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Library, N.W. Bldg
NOV 1 1965
^ecUnical vtete 92c. 3/7
BOILING HEAT TRANSFER FOR OXYGEN, NITROGEN,
HYDROGEN, AND HELIUM
E,G. BRENTARI, P. J. GIARRATANO, AND R.V.SMITH
U. S. DEPARTMENT OF COMMERCENATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS
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maximum application of the physical and engineering sciences to the advancement of technology in
industry and commerce. Its responsibilities include development and maintenance of the national stand-
ards of measurement, and the provisions of means for making measurements consistent with those
standards; determination of physical constants and properties of materials; development of methodsfor testing materials, mechanisms, and structures, and making such tests as may be necessary, particu-
larly for government agencies; cooperation in the establishment of standard practices for incorpora-
tion in codes and specifications; advisory service to government agencies on scientific and technical
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listing of its four Institutes and their organizational units.
Institute for Basic Standards. Applied Mathematics. Electricity. Metrology. Mechanics. Heat.
Atomic Physics. Physical Chemistry. Laboratory Astrophysics.* Radiation Physics. Radio StandardsLaboratory:* Radio Standards Physics; Radio Standards Engineering. Office of Standard Reference
Data.
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rials. Reactor Radiations. Cryogenics.* Materials Evaluation Laboratory. Office of Standard Refer-
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NATIONAL BUREAU OF STANDARDS
*€ecltnical ^ote 3/7
ISSUED September 20, 1965
BOILING HEAT TRANSFER FOR OXYGEN,NITROGEN, HYDROGEN, AND HELIUM
E. G. Brentari, P. J. Giarratano, and R. V. Smith
Institute for Materials Research
National Bureau of Standards
Boulder, Colorado
NBS Technical Notes are designed to supplement the Bu-reau's regular publications program. They provide ameans for making available scientific data that are oftransient or limited interest. Technical Notes may belisted or referred to in the open literature.
For sale by the Superintendent of Documents, IJ. S. Government Printing OfficeWashington, D.C. 20402
Price: 60?
FOREWORD
This Technical Note represents a collection of several essentially separate works some of
which have been reported in less detail elsewhere (see Advances in Cryogenic Engineering, Volumes10 and 11). It may be well to specifically identify the contributions of the various authors. The first
author did a major portion of the work on pool boiling, section 2. The second author was the majorcontributor to forced convection, section 3, and the properties, section 6. The third author madesome contributions to all sections and wrote section 4 on boiling variables. Also special credit is due
to Mrs. Dorothy Johnson who assisted in the final editing and the typing of several drafts of the manu-script.
The work in section 3 was partially supported by NASA Contract R-45, and that assistance is
gratefully acknowledged.
11
CONTENTS
PAGE
FOREWORD ii
ABSTRACT 1
1. INTRODUCTION 1
Z. NUCLEATE AND FILM POOL BOILING DESIGN CORRELATIONS FOR O ,
N , H AND he 1
Z. 1 . Presentation of experimental data 1
Table Z. 1 Experimental oxygen pool boiling systems 6
Table Z. Z Experimental nitrogen pool boiling systems 7
Table 2. 3 Experimental hydrogen pool boiling systems 9
Table Z.4 Experimental helium pool boiling systems 10
Z. Z. Presentation of predictive nucleate correlations 16
Z. 3. Presentation of predictive film correlations 17
Z.4. Conclusions - pool boiling 19
3. FORCED CONVECTION BOILING ZZ
3.1. Introduction ZZ
3.1.1. Boiling regimes 2Z
3.1.2. Transition points ZZ
3.1.2.1. Bubble inception ZZ
3. l.Z.Z. Transition to film boiling ZZ
3.1.3. Predictive expressions Z3
3.2. Experimental data 23
3.3. Correlations Z3
3.3.1. Film boiling Z3
Table 3. 1 Data used by this study 24
Table 3.2 Data considered but not used 25
3.3.2. Film boiling - Nusselt number ratio vs x( quality) 29
3.3.3. Film boiling - superposition 34
3.3.4. Nucleate boiling - superposition 36
3. 3. 5. Burnout or transition point 39
3.4. Discussion of correlations and results 43
3.4.1. X and x correlations without boiling number 43
Table 3.3 Summary of results 44
3. 4. Z. X and x correlations with boiling number 5Ztt
B
3.4.3. vonGlahn correlation 5Z
3.4.4. Film boiling superposition 54
3.4.5. Nucleate boiling correlations 54
3. 4. 6. Burnout 54
3.5. Conclusions . 54
iii
4. BOILING VARIABLES 56
4. 1. Saturation fluid properties (pressure) 56
4.2. Subcooling 56
4.3. External force field (gravity) 57
4.4. Surface orientation and size 62
4.5. Surface condition 62
4.5.1. Surface history 62
4. 5. 2. Surface temperature variations 62
4.5.3. Surface roughness 63
4.6. Surface-fluid chemical composition 63
5. SUMMARY AND CONCLUSIONS 63
6. PROPERTIES DATA AND X CURVES 64
7. NOMENCLATURE Ill
8. LITERATURE REFERENCES - - 115
IV
Boiling Heat Transfer for Oxygen, Nitrogen, Hydrogen, and Helium
E. G. Brentari, P, J. Giarratano, and R. V. Smith
This study has been conducted to provide an orderly examination of the information relative to
boiling heat transfer for four cryogenic fluids. The general approach has been to examine experi-
mental data with respect to the predictive correlations which would appear to have probable successand which would be likely to be used by design engineers. These correlations were graphically andstatistically compared. The results are discussed, and when it appears a best or acceptable recom-mendation can be made, computation aids for designers are included. These aids are in the form of
graphical presentations for preliminary studies and equations for computer studies. The authorshave also indicated the apparent limits for the use of these correlations, when possible. The effect
of many variables which would often be significant are not included in the predictive correlations.
The influence of these variables is discussed in a separate section on boiling variables.
Keywords: boiling, cryogenic, film, forced-convection, free-convection, helium, hydrogen, nitrogen,
nucleate, oxygen
1. Introduction
Although the boiling phenomenon has been studied for at least a hundred years and has been the
subject of many papers, it is not well understood. All this work has not been a failure, however,even though complete understanding has not been achieved. From a designer's point of view, the pre-dictive expressions have become increasingly more reliable but also more abundant. The purpose of
this study is to test these predictive expressions with the available experimental data in order that a
better choice of expression can be made.
2. Nucleate and Film Pool Boiling Design Correlations
for O , N , H , and He
The purposes of this section are to present the available cryogenic pool boiling heat transfer
data from the literature and selectively to evaluate and present the best available predictive correla-
tions as functions of significant system parameters for use by the cryogenic design engineer. Thebasic approach was to search the literature to gather experimental cryogenic boiling data which werethen compared with existing correlations evaluated for the cryogenic fluids. In presenting the data,
effects of heating surface geometry, finish, and material were not discriminated from each other.
From these correlations, the ones which best appeared to describe the data were selected for presen-tation in the cryogenic range. Generally, the selected correlations fell within the spread of the
experimental data and thus should provide engineering utility for many design studies.
2.1. Presentation of Experimental Data
The experimental data for oxygen, nitrogen, hydrogen, and helium are respectively presentedby figures 2.1, 2.2, 2.3, and 2.4, for one atmosphere, with the exception of the nucleate regime for
helium, which is shown at 1/2 atmosphere because more data were available at that pressure. Thecorrelations appear to describe the data adequately, but the general correlations show a tendency to
be slightly lower than the visual average of the data. The circles represent the maximum experimen-tal nucleate pool boiling heat fluxes. No minimum film boiling data were found for the cryogenicrange.
In general, the widths of the bands of data indicate the spread of each respective investigator
under constant experimental conditions. However, this is not true for the oxygen and nitrogen nucle-
ate boiling data of Lyon [ 1964 ] and Lyon, et al. [ 1964 ] . These deceptively wide bands are due
to his careful investigations of extensive ranges of heater geometries, orientations, and surfaces.
As previously stated, there is no discriminiation between these effects in this paper. Tables 2.1 -
2.4 summarize by fluid the experimental systems shown in the figures.
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14
AT,(°K)
FIGURE 2.. 9
Comparison of Elevated Pressure Data with the General Kutateladze Correlation
15
2.2. Presentation of Predictive Nucleate Correlations
One experimental difficulty which could contribute significantly to the experimental scatter is
that of accurately measuring AT. These inaccuracies would then introduce large fluctuations in
— for the higher heat fluxes, when compared to the correlations where AT is cubed. It should be
noted that the correlations and experimental data as presented were evaluated at a standard gravita-tional acceleration of 980 cm/ sec .
For the nucleate pool boiling regime, the general nucleate heat transfer correlation and the
maximum nucleate heat transfer correlation of Kutateladze [1952] were selected. Empirical evidenceindicates that the nucleate heat transfer flux is approximately a function of AT cubed at one atmos-phere. Other correlations which were functions of AT raised to powers from 2. 5 to 3. 3 were those of
Rohsenow [1952] , McNelly [1953] , Gilmour [1958] , Tolubinskii [1959] , Labountzov [i960] ,
Michenko [i960] , and a second correlation by Kutateladze [1952] . All of these were compared withthe experimental data.
In his text on condensation and boiling, Kutateladze [1952] developed a set of basic equationsfrom basic fluid flow considerations which described the nucleate pool boiling phenomena and byapplying similarity considerations to these equations he produced a set of dimensionless groups.Groups of negligible influence were eliminated, and the remaining variables were formulated into
groups which could be evaluated from available theoretical and experimental property data. This is
shown below as (2.1).
