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4* Analogy Between Heat and Momentum Transfer in Turbulent Flow

# 4* Analogy Between Heat and Momentum Transfer in Turbulent Flow

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Similarly, the shearing stress caused by the combined action of the viscous forces

and the turbulent momentum transfer is given by

m

t

du

= a

+ eM b

r

r

dy

(6.48)

According to the Reynolds analogy, heat and momentum are transferred by analogous processes in turbulent flow. Consequently, both q and ␶ vary with y, the distance 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

y

t

r

=

= 1 ts

rs

rs

(6.49)

and

qc>A

(qc>A)s

=

y

r

= 1 rs

rs

(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

y

ts

m

du

a1 + eM b

b = a

rs

r

r

dy

(6.51)

and

qc,s

Asrcp

a1 -

y

k

dT

b = -a

+ eH b

rs

rcp

dy

(6.52)

If eH = eM, the expressions in parentheses on the right-hand sides of Eqs. (6.51) and

(6.52) are equal, provided the molecular diffusivity of momentum ␮> ␳ equals the

molecular diffusivity of heat k> rcp, that is, the Prandtl number is unity. Dividing Eq.

(6.52) by Eq. (6.51) yields, under these restrictions,

qc,s

Ascpts

du = - dT

(6.53)

Integration of Eq. (6.53) between the wall, where u ϭ 0 and T ϭ Ts, and the bulk of

the fluid, where u = Uq and T ϭ Tb, yields

qsUq

= Ts - Tb

Ascpts

which can also be written in the form

ts

qs

hqc

1

=

=

2

As(Ts - Tb) cprUq

cprUq

rUq

(6.54)

since qhc is by definition equal to qs>As(Ts - Tb). Multiplying the numerator and the

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

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Chapter 6 Forced Convection Inside Tubes and Ducts

hqc

qhcDH

DHmk

m

k

Nu

= a

ba

ba

b =

= St

cpm UqDHr

k

RePr

cprUq DHmk

where St is the Stanton number.

To bring the left-hand side of Eq. (6.54) into a more convenient form, we use

Eqs. (6.13) and (6.14):

ts = f

rUq 2

8

Substituting Eq. (6.14) for ␶s in Eq. (6.54) finally yields a relation between the

Stanton number St and the friction factor

St =

f

Nu

=

RePr

8

(6.55)

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

empirical relation [17]

-0.2

f = 0.184ReD

(6.56)

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

St =

Nu

-0.2

= 0.023ReD

RePr

(6.57)

Since Pr was assumed unity,

Nu = 0.023Re0.8

D

(6.58)

m -0.8

qhc = 0.023Uq 0.8D-0.2k a b

r

(6.59)

or

Note that in fully established turbulent flow, the heat transfer coefficient is

directly proportional to the velocity raised to the 0.8 power, but inversely proportional 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 hqc proportional to 1> D1.8. The use of small tubes and high velocities is therefore conducive to

large heat transfer coefficients, but at the same time, the power required to overcome

the frictional resistance is increased. In the design of heat exchange equipment, it is

therefore necessary to strike a balance between the gain in heat transfer rates

achieved by the use of ducts having small cross-sectional areas and the accompanying 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 relative

roughness, 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

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6.4 Analogy Between Momentum and Heat Transfer in Turbulent Flow

385

0.1

0.09

0.08

0.05

0.04

0.07

0.06

0.03

0.05

f, friction factor

0.04

0.01

0.008

0.006

0.03

0.004

0.025

0.02

Laminar flow

f = 64

ReD

Equation 6.56

Laminar

flow

Transition

zone

0.002

0.001

0.0008

0.0006

0.0004

0.015

0.01

0.009

0.008

0.0002

0.0001

Complete turbulence, rough pipes

0.00005

Critical

zone

103

2

Relative roughness ε

D

0.02

0.015

3 4 5 6789104

2

3 4 5 6789105 2 3 4 5 6789106 2

Reynolds number ReD = ρuD/µ

0.0001

2 3 4 5 6789108

ε = 0.000005

D

ε = 0.000001

D

3 4 5 6789107

FIGURE 6.17 Friction factor versus Reynolds number for laminar and turbulent flow in tubes with various

surface roughnesses.

Source: Courtesy of L. F. Moody, “Friction Factor for Pipe Flow,” Trans. ASME, vol. 66, 1944.

increases the friction coefficient, also increases the convection conductance.

