A solution to the turbulent Graetz problem—III Fully developed and entry region heat transfer rates

Chemical

Engineering

Science.

1972, Vol. 27, pp. 2073-2093.

Pergamon Press.

Printed in Great Britain

A solution to the turbulent Graetz problem-III Fully developed and entry region heat transfer rates R. H. N0TTER-F Department

and C. A. SLEICHER

of Chemical Engineering, University of Washington, Seattle, Washington 98 105, U.S.A. (Received

19 April 1971; in revisedform

7 March

1972)

Abstract-The equation describing the turbulent Graetz problem (heat transfer to a fluid in a pipe) is solved numerically for the lower eigenvalues and constants for Reynolds numbers in the range lo4 < Re < lo6 and for Prandtl numbers in the range 0 < Pr < 104. The numerical calculations incorporate new information on eddy diffusivities and are supplemented by asymptotic calculations of the higher eigenvahres and constants of the problem. Heat transfer rates are predicted for both the entry and fully developed regions of the pipe. The numerical results of this study are shown to be in good agreement with experimental data on fully developed heat transfer rates for Prandtl numbers between 0.01 and 105. The numerical predictions are described as follows:

Wall thermal condition Uniform temp. uniform flux Uniform flux or temp.

Correlation

Minimum range of validity

Nu, = 4.8 + 0.0156 PeO.& PPo8

-t-5%

Nu, = 6.3 +0.0167 Pews5 PF’*

0.004 < Pr < 0.1 lo4 < Re < lo6

Nu,=5+0.016RenPP a = 0.88-0024/(4+Pr)

b = 0.33 + 0.5 e-@6pr

2 10%

0.1 < Pr < 104 lo4 < Re < lo6

Tables of eigenvalues and constants are provided for Nusselt number calculations in the entry region. Calculations of entry region Nusselt numbers are compared with experimental data, and five per cent heat transfer entry lengths are calculated for both wall temperature boundary conditions. The effect of the wall temperature boundary condition on heat transfer rate is discussed. 1. INTRODUCTION

considered here is heat transfer to an incompressible, constant property fluid in steady, fully developed flow in a smooth, straight circular pipe. Much research, both experimental and theoretical, has been directed at determining heat transfer rates and temperature profiles for this type of system. Temperature profiles for the turbulent flow of liquid metals and of air have been the object of extensive experimental study, for example, in Refs. [4,5,12-14,26,28,291. From such profiles it is possible to obtain eddy diffusivity profiles which may then be used to THE PROBLEM

calculate numerical solutions at values of the Reynolds and Prandtl numbers different from those measured in experiments. The solutions are usually in the form of an infinite series of eigenfunctions. Solutions of this type were obtained for turbulent flow by Sleicher and Tribus [30] for the boundary condition of uniform wall temperature and by Sparrow et al.[32] for the boundary condition of uniform wall temperature. In addition to these numerical studies, investigators have studied specific aspects of the heat transfer problem in both laminar and turbulent systems. It is not our aim to review these works

tPresent address: Dept. of Chemical Engineering, Pennsylvania State University, University Park, Pa. 16802, U.S.A.

2073

R. H. NO-ITER

and C. A. SLEICHER

here as this is done periodically in the literature. comprehensive heat transfer For example, reviews are published periodically in the Infernational Journal of Heat and Mass Transfer. The

primary objective of this work is- to present calculations which allow the prediction of heat (and mass) transfer rates for the turbulent Graetz problem over a wide range of Reynolds and Prandtl numbers, but especially in liquid metal systems. It is necessary, therefore, to discuss the studies of Sleicher and Tribus [30] and of Sparrow et al.[32] in order to show the motivation for the present work. These two studies display clearly the nature and mathematics of the problem, but they are limited numerically in several ways. First, the study of Sparrow et al.[32] covers only three Reynolds numbers and three Prandtl numbers, and it includes no Prandtl numbers in the liquid metal range, Pr s 0.04. The numerical results of Sparrow et al. have since been extended by Malina and Sparrow [21], but only high Prandtl number results were calculated. The study of Sleicher and Tribus [30], on the other hand, is primarily for Pr < 1, although a few calculations are given at Prandtl numbers of 3 and 7.5. Many Reynolds numbers covering a wide range were investigated for systems with Pr< 1. A second important limitation of these studies concerns the choice of eddy diffisivity profiles used. The solution of the energy equation is affected by but is insensitive to the choice of velocity profile whereas it is quite sensitive to the choice of eddy diffisivity profile. The velocity distributions used by Sleicher and Tribus and by Sparrow et al. are accurate enough. Both groups of investigators, however, made approximations concerning the eddy diffusivity profile which can now be refined. Sparrow and his colleagues made use of an eddy diffisivity profile which gives eH/u proportional to y4 near the wall, which at the time was the best distribution available. It is now clear, however, that near the wall +, is proportional to y3, and the difference is significant at high Prandtl number. They also took the eddy diffusivities for heat and for momentum to be identical, an assumption which can be improved.

The study of Sleicher and Tribus, on the other hand, suffers from obsolete eddy diffusivity profiles at very low Prandtl number. Their study was done at a time when only a few experiments had been reported on temperature distribution on liquid metals, and the data that were available are scattered and of questionable accuracy. ,A third factor affecting the studies of Sleicher and Tribus and of Sparrow et al. is the nature of the infinite series which form the solutions. While these series are convergent, a particular problem may require more terms than have been tabulated. This is especially true of liquid metal systems, where many terms may be required even for moderate values of the axial coordinate. The study of Sparrow et al. is not seriously limited in this regard since the first six (n = O-5) sets of terms were tabulated for moderate or high Prandtl number (Pr = O-7, 10, 100). The study of Sleicher and Tribus, however, is primarily in the low Prandtl number range, and they present only the first three (n = 0, 1,2) sets of terms. Some of the difficulties discussed above are eased if analytical rather than numerical solutions can be obtained. Analytical solutions, however, are apparently possible only when the eigenvalues of the problem are large. Such solutions have been obtained for the turbulent Graetz problem by Sleicher et a1.[3 I] for the uniform wall temperature case and by Notter and Sleicher [25] for the uniform wall heat flux case. In this paper we combine these asymptotic solutions with numerical solutions found with a digital computer for a wide range of Reynolds and Prandtl numbers. Solutions are presented for a Reynolds number range of 104- 106, a Prandtl number range of 0- 104, and for the boundary conditions of uniform wall temperature and uniform wall heat flux. The results may be extended to the general boundary condition of an arbitrary wall temperature or wall heat flux distribution by the superposition technique of Tribus and Klein [3 51. 2. SOLUTION FOR UNIFORM TEMPERATURE

The dimensionless the problem is

2074

WALL

energy equation describing

A solutionto the turbulentGraetz problem-III

(1) The variables x and r are the dimensionless axial and radial coordinates, fand g are the dimensionless velocity and total diffisivity, and 8 is the dimensionless temperature. The temperature distribution is given by 8=

5

C,R, (r) epAnzx.

