DIONYSIUS C. GROENEVELD, ANALYTICAL AND EXPERIMENTAL
PROGRAM OF SUPERCRITICAL HEAT TRANSFER RESEARCH AT THE
UNIVERSITY OF OTTAWA, NUCLEAR ENGINEERING AND TECHNOLOGY,
VOL.40 NO.2 SPECIAL ISSUE ON THE 3RD INTERNATIONAL
SYMPOSIUM ON SCWR, 2007
Fluid-to-fluid modelling of SCHT
Successful fluid-to-fluid modelling or scaling of SCHT requires the use of appropriate similarity
relationships. It is proposed to apply fluid-to-fluid modelling of SCHT using the following
dimensionless groups:


P/Pc and T/Tc
For the subcritical region, the saturation lines of CO2, water and R-134a nearly coincide on a
P/Pc vs. T/Tc (absolute temperatures) diagram, as can be seen in Figure 2.
For SC conditions, we hypothesized that the dependence of the pseudocritical temperature
Tpc on pressure might be similar to the dependence of the saturation temperature on
pressure, because the enthalpy gradient dh/dT reaches a maximum at both temperatures.
This was confirmed in Figure 2 where a remarkable degree of similarity in the SC behaviours
of these three fluids is noted. On a P/Pc vs. T/Tc plot, the pseudocritical lines for all three
fluids nearly coincide and the pseudocritical line appears to be an extension of the saturation
line.
Reynolds number and Nusselt number
At SC conditions that are well beyond the critical or pseudo critical points, single-phase-like
flow characteristics prevail and the conventional Nu = f (Re, Pr) relationship is applicable for
predicting the heat transfer. The heat transfer mode at these SC conditions is labeled as
“normal” (Pioro and Duffey [14]). Thus, the product Re.Pr0.5 can be used as a first
approximation to determine equivalent mass flux conditions, especially when the Prandtl
number is not far from unity.
Not surprisingly, for near-pseudocritical conditions, at which the fluid properties change
drastically, the heat transfer also displays an atypical behaviour, as shown in Figure 1.
Because the atypical behaviour appears to be similar for both CO2 and water, when
normalizing Nu by NuDittus-Boelter (Figure 1), this methodology will be used also in our
preliminary fluid-to-fluid modelling approach. We will therefore scale test conditions for all
three fluids by maintaining the same value of Re.Pr0.5. We also expect that the similarity
would apply equally to heat transfer in the deteriorated heat transfer region, which,
compared to the normal heat transfer mode, is characterized by lower values of the heat
transfer coefficient (see again Figure 1) and hence higher values of wall temperature within
parts of a test section at high heat fluxes and low mass fluxes.

Mechanisms responsible for this deterioration in heat transfer have been described by
various authors (e.g., Jackson and Hall [3]). This anomalous behaviour has been observed in
various fluids operating at SC conditions. Additional confirmation that the fractional decrease
in heat transfer (or the ratio Nuexp /NuDB) is the same in all three fluids of interest is
required for a wider range of fluids, and values of P/Pc and Re within the entire ranges of
interest.
Heat flux
Jackson and Hall [3] examined the governing SC heat transfer equations and suggested that
the values of the heat flux parameter q.D/(k.Tb) should be kept the same in the prototype
and modelling fluid; in this expression, D is the tube inside diameter, k is the thermal
conductivity and Tb is the absolute bulk coolant temperature. Yang and Khartabil [15] found
that the heat flux parameter q/(G.Hb), when included in a Nu = f(Re, Pr, Tb/Tc, P/Pc) type
correlation, provided an improved prediction for the deteriorated heat transfer region for
the AECL SC CO2 data and the SC water data by Yamagata et al. [16]. We will therefore
explore both heat flux parameters for SC fluid-to-fluid modelling.

S.K. Yang and H.F. Khartabil, “Normal and deteriorated heat transfer correlations for
supercritical fluids”, Trans. ANS Meeting, 95, Washington DC, USA (2005).
Geometry
Both Re and Nu contain an equivalent diameter, which should, in principle, account for
differences in geometries in scaled tests. However, in previous CHF and film boiling
modelling studies (e.g., Groeneveld et al. [11]), it was found that the accuracy of fluid-to-fluid
modelling could be adversely affected by significant geometrical differences. To remove this
uncertainty, SC fluid-to-fluid modelling experiments at the University of Ottawa will be based
on identical test section geometries.
Experimental studies

The ranges of similarity parameters used to scale the tests are listed in Table 3.

Reference heat transfer and pressure drop measurements:
Surface temperature and pressure drop of CO2 flow in a 2 m long, 8 mm ID, vertical tube will
be measured for the conditions listed in Table 4.
These measurements will serve as reference and will be compared to CO2 measurements
from the literature. Heating will be applied such as to generate conditions for both normal
and deteriorated heat transfer.
Yang and Khartabil [15] proposed a criterion for the onset of deteriorated heat transfer for
CO2 in 8 mm ID tubes as q > 0.27 G0.94, where q is the heat flux in kW/m2 and G is the mass
flux in kg/m2s. This is analogous to the condition q > 0.20 G1.2, suggested by Yamagata et al.
[16] for water in 10 mm tubes.
Cheng and Schulenberg [44] have reviewed additional criteria for deteriorated heat transfer
and demonstrated that they provide vastly different estimates. This issue will be examined in
detail in the future. For planning purposes, the range of heat fluxes for the present tests was
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estimated to extend from one order of magnitude lower to one order of magnitude higher
than the value given by the criterion of Yang and Khartabil.
Effect of fluid type:
Tests similar to those in CO2 will be performed in Freon R-134a to facilitate the development
and validation of fluid-to-fluid scaling laws for SC heat transfer and pressure drop.
Effect of orientation:
The proposed Canadian Generation IV reactor design uses horizontal fuel channels. Although
some SCHT tests have been performed in horizontal channels (see Table 2), no systematic
investigation of the orientation effect has yet been performed. It is proposed to conduct heat
transfer and pressure drop measurements in horizontal tubes over the complete range of
conditions of interest and in both fluids. These results will be compared to corresponding
measurements in vertical tubes for an assessment of the orientation effect.
Effect of flow geometry:
Upon the completion of the circular-tube tests, rod-bundle subassemblies will be tested as
part of a systematic study of the effects of equivalent diameter, heater curvature, interelement gap size, and rod-wall gap size. A three-rod subassembly is already available for
these tests.
Effect of flow obstructions:
Nuclear fuel bundles require spacers between fuel rods and between fuel rods and pressure
tubes or containment channels. Spacers affect both pressure drop and heat transfer
significantly (Yao et al. [45], Groeneveld et al. [11]), depending on the flow blockage ratio,
their shape and their axial pitch. Spacer effects will be investigated initially by inserting
simple obstructions in a heated tube and will be extended later to include more realistic
obstructions in the rodbundle subassembly
Measurements of mean and turbulent velocity and temperature:
Traverses of Pitot-tubes and micro thermocouples will be made across different test sections
to measure the average velocity and temperature profiles. In addition, cold-wire/hot-wire
probe combinations will be used to measure simultaneously the velocity and temperature
fluctuations at selected locations, including narrow gaps. These results will be valuable in
understanding SCHT phenomena, for developing phenomenological models and for
validating SC subchannel analysis codes and CFD studies.
X. Cheng, Y.H. Yang, S.F. Huang, A simplified method for heat transfer
prediction of supercritical fluids in circular tubes, Annals of Nuclear
Energy 36 (2009) 1120–1128
Abstract
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New method for heat transfer prediction via simple correlation structure
Explicit coupling with physical phenomena
Introduction of 1 dimensionless number, the acceleration number’, to correct the deviation
of the supercritical fluids to that of conventional fluids
New correlation excludes direct dependence of the HTC on T_wall and eliminates possible
numerical instability
It gives a reasonable prediction accuracy over a wide parameter range + capable of
predicting HT behaviour in the HTD region
Heat transfer prediction
General features of heat transfer
 Strong dependence on heat flux especially as T_b approaches T_pc.
o At low heat fluxes (approaching ‘0’): HTC well predicted by Dittus-Boelter equation
o For increasing heat flux: peak shifts to lower T_b and also decreases

Ratio of HTC to HTC at zero flux
o Starts at ‘1’
o Then reaches a maximum at T_b still far below T_pc
o After maximum it decreases as T_b approaches T_pc
o Minimum at T_b around T_pc
o At T_b >> T_pc  HTC ration approaches ‘1’ again


Region where ration >1 = HT ENHANCEMENT
Region where ration <<1 = HT DETERIORATION
Selection of dimensionless parameters

The new correlation is of the same type:
o The goal is to develop a correlation for the factor ‘F’ which accounts for the deviation
of heat transfer from the Dittus-Boelter correlation.

The deviation is mainly due to 3 issues:
o Property variation
o Buoyancy effect
o Acceleration effect
1. Property variation
 Small heat flux  HTC prediction is good with Dittus-Boelter correlation
 Increasing heat flux  deviation is more significant
o Due to strong dependence of thermo-physical properties on temperature, especially
Cp
o The near wall sub-layer properties deviate from the ones in the bulk region 
deviation increases as q↑
o
Many researchers accepted the use of the effective specific heat Cp,a:
to account for the effect of property variation
o
o
: ration of effective cp to cp at T_bulk
The behaviour of cp-ratio is similar to behaviour of HTC-ratio
  abnormal behaviour could be caused by cp-variation also without any
significant change in flow structure!
  cp-ratio plays an dominant roll amongst all thermo-physical properties
2. Acceleration effect
 The acceleration effect can be characterized by the density gradient in the main flow
direction:
o
 introduction of dimensionless parameter:
number’)
(called here: ‘acceleration
 
3. Buoyancy effect

Characterized by density gradient in the radial direction:

Simplification: property parameters based on T_bulk 
(called here: ‘buoyancy number’)
Correlation structure of correction factor
 Criteria for deriving structure of correlation
o Based on dimensionless numbers  so to extend to other SC fluids
o As few as parameters
o Cover normal and HTD conditions
o No T_wall’s or parameters depending of T_w to avoid numerical instability
  correlations containing the wall temperature require an iterative solution
procedure, which might not only lead to convergence problems, but also to
numerical instability, especially near the pseudo-critical point!


