Internal Flow:
Heat Transfer Correlations
Chapter 8
Sections 8.4 through 8.8
Fully Developed Flow
Fully Developed Flow
• Laminar Flow in a Circular Tube:
The local Nusselt number is a constant throughout the fully developed
region, but its value depends on the surface thermal condition.
– Uniform Surface Heat Flux (qs ) :
NuD  hD  4.36
k
– Uniform Surface Temperature (Ts ):
NuD  hD  3.66
k
• Turbulent Flow in a Circular Tube:
– For a smooth surface and fully turbulent conditions  Re D  10,000  , the
Dittus – Boelter equation may be used as a first approximation:
n  0.3 Ts  Tm 
NuD  0.023Re4D/ 5 Pr n
n  0.4 Ts  Tm 
– The effects of wall roughness and transitional flow conditions  Re D  3000 
may be considered by using the Gnielinski correlation:
 f / 8 Re D  1000  Pr
NuD 
1/ 2
1  12.7  f / 8   Pr 2 / 3  1
Fully Developed (cont.)
Smooth surface:
f   0.790 1n Re D  1.64 
2
Surface of roughness e  0 :
f  Figure 8.3
• Noncircular Tubes:
– Use of hydraulic diameter as characteristic length:
4A
Dh  c
P
– Since the local convection coefficient varies around the periphery of a tube,
approaching zero at its corners, correlations for the fully developed region
are associated with convection coefficients averaged over the periphery
of the tube.
– Laminar Flow:
The local Nusselt number is a constant whose value (Table 8.1) depends on
the surface thermal condition Ts or qs  and the duct aspect ratio.
– Turbulent Flow:
As a first approximation, the Dittus-Boelter or Gnielinski correlation may be used
with the hydraulic diameter, irrespective of the surface thermal condition.
Entry Region
Effect of the Entry Region
• The manner in which the Nusselt decays from inlet to fully developed conditions
for laminar flow depends on the nature of thermal and velocity boundary layer
development in the entry region, as well as the surface thermal condition.
Laminar flow in a
circular tube.
– Combined Entry Length:
Thermal and velocity boundary layers develop concurrently from uniform
profiles at the inlet.
Entry Region (cont)
– Thermal Entry Length:
Velocity profile is fully developed at the inlet, and boundary layer development
in the entry region is restricted to thermal effects. Such a condition may also
be assumed to be a good approximation for a uniform inlet velocity profile if
Pr  1. Why?
• Average Nusselt Number for Laminar Flow in a Circular Tube with Uniform
Surface Temperature:
– Combined Entry Length:
 Re D Pr/  L / D  
1/ 3
  / s 0.14  2 :
1/ 3
 Re Pr 
Nu D  1.86  D 
 L/ D 
 Re D Pr/  L / D  
1/ 3

