ptinciples of heat transfer 2 1684

in average bulk temperature between two sections of a conduit is a direct measure ofthe rate of heat transfer:6.5where q c⫽ rate of heat transfer to fluid, W ⫽ flow rate, kg/s c p⫽ speci

CHAPTER 6

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 crosssection, the growth and development of boundary layers are also confined

In such flows, the hydraulic diameter of the duct, rather than its length, isthe characteristic length for scaling the boundary layer as well as fordimensionless representation of flow-friction loss and the heat transfercoefficient 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 cient in a duct, and its dependence on flow properties and tubegeometry

coeffi-• How to mathematically model forced-convection heat transfer in along circular tube for laminar fluid flow

• How to determine the heat transfer coefficient in ducts of differentgeometries from different theoretical and/or empirical correlations inboth laminar and turbulent flows

• How to model and employ the analogy between heat and momentumtransfer in turbulent flow

• How to evaluate heat transfer coefficients in some examples whereenhancement techniques, such as coiled tubes, finned tubes, andtwisted-tape inserts, are employed

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.

6.1 Introduction

Heating and cooling of fluids flowing inside conduits are among the most importantheat transfer processes in engineering The design and analysis of heat exchangersrequire a knowledge of the heat transfer coefficient between the wall of the conduitand the fluid flowing inside it The sizes of boilers, economizers, superheaters, andpreheaters 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 refrigerationequipment, it is necessary to evaluate heat transfer coefficients for fluids flowinginside ducts Once the heat transfer coefficient for a given geometry and specifiedflow conditions is known, the rate of heat transfer at the prevailing temperaturedifference can be calculated from the equation

(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 afluid inside a conduit, the fluid temperature varies along the conduit and at any crosssection The fluid temperature for flow inside a duct must therefore be defined withcare and precision

The heat transfer coefficient can be calculated from the Nusselt number, 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, D H,defined as

sec-tion and (b) annulus

diameter For an annulus formed between two concentric tubes (Fig 6.1b), wehave

(6.3)

In engineering practice, the Nusselt number for flow in conduits is usually uated from empirical equations based on experimental results The only exception islaminar flow inside circular tubes, selected noncircular cross-sectional ducts, and afew 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

eval-in Section 6.2 From a dimensional analysis, as shown eval-in Section 4.5, the tal results obtained in forced-convection heat transfer experiments in long ducts andconduits can be correlated by an equation of the form

experimen-(6.4)where the symbols ␾ and ␺ denote functions of the Reynolds number and Prandtlnumber, respectively For short ducts, particularly in laminar flow, the right-handside of Eq (6.4) must be modified by including the aspect ratio :

where 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 heattransfer to a fluid flowing in a conduit is defined by Eq (6.1) The numerical

value of h

-c, as mentioned previously, depends on the choice of the reference perature in the fluid For flow over a plane surface, the temperature of the fluidfar away from the heat source is generally uniform, and its value is a naturalchoice for the fluid temperature in Eq (6.1) In heat transfer to or from a fluidflowing in a conduit, the temperature of the fluid does not level out but variesboth along the direction of mass flow and in the direction of heat flow At a givencross section of the conduit, the temperature of the fluid at the center could beselected as the reference temperature in Eq (6.1) However, the center tempera-ture is difficult to measure in practice; furthermore, it is not a measure of thechange 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 perature, Tb, as the reference fluid temperature in Eq (6.1) The average fluid

tem-temperature at a station of the conduit is often called the mixing-cup tem-temperature

because it is the temperature which the fluid passing a cross-sectional area of theconduit during a given time internal would assume if the fluid were collected andmixed 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

in average bulk temperature between two sections of a conduit is a direct measure ofthe rate of heat transfer:

(6.5)where q c⫽ rate of heat transfer to fluid, W

⫽ flow rate, kg/s

c p⫽ specific heat at constant pressure, kJ/kg K

⫽ difference in average fluid bulk temperature between cross tions in question, K or °C

sec-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 ers is taken up For preliminary calculations, it is common practice to use the bulktemperature halfway between the inlet and the outlet section of a duct as the refer-ence temperature in Eq (6.1) This procedure is satisfactory when the wall heat flux

exchang-of the duct is constant but may require some modification when the heat is rred between two fluids separated by a wall, as, for example, in a heat exchangerwhere 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 notconcern us in this chapter, where the emphasis is placed on the evaluation of con-vection heat transfer coefficients, which can be determined in a given flow systemwhen the pertinent bulk and wall temperatures are specified

transfer-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 thehydraulic diameter, and the velocity to be used is the average over the flow cross-sectional area, , or

(6.6)

In long ducts, where the entrance effects are not important, the flow is laminar whenthe Reynolds number is below about 2100 In the range of Reynolds numbersbetween 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 nomixing of warmer and colder fluid particles by eddy motion, and the heat transfertakes place solely by conduction Since all fluids with the exception of liquid metalshave small thermal conductivities, the heat transfer coefficients in laminar flow arerelatively small In transitional flow, a certain amount of mixing occurs througheddies that carry warmer fluid into cooler regions and vice versa Since the mixing

ReD H =

U qD Hr

U qD H v

Uq

¢T b

m#

q c = m#c p¢T b

100 1.0 2.0 5.0 10 20 50 100 200

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 ably, a marked increase in the heat transfer coefficient occurs above

consider-(it should be noted, however, that this change, or transition, can generally occur over

Fig 6.2, where experimentally measured values of the average Nusselt number foratmospheric air flowing through a long heated tube are plotted as a function of theReynolds number Since the Prandtl number for air does not vary appreciably,