Assume a HT correlation which depends on T_w:

Also the following relation has tob e fulfilled:



This equation could have 1 solution or more than 1 (see figure) 
convergence problems! The case with more than 1 solution would lead to
numerical instability
  eliminate cp-ratio
The effect of T_w and cp ratio on heat transfer will be taken into account indirectly with
other parameters: e.g. acceleration parameter (mathematically contains the heat flux and
affects T_w and thus the property variation)
Acceleration and buoyancy parameter are tightly related to each other:
o


eliminate one for simplicity and a systematic evaluation of the effect of both
parameters showed that the selction of the acceleration parameter would be more
favourable
PROPOSED FINAL CORRELATION STRUCTURE of ‘F’:
Experimental database


The correlation was validates with the experimental data of Herkenrath et al. [1] (1967).
Error analysis of test data:
New heat transfer correlation

Extensive analysis is done for the effects of the various parameters on the correction factor
o  Acceleration factor has a strong influence and is unique

Relation between F and acceleration factor  2 regions
Region 1: Small values of acc. factor  F↑ as acc. factor↑
  relationship described with ONE single curve for different experimental
conditions
o Region 2: large acc. factor  F↓ as acc. factor↑
  different curves required for different parameters combinations
 For each combination of pressure, mass flux and heat flux, the acc. factor
depends on T_fluid
Ratio thermal expansion coefficient and cp  maximum at T_pc  gives maximum
acceleration number at T_pc
o


Correlation
o
Region 1:
o
o
Region 2:
Constants determined based on the criterion that the error parameter
has its minimum value
o

Assessment of the new HT correlation
 For larger values of the acc. factor  large scatter
 70% of data points: deviation between calculated and measured Nu fall into 20% error band

Comparison correlation and measurements for the HTC-ratio
o
o

As T_b approaches T_pc: ratio ↓ to values lower dan 0.2 = HTD
The correlation predicts well the behaviour of HTD, but quantitatively it needs more
accuracy
Comparison low heat flux

o No HTD, maximum of HTC at T_pc after first a slow decrease
o Correlation of Griem shows a good agreement
o New correlation: stronger increase, but peak value well predicted
Comparison high heat flux
o

Besides the new correlation and the one from Griem, the rest fails to predict
correctly the test data near T_pc
Other test data for comparison
o
Average value:
o
o
Standard deviation:
The new correlation shows the best agreement with all the selected data (except the
test data of Xu)
ONSET OF HTD
 Smoother behaviour of T_w at HTD compared to a much sharper increase in T_W at voiling
crisis at subcritical pressure conditions  no unique definition for the onset of HTD
 Criterion if Koshizuka et al (1995)
o Koshizuka, S., Takano, N., Oka, Y., 1995. Numerical analysis of deterioration
phenomena in heat transfer to supercritical water. Int. J. Heat Mass Transfer 38 (16),
3077–3084.
o
o

 onset
According to
, the correction factor has a minimum
value as T_b approaches T_pc
o  criterion for HTD onset at a given parameter combination of pressure, heat flux an
o
mass flux:
Relationship between heat flux and mass flux at onset of HTD
o
The heat flux at onset of HTD Increases with increasing pressure
V.A. Kurganov, Yu.A. Zeigarnik, I.V. Maslakova, Heat transfer and
hydraulic resistance of supercritical-pressure coolants. Part I:
Specifics of thermophysical properties of supercritical pressure fluids
and turbulent heat transfer under heating conditions in round tubes
(state of the art), International Journal of Heat and Mass Transfer 55
(2012) 3061–3075
Abstract
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Objective: present a systematized picture of the main results of studies of HT regularities and
the specifics of the hydrodynamics of SCP fluid flows under heating in channels of a standard
form (tubes).
 These studies were conducted at different scientific centers and the results constitute the
basis of the current knowledge on the heat transfer mechanism at SCP. We assume that a
compact representation of these data, as well as the main results of their application in the
thermohydraulic design, will be useful for competent planning and comprehensive arranging
of new-generation studies for new fields of SCP coolant application.
PART I: Specifics of thermophysical properties of supercritical pressure fluids and turbulent
heat transfer under heating conditions in round tubes
PART II: Results of hydraulic and flow-sounding studies
PART III: Discussion of practical problems  methods of calculating normal and
deteriorated heat transfer using new standards for fluid heat conductivity, assessing the
“boundaries” of the normal heat transfer region, and enhancing HT to prevent its
deterioration.
Specific features of typical heat transfer modes are pointed out: normal, deteriorated and
improved HT
Discussion of the existing concept of HTD
Proposal of a simple classification of the heat transfer regimes under high heat loads 
makes it possible to determine the reasons for and assess the degree of danger of HTD
Introduction
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
Very few studies have been conducted within the range of parameters and geometric
characteristics of cooling channels typical of future SCP reactors  same as for ORCs using
waste heat!!!
Many “old” studies carried out with relatively poor developed and inadequate old-fashioned
measurements devices compared with the present state, computational base, that ensures
high-quality of experiments
During the entire second half of 20th century, intense investigations and refinement of the
thermophysical properties of substances in the near critical range of parameters took place.
At the end of the century, even for such thoroughly studied SCP coolants as water and
carbon dioxide, the necessity of considerably correcting standardized tabular data on
transfer properties, viscosity, and especially conductivity, was revealed
o A.A. Aleksandrov, A.I. Ivanov, A.B. Matveev, Dynamic viscosity of water and steam
within a wide range of pressures and temperatures, Therm. Eng. 22 (4) (1975) 77–
83
o

A.A. Aleksandrov, International tables and equations for the thermal conductivity
of water and steam, Therm. Eng. 27 (4) (1980) 235–240
o V. Vesonic, W.A. Wakeham, G.A. Olchowy, J.V. Sengers, J.T.R. Watson, L. Millat,
The transport properties of carbon dioxide, J. Phys. Chem. Ref. Data 19 (3) (1990)
763–808
 Meanwhile, the old experimental and calculated data and empiric correlations for heat
transfer under SCP that constitute almost 100% of the available reference material are based
on the old standards of properties. With the transition to the new standard IAPWS-97 for
water properties, the question arises, whether many widely used empiric correlations will
retain their workability!!!
Also a large volume of experimental, calculation and theoretical studies of heat transfer at
SCP were caused by difficulties and failures that occurred in the first phase as SCP
apparatuses were introduced in power engineering, rocket building and other advanced
technology fields.
o As revealed post facto, they stemmed from insufficient scientific exploration of the
new problem and imperfection in the scientific and methodological concepts of the
problem at the time.
o Back then, Powell’s experimental data were already known [10], obtained in the
heating of SCP oxygen under parameters typical of rocket technologies. These data
showed the possibility of an extremely deep drop in heat transfer coefficient values
in the vicinity of tb = tm under large t values.
 It is presently known that this is determined by the effect of thermal acceleration
of the flow. However, at that time, such a drop in the heat transfer coefficient was
explained as a consequence of the peculiar value and specific behavior of the
complex of oxygen, as compared to those of water and CO2 [11].

Powell’s work did not provoke any anxiety concerning the possibility of heat transfer
deterioration while operating with water and CO2. Persuasive arguments were later
obtained for the fact that the main role in these phenomena is played by the change
in fluid density over the tube cross section and along its length, rather than the
specific behavior of  or even Cp.
Thus very often it can be said even as a rule that the all the existing knowledge appears
insufficient for introducing new technical ideas without certain problems. Therefore, the long
list of works on heat transfer of SCP coolants should not make us overconfident that all of
the main problems in this field have been solved to the proper extent and that there is no
acute need to regenerate large-scale experimental and theoretical studies in advance to
create nuclear reactors with SCP fluid cooling.
Specifics of the behaviour of thermophysical properties of coolants in the
single-phase near-critical region: effect of gas admixtures on the properties
of SCP CO2 and water

Typical behaviour of thermophysical properties of SCP fluids with changes in T and enthalpy
(In SCP fluid flow and heat transfer, changes in pressure are most often small as compared to
the absolute value and do not considerably affect the properties of a fluid. Therefore, as a
rule, pressure is considered only as a parameter of the temperature (enthalpy) dependence
of fluid properties)
o
o
o
Presently: generally accepted to consider SCP fluids as single- phase media with
variable physical properties and correlate the specifics of SCP fluid heat transfer just
with peculiarities in the behavior of the thermophysical properties of such a fluid
[16].
B.S. Petukhov, Heat transfer in a single-phase medium under supercritical
conditions (survey), High Temp. 6 (4) (1968) 696–709
The attempts of some authors to consider the region of maximum specific heat
capacity at SCP as a special kind of phase transition zone have not been sufficiently
substantiated [17].
A.M. Sirota, On supercritical transitions in single-component systems,
Teploenergetika 19 (8) (1972) 73–78 (in Russian)
Nevertheless, this zone is very often called the ‘‘pseudophase transition region’’; this
has certain sense, because on both sides of this zone, the dependences of
thermophysical properties on temperature and pressure are quite different.
FIG 1-4
 Left of critical isotherm  dependence of thermophysical properties
remains qualitatively the same as that for dropletlike fluids at subcritical
pressures (liquid phase)  the specific volume, as well as specific heat and