K,.„, / „ vl/2 , r ('!/A)„.„,(C) lP l/ „ 1/2 0.6— ) =3.25
( io)-4
r*ucl
;*'* i (-g-) 1
gP. ' L Xp k. \ gp, / J
Nuclki
(2.1)
Wf a ,
3/2
]
- 125r p i
- 7
Uj' ^gPjg / -I L(a gPi )r72-J
Finally, these groups in (2. 1) were rearranged into the conventional form of heat flux as a function of
difference between the wall and the saturated liquid temperature. Since all the properties were func-
tions of the saturation pressure, a family of pressure curves was generated for each fluid on a
digital computer and compared with the elevated pressure data of Lyon, et al. [1964] , Roubeau[i960] , Class, et al. [i960] , and Graham, et al. [1965] . These curves appear as the pressuredependent lines in the nucleate boiling regime of figures 2.5, 2.6, 2.7, and 2.8. Figure 2.9 showsa representative sample of dimensionless elevated pressure data as a function of AT to illustrate
general agreement or deviation from (2. 1).
As the heat flux is increased, a point is reached where any increase in AT will decrease the
total heat flux. This nucleate boiling limit is known as the maximum or peak nucleate boiling heat
transfer flux. It is of interest to designers since heat fluxes are markedly reduced at this point and
fluxes greater than this maximum will occur only in the film boiling region at relatively high values
of AT. It is also of interest to researchers because some of the boiling variables appear to be negli-
gible at this point, making the data more reproducible. The exact mechanism of this transition
remains controversial, but the maximum flux is associated with the inception of transition fromnucleate to film boiling. However, the correlations generally in use which predict the maximum heat
flux are of similar form, being functions of fluid properties only and not being dependent on AT.
Kutateladze [1952] proposed that the maximum heat flux occurs when the stability of the liquid
films penetrating the two-phase boundary layer is destroyed. From this analysis, his maximumnucleate pool boiling correlation is
(q/A)Max.Nucl. K> {2Z)
1/2, , ,1/4 1
\p agAp )
v v
16
A value of K = 0. 16 was chosen as the numerical average of the values presented in his text. Also,
he recommended this value when the exact evaluation of the constant was unknown. More recently,
Zuber, et al. [1959] have also developed a correlation based on a hydrodynamic analysis similar to
that of Kutateladze. After negligible groups were eliminated, their expression was essentially the
same as (2. 2)
.
Since the maximum is not a function of AT, the intersection between (2.1) and (2.2) is required
to determine approximate AT values at which the maximum will occur. These are shown in the nucle-
ate regime as the circles in figures 2.5, 2.6, 2.7, and 2.8 and as the horizontal lines in figures 2. 1,
2. 2, 2. 3, and 2. 4. In this respect, Lyon, et al. [1964) indicate that the data begin to deviate fromthe maximum correlation predictions above 0. 6 of their critical pressures and that they becomevirtually useless above 0.8. Thus, both of the Kutateladze correlations in the nucleate regime are
shown only up to 0.6 of critical pressure for their respective fluids, because the general boiling
correlation uses the same parameters as the peak correlation and thus would also appear to be
incorrect at the higher reduced pressures. Figure 2. 10 illustrates the agreement of data from vari-
ous investigators with the Kutateladze maximum nucleate boiling correlation, (2. 2), where the dashed
portion indicates pressures exceeding 0.6 of critical. It is of interest to note that the maxima for the
predicted peak values occur at a reduced pressure of 0. 3, which is in agreement with experimental
evidence for the non-cryogenic liquids and for the cryogenic data of Lyon, et al. [1964, 1965] andRoubeau [i960] .
2.3. Presentation of Predictive Film Correlations
For the film pool boiling regime the general film heat transfer correlation of Breen andWestwater [1962] and the minimum film heat transfer correlations of Linehard and Wong [1963] andof Berenson [i960] (which is an empirical adjustment of the Zuber, et al. [1959] analysis) wereselected. These correlations were chosen on a basis of empirical agreement with the available data,
theoretical exploration by the respective investigators, and inclusion of previous work. In all, eight
correlations were investigated, some of the major ones being those of Bromley [1950] , Hsu andWestwater [i960] , Chang [1959] , Berenson [i960] , and Breen and Westwater [1962] .
Breen and Westwater [1962] base their correlation upon the Taylor instability theory by con-
sidering the maintenance of the minimum wavelength on the bottom surface of the boiling liquid whichwill release vapor bubbles into the liquid from the supporting vapor film. A significant parameter in
this analysis which was neglected by some previous authors is surface tension. Also, for cylinders,
heater diameters up to 1.0 cm play a significant role in film boiling, as opposed to nucleate boiling
where that effect appears negligible. In dimensionless form, the Breen and Westwater correlation is
FilmCond.
(i^)1/8
(^f-)1/4— o-»(^f-)
B Hf kjP
fAp
fgXi gD A
Pf
For this correlation, X' is an effective heat of vaporization, given by
[X + 0.340(C ) (AT)]2
X» = 1—E-* . (2.4)
It is of interest to note that this correlation coincides with the Bromley [1950] correlation over the
range of diameters from which the Bromley correlation was developed. Figures 2.5, 2.6, 2.7, and2.8 show this correlation as a function of varying heater diameters at a constant pressure of oneatmosphere, only, since the available cryogenic data indicate somewhat contradictory results at
elevated pressures.
As the heat flux is decreased, a point is reached where a further decrease in AT will markedlyincrease the total heat flux. This film boiling limit is known as the minimum film boiling heat trans-
fer flux. At this point, the transition from film boiling to nucleate boiling occurs, but not
17
100
60
40
20
6
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AyO em
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A A
/EXPERIMENTAL DATA
O Oxygen, Lyon
/ A 9O
Oxygen, Banchero et al
Oxygen, Weil
Nitrogen, Lyon —Nitrogen, Roubeau
C )
B Nitrogen, Cowley et al
Ql Nitrogen, Flynn et al
ffl Nitrogen, Weil & Lacaze
CD Hydrogen, Class et al
© Hydrogen, Roubeau
9 Hydrogen, Weil & Lacaze
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©A
Hy<
Hel
drogen, Wei
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r.o 4 6 10 20 40 60
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100 200
cm /sec
400600
FIGURE 2. 10
Comparison of the Maximum Nucleate Heat Transfer Fluxes with the Kutateladze
Maximum Correlation
18
necessarily at the same value of AT as the maximum nucleate boiling flux. However, the correlations
predicting this minimum are again functions of fluid properties only, being independent of AT.
Lienhard and Wong [1963] also utilize Taylor instability theory by considering the critical
surface wavelength that creates surface waves of sufficient amplitude to break down the vapor film andcarry liquid to the heater surface. Their expression, after empirical adjustment to the same units
as shown in the nomenclature is
/„\ /P
fX\r
2 SAp f 4a l1/2 /§Apf ? x"
3/4
Film *+
f D (P +p) DCond. x t
In essence, (2. 5) is an extension of the Zuber, et al. analysis by Lienhard and Wong [1963] for a flat
plate, to account for small diameters. Thus, as D was increased, the Zuber prediction was used as
the absolute minimum heat flux for film boiling. After adjustment by an empirical constant, this is
.gCTAp 1/4
(lJAbs.= °- l6XPf[——Z-] <
2 ' 6 >
Min. (Pi+P
f)
FilmCond.
Since these minima are not functions of AT, the intersections between (2. 3) and either (2. 5) or
(2.6) are required to determine the approximate AT values at which the minima will occur. These are
shown in the film regime as the circles in figures 2. 5, 2. 6, 2. 7, and 2. 8 and as the horizontal lines
in figures 2.1, 2. 2, 2. 3, and 2.4. As an example of the other correlations which were investigated,
Figure 2.11 shows the relationships between various nucleate and film correlations with respect to the
body of hydrogen data. A similar analysis was performed for each of the other three fluids. A morerigorous comparison such as the statistical methods of section 3 was not used because much of the
data were shown in general curves and not reported as data points which are necessary for such a
study. In general, these additional nucleate correlations were taken from Drayer [1965] while the
additional film correlations were taken from Richards, et al. [1957] . Also, a recent report by
Seader, et al. [1964] provides an excellent survey of oxygen, nitrogen, and hydrogen pool boiling.
It should be noted that (2. 3), (2. 5), and (2. 6) do not include the heat transfer due to radiation.
These equations predict the film boiling flux by conduction through the vapor film only. However, if
ideal radiative transfer is assumed ("black body" emissivity and absorptivity) then heater surface
temperatures of the order of 400 to 425 C K can be tolerated for all of the four fluids without exceeding
5% error from the radiation even if it is neglected in the calculations. Figure 2.12 illustrates the
heat flux due to radiation under the "black body" assumptions which would, of course, yield somewhatlarger values than would actually be experienced in practice.
2. 4. Conclusions - Pool Boiling
For nucleate boiling, the correlation of Kutateladze [19 52] appeared to fit the available experi-
mental data best. At higher pressures the agreement was poorer but perhaps acceptable for somedesign studies. For film boiling the correlation of Breen and Westwater [1962] was chosen for the
best fit of experimental data. At higher pressures there were insufficient experimental data for
comparison. For maximum and minimum heat fluxes in the region of transition from nucleate to film
boiling the correlations of Kutateladze [1952] and Lienhard and Wong [1963] were selected for pres-
entation.
It would appear from the rather wide range of experimental data that some variables, such as
surface conditions, orientation, etc. , are missing and others are perhaps not properly taken into
account in the correlations. There is a further discussion of some of these variables in section 4.
Finally, the predictive correlations mentioned above were evaluated in the figures at one standard"g" only.
19
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21
3. Forced Convection Boiling
3.1. Introduction
The purpose of this study was to investigate the relative merits of proposed predictive methodswhich may be used to determine required design information from data normally available. Therequired design information was assumed to be values of heat flux in the nucleate and film boiling
regions and conditions for the transition between the two regions often referred to as the critical orburnout point. It has been assumed that flow rate, quality, fluid properties, and system geometrydata are available or may be determined in most design studies; thus these quantities constitute the
independent variables for the equations or correlations presented.
Since both the forced convection and the boiling phenomena when considered separately arepoorly understood, one may expect only broad treatments of a subject involving their interaction
rather than rigorous and specific studies.
3. 1. 1. Boiling Regimes
Some general divisions for the forced convection boiling regions have been proposed, for
example, by Davis [i960] :
(1) Region I - The nucleate boiling (wet wall) region where the nucleate boiling contribution is
significant, as for low velocity flows.
(2) Region II - The wet wall region where the forced convection effects are more significantly
controlling for high velocity flows.
