7. Forced convection in a variety of configurations ·...

7. Forced convection in a variety of configurations

John Richard Thome

20 avril 2008

John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 1 / 64

7.2 Heat transfer to and from laminar flows in pipes

Figure 7.1 The development of a laminar velocity profile in a pipe.


4.1 Heat transfer to and from laminar flows in pipesDevelopment of a laminar flow

m : The mass flow rate [kg/s]uav : The average velocityAc : The cross-sectional area of the pipe

m = ρuavAc (7.1)

Fully developed flow∂u∂x = 0 or v = 0 (7.2)

The notion of the mixing-cup, or bulk are important. We define enthalpy andtemperature, hb and Tb. The bulk enthalpy for the fluid flowing through a crosssection of the pipe :

mhb ≡∫

Ac

ρuhdAc (7.3)

If we assume that fluid pressure variation in the pipe are too small to affect thethermodynamic state much and if we assume a constant value ofh = cp(T − Tref )dAc .

Tb =

∫AcρucpTdAmCp

(7.5)



In other word.

Tb =rate of flow of enthalpy through a cross section

rate of flow of heat capacity through a cross section

A fully developed flow, from the thermal standpoint, is one for which the relativeshape of the temperature profile does not change with x. We state thismathematically as.

∂

∂x

(Tw − TTw − Tb

)= 0 (7.7)

Where T generally depends on x and r.



Figure 7.2 The thermal development of flows in tubes with a uniform wall heatflux and with a uniform wall temperature (the entrance region)John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 5 / 64


The entrance regionIf we consider a small length of pipe, dx long with perimeter P, then its surfacearea is Pdx and an energy balance on it is.

dQ = qwPdx = mdhb = mcpdTb (7.8)− (7.9)

so thatdTbdx =

qwPmcp

(7.10)



Figure 7.3 The thermal behavior of flows in tubes with a uniform wall heat fluxand with a uniform temperature (the thermally developed region)John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 7 / 64


The thermally developed regionThe condition to obtain a developed profile. The convection coefficient, h, isconstant in the fully developed flow. When qw is constant, then Tw − Tb will beconstant in fully developed flow.

∂T∂x =

dTbdx =

qwPmcp

= cst (7.11)

In the uniform wall temperature case, the temperature profile keeps the sameshape, but its amplitude decreases with x, as does qw .


7.2 Heat transfer to and from laminar flows in pipesThe velocity profile in laminar tube flows

ReD ≡uavDν

For a fully developed velocity profile, the hydrodynamic entry length xe .xeD ' 0.03ReD (7.12)

The velocity profile for a fully developed laminar incompressible pipe flow can bederived from the momentum equation for an axisymetric flow. It turns out that theboundary layer assumption all happen te be valid for a fully developed pipe flow :

The pressure is constant across any section.∂2/∂x2 is exactly zero.The radial velocity is not just small, but it is zero.The term ∂u/∂x is not just small, but it is zero.

The boundary layer equation for cylindrically symmetrical flows.

u ∂u∂x + v ∂u

∂r = −1ρ

dpdx +

ν

r∂

∂r

(r ∂u∂r

)(7.13)



The solution for the velocity profile in laminar tube flows is

u(r) =R2

4µ

(−dpdx

)(1−

( rR

)2)

(7.14)

Average velocityuav = −R2

8µ

(dpdx

)In accordance with the conservation of mass, 2uav = umax , so

uuav

= 2(1−

( rR

)2)

(7.15)


7.2 Heat transfer to and from laminar flows in pipesThermal behavior of a flow with a uniform heat flux at the wallFully developed laminar region. In cylindrical coordinates, the energy equation is(the radial velocity is equal to zero) :

u ∂T∂x = α

1r∂

∂r

(r ∂T∂r

)(7.16)

∂T∂x =

dTbdx =

qwPmcp

(7.17b)

For the constant surface heat flux boundary condition, i.e. (∂2T/∂x2) = 0.1rddr

(r dTdr

)= 4qw

Rk

(1−

( rR

)2)

(7.18)

T =4qwRk

(r2

4 −r4

16R2

)+ C1lnr + C2 (7.19)

Since the temperature remains finite at r=0, then C1 = 0. From the requirementthat T (R) = Tb where Tb varies with x, it follows that C2 is :

C2 = Tb −724

qwRk



The temperature profile for fully developed laminar flow for constant surface heatflux is thus :

T = Tb +qwRk

(( rR

)2− 1

4( rR

)4− 7

24

)(7.20)

And at r = R, the last equation gives.