heat conductivity, change slightly with temperature and almost do not
depend on pressure, while viscosity and, correspondingly, the Prandtl
number considerably decrease with temperature
Right of T_crit at a certain distance from it: pattern of changes in SCP fluid
properties qualitatively correspond to that of gases with variable properties
 when an increase in the  and  values with temperature and constancy
or a slight growth in Cp are observed, while the density is nearly proportional
to the pressure and inverse temperature 1/T. The Prandtl number is on the
order of unity and slightly depends on temperature and pressure.
[12] M.P. Vukalovich, S.L. Rivkin, A.A. Aleksandrov, Tables of Thermophysical Properties of Water and
Steam, Izd. Standartov, Moscow, 1969 (in Russian).
[13] S.L. Rivkin, Thermophysical Properties of Water in Critical Region, Izd. Standartov, Moscow,
1970. p.635 (in Russian).
[14] A.A. Aleksandrov, B.A. Grigor’ev, Tables of Thermophysical Properties of Waterand Steam,
second ed., MEI Publishing House, Moscow, 2006 (in Russian).
[15] V.V. Altunin, Thermophysical Properties of Carbon Dioxide, Izd. Standartov, Moscow, 1975. p.
551 (in Russian).
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When the passage through the region of the specific-heat maximum occurs, we observe a
considerable change in such an important parameter as the relative work of expansion Eq =
(pdV/ dq)p = p/(Cp), which is performed by a substance in the course of thermal expansion
against external pressure forces (see Figs. 2a and 3a).
o At t<<tm, the values of Eq are small and have an order of magnitude of 10 -2, which is
typical of dropletlike liquids.
o the right of tm, Eq increases to rather high values of 0.2–0.4, which are characteristic
of gases.
 In this connection, we propose using Eq to determine the boundaries of the
pseudophase transition region.
o Beyond these boundaries, we can consider an SCP fluid as a certain analog of a
dropletlike liquid or as a gas with variable physical properties
o It is clear from Figs. 2a and 3a that it is expedient to make such a definition using the
dependence of Eq on enthalpy h. In so doing, the region of the pseudoliquid state (I)
can be determined from the condition Eq ≤ 0.02–0.03. We designate the lower
boundary of the pseudophase transition region (II), which corresponds to this
condition, as hm0. Then, the values of hm0 will be as follows: for water, hm0 ≈ 1500
kJ/kg, and for CO2 ≈ 500 kJ/kg.
o It is expedient to designate the upper boundary of the pseudophase transition
region, from which SCP fluid can be considered as a gas (region III), as hm1, the value
of enthalpy at which Eq reaches the level E0q = R=C0p typical of a particular
substance in an ideal gas state. For water, hm1 = 2800–3000 kJ/kg (2900 kJ/kg, on
average), and that for CO2 = 750–800 kJ/kg (780 kJ/kg). Note that the difference hm1
- hm0, which is ≈1400 kJ/kg, for water and ≈280 kJ/kg for CO2, corresponds to the
heat of evaporation of water at p ≈ 8.5 MPa and that of CO2 at p ≈ 1.97 MPa (in both
cases p/ pcr ≈ 0.27). This gives additional grounds to call the range hm0 < h < hm1 the
pseudophase transition region and to apply the customary thermal-engineering
terminology to characterize the stages of SCP-flow heating, i.e., economizer-type
heating (hb < hm0), steam generation (hm0 ≤ hb ≤ hm1), and steam superheating (hb >
hm1).
In cases when the temperature parameters of the heat transfer process (tb and tw) at SCP do
not fall beyond the boundaries of the regions of the liquid (I) and gaseous (III) states, we
should anticipate that the hydraulic-resistance and heat-transfer characteristics will satisfy
the regularities obtained for viscous liquids or gases with variable physical properties. This
has been confirmed by numerous experimental data. The specific features of SCP heat
transfer manifest themselves in cases when the temperature range within which the process
takes place fully or partially falls in the pseudophase transition region (II).
While studying hydrodynamic and heat transfer processes of SCP coolants experimentally,
especially if we are dealing with the pseudophase transition region, it is of great importance
to be able to quite accurately determine the thermophysical properties of the studied fluid.
This primarily concerns the thermodynamic parameters of a state. Such a possibility depends
on the availability of sufficiently accurate and detailed tables of the properties of a
substance, as well as on properly arranging of primary measurements, which account for the
specifics of the near-critical region.
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Presently, there are few substances, for which rather detailed and reliable data on
thermophysical properties in the near-critical region are available. Water and carbon dioxide
have been investigated better than other fluids. Recently, the near-critical region of helium
was rather intensely explored. For water and pure carbon dioxide (99.9% CO2 or more)
mutually consistent tables of the thermodynamic and calorific parameters of state have been
elaborated [14,15]. These tables are based on the vast experimental material on p-V-T data,
as well as those on enthalpy and specific heat.
Transport properties ( and ) in the near-critical region remain insufficiently studied, even
for water and CO2. Recently, the data on viscosity and heat conductivity of water and steam
were considerably refined (see [7,8]). Since 1997, the new standard on the thermophysical
properties of water IAWPS-97 (briefly IF-97) has been in force. All calculations for industrial
equipment must use IF-97 data [14]. The new tables on the heat conductivity of water have
constructed with allowance for the existence of peak k values in the pseudophase transition
region, which were recognized in many experiments (Fig. 2). The region of elevated  values
encompasses a rather large interval of pressures and enthalpies. As is clear from Fig. 2, the
presence of  peaks considerably changes the value of the Prandtl number in the vicinity of
tm. In the range of p/pcr from 1.02 to 1.12, the decrease in the Pr(tm) value is by a factor of
~2.5–1.9; in the range of p/pcr from 1.2 to 1.35, by ~1.5–1.35.
It is easy to obtain, with the use of the known McAdams formula [18], for example, that such
changes in the Prandtl number lead to a considerable increase in the calculated heat transfer
coefficients, in the vicinity of tm, by a factor of ~1.75–1.18. The consequences of these
changes in the standard tabular thermophysical properties on heat transfer calculations will
be discussed in detail in the third part of the paper.
[18] W.H. MacAdams, Heat Transmission, McGraw-Hill, New York, 1942
Analysis of the experimental data on the heat conductivity of CO2 conducted in [15] also
shows the presence of  peaks in the pseudocritical region. For CO2, however, the
discrepancy in the experimental  values in these peaks is very large. Because of this, the
interpolating equation for heat conductivity of CO2, which was used in constructing the
tables [15], does not allow for the presence of  peaks in the near-critical region. Proceeding
from the well-acknowledged experimental data, the authors of [9], which was published later
than [15], consider the presence of peaks on (t) isobars doubtless and propose coordinating
correlations to introduce proper corrections to the canonic heat conductivity values of SCP
CO2. Fig. 4a shows the (t) values calculated with the use of correlations from [9] and [15] at
p = 7.7 and 9.0 MPa (p/pcr = 1.05 and 1.23, respectively). It is clear that allowance for the
excesses in k at pressures close to critical also considerably changes the commonly used
Prandtl number values in the vicinity of tm, Fig. 4b. For example, with p/pcr = 1.05, the
Prandtl number decreases by a factor of ~1.8, and with p/pcr = 1.23, by ~15%.
When the hydraulic resistance coefficients and velocity fields are determined experimentally
by measuring the distribution of static and dynamic pressures in the flow, it is necessary to
exactly know the fluid density under the conditions of the experiment. In this connection, we
have always paid special attention to the problem of determining the actual state and
density of an SCP fluid at temperatures corresponding to the pseudophase transition region.
We consider this problem as it relates to carbon dioxide, which was used in our experiments
as the working substance. Naturally, the main conclusions are applicable to the other SCP
fluids.
Experimental data on regularities of turbulent heat transfer in tubes under
SCP conditions
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Large amount of experimental works on HT to near-critical pressure coolants in tubes and
channels  Reviews:
o V.S. Protopopov, Study of heat transfer under turbulent flow of
supercriticalpressure carbon dioxide. Part 1. Heat transfer and hydraulic resistance
under turbulent flow of a supercritical-pressure fluid in tubes (analysis of the
stateof-the-art), Report B376948, MEI, Moscow, 1975
o I.L. Pioro, R.B. Duffey, Heat Transfer and Hydraulic Resistance at Supercritical
Pressures in Power Engineering Applications, ASME Press, New York, 2006. p. 334
o B.S. Petukhov, Heat transfer in a single-phase medium under supercritical
conditions (survey), High Temp. 6 (4) (1968) 696–709
o B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable
Physical Properties, Advances in Heat Transfer, Vol. 6, Academic Press, New York,
1970. pp. 503–564.
o R.C. Hendricks, R.J. Simoneau, R.V. Smith, Survey of heat transfer to nearcritical
fluids, Adv. Cryogenic Eng., vol. 15, Plenum Press, USA, 1970. pp. 197–237.
o G.V. Alekseev, A.M. Smirnov, Heat transfer in turbulent flow of liquids at
supercritical pressures in channels, FEI, Obninsk, 1973, p. 83 (in Russian).
o W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int. Heat Trans
fer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392.
o J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical
pressure, in: S. Kakacˇ, D.B. Spalding (Eds.), Turbulent Forced Convection in
Channels and Bundles, Vol. 2, Hemisphere, Washington, 1979, pp. 563–612.
o A.F. Polyakov, Heat transfer under supercritical pressures, Adv. Heat Transfer 21
(1991) 1–53.
o V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical
pressure of the coolant. Part I: specifics of thermophysical properties,
hydrodynamics, and heat transfer of the liquid. Regimes of normal heat transfer,
Therm. Eng. 45 (3) (1998) 177–185.
o V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical
pressure of the coolant. Part II: Heat transfer and friction at high heat fluxes. The
influence of additional factors. Enhancement of deteriorated heat transfer, Therm.
Eng. 45 (4) (1998) 301–310.
o S. Yoshida, H. Mori, Heat Transfer to Supercritical Fluids Flowing in Tubes. SCR–
2000, University of Tokyo, Japan, 2000.
H2O, CO2, O2, N2, H, NH3, He and different kind of refrigerants
To simulate HT in the pseudoliquid region and in the vicinity of T_pc  refrigerants can be
applied! (CO2 not because at t=20°C h is already > hmo)
Investigations revealed the extraordinary complexity of the regularities of SCP HT!
o Sorting of HT regimes depending on heat load and thermodynamic state of a fluid:
normal, deteriorated and improved HT
o B.S. Petukhov, Heat transfer in a single-phase medium under supercritical
conditions (survey), High Temp. 6 (4) (1968) 696–709
o

