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CHAPTER 6
Forced Convection
Inside Tubes and Ducts
Typical tube bundle of
multiple circular tubes and
cutaway section of a mini
shell-and-tube heat
exchanger.
Source: Courtesy of Exergy, LLC.
Concepts and Analyses to Be Learned
The process of transferring heat by convection when the fluid flow is driven
by an applied pressure gradient is referred to as forced convection. When
this flow is confined in a tube or a duct of any arbitrary geometrical cross
section, the growth and development of boundary layers are also confined.
In such flows, the hydraulic diameter of the duct, rather than its length, is
the characteristic length for scaling the boundary layer as well as for
dimensionless representation of flow-friction loss and the heat transfer
coefficient. Convective heat transfer inside tubes and ducts is encountered
in numerous applications where heat exchangers, made up of circular tubes
as well as a variety of noncircular cross-sectional geometries, are employed.
A study of this chapter will teach you:
• How to express the dimensionless form of the heat transfer coefficient in a duct, and its dependence on flow properties and tube
geometry.
• How to mathematically model forced-convection heat transfer in a
long circular tube for laminar fluid flow.
• How to determine the heat transfer coefficient in ducts of different
geometries from different theoretical and/or empirical correlations in
both laminar and turbulent flows.
• How to model and employ the analogy between heat and momentum
transfer in turbulent flow.
• How to evaluate heat transfer coefficients in some examples where
enhancement techniques, such as coiled tubes, finned tubes, and
twisted-tape inserts, are employed.
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Introduction
Heating and cooling of fluids flowing inside conduits are among the most important
heat transfer processes in engineering. The design and analysis of heat exchangers
require a knowledge of the heat transfer coefficient between the wall of the conduit
and the fluid flowing inside it. The sizes of boilers, economizers, superheaters, and
preheaters depend largely on the heat transfer coefficient between the inner surface
of the tubes and the fluid. Also, in the design of air-conditioning and refrigeration
equipment, it is necessary to evaluate heat transfer coefficients for fluids flowing
inside ducts. Once the heat transfer coefficient for a given geometry and specified
flow conditions is known, the rate of heat transfer at the prevailing temperature
difference can be calculated from the equation
qc = qhc A(Tsurface - Tfluid)
(6.1)
The same relation also can be used to determine the area required to transfer heat at
a specified rate for a given temperature potential. But when heat is transferred to a
fluid inside a conduit, the fluid temperature varies along the conduit and at any cross
section. The fluid temperature for flow inside a duct must therefore be defined with
care and precision.
The heat transfer coefficient hqc can be calculated from the Nusselt number
hqc DH> k , as shown in Section 4.5. For flow in long tubes or conduits (Fig. 6.1a),
the significant length in the Nusselt number is the hydraulic diameter, DH,
defined as
DH = 4
flow cross-sectional area
wetted perimeter
(6.2)
For a circular tube or a pipe, the flow cross-sectional area is pD2> 4 , the wetted
perimeter is D, and therefore, the inside diameter of the tube equals the hydraulic
Wetted perimeter
D2
Flow cross-sectional area
(a)
D1
(b)
FIGURE 6.1 Hydraulic diameter for (a) irregular cross section and (b) annulus.
351
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Chapter 6 Forced Convection Inside Tubes and Ducts
diameter. For an annulus formed between two concentric tubes (Fig. 6.1b), we
have
DH = 4
(p> 4)(D22 - D21)
= D2 - D1
p(D1 + D2)
(6.3)
In engineering practice, the Nusselt number for flow in conduits is usually evaluated from empirical equations based on experimental results. The only exception is
laminar flow inside circular tubes, selected noncircular cross-sectional ducts, and a
few other conduits for which analytical and theoretical solutions are available [13].
Some simple examples of laminar-flow heat transfer in circular tubes are dealt with
in Section 6.2. From a dimensional analysis, as shown in Section 4.5, the experimental results obtained in forced-convection heat transfer experiments in long ducts and
conduits can be correlated by an equation of the form
Nu = f(Re)c(Pr)
(6.4)
where the symbols and denote functions of the Reynolds number and Prandtl
number, respectively. For short ducts, particularly in laminar flow, the right-hand
side of Eq. (6.4) must be modified by including the aspect ratio x/DH:
Nu = f(Re)c(Pr )f a
x
b
DH
where f(x> DH) denotes the functional dependence on the aspect ratio.
