 2001, W. E. Haisler
Chapter 6: Conservation of Energy
1
Conservation of Energy (chapter 6)
The total energy must be conserved for a volume element x
y z during a time period t. Energy may take on several
forms which are generally associated with mechanical and
thermal terms such as
 internal energy (due to deformation; pressure,
temperature or volume change, etc.),
 kinetic energy (due to motion),
 external energy or work (due to external forces, tractions
or body forces),
 energy flow due to temperature gradients,
 internal energy generation (due to inelastic deformation,
chemical reactions, point thermal sources, etc).
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
2
Consider the various forms of energy flux associated
with the continuum differential volume :
Only the x component listed below (flow through dy dz):
 vxU J/ m2 /s
Internal energy of the mass flux:
U  Energy/mass, Energy Joule  0.74 ft-lb, watt=Joule/sec
Kinetic energy of the mass flux:  vx KE
J/ m2 /s
Work done by boundary tractions:
J/ m2 /s
t(i )  v
Heat energy flux by conduction:
J/ m2 /s
qx
Work done by the body forces:
 g xvx J/ m3 /s
Radiation source or heat sink in volume: 
J/ m3 /s
Note: Other heat energy flux terms which could be
considered include convection and radiation.
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
3
Considering the x, y and z components, we have


[  v (U  KE )]  [  v (U  KE )]
 yzt
x
x

x
x x 



 [  v (U  KE )]  [  v (U  KE )]
 zxt
y
y

y y 
y



 [  v (U  KE )]  [  v (U  KE )]
 xyt
z
z

z
z z 
(  g  v )xyzt





 (t v )
 (t v )  yzt  (t v )
 (t v )  zxt
( j)
 (i)
 ( j)
x x (i)
x 
y

y
y 



 (t v )
 (t v )  xyt
(
k
)

z z (k )
z 
 2001, W. E. Haisler
Chapter 6: Conservation of Energy


 q  q
xyt
zz
z z  z 


 q
q
yzt  q
q
 zxt
xx
x x  x
y
y
y  y 
 y
 xyzt

 [  (U  KE )]
t  t
 [  (U  KE )]
t xyz
4
 2001, W. E. Haisler
In the limit as
Chapter 6: Conservation of Energy
xyz  0
t  0
5
, we obtain Conservation of Energy
  [  v (U  KE )]  [  v (U  KE )]  [  v (U  KE )] 
y
x
z
  g v





x
y
z


q
  (t(i )  v )  (t( j )  v )  (t(k )  v )    q
q 
y
 x 



 z   
y
z
y z 
  x
   x

 [  (U  KE )]

t
Using the product rule on the first and last terms gives
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
6

 (U  KE )
 (U  KE )
 (U  KE ) 
  v
 v
 v

x
y
z
x
y
z


  (  vx )  (  v y )  (  vz ) 



(U  KE )   g  v

y
z 
 x
 v )  (t  v )    q  q
  (t  v )  (t
q 
(
i
)
(
j
)
(
k
)
y
 x 



 z
x
y
z

y

z 

  x


 
 (U  KE ) 
   

(U  KE )
t
t
The underlined terms [multiplied by (U  KE )] sum to zero
by Conservation of Mass.
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
7
Thus we have the Conservation of Total Energy (Mechanical
+ Thermal Energy)
  (U  KE )
 (U  KE )
 (U  KE ) 
  v
v
v

x
y
z
x
y
z


 v )  (t  v ) 
  (t  v )  (t
(i)
( j)
(k )


 g  v 


x
y
z




q q 
 q
y
x



 z   

y

z 
 x


 (U  KE )

t
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
8
We can rewrite the energy, convection and traction terms in
vector form:
  (U  KE )
 (U  KE )
 (U  KE ) 
   (v )(U  KE )
 v
v
v
y y
z z 
 x

x


q
q
q
y
x
 z q
x y z
 (t
 v )  (t  v )  (t  v )
(i)
( j)
(k )


   (  v )
x
y
z
Each of the above vector products is a scalar quantity (energy)!
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
The final result in vector form for Conservation of Total
Energy (Mechanical + Thermal Energy) is
  (U  KE )


 (v  )(U  KE ) 
t


  g  v      q    (  v )
9
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
10
It is useful to separate the total energy into mechanical
energy (that due to tractions and body forces) and thermal
energy. The mechanical energy can be thought of as being
associated with the linear momentum. Consider the
conservation of linear momentum equation
 v

   (v )v    g 
 t

Take the dot product of both sides with the velocity vector
 v

   (v )v   v
 t

 [  g  ( )]  v
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
11

Note: KE =kinetic energy per unit mass
=1/2 mass v  v /mass = 1/2 v  v .

