GAZİANTEP ÜNİVERSİTESİ

HAVACILIK VE UZAY BİLİMLERİ FAKÜLTESİ

UÇAK VE UZAY MÜHENDİSLİĞİ BÖLÜMÜ

Dr.Öğr.Üyesi MOHAMMAD MUNIR ALHAMWI

AE 301 Heat transfer

6 CHAPTER
Empirical and Practical Relations
for Forced-Convection Heat Transfer
Empirical and Practical
Relations for
Forced-Convection
Heat Transfer
EMPIRICAL RELATIONS FOR PIPE AND TUBE FLOW
 A traditional expression for calculation of heat transfer in fully developed turbulent
flow in smooth tubes is that recommended by Dittus and Boelter

𝑇𝑏

Prandtl numbers ranging from about 0.6 to 100 and with moderate temperature differences
More recent information by Gnielinski [45] suggests that better results for
turbulent flow in smooth tubes may be obtained from the following:

𝑇𝑏

𝑇𝑏
A power function for each of these parameters is a simple
type of relation to use, so we assume

𝑇𝑏

where C, m, and n are constants to be determined from the experimental data

the fact that the viscosity of gases increases with an increase in
temperature, while the viscosities of liquids decrease with an increase in
temperature.
take into account the property variations, Sieder and Tate [2]
recommend the following relation:

𝑇𝑏 𝑇𝑤

All properties are evaluated at bulk-temperature conditions,

except μ , which is evaluated at the wall temperature
w
In the entrance region the flow is not developed, and Nusselt [3]
recommended the following equation:

𝑇𝑏

where L is the length of the tube and d is the tube diameter. The
properties in Equation (6-6) are evaluated at the mean bulk
temperature.
Petukhov [42] has developed a more accurate, although more complicated,
expression for fully developed turbulent flow in smooth tubes:

𝑇ഥ𝑓 𝑇𝑏
𝑇𝑤

where n=0.11 for T >T , n=0.25 for T <T , and n=0 for constant heat flux or
w b w b

for gases. All properties are evaluated at T =(T +T )/2 except for μ and μ .
f w b b w

The friction factor may be obtained either from Figure 6-4 or from the
following for smooth tubes:
Hausen [4] presents the following empirical relation for fully developed
laminar flow in tubes at constant wall temperature:

𝑇𝑏

The heat-transfer coefficient calculated from this relation is the average

value over the entire length of tube. Note that the Nusselt number
approaches a constant value of 3.66 when the tube is sufficiently long.
A some what simpler empirical relation was proposed by Sieder and Tate
[2] for laminar heat transfer in tubes:

𝑇𝑏
𝑇𝑤

In this formula the average heat-transfer coefficient is based on the

arithmetic average of the inlet and outlet temperature differences, and all
fluid properties are evaluated at the mean bulk temperature of the fluid,
except μw, which is evaluated at the wall temperature.
rather sparse where rough tubes are concerned, and it is sometimes appropriate that
the Reynolds analogy between fluid friction and heat transfer be used to effect a
solution under these circumstances. Expressed in terms of the Stanton number,

for 10−6 <ε/d<10−3 and 5000<Red <108.