(3) Region III - The dry wall or liquid deficient region where the heat transfer is through a
vapor film to a liquid or two-phase core.
Since almost all of the cryogenic data available were in Region III, most of this study wasoriented toward that region. The data used are entirely hydrogen since these were the only completeand tabulated cryogenic data available.
3. 1. 2 Transition Points
The transition points from single-phase flow to flow with bubble inception and the transitions
between the three regions previously described are often of prime design interest; studies of these
transition points may produce a better understanding of the phenomena. In this study very few tran-
sition data points were found; however, the boiling inception point and the transition between Regions
II and III (burnout) will be briefly discussed.
3. 1.2. 1. Bubble Inception
Bubble inception will be defined as the point of discontinuous slope change of a heat flux vs ATcurve. This change of slope may be attributed to the change in heat transfer mechanism as boiling
occurs. For the forced convection case this change of slope may occur for low velocity flow (Region
I) where the boiling effect is not negligible. The only cryogenic fluid data are for pool boiling
[Mikhail 1952, Tuck 1962, Ehricke 1963, Graham, et al. 1965] , and these are discussed in the
Boiling Variables section of this report. It would appear that this inception point is quite sensitive to
surface conditions.
3. 1. 2. 2. Transition to Film Boiling
The transition from Region II to Region III may not produce an actual burnout in many cryogenic
systems because, although the solid surface temperature may be markedly increased it may still be
below the temperature for failure conditions for the heated solid surface. This discontinuity in the
heat flux vs AT curve does mark the transition, however, and essentially all work has been for flow
inside conduits.
Most of the analytical studies for the' transition from Region II to Region III use a general
superposition approach[ Levy 1962, Gambill 1963a] since the cryogenic burnout data were for a change
from a wetted wall to a dry wall condition. Gambill [1963b] has prepared an excellent review of the
subject of forced-convection burnout in which he discusses in detail the primary variables which are:
22
(1) Liquid properties.
(2) Velocity past the heated surface.
(3) System pressure [closely related to (1)] .
(4) Subcooling or quality.
(5) Local acceleration forces (gravity).
For the case of flow over surfaces, Vliet and Leppert [1962] have reported studies on critical
heat flux for saturated and subcooled water flowing normal to a cylinder. These results showed that
the primary variables were (T ,,-T ), velocity, and cylinder diameter,wall sat
3.1.3. Predictive Expressions
Generally, the predictive expressions fall into two categories:
(1) Correlating a simple or modified Nusselt number ratio with the Martinelli correlating
parameter or a similar term, primarily reflecting quality. The Nusselt number ratio usedis the ratio of the experimental (or actual) Nusselt number to that value obtained by use of
a Dittus-Boelter or Sieder-Tate type of equation with either single-phase properties or
some modifications involving two-phase properties.
(2) Simple superposition, that is, adding the separately determined pool boiling and forced
convection (without boiling) contributions.
Many other predictive systems have been proposed, but many of them include empirical con-
stants (previously determined for water data), values of which for cryogenic fluids the authors
assumed would not be generally available to designers. (For instance, see Tippets, 1964).
3. 2. Experimental Data
The experimental data considered by this study are shown in table 3. 1. Other experimentaldata found are shown in table 3. 2 but were not included in this study because of lack of complete tabu-
lated data which were necessary for calculations, or because of complex system geometry.
None of the experimental data used were culled for experimental accuracy.
3. 3. Correlations
(See Appendix for Nomenclature)
3. 3. 1 Film Boiling - Nusselt No. Ratio vs X..
A X correlation for the film boiling regime or Region III (as proposed by Hendricks, et al.
1961) used in this study was
Nu
Nu-f^l = f(V' <>>"calc, f, t. p.
where
0.9 ,P 0.5 , u 0. 1
tt =(^) (17) (3
Nu . . . = 0.023 Re°'8
Pr°*4
(3.3)calc, f, t.p. f, m, t. p. f, v
P, . U DRe, = f.m.t.p. avg
(3<4)f, m, t.p. Uf v
23
Q
Hto
to
cq
QWto
<
a
w
pq
i—
1
i—
i
r-H
rtj ni ai nj
a
o• IH •iH
o OH
u u u Ih
« -rH 4-> -J-J +-> +->(U 4-1 o o o uft <$
>> 4)4)
1—
I
41I-H i-H
ai
H ffi W W w W
oo
<V _ o o o oMH o o o oh
*" o vD o ° o ° ^ °rt 3 -^ o o •* +-> „ o a o o 5 -
01 O vO CO xt< o ^ O ro
Shfi LTl CM rt in ** oin 00 i-H
'+j
fl
U oen >-<v O o o o
o o o0) H ^ O o o n o o oH to y in °- 2
o ° o ° o o o ow ° -d M •» +j rj o a o * "*H n
i-H j H^ 00 LTl in (NJ 00 ^ ko mrf 2 > £ t>- CM cm n ^ r^ O xO
PO r~ * i-H (SJ 1-1 <n 00
CD
oo
3 « 41 4)o
0)
tu ^3 O ° a aft a o 4J
K to u Z 2 o Zw
i-H X3 nj o
,£> 2o
o
00
<1 1 °o o^ ID ^ oO CM o o o -M
H in ^h in r- o -H
01
+J m,H o
CJ (D t« ~ 00O 00 • o oro r-l csj r~ ro r- OJ HH
(D
c od
niAi
•H.,H
Ih soo <u
r-
1
NucL
andFilm
cq a £ h cq
01 01 ?H
01
00r—
(
01 Q4) I—
l
0)
ft
.24-» ^
3 *>01 vO
1—
1
oi-H
oin
oo .
h QOBao
i-H •
a "-•|H „
to inin
boc
i-H
ft
u
rH Q 00
<n +j CD•-H —
T
ni ° CM*i-H h-1 r—
1
i-H ^-TnJ ° -
T i—
1
w ^ "fiu o - z u in o =
* -rH 00 o (M N O r-M i—l o •ih m o
oi o +j0) <u ™h a 2
Q h (v-1
> O<\J (U «>
> 01
s£>
ffi O vO
C3 C fl fl
S01 V <p 0)
60 00 00 003 O o O
xj "H M ^ ^H ^01 XI XI T3 T) X)D <u >. ^ >s >>
H 2 X ffi X a
S o N01
it! "3 fi h
o 5 fi '—
'
1)
-as*3 xi
s
i-H
ni
(U uO^ -a
oT^
O CI
o —a g
XIa
5to <;
- rdI ,
H «,,—
,
iS S '—
i
a» h O T3i-H oo o»
^1 01 m fl d •-o
1-1 rtj
.in mO -H O ai oi
0^ u oo 2 ill X X 1—
t
^i24
TABLE 3.2 DATA CONSIDERED BUT NOT USED
Source and/orAuthor
TestMedium
Boiling
Region Reason for not using
Dean and Thompson[1955]
NitrogenNucleate,
Film, &Burnout
Geometry
Richards, Robbins,Jacobs, Holten
[ 1957]
NitrogenHydrogen
Nucleate Geometry
Sydoriak and
Roberts [ 1957]
NitrogenHydrogen
Nucleate Geometry
von Glahn and
Lewis [ 1959]
NitrogenHydrogen
Film andBurnout
Lack of complete tabulated
data
Hendricks and
Simon [ 1963]
Nitrogen Film Boiling Geometry
Aerojet [ 1963] Hydrogen Film Boiling Geometry
Chi [ 1964] Hydrogen NucleateFilm
Lack of complete tabulated
data
25
f, v
f, m, t. p. x 1 - x (3.6)
Pf
P,
Uavg=ir^ < 3 - 7 >
b c
and p = - (3.8)b x 1 - x
From the definition of the Reynolds number it is apparent that Nuf
is a two-phase type ofcaxc, i, t. p.
Nasselt number which will, however, approach the value of the single gaseous phase Nusselt numberas the quality approaches 1.
A variation of the above correlation was also considered. The correlation used was
Nu
Nu-f^ = f<V' (3 ' 9)
calc, v, s„ p.
where Nu , = 0.026 Re°'8 ° Pr°"
33(—
)
(3.10)calc.v, s.p. v, s.p. v \|i /
,P U DRe = (
V aVgJ (3.11)
v, s.p. \ u
C u
IPv
V\
and Pr =(^—
^
J(3.12)
v
Since the heat transfer from the wall is assumed to be through a gaseous film adjacent to the
wall for film boiling (Region III), the Reynolds number was defined as a gaseous single-phaseReynolds number based on the average velocity of the mixture. The Sieder-Tate equation was chosenand properties of the gas were evaluated at bulk temperatures of the stream, with the exception of
u which is the viscosity of the gas at the temperature of the inner wall. The general form of the
Sieder-Tate equation was taken from Bird, et al. [1962] . The results of plotting (3. 1) and (3.9) areshown in figures 3. 1 and 3. 2 respectively.
Finally, as suggested by Ellerbrock, et al. [1962] , (3. 1) was modified by the use of the
dimensionless boiling number in the following manner:
26
x 1I
11 1
O- -
G )
o
r ^<] <3
_ Xc
_ «*
00oo
- r
Xcex
i
CVJ- m- Q
X- *ci
II —
3< /
<1
G o\T O
<3
G V KG3
b r
aQo^
li
<H > eGjq
GtQ) >o O oO — —
3<l , x G#W
~<b<d
.1 i<l
<-] —<3_ <J^L-Q_
Q.X
H-T"
ooo3 -
r
<3°
fern«a
<3
<3 r% 4
<
^ /o
gP
c9>-
" o
o
Hendricks
et
al
(1961)
Wright
S
Walters
(1959)
n-
T)- q
- -
a
a>
o
JJ 1 1 1
i-i|
G [D «SI
oo
X
x
>
u
u
J3
<u
in
03
0)
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ex
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Eh
on
cu
u
•i-i
Q.Xa>
3
Ooo3
27
4 1I
I 11
- -
o
o
1 4 4
1 <
14
loI (X
1 °
4
G
Go
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-< In 4 O
o- -
o
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IJlo*^ n
U
€1
' © oI 4 G
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4
[fa°*
,? G
*'i
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4
44
- © o $§•$ 4 --
4 /e CW
'
fc^ -<*Is? n \m> nr
4irbn
4
C\J
><cex
00oor-*-
cof
OC0
+CMCMCM
QlX<u
ii
4 4 Kj, c 1
« I©!
