CHAPTER 8: INTERNAL FLOW
Objectives:
1. To develop an appreciation for the physical phenomena associated with internal flow.
2. To obtain convection coefficients for flow conditions of practical importance.
8.0 Introduction
 An internal flow is flow for which the fluid is confined by a surface, e.g. flow in a pipe.
 Internal flow configuration is widely employed in heating and cooling of fluids used in
chemical processing, environmental control and conversion technologies.
8.1 Hydrodynamic Considerations
8.1.1 Flow Conditions
 The boundary layer development for laminar flow in a circular tube is as shown below.

The boundary layer develops with increasing x until it reaches the centerline. The flow is
then said to be hydrodynamically fully developed (HFD).

The distance from the entrance at which HFD condition is achieved is termed the
hydrodynamic entry length, xfd,h.

The entry length for flow in a tube can be estimated as follows:
x fd , h
D
 0.05 Re D , la min ar
 
, turbulent
 10
(1)

The Reynolds number for flow in a circular tube is defined as
Re D 
 um D
u D
 m


(2)
where um is the mean fluid velocity and D is the tube diameter.

The critical Reynolds number corresponding to the onset of turbulence is
ReD, c  2300
(3)
8.1.2 The Mean Velocity
 The mass flow rate through the tube is
m 
 Ac  u r, xdAc
(4)
where Ac is the tube cross-sectional area.
 Defining a mean velocity over the cross section, the mass flow rate may also be expressed as
   um A c
m
(5)
 Equating (4) and (5), it follows that the mean velocity for incompressible flow in a circular
tube is
um 
1
2 ro
u r , x  dAc 
u r , x  r dr
2
Ac Ac
ro 0


(6)
 From equations (2) and (5), the Reynolds number can also be expressed as
Re D 
4 m
 D
(7)
8.1.3 Velocity Profile in the Fully Developed Region
 In the fully developed region, the following condition prevails:
v  0
and
u
 0
x
(8)
Hence, the axial velocity component depends only on r, u(r,x) = u(r).
 The velocity profile for laminar flow in a circular tube is
2

1  dp  2   r  
u r   
  ro 1 
4   dx    r o  
   
 The mean velocity in the tube is
(9)
2
r o dp
um  
8  dx
(10)
8.1.4 Pressure Gradient and Friction Factor in Fully Developed Flow
 The pressure drop needed to sustain an internal flow determines the pump or fan power
required.
 A dimensionless parameter called the friction factor is defined as a function of pressure drop
as
 dp dx  D
2 2
 um
f 
(11)
 For laminar flow, the friction factor is
f 
64
Re
(12)
 For turbulent flow, the friction factor is obtained from the Moody diagram (see figure 8.3 of
the textbook).
 In the fully developed region, the pressure gradient is constant. Hence, the pressure drop
between two axial positions in the tube can be obtained from equation (11) as
p  p1  p2  f
2
 um
2D
x2  x1 
(13)
 The pump or fan power required is
P  p V
(14)
where V is the volumetric flow rate.
 Note: The friction factor, f, should not be confused with the friction coefficient, Cf, (defined in
chapter 6), which is the dimensionless shear stress. The relationship between them is
Cf 
f
4
(15)
8.2 Thermal Considerations
 The thermal boundary layer development in a heated circular tube is as shown below. Here,
the fluid enters the tube with a uniform temperature T(r, 0) which is less than the tube surface
temperature.

The thermal boundary layer develops with increasing x until it reaches the centerline.

If the tube surface is maintained at a uniform temperature (Ts is constant) or a uniform
heat flux (q''s is constant), thermally fully developed (TFD) condition is eventually
reached.

The thermal entry length can be estimated as follows:
x fd , t
D
 0.05 Re D Pr , la min ar
 
, turbulent
 10
(16)
8.2.1 The Mean Temperature
 The temperature of the fluid varies over the cross section of the tube.
 The mean temperature at any axial location in the tube is the temperature that would be
measured if the fluid flowing through the cross-sectional area were collected and mixed in a
cup. It is also commonly referred to as the bulk-stream temperature or mixing-cup
temperature. It is obtained from
m c p Tm 

 Ac  u c p T dAc
 Ac  u c p T dAc
Tm 
m c p
(17)
(18)
 For flow in a circular tube with constant ρ and cp, equation (18) reduces to
Tm 
2

ro
u m ro2 0
where um is the mean velocity.
8.2.2 Newton’s Law of Cooling
u T r dr
(19)
 For internal flows, the mean temperature Tm is the reference temperature (whereas the free
stream temperature T is the reference temperature for external flows).
 The Newton’s law of cooling for internal flow is expressed as
qs  h Ts  Tm 
(20)
where h is the local convection heat transfer coefficient.
 Note that Tm is not constant. It varies in the flow direction since heat transfer is occurring.
8.2.3 Fully Developed Conditions
 Whether the flow is fully developed or not, the fluid temperature continues to change in the
axial direction due to convection heat transfer. Therefore, the temperature profile T(r) is not
suitable for defining the TFD condition.
 The thermally fully developed (TFD) condition is defined in terms of a dimensionless
temperature difference. The requirement for TFD condition is
  Ts x   T r , x  

  0
x  Ts x  Tm x  
(21)
 An important feature of TFD flow is that the local convection coefficient is a constant,
independent of x, i.e.
hx  h  cons tan t
(22)
(This will be shown in class.)
 The axial variation of hx for flow in a tube is shown in figure 8.5 of the textbook.
8.3 The Energy Balance
8.3.1 General Considerations
 Consider the tube flow shown below. Here, fluid moves at a constant flow rate m and
convection heat transfer occurs at the inner surface.

