Chapter 8: Internal Forced
Convection
Yoav Peles
Department of Mechanical, Aerospace and Nuclear Engineering
Rensselaer Polytechnic Institute
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives
When you finish studying this chapter, you should be able to:
• Obtain average velocity from a knowledge of velocity profile, and
average temperature from a knowledge of temperature profile in
internal flow,
• Have a visual understanding of different flow regions in internal flow,
such as the entry and the fully developed flow regions, and calculate
hydrodynamic and thermal entry lengths,
• Analyze heating and cooling of a fluid flowing in a tube under
constant surface temperature and constant surface heat flux
conditions, and work with the logarithmic mean temperature
difference,
• Obtain analytic relations for the velocity profile, pressure drop,
friction factor, and Nusselt number in fully developed laminar flow,
and
• Determine the friction factor and Nusselt number in fully developed
turbulent flow using empirical relations, and calculate the pressure
drop and heat transfer rate.
Introduction
•
•
•
•
Pipe ─ circular cross section.
Duct ─ noncircular cross section.
Tubes ─ small-diameter pipes.
The fluid velocity changes from zero at the surface
(no-slip) to a maximum at the pipe center.
• It is convenient to work with an
average velocity, which remains
constant in incompressible flow
when the cross-sectional area
is constant.
Average Velocity
• The value of the average velocity is determined from
the conservation of mass principle
m  Vavg AC 
 u  r  dAC
(8-1)
Ac
• For incompressible flow in a circular pipe of radius R
 u  r  dA
C
Vavg 
Ac
 AC


R
0
 u  r  2 rdr
 R 2
R
2
 2  u  r  rdr
R 0
(8-2)
Average Temperature
• It is convenient to define the value of the mean
temperature Tm from the conservation of
energy principle.
• The energy transported by the fluid through a
cross section in actual flow must be equal to
the energy that would be transported through
the same cross section if the fluid were at a
constant temperature Tm
E fluid  mc pTm   c pT  r   m 
m
  c T  r  u  r VdA
p
Ac
c
(8-3)
• For incompressible flow in a circular pipe of radius R
Tm 
 c pT  r   m
m
mc p
 c T  r  u  r  2 rdr
p

Ac
Vavg  R 2  c p
(8-4)
R
2

T  r  u  r  rdr
2 
Vavg R 0
• The mean temperature Tm of a fluid changes during
heating or cooling.
Idealized
Actual
Laminar and Turbulent Flow in
Tubes
• For flow in a circular tube, the Reynolds number is
defined as
Vavg D Vavg D
Re 



(8-5)
• For flow through noncircular tubes D is replaced by
the hydraulic diameter Dh.
4 Ac
Dh 
(8-6)
P
• laminar flow: Re<2300
• fully turbulent: Re>10,000.
The Entrance Region
• Consider a fluid entering a circular pipe at a uniform
velocity.
• Because of the no-slip condition a velocity gradient
develops along the pipe.
• The flow in a pipe is divided into two regions:
– the boundary layer region, and
– the and the irrotational (core) flow region.
• The thickness of this
boundary layer
increases in the flow
direction until it
reaches the pipe
center.
Irrotational Boundary
layer
flow
• Hydrodynamic entrance region ─ the region from
the pipe inlet to the point at which the boundary layer
merges at the centerline.
• Hydrodynamically fully developed region ─ the
region beyond the entrance region in which the
velocity profile is fully developed and remains
unchanged.
• The velocity profile in the fully developed region is
– parabolic in laminar flow, and
– somewhat flatter or fuller in turbulent flow.
Thermal Entrance Region
• Consider a fluid at a uniform temperature entering a circular
tube whose surface is maintained at a different temperature.
• Thermal boundary layer along the tube is developing.
• The thickness of this boundary layer increases in the flow
direction until the boundary layer reaches the tube center.
• Thermal entrance region.
• Thermally fully developed region ─ the region beyond the
thermal entrance region in which the dimensionless
temperature profile
expressed as
(Ts-T)/(Ts-Tm)
remains unchanged.
– Hydrodynamically fully developed:
u  r , x 
 0  u  u r 
x
– Thermally fully developed:
  Ts  x   T  r , x  