6.1.1 Reference Fluid Temperature
The convection heat transfer coefficient used to build the Nusselt number for heat
transfer to a fluid flowing in a conduit is defined by Eq. (6.1). The numerical
value of hc, as mentioned previously, depends on the choice of the reference temperature in the fluid. For flow over a plane surface, the temperature of the fluid
far away from the heat source is generally uniform, and its value is a natural
choice for the fluid temperature in Eq. (6.1). In heat transfer to or from a fluid
flowing in a conduit, the temperature of the fluid does not level out but varies
both along the direction of mass flow and in the direction of heat flow. At a given
cross section of the conduit, the temperature of the fluid at the center could be
selected as the reference temperature in Eq. (6.1). However, the center temperature is difficult to measure in practice; furthermore, it is not a measure of the
change in internal energy of all the fluid flowing in the conduit. It is therefore a
common practice, and one we shall follow here, to use the average fluid bulk temperature, Tb, as the reference fluid temperature in Eq. (6.1). The average fluid
temperature at a station of the conduit is often called the mixing-cup temperature
because it is the temperature which the fluid passing a cross-sectional area of the
conduit during a given time internal would assume if the fluid were collected and
mixed in a cup.
Use of the fluid bulk temperature as the reference temperature in Eq. (6.1)
allows us to make heat balances readily, because in the steady state, the difference
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6.1 Introduction
353
in average bulk temperature between two sections of a conduit is a direct measure of
the rate of heat transfer:
#
(6.5)
qc = mcp ¢Tb
where
qc ϭ rate of heat transfer to fluid, W
#
m ϭ flow rate, kg/s
cp ϭ specific heat at constant pressure, kJ/kg K
¢Tb ϭ difference in average fluid bulk temperature between cross sections in question, K or °C
The problems associated with variations of the bulk temperature in the direction
of flow will be considered in detail in Chapter 8, where the analysis of heat exchangers is taken up. For preliminary calculations, it is common practice to use the bulk
temperature halfway between the inlet and the outlet section of a duct as the reference temperature in Eq. (6.1). This procedure is satisfactory when the wall heat flux
of the duct is constant but may require some modification when the heat is transferrred between two fluids separated by a wall, as, for example, in a heat exchanger
where one fluid flows inside a pipe while another passes over the outside of the pipe.
Although this type of problem is of considerable practical importance, it will not
concern us in this chapter, where the emphasis is placed on the evaluation of convection heat transfer coefficients, which can be determined in a given flow system
when the pertinent bulk and wall temperatures are specified.
6.1.2 Effect of Reynolds Number on Heat Transfer and
Pressure Drop in Fully Established Flow
For a given fluid, the Nusselt number depends primarily on the flow conditions,
which can be characterized by the Reynolds number, Re. For flow in long conduits,
the characteristic length in the Reynolds number, as in the Nusselt number, is the
hydraulic diameter, and the velocity to be used is the average over the flow crosssectional area, Uq , or
ReDH =
UqDHr UqDH
=
v
m
(6.6)
In long ducts, where the entrance effects are not important, the flow is laminar when
the Reynolds number is below about 2100. In the range of Reynolds numbers
between 2100 and 10,000, a transition from laminar to turbulent flow takes place.
The flow in this regime is called transitional. At a Reynolds number of about 10,000,
the flow becomes fully turbulent.
In laminar flow through a duct, just as in laminar flow over a plate, there is no
mixing of warmer and colder fluid particles by eddy motion, and the heat transfer
takes place solely by conduction. Since all fluids with the exception of liquid metals
have small thermal conductivities, the heat transfer coefficients in laminar flow are
relatively small. In transitional flow, a certain amount of mixing occurs through
eddies that carry warmer fluid into cooler regions and vice versa. Since the mixing
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Chapter 6 Forced Convection Inside Tubes and Ducts
200
100
50
hc D
k
NuD ∝ ReD0.8
20
NuD =
354
5/14/10
10
Laminar
Transitional
Turbulent
5.0
NuD ∝ ReD0.3
2.0
1.0
100
200
500
1000 2000
5000 10,000 20,000 50,000
ReD = U∞ D/v
FIGURE 6.2 Nusselt number versus Reynolds number for air
flowing in a long heated pipe at uniform wall temperature.
