2.3 Differential Energy-Balance Equation
2.3.1 Derivation
In materials processing, the kinetic and potential energy are negligible as
compared to the thermal energy, that is the total energy per unit mass et=CvT.
Furthermore, the pressure, viscosity, and shaft work are usually negligible.
According to Gauss’ divergence theorem, we have

A
and
CvTv  ndA     ( CvTv)d

 q  ndA     qd
A
[2.3-1]

[2.3-2]
Substituting Eqs.[2.3-1] and [2.3-2] into Eq.[2.2-6]

CvTd     ( CvTv)d     qd   sd  0


A
A

t
If the control volume Ω does not change with time,  t in Eq.[2.3-3] can be
moved inside the integration sign:

 {t ( CvT    ( CvTv)    q - s}d  0
19
The integrand, which is continuous, must be zero everywhere since the
equation must hold for any arbitrary region.

( CvT )    ( CvTv)    q - s  0
t
[2.3-5]
The first two terms can be expanded and the equation becomes


CvT [( )    (  v)]  0
( CvT )    ( CvTv)
t
t


  (CvT )  (CvT )
 (CvT )  ( v)  v  (CvT )
t
t

  (CvT )  v  (CvT )
t
0
[2.3-6]
Note that q=-k▽T, substituting the Equation into Eq.[2.3-5], we have

 (CvT )  v  (CvT )    (kT )  s
t
[2.3-7]
20
Assuming constant Cv and k
T
Cv [  v  T ]  k 2T  s
t
[2.3-8]
Eq. [2.3-8] is the differential energy-balance equation, or the equation of energy.
2.3.2 Dimensionless form
The equation of energy can be presented in the dimensionless form to
make the solutions more general. For forced convection the following
dimensionless variables can be defined:
Dimensionless temperature
T* 
T  T0
T1  T0
[2.3-9]
v
[2.3-11]
V
p - p o [2.3-12]

dimensionless pressure p 
V 2
tV
x y z




Dimensionless timet 
[2.3-10] dimensionless coordinates x , y , z  , ,
L
L L L

Dimensionless velocity v 
[2.3-13]
Dimensionless operator  , *2  L, L2 2 [2.3-14]
21
In the absence of a heat source, the energy equation Eq.[2.3-8] reduces to
T
 v  T   2T
t
[2.3-15]
Substituting Eqs.[2.3-9] through [2.3-14] into Eq. [2.3-15]
V 
1 2 


  1


T T1  T0    Vv  T
T1  T0    2  T T1  T0 
 
L t
L
L
[2.3-16]
Multiplying Eq.[2.3-16] by L/[V(T1-T0]
T 
 2 

 

v

T

 T

t
LV
[2.3-17]
By combining Eq. [2.3-17] with Eq.[1.5-25] through Eq.[1.5-28], the following
equations can be obtained for heat transfer in forced convection:
Continuity:
  v   0
[2.3-18]
Motion:
v
1 2  1

 
 

v

v


p

 v 
eg

t
Re
Fr
[2.3-19]
Energy:
T 
1
1 2 

 
2 

v

T


T

 T
t 
Re Pr
PeT
[2.3-20] 22
Where
LV 
inertia force V 2 L  [2.3-21]
Re 
 Reynold number =

v 
viscous force  V L2 
[2.3-22]
V2 
inertia force V 2 L 
Fr 
 Froude number =

gL 
gravity force  g 
v 
viscous diffusivity v 
[2.3-23]
Pr 
Prandtl
number
=

 
thermal diffusivity  
PeT  Re Pr 
LV


convection heat transport  CvV T1  T0  
  thermal Peclet number 

conduction
heat
transport
k
T

T
L


1
0


[2.3-24]
23
2.3.3 Boundary conditions
Heat flow boundary conditions in rectangular coordinates
24
25
Heat flow boundary conditions in cylindrical coordinates
26
2.3.4 Solution procedures
T
 v  T   2T
t
27
Example 2.3.1 Heat conduction in a resistance heated rod
0
 Cv [
0
0
0
T
T v T
T
 vr

