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5* Finned Tube Bundles in Cross-Flow

5* Finned Tube Bundles in Cross-Flow

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7.5 Finned Tube Bundles in Cross-Flow



459



thermal resistance in flow over tube bundles. This, as is evident from the heat transfer rate equation,

qc ϭ hq cA⌬T

results in either increased qc for a fixed temperature difference ⌬T or a reduction in

the required ⌬T for a fixed heat load qc. The most widely used method to meet these

enhancement objectives is to employ externally finned tubes. A typical example of

such tubes for a variety of industrial heat exchangers are shown in Fig. 7.29.

For cross-flow over finned tube banks, a large set of experimental data and correlations for tubes with circular or helical fins have been reviewed by ZI ukauskas

[42]. In calculating the pressure drop and heat transfer, recall that the Reynolds

number is based on the maximum flow velocity in the tube bank, and it is given by

Umax = Uq * max c



(ST> 2)

ST

, 2

d

ST - D [SL + (ST> 2)2]1/2 - D



and

Re = (rU max D>m)



(7.39)



where ST and SL are the transverse and longitudinal pitch, respectively, of the tube

array. Also, based on the analysis and results of Lokshin and Fomina [43] and

Yudin [44], the friction loss is given in terms of the Euler number Eu, and the pressure drop is obtained from

¢p = Eu1rV2qNL2Cz



FIGURE 7.29 Typical tube

with fins on its outer surfaces

that are used in industrial

heat exchangers.



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(7.40)



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Chapter 7 Forced Convection Over Exterior Surfaces

where Cz is a correction factor for tube bundles with NL Ͻ 5 rows of tubes in the

flow direction, and it can be obtained from the following table:



NL

Aligned

Staggered



1



2



3



4



Ն5



2.25

1.45



1.6

1.25



1.2

1.1



1.05

1.05



1.0

1.0



In flows across inline (aligned) tube banks with circular or helical fins, where

␧ is the finned surface extension ratio (␧ ϭ ratio of total surface area with fins to the

bare tube surface area without fins), the Euler number and the Nusselt number,

respectively, are given by the following equations:

Eu = 0.068e0.5 a



ST - 1 - 0.4

b

SL - 1



(7.41)



for 103 Յ ReD Յ 105, 1.9 Յ ␧ Յ 16.3, 2.38 Յ (ST> D) Յ 3.13, and 1.2 Յ (SL> D) Յ 2.35,

NuD ϭ 0.303␧Ϫ0.375 ReD0.625 Pr0.36 a



Pr 0.25

b

Prw



(7.42)



for 5 ϫ 103 Յ ReD Յ 105, 5 Յ ␧ Յ 12, 1.72 Յ (ST> D) Յ 3.0, and 1.8 Յ (SL> D) Յ 4.0,

Likewise for cross-flow over staggered tube bundles with circular or helical fins, the

recommended correlation for Euler number is

Eu ϭ C1 ReaD ␧0.5 (ST> D) - 0.55 (SL > D) - 0.5



(7.43)



where

C1 ϭ 67.6, a ϭ Ϫ0.7 for 102 Յ ReD Ͻ 103, 1.5 Յ ␧ Յ 16, 1.13 Յ ST> D Յ 2.0, 1.06 Յ

SL > D Յ 2.0



C1 ϭ 3.2, a ϭ Ϫ0.25 for 103 Յ ReD Ͻ 105, 1.9 Յ ␧ Յ 16, 1.6 Յ ST> D Յ 4.13, 1.2 Յ

SL > D Յ 2.35



C1 ϭ 0.18, a ϭ 0 for 105 Յ ReD Ͻ 1.4 ϫ 106, 1.9 Յ ␧ Յ 16, 1.6 Յ ST> D Յ 4.13, 1.2 Յ

SL > D Յ 2.35



and the Nusselt number is given by

Nu ϭ C2 ReaD Prb (ST > SL )0.2 ( pf >D)0.18 (hf >D) - 0.14 (Pr> Prw)0.25



(7.44)



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7.6 Free Jets



461



where pf is the fin pitch, hf is the fin height, and

C2 ϭ 0.192, a ϭ 0.65, b ϭ 0.36 for 102 Յ ReD Յ 2 ϫ 104

C2 ϭ 0.0507, a ϭ 0.8, b ϭ 0.4 for 2 ϫ 104 Յ ReD Յ 2 ϫ 105

C2 ϭ 0.0081, a ϭ 0.95, b ϭ 0.4 for 2 ϫ 105 Յ ReD Յ 1.4 ϫ 106

Also, Eq. (7.44) is valid for the general range of the following fin-and-tube pitch

parameters:

0.06 Յ (pf /D) Յ 0.36, 0.07 Յ hf >D Յ 0.715, 1.1 Յ (ST> D) Յ 4.2, 1.03 Յ (SL> D) Յ 2.5

In evaluating the Euler number Eu and the Nusselt number Nu given by the correlations in Eqs. (7.41) through (7.44), and hence the pressure drop and heat transfer coefficient in cross-flow over finned tube banks, it would be instructive to

compare the results with those for plain or unfinned tubes. To this end, the student

should repeat as a home exercise the problems of Examples 7.5 and 7.6 (Section 7.4)

by using finned tubes instead of plain tubes.



7.6*



Free Jets

One method of expending high convective heat flux from (or to) a surface is with

the use of a fluid jet impinging on the surface. The heat transfer coefficient on an

area directly under a jet is high. With a properly designed multiple jet on a surface

with nonuniform heat flux, a substantially uniform surface temperature can be

achieved. The surface on which the jet impinges is termed the target surface.

Confined and Free Jets The jet can be either a confined jet or a free jet. With a

confined jet, the fluid flow is affected by a surface parallel to the target surface

[Fig. 7.30(a)]. If the parallel surface is sufficiently far away from the target

surface, the jet is not affected by it, and we have a free jet [Fig. 7.30(b)].

Heat transfer from the target surface may or may not lead to a change in phase

of the fluid. In this section, only free jets without change in phase are considered.

Classification of Free Jets Depending on the cross section of the jet issuing from

a nozzle and the number of nozzles, jets are classified as

Single Round or Circular Jet (SRJ)

Single Slot or Rectangular Jet (SSJ)

Array of Round Jets (ARJ)

Array of Slot Jets (ASJ)



Target

surface



Confined jet

(a)



Nozzle exit



Free jet

(b)



FIGURE 7.30 Confined and free jets.

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Chapter 7 Forced Convection Over Exterior Surfaces

Free jets are further classified as free-surface or submerged jets. In the case of

a free-surface jet, the effect of the surface shear stress on the flow of the jet is negligible. A liquid jet surrounded by a gas is a good example of a free-surface jet. In

the case of a submerged jet, the flow is affected by the shear stress at the surface. As

a result of the surface shear stress, a significant amount of the surrounding fluid is

dragged by the jet. The entrained fluid (that part of the surrounding fluid dragged by

the jet) affects the flow and heat transfer characteristics of the jet. A gaseous jet issuing into a gaseous medium (e.g., an air jet issuing into an atmosphere of air) or a liquid jet into a liquid medium are examples of submerged jets. Another difference

between the two is that gravity usually plays a part in free-surface jets; the effect of

gravity is usually negligible in submerged jets. The two types of jets are illustrated

in Fig. 7.31.

In a free-surface round jet, the liquid film thickness along the target surface

continuously decreases [Fig. 7.31(a)]. With a slotted free-surface jet, the thickness of the liquid film attains a constant value some distance from the axis of

the jet [Fig. 7.31(b)]. With a submerged jet, because of the entrainment of the

surrounding fluid, the fluid thickness increases in the direction of flow

[Fig. 7.31(c)].

Flow with Single Jets Three distinct regions are identified in single jets (Fig. 7.32).

For some distance from the nozzle exit, the jet flow is not significantly affected by

the target surface; this region is the free-jet region. In the free-jet region, the velocity

component perpendicular to the axis of the jet is negligible compared with the axial

component. In the next region, the stagnation region, the jet flow is influenced by the

target surface. The magnitude of the axial velocity decreases while the magnitude of

the velocity parallel to the surface increases. Following the stagnation region is the

wall-jet region where the axial velocity component is negligible compared with the

velocity component parallel to the surface.

d



w



Nozzle exit



Nozzle exit



Free-surface round jet

(a)



Free-surface slotted jet

(b)



Nozzle exit



Submerged jet

(c)



FIGURE 7.31 Free surface and submerged jets.

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7.6 Free Jets



463



d,w

Nozzle exit



Free jet



zo



z

r, x

Stagnation



Wall jet



FIGURE 7.32 The three regions in a jet and definition of coordinates.



7.6.1 Free-Surface Jets—Heat Transfer Correlations

Unless the turbulence level in the issuing jet is very high, a laminar boundary layer

develops adjacent to the target surface. The laminar boundary layer has four regions,

as shown in Fig. 7.33.

