Int. .I. Heat Moss Trms/er, Vol. 23, pp. 73-87 CNll7-93lO/8O/OlOl-oO73 $02.

00/O
0 Pergamon Press Ltd. 1980. Prmted m Great Britam

HEAT-TRANSFER CORRELATIONS FOR

NATURAL CONVECTION BOILING
K. STEPHAN and M. ABDELSALAM
Universitgt Stuttgart, Stuttgart, Germany

(Received 14 December 1978)

Abstract+To-date there exists no comprehensive theory allowing the prediction of heat-transfer

coefficients in natural convection boiling, in spite of the many efforts made in this field. In order to
establish correlations with wide application, the methods of regression analysis were applied to the nearly
5000 existing experimental data points for natural convection boiling heat transfer. As demonstrated by
the analysis, these data can best be represented by subdividing the substances into four groups (water,
hydrocarbons, cryogenic fluids and refrigerants) and employing a different set of dimensionless numbers
for each group of substances, because certain dimensionless numbers important for one group of
substances are unimportant to another. One equation valid for all substances could be built up, but its
accuracy would be less than that obtained for the individual correlations without adding undesirable
complexity.

NOMENCLATURE intensively for many years and many phenomena

thermal diffusivity[m’/s] ; have been explained. However, at present it is still
heater surface [m’] ; difficult or even impossible to predict heat-transfer
[20/g@-p”)]‘!* Laplace constant; coefficients with satisfactory accuracy. Many of the
specific heat capacity at constant pressure existing results on simple phenomena are incon-
in [kJ/(kg K)] ; sistent with each other and should be critically
equilibrium break-off-diameter reviewed. One of the tasks still to be completed is the
d = 0.146bb [m] ; review in a comprehensive and critical way of the
bubble frequency [l/s] ; existing data on heat transfer in natural convection
acceleration of gravity [m/s*] ; boiling and the correlation of these data by
pressure [bar] ; equations. The present paper is directed towards this
critical pressure [bar] ; object.
heat flux density [W/m’] ; When attempting a general correlation of many
enthalpy of evaporation [kJ/kg] ; experimental data, various procedures are conceiv-
mean roughness according to DIN able: one can, for example, determine what cor-
(Deut. Ind. Norm) 4762; relation among the many known from the literature
thermodynamic temperature [K] ; represents best the entity of all existing data and then
wall temperature [K] ; if necessary improve this correlation, or, one can
saturation temperature [K] ; build up a model, derive the general form of an
T, - T,, difference between wall and equation from it and then adapt the constants and
saturation temperature [K]. exponents in this equation to the experimental data.
However, taking into account the present knowledge
Greek symbols on heat transfer in natural convection boiling, both
*, q/AT, heat-transfer coefficient of these procedures are likely to prove unsatisfactory.
[W/(K m*)l ; All of the existing equations are based on models
83 contact angle [deg] ; which prove nonsuitable for some substances, ap-
A, heat conductivity [W/(K m)] ; parently none take into account all of the processes
V, kinematic viscosity [m*/s] ; important to boiling heat transfer, and present
PY mass density [kg/m31 ; knowledge is not sufficient for the building up of a
0, surface tension [N/m]. valid general model. Consequently, it does not seem
appropriate to start from a given model, but
Subscripts and superscripts correlate instead the existing experimental data by
saturated liquid ; means of more mathematical methods.
II saturated vapour ;
c, cover, or heater surface if surface is 2. CORRELATIONS BY MEANS OF
unprotected ; THE REGRESSION ANALYSIS

s, solid material behind cover, at saturation In order to arrive at equations for heat transfer in
temperature. natural convection boiling it is reasonable to start
from the fact that a certain number of physical
1. INTRODUCTION properties and variables characterize the heat-
HEAT transfer in boiling has been investigated transfer process. Such properties and variables are.
73
74

103
8
6

8
6

10’
8
6 I
1 1,

10' 2 4 6 8 10’ 2 4 6 8 lo3

- Nu exp.
FIG. I. Nusselt number for water after first step of regression analysis.