Thus, terms on left side can be written in terms of KE as follows
 ( 12 v  v )
 KE

t
t
 ( 12 v  v )
 KE

v
v  v
v
Then: v
xx
x
x x
x
Note:  v  v 
t
The mechanical energy portion of Conservation of
Energy becomes


   KE  v  KE    g  v  ( )  v
 t



 2001, W. E. Haisler
Chapter 6: Conservation of Energy
12
The Thermal Energy portion of Conservation of Energy
may be obtained by subtracting the mechanical energy
portion from the total energy: We obtain
U


 (v  )U     q      (  v )  (   )  v
 t

   q    tr ( v )
   q    tr (S v )  P(  v )
where
P is the average compressive pressure (hydrostatic stress),
S is the deviatoric or extra stress tensor ([] = [S] - P [I]
where [I] = identity matrix), as defined before, and
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
13
tr (trace) is a scalar quantity and is defined as the sum of the
diagonal terms of a square matrix.
Thus tr (T  v ) is the sum of the diagonal terms of the 3x3
matrix formed by the vector operation (T  v ), where T and
v are each 3x3 matrices.
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
Conservation of Thermal (internal) Energy
We previously obtained the thermal energy equation:
U


 (v  )U     q      (  v )  (   )  v
 t

   q    tr ( v )
   q    tr (S v )  P(  v )
14
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
15
Problem: Internal Energy, U ,is not a directly observable
quantity. If we wish to solve real problems, we must relate
U to observable (measurable) quantities like




heat capacity, C (at either constant volume or pressure)
pressure, P
volume, V
temperature, T
This requires the use of the thermodynamics equations
which relate U and other thermodynamic properties to the
measurable variables of P, V and T.
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
16
The thermodynamic energy functions of current interest are:
U = internal energy
S = entropy
H =enthalpy
From your previous work in thermodynamics, we can write
thermodynamic functions in terms of the PVT measurable
variables:
dU  Td S - PdV
d H  Td S  Vd P
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
17
Consider the internal energy written in terms of two of the
three measurable variables (say, T and V):
U  U (T ,V )
then the total differential of the internal energy is given by
U 
U 
dU  
 dT  
 dV
  T V
  V T
U 
The term 
 implies change in U  U (T ,V ) with respect to
  T V
T at constant volume,V .
 2001, W. E. Haisler
18
Chapter 6: Conservation of Energy
We assume that that U is related to T (during a constant
U 
volume process) by a constant C such that 
C

V
V
 T 
V
and, from previous thermodynamics,
U 
 P 

 T 
 P
  T V
  V T
Thus, the total change in internal potential energy becomes
  P 

dU  C dT   T 
 P  dV

V
   T V

 2001, W. E. Haisler
19
Chapter 6: Conservation of Energy
Note that since U  U (t , x, y, z ) , the total change, dU ,is
U 
U 
U 
U 
dU  







  t  x, y , z   x  y , z , t   y  z , x, t   z  x, y , t
The four partial derivatives above are given by
U
  P
V
C
 T 
 P

V  t    T V
t
 t
U
x
U
y
U
T
C
V
C
V
T
  P
V
 T 
  P
 x    T V
x
T   P 
V
 T 
  P
 y    T V
y
T
  P
V
C
 T 
 P

V  z    T V
z
z

 2001, W. E. Haisler
20
Chapter 6: Conservation of Energy
The sum of the last three equations is the gradient of U , i.e.,
  P 

U  C T  T 
 P  V

V
   T V


Substituting the last two equations (in boxes) into the
thermal energy equation (in terms of deviatoric or extra
stress) yields the following Conservation of Thermal Energy
Equation (for a constant volume process):
 P
T


C 
 v  T     q    T 
   v  tr ( S  v )
V  t
  T V

 2001, W. E. Haisler
Chapter 6: Conservation of Energy
21
For incompressible continua
  function of time,   v  0 (from conservation of
mass), C  C  C , and the thermal energy equation is
V
P
T
C 
 v  T     q    tr (S  v )
 t

For the static case (no mass velocity, v  0 )
T
C
   q  
t
Heat Conduction in Solid
Later, we will show that q  kT (Fourier's Law where k
T
depends on the material). In 1-D, q  k
, where k =
x
x
thermal conductivity (for an isotropic material).
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
Second Law of Thermodynamics
The second law of thermodynamics states that entropy is
related to the amount of reversible heat transfer and that
entropy is not conserved.
d Qrev
dS 
0
T
T = absolute temperature (R or K)
S = entropy/unit mass
Q = reversible heat transfer/unit mass
rev
22
 2001, W. E. Haisler
23
Chapter 6: Conservation of Energy
The second law states that entropy is not conserved, but only
generated. Considering heat and entropy flow through a
control volume, we obtain
 S

 q  

 v  S       
S
0
gen
T  T
 t

where S
is the entropy generation rate. If we use the
gen
energy conservation equation from chapter 7, we can show
that the 2nd Law becomes
 S

 q  
q T
   v  S      
S

t

T  T
gen


 
T2

tr (S v )
0
T
 2001, W. E. Haisler
Chapter 6: Conservation of Energy
24
The entropy generation terms from the right side of the
previous equation state:
q  T tr ( S  v )
S


0
gen
T
T2
Thus we conclude that entropy generation may occur as a
result of the heat flux by conduction or the conversion of
mechanical energy of tractions into heat energy.
q T  0
Due to heat flux by conduction
tr (S v )  0 Conversion of mechanical energy to heat energy