Experiments performed by Cope [28] and Nunner [29] are qualitatively in agreement with this prediction, but a considerable increase in surface roughness is

required to improve the rate of heat transfer appreciably. Since an increase in the

surface roughness causes a substantial increase in the frictional resistance, for the

same pressure drop, the rate of heat transfer obtained from a smooth tube is larger

than from a rough one in turbulent flow.

Measurements by Dipprey and Sabersky [30] in tubes artificially roughened

with sand grains are summarized in Fig. 6.18 on the next page. Where the

Stanton number is plotted against the Reynolds number for various values of the

roughness ratio e> D. The lower straight line is for smooth tubes. At small

Reynolds numbers, St has the same value for rough and smooth tube surfaces.

The larger the value e> D, the smaller the value of Re at which the heat transfer

begins to improve with increase in Reynolds number. But for each value of e> D,

the Stanton number reaches a maximum and, with a further increase in Reynolds

number, begins to decrease.

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Chapter 6 Forced Convection Inside Tubes and Ducts

4 × 10–3

St = NuD / ReD Pr

3

0.08

ε /D = 0.04

0.02

2

0.01

ε /D = 0.005

0.002

10–4

5 × 103 8 104

0.0005

Smooth pipe

6

ε /D = 0.001

10–3

8

2

4

6 8 105

ReD

2

4

6 8 × 105

FIGURE 6.18 Heat transfer in artificially roughened tubes, St versus

Re for various values of e/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.

6.5

Empirical Correlations for Turbulent Forced Convection

The Reynolds analogy presented in the preceding section was extended semi-analytically to fluids with Prandtl numbers larger than unity in [31–34] and to liquid metals with very small Prandtl numbers in [31], but the phenomena of turbulent forced

convection are so complex that empirical correlations are used in practice for engineering design.

6.5.1 Ducts and Tubes

The Dittus-Boelter equation [35] extends the Reynolds analogy to fluids with

Prandtl numbers between 0.7 and 160 by multiplying the right-hand side of

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

NuD =

hqcD

n

= 0.023Re0.8

D Pr

k

(6.60)

where

n = e

0.4 for heating (Ts 7 Tb)

0.3 for cooling (Ts 6 Tb)

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

Eq. (6.60) has been confirmed experimentally to within Ϯ25% for uniform wall

temperature as well as uniform heat-flux conditions within the following ranges of

parameters:

0.5 6 Pr 6 120

6000 6 ReD 6 107

60 6 (L> D)

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6.5 Empirical Correlations for Turbulent Forced Convection

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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 situations with moderate temperature differences (Ts - Tb).

For situations in which significant property variations due to a large temperature difference (Ts - Tb) exist, a correlation developed by Sieder and Tate [16] is

recommended:

1/3

NuD = 0.027Re0.8

D Pr a

mb 0.14

b

ms

(6.61)

In Eq. (6.61), all properties except ms are evaluated at the bulk temperature. The viscosity ms is evaluated at the surface temperature. Equation (6.61) is appropriate for

uniform wall temperature and uniform heat flux in the following range of conditions:

0.7 6 Pr 6 10,000

6000 6 ReD 6 107

60 6 (L> D)

To account for the variation in physical properties due to the temperature gradient in

the flow direction, the surface and bulk temperatures should be the values halfway

between the inlet and the outlet of the duct. For ducts of other than circular crosssectional shapes, Eqs. (6.60) and (6.61) can be used if the diameter D is replaced by

the hydraulic diameter DH.

A correlation similar to Eq. (6.61) but restricted to gases was proposed by Kays

and London [17] for long ducts:

0.3

NuDH = CRe0.8

DH Pr a

Tb n

b

Ts

(6.62)

where all properties are based on the bulk temperature Tb. The constant C and the

exponent n are:

0.020 for uniform surface temperature Ts

C = e

0.020 for uniform heat flux q s–

n = e

0.020 for heating

0.150 for cooling

More complex empirical correlations have been proposed by Petukhov and

Popov [38] and by Sleicher and Rouse [37]. Their results are shown in Table 6.3 on

the next page, which presents four empirical correlation equations widely used by

engineers to predict the heat transfer coefficient for turbulent forced convection in

long, smooth, circular tubes. A careful experimental study with water heated in

smooth tubes at Prandtl numbers of 6.0 and 11.6 showed that the Petukhov-Popov

and the Sleicher-Rouse correlations argeed with the data over a Reynolds number

range between 10,000 and 100,000 to within Ϯ5%, while the Dittus-Boelter and

Sieder-Tate correlations, popular with heat transfer engineers, underpredicted the

data by 5 to 15% [38]. Figure 6.19 on the next page shows a comparison of these

equations with experimental data at Pr ϭ 6.0 (water at 26.7°C). The following

example illustrates the use of some of these empirical correlations.