e.g. [I], the variables cj, TX,- 7;. 1. q, K,,, tl. TS, the for heat transfer in natural convection boiling be
fluid physical properties i.‘. p’ c;, [J”. r. q. 0 the thermal included in the above dimensionless numbers, an
properties of the heater /J,%.cPs. i,, and also those of a assumption that seems to be fulfilled since the list of
cover material /J<, cPC, I.,. that protects the heater thermal properties certainly contains all those
surface. These properties may be combined in the properties that have proved to be relevant for heat
usual way to yield a set of dimensionless numbers. A transfer in natural convection boiling.
possible set is [I] : A very powerful tool for finding a correlation
between the Nusselt number and the values Xi is
X, = (Cjrl)j(,.‘T,): x2 = (rPll’),i(orl);
given by regression analysis, which proved to be very
x, = &,T,P )/U”; X, = (rd’)/~l” ; X, = p”,‘p’; useful in statistical economics [2,3]. Recently Wag-
X, = I~‘:LI’: X, =a”/(d”g); X, = R,,‘ti: ner [4] applied this method to obtain a vapour
X, = (pc,i),/(p’c,i’): pressure equation from experimental data. The
regression analysis represents a method for deriving
x,, = (pc,/l),/(p’c’;i’); x,, = tr,.;a’; x,2 = m.,/a’
a correlation between an independent and several
and Y = (jiP[(7;, - T,)i’]. dependent variables. It is based on two assumptions:

(i) A sufficiently large number of experimental

The Nusselt number Y=(ctd)/i,‘=Nu and the di-
data describing the influence of the essential vari-
mensionless numbers X, depend on each other
ables over a wide range must be available. The
Y = ,/‘(X,, x2, . ..). (1) quality of the correlation depends decisively on the
number and accuracy of the experimental data.
For the following considerations. this set of dimen-
sionless numbers need not necessarily be complete. It (ii) A general form of equation (I) must be known
is required only that the physical propertics essential including all essential variables.
 Heat-transfer correlations for natural convection boiling

lo3
8
6

lo2
8
6

10’
8

FIG. 2. Nusselt number for water after second step of regression analysis

In fact there exists a great number of experimental exert the most significant influence on the dependent
data on heat transfer in natural convection boiling variable Y. This selection may be achieved in
for many substances, especially for substances which different steps according to the following scheme:
are often used in technical applications such as In a first step for each of the independent variables
water, hydrocarbons, cryogenic fluids and refri- an equation of the form
gerants. A possible form of equation (1) is a power p = e’“X; I
(3)
law which has proved to be very useful in many heat-
transfer problems. However, such a form has the is assumed, where Xi now stands for each of the 13
disadvantage that the pressure dependency of heat- variables. For each of them, the exponents p0 and pi
transfer coefficients then is mainly represented by a are evaluated according to the method of least
power of X, = p”/p’ which is not adequate over a squares. Of all the different equations employed in
wide pressure range. The pressure dependency can be the analysis the one which contains the most
much better described, as confirmed by the following essential dimensionless number XT is the one which
results, by introducing an additional term yields the smallest square error sum
X,, = (p’ -p”)/p’ in the power law. We therefore use
the following form QI = fl (ri- Ei)*. (4)

Y = eDUX{lXg2.. . Xfy. (4 In a second step for each of the remaining 12

Y can be whatever one defines it to be. independent variables an equation of the form
The regression analysis does not aim at estimating p = e,r”X:l~,X;* (5)
all exponents pi. This could be done by a mere
adjustment to the experiments. The regression ana- is introduced, where XT is the most essential variable
lysis rather allows to select those values Xi, which from the first step and X, stands for the other
76

8
6

” 10' 2 4 6 8 10’ 2 1 6 9 IO3

- Nu ev
Flc;. 3. Nusselt number for water after third step of regression analysis.

remaining variables. For each of these remaining reduced to 13.965,. if one includes a second variable
variables the square error sum Qz is calculated and XT=U’~/(~&) in the analysis, Fig. 2. A further
yields the next essential variable Xy. The procedure reduction of the average error to 12.2”b is obtained.
is continued until the experimental accuracy is well Fig. 3, when adding as a third variable
represented by the power law. x: = (c;?$P)/LP and finally, Fig. 4. the average
Some specific procedures, however, must be error decreases to 11.3j’;, by taking up as a fourth
observed in the course of the analysis. From the variable XX = (p’-p”)ip’. Introduction of further
second step on, the significance of the individual dimensionless numbers does not improve the result.
terms and also that of the actual equation must be In these calculations the methods of linear and those
tested again. One has to determine whether the of nonlinear regression analysis [S] were applied and
independent variables in the actual equation, except it turned out that the results from the linear analysis
the variable from the last step, may be replaced by represented experimental data better than those from
one of the variables not yet included in the actual nonlinear analysis.
equation. Thereby the actual equation and the order
of the independent variables may be changed. 3. SELECTION OF SUBSTANCES AND DATA

