Literature Study
Supercritical heat transfer
Department of Flow, Heat and Combustion Mechanics
Catternan Tom
Table of Content
Table of Content ...................................................................................................................................... 2
Nomenclature.......................................................................................................................................... 4
Chapter 1 Introduction ............................................................................................................................ 5
1.
Supercritical state .................................................................................................................... 5
2.
Thermophysical fluid properties ............................................................................................. 7
Chapter 2 Forced convection heat transfer in supercritical fluids ........................................................ 11
1.
Introduction ........................................................................................................................... 11
2.
Literature review ................................................................................................................... 11
3.
Review experimental studies ................................................................................................ 16
4.
Data presentation [4] ............................................................................................................ 19
5.
6.
4.1
Description in terms of local conditions only ................................................................ 20
4.2
Presentation in terms of a heat transfer coefficient ..................................................... 21
4.3
Presentation in terms of dimensionless groups ............................................................ 22
General characteristics for supercritical heat transfer ......................................................... 23
5.1
Heat transfer enhancement .......................................................................................... 25
5.2
Heat transfer deterioration ........................................................................................... 26
5.3
Influence of the heat flux .............................................................................................. 29
5.4
Influence of the mass flow flux ..................................................................................... 32
5.5
Influence of the direction of flow .................................................................................. 33
5.6
Influence of the diameter of the pipe ........................................................................... 34
5.7
Influence of buoyancy ................................................................................................... 35
Summary and future experimental works ............................................................................ 36
Chapter 3 Heat transfer regimes and mechanisms ............................................................................... 38
3.1
Normal heat transfer ..................................................................................................... 40
3.2
Enhanced heat transfer ................................................................................................. 41
3.3
Heat transfer deterioration ........................................................................................... 42
Chapter 4 Correlations for forced convection supercritical heat transfer ............................................ 47
Chapter 5 Friction and pressure drop in supercritical fluids ................................................................. 68
Chapter 6 Experimental system and data reduction............................................................................. 69
1.
Experimental setup ............................................................................................................... 69
2.
Procedure and conditions ..................................................................................................... 69
3.
Data reduction ....................................................................................................................... 69
4.
Uncertainty analysis .............................................................................................................. 69
Chapter 7 Numerical analysis ................................................................................................................ 70
Chapter 8 Free convection heat transfer in supercritical fluids ............................................................ 71
References ............................................................................................................................................. 72
Nomenclature
Greek symbols
Sub- and superscripts
Acronyms
Chapter
Introduction
1
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 fossil-fired 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. Besides 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, TS-1, 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 physical properties with
temperature, with as a result that theoretical 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).
C
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 liquidvapour 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.
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,
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 [63].
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 transcritical 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.
Figure 2: Isothermal lines in a p,v-diagram
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 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).
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
(), 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 pseudo-critical 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].
The strong dependence of the thermodynamic properties on temperature and pressure leads to
different heat transfer regime.
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 thermophysical properties of water at different pressure and temperature can be calculated
using the NIST software (1996, 1997). Also, the latest NIST software (2002) 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.
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.
Experimental works carried out for supercritical water are summarized in Table 1 for carbon dioxide
in
Table 2, for cryogens in
Table 3 and for refrigerants in
Table 4 with their test conditions.
Table 1: Summary of the test condition for supercritical water.
p (MPa)
Dickinson (1958) [11]
G (Mg/m²s) Q (MW/m²) D (mm)
L/D
TB (°C)
T (°C)
Remarks
Subject
0,88-1,8
7,6
1600
-
-
-
-
Heat transfer
Shitsman (1959, 1963) 22,0-25,0 0,3-1,5
<1,16
8
1500
-
=<450
-
-
Heat transfer, heat transfer deterioration,
oscillation
Domin (1963) [12]
0,58-4,5
2,0; 4,0
1075; 1233 -
=<450
-
-
Heat transfer, oscillation
Bishop (1962, 1965)
22,6-27,5 0,68-3,6
[13]
0,31-3,5
2,5-5,1
-
30-565 294-525 16-216
-
Heat transfer
Swenson (1965) [14]
22,7-41,3 0,2-2,0
0,2-2,0
9,4
1830
-
70-575
6,0-285 -
Heat transfer, Heat transfer deterioration
Ackermann
[15]
22,7-44,1 0,135-2,17
0,12-1,7
9,4-24,4
-
-
77-482
-
Heat transfer, pseudo-boiling phenomena
Yamagata (1972) [16]
22,6-29,4 0,31-1,83
0,116-0,930 7,5; 10,0 1500-2000
-
230-540 -
Vertical
horizontal
Griem (1999)
22,0-27,0 0,3-2,5
0,20-0,70
-
-
-
(1970)
25,0-32,1 2,1-3,4
L (mm)
22,0-26,0 0,6-5,1
10-24
-
-
and
Heat transfer, Heat transfer deterioration
Heat transfer
Table 2: Summary of the test condition for supercritical carbon dioxide.