Tw − Tb =1124

qwRk =

1145

qwDk (7.21)

So the local NuD for fully developed flow, based on h(x) = qw/[Tw (x)−Tb(x)], is

NuD ≡qwD

(Tw − Tb)k =1148 = 4.364 (7.22)

for q′′s constant.



Thermal behavior of a flow in a isothermal pipeThe dimensional anlysis that showed NuD = cst for flow with a uniform heat fluxat the wall is unchanged when the pipe walll is isothermal. For fully developedflow and for Tw = constant, the convective heat transfer coefficient is :

NuD = 3.66 (7.23)



The thermal entrance regionThe entry-length equation takes the following form for the thermal entry region,where the velocity profile is assumed to be fully developed before heat transferstarts at x=0. xet

D ' 0.034ReDPr (7.24)

Graetz number : Gz ≡ ReDPrDx

For an isothermal wall, the following curve ffits are available for the Nusseltnumber in thermally developing flow.

NuD = 3.657 +0.0018Gz1/3

(0.04 + Gz−2/3)2 (7.28)

NuD = 3.657 +0.0668Gz1/3

0.04 + Gz−2/3 (7.29)



For fixed qw , a more complicated formular reproduces the exact result for localNusselt number (7.30).

NuD = 1.302Gz1/3 − 1 for 2x104 ≤ GzNuD = 1.302Gz1/3 − 0.5 for 667 ≤ Gz ≤ 2x104

NuD = 4.364 + 0.263Gz0.506e−41/Gz for 0 ≤ Gz ≤ 667



Figure 7.4 Local and average Nusselt numbers for the thermal entry region in ahydrodynamically developed laminar pipe flow.


7.3 Turbulent pipe flow

Turbulent entry lengthThe entry length xe and xet are generally shorter in turbulent flow than in laminarflow.

xetD ≈ 10

Figure 7.1 : Thermal entry lengths, xet/D, which NuD will be no more than5 per cent above its fully developed value in turbulent flow.



When heat transfer begins at the inlet to a pipe, the velocity and temperatureprofiles develop simultaneously. The entry length is then very strongly affected bythe shape of the inlet. For example, an inlet that induces vortices in the pipe, suchas a sharp bend or contraction, can create a much longer entry length than occursfor a thermally developing flow. These vortices may require 20 to 40 diameters todie out.

For various types of inlets, Bhatti and Shah provide the following correlation(value in Figure 7.2) :

NUDNUoo

= 1 +C

(L/D)n for Pr = 0.7 (7.31)



Figure 7.2 : Constants for the gas-flow simultaneous entry length correlation,eqn. (7.31), for various inlet configurations.



Illustrative experiment

Figure 7.5 : Heat transfer to air flowing in a 1 in. I.D., 60 in. long pipe



Illustrative experiment

Figure 7.5 shows average heat transfer data given by Kreith for air flowing in a 1in. I.D. isothermal pipe 60 in. in length. Let us see how these data compare withwhat we know about pipe flows thus far.

For laminar flow, NUD ≈ 3.66 at ReD = 750. This is the correct value for anisothermal pipe. However, the pipe is too short for flow to be fully developed overmuch, if any, of its length. Therefore NUD is not constant in the laminar range.The rate of rise of NUD with ReD becomes very great in the transitional range,which lies between ReD = 2100 and about 5000 in this case. Above ReD ≈ 5000,the flow is turbulent and it turns out that NUD ≈ Re0.8

D .