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Heat transfer correlations for liquids and gases in incompressible flow through tubes and pipes

Formulaa

Name (reference)

Dittus-Boelter [35]

NuD =

ne

Sieder-Tate [16]

Petukhov-Popov [36]

n

0.23Re0.8

D Pr

= 0.4  for heating

= 0.3  for cooling

mb 0.14

b

ms

where

0.5 6 Pr 6 120

(6.60)

6000 6 ReD 6 107

0.7 6 Pr 6 104

( f/8)ReDPr

K1 + K2( f/8)

Equation

6000 6 ReD 6 107

0.3

NuD = 0.027Re0.8

D Pr a

NuD =

Conditions

1/2

(Pr

2/3

0.5 6 Pr 6 2000

104 6 ReD 6 5 * 106

- 1)

(6.61)

(6.63)

f = (1.82 log10 ReD - 1.64)-2

K1 = 1 + 3.4f

1.8

K2 = 11.7 +

Sleicher-Rouse [37]

Pr 1/3

where

a = 0.88 -

0.1 6 Pr 6 105

104 6 ReD 6 106

(6.64)

0.24

4 + Prs

b ϭ 1/3 ϩ 0.5eϪ0.6Prs

a

All properties are evaluated at the bulk fluid temperature except where noted. Subscripts b and s indicate bulk and surface temperatures,

respectively.

103

9

8

Range of experimental data

7

6

Petukhov-Popov

5

Nusselt number, NuD

Sleicher-Rouse

4

Sieder-Tate

3

Dittus-Boelter

2

102

3 × 104

388

105

Reynolds number, ReD

2 × 105

FIGURE 6.19 Comparison of predicted and measured Nusselt number for turbulent flow of water in

a tube (26.7°C; Pr ϭ 6.0).

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6.5 Empirical Correlations for Turbulent Forced Convection

EXAMPLE 6.5

SOLUTION

389

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 and

compare the results obtained from all four equations in Table 6.3. The properties of

water are given below in engineering units.

T

(°F)

m

(lbm /h ft)

k

(Btu/h ft °F)

r

(lbm/ft3)

c

(Btu/lbm °F)

100

140

180

1.67

1.14

0.75

0.36

0.38

0.39

62.0

61.3

60.8

1.0

1.0

1.0

The hydraulic diameter DH for this geometry is 0.5 in. The Reynolds number based

on the hydraulic diameter and the bulk temperature properties is

ReDH =

(10 ft/s)(0.5/12 ft)(60.8 lbm/ft3)(3600 s/h)

rUqDH

=

m

0.75 lbm/h ft

= 125,000

The Prandtl number is

Pr =

cpm

k

=

(1.0 Btu/lbm °F)(0.75 lbm/h ft)

= 1.92

(0.39 Btu/h ft °F)

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

0.3

NuDH = 0.023 Re0.8

= (0.023)(11,954)(1.22) = 334

DH Pr

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

0.3

NuDH = 0.27Re0.8

DH Pr a

mb 0.14

b

ms

= (0.027)(11,954)(1.24) a

0.75 0.14

= 358

b

1.67

1.5 in.

1 in.

Water in

annulus

180°F

10 ft/s

Insulation

Inner wall temperature = 100°F

FIGURE 6.20 Schematic diagram of annulus for

cooling of water in Example 6.5.

