406
HEAT TRANSFER
which is not bad for a day’s worth of work. The plant manager is also a big win-
ner in this venture since the heat losses will cost him only $320/yr during the
second and consequent years instead of $8093/yr. A thicker insulation could
probably be justified in this case if the cost of insulation is annualized over the
lifetime of insulation, say 20 years. Several energy conservation measures are
being marketed as explained above by several power companies and private
firms.
SUMMARY
The force a flowing fluid exerts on a body in the flow direction 1.328
Laminar: Cf Re L 5 10 5
is called drag. The part of drag that is due directly to wall shear Re 1/2
L
stress w is called the skin friction drag since it is caused by
0.074
frictional effects, and the part that is due directly to pressure is Turbulent: Cf 5 10 5 Re L 10 7
called the pressure drag or form drag because of its strong de- Re 1/5
L
0.074 1742
pendence on the form or shape of the body. Combined: Cf 5 10 5 ReL 10 7
The drag coefficient CD is a dimensionless number that rep- Re1/5
L ReL
resents the drag characteristics of a body, and is defined as
2.5
Rough surface, turbulent: Cf 1.89 1.62 log
FD L
CD 1 2
2
A
The average Nusselt number relations for flow over a flat plate
where A is the frontal area for blunt bodies, and surface area are:
for parallel flow over flat plates or thin airfoils. For flow over
a flat plate, the Reynolds number is hL
Laminar: Nu 0.664 Re 0.5
L Pr
1/3
ReL 5 10 5
x x k
Rex
Turbulent:
hL 0.6 Pr 60
Transition from laminar to turbulent occurs at the critical Nu 0.037 Re0.8
L Pr
1/3
Reynolds number of k 5 10 5 ReL 10 7
xcr Combined:
Re x, cr 5 10 5
hL 0.6 Pr 60
Nu (0.037 Re0.8
L 871) Pr ,
1/3
k 5 10 5 ReL 10 7
For parallel flow over a flat plate, the local friction and con-
vection coefficients are
For isothermal surfaces with an unheated starting section of
0.664 length , the local Nusselt number and the average convection
Laminar: C f, x Re x 5 10 5
Re 1/2
x coefficient relations are
hx x
Nu x 0.332 Re0.5
x Pr
1/3
Pr 0.6 Nux (for 0)
k 0.332 Re0.5
x Pr
1/3
Laminar: Nux
0.0592 [1 ( /x) 3/4 ]1/3 [1 ( /x) 3/4 ]1/3
Turbulent: Cf, x , 5 10 5 Re x 10 7 Nu x (for 0)
Re 1/5 0.0296 Re0.8x Pr
1/3
x
Turbulent: Nu x
hx x 0.6 Pr 60 [1 (/x)9/10 ]1/ 9 [1 (/x) 9/10 ]1/9
Nu x 0.0296 Re 0.8
x Pr
1/3
2[1 ( /x) 3/4]
k 5 10 5 Re x 10 7 Laminar: h hxL
1 /L
The average friction coefficient relations for flow over a flat 5[1 ( /x) 9/10
plate are: Turbulent: h hxL
(1 / L)
, 407
CHAPTER 7
These relations are for the case of isothermal surfaces. When a hD
Nu D C Re Dm Pr n(Pr/Prs) 0.25
flat plate is subjected to uniform heat flux, the local Nusselt k
number is given by
where the values of the constants C, m, and n depend on value
Laminar: Nux 0.453 Re 0.5
x Pr
1/3
Reynolds number. Such correlations are given in Table 7–2.
Turbulent: Nux 0.0308 Re 0.8
x Pr
1/3 All properties except Prs are to be evaluated at the arithmetic
mean of the inlet and outlet temperatures of the fluid defined as
The average Nusselt numbers for cross flow over a cylinder Tm (Ti Te)/2.
and sphere are The average Nusselt number for tube banks with less than
16 rows is expressed as
0.62 Re1/2 Pr1/3
hD Re 5/8 4/5
Nucyl 0.3 1 Nu D, NL F NuD
k [1 (0.4/ Pr)2/3]1/4 282,000
where F is the correction factor whose values are given in
which is valid for Re Pr 0.2, and Table 7-3. The heat transfer rate to or from a tube bank is de-
termined from
hD 1/4
Nu sph 2 [0.4 Re1/2 0.06 Re 2/3]Pr 0.4
k s Q̇ h A s
Tln ṁCp (Te Ti)
which is valid for 3.5 Re 80,000 and 0.7 Pr 380. where
Tln is the logarithmic mean temperature difference de-
The fluid properties are evaluated at the film temperature fined as
Tf (T Ts)/2 in the case of a cylinder, and at the free- (Ts Te ) (Ts Ti )
Te
Ti
stream temperature T (except for s , which is evaluated at the
Tln
ln[(Ts Te )/(Ts Ti )] ln(
Te /
Ti)
surface temperature Ts) in the case of a sphere.