As an example, Figs. f--4 present the results for In order to apply the method of regression
boiling of water in natural convection. The deviation analysis to experimental data in natural convection
between the Nusselt number NIL,,~ calculated from boiling, the existing data had to be collected and
experimental data and the Nusselt number Nu,,iC critically reviewed. This was done under the follow-
from the regression analysis is considerable, if the ing criteria:
Nusselt number is assumed to depend only on the Only data concerning pool boiling on horizontal
most essential independent variable XT = (@)/(i’TJ, surfaces in the range of fully established nucleate
Fig. I. The average deviation is about 74.06”,. It is boiling under the influence of the gravity field were
 Heat-transfer correlations for natural convection boiling 17

103
8
6

6 ’ I , 1 I 1 1 1 1
10’ 2 4 6 8 lo2 2 4 6 8 lo3
-- Nu ew
FIG. 4. Nusselt number for water after fourth step of regression analysis.

considered and, in order to permit conclusions on number of about 400 characteristic points. It seemed
the influence of wall material on heat transfer, the reasonable therefore to study these groups sep-
data were further limited to those for which the arately, all the more so, as the accuracy of
heating surface material was indicated. These con- measurement is different for the groups. Heat-
ditions were fulfilled by about 5000 data from 72 transfer data on pool boiling of water are, for
papers. Only a few of the above mentioned 5000 data instance, more reliable than those on pool boiling of
from the literature gave information on the rough- cryogenic liquids. Also, the transport properties of
ness of the heating wall. In cases where these one group of substances may differ considerably
specifications were missing, a mean surface rough- from those of another group, whereas the deviations
ness of 1 urn, as is often met in technical appli- between substances within one group are usually
cations, was assumed. Very often the experimenters smaller, so some of the dimensionless parameters are
reported their results only with fitted curves accom- different for the various groups of substances. Some
panied by some of their raw data. In order to have a of them, important for one group of substances, are
common basis for the analysis therefore, all the expected to be unimportant for another group, an
experimental data were fitted by curves CC(~)and effect which indeed was confirmed by the regression
each of these curves represented by a certain number analysis. By considering the groups of substances
(usually four) of characteristic points. Thus the total separately first, equations can be developed with a
of 5000 original measuring points were replaced by minimum number of dimensionless variables repre-
about 1553 characteristic points. A great number of senting the experimental data within the scope of
experimental data are available for water, hy- their accuracy. Eventually an overall-correlation
drocarbons, cryogenic liquids and for refrigerants. valid for all substances of the four groups was
For each of these substances, or respectively, groups established.
of substances there exists approximately the same Due to the experimental difficulties, none of the
78 K. STEPHAN and M. ABDELSALAM

experimenters, when measuring heat-transfer coef- completely wetted. In the analysis the contact angle
ficients, simultaneously measured contact angles of was assumed arbitrarily to be /i = I When more
the vapour bubbles, so average values of the contact reliable contact angle data become available the effect
angle /I were used for the analysis. It was assumed of this assumption should be reviewed. A different
for water /j = 45 , for refrigerants and hydrocarbons contact angle leads only to a different constant in
/l=35. and for cryogenic liquids fl = I These the correlation. It does not change the heat-transfer
values were taken from the literature. Contact angles coefficient to be evaluated from the correlation.
of cryogenic liquids are known to be extremely low. In a first run, all the 1553 characteristic points
According to Good and Ferry [58] contact angles were used in the regression analysis. Upon com-
between liquid hydrogen and stainless steel, inconel, parison of the correlation thus obtained with these
titanium. aluminium, or teflon are zero, whereas characteristic points it was apparent that a certain
Brennan and Skrabek [59] obtained, according to number of characteristic points deviated con-
temperature and material of the heating wall. contact siderably from the results of the correlation and
angles between 7’ and 10” for nitrogen and between also from results of other authors.
1.5’ and 7. for oxygen. They stated, however, that Eliminating these characteristic points reduced the
the accuracy of contact angles below 10 is question- total to be used to 9X3 characteristic points
able. Bald [60] and Grigorev [40] assumed the representing 2806 original experimental data in a
contact angles of cryogenic liquids to be zero. As a wide pressure range between 0.0001 < JI:II~ < 0.97.
matter of fact, they are extremely low. However, they Details on these experimental data are given in Table
cannot vanish. because vapour bubbles form along a 1. From a second analysis with these characteristic
heater surface and the surface therefore is not points the final correlations were established.