0
x  Ts  x   Tm  x  
(8-7)
(8-8)
  T r  r  R
  Ts  T 

 f  x  (8-9)


r  Ts  Tm  r  R
Ts  Tm
• Surface heat flux can be expressed as
k  T r  r  R
T
qs  hx Ts  Tm   k
 hx 
(8-10)
r r=R
Ts  Tm
• For thermally fully developed region From (Eq. (8-9))
 T
r  r  R
Ts  Tm
 f  x
hx  f  x 
Fully developed flow
hx  constant Fully developed flow
The Heat Transfer coefficient and
Friction factor
Developing
region
Fully
developed
region
Entry Lengths
Laminar flow
– Hydrodynamic
Lh,laminar  0.05Re D
(8-11)
– Thermal
Lt ,laminar  0.05Re Pr D  Pr Lh,laminar
(8-12)
Turbulent flow
– Hydrodynamic
Lh,turbulent  1.359D  Re1 4
(8-13)
– Thermal (approximate)
Lh,turbulent  Lt ,turbulent  10D
(8-14)
Turbulent flow Nusselt Number
• The Nusselt numbers are much
higher in the entrance region.
• The Nusselt number reaches
a constant value at a distance
of less than 10 diameters.
• The Nusselt numbers for the
uniform surface temperature and uniform surface heat
flux conditions are identical in the fully developed
regions, and nearly identical in the entrance regions.
 Nusselt number is insensitive to the type of
thermal boundary condition.
General Thermal Analysis
• In the absence of any work interactions, the conservation
of energy equation for the steady flow of a fluid in a tube
Q  mcp Te  Ti 
(W)
(8-15)
• The thermal conditions at the surface can usually be
approximated as:
– constant surface temperature, or
– constant surface heat flux.
• The mean fluid temperature Tm must
change during heating or cooling.
• Either Ts= constant or qs = constant at the surface of a
tube, but not both.
Constant Surface Heat Flux
• In the case of constant heat flux, the rate of heat transfer can
also be expressed as
(8-17)
Q  qs As  mcp Te  Ti  (W)
• Then the mean fluid temperature at the tube exit becomes
qs As
(8-18)
Te  Ti 
mc p
• The surface temperature in the case of constant surface heat
flux can be determined from
qs
(8-19)
qs  h Ts  Tm   Ts  Tm 
h
• In the fully developed region, the
surface temperature Ts will also
increase linearly in the flow direction
• Applying the steady-flow energy
balance to a tube slice of thickness
dx, the slope of the mean fluid
temperature Tm can be determined
dTm qs p
mc p dTm  qs  pdx  

 constant (8-20)
dx mc p
• Noting that both the heat flux and h
(for fully developed flow) are
constants
dTm dTs

(8-21)
dx
dx
• In the fully developed region (Ts-Tm=constant)
  Ts  T 
1  Ts T 
T dTs

0


0


x  Ts  Tm 
Ts  Tm  x x 
x dx
(8-22)
• Combining Eqs. 8–20, 8–21, and 8–22 gives
T dTs dTm qs p



 constant
x dx
dx mc p
(8-23)
• For a circular tube
2qs
T dTs dTm



 constant (8-24)
x dx
dx Vavg c p R
Constant Surface Temperature
• The energy balance on a differential control volume
 Q  mcp dTm  h Ts  Tm  dAs
(8-27)
• Since the mean temperature of the fluid Tm increases in
the flow direction the heat flux decays with x.
• The surface temperature is constant (dTm=-d(Ts-Tm)) and
dAs=pdx, therefore,
d Ts  Tm 
Ts  Tm
hp

dx
mc p
(8-28)
• Integrating Eq. 6-28 from x=0 (tube inlet
where Tm=Ti) to x=L (tube exit where Tm=Te)
gives
Ts  Te
hAs
ln