In tube banks, the Reynolds number is based on the maxi- and the exit temperature of the fluid Te is
mum velocity max that is related to the approach velocity as
ṁC
Ash
Te Ts (Ts Ti ) exp
In-line and Staggered with SD (ST D)/2: p
ST
max where As NDL is the heat transfer surface area and m·
ST D (NT ST L) is the mass flow rate of the fluid. The pressure
Staggered with SD (ST D)/2: drop
P for a tube bank is expressed as
ST 2max
max
P NL f
2(SD D) 2
where ST the transverse pitch and SD is the diagonal pitch. The where f is the friction factor and is the correction factor, both
average Nusselt number for cross flow over tube banks is ex- given in Figs. 7–27.
pressed as
REFERENCES AND SUGGESTED READING
1. R. D. Blevin. Applied Fluid Dynamics Handbook. 5. W. H. Giedt. “Investigation of Variation of Point Unit-
New York: Van Nostrand Reinhold, 1984. Heat Transfer Coefficient around a Cylinder Normal to an
2. S. W. Churchill and M. Bernstein. “A Correlating Air Stream.” Transactions of the ASME 71 (1949),
Equation for Forced Convection from Gases and Liquids pp. 375–381.
to a Circular Cylinder in Cross Flow.” Journal of Heat 6. M. Jakob. Heat Transfer. Vol. l. New York: John Wiley &
Transfer 99 (1977), pp. 300–306. Sons, 1949.
3. S. W. Churchill and H. Ozoe. “Correlations for Laminar 7. W. M. Kays and M. E. Crawford. Convective Heat and
Forced Convection in Flow over an Isothermal Flat Plate Mass Transfer. 3rd ed. New York: McGraw-Hill, 1993.
and in Developing and Fully Developed Flow in an
Isothermal Tube.” Journal of Heat Transfer 95 (Feb. 8. F. Kreith and M. S. Bohn. Principles of Heat Transfer, 6th
1973), pp. 78–84. ed. Pacific Grove, CA: Brooks/Cole, 2001.
4. W. M. Edmunds. “Residential Insulation.” ASTM 9. H. Schlichting. Boundary Layer Theory, 7th ed. New
Standardization News (Jan. 1989), pp. 36–39. York, McGraw-Hill, 1979.
HEAT TRANSFER
which is not bad for a day’s worth of work. The plant manager is also a big win-
ner in this venture since the heat losses will cost him only $320/yr during the
second and consequent years instead of $8093/yr. A thicker insulation could
probably be justified in this case if the cost of insulation is annualized over the
lifetime of insulation, say 20 years. Several energy conservation measures are
being marketed as explained above by several power companies and private
firms.
SUMMARY
The force a flowing fluid exerts on a body in the flow direction 1.328
Laminar: Cf Re L 5 10 5
is called drag. The part of drag that is due directly to wall shear Re 1/2
L
stress w is called the skin friction drag since it is caused by
0.074
frictional effects, and the part that is due directly to pressure is Turbulent: Cf 5 10 5 Re L 10 7
called the pressure drag or form drag because of its strong de- Re 1/5
L
0.074 1742
pendence on the form or shape of the body. Combined: Cf 5 10 5 ReL 10 7
The drag coefficient CD is a dimensionless number that rep- Re1/5
L ReL
resents the drag characteristics of a body, and is defined as
2.5
Rough surface, turbulent: Cf 1.89 1.62 log
FD L
CD 1 2
2
A
The average Nusselt number relations for flow over a flat plate
where A is the frontal area for blunt bodies, and surface area are:
for parallel flow over flat plates or thin airfoils. For flow over
a flat plate, the Reynolds number is hL
Laminar: Nu 0.664 Re 0.5
L Pr
1/3
ReL 5 10 5
x x k
Rex
Turbulent:
hL 0.6 Pr 60
Transition from laminar to turbulent occurs at the critical Nu 0.037 Re0.8
L Pr
1/3
Reynolds number of k 5 10 5 ReL 10 7
xcr Combined:
Re x, cr 5 10 5
hL 0.6 Pr 60
Nu (0.037 Re0.8
L 871) Pr ,
1/3
k 5 10 5 ReL 10 7
For parallel flow over a flat plate, the local friction and con-
vection coefficients are
For isothermal surfaces with an unheated starting section of
0.664 length , the local Nusselt number and the average convection
Laminar: C f, x Re x 5 10 5
Re 1/2
x coefficient relations are
hx x
Nu x 0.332 Re0.5
x Pr
1/3
Pr 0.6 Nux (for 0)
k 0.332 Re0.5
x Pr
1/3
Laminar: Nux
0.0592 [1 ( /x) 3/4 ]1/3 [1 ( /x) 3/4 ]1/3
Turbulent: Cf, x , 5 10 5 Re x 10 7 Nu x (for 0)
Re 1/5 0.0296 Re0.8x Pr
1/3
x
Turbulent: Nu x
hx x 0.6 Pr 60 [1 (/x)9/10 ]1/ 9 [1 (/x) 9/10 ]1/9
Nu x 0.0296 Re 0.8
x Pr
1/3
2[1 ( /x) 3/4]
k 5 10 5 Re x 10 7 Laminar: h hxL
1 /L
The average friction coefficient relations for flow over a flat 5[1 ( /x) 9/10
plate are: Turbulent: h hxL
(1 / L)
, 407
CHAPTER 7
These relations are for the case of isothermal surfaces. When a hD
Nu D C Re Dm Pr n(Pr/Prs) 0.25
flat plate is subjected to uniform heat flux, the local Nusselt k
number is given by
where the values of the constants C, m, and n depend on value
Laminar: Nux 0.453 Re 0.5
x Pr
1/3
Reynolds number. Such correlations are given in Table 7–2.
