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```AME 513
Principles of Combustion
Lecture 7
Conservation equations
Outline
 Conservation equations




Mass
Energy
Chemical species
Momentum
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
2
Conservation of mass
 Cubic control volume with sides dx, dy, dz
 u, v, w = velocity components in x, y and z directions
æ
¶ ( rv) ö
dy ÷ dxdz
ç rv +
¶y
è
ø
( ru) dydz
æ
¶ ( ru) ö
dx ÷ dydz
ç ru +
¶x
è
ø
dy
dx
( rv) dxdz
 Mass flow into left side & mass flow out of right side
mleft = ruA = ru(dydz)
mright
æ
¶ ( r u) ö
= - ç ru +
dx ÷ dydz
¶x
è
ø
 Net mass flow in x direction = sum of these 2 terms
æ
¶ ( r u) ö
¶ ( r u)
mx = rudydz - ç ru +
dx ÷ dydz = dxdydz
¶x
¶x
è
ø
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
3
Conservation of mass
 Similarly for y and z directions
¶ ( rv)
¶ ( r w)
my = dxdydz;mz = dxdydz
¶y
¶z
 Rate of mass accumulation within control volume
¶m ¶ ( rV ) ¶r
=
= dxdydz;V = volume
¶t
¶t
¶t
 Sum of all mass flows = rate of change of mass within
control volume
¶ ( ru)
¶ ( rv)
¶ ( r w)
¶r
-r
dxdydz - r
dxdydz - r
dxdydz =
dxdydz
¶x
¶y
¶z
¶t
æ ¶ ( ru) ¶ ( r v) ¶ ( r w) ö ¶r
¶r
Þ
= -ç
+
+
+ Ñ × ( ru ) = 0
÷Þ
¶t
¶y
¶z ø
¶t
è ¶x
Note u = velocity vector = uiˆ + viˆ + wiˆ
x
y
z
iˆx , iˆy , iˆz = unit vectors in x, y, z directions
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
4
Conservation of energy – control volume
 1st Law of Thermodynamics for a control volume, a fixed volume
in space that may have mass flowing in or out (opposite of control
mass, which has fixed mass but possibly changing volume):
2
dE
vin2
vout
= Q - W + min (hin +
+ gzin ) - mout (hout +
+ gzout )
2
2
dt



E = energy within control volume = U + KE + PE as before
Q˙ ,W˙ = rates of heat & work transfer in or out (Watts)
Subscript “in” refers to conditions at inlet(s) of mass, “out” to
outlet(s) of mass
˙ = mass flow rate in or out of the control volume
 m
 h  u + Pv = enthalpy
 Note h, u & v are lower case, i.e. per unit mass; h = H/M, u = U/M, V =
v/M, etc.; upper case means total for all the mass (not per unit mass)
 v = velocity, thus v2/2 is the KE term
 g = acceleration of gravity, z = elevation at inlet or outlet, thus gz is
the PE term
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
5
Conservation of energy
 Same cubic control volume with sides dx, dy, dz
 Several forms of energy flow
 Convection
 Conduction
 Sources and sinks within control volume, e.g. via chemical
reaction & radiative transfer = q’’’ (units power per unit volume)
 Neglect potential (gz) and kinetic energy (u2/2) for now