3
4
—- u -
Hendricks
et
al
(1961)
Wright
a
Walters
(1959)
CT>
lOCD -
- —
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-
aX
7
a
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co
2
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1| I | 1 1
c> <
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a)
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ft
tn
en
d
CD
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I
<u1—1
bCd-H</3
<D
dCuO•rH
ds '»'°l°3nMdxaon
28
Nu^2 )(Bo.No.)-°-
40=f(X
tt), (3.13)
calc, f, t.p.
where
Bo. No. = ^-£t (3. 14)
mix
Examination of the factors involved in the definition of the boiling number leads to the interpretation
of the boiling number as being a ratio of the vapor generation to the mass flow rate in the tube or in a
sense an index of the boiling-induced flow normal to the axis relative to the axial flow.
Thus, for a fixed power input to the tube, a large boiling number would indicate well developedboiling where velocity is not influential and a small boiling number would indicate forced convectioneffects are controlling (high velocity).
The results of this plot are shown in figure 3. 3.
3.3.2. Film Boiling-Nusselt Number Ratio vs x(quality)
Since quality appeared to be the controlling parameter in X. . and, perhaps, in the heat transfer
mechanism, (3. 1), (3.9), and (3. 13) were all plotted as a function of quality (f(x)) in place of f(X ).
Results of these plots are shown in figures 3. 4, 3. 5, and 3. 6.
Another type of quality correlation proposed in a recent publication by von Glahn [1964] wasalso included herein:
Nu~^S- F = f(XJ (3.15)
where
Nu tpv
fv, calc
Nu . = 0.023 Re0,8
Pr0,4
(3.16)v, calc v v
DG .
Re = ^2i?S( 3.17)V %
C uP v
Pr = —-1 (3.18)v k
v
xf= t^t < 3 - 19)
F =2.0xl0- 10a
i(3)
- l67( Y )
[l ' 8 - (Xf)a] (0.005)t 1 - (X^(N f" 667
(3.20)tp 1 bo. no.
29
1 1 I 1i
—
11
01
.01
.1
1
10
100
Xtt
-
'o
4 4 <
1
+->
4 4 /
'
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3
en
>
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< io o
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S
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et
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(1959)
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<-
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jp\
o<J
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en
^(
\L<<J-
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(1961)
Waltei
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(1959
-
Hendricks
Wright
S
&
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et
c
iii
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et
a\
(I960
i
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a
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(1959)
\eG
O 0>
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-__
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L-.-ON ogV di y '3103 nisi
'
33
l-[Re
4.2 1 -
Re + 5.85 x 10v
n) +- 92
](3.21)
C-^l (3.22)
Y = Re 1 +2500
)] (3.23)
C^r) (3.24)
N "y^i-'v 1
bo. no. , .1.5P CT Jv i
and (3.25)
{..,[ 1 -
(L /D)
J+ 0.
13J- = 0. 13 for L /D> 3. 5
(L /D) + 0.05'(3.26)
Results for the von Glahn correlation are shown in figure 3.7 for the data of this study.
3. 3. 3. Film Boiling - superposition
A simple superposition correlation was tested with the available cryogenic data. This proce-dure has been suggested by numerous papers and textbooks. The heat transfer coefficient calculated
from Breen and Westwater' s [1962] film pool boiling correlation (2.3) was added directly to the heat
transfer coefficient calculated from the modified Sieder-Tate single-phase forced convection correla-
tion. That is
h = h + hpred conv film
s.p.v. poolSoil
(3.27)
where
h =0.026(Re )°' 8(Pr )°-
33(
U_Z)°- 14 (il)
conv \ v. s.p./ \ v/ \ u / \ D/s. p. v,
(3.28)
.(APJ0. 375
0. 375 _, , kTX V_, 0.250w (— [~bld <> >» [-^ 1 [-T71]
(AT)-.250
(3.29)
)ool>oil
D(Apf)
34
=f ! 11
1
1
-
CSJ-«— -
X- 4'ar
4" c OOqi "- 4
<s> vS4
rOO o
n^4 4
1 4
« 4
1
«•-
X OO
3 «
P1>
4 «
4
•34
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C\J
1
O
3
O G
c
9?4
I
4
4-
4
4
1
<4
- (0 O W c>^^" < 4 « 1-
- O™ 4 4 1
-
L~~ D D
X j
II
a.
u
a D W( 1
c
30
Hendricks
et
al
(1961)
Wright
a
Walters
(1959)
a.X«3
I
t
3
>
! o
« l>
>
- -3
i>
i>
1 | 1 1 | 1
c> J <1
X
ooo
oo
ao
• iH+->
aJsi
•rH
Uoa>
0)
oo
<u
en
04
0)
a
s
r—
I
o>
b0
X
+->
(U
s
0<
d4j Q|P3'AnN'
dX9nN
35
_X + 0.340(C ), (AT)., 2
*'=[ ^ ]• (3.29a)
Results of plotting h , vs h are shown in figure 3.8.pred exp
3.3.4. Nucleate Boiling - superposition
Simple superposition (h based on the liquid phase) and the correlations of Chen [1963] and
Kutateladze [1963] were tested for the nucleate boiling region.
The simple superposition correlation is
h j = h + h (3. 30)pred conv nucleates. p. £ . pool
boil
where
hconv
=0.023(Rei
)-8(Pr
i)-
4
(4) (3.31)
s.p.-?.
Re, = DG . /u*
(3.31a)£ mix i
and
1.282 1.750 1.500
w-^w-TfwA ]<' t »
1 - 500•
pool 0-P )cr u
boil v *
Chen' s correlation is
where
h = (h, )F+ (h, )S, (3.33)pred fc f-z
hfc =(4) <°- 023
> (Re
i)°'
8(Pr
i)0 ' 4
(3 ' 34)
ki' 79 (C)/-
45pi
- 49(AT)
- 24(AP)
- 75
hf- z = <°' °° 122 >
P0.5 0.29,0.24 0.24
(3'35)
F = correction factor which is a function of X and accounts for increased convective turbulence due to
the presence of vapor.
/ 125S = correction factor which is a function of Re. F ' and accounts for suppression of bubble growth
due to flow:
36
2.2~
2.0
.8
I .6
I .4
I .2
ON
I .0
.8
.6
o r
n \
lendricks
Vright a
^ore et a
et al (IS
Walters (
(1959)
>6I)
1959)
* c
A
O
o
A /
o
AO
4
A /
A
*
A
oA C
A Go
<£AJ ft
00 A* * *
O */O Of A
AA
A
A
A
A A i
** cA n
Z. j,., A -
* oO
A
A
oA
A
AO
o
o
/aA
A
.2 .8 1.0
watts'ex P cm2 °K
1.2 1.4 1.6
Figure 3.8 h vs h (film boiling superposition correlation)to pred exp
37
(Re,)
DG . (1-x)mix
Kutateladze' s correlation is
>rednucleate, npool DOll \
h /1 +
convs.p.i.
(3.36)
>red
s.p.
nucleatepool boil
iconvs.p.i.
:le1
ate1
>I Doilnucl
Figure 3.9 Functional dependence of h , on _— according to Kutateladze (1963).pred h
convs.p.i.
38
h nucleater- -J i .. c Pred pOOl DOll , . r . _ -Consider a plot of r-^ vs 7-*- as shown in figure 3.9,
convs.p.i.
convs.p.i.
pred
s.p.
nucleat^
conv
pool boiJ
s.p.i.
Asnucleatepool Don
1convs.p.i.
-, i.e., for no boiling, where velocity effects are controlling,
pred , , „/then -r^ = 1 and f = 0.
hconvs.p.i.
Asnucleate,pool DOll
s.p.
i.e., for well developed boiling where velocity effects arenot influential,
thenpred1convs.p.i.
nucleate,P°°Ib°fl
and f'
convs.p.i.
A function which satisfies the above criteria is (3. 36). Values of n from 0. 7 to 2. were used in this
study in an attempt to correlate the limited data available. Results of plots of h , vs h forpred exp
(3.30), (3.33), and (3. 36) are shown in figures 3. 10, 3.11, and 3. 12, respectively. Figure 3. 12
reflects results when n = 2 in (3. 36), since that value of n produced the best results.
3. 3. 5. Burnout or Transition Point
Since the only burnout data for cryogenic liquids found was that of Lewis, et al. [1962] , this
paper has merely reproduced their curve, showing a comparison of burnout heat flux for cryogenic
liquids with the water correlation of Lowdermilk, et al. [1958] .
Lowdermilk' s correlation is
(q/A)270 G0.85
burnout 0. 2, T ,.0.85 '
D (L/D)1 < < 150
(L/D)(3.37)
(q/A)1400 G 0. 5
burnout ^0.2, T ,.0.15 '
D (L/D)
150< < 10000
(L/D)
39
8
o
CM
Eo
o> 6
a \Vright S Waiters ( 1959)
LJ
Q
D
P y^
Q '5f
f P
P
/ m
3
I 2345678. watts" e*P cm2°K
Figure 3. 10 h vs h^
(nucleate boiling superposition correlation)
40
o
eg
eo
a>
10
G>iVright S Walters M959)
9
8
7
6
5
4
D
3D Q
3 n
2
1
/ D
DD
I 4 5
wattsW cm2 °K
Figure 3.11 h , vs h (Chen correlation)& pred exp
41
10
«o
06
0)
a\Vright 8 Walters ( 1959)
D
-
n
n/
/ &
D
y' D
§
rn
D
4 5watts
Jexp cm 2 °K
8
Figure 3. 12 h vs h (Kutateladze correlation)prect exp
42
3.4. Discussion of Correlations and Results
The data used in this study are limited in numbers of points, ranges of fluid properties, and in
experimental systems. Figures 3. 1 through 3.7 illustrate significant differences in the three sets of
data considered; furthermore, no attempt was made to cull the data for accuracy. Thus no conclu-
sions regarding the absolute reliability of the correlations are implied; however it is felt that somegeneral guides are indicated by this comparative study.