(2)

a=0

Here the R,(r) are eigenfunctions the ordinary differential equation

The velocity function f (r) Measurements of turbulent velocity distributions in pipes have been reported by many authors. The distributions chosen for this work, and a review of other investigations, are discussed in detail by Notter and Sleicher[241. Although the various velocity distribution studies are not in complete agreement, the discrepancies are not sufficient to affect significantly the solution of Eq. (3). The eddy di$usivity function, g(r) The dimensionless total dilfusivity is

which satisfy

g(r) = l+Pr----4r) u

(3) with boundary conditions R: (0) = 0 and R,( 1) = 0 and the normalizing condition R,(O) = 1. The C, are constants and are evaluated from

”

2 = X,(aR,( 1)/a&)-

e(r) -_=--=--

D

Tw> m C A.ephnzx n=o

(5)

and

TiAn

e-hn5x

fv!l(X)

=

n=o

c&ID K(Tw- Tmm)= 2 g

(AJXn2) ephnzz

n=o

(6) where the constants A,, are defined by

To determine heat transfer rates from Eqs. (5) and (6), it is necessary to solve Eq. (3), which requires expressions for the dimensionless velocity distribution f(r) and the dimensionless total diffisivity g(r). These quantities will be discussed in turn.

ii%

1-Y

du/dy

V

4(X) = _4K(To-

lJ

(8)

where y is the ratio of the eddy diffusivity lH to the eddy viscosity E. The eddy viscosity is not a function of the Prandtl number and may be calculated from the turbulent velocity distribution by the relation

(4)

The heat flux and Nusselt number are given by

_ = l+yPrI.(L)

1

du+ldy+

(9)

where y = 1 - r is the dimensionless distance from the wall. The eddy viscosity distributions used in this work are those recommended by Notter and Sleicher [24] after an extensive review of turbulent velocity distribution data. The dimensionless total dilfusivity function can be calculated from Eq. (8) if values of eU or y = E~/E are known. In discussing the form of y used in this work, it is convenient to divide the heat transfer problem into the two cases of high Prandtl number (Pr > 1) and of low or moderate Prandtl number (Pr < 1). Case of large Prandtl number In systems where the Prandtl number is large, the form of the eddy diffusivity near the pipe wall is the determining factor in the calculation of heat transfer rates. This fact is discussed in detail in Ref. [24], where a relation for EJV in the region of the wall is given:

2075

Gf -v

04009y+3 (1

+

0.0067y+‘)

l/2

’

<

y+

<

45*

(lo)

R. H. NOTTER

and C. A. SLEICHER

This relation was determined from heat and mass transfer data, and so the question of the value of y in the wall region does not arise here. Equation (IO) is used for the heat transfer calculations of this work for the case of large Prandtl number. The eddy diffisivity profile over the rest of the pipe, from y+ = 45 to the pipe center, was determined by choosing y so that a smooth match was obtained between Eq. (10) and the eddy viscosity distributions given in Ref. [24] at y+ = 45. For high Prandtl number systems any errors in E,& introduced by this matching procedure have a negligible effect on the heat transfer results, for these results are determined almost exclusively by +Iv in the region y+ < 4.5. Case of low or moderate Prandtl number Experimental investigations of temperature profiles and eddy difhisivities in liquid metal systems have been undertaken both by American and Russian investigators. The most applicable investigations are those of Isakoff [ 121, Brown et al.[41, Buyco[51 and Schrock[26]. In addition, a group of Russian investigators headed by V. Subbotin and P. Kirillov has published a series of papers on liquid metal heat transfer [ 17,18,33, 341. Although this rather substantial amount of data exists, knowledge of y is still meagre. This is because, almost without exception, the experimental studies mentioned above are in conflict [6]. Experimental values of y at a given radial position vary by as much as a factor of four for a given Reynolds number. Moreover, while all theoretical analyses of the eddy diffusivity in liquid metal systems predict y < 1, some experimental results yield y > 1. Many of these anomalies can be explained by experimental limitations or imperfections. The results of the pioneering investigations of Isakoff [121 are subject to uncertainty because his nonretractable temperature probes may have caused large disturbances in the flow. The study of Brown et a/.[41 is not comparable because the wall of their heat transfer test section was not hydraulically smooth for most of their runs. The remaining experimental studies, those of Buyco[51 and Schrock[26] and of Subbotin and

Kirillov [17, l&33,34] are also open to criticism. The former two studies report asymmetrical temperature profiles, and since the magnitude of the eddy diffusivity is highly dependent on the shape of the temperature profile, the applicability of these studies to the present analyses is doubtful. The Soviet Union studies are criticized on two points. First, the test section length was only 32 diameters, which implies that the temperature profiles probably were not fully developed. Second, the temperature probe used in the studies was large enough to cause appreciable flow disturbance at the point of measurement. In summary, it is clear that most of the reported values of eddy diffisivity in liquid metal systems are questionable. The disagreement among previous studies was the primary motivation for the investigation of Sleicher et a1.[29]. This study was designed to avoid the defects referred to above, and the internal consistency of the results was verified by heat balances. Their results are for Pr = 0.020 O-025 and were used in the present study. Heat transfer rates and temperature profiles for air (Pr = 0.72) in turbulent flow in pipes are available from several authors. The data of different investigators are in much better agreement for air flow than for liquid metal flows, but there is a discrepancy of about 10 per cent in the fully developed Nusselt numbers obtained from different studies. The discrepancy is reflected in the difference between the well-known DittusBoelter correlation Nu, = O-023 Re0’8Pr0’1

(11)

and the correlation of Drexel-McAdams Nu a = O-021 Re”.sPro.4.

(12)

For example, Sleicher[28] and Abbrecht and Churchill [ 11 obtained Nusselt numbers in agreement with Eq. (1 l), while Johnk and Hanratty [ 13,141 and Deissler and Eian[7] report results described by Eq. (12). One reason for the variation among experimental Nusselt numbers is variation in T, - T,,.

2076

A solution to the turbulent Graetz problem-III

A large value of this temperature difference leads to spatial variation in viscosity, which in turn leads to increased Nusselt numbers for liquids and decreased Nusselt numbers for gases. However, even when corrections for finite (T, - T,& are applied to the data of Johnk and Hanratty [ 131, the resulting 3-4 per cent rise in Nu, is not enough to bring the results into agreement with Eq. (11). Similarly, extrapolation of Deissler and Eian’s data to (T, - T,,) = 0 raises the value of Nu, but not to the extent predicted by Eq. (11). These observations suggest that the fully developed Nusselt numbers of Sleicher[28] are probably hii by about 5 per cent. Therefore, although the variation of y with pipe radius was taken from this work, the magnitude of y = E& was lowered slightly (about 10 per cent) from Sleicher’s original results because of the Nusselt number findings discussed above. To determine values of y at Prandtl numbers other than 0~02-0~025 and 0.72, additional information is needed. A number of theoretical analyses of y have been made, but all incorporate rather untenable assumptions applied to a model of an “eddy”. No convincing theoretical analysis for y is available, and the procedure used in the numerical calculations here was to construct a purely empirical interpolation formula for y. This relation, described elsewhere [23], was constructed under the constraints (i) y+ (ii) y +

OasPr+

0

1 as e/v + w,

which follow from experimental and theoretical analyses of y. These constraints on y have been refined since the numerical calculations of this work by the analysis of Hill[ 111, who calculated the rate of heat transfer through an isotropically turbulent fluid with a uniform temperature gradient. His calculations with both the direct interaction and cumulant-discardapproximations to the dynamics of turbulence can be used to show that in the limit Pr + 0, y is proportional to Pre and that at Pr = 0.1, y is within about 10 per cent of its

asymptotic value for high Pr. With the assumption that this Prandtl number dependence of y can be extended to shear flow, we have constructed an empirical equation which exhibits this dependence and which also fits very well the experimental data at Pr = 0~02-0~025 and O-72: y = 0~025Pr~/v+90P$‘2(e/v)1/4 1+90P13’2(e/v)1’4

(

10 35+e/v > * (13)

This interpolation formula gives values of y that differ only slightly from those used here and produces insignificant changes in the calculated Nusselt numbers. Its form is both simple and credible. Consequently we recommend Eq. (13), together with values of e/v described in Ref. [24], to predict values of E,, for Pr < 1.

Computer calculations of h,2, A,, and C, Equation (3), with the velocity and eddy diffusivity profiles discussed above, was solved for the eigenfunctions R,,(r) and on an IBM 7040/ 7094 DCS for lo4 G Re G lo6 and 0 G Pr s 104. The resulting values of h,2, A, and C, are tabulated in Appendix 1, and graphs of these quantities in a form convenient for interpolation are given by Notter[23].t These data allow the calculation of fully-developed Nusselt numbers and thermal entry lengths. In order to calculate entry region Nusselt numbers, however, the determination of additional values of A,2, A, and C, from the asymptotic results of Sleicher et al. [3 11 is necessary.