motion, even if it is only on a small scale, accelerates the transfer of heat considerably, a marked increase in the heat transfer coefficient occurs above ReDH = 2100
(it should be noted, however, that this change, or transition, can generally occur over
a range of Reynolds number, 2000 6 ReDH 6 5000). This change can be seen in
Fig. 6.2, where experimentally measured values of the average Nusselt number for
atmospheric air flowing through a long heated tube are plotted as a function of the
Reynolds number. Since the Prandtl number for air does not vary appreciably,
Eq. (6.4) reduces to Nu = f(ReDH), and the curve drawn through the experimental
points shows the dependence of Nu on the flow conditions. We note that in the laminar regime, the Nusselt number remains small, increasing from about 3.5 at
ReDH = 300 to 5.0 at ReDH = 2100. Above a Reynolds number of 2100, the Nusselt
number begins to increase rapidly until the Reynolds number reaches about 8000. As
the Reynolds number is further increased, the Nusselt number continues to increase,
but at a slower rate.
A qualitative explanation for this behavior can be given by observing the fluid
flow field shown schematically in Fig. 6.3. At Reynolds numbers above 8000, the
flow inside the conduit is fully turbulent except for a very thin layer of fluid adjacent to the wall. In this layer, turbulent eddies are damped out as a result of the viscous forces that predominate near the surface, and therefore heat flows through this
layer mainly by conduction.* The edge of this sublayer is indicated by a dashed line
*Although some studies [1] have shown that turbulent transport also exists to some extent near the
wall, especially when the Prandtl number is larger than 5, the layer near the wall is commonly referred
to as the “viscous sublayer.”
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6.1 Introduction
355
Edge of viscous
sublayer
Edge of buffer or
transitional layer
Turbulent core
FIGURE 6.3 Flow structure for a fluid in turbulent flow through
a pipe.
in Fig. 6.3. The flow beyond it is turbulent, and the circular arrows in the turbulentflow regime represent the eddies that sweep the edge of the layer, probably penetrate
it, and carry along with them fluid at the temperature prevailing there. The eddies
mix the warmer and cooler fluids so effectively that heat is transferred very rapidly
between the edge of the viscous sublayer and the turbulent bulk of the fluid. It is thus
apparent that except for fluids of high thermal conductivity (e.g., liquid metals), the
thermal resistance of the sublayer controls the rate of heat transfer, and most of the
temperature drop between the bulk of the fluid and the surface of the conduit occurs
in this layer. The turbulent portion of the flow field, on the other hand, offers little
resistance to the flow of heat. The only effective method of increasing the heat transfer coefficient is therefore to decrease the thermal resistance of the sublayer. This
can be accomplished by increasing the turbulence in the main stream so that the turbulent eddies can penetrate deeper into the layer. An increase in turbulence, however, is accompanied by large energy losses that increase the frictional pressure drop
in the conduit. In the design and selection of industrial heat exchangers, where not
only the initial cost but also the operating expenses must be considered, the pressure
drop is an important factor. An increase in the flow velocity yields higher heat transfer coefficients, which, in accordance with Eq. (6.1), decrease the size and consequently the initial cost of the equipment for a specified heat transfer rate. At the same
time, however, the pumping cost increases. The optimum design therefore requires
a compromise between the initial and operating costs. In practice, it has been found
that increases in pumping costs and operating expenses often outweigh the saving in
the initial cost of heat transfer equipment under continuous operating conditions. As
a result, the velocities used in a majority of commercial heat exchange equipment
are relatively low, corresponding to Reynolds numbers of no more than 50,000.
Laminar flow is usually avoided in heat exchange equipment because of the low heat
transfer coefficients obtained. However, in the chemical industry, where very viscous liquids must frequently be handled, laminar flow sometimes cannot be avoided
without producing undesirably large pressure losses.
It was shown in Section 4.12 that, for turbulent flow of liquids and gases over a
flat plate, the Nusselt number is proportional to the Reynolds number raised to the
0.8 power. Since in turbulent forced convection the viscous sublayer generally controls the rate of heat flow irrespective of the geometry of the system, it is not surprising that for turbulent forced convection in conduits the Nusselt number is related
to the Reynolds number by the same type of power law. For the case of air flowing
in a pipe, this relation is illustrated in the graph of Fig. 6.2.
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Chapter 6 Forced Convection Inside Tubes and Ducts
6.1.3 Effect of Prandtl Number
The Prandtl number Pr is a function of the fluid properties alone. It has been defined
as the ratio of the kinematic viscosity of the fluid to the thermal diffusivity of the
fluid:
Pr =
cpm
n
=
a
k
The kinematic viscosity v, or m> r, is often referred to as the molecular diffusivity of
momentum because it is a measure of the rate of momentum transfer between the
molecules. The thermal diffusivity of a fluid, k> cpr, is often called the molecular diffusivity of heat. It is a measure of the ratio of the heat transmission and energy storage capacities of the molecules.