Eq (6.4) reduces to , and the curve drawn through the experimentalpoints shows the dependence of Nu on the flow conditions We note that in the lam-inar regime, the Nusselt number remains small, increasing from about 3.5 at

to 5.0 at Above a Reynolds number of 2100, the Nusseltnumber begins to increase rapidly until the Reynolds number reaches about 8000 Asthe 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 fluidflow field shown schematically in Fig 6.3 At Reynolds numbers above 8000, theflow inside the conduit is fully turbulent except for a very thin layer of fluid adja-cent to the wall In this layer, turbulent eddies are damped out as a result of the vis-cous forces that predominate near the surface, and therefore heat flows through thislayer mainly by conduction.* The edge of this sublayer is indicated by a dashed line

Edge of viscous sublayer

Edge of buffer or transitional layer Turbulent core

in this layer The turbulent portion of the flow field, on the other hand, offers littleresistance to the flow of heat The only effective method of increasing the heat trans-fer coefficient is therefore to decrease the thermal resistance of the sublayer Thiscan be accomplished by increasing the turbulence in the main stream so that the tur-bulent eddies can penetrate deeper into the layer An increase in turbulence, how-ever, 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 notonly the initial cost but also the operating expenses must be considered, the pressuredrop is an important factor An increase in the flow velocity yields higher heat trans-fer coefficients, which, in accordance with Eq (6.1), decrease the size and conse-quently the initial cost of the equipment for a specified heat transfer rate At the sametime, however, the pumping cost increases The optimum design therefore requires

a compromise between the initial and operating costs In practice, it has been foundthat increases in pumping costs and operating expenses often outweigh the saving inthe initial cost of heat transfer equipment under continuous operating conditions As

a result, the velocities used in a majority of commercial heat exchange equipmentare 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 heattransfer coefficients obtained However, in the chemical industry, where very vis-cous liquids must frequently be handled, laminar flow sometimes cannot be avoidedwithout producing undesirably large pressure losses

It was shown in Section 4.12 that, for turbulent flow of liquids and gases over aflat plate, the Nusselt number is proportional to the Reynolds number raised to the0.8 power Since in turbulent forced convection the viscous sublayer generally con-trols the rate of heat flow irrespective of the geometry of the system, it is not sur-prising 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

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 thefluid:

The kinematic viscosity v, or , is often referred to as the molecular diffusivity ofmomentum because it is a measure of the rate of momentum transfer between themolecules The thermal diffusivity of a fluid, , is often called the molecular dif-fusivity of heat It is a measure of the ratio of the heat transmission and energy stor-age capacities of the molecules

The Prandtl number relates the temperature distribution to the velocity bution, as shown in Section 4.5 for flow over a flat plate For flow in a pipe, just

distri-as over a flat plate, the velocity and temperature profiles are similar for fluidshaving a Prandtl number of unity When the Prandtl number is smaller, the tem-perature gradient near a surface is less steep than the velocity gradient, and forfluids whose Prandtl number is larger than one, the temperature gradient issteeper than the velocity gradient The effect of Prandtl number on the tempera-ture gradient in turbulent flow at a given Reynolds number in tubes is illustratedschematically in Fig 6.4, where temperature profiles at different Prandtl numbersare shown at These curves reveal that, at a specified Reynoldsnumber, the temperature gradient at the wall is steeper in a fluid having a largePrandtl number than in a fluid having a small Prandtl number Consequently, at agiven Reynolds number, fluids with larger Prandtl numbers have larger Nusseltnumbers

Liquid metals generally have a high thermal conductivity and a small specificheat; their Prandtl numbers are therefore small, ranging from 0.005 to 0.01 ThePrandtl numbers of gases range from 0.6 to 1.0 Most oils, on the other hand, havelarge Prandtl numbers, some up to 5000 or more, because their viscosity is large atlow 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 caninfluence heat transfer by forced convection in a duct For example, when the con-duit is short, entrance effects are important As a fluid enters a duct with a uniformvelocity, the fluid immediately adjacent to the tube wall is brought to rest For ashort distance from the entrance, a laminar boundary layer is formed along the tubewall If the turbulence in the entering fluid stream is high, the boundary layer willquickly become turbulent Irrespective of whether the boundary layer remains lam-inar 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 essentiallyunchanged

Viscous layer Buffer layer

Pr = 1 100

u(r)

ReD = 10,000

10

pro-file for turbulent flow in a long pipe (y is the distance from the tube wall and r0is 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 orcooled 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 fluidflows along the duct, the heated or cooled layer increases in thickness until heat istransferred to or from the fluid in the center of the duct Beyond this point, thetemperature profile remains essentially constant if the velocity profile is fullyestablished

The final shapes of the velocity and temperature profiles depend on whetherthe fully developed flow is laminar or turbulent Figures 6.5 on the next page andFigure 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 theentrance of a tube for laminar and turbulent conditions, respectively Inspection ofthese figures shows that the convection heat transfer coefficient varies consider-ably near the entrance If the entrance is square-edged, as in most heat exchang-ers, the initial development of the hydrodynamic and thermal boundary layersalong the walls of the tube is quite similar to that along a flat plane Consequently,

δ –hydrodynamic boundary layer

cooled (T s = 0)

local heat transfer coefficient near the inlet of a tube for air being cooled in

laminar flow (surface temperature T suniform)

the heat transfer coefficient is largest near the entrance and decreases along theduct until both the velocity and the temperature profiles for the fully developedflow have been established If the pipe Reynolds number for the fully developedflow 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 cular tube, the hydraulic entry length at which the velocity profile approaches itsfully developed shape can be obtained from the relation [3]

q q q

Growth of boundary layers

Variation of velocity distribution

x/D

Laminar flow behavior Laminar

boundary layer Turbulent boundary layer

Fully established velocity distribution

Turbulent flow behavior

transfer coefficient near the entrance of a uniformly heated tubefor a fluid in turbulent flow