 vz
]
t
r
r 
z
0
0
1  T
1  2T  2T
k[
(r
) 2
 2 ] s
2
r r r
r 
z
T
0
t
No convection inside the rod
Axisymmetry T  0
 T
Neglect end effect
0
z
Steady state
Governing Eq.
B.Cs.
vr  v  vz  0
1 d dT
s
(r
)
r dr dr
k
dT
 0 at r = 0
and
dr
dT
-k
 h(T  T f ) at r = R
28
(A)
Example 2.3.2 Heat conduction in a cooling fin
qz+dz
Given:
No temp. variation in thickness dir.
wall temp. Tw
dz
ambient temp. Ta
heat transfer coefficient h
w
q=hA(T-Ta)
f
qz
Find: T and Q under steady state
T
T
T
T
 2T  2T  2T
Governing Eq. Cv [
 vx
 vy
 vz
]  k[ 2  2  2 ]  s
t
x
y
z
x
y
z
0
Steady
state
0
No convection
0
T varies with z only
Consider the volume of a C.V. with the length of dz in the fin: f w dz, f : fin thickness
Arte of convection heat loss from the wall: 2w dz h (T-Ta) (heat sink loss heat)
2h
(T  Ta )
dT
f
2
 0 at z = L and
dT
2h
Governing Eq.
B.Cs. dz
 a (T  Ta ) where a=
2
dz
fk
T  Tw at z = 0
f w dz  s  2w dz h(T  Ta )
s
29
Example 2.3.2 Heat conduction in a cooling fin
qz+dz
qz f w  qz  dz f w 
w
q=hA(T-Ta)
dz
f
2h (w dz)(T-Ta )
qz
dqz
dz  2(h wdz )(T -Ta )
dz
d
dT
 fw ( k
)dz  2(h wdz )(T -Ta )
dz
dz
d 2T
fk 2  (2h)(T -Ta )
dz
d 2T 2h

(T -Ta )  a (T  Ta )
2
dz
fk
 fw
qz f w  qz  dz f w  2h (w dz )(T -Ta )
dqz
dz
dz
dq
qz f w  (qz  z dz ) f w  2h (w dz )(T -Ta )
dz
qz  dz  qz 
dT
 0 at z = L and
B.Cs. dz
T  Tw at z = 0
30
Example 2.3.3 Heat conduction into a semiinfinite solid
No source
term
T
T
T
T
 2T  2T  2T
Cv [  v x
 vy
 vz
]  k[ 2  2  2 ]  s
t
x
y
z
x
y
z
0
No convection
0
0
T varies
with x only
T
d 2T
Governing Eq.
 2
t
dx
T ( x, 0)  Ti
B.Cs.:

T (0, t )  Ts
T (, t )  Ti
31
Example 2.3.4 Heat loss from a rising film
Given: Film thickness L
Belt velocity V
Temp. of liquid bath Ti
Temp. at infinite Tf
Find: T under steady state
No source
term
0
Governing Eq.
T
T
T
T
 2T  2T  2T
Cv [  v x
 vy
 vz
]  k[ 2  2  2 ]  s
t
x
y
z
x
y
z
0
Steady
state
0
Flow in the z-dir.
only, Vx=vy=0
0
No Temp. variation
In the x-dir
0
No conduction,
Heat transfer mainly by
convection in the z-dir
Governing Eq. reduces to
T
 2T
vz
 2
z
y
32
T
 2T
vz
 2
z
y
Velocity distribution
vz ( y)  V 
g 2 2
( y  L ) (Example 1.5-3)
2
The temp. drops significantly only near the free surface
(y<<L), especially for high-Pr liquid (heat conduction is slow)
v z ( y)  v min  V 
g 2
L
2
Then governing Eq. reduces further to
B.Cs.:

T ( y, 0)  Ti
T
  2T

z v min y 2
T ( y, z )  Ti as y  
k
T (0, z )
 h(T (0, z )  T f )
y
Can the condition T(0,z)=Tf be used ?
33
Example 2.3.5 Heat transfer in laminar flow over a flat plate
Given: Steady state, laminar flow, incompressible,
Newtonian, uniform temp. Ts. Physical properties is
not a function of temp., no source term and viscous
dissipation
Governing Eq.
Steady
state
0
No source
term
0
0
T
T
T
T
 2T  2T  2T
Cv [  v x
 vy
 vz
]  k[ 2  2  2 ]  s
t
x
y
z
x
y
z
0
No temp. gradient
in the x-dir.
0
0
No Temp. gradient
In the z-direction,
In the x-dir
heat transfer mainly due to
convection but not conduction
T
T
 2T
vy
 vz
  2 (energy Eq.) (2.3-70)
y
z
y
v z
v z
2vz
vy
 vz