B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable
Physical Properties, Advances in Heat Transfer, Vol. 6, Academic Press, New York,
1970. pp. 503–564
NORMAL HT
o  corresponds qualitatively with the existing ideas on turbulent HT with constant or
slightly variable physical properties
o For boundary conditions (q_w = cte, like in most experiments), for any h_in, T_w
changes MONOTONIC along the heated tube (common for linear HT problems)
o At a small distance from the inlet (x/d>20-40)  stabilization of the HT intensity
 Depends only on the local parameters and hardly on the temperature
(enthalpy) at the inlet and the geometric characteristics of the inlet
 For identical q_w, d and 𝜌𝑢
̅̅̅̅  stabilized values of T_w are described by a
single monotonic curve IIRESPECTIVE of h_in
 See FIG 8 for q_w < 300 kW/m²
 See FIG 9 curve 1
[36] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow in tubes at
supercritical pressure, Heat Transfer-V, in: Proceedings of the fifth All- Union conference on heat
mass transfer, Nauka i Tekhnika, Minsk, Belarus’, 1976, 1(1) pp. 261–269 (in Russian).
[37] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under water flow at
supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–2349.
[38] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the
turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89.
o

Effect of the starting section remains qualitatively the same as for constant fluid
properties  within 15-20 gages of the tube there is a relative decrease in HT rate
caused by the formation of the temperature field in the liquid flow.
 B.S. Petukhov, V.S. Protopopov, V.A. Silin, Experimental investigation of
worsened heat transfer conditions with the turbulent flow of carbon dioxide
at supercritical pressure, High Temp. 10 (2) (1972) 304–310
o The influence of the type of boundary condition (e.g. a change in q_w value along
the tube) and of the wall roughness remains in within the limits of typical developed
turbulent flows
o Effect of variable properties on the local characteristics of normal HT is significant 
HTC and Nu numbers differ with a factor 1.5 to 0.2 compared to the correlations for
constant properties
 The different between Nu_b and Nu_b,0 (ref cte properties) are very high in
the pseudophase transition region (< or > 1)!
 Nu_b / Nu_b,0 > 1 for T_b < T_pc and T_w≈T_pc: condition of increased heat
capacity in the wall layer compared to the flow core.
 Nu_b / Nu_b,0 << 1 for T_b ≈ T_pc and T_w>>T_pc: decrease in density and
c_p in the wall layer flow
NORMAL HT regimes  meets requirements for reliable and safe operation of SCP HX
o
o
BUT, this regime is limited to relative LOW HEAT LOADS 𝑞/ 𝜌𝑢
̅̅̅̅
At HIGH heat loads  more complex HT  unfavourable phenomena e.g. HTD =
sharp reduction in HTC at certain limited sections of the tube!
 For boundary condition q_w = cte  peaks in T_w appear (FIG 8 and 9) 
superheating of the wall can occur  dangerous to wall strength
 A significant HTD due to a small increase in heat load (1-10%) is
UNFAVOURABLE


M.E. Shitsman, Impairment of heat transmission at supercritical pressures,
High Temp. 1 (2) (1963) 237–244
o  THE DETERMINATION OF THE REASONS AND CONDITIONS AT WHICH THE
TRANSITION O THE REGIME OF HTD AND THE OOCURS ARE THE MOST ACUTE
PROBLEMS!!!
HEATTRANSFER DETERIORATIONS
o Numerous studies: depending on d, mass flow rate and h_in  non-monotonic T_wdistribution under heating of tube according to the q_w=cte law can originate near
the tube inlet (“inlet-peaks in T_w) + in the range of flow enthalpies h_b which
correspond to the pseudophase transition
o
Inlet peaks in T_w for different dia (5.7-32.2mm) for upward flow for moderate mass
flow rates (200-1000 kg/m²s) for water and CO2
 S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under
water flow at supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–
2349.
 [43] M.E. Shitsman, Peculiarities of a temperature regime in tubes under
supercritical pressures, Teploenergetika 15 (5) (1968) 57–61 (in Russian).




[44] Yu.V. Vikhrev, Yu.D. Barulin, A.S. Kon’kov, Study of heat transfer in
vertical tubes under supercritical pressures, Therm. Eng. 14 (9) (1967) 116–
119.
[45] I.S. Alferov, R.A. Rybin, B.F. Balunov, Heat transfer under turbulent
flow of water in vertical tubes with significant effect of free convection,
Teploenergetika 16 (12) (1969) 66–70 [in Russian]..
[46] P.J. Bourke, D.J. Pulling, L.E. Gill, W.H. Denton, Forced convective heat
transfer to turbulent CO2 in the supercritical region, Int. J. Heat Mass
Transfer 13 (8) (1970) 1339–1348.
[47] I.I. Belyakov, L.Yu. Krasyakova, A.V. Zhukovskii, N.D. Fefelova, Heat
transfer in vertical and horizontal tubes under supercritical pressure,
Teploenergetika 18 (11) (1971) 39–43 (in Russian).



[48] J.N. Ackerman, Heat transfer during pseudoboiling of water in
supercritical region in smooth and finned tubes, Trans. ASME J. Heat
Transfer 3 (1970) 490–498.
 [49] N.P. Ikryannikov, B.S. Petukhov, V.S. Protopopov, Calculation of heat
transfer in single phase near-critical region under viscous-inertialgravitational flow, High Temp. 11 (5) (1973) 949–955.
 [50] D.J. Brassington, D.N.H. Cairns, Measurements of forced convective
heat transfer to supercritical helium, Int. J. Heat Mass Transfer 20 (8)
(1977) 207– 214.
 [51] V.A. Bogachev, V.M. Eroshenko, L.A. Yaskin, Heat transfer in upward
flow of supercritical-pressure helium in a transition-flow regime in a round
tube, High Temp. 21 (3) (1983) 611–619.
 [52] M.J. Watts, C.T. Chou, Mixed convective heat transfer to supercritical
pressure water, Proc. 7th Int. Heat Transfer Conf. Munchen 3 (1982) 495–
500.
 [53] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance
and heat transfer in vertical heated tubes under supercritical pressure of a
coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed
convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in
Russian].
 [54] V.A. Kurganov, Heat transfer of gases in turbulent flow in tubes,
Teploenergetika (Thermal Engineering) 39 (5) (1992) 2–9 (in Russian).
o These peaks are more typical for tubes with larger diameter (16-32.2mm)
 Located at inlet section of 0≤x/d≤30-50 (initial thermal section)
o Inlet peaks recognized for different values of liquid enthalpy at inlet h_in up to h_m
and more
 [53] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance
and heat transfer in vertical heated tubes under supercritical pressure of a
coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed
convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in
Russian].
o The T_w in the inlet peaks can be higher or lower than T_pc
o As a rule: DOWNSTREAM of the inlet maximum T_w  the section of normal or
increased HT was observed (especially for h_in<<h_m)
o Very long tube with h_b>h_m0  secondary HTD occurs  increase in h_in  2
regions of HTD (2 maxima in T_w or even more)
DOWNWARD AND HORIZONTAL FLOW (under same conditions as UPWARD flow)
o No inlet peaks of T_w!  inlet peaks in T_w originate as a result of the considerable
effect of buoyancy forces on turbulent flow in the region of the initial thermal
section!
Evolution of T_w at the initial section of large diameter tubes with ↑ heat flow rate q (rest of
conditions equal)  complex nature depending of values of 𝜌𝑢
̅̅̅̅ and h_in
o Typical if q_w ↑:
 T_w peak shifts to inlet of the tube


For small mass flows (200 kg/m²s) the inlet peak can degrades as q_w↑ 
T_w distribution monotonic again (like at small heat flow rates)
 [49] N.P. Ikryannikov, B.S. Petukhov, V.S. Protopopov, Calculation
of heat transfer in single phase near-critical region under viscousinertial-gravitational flow, High Temp. 11 (5) (1973) 949–955.
 [53] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic
resistance and heat transfer in vertical heated tubes under
supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent
heat transfer under mixed convection in vertical tubes, IVTAN,
Moscow, 1989, pp. 95–160 [in Russian].
HTD for h_b in the pseudophase transition is most typical
o This KIND of HTD (for h_b in pseudophase transition) can occur in tubes of different
DIAMETER!
 S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow
in tubes at supercritical pressure, Heat Transfer-V, in: Proceedings of the
fifth All-Union conference on heat mass transfer, Nauka i Tekhnika, Minsk,
Belarus’, 1976, 1(1) pp. 261–269 (in Russian).
 B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow
resistance in the turbulent pipe flow of a fluid with near-critical state
parameters, High Temp. 21 (1) (1983) 81–89.
 V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and
heat transfer in vertical heated tubes under supercritical pressure of a
coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed
convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in
Russian].
 Yu.D. Barulin, Yu.V. Vikhrev, B.V. Dyadyakin, et al., Heat transfer in
turbulent flow of supercritical-state water in vertical and horizontal tubes,
Inzh. Fiz. Zhurnal (Eng. Phys. Journal) 20 (5) (1971) 929–930 (in Russian).
 REVIEWS
 V.S. Protopopov, Study of heat transfer under turbulent flow of
supercriticalpressure carbon dioxide. Part 1. Heat transfer and hydraulic
resistance under turbulent flow of a supercritical-pressure fluid in tubes
(analysis of the stateof-the-art), Report B376948, MEI, Moscow, 1975
 I.L. Pioro, R.B. Duffey, Heat Transfer and Hydraulic Resistance at
Supercritical Pressures in Power Engineering Applications, ASME Press,
New York, 2006. p. 334
 B.S. Petukhov, Heat transfer in a single-phase medium under supercritical
conditions (survey), High Temp. 6 (4) (1968) 696–709
 B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with
Variable Physical Properties, Advances in Heat Transfer, Vol. 6, Academic
Press, New York, 1970. pp. 503–564.
 R.C. Hendricks, R.J. Simoneau, R.V. Smith, Survey of heat transfer to
nearcritical fluids, Adv. Cryogenic Eng., vol. 15, Plenum Press, USA, 1970.
pp. 197–237.