6.1.5 Variation of Physical Properties

Another factor that can influence the heat transfer and friction considerably is thevariation of physical properties with temperature When a fluid flowing in a duct

is heated or cooled, its temperature and consequently its physical properties varyalong the duct as well as over any given cross section For liquids, only the tem-perature dependence of the viscosity is of major importance For gases, on theother hand, the temperature effect on the physical properties is more complicatedthan for liquids because the thermal conductivity and the density, in addition to theviscosity, vary significantly with temperature In either case, the numerical value

of the Reynolds number depends on the location at which the properties areevaluated It is believed that the Reynolds number based on the average bulk tem-perature is the significant parameter to describe the flow conditions However,considerable success in the empirical correlation of experimental heat transfer data

has been achieved by evaluating the viscosity at an average film temperature,

defined as a temperature approximately halfway between the wall and the averagebulk temperatures Another method of taking account of the variation of physicalproperties with temperature is to evaluate all properties at the average bulk tem-perature 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 orbulk 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 alsodepends on the thermal boundary condition For example, in geometrically similarliquid metal or gas heat transfer systems, a uniform wall temperature yields smallerconvection heat transfer coefficients than a uniform heat input at the same Reynoldsand Prandtl numbers [5–7] When heat is transferred to or from gases flowing at veryhigh velocities, compressibility effects influence the flow and the heat transfer.Problems associated with heat transfer to or from fluids at high Mach numbers arereferenced 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 lems, it is important to bear in mind that the predicted values of the heat transfercoefficient are not exact The results obtained by various experimenters, even undercarefully controlled conditions, differ appreciably In turbulent flow, the accuracy of

prob-a heprob-at trprob-ansfer coefficient predicted from prob-any prob-avprob-ailprob-able equprob-ation or grprob-aph is nobetter than ⫾20%, whereas in laminar flow, the accuracy may be of the order of

⫾30% In the transition region, where experimental data are scant, the accuracy ofthe Nusselt number predicted from available information may be even lower Hence,the number of significant figures obtained from calculations should be consistentwith these accuracy limits

To illustrate some of the most important concepts in forced convection, we will lyze a simple case and calculate the heat transfer coefficient for laminar flowthrough a tube under fully developed conditions with a constant heat flux at the wall

ana-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 forcesare balanced by the viscous shear forces acting over the surface:

pr2[p - (p + dp)] = t2pr dx = -am du

dr b2pr dx

τ (2πr dx) = –µ r

x dx

radius r s

From this relation, we obtain

where dp dx is the axial pressure gradient The radial distribution of the axial

velocity is then

where C is a constant of integration whose value is determined by the boundary

con-dition that at Using this condition to evaluate C gives the velocity

consid-through the conduit The pressure loss in a tube of length L is obtained from a force

balance on the fluid element inside the tube between and (see Fig 6.7):

(6.12)

ts = wall shear stress (ts = -m(du>dr)| r = r s)

The pressure drop also can be related to a so-called Darcy friction factor f according to

(6.13)

where is the average velocity in the tube

It is important to note that f, the friction factor in Eq (6.13), is not the same quantity as the friction coefficient C f, which was defined in Chapter 4 as

(6.14)

C f is often referred to as the Fanning friction coefficient Since

it is apparent from Eqs (6.12), (6.13), and (6.14) that

For flow through a pipe the mass flow rate is obtained from Eq (6.9)

Comparing Eq (6.17) with Eq (6.13), we see that for fully developed laminar flow

in a tube the friction factor in a pipe is a simple function of Reynolds number

(6.18)

The pumping power, P p, is equal to the product of the pressure drop and the metric flow rate of the fluid, , divided by the pump efficiency, ␩p, or

volu-(6.19)The analysis above is limited to laminar flow with a parabolic velocity distribu-tion in pipes or circular tubes, known as Poiseuille flow, but the approach taken toderive this relation is more general If we know the shear stress as a function of thevelocity and its derivative, the friction factor also could be obtained for turbulentflow However, for turbulent flow, the relationship between the shear and the averagevelocity is not well understood Moreover, while in laminar flow, the friction factor

is independent of surface roughness; in turbulent flow, the quality of the pipe surfaceinfluences the pressure loss Therefore, friction factors for turbulent flow cannot bederived analytically but must be measured and correlated empirically

=

-¢prs28Lm

Uq

m# = rL

6.2.1 Uniform Heat Flux

For the energy analysis, consider the control volume shown in Fig 6.8 In laminarflow, heat is transferred by conduction into and out of the element in a radial direc-tion, whereas in the axial direction, the energy transport is by convection Thus, therate of heat conduction into the element is

while the rate of heat conduction out of the element is

The net rate of convection out of the element is

Writing a net energy balance in the form

net rate of conduction

=net rate of convectioninto the element out of the element

we get, neglecting second-order terms,

which can be recast in the form

(6.20)1

The fluid temperature must increase linearly with distance x since the heat flux over

the surface is specified to be uniform, so

(6.21)

When the axial temperature gradient is constant, Eq (6.20) reduces from a

partial to an ordinary differential equation with r as the only space coordinate.

The symmetry and boundary conditions for the temperature distribution in

Eq (6.20) are

To solve Eq (6.20), we substitute the velocity distribution from Eq (6.11).Assuming that the temperature gradient does not affect the velocity profile, that is,the properties do not change with temperature, we get

But note that since and that the second boundary condition

is satisfied by the requirement that the axial temperature gradient is constant

If we let the temperature at the center (r ⫽ 0) be T c, then and the ture distribution becomes

Since the heat flux from the tube wall is uniform, the enthalpy of the fluid in the tube

must increase linearly with x, and thus We can calculate the bulk

temperature by substituting Eqs (6.25) and (6.11) for T and u, respectively, in Eq.