(Eq. of motion)
2
y
z
y
v y v z
y

 0 (Continuity Eq. )  v y   v z dy (2.3-71)
0 z
y
z
34
Substituting (2.3-71) into (2.3-70) to give
( 
y
0
vz
T
T
 2T
dy)
+vz
= 2
z
y
z
y
(2.3-72)
Using the temp. profile of
T  Ts 3 y
1 y 3
 ( )  ( ) (2.3-73)
T  Ts 2  T
2 T
Subjective to the B.Cs.
T  Ts at y = 0
(2.3-74)
T
T  T and
 0 at y =  T
y
(2.3-75)
Take the derivation of
T
T
 2T
(2.3-76) ,
(2.3-77), and
(2.3-78)
2
z
y
y
Substitute Eqs. (2.3-76) through (2.3-78) to Eq. (2.3-72) and substitute
Eq. (1.5-78) for vz and Eq. (1.5-81) for v z z
35
We have
8
T
5
1  T 2  d T 6 4  2 1  T 2  d
8 2


1

(
)



(
)


 T (2.3-79)
T 



7   dz 5  3 7   dz
v
According to Eq. (1.5-87)
vz
  4.64
(2.3-80)
v
vz
280 v
 2  (4.64) 2
and
2 d 
dz (2.3-81)
v
13 v 

Let T =a
(2.3-82)

where a is a constant depending only on the physical properties of the fluid.
Substituting Eqs. (2.3-80) and (2.3-82 into Eq. (2.3-79), we have
13
a -14a + =0
Pr
5
3
(2.3-83)
Let consider the case where a<1, that is
a 5 <<14a 3
Equation (2.3-83) reduces approximately to a=Pr  (1/3)
That means
vz
T
(2.3-87)
=Pr ( 1/3) (2.3-86) and T  4.64 Pr  (1/3)

v
36
Example 2.3.6 Heat transfer with laminar in a tube
Given: Steady state, laminar flow, incompressible,
Newtonian, uniform heat flux qR at the wall
T
qR  h(TR  Tav )  k
r
r R
Define average fluid temperature
R
Tav 
2   rv zTdr
0
m
R

2  rv zTdr
0
(2.3-89)
Q(volume flow rate)
Fluid flow is significant enough that heat transfer in
the z-dir is dominated by convection
Find: h
According to B.Cs.
h(TR  Tav )  k
In order to find h, we must first find
T
r
T
r
r R
r R
h
k
T
(TR  Tav ) r
r R
and (TR  Tav )
37
Governing Eq.
0
 Cv [
0
0
0
T
T v T
T
 vr

 vz
]
t
r
r 
z
0
0
1  T
1  2T  2T
k[
(r )  2
 2 ] s
2
r r r
r 
z
Governing Eq. reduces to
vz
T
 1  T 
 
(r
)
z
r

r

r


From Eq. [1.5-54]
2Q
r 2
[1

(
) ] and thus
2
R
R
1  T
r 2
 2Q T  
(r )  
 1  ( ) 
2
r r r
R 
  R z  
vz 
(2.3-94)
B.Cs.
T
 0 at r = 0 and T=TR (z) at r=R
r
38
T  TR 
T
4
2 2
4
(3
R

4
R
r

R
)
4
2 R z
Q
(2.3-99)
T
Q T
2
3

(

8
R
r

4
r
)
4
r
8 R z
and
T
r
rR

T
2 R z
Q
(2.3-101)
Substitution Eqs. (2.3-94) and (2.3-99) into Eq.(2.3-89), we have
Tav  TR 
11Q T
48 z
(2.3-102)
Substitution Eqs. (2.3-101) and (2.3-102) into Eq.(2.3-90),
we have
NuD 

hD D
k
T

k
k (TR  Tav ) r
r R

D
Q T
(TR  Tav ) 2 R z
DQ T 48 T 1 48
( ) 
 4.36
2 R z 11Q z
11
39
2.5 Turbulence
2.5.1 Time-smoothed variables
The velocity fluctuations arising in turbulent flow affect the local pressure. The
velocity fluctuations also affect the local temperature.
T  T T '
1
T
t0
1
T 
t0
'

t t0
t

t t 0
t
[2.5-1]
Tdt
T ' dt  0
[2.5-2]
[2.5-3]
40
2.5.2 Time-smoothed governing equations
Substituting

v  v  v'
[1.7-1] and T  T  T [2.5-1] into [2.3-7]
'