Ts  Ti
mc p
(8-29)
• Taking the exponential of both sides and
solving for Te
• or
Te  Ts  Ts  Ti  exp   hpL mc p 
Tm  x   Ts  Ts  Ti  exp   hpx mc p 
(8-30)
• The temperature difference between the fluid and the
surface decays exponentially in the flow direction, and the
rate of decay depends on the magnitude of the exponent
hAs mc p
• This dimensionless parameter is
called the number of transfer
units (NTU).
– Large NTU value – increasing tube
length marginally increases heat
transfer rate.
– Small NTU value – heat transfer increases
significantly with increasing tube length.
• Solving Eq. 8–29 for mcp gives
hAs
mc p 
ln Ts  Te  Ts  Ti 
(8-31)
• Substituting this into Eq. 8–15
Q  mcp  hAs DTln
where
(8-32)
Q  mcp Te  Ti 
Ti  Te
DTe  DTi
DTln 

ln Ts  Te  Ts  Ti  ln  DTe DTi 
(8-33)
DTln is the logarithmic mean temperature
difference.
(W)
Laminar Flow in Tubes
•
•
•
•
•
Assumptions:
steady laminar flow,
incompressible fluid,
constant properties,
fully developed region,
and
straight circular tube.
• The velocity profile u(r)
remains unchanged in
the flow direction.
• no motion in the radial
direction.
• no acceleration.
• Consider a ring-shaped
differential volume element.
• A force balance on the volume
element in the flow direction
gives
 2 rdrP  x   2 rdrP  xdx
  2 rdr r   2 rdr r  dr  0
(8-34)
• Dividing by 2drdx and rearranging
Px  dx  Px  r r  dr   r r
r

0
dx
dr
(8-35)
• Taking the limit as dr, dx → 0 gives
dP d  r 
r

0
dx
dr
(8-36)
• Substituting =(du/dr) gives
 d  du 
dP
r  
r dr  dr  dx
(8-37)
1  dP 
u r  

  C1 ln r  C2
4  dx 
(8-38)
• Rearranging and integrating it twice to give
• Boundary Conditions:
– symmetry about the centerline ∂u/∂r=0 at r=0,
– no-slip condition u=0 at r=R.
• Eq. 6-38 with the boundary conditions
R 2  dP   r 2 
(8-39)
u r   
1





4  dx   R 2 
• Substituting Eq. 8–39 into Eq. 8–2, and performing the
integration gives the average velocity
R
R
2
2

2
2 R  dP 
r 
Vavg  2  u  r  rdr   2 

 1  2  rdr
R 0
R 0 4   dx   R 
R 2  dP 
(8-40)



8  dx 
• Combining the last two equations, the velocity profile is
rewritten as
 r2 
u  r   2Vavg 1  2  ; umax  2Vavg (8-41)
 R 
Pressure Drop
• One implication from Eq. 8-37 is that the
pressure drop gradient (dP/dx) must be constant
(the left side is a function only of r, and the right
side is a function only of x).
• Integrating from x=x1 where the pressure is P1 to
x=x1=L where the pressure is P2 gives
dP P2  P1

dx
L
(8-43)
• Substituting Eq. 8–43 into the Vavg expression in
Eq. 8–40
8 LV
32 LV
DP  P1  P2 
avg
R
2

avg
D
2
(8-44)
• A pressure drop due to viscous effects represents an
irreversible pressure loss.
• It is convenient to express the pressure loss for all
types of fully developed internal flows in terms of the
dynamic pressure and the friction factor
dynamic pressure
friction factor
DPL 
f
L
 
