−=
∑σ
&&&
Control volume, σσσσ
System
σ iP
σ eP σ d P
σ sP
( )
i ,eee
ss
id d
PPP
PP
PPP
ϑ ∆σ
σ
σ
+=
=
+=Include in internalEnergy flow acrossThe CVS.
Becomes usual expressionFor “flow work” underAssumption of normal forcesAt CVS, i.e., no shear at CVS.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-14
Conservation of energy
Include in the internalenergy flow for the CV
i
i
ii
qqq
Pqwedt
dE
ϑ ∆σ
σ
+=
−=−∑
Control volume, σσσσ
System
σ iP
σ eP σ d P
σ sP
Combining terms for q and P,
( )σ σ
σ
ρ esd i
ii
PPPqw p
edt
dE ++−+
+= ∑
0ih=
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-17
Integral formulation
of general laws
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-18
System and fixed control
volume at t
x
z
y
System at time t
Fixed controlVolume, σσσσ
Flow field,
V(x,t)
Properties and the velocity field are specified in time and space.
The system can move and deform in time and space with V(x,t)
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-19
System and fixed control
volume at t + ∆∆∆∆t
x
System at time t+∆∆∆∆T
V
n
z
y
Fixed control
Volume, σσσσ
x
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-20
V
n
System at time t+∆∆∆∆T
2
3
z
y
Fixed control
Volume, σσσσ
x
System and fixed control
volume at t + ∆∆∆∆t
1
Determine change of B in
system regarding regions
1 to 3 via finite calculus.
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-21
Determine change of B insystem regarding regions
1 to 3 via finite calculus.
Then take the limit as ∆∆∆∆t →→→→ 0.
( ) ( )
( ) ( ) ( ) ( )
t
t B
t
t t B
t
t Bt t B
dt
dB
t
B B B B
dt
dB
system
t t t
system
∆∆
∆
∆
∆
∆
∆
2311
2131
−+
+−+
≈
+−+≈
+
V
n
2
3
z
y
1
x
( ) ( ) ( )ϑ
∂
ρ∂=ϑρ=
=
∆
−∆+∫ ∫ →∆
dt
bdb
dt
d
dt
dB
t
tBttBlim
CVCVCV
11
0t
( ) ( ) ∫∫ ∫∫ ∫∫ σσσ
→∆σ⋅ρ=σ⋅ρ−σ⋅ρ=
∆
−∆
∆+ dnVbdnVbdnVbttB
tttBlim
in,out,
23
0t
Rate of change of B in the CV
Rate of change of B in the CV due to material flow at the boundary, σσσσ
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-22
( ) ( ) ( ) ( )
( ) ( )CVsystem
2311
tsystem
dnVbdtb
dtdB
t
tB
t
ttB
t
tBttBlim
dt
dB
σ⋅ρ+ϑ∂ρ∂=
∆−
∆
∆++
∆
−∆+=
∫∫∫ ∫∫
→∆
V
n
2
3
z
y
1
x
Combine terms to obtain the
Relation between the rate ofchange of B in the system andcontrol volume.
This development is heuristic, or“physically based”. The resultis the same as for the Reynolds
Transport Theorem.
This result is used to produce the general laws as in the case of thelumped formulation (mass, energy, entropy).
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-23
Conservation of mass
( )( )
CVsystem
CVsystem
dnVdt
0dtdm
1b,mB
dnVbdt
b
dt
dB
σ⋅ρ+ϑ∂ρ∂==
==
σ⋅ρ+ϑ
∂
ρ∂=
∫∫∫ ∫∫
∫∫∫ ∫∫
System mass stays the same following fluid motion in
the assumed flow field.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-24
Conservation of energy – First
law of thermodynamics
( )( )
( )CVsystem
CVsystem
dnVedt
e
dt
de
eb,EB
dnVbdt
b
dt
dB
σ⋅ρ+ϑ
∂
ρ∂=
==
σ⋅ρ+ϑ
∂
ρ∂=
∫∫∫ ∫∫
∫∫∫ ∫∫
Begin by consider the system statement of the first law of thermodynamics.
Note that this “general law” remains valid for stationary systems and systems
that follow fluid motion.
systemsystemsystem dt
W
dt
Q
dt
dE
δ−
δ=
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-25
Conservation of energy – First
law of thermodynamics
( )
∫∫
∫∫∫ ∫∫
σ
σ⋅−=
δ
δ−
δ=
σ⋅ρ+ϑ
∂ρ∂
dnqdt
Q
dtW
dtQdnVed
te
system
systemsystemCV
Here the heat flux, q, is introduced
at the control surface. In the limit
of ∆∆∆∆t →→→→ 0, the integration approachesthat for the system derivative following
fluid motion.