(CvT )  v  (CvT )    (kT )  s
t
[2.3-7]
and taking the time average, we get the time-smoothed equation of energy
 T

Cv   v T     q    q '  s
 t

where
q  kT
[2.5-4]
and the turbulent heat flux

q '   Cv v x ' T '  v y ' T '  vz ' T '

[2.5-5]
Eq. [2.5-4] is the same as the equation of energy for laminar flow Eq.[2.3-7],
except that the time-smoothed velocity and temperature replace the instantaneous
velocity and temperature, and that one new terms  q ' arises.
41
2.5.3 Turbulent heat flux
Several semiempirical relations have been proposed for the turbulent heat flux
To solve Eq. [2.5-4] for temperature distributions in turbulent flow.
2.5.3.1 Eddy thermal conductivity
By analogy with Eq. [2.1-1] Fourier’s law of conduction, one may write
dT
q y '  k '
dy
[2.5-6]
The coefficient K’ is a turbulent or eddy thermal conductivity and is usually
position-dependent.
2.5.3.2 Prandtl’s Mixing length
By analogy with Eq. [1.7-13]
2 dvz dT
q y '   Cvl
dy dy
[2.5-7]
And from Eq. [2.5-7]
dvz
k '   Cvl
dy
2
[2.5-8]
42
In the turbulent boundary-layer energy equation in time-averaged values
for small velocities
T
T
 2T
 (v 'T ' )
 Cv (u
v
)  k 2   Cv
x
y
y
y
This equation differs from the corresponding laminar equation for steady flow by
the last term on the right-hand side, which is an expressed for the turbulent
exchange of heat equivalent to the following Eq.
qt  m ' C p (T  T ')
Boussinesq introduced for this
turbulent heat flow the expression.
T
qt    C p v T    C p q
y
'
'
With q called the turbulent diffusivity for heat.
This expression, together with the turbulent diffusivity for momentum, has
become very useful for a calculation of heat transfer from flow information,
since even the simplest assumption for the ratio m/q, namely, that is a constant.
43
Heat and Mass Transfer, by Eckert & Drake, McGraw-Hill 1950, p.219
turbulent diffusivity for momentum
 
u
 t    u v   m
y
' '
T
qt    C p q
y
By dividing the equations for the turbulent heat flow and the turbulent shear
stress, The result is
q
T
  Cp
t
m
u
qt
The ratio of m/q, which is called turbulent Prandtl number (Prt), has a value
of approximately 0.7 in boundary-layer flow and approximately 0.5 for wake
flow behind blunt objects and for vortex flow.
44
Heat and Mass Transfer, by Eckert & Drake, McGraw-Hill 1950, p.219
2.6 Heat transfer correlations
Convection heat transfer can be determined by solving the governing equation
for fluid flow and heat transfer. Correlations that are derived theoretically but verified
experimentally or are based on experimental data alone, are useful for studying
convection heat transfer.
2.6.1 External Flow
2.6.1.1 Forced convection over a flat plate
Flow over a flat plate is laminar for local Reynolds number Rez < 2 x 105. The
following correlation can be used for laminar flow of fluid with a bulk temperature
T∞ over a flat plate with a surface temperature Ts.
Nu z  0.332 Re z1 2 Pr1 3
0.6 < Pr < 50
Nu z (local Nusselt number) 
hz 
v

[2.6.2]
k
k
Nu z  0.332 Re1/ 2 Pr1/ 3
z
z
Re z (local Reynolds number) 
Pr (Prandtl number) 
hz z
k
[2.6.1]

Cv 
k
zv z  v

v

[2.6.3]
[2.6.4]
45
From these equations, the heat transfer coefficient averaged over a distance L
from the leading edge of the plate is
1 L
hL   hz dz
L 0
[2.6.5]
Substituting Eq. [2.6-1] into Eq. [2.6-5]
1 L k
hz  
0.332 Re1/ 2 Pr1/ 3 dz
L 0 z
or

1
2
k
1/ 3 v 
hz  0.332( ) Pr ( )  z dz
z
 0
L
NuL  0.664 Re L1 2 Pr1 3
(0.6<Pr<50)
[2.6.6]
[2.6.7]
where
hL L
NuL (average Nusselt number) 
k
Re L (average Reynolds number) 
Lv L v

v

[2.6.8]
[2.6.9] 46
The fluid properties in Eqs. [2.6-1] and [2.6-7] are evaluated at the film temperature
T  Ts
Tf 
2
For liquid metals and semiconductors Pr<<1 and Eqs. [2.6-1] and [2.6-2] cannot be
applied. For these materials the following theoretical correlation has been suggested:
Nu z  0.565  Re z Pr 
12
 0.565 Pez1 2
[2.6-11]
where
Pez (local Peclet number)  Re z Pr 
zv

