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4* Rotating Cylinders, Disks, and Spheres

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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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