report Nr. 00000
Literature Study
Convective heat transfer to fluid operating at a supercritical pressure
Author(s):
Catternan Tom
University/department:
Ghent University - Department of Flow, Heat and
Combustion Mechanics
Address:
Sint-Pietersnieuwstraat 41, 9000 Gent, Belgium
09/02/2016
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SBO project funded by
Frame
This report is composed in the frame of the IWT SBO-110006 project The Next
Generation Organic Rankine Cycles (www.orcnext.be), funded by the Institute for
the Promotion and Innovation by Science and Technology in Flanders (IWT).
The presented work is part of WP4 ‘Development of supercritical technologies’. In
particular a literature survey is made in agreement with subtask D4.2. In this
report the possible benefits of using SC fluids in ORCs and acceptable ranges of
the different operational parameters are presented.
The goal of this report is to communicate the advantages using supercritical
fluids and recent progress in research about convective heat transfer to fluids
working at a supercritical pressure towards the research partners and advisory
board of the ORCNext project. As such, this work should not be considered a
scientific article.
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Content
Frame ....................................................................................................... 2
Content ..................................................................................................... 3
Chapter 1 Introduction ................................................................................ 4
1.
Supercritical state ........................................................................... 5
2.
Thermophysical fluid properties......................................................... 6
Chapter 2 Forced convection heat transfer in supercritical fluids ..................... 10
1.
Introduction ................................................................................. 10
2.
Literature review ........................................................................... 10
3.
Review of a selected group of experimental studies ........................... 13
4.
Data presentation [4] .................................................................... 16
4.1
Description in terms of local conditions only ................................... 17
4.2
Presentation in terms of a heat transfer coefficient ......................... 18
4.3
Presentation in terms of dimensionless groups ............................... 20
5.
General characteristics for supercritical heat transfer – Heat transfer
regimes ................................................................................................ 20
5.1
Heat transfer enhancement ......................................................... 22
5.2
Heat transfer deterioration .......................................................... 23
5.3
Influence of the heat flux ............................................................ 28
5.4
Influence of the mass flux ........................................................... 31
5.5
Influence of the direction of flow .................................................. 32
5.6
Influence of the diameter of the pipe ............................................ 33
5.7
Influence of buoyancy ................................................................. 33
6.
Summary and future experimental work .......................................... 35
Chapter 3 Correlations for forced convection supercritical heat transfer ........... 36
1.
Introduction ................................................................................. 36
2.
Correlations .................................................................................. 36
3.
Conclusion.................................................................................... 47
References .............................................................................................. 52
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Chapter 1
Introduction
Investigation of the heat transfer process and heat transfer coefficients are of
major importance as it reflects to the efficiency and the cost of the heat
exchanger design. The sizing of heat exchangers for supercritical fluid
parameters with existing models for subcritical parameters can lead to inaccurate
results and false conclusions.
Compared to a subcritical organic Rankine cycle, the temperature profiles of the
heat source and the supercritical organic working fluid are closer to each other,
resulting in a smaller logarithmic temperature difference (LMTD) and so a lower
heat exchanger thermal efficiency is expected. In order to achieve the same
efficiency, a much larger heat exchanger surface is needed. So, it is very
important to study the relatively unknown heat transfer mechanisms around the
critical point to improve the heat exchanger surface and the design algorithms.
Studies concerning heat transfer to supercritical fluids have been widely
investigated since the 1950’s and have been practically used in the field of fossilfired power plants, where supercritical water is used in steam generators to
increase the thermal efficiency. At the beginning of the 1960s, the use of
supercritical fluids as coolant in nuclear reactors has been broadly studied in the
USA and the former USSR. This idea regained potential in the 1990s when the
SCWRs (Supercritical Water Reactor) as the next generation nuclear reactors
were developed. Superconductivity effects are achieved by cooling the conductor
with fluids that are close to their critical points. Rockets and military aircraft are
cooled using fuel at supercritical pressure as an on-board coolant. Highly charged
machine elements such as gas turbine blades, supercomputer elements, magnets
and power transmission cables are cooled with supercritical fluids.
The fluids used in all studies dealing with heat transfer and hydraulic resistance
are water, carbon dioxide and cryogens like hydrogen and helium, and this
almost only in circular tubes. Beside these commonly used fluids, there were also
some experiments using liquefied gases (air, argon, hydrogen, nitrogen, nitrogen
tetraoxide, oxygen, and sulphur hexafluoride), alcohols (ethanol and methanol),
hydrocarbons (n-heptane, n-hexane, di-isopropyl-cyclohexane, n-octane,
isobutane, isopentane, and n-pentane), aromatic hydrocarbons (benzene,
toluene, and poly-methyl-phenyl-siloxane), hydrocarbon coolants (kerosene, TS1, RG-1, and jet propulsion fuels RT and T-6) and refrigerants [1]. Only a few
studies were done in annuli, rectangular channels and bundles.
Heat transfer experiments are complex due to the extreme variation of the
thermo-physical properties with temperature, with as a result that theoretical
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Report no. 00000
and empirical models become useless. Also difficulties occur concerning high
operating pressures, high compressibility which makes the density sensitive to
relatively small pressure variations and the high specific heat which can prevent
the achievement of a thermal equilibrium.
1. Supercritical state
A supercritical fluid is a fluid at pressures and temperatures that are higher than
the thermodynamic critical values. A fluid that is at a pressure above the critical
pressure, but at a temperature below the critical temperature is also known as a
“compressed fluid”. Mostly, the term “supercritical fluid” refers to both a
supercritical fluid and a compressed fluid, which is also the case in this literature
study (Figure 1).
Figure 1: Different fluid phases in p,T-diagram.
The critical point can be defined as the pressure and temperature at which no
distinction between the liquid and the vapour phase of a fluid can be made (point
c in Figure 1 and Figure 2). The supercritical state then can be defined as the
region in which the fluid pressure is slightly above this critical value. The critical
point is characterized by the state parameters Tcrit, Vcrit, and pcrit, which have
unique values for each pure substance and must be determined experimentally.
As there is no liquid-vapour phase transition, a critical heat flux1 or dry-out does
not occur. A decline in heat transfer does occur, but only in a limited range of
parameters, also known as heat transfer deterioration. This is a steady
deterioration and does not result in a drastic drop in heat transfer compared with
the dry-out phenomenon.
1
Critical heat flux describes the thermal limit of a phenomenon where a phase change occurs during heating
(such as bubbles forming on a metal surface used to heat water), which suddenly decreases the efficiency of
heat transfer, thus causing localised overheating of the heating surface [77].
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
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Figure 2: Isothermal lines in a p,v-diagram
The major difference in behaviour between a subcritical and a supercritical fluid
is shown in Figure 2. Below the critical temperature, Tcrit, the variation of
pressure and volume along an isotherm shows discontinuities where the isotherm
intersects the saturation line. At this line, phase-change occurs at a constant
pressure and temperature. Along this isotherm the vapour and liquid fraction is
changing from 100% vapour to 100% liquid. At the critical temperature the
isotherm has a zero slope at only one point, there where the pressure is equal to
the critical pressure. Above the critical temperature, the isotherms have no
discontinuities anymore and there is a continuous transition from a liquid-like
fluid to a gas-like fluid.
2. Thermophysical fluid properties
One of the challenges in the design process of a supercritical heat exchanger is
to determine the value of the overall heat transfer coefficient U as well as the
necessary heat exchanger area. As the value of the heat transfer coefficient
depends on the thermophysical properties of the working fluid, it is important to
study and understand the behaviour of these properties transferring from
subcritical to supercritical state.
The thermophysical properties of a fluid going from subcritical to supercritical
state are strongly dependent on temperature, especially in the critical and
pseudo-critical temperature range where thermodynamic and transport
properties show rapid variations [2]. For a supercritical pressure there is a
temperature where the specific heat capacity cp rises to a peak and then falls
steep. This temperature is the so-called pseudo-critical temperature, Tpc (Figure
3). Below the pseudo-critical temperature, the fluid has liquid-like properties
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Report no. 00000
while above, it resembles more to a vapour. As the pressure increases, the
pseudo-critical temperature also increases (Figure 4), the maximum value of the
specific heat cp becomes smaller and the variations of the other fluid properties
are less severe.
When a fluid at supercritical pressure in a turbulent flow is heated from a
subcritical to a supercritical temperature, it changes gradually from a liquid to a
gaseous state. At positions further away from the critical and pseudo-critical
region, the forced convection heat transfer is nearly the same, correlated by the
usual single phase correlations.
As a result, the heat transfer coefficient cannot be considered constant through
the complete heat transfer process.
Figure 3: The variation of specific volume v, specific heat cp, absolute viscosity η, thermal
conductivity λ and specific enthalpy h for water at pressure of 245 bar.
At each pressure, a local maximum of the specific heat capacity occurs. The line
connecting the maximum values (in the supercritical pressure range) is called the
pseudo-critical line (Figure 4).
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Figure 4: Pseudo-critical line of water in a p,T-diagram (left) and specific heat of water
at the pseudo-critical line (right) [3].
Besides the specific heat capacity, other thermophysical and transport properties
such as the density (𝜌), Prandtl number (Pr), the dynamic viscosity (𝜇) and the
thermal conductivity (𝜆) also vary with the temperature and pressure (Figure 5).
Within a very narrow temperature range near the pseudo-critical line the density
and the dynamic viscosity experience a significant drop. The Prandtl number
(𝑃𝑟 = 𝑐𝑝 𝜇⁄𝜆) shows the same behaviour as the specific heat capacity cp, having a
large peak at the pseudo-critical point. The thermal conductivity λ decreases as
the bulk temperature of the fluid rises, showing a local peak near the pseudocritical point, therefore not at the pseudo-critical point. With temperatures above
the pseudo-critical temperature, the thermal conductivity drops very fast.
As mentioned before, as the supercritical pressure increases, the pseudo-critical
temperature rises and the variations of the thermophysical properties with the
temperature are less severe and the existing theoretical and empirical methods
become generally more acceptable. Severe property variations with significant
heat transfer effects as a result occur in the pressure region from the critical up
to about 1.2 times the critical pressure [4].
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The strong dependence of the thermodynamic properties on temperature and
pressure leads to different heat transfer regimes.
Figure 5: Variation of density, Prandtl number, dynamic viscosity and thermal
conductivity in supercritical water with T and p (pcrit = 22.03 MPa, Tcrit =374°C) [3].
The latest versions of NIST software calculates the thermophysical properties of
ammonia, argon, butane, carbon dioxide, ethane, isobutane, methane, nitrogen,
oxygen, propane, propylene, refrigerants R-11–14, 22, 23, 32, 41, 113–116,
123–125, 134a, 141b, 142b, 143a, 152a, 218, 227ea, 236ea, 236fa, 245ca,
245fa and RC318, and water within wide ranges of pressures and temperatures.