The delineation of the four regions for an SRJ with Pr Ͼ 0.7 are

Region I

Region II

Region III

Region IV



Stagnation layer: The velocity and temperature boundary layer thicknesses

are constant, ␦ Ͼ ␦t

The velocity and temperature boundary layer thicknesses increase with r

but neither has reached the free surface of the fluid film.

The velocity boundary layer has reached the free surface but the temperature boundary layer has not.

Both velocity and temperature boundary layers have reached the free surface.



d



z

zo



r

I



II





b



δ δt

III



IV



rt



Laminar boundary layer



Turbulent boundary layer



FIGURE 7.33 Definitions of the four regions in the laminar

boundary layer.

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Chapter 7 Forced Convection Over Exterior Surfaces

Heat Transfer Correlations with a Free-Surface SRJ Uniform Heat Flux

(Liu et al. [45])

Region I: r Ͻ 0.8 d

Pr 7 3



1/3

Nud = 0.797 Re1/2

d Pr



(7.45)



0.15 … Pr … 3



2/5

Re1/2

d Pr



(7.46)



Region II: 0.8 Ͻ r>d Ͻ rv >d



Nud = 0.715



rv

= 0.1773 Re1/3

d

d



(7.47)



1/2

1/3 d

Nud = 0.632 Re1/2

d Pr a b

r



(7.48)



The Reynolds number in this section is based on the jet velocity, vj.

Region III: rv Ͻ r Ͻ rt (from Suryanarayana [46])

rt

p 3 1/2 1/3

p 3 1/2 1/3

s

s 2

s

s 2

= e+ ca b + a b d f

+ e - + ca b - a b d f

(7.49)

d

2

2

3

2

2

3

-2c

0.2058 Pr - 1

0.00686 RedPr

s =

0.2058 Pr - 1



p =



c = - 5.051 * 10-5 Re2/3

d



Nud =



1/3

0.407 Re1/3

d Pr a



d 2/3

b

r



1/3

d 2

5.147 r 2/3 1 r 2

c0.1713a b +

a b d c a b + cd

r

Red d

2 d



(7.50)



Region IV: r Ͼ rt

Nud =



where



0.25

rt 2 r 2

bt

b

1

c1 - a b d a b + 0.13a b + 0.0371a b

r

RedPr

d

d

d



b

d

5.147 r 2

= 0.1713a b +

a b

r

d

Red d



bt ϭ b



(7.51)



at rt



Region IV occurs only for Pr Ͻ 4.86 and is not valid for Pr Ͼ 4.86. Values of

rv >d and rt >d are given in Table 7.4.

Equations (7.45) through (7.51) are applicable for laminar jets. With a round

nozzle, the upper limit of Reynolds number for laminar flow is between 2000 and

4000. In the experiments leading to the correlations, specially designed sharp-edged



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7.6 Free Jets

TABLE 7.4



465



Values of rv/d [Eq. (7.47)] and rt /d [Eq. (7.49)]

rt/d



Red



rt /d



Pr ‫ ؍‬1



Pr ‫ ؍‬2



Pr ‫ ؍‬3



Pr ‫ ؍‬4



1,000

4,000

10,000

20,000

30,000

40,000

50,000



1.773

2.81

3.82

4.82

5.5

6.1

6.5



4.1

6.51

8.8

11.1

12.8

14.0

15.1



5.71

9.07

12.3

15.5

17.8

19.5

21.0



7.55

11.98

16.3

20.5

23.5

25.8

27.8



10.75

17.06

23.2

29.2

33.4

36.8

39.6



Sharp edged nozzle



FIGURE 7.34 Sharp-edged

orifice.

nozzles (with an inlet momentum break-up plate), as shown in Fig. 7.34, were

employed. In those experiments, even with Reynolds numbers as high as 80,000,

there was no splattering. Usually, pipe-type nozzles are used, and it is recommended

that Eqs. (7.45) through (7.51) be used for laminar flow in pipes. With turbulent flows

in pipe nozzles, splattering results. For information on heat transfer with splattering,

refer to Lienhard et al. [47].



EXAMPLE 7.7



SOLUTION



A jet of water (at 20°C) issues from a 6-mm-diameter (1/4-inch) nozzle at a rate of

0.008 kg/s. The jet impinges on a 4-cm-diameter disk which is subjected to a uniform heat flux of 70,000 W/m2 (total heat transfer rate of 88 W). Find the surface

temperature at radial distances of (a) 3 mm and (b)12 mm from the axis of the jet.