The Prandtl number relates the temperature distribution to the velocity distribution, as shown in Section 4.5 for flow over a flat plate. For flow in a pipe, just
as over a flat plate, the velocity and temperature profiles are similar for fluids
having a Prandtl number of unity. When the Prandtl number is smaller, the temperature gradient near a surface is less steep than the velocity gradient, and for
fluids whose Prandtl number is larger than one, the temperature gradient is
steeper than the velocity gradient. The effect of Prandtl number on the temperature gradient in turbulent flow at a given Reynolds number in tubes is illustrated
schematically in Fig. 6.4, where temperature profiles at different Prandtl numbers
are shown at ReD = 10,000. These curves reveal that, at a specified Reynolds
number, the temperature gradient at the wall is steeper in a fluid having a large
Prandtl number than in a fluid having a small Prandtl number. Consequently, at a
given Reynolds number, fluids with larger Prandtl numbers have larger Nusselt
numbers.
Liquid metals generally have a high thermal conductivity and a small specific
heat; their Prandtl numbers are therefore small, ranging from 0.005 to 0.01. The
Prandtl numbers of gases range from 0.6 to 1.0. Most oils, on the other hand, have
large Prandtl numbers, some up to 5000 or more, because their viscosity is large at
low temperatures and their thermal conductivity is small.
6.1.4 Entrance Effects
In addition to the Reynolds number and the Prandtl number, several other factors can
influence heat transfer by forced convection in a duct. For example, when the conduit is short, entrance effects are important. As a fluid enters a duct with a uniform
velocity, the fluid immediately adjacent to the tube wall is brought to rest. For a
short distance from the entrance, a laminar boundary layer is formed along the tube
wall. If the turbulence in the entering fluid stream is high, the boundary layer will
quickly become turbulent. Irrespective of whether the boundary layer remains laminar or becomes turbulent, it will increase in thickness until it fills the entire duct.
From this point on, the velocity profile across the duct remains essentially
unchanged.
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6.1 Introduction
357
Viscous layer
Buffer layer
0.1
=1
Pr
u(r)
umax
0
0.6
0.0
01
TS – T
TS – Tcenter
0.8
10
0.0
1
10
0
1.0
0.4
ReD = 10,000
0.2
0
0.2
0.4
0.6
0.8
1.0
y
r0
FIGURE 6.4 Effect of Prandtl number on temperature profile for turbulent flow in a long pipe (y is the distance from
the tube wall and r0 is the inner pipe radius).
Source: Courtesy of R. C. Martinelli, “Heat Transfer to Molten Metals”, Trans.
ASME, Vol. 69, 1947, p. 947. Reprinted by permission of The American
Society of Mechanical Engineers International.
The development of the thermal boundary layer in a fluid that is heated or
cooled in a duct is qualitatively similar to that of the hydrodynamic boundary layer.
At the entrance, the temperature is generally uniform transversely, but as the fluid
flows along the duct, the heated or cooled layer increases in thickness until heat is
transferred to or from the fluid in the center of the duct. Beyond this point, the
temperature profile remains essentially constant if the velocity profile is fully
established.
The final shapes of the velocity and temperature profiles depend on whether
the fully developed flow is laminar or turbulent. Figures 6.5 on the next page and
Figure 6.6 on page 359 qualitatively illustrate the growth of the boundary layers
as well as the variations in the local convection heat transfer coefficient near the
entrance of a tube for laminar and turbulent conditions, respectively. Inspection of
these figures shows that the convection heat transfer coefficient varies considerably near the entrance. If the entrance is square-edged, as in most heat exchangers, the initial development of the hydrodynamic and thermal boundary layers
along the walls of the tube is quite similar to that along a flat plane. Consequently,
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Chapter 6 Forced Convection Inside Tubes and Ducts
u/U∞
u/U∞
u/U∞
x
Velocity profile
δ –hydrodynamic boundary layer
Ts
T/Tb
T/Tb
T/Tb
Temperature profile
for fluid being
cooled (Ts = 0)
δr – thermal boundary layer
Ts
hcx
hc∞
1.0
x/D
FIGURE 6.5 Velocity distribution, temperature profiles, and variation of the
local heat transfer coefficient near the inlet of a tube for air being cooled in
laminar flow (surface temperature Ts uniform).