Table 3. 3 gives the equations of the best fit curves of the plots given in column 2. The equationswere determined using the method of least squares by means of a digital computer. By applying the
resulting least squares fit curve equations and the superposition type predictive equations, an hpred
was then calculated for each data point. The figures given in column 5 of table 3. 3 are the results of
plotting h vs h for each correlation. As a means of comparing the relative success or
failure of all the film boiling correlations and all the nucleate boiling correlations for the data con-sidered, a root mean square fractional deviation, d, of the predicted value, h
n , from the— predmeasured value, h , is given in column 7. The d is defined as
exp —
where n is the number of data points. If the data population followed a normal distribution pattern,
±d would be the spread within which about 68% of the data occur and ± 2d would be the spread within
which about 95% of the data are found. Even if the distribution varies considerably from a normaldistribution, it is felt that d may be used as a measure of the relative reliability of all the correlations.
Although all results for d were based on the least squares equations shown in columns 4 and 6
of table 3. 3, it is felt that, for the range of X and x shown in figures 3. 1 through 3. 7, there would
be no significant loss of accuracy if the last term in each of the least squares equations is ignored.
This study also performed the aforementioned calculations for h , and d, using Hendricks,pred —
et al. [l96l] data only. These results also are shown in column 7 of table 3. 3. It is interesting to
note that in this case the single-phase Reynolds number and superposition systems do not correlate
the data as well as do the correlations of von Glahn and Nu, T
6XP- f(X ) or f(x).
Nu tt'calc, i, t. p.
The converse is true when all the data are considered. Furthermore, for Hendricks, et al. [ 196 l]
data only, the boiling number correlations are not as effective in reducing the data spread as they
are when all data are used.
However, unless specifically stated otherwise, the following remarks are made with reference
to the results obtained when all data [Core, et al. 1959, Wright and Walters, 1959, and Hendricks,et al. 196l] were used in the calculations of h , and d.
pred —
3. 4. 1. X and x Correlations without Boiling Numbertt
e
The first four correlations shown in table 3. 3 may be generally considered together. The X
correlations appear to be based on the analogy between heat and momentum transport. This systemhas been used extensively [Guerrieri and Talty, 1956; Dengler and Addoms, 1956; Hendricks, et al.
196l] and a further use of the analogy between momentum and mass transport has been proposed and
used[Wicks III, and Dukler, I960] . It would appear, however, for hydrogen that x is the controlling
factor in X and that a correlation with x might be simpler and just as satisfactory. Also, the means
43
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46
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47
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48
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49
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50
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51
of computing Reynolds number varied with the investigators. Hendricks, et al. [ 196 1 ] use a meanfilm density with the average velocity of the mixture and the viscosity of the gas evaluated at the mean
,P£ m t
Uav
Dfilm temperature (
—
'-—'-*— —J
anc[ von Glahn [1964] evaluates the Reynolds number using the
f ' V/DG
mixXsaturated vapor properties and the mass velocity of the mixture (
J. Since, for film boiling,
u vthe process is visualized as heat transfer through a vapor film to a liquid or liquid-vapor core, it was
PvU^ D
felt that perhaps a single-phase gas Reynolds number ( — ) , rather than the two-phase gasV U v ^
A m tUav
D\Reynolds number (
—
'—'-^ -— J used in the calculation of Nusselt number, might also produce
a successful correlation. All of these considerations (except the von Glahn approach) variouslycombined have produced the first four correlations shown in figures 3. 1, 3.2, 3. 4, and 3. 5. The d
values for these correlations indicate that for these data the simpler system of figures 3. 5 and 3. 16
has provided the best correlation. This is not true when the data of Hendricks, et al. [1961] are usedalone. It appears then, that for hydrogen, accuracy is not lost by the use of quality rather than X
as a correlating parameter and may not be lost by use of the single-phase gas at average velocity in
the determination of the calculated Nusselt number.
3.4.2. X and x Correlations with Boiling Number
Several investigators [Schrock and Grossman, 1959; Ellerbrock, et al. , 1962] have found that
by use of the boiling number the correlations have been markedly improved. Figure 3. 20, which is
Nua plot of — vs X for some of the data of Hendricks, et al. [l96l] shows, for example,p Nu , tt
L J ' * '
calc, f, t. p.
how the data tend to separate according to boiling number. Thus, as shown in figures 3. 3 and 3. 6,
Nuwhen the ratio tt is multiplied by (Bo. No.)-0.4, the correlation is improved. The
calc, f, t. p.
improvement in d values is quite significant, reducing the data spread by a factor of 3 for all the data
but less for the Hendricks, et al. [ 196 1] data alone. As discussed previously, perhaps an explana-
tion for the effectiveness of the boiling number in the correlation is that this provides a measure of
the ratio of the velocity of the vapor formed to the normal stream velocity. It seems quite conceivable
that this ratio would have a considerable effect on the flow pattern and subsequently on the heat trans-
fer coefficient. No plots are shown using the boiling number with the single-phase Nusselt numberratio as it would have been necessary to make appropriate changes in the exponent on the boiling
number to optimize the correlation using that Nusselt number ratio.
A disadvantage of the boiling number correlation for design purposes is that the heat flux mustbe known in order to use the correlation.
3. 4. 3 von Glahn Correlation
The value of d for the correlation proposed by von Glahn [1964] is relatively high. The best
equation for this correlation given in table 3. 3 was determined only from figure 3.7, which shows the
data of Hendricks, et al. [l96l] , Wright and Walters [1959] , and Core, et al. [1959] . The data of
Core, et al. appear to plot at a reduced slope compared with the data of Hendricks, et al. and Wrightand Walters, von Glahn points out in his report that for 0. 7< Le/D< 3. 5 such a reduced slope wasobtained when the exponent a was taken to be 0. 13. These authors used a = 0. 13 for the data of Core,
et al. The dimension L was not recorded, but figure 4, Appendix I of Core's report indicates an
L /D ratio > 3. 5. However, it is conceivable that L, /D for Core' s test section was in fact less thane e
3. 5 which may account for the reduced slope plot of their data which subsequently would produce a
higher d value for von Glahn1 s correlation. For the Hendricks, et al. [l96l] data alone, the von
Glahn correlation compares favorably with the boiling number correlation. Like the boiling numbercorrelation, the von Glahn correlation requires a knowledge of the heat flux in order to determine F .
52
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Another consideration present in the von Glahn correlation is that the correlating parameterswere developed using both hydrogen and nitrogen data and further tested for applicability to Freon 113.
Since these fluids present wider ranges of property data one might conclude that this correlation maybemused over a greater range than those developed for hydrogen data only. This would be true,
however, only if the properties data are accounted for properly. It may be open to question whetheror not any of the correlations used in this study have reached that degree of development.
3.4.4. Film Boiling Superposition
The superposition approach is an extremely simple one, and its d value indicates that it is
comparable to the other film boiling correlations. Although the results are not included herein aninteresting sidelight came about when Breen and Westwater' s [ 1962 ] film pool boiling equation wasevaluated with saturation vapor properties rather than the film temperature properties for which the
equation was developed. The result was an improved d for the superposition equation,
h , = h + h . Since using the saturation properties reduced the predicted pool boiling flux,pred conv film
s.p.v. poolboil
it may be that this lower value provided partial correction for the interaction between the forced
convection and boiling heat transfer mechanisms.
Since the superposition approach does not consider any of the interactions of the boiling andconvective heat transfer mechanisms which would appear certain to be present and significant, the
comparable success of this correlation raises the question as to whether or not any of the other film
boiling correlations of this study have properly accounted for such interrelationship.
3.4.5. Nucleate Boiling Correlations
It might be noted here that some researchers have proposed use of the pool boiling equations
alone for the forced convection boiling case. In a review, paper Zuber and Fried [1962] report the
following in this category:
(1) Kutateladze [1949]
(2) Michenko [i960]
(3) Gilmour [ 1958]
(4) Labountzov [ I960 ]
(5) Forster and Grief [1959]
The forced convection contributions for the data of this study, however, were significant and varied;
on this basis it is assumed that the use of the boiling component alone would constitute a relatively
poor predictive system.
For the limited data considered, correlations applied for nucleate boiling, Region I and RegionII, appear to be as good as or better than the film boiling correlations.
All three of these predictive equations employ a superposition approach but the correlation of
Chen does attempt to account for the interaction of the boiling and convective heat transfer mecha-nisms by the factors F and S. However, the d values indicate that the simple superposition equation
and Kutateladze' s equation, neither of which attempt to account for these interactions, are just as
successful in correlating the data used in this study. Unfortunately, limited data from only one
source (Wright and Walters [1959] ) were available for this comparison.
3.4.6. Burnout
As might be expected and as substantiated by figure 3.21, the burnout correlation of Lowdermilk,et al. [1958] cannot be applied to cryogenic fluids without modification to the constants and/or
exponents in the predictive equation. Due to lack of other cryogenic fluid burnout data, no attempt
was made to determine the necessary modifications.
3. 5. Conclusions
The general conclusions of this study are:
(1) Within the range of fluid property variables and systems of the data of this study, the
boiling number correlation appears to be more successful in predicting heat transfer
54
dQ
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Liquid
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Figure 3.21 Comparison of maximum critical heat flux for cryogenicliquids with water correlation of Lowdermilk, et al. (1958)
55
coefficients in the film boiling region (Region III). However, for design purposes, little
accuracy is lost by using the simpler systems such as the correlation using the Nusseltnumber ratio versus quality. It may be that further simplification, as the use of single-
phase properties in determining the Nusselt number (calc) for the correlations vs x andX , and use of the superposition method will not result in an objectionable loss of accuracy.
The use of all data considered in this report indicates this to be true, but the use of the
Hendricks, et al. [ 196 1] data only does not.
Correlations for the nucleate boiling region (Region I and II) generally are better thanthe film boiling correlations, but on the basis of the small number of data points available
for consideration, these results are not considered very significant. The Kutateladzeapproach appears to be the better of the nucleate boiling correlations.