Properties of water (from Appendix 2, Table 13):



␮ ϭ 993 ϫ 10Ϫ6 N s/m2

k ϭ 0.597 W/m K

Pr ϭ 7.0

Red =



#

4m

4 * 0.008

=

= 1709

p dm

p * 0.006 * 993 * 10-6



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Chapter 7 Forced Convection Over Exterior Surfaces

(a) For r ϭ 3 mm, r> d ϭ 0.003/0.006 ϭ 0.5 (Ͻ0.8).

From Eq. (7.45),

Nud =



qhcd

= 0.797 * 17091/2 * 7.01/3 = 63.0

k



qhc = 63.0 * 0.597 = 6269 W/m2 °C

0.006

qœœ

70,000

= 20 +

= 31.2 °C

Ts = Tj +

qhc

6269

(b) For r ϭ 12 mm, rv ϭ 0.1773 ϫ 17091/3 ϫ 0.006 ϭ 0.013 m and r Ͻ rv .

From Eq. (7.48) for Region II,

Nud = 0.632 * 17091/2 * 7.01/3 * a



0.006 1/2

b

= 35.3

0.012



35.3 * 0.597

hqc =

= 3512 W/m2 °C

0.006



Ts = 20 +



70,000

= 39.9 °C

3512



The boundary layer becomes turbulent at some point downstream. Different criteria for the transition to turbulent flow have been suggested. Denoting the radius at

which the flow becomes turbulent by rc, rc> d = 1200 Red-0.422. The criterion of Liu

et al. [45] for the radius rh at which the flow becomes fully developed turbulent and

the heat transfer correlation in that region are given here.

Fully developed turbulent flow:

rh

28,600

=

d

Re0.68

d

Nud =



8 RedPrf

b

r 2

49a b + 28a b f

d

d



(7.52)



where

f =



Cf >2



1.07 + 12.7(Pr2/3 - 1)3Cf >2



0.02091 r 5/4

d

b

=

a b

+ Ca b

1/4

r

d

d

Red



r 1/4

Cf = 0.073 Red-1/4 a b

d



C = 0.1713 +



5.147 rc

0.02091 rc 1/4

a ba b

Red d

d

Re1/4

d



Although the stagnation region is limited to less than 0.8d from the axis of the

jet, one can take advantage of the high heat transfer coefficient for cooling in regions

of high heat fluxes.

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7.6 Free Jets

Heat Transfer Correlations with a Free-Surface SRJ

Temperature (Webb and Ma [48]) Pr Ͼ 1.



467



Uniform Surface



Region I: r> d Ͻ 1

1/3

Nud = 0.878 Re1/2

d Pr



Region II: ␦ Ͻ b



r Ͻ rv



rv

= 0.141 Re1/3

d

d



(7.53)

rN =



r 1

d Re1/3

d



1/3 -1/2

Nud = 0.619 Re1/3

d Pr (rN )



Region III: ␦ ϭ b

Nud =



␦t Ͻ b



rv Ͻ r Ͻ rt



rN =



(7.54)



r 1

d Re1/3

d



1/3

2 Re1/3

d Pr



(6.41rN 2 + 0.161> rN )[6.55 ln(35.9rN 3 + 0.899) + 0.881]1/3



(7.55)



In general, the convective heat transfer coefficients with uniform surface temperature are less than those with uniform surface heat flux.

Heat Transfer Correlations with a Free-Surface SSJ Local convective heat transfer

coefficient—Uniform Heat Flux (Wolf et al. [49], valid for 17,000 Ͻ Rew Ͻ 79,000,

2.8 Ͻ Pr Ͻ 5:

0.4

Nuw = Re0.71

w Pr f(x> w)



For 0 …



(7.56)



x

… 1.6, use

w



x 2

x 2

x

f (x>w) = 0.116 + a b c0.00404a b - 0.00187a b - 0.0199 d (7.57)

w

w

w

x

For 1.6 … a b … 6, use

w

x

x 2

f (x>w) = 0.111 - 0.02a b + 0.00193a b

w

w



(7.58)



Figure 7.32 defines x and w.

Turbulent Flow Correlation Equation (7.56) is valid for laminar flows. Transition

to turbulence is affected by the free-stream turbulence level. Turbulent flow occurs

for Rex in the range of 4.5 ϫ 106 (low free-stream turbulence of 1.2%) to 1.5 ϫ 106

(high turbulence of 5%). In the turbulent region for the local convective heat transfer

coefficient, McMurray et al. [50] proposes

1/3

Nu x = 0.037 Re4/5

x Pr



(7.59)



where Nux ϭ (hc x> k) and Rex ϭ yJ x> n. Equation (7.59) is valid to a local Reynolds

number Rex ϭ 2.5 ϫ 106.