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Chapter 6 Forced Convection Inside Tubes and Ducts

The Petukhov-Popov correlation [Eq. (6.63)] gives

f = (1.82 log10ReDH - 1.64)-2 = (9.276 - 1.64)-2 = 0.01715

K1 = 1 + 3.4f = 1.0583

K2 = 11.7 +

NuDH =

1.8

= 13.15

Pr0.33

f ReDHPr/8

K1 + K2(f/8)1/2( Pr 0.67 - 1)

(0.01715)(125,000)(1.92/8)

=

1.0583 + (13.15)(0.01715/8)1/2(0.548)

= 370

The Sleicher-Rouse correlation [Eq. (6.64)] yields

a = 0.88 b =

0.24

= 0.88 - 0.0278 = 0.852

4 + 4.64

1

0.5

0.5

+ 0.6 Pr = 0.333 +

= 0.364

s

3

16.17

e

ReD = 82,237

NuDH = 5 + (0.015)(82,237)0.852(4.64)0.364

= 5 + (0.015)(15,404)(1.748) = 409

Assuming that the correct answer is NuDH = 370, the first two correlations underpredict NuDH by about 10% and 3.5%, respectively, while the Sleicher-Rouse

It should be noted that in general, the surface and film temperatures are not

known and therefore the use of Eq. (6.64) requires iteration for large temperature

differences. The main difficulty in applying Eq. (6.63) for conditions with varying

properties is that the friction factor f may be affected by heating or cooling to an

unknown extent. Thus, to account for variable property effects in the flow cross section due to a significant temperature difference between the tube surface and bulk

fluid, a correction factor is commonly employed. This is usually in the form of a

bulk-to-surface viscosity ratio or temperature ratio raised to some power, depending

on whether the fluid is heated or cooled in the tube; two examples are given in

Eqs. (6.61) and (6.62)

For gases and liquids flowing in short circular tubes (2 Ͻ L> D Ͻ 60) with abrupt

contraction entrances, the entrance configuration of greatest interest in heat exchanger

design, the entrance effect for Reynolds numbers corresponding to turbulent flow

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6.5 Empirical Correlations for Turbulent Forced Convection

391

becomes important [40]. An extensive theoretical analysis of the heat transfer and

the pressure drop in the entrance regions of smooth passages is given in [41], and

a complete survey of experimental results for various types of inlet conditions is

given in [40].

The most commonly used and widely accepted correlation in current practice

for turbulent flows in circular tubes, however, and one that accounts for both variable property and entrance length effects is the Gnielinski correlation [42]. It is a

modification of the Petukhov and Popov [36] equation, is valid for the transition

flow and fully developed turbulent flow regimes (2300 … ReD … 5 * 106) as well

as a broad spectrum of fluids (0.5 6 Pr … 200), and is expressed as follows:

NuD =

(f> 8)(ReD - 1000) Pr

C 1 + (D> L)2/3 D K

1 + 12.7(f> 8)1/2( Pr 2/3 - 1)

(6.65)

where

(Prb> Pr s)0.11 for liquids

(Tb> Ts)0.45 for gases

and the friction factor f is calculated from the same expression used in the PetukhovPopov correlation of Eq. (6.65), as listed in Table 6.3. Note that instead of a viscosity ratio, the ratio of Prandtl number at bulk fluid and tube surface temperatures has

been used to account for variable property effects. This same correction factor can

be used as a multiplier to calculate f as well.

K = e

6.5.2 Ducts of Noncircular Shape

In many heat exchangers, rectangular, oval, trapezoidal, and concentric annular flow

passages, among others, are often employed. Some examples include plate-fin, ovaltube-fin, and double-pipe heat exchangers. The generally accepted practice in most

such cases, to a fair degree of accuracy as verified with experimental data [43], is to

use the circular-tube correlations with all dimensionless variables based on the

hydraulic diameter to estimate both the convective heat transfer coefficient and friction factor in turbulent flows. Thus, any of the correlations listed in Table 6.3 could

be employed, although the more popular recommendation in many handbooks is for

the Gnielinski correlation of Eq. (6.65)

The exception to this rule is the case of turbulent flows in concentric annuli

where the curvatures of the inner and outer diameters, or Di and Do, tend to have an

effect on the convective behavior, particularly when the ratio (Di> Do) is small [44,

45]. Based on experimental data and an extended analysis [44], the following correlation has been proposed:

NuDH = Nuc C1 + {0.8(Di> Do)-0.16}15 D

1/15

(6.66)

where Nuc is calculated from Eq. (6.65), again by using the hydraulic diameter of

the annular cross section, DH = (Do - Di), as the length scale. The duct-wall curvature effect, represented by the diameter ratio used in Eq. (6.66) is a modified

form of the correction factor considered by Petukhov and Roizen [45].