 
 s
0.14
  / s 0.14  2 :
Nu D  3.66
– Thermal Entry Length:
Nu D  3.66 
0.0668  D / L  Re D Pr
1  0.04  D / L  Re D Pr 
2/3
Entry Region (cont)
• Average Nusselt Number for Turbulent Flow in a Circular Tube :
– Effects of entry and surface thermal conditions are less pronounced for
turbulent flow and can be neglected.
– For long tubes  L / D  60  :
Nu D  NuD, fd
– For short tubes  L / D  60  :
Nu D  1 
C
NuD , fd
 L / D m
C 1
m  2/3
• Noncircular Tubes:
– Laminar Flow:
Nu Dh depends strongly on aspect ratio, as well as entry region and surface
thermal conditions. See references 11 and 12.
Entry Region (cont)
– Turbulent Flow:
As a first approximation, correlations for a circular tube may be used
with D replaced by Dh .
• When determining Nu D for any tube geometry or flow condition, all
properties are to be evaluated at
T m  Tm,i  Tm,o  / 2
Why do solutions to internal flow problems often require iteration?
Annulus
The Concentric Tube Annulus
• Fluid flow through
region formed by
concentric tubes.
• Convection heat transfer
may be from or to inner
surface of outer tube and
outer surface of inner tube.
• Surface thermal conditions may be characterized by
uniform temperature Ts ,i , Ts ,o  or uniform heat flux  qi, qo  .
• Convection coefficients are associated with each surface, where
qi  hi Ts ,i  Tm 
qo  ho Ts ,o  Tm 
Annulus (cont)
Nui 
hi Dh
k
Nuo 
ho Dh
k
Dh  Do  Di
• Fully Developed Laminar Flow
Nusselt numbers depend on Di / Do and surface thermal conditions (Tables 8.2, 8.3)
• Fully Developed Turbulent Flow
Correlations for a circular tube may be used with D replaced by Dh .
Problem: Solar Collector
Problem 8.30: Determine the effect of flow rate on outlet temperature
and heat rate for water flow through the tube of a
flat-plate solar collector.
KNOWN: Diameter and length of copper tubing. Temperature of collector plate to which
tubing is soldered. Water inlet temperature and flow rate.
FIND: (a) Water outlet temperature and heat rate for m  0.01 kg / s, (b) Variation of outlet
temperature and heat rate with flow rate.
ASSUMPTIONS: (1) Straight tube with smooth surface, (2) Negligible kinetic/potential energy
and flow work changes, (3) Negligible thermal resistance between plate and tube inner surface,
(4) ReD,c = 2300.
PROPERTIES: Table A.6, water (assume Tm = (Tm,i + Ts)/2 = 47.5C = 320.5 K):  = 986
kg/m3, cp = 4180 J/kgK,  = 577  10-6 Ns/m2, k = 0.640 W/mK, Pr = 3.77. Table A.6, water
(Ts = 343 K): s = 400  10-6 Ns/m2.
Problem: Solar Collector (cont)
ANALYSIS: (a) For m = 0.01 kg/s, ReD = 4 m  D = 4(0.01 kg/s)/(0.01 m)577  10-6
Ns/m2 = 2200, in which case the flow may be assumed to be laminar.
With x fd,t D  0.05ReDPr = 0.05(2200)(3.77) = 415 and L/D = 800, the flow is fully
developed over approximately 50% of the tube length.
With ReD Pr  L D    s 
average convection coefficient
1/ 3
0.14
1/ 3
 Re Pr 
Nu D  1.86  D 
 LD 
= 2.30, Eq. 8.57 may therefore be used to compute the
  
 
 s 
0.14
 4.27
h   k D Nu D  4.27  0.640 W m  K  0.01m  273W m2  K
From Eq. 8.42b,
   0.01m  8m  273W m 2  K 
  DL 