The overall energy balance gives

 c p T m, o  T m, i
qconv  m


Energy balance applied to the differential control volume shown in the figure gives
 c p dTm
dqconv  m

(23)
(24)
Note that
dqconv  qs P dx
(25)
where P is the surface perimeter (P = πD for a circular tube).

Substitute equation 25 and the Newton’s law of cooling (equation 20) into equation 24 to
obtain
dTm
q P
P
 s

h Ts  Tm 
dx
m c p
m c p

(26)
Equation 26 can be used to determine the axial variation of Tm. The solution depends on
the surface thermal condition. Two special cases of interest are constant surface heat flux
(q''s = constant) and constant surface temperature (Ts = constant).
8.3.2 Constant Surface Heat Flux
 For constant surface heat flux in a tube of perimeter P and length L, the total heat transfer rate
is
qconv  qs P  L
(27)
 For constant q''s, the middle expression in equation 26 is a constant independent of x. Hence,
dTm
dx

q s P
 f x 
m c p
(28)
 Integrating 28 from x = 0, it follows that
q P
Tm x   Tm, i  s x
m c p
(29)
 Equation 29 shows that Tm varies linearly with x along the tube.
 The variation of (Ts – Tm) with x is shown in figure 8.7(a) of the textbook. The variation can
be explained with equations 20, 22 and 29 of this note.
Note: If q''s is not constant but is a known function of x, equation 26 may still be integrated to
obtain the variation of Tm with x.
8.3.3 Constant Surface Temperature
 Defining ΔT = Ts – Tm, equation 26 may be expressed as
dTm
dx
 
d T 
P

h T
dx
m c p
(30)
 Separating variables, integrating from the tube inlet to the outlet, we obtain
To
Ti

Ts  Tm, o
Ts  Tm, i
 PL 
 exp  
h
 m c p 


(31)
where h is the average value of h over the entire tube.
 Equation 26 may also be integrated from the tube inlet to some axial position x within the
tube. In this case we get the following general result:

T s  Tm  x 
Px 
 exp  
h
 m c p 
Ts  Tm, i


(32)
where h is the average value of h from the tube inlet to x.
 Equation 32 shows that the temperature difference (Ts – Tm) decays exponentially with
distance along the tube axis. This is displayed in figure 8.7b of the textbook.
 The expression for the total heat transfer rate is obtained by using equation 31 to eliminate
m c p from equation 23. The result is
qconv  h As Tlm
(33)
where As is the tube surface area (As = P.L) and ΔTlm is the log mean temperature difference,
Tlm 
To  Ti
ln To Ti 
(34)
Equation 33 is a form of Newton’s law of cooling for the entire tube.
 In many applications, it is the temperature of an external fluid, rather than the tube surface
temperature Ts that is fixed (as in the figure below). In such cases, equations 31 and 33 may
still be used if Ts is replaced by T and h is replaced by U (the average overall heat transfer
coefficient). It then follows that
To
Ti

T  Tm, o
T  Tm, i
 U As 

 exp  
 m c p 


(35)
and
q  U As Tlm
(36)
[Check chapter 3 for the definition of overall heat transfer coefficient.]
8.4 Laminar Flow in Circular Tubes: Convection Correlations
 For laminar fully developed flow in circular tubes, the Nusselt number is obtained
theoretically as
NuD 
4.36
hD
 
k
3.66
qs  cons tan t
Ts  cons tan t
(37)
 For laminar flow in the entry region, the applicable Nusselt number correlations are presented
in equations 8.56 and 8.57 of the textbook.
Note: to use these Nusselt number correlations, properties, except those referred to the
surface, should be evaluated at the average mean temperature T m  Tm, i  Tm, o  2 .
8.5 Convection Correlations: Turbulent Flow in Circular Tubes
 Many correlations for local Nusselt number are available in the literature. One of such
correlations is as follows:
45
Nu D  0.023 Re D Pr n
(38)
where n = 0.4 for heating (Ts > Tm) and 0.3 for cooling (Ts < Tm).
Equation 38 applies under these conditions: 0.7  Pr  160 ; Re D  10, 000 ; L D 10 .
8.6 Convection Correlations: Noncircular Tubes and the Concentric Tube
Annulus
Noncircular Tubes
 For noncircular tubes, an effective diameter, termed the hydraulic diameter, is used as the
characteristic length.
 The hydraulic diameter is defined as
Dh 
4 Ac
P
(39)
where Ac is the cross-sectional area and P is the wetted perimeter.
 It is the hydraulic diameter that should be used in calculating parameters such as ReD and
NuD.
 For fully developed laminar flow, the Nusselt numbers and friction factors for various cross
sections are presented in Table 8.1.
 For turbulent flow in noncircular tubes, the convection correlations of circular tubes may be
used.
Annulus
 Many internal flow problems involve heat transfer in a concentric tube annulus (as shown
below). Here, fluid passes through the space (annulus) formed by the concentric tubes.
 The ReD and NuD for the annulus are calculated using the hydraulic diameter, Dh, where Dh is
4 Ac
Dh 

P
4 4 Do2  Di2 

  D D
o
i
 Do   Di
(40)