(2) The conditions at which properties are to be evaluated and the interrelationship betweenboiling and forced convection phenomena is quite different in all of the correlations of this
study. Yet their reliability as evidenced by the d values is generally the same; it wouldappear that all systems of approach are lacking in arrangement and treatment of the sig-
nificant variables.
(3) The predictive quality of the systems of this study does not appear to fall in a respectable
range for correlation, but certainly for a thorough test of the predictive systems moreexperimental data are needed. Further work should be in the direction of a more funda-
mental analytical study of the fluid mechanics and heat transfer phenomena and the
acquiring of data from very carefully controlled, experimental systems.
4. Boiling Variables
4. 1. Saturation Fluid Properties (Pressure)
The predictive equation of Kutateladze [1952] for nucleate pool boiling appears to account
generally for the changes in boiling characteristics brought about by pressure changes which, in turn,
change the saturated fluid properties. The increasing steepness of the experimental curves with
pressure, however, is not reflected in the Kutateladze correlation or in other correlations proposed,
as essentially all use a constant exponent for AT. Deviation of the peak flux from predictive expres-
sions at pressures above 0.6 of the critical pressure as shown by Lyon, et al. [1964] would tend to
indicate that the general relationships for boiling do not hold at high reduced pressures. This seemsreasonable as the apparent primary factors in pool boiling heat transfer are the stirring and the
vaporization effects and both of these would be greatly diminished as the critical pressure is
approached. Graham, et al. [1964] show that at a pressure of 170 psig hydrogen exhibits a nucleate
boiling type of curve over a very short interval and then apparently enters a film boiling regime,
indicating that conventional nucleate boiling behavior does not occur as the critical pressure is
approached.
It is not possible to discuss the effect of pressure on pool film boiling because there are
insufficient data for comparison. In the forced-convection correlations considered, the predictions
at higher pressures appeared to be as good as those at one atmosphere.
4. 2. Subcooling
Apparently the temperature difference between the warm surface and the saturated liquid
represents the most significant variable affecting heat flux. Therefore, this temperature difference
is conventionally used in the Leidenfrost plot as shown in figures 2. 1, 2. 2, 2. 3, and 2. 4. It may be
that subcooling effects are rather small, at least, when the subcooling is no larger than roughly the
temperature difference between the wall and saturation necessary to produce burnout. Graham, et al.
[1964] have reported hydrogen data at 175 psia which indicate subcooling temperature differences have
a smaller effect on the nucleate boiling curve than temperature differences above saturation. It
would seem that a major subcooling effect may occur in the normal film boiling or transition region,
as a result of the increased rate of condensation. Vliet and Leppert [1964] report such changes in
studies of forced convection boiling on a cylinder with water crossflow.
Most of the predictive expressions which have considered subcooling have been for pool-boiling,
peak-flux studies. Among these are Griffith [1957] , Zuber, et al. [l96l] , and three reported by
56
Gambill [1963a] in his peak flux survey. Results of the latter four are shown in figures 4.1, 4. 2,
4.3, and 4.4. The expressions are:
p 0.923 CKutateladze I [1952] F =1.0+0. 040
(^
—
) \ir) (AT) (4.1)
v
Kutateladze II p. 0.800 C(from Gambill 1963a) F , = 1.0+0. 065 (—
)
(^-£)(AT) ,. , „ „,v ' sub \p / \ X /v 'sub
, 4.2v
Ivey and Morris P, .0.750 C(from Gambill 1963a): F , =1.0+0.102(—
)
( _E)(AT)
, , (4.3)sub \p / V X I sub
lr, , , 0.500,, ,0. 125(C pk) (AP)
Zuber, etal. [l 9 6l] F = 1.0+12.326[ E- > T > sub- <
4 " 4>
Xp av
Considerably more work will be required before the effect of subcooling is finally determined. Thegeneral effect, however, should be indicated by these expressions.
4. 3. External Force Field (Gravity)
Because buoyancy forces are significantly present in boiling and are functions, in turn, of someexternal force fields (viz. gravity and other accelerations, magnetic and electrostatic fields) onewould expect to find the boiling phenomena changing with changes in the force field. Such changeshave been reported, but the influence has been established only generally at this time.
The lack of resolution results from uncertainties in the effect of other boiling variables. As the
external force field (gravity) is reduced, the normal density separation pattern is replaced by one in
which the vapor tends to collect as a centrally-located sphere for wetting liquids (most cryogenicsystems). Therefore, system geometry may play a major role in determining the influence of
gravity forces on heat flux, since this could influence whether or not vapor will be removed from the
surface.
For the pool nucleate boiling curves, the changes appear small (i.e., in the slope of a
Leidenfrost plot) as evidenced by the work of Graham, et al. [1964] and Sherley [1963] except,
perhaps, very near zero gravity. The inception point and the peak flux are, however, markedlyinfluenced by the force field. Graham, et al. [1964] report that the Leidenfrost curves at one g andat seven g
1 s separate considerably near the inception point for hydrogen and then tend to become the
same curve at higher fluxes and temperature differences.
For the peak flux, (2. 2) shows a one-fourth power gravitational dependence. The works of
Usiskin and Siegel [l96l] and Lyon, et al. [1965] using water, indicate the 1/4 power influence maybe generally correct for force fields down to about 1/4 normal gravity, but then the power influence
is substantially reduced.
For film boiling, one might expect larger influences on the boiling phenomena because the
interface behavior may change markedly. The data of Graham, et al. [1964] for hydrogen indicate
that changes in the external force field have a greater effect on the film boiling flux. The authors
are not aware of any studies which consider the effects of a gravitational force field on forced con-
vection boiling. The effect of electric fields have also been investigated by Markels and Durfee [1964]
who found that the peak nucleate boiling heat flux with isopropyl alcohol and water was substantially
57
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increased under the influence of an electrostatic field. This was attributed to de stabilization of the
vapor film by the electrostatic field. Choi [1962] proposed an expression for an effective gravitationalforce of the electrostatic field.
4. 4. Surface Orientation and Size
Pool-nucleate boiling is generally regarded as insensitive to system geometry. Several textbookauthors make this observation. The work of Class, et al. [i960] and Lyon [1964] do show significant
variations when changing from horizontal to vertical orientation with no other changes in the system.Unfortunately, the results of these works are somewhat contradictory regarding orientation effect,
so no general conclusions can be drawn. Lyon also reports a significant orientation effect on the peakheat flux. As mentioned previously, surface orientation with respect to an external force field
(gravity) may have a major effect. For example, Lyon [ 1 965] reports marked reductions in heliumboiling fluxes for horizontal surfaces facing downward, where the influence of vapor removal is
significant.
Costello, et al. [1964] have reported that for pool boiling burnout heater size is quite signifi-
cant. They found that one- sixteenth-inch diameter semi-cylinders burned out at fluxes 2.7 timesgreater than for flat plate heaters with no liquid in-flow from the sides. This suggested to the authorsthat such difference may be a result of different convective effects for the various heated surfaces.
They show that the convective component may be approximately one-half the total flux in some cases.
4. 5. Surface Condition
While it is known that surface conditions have a significant effect on the boiling phenomena, the
specific influence of any given surface variable is not well understood. Enough data have beenacquired, however, to permit a qualitative discussion of the effect of specific surface variations.
Considered here are surface history, surface temperature variations as affected by the heatedmaterial mass and properties or by the type of heat source, surface roughness, and surface-fluid
interface phenomena as influenced by the surface and fluid chemical composition.
4. 5. 1. Surface History
A heating surface immediately after immersion in liquid will produce a higher heat transfer
coefficient than one which has been immersed for a reasonable period of time [Kutateladze 1952] .
This is presumably due to additional nucleation centers provided by factors such as dissolved air andoxidation of the heating surface. Graham, et al. [1964] reported that for nucleate boiling of
hydrogen, boundary layer history has a significant effect on the boiling incipient point. The apparent
incipient point for nucleate boiling occurred at a much lower AT when a boiling run was immediatelyrepeated rather than begun with a fresh supply of hydrogen surrounding the heating surface; the twocurves then join at higher heat fluxes. The authors speculate that some residue of the thermal layer
remained to change, the incipient point for the succeeding test.
Vliet and Leppert [1964] studied the effect of aging or boiling for a period of time at about half
peak flux. They found that with water flowing over a stainless steel tube, aging of about 90 minuteswas necessary before reproducible peak fluxes could be obtained. Aging for about one-third that
time produced peak fluxes only slightly greater than half the fluxes produced using the longer aging
procedure.
4. 5. 2 Surface Temperature Variations
It is possible that surface temperature differences can occur which may be attributed to the
heater surface and not to the boiling phenomena. Heaters with a small mass per unit of heater
surface such as very thin materials may produce such temperature variations and, subsequently, a
lower peak flux. Vliet and Leppert [1964] , however, reported that there were no surface effects
down to a thickness of 0. 006 inch for a cylinder with water crossflow.
The source of energy for the heater may also influence surface temperature variations.
Kutateladze [1952] reports that electrically heated surfaces have slightly different heat transfer
characteristics than those heated by vapor condensation, probably because condensation droplets
cause surface temperature differences.
62
Of course, the boiling phenomena also produce temperature variations at the surface, and theseare reported, for example, by Kutateladze [1952] , Hendricks and Sharp [1964] , and Moore andMesler [ 196 1] . Sharp [1964] studied the microlayer film at the base of nucleate bubbles and foundthat the flux from this microlayer appeared to vary with k/Vct for the surface material.
4.5.3. Surface Roughness
Rougher surfaces generally produce higher fluxes for the same AT. Mikhail [1952] reportedwork with oxygen using nickel surfaces with different roughness values, and his data were similar to
others who investigated higher temperature fluids. Rougher surfaces cause incipient nucleate
boiling to occur at a lower AT, and then the h vs AT curves rise abruptly from that point. Thus the
rougher surfaces produce markedly higher coefficients for the same AT, in Mikhail' s work as high asa factor of 4. Lyon [1964] .working with cryogenic and other liquids, indicates, however, that
although the nucleate boiling flux curve is changed by roughness, the peak flux does not change.Surface roughness would be expected to show a much smaller effect in the film boiling region, andwork such as that of Class, et al. [i960] indicate essentially no effect of roughness for film boiling.