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Report no. 00000
Chapter 2
Forced convection heat transfer in
supercritical fluids
1. Introduction
Forced convection heat transfer measurements in pipes to fluids at supercritical
pressure have been made using a wide range of fluids (water, carbon dioxide,
nitrogen, hydrogen, helium, ethane, R22 and R134a), with the majority of data
for water and carbon dioxide. Carbon dioxide is an easier fluid to handle because
of its lower critical temperature and pressure and so most of the experiments in
literature are about supercritical CO2. Most of the data obtained for forced
convection near the critical point has been obtained for pipes and channels with
uniform cross section. In recent years also non-circular sections have been
investigated, like triangular and square cross-sections. Mostly a uniform heat flux
is used to heat the supercritical fluid. Even with these simplified conditions, the
obtained experimental results are quite different even for the same sets of data
and each set of data is matched with their own correlations.
2. Literature review
In literature more than one hundred papers are found about heat transfer at
supercritical pressures. Several correlations have been proposed, but most of
them are limited to a certain parameter range and working fluid.
Several review studies about forced convection heat transfer at supercritical
pressure have been written. Petukhov [5] made in 1970 a review of experimental
works and correlations for heat transfer and pressure drop for supercritical water
and CO2. Jackson and Hall [6] [7] [8] (1975 and 1979) investigated the heat
transfer phenomena at supercritical pressure, compared several correlations with
test data and a semi-empirical correlation was proposed to account the effect of
buoyancy on the heat transfer at supercritical pressure. Polyakov [9] updated
this review in 1991 and added a numerical analysis. The heat transfer
mechanism and the trigger of heat transfer deterioration were discussed in his
review. In 2000, Kirillov [10] reviewed the researches done in Russia about heat
and mass transfer at supercritical parameters of water and a new correlation was
discussed. Prioro et al. [1] made a literature survey in 2004, giving an overview
of almost all correlations.
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Report no. 00000
Some experimental works carried out for supercritical water and carbon dioxide
are summarized in Table 1 with their test conditions, this is not a complete list.
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Table 1: Summary of the test condition for supercritical water and CO2.
p (MPa)
Dickinson (1958) [11]
25,0-32,1
G (Mg/m²s)
2,1-3,4
Q (MW/m²)
0,88-1,8
D (mm)
7,6
L (mm)
L/D
1600
-
TB (°C)
Δ𝑇 (°C)
-
-
Remarks
-
Subject
Heat transfer
Heat
transfer,
heat
transfer
deterioration,
oscillation
Heat
transfer,
oscillation
Shitsman (1959, 1963)
22,0-25,0
0,3-1,5
<1,16
8
1500
-
=<450
-
-
Domin (1963) [12]
22,0-26,0
0,6-5,1
0,58-4,5
2,0; 4,0
1075;
1233
-
=<450
-
-
Bishop (1962, 1965) [13]
22,6-27,5
0,68-3,6
0,31-3,5
2,5-5,1
-
30565
294-525
16-216
-
Swenson (1965) [14]
22,7-41,3
0,2-2,0
0,2-2,0
9,4
1830
-
70-575
6,0-285
Ackermann (1970) [15]
22,7-44,1
0,135-2,17
0,12-1,7
9,4-24,4
-
-
77-482
-
Yamagata (1972) [16]
22,6-29,4
0,31-1,83
0,116-0,930
7,5; 10,0
15002000
-
230-540
-
Griem (1999) [17]
22,0-27,0
0,3-2,5
0,20-0,70
10-24
-
-
-
-
-
Heat transfer
Sabersky (1967) [18]
7.24-7.588.27
-
0,437
-
-
-
24.9- 25.640.5
-
Horizontal
Visualisation,
turbulence
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Heat transfer
Heat
transfer,
Heat
transfer
deterioration
Heat
transfer,
pseudo-boiling
phenomena
Vertical Heat
transfer,
and
Heat
transfer
horizontal deterioration
-
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Report no. 00000
3. Review of a selected group of experimental studies
As can be seen, experimental studies have been performed since 50’s. The
experiments of Dickinson (1958) [11], Ackermann (1970) [15], Yamagata
(1972) [16] and Griem (1995) [17] were mainly related to the design of
supercritical pressure fossil power plants. The tube diameter ranges from 7.5
mm up to 24 mm. A good agreement was obtained between the test data of
Dickinson [11] and the Dittus-Boelter equation at a wall temperature below
350°C. Large deviation was obtained at a wall temperature between 350°C and
430°C. In both the experiments of Domin (1963) [12] and of Dickinson [11], no
heat transfer deterioration was observed, whereas heat transfer deterioration
occurs in the tests of Yamagata [16] and of Ackermann [15]. It was shown by
Yamagata [16] that at low heat fluxes, heat transfer is enhanced near the
pseudo-critical line. Heat transfer deterioration happened at high heat fluxes.
Ackermann [15] observed boiling like noise at the onset of heat transfer
deterioration, which was, therefore, treated as a similar phenomenon like boiling
crisis under sub-critical pressures. The test data indicated that pseudo-critical
heat flux (CHF), at which heat transfer deterioration occurs, increases by the
increasing pressure, increasing mass flux and decreasing tube diameter.
The experimental works of Bishop (1964) [13] and Swenson (1965) [14] were
performed in the frame of designing supercritical light water reactors. In the
work of Bishop [13], small diameter tubes were used, whereas in the work of
Swenson [14], circular tubes of a larger diameter 9.4 mm were applied. In
addition to smooth circular tubes, whistled circular tubes and annular channels
were also used by Bishop [13]. Nevertheless, no experimental data in annular
channels are available in the open literature. Both tests showed the entrance
effect on heat transfer coefficient. In the experiments of Swenson [14], no heat
transfer deterioration was observed. Empirical correlations were derived based
on the test data achieved.
Many tests were performed in former Soviet Union in supercritical water, carbon
dioxide and Oxygen [19] [20]. The phenomenon of heat transfer deterioration
was first observed by Shitsman et al. (1963) [19] at low mass fluxes. During the
tests pressure pulsation took place, when the bulk temperature approached the
pseudo-critical value. Based on the test data, several correlations were
developed for predicting heat transfer coefficient, onset of heat transfer
deterioration and friction pressure drop.
The main conclusions drawn from the experimental works mentioned above are
summarized as follows:

The experimental studies in the literature covers a large parameter range:
o P: 22.0 – 44.1 MPa
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o
o
o
o
G: 0.1 – 5.1 Mg/m²s
Q: 0.0 – 4.5 MW/m²
D: 2.0 – 32.0 mm
TB: ≤ 575°C
However, it has to be kept in mind that this parameter matrix is not
completely filled with test data. Further check is necessary to find out
parameter combination at which no test data are still available.

Heat transfer deterioration is only observed at low mass fluxes and high
heat fluxes with the following temperature condition:
𝑇𝐵 ≤ 𝑇𝑝𝑐 ≤ 𝑇𝑤

At low heat fluxes a heat transfer enhancement was obtained as the bulk
temperature approaching the pseudo-critical point.

The experimental works are mainly restricted to circular tube geometry.

Some special effect has been studies, i.e. entrance effect, channel inserts,
flow channel orientation and heat flux distribution.

Large deviation was obtained between the Dittus-Boelter equation and the
test data with the bulk temperature or the wall temperature near the
pseudo-critical value.

Several empirical correlations have been derived based on the test data.
Due to its lower critical pressure (7.4 MPa) and critical temperature (31°C),
experiments in supercritical carbon dioxide require much less technical
expenditure. However, some results have been well extrapolated to water
equivalent conditions. Based on the test data in CO2, Krasnoshchekov (1966)
[21] proposed an empirical correlation of heat transfer, which was also
successfully applied to heat transfer in supercritical water [6]. Several authors
have performed tests with carbon dioxide studying systematically the effect of
different parameters on heat transfer [6] [8] and on the behaviour of heat
transfer deterioration [22].
Flow visualization and more comprehensive measurement have been realized in
experiments with carbon dioxide, to study the physical phenomena involved in
heat transfer at supercritical pressure [18] [23] [24] [25]. By measuring the
velocity profile and turbulence parameters of fluid near the heated wall, the
mechanisms affecting heat transfer have been investigated.
Adebiyi and Hall (1976) [26] performed heat transfer experiments in horizontal
flow of carbon dioxide at supercritical and subcritical pressures. Axial (Figure 6a
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and b) and circumferential (Figure 6c) temperature profiles were obtained. It was
found that non-uniform cross-section temperature profile exists in horizontal flow
(Figure 6c). Comparison with buoyancy free data showed that heat transfer on
the bottom of a tube was enhanced by buoyancy forces, but heat transfer on the
top was reduced by buoyancy forces (hotter fluid is at the top of a tube). Figure
7 shows a comparison between temperature profiles along horizontal and vertical
tubes with upward and downward flow. The data showed that the horizontal flow
temperature profiles are more gradual compared to those for vertical upward
flows.
Figure 6: Temperature profiles along horizontal circular tube: T pc = 32.3°C, Hpc = 337
kJ/kg [26].
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Figure 7: Temperature profiles along horizontal and vertical circular tubes (comparison
with data from Weisberg, 1972): Tpc = 32.3°C, Hpc = 337 kJ/kg [26].
Ko et al. (2000) [27] performed flow visualization experiments in a vertical oneside heated rectangular test section cooled with forced flow of supercritical
carbon dioxide. They calculated temperature and density profiles of the heated
carbon dioxide inside the test section from measured interferometry projections.
A similar investigation was reported by Sakurai et al. (2000) [28].
4. Data presentation [4]
The presentation of experimental data in tables and figures is very important and
has to be accurate and meaningful. In this section, based on the review of Hall
[4], some methods will be discussed in which experimental data is being
presented.
For constant property fluids, the heat transfer is proportional to the temperature
difference between the surface and the fluid, and is a consequence of the fact
that the energy equation is linear in temperature. The heat transfer process does
not affect the flow process. The presentation of the experimental data is then
mostly in a form which neither the temperature of the heat transfer surface nor
that of the fluid is explicitly given. For fluids near the critical point, such a
presentation is wrong because of the non-proportionality with variable property
fluids.
To illustrate this, the same data is presented in different forms using carbon
dioxide at a pressure of 75.8 bar (pcrit = 73.8 bar) flowing downward in a
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heated vertical tube with a diameter of 1.9cm (Evans et al. PhD thesis [29]. The
behaviour of the fluid is usually related to the pseudo-critical temperature (32°C
at 75.8bar), rather than the critical temperature (31.04°C at 73.8bar).
The measured parameters were the mass flow, the fluid inlet temperature, the
heat input (nearly uniform wall heat flux) and the temperature of the pipe wall
which was measured at intervals of one pipe diameter along the length of the
test section.
Figure 8 shows the variation of the wall temperature Tw along the vertical pipe
(downward flow) for three different heat fluxes, with the same mass flow and
fluid inlet temperature.
Figure 8: Temperature distribution along a 1.9cm diameter vertical pipe for downward
flow. Carbon dioxide at a pressure of 75.8bar and a mass flow of 160gm/s [29].
4.1 Description in terms of local conditions only
For constant property fluids at a certain point after the inlet section, the velocity
and the temperature distribution across the pipe becomes invariant and a fully
developed fluid flow has been set. In literature sufficient data is available and it
is common that this condition sets in about 10 to 20 pipe diameters after the
inlet section. As the properties of the fluid near the critical region vary with
temperature and thus also with the distance along the pipe, a hypothesis of a
fully developed is less reliable.