Furthermore, if effects of temperature-dependent fluid property variations in the

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Chapter 6 Forced Convection Inside Tubes and Ducts

flow cross section have to be included in the analysis, then the same correction

factor K recommended in Eq. (6.65) may be employed for liquids or gases, as the

case may be.

6.5.3 Liquid Metals

Liquid metals have been employed as heat transfer media because they have certain advantages over other common liquids used for heat transfer purposes.

Liquid metals, such as sodium, mercury, lead, and lead-bismuth alloys, have relatively low melting points and combine high densities with low vapor pressures

at high temperatures as well as with large thermal conductivities, which range

from 10 to 100 W/m K. These metals can be used over wide ranges of temperatures, they have a large heat capacity per unit volume, and they have large convection heat transfer coefficients. They are especially suitable for use in nuclear

power plants, where large amounts of heat are liberated and must be removed in

a small volume. Liquid metals pose some safety difficulties in handling and

pumping. The development of electromagnetic pumps has eliminated some of

these problems.

Even in a highly turbulent stream, the effect of eddying in liquid metals is of

secondary importance compared to conduction. The temperature profile is established much more rapidly than the velocity profile. For typical applications, the

assumption of a uniform velocity profile (called “slug flow”) may give satisfactory

results, although experimental evidence is insufficient for a quantitative evaluation

of the possible deviation from the analytic solution for slug flow. The empirical

equations for gases and liquids therefore do not apply. Several theoretical analyses

for the evaluation of the Nusselt number are available, but there are still some unexplained discrepancies between many of the experimental data and the analytic

results. Such discrepancies can be seen in Fig. 6.21, where experimentally measured

Nusselt numbers for heating of mercury in long tubes are compared with the analysis of Martinelli [2].

Lubarsky and Kaufman [46] found that the relation

NuD = 0.625(ReDPr)0.4

(6.67)

empirically correlated most of the data in Fig. 6.21, but the error band was substantial. Those points in Fig. 6.21 that fall far below the average are believed to

have been obtained in systems where the liquid metal did not wet the surface.

However, no final conclusions regarding the effect of wetting have been reached

to date.

According to Skupinski, et al. [47], the Nusselt number for liquid metals flowing in smooth tubes can be obtained from

NuD = 4.82 + 0.0185(ReDPr )0.827

(6.68)

if the heat flux is uniform in the range ReDPr 7 100 and L> D 7 30, with all properties evaluated at the bulk temperature.

According to an investigation of the thermal entry region for turbulent flow of

a liquid metal in a pipe with uniform heat flux, the Nusselt number depends only on

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6.5 Empirical Correlations for Turbulent Forced Convection

Lyon (theoretical)

Trefethen (mercury)

Johnson, Harinett, and Clabaugh (mercury

Johnson, Clabaugh, and Hartnett (mercury)

Stromquist (mercury)

English and Barrett (mercury)

Nusselt number, NuD = hcD/k

102

393

Isakoff and Drew (mercury: inside wall temperatures

calculated from fluid temperature profiles)

Isakoff and Drew (mercury: inside wall temperatures

calculated from outside wall temperature)

Styrikovich and Semenovker (mercury)

MacDonald and Quittenton (sodium)

Elser (mercury)

10

1

10

102

103

Peclet number, Pe = ReDPr

104

105

FIGURE 6.21 Comparison of measured and predicted Nusselt numbers for liquid metals heated in long tubes

with uniform heat flux.

Source: Courtesy of the National Advisory Committee for Aeronautics, NACA TN 3363.

the Reynolds number when ReDPr 6 100. For these conditions, Lee [48] found that

the equation

NuD = 3.0Re0.0833

D

(6.69)

fits data and analysis well. Convection in the entrance regions for fluids with

small Prandtl numbers has also been investigated analytically by Deissler [41],

and experimental data supporting the analysis are summarized in [49] and

[50]. In turbulent flow, the thermal entry length (L> DH)entry is approximately

10 equivalent diameters when the velocity profile is already developed and 30

equivalent diameters when it develops simultaneously with the temperature

profile.

For a constant surface temperature the data are correlated, according to Seban

and Shimazaki [51], by the equation

NuD = 5.0 + 0.025(ReDPr)0.8

in the range RePr 7 100, L> D 7 30.

(6.70)

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