o
G.V. Alekseev, A.M. Smirnov, Heat transfer in turbulent flow of liquids at
supercritical pressures in channels, FEI, Obninsk, 1973, p. 83 (in Russian).
 W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int.
Heat Trans fer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392.
 J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at
supercritical pressure, in: S. Kakacˇ, D.B. Spalding (Eds.), Turbulent Forced
Convection in Channels and Bundles, Vol. 2, Hemisphere, Washington,
1979, pp. 563–612.
 A.F. Polyakov, Heat transfer under supercritical pressures, Adv. Heat
Transfer 21 (1991) 1–53.
 V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical
pressure of the coolant. Part I: specifics of thermophysical properties,
hydrodynamics, and heat transfer of the liquid. Regimes of normal heat
transfer, Therm. Eng. 45 (3) (1998) 177–185.
 V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical
pressure of the coolant. Part II: Heat transfer and friction at high heat
fluxes. The influence of additional factors. Enhancement of deteriorated
heat transfer, Therm. Eng. 45 (4) (1998) 301–310.
 S. Yoshida, H. Mori, Heat Transfer to Supercritical Fluids Flowing in Tubes.
SCR–2000, University of Tokyo, Japan, 2000.
For higher MASS FLOW RATES (1500 kg/m²s or more for CO2 and H2O) in SMALL
DIAMETER (up to ~10mm) an INCREASE in HEAT FLOW RATE q_w  HTD regardless
of FLOW DIRECTION!! (FIG 9)
 Here: ARCHIMEDES EFFECT is SECONDARY, but it can causes a considerable
difference in local HT under upward and downward flows as well as a large
temp maldistribution over the circumference in a horizontal tube, as well as
a change in the location, height and configuration of the T_wall peak under
equal heating conditions!!!
 THIS KIND of HTD occurs at h_in < and > than h_m (FIG 9)
 Hydraulic measurements (SEE PART II)  in HTD regimes like in THIS CASE:
pressure drop p_i due to flow acceleration is considerably higher than the
friction resistance p_
  the flow is significantly gradient in nature in contrast to the
normal HT regimes (FIG 12)
B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent
pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89

MODERATE MASS FLOW RATES (200<𝜌𝑢
̅̅̅̅<1000 kg/m²s for CO2 and H2O) + h_b in
PSEUDOPHASE TRANSITION
o HT depends on FLOW ORIENTATION in gravity field (FIG 8)
 As q_w ↑  greatest danger for HTD is for UPWARD FLOW
 Same conditions:
 Downward: strong T_w peaks
 Upward: T_w distributions remains monotonically
o  local HTC differ a lot!
o LOW values of h_in<<h_m in LONG tubes + rather LARGE diameter + UPWARD FLOW
 TWO T_w peaks = inlet peak + peak in region of PSEUDOPHASE TRANSITION


[44] Yu.V. Vikhrev, Yu.D. Barulin, A.S. Kon’kov, Study of heat
transfer in vertical tubes under supercritical pressures, Therm. Eng.
14 (9) (1967) 116–119.
 [45] I.S. Alferov, R.A. Rybin, B.F. Balunov, Heat transfer under
turbulent flow of water in vertical tubes with significant effect of
free convection, Teploenergetika 16 (12) (1969) 66–70 [in Russian]..
 [46] P.J. Bourke, D.J. Pulling, L.E. Gill, W.H. Denton, Forced
convective heat transfer to turbulent CO2 in the supercritical
region, Int. J. Heat Mass Transfer 13 (8) (1970) 1339–1348.
 [47] I.I. Belyakov, L.Yu. Krasyakova, A.V. Zhukovskii, N.D. Fefelova,
Heat transfer in vertical and horizontal tubes under supercritical
pressure, Teploenergetika 18 (11) (1971) 39–43 (in Russian).
 [48] J.N. Ackerman, Heat transfer during pseudoboiling of water in
supercritical region in smooth and finned tubes, Trans. ASME J.
Heat Transfer 3 (1970) 490–498.
 As h_in increases  TWO region come together  T_w distribution show
two peaks (or more) each following directly one another
 = TYPICAL for CO2, because h_in<<h_m0!!!
 M.E. Shitsman, Peculiarities of a temperature regime in tubes under
supercritical pressures, Teploenergetika 15 (5) (1968) 57–61 (in
Russian).
 V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance
and heat transfer in vertical heated tubes under supercritical
pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer
under mixed convection in
vertical tubes, IVTAN, Moscow, 1989,
pp. 95–160 [in Russian].
 A. Watson, The influence of axial wall conduction in variable
property convection-with particular reference to supercritical
pressure fluids, Int. J. Heat Mass Transfer 20 (1) (1977) 65–71
 B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int.
Heat Transfer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–
392.
HORIZONTAL AND INCLINED tubes with MODERATE MASS FLOW RATES
o In region of HIGH HEAT LOADS q_w/𝜌𝑢
̅̅̅̅  HT has some specific features!
 Due to presence of SECONDARY FREE-CONVECTIVE FLOWS + STEADY
DENSITY DISTRIBUTION NEAR THE UPPER GENERATRIX
 [59] M.E. Shitsman, The effect of natural convection on
temperature conditions in horizontal tubes at supercritical
pressures, Therm. Eng. 13 (7) (1966) 69–75.
 [60] A.V. Zhukovskii, L.Yu. Krasyakova, I.I. Belyakov, N.D. Fefelova,
Heat transfer in a horizontal tube at SCP, Energomashinostroenie 2
(1971) 23–26 (in Russian).
 [61] V.M. Solomonov, V.A. Lokshin, Temperature conditions and
heat transfer in horizontal and inclined tubes of steam generators


at supercritical pressure under conditions of joint free and forced
convection, Therm. Eng. 22 (7) (1975) 74–77.
 [62] J.A. Adebiyi, W.B. Hall, Experimental investigation of heat
transfer to supercritical pressure carbon dioxide in a horizontal
tube, Int. J. Heat Mass Transfer 19 (8) (1976) 715–720.
 Region of HTD localized NEAR the UPPE GENERATRIX of the tube, while NEAR
the LOWER GENERATRIX  HIGH level of HT!!!
 Temperature difference between upper and lower generatrix can reach
200K for CO2 and H2O!!!
One of the main difficulties on generalizing the experimental data on HTD is the fact that all
T_w(h_b) curves corresponding to different h_in (the rest of the conditions equal) have an
individual pattern which coincide with one another only at h_b ≈ h_m1!
o BUT, for water and possible refrigerants, a certain REGULARITY exists in the HTD in
certain enthalpy regions within h_m0<h_b<h_m, regardless of h_in, if a fluid is
heated from the tube section with h_in<<h_m0 (the tube has an economizer section
of considerable length)
 H. Komita, S. Morooka, S. Yoshida, H. Mori, Study on the heat transfer to
the supercritical water cooled power reactor development, NURETH-10,
Seoul, Korea, October 5–9, 2003
o EXPLANATION REGULARITY: in these cases, a flow enters the “vapour-generation”
section with ± appr the same hydrodynamic structure, because t_w<t_pc (in the
heating zone) and the scale of variation of density and other physical properties over
the tube cross section and length is relatively small.
STUDIES about HTD:
o THE ULTIMATE HEAT LOAD VALUE, where the criteria of normal HT and qualitative
description of the HTD process are maintained, depend strongly on ADDITIONAL
FACTORS that do NOT affect Normal HT!!
 Conditions at the tube inlet
 Wall roughness
The data in [65] also showed that wall roughness DELAYED the transition to
HTD regime  possible to increase the allowable heat load level by 15-20%!!
 [65] H. Tanaka, N. Nishiwaki, M. Hirata, A. Tsuge, Forced
convection heat transfer to fluid near critical point flowing in
circular tube, Int. J. Heat Mass Transfer 14 (6) (1971) 739–750
 BUT, as q_w increased, eventually HTD occurred and the increase in
T_W was even more sharply and to a greater value!
 The used test section with uncontrolled roughness of
commercial-grade smooth tubes COULD BE the source of VARIATION
between the experimental data on HTD
 FEW CASES where hydraulic resistance was preliminarily measured
under adiabatic conditions  considerable influence of wall
roughness on hydraulic resistance was recognized in the same rang
of Re numbers, within which the abnormal HT data were than later
obtained.


E.g. Water: salt deposits and fouling on heated walls  affect HT
deterioration.
o [67] G.V. Alekseev, V.A. Silin, A.M. Smirnov, V.I. Subbotin,
Study of thermal conditions on the wall of a pipe during the
removal of heat by water at asupercritical pressure, High
Temp. 14 (4) (1976) 683–687.
o [68] M.E. Shitsman, L.S. Midler, A.V. Firsov, Temperature
regime of tubes and critical phenomena in solutions, Heat
Transfer-IV, in: Proceedings of the fourth All-Union
conference on heat mass transfer, Nauka i Tekhnika, Minsk,
Belarus’, 1972, 2(1) pp. 30–36. (in Russian).
o [69] O.K. Smirnov, Yu.P. Michurov, Study of
thermohydraulic characteristics of a steam generating tube
with formation of a solid phase of admixtures in
supercritical parameters medium, Teploenergetika (Therm.
Eng.) 22 (7) (1975) 83–86 (in Russian).
 Also water chemistry of the coolant (admixtures) can change the
conditions and the pattern of HTD.
o E.g. dissolved gases in the SCP fluid  convert the solution
in a subcritical state  the gas admixture can provoke a
SHARP HTD!! (compared to HTD in the SC region) 
resembles the HT CRISIS under subcooled liquid boiling.
o E.g. CO2-N2 mixture (0.5 to 4.0 mol% N2 concentration)
 B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, A.G.
Kaptil’nyi, Dissolved gas effect on heat transfer
under supercritical-pressure carbon dioxide
turbulent flow in a tube, High Temp. 23 (4) (1985)
742–747.
Forced pressure oscillations (flow rate) due to pump operation
Artificial damping of pressure oscillations downstream of the pump under
water heating in a rising tube  sharp change in temperature regime of the
wall tube  HTD!!
K. Yamagata, K. Nishikawa, S. Hasegava, T. Fujii, S. Yoshida, Forced
convective heat transfer to supercritical water flowing in tubes, Int. J. Heat
Mass Transfer 15 (12) (1972) 2575–2593
For HT of SC CO2 in a smooth tube (d=6mm and 𝜌𝑢
̅̅̅̅ = 1180-2350 kg/m²s)
[65]
 For a gear-type pump  pressure oscillations generated an increase
in heat flow rate in the deteriorated HT regimes (T_w same level as
in [38,66]) + shift of the T_w maximum to the region h_b>h_m
 [38] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer
and flow resistance in the turbulent pipe flow of a fluid with nearcritical state parameters, High Temp. 21 (1) (1983) 81–89.