 exp  
h   exp  
 mcp 


Ts  Tm,i
0.01kg
s

4180
J
kg

K




Ts  Tm,o


Tm,o  Ts  0.194 Ts  Tm,i  70 C  8.7 C  61.3 C


q  mcp Tm,o  Tm,i  0.01kg s  4186J kg  K 36.3K   1519 W
Problem: Solar Collector (cont)
The effect of m is considered in two steps, the first corresponding to m < 0.011 kg/s (ReD <
2300) and the second for m > 0.011 kg/s (ReD > 2300). In the first case, Eq. 8.57 is used to
determine h , and in the second case Eq. 8.60 is used. The effects of m are as follows.
70
67
Outlet temperature, Tmo(C)
Outlet temperature, Tmo(C)
66
65
64
63
62
69.8
69.6
69.4
69.2
61
69
60
0.005
0.006
0.007
0.008
0.009
0.01
0.01
0.011
0.02
0.04
0.05
0.04
0.05
Mass flowrate, mdot(kg/s)
Mass flowrate, mdot(kg/s)
1700
0.03
T urbulent flow (ReD>2300)
Laminar flow (ReD < 2300)
9500
1600
7500
1400
Heat rate, q(W)
Heat rate, q(W)
1500
1300
1200
1100
5500
3500
1000
900
1500
800
0.005
0.006
0.007
0.008
0.009
Mass flowrate, mdot(kg/s)
Laminar flow (ReD < 2300)
0.01
0.011
0.01
0.02
0.03
Mass flowrate, mdot(kg/s)
T urbulent flow (ReD>2300)
Problem: Solar Collector (cont)
The outlet temperature decreases with increasing m , although the effect is more pronounced for
laminar flow. If q were independent of m , (Tm,o - Tm,i) would decrease inversely with increasing
m . In turbulent flow, however, the convection coefficient, and hence the heat rate, increases
approximately as m0.8 , thereby attenuating the foregoing effect. In laminar flow, q ~ m0.5 and
this attenuation is not as pronounced.
Problem: Oven Exhaust Gases
Problem 8.54: Determine effect of ambient air temperature and wind
velocity on temperature at which oven gases are discharged
from a stack.
KNOWN: Thin-walled, tall stack discharging exhaust gases from an oven into the environment.
Problem: Oven Exhaust Gases (cont)
FIND: (a) Outlet gas and stack surface temperatures, Tm,o and Ts,o, for V=5 m/s and T  4 C ;
(b) Effect of wind temperature and velocity on T m,o.
ASSUMPTIONS: (1) Steady-state conditions, (2) Negligible wall thermal resistance, (3)
Exhaust gas properties approximate those of atmospheric air, (4) Negligible radiative exchange
with surroundings, (5) Negligible PE, KE, and flow work changes, (6) Fully developed flow, (7)
Constant properties.
PROPERTIES: Table A.4, air (assume Tm,o = 773 K, Tm = 823 K, 1 atm): cp = 1104 J/kgK, 
= 376.4  10-7 Ns/m2, k = 0.0584 W/mK, Pr = 0.712; Table A.4, air (assume Ts = 523 K, T =
4C = 277 K, Tf = 400 K, 1 atm):  = 26.41  10-6 m2/s, k = 0.0338 W/mK, Pr = 0.690.
Problem: Oven Exhaust Gases (cont)
ANALYSIS: (a) From Eq. 8.46a,
 PL 
Tm,o  T  T  Tm,i exp  
U
 mcp 


 1
1 
U 1  

 hi h o 
Internal flow: With a Reynolds number of
4  0.5 kg s
4m
 33,827

ReDi 
 D   0.5 m  376.4 107 N  s m 2
The flow is turbulent, and assuming fully developed conditions throughout the stack, the DittusBoelter correlation may be used to determine hi .
hD
Nu D  i  0.023Re4 / 5 Pr 0.3
Di
k
58.4  103 W m  K
4/5
hi 
 0.023  33,827 
 0.712 0.3  10.2 W m2  K
0.5m
External flow: Working with the Churchill/Bernstein correlation and
ReDo 
VD

Nu D  0.3 

5 m s  0.5 m
26.41106 m 2 s
 94, 660
2 1/ 3
0.62 Re1/
D Pr
2 / 3 1/ 4
1   0.4 Pr 



5/ 8
  Re
 
D
1  


  282, 000  


4/5
 205
Problem: Oven Exhaust Gases (cont)
Hence,
ho   0.0338W m  K 0.5m   205  13.9 W m2  K
The outlet gas temperature is then



  0.5 m  6 m
1
2
Tm,o  4 C   4  600  C exp  
W
m

K

  543 C
0.5
kg
s

1104
J
kg

K
1
10.2

1
13.9



The outlet stack surface temperature can be determined from a local surface energy balance of
the form,
hi(Tm,o - Ts,o) = ho(Ts,o - T),
which yields
Ts,o 
hiTm,o  h oT
hi  h o
10.2  543  13.9  4  W m2


 232
2 K
10.2

13.9
W
m


C
Problem: Oven Exhaust Gases (cont)
b) The effects of the air temperature and velocity are as follows
Gas outlet temperature, Tmo(C)
560
550
540
530
520
2
3
4
5
6
7
8
9
10
Freestream veloci ty, V(m /s)
T inf = 35 C
T inf = 5 C
T inf = -25C
Due to the elevated temperatures of the gas, the variation in ambient temperature has only a small
effect on the gas exit temperature. However, the effect of the freestream velocity is more
pronounced. Discharge temperatures of approximately 530 and 560C would be representative of
cold/windy and warm/still atmospheric conditions, respectively.
COMMENTS: If there are constituents in the gas discharge that condense or precipitate out at
temperatures below Ts,o, related operating conditions should be avoided.