Calculation of asymptotic A,,“,A,,, and C, The formulas for the calculation of asymptotic values for the eigenvalues and constants are [3 11: A, = (n+$)/G

(14)

tThe calculations in Ref. [23] are based on the original eddy dXusivity relations of Sleicher[ZI], so they differ by a few per cent from the values given in this paper at Prandtl numbers of O-4,0.72 and 1.

2077 CESVol.27.NalI-L

1+

R. H. NO-ITER

and C. A. SLEICHER

C = (- l)n(0*897)H”6 n G6 A,2’3

standard tables for the Moody friction factor. These four parameters may be used in Eqs. (15) and (16) to calculate numerical values of A,,, C,, and A, for values of II greater than those for which computer solutions are available.

(16) The parameters G, go and c in functions of the Reynolds and and typical values are given value of H = Ref,/32 may be

Eqs. (14)-( 16) are Prandtl numbers, in Table 1. The obtained from the

Nusselt number and entry length calculations Fully-developed Nusselt numbers. Fullydeveloped Nusselt numbers for the case of uniform wall temperature are given by the first term of Eq. (6), Nu, (T,) = ho2/2 and so are easily determined

Table 1. Values of G, Reynolds number

G

go

c

0.219 0.206 0.186 0.148 0.117

1.145 1.31 1.65 2.70 4.54

0.225 0.217 0.197 0.173 0.143 0.104 0.0776

1.08 1.17 1.45 1.93 2.92 5.88 10.9

10,000 50,000 100,000 200,000 500,000 1,000,000

0,216 0.200 0.169 0.139 0.109 0.0752 0.0552

1.19 1.40 2.05 3.12 5.28 11.6 22.3

10,000 20,000 50,000 100,000 200,000 500,000

0.206 0.185 0.149 0.119 o#xw 0.0616 O&l49

1.32 1.67 2.70 4.41 7.81 17.7 34.1

C

0.197 0.134 0.105 0.0792 O-0532 0.0388

1.47 3.40 5.77 10.4 23.9 46.3

2.4 11 16 19 22 24

0.180 O-152 0.113 0.0867 0.0648 O-0432

1.79 2.58 4.90 8.64 15.9 36.8

4-o 6.4 11 14 17 19

Prandtl number = 0.10 0.96 4.9 12 18 22 26 29

10,000 20,000 50,000 100,000 200,000 500,000

Prandtl number = 0.03 10,000 20,000 50,000 100,000 200,000 500,000 1,000,~

&?a

Prandtl number = 0.06 0.43 5.1 13 19 24 29 32

Prandtl number = 0.02 10,000 20,000 50,000 100,000 200,000 500,000 1,oOO,OOO

G

Prandtl number = 0.04 14 21 26 32 35

Prandtl number = 0.01 10,000 20,000 50,000 100,000 200,000 500,000 1,000,000

g, and c Reynolds number

Prandtl number = 0.004 50,000 100,000 200,000 500,000 1,000,000

from the tables pro-

0.154 0.125 0~0891 o+l671 0.0497 0.033 1

2.51 3.98 8.16 14.8 27.6 63.7

7.0 9.0 11 13 14 15

PrandtI number = 0.72 1.6 5.0 12 16 20 24 26

10,000 20,000 50,000 100,000 200,000

2078

O-0609 0.0456 0.0311 0.0232 0.0172

19.1 33.9 72.7 131 238

15 15 11 7.8 3.9

A solution to the turbulent Graetz problem- III

to heat the test section. Because of these uncertainties, the results of Gilliland et al. are not compared with the numerical results on Fig. 1. It is apparent that the calculations are in excellent agreement with the data of Ref. [29] as indeed they should be since difhtsivity data from that reference were used in the calculations. The calculations are correlated by

vided. These results are best discussed separately for different ranges of Prandtl number. In the liquid metal range, Pr < O-06, results are shown by the solid lines on Fig. 1. The dashed line on the figure is the theoretical prediction of Seban and Shimazaki[27], while the dotted line is the prediction of Sleicher and-Tribus[30] at Pr = 0.024. The experimental data on Fig. 1 are those of Sleicher et a1.[29]. This study and the one of Gilliland et al.[9] are the only ones we can find which approximate the uniform wall temperature Graetz problem for liquid metals. However, Lubarsky and Kaufman [ 191, in their review of liquid metal heat transfer data taken prior to 1955, note that the data of Gilliland et al. are subject to experimental inaccuracies. Their test section was rather short, and average heat transfer coefficients over the entire test section length were measured rather than fully-developed coefficients. Moreover, it was necessary by an empirical procedure to separate the mercury heat transfer coefficients from those of the steam used

Nu,( T,) = 4.8 + O-0156 Pe”**5Proos.

This empirical equation fits the calculated results within 5 per cent for 0.004 < Pr < 0.10 and Re < 500,000 and within 10 per cent for 0 < Pr < 0.10 and Re < 106. Equation (17) is recommended for the calculation of fullydeveloped Nusselt numbers for heat transfer to liquid metals in a smooth pipe with uniform wall temperatures at low values of T,, - T,. The values of Nu, for high Prandtl numbers (Pr B 1) have been discussed previously by Notter and Sleicher[24], who report that numeri-

IO' A -

-

SLEICHER , AWAD ( 8 NOTTER (29) , P~0.022 SEBAN 8 SHIMAZAKI

(2-7 , THEORETICAL

---------

SLEICHER 8 TRIBUS (301, THEORETICAL, Pr = 0.024

-

THIS STUDY, THEORETICAL

NU,

IO -

I

IO2

I

I

(17)

Illll

I

IO3

I

I

IllIll IO4

PECLET NUMBER Fig. 1. Fully developed Nusselt number in the liquid metal region for uniform wall temperature.

2079

R. H. NO’ITER and C. A. SLEICHER

cal calculations and data are correlated per cent for Pr > 100 by

within 4

Nu, = 0.0149 Re0’88Pr1’3 Pr > 100.

(18)

In the intermediate range of Prandtl number, O-1 < Pr < 100, it is probably not possible to find one simple equation that will correlate Nusselt numbers accurately. We have found, however, that the results can be well correlated by but a small sacrifice in simplicity; the equation proposed is Nu = 5 + O-016 Re” PP

(19)

with a = 0.88 --!!&

b = 0.33 +O.‘je-O@‘r.

This equation correlates the calculations within 10 per cent for O-1 < Pr < lo4 and lo4 < Re < 106, which exceeds the range of validity of the Dittus-Boelter equation. It is in excellent agreement with the best data for air and within 10 per cent of the best data at higher Prandtl number including Prandtl numbers as high as 105. The accuracy and wide range of validity of this equation should make it useful for design purposes.

presented in the form of plots of Nu/Nu, vs. RID for a given Reynolds and Prandtl number, and Fig. 2 shows a plot of this type at Prandtl numbers of 0.72 (air). The data points are from Abbrecht and Churchill [ I]. Thermal entry lengths The thermal entry length is defined here to be that distance downstream of the thermal entrance necessary for the local Nusselt number, Nu(x), to fall to within five per cent of its fully-developed value. Calculations of this quantity were carried out for a wide range of Reynolds and Prandtl numbers, and the results are shown in Fig. 3. They appear to be in good agreement with experimental data, as can be seen from Fig. 2 for Prandtl number of O-72. The X/D at which Nu/Nu, = l-05 closely matches the experimental results shown. 3. SOLUTION

?I=1

where

ABBRECHT & CHURCHILL Re = 6.5 x IO’ Pr = 0.72

2

I 4

I 6

I 6

WALL HEAT

When the wall heat flux is uniform, it is convenient to divide the temperature profile, 0(x, r), into the sum of a fully developed temperature profile 8, (x, r) and an entry region proiile &(x, r) [32]. For the present analysis we are concerned with the entry region profile, which is given by - (20) 02(x, r) = 5 CnR,,(r)e-X*z

Entry region Nusselt numbers The local Nusselt number, Nu(x), is given by Eq. (6) for the case of uniform wall temperature. Entry region heat transfer rates are conveniently

1.0 0

FOR UNIFORM FLUX

Y IO

the R,(r)

are eigenfunctions

(I)

I 12

I 14

I 16

WI

Fig. 2. Entry region Nusselt number for uniform wall temperature.