Table I. Experimental results

(a) Water

Heater

Size
(diameter) Material Pressure
Author [Reference] (cm 1 roughness (bar)

Raben er al. [5] flat plate D = 3.77 pure copper, 0.0133, 0.0266,
smoothed and 0.0665. 0.2667
polished
Cryder and cylinder D = 3.81 brass, treated 0.0373,
Finalborgo [6] with emery 0.141
paper
Nishikawa wire D = 0.03 platinum 0.0971. 0.179,
et crl. [7] L = 4.5 0.318, 1.01,
= 3.6 5.066, IO.1 32,
20.264, 30.396,
40.528
Styushin and cylinder D = 0.08 brass, 0. I I, 0.204,
Elinzon [8] L = 25.0 polished 0.296, 0.48
Akin and cylinder D = 1.6 nickel coated 0.157, 0.2919,
McAdams [9] L = 21.59 coppertube 0.442. 0.657
Konig [lo] flat plate stainless steel 0.4, 0.67
electrolyte 1.0133
copper brass
MS 58 coated with
nickel
Rallis and wire D = 0.051 nickel 0.8268
Jawurek [ 1 I]
Magrini and cylinder D= 1.0 layer of zinc, 1.01
Nannei [ 121 L = 19.0 nickel and tin,
polished with
emery paper
Addoms [ 131 wire D = 0.03 platinum 1.01.53.1,
= 0.061 83.0, 110.4,
= 0.122 136.9
Fedders [ 141 D= 1.0 stainless I .51, 2.65,
r = 0.02 steel 4.42. 7.45.
L = 15.0 R, = 0.475 12.74, 19.6,
R, = 3.63 urn 33.3. 56.8,
98.5
 Heat-transfer correlations for natural convection boiling 79

(Table I.---continued)
(a) Water

Heater

Size
(diameter) Material Pressure
Author [Reference] Geometry (cm) roughness (bar)
Borishanskii cylinder D = 0.694 stainless 4.51, 73.05.
et al. [Is] L = 26.0 steel 98.1, 147.1.
lxl8H9T 196.1
Borishanskii cylinder stainless 5.88, 9.81.
et ul. [ 161 steel 22.6,31.4.
I Kh 18N9T 42.2, 55.0.
clean 99.1. 128.5,
147.0, 169.0,
178.0
Cichelli and flat plate D = 9.5 chromium plated 7.93, 18.28,
Bonilla [ 171 copper, clean 35.48, 52.76.
and polished 70.0
Elrod et al. [ 1S] cylinder D = 1.91 carbon steel 36.86, 70.0,
r = 0,124 monel and inconel 206.8
L = 17.78 commercial
material

(b) Hydrocarbons

Heater
Size
Boiling (diameter) Material, Pressure
Author [Reference] liquid Geometry (cm) roughness (bar)
--.
Mesler and benzene cylinder D = 0.163 stainless 25.294,
Banchero [ 191 ethanol f = 0.021 steel 7.98, 18.588

Berenson [20] it-pentane flat plate D = 5.08 copper 1.013

Bonilla and n-heptane flat plate D = 7.62 chromium 0.667,
Eisenberg [21] plated copper, 1.013
0.002 in,
clean
Kurihara and n-hexane cylinder D = 7.62 copper, 1.013
Myers [22] polished
Bonilla and ethanol Bat plate D = 9.093 chromium 0.35Y. 0.56,
Perry [23] n-butanol plated copper, 1.172,
0.002 in, 1.013
clean
Akin and n-butanol cylinder D = 1.905 nickel 1.013
McAdams [9] 1, = 21.59 plated copper,
polished

Fastovskii [24] benzene D = 0.08 nickel 1.013

L = 20.0
Ratiani and benzene cylinder D = 0.05 nickel 1.013
Shekriladze [ZS] ethanol L = 6.0 I.013
Miyauchi and benzene flat plate D = 6.8 brass 1.013
Yagi [26] n-hexane l.OI?
Borishanskii, ethanol cylinder D = 0.694 stainless 1.013, 3.01,
Bobrovich and L = 26.0 steel 5.0, 14.71,
Michenko [ 151 1x IXHOT 39.41, 49.22
Cichelli and benzene plate chromium 3.45, 7.92,
Bonilla [ 171 n-pentane plated copper, 18.24, 3 1.96,
n-heptane 0.002 in, 44.51,4.137,
ethanol polished 7.922, 14.8,
clean 2 1.96,28.6,
0.454, 1.013,
3.446, 7.941,
7.931, 1.013,
3.798, 18.24,
35.69. 52.75
80 K. STEPHAN and M. AHIXLSALAM