[66] B.S. Shiralkar, P. Griffith, Deterioration in heat transfer to
fluids at supercritical pressure and high heat fluxes, Trans. ASME J.
Heat Transfer 91 (1) (1969) 27–36.
o ALL these factors should be taken into account when comparing the QUANTITATIVE
DATA on HTD in literature!!!
During transition to HTD regime  excitation of a self-oscillating process is often observed!
o Acoustic instability on the flow is also a feature of heating turbulent SCP fluids in
tubes especially of small diameter with h_b<h_m0 leads often to resonance
thermoacoustic oscillations (TAOs) of pressure in tubes at a natural frequency and
overtones
o SPAN of pressure oscillations in the antinodes of standing waves can be very high (up
to 50% of the absolute pressure)  DANGEROUS FOR THE TUBE STRENGTH!
 E. Stewart, P. Stewart, A. Watson, Thermo-acoustic oscillations in forced
convection heat transfer to supercritical pressure water Int. Journal of Heat
and Mass Transfer Vol. 16, pp. 257-270. 16 (2) (1973) 257–270.
o TAOs contribute to HT ENHANCEMENT!
 V.V. Sevastyanov, A.T. Sinitsin, F.L. Yakaitis, Study of heat exchange
process in supercritical region of parameters under conditions of highfrequency pressure oscillations, High Temp. 18 (3) (1980) 433–439
  also called regimes of enhanced HT [16] or HT with pseudoboiling
 [16] B.S. Petukhov, Heat transfer in a single-phase medium under
supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709
o Conditions for TAOs development:
 Existence of turbulent mode of the flow
 Considerable subcooling of the fluid relative to T_pc
 A level of heat flow rates measuring T_w>T_pc
 an ultimate change in density and other properties of the fluid in the wall
layer
o LOWER and UPPER boundaries of the acoustic instability as function of the heat flow
rate  V.I. Vetrov, V.A. Gerliga, V.G. Razumovskii, Experimental study of
thermoacoustic oscillations in heated channels under supercritical pressures of
water, Voprosy Atomnoi nauki i Tekhniki, Ser. Dinamika Yadernykh
Energeticheskikh Ustanovok (Problems of Nuclear Sci. Techn., Ser. Dynamics of
Nuclear Power Plants), 2 (12) (1977) pp. 51–57 (in Russian).
o Attemps for theoretical models for this phenomenon 
 [72] V.V. Sevastyanov, A.T. Sinitsin, F.L. Yakaitis, Study of heat exchange
process in supercritical region of parameters under conditions of highfrequency pressure oscillations, High Temp. 18 (3) (1980) 433–439.
 [74] V.I. Vetrov, Mechanism of thermoacoustic instability under forced
flow of a coolant, Izv. AN SSSR, ser. Energetika i Transport (J. USSR Acad. of
Sciences, ser. Energy and Transport), No. 1, (1987), pp. 119–127 (in
Russian).
o E.g. TAO development in tests on HTD in steam-generating tubes
 Also in V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance
and heat transfer in vertical heated tubes under supercritical pressure of a

coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed
convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in
Russian].
For CO2 (p = 7.7 and 9 MPa, tin = 20°C) 8 mm diameter tube  TAO at
̅̅̅̅=1000 and 1350 kg/m²s with the transition to HTD regime
𝜌𝑢
 Freq. of 50 an 350 Hz – amplitude = 0.5MPa!!
 distribution of static pressure
[38] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the
turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89


As h_out approached h_m1  TAO stopped
No significant effect of these oscillations in HT were recognized, only
in a 22.7mm tube TAO improved the HT in the inlet peak region of
T_w!
The nature and classification of deteriorated heat transfer regimes


GOLDMAN (1954) and SHLYKOV (1971)
o Reason for turbulent HTD: localization of the T_pc region in certain zones of the flow
 causes difficulties in HT through the zone with higher values of  and c_p.
o Proposed generalized variables, but local analysis of the problem based on the this
concept appeared UNSUCCESSFUL
POLYAKOV (1973-1975)
o
o
o
In non-isothermal fluid flow with variable properties  turbulent HT changes due to
the action of:
 Mass forces induced by thermal acceleration of the flow and
 Archimedes forces originated due to a non-uniform density distribution over
the flow cross section
The assumed that these forces act immediately on the intensity of turbulent HT.
̅̅̅̅̅ in the equation of the turbulent energy balance (in the term
The correlation 𝜌′𝑢′
𝜕𝑢
̅̅̅̅̅
𝜌′𝑢′ [±𝑔 + 𝑢 ( )]) was considered an agent of this effect  which can considerably
𝜕𝜒

changer the local parameters of turbulent transfer, fluid flow and heat transfer. This
happens PRIOR to the distortion of the averaged flow according to the equation of
motion.
o On this basis, Polyakov did estimations for the boundaries of the effect of mass
forces in friction and heat transfer in tubes and expressions were presented for
specific criteria of the thermal acceleration and the effects of Archimedes forces (SEE
PART III)
 A.F. Polyakov, Mechanism and limits on the formation of conditions for
impaired heat transfer at a supercritical coolant pressure, High Temp. 13
(6) (1975) 1119–1126
At the same time (1970-1971)  calculations and experimental works that demonstrated
that in DHT regimes COMPLETE REARRANGEMENT of the AVERAGED FLOW takes place 
velocity profile becoming M-SHAPED!
o HTD develops at the phase of velocity profile rearrangement when the zone with
𝜕𝑢
very small values of the product 𝜏𝑡 (𝜕𝑦) appears, which is responsible for the

gradient generation of turbulent energy = BARRIER LAYER formed in the flow
o [76] C.A. Bankston, D.M. McEligot, Turbulent and laminar heat transfer to gases
with varying properties in the entry region of circular ducts, Int. J. Heat Mass
Transfer 13 (2) (1970) 319–344.
o [77] P.J. Bourke, D.J. Pulling, Experimental explanation of deterioration in heat
transfer to supercritical carbon dioxide, Paper ASME, No. 71-HT-24, 1971, p. 7
ESTIMATE describing the above ideas of POLYAKOV
o
o
Prandtl and Boussinesq hypotheses  correlation

Expression
for
turbulent
momentum
transfer
coefficient:
 a decrease in shear stresses in the wall region of the flow  reduction in
turbulent transfer and heat transfer
 THE FACTORS THAT DETERMINE THESE TRANDS are (HALL AND JACKSON):
1. THERMAL ACCELARATION (for SCP fluid and gas)
2. LIFTING ARCHIMEDES FORCES (BUOYANCY FORCES) (for vertical tubes)
W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int.
Heat Transfer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392
  the pressure gradient that provides fluid motion is determined NOT ONLY
by FRICTION RESISTANCE (as in case of constant properties fluid), but also
ADDITIONAL DYNAMIC FACTORS that appear due to DENSITY CHANGES over
the tube cross section and along its length:

With ‘+’ in gravitational term = upward flow and ‘-‘ = downward flow

With:
With:
inertial flow resistance


= coefficient of
In
, in the inertial resistance of nonisothermal SCP flows  the momentum coefficient S_b and its
changes along the tube length play an ACTIVE ROLE!
 actual 𝜉𝑖 -values can considerably differ from mass-averaged
estimates (SEE PART II)
The
solution
to
the
equation
of
motion
(
) of an axisymmetric
VERTICAL flow, in the wall region (R1), where convective terms
can be neglected 
Where ‘C’ = averaged value of the ratio
in the interval from R to 1
o Roughest estimate  C≈0.5
 asymptotic formula:
o
o

The smaller K_g/K_i, the closer to reality
Analysis of the results of hydraulic and sounding
measurements (SEE PART II) showed that in NORMAL HT
regimes, the shear-stress profile satisfied the following
condition:
CONCLUSION: from the correlations it is clear that for K_ig>>1 
and
Eq.
and
 radical changes in the
shear-stress profiles and in the values of the turbulent transfer
coefficient  which can lead to HTD!!!!


The experimental data on HT in VERTICAL heated tubes in literature showed that the general
behaviour of the T_w at boundary condition q_w≈cte + rather high heat loads q_w/𝜌𝑢
̅̅̅̅ (thus
beyond the limits of NHT), DEPEND on the PARAMETER:
 this parameter specifies the potential scale of BUOYANCY EFFECT on the DYNAMICS of a
fluid IN THE WALL LAYER.
CLASSIFICATION OF DHT REGIMES UNDER MIXED CONVECTION IN VERTICAL HEATED TUBED
OF SCP FLUIDS  6 typical groups
o
Schematic presentation of heat transfer in vertical tubes at q_w≈const and high heat
loads depending on the scale of Archimedes forces
o




Supposes tubes that are sufficiently long L/d~100 or more!
It was recognized that the boundary values of
for different groups of
regimes, which were specified from the experimental data, form a straight line with
equidistant nodes in logarithmic coordinates  allows to determine the number of the
group Ngr into which the regime under consideration falls using the simple formula:
 It is advised to perform calculations with an accuracy of 0.1
If Ngr ↑  effect of Archimedes forces ↓and the influence of thermal acceleration ↑
In very long tubes (thermodynamic state can change considerably along the tube)  the HT
regime can pass into another group in accordance with the local values of parameter
Also possible to introduce the buoyancy effect parameter F_g (= the value which will give an
idea on the relative role of this effect on the HT pattern)
o Consider between 3rd and 4th group an equal effect of buoyancy and thermal
acceleration + consider in groups 1 and 2: F_g 1, while in regimes of 5th and 6th
groups (where thermal acceleration rules): F_g  0
o
 simple expression for buoyancy effect parameter F_g:
THIS EXPRESSION CAN BE USED IN THE DEVELOPMENT OF CORRELATIONS FOR
FORECASTING HTD
Conclusions