Next, we consider the work term, where displacement, shaft and electricalwork are formally present. In addition, work done at the surface against
viscous forces (shear) is included. Thus the work term is re-written as,
τ
δ+
δ+
δ+
δ=
δ
dtW
dtW
dtW
dtW
dtW
esdsystem
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-26
Conservation of energy – First
law of thermodynamics
τσ
σσ
τ
τ
δ+ϑ′′′−+
δ+σ⋅ρ
ρ−=
δ
ϑ′′′−=
σ⋅ρ
ρ−=σ⋅=
−−−−=
δ+
δ+
δ+
δ=
δ
∫∫∫ ∫∫
∫∫∫
∫∫ ∫∫
dt
Wdu
dt
WdnV
p
dt
W
duP
dnVp
dnVpP
PPPP
dt
W
dt
W
dt
W
dt
W
dt
W
CVssystem
CV
e
d
esd
esdsystem
(Displacement work)
(Electrical work drawn to system; generation term)
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-27
Conservation of energy – First
law of thermodynamics
( )( )
( ) ( ) ϑ′′′+−−σ⋅−=σ⋅ρ+ϑ∂
ρ∂
ϑ′′′+−−σ⋅ρ
ρ−σ⋅−=σ⋅ρ+ϑ
∂
ρ∂
δ+ϑ′′′−+
δ+σ⋅ρ
ρ
−=
δ
∫∫∫ ∫∫ ∫∫ ∫∫∫
∫∫∫ ∫∫ ∫∫ ∫∫ ∫∫∫
∫∫∫ ∫∫
τ
σσ
τ
σσσ
τσ
duPPdnqdnVhdt
e
duPPdnVp
dnqdnVedt
e
dtWdu
dtWdnVp
dtW
CV
s
0
CV
CV
s
CV
CVssystem
Combine all terms and substitute into the first law statement to obtain
the integral energy balance.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-28
Second law of thermodynamics
T
qS
T
QS
0T
Q
21
2121
cycle
≥
δ≥∆
≤δ
−
−−
∫
&
( ) ( )
( )( ) ∫∫∫ ∫∫ ∫∫ ∫∫∫
∫∫ ∫∫ ∫∫∫
∫∫
ϑ′′′+σ⋅
−=σ⋅ρ+ϑ
∂
ρ∂
σ⋅
−≥σ⋅ρ+ϑ
∂ρ∂=
σ⋅
−=
σσ
σσ
σ
CVCV
CVsystem
dsdnT
qdnVsd
t
s
dnT
qdnVsdt
s
dt
dS
dnT
q
T
q
Begin with the Clausius inequality andapply it to a process in the usual way.
the heat flux is evaluated at the surfaceof the C.V.
We then apply the system derivative anduse the result to relate system changes
to that in the C.V.
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-29
Differential
formulation of
general laws
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-30
Methods
• Use integral formulation with thedivergence theorem – Reduces integral expression to a P.D.E.
• Use differential volume elementswith mean value theorems of the
calculus – Produces a P.D.E. from general principles.
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-31
Conservation of mass
( )
( ) Vdt
d0,Vt0
dVt
0
dnVdt
0
CV
CV
⋅∇ρ+ρ
=ρ⋅∇+∂
ρ∂=
ϑ
ρ⋅∇+
∂
ρ∂=
σ⋅ρ+ϑ∂ρ∂=
∫∫∫
∫∫∫ ∫∫
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-32
Conservation of energy – First law
of thermodynamics
( ) ( )
( )[ ]
( ) uVpqdt
de
duVpq
duPPdnVpdnqddt
de
duPPdnqdnVhdt
e
CV
CV
s
CV
CV
s
0
CV
′′′=⋅∇+⋅∇+ρ
ϑ′′′−⋅∇+⋅∇−=
ϑ′′′+−−σ⋅−σ⋅−=ϑρ
ϑ′′′+−−σ⋅−=σ⋅ρ+ϑ∂
ρ∂
∫∫∫
∫∫∫ ∫∫ ∫∫ ∫∫∫
∫∫∫ ∫∫ ∫∫ ∫∫∫
τ
σσ
τ
σσ
Here we specialize the
energy balance to ideal fluids with no
shaft work.
eVt
e
dt
de∇⋅+
∂
∂=
(rearranging and combining)
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-33
Conservation of energy – First
law of thermodynamics
( )
upVqdt
de
.Const
uVpqdt
de
′′′=∇⋅+⋅∇+ρ
=ρ
′′′=⋅∇+⋅∇+ρ
From this form for constantdensity, we can get two
important results: themechanical energy equation
and the thermal energy
equation.
The general form of theconservation of energy
equation, or first law ofthermodynamics, for adifferential control volume.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-34
Conservation of energy – The
mechanical energy equation
• Absence of thermal, chemical, and nucleareffects
• Constant density.