the heat transfer coefficient is largest near the entrance and decreases along the
duct until both the velocity and the temperature profiles for the fully developed
flow have been established. If the pipe Reynolds number for the fully developed
flow UqDr> m is below 2100, the entrance effects may be appreciable for a length
as much as 100 hydraulic diameters from the entrance. For laminar flow in a circular tube, the hydraulic entry length at which the velocity profile approaches its
fully developed shape can be obtained from the relation [3]
a
xfully developed
D
b
lam
= 0.05ReD
(6.7)
whereas the distance from the inlet at which the temperature profile approaches its
fully developed shape is given by the relation [4]
a
xfully developed
D
b
lam,T
= 0.05ReD Pr
(6.8)
In turbulent flow, conditions are essentially independent of Prandtl numbers, and for
average pipe velocities corresponding to turbulent-flow Reynolds numbers, entrance
effects disappear about 10 or 20 diameters from the inlet.
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6.1 Introduction
q
q
q
q
q
q
359
Growth of
boundary layers
Variation of
velocity
distribution
hcx
hc∞
Turbulent flow
behavior
Laminar flow
behavior
Laminar
boundary
layer
Turbulent boundary layer
Fully established
velocity distribution
x/D
FIGURE 6.6 Velocity distribution and variation of local heat
transfer coefficient near the entrance of a uniformly heated tube
for a fluid in turbulent flow.
6.1.5 Variation of Physical Properties
Another factor that can influence the heat transfer and friction considerably is the
variation of physical properties with temperature. When a fluid flowing in a duct
is heated or cooled, its temperature and consequently its physical properties vary
along the duct as well as over any given cross section. For liquids, only the temperature dependence of the viscosity is of major importance. For gases, on the
other hand, the temperature effect on the physical properties is more complicated
than for liquids because the thermal conductivity and the density, in addition to the
viscosity, vary significantly with temperature. In either case, the numerical value
of the Reynolds number depends on the location at which the properties are
evaluated. It is believed that the Reynolds number based on the average bulk temperature is the significant parameter to describe the flow conditions. However,
considerable success in the empirical correlation of experimental heat transfer data
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Chapter 6 Forced Convection Inside Tubes and Ducts
has been achieved by evaluating the viscosity at an average film temperature,
defined as a temperature approximately halfway between the wall and the average
bulk temperatures. Another method of taking account of the variation of physical
properties with temperature is to evaluate all properties at the average bulk temperature and to correct for the thermal effects by multiplying the right-hand side
of Eq. (6.4) by a function proportional to the ratio of bulk to wall temperatures or
bulk to wall viscosities.
6.1.6 Thermal Boundary Conditions
and Compressibility Effects
For fluids having a Prandtl number of unity or less, the heat transfer coefficient also
depends on the thermal boundary condition. For example, in geometrically similar
liquid metal or gas heat transfer systems, a uniform wall temperature yields smaller
convection heat transfer coefficients than a uniform heat input at the same Reynolds
and Prandtl numbers [5–7]. When heat is transferred to or from gases flowing at very
high velocities, compressibility effects influence the flow and the heat transfer.
Problems associated with heat transfer to or from fluids at high Mach numbers are
referenced in [8–10].
6.1.7 Limits of Accuracy in Predicted Values
of Convection Heat Transfer Coefficients
In the application of any empirical equation for forced convection to practical problems, it is important to bear in mind that the predicted values of the heat transfer
coefficient are not exact. The results obtained by various experimenters, even under
carefully controlled conditions, differ appreciably. In turbulent flow, the accuracy of
a heat transfer coefficient predicted from any available equation or graph is no
better than Ϯ20%, whereas in laminar flow, the accuracy may be of the order of
Ϯ30%. In the transition region, where experimental data are scant, the accuracy of
the Nusselt number predicted from available information may be even lower. Hence,
the number of significant figures obtained from calculations should be consistent
with these accuracy limits.
6.2*
Analysis of Laminar Forced Convection in a Long Tube
To illustrate some of the most important concepts in forced convection, we will analyze a simple case and calculate the heat transfer coefficient for laminar flow
through a tube under fully developed conditions with a constant heat flux at the wall.
We begin by deriving the velocity distribution. Consider a fluid element as shown
in Fig. 6.7. The pressure is uniform over the cross section, and the pressure forces
are balanced by the viscous shear forces acting over the surface:
pr 2[p - (p + dp)] = t2pr dx = - am
du
b2pr dx
dr
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