Table 1. -continued
(b) Hydrocarbons

Heater

Size
Boiling (diameter) Material Pressure
Author [Reference liquid Geometry (cm) roughness (bar)

Huber and benzene cylinder D = 0.953 stainless 3.48, 3.5 I,

Hoehne [27] L = 15.24 steel 7.94, 13.73.
20.78
Cryder and n-butanol cylinder D = 2.64 brass, clean 1.013
Gilliland [28] L = 0.833 polished
Jordan and diphenyl cylinder D = 0.343 stainless 1.013
Leppert [29] meta-terphenyl, L = 12.7 steel, clean
o&o-terphenyl
Cryder and n-butanol cylinder brass, 1.013
Finalborgo [6] roughened
with emery
paper

(c) Cryogenic fluids

Lyon [ 301 helium ring D, = 6.861 platin 0.063, 1.013

D, = 6.45 clean and
polished
Grigoriev et al. [31] helium end of a D = 0.8 nickel, 1.0
vertical L = 4.0 bronze (Cu + 1,5Fe),
rod brass (Cu + 30 Zn).
copper mean
roughness
5%lOpm
Jergel and cylinder aluminium 1.013
Stevenson [ 321 (99.99991;,)
Bewilogua hydrogen D = 0.3 stainless 1.013
l?Tul. [33] nitrogen steel
Bland et a/. [34] nitrogen rod D = 0.2 copper with 1.013, 2.05,
L=2 artificial 3.1
cavities.
radius
80_190pm, depth
150 pm 500 pm
Lyon et al. [35] nitrogen ring D, = 6.858 platinum, 0.432, 2.05.
oxygen D,= 6.452 clean and 3.89, 7.822,
smoothed 15.95, 0.228,
1.018. 2.05.
4.13. 8.04,
15.7
Akhmedov nitrogen end of a D = 1.0 Cr18NiYTi I .o, 10.0.
et al. [36] vertical steel, copper, 20.0.25.0,
rod smoothed 30.0. 32.0
with emery
paper
Marto et (11.[37] nitrogen flat D = 2.54 copper, 1.013
mirrorfinish
Frederking [38] nitrogen wire D = 15--2OOpm platinum 1.013
Astruc et ul. [39] wire D = 0.015 platinum 1.0, 10.0,
coiled L = 49.0 (pure), 28.0
in a A = 2.3cm2 smooth
spiral
O.D.
9.5 cm
Grigorev nitrogen copper 1.013
et al. [40] oxygen (Cu + 0.56 Fe) I.013
stainless
steel
(lCrlSNi9Ti)
bronze, brass.
roughness 5 pm
 Heat-transfer correlations for natural convection boiling 81

Table l.--continued
(c) Cryogenic Fluids

Heater

Size
Boiling (diameter) Material Pressure
Author [Reference] fluid Geometry (cm) roughness (bar)

Kosky and nitrogen circular D= 1.9 platinum 3.58, 7.587,

Lyon [41] oxygen end of a coated with 15.7, 23.0.
methane cylinder ETP-copper, 29.78, 32.82,
argon clean, 42.85,
polished 1.08. 16.4,
1.08. 33.33
Ackermann nitrogen German silver 1.013
et a[. [42] smooth,
depth = 0.2 urn
Lyon [43] nitrogen cylinder D = 0.952-6.98 copper, 1.013
oxygen L = 4.4-10.4 coated with
gold, clean
polished,
(I-4um)
Haselden and oxygen cylinder D = 1.588 copper, 1.013
Peters [44] L = 1.62 clean
Sciance methane cylinder D = 2.06 ARMCO-iron, 41, 76
et al. [45] L = 10.16 coated with
gold
Sciance ethane cylinder D = 2.06 ARMCO-iron, 4.89. 14.67
et al. [46] L = 10.16 coated with
gold
Bewilogua nitrogen flat A = 2.9 cm2 copper, 0.983. 2.94
et al. [47] helium A = 4.9 cm2 smoothed 1.0
hydrogen A = 2.9cm’ with emery 1.013, 9.8
paper, depth
z 0.2 pm