Evaluating the radial temperature gradient at r ⫽ r sfrom Eq (6.23) and substituting

it with Eqs (6.27) and (6.28) in the above definition yields

(6.30)

or

(6.31)

EXAMPLE 6.1 Water entering at 10°C is to be heated to 40°C in a tube of 0.02-m-ID at a

mass flow rate of 0.01 kg/s The outside of the tube is wrapped with aninsulated electric-heating element (see Fig 6.9) that produces a uniform flux

of 15,000 W m> 2over the surface Neglecting any entrance effects, determine

48k 11D

h

qc =

qc A(T s - T b) =

Water out 40°C

Electric power supply

L = ?

heated tube, Example 6.1

(a) the Reynolds number(b) the heat transfer coefficient(c) the length of pipe needed for a 30°C increase in average temperature(d) the inner tube surface temperature at the outlet

(e) the friction factor(f) the pressure drop in the pipe(g) the pumping power required if the pump is 50% efficient

SOLUTION From Table 13 in Appendix 2, the appropriate properties of water at an average

tem-perature between inlet and outlet of 25°C are obtained by interpolation:

(a) The Reynolds number is

This establishes that the flow is laminar

(b) Since the thermal-boundary condition is one of uniform heat flux, NuD⫽ 4.36from Eq (6.31) and

(c) The length of pipe needed for a 30°C temperature rise is obtained from a heatbalance

Solving for L when gives

Eq (6.7) Note that if L D had been significantly less than 33.5, the calculations

would have to be repeated with entrance effects taken into account, using relations

(e) The friction factor is found from Eq (6.18):

(f) The pressure drop in the pipe is, from Eq (6.17),

Since

we have

(g) The pumping power P pis obtained from Eq 6.19 or

6.2.2* Uniform Surface Temperature

When the tube surface temperature rather than the heat flux is uniform, the analysis

is more complicated because the temperature difference between the wall and bulkvaries along the tube, that is, Equation (6.20) can be solved subject

to the second boundary condition that at , but an iterativeprocedure is necessary The result is not a simple algebraic expression, but theNusselt number is found (for example, see Kays and Perkins [11]) to be a constant:

(6.32)

In addition to the value of the Nusselt number, the constant-temperatureboundary condition also requires a different temperature to evaluate the rate ofheat transfer to or from a fluid flowing through a duct Except for the entranceregion, in which the boundary layer develops and the heat transfer coefficientdecreases, the temperature difference between the surface of the duct and the bulkremains constant along the duct when the heat flux is uniform This is apparent

=

4a0.01 kg

s ba997 kg

T Entrance

region

and constant wall temperature: (a) constant heat flux, q s (x)⫽ constant;

(b) constant surface temperature, T s (x)⫽ constant

from an examination of Eq (6.20) and is illustrated graphically in Fig 6.10 For aconstant wall temperature, on the other hand, only the bulk temperature increasesalong the duct and the temperature potential decreases (see Fig 6.10) We firstwrite the heat balance equation

where P is the perimeter of the duct and q s⬙ is the surface heat flux From the preceding

we can obtain a relation for the bulk temperature gradient in the x-direction

(6.33)

variables, we have

(6.34)

where and the subscripts “in” and “out” denote conditions at the inlet

(x ⫽ 0) and the outlet (x ⫽ L) of the duct, respectively Integrating Eq (6.34) yields

(6.35)

where

h

qc =1

Rearranging Eq (6.35) gives

(6.36)

The rate of heat transfer by convection to or from a fluid flowing through a duct with

T s⫽ constant can be expressed in the formand substituting from Eq (6.35), we get

(6.37)

The expression in the square bracket is called the log mean temperature difference (LMTD).

EXAMPLE 6.2 Used engine oil can be recycled by a patented reprocessing system Suppose that

such a system includes a process during which engine oil flows through a 1-cm-ID,0.02-cm-wall copper tube at the rate of 0.05 kg/s The oil enters at 35°C and is to beheated to 45°C by atmospheric-pressure steam condensing on the outside, as shown

in Fig 6.11 Calculate the length of the tube required

SOLUTION We shall assume that the tube is long and that its temperature is uniform at 100°C

The first approximation must be checked; the second assumption is an engineeringapproximation justified by the high thermal conductivity of copper and the large heattransfer coefficient for a condensing vapor (see Table 1.4) From Table 16 inAppendix 2, we get the following properties for oil at 40°C:

0.02 cm Copper tube

Condensing steam

Oil out 45°C

1 cm

The Reynolds number is

The flow is therefore laminar, and the Nusselt number for a constant surface ature is 3.66 The average heat transfer coefficient is

temper-The rate of heat transfer is

Substituting the preceding information in Eq (6.37), where , gives

Checking our first assumption, we find L D⬃ 1000, justifying neglect of entranceeffects Note also that LMTD is very nearly equal to the difference between the sur-face temperature and the average bulk fluid temperature halfway between the inletand outlet The required length is not suitable for a practical design with a straightpipe To achieve the desired thermal performance in a more convenient shape, onecould route the tube back and forth several times or use a coiled tube The firstapproach will be discussed in Chapter 8 on heat exchanger design, and the coiled-tube design is illustrated in an example in the next section

This section presents empirical correlations and analytic results that can be used inthermal design of heat transfer systems composed of tubes and ducts containinggaseous or liquid fluids in laminar flow Although heat transfer coefficients in lam-inar flow are considerably smaller than in turbulent flow, in the design of heatexchange equipment for viscous liquids, it is often necessary to accept a smaller heattransfer coefficient in order to reduce the pumping power requirements Laminar gasflow is encountered in high-temperature, compact heat exchangers, where tubediameters are very small and gas densities low Other applications of laminar-flowforced convection occur in chemical processes and in the food industry, in electroniccooling as well as in solar and nuclear power plants, where liquid metals are used asheat transfer media Since liquid metals have a high thermal conductivity, their heattransfer coefficients are relatively large, even in laminar flow