The Reynolds analogy and heat transfer

St =h

ρcpuav=

Cf /21 + 12.8 (Pr0.68 − 1)

√Cf /2

(7.32)

This should not be used at very low Pr’s, but it can be used in either uniform qwor uniform Tw situations.

The frictional resistance (coefficient de perte de charge ≤ coefficient defrottement) to flow in a pipe is normally expressed in terms of the Darcy-Weisbachfriction factor.

f ≡ head loss(pipe lenght

Du2

av2

) =4p(

LDρu2

av2

) (7.33)

where ∆p is the pressure drop in a pipe of length L.



Sof =

8τwρu2

av= 4Cf (7.34)

For a fully developed laminar flow, we obtain.

NUD =(f /8)ReDPr

1 + 12.8(Pr0.68 − 1)√f /8

(7.35)

For turbulent flows, the frictional resistance depends of the roughness. Equation(7.35) can be used directly along with Fig. 7.6 to calculate the Nusselt number forsmooth-walled pipes.



Figure 7.6 : Pipe friction factors.John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 24 / 64


Historical formulationThe local Nusselt number for fully developed turbulent flow in a smooth circulartube is obtained from the Chilton-Colburn analogy.

Cf2 =

f8 = StPr2/3 =

NuDReDPr

Pr2/3 (7.36/37)

For, 20000 < ReD < 300000, substituting the friction factor, we can find.

f4 = Cf =

0.046Re0.2

D(7.38)

So equation becomesNuD = 0.023Re4/5

D Pr1/3



A more widely used correlation however is the Dittus-Boelter equation.

NuD = 0.0243Re0.8D Pr0.4 (7.39)

These equations are intended for reasonably low temperature differences underwhich properties can be evaluated at a mean temperature Tb + Tw/2. TheColburn equation can be modified in the following way for liquids.

NuD = 0.023Re0.8D Pr1/3

(µbµw

)0.14(7.40)

The above correlations are applicable to constant temperature and constant heatflux wall condition with accuracies of about 25 percent.



Entrance region

Typically is very short for turbulent flow, 10 ≤( xxd

D)≤ 60. For short tube

( LD > 60), use

NuDNuD,fd

= 1 +C

(x/D)m

Where C and m depend on type of inlet (sharp or rounded).



Modern formulation

Petukhov recommends the following equation for the local Nusselt number in fullydeveloped flow in smooth pipes where all properties are evaluated at Tb

NuD =(f /8)ReDPr

1.07 + 12.7√f /8

(Pr2/3 − 1

) (7.41)

Where104 < ReD < 5 · 106 and 0.5 < Pr < 200

Where the friction factor for smooth pipes is given by

f =1

(1.82log10ReD − 1.64)2 (7.42)



Gnielinski later showed that the range of validity could be extended down to thetransition Reynolds number by making a small adjustment.

NuD =(f /8) (ReD − 1000)Pr

1 + 12.7√f /8

(Pr2/3 − 1

) (7.43)

For 2300 ≤ ReD ≤ 5 · 106.



Variations in physical propertiesThe effect of variable physical properties is dealt with differently for liquids andgases. In both cases, the Nusselt number is first calculated with all propertiesevaluated at Tb using eqn. (7.41) or (7.43). For liquids, one then corrects bymultiplying with a viscosity ratio. Over the interval 0.025 ≤ µb

µw≤ 12.5,

NUD = NUD |Tb

(µbµw

)n(7.44)

For gases a ratio of temperatures in kelvins is used, with 0.27 ≤ TbTw≤ 2.7

NUD = NUD |Tb

(TbTw

)n(7.45)


7.3 Example 7.3

A 21.5 kg/s flow of water is dynamically and thermally developed in a 12 cm I.D.pipe. The pipe is held at 90C and ε/D = 0. Find h and f where the bulktemperature of the fluid has reached 50C.Solution

uav =mρAc

= 1.946 m/s

SoReD =

uavDv = 573, 700

AndPr = 2.47, µb

µw= 1.74

From equation (7.42), f=0.0128 at Tb and n=0.11, we have NUD = 1617. Or

h = NUDkD = 8.907 W /m2K

The corrected friction factor is f=0.0122.