Figure 9 shows the same set of results presented in the form of heat flux against
wall temperature, with the fluids bulk temperature as parameter. The bulk fluid
temperature was calculated by applying a heat balance from the pipe inlet to the
point in question by knowledge of the enthalpy as a function of the temperature.
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Bulk temperature
(●) 19°C
(+) 22°C
(∆) 25°C
(x) 28°C
(□) 31°C
Figure 9: Heat flux versus wall temperature for various bulk temperatures [29].
The dotted lines are fitted because they were not measured. The point where
they intersect the Tw-axis, is the point where the heat flux q = 0 and Tw = Tb.
The slope of the curves at this point gives the limiting value of the heat transfer
coefficient as the temperature difference tends to zero.
4.2 Presentation in terms of a heat transfer coefficient
If the same results are presented in terms of a heat transfer coefficient versus
wall temperature for various bulk temperatures (Figure 10), one might think that
high heat fluxes are possible with small temperature differences, while in Figure
9 it can be seen that is not possible.
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Bulk temperature
(●) 19°C
(+) 22°C
(∆) 25°C
(x) 28°C
(□) 31°C
Figure 10: Heat transfer
temperatures [29].
coefficient
versus
wall
temperature
for
various
bulk
The use of the heat transfer coefficient for supercritical fluids has been
questioned by Goldman [30]. Generally the heat transfer coefficient is expressed
as a relation between the dimensionless parameter of Nusselt, Reynolds and
Prandtl, as show in below equation.
𝑁𝑢 = 𝑐 𝑅𝑒 𝑛 𝑃𝑟 𝑠
[Eq. 1]
with c, n and s constants.
Goldman, however, suggested collecting all the temperature dependent terms in
the dimensionless groups.
𝑞0 𝑑 1−𝑛
(𝜌𝑢)𝑛
= 𝑓(𝑇0 , 𝑇𝑚 )
[Eq. 2]
This presentation resembles more to the data presented in Figure 9, but it
suggests that there is a variation with the pipe diameter d and mass velocity 𝜌𝑢.
However, the latter equation is as valid as the former equation, because it is
derived from that one.
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4.3 Presentation in terms of dimensionless groups
Using the correlation developed by Miropolsky and Shitsman [31], the same data
as in Figure 10 is presented in Figure 11.
𝑁𝑢𝑚 = 𝑐 (𝑅𝑒𝑚 )𝑛 (𝑃𝑟𝑚𝑖𝑛 )𝑠
[Eq. 3]
Figure 11: Correlation of the data of Figure 10.
The Nusselt and Reynolds number are evaluated at the bulk temperature, while
the Prandtl number is evaluated at the lower of the bulk and wall temperature.
The constant n = 1.4 gives the best fit for the results.
The problem with such a representation is that the scatter shows a better
correlation than in the original data presented in Figure 9, and also the fact that
it is impossible to recover the original data from such a presentation.
5. General characteristics for supercritical heat transfer –
Heat transfer regimes
Convective heat transfer near the critical point is characterized by properties
having rapid variation with temperature. As a consequence, the flow and heat
transfer processes are linked. The equation describing the temperature
distribution in the fluid is essentially nonlinear, so that the proportionality
between heat flux and temperature difference no longer exists.
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As already stated by Hall [4], the heat transfer coefficient then becomes a
parameter of doubtful utility which can take widely differing values depending on
the conditions.
In the following section, the general characteristics for heat transfer to a
supercritical fluid are discussed. Phenomena, such as heat transfer enhancement
and heat transfer deterioration are described and the influence of the heat flux,
mass flux, tube diameter, flow direction and buoyancy are demonstrated.
As mentioned before most of the data exist for circular pipe cross sections with a
uniform heat flux boundary condition. Even with such a large amount of data,
still in some cases it is not possible to correlate the results due to occurring
physical phenomena.
Figure 12 presents examples of variation between experiments, this in all cases
for supercritical water in a circular pipe with a uniform heat flux. For similar entry
conditions, the wall temperature is expected to be a function of the bulk
enthalpy, the mass velocity, the pipe diameter and the wall heat flux. The
conditions are given in Table 2 and Table 3.
p = 1.15 pcrit
p = 1.05 pcrit
Figure 12: Experimental wall temperature distributions as a function of local bulk
enthalpy along a pipe: p = 1.05 pcrit and p = 1.15 pcrit [32].
Table 2: Experimental conditions for supercritical water at p = 1.05 pcrit [32].
a
b
c
d
e
Shitsman [19]
Shitsman [19]
Shitsman [19]
Domin [12]
Domin [12]
𝒒 (𝑾/𝒄𝒎𝟐 )
34
28.5
28.0
72.5
72.5
𝒎̇⁄𝑨 (𝒈𝒎⁄𝒔 𝒄𝒎²)
43
43
43
68.6
72.4
𝒅 (𝒄𝒎)
0.8
0.8
0.8
0.2
0.2
Flow direction
vertical upward
vertical upward
vertical upward
horizontal
horizontal
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Table 3: Experimental conditions for supercritical water at p = 1.15 pcrit [32].
a
b
c
d
e
f
Vikrev and Lokshin [33]
Vikrev and Lokshin [33]
Schmidt [34]
Schmidt [34]
Domin [12]
Shitsman [19]
𝒒 (𝑾/𝒄𝒎𝟐 )
𝒎̇⁄𝑨 (𝒈𝒎⁄𝒔 𝒄𝒎²)
𝒅 (𝒄𝒎)
69.9
69.9
58
82
91
39.6
100
40
61
61
101
44.9
0.8
0.8
0.5
0.5
0.2
0.8
Flow
direction
horizontal
horizontal
horizontal
horizontal
horizontal
vertical upward
It is very difficult to compare the different experiments and find a pattern in
them, but several general trends can be found.



The unusual behaviour of the wall temperature occurs just before the bulk
temperature reaches its critical value.
The heat transfer coefficient is strongly dependent on the heat flux, as can be
seen in Figure 12 curves a, b and c for p = 1.05 pcrit.
When 𝑇𝐵𝑢𝑙𝑘 ≤ 𝑇𝑐𝑟𝑖𝑡 ≤ 𝑇𝑤𝑎𝑙𝑙 , local enhancement (Figure 12 for p = 1.15 pcrit –
curve e) and deterioration (Figure 12 e.g. for p = 1.05 pcrit – curves a and b)
can occur in the heat transfer.
From the experimental data in Figure 12 is it clear that the orientation of the
heated pipe is from major importance.
5.1 Heat transfer enhancement
On Figure 9 and Figure 10 (supercritical CO2 – vertical downward flow – d =
1.095cm), heat transfer enhancement is visible for small heat fluxes and the
condition where 𝑇𝐵𝑢𝑙𝑘 ≤ 𝑇𝑐𝑟𝑖𝑡 ≤ 𝑇𝑤𝑎𝑙𝑙 . As the heat flux increases, the heat transfer
enhancement reduces. The results for a vertical upward flow are very different.
From the data presented by Tanaka, Nishiwaki and Hirate [35] (supercritical CO2
– vertical upward flow – d = 1.0cm) in Figure 13, it is noticed that a maximum
occurs for the heat transfer coefficient for a condition where bulk temperature
𝑇𝐵𝑢𝑙𝑘 is slightly below the pseudo-critical temperature 𝑇𝑝𝑐 and when the wall
temperature 𝑇𝑊𝑎𝑙𝑙 is slightly above 𝑇𝑝𝑐 . The peak is, as also observed in Figure
10, higher for lower values of the heat flux. Furthermore, it can also be seen that
as the mass flux increases, the heat transfer coefficient increases.
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(1) Theory
(∆) Experimental:
𝑚̇ = 140±4.4 kg/h;
q = 1.44 W/cm²
(2) Theory
(x) Experimental:
𝑚̇ = 140±3.1 kg/h;
q = 2.73 W/cm²
(3) Theory
(○) Experimental:
𝑚̇ = 280±5.6 kg/h;
q = 3.32 W/cm²
(4 Theory
(●) Experimental:
𝑚̇ = 280±7.8 kg/h;
q = 5.20 W/cm²
Figure 13: Variation of the heat transfer coefficient with bulk temperature for forced
convection in a heated pipe for carbon dioxide of 78.5bar flowing upwards in a 1.0
diameter vertical pipe [35].
5.2 Heat transfer deterioration
In Figure 12, it can be seen that the experiments with horizontal pipes show
broad wall temperature peaks at higher heat fluxes. For a vertical upward flow,
sharp temperature peaks are observed.
Shitsman et al. [36] compared an upward and downward supercritical water flow
for several uniform heat fluxes (Figure 14) and found that there is no unusual
behaviour for a downward flow, but that for an upward flow a sharp peak occurs
for the wall temperature as the heat flux exceeds a certain value. As the heat
flux rises, the peak in wall temperature occurs more to the inlet section of the
pipe.
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Table 4: Experimental conditions for supercritical water at 245 bar in a vertical upward
and downward 1.6 cm diameter heated pipe ( 1.11 pcrit) [36].
1
2
3
4
5
6
7
8
𝒎̇⁄𝑨 (𝒈𝒎⁄𝒔 𝒄𝒎²)
382
382
400
375
400
400
393
381
𝒒 (𝑾/𝒄𝒎𝟐 )
27
37
45
52
27
36
43
50
Flow direction
Vertical upward
Vertical upward
Vertical upward
Vertical upward
Vertical downward
Vertical downward
Vertical downward
Vertical downward
Figure 14: Wall and bulk temperature as a function of the distance along a vertical
heated 1.6 cm diameter pipe for water at 245 bar (1.11 pcrit): (left) upward flow; (right)
downward flow [36].
Jackson et al. [37] performed a similar experiment with carbon dioxide for an
upward flow and found that severe heat transfer deterioration occurs when a
certain value of the heat flux is exceeded. It is to be noted that the
deteriorations for CO2 occur for 𝑇𝑊𝑎𝑙𝑙 > 𝑇𝑝𝑐 , while the deteriorations in water from
Shitsman [36], occurred below 𝑇𝑝𝑐 as well as above 𝑇𝑝𝑐 . Tanaka et al. [35]
(Figure 13) performed experiments under almost the same conditions as Jackson
et al. but no deterioration was noticed. The only difference was that Tanaka used
a 1 cm diameter tube instead of a 1.905 cm diameter from Jackson. From this
comparison, it can be concluded that the diameter could be an important factor
in the heat transfer behaviour.
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Evans et al. [29] performed in his PhD thesis, experiments with carbon dioxide at
a pressure of 75.8 bar (pcrit = 73.8 bar) flowing downward and upward in a
heated vertical tube with a diameter of 1.9cm (Figure 15). The same conclusion
can be drawn about the deterioration of the heat transfer of a vertical upward
flow, which increases as the heat flux increases.
(a) q = 3.09 W/cm²
(b) q = 4.05 W/cm²
(c) q = 5.19 W/cm²
(d) q = 5.67 W/cm²
(a) q = 3.09 W/cm²
(b) q = 4.05 W/cm²
(c) q = 5.19 W/cm²
Figure 15: Temperature distribution along a 1.9cm diameter vertical pipe as a function of
the distance along the pipe for carbon dioxide at a pressure of 75.8bar and a mass flow
of 160gm/s: (above) upward flow, (below) downward flow [29].