 𝜌 .𝑢.𝐷 𝑢.𝐷 4,𝑚ሶ 𝐷.mሶ 4.𝜌 .𝑉 4.𝑉
𝑅𝑒 = = = = = =
𝜇 𝜈 𝜋.𝐷.𝜇 𝐴𝑐 .𝜇 𝜋.𝐷.𝜇 𝜋.𝜈
NuH = average Nusselt number for uniform heat flux in flow direction and
uniform wall temperature at particular flow cross section
NuT = average Nusselt number for uniform wall temperature
fReDH /4 = product of friction factor and Reynolds number based on
hydraulic diameter
 Turbulent Heat Transfer in a Tube EXAMPLE
EXAMPLE 6-1
6-1
Air at 2 atm and 200◦C is heated as it flows
through a tube with a diameter of 1 in (2.54 cm)
at a velocity of 10 m/s. Calculate the heat transfer
per unit length of tube if a constant-heat-flux
condition is maintained at the wall and the wall
temperature is 20◦C above the air temperature, all
along the length of the tube. How much would
the bulk temperature increase over a 3-m length
of the tube?
Solution
We first calculate the Reynolds number to
determine if the flow is laminar or turbulent, and
thenselect the appropriate empirical correlation to
calculate the heat transfer. The properties of air at
abulk temperature of 200◦C are
Heating of Water in Laminar Tube Flow
Water at 60◦C enters a tube of 1-in (2.54-cm) diameter at a mean flow velocity of 2 cm/s.
Calculate the exit water temperature if the tube is 3.0 m long and the wall temperature is constant at 80◦C.
Solution
We first evaluate the Reynolds number at the inlet bulk temperature to determine the flow regime.
The properties of water at 60◦C are
Heating of Air in Laminar Tube Flow for Constant Heat Flux
EXAMPLE 6-3
Air at 1 atm and 27◦C enters a 5.0-mm-diameter smooth tube with a velocity of 3.0 m/s. The length of the tube is 10
cm. A constant heat flux is imposed on the tube wall. Calculate the heat transfer if the exit bulk temperature is 77◦C.
Also calculate the exit wall temperature and the value of h at exit.
Solution
We first must evaluate the flow regime and do so by taking properties at the average bulk temperature
EXAMPLE 6-4 Heating of Air with Isothermal TubeWall
Repeat Example 6-3 for the case of constant wall temperature.
Solution
We evaluate properties as before and now enter Figure 6-5 to determine Nud for Tw = constant.
For Gz−1 =0.0346 we read
Heat Transfer in a Rough Tube EXAMPLE 6-5
A 2.0-cm-diameter tube having a relative roughness of 0.001 is maintained at a constant Wall temperature of
90◦C.Water enters the tube at 40◦C and leaves at 60◦C. If the entering velocity is 3 m/s, calculate the length of
tube necessary to accomplish the heating.
Solution
We first calculate the heat transfer from
EXAMPLE 6-6 Turbulent Heat Transfer in a Short Tube
Air at 300 K and 1 atm enters a smooth tube having a diameter of 2 cm and length of 10 cm.
The air velocity is 40 m/s. What constant heat flux must be applied at the tube surface to
result in an air temperature rise of 5◦C? What average wall temperature would be necessary for
this case?
Solution
Because of the relatively small value of L/d =10/2=5 we may anticipate that thermal entrance
effects will be present in the flow. First, we determine the air properties at 300 K as
 The resulting correlation for average heat-transfer coefficients in cross flow over circular cylinders is

where the constants C and n are tabulated in Table 6-2. Properties for use with Equation (6-17) are evaluated at the film
temperature as indicated by the subscript f .
Fand [21] has shown that the heat-transfer coefficients from liquids to cylinders in
cross flow may be better represented by the relation

This relation is valid for 10−1 <Ref <105 provided excessive free-stream turbulence is
not encountered.
following relations for heat transfer from tubes in cross flow, based on the
extensive study of References 33 and 39:

For gases the Prandtl number ratio may be dropped, and fluid properties are evaluated
at the film temperature. For liquids the ratio is retained, and fluid properties are
evaluated at the free-stream temperature.
Still a more comprehensive relation is given by Churchill and
Bernstein [37] that is
applicable over the complete range of available data:
This relation underpredicts the data somewhat in the midrange of
Reynolds numbers between 20,000 and 400,000, and it is suggested
that the following be employed for this range

Properties in Equations are evaluated at the film temperature.

The heat-transfer data that were used to arrive at Equations (6-21) and (6-22)
include fluids of air, water, and liquid sodium. Still another correlation
equation is given by Whitaker [35] as

Properties in Equations are evaluated at the film temperature.

for 40<Re<10 , 0.65<Pr <300, and 0.25<μ /μ <5.2. All properties are valuated
5 ∞ w

at the free-stream temperature except that μ is at the wall temperature.

w
Below Pe =0.2, Nakai and Okazaki [38] present the following relation:
d

Properties in Equations are evaluated at the film temperature.

∆ Tb = small Tb

Tb
Tb

Tb
Tw
Tb
 Tb
∆ Tb = large Tb
Tw
6-1
Engine oil enters a 5.0-mm-diameter tube at 120◦C. The tube wall is maintained at
50◦C, and the inlet Reynolds number is 1000.
• Calculate the heat transfer, average heat-transfer coefficient, and exit oil temperature for tube
lengths of 10, 20, and 50 cm.

120◦C
d=5.0-mm
Reynolds number is 1000.

Engine oil 50◦C

L= 10, 20, and 50 cm.

6-2

Water at an average bulk temperature of 80◦F flows inside a horizontal smooth tube with wall temperature
maintained at 180◦F. The tube length is 2 m, and diameter is 3 mm. The flow velocity is 0.04 m/s.

• Calculate the heat-transfer rate.

d=3-mm
180◦F
80◦F
Water 50◦C
0.04 m/s.
L= 2 m.
6-3

• Calculate the flow rate necessary to produce a Reynolds number of 15,000 for the flow of air at 1 atm and 300 K
in a 2.5-cm-diameter tube.