p (MPa)
Sabersky (1967) [17]
7.247.58-8.27
G (Mg/m²s) Q (MW/m²) D (mm)
0,437
L (mm)
L/D
TB (°C)
24.925.640.5
T (°C)
Remarks
Subject
Horizontal
Visualisation, turbulence
Jackson (1966, 1968)
Heat transfer, buoyancy effect
Petukhov (1979)
Heat transfer, pressure drop
Kurganov (1985, 1993)
Flow structure
Sakurai (2000) [18]
Flow visualisation
Table 3: Summary of the test condition for supercritical cryogens.
p (MPa)
G (Mg/m²s) Q (MW/m²) D (mm)
L (mm)
L/D
TB (°C)
T (°C)
Remarks
Subject
Table 4: Summary of the test condition for supercritical refrigerants.
p (MPa)
G (Mg/m²s) Q (MW/m²) D (mm)
L (mm)
L/D
TB (°C)
T (°C)
Remarks
Subject
3. Review 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) [19] 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 DittusBoelter 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
[20] [21]. The phenomenon of heat transfer deterioration was first observed by Shitsman et al.
(1963) [20] 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
o G: 0.1 – 5.1 Mg/m²s
o Q: 0.0 – 4.5 MW/m²
o D: 2.0 – 32.0 mm
o 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) [22] 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 [23].
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
[17] [24] [25] [18]. 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) performed heat transfer experiments in horizontal flowof carbon dioxide at
supercritical and subcritical pressures. Axial (Fig. 9a and b) and circumferential (Fig. 9c) temperature
profiles were obtained. It was found that non-uniform cross-section temperature profile exists in
horizontal flow (Fig. 9c). 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). Fig. 10 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.
Ko et al. (2000) performed flow visualization experiments in a vertical one-side 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).
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 heated vertical tube with a diameter of 1.9cm
(Evans et al. PhD thesis [26]. 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 6 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 6: 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 [26].
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 7 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.
Bulk
temperature
(●) 19°C
(+) 22°C
(∆) 25°C
(x) 28°C
(□) 31°C
Figure 7: Heat flux versus wall temperature for various bulk temperatures [26].
The dotted lines are fitted because they were not measured. The point where they intersect the Twaxis, 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 8), one might think that high heat fluxes are possible with small
temperature differences, while in Figure 7 it can be seen that is not possible.
Bulk
temperature
(●) 19°C
(+) 22°C
(∆) 25°C
(x) 28°C
(□) 31°C
Figure 8: Heat transfer coefficient versus wall temperature for various bulk temperatures [26].
The use of the heat transfer coefficient for supercritical fluids has been questioned by Goldman [27].
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
with c, n and s constants.
Goldman, however, suggested collecting all the temperature dependent terms in the dimensionless
groups.
𝑞0 𝑑1−𝑛
= 𝑓(𝑇0 , 𝑇𝑚 )
(𝜌𝑢)𝑛
This presentation resembles more to the data presented in Figure 7, 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.
4.3
Presentation in terms of dimensionless groups
Using the correlation developed by Miropolsky and Shitsman [28], the same data as in Figure 8 is
presented in Figure 9.
𝑁𝑢𝑚 = 𝑐 (𝑅𝑒𝑚 )𝑛 (𝑃𝑟𝑚𝑖𝑛 )𝑠
Figure 9: Correlation of the data of Figure 8.
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 7, and also the fact that it is impossible to recover the original data
from such a presentation.