Tuck [1962] , in experimental work with hydrogen, found that the AT for inception was less than
0. 1°K for a rough surface but could be as great as 3°K for a surface finished to 1. 25 micro-inchRMS. The Tuck experiments were at zero gravity condition; however, the results would be expectedto be generally applicable under other gravitational fields, although at or near zero gravity the
inception point seems to be time dependent and that time a function of the gravitational field.
4.6. Surface-Fluid Chemical Composition
The surface chemical effect is often difficult to separate from other surface effects such asroughness. Wetting characteristics would appear to be a major influence. Cryogenic fluids will wetalmost all surfaces except those with a very low surface energy, illustrated, for example, in a
hydrogen study by Good and Ferry [1963] and perhaps further substantiated by the reasonably effective
use of a single wetting coefficient in the Rohsenhow [1952] correlation for cryogenic fluids. Lyon[1964] studied nucleate boiling with oxygen and nitrogen using clean copper and gold surfaces andsurfaces with various chemical films. He found that the different surfaces produced somewhatdifferent nucleate boiling curves and differences as much as 25% in the peak flux.
Young and Hummel [1964] have shown that higher coefficients in the lower region of the
nucleate boiling regime are made possible by providing poorly wetted spots on the metal surface.
Sharp [1964] has studied the microlayer at the base of bubbles and has found that non-wetting surfaces
tend to destabilize the layer. Costello, et al. [1964] found that the burnout heat flux was increasedby a factor of 2. 3 if tap water rather than distilled water was used.
5. Summary and Conclusions
For the pool boiling case, the correlations of Kutateladze [ 1952 ] (nucleate) and Breen andWestwater [1962] (film) have been indicated as those which apparently have the greatest reliability.
Although agreement with experimental data is not particularly good, the accuracy of the predictive
expressions is perhaps sufficient for many design studies. Maximum nucleate flux data are
reasonably well predicted by the Kutateladze [1952] expression.
For the forced convection boiling case, no correlation emerged as distinctly better; however,several simple predictive schemes are shown to be as accurate as some of the more complexsystems when tested with the limited experimental data available.
For all boiling cases it is questionable as to whether or not the predictive correlations include
all of the significant variables and whether the variables considered reflect their proper influence in
the final expression. It would appear that new analytical approaches considering phenomena not
included in the present correlations, such as a treatment of the process near the heated wall as
reviewed by Zuber [1964] , would be desirable. Recent experimental work indicates that to properlyaccount for all the variables, more detailed and better controlled experiments are required. This
would be particularly true for surface and geometry effects which are often averaged or neglected andin detailed studies of the boiling flow pattern or mechanism.
63
6. Properties Data and X Curvesptt
As an aid to designers, available properties data for hydrogen, nitrogen, oxygen, and heliumand values of X for hydrogen, nitrogen, and oxygen are presented in graphical form in figures 6. 1
through 6. 46. Values of C for the saturated liquid and vapor were not available for all of the cryo-
genie fluids included in this study. Therefore C at saturation was approximated by
AH Hl
' H2
PSat~AT
~Tl-
T2
'
where H = enthalpy of liquid or vapor at saturation temperature T
H = enthalpy of liquid or vapor at a temperature T slightly higher than T for the vapor andslightly lower than T for the liquid (all at the same pressure).
Curves obtained in this manner are so indicated on the graph. The values for H , H and corres-1 £
ponding temperatures T and T were obtained from NBS Technical Notes or private communications
cited in the bibliography for thermodynamic properties of the various cryogenic fluids included in
this study. Because of uncertainties in the data used in obtaining an equation of state for oxygen in
the temperature range greater than 98 "K for liquid [Stewart, Hust, and McCartyl] , the C values for
saturated liquid oxygen are not considered reliable above 90°K (or above the corresponding saturation
pressure of 1 atm). Furthermore, it is pointed out that the C values for saturated liquid heliumP
were obtained by extrapolation of the data of Lounasmaa [1958] , and the C values may be inaccurate
by as much as 20%. p
64
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6/oo'ainon jo 3iAimoA ouio3ds
1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i 1 i i i
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u
84
qi/£ u * svo do aiAimoA oiaioaas
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oIT)
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qi/e.u'ainon ao awrnoA oidioaaso o o o o O or- to m <3". ro cvj _d o o' o o" o d o
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X)
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Roder
,
Dil
ler
,
Weber,
and
Goodwin
(1963)
—en
O
to
- - bo
- TS
LU +j
or ctf
00 3 ^w 5ft 1cc
m
,n
VAPOR olumegen
o'"2\
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^ £tf
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co a\
\
o *
d0D
momomoioomc<3-'frrorO(\jc\j — —•6/oo 4 ainon ao awmoA oiaioaas
1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i 1 i i i i
CD ro h
<n 6/00* svo ao aiMnnoA oiaioaas
85
VAPOR PRESSURE, psia
100 200 300
220
200
180
160
— 10
40
20
\ uou
o — "
"140 — OCO -)< — O
_ _J
&-I20 — u.— O
Ld _2 LiJ
2_|I00 =>
o _l
> O>
ouL 80 oc_>
U.
UJ oa.CO
UJn
60 CO
4001 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
Strobridge (1962)
-
i
I
-
\
I
\ - —]
[
\\\{\
\
r\\\\ nIj"u "'-
^GA§_
0.16
0.14
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u. -1o
0.08=) -_lo>
o0.06 U.
oUJa. .
CO
0.04
0.02
3.5
3.0
2.5 x>
2.0
CO
<ou.
oUJ
-1
> p>
oUl
1.0 CO
0.5
10 15 20
VAPOR PRESSURE , atm.
25 30
Figure 6.22 Specific volume of saturated gaseous and saturated liquid nitrogen
86
x 10
40i—
36
32
28
to
24
to<
u_20
o>-
</> 16ZUJQ
2 —
0>—
75 —
25 —I
x 10
250
225
200
175
to
150
125
CO<
ioo TnzUJQ
75
50
25
24 32 40
PRESSURE, atm
Figure 6.23 Density of saturated gaseous and saturated liquid oxygen
87
0.7
0.6
ob 0.5
OCO 0.42Id
UJO<Ll
IDCO
0.3
QLJO2 0.2
UJ
0.1
Stewart, Germann and McCarty
(1962)
He-3 : <r = 0.2185 dyne/cm, Tc = 3.33°K
He-4:<r = 0.5308 dyne /cm, Tc =5.20°K
0.4 0.5 0.6 0.7 0.8 0.9
REDUCED TEMPERATURE, T/T c
Figure 6.24 Surface tension for helium- 3, and helium 4
1.0
88
20
TEMPERATURE, °R30 40 50 60
3.0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 I
V i i 1
Corruccini
(1965)
\\
POINT
2.5 \
sw-NORMAL H 2
1k.
Eo 2.0v.
\
\v
c
"O PARA H 2—2
VS.
Ow 1-5
2A
UJ
UJ \<
>
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i 1.0
CO
^\
N\\
i
v0.5
n
Vv
N\
CRl lUfl
POINT-L ^^
n10
xlO
20
10
oCOzUJHUJo<U_
IDCO
18 22 26
TEMPERATURE, °K
30 34
Figure 6.25 Surface tension for normal hydrogen and parahydrogen
89
Eo
c>*T3
OCOzUJ
UJo<o:=>
CO
60
120 140
TEMPERATURE, °R
160 180 200 220 240
70
1 1
II III III III 1 II III III II1 1 i
\
- -
i
M cCarty(l 965 a)
10
r>j
8
7
pD
—R
—
—
/IH
*
o
1
o80 120 13090 100 110
TEMPERATURE , °K
Figure 6.26 Surface tension of nitrogen
70xl0" 5
60
50 Z
40 2COz
—I UJ
UJ
30 O<u.tr
CO
20
140
90
140 160
TEMPERATURE , ° R
180 200 220 240 260 280
20
18
E 14o\0)c•>,
TJ \Z
CO
l±JO<li-
ar3CO
10
1 1 III III III III 1 1 1 III III 1
M c Carty (1965b)
-
•
•-
_
-
-
70 80 90 100 110 120 130
TEMPERATURE , °K
140 150
Figure 6„ 27 Surface tension of oxygen
I 20 x 10
00
80
60
40
OCOzUJ
LJo<u.or
3CO
20
160
91
CM
XO
'AiiAiionaNoou- rugO) 00 CD
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ro
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IT)
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O00ro
O
—
(X)
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<ro
LU
3r-
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—
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Lir- oA
in
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a
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co
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92
28
TEMPERATURE, °R30 32 34 36
Eu
V)*-
o$
>-"
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o3Ozoo
tri±j
i
JOl ! I | I | I | I
;1 | 1 | 1 | 1 |
- 1 | 1
CO
Cook (1961)-
26
—
pJr\K *h
l \<y1<
24*>*J>
22
15 16 17 18 19
TEMPERATURE, °K
TEMPERATURE, °R
20
xlO'16
15
— 14
— 13
21
— 4
(
xl(
) 2 A ^ 6 8 10
5=]-
| ii
1
J
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— 2
12 3 4 5 6
TEMPERATURE, °KFigure 6.29 Thermal conductivity of gaseous helium from
1.5°K.to 21°K
93
I
oxr-
n _ nig•AiiAiionaNOO -iviaiu3h±
co m ^J: ro cm
oc
LU
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94
200x10
180
oi
Eu
O3
160
140
120
>r ioo
oz>Q-z.