• Repeat for liquid water at 300 K.

6-4
Liquid ammonia flows in a duct having a cross section of an equilateral triangle 1.0 cm on a
side. The average bulk temperature is 20◦C, and the duct wall temperature is 50◦C. Fully
developed laminar flow is experienced with a Reynolds number of 1000.

• Calculate the heat transfer per unit length of duct.

6-5
Water flows in a duct having a cross section 5×10 mm with a mean bulk temperature of 20◦C. If the
duct wall temperature is constant at 60◦C and fully developed laminar flow is experienced,
• calculate the heat transfer per unit length.

20◦C.

Water 5×10mm
60◦C

𝒌 𝒒
𝒉= . 𝑵uT = 𝑵uT . P. dT
𝑫𝑯 𝑳
6-6
Water at the rate of 3 kg/s is heated from 5 to 15◦C by passing it through a 5-cm-ID copper
tube. The tube wall temperature is maintained at 90◦C.

What is the length of the tube ?

4195
5◦C
15◦C
d=5.0-cm

3 kg/s
Water 90◦C

L= ????? .
6-7
Water at the rate of 0.8 kg/s is heated from 35 to 40◦C in a 2.5-cm-diameter tube whose surface
is at 90◦C.
• How long must the tube be to accomplish this heating?

2.5-cm ID
35◦C
0.8 kg/s
90 ◦C 40◦C
Water
L= ?????

𝑅𝑒 = 𝑥. 𝑈/𝜇
𝑚ሶ
=𝑈
𝐴
6-8
Water flows through a 2.5-cm-ID pipe 1.5 m long at a rate of 1.0 kg/s. The pressure drop is 7 kPa
through the 1.5-m length. The pipe wall temperature is maintained at aconstant temperature of 50◦C by
a condensing vapor, and the inlet water temperature is 20◦C.

• Estimate the exit water temperature.

20◦C
2.5-cm ID

1 kg/s
50 ◦C ??◦C
7 kPa
L= 1.5 m

𝐾𝑔 2 𝑚 𝐾𝑔
= 𝑚 . . 3
𝑠 𝑠 𝑚
6-9
Water at the rate of 1.3 kg/s is to be heated from 60◦F to 100◦F in a 2.5-cm-diameter tube. The tube wall
is maintained at a constant temperature of 40◦C.
• Calculate the length of tube required for the heating process.

60◦F
2.5-cm ID

1.3 kg/s
40 ◦C 100◦F

L= ?????

𝑅𝑒 = 𝑥. 𝑈/𝜇
𝑚ሶ
=𝑈
𝐴
6-10
Water at the rate of 1 kg/s is forced through a tube with a 2.5-cm ID. The inlet water temperature is
15◦C, and the outlet water temperature is 50◦C. The tube Wall temperature is 14◦C higher than the
water temperature all along the length of the tube.
• What is the length of the tube?

Cp 15◦C
2.5-cm ID

1 kg/s
Water 14+ Tw ◦C 50◦C

L= ?????

𝑅𝑒 = 𝑥. 𝑈/𝜇
𝑚ሶ
=𝑈
𝐴
6-11
Engine oil enters a 1.25-cm-diameter tube 3 m long at a temperature of 38◦C. The tube wall temperature
is maintained at 65◦C, and the flow velocity is 30 cm/s.
• Estimate the total heat transfer to the oil and the exit temperature of the oil.

38◦C
1.25-cm

30 cm/s
Engine oil 65◦C

L= 3 m
6-12
Air at 1 atm and 15◦C flows through a long rectangular duct 7.5 cm by 15 cm. A 1.8-m section of the
duct is maintained at 120◦C, and the average air temperature at exit from this section is 65◦C.
• Calculate the airflow rate and the total heat transfer.

120◦C 7.5 cm

Air 65◦C

1.8-m
6-13
Water at the rate of 0.5 kg/s is forced through a smooth 2.5-cm-ID tube 15 m long. The inlet water
temperature is 10◦C, and the tube wall temperature is 15◦C higher than the water temperature all along
the length of the tube.
• What is the exit water temperature?
10◦C
2.5-cm

0.5 kg/s
Water 15+ Tw ◦C

L= 15 m
6-14
Water at an average temperature of 300 K flows at 0.7 kg/s in a 2.5-cm-diameter tube 6 m long. The pressure
drop is measured as 2 kPa. A constant heat flux is imposed, and the average wall temperature is 55◦C.
• Estimate the exit temperature of the water.