Heat transfer to fully developed liquid-metal in tubes A dimensional analysis

of the forced convection flow of a liquid metal over a flat surface showed that

Nu = fn (Pe) (7.51)

because viscous infuence were confined to a region very close to the wall.During heat transfer to liquid metals in pipes, the same thing occurs as isillustrated in Fig. 7.7 (next slide).The region of thermal influence extends far beyond the laminar sublayer, when Pris very small, and the temperature profile is not influenced by the sublayer.Conversely, when Pr is very high, the temperature profile is largely shaped withinthe laminar sublayer. At high or even moderate Pr’s, v is therefore very important,but at low Pr’s it vanishes from the functional equation. Equation (7.51) thusapplies to pipe flows as well as to flow over a flat surface.



Figure 7.7 : Velocity and temperature profiles during fully developed turbulentflow in a pipe.John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 33 / 64


Figure 7.8 : Comparison of measured and predicted Nusselt numbers for liquidmetals heated in long tubes with uniform wall heat flux, qw .John Richard Thome (LTCM - SGM - EPFL) Heat transfer - Convection 20 avril 2008 34 / 64


Another problem that besets liquid metal heat transfer measurements is the verygreat difficulty involved in keeping such liquids pure. There is a body of theory forturbulent liquid metal heat transfer that yields a prediction of the form

NUD = C1 + C2Pe0.8D (7.52)

Using the few reliable data sets available for uniform wall temperature conditions,Reed recommends

NUD = 3.3 + 0.02Pe0.8D (7.53)



For uniform wall heat flux, many more data are available, and Lyon recommendsthe following equation, shown in Fig. 7.8.

NUD = 7 + 0.025Pe0.8D (7.54)

Although equations. (7.53) and (7.54) are probably correct for pure liquids, wecannot overlook the fact that the liquid metals in actual use are seldom pure.Lubarsky and Kaufman put the following line through the bulk of the data in Fig.7.8.

NUD = 0.625Pe0.8D (7.55)


7.4 Heat transfert surface viewed as a heat exchanger

Let us reconsider the problem of a fluid flowing through a pipe with a uniform walltemperature. Suppose that we need to know the net heat transfer to a pipe ofknown length once h is known. This problem is complicated by the fact that thebulk temperature, Tb, is varying along its length.

Thus, if we wish to know how much pipe surface area is needed to raise the bulktemperature from Tb in to Tbout , we can calculate it as follows

Q = (mcp)b(Tbout − Tbin) = hA(LMTD)

Or

A =(mcp)b(Tbout − Tbin)

h

ln(

Tbout−TwTbin−Tw

)(Tbout − Tw )− (Tbin − Tw )

(7.56)


7.4 Heat transfert surface viewed as a heat exchangerSuppose that we do not know Tbout in the example above. Then we can write anenergy balance at any cross section

dQ = qwPdx = hP(Tw − Tb)dx = mcpdTb

Integration can be done∫ L

0

hPmcp

dx = −∫ Tbout

Tb in

d(Tw − Tb)

Tw − Tb

Pmcp

∫ L

0hdx = −ln

(Tw − TboutTw − Tbin

)Hence

hPLmcp

= −ln(Tw − TboutTw − Tbin

)which can be rearranged as

Tbout − TbinTw − Tbin

= 1− exp(−hPLmcp

)(7.57)


7.4 Heat transfert surface viewed as a heat exchanger

The left-hand side of equation (7.57) is the heat exchanger effectiveness. On theright-hand side we identify the argument of the exponential as NTU = UA/Cmin,and equation (7.57) becomes

ε = 1− exp(−NTU) (7.58)

Equation (7.57) applies to ducts of any cross-sectional shape. We can cast it interms of the hydraulic diameter, Dh = 4Ac/P, by substituting


= 1− exp(− hPLρuavcpAc

)(7.59a)


= 1− exp(− hρuavcp

4LDh

)(7.59b)


7.4 Example 7.5

Air at 20C is hydrodynamically fully developed as it flows in a 1 cmI.D. pipe. Theaverage velocity is 0.7 m/s. If it enters a section where the pipe wall is at 60C,what is the temperature 0.25 m farther downstream ?