0pVgzV2
1
dt
d 2 =∇⋅+
+ρ
This equation is reducible to the Bernoulli equationfor steady flows along a streamline
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-35
Conservation of energy – The
thermal energy equation
•
Subtract the mechanical energy equation from thefull energy equation.
• Result is the thermal energy equation.
uqdt
dTc
uq
dt
du
0pVgzV2
1
dt
d
upVqdt
de
2
′′′=⋅∇+ρ
′′′=⋅∇+ρ
=∇⋅+
+ρ
′′′=∇⋅+⋅∇+ρ
Thermal energy equation for
a differential C.V.
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-36
Second law of thermodynamics
( )( )
( )
( )
sT
q
dt
ds
dsdnT
q
dnVsdt
s
dnT
qdnVsd
t
s
dt
dS
dnT
q
T
q
CVCV
CVsystem
′′′=
⋅∇+ρ
ϑ′′′+σ⋅
−=σ⋅ρ+ϑ∂
ρ∂
σ⋅
−≥σ⋅ρ+ϑ
∂
ρ∂=
σ⋅
−=
∫∫∫ ∫∫ ∫∫ ∫∫∫
∫∫ ∫∫ ∫∫∫
∫∫
σσ
σσ
σ
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© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-37
Second law of thermodynamics
uqdt
du,
dt
du
dt
dsT
.Const
1pdduTds
sTq
dtds
′′′=⋅∇+ρ=
=ρ
ρ+=
′′′=
⋅∇+ρ ( )
T uT q
T s ′′′+∇⋅−=′′′
21
This equation governs the rate
of entropy generation owing toconduction of heat. Note that the
entropy generation rate will always begreater than zero because of the
increase of entropy principle. Thus, a
relation between the heat flux andtemperature gradient is suggested here
Appendix. Differentiation
of Multivariable functions
Partial Time Derivative
Substantial Time Derivative
© 2006, 2009 F. A. Kulacki Conduction Heat TransferME8341-02-38
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Multivariate Function
• General description of a scalar as afunction of position and time.
– f = f(x,t), where t = time
– All particles have a location at somestarting point x0. Thus, generally, x
= x(x0,t)
– The position x0 identifies a particle at
some time t0.
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer39
The general flow field with
scalar transport
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer40
( )t),t,x(x 00
x
y
z
( )00 t,x
the along the particle path for The initial location of the along the particle path for
Particle path in time and space.
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Partial time derivative
•
Consider a fixed position for all time, (ξξξξ).• The change in the scalar field at this
fixed position is only with time.
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer41
x
y
z
ξ
volume in formulati e quations and appl
Many particle path of observation ove extension is to the
volume in formulati equations and appl
3t2t1t
Partial time derivative
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer42
If the scalar field is denoted by g(x,t), then for the fixed observation point the time variation of the field is,
z,y,xt
g
dt
dg
∂
∂=
i This derivative is (physically) the ti
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Substantial time derivative
• If the observation of time variation ofthe scalar field is made “following theparticle” along its path, the changeobserved owing to the “advected”motion of the observation must beincluded.
• The derivative with respect to timeunder this physical description isdenoted a Dg/Dt, or as in this course,
dg/dt when no ambiguity arises.
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer43
Substantial time derivative
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer44
( )t),t,x(x 00
x
y
z
( )00 t,x
the part along the particle path for t > The initial location of the part along the particle path for t >
Particle path in time and space and observation path in time.
Along the particle path, the path on which
the time variation of the scalar field is to be observed, we have g(x,t) = g[x(x 0 ,t),t].
Note that g(x,t) is identified with a particular Starting location (x 0 ,t 0 ), but now we must
Consider x = x(x 0 ,t) where x 0 is taken as a parameter.
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Substantial time derivative
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer45
( ) ( )
∂
∂+
∂
∂+
∂
∂+
∂
∂=≡
==
z
g
dt
dz
y
g
dt
dy
x
g
dt
dx
t
g
Dt
Dg
dt
dg
.timeof functionaformallyisx
.motionparticlefollowingtime.t.r.wateDifferenti
]t,t,tx[gt,xgg 0
Time variation independent
of advection.
Variation due to the spatial variation because of advection
along the particle path where time dependence of x must be included.
Substantial time derivative
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer46
( ) ( )
gvt
g
x
gv
t
g
z
g
dt
dz
y
g
dt
dy
x
g
dt
dx
t
g
Dt
Dg
]t,t,tx[gt,xgg
ii
i
0
∇⋅+∂
∂=
∂
∂+
∂
∂=
∂
∂+
∂
∂+
∂
∂+
∂
∂=
==
∑
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End of Mod-02
© 2006, 2009 F. A. Kulacki Conduction Heat Transfer47