Turbulent: Nux 0.0308 Re 0.8
x Pr
1/3 All properties except Prs are to be evaluated at the arithmetic
mean of the inlet and outlet temperatures of the fluid defined as
The average Nusselt numbers for cross flow over a cylinder Tm (Ti Te)/2.
and sphere are The average Nusselt number for tube banks with less than
16 rows is expressed as
0.62 Re1/2 Pr1/3
hD Re 5/8 4/5
Nucyl 0.3 1 Nu D, NL F NuD
k [1 (0.4/ Pr)2/3]1/4 282,000
where F is the correction factor whose values are given in
which is valid for Re Pr 0.2, and Table 7-3. The heat transfer rate to or from a tube bank is de-
termined from
hD 1/4
Nu sph 2 [0.4 Re1/2 0.06 Re 2/3]Pr 0.4
k s Q̇ h A s
Tln ṁCp (Te Ti)
which is valid for 3.5 Re 80,000 and 0.7 Pr 380. where
Tln is the logarithmic mean temperature difference de-
The fluid properties are evaluated at the film temperature fined as
Tf (T Ts)/2 in the case of a cylinder, and at the free- (Ts Te ) (Ts Ti )
Te
Ti
stream temperature T (except for s , which is evaluated at the
Tln
ln[(Ts Te )/(Ts Ti )] ln(
Te /
Ti)
surface temperature Ts) in the case of a sphere.
In tube banks, the Reynolds number is based on the maxi- and the exit temperature of the fluid Te is
mum velocity max that is related to the approach velocity as
ṁC
Ash
Te Ts (Ts Ti ) exp
In-line and Staggered with SD (ST D)/2: p
ST
max where As NDL is the heat transfer surface area and m·
ST D (NT ST L) is the mass flow rate of the fluid. The pressure
Staggered with SD (ST D)/2: drop
P for a tube bank is expressed as
ST 2max
max
P NL f
2(SD D) 2
where ST the transverse pitch and SD is the diagonal pitch. The where f is the friction factor and is the correction factor, both
average Nusselt number for cross flow over tube banks is ex- given in Figs. 7–27.
pressed as
REFERENCES AND SUGGESTED READING
1. R. D. Blevin. Applied Fluid Dynamics Handbook. 5. W. H. Giedt. “Investigation of Variation of Point Unit-
New York: Van Nostrand Reinhold, 1984. Heat Transfer Coefficient around a Cylinder Normal to an
2. S. W. Churchill and M. Bernstein. “A Correlating Air Stream.” Transactions of the ASME 71 (1949),
Equation for Forced Convection from Gases and Liquids pp. 375–381.
to a Circular Cylinder in Cross Flow.” Journal of Heat 6. M. Jakob. Heat Transfer. Vol. l. New York: John Wiley &
Transfer 99 (1977), pp. 300–306. Sons, 1949.
3. S. W. Churchill and H. Ozoe. “Correlations for Laminar 7. W. M. Kays and M. E. Crawford. Convective Heat and
Forced Convection in Flow over an Isothermal Flat Plate Mass Transfer. 3rd ed. New York: McGraw-Hill, 1993.
and in Developing and Fully Developed Flow in an
Isothermal Tube.” Journal of Heat Transfer 95 (Feb. 8. F. Kreith and M. S. Bohn. Principles of Heat Transfer, 6th
1973), pp. 78–84. ed. Pacific Grove, CA: Brooks/Cole, 2001.
4. W. M. Edmunds. “Residential Insulation.” ASTM 9. H. Schlichting. Boundary Layer Theory, 7th ed. New
Standardization News (Jan. 1989), pp. 36–39. York, McGraw-Hill, 1979.