2080

satisfying

A solution to the turbulent Graetz problem-III

Eq. (3) but with the wall boundary condition d;(l) =0 rather than R,(l) =O. The 6, are constants and as shown on Ref. [25] are given by

c’,=_

2

(23)

2/Nu, -t 2 &, eeXzx ?I=1

which follows from Eq. (20). Here the &, are

(21) related to the constants C, in Eq. (2 1) by

A(aR'(i)/aX,)'

- A;, = C,&(l).

The fully developed Nusselt number for the boundary condition of uniform wall heat flux may be found from the eigenvalues h,2 and constants A, of the uniform wall temperature case by the relation [ 301 1

Nu, (uniform flux) =

2

Nu(x) =

(24)

In order to determine Nusselt numbers from Eq. (23), Xn2 and & must be evaluated. This is accomplished by numerical integration of Eq. (3), as before, but with boundary conditions Rk( 1) = 0 and R,(O) = 1. The important points in the solution procedure are discussed below.

(22)

16 5 A,/h,4’ n=o This series converges so rapidly that only a few terms are needed to calculate the fully developed Nusselt number. In fact, for Prandtl numbers slightly above one, only one term is sufficient. Whereas fully developed Nusselt numbers for uniform wall flux can be calculated from Eq. (22), entry region Nusselt numbers must be calculated from

Velocity and eddy diffusivity distributions, f(r)

and&) The velocity and eddy diffusivity distributions used for uniform wall heat flux are the same as those used for uniform wall temperature. This is clearly valid for the velocity distribution but involves some approximation for the eddy

35

25 -

10 -

5LAUINAR

01

’

,oo

’

IllI’

I

I

$

I

I111l

IO'

I

1

I

I1111l

105

Re Fig. 3. Five per cent thermal entry length for uniform wall temperature.

208 1

Id

1

R. H. NOTTER

and C. A. SLEICHER

diffusivity distribution. Since in general the eddy diffusivity is a function of the entire temperature field, it is a function of the thermal boundary condition at the wall. It is reasonable, however, to expect the eddy diffusivities to be similar. Moreover, it has been demonstrated [22] that for fluids of high Prandtl number the eddy diffisivity near a wall is identical for the boundary conditions of uniform wall temperature and uniform wall flux. Hence we assume that any difference in the eddy diffusivities for the two systems will be small even in low Prandtl number systems. It is also relevant to note that a large change in the eddy diffisivity will effect a relatively small change in calculated heat transfer rates for liquid metals (Pr < 0.04). Computer calculations of in2 andA_,, The solution of Eq. (3) with the boundary condition of uniform wall heat flux was carried out on an IBM 7040/7094 DCS. The values of X,2 and & for the turbulent flow case are presented in Appendix 2 and are also given in graphical form in Ref. [23]. As for the case of uniform wall temperature, only a few calculations were carried out for Prandtl numbers greater than one because for high Prandtl number the entry lengths are quite short and only the fully developed Nusselt numbers are of practical interest. Calculation of asymptotic ii,2 and&, The formulas for the calculation of asymptotic values for the eigenvalues and constants are r23,251 0. 189G2’3 “+f-H1,3(n+4)2,3

A7z=

0.762 GH”3%,5’3

G

(25)

(1 + 0.343/H”3r;,2’3) ( 1 -

().343/H1/3j;n2/3)

’

(26)

The values of the constants G and H are the same as for the uniform wall temperature case. The asymptotic results of Eqs. (25) and (26), coupled with computed values of iin and &, for low values of n, allow calculations of heat trans-

fer rates to be made in the thermal entry region of the pipe. Nusselt number and entry length calculations Fully developed Nusselt numbers. The asymptotic Nusselt number for the boundary condition of uniform wall heat flux is calculated from Eq. (22). Representative values of Nu,(QW) are given in Table 2 as a function of the Reynolds and Prandtl numbers. Figure 4 is a plot of the ratio Nu, (QW)/Nu, (I’,), and it shows the effect of the wall temperature boundary condition on the heat transfer rate. As shown in Fig. 4, there is a significant difference, as much as 30 per cent, between the fully developed Nusselt numbers for uniform wall heat flux and for uniform wall temperature when the Prandtl number is low. The difference is less than 5 per cent when the Prandtl number is greater than 0.72. For Prandtl numbers in the liquid metal range the numerical calculations are described by Nu, (Qw) = 6.3 + 0.0167 Pe”‘85Pro’08,

(27)

which correlates the results within 5 per cent for 0.004 6 Pr G 0.10 and Re G 500,000 and within 10 per cent for 0 < Pr < 0~10 and Re < 106. Equation (27) is recommended for the calculation of the fully developed Nusselt number for heat transfer to liquid metals in a smooth pipe at uniform wall heat flux when T,- T,, is not large. Figure 5 is a plot of Nu,(Qw) vs. P&let number for a Prandtl number of O-02. The Nusselt numbers are calculated from Eq. (22). The dashed line gives the theoretical predictions of Lyon[20], while the dotted line gives the predictions of Sleicher and Tribus [30]. The empirical correlation of Lubarsky and Kaufman[ 191, which is based primarily on uniform heat flux data, is also shown on Fig. 5 as is the correlation suggested by Herrick[lO]. Herrick pointed out that although Lubarsky and Kaufman’s equation is a good representation of experimental data obtained prior to 1955, more recent experiments indicated that earlier heat transfer coefficients were too low. Her-rick cited several reasons why

2082

A solution to the turbulent Graetz problem-

III

Table 2. Asymptotic Nusselt number uniform wall flux Reynolds number

Nu&Q ,)

Reynolds number

Prandtl number = 0+04 50,000 100,000 500,000

Prandtl number = 0.06

7.31 8.07 13.36

10,000 50,000 100,000 500,000

10,000 50,000 100,000 500,000

6.75 8.51 10.39 22.62

10,000 50,000 100,000 500,000

7.17 10.65 14.27 36.8

Prandtl number = 0.03 10,000 50,000 100,000 500,000

34.0 112.5 192.5 692

Prandtl number = 3

7.62 12.77 18.04 50.0

10,000 50,000 100,000 500,000

Prandtl number = 0.04 10,000 50,000 100,000 500,000

11.07 27.73 41.9 133.7

Prandtl number = 0.72

Prandtl number = 0.02 10,000 50,000 100,000 500,000

9.07 18.94 28.78 87.2

Prandtl number = 0.10

Prandtl number = 0.01 10,000 50,000 100,000 500,000

Num(Q zJ

61.2 224.3 399 1,583

Prandtl number = 8

8.11 14.89 21.68 62.6

10,000 50,000 100,000 500,000

90.3 344 617 2,520

Fig. 4. Ratio of fully developed Nusselt numbers for uniform wall heat flux and uniform wall temperature.