100.167 = 59.9 K

= 30.3

6.3.1 Short Circular and Rectangular Ducts

The details of the mathematical solutions for laminar flow in short ducts withentrance effects are beyond the scope of this text References listed at the end ofthis chapter, especially [4] and [11], contain the mathematical background for theengineering equations and graphs that are presented and discussed in this section.For engineering applications, it is most convenient to present the results of ana-lytic and experimental investigations in terms of a Nusselt number defined in the

conventional manner as h c D k However, the heat transfer coefficient h ccan varyalong the tube, and for practical applications, the average value of the heat transfercoefficient is most important Consequently, for the equations and charts presented

in this section, we shall use a mean Nusselt number, , averaged with

respect to the circumference and length of the duct L:

where the subscript x refers to local conditions at x This Nusselt number is often called the log mean Nusselt number, because it can be used directly in the log mean

rate equations presented in the preceding section and can be applied to heat ers (see Chapter 8)

exchang-Mean Nusselt numbers for laminar flow in tubes at a uniform wall temperature havebeen calculated analytically by various investigators Their results are shown in Fig 6.12

NuD = hqc D >k

>

0.2 0.1 2 5 10 20 50 100

Parabolic velocity

Region of interest in gas flow heat exchangers

Noris and streid interpolation

Short duct approximation

Uniform velocity Boundary-layer analysis

modified for tube Very “long” tubes Very “short” tubes

transfer in laminar flow through circular tubes at constant walltemperature, versus ReD PrD/L The dots represent Eq (6.38).

Source: Courtesy of W M Kays, “Numerical Solution for Laminar Flow Heat Transfer in Circular Tubes,” Trans ASME, vol 77, pp 1265–1274, 1955.

NuD

for several velocity distributions All of these solutions are based on the idealizations of

a constant tube-wall temperature and a uniform temperature distribution at the tube inlet,and they apply strictly only when the physical properties are independent of temperature.The abscissa is the dimensionless quantity * To determine the mean value of

the Nusselt number for a given tube of length L and diameter D, one evaluates the

Reynolds number, ReD, and the Prandtl number, Pr, forms the dimensionless parameter

, and enters the appropriate curve from Fig 6.12 The selection of the curverepresenting the conditions that most nearly correspond to the physical conditionsdepends on the nature of the fluid and the geometry of the system For high Prandtl num-ber fluids such as oils, the velocity profile is established much more rapidly than the tem-perature profile Consequently, application of the curve labeled “parabolic velocity”does not lead to a serious error in long tubes when is less than 100 For verylong tubes, the Nusselt number approaches a limiting minimum value of 3.66 when thetube temperature is uniform When the heat transfer rate instead of the tube temperature

is uniform, the limiting value of is 4.36

For very short tubes or rectangular ducts with initially uniform velocity andtemperature distribution, the flow conditions along the wall approximate thosealong a flat plate, and the boundary Layer analysis presented in Chapter 4 isexpected to yield satisfactory results for liquids having Prandtl numbers between

0.7 and 15.0 The boundary layer solution applies [14, 15] when L D is less than

0.0048ReD for tubes and when L D His less than for flat ducts of tangular cross section For these conditions, the equation for flow of liquids andgases over a flat plate can be converted to the coordinates of Figs 6.12, leading to

NuD = e1.953[L >(DRe DPr)]1/3 for [L >(DRe DPr)] … 0.03

4.364 + (0.0722(DRe DPr)]>L for [L>(DRe DPr)] … 0.03

Note that when L is very large (: ⬁), the values of are obtained as 4.364and 3.657, respectively, for the mean Nusselt number with the two boundaryconditions from Eqs (6.39) and (6.40).

6.3.2 Ducts of Noncircular Cross Section

Heat transfer and friction in fully developed laminar flow through ducts with a ety of cross sections have been treated analytically [13] The results are summarized

vari-in Table 6.1 on the next page, usvari-ing the followvari-ing nomenclature:

A duct geometry encountered quite often is the concentric tube annulus shownschematically in Fig 6.1(b) Heat transfer to or from the fluid flowing through thespace formed between the two concentric tubes may occur at the inner surface, theouter surface, or both surfaces simultaneously Moreover, the heat transfer surfacemay be at constant temperature or constant heat flux An extensive treatment of thistopic has been presented by Kays and Perkins [11], and includes entrance effects andthe impact of eccentricity Here we shall consider only the most commonly encoun-tered case of an annulus in which one side is insulated and the other is at constanttemperature

Denoting the inner surface by the subscript i and the outer surface by o, the rate

of heat transfer and the corresponding Nusselt numbers are

The Nusselt numbers for heat flow at the inner surface only with the outersurface insulated, , and the heat flow at the outer surface with the innersurface insulated, , as well as the product of the friction factor and theReynolds number for fully developed laminar flow are presented in Table 6.2 onpage 375 For other conditions, such as constant heat flux and short annuli, thereader is referred to [13]

Nui =

h

qc,iDH k

q c,o = hqc,o pD o L(T s,o - T b)

q c,i = hqc,i pD i L(T s,i - T b)

f Re D H = product of firction factor and Reynolds number

NuT = average Nusselt number for uniform wall temperature and circumferentially

NuH2 = average Nusselt mumber for uniform heat flux both axially direction and uniform wall temperature at any cross section