SolutionReD =

uavDv = 422

The flow is therefore laminar. To account for the thermal entry region, wecompute the Graetz number

Gz =ReDPrD

x = 12.0


7.4 Example 7.5

Substituting this value into equation (7.29), we find NUD = 4.32. Thus

h = 11.6 W /m2K

Then, using equation (7.59b)


= 1− exp(− 11.61.14(1007)(0.7)

4(0.25)

0.01

)So that

Tb − 2060− 20 = 0.764

Or Tb = 50.6C .


7.5 Heat transfer coefficients for noncircular ducts

So far, we have focused on flows within circular tubes, which are by far the mostcommon configuration. Nevertheless, other cross-sectional shapes often occur. Forexample, the fins of a heat exchanger may form a rectangular passage throughwhich air flows.

In situations like these, all the qualitative ideas that we developed in Sections7.17.3 still apply, but the Nusselt numbers for circular tubes cannot be used incalculating heat transfer rates. The hydraulic diameter, which was introduced inconnection with eqn. (7.59b), provides a basis for approximating heat transfercoefficients in noncircular ducts. Recall that the hydraulic diameter is defined as

Dh ≡4AcP (7.60)

where Ac is the cross-sectional area and P is the passages wetted perimeter. Thehydraulic diameter measures the fluid area per unit length of wall.



The factor of four in the definition of Dh ensures that it gives the actual diameterof a circular tube. We noted in the preceding section that, for a circular tube ofdiameter D, Dh = D. Some other important cases include :

A rectangular duct of width a and height b :

Dh =4ab

2a + 2b =2aba + b (7.61a)

An annular duct of inner diameter Di and outer diameter Do :

Dh =4(πDo

2/4− πDi2/4)

π(Do + Di )= (Do − Di ) (7.61b)

And, for very wide parallel plates, eqn. (7.61a) with a » b gives :

Dh = 2b (7.61c)



Turbulent flow in noncircular ducts

With some caution, we may use Dh directly in place of the circular tube diameterwhen calculating turbulent heat transfer coefficients and bulk temperaturechanges.

Specifically, Dh replaces D in the Reynolds number, which is then used tocalculate f and NuDh from the circular tube formulas. The mass flow rate and thebulk velocity must be based on the true cross-sectional area, which does notusually equal πDh

2/4.

The following example illustrates the procedure.


7.5 Heat transfer coefficients for noncircular ducts -Example 7.6An air duct carries chilled air at an inlet bulk temperature of Tbin = 17°C and aspeed of 1 m/s. The duct is made of thin galvanized steel, has a squarecross-section of 0.3 m by 0.3 m, and is not insulated. A length of the duct 15 mlong runs outdoors through warm air at T∞ = 37°C.

The heat transfer coefficient on the outside surface, due to natural convection andthermal radiation, is 5 W /m2K . Find the bulk temperature change of the air overthis length.

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

Dh = a = 0.3 m

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

ReDh =uavDhv =

(1)(0.3)

(1.578× 10−5)= 19, 011


7.5 Heat transfer coefficients for noncircular ducts -Example 7.6

The Reynolds number for turbulent transition in a noncircular duct is typicallyapproximated by the circular tube value of about 2300, so this flow is turbulent.