New recent data on specifics of the behaviour of thermophysical properties
o Advised to DIVIDE the entire region of possible states in 3 typical zones
 Pseudoliquid state
 Pseudophase transition: with in here T_pc
 Pseudogas state (superheated vapour)
o


Determine boundaries of these zones via
parameter (characterises the
specific work of fluid expansion per unit of heat added
o The new standards contain sharp changes in (T) and Pr(T)  must be taken into
account when using the known correlations and constructing new formulas
o Effect that small admixtures of dissolved gases have on (T) and T(h) dependences
and other properties of SCP water and CO1 are considered
The reason for HTD is WEAKENING of turbulent transfer in the wall region caused by a
DECREASE in SHEAR STRESS due to the effect of THERMAL ACCELERATION and ARCHIMEDES
FORCED (buoyancy – in vertical tubes)  this concept is in agreement with Hall and Jackson’s
ideas
Results of experimental studies of HT regularities for q_w≈cte considered  existing data
base on DHT was classified (for high heat loads q_w/𝜌𝑢
̅̅̅̅) using the PARAMETER
which characterises the scale of the effect of ARCHIMEDES FORCES on turbulent flow and
heat transfer.
 6 representative groups were distinguished with this parameter + corresponding typical
modes of T_w behaviour within these groups were described.
V.A. Kurganov, Yu.A. Zeigarnik, I.V. Maslakova, Heat transfer and
hydraulic resistance of supercritical-pressure coolants. Part II:
Experimental data on hydraulic resistance and averaged turbulent
flow structure of supercritical pressure fluids during heating in round
tubes under normal and deteriorated heat transfer conditions,
International Journal of Heat and Mass Transfer 58 (2013) 152–167
Abstract




Experimental results on hydraulic and friction resistances in round tubes under adiabatic and
heating conditions
Experimental technique + results sounding studies under normal and deteriorated HT
Possible errors in hydraulic resistance calculations using 1-D flow model are pointed out
Analysis between HTD and significant changes in the averaged flow structure (due to fluid
thermal acceleration and Archimedes forces effects)
Introduction

PART I:
o Divide range of SCP fluid states in 3 regions:
 pseudoliquid (h<h_m0)
 pseudophase transition h_m0<h<h_m1
 pseudogas (superheated vapour) (h>h_m1)
o
o
the reference enthalpy values (h_m0, h_m and h_m1) determined with 
E_q=p./(c_p.)
 In the range of p/p_crit ≈1.05 to 1.30  values remain unchanged for water
and CO2
Classification of HT regimes in vertical round tubes at the boundary condition of q_w
≈cte and high heat loads q_w/𝜌𝑢
̅̅̅̅.
 Classification depends on buoyancy force and thermal acceleration effects.
 6 typical groups that differs from the T_wall behaviour and its dependence
on the flow direction in the gravity field

Group number via


(for L<100-150dia)
Where:
= the
potential parameter of the effect of Archimedes forces
For very long tubes  additionally account the local value of K_rx
parameter
the regime number far from the tube inlet.
 determines

Buoyancy effect parameter
 value gives an
understanding on the relative contribution of the buoyancy and thermal
acceleration effects in building-up the HT pattern typical of the given group.



Group1  3: 1≥F_g≥0.6  the flow pattern is primarily determined by the
buoyancy effects
 Group 4: Archimedes forces retain their triggering role in HTD in an upward
flow and damp this process in a downward flow
 Regimes that fall in this group during HTD  thermal acceleration
effect increases a lot and provokes unfavourable T_w changes!!
 Group 5 and 6: F_g  0  HTD is initiated by the thermal acceleration,
regardless the flow direction. Buoyancy effects are of the second order of
significance and can either intensify or weaken the deterioration rate!!
COMPLEX THERMOHYDRAULIC INVESTIGATIONS = HT studies + measurements of
hydrodynamic parameters of the process (hydraulic resistance + components)
o  very important for understanding the nature of SCP fluid turbulent HT specifics +
developing methods for its forecasting and calculating!
 EXPERIMENTAL DATA on SCP TURBULENT FLOW INTERNAL STRUCTURE is needed by
SOUNDING MEASUREMENTS!
Hydraulic resistance in tubes under SCP heating



VERY LIMITED experimental data on SCP fluids hydraulic resistance in tubes under heating
conditions!!!  COSTING
Experimental technique = measuring p along the tube length  separated into components
caused by friction, flow thermal acceleration and hydrostatic head.
‘integral equation of motion’ = main correlation for this.
o Within the frames of the usual boundary layer theory assumptions for an axially
symmetric flow in a rather long straight round vertical tubes  equation
o
o
‘+’ = upward flow
‘-‘ = downward flow
 correlation
o
Dividing the terms by the local conditional dynamic head
o
between the resistance coefficients:
Take a tube segment of length L (first cross section of this segment has the origin of
the x-coordinate)



Friction resistance:

The inertial resistance:

Hydraulic resistance due to fluid flow:
 Effective hydrostatic head:
Flow momentum and its inertial resistance using the momentum coefficient
(Boussinesq coefficient):

o
 the momentum coefficient S_b plays a significant role in the dynamics of
non-isothermal SCP flow, because the inertial resistance depends on both its
absolute value and its changing along the tube length!! (see later)
To separate p in components  important to have DETAILES INFORMATION on the
AVERAGED FLOW STRUCTURE and its TEMPERATURE FIELD in different cross sections
of the tube
 Early works  no such info  estimations of p_i and p_g  correspond
to the 1D homogeneous flow model (flow with uniform distribution of
temperature T_b over the tube cross section and constant momentum
coefficient value S_b = S_0 = cte)




With:
In turbulent flow  S_0 = S_0t = 1.02-1.03 ≈1
In developed laminar flow with Poiseuille velocity profile  S_0l=1.33
 1D Darcy expression for local and averaged resistance coefficients:
MOST EXPERIMENTAL STUDIES  MEAN HYDRAULIC RESISTANCE in the heating zone of
round tubes measured
o Main regime parameter in existing studies:
When processing these result  1D correlations have been used (similar to above
Darcy expressions)
o The inner structure of the flows could also differ!
 E.g. tube of 12Kh1MF steel  resistance=f(T)  q_w = var
 Horizontal tubes  difference in q_w values at upper and lower part (up to
30-50%)
 Tube roughness has also an important affect
 due to this the results from these studies are for qualitative assessment
and were instructive for further studies
In the existing experiments + special measurement  NO specifics in the friction resistance
in the region of T_pc!!
o [5] N.V. Tarasova, A.I. Leont’ev, Hydraulic resistance during flow of water in heated
pipes at supercritical pressure, High Temp. 6 (4) (1968) 721–722.
o [8] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow in
tubes at supercritical pressure, heat transfer – V, in: Proc. Fifth All-Union Conf. on
Heat Mass Transfer, Nauka i Tekhnika, Minsk, Belarus’, vol. 1, issue no. 1, 1976, pp.
261–269 (in Russian).
o [9] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under water
flow at supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–2349.
o [10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic resistance and heat
transfer of smooth channels with turbulent flow of water of supercritical pressure,
Thermal Eng. 31 (2) (1984) 109–113.
o [14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental study of heat
transfer and hydraulic resistance in turbulent flow of CO2 of supercritical pressure,
Report B914102, IVTAN, Moscow, 1979 (in Russian).
o [15] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, V.S. Grigor’ev, Experimental
investigation of drag and heat transfer in a turbulent flow of fluid at supercritical
pressure, High Temp. 18 (1) (1980) 90–99.
o [16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New experimental data on
pressure drop and heat transfer in a round tube under heating carbon dioxide of
near-critical parameters of the state, in: Convective Heat Transfer. Method and
Results from the Studies, IVTAN, Moscow, 1982, pp. 29–68 (in Russian).
o

o
o
o

[17] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow
resistance in the turbulent pipe flow of a fluid with near-critical state parameters,
High Temp. 21 (1) (1983) 81–89.
[18] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Total flow resistance and fluid
friction associated with ascending and descending supercritical fluid flow in heated
pipes, High Temp. 27 (1) (1989) 87–94.
[19] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat
transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F.
Polyakov (Ed.), Turbulent Heat Transfer under Mixed Convection in Vertical Tubes,
IVTAN, Moscow, 1989, pp. 95–160 (in Russian).
Data from [5.839] show a DECREASE in
under heating conditions
o
Measurements in short test sections [10,11] at h_in ≥ h_m0 (typical CO2)  show
that 1D analysis of the measured pressure drops at high heat loads under HTD gives

Due to actual p_i values are much larger than p_io.d. under the conditions
of [10,11]
 [10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic resistance
and heat transfer of smooth channels with turbulent flow of water of
supercritical pressure, Thermal Eng. 31 (2) (1984) 109–113.
 [11] I.V. Kuraeva, V.S. Protopopov, Mean friction coefficients for turbulent
flow of a liquid at a supercritical pressure in horizontal circular tubes, High
Temp. 12 (1) (1974) 194–196.
Data from Table 1  generalized by empirical correlations
o Tarasova and Leont’ev




With:


_w and _b = the mean integral viscosity values along the
measurement section
= The friction coefficient at constant physical properties,
calculated via Filinenko formula
o
o



[24] G.K. Filonenko, Hydraulic resistance in pipelines,
Teploenergetika (Thermal Engineering) 1 (4) (1954) 40–44
(in Russian).
[5] N.V. Tarasova, A.I. Leont’ev, Hydraulic resistance during flow of water
in heated pipes at supercritical pressure, High Temp. 6 (4) (1968) 721–722.
At Re~105 
correlates well with Lafay’s and
Razumovskii experimental data
 [25] J. Lafay, Mesure du coefficient de frottement avec transfert de
chaleur en convection forcee dans un canal circulaire, Centre
d’Etudes nucleairs de Grenoble, 1970, Report CEA-R-3896, pp. 52.
 [10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic
resistance and heat transfer of smooth channels with turbulent
flow of water of supercritical pressure, Thermal Eng. 31 (2) (1984)
109–113
 [26] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii,
Determination of resistance factors under turbulent flow of
supercritical-pressure water in smooth channels, Prom. Teplotechn.
(Indust. Ther. Eng.) 7 (5) (1985) 24–28 (in Russian).
In vicinity of T_pc at HIGH MASS FLOW RATES and Re numbers: p_io.d. =
several percents of p_o.d. 
friction coefficient even at SMALL HEAT LOADS
overestimates the