2083

R. H. NOTTER and C. A. SLEICHER

--_-

LIJ~~~KY

a KAUFMAN

(19),

EMPIRICAL

LYON (20), THEORETICAL SLElcHER a TRleus eo), THEORETICAL , pr = 0.02 HERRICK (IO),

EMPIRICAL

THIS STUOY, THEORETICAL,

Pr=002

LAMINAR

I-

I4 PECLET NUMBER Fig. 5. Fully developed Nusselt number in the liquid metal region for uniform wall heat flux.

earlier experiments may have been in error, and he based his correlation on the experiments of Kirillov el a/.[ 171 and Subbotin et al.[34]. As discussed earlier, the Nusselt numbers found by these investigators are expected to be high because of entry length and temperature probe effects. In summary, the correlation of Lubarsky and Kaufman probably underestimates the value of the asymptotic Nusselt number for liquid metal systems while the correlation of Herrick probably overestimates it. In any case, the heat transfer calculations of this study are between the correlations of Lubarsky and Kaufman and of Henick. For systems of moderate Prandtl number, O-1 < Pr < 100, the value of Nu, (Qw) may be found from Table 2 or from Eq. (22), with A, and hn2 found from Appendix 1. Like the case of uniform wall temperature, values of Nu,( QW) are correlated within 10 per cent by Eq. (19) for O-1 < Pr < 1O4 and lo4 < Re < lo6 except for Pr < 0.2 and Re < 100,000, where the correlation is about 12 per cent low. For Pr > 100, fully developed heat transfer rates for the boun-

dary conditions of uniform wall heat flux and of uniform wall temperature are essentially identical and are described within 5 per cent by Eq. (18). Entry region Nusselt numbers. The local Nusselt number is given by Eq. (23). Values of An and i,,” given in Appendix 2 for low n, and from Eqs. (25) and (26) for high n, may be substituted in Eq. (23) to determine entry region heat transfer rates. Figure 6 is a plot of Nu(x)/Nu, vs. X/D for a Prandtl number of 0.02 and a Reynolds number of 50,000. The experimental points are from Johnson et al. [ 15,161. The data of these investigators at Pe = lo3 has been averaged to obtain the points shown on Fig. 6. The experimental points at X/D = 4-6 showed considerable scatter. Thermal entry lengths. Calculations of the five per cent thermal entry length for the boundary condition of uniform wall heat flux were carried out for a wide range of Reynolds and Prandtl numbers. The results are shown graphitally in Fig. 7. In general, the five per cent thermal entry lengths for the case of uniform wall heat flux are slightly longer (lo-20 per cent) than

2084

A solution to the turbulent Graetz problem- III

2.4

-

0

JOHNSON, HARTNETT R CLABAUGH LEAD-BISMUTH, Pe = ICOO

a

JOHNSON, MERCURY.

CLABAUGH P8= IO00

8

HARTNETT

THEORY,

Pr = 01)2 , Pe = 1300

I 12

I 16

(151, (161,

-Nu

N”,

I 4

1.0 0

I e

I 20

a (II 24

26

32

WQ

Fig. 6. Entry region Nusselt number for uniform wall heat flux.

35-

30-

flz.

I5 -

Oa6 IO-

cir2 a03 0.02

5-LAYINAR

,oo

300 0.01

Fig. 7. Five per cent thermal entry length for uniform wall heat flux.

2085

R. H. NOTTER

and C. A. SLEICHER

those for the case of uniform wall temperature in the liquid metal region. For Pr 5 0.72, however, the entry lengths for the two cases are essentially identical. The entry lengths found here are in good agreement with experimental data over a wide range of Prandtl number. The five per cent entry length for Pr = O-02 agrees very well with that found by Johnson et a1.[15,16], as is evident from Fig. 6. For Pr = 3 the data of Malina and Sparrow [2 1I show a five per cent entry length of four diameters at a Reynolds number of 14,000 and an entry length of 2-3 diameters at a Reynolds number of 50,000. The data of Allen and Eckert [2] show an entry length of about 2-3 diameters for Pr = 8 and Re = 50,000. These results are in agreement with the calculated results of this study.

k NU NU, Pe Pr

Acknowledgment-This work was supported by a grant from the National Science Foundation.

Y+ Y

40 r0 L

R:

R2

Re TO TUl T mm f x

GA,,, redefined eigenvalue Nusselt number, hD/K fully-developed Nusselt number P&let number, Pe = RePr Prandtl number, Pr = v/a constant heat flux at the pipe wall pipe radius dimensional radial coordinate f/ro, dimensionless radial coordinate nth eigenfunction for uniform wall temperature nth eigenfunction for uniform wall heat thlx Reynolds number, Re = Du,,.& inlet temperature, x < 0 wall temperature, x > 0 mixed mean or “bulk” fluid temperature dimensional axial variable 2R/RePrD, dimensionless axial variable W/2) my l-r, dimensionless distance from wall

Greek symbols

NOTATION

constants for uniform wall temperature A;, constants for uniform wall heat flux C, constants for uniform wall temperature c parameter in asymptotic relations for A, and C,. Defined in Ref. [32] D 2ro, diameter f iilu,,p = local average velocity/bulk average velocity Moody friction factor = APIA (L/D) /3pu& fm 1 + Ptf&) = dimensionless total diffisivg ity go value of g in pipe center region G (1/7r)S~~(fl2g)dr, dimensionless constant H tidfldz),=o = RefJ32, dimensionless constant

A,

thermal diffusivity, K/PC* ratio of eddy diffisivity to eddy viscosity eddy viscosity eddy diffusivity thermal conductivity nth eigenvalue, uniform T, nth eigenvalue, uniform Q, kinematic viscosity = p/p dimensionless temperature profile, 0 = (T - T,) / (To - T,) for uniform T, and 8 = (T - To) lqorolk for uniform Q, fully-developed temperature profile for uniform Q, entry region temperature profile for uniform Q,

REFERENCES Ill ABBRECHT P. H. and CHURCHILL S. W.,A.I.Ch.E. JI 1960 6 268. [21 ALLEN R. W. and ECKERT E. R. G., Trans. Am. Sm. Mm-h. Engrs 1964 86 301. [31 BROWN G. M.,A.Z.Ch.E. Jl 1960 6 179. 141 BROWN H. E., AMSTEAD E. B. H. and SHORT B. E., Trans. Am. Sm. Mech. Engrs 151 BUYCO E. H., Ph.D. Thesis, Purdue University 1961. [61 CARR A. D. and BALZHISER R. E., &it. Chem. Engng 1967 12 53. [71 DEISSLER R. G. and EIAN C. S., NACATN 2629,1952. [81 DREXEL R. E. and McADAMS W. H., NACA Wartime Report W-108,1945.

2086

1957 79 279.

A solution to the turbulent Graetz problem-

III

E. R., MUSSER R. J. and PAGE W. R., General Discussion on Heat Transfer, Znsm. Me& Engrs 191 GILLILAND ASME 195 1402. Report 546 (R), U.D.C. Nos. 669-154; 536.24, November 1963. [lOI HERRICK R., UnclassifiedTRG t111 HILL J. C., Estimation of Eddy Diffusivities in Isotropic Turbulence 64th Annual AIChE Meeting, San Francisco, Nov. 1972. (Also personal communication.) H21 ISAKOFF S. E., Ph.D. Thesis, Columbia Univ. 1952, Univ. Microfilms, Ann Arbor, Mich., Microfilm No. 4200. See also ISAKOFF S. E. and DREW T. B., General Discussion on Heat Transfer, 405-409, Insrn. Mech. Engrs Land. 1951. T. J., Chem. Engng Sci. 1962 17 867. [I31 JOHNK R. J. and HANRAPY T. J., Chem. Engng t141 JOHNK R. J. and HANRATTY _ - Sci. 1962 17 881. J. P. and CLABAUGH W. J., Trans.Am. Sot. Mech. Engrs 1953 75 1191. v51 JOHNSON H. A., HARTNETT W. J. and HARTNETT J. P., Institute Engineerma Research, University of Cali[I61 JOHNSON H. A., CLABAUGH fomia, July 1953. M. F.,J. Nucl. Engng B 1959 1123. u71 KIRILLOV P. L., SUBBOTIN V. I., SUVOROV M. Ya., and TROYANOV U81 KIRILLOV P. L., Sou. J. Afom. Energy 1963 13 1103. B. and KAUFMAN S. J., NACA Report 1270 1956. t191 LUBARSKY PO1 LYON R. N., Chem. Engng Prog. 195 147 (No. 2) 75. PII MALINA J. A. and SPARROW E. M., Chem. Engng Sci. 1964 19 953. WI NOTTER R. H. and SLEICHER C. A.,A.I.Ch.E. JI 1969 15 936. P31 NOTTER R. H., Ph.& Dissertation, University of Washington 1969. ~241 NOTTER R. H. and SLEICHER C. A., Chem. Engng Sci. 197126 161. P51 NOTTER R. H. and SLEICHER C. A., Chem. Engng Sci. 197 126 559. WI SCHROCK S. L., Ph.D. Thesis, Purdue University 1964. T. T., Trans. Am. Sot. Mech. Engng 195173 803. t271 SEBAN R. A. and SHIMAZAKI WI SLEICHER C. A., Trans. Am. Sot. Mech. Engrs 1958 80693. 1291 SLEICHER C. A., AWAD A. S., and NOTTER R. H., submitted to Inrn. J. Heat Mass Transfer. SLEICHER C. A. and TRIBUS M., Trans. Am. Sot. Mech. Engrs 1957 79 789. ;;:; SLEICHER C. A., NOTTER R. H. and CRIPPEN M.. Chem. Engng Sci. 1970 25 845. T. M. and SIEGEL R,Appl. ScieitrRes. 1957 A7 37. ~321 SPARROW E. M.; HALLMAN V. I., IBRAGIMOV M. K., IVANOVSKII N. N., ARNOLDOV M. N., and NOMOFILOV E. V., [331 SUBBOTIN Zntn. J. Heat Mass Trans. 19614 79. A. K., KIRILLOV P. L. and IVANOVSKII N. N., Sov. J. Atom. Energy 1963 13 [341 SUBBOTIN V. I., PAPOVYANTS 991. 1351 TRIBUS M. and KLEIN J., Heat Transfer Symposium held at the Univ. of Mich. 1952, Engng Res. Inst. Univ. of Mich., pp. 21 l-235.