 Energy flow in from left side of CV
Eleft = mleft h + qleft = ruAh - kA
æ ¶T ö
¶T
= ruh(dy)(dz) - (dy)(dz)ç k ÷
è ¶x ø
¶x
 Energy flow out from right side of CV
é
é ¶T ¶ æ ¶T ö ù
¶ ( r u) ù
é ¶h ù
Eright = mright h + qright = êru +
dx ú(dy)(dz)êh + dx ú - (dy)(dz)êk
+ ç k ÷ dx ú
ë ¶x û
¶x
ë ¶x ¶x è ¶x ø û
ë
û
ì
é ¶T ¶ æ ¶T ö ùü
¶ ( r u)
¶ ( ru) ¶h
¶h
2
= í ruh + ru dx + h
dx +
( dx ) - êk + çè k ÷ø dxúý (dy)(dx)
¶x
¶x
¶x
¶x
ë ¶x ¶x ¶x
ûþ
î
 Can neglect higher order (dx)2 term
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
6
Conservation of energy
 Net energy flux (Ex) in x direction = Eleft – Eright
ì
¶ ( ru) ¶ æ ¶T öü
¶h
Ex = í-ru - h
+ ç k ÷ý (dy)(dz)(dx)
¶x
¶x
¶x è ¶x øþ
î
 Similarly for y and z directions (only y shown for brevity)
ì ¶h
¶ ( rv) ¶ æ ¶T öü
Ey = í r v + h
- ç k ÷ý (dx)(dz)(dy)
¶y
¶y è ¶y øþ
î ¶y
 Combining Ex + Ey
ìï
¶h
¶h ¶ æ ¶T ö ¶ æ ¶T ö æ ¶ ( ru) ¶ ( rv) öüï
E x + E y = í- r u - r v + ç k ÷ + ç k ÷ - h ç
+
÷ý (dx)(dz)(dy)
¶x
¶y ¶x è ¶x ø ¶y è ¶y ø è ¶x
¶y øïþ
ïî
{ (
(
)
)}
= -r u × Ñh - hÑ × ( ru ) + Ñ × kÑT (dx)(dz)(dy)
 dECV/dt term
¶ECV ¶éëm ( h - P r )ùû
=
=
¶t
¶t
(h - P r )
æ ¶h ¶ ( P r ) ö ì ¶r P ¶r
¶(P r) ü
¶( rV )
¶h
+ rV ç =
h
+
r
r
ýV
÷ í
¶t
¶t ø î ¶t r ¶t
¶t
¶t þ
è ¶t
ì
¶P
¶r ü
r
P
ï ¶r P ¶r
ï
ì ¶r
¶h
¶h ¶P ü
¶t
¶t
= íh +r -r
V
=
h
+
r
- ý (dx)(dy)(dz)
ý
í
2
î
¶t
r
¶t
¶t
r
¶t
¶t
¶t þ
ï
ï
î
þ
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
7
Conservation of energy
 dECV/dt = Ex + Ey + heat sources/sinks within CV
ì ¶r
¶h ü
íh + r ý (dx)(dy)(dz) = {-r ( u × Ñh) - hÑ × ( ru ) + Ñ × ( kÑT ) + q'''} (dx)(dz)(dy)
î ¶t
¶t þ
ì ¶r
ü é ¶h
Þ h í + Ñ × ( ru )ý + r ê + u × Ñh
î ¶t
þ ë ¶t
(
)
ù
úû - Ñ × kÑT = q'''
(
)
 First term = 0 (mass conservation!) thus (finally!)
é ¶h
r ê + u × Ñh
ë ¶t
(
)
ù
úû - Ñ × kÑT = q'''
(
)
 Combined effects of unsteadiness, convection, conduction
and enthalpy sources
 Special case: 1D, steady (∂/∂t = 0), constant CP (thus ∂h/∂T =
CP∂T/∂t) & constant k:
dT
k d 2T q'''
u
=
2
dx rCP dx
rCP
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
8
Conservation of species
 Similar to energy conservation but
 Key property is mass fraction of species i (Yi), not T
 Mass diffusion rD instead of conduction – units of D are m2/s
 Mass source/sink due to chemical reaction = Miwi (units kg/m3s)
é ¶Yi
r ê + u × ÑYi
ë ¶t
(
)
ù
úû - Ñ × r DÑYi = M iwi
(
)
 Special case: 1D, steady (∂/∂t = 0), constant rD
dYi
d 2Yi M iwi
u
-D 2 =
dx
dx
r
 Note if rD = constant and rD = k/CP and there is only a single
reactant with heating value QR, then q’’’ = -QRMiwi and the
equations for T and Yi are exactly the same!