The friction factor is obtained from eqn. (7.42)

f = [1.82log10(19, 011)− 1.64]−2 = 0.02646

and the Nusselt number is found with Gnielinskis equation, (7.43)

NuDh =(0.02646/8)(19, 011− 1, 000)(0.713)

1 + 12.7√0.02646/8[(0.713)2/3 − 1]

= 49.82

The heat transfer coefficient is

h = NuDh

kDh

=(49.82)(0.02623)

0.3 = 4.371 W /m2K



The remaining problem is to find the bulk temperature change. The thin metalduct wall offers little thermal resistance, but convection resistance outside theduct must be considered. Heat travels first from the air at T∞ through theoutside heat transfer coefficient to the duct wall, through the duct wall, and thenthrough the inside heat transfer coefficient to the flowing air - effectively throughthree resistances in series from the fixed temperature T∞ to the risingtemperature Tb. We have seen in Section 2.4 that an overall heat transfercoefficient may be used to describe such series resistances.

Here, with Ainside ∼=Aoutside , we find U based on inside area to be

U =1

Ainside

[1

(hA)inside+ Rtwall +

1(hA)outside

]−1

=

(1

4.371 +15

)−1= 2.332 W /m2K



We then adapt eqn. (7.59b) by replacing h by U and Tw by T∞ :

Tbout − TbinT∞ − Tbin

= 1− exp(− Uρuavcp

4LDh

)

= 1− exp[− 2.332

(1.217)(1)(1007)

4(15)

0.3

]= 0.3165

The outlet bulk temperature is therefore

Tbout = [17 + (37− 17)(0.3165)] = 23.3 C



Laminar flow in noncircular ducts

Laminar velocity profiles in noncircular ducts develop in essentially the same wayas for circular tubes, and the fully developed velocity profiles are generallyparaboloidal in shape. For example, for fully developed flow between parallel plateslocated at y = b/2 and y = −b/2,

uuav

=32

[1− 4

(yb

)2]

(7.62)

for uav the bulk velocity.

An analysis of the temperature profiles between parallel plates leads to constantNusselt numbers, which may be expressed in terms of the hydraulic diameter forvarious boundary conditions :



Cross-section Twfixed qwfixedCircular 3.657 4.364Square 2.976 3.608

Rectangulara=2b 3.391 4.123a=4b 4.439 5.331a=8b 5.597 6.490

Parallel plates 7.541 8.235

Figure 7.9 Laminar, fully developed Nusselt numbers based on hydraulicdiameters given in eqn. (7.61)


7.6 Heat transfer during cross flow over cylinders

Fluid flow pattern

First look in detail at the fluid flow patterns that occur in one cross-flowconfiguration : a cylinder with fluid flowing normal to it. Figure 7.10 shows howthe flow develops as

Re ≡ u∞Dv

is increased from below 5 to near 107.

An interesting feature of this evolving flow pattern is the fairly continuous way inwhich one flow transition follows another.



Figure 7.10 Regimes of fluid flow across circular cylinders



It can be seen on figure 7.10 that

flow field degenerates to greater and greater degrees of disorder with eachsuccessive transitionit regains order at the highest values of ReD

Heat transfer

Giedts data in Fig. 7.11 show how the heat removal changes as the constantlyfluctuating motion of the fluid to the rear of the cylinder changes with ReD .



Figure 7.11 Giedts local measurements of heat transfer around a cylinder in anormal cross flow of air.



It can be noticed on figure 7.11 that

NuD is near its minimum at 110° when ReD = 71, 000it maximizes at the same place when ReD = 140, 000

Direct prediction by methods discussed in Chapter 6 is out of the question.However, a great deal can be done with the data using relations of the form

NuD = fn(ReD ,Pr)

Indeed, for the entire range of the available data, Churchill and Bernstein offer thefollowing

NuD = 0.3 +0.62Re1/2

D Pr1/3

[1 + ( 0.4Pr )2/3]1/4

[1 +

(ReD

282, 000

)5/8]4/5

(7.63)



Breaking the correlation into component equations :

Below ReD = 4000, the bracketed term[1 +

(ReD

282, 000

)5/8]4/5

is ∼= 1. So

NuD = 0.3 +0.62Re1/2

D Pr1/3

[1 + ( 0.4Pr )2/3]1/4 (7.64)

Below Pe = 0.2, the Nakai-Okazaki relation

NuD =1

0.8237− ln(Pe1/2)(7.65)

should be used.