The deteriorations in horizontal pipes are less prompt than vertical upward flow
pipes. Miropolsky and Shitsman [31] measured the temperature distribution for
supercritical water around a horizontal and vertical 1.6 cm diameter pipe (Figure
16). The temperature difference between the bulk temperature and the upper
surface is a lot bigger than the difference between the lower surface and the bulk
temperature. In the conditions presented in Figure 16, this leads to a reduction
in the heat transfer coefficient of about a factor 4 compared to the lower surface.
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(1) Horizontal pipe – upper surface
(2) Horizontal pipe – lower surface
(3) Vertical pipe – upward flow
(4) Bulk fluid temperature
Figure 16: Temperature distribution as a function of local bulk enthalpy along heated
vertical and horizontal pipes (1.6 cm diameter) for water at 245 bar (= 1.11 pcrit):
𝒎̇⁄𝑨 = 𝟔𝟎 𝒈𝒎⁄𝒔 𝒄𝒎² and 𝒒 = 𝟓𝟐 𝑾/𝒄𝒎𝟐 [31].
Hall compared in his review [4] three sets of data for supercritical CO2 with both
an upward and a downward flow in a vertical pipe. The comparison was between
the data of Shiralkar and Griffith [38], Jackson et al. [37] and Bourke et al.
[39], where only the test section diameter differs (Table 5).
Table 5: Comparison of three sets of data for supercritical CO 2 flowing up- and
downwards in a vertical pipe [4].
𝑹𝒆𝒇𝒆𝒓𝒆𝒏𝒄𝒆
Shiralkar and Griffith [38]
Jackson and Evans-Lutterodt [37]
Bourke et al. [39]
𝒅 (𝒄𝒎)
0.635
1.905
2.285
𝒒
(𝑾/𝒄𝒎𝟐 )
15.8
5.67
5.1
𝑹𝒆
𝑮𝒓
𝒒. 𝒅
(𝑾/𝒄𝒎)
1.0
1
10.0
1.24 27
10.8
0.82 46.5 11.6
𝒑
(𝒃𝒂𝒓)
75.8
75.8
74.5
Figure 17 shows the wall temperature as a function of the bulk enthalpy for a
downward and upward flow.
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Legend:
_.__._: Shiralkar and Griffith [38]
_____: Jackson and Evans-Lutterodt [37]
_ _ _ _ : Bourke et al. [39]
Figure 17: Comparison of the data of Shiralkar and Griffith [38], Jackson and EvansLutterodt [37] and Bourke et al. [39] for forced convection of carbon dioxide flowing upand downwards in vertical heated pipes [4].
No significant difference was found between an upward and downward flow for
the data of Shiralkar and Griffith [38], while for larger pipe diameters, Jackson et
al. [37] and Bourke et al. [39] observed sharp peaks for an upward flow, as
already seen in experiments by Shitsman [19] in Figure 12. Furthermore, the
wall temperatures for an upward flow are lower than the ones for a smaller
diameter. For a downward flow, no significant peaks are noticed and the wall
temperatures are lower than those for the small pipe.
From the results of Shitsman [19] (Figure 12) it is also clear that, besides an
increasing heat flux, the heat transfer deterioration phenomena becomes also
more outspoken for lower mass flow fluxes.
In literature there is no unique definition for the start of heat transfer
deterioration, because the increase in wall temperature (see Figure 14, Figure 15
and Figure 16) is smoother compared to the much sharper increase for the
boiling phenomenon at subcritical pressures.
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5.3 Influence of the heat flux
The heat flux is not the only incentive which influences the heat transfer, but for
a certain configuration (orientation and diameter of the pipe, mass flow flux), the
heat flux has a key role in the heat transfer phenomena. However, the
orientation of the pipe is also very important and distinctive results are found
under certain conditions between a horizontal and vertical upwards and
downwards flow direction. In this section the influence of the heat flux will be
more examined.
As mentioned before, a lower heat flux reduces the deterioration or even
improves the heat transfer. At very low heat fluxes, the temperature variations
in the fluid are small and constant properties, with actual values dependent of its
location to the critical temperature, can be approached in this small range. The
correlations for constant properties could be adopted. Consider a general form of
the Dittus-Boelter correlation:
𝑁𝑢 = 𝑐𝑡𝑒 𝑅𝑒 0.8 𝑃𝑟 𝑛
[Eq. 4]
Where:



𝑁𝑢, the Nusselt Number (= ℎ 𝐿⁄𝜆) [−];
𝑅𝑒, the Reynolds Number (= 𝑚̇ 𝐿⁄𝜇) [−];
𝑃𝑟, the Prandtl Number (= 𝜇 𝑐𝑝 ⁄𝜆) [−];






ℎ, the heat transfer coefficient 𝑊 ⁄𝑚2 𝐾;
𝜇, the dynamic viscosity in 𝑁𝑠⁄𝑚2;
𝜆, the thermal conductivity in 𝑊 ⁄𝑚𝐾;
𝐿, the characteristics length (e.g. diameter D) in 𝑚;
𝑛=0.4 for heating and 𝑛=0.3 for cooling of the fluid;
𝑚̇, the mass flow rate per unit area in 𝑘𝑔⁄𝑠.
From this it follows that for heating of the fluid, the heat transfer coefficient can
be written as:
ℎ = 𝑐𝑡𝑒 𝑚̇0.8
𝜆0.6 𝑐𝑝 0.4
𝐿0.2 𝜇0.4
[Eq. 5]
Figure 4 and Figure 5 showed the variations of the thermophysical properties
with the temperature near the critical region. As the thermal conductivity λ and
the dynamic viscosity 𝜇 show a similar trend, these will not have a dominant
effect on the heat transfer coefficient. The variation of the specific heat cp is
severe near the pseudo-critical temperature and this will have a major influence
on the value of the heat transfer coefficient. This can be seen in the experiments
performed by Yamagata et al [16] for supercritical water at a pressure of 245 bar
(= 1.11xpcrit) (Figure 18).
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Figure 18: Experimental heat transfer coefficient by the data of Yamagata et al [16].
As the heat flux increases, the temperature gradient increases and so the region
of the fluid at high Prandtl number will reduce with as a result that the peak of
the heat transfer coefficient will decrease.
Figure 19 shows the calculated heat transfer coefficient by Cheng X. et al. [3] for
water according to the Dittus-Boelter equation at a mass flux of 1.1 Mg/m²s,
pressure 250bar (= 1.13xpcrit), heat flux of 0.8MW/m² and a tube diameter of
4.0 mm. The value of the heat transfer coefficient at the pseudo-critical point is
about two times the value of that at low temperatures and five times of that at
high temperatures. The peak decreases for pressure values further away of the
critical point.
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Figure 19: Heat transfer coefficient as a function of the fluids bulk temperature
according to the Dittus-Boelter equation [3].
Figure 20: (left) Ratio of the experimental heat transfer coefficient to the value
calculated via the Dittus-Boelter equation; (right) Wall temperature behaviour for low
and high heat fluxes [3].
Comparing the heat transfer coefficient values of experiments (𝛼) and those
calculated via the Dittus-Boelter equation (𝛼0 ), presented as the ratio by 𝛼 ⁄𝛼0 in
Figure 20, it was noticed that the heat transfer coefficients at low heat fluxes
were higher than the values calculated via the equation. This phenomenon is
called heat transfer enhancement. The heat transfer coefficients at high heat
fluxes were lower than the values calculated via the Dittus-Boelter equation.
Under some specific conditions even a very low heat transfer coefficient ratio was
obtained.
Comparing the behaviour of the wall temperature at low and high heat fluxes, as
seen in Figure 20, it is noticeable that the wall temperature at low heat fluxes
behaves smoothly and increases with the bulk temperature. For high heat fluxes
the behaviour is similar, but when the bulk fluid temperature approaches the
pseudo-critical temperature, a sudden increase in wall temperature can occur.
When the bulk temperature exceeds the value of the pseudo-critical temperature
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the wall temperature decreases again and the heat transfer coefficient is
restoring again. The sudden increase in wall temperature is also known as heat
transfer deterioration.
5.4 Influence of the mass flux
From the data presented by Vikrev and Lokshin [33] in Figure 12 and from
Tanaka, Nishiwaki and Hirate [35] in Figure 13, it was clear that as the mass flux
increases, the heat transfer coefficient increases.
As mentioned before, the enhancement of the heat transfer coefficient for small
heat fluxes (small temperature difference) when the bulk fluid temperature is
near (slightly lower than) the pseudo-critical temperature is attributed to the
large value of the specific heat in this region. For higher heat fluxes (higher
temperature difference), the proportion of the flow experiencing this high specific
heat is smaller. Lokshin [33] uses the ration 𝑞̇ ⁄𝑚̇ as a parameter to compare the
heat transfer coefficient to that for constant properties. Generalized curves for
supercritical water at 250 bar can be found in Figure 21 and it can be seen that
above a value of 𝑞. 10̇ −3 ⁄𝑚̇ ≈ 0.7, no heat transfer enhancement occurs anymore
and there is a monotonic deterioration in heat transfer coefficient as the fluid
bulk temperature crosses the pseudo-critical temperature.
Figure 21: Generalized curves for water at 250bar (Lokshin et al. [33])
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5.5 Influence of the direction of flow
Shitsman et al. [36] (Water: Figure 14) and Evans et al. [29] (CO2: Figure 15)
performed experiments for an upward and downward flow for several uniform
heat fluxes and found for an upward flow that severe heat transfer deterioration
(sharp peak occurs for the wall temperature) occurs when a certain value of the
heat flux is exceeded, while for a downward flow no unusual behaviour occurs.
This phenomenon can also be seen Figure 22, from experiments by Jackson and
Evans-Lutterodt [37] performed a similar experiment with carbon dioxide for an
upward flow and came to the same conclusion.
Figure 22: Comparison of heat transfer between an upward and downward flow for CO 2
by Jackson and Evans-Lutterodt [37].
The deteriorations in horizontal pipes are less prompt than vertical upward flow
pipes (Figure 16). For a horizontal setup, a temperature difference occurs
between the upper and lower surface of the pipe, caused by the buoyancy. This
temperature difference leads to a reduction in the heat transfer coefficient at the
upper surface compared to the lower surface.
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5.6 Influence of the diameter of the pipe
Tanaka et al. [35] (Figure 13) and Jackson et al. [37] performed experiments
with carbon dioxide for an upward flow under almost the same conditions with
the only difference that Tanaka used a 1cm diameter tube and Jackson a 1.905
cm tube. The results showed that with the smaller diameter no deterioration was
observed, while with the bigger diameter severe heat transfer deterioration
occurs when a certain value of the heat flux is exceeded. For larger diameters
buoyancy will have a bigger influence.
Cheng X. et al. [3] investigated the effect of increasing the tube diameter for
different existing correlations and it was noticed that the heat transfer coefficient
decreases by increasing the tube diameter (Figure 23). A slightly stronger effect
of the tube diameter was found using the correlation of Bishop [13] and of
Krasnoshchekov [21].