NuH1 = average Nusselt number for uniform heat flux in flow

NuD

TABLE 6.1 Nusselt number and friction factor for fully developed laminar flow

of a Newtonian fluid through specific ductsa

2b 2a

Insulation

2b 2a = 0

2b 2a =

1 8

2b 2a

2b 2a

= 0.9

2b 2a

2b 2a =

1 4

2b

Insulation

2a

2b 2a =

1 4

2b 2a

2b 2a =

1 2

2b 2a

a

a

a a

a a

2b 2a

= 1

2b 2a

2b 2a =

13 2

TABLE 6.2 Nusselt number and friction factorfor fully developed laminar flow in an annulusa

heating duct for Example 6.3

EXAMPLE 6.3 Calculate the average heat transfer coefficient and the friction factor for flow of

n-butyl alcohol at a bulk temperature of 293 K through a 0.1-m⫻ 0.1-m-square duct,

5 m long, with walls at 300 K, and an average velocity of 0.03 m/s (see Fig 6.13)

SOLUTION The hydraulic diameter is

Physical properties at 293 K from Table 19 in Appendix 2 are

The Reynolds number is

Hence, the flow is laminar Assuming fully developed flow, we get the Nusselt ber for a uniform wall temperature from Table 6.1:

num-This yields for the average heat transfer coefficient

Similarly, from Table 6.1, the product and

Recall that for a fully developed velocity profile the duct length must be at least

, but for a fully developed temperature profile, the duct must

be 172 m long Thus, fully developed flow will not exist.

This value is five times larger than that for fully developed flow

Note that for this problem the difference between bulk and wall temperature issmall Hence, property variations are not significant in this case

6.3.3 Effect of Property Variations

Since the microscopic heat-flow mechanism in laminar flow is conduction, the rate

of heat flow between the walls of a conduit and the fluid flowing in it can beobtained analytically by solving the equations of motion and of conduction heat flowsimultaneously, as shown in Section 6.2 But to obtain a solution, it is necessary toknow or assume the velocity distribution in the duct In fully developed laminar flowthrough a tube without heat transfer, the velocity distribution at any cross section isparabolic But when appreciable heat transfer occurs, temperature differences arepresent, and the fluid properties of the wall and the bulk may be quite different.These property variations distort the velocity profile

In liquids, the viscosity decreases with increasing temperature, while in gasesthe reverse trend is observed When a liquid is heated, the fluid near the wall is lessviscous than the fluid in the center Consequently, the velocity of the heated fluid islarger than that of an unheated fluid near the wall, but less than that of the unheatedfluid in the center The distortion of the parabolic velocity profile for heated orcooled liquids is shown in Fig 6.14 For gases, the conditions are reversed, but thevariation of density with temperature introduces additional complications

= 824

C C

C

A

fully developed laminar flow through a pipe Curve A, isothermal flow; curve B, heating of liquid or cooling of gas; curve C, cooling of liquid or heating of gas.

Empirical viscosity correction factors are merely approximate rules, and recentdata indicate that they may not be satisfactory when very large temperature gradientsexist As an approximation in the absence of a more satisfactory method, it is suggesed[16] that for liquids, the Nusselt number obtained from the analytic solutions presented

in Fig 6.12 be multiplied by the ratio of the viscosity at the bulk temperature ␮bto theviscosity at the surface temperature ␮s, raised to the 0.14 power, that is, ,

to correct for the variation of properties due to the temperature gradients For gases,Kays and London [17] suggest that the Nusselt number be multiplied by the tempera-ture correction factor shown below If all fluid properties are evaluated at the averagebulk temperature, the corrected Nusselt number is

where n⫽ 0.25 for a gas heating in a tube and 0.08 for a gas cooling in a tube.Hausen [18] recommended the following relation for the average convection coeffi-cient in laminar flow through ducts with uniform surface temperature:

(6.41)

A relatively simple empirical equation suggested by Sieder and Tate [16] hasbeen widely used to correlate experimental results for liquids in tubes and can bewritten in the form

of the temperature variation on the physical properties Equation (6.42) can beapplied when the surface temperature is uniform in the range 0.48⬍ Pr ⬍ 16,700

For laminar flow of gases between two parallel, uniformly heated plates a

dis-tance 2y0apart, Swearingen and McEligot [20] showed that gas property variationscan be taken into account by the relation

(6.43)where

and the subscript b denotes that the physical properties are to be evaluated at T b.The variation in physical properties also affects the friction factor To evaluatethe friction factor of fluids being heated or cooled, it is suggested that for liquids theisothermal friction factor be modified by

(6.44)

and for gases by

(6.45)

EXAMPLE 6.4 An electronic device is cooled by water flowing through capillary holes drilled in the

casing as shown in Fig 6.15 The temperature of the device casing is constant at 353 K.The capillary holes are 0.3 m long and 2.54⫻ 10⫺3m in diameter If water enters at atemperature of 333 K and flows at a velocity of 0.2 m/s, calculate the outlet tempera-ture of the water

SOLUTION The properties of water at 333 K, from Table 13 in Appendix 2, are

To ascertain whether the flow is laminar, evaluate the Reynolds number at the inletbulk temperature,

ReD =

rU qD

(983 kg/m3)(0.2 m/s)(0.00254 m)4.72 * 10- 4 kg/ms

Q+

= q s–y0>(KT)entrance

Nu = Nuconstant properties + 0.024Q+ 0.3Gz b0.75

(ReD PrD>L)0.33(mb>ms)0.140.0044 6 (mb>ms) 6 9.75

0.3 m

Capillary holes

Water

333 K 0.2 m/s 2.54 × 10 –3 m

Surface temperature = 353 K

Single capillary Water

The flow is laminar and because

Eq (6.42) can be used to evaluate the heat transfer coefficient But since the meanbulk temperature is not known, we shall evaluate all the properties first at the inlet

bulk temperature T b1, then determine an exit bulk temperature, and then make a ond iteration to obtain a more precise value Designating inlet and outlet conditionwith the subscripts 1 and 2, respectively, the energy balance becomes

sec-(a)