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Chapter 7 Forced Convection Over Exterior Surfaces

Heat Transfer Correlations with an Array of Jets With single jets, the heat transfer

coefficient in the stagnation zone is quite high but decreases rapidly with r> d or x> w.

High heat transfer rates from large surfaces can be achieved with multiple jets by

taking advantage of the high heat transfer coefficients in the stagnation zone. If the

separation distance between two jets is approximately equal to the stagnation zone, one

may expect such a high heat transfer coefficient. However, unless the fluid is removed

rapidly, the presence of the spent fluid leads to a degradation in heat transfer rate and

the average heat transfer coefficient may not reach the high values obtained in the

stagnation region with single jets.

The number of variables with an array of jets is quite large, and it is unlikely that

a single correlation can be developed to encompass all possible variables. Some of

the variables are the spacing between the jets and the target surface, the jet Reynolds

number, fluid Prandtl number, the pitch of the jets (distance between the axis of two

adjacent jets), and arrangement of the array [square or triangular—see Fig. (7.35)]. In

most cases, it is expected that the Reynolds number for each jet has the same value;

although with nonuniform heat flux, employing different jet Reynolds numbers may

lead to a more uniform surface temperature.

From experimental data with in-line and triangular jets, Pan and Webb [51] suggest the following correlation.

1/3 -0.095(S/d)

Nud = 0.225 Re2/3

d Pr e



(7.60)



Equation (7.60) is valid for

2 …



zo

… 5

d



2 …



S

… 8

d



5000 … Red … 22,000



For larger values of S> d, based on experimental results, Pan and Webb [51] recommend

1/3

Nud = 2.38 Re2/3

d Pr a



d 4/3

b

S



(7.61)



Equation (7.61) is valid for 13.8 Ͻ S> d Ͻ 330 and 7100 Ͻ Red Ͻ 48,000. For other

configurations, refer to the review by Webb and Ma [48].



S



d

In-line arrangement



Triangular arrangement



FIGURE 7.35 Definition of in-line and triangular arrangements

of jet arrays.



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7.6 Free Jets



469



It must be noted that with a vertical nozzle the fluid velocity increases (or

decreases) as the fluid issuing from the nozzle approaches the target surface. If such an

increase (or decrease) in the jet velocity is significant, the jet velocity and diameter or

width used in the computations of the Reynolds number and Nusselt number must

reflect the change in the velocity. The modified velocity is ym = yj ; 12gzo, where

yj is the jet velocity at the nozzle exit and zo is the distance between the nozzle exit and

the target surface. The jet velocity is increased if the target surface is below the nozzle

and decreased if the surface is above the nozzle. The corresponding diameter and width

are dj 1yj>ym, or wjyj >ym where the subscript j denotes the values at exit of the nozzle.



7.5.2 Submerged Jets—Heat Transfer Correlations

When the jet fluid is surrounded by the same type of fluid (liquid jet in a liquid or

gaseous jet in a gas) we have a submerged jet. Most engineering applications of submerged jets involve gaseous jets, usually air jets into air. The surrounding fluid is

entrained by the jet both in the free-jet and the wall-jet regions. Because of such

entrainment, the thickness of the fluid in motion increases in the direction of flow.

With free jets, the thickness is substantially constant for slotted jets and decreases

for round jets in the wall-jet region. Consequently, both fluid mechanical and heat

transfer characteristics of submerged jets are different from those of free surface jets.

Single Round Jets For local heat transfer with uniform heat flux, Ma and Bergles

[52] proposed

Nud = Nud,o c

Nud =

where



tanh(0.88r>d) 1/2

d

(r >d)



1.69 Nud,o

(r> d)



1.07



r

6 2

d



(7.62)



r

7 2

d



(7.63)



0.4

Nud,o = 1.29 Re0.5

d Pr



(7.64)



For liquid jets, replace the exponent of 0.4 for Pr in Eq. (7.64) by 0.33.

A composite equation for both the stagnation and wall jet regions by Sun et al.

[53] is

-1/17



Nu d = Nu d,o c c



1tanh(0.88r>d)

1r>d



d



-17



-17

1.69

+ c

d

s

(r>d)1.07



(7.65)



where Nud,o is given by Eq. (7.64).

A correlation for the average heat transfer coefficient to radius r with uniform

surface temperature by Martin [54] is

Nud = 2



d

r



0.5

1 - 1.1(d> r)

Re0.55

d

cRed a1 +

b d Pr0.42

zo

d

200

1 + 0.1 a

- 6b

r

d



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(7.66)



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