(d) Refrigerants

Hesse [48] R 12 tube D = 1.4 pure nickel 7.0, 14.0,

R 114 L = 35.0 (99.8%) 30.0, 3.0,
R113 R, = 0.61 6.0, 9.0, 12.0,
15.0, 20.0,
0.5, 1.0
Wickenhauser [49] R 113 tube D = 0.08 copper, 1.02, 1.55,
RC318 L = 27.0 R, = 0.4 3.21, 3.65,
R, = 0.9 7.03, 13.82
Stephan [l] RI1 tube and D=3 copper, 1.31
flat plate L = 50 R, = 7.9,
4.4, 1.4,
0.51,0.15um
Gorenflo [50] R 11 tube D = 1.2 copper 1.3,2.0, 3.0,
R 113 commercial, 0.1,0.4
R, = 0.4 urn
smooth
Hesse [51] R 114 tube D=3 copper, 0.37, 1.28,
L = 47.0 R, = 0.2 urn 2.52
polished
Schroth 1521 Rll tube D = 2.5 steel, 0.537, 1.22,
R 12 L = 40.0 R, = 9.0 pm 2.24, 2.0,
2.67, 5.02
Stephan [53] R12 flat plate D = 13.0 copper 1.63, 2.35,
t = 2.5 R, = l.Oum 3.51, 5.02

Happel [54] R113 tube D= 1.4 99.8 nickel 1.0

t = 0.075 R, = 0.43 urn
L=40.0
82 K. STEPHAN md M. ARIXXSAI.+%l

Table I. -continued
(cl) Refrigerants
Heater
Site
Boiling (diameter) Material Prcssurc
Author [Reference] fluid Geometry (cm) roughness (bar)

Danilova and R2l tube D = 0.51 stainless

Kupriyonova [SS] RC3iX L = 9.1 steel IIS, 1.65,
R, = 3.84. 13.82
9.1 ,ml

Danilova [56] Kl2 tube D = 1.25 stainless 5.68. 9.59.

R22 r = 0.02 steel, X55.4.99.
R113 L = 25.3 commercial 6.X1.9.104,
material 15.32. 24.24,
I .OY
Sciance YI cti. [45] propane cylinder D = 2.06 ARMCO-iron 213.426,
Ii-butane L = 10.16 coated with 8.52. 12.78,
gold 71.3, 25.56.
1.0. 5.7. 11.4
Abadzic [57] carbon- wire D = 0.01 pure 55.7 57.4
dioxide L = 10.0 ~~dtinum

4. RESULTS X, = (cjd),/(i’TJ; x2 = (d’p’)/(d):

On this basis, the following equations were obtained,

x, = (r;ly)&‘2 : X, = (rd’ )id2 :
the dimensionless parameters being arranged in the
x, = ,r;pr; x, = I”.,d :
order of their influence on the Nusselt number. X, = (Ijc.pj.),i(p’cb3.‘): Xl:, LT- (I)‘-p”)!p’.