Appendix 2 From Eq (6.42), we can calculate the average Nusselt number

and thus

The mass flow rate is

Inserting the calculated values for and into Eq (a), along with and

Solving for T b2gives

For the second iteration, we shall evaluate all properties at the new average bulktemperature

At this temperature, we get from Table 13 in Appendix 2:

Recalculating the Reynolds number with properties based on the new mean bulktemperature gives

With this value of ReD, the heat transfer coefficient can now be calculated One obtains

Substituting the new value of in Eq (b) gives Further iterations willnot affect the results appreciably in this example because of the small differencebetween bulk and wall temperature In cases where the temperature difference is large,

a second iteration may be necessary

It is recommended that the reader verify the results using the LMTD methodwith Eq (6.37)

6.3.4 Effect of Natural Convection

An additional complication in the determination of a heat transfer coefficient inlaminar flow arises when the buoyancy forces are of the same order of magnitude

as the external forces due to the forced circulation Such a condition may arise inoil coolers when low-flow velocities are employed Also, in the cooling of rotat-ing parts, such as the rotor blades of gas turbines and ramjets attached to the pro-pellers of helicopters, the natural-convection forces may be so large that theireffect on the velocity pattern cannot be neglected even in high-velocity flow.When the buoyancy forces are in the same direction as the external forces, such asthe gravitational forces superimposed on upward flow, they increase the rate ofheat transfer When the external and buoyancy forces act in opposite directions,the heat transfer is reduced Eckert, et al [14, 15] studied heat transfer in mixedflow, and their results are shown qualitatively in Fig 6.16(a) and (b) In the darklyshaded area, the contribution of natural convection to the total heat transfer is less

1 10

Forced convection turbulent flow

Mixed convection turbulent flow

laminar flow Forced convection laminar flow

ReD

Forced convection turbulent flow

Mixed convection turbulent flow Laminar turbulent transition

(a) horizontal pipe flow and (b) vertical pipe flow

Source: Courtesy of B Metais and E R G Eckert, “Forced, Free, and Mixed Convection Regimes,” Trans ASME Ser C J Heat Transfer, Vol 86, pp 295–298, 1964.

than 10%, whereas in the lightly shaded area, forced-convection effects are lessthan 10% and natural convection predominates In the unshaded area, natural andforced convection are of the same order of magnitude In practice, natural-convec-tion effects are hardly ever significant in turbulent flow [21] In cases where it isdoubtful whether forced- or natural-convection flow applies, the heat transfercoefficient is generally calculated by using forced- and natural-convection rela-tions separately, and the larger one is used [22] The accuracy of this rule is esti-mated to be about 25%.

The influence of natural convection on the heat transfer to fluids in horizontalisothermal tubes has been investigated by Depew and August [23] They found that theirown data for as well as previously available data for tubes with L D⬎ 50could be correlated by the equation

(6.46)

In Eq (6.46), Gz is the Graetz number, defined by

The Grashof number, GrD, is defined by Eq (5.8) Equation (6.46) was developedfrom experimental data with dimensionless parameters in the range 25⬍ Gz ⬍ 700,

5⬍ Pr ⬍ 400, and 250 ⬍ GrD⬍ 105 Physical properties, except for ␮s, are to beevaluated at the average bulk temperature

Correlations for vertical tubes and ducts are considerably more complicatedbecause they depend on the relative direction of the heat flow and the natural con-vection A summary of available information is given in Metais and Eckert [24] andRohsenow, et al [25]

To illustrate the most important physical variables affecting heat transfer by bulent forced convection to or from fluids flowing in a long tube or duct, weshall now develop the so-called Reynolds analogy between heat and momentumtransfer [26] The assumptions necessary for the simple analogy are valid onlyfor fluids having a Prandtl number of unity, but the fundamental relationbetween heat transfer and fluid friction for flow in ducts can be illustrated forthis case without introducing mathematical difficulties The results of the simpleanalysis can also be extended to other fluids by means of empirical correctionfactors

tur-The rate of heat flow per unit area in a fluid can be related to the temperaturegradient by the equation developed previously:

(6.47)

q c Arc p = -arc k

Similarly, the shearing stress caused by the combined action of the viscous forcesand the turbulent momentum transfer is given by

(6.48)

According to the Reynolds analogy, heat and momentum are transferred by

analo-gous processes in turbulent flow Consequently, both q and ␶ vary with y, the

dis-tance from the surface, in the same manner For fully developed turbulent flow in a

pipe, the local shearing stress increases linearly with the radial distance r Hence, we

can write

(6.49)

and

(6.50)

where the subscript s denotes conditions at the inner surface of the pipe Introducing

Eqs (6.49) and (6.50) into Eqs (6.47) and (6.48), respectively, yields

since is by definition equal to Multiplying the numerator and the

denominator of the right-hand side by D H ␮k and regrouping yields

r

rs = 1

-y rs

t

r = amr + eMb du dy

where is the Stanton number.

To bring the left-hand side of Eq (6.54) into a more convenient form, we useEqs (6.13) and (6.14):

Substituting Eq (6.14) for ␶s in Eq (6.54) finally yields a relation between theStanton number and the friction factor

(6.55)

known as the Reynolds analogy for flow in a tube It agrees fairly well with

experi-mental data for heat transfer in gases whose Prandtl number is nearly unity

According to experimental data for fluids flowing in smooth tubes in the range

of Reynolds numbers from 10,000 to 1,000,000, the friction factor is given by theempirical relation [17]

(6.56)Using this relation, Eq (6.55) can be written as

(6.57)

Since Pr was assumed unity,

(6.58)or

(6.59)

Note that in fully established turbulent flow, the heat transfer coefficient isdirectly proportional to the velocity raised to the 0.8 power, but inversely propor-tional to the tube diameter raised to the 0.2 power For a given flow rate, an increase

in the tube diameter reduces the velocity and thereby causes a decrease in

propor-tional to 1 D1.8 The use of small tubes and high velocities is therefore conducive tolarge heat transfer coefficients, but at the same time, the power required to overcomethe frictional resistance is increased In the design of heat exchange equipment, it istherefore necessary to strike a balance between the gain in heat transfer ratesachieved by the use of ducts having small cross-sectional areas and the accompany-ing increase in pumping requirements

Figure 6.17 shows the effect of surface roughness on the friction coefficient

We observe that the friction coefficient increases appreciably with the relativeroughness, defined as ratio of the average asperity height ␧ to the diameter D.