D
2
Vavg
(8-45)
2
• Setting Eqs. 8–44 and 8–45 equal to each other and
solving for f gives
– Circular tube, laminar:
64
64
f 

 DVavg Re (8-46)
Temperature Profile and the Nusselt
Number
• Energy is transferred by mass in the
x-direction, and by conduction in the
r-direction.
• The steady flow energy balance for a
cylindrical shell element can be
expressed as mcpTx  mcpTxdx  Qr  Qr dr  0
• Substituting
m  uAc  u  2 rdr 
and dividing by 2rdrdx gives, after rearranging
Tx  dx  Tx
1 Qr  dr  Qr
 c pu

dx
2 rdx
dr
(8-49)
(8-50)
• Or
T
1
Q
u

x
2 c p rdx r
(8-51)
Q  
T 
  T 
• Since
  k 2 rdx
  2 kdx  r

r r 
r 
r  r 
(8-52)
Eq 8-51 becomes
T    T 
u

r

x r dr  r 
;
k

cp
(8-53)
Constant Surface Heat Flux
• Substituting Eqs. 8-24 and 8-41 into Eq. 8.53
 r2 
u  r   2Vavg 1  2 
 R 
2qs
T

 constant
x Vavg c p R
(8-41)
(8-24)
T    T 
u

r

x r dr  r 
4 qs
kR

r2
1  2
 R
(8-53)
 1 d  dT  (8-55)

r

 r dr  dr 
• Separating the variables and integrating twice
4

qs 2 r 
T
 r  2   C1r  C2
kR 
4R 
(8-56)
• Boundary conditions
– Symmetry at r=0:
– At r=R:
T  r  0 
r
0
T(r=R)=Ts
qs R  3 r 2
r4 
T  Ts 
  2 4
k  4 R 4R 
C1=0
C2
(8-57)
• The bulk mean temperature Tm is determined by substituting
the velocity and temperature profile relations (Eqs. 8–41
and 8–57) into Eq. 8–4 and performing the integration
(8-58)
11 qs R
Tm  Ts 
24 k
qs  h Ts  Tm 
24 k 48 k
k
h

 4.36
11 R 11 D
D
(8-59)
Constant heat flux (circular tube, laminar)
hD
Nu 
 4.36
k
(8-60)
Constant Surface temperature (circular tube, laminar)
hD
Nu 
 3.66
k
(8-61)
Laminar Flow in Noncircular Tubes
• The friction factor (f)
and the Nusselt number
relations are given in
Table 8–1 for fully
developed laminar flow
in tubes of various cross
sections.
Developing Laminar Flow in the
Entrance Region
• For a circular tube of length L subjected to constant
surface temperature, the average Nusselt number for
the thermal entrance region (hydrodynamically
developed flow)
0.065  D L  Re Pr
(8-62)
Nu  3.66 
23
1  0.04  D L  Re Pr 
• For flow between isothermal parallel plates
Nu  7.54 
0.03  Dh L  Re Pr
1  0.016  Dh L  Re Pr 
23
(8-64)
Turbulent flow in Tubes
• Most correlations for the friction and heat transfer
coefficients in turbulent flow are based on experimental
studies.
• For smooth tubes, the friction factor in turbulent flow can
be determined from the explicit first Petukhov equation
f   0.79 ln Re 1.64 
2
3000<Re<5 106 (8-65)
• For fully developed turbulent flow the Nusselt number
(Dittus–Boelter equation)
n  0.4 heating 
0.8
n Re  10, 000
Nu  0.023Re Pr 

0.7  Pr  160 n  0.3 cooling 
(8-68)
• Modified correlations are available for/due to :
– liquid metals (Pr<<1),
– large variation in fluid properties due to a large
temperature difference,
– surface roughness,
– flow through tube annulus.
• Original correlations are also approximately
valid for:
– developing Turbulent Flow in the Entrance
Region,
– turbulent Flow in Noncircular Tubes.