Finally, in the range 20,000 < ReD < 400,000, somewhat better results are givenby

NuD = 0.3 +0.62Re1/2

D Pr1/3

[1 + ( 0.4Pr )2/3]1/4

[1 +

(ReD

282, 000

)1/2]

(7.66)

than by (7.63).

Heat transfer during flow across tube bundles

A rod or tube bundle is an arrangement of parallel cylinders that heat, or arebeing heated by, a fluid that might flow normal to them, parallel with them, or atsome angle in between. The flow of coolant through the fuel elements of allnuclear reactors being used in this country is parallel to the heating rods. The flowon the shell side of most shell-and-tube heat exchangers is generally normal to thetube bundles.

See figure 7.12 .



Figure 7.12 Aligned and staggered tube rows in tube bundles.



Figure 7.12 shows the two basic configurations of a tube bundle in a cross flow. Inone, the tubes are in a line with the flow ; in the other, the tubes are staggered inalternating rows. For either of these configurations, heat transfer data can becorrelated reasonably well with power-law relations of the form

NuD = CRenDPr1/3 (7.67)

but in which the Reynolds number is based on the maximum velocity,

umax = uav

in the narrowest transverse area of the passage.

Thus, the Nusselt number based on the average heat transfer coefficient over anyparticular isothermal tube is

NuD =hDk ReD =

umaxDv



Zukauskas was able to correlate data on tube-bundle heat transfer over very largeranges of Pr , ReD , ST/D, and SL/D (see Fig. 7.12) with an expression of theform

NuD = Pr0.36(

PrPrw

)nfn(ReD) (7.68)

with n=0 for gases and 1/4 for liquids.

Properties are to be evaluated at the local fluid bulk temperature, except for Prw ,which is evaluated at the uniform tube wall temperature, Tw .

The function fn(ReD) takes the following form for the various circumstances offlow and tube configuration : see figure 7.13.



Figure 7.13 Various forms of the function fn(ReD).


7.6 Heat transfer during cross flow over cylindersAll of the preceding relations apply to the inner rows of tube bundles. The heattransfer coefficient is smaller in the rows at the front of a bundle, facing theoncoming flow. The heat transfer coefficient can be corrected so that it will applyto any of the front rows using Fig. 7.14.

Figure 7.14 Correction for the heat transfer coefficients in the front rows of atube bundle.


7.7 Other configurations

Do not forget that even the most completely empirical relations in Section 7.6were devised by people who were keenly aware of the theoretical framework intowhich these relations had to fit.

Thus, the theoretical considerations in Chapter 6 guide us in correlating limiteddata in situations that cannot be analyzed. Such correlations can be found for allkinds of situations, but all must be viewed critically. Many are based on limiteddata, and many incorporate systematic errors of one kind or another.

When you find a correlation, there are many questions that you should askyourself :

Is my case included within the range of dimensionless parameters upon whichthe correlation is based, or must I extrapolate to reach my case ?What geometric differences exist between the situation represented in thecorrelation and the one I am dealing with ? (inlet flow conditions for example)


7.7 Other configurations

Does the form of the correlating equation that represents the data, if there isone, have any basis in theory ? (If it is only a curve fit to the existing data,one might be unjustified in using it for more than interpolation of those data.)What nuisance variables might make our systems different ? (for examplesurface roughness, fluid purity, problems of surface wetting etc.)To what extend do the data scatter around the correlation line ? Are errorlimits reported ? Can I actually see the data points ? (In this regard, you mustnotice whether you are looking at a correlation on linear or logarithmiccoordinates. Errors usually appear smaller than they really are on logarithmiccoordinates.Are the ranges of physical variables large enough to guarantee that I can relyon the correlation for the full range of dimensionless groups that it purportsto embrace ?Am I looking at a primary or secondary source (i.e., is this the authorsoriginal presentation or someones report of the original) ?Has the correlation been signed by the persons who formulated it ? Has itbeen subjected to critical review by independent experts in the field ?


7. Forced convection in a variety of configurations ·...

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