 k/rCPD is dimensionless, called the Lewis number (Le) –
generally for gases D ≈ k/rCP ≈ n, where k/rCP = a = thermal
diffusivity, n = kinematic viscosity (“viscous diffusivity”)
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
9
Conservation equations
 Combine energy and species equations
é ¶Yi
+ u × ÑYi
êë
¶t
ù
é ¶Yi
ù a
M iw i
M iw i
DÑ
×
ÑY
=
Þ
+
u
×
ÑY
Ñ
×
ÑY
=
i
i ú
i
úû
êë
û Le
r
¶t
r
é ¶T
ù k
M iwiQR
T - T¥
T - T¥
Yi
+
u
×
ÑT
Ñ
×
ÑT
=
q'''
=
;
Let
T
=
=
,Y
=
êë
úû
¶t
r CP
rCP
Yi,¥QR / CP Tad - T¥
Yi,¥
(
)
(
)
é ¶T
Þ ê + u × ÑT
ë ¶t
( )
(
)
( )
( )
ù
ù a
M iwi é ¶Y
; ê + u × ÑY ú - Ñ × ÑY
ú - aÑ × ÑT = rYi,¥ ë ¶t
û
û Le
é ¶ T +Y
Add species & energy equations for Le = 1: ê
+ u × Ñ T +Y
êë ¶t
(
)
( )
(
(
( )
)
)
(
(
))
=
M iwi
rYi,¥
ù
ú - aÑ × Ñ T +Y
úû
( (
)) = 0
 T +Y is constant, i.e. doesn’t vary with reaction but
 If Le is not exactly 1, small deviations in Le (thus T) will have
large impact on w due to high activation energy
 Energy equation may have heat loss in q’’’ term, not present in
species conservation equation
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
10
 Outside of a thin reaction zone at x = 0
dT
k d 2T
m
=
0;
r
u
=
= constant; 2nd order O.D.E
2
dx ruCP dx
A
Boundary conditions upstream of reaction zone: x = 0,T = Tad ; x ® ¥,T ® T¥
-k
k
=
ruCP r¥SLCP
Boundary conditions downstream of reaction zone: x = 0,T = Tad ; x ® -¥,T ® Tad
Þ T (x) = Tad = constant
Þ T (x) = T¥ + (Tad - T¥ ) e- x/d ; d º
 Temperature profile is exponential in this convectiondiffusion zone (x ≥ 0); constant downstream (x ≤ 0)
 u = -SL (SL > 0) at x = +∞ (flow in from right to left); in
premixed flames, SL is called the burning velocity
 d has units of length: flame thickness in premixed flames
 Within reaction zone – temperature does not increase despite
heat release – temperature acts to change slope of
temperature profile, not temperature itself
11
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
Schematic of deflagration (from Lecture 1)
 Temperature increases in convection-diffusion zone or preheat
zone ahead of reaction zone, even though no heat release occurs
there, due to balance between convection & diffusion
 Temperature constant downstream (if adiabatic)
 Reactant concentration decreases in convection-diffusion zone,
even though no chemical reaction occurs there, for the same
reason
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
12
 In limit of infinitely thin reaction zone, T does not change but
dT/dx does; integrating across reaction zone
0+
0+
0+
dT
d 2T
q'''
dT ù
q'''
0+
u
dx
a
dx
=
dx
Þ
uT
a
=
dx
]
ò dx ò dx 2
ò rC
ò
úû
0dx 0- 0- rCP
P
0000+
æ dT
ö 0+ q'''
dT
Þ -ç
dx = ò
=ò
÷= ò
kA
kA
è dx x=0+ dx x=0- ø 0- k
00æ dT
ö
dT
mQR
r¥ SL ACP (Tad - T¥ )
Þç
=
=
=
÷
dx
dx
kA
kA
d
è
x=0+
x=0- ø
0+
0+
 Note also that from temperature profile:
T(x) = T¥ + (Tad - T¥ ) e-x/d (x ³ 0)üï æ dT
ýÞç
ïþ è dx
T(x) = Tad = constant (x £ 0)
dT
dx
x=0+
ö
=
÷
d
x=0- ø
 Thus, change in slope of temperature profile is a measure of
the total amount of reaction – but only when the reaction
zone is thin enough that convection term can be neglected
compared to diffusion term
13
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
Conservation of momentum
 Apply conservation of momentum to our control volume
results in Navier-Stokes equations:
¶u
r + r u × Ñ u = -ÑP + r g + µÑ 2u
¶t
or written out as individual components
(
)
æ ¶ 2u ¶ 2 v ö
¶u
¶u
¶u ¶ P
r + ru + r v =- +r gx +m ç 2 + 2 ÷ (x momentum)
¶t
¶x
¶y ¶x
è ¶x ¶y ø
æ ¶ 2u ¶ 2 v ö
¶v
¶v
¶v ¶ P
r + ru + r v =- +r gy +m ç 2 + 2 ÷ (y momentum)
¶t
¶x
¶y ¶y
è ¶x ¶y ø
 This is just Newton’s 2nd Law, rate of change of momentum =
d(mu)/dt = S(Forces)
 Left side is just d(mu)/dt = m(du/dt) + u(dm/dt)
 Right side is just S(Forces): pressure, gravity, viscosity
AME 513 - Fall 2012 - Lecture 7 - Conservation equations
14
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