According to Eq (6.55), one would expect that roughening the surface, which

- 0.2

f = 0.184Re D- 0.2

St = NuRePr =

f

8St

increases the friction coefficient, also increases the convection conductance.Experiments performed by Cope [28] and Nunner [29] are qualitatively in agree-ment with this prediction, but a considerable increase in surface roughness isrequired to improve the rate of heat transfer appreciably Since an increase in thesurface roughness causes a substantial increase in the frictional resistance, for thesame pressure drop, the rate of heat transfer obtained from a smooth tube is largerthan from a rough one in turbulent flow.

Measurements by Dipprey and Sabersky [30] in tubes artificially roughenedwith sand grains are summarized in Fig 6.18 on the next page Where theStanton number is plotted against the Reynolds number for various values of theroughness ratio The lower straight line is for smooth tubes At smallReynolds numbers, St has the same value for rough and smooth tube surfaces.The larger the value , the smaller the value of Re at which the heat transferbegins to improve with increase in Reynolds number But for each value of the Stanton number reaches a maximum and, with a further increase in Reynoldsnumber, begins to decrease

e >D,

e >D

e >D

Critical zone

10 3

0.008 0.009 0.01 0.015 0.02

0.025

actor 0.03

0.04 0.05 0.06 0.07 0.08 0.090.1

10 4

2 3 4 5 6789 2 3 4 5 6789 10 5 2 3 4 5 6789 10 6 2 3 4 5 6789 10 7 2 3 4 10 8

= 0.000001 ε

D

= 0.000005

5 6789

Transition zone Laminar

0.02 0.01 0.08

0.002 /D = 0.001

/D = 0.005

0.0005 Smooth pipe

/D = 0.04

ε

ε

ε 3

6 4 2 8

ReD

Re for various values of /D according to Dipprey and Sabersky [30].

Source: Courtesy of T von Karman, “The Analogy between Fluid Friction and Heat Transfer,” Trans ASME, vol 61, p 705, 1939.

e

St

The Reynolds analogy presented in the preceding section was extended ically to fluids with Prandtl numbers larger than unity in [31–34] and to liquid met-als with very small Prandtl numbers in [31], but the phenomena of turbulent forcedconvection are so complex that empirical correlations are used in practice for engi-neering design

semi-analyt-6.5.1 Ducts and Tubes

The Dittus-Boelter equation [35] extends the Reynolds analogy to fluids withPrandtl numbers between 0.7 and 160 by multiplying the right-hand side of

Eq (6.58) by a correction factor of the form Prn:

(6.60)

where

With all properties in this correlation evaluated at the bulk temperature T b,

Eq (6.60) has been confirmed experimentally to within ⫾25% for uniform walltemperature as well as uniform heat-flux conditions within the following ranges ofparameters:

60 6 (L >D)

6000 6 ReD 6 1070.5 6 Pr 6 120

Since this correlation does not take into account variations in physical properties due

to the temperature gradient at a given cross section, it should be used only for ations with moderate temperature differences

situ-For situations in which significant property variations due to a large ture difference exist, a correlation developed by Sieder and Tate [16] isrecommended:

tempera-(6.61)

In Eq (6.61), all properties except are evaluated at the bulk temperature The cosity is evaluated at the surface temperature Equation (6.61) is appropriate foruniform wall temperature and uniform heat flux in the following range of conditions:

vis-To account for the variation in physical properties due to the temperature gradient inthe flow direction, the surface and bulk temperatures should be the values halfwaybetween the inlet and the outlet of the duct For ducts of other than circular cross-

sectional shapes, Eqs (6.60) and (6.61) can be used if the diameter D is replaced by the hydraulic diameter D H

A correlation similar to Eq (6.61) but restricted to gases was proposed by Kaysand London [17] for long ducts:

n = e0.020 for heating 0.150 for cooling

C = e0.020 for uniform surface temperature T s

0.020 for uniform heat flux qs–

NuD H = CRe D0.8H Pr0.3aTb T

sbn

60 6 (L>D)

6000 6 ReD 6 1070.7 6 Pr 6 10,000

Name (reference) Formulaa Conditions Equation

Sieder-Tate

10 2

10 3

2 3 4 5 6 7 8 9

meas-ured Nusselt number for turbulent flow of water in

a tube (26.7°C; Pr⫽ 6.0)

388

1.5 in.

1 in.

Water in annulus 180°F

10 ft/s

Insulation Inner wall temperature = 100°F

cooling of water in Example 6.5

EXAMPLE 6.5 Determine the Nusselt number for water flowing at an average velocity of 10 ft/s in

an annulus formed between a 1-in.-OD tube and a 1.5-in.-ID tube as shown in Fig.6.20 The water is at 180°F and is being cooled The temperature of the inner wall

is 100°F, and the outer wall of the annulus is insulated Neglect entrance effects andcompare the results obtained from all four equations in Table 6.3 The properties ofwater are given below in engineering units

SOLUTION The hydraulic diameter D Hfor this geometry is 0.5 in The Reynolds number based

on the hydraulic diameter and the bulk temperature properties is

The Prandtl number is

The Nusselt number according to the Dittus-Boelter correlation [Eq (6.60)] is

Using the Sieder-Tate correlation [Eq (6.61)], we get