2 kPa
2.5-cm Tav=300 K

0.7 kg/s
Water 55 ◦C
6-15
An oil with Pr =1960, ρ=860 kg/m3, ν=1.6×10−4 m2/s, and k =0.14W/m· ◦C enters a 2.5-mm-diameter
tube 60 cm long. The oil entrance temperature is 20◦C, the mean flow velocity is 30 cm/s, and the tube wall
temperature is 120◦C.
• Calculate the heat-transfer rate.

2.5-cm

30 cm/s
oil 120 ◦C
20◦C
60 cm
6-16
Liquid ammonia flows through a 2.5-cm-diameter smooth tube 2.5 m long at a rate of 0.4 kg/s. The
ammonia enters at 10◦C and leaves at 38◦C, and a constant heat flux is imposed on the tube wall.
• Calculate the average wall temperature necessary to effect the indicated heat transfer.

2.5-cm

30 cm/s
oil 120 ◦C
20◦C
60 cm
6-17
Liquid Freon 12 (CCl2F2) flows inside a 1.25-cm-diameter tube at a velocity of 3 m/s. the heat-
transfer coefficient for a bulk temperature of 10◦C.
• Calculate How does this compare with water at the same conditions?

1.25-cm

3 m/s
Freon 12 550 ◦K

10◦C
6-18
Water at an average temperature of 10◦C flows in a 2.5-cm-diameter tube 6 m long at a rate of 0.4 kg/s.
The pressure drop is measured as 3 kPa. A constant heat flux is imposed, and the average wall
temperature is 50◦C.
• Estimate the exit temperature of the water.

2.5-cm

0.4 kg/s
water 50 ◦C

3 kPa 6m
Tav= 10◦C
6-19
Water at the rate of 0.4 kg/s is to be cooled from 71 to 32◦C.
• Which would result in less pressure drop—to run the water through a 12.5-mm-diameter pipe
at a constant temperature of 4◦C or through a constant-temperature 25-mm-diameter pipe at
20◦C?

12.5-cm
0.4 kg/s
Water 4 ◦C

T=71◦C
T=32◦C
6-20
Air at 1400 kPa enters a duct 7.5 cm in diameter and 6 m long at a rate of 0.5 kg/s The duct wall is
maintained at an average temperature of 500 K. The average air temperature in the duct is 550 K.
• Estimate the decrease in temperature of the air as it passes through the duct.

7.5-cm

0.5 kg/s
Air 550 ◦K

1400 kPa 6m
Tav=500◦K
6-103

Glycerin at 10◦C flows in a rectangular duct 1 cm by 8 cm and 1 m long. The flow rate is such
that the Reynolds number is 250.

• Estimate the average heat transfer coefficient for an isothermal wall condition.
6-104
Air at 300 K blows normal to a 6-mm heated strip maintained at 600 K. The air velocity is such
that the Reynolds number is 15,000.
• Calculate the heat loss for a 50-cm-long strip.
6-105

• Repeat Problem 6-104 for flow normal to a square rod 6 mm on a side.

6-106
Repeat Problem 6-104 for flow parallel to a 6-mm strip.
• (Calculate heat transfer for both sides of the strip.)
6-107
Air at 1 atm flows normal to a square in-line bank of 400 tubes having diameters of 6 mm and lengths of
50 cm. Sn =Sd =9 mm. The air enters the tube bank at 300 K and at a velocity such that the Reynolds
number based on inlet properties and the maximum velocity at inlet is 50,000. If the outside wall
temperature of the tubes is 400 K,
• calculate the air temperature rise as it flows through the tube bank.

Sn =Sd =9 mm
V=50,000 L=50 cm d=6 mm
Air

400 K
1 atm
300 K 400 tubes
6-108
• Repeat Problem 6-107 for a tube bank with a staggered arrangement, the same
dimensions, and the same free-stream inlet velocity to the tube bank. ?
6-109
Compare the Nusselt number results for heating air in a smooth tube at 300 K and Reynolds numbers
of 50,000 and 100,000, as calculated from Equation (6-4a), (6-4b), and (6-4c).
• What do you conclude from these results?
6-110

Repeat Problem 6-109 for heating water at 21◦C.

6-111
Compare the results obtained from Equations (6-17), (6-21), (6-22), and (6-23) for air at 1 atm and
300 K flowing across a cylinder maintained at 400 K, with Reynolds numbers of 50,000 and 100,000.
• What do you conclude from these results?
6-112
Repeat Problem 6-111 for flow or water at 21◦C across a cylinder maintained at 32.2◦C.
• What do you conclude from the results