APPENDIX 1. Eigenvalues and constants-uniform

T,

Reynolds number Prandtl number = 0 10,000 20,008 50,000 100,000 200,008 500,008

9.87 10.03 10.16 10.25 10.35 10.48

54.27 54.95 5548 55.86 56.27 56.82

135.1 136.4 137.5 138.4 139.3 140.5

252.4 254.5 256.4 257.8 259.4 261.7

10.61 11.23 1244 15.91

58.19 61.74 68.90 90.19

144.6 153.6 172.0 227.4

269.7 286.7 321.4 426.8

1.559 l-564 1.568 1.571 1.573 1.575

0.973 0.986 0996 1.001 1.006 1.011

0.750 0.769 0.782 0.787 0.793 0.798

0.623 0647 0661 0668 0.673 0.678

0.910 0.918 0.926 0.931 0.935 0.940

0.824 0849 0.867 0.876 0.886 O-895

0.769 0.812 0.839 0.852 0.865 0.876

0.726 0.786 0.822 0.837 0.853 0.865

0.650 0644 0.627 0.576

0.975 1.036 1.159 1.527

0.897 0.943 1.027 1.246

0.862 0903 0972 1.137

0.841 0.880 0.942 1.080

0.636 0.618 0.582 0.506 0.439

1.041 1.175 1441 2.203 3.41

0.939 1.028 1.191 1.589 2.130

0.894 0.967 1.092 1.376 1.750

0.867 0.932 1.040 1.277 1.594

Prandtl number = O-002 50,000 100,008 200,000 500,000

1.564 1.561 1.554 1.531

0.986 0.980 0.%3 0.913

0.770 0.763 0.745 0.691

Prandtl number = 0.004 50,000 100,000 200,008 500,008 l,~,~

11.23 12.52 15.03 21.96 32.57

61.94 69.57 84.91 129.6 202.6

154.3 173.9 213.8 331.9 527.7

288.1 325.3 400.9 626.8 1003.0

1.558 1.550 1.534 1.497 1.457

0972 0.955 0.920 0.840 0.760

2087

0.755 0.735 0.698 0.616 0.540

R. H. NOTTER

and C. A. SLEICHER

Appendix 1 (cont.) Reynolds number

h2

C,

--c,

C,

-C,

Ao

A,

A,

A,

0606 0.616 0,593 0.550 0.491 0.405 0.345

0,970 1.050 1.277 1642 2.34 4.20 7.01

0,857 0.928 1.079 1.290 1644 2.42 340

0.788 0.868 0.996 1.154 1406 1.93 2.59

0.738 0.829 0.949 1.085 1.2% 1.752 2.34

0.561 0.509 0446 0.310

1.491 2.050 3.09 5.81 9.84

l-198 1.495 1.972 2.98 4.26

1.079 1.295 1.625 2.30 3.16

1.015 1.198 1.478 2.85

0.583 0.580 0.534 0.477 0.414 0.337 0.288

1.055 1.228 1.713 246 3.83 7.36 12.6

0901 1.029 1.314 1685 2.27 3.49 5-03

0.814 0.937 1.158 1.422 1.819 264 366

0.754 0.879 1.077 1.302 1.643 2.37 3.30

0.560 0.547 0489 0.430 0.371 0.303 0.260

1.152 1.423 2.17 3.28 5.29 10.38 17.9

0949 1.132 1.532 2.03 2.79 4.39 6.42

0.840 1.002 1.320 1649 2.16 3.22 4.57

0.769 0.924 1.189 1488 1.938 2.90 4.12

0.537 0.517 0.454 0.397 0.342 0.281 0.242

1.255 1.627 2.63 4.10 6.71 13.30 23.0

0996 1.230 1.731 2.34 3.26 5.19 7.67

0.863 1.062 1.428 1.847 ::: 5.38

0.782 0.964 1.287 1.651 2.20 3.37 4.81

0.4% 0.466 0.404 0.352 0.305 0.253 0.221

1.473 2.05 3.54 5.71 9.48 19.0 33.0

1.086 1.411 2.08 2.88 4.09 666 9.93

0905 1.162 1642 2.18 299 4.70 6.88

0.806 1.028 1.451 1.932 2.65 4.18 6.12

Prandtl number = 0.01 10,000 20,000 50,000 100,000 200,000 500,000 1,000,000

10.41 11.24 13.39 16.75 23.00 39.31 63.20

57.61 62.33 75.23 96.15 137.2 252.9 438.6

143.8 155.6 188.9 243.8 352.4 665.6 1176

269.2 290.9 353.9 458.5 666.3 1271-O 2260

50,000 100,000 200,000 500,000 1&WOO

15.33 20.37 29.56 53.03 86.99

87.37 119.9 182.5 357.8 638.0

220.9 306.7 474.3 953.5 1731

415.0 579.0 901.3 1829.0 3339

1.554 1.552 1.540 1.520 1.490 l-436 1.390

0.957 0.958 0.932 0.888 0.824 0.717 0.633

0.732 0.738 0.710 0.665 0600 0.500 0.428

Prandtl number = 0.015 1,526 1.499 1463 1406 1.362

0901 0.844 0.771 0660 0.581

0.677 0.620 0.548 0.451 0.386

Prandtl number = 0.02 10,000 11.18 20,000 12.86 50,000 17.31 100,000 2397 200,000 35.92 500,000 66.17 l,OOO,OOO 109.8

62.33 72.30 100.1 144.2 228.5 463.8 838.7

156.4 181.9 254.6 371.9 599.5 1247.0 2293

293.4 341.0 479.3 704.3 1143.0 2399.0 4437

1.544 1.537 1.513 1.483 1443 1.384 1.342

0.937 0.925 0.873 0.809 0.730 0.622 0.546

0.709 0,701 0648 0.584 0.511 0.419 0.359

Prandtl number = 0.03 10,000 12~03 20,000 14.60 50,000 21.28 100,000 31.06 200,000 48.32 500,000 91.50 l,OOO,OOO 154.0

67.76 83.37 126.7 194.2 322.3 678.1 1244

170.8 211.0 325.2 506.9 856.3 1844 3435

321-l 3%.8 614.8 964.8 1641.0 3562 6667

1.534 1.523 1.491 1.454 1.412 1.356 1.318

0.915 0.894 0.826 0.753 0.673 0.571 0.503

0.685 0666 0.600 0.531 0461 0.377 0.325

Prandtl number = 0.04 10,000 12.94 20,000 16.38 50,000 25.27 100,000 38.01 200,000 60.29 500,000 115.8 l,OOO,OOO 196.5

73.62 95.06 154.0 245.2 417.5 894.5 1652

186.5 242.1 398.7 646.2 1120 2452 4593

351.3 456.4 756.7 1234 2153 4750 8928

1.524 1.510 1.472 1.434 1.391 1.337 1.301

0.893 0.864 0.787 0.712 0.633 0.536 0.475

0660 O-635 0.562 0.494 0.427 0.350 0.304

Prandtl number = 0.06 10,000 20,000 50,000 100,000 200,000 500,000 1,000,000

14.80 20.00 33.02 51.57 83.45 163.4 279.3

86.22 119.6 210.2 349.7 611.3 1334 2475

220.5 308.0 552.3 933.7 1660 3695 6947

417.3 583.9 1054.0 1793 3205 7173 13540

1.505 1485 1442 1402 1.361 1.313 1.281

0.850 0.812 0.726 0.652 0.579 0.493 0.438

2088

0.615 0.581 0.504 0441 0.382 0.318 0.278

A solution to the turbulent Graetz problemAppendix

III

1 (cont.)