5. General characteristics for supercritical heat transfer
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.
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 10 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 5 and Table 6.
p = 1.05 pcrit
p = 1.15 pcrit
Figure 10: Experimental wall temperature distributions as a function of local bulk enthalpy along a pipe: p = 1.05 pcrit
and p = 1.15 pcrit [29].
Table 5: Experimental conditions for supercritical water at p = 1.05 pcrit [29].
a
b
c
d
e
Shitsman [20]
Shitsman [20]
Shitsman [20]
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
Table 6: Experimental conditions for supercritical water at p = 1.15 pcrit [29].
a
b
c
d
e
f
Vikrev and Lokshin [30]
Vikrev and Lokshin [30]
Schmidt [31]
Schmidt [31]
Domin [12]
Shitsman [20]
𝒒 (𝑾/𝒄𝒎𝟐 )
69.9
69.9
58
82
91
39.6
𝒎̇⁄𝑨 (𝒈𝒎⁄𝒔 𝒄𝒎²)
100
40
61
61
101
44.9
𝒅 (𝒄𝒎)
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 10
curves a, b and c for p = 1.05 pcrit.

When 𝑇𝐵𝑢𝑙𝑘 ≤ 𝑇𝑐𝑟𝑖𝑡 ≤ 𝑇𝑤𝑎𝑙𝑙 , local enhancement (Figure 10 for p = 1.15 pcrit – curve e) and
deterioration (Figure 10 e.g. for p = 1.05 pcrit – curves a and b) can occur in the heat transfer.
From the experimental data in Figure 10 is it clear that the orientation of the heated pipe is from
major importance.
5.1
Heat transfer enhancement
On Figure 7 and Figure 8 (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 [32] (supercritical CO2 – vertical upward
flow – d = 1.0cm) in Figure 11, 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 8,
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.
(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 11: 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 [32].
5.2
Heat transfer deterioration
In Figure 10, 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. [33] compared an upward and downward supercritical water flow for several uniform
heat fluxes (Figure 12) 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.
Table 7: Experimental conditions for supercritical water at 245 bar in a vertical upward and downward 1.6 cm diameter
heated pipe ( 1.11 pcrit) [33].
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 12: 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 [33].
Jackson et al. [34] 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 [33], occurred below 𝑇𝑝𝑐 as well as above 𝑇𝑝𝑐 . Tanaka et al. [32] (Figure 11)
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.
Evans et al. [26] 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 13). 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 13: 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 [26].
The deteriorations in horizontal pipes are less prompt than vertical upward flow pipes. Miropolsky
and Shitsman [28] measured the temperature distribution for supercritical water around a horizontal
and vertical 1.6 cm diameter pipe (Figure 14). 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 14, this leads to a reduction in the heat
transfer coefficient of about a factor 4 compared to the lower surface.
(1) Horizontal pipe – upper surface
(2) Horizontal pipe – lower surface
(3) Vertical pipe – upward flow
(4) Bulk fluid temperature
Figure 14: 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 [28].
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
[35], Jackson et al. [34] and Bourke et al. [36], where only the test section diameter differs (Table 8).
Table 8: Comparison of three sets of data for supercritical CO2 flowing up- and downwards in a vertical pipe [4].
𝑹𝒆𝒇𝒆𝒓𝒆𝒏𝒄𝒆
Shiralkar and Griffith [35]
Jackson and Evans-Lutterodt [34]
Bourke et al. [36]
𝒅 (𝒄𝒎)
0.635
1.905
2.285
𝒒 (𝑾/𝒄𝒎𝟐 )
15.8
5.67
5.1
𝑹𝒆
1.0
1.24
0.82
𝑮𝒓
1
27
46.5
𝒒. 𝒅 (𝑾/𝒄𝒎)
10.0
10.8
11.6
𝒑 (𝒃𝒂𝒓)
75.8
75.8
74.5
Figure 15 shows the wall temperature as a function of the bulk enthalpy for a downward and upward
flow.
Legend:
__.__.__: Shiralkar and Griffith [35]
_______: Jackson and Evans-Lutterodt [34]
_ _ _ _ _ : Bourke et al. [36]
Figure 15: Comparison of the data of Shiralkar and Griffith [35], Jackson and Evans-Lutterodt [34] and Bourke et al. [36]
for forced convection of carbon dioxide flowing up- and downwards in vertical heated pipes [4].