[2.6-12]
For turbulent flow over a flat plate, the following theoretical correlation has
been suggested:
Nu z  0.0288 Re z 4 5 Pr1 3 (0.6<Pr<60)
[2.6-13]
and from Eq. [2.6-5]
Nu z  0.036 Re L 4 5 Pr1 3
(0.6<Pr<60)
[2.6-14]
47
2.6.1.2 Forced convection normal to a cylinder
For the flow of air normal to a cylinder of diameter D. An empirical correlation for
the forced convection heat transfer is
NuD  a Re D b Pr1 3
 0.1  Re
D
 3 105 , Pr  0.7 
where
hD
Dv D  v
Re


[2.6-16]
D
k
v

The constants a and b are listed in Table 2.6-1
NuD 
[2.6-15]
[2.6-17]
The following empirical correlation has also been suggested:
NuD  0.3 
12
D
13
0.62 Re Pr
[1  (0.4 / Pr) 2 3 ]1 4
58
  Re


D
1  


  282, 000  
45
[2.6-18]
48
2.6.1.3 Forced convection past a sphere
For the flow of air past a sphere of diameter D, an empirical correlation for
the forced convection heat transfer is
14
12
23
0.4    
NuD  2   0.4 Re D  0.06 Re D  Pr 


 s
[2.6-19]
(0.71 < Pr < 380, 3.5 < ReD < 7.6 X 104, 1 < /s < 3.2)
All fluid properties are evaluated at bulk T∞ except for s, which is evaluated at
the surface temperature of the sphere. This correlation is accurate to within ±30%
for the range of parameter values specified.
A special case of convection heat transfer from sphere is that of a freely falling
drop. The following theoretical correlation has been proposed:
NuD  2  0.6 Re D1 2 Pr1 3
[2.6-20]
49
2.6.2 Internal flow
2.6.2.1 Forced convection inside a circular tube
For laminar fully developed flow inside a circular tube of diameter D and uniform
surface heat flux. The following theoretical correlation has been proposed:
NuD  4.36  Pr  0.6
[2.6-21]
For a uniform surface temperature rather than heat flux, the following theoretical
correlation has been proposed:
[2.6-22]
NuD  3.66  Pr  0.6
For turbulent fully developed flow inside a circular tube of diameter D and
length L, the following theoretical correlation has been suggested:
NuD  0.023Re D 4 5 Pr1 3
[2.6-23]
A slightly different and preferred correlation is as follows:
NuD  0.023Re D 4 5 Pr n
L
(0.6  Pr  160, Re D  10, 000,  10)
D
where n is 0.4 if the fluid is being heated and 0.3 if it is being cooled.
[2.6-24]
50
Eq.[2.6-24] is good for small to moderate temperature difference between the
wall and the bulk fluid. The following equation is preferred for flows characterized
by large property variations:
0.14
 
NuD  0.027 Re D 4 5 Pr1 3  b 
 w 
[2.6-25]
L
(0.7  Pr  16, 700, Re D  10, 000,  10)
D
All fluid properties are evaluated at the bulk temperature except w, which is
evaluated at the wall temperature.
Equations [2.6-23] through [2.6-25] are ReD>104. The following correlation
can be used even if ReD is below 104.
 f 8 Re D  1000  Pr
NuD 
12
1  12.7  f 8  Pr 2 3  1
[2.6-26]
6
(0.5  Pr  2000, 2300  Re D  5 10 )


Where the friction factor f can be obtained from the Moody chart or, for smooth
pipes, from.
f   0.79 ln Re D  1.64 
2
[2.6-27]
51
Example
Air at 200C and 1 atm flows over a flat plate at 35 m/s. The plate is 75 cm long
and is maintained at 600C. Assuming unit depth in the z direction, calculate the
heat transfer from the plate.
Given: P=1.0132x105 N/m2, R=287, =2.007x10-5kg/ms, Cp=1.007 kJ/kg0C
k=0.02723 W/m0C
Eq. for forced convection heat transfer
1
hL
0.8
Nu L 
 Pr 3 (0.037 Re L  850)
k
52