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<orUJX
80
60
40
20
100
TEMPERATURE , °R
200 300 400 500n n n i n i r~r I I TT
-
Scott , Denton , and
Nicholls (1964)
-\ y
<^y>/;^;
<^t
^/
V.N<y
-
^
40 80 280
.10
09
08
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oro
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02
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320120 160 200 240
TEMPERATURE , ° K
Figure 6. 31 Thermal conductivity of gaseous normal hydrogen
95
30
140x10'
1.35
°' 1.30
o
o 1.25
> 1.20
oQOO
or
X
1.15
1. 10
1.05
.00
0.50
TEMPERATURE , °R
35 40 45 50I I I I I I I I I I I I I I I I
—
Scott , Denton , and
Nicholls (1964)
8.0x10
— 75
i
I-00
oro
i
7.0
— 6.5
>-
oazoo
orUJX
6.0
16 18 20 22 24 26
TEMPERATURE , °K
28 30
Figure 6. 32 Thermal conductivity of saturated liquid normal hydrogenand saturated liquid parahydrogen
96
100 120 140
TEMPERATURE , °R160 180 200 220 240 260 280
200x10
180
o
Eu
>-
>
oaoo
160
140
120
100
< 80SorLlI
XI-
60
40
20
III 1 1 1
1
1 1 1
1
1 1 1 1 II III III Ml III
Brewer (1961)
^K.^^
„/h.3>
?*>
V'
Critical Point\
C ritical Point
126.16 °K 154.77 °K\^
- -- i
1
0.1
1
0.10
0.09
0.08CD
oro
.1
0.07
o0.06 Q
z.oo
0.05
004
0.03
0.02
<orUJX
55 65 75 85 95 105 115 125 135 145 155
TEMPERATURE , ° K
Figure 6. 33 Thermal conductivity of oxygen and nitrogen saturated liquid
97
150 250 350
TEMPERATURE ,
450 550 650 750 850
50x10
I I I I I I I I I I I I I I I I I II I I I I I I I II I I
950
45
40
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eo
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orLUX
35
30
25
20
15
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1 Brewer
(1961)
1 1 1
Q!
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PointI26.I6°K
\
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r
TEMPER ATURE,°R260 360
T r
I -5 v1
/P = 3500 psi -..7,nnn
-2-8x10
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1- °l
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O £4 U s-J £< i
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F 140x10 ^
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100 150 200 25C
3Z TEMPERATURE, °K
-
10 —
100 200 300 400
TEMPERATURE , °K
500
Figure 6. 34 Thermal conductivity of nitrogen
28x10
26
,-3
24
22
i
20 m
>H
18 >1-o3
16Qoo
_l
14 <2oruii-
12
98
150 250 350
TEMPERATURE , °R
450 550 650 750 850 950
100 200 300 400 500
TEMPERATURE , °K
Figure 6. 35 Thermal conductivity of oxygen
50xl0"5II 1 1 1 1 I 1 I 1 1 II 1 II 1 1 MM II 1 1 MM MM
Q3O_j
\28xlC
\\ ^P = 4000^2^-^
\1 1
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100
ua>w 80Eu
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60
40
20
TEMPERATURE, °R50 100 150 180
i r
J
Cook
(1961)
*^/
4\^i«
1
\y
9*
*lExperimental results indicate viscosity
independent of pressure at these
moderate pressures.
irve was drawn thru experimental dataCi
points obtained at different pressures by
different authors as reported by
Cocik (1961)
-4xlO
250
200
150 >-
COOoCO
100
50
20 40 60 80
TEMPERATURE, °K
100
Figure 6.37 Viscosity of gaseous helium from 0* to 100 6K
101
200
200 x
TEMPERATURE, °R300 400 500
-4xlO
450
400 v
350 „
>-
oo
300 2
250
200
140 180 220 260
TEMPERATURE, °K
300
Figure 6.38 Viscosity of gaseous helium from I00 e to 300°K
102
JM-U/qi 'A1ISO0SIAo o o oo o o o00 N 10 If)
oo
I I II I I I I I I I I I
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103
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125
TEMPERATURE, °R
150 175 200 225
O.OX MJ
Forster (1963)
3.2
2.8
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800 x I03
700
600
500
400 ooOOCO
300
200
100
70 80 12090 100 110
TEMPERATURE, °K
Figure 6.40 Viscosity of saturated liquid nitrogen
130
104
TEMPERATURE, °R200 300 400 500 600 700 800 900
5.5x1 6 4
1 1 1 1 1 1 1 1
u
Brewer
( 1961 )
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5.013 ]
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1.5
?
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hr{ fW \w .03
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// 02//
n ^
100 200 300 400
TEMPERATURE, °K
Figure 6.41 Viscosity of nitrogen
500
105
JM-U/ Ql 'A1IS00SIA
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106
TEMPERATURE, °R
200 300 400 500 600 700 800 900
5.5x1 n4
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Brewer
( I960\ \ .13i
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100 200 300 400
TEMPERATURE, °K
Figure 6 43 Viscosity of oxygen
500
107
4 6 8
PRESSURE , atmospheres
10
Figure 6.44 Martinelli-Nelson correlating term ^ as a function of quality
and pressure for H
108
Figure 6. 45
4 8 12 16 20
PRESSURE , atmospheres
Martinelli-Nelson correlating term x as a function of quality
and pressure for N_
109
10
8
tt
.8
.6
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PRESSURE, atmospheres
2.5 3.0
Figure 6. 46 Martinelli-Nelson correlating term X as a function of quality
and pressure for O
110
7. Nomenclature
English Letters
d
D
sub
tp
g
G
h
hconvs. p.i.
= specific heat capacity at constant pressure,joules
g°K
= root mean square fractional deviation, dimensionless
= diameter of cylindrical heater in the film boiling regime for pool boiling, andinside diameter of tube for forced convection boiling, cm
= multiplying factor for peak heat flux due to subcooling, (q/A) =(q/A) Fsub sat sub
= von Glahn two-phase modification factor, dimensionless
acceleration of gravity, 3-sec
= mass velocity,sec cm
rr- r , r WattScoefficient of heat transfer, 5—rr-rcm3 c K
1 1 r rr- .Watt S
single-phase liquid convective heat transfer coefficient, ?cm °K
watts
convs. p. v.
fc
= single-phase vapor convective heat transfer coefficient, 3cm3 K
watts= forced convection heat transfer coefficient from Chen's correlation, 3
cm3 °K
f-z
watt spool boiling heat transfer coefficient from Chen' s correlation, 3-rrr
cnrr °K
filmpoolboil
wattsfilm pool boiling heat transfer coefficient, 5-777
cmr K
nucleatepoolboil
nucleate pool boiling heat transfer coefficient, 5-777r 6 cm3 °K
pred
k
Le
(WD)
, . , , , rr . WattSpredicted heat transfer coefficient, ^ —v cm3 °K
thermal conductivity,watts
cm°K
sum of distance 1 and unheated upstream hydrodynamic portion of tube, where 1 is
the critical length measured from beginning of heated portion of tube to burnoutlocation, (cm)
critical-length to diameter ratio in Lowdermilk' s burnout correlation,
dimensionle s s
m = mass flow rate,
bo. no
Nu
Nucalc, f, t. p.
= von Glahn correlation boiling number, dimensionless
= Nusselt Number, dimensionless
= two-phase calculated Nusselt Number with vapor properties evaluated at the film
T +T.w itemperature , dimensionless
111
Nu = single-phase calculated Nusselt Number with vapor properties evaluated atcalc, v, s. p saturation conditions, dimensionless
Nu ,= calculated Nusselt Number with vapor properties evaluated at saturation conditions,
v, calc .
dimensionless
dynesP = pressure of the boiling system, —-—5—
r crrr
Pr = Prandtl number, dimensionless
wattsq
A
Ref, m t. P
Re,£
Re'i
ReV
Rev, s
.
P.
rate of heat transfer per unit area,
= two-phase Reynolds number based on the average velocity of the mixture and with
vapor viscosity evaluated at mean film conditions, dimensionless
= two-phase Reynolds number based on the average velocity of the mixture and with
liquid viscosity evaluated at bulk saturation conditions, dimensionless
- single-phase Reynolds number based on the average velocity of the mixture and with
liquid viscosity evaluated at bulk saturation conditions, dimensionless
- two-phase Reynolds number based on the average velocity of the mixture and with
vapor viscosity evaluated at bulk saturation conditions, dimensionless
= single-phase Reynolds number based on the average velocity of the mixture andwith vapor viscosity evaluated at bulk saturation conditions, dimensionless
AT = T - T„, °Kw £'
cmU = averag e fluid velocity,avg sec
gvapor .
x = quality, *, dimensionless
g .
mixture
x = thermodynamic fluid quality at burnout location, dimensionless
x = quality in film boiling regime, dimensionless
X = film boiling vaporization parameter, dimensionless
112
Greek Letters
Oj, p.y, r
x
X'
C P
thermal diffusivity,, , dimensionlessk
von Glahn film boiling correlation parameters, dimensionless
jouleslatent heat-of vaporization at saturation,
"effective" latent heat of vaporization, defined by equation 2. 4,joules
Newtonian coefficient of viscosity, cm sec.
= density,
APf
f, m, t. p.
APv
a
Abs.Min.FilmCond.
avg
b
exp
for f , v
£ f cm3
- two-phase mean film density, —2-g-
FilmCond.
MaxNucl.
Min.FilmCond.
mix
Nucl.
sat
sub
t.p.
v
dynescm
i v cm3
surface tension between the liquid and its own vapor, evaluated at T.,
Martinelli parameter, dimensionless *
Subscripts
indicates heat transfer conditions at the absolute minimum value theoretically
possible for film pool boiling. (Some authors call this the second critical.)
average
indicates bulk property
indicates the subscripted h or Nu is the experimental value
indicates that the subscripted vapor property is to be evaluated at the vapor film
temperature, (1/2) (T + T.)w £
indicates heat transfer conditions, by conduction only through the vapor film, for
film pool boiling
indicates that the subscripted liquid property is to be evaluated at the saturation
temperature of the boiling fluid
indicates heat transfer conditions at the maximum value possible for nucleate pool
boiling. (Some authors call this the first critical, or burnout.)
indicates heat transfer conditions at the minimum value correlated for film pool
boiling. (Some authors call this the second critical.)
mixture
indicates heat transfer conditions for nucleate pool boiling
saturation conditions
subcooling
indicates two phase
indicates that the subscripted vapor property is to be evaluated at the saturation
temperature of the boiling fluid
113
indicates that the subscripted property is to be evaluated at the temperature of the
heater surface
114
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