No significant difference was found between an upward and downward flow for the data of Shiralkar
and Griffith [35], while for larger pipe diameters, Jackson et al. [34] and Bourke et al. [36] observed
sharp peaks for an upward flow, as already seen in experiments by Shitsman [20] in Figure 10.
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 [20] (Figure 10) 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 12, Figure 13 and Figure 14) is smoother compared to the
much sharper increase for the boiling phenomenon at subcritical pressures.
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
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 𝜇 0.4
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  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 16).
Figure 16: 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 17 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 pseudocritical 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.
Figure 17: Heat transfer coefficient as a function of the fluids bulk temperature according to the Dittus-Boelter equation
[3].
Figure 18: (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 18, 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 18, 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 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 [30] in Figure 10 and from Tanaka, Nishiwaki and
Hirate [32] in Figure 11, 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 pseudocritical 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 [30] 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 19 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 19: Generalized curves for water at 250bar (Lokshin et al. [30])
5.5
Influence of the direction of flow
Shitsman et al. [33] (Water: Figure 12) and Evans et al. [26] (CO2: Figure 13) 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 20, from an experiments by Jackson and Evans-Lutterodt
[34] performed a similar experiment with carbon dioxide for an upward flow and came to the same
conclusion.
Figure 20: Comparison of heat transfer between an upward and downward flow for CO 2 by Jackson and Evans-Lutterodt
[34].
The deteriorations in horizontal pipes are less prompt than vertical upward flow pipes (Figure 14).
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.
5.6
Influence of the diameter of the pipe
Tanaka et al. [32] (Figure 11) and Jackson et al. [34] 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 21). A slightly stronger effect of the tube diameter was found using the correlation
of Bishop [13] and of Krasnoshchekov [22].
Figure 21: 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.
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. [37] performed some measurements for heat transfer to supercritical water in
horizontal pipes (Figure 22). 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 22: Heat transfer in a horizontal supercritical flow for different values of 𝐪̇ ⁄𝐦̇ (Belyakov et al. [37]).
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
̅̅̅̅𝑏
𝐺𝑟
𝑅𝑒𝑏
̅̅̅𝑏 is the Grashof number (=
where ̅𝐺𝑟
2.7
̅ )𝐷3
𝑔 (𝜌𝑏 −𝜌
),
𝜈𝑏 2
< 10−5
in which 𝜌̅ is the integrated mean density and the
subscript b indicates physical properties evaluated at the local bulk temperature.
The buoyancy parameter Bo, defined as
, can be used to determine
whether the flow is in the forced convection or mixed convection regime (Hall and Jackson, 1969
[38]).
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 [38], the effects of radial differences
in viscosity [39] and the effects of rapid changes in density in the flow [40] 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. [34] used 200 thermocouples on a 1.9 cm diameter pipe
over a length of 3m).
More detailed work is necessary for horizontal pipes.
Chapter
3
Heat transfer regimes and mechanisms
At supercritical pressure, despite non-existence of tangible phase change, the working fluid
undergoes a transition from liquid- like substance to gas-like one without any of discontinuities
associated with two phases being present when the fluid temperature rises up and passes the
pseudo-critical temperature. Depending on the applied heat flux and the mass flux of flow, the heat
transfer regime can be categorized into three types of enhanced, normal and deteriorated heat
transfer at supercritical pressure. In general, deviations from normal heat transfer have been found
to occur when the wall temperature is greater than the pseudo-critical temperature and the bulk
fluid temperature is less than the pseudo-critical temperature, i.e., Tw > Tpc > Tb. This criterion
indicates the condition of large property variations occurring within the near wall region.
The heat transfer in a supercritical flow is strongly affected by the property variation near
pseudocritical temperature and buoyancy. When the fluid temperature in the near-wall region
exceeds the pseudocritical temperature, the fluid density decreases sharply, which causes flow
acceleration. Since heat transfer depends on the energy transport from the boundary layer to the
core of the turbulent flow, the change in the velocity distribution due to the buoyancy force and flow
acceleration in the near-wall region plays a dominant