For wattv-: In order to facilitate the practical application of

~~~=0.246.107X~.“73X1 i.5~X;.‘“X;f’ equations (6) to (8), we employ the simpler form
(6) 2 = 1’4”. where c depends on the thermal properties
lo- s < pip,. < 0.886, contact angle /j = 45‘ . of the substances and may be represented as a
mean absolute error 1I.?“,,. function of pressure. The value of 17 is different for
For ~~~~ru~u~~?~i?s: each group of substances. We have therefore with 1
Nu=O.O~~~(X’:.~.X,)O.~‘X;:.~~X~-~~~ in W/m’. K. 4 in W!m’ :
(7)
5.7 IO-- 3 < p/p, ,< 0.9, contact angle /i = 35”, For water’:
mean absolute error IX?, I>. y: = (‘, p-3 111)
For cr~~~~~rnic~fluids: For hydrocurhons.
Nu =4,X2#.h24X..‘, ?X;.2’7X;,“7”& 0..?2’-)
(8) 3 _ (.2‘jo.hx), (12)
4. 10m3 4 pip, < 0.97, contact angle fl = 1 .
For cryoge~7ic ,~~~~~, where the heist-transfer coef-
mean absolute error 14.3”,,. ficients proved to depend also on the material of the
For rqfkigrran~a: cover or heater surface
h,u = ~()7‘~:‘.“5‘~~.5”l~~,O.s33 ,x = ‘.3’i0.h24(p’.p,)i’.’ 17 (13)
(9)
3- lo-” < pip, < 0.78. contact angle /I = 35 , where p in kg/m”, c,, in kJ/kg K) and i. in W/(Km)
mean absolute error 10.57”,,. are evaluated at the saturation temperature of the
boiling liquid. R’eplacing equation (13) by
For till suhsrrrn~es used in the analysis:
Mu = o~~~x~_"'J~~~.'97~~."~~~~~.~3~~.3s x = ,.;3g”.“‘” (13a)
(10)
10-j <p/p,. d 0.97, with c; = c (pc ;_)“.“7 we find for atmospheric
mean absolute error 22.3’:,. pressure p =31 b&, gnd for different combinations of
heater surface and boiling liquid. the values C; listed
The mean absolute error gives the mean absolute
in Table 2.
deviation from the characteristic points used for the
final analysis. For r~fGpmt.s we have:
For many applications the accuracy of equation ~ = (j$>.iss
(14)
(9) is not adequate. One should prefer therefore the
individual equations (5)-(8). The pressure dependent values c’,, c2, cj, L’~ are
Only some of the 13 original dimensionless plotted in Figs. 5 8 for different substances, thus
numbers appear in the above equations, namely: permitting a simple and rapid evaluation of heat-
 Heat-transfer correlations for natural convection boiling 83

Table 2. Values c; for different com- transfer coefficients of many substances important
binations of heater surface and boiling for technical applications.
liquid;p = 1bar
As already mentioned the mean surface roughness
Combination R, in the equations (5).--(13) was assumed to be
Heater surfaceiboiting liquid c; R, = I urn. As shown in an earlier paper [I], in a
Copper/nitrogen 12.65 first approximation the heat-transfer coefficient r is
Stainless steel/nitrogen 1.6 proportional to R;” ‘33 for surfaces with a regular
Coppeqoxygen 12.3 roughness distribution as prepared for example with
Stainless steeljoxygen 1.46 emery paper, on a lathe or on a drawing bench. The
Copper/hydrogen 38.9
surface roughness therefore may be taken into
Stainless steel/hydrogen 21.0
account by multiplying the heat-transfer coefficients

lo*
8
6

10’
8
6
4
I I I I I llllli I I I I llllll I
loo 2 4 6 8 10’ 2 4 6 8 10’ bar 2 3’
4

loo
lo-* 2 4 6 8 10“ 2 bar 4 6 8 10’
-P
Frti. 5. Constant c, in equation (10).

IO2
8
6

‘I q meto-Terphenyl
q ortho-Ierphenyi
6

10-l 2 4 6 810’ 2 4 6 810’ 2 bar 4 6 0 lo2

-P
FIG. 6. Constant c2 in equation (11).
 K.ST~PHAN and M. h3DELSALAhl

ftonrP

FTnylene
Methope
$rgor
Ox)gefi
Yitwgen
We7n
Hydrogen
&‘lJrr :
, ,, Ll
IO-' 2 I 6 8 10' 2 4 5 8 10' 2 bar 4 6 8 10'

Ftc,. 7. Constant c3 in equation (I 2).

. NH, v
oR40 ~7
sRZ1 a
x RI13 A
x Rll!, D
+R13 v
0 CO? 8
o Rt31B

1--l-L
10“ 2 4 6 a IO0 2 4 6 8 10' 2 bar 4 6 8 10'
-P
Frc;.
8. Constant uJ in equation (13).

from equations (5).-(IS). for 0.1 < R,, < iOpm, with different metal surfaces difTer by more than a factor
a factor RF.'"", R, in pm. of 10, and with boiling of helium by more than a
It is noteworthy that in the above equations only factor of 40, whereas a much lower influence of the
the equations for boiling of cryogenic liquids include heater surface has been noted in boiling of normal
a term for the thermal properties of the heater liquids. Grigorev demonstrated that this effect may
surface or the cover protecting the wall of the heater. be explained by different factors: The thermophysical
Grigorev rt ui. [40] stated already that the wail properties of various metals, such as heat con-
material has a pronounced ~n~uen~e on heat transfer ductivity and heat capacity. differ significantly more
in boiling of cryogenic liquids. Th.eT point out that than at normal temperature. A small change in the
heat-transfer coefficients with boiling nitrogen on boiling heat flux at cryogenic temperature and hence
 Heat-transfer correlations for natural convection boiling 85