Reynolds number

Prandtl number = 0.1 10,000 20,000 50,000 100,000 200;OO0 500,000 l,~,ooO

18-66 27.12 48.05 77.13 127.4 253.6 437.3

113.6 171.6 327.5 564.7 1,007 2,226 4,150

2%-O 450.7 876.1 1.534 235; 6,239 11,750

1468 1444 1.398 1.361 1.325 1.284 1.257

0.774 0.728 0644 0.577 0.515 0.444 0400

0.540 0.499 0.431 0.378 0.332 0.280 0.249

1928 2.89 5.34 8.79 14.79 29.9 52.3

1.235 1.701 2.65 3.77 546 9.16 14.05

O-%5 1.304 l-959 2.71 3.84 6.27 9.45

3.06 4.93 9.54 16.02 27.3 55.5 97.2

1.472 2.17 3.62 5.35 8.09 14.21 22.5

1.049 1.493 2.45 360 5.38 9.32 14.58

4.907 8.215 1648 28.0 48.2 99.2

1669 2.58 4.58 7.17 11.32 20.9

1.127 1641 2.90 4.52 7.13 13.07

Prandtl number = 0.2 10,000 20,000 50,000 100,000 200,000 500,000 1,000,000

27.92 43.78 82.45 136.3 229.3 462.6 804.6

190.4 312.5 636.8 1,126 2,033 4,515 8,416

514.3 849.1 1,754 3,134 5,715 12,840 24,090

10,000 20,000 50,000 100,000 200,000 500,000

42.69 70.25 138.0 232.9 398.1 817.9

346.9 594.2 1,249 2,234 4,057 9,03 1

972.2 1,672 3,537 6,363 11,620 26,040

100,000 200,000 500,000

64.38 109.0 219.0 375.9 651.2 1,357

646.8 1.119 2;350 4,183 7,539 16,630

1,870 3.240 6,808 12,130 21,940 48,540

10,000 20,000 50,000 100,000 200,000 500,000

81.45 139.9 285.2 493.1 859.4 1,802

940.4 1,624 3,385 5,987 10,730 23,480

2,758 4,768 9,923 17,540 31,430 68,980

1.395 1.371 1.334 1.306 1.279 1.249 1.229

0.634 0.591 0.527 0.478 0.435 0.385 0.353

0.417 0.381 0.334 0.301 0.270 0.236 0.214

Prandtl number = 0.4 1.316 1.295 1.273 1.252 1.235 1.213

0.488 0.458 0.416 0.387 0.360 0.327

0.308 0.280 0.252 0.233 0.216 0.195

Prandtl number = 0.72 10,000 20.000

jo;ooo

1.239 1.231 1.220 1.21 1.200 1.19

0.369 0.352 0.333 o-319 0.302 0.282

0.227 0.208 0.193 0,185 0.177 0.165

7.5% 13a6 26.6 45.8 79.6 166-O

1.829 2.95 5.63 9.25 15.05 28.9

1.217 1.784 3.32 548 9.10 17.5

0.191 0.1745 0.166 0.161 0.155 0.148

9.69 16.89 34.8 60.4 105.5 221.0

1.915 3.16 6.24 10.54 -

1.28 1.87 3.57 6.04 10.03 20.2

Prandtl number = 1.0 1.200 1.199 1.194 1.189 1,183 1.174

0.311 0.301 0.291 0.283 0.258

34.3

Prandtl number = 3.0 10,000 20,000 50,000 100,000 200,000 500,000 1,000,000

119.1 209.4 445.6 794.9 1,429 3,147 5,733

2,736 4,680 10,250 18,240 33,980 77,120 144,200

-

1.097 1.097 1.093 1a93 la)0 1.087 1.085

2089

0.151 0.146 0.139 0.137 0.133 0.127 0.125

-

14.5 25.8 55-3 98.9 178.0 391-o 713.0

1.37 2.20 4.45 7.71 13.7 28.0 51-l

-

R. H. NOTTER

and C. A. SLEICHER

Appendix 1 (conl.) Reynolds number

ho2

AI2

AZ2

h2

C,

-C,

Cz

-C,

Ao

Prandtl number = 8 10,000 20,000 50,000 100,000 200,000 500,000 1,000,~

176.6 313.5 685.6 1,232 2,271 5,020 9,369

1.056 l-056 1.054 1.054 1.054 1.052 1.052

21.6 38.7 85.4 154.0 284.0 625.0 1,170

Prandtl number = 20 10,000 20,000 50,000 100,000 200,000 500,000 l,~,~

247.9 448.2 990.6 1,799 3,346 7,509 14,090

1.033 1.033 1.032 1.032 1.032 1.031 1.031

30.3 55.4 124.0 225.0 418.0 936.0 1,760

Prandtl number = 50 348.0 10,000 631.1 20,000 1,393 50,000 2,570 100,000 4,778 200,000 10,800 500,000 l,OOO,OOO 20,420

1,019 1.019 1.018 1.018 1.018 1.018 1.018

42.6 78.1 174.0 321.0 598.0 1,350.o 2,550.O

Prandtl number = 100 10,000 20,000 50,000 100,000 200,000 500,000 l,OOWOO

444.6 811.1 1,788 3,317 6,129 14,040 26,220

1.012 1.012 1.012 1.012 1.012 1.012 1.012

54.5 loo*0 223.0 415.0 766.0 1,750 3,270

Prandtl number = 1000 10,000 20,000 50,000 100,000 200,000 500,000

988.4 1,794 4,012 7,383 13.820 31:380

1.003 lXm3 l-003 1.003 1.003 1.003

121.0 222.0 501.0 923.0 1,730 3,910

Prandtl number = 10,000 10,ooo 20,000 50,000 100,000 200,000

2,132 3,907 8,695 16,140 29,840

1XlOl 1Tml 1.001 1.001 1.001

2090

261.0 484.0 1,090 2,020 3,732

A,

A,

A3

A solution to the turbulent Graetz problemAPPENDIX 2 Eigenvalues and constants-uniform Reynolds number

III

Qw

_ Al2 Prandtl number = 0

10,000 50,000 100,000 500,000

28.31 28.04 28.09 28.40

94.26 93.34 93.45 94.43

197.5 195.8 196.1 198.3

337.9 335.3 335.7 339.3

0.156 0.153 0.152 0.150

0.4091 0.0474 0.0469 0.0459

0.0243 0.0229 0.0227 0.0221

0.0147 0.0136 O-0134 0.0130

0.0457 0.0437 0.0329

0.0223 0.0213 0.0167

0.0132 0.0127 0.0102

0.0438 o+wO 0.0249

0.0214 0.0198 0.0133

0.0128 0.0119 O+NW

0.0471 0.0379 0.0314 0.0144

0.0235 0.0143 0.0190 0.0115 0.0162 OTKa34 0*0056

0.0337 0.0264 0.0106

0.0173 0.0140 oM65

0.0106 OMl88 0*0044

0.0445 0.0303 0.0227 0.0084

0.0225 0.0158 0.0124 0.0053

0.0139 0.0098 0.0079 0.0037

Prandtl number = OTtO2 50,000 100,000 500,006

29-66 31.61 48.83

98.51 104.7 159.0

50,OOll 100,000 500,000

31.93 36.37 74.41

105.7 119.7 239.1

10,000 50,000 100,000 500,000

30.45 40.12 53.11 159.8

101.1 131.8 172.7 504.6

50,000 100,000 500,000

47.84 68.58 235.9

156.2 221.3 740.5

10,000 50,000 100,000 500,000

33.55 56.08 84.92 314-7

Ill.0 182.3 272.5 984.5

206.5 219.2 331.0

353.4 374.9 564.8

0.147 0.138 0.096

Prandtl number = 0.004 221.4 250.3 494.5

378.7 427.5 839.8

0.138 0.123 0.067

Prandtl number = 0.01 211.6 275.0 358.8 1032

361.6 469.3 1737

0.147 0.114 0.090 0.034

Prandtl number = 0.015 325.1 458.1 1506

554.0 778.6 2528

0.0982 0.0718 0.0237

Prandtl number = 0.02 231.8 378.4 562-3 1997

395.8 643.8 953.6 3342

0.136 0.0857 0.0595 0.0181

Prandtl number = 0.03 10,000 50,000 100,000 500,000

37.13 73.68 119.4 477.7

122.4 237.8 380.5 1490

255.2 491.7 781.4 3011

435.1 835.3 1320 5027

0.125 0.0676 0.0439 0.0121

0.0419 0.0251 0.0177 0.0059

0.0215 0.0135 0.0101 0.0038

O-0134 0.0086 0.0066 0.0027

0.0394 0.0213 0.0145 0.0045

0.0206 0.0118 0*0085 ow30

0.0130 0.0077 O+lO57 0~0021

Prandtl number = 0.04 10,000 50,000 100,000 500,000

41% 92-42 155.8 646.0

134.9 2%.7 494.1 2011

49.74 132.4 232.4 993.6

162.4 422.3 733.3 3090

280.8 611.7 1011 4059

478.0 1037 1704 6765

0.115 0.0553 0.0345 0@091

Prandtl number = 0% 10,000 50,000 100,000 500,000

336.8 866-7 1494 6225

209 I

0.0979 o+wO 0.0238 oTlO594

0.0351 0.0190 0.0163 OW940 0.0105 0.00642 o$Kl3OO 0~00204

R. H. NO-ITER

and C. A. SLEICHER

Appendix 2 (conr.) Reynolds number

i,’

h3P

r;*2

-A;

h,P

-2,

--A,

0.0286 0.0109 OW663 OW176

0.0165 o+lO667 OW427 0.00122

-A,

PrandtI number = 0.10 10,000 50,000

224.9 695.9 1247 5341

463.0 1421 2531 10750

0.737 0.0250 0.0143 0.00344

100,ooO 500,000

69.52 219.6 396.9 1718

10,000 50,000 100,000 500,000

128.7 465.1 849.7 3633

Prandtl number = 0.20 410.4 833.1 0.0433 1467 2980 0.0122 2664 5392 0.00679 11310 22780 0.00162

0.0194 0@0570 0.00329 0.0083

0.0128 oXrO377 OWJ222 0.00058

10,000 50,000 100,000 500,000

258.1 982.3 1793 7551

Prandtl number = 040 812.8 1623 O-0234 3093 6268 0.00589 5621 11370 OM323 23530 47450 0.00078

0.0123 0.00288 o+O160 0*00039

O+lO946 0+0202 OGIlll 0*00027

10,000 50,000 100,000 500,OcO

519.5 1952 3510 14310

10,000 50,000 100,000 500,000

2483 9507 16980 72190

-

Prandtl number = 3 0.00338 owO646 0.000349 oXlOOO83

-

-

10,000 50,000 100,000 500,000

6564 25300 45100 192000

-

Prandtl number = 8 oJlO144 -

-

-

1624 6154 11030 44690

Prandtl number = 0.72 3202 0.0123 12480 0.002% 22340 00Ml4 89830 0.000405

0.00738 0.00653 0.00147 0~00106 O~ooO8 1 OMO56 0+)0020 -

R&urn& Les auteurs trouvent une solution numkrique a l’equation dtkrivant le probleme de turbulence de Graetz (transfert de chaleur a un fluide dans un tube) pour des valeurs eigen faibles et des constantes des nombres de Reynolds lo4 Re lo6 et pour des nombres de Prandtl 0 Pr 10’. Les calculs numeriques comprennent des don&es nouvelles sur les ditfusivites des tourbillons et les auteurs presentent les calculs asymptotiques des valeurs eigen et des con&antes les plus &eves du probleme. Une prediction est faite des taux du transfert de chaleur a l’entree et des regions pleinement developpees du tube. Les resultats numeriques de cette etude sont en bon accord avec les don&es experimentales des taux de transfert de chaleur pleinement developpes pour les nombres de Prandtl entre 0,Ol et 105. Les predictions numeriques sont: Condition thermique de la paroi

Correlation Nu, = 4.8 + 0.0156 PeO~ssPr”~Os

Intervalle minimum de validite

Flux uniforme

Nu m= 6.3 + 0.0167 Peo’85PPo8

-c 5% oGo4 < Pr < 0.1 lo4 < Re < lo6

Flux unifonne ou temperature

Nu, = 5 +0.016 Re”PP a = 0.88 -0.24/(4 + Pr) b = 0.33 + 5e-“‘6rr

k 10% 0.1 < Pr < lo4 lo4 < Re < lo6

Temperature

uniforme

2092

A solution to the turbulent Graetz problem-

III

Les auteurs donnent un tableau des valeurs eigen et des constantes pour de Nusselt a l’entree de la region. Ces calculs sont compares aux donnees calcule une entree du transfert de chaleur de 5 pour cent dans les conditions des parois. L’effet de la condition limite de temperature de la paroi sur le taux est discud.

le calcul des nombres experimentales et l’on limites de temperature de transfert de chaleur

Zusammenfassung-Die das turbulente Graetz Problem (Warrneilbertragung an eine Fliissigkeit in einem Rohre) beschreibende Gleichung wird numerisch gel&t fur die niedrigeren Eigenwerte und Konstanten fur Reynoldssche Zahlen im Bereich lo4 < Re < lo6 und fur Prandtlsche Zahlen im Bereiche 0 < Pr # 104. Die numerischen Berechnungen schliessen neue Informationen in bezug auf Wirbelstrom-DitIusionsverrnogen ein und werden erganzt durch asymptotische Berechnungen der hiiheren Eigenwerte und Konstanten des Problems. Es werden Warrneilbergangsgeschwindigkeiten sowohl fur den Eintritts-als such den Entfaltungsbereich des Rohres vorausgesagt. Man stellt fest, dass die Ergebnisse dieser Untersuchung gut iibereinstimmen mit Versuchsdaten iiber voll entfaltete Warmeiibergangsgeschwindigkeiten fur Prandtlsche Zahlen zwischen 0,Ol und 105. Die numerischen Vorhersagen werden wie folgt beschrieben:

Warmezustand

der Wand

Gleichformige Temperatur . Gleichformiger

Fluss

Gleichformiger Fluss oder Temperatur

Korrelation Nu m= 4 t8+0 ,0156 PeO~SSPP*Os Nu, = 6,3 +0,0167Pe0= PF8 Nu, = 5 +O,016Re”PP a=0,88-0,24/(4+Pr) b=033+05e-0.6Pr , 1

Minimum Geltungsbereich r5% 0,004 < Pr < 1 lo4 < Re < lo6 * 10% 0,l < Pr C lo4 lo4 < Re < lo6

Es werden Tabellen fur Eigenwerte und Konstanten zur Berechnung der Nusseltschen Zahlen im Eintrittsbereich angefuhrt. Die Berechnungen fur Nusseltsche Zahlen im Eintrittsbereich werden mit Versuchsergebnissen verglichen und es werden funf Prozent WarmeilbergangseintrittslSngen fur beide Wandtemperature-Grenzbedingungen errechnet. Die Wirkung der Wandtemperatur-Grenzbedingung auf die Warmeilbergangsgeschwindigkeit wird erortert.

2093 CESVd.27.No.

11-M