Figure 23: Effect of tube diameter on heat transfer coefficient [3].
5.7 Influence of buoyancy
For a downward heated flow there is a continuous enhancement in heat transfer
as buoyancy becomes relatively stronger. This behaviour has been found with
many fluids at supercritical pressure and also with other fluids. Not only is the
heat transfer improved, but wall temperatures are less sensitive to heat flux.
Hall and Jackson [4] proposed a mechanism for which buoyancy will affect the
heat transfer. The dominant factor is the modification of the shear stress
distribution across the pipe, with a consequential change in turbulence
production.
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As mentioned before, buoyancy effects are also noticed in horizontal flows. Due
to a stratification of the flow, the hotter (less dense) fluid can be found in the
upper part of the pipe. There may also be an effect due to the damping effect of
the stabilizing density gradient on turbulence near the upper surface of the pipe.
At the lower surface heat transfer is frequently better than for forced convection
alone, suggesting that there may be some amplification of turbulence by the
destabilizing density gradient in this region.
Belyakov et al. [40] performed some measurements for heat transfer to
supercritical water in horizontal pipes (Figure 24). The deterioration of the upper
surface occurs progressively along the pipe and does not show the sharp peaks
that are obtained with upward flow. As the ratio of the heat flux to the mass flow
flux increases, the wall temperature and thus deterioration at the upper surface
increases.
Figure 24: Heat transfer in a horizontal supercritical flow for different values of 𝐪̇ ⁄𝐦̇
(Belyakov et al. [40]).
In forced convection the Reynolds number describes the fluid flow; however in
natural convection the Grashof number is the dimensionless parameter that
describes the fluid flow. The Grashof number is a dimensionless parameter which
approximates the ratio of the buoyancy to viscous force acting on a fluid.
It can be shown that a criterion for negligible buoyancy effects for horizontal flow
is
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̅̅̅̅𝑏
𝐺𝑟
𝑅𝑒𝑏 2.7
< 10−5 [Eq. 6]
3
̅ )𝐷
̅̅̅̅𝑏 is the Grashof number (= 𝑔 (𝜌𝑏 −𝜌
where 𝐺𝑟
) [Eq. 7], in which 𝜌̅ is the integrated
2
𝜈
𝑏
mean density and the subscript b indicates physical properties evaluated at the
local bulk temperature.
The buoyancy parameter Bo, defined as 𝐵𝑜 = 8 𝑥 104
𝐺𝑟
𝑅𝑒 3.425 𝑃𝑟 0.8
[Eq. 8], can be
used to determine whether the flow is in the forced convection or mixed
convection regime (Hall and Jackson, 1969 [41]).
6. Summary and future experimental work
The results obtained from experimental data presented by several researchers
can sometimes conflict with each other. Mostly this is because of the differences
in experimental arrangement.
Summarising the results from previous experiments, it was found that heat
transfer deterioration occurs with upward flow only and that this deterioration
can be reduced by applying a lower heat flux or using a smaller pipe diameter.
Buoyancy has a big influence in the heat transfer differences between an upward
and a downward flow, and the Archimedes forces enlarge the heat transfer
deterioration for an upward flow.
Possible explanations of the heat transfer improvement and deterioration
phenomena have been suggested, e.g. the effect of buoyancy due to density
gradients [42], the effects of radial differences in viscosity [43] and the effects of
rapid changes in density in the flow [44] on heat transfer by turbulent
convection.
Most of the data presented in papers are for carbon dioxide. Data for
supercritical water is less available because of the large pressures needed to
work with supercritical parameters.
Hall did some suggestions in his review [4] for further experimental research.




Experiments should be done for upward and downward flow.
Different pipe diameters must be used during the experiments.
Detailed pipe wall temperature measurements in axial as well as
circumferential directions are necessary (e.g. Jackson et al. [37] used 200
thermocouples on a 1.9 cm diameter pipe over a length of 3m).
More detailed work is necessary for horizontal pipes.
_____________________________________________________________________________________________________
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
Chapter 3
Correlations for forced convection
supercritical heat transfer
1. Introduction
Due to the radial variations with the temperature of the thermophysical
properties near the wall, it is not easy to describe the heat transfer behaviour of
a supercritical pressure fluid with a standard correlation for constant properties,
like the Dittus-Boelter correlation. For constant property conditions, most of the
correlations describing the Nusselt number are expressed as a simplified function
of Reynolds and Prandtl. The advantage is that a small number of dimensionless
parameters can describe a certain situation. For situations where the property
variations are large, extra property ratio terms have to be added to take their
influence into account. The problem then can occur that the dimensionless
correlation can become bigger than the original number of influence parameters.
Hall stated in his review [4] that the effect of dissipation is negligible,
acceleration effects can be important and that buoyancy effect is a major factor
at any rate when the flow is vertically upward. Existing correlations don’t take
the acceleration and buoyancy effects into consideration. In most cases the
influence of these effect are neglected, which reduces the range of its
applicability.
Until now, adequate analytical methods have not been developed due to the
difficulty in dealing with the extreme variations of the thermophysical properties.
Various empirical correlations, based on experimental data, have been developed
for normal heat transfer calculations at supercritical pressures, using
experimental data of water, carbon dioxide, Freon and some cryogens. As
mentioned here above, most of these correlations are expressed in the form of a
constant properties heat transfer correlation added with extra terms (mostly
ratios of properties between the bulk and wall temperature) to take the property
variations into account.
2. Correlations
Table 6 gives an overview of existing correlations for supercritical heat transfer
(this table cannot be considered complete).
_____________________________________________________________________________________________________
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
Table 6: Summary of the correlations for supercritical fluids.
Bringer
Smith
[45]
Fluid
Correlation
Water
𝑁𝑢𝑋
𝑇𝑥
= 0.0266𝑅𝑒𝑋0.77 𝑃𝑟𝑤 0.55
and
(1957)
CO2
Dickinson (1958)
Water
[11]
Miropolsky
and
Shitsman (1959, Water
1963) [31]
G (Mg/m²s)
Q
(MW/m²)
D
L (mm)
(mm)
TB (°C)
Remarks
-
-
-
-
-
-
25,0-32,1 2,1-3,4
0,88-1,8
7,6
1600
-
-
22,0-25,0 0,3-1,5
<1,16
8
1500
=<450
-
-
-
-
-
-
-
p (MPa)
𝑁𝑢𝑋
= 0.0375𝑅𝑒𝑋0.77 𝑃𝑟𝑤 0.55
𝑇𝑝𝑐 − 𝑇𝑏
<0
𝑇𝑤 − 𝑇𝑏
𝑇𝑝𝑐 − 𝑇𝑏
<34.5
= 𝑇𝑝𝑐 𝑖𝑓 0 ≤
≤1
𝑇𝑤 − 𝑇𝑏
𝑇𝑝𝑐 − 𝑇𝑏
𝑇𝑤 𝑖𝑓
>1
{
𝑇𝑤 − 𝑇𝑏
𝑇𝑏 𝑖𝑓
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑚𝑖𝑛 0.8
𝑤ℎ𝑒𝑟𝑒 𝑃𝑟𝑚𝑖𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑠𝑠𝑒𝑟 𝑜𝑓 𝑃𝑟𝑏 𝑎𝑛𝑑 𝑃𝑟𝑤
0.35
𝑐̅𝑝
𝑁𝑢𝑏 = 𝑁𝑢0,𝑏 (
)
𝑐𝑝,𝑏
𝜆𝑏 −0.33 𝜇𝑏 0.11
( )
( )
𝜆𝑤
𝜇𝑤
𝑓𝑏
𝑅𝑒𝑏 ̅̅̅̅
𝑃𝑟
8
𝑁𝑢0,𝑏 =
(
2
𝑓 0.5
12.7 ( 𝑏 ) (̅̅̅̅
𝑃𝑟 3 − 1) + 1.07
8
)
𝑓 = (1.82𝑙𝑜𝑔10 (𝑅𝑒𝑏 ) − 1.64)−2
Petukhov,
Krasnoshchekov
Water
and
Protopopov and
(1959,
1961) CO2
[46] [47]
Valid within:
2𝑥104 < 𝑅𝑒𝑏 < 8.6𝑥105
-
0.85 < ̅̅̅
𝑃𝑟𝑏 < 65
0.90 <
𝜇𝑏
< 3.60
𝜇𝑤
1.00 <
𝑘𝑏
< 6.00
𝑘𝑤
0.07 <
𝑐̅𝑝
< 4.50
𝑐𝑝,𝑏
_____________________________________________________________________________________________________
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
Domin
[12]
(1963)
Water
Bishop
(1962,
Water
1965) [13]
Kutateladze
Leontiev
(1964)
Swenson
[14]
Sabersky
[18]
and
[48] -
(1965)
Touba
McFadden
(1966)
𝑁𝑢𝑏 = 0.1𝑅𝑒𝑏0.66 𝑃𝑟𝑏1.2 𝑓𝑜𝑟 𝑇𝑤 ≥ 350°𝐶
𝜇𝑤
𝑁𝑢𝑏 = 0.036𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 ( ) 𝑓𝑜𝑟 𝑇𝑤
22,0-26,0 0,6-5,1
𝜇𝑏
= 250 − 350°𝐶
0.43
2.4 𝐷
0.90 ̅̅̅̅̅̅0.66 𝜌𝑤
𝑁𝑢𝑏,𝑥 = 0.0069𝑅𝑒𝑏,𝑥 𝑃𝑟𝑏,𝑥
( )
(1 +
)
𝜌𝑏 𝑥
𝑥
𝑥 = 𝑎𝑥𝑖𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 ℎ𝑒𝑎𝑡𝑒𝑑 𝑡𝑢𝑏𝑒
22,6-27,5 0,68-3,6
𝑐̅𝑝 𝜇𝑏
ℎ𝑤 − ℎ𝑏
̅̅̅
𝑃𝑟𝑏 =
𝑎𝑛𝑑 𝑐̅𝑝 =
𝜆𝑏
𝑇𝑤 − 𝑇𝑏
2
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 [2⁄√𝜌𝑤 ⁄𝜌𝑏 + 1]
Water
̅̅̅𝑤 =
𝑃𝑟
and
[49] Water
(1967)
𝜌𝑤 0.231
)
𝜌𝑏
ℎ𝑤 − ℎ𝑏
𝑎𝑛𝑑 𝑐̅𝑝 =
𝑇𝑤 − 𝑇𝑏
0.923 ̅̅̅
𝑁𝑢𝑤 = 0.00459𝑅𝑒𝑤
𝑃𝑟𝑤
CO2
𝑐̅𝑝 𝜇𝑤
𝜆𝑤
0.613
0,58-4,5
2,0;
4,0
1075;
1233
=<450
Horizontal
tubes
0,31-3,5
2,55,1
-
294-525
Upward
inside tube
and annulus
-
-
-
-
-
22,7-41,3 0,2-2,0
0,2-2,0
9,4
1830
70-575
-
-
-
-
-
-
-
0,437
-
-
24.925.640.5
Horizontal
260-560
Vertical
tubes
-
-
(
̅̅̅ 𝑒 [2.19(ℎ𝑏⁄ℎ𝑝𝑐−0.801)]
𝑁𝑢𝑏 = 0.0068𝑅𝑒𝑏0.80 𝑃𝑟
-
7.247.58-8.27
-
22.8-30.4
Kondrat’ev
(1969) [50]
Water
𝑁𝑢𝑏 = 0.020𝑅𝑒𝑏0.80
Valid within the range of:
104 < 𝑅𝑒 < 4𝑥105 𝑎𝑛𝑑 𝑇𝑏 = 130 − 600°𝐶.