the wall temperature of the heater, therefore causes a nykhsred, in Symposium on Problem of Heat Transjtir
considerable change of the thermal properties of the and Hydraulics in Two-Phase Media, edited by S. S.
Kutateladze. Gosenergoizdat, Moscow (1961).
heater.
16. V. Borishanskii, A. Kozyrev and L. Svetlova, Heat
Furthermore the thermal properties of different transfer in the boiling of water in a wide range of
heater materials differ much more at low tempera- saturation pressure, High Temperature 2(l), 119-121
tures. Another effect may come from the extremely (1964).
small contact angles between boiling liquid and 17. M. T. Cichelli and C. F. Bonilla, Heat transfer to
liquids boiling under pressure, Trans. Am. Instn Chem.
heater wall, which though very small, may also differ
Engrs 41,755-787 (1945).
considerabiy for different materials. Finally one 18. W. Elrod, J. Clark. E. Lady and H. Merte, Boiling
should also keep in mind that most of the cryogenic heat-transfer data at low heat flux, J. Heut Transfir
liquids exhibit a higher heat conductivity than 87C. 235-243 (1967).
R. Mesler and J. Banchero, Effect of superatmospheric
liquids with a higher boiling point. The thermal 19.
pressures on nucleate boiling of organic liquids,
resistance of the heater therefore is more important A.1.Ch.E. JI 411). 102-113 (1958).
in boiling of cryogenic liquids. 20. P. Berenson. Experiments on pool-boiling heat transfer,
As a concluding remark one should notice that the Inc. J. Heat Mass Trumfer 5, 985 -999 (1962).
cited equations allow a fairly good representation 21. C. Bonilla and A. Eisenberg, Heat transfer to boiling
styrene and butadiene and -their mixtures with water,
of the existing experimental data. They should,
I&. Enana Chem. 4016). I 113-I 122 119481.
however, not be regarded as conclusive but be 22. .I ,, .,

H. Kurihara and J. Myers, The effects of superheat and

improved as soon as a sufficient number of new and surface roughness on boiling coefficients. A.Z.Ck.E. JI
more accurate data become available. 6(l). 83-91 (1961).
C. F. BoniIla and C. W. Perry, Heat transmission to
boiling binary liquid mixtures, Truns. Am. Iastn Chem.
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Heat Transfer 97C, 173-178 (1975). platinum surfaces from below 0.5 atmosphere to the
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der verfli&gten Gasc Helium und Stickstoff, f;r~rs&. maxim&r W~rmestr~)mdich~c untl rm i! hergang-
Geh. IngWrs. 27(l), 17-30 (1961). sbereich Lur Fill~verdampfung, DISS. TU Berlin ( 1972L
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EXPRESSIONS DU TRANSFERT THERMIQUE EN EBULLITION AVEC CONVECTION

NATURELLE

R&sum& I1 n’existe pas actuellement de thkorie explicative perm~ttant la prevision dcs coefficients de
transfert thermique pour l’ibuliition avec convection naturelie, malgrk de nombreux efforts dans ce
domaine. Afin d’itablir des formules ayant une large application, on applique les miithodes d’analyse de
rtgression i 5000 points expPrimentaux pour I’&bullition avec convection naturelie.
Ces donnkes peuvent itre regroupies en quatre families (eau, hydrocarbures. fluides cryogeniques et
rtfrigtrants) et en employant un systeme diffirent de nombres sans dimension pour chaque groupe de
substances. On peut ktablir une Cquation valable pour toutes les substances mais sa pricision cst moindre
que celle des formules individuelles sans ajouter une quelconque complexite.

GLEiCHUNGEN FijR DEN W~RM~~BERGANG BEIM VERDAMPFEN IN NAT~RL~CH~R

STRiiMUNG

Zusammenfassung--- Trotz vieler Bermiihungen ist es bisher nicht gelungen, eine umfassende Theorie zur
Vorausberechnung des Wlrmeiibergangs beim Verdampfen in natiirlicher StrBmung zu entwickeln. Urn
Korrelationen mit miiglichst breitem Giiltigkeitsbereich zu erhalten, wurden die Methoden der
Regressionsanalyse auf die etwa 5000 bisher bekannten MeDdaten iiber den WCrmelbergang beim
Verdampfen in natiirlicher Striimung angewandt. Wie sich dabei zeigte, lassen sich diese Daten am besten
wiedergeben, wenn man die Stoffe in vier Gruppen (Wasser, Kohlenwasserstoffe. tiefsiedende Fluide und
Kgltermittel) einteilt und einen unterschiedlichen Satz dimensionsloser GraBen fiir jede dieser Stoffgruppen
verwendet, da einige der dimensionslosen GrBDen fiir eine Stoffgruppe wichtig, fiir eine andere hingegen
unbedeutend sein konnen. Es konnte aunerdem eineeinzige Gleichungfiir alleStoffeangegeben werden, deren
Genauigkeit jedoch geringer ist als die der Gleichungen fiir die einzelnen Stoffklassen, solange man such fiir
diese allgemeine G&hung keinen unerwsnscht komplizierten Ansatz wiihlt.
 Heat-transfer correlations for natural convection boiling

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