In [8,9], the o.d. experimental values for HORIZONTAL tubes (FIG. 3) 
acceptable (+-20%) for formula



0.6<
<4
HORIZONTAL TUBES  NATURAL CONVECTION EFFECT  difference
between upper and lower temperature (up to 70°C)
At 𝜌𝑢
̅̅̅̅ ≅ 1000 𝑘𝑔/𝑚²𝑠  𝐺𝑟𝜌 /𝑅𝑒𝑏2 ≈ (1.3 − 3.5)10−2
o
o


Using this in the correlations in [11]  o.d. values are 1.4-1.5
times HIGHER than via formula
[11] I.V. Kuraeva, V.S. Protopopov, Mean friction
coefficients for turbulent flow of a liquid at a supercritical
pressure in horizontal circular tubes, High Temp. 12 (1)
(1974) 194–196
In an UPWARD FLOW 
does NOT describe
the experimental data corresponding to 0.9<
<3
o The data lays lower!
o.d.
The  /values for a VERTICAL TUBE at MODERATE HEAT LOADS (q_w/
̅̅̅̅≤0.6 kJ/kg) (UPWARD AND DOWNWARD)  correlations
𝜌𝑢




With
as governing enthalpy
Most of the experimental results follow this correlation within +-15%
HOWEVER: at _b/_w < 2.5  the effect of changes in fluid density
is HIGHER than predicted with the correlation
o

More at a level of
At 𝜌𝑢
̅̅̅̅ ≅ 1500 𝑘𝑔/𝑚²𝑠 and
𝑞𝑤
̅̅̅̅
𝜌𝑢
= 0.75 − 0.9 𝑘𝐽/𝑘𝑔 (_b/_w ≈3-5)
 experimental points have a lot of scatter and the trend is an
INCREASING ̅̅̅̅̅
𝜉𝑜.𝑑. compared to the correlation

CONCLUSION:
o Contradictive results from Table 1 experiments  insufficient data on the averaged
hydraulic resistance for understanding the reasons and mechanism of HTD!!
o Necessary to develop NEW measurements techniques that allows us to analyse the
structure of LOCAL hydraulic resistances
 Institute for High Temperatures (IVTRAN, Russia)  small diameter tubes 
unreal to perform sounding measurements  proposed the unique 2
pressure drops method (method for high mass flow rate  weak
Archimedes forces effect)
 [14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental study of
heat transfer and hydraulic resistance in turbulent flow of CO2 of
supercritical pressure, Report B914102, IVTAN, Moscow, 1979 (in Russian).


[15] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, V.S. Grigor’ev,
Experimental investigation of drag and heat transfer in a turbulent flow of
fluid at supercritical pressure, High Temp. 18 (1) (1980) 90–99.
 [16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New experimental data
on pressure drop and heat transfer in a round tube under heating carbon
dioxide of near-critical parameters of the state, in: Convective Heat
Transfer. Method and Results from the Studies, IVTAN, Moscow, 1982, pp.
29–68 (in Russian).
The ‘2 pressure drops method’
o
o
o
Simultaneous measurement of pressure drops at the heated section of the tube with
length L (FULLY DEVELOPED FLOW AT THE SECTION INLET) and unheated (adiabatic)
section with length Lad ≈ 50- 60 dia.
At adiabatic section: velocity and temperature profiles levelling occur = heated fluid
flow
recovers
its
usual
turbulent
flow
structure!

Further, due to INCOMPRESSIBILITY of SCP flow + SMALL DISSIPATIVE HEAT RELEASE
 the sale mass averaged fluid parameters are retained along the adiabatic section
(= parameters at outlet cross section)
o
 the
and
values can be determined from the results of the
thermal measurements
Experimental data showed that the T_w relaxes at <10 dia after heated zone end
o
(x=L) 
and
with
in an UPWARD FLOW, pi/2
in a HORIZONTAL FLOW and pi in a DOWNWARD FLOW.
MOMENTUM BAMANCE in heated zone and adiabatic section:
o
o
A change in friction coefficient value at the adiabatic section can be described as the
exponential correlation:
 a = 9 (average)
o
At

with
via
(Filonenko)
with Re=Re_bL (known from the thermal measurements of the heated tube)

o
ESTIMATIONS of

Assuming
in
flow rates  hydrostatic heads are relatively small)

(at large fluid
in the heated zone (clos to reality FIG. 5)

Calculation of the integral via the thermal measurement data

Solving system to the unknown

and
In [16,17]  series of p and pad measurements with different L_i (0<L_i≤L)
have been conducted.

and
via
Constructed the curves
via ITERATIVE
LOOP starting from the approximation for the first section 0-L1 (
).

 differentiating these curves  local values of

and
Details
of


coefficients
the
iterative
technique
+
for
solving
see [14,19]
[14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental
study of heat transfer and hydraulic resistance in turbulent flow of
CO2 of supercritical pressure, Report B914102, IVTAN, Moscow,
1979 (in Russian).
[16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New
experimental data on pressure drop and heat transfer in a round
tube under heating carbon dioxide of near-critical parameters of
the state, in: Convective Heat Transfer. Method and Results from
the Studies, IVTAN, Moscow, 1982, pp. 29–68 (in Russian).


[17] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer
and flow resistance in the turbulent pipe flow of a fluid with nearcritical state parameters, High Temp. 21 (1) (1983) 81–89.
[19] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic
resistance and heat transfer in vertical heated tubes under
supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent
Heat Transfer under Mixed Convection in Vertical Tubes, IVTAN,
Moscow, 1989, pp. 95–160 (in Russian).

Results

NHT  flow direction doesn’t influence
 inertial resistance coefficient


values
does not exceed the friction
resistance coefficient 
In this regimes, the momentum coefficient S_b ≈ S_0t (FIG. 8)
CONCLUSION: in NHT regime, the calculation of the SCP flow hydraulic
characteristics is not a problem and can be performed using the correlation
for 1D flow model:

Data on LOCAL HYDRAULIC RESISTANCE structure in DT regimes
 at HIGH MASS FLOW RATES (= rel. small Archimedes forces effects) in DHT
region (located in the vicinity of T_pc regardless of tube orientation) 
INERTIAL resistance >> FRICTION resistance
 CONCLUSION: at high mass flow rates 𝜌𝑢
̅̅̅̅, HTD is first connected with
THERMAL ACCELERATION! + exp. Data shows that the 1D-model is NOT
suitable for analysing the local inertial resistance in the HTD zone.

See FIG. 7: values of local
relative coefficients + 1D
analogs in the regimes with peaks in T_w
In SHORT tubes + HIGH HEAT FLOW RATES +T wall-fluid  the behaviour of
the local and mean
(1D) values create an illusion of the friction
resistance growth, the actual p_i values are much LARGER ( 1D) due to
strong increase in the momentum coefficient S_b (FIG 8) and the actual p_
values are much SMALLER

REASON of large difference between actual and 1D INERTIAL resistance in
DHT regimes
 Differences connected with behaviour of the momentum coefficient
S_b along the tube length  see

FIG 7
o

For fast INCREASE of S_b (left side FIG 8) 
o For a DECREASE of S_b 
PHYSICAL REASONS of INERTIAL RESISTANCE INSTABILITY  calculations
 Heating SCP fluid  turbulent velocity profile becomes MORE
FILLED (flat in the main part of the flow)
o Flow core velocity u_c

o
With
= relative momentum thickness (several %)
Mean flow velocity over the tube cross section =
With
= relative displacement thickness (~10-2)

where
o
Similar 
o
Comparing
with usual
definition
o
o

(these correlations have also been used in the 2 pressure
drops technique for determining hydrostatic head in vertical
tubes)
Call
= “actual” momentum coefficient





For  = cte 
Kurganov simulated this S_00 for water and CO2
using the following exponential functions
with
m and n varying from 69
Results of simulation:
In contrast to S_b coefficient (increases to 1.2-1.3 at
high h and b/w  see FIG 8), the S_00 value
changes weakly and as first approximation 
assume
Sounding measurements validate the correlations
and
confirm
the
conservative nature of the actual momentum
coefficient (see later)
On the basis of the correlations and the equations of
flow,
continuity
and
energy


In gases (and pseudo-gas) when Eq≈cte 
inertial
resistance
coefficient
, regardless of the HT
regime

Under stabilized NHT conditions + uniformly
heated fluid over the cross section + heat
flow rate dissipation described by:
(Petukhov)
B.S. Petukhov, L.G. Genin, S.A. Kovalev,
Heat
Transfer
in
Nuclear
Power
Installations, Energoatomizdat, Moscow,
1986. 472 p. (in Russian)

 at small h (typical for NHT regime)  a
thin wall region with elevated Eq values but
with small mass flow rate
doesn’t
significantly contribute to the integral sum


However if
+
order of a magnitude higher than





are an

possible that
 esp. in HTD development (q/qw profile
differs from
)
 a large amount of heat is retained in the
wall gas layer
Due to the high expansion work  this layer
forces back the dense flow core from the
wall  a kind of gas-nozzle appears in the
tube that causes sharply accelerating of the
dense flow core, which carries the main part
of the flow momentum + fast increasing of
and values along the tube length
The latter is accompanied by an INCREASE in
INERTIAL RESISTANCE COEFFICIENT I and in
the gradient
 consequences in eddy
diffusivity values and heat transfer intensity
These ideas are confirmed by numerical
simulations of
Simulation with formula
,
power velocity and enthalpy profiles and the
following q/qw profile (reflect specifics of HT
at stages of HTD and HTE (downstream of
2peaks))




Where -2≤A≤2
Corresponding q/qw curves  FIG 9a
FIG 9b  calculated values of inertial
resistance coefficient