12.02
25.2-32.0 -
-
Ornatsky et
(1970) [51]
Water
al.
Water
Yamagata (1972)
Water
[16]
𝜌𝑤 0.3
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑚𝑖𝑛 0.8 ( )
𝜌𝑏
𝑤ℎ𝑒𝑟𝑒 𝑃𝑟𝑚𝑖𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑙𝑒𝑠𝑠𝑒𝑟 𝑜𝑓 𝑃𝑟𝑏 𝑎𝑛𝑑 𝑃𝑟𝑤
𝑁𝑢𝑏 = 0.0135𝑅𝑒𝑏0.85 𝑃𝑟 0.8 𝐹𝐶
𝐹𝐶 = 1.0 𝑓𝑜𝑟 𝐸 > 1
-
9.73;
6.35
≤24.3
Ackermann
(1970) [15]
7.62
105-537
220-545
Horizontal
tubes
Vertical
annular
channel
22,7-44,1 0,135-2,17
0,12-1,7
9,424,4
-
77-482
-
-
-
-
-
-
Inside
parallel
tubes
0,1160,930
7,5;
10,0
15002000
230-540
Vertical and
horizontal
-
22,6-29,4 0,31-1,83
_____________________________________________________________________________________________________
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5
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
𝐹𝐶 = 0.67𝑃𝑟𝑚−0.05 (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 )
𝑛1
𝑓𝑜𝑟 0 ≤ 𝐸 ≤ 1
𝑛2
Yaskin
et
(1977) [52]
𝐹𝐶 = (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 ) 𝑓𝑜𝑟 𝐸 < 0
𝑇𝑝𝑐 − 𝑇𝑏
𝐸=
𝑇𝑤 − 𝑇𝑏
𝑛1 = −0.77(1 + 1⁄𝑃𝑟𝑝𝑐 ) + 1.49
𝑛2 = 1.44(1 + 1⁄𝑃𝑟𝑝𝑐 ) − 0.53
2
𝑁𝑢
𝑁𝑢
= [1 − 0.2
𝛽(𝑇𝑤 − 𝑇𝑏 )]
al.
𝑁𝑢0
𝑁𝑢0
Helium
Where 𝑁𝑢0 is calculated with the DittusBoelter equation.
-
-
-
-
-
-
22,5-26,5 0,7-3,6
≤602 G
1.620
-
-
-
(1.061.33)xpcrit
4.6x104
4.1
< q < 2.6
2000
-
ReB=8 1045 105
(x/D)≥15
𝑛
𝜌𝑤 0.3 𝑐̅𝑝
𝑁𝑢𝑏 = 𝑁𝑢0,𝑏 ( ) (
)
𝜌𝑏
𝑐𝑝,𝑏
𝑓𝑏
𝑅𝑒𝑏 ̅̅̅̅
𝑃𝑟
8
𝑁𝑢0,𝑏 =
Water
(
2
𝑓 0.5
12.7 ( 𝑏 ) (̅̅̅̅
𝑃𝑟 3 − 1) + 1.07
8
)
𝑓 = (1.82𝑙𝑜𝑔10 (𝑅𝑒𝑏 ) − 1.64)−2
𝑛 = 0.4 𝑓𝑜𝑟 𝑇𝑏 ≤ 𝑇𝑤 ≤ 𝑇𝑝𝑐 𝑎𝑛𝑑 1.2𝑇𝑝𝑐 ≤ 𝑇𝑏 ≤ 𝑇𝑤
Petukhov,
Krasnoshchekov
and
Protopopov(1966)
(1979) [21]
𝑇𝑤
𝑛 = 0.4 + 0.2 (
− 1) 𝑓𝑜𝑟 𝑇𝑏 ≤ 𝑇𝑝𝑐 ≤ 𝑇𝑤
𝑇𝑝𝑐
𝑇𝑤
𝑇𝑏
𝑛 = 0.4 + 0.2 (
− 1) (1 − 5 (
− 1))
𝑇𝑝𝑐
𝑇𝑝𝑐
CO2
𝑓𝑜𝑟 𝑇𝑝𝑐 ≤ 𝑇𝑏 ≤ 1.2𝑇𝑝𝑐 𝑎𝑛𝑑 𝑇𝑏 < 𝑇𝑤
Valid within:
8𝑥104 < 𝑅𝑒𝑏 < 5𝑥105
̅̅̅ < 65
0.85 < 𝑃𝑟
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
0.09 <
𝜌𝑏
< 1.0
𝜌𝑤
0.02 <
𝑐̅𝑝
< 4.0
𝑐𝑝,𝑏
0.9 <
𝑇𝑤
< 2.5
𝑇𝑝𝑐
𝑛
𝜌𝑤 0.3 𝑐̅𝑝
𝑁𝑢𝑏 = 0.0183𝑅𝑒𝑏0.82 𝑃𝑟𝑏 0.5 ( ) (
)
𝜌𝑏
𝑐𝑝,𝑏
𝑓 = (1.82𝑙𝑜𝑔10 (𝑅𝑒𝑏 ) − 1.64)−2
CO2
𝑛 = 0.4 𝑓𝑜𝑟 𝑇𝑏 ≤ 𝑇𝑤 ≤ 𝑇𝑝𝑐 𝑎𝑛𝑑 1.2𝑇𝑝𝑐 ≤ 𝑇𝑏 ≤ 𝑇𝑤
𝑇𝑤
𝑛 = 0.4 + 0.2 (
− 1) 𝑓𝑜𝑟 𝑇𝑏 ≤ 𝑇𝑝𝑐 ≤ 𝑇𝑤
𝑇𝑝𝑐
Jackson
(1979,
2002) [7]
𝑇𝑤
𝑇𝑏
𝑛 = 0.4 + 0.2 (
− 1) (1 − 5 (
− 1))
𝑇𝑝𝑐
𝑇𝑝𝑐
Water
-
-
-
-
-
-
-
-
-
-
-
-
-
-
𝑓𝑜𝑟 𝑇𝑝𝑐 ≤ 𝑇𝑏 ≤ 1.2𝑇𝑝𝑐 𝑎𝑛𝑑 𝑇𝑏 < 𝑇𝑤
Simplified form (Jackson and Fewster):
0.5
𝑁𝑢𝑏 = 0.0183𝑅𝑒𝑏0.82 ̅̅̅̅̅
𝑃𝑟𝑏
𝜌𝑤 0.3
( )
𝜌𝑏
2
Yeroshenko and
Yaskin
(1981) [53]
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 [2⁄√(0.8𝜓 + 0.2) + 1] 𝐹
0,28
𝐹 = (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 )
𝑎𝑡 𝑐̅𝑝 > 𝑐𝑝,𝑏
𝐹 = 1 𝑎𝑡 𝑐̅𝑝 ≤ 𝑐𝑝,𝑏
𝜓 = 1 + 𝛽𝑏 (𝑇𝑤 − 𝑇𝑏 )
_____________________________________________________________________________________________________
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
Watts
[54]
(1982)
Water
al.
Gorban’ et
(1990) [56]
al. Water
R-12
-
Griem
(1995,
Water
1999) [17]
(1999)
Water
𝑐̅𝑝
𝑁𝑢
=(
)
𝑁𝑢0
𝑐𝑝,𝑏
𝑁𝑢𝑏 = 0.0059𝑅𝑒𝑏0.9 𝑃𝑟𝑏 −0.12 𝑓𝑜𝑟 𝑇 > 𝑇𝑐𝑟𝑖𝑡
𝑁𝑢𝑏 = 0.0094𝑅𝑒𝑏0.86 𝑃𝑟𝑏 −0.15 𝑓𝑜𝑟 𝑇 > 𝑇𝑐𝑟𝑖𝑡
𝜌𝑤 0.231
̅̅̅̅
𝑁𝑢 = 0.0169𝑅𝑒 0.8356 𝑃𝑟 0.432 ( )
𝜔
𝜌𝑏
𝜆 = 0.5(𝜆𝑏 + 𝜆𝑤 )
𝜇 = 𝜇𝑏
-
-
𝑁𝑢 = 0.015𝑅𝑒 0.85 𝑃𝑟 𝑚
8100
𝑚 = 0.69 −
+ 𝑓𝑐 𝑞
𝑞𝑑ℎ𝑡
1.2
𝑞𝑑ℎ𝑡 = 200𝐺
-
-
Buoyancy
effect
0.23-0.3
(0.1(0.190.26)x101,85)
3 kg/s
x10-3
1.8
-
-
-
-
-
-
-
0,20-0,70
1024
-
-
-
0-1.8
MW/m²
-
-
Tb: 20550°C
5
1
𝑐𝑝 = [∑ 𝑐𝑝,𝑖 − 𝑐𝑝,𝑚𝑎𝑥 − 𝑐𝑝,2,𝑚𝑎𝑥 ]
3
𝑖=1
1.0
0.82
𝜔 = 𝑚𝑖𝑛 {
𝑚𝑎𝑥 {
0.82 + 9. 10−7 (ℎ − 1.54. 106 )
-
Local values
of Re: (36–
Tin=4.210.4
90)×103
4.24
<
(Ltot=0.51)
and
Tpc
(Gr/Re2) <
10−2
0.35
Bogachev et
[55] (1983)
Kitoh
[57]
Where 𝑁𝑢0 is calculated with the DittusBoelter equation and 𝛽𝑏 , the volumetric
thermal expansion coefficient.
̅̅̅̅𝑏 0.295
̅̅̅̅𝑏
3000𝐺𝑟
𝐺𝑟
𝑁𝑢 = 𝑁𝑢𝑣𝑎𝑟 𝑝 [1 − 2.7 0.5 ]
𝑓𝑜𝑟
2.7 ̅̅̅ 0.5
̅̅̅
𝑅𝑒𝑏 𝑃𝑟𝑏
𝑅𝑒𝑏 𝑃𝑟𝑏
< 10−4
̅̅̅̅𝑏 0.295
̅𝐺𝑟
̅̅̅𝑏
7000𝐺𝑟
𝑁𝑢 = 𝑁𝑢𝑣𝑎𝑟 𝑝 [1 − 2.7 0.5 ]
𝑓𝑜𝑟
2.7
̅̅̅𝑏
̅̅̅𝑏0.5
𝑅𝑒𝑏 𝑃𝑟
𝑅𝑒𝑏 𝑃𝑟
−4
> 10
With:
̅̅̅̅𝑏
𝜌 0.35
𝐺𝑟
̅̅̅𝑏0.55 ( 𝑤 )
𝑁𝑢𝑣𝑎𝑟 𝑝 = 0.021𝑅𝑒𝑏0.8 𝑃𝑟
𝑓𝑜𝑟
2.7 ̅̅̅ 0.5
𝜌𝑏
𝑅𝑒𝑏 𝑃𝑟𝑏
< 10−4
22,0-27,0 0,3-2,5
-
100-1750
kg/m²s
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
𝑓𝑐
0.11
𝑘𝐽
𝑓𝑜𝑟 0 ≤ 𝐻𝑏 ≤ 1500
𝑞𝑑ℎ𝑡
𝑘𝑔
0.65
𝑘𝐽
= −8.7 𝑥 10−8 −
𝑓𝑜𝑟 1500 ≤ 𝐻𝑏 ≤ 3300
𝑞𝑑ℎ𝑡
𝑘𝑔
1.30
𝑘𝐽
−9.7 𝑥 10−7 +
𝑓𝑜𝑟 3300 ≤ 𝐻𝑏 ≤ 4000
𝑞𝑑ℎ𝑡
𝑘𝑔
{
2.9 𝑥 10−8 +
Komita
[58]
(2003)
Sakurai
[25]
(2000)
Water
-
CO2
-
-
-
-
-
-
-
Deteriorated
heat
transfer
+
Buoyancy
effect
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
0.6
Kim et
(2007)
al
[59]
CO2
Mokry et al [60]
Water
(2009)
𝑐̅𝑝
𝜉𝑀
𝜌𝑤 𝑛
𝑁𝑢 = 𝑁𝑢𝐹 ( ) (
) ( )
𝜉𝐹 𝑐𝑝,𝑏
𝜌𝑏
𝑁𝑢𝐹 = 0.0243𝑅𝑒𝑏0.8 𝑃𝑟𝑏0.4
For circular tubes :
𝑞
𝑞 2
𝑛 = 0.955 − 0.0087 ( ) + 1.30𝑥10−5 ( )
𝐺
𝐺
0.564
0.904 ̅̅̅ 0.684 𝜌𝑤
𝑁𝑢𝑏 = 0.0061𝑅𝑒𝑏 𝑃𝑟𝑏
( )
𝜌𝑏
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
When comparing these correlations, it has to be noted that a lot of scatter exist
between them. Most of them, but not all, show the trends of enhanced heat
transfer when the wall temperature approaches the pseudo-critical temperature.
The problem with these correlations is that almost all of them do not even take
the orientation of the flow and the interaction of this with buoyancy into
consideration. Furthermore all correlations are applicable only to cases without
heat transfer deterioration.
The Dittus-Boelter equation [61] for forced convective heat transfer in turbulent
flows at subcritical pressures was also used for supercritical heat transfer. This
equation shows a relative good agreement with the experimental data for water
flowing inside circular tubes at 310 bar and low heat fluxes, but is completely
unsuitable near the critical and pseudo-critical points.
The correlation developed by Bringer and Smith (1957) [45] for supercritical
water and carbon dioxide did not take the peak in thermal conductivity into
account near the pseudo-critical temperature. Miropolsky and Shitsman (1959,
1963) [31], on the other hand, assumed that the thermal conductivity was a
smooth decreasing function of the temperature near the critical ad pseudocritical points.
Krasnoshchekov, Protopopov and Petukhov (1959, 1960, 1961) [46] [47] took
the variations of the thermophysical properties into account by using the
averaged Prandtl number and specific heat. About 85% of their data and data of
former experiments by other researchers ( [11], [31]) matches with their
proposed correlation and showed a deviation within ±15%. The proposed
correlation for forced convection heat transfer in carbon dioxide and water is
valid within the ranges as specified in Table 6.
Domin (1963) [12] performed experiments with supercritical water flowing inside
horizontal tubes and proposed two correlations according to the temperature
range. Bishop et al. [13] (1965) performed experiments for supercritical water
flowing upward inside a tube and an annulus. The proposed correlation has been
found to correlate their data with an accuracy of ±15% and they’ve also
considered the entrance effect in the heat transfer correlation.
In the correlations of Swenson (1965) [14] and Griem (1995) [17], the fluid
properties are not calculated on the bulk temperature, compared to most
correlations. Swenson et al. use the wall temperature as reference and Griem
chooses a temperature to avoid a severe variation in heat transfer coefficient. It
correlated 80% of the data points to within ± 15% and 91% to within ± 20%.
The correlation of Swenson predicted the data of carbon dioxide with a very good
accuracy. However, Swenson et al. also assumed that the thermal conductivity
near the critical and pseudo-critical temperature was a gradually decreasing
function of temperature.
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Report no. 00000
In 1966, Petukhov, Krasnoshchekov and Protopopov [21] adapted their earlier
proposed correlation for supercritical water using a Dittus-Boelter form with
additional terms involving wall to bulk density ratio and integrated to bulk
specific heat ratio, each raised to suitable powers. Later in 1971,
Krasnoshchekov [62] added a correction factor to the correlation to take the
entrance effect into account in the form of 𝑓(𝑥 ⁄𝐷) = 0.95 + 0.95(𝑥 ⁄𝐷)0.8 . This
correction factor can also be used for a heated tube with an abrupt inlet, valid in
the range of 2 ≤ (𝑥 ⁄𝐷) ≤ 15.
The very simple correlation proposed by Kondrat’ev (1969) [50] (1969), is valid
for supercritical heat transfer inside vertical and horizontal tubes and inside
vertical annular channels within the range of 104 < 𝑅𝑒 < 4𝑥105 𝑎𝑛𝑑 𝑇𝑏 = 130 − 600°𝐶.
Most of the experimental data corresponds with the correlation (within ±10%),
but this is not valid in the pseudo-critical range.
Ornatsky et al. (1970) [51] modified the correlation proposed by Miropolsky and
Shitsman (1959, 1963) [31], taking the density ration between the bulk and wall
temperature into account.
Further in 1972, Yamagata et al. [16]proposed a correlation for the forced
convection heat transfer to supercritical water flowing inside tubes.
In 1979 and 2002, Jackson and Hall [7] reviewed and adjusted the correlation of
Petukhov, Krasnoshchekov and Protopopov [21] (1966) by using approximately
2000 experimental data for water and carbon dioxide. They had also excluded
data that may have been affected by buoyancy, which gave an essential advance
over earlier attempts to correlate forced convection data. In this form the
equation correlated 77% of the data points to ± 15% and 90% to within ± 20%.
Approximately 2000 data points were tested against the correlation. A simplified
correlation was proposed by Jackson and Fewster [63] (1975), making the
correlation similar to the correlation of Bishop without the effect of geometrical
parameters and with different values of constant and exponents. Furthermore,
Jackson tested the Krasnoshchekov correlation (originally for CO2) also for water
and found that this gives rather good results for a certain parameter range, as
specified in Table 6.
Yaskin et al. (1977) [52] found that available data on heat transfer to
supercritical helium in a purely forced convection flow regime can be correlated
on the basis of an analogy with the heat transfer process accompanying gas
injection at a heated wall.
Yeroshenko and Yaskin (1981) [53] proposed a correlation by analysing the
correlating equations of Miropol’skii and Shitsman (1957), Krasnoshchekov and
Protopopov (1966), Pron’ko et al. [64] (1976), Petukhov et al. (1976). In this
correlation, a correction factor F is added, which account for the possible heat
transfer enhancement (ℎ⁄ℎ0 > 1).
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Report no. 00000
Bogachev et al. [55] (1983) gave special attention to the conditions of heat
transfer increase during turbulent flow of helium, where free convection effect
can be neglected. The experiments were carried out in a vertical tube with a
constant uniform heat flux. Local values of Reynolds number were 36000–90000
and the parameter (Gr/Re2) < 10−2, which allowed the consideration of these
flow regimes as regimes without the effect of natural convection. The values for
(Nu/Nu0) > 1 were described with an accuracy of about ±20% by the Protopopov
equation.
Kirillov et al. [65] (1990) also showed that the role of free convection in heat
transfer near the critical point can be taken into account via the parameter
𝜌
𝐺𝑟
(Gr/Re2) or 𝑘 ∗ = (1 − 𝜌𝑤) 𝑅𝑒 2 [Eq. 9]. For k*<0.4 or (Gr/Re2) < 0.6, deteriorated
𝑏
heat transfer exists. At larger values of these terms, improved heat transfer
occurs.
For heating of a supercritical fluid flowing inside a circular tube at a constant
uniform heat flux, Kirillov et al [65] (1990) proposed to use the following
equations:
𝑐̅𝑝
𝑁𝑢
𝑁𝑢0
𝑁𝑢
𝑁𝑢0
𝑛
𝑚
𝜌
= (𝑐 ) ( 𝜌𝑤)
𝑝,𝑏
=(
𝑐̅𝑝
𝑐𝑝,𝑏
𝑛
𝑏
(for k*< 0.01) [Eq. 10]
𝜌 𝑚
𝜌𝑏
) ( 𝑤) 𝜑(𝑘 ∗ ) (for k*> 0.01) [Eq. 11]
Where the value of 𝜑(𝑘 ∗ ) can be found in Table 7.
Table 7: Values for 𝝋(𝒌∗ )
k*
𝜑(𝑘 ∗ )
0.01 0.02 0.04 0.06 0.08 0.1
0.2
1
0.88 0.72 0.67 0.65 0.65 0.74
0.4
1
The local values of 𝑁𝑢0 for smooth circular tubes under turbulent flow can be
calculated as follows:
̅̅̅̅
(𝑓⁄2)𝑅𝑒𝑃𝑟
̅̅̅̅ 2/3 −1)
(𝑃𝑟
𝑁𝑢0 = 𝑏+4.5𝑓1/2
̅̅̅ = 1 − 2000 and 𝑅𝑒 = (4 − 5000)103) [Eq. 12]
(𝑃𝑟
Where
𝑇𝑏
is
the
characteristic
−2
𝑓 = (1.82𝑙𝑜𝑔10 (𝑅𝑒𝑏 ) − 1.64) .
temperature,
𝑏 = 1 + (900⁄𝑅𝑒)
and
For k*< 0.01, Eq. 10 can be used to calculate the deteriorated heat transfer for
any value of k*. A peak in wall temperature usually appears in the tube cross
section, where the fluid temperature is lower than the pseudo-critical
temperature by several degrees. Possibly the deteriorated heat transfer at k* <
0.01 is associated with the effects of acceleration and variability of phusical
properties over the flow cross section in the process of turbulent transport.
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Report no. 00000
At k* = 0.01 – 0.4, additional deterioration of heat transfer occurs due to the
effect of natural convection. Maxima in wall temperature appear in the tube cross
section, where the average flow temperature is lower than the pseudo-critical
temperature by 15-20°C or more.
At k* > 0.4, heat transfer decreases when the effects of natural convection
disappears and the regime with improved heat transfer starts.
In Eq. 10 and Eq. 11, 𝑁𝑢 and 𝑁𝑢0 are calculated based on the average bulk
temperature and 𝑐̅𝑝 =
ℎ𝑏 −ℎ𝑤
𝑇𝑏 −𝑇𝑤
is the integral average specific heat in the range of
(𝑇𝑏 − 𝑇𝑤 ). The exponent m is 0.4 for upward flow in vertical tubes; m = 0.3 for
horizontal tubes. For horizontal tubes, the exponent n is calculated using the
ratios 𝑇𝑏 ⁄𝑇𝑝𝑐 and 𝑇𝑤 ⁄𝑇𝑝𝑐 , where all temperatures are in Kelvin. For downward flow
in vertical tubes, the exponents m and n are calculated in the same way as those
for horizontal tubes. For an upward flow in vertical tubes at (𝑐̅𝑝 ⁄𝑐𝑝𝑏 ) ≥ 1, 𝑛 = 0.7;
for (𝑐̅𝑝 ⁄𝑐𝑝𝑏 ) < 1, the value of n is determined according to Table 8, the same as
for horizontal tubes.
Table 8: Values of exponent n
Gorban’ et al. (1990) [56] proposed a correlation for the forced heat transfer to
R-12 and water at temperatures above the critical temperature.
In 1999, Kitoh [57] proposed a correlation for forced convection in supercritical
water, taking the heat flux at which deteriorated heat transfer occurs, into
account.
Cheng X. et al. [3] compared the most important correlations for a certain
condition, applicable for a High Performance Light Water Reactor as can be seen
in Figure 25.
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
Figure 25: Heat transfer coefficient for supercritical water according to different
correlations [3].
All correlations show a maximum value at a bulk temperature near (or lower
than) the pseudo-critical temperature (384°C). For the bulk temperature far
away from the pseudo-critical temperature, a satisfied agreement is obtained
between different correlations, whereas a big deviation is observed as the fluid
bulk temperature approaching the pseudo-critical value. For the parameter
combination considered, the Dittus-Boelter equation gives the highest heat
transfer coefficient which occurs when the fluid bulk temperature is equal to the
pseudo-critical value. The correlation of Swenson (1965) [14] shows the lowest
peak of heat transfer coefficient. At the pseudo-critical temperature, the heat
transfer coefficient determined by the Swenson correlation is about 5 times lower
than that of Dittus-Boelter equation, about 3 times lower than that of Yamagata
and is about 50% of that of Bishop.
3. Conclusion
A general form of a modified Dittus-Boelter correlation which can be used for
designing an own correlation is written as:
𝑁𝑢𝑋 = 𝐶 𝑅𝑒𝑋𝑛 𝑃𝑟𝑋𝑚 𝐹
The subscript indicates the reference temperature which is used for the
calculation of the properties (b, w, ps and x are used, respectively for bulk, wall,
pseudo-critical and mixed temperature). The coefficient C, as well as the
exponents n and m are experimentally determined. Due to the severe property
variations an additional term F is added which takes the property variation,
buoyancy and entrance effect into account.
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Report no. 00000
When comparing several correlations, it is noticed that the calculated heat
transfer coefficients are quite different, especially with high heat fluxes. Some of
them show similar results and correlate with the experimental data for normal
heat transfer in water and carbon dioxide. However, none of them is able to
accurately predict the onset and magnitude of heat transfer enhancement and
deterioration. The major reason why these correlations predict different values
for the heat transfer coefficient is that these are closely related to the significant
changes of the thermophysical properties near the critical and pseudo-critical
points. One of the design criteria of a heat exchanger is also the maximum
allowable surface temperature, which means that the prediction of a heat
transfer coefficient in the deteriorated region is of major importance.
An alternative, instead of using dimensionless parameters, semi-empirical
correlations were developed by solving the equations of motions and energy
using empirical data on turbulent diffusion. The problem here is that appropriate
mathematical functions have to be developed to fit an empirical result to the
disadvantage of the physical understanding of the phenomena, making these
correlations deviating strongly from a modified Dittus-Boelter form. These
correlations will not be discussed in detail in this literature study, as the focus of
this study is on experimental research. An overview of some semi-empirical
correlations is given in Table 9.
Due to the further development of supercritical water cooled reactors, new
prediction methods have been developed, taking more parameters into account.
This makes the correlation even more complex and therefore not always more
accurate.
Jackson et al. [66] (2008) proposed a correlation of the following form:
𝑁𝑢𝑏 = 𝐶. 𝑅𝑒𝑏𝑛 . 𝑃𝑟𝑏𝑚 . 𝐹
With
𝐹 = 𝑓1 (
𝑞.𝛽 𝐷
𝜌𝑤 𝑐𝑝,𝐴
𝜌𝑤 𝜇𝑤
,
, 𝑅𝑒𝑏 , 𝑃𝑟𝑏 , 𝑏 )
) . 𝑓2 ( ,
𝜌𝑏 𝑐𝑝,𝐵
𝜌𝑏 𝜇𝑏
𝜆𝑏
Kuang et al. [67] (2008) used the following correction factor:
𝐹 = 𝑓 (𝐺𝑟,
𝜌𝑤 𝑐𝑝,𝐴 𝑞.𝛽𝑏 𝐷 𝜇𝑤 𝜆𝑤
,
,
,
, )
𝜌𝑏 𝑐𝑝,𝐵 𝑐𝑝,𝐵 𝐺 𝜇𝑏 𝜆𝑏
In 2009, Cheng X. et al. [68] presented a new approach to derive a prediction
correlation, where the emphasis is placed on the simplicity of the structure and
its explicit connection with the physical phenomena. The correlation only consists
of 1 dimensionless number to correlate the correction factor and excludes the
direct dependence of the heat transfer coefficient on the wall temperature. The
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Report no. 00000
correlation was validates with the experimental data of Herkenrath et al. [69]
(1967).
It is generally agreed that the correlations do not show sufficient agreement with
experiments to justify their use except in very limited conditions.
At bulk temperatures well above the critical temperature, the heat transfer
resembles more to a normal single phase heat transfer to a gas, which can be
predicted with a conventional Dittus-Boelter type of correlation.
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Report no. 00000
Table 9: Summary of the semi-empirical correlations for supercritical fluids.
Fluid
Correlation
p
(MPa)
G
(Mg/m²s)
Q
(MW/m²)
D
(mm)
L (mm)
L/D
TB
(°C)
Remarks
𝑁𝑢
̃≤1
1,
𝐾
= { −𝑚
̃
̃>1
𝑁𝑢𝑛
𝐾
,
𝐾
𝜀0
𝐺𝑟
1
̃ = [ 𝑢 + 𝑔𝑥 𝑚 ]
𝐾
𝐹
𝑅𝑒𝐵2 𝜀𝑛 [1 − 𝑒𝑥𝑝(− 𝑅𝑒𝑓 ⁄30000)]
Kurgan
ov
(1985,
1993)
[70]
CO2,
Water,
He
̃ accounts the effect of buoyancy
The parameter 𝐾
and the effect of acceleration induced by the
density variation near the heated wall.
The friction at supercritical condition is computed
by
𝜌𝑊 0.4
𝜀𝑛 = 𝜀0 ( ) , 𝜀0 = [0.55⁄𝑙𝑜𝑔(𝑅𝑒𝐵 ⁄8)]2
𝜌𝐵
̅̅̅𝐵
𝜀𝑛 𝑅𝑒𝐵 𝑃𝑟
𝑁𝑢𝑛 =
⁄
1 + (900⁄𝑅𝑒𝐵 ) + 12.7(𝜀𝑛 ⁄8)0.5 (𝑃𝑟𝐵2 3 − 1)
Nun represents the Nusselt number at normal heat
transfer conditions, i.e. without heat transfer
deterioration.
8𝑞𝑊 𝛽𝐵
𝑔 𝑑3
𝜌𝑊
𝜀𝑢 =
, 𝐺𝑟𝑢 = 2 (1 −
)
𝐺 𝑐𝑝,𝐵
𝜌𝐵
𝑣𝑏
In case of a strong effect of buoyancy and
̃ ≥1), a correction factor is
acceleration (𝐾
introduced to account the heat transfer reduction.
The exponent m is dependent on the heated
length and expressed as:
𝑥⁄
𝑚 = 0.55 [1 − 𝑒𝑥𝑝 (− 𝑑 )]
50
2
𝑞𝑊 𝛽𝐵 𝑑 ⁄4𝑥
𝐹 = 1 − 0.8𝑒𝑥𝑝 [−3 (
) ]
𝐺 𝑐𝑝,𝐵 𝑙𝑛(𝜌𝑖𝑛 ⁄𝜌𝑩 )
Based
on
mechanistic
analysis.
-
-
-
-
-
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≥40
-
a
Application:
Circular tubes,
downward,
upward
and
horizontal
𝐷
𝑔 2 ≤ 0.015
𝑢𝑖𝑛
𝑅𝑒𝑖𝑛 ≥ 2. 104
+
no
considerable
change in wall
heat flux over
the length
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Literature Study - Convective heat transfer to fluid operating at a supercritical pressure
Report no. 00000
0.69−81000 ⁄𝐶𝐻𝐹 +𝑓 𝑞
Koshizu
ka
(2000)
[71]
𝑐
𝑁𝑢𝐵 = 0.015𝑅𝑒𝐵0.85 𝑃𝑟𝐵
0.11
𝑓𝑐 = 2.9. 10−8 +
𝑓𝑜𝑟 ℎ ≤ 1.5 𝑀𝐽⁄𝑘𝑔
𝐶𝐻𝐹
0.65
𝑓𝑐 = −8.7. 10−8 −
𝑓𝑜𝑟 1.5 ≤ ℎ ≤ 3.3 𝑀𝐽⁄𝑘𝑔
𝐶𝐻𝐹
1.30
−7
⁄
{ 𝑓𝑐 = −9.7. 10 + 𝐶𝐻𝐹 𝑓𝑜𝑟 3.3 ≤ ℎ ≤ 4 𝑀𝐽 𝑘𝑔
1.2
𝐶𝐻𝐹 = 200𝐺
-
1.0-1.75
00-1.8
-
-
-
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20550
-
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Report no. 00000
References
[1] Igor L. Pioro, Hussam F. Khartabil, Romney B. Duffey, Heat transfer to
supercritical fluids flowing in channels—empirical correlations (survey),
Nuclear Engineering and Design 230 (2004) 69–91.
[2] Kyoung-Ho Kang, Soon-Heung Chang. Experimental study on the heat
transfer characteristics during the pressure transients under supercritical
pressures, International Journal of Heat and Mass Transfer 52 (2009) 4946–
4955.
[3] Cheng X, Schulenberg T; Heat transfer at supercritical pressure literature
review and application to an HPLWR, Forschungszentrum Karlsruhe, 2001.
[4] W.B. Hall, Heat Transfer near the Critical Point, Advances in Heat Transfer,
Volume 7, 1971 , p.1-86.
[5] Petukhov, B. S. Heat transfer and friction in turbulent pipe flow with variable
physical properties, Advances in Heat Transfer, Vol. 6, Academic Press, New
York, 1970, 511-564.
[6] Jackson, J D, Hall, W B, Heat Transfer to Supercritical Pressure Fluids,
H.T.F.S. Design Report No. 34, Part 1—Summary of design
recommendations and equations, Part 2—Critical reviews with design
recommendations, AERE-R8157 and R8158, 1975.
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