A HEAT TRANSFER TEXTBOOK - THIRD EDITION Episode 2 Part 6 pptx

7.4 Heat transfer surface viewed as a heat exchanger Let us reconsider the problem of a fluid flowing through a pipe with a uniform wall temperature.. However, we need only recognize that

The heat transfer coefficient on a rough wall can be several timesthat for a smooth wall at the same Reynolds number The friction fac-tor, and thus the pressure drop and pumping power, will also be higher.Nevertheless, designers sometimes deliberately roughen tube walls so as

to raise h and reduce the surface area needed for heat transfer

Sev-eral manufacturers offer tubing that has had some pattern of roughnessimpressed upon its interior surface Periodic ribs are one common con-figuration Specialized correlations have been developed for a number

§7.3 Turbulent pipe flow 365

Figure 7.7 Velocity and temperature profiles during fully

de-veloped turbulent flow in a pipe

Heat transfer to fully developed liquid-metal flows in tubes

A dimensional analysis of the forced convection flow of a liquid metal

over a flat surface [recall eqn (6.60) et seq.] showed that

because viscous influences were confined to a region very close to the

wall Thus, the thermal b.l., which extends far beyond δ, is hardly

influ-enced by the dynamic b.l or by viscosity During heat transfer to liquid

metals in pipes, the same thing occurs as is illustrated in Fig.7.7 The

re-gion of thermal influence extends far beyond the laminar sublayer, when

Pr  1, and the temperature profile is not influenced by the sublayer.

Conversely, if Pr 1, the temperature profile is largely shaped within

the laminar sublayer At high or even moderate Pr’s, ν is therefore very

important, but at low Pr’s it vanishes from the functional equation

Equa-tion (7.51) thus applies to pipe flows as well as to flow over a flat surface

Numerous measured values of NuDfor liquid metals flowing in pipes

with a constant wall heat flux, q w, were assembled by Lubarsky and

Kauf-man [7.18] They are included in Fig.7.8 It is clear that while most of the

data correlate fairly well on NuD vs Pe coordinates, certain sets of data

are badly scattered This occurs in part because liquid metal experiments

are hard to carry out Temperature differences are small and must often

be measured at high temperatures Some of the very low data might

pos-sibly result from a failure of the metals to wet the inner surface of the

pipe

Another problem that besets liquid metal heat transfer measurements

is the very great difficulty involved in keeping such liquids pure Most

Figure 7.8 Comparison of measured and predicted Nusselt

numbers for liquid metals heated in long tubes with uniform

wall heat flux, q w (See NACA TN 336, 1955, for details anddata source references.)

impurities tend to result in lower values of h Thus, most of the

Nus-selt numbers in Fig.7.8have probably been lowered by impurities in theliquids; the few high values are probably the more correct ones for pureliquids

There is a body of theory for turbulent liquid metal heat transfer thatyields a prediction of the form

§7.4 Heat transfer surface viewed as a heat exchanger 367

and Lyon [7.21] recommends the following equation, shown in Fig.7.8:

NuD = 7 + 0.025 Pe 0.8

In both these equations, properties should be evaluated at the average

of the inlet and outlet bulk temperatures and the pipe flow should have

L/D > 60 and Pe D > 100 For lower Pe D, axial heat conduction in the

liquid metal may become significant

Although eqns (7.53) and (7.54) are probably correct for pure liquids,

we cannot overlook the fact that the liquid metals in actual use are seldom

pure Lubarsky and Kaufman [7.18] put the following line through the

bulk of the data in Fig.7.8:

NuD = 0.625 Pe 0.4

The use of eqn (7.55) for qw = constant is far less optimistic than the

use of eqn (7.54) It should probably be used if it is safer to err on the

low side

7.4 Heat transfer surface viewed as a heat exchanger

Let us reconsider the problem of a fluid flowing through a pipe with a

uniform wall temperature By now we can predict h for a pretty wide

range of conditions Suppose that we need to know the net heat transfer

to a pipe of known length onceh is known This problem is complicated

by the fact that the bulk temperature, T b, is varying along its length

However, we need only recognize that such a section of pipe is a heat

exchanger whose overall heat transfer coefficient, U (between the wall

and the bulk), is justh Thus, if we wish to know how much pipe surface

area is needed to raise the bulk temperature from T bin to T bout, we can

By the same token, heat transfer in a duct can be analyzed with the

ef-fectiveness method (Sect.3.3) if the exiting fluid temperature is unknown

Suppose that we do not know T bout in the example above Then we canwrite an energy balance at any cross section, as we did in eqn (7.8):

dQ = q w P dx = hP (T w − T b ) dx = ˙ mc P dT b Integration can be done from T b (x = 0) = T bin to T b (x = L) = T bout

L0

This equation can be used in either laminar or turbulent flow to

com-pute the variation of bulk temperature if T bout is replaced by T b (x), L is replaced by x, and h is adjusted accordingly.

The left-hand side of eqn (7.57) is the heat exchanger effectiveness

On the right-hand side we replace U with h; we note that P L = A, the exchanger surface area; and we write Cmin= ˙ mc p Since T w is uniform,the stream that it represents must have a very large capacity rate, so that

Cmin/Cmax = 0 Under these substitutions, we identify the argument of

the exponential as NTU= UA/Cmin, and eqn (7.57) becomes

which we could have obtained directly, from either eqn (3.20) or (3.21),

by setting Cmin/Cmax = 0 A heat exchanger for which one stream is isothermal, so that Cmin/Cmax = 0, is sometimes called a single-stream

heat exchanger

Equation7.57 applies to ducts of any cross-sectional shape We can

cast it in terms of the hydraulic diameter, D h = 4A c /P , by substituting

§7.4 Heat transfer surface viewed as a heat exchanger 369

For a circular tube, with A c = πD2/4 and P = πD, D h = 4(πD2/4) (π D)

= D To use eqn (7.59) for a noncircular duct, of course, we will need

the value ofh for its more complex geometry We consider this issue in

the next section

Example 7.5

Air at 20◦C is fully thermally developed as it flows in a 1 cm I.D pipe

The average velocity is 0.7 m/s If the pipe wall is at 60 ◦C , what is

the temperature 0.25 m farther downstream?

so that

T b − 20

60− 20 = 0.698 or T b = 47.9 ◦C

7.5 Heat transfer coefficients for noncircular ducts

So far, we have focused on flows within circular tubes, which are by far themost common configuration Nevertheless, other cross-sectional shapesoften occur For example, the fins of a heat exchanger may form a rect-angular passage through which air flows Sometimes, the passage cross-section is very irregular, as might happen when fluid passes through aclearance between other objects In situations like these, all the qual-itative ideas that we developed in Sections 7.1–7.3 still apply, but theNusselt numbers for circular tubes cannot be used in calculating heattransfer rates

The hydraulic diameter, which was introduced in connection witheqn (7.59), provides a basis for approximating heat transfer coefficients

in noncircular ducts Recall that the hydraulic diameter is defined as

D h ≡ 4 A c

where A c is the cross-sectional area and P is the passage’s wetted

perime-ter (Fig 7.9) The hydraulic diameter measures the fluid area per unitlength of wall In turbulent flow, where most of the convection resis-tance is in the sublayer on the wall, this ratio determines the heat trans-fer coefficient to within about±20% across a broad range of duct shapes.

In fully-developed laminar flow, where the thermal resistance extendsinto the core of the duct, the heat transfer coefficient depends on the

details of the duct shape, and D h alone cannot define the heat transfercoefficient Nevertheless, the hydraulic diameter provides an appropriatecharacteristic length for cataloging laminar Nusselt numbers

Figure 7.9 Flow in a noncircular duct.

§7.5 Heat transfer coefficients for noncircular ducts 371

The factor of four in the definition of D h ensures that it gives the

actual diameter of a circular tube We noted in the preceding section

that, for a circular tube of diameter D, D h = D Some other important

and, for very wide parallel plates, eqn (7.61a) with a b gives

two parallel plates

a distance b apart D h = 2b (7.61c)

Turbulent flow in noncircular ducts

With some caution, we may use D h directly in place of the circular tube

diameter when calculating turbulent heat transfer coefficients and bulk

temperature changes Specifically, D h replaces D in the Reynolds

num-ber, which is then used to calculate f and Nu D h from the circular tube

formulas The mass flow rate and the bulk velocity must be based on

the true cross-sectional area, which does not usually equal π D h2/4 (see

Problem7.46) The following example illustrates the procedure

Example 7.6

An air duct carries chilled air at an inlet bulk temperature of T bin =

17◦C and a speed of 1 m/s The duct is made of thin galvanized steel,

has a square cross-section of 0.3 m by 0.3 m, and is not insulated

A length of the duct 15 m long runs outdoors through warm air at

T ∞ = 37 ◦C The heat transfer coefficient on the outside surface, due

to natural convection and thermal radiation, is 5 W/m2K Find the

bulk temperature change of the air over this length

Solution. The hydraulic diameter, from eqn (7.61a) with a= b, is

simply

D h = a = 0.3 m

Using properties of air at the inlet temperature (290 K), the Reynoldsnumber is

ReD h = uavD h

ν = (1)(0.3)

(1.578 × 10 −5 ) = 19, 011

The Reynolds number for turbulent transition in a noncircular duct

is typically approximated by the circular tube value of about 2300, sothis flow is turbulent The friction factor is obtained from eqn (7.42)

tion must be considered Heat travels first from the air at T ∞throughthe outside heat transfer coefficient to the duct wall, and then throughthe inside heat transfer coefficient to the flowing air — effectively

through two resistances in series from the fixed temperature T ∞ to

the rising temperature T b We have seen in Section2.4that an overallheat transfer coefficient may be used to describe such series resis-tances Here,

= 0.3165

The outlet bulk temperature is therefore

T b = [17 + (37 − 17)(0.3165)] ◦C= 23.3 ◦C

§7.5 Heat transfer coefficients for noncircular ducts 373

The accuracy of the procedure just outlined is generally within±20%

and often within±10% Worse results are obtained for duct cross-sections

having sharp corners, such as an acute triangle Specialized equations

for “effective” hydraulic diameters have been developed in the literature

and can improve the accuracy of predictions to 5 or 10% [7.8]

When only a portion of the duct cross-section is heated — one wall of

a rectangle, for example — the procedure is the same The hydraulic

di-ameter is based upon the entire wetted perimeter, not simply the heated

part One situation in which one-sided or unequal heating often occurs

is an annular duct, for which the inner tube might be a heating element

The hydraulic diameter procedure will typically predict the heat transfer

coefficient on the outer tube to within ±10%, irrespective of the heating

configuration The heat transfer coefficient on the inner surface,

how-ever, is sensitive to both the diameter ratio and the heating configuration

For that surface, the hydraulic diameter approach is not very accurate,

especially if D i  D o; other methods have been developed to accurately

predict heat transfer in annular ducts (see [7.3] or [7.8])

Laminar flow in noncircular ducts

Laminar velocity profiles in noncircular ducts develop in essentially the

same way as for circular tubes, and the fully developed velocity profiles

are generally paraboloidal in shape For example, for fully developed flow

between parallel plates located at y = b/2 and y = −b/2, the velocity

profile is

u uav = 3

for uav the bulk velocity This should be compared to eqn (7.15) for a

circular tube The constants and coordinates differ, but the equations

are otherwise identical Likewise, an analysis of the temperature profiles

between parallel plates leads to constant Nusselt numbers, which may

be expressed in terms of the hydraulic diameter for various boundary

7.541 for fixed plate temperatures

8.235 for fixed flux at both plates

5.385 one plate fixed flux, one adiabatic

(7.63)

Some other cases are summarized in Table7.4 Many more have been

considered in the literature (see, especially, [7.5]) The latter include

Table 7.4 Laminar, fully developed Nusselt numbers based on

hydraulic diameters given in eqn (7.61)

Cross-section T w fixed q w fixed

is small (so that h will be large): if the conduction resistance of the tube

wall is comparable to the convective resistance within the duct, then perature or flux variations around the tube perimeter must be expected.This will significantly affect the laminar Nusselt number The rectangu-lar duct values in Table 7.4 for fixed wall flux, for example, assume auniform temperature around the perimeter of the tube, as if the wall has

tem-no conduction resistance around its perimeter This might be true for acopper duct heated at a fixed rate in watts per meter of duct length.Laminar entry length formulæ for noncircular ducts are also given byShah and London [7.5]

7.6 Heat transfer during cross flow over cylindersFluid flow pattern

It will help us to understand the complexity of heat transfer from bodies

in a cross flow if we first look in detail at the fluid flow patterns that occur

in one cross-flow configuration—a cylinder with fluid flowing normal to

it Figure7.10shows how the flow develops as Re≡ u ∞ D/ν is increased

from below 5 to near 107 An interesting feature of this evolving flowpattern is the fairly continuous way in which one flow transition follows

§7.6 Heat transfer during cross flow over cylinders 375

Figure 7.10 Regimes of fluid flow across circular cylinders [7.22]

Figure 7.11 The Strouhal–Reynolds number relationship for

circular cylinders, as defined by existing data [7.22]

another The flow field degenerates to greater and greater degrees ofdisorder with each successive transition until, rather strangely, it regainsorder at the highest values of ReD

An important reflection of the complexity of the flow field is the

vortex-shedding frequency, f v Dimensional analysis shows that a mensionless frequency called the Strouhal number, Str, depends on theReynolds number of the flow:

di-Str≡ f v D

Figure7.11defines this relationship experimentally on the basis of about

550 of the best data available (see [7.22]) The Strouhal numbers stay alittle over 0.2 over most of the range of ReD This means that behind

a given object, the vortex-shedding frequency rises almost linearly withvelocity

Experiment 7.1

When there is a gentle breeze blowing outdoors, go out and locate alarge tree with a straight trunk or the shaft of a water tower Wet your

§7.6 Heat transfer during cross flow over cylinders 377

Figure 7.12 Giedt’s local measurements

of heat transfer around a cylinder in anormal cross flow of air

finger and place it in the wake a couple of diameters downstream and

about one radius off center Estimate the vortex-shedding frequency and

use Str 0.21 to estimate u ∞ Is your value of u ∞reasonable?

Heat transfer

The action of vortex shedding greatly complicates the heat removal

pro-cess Giedt’s data [7.23] in Fig.7.12show how the heat removal changes

as the constantly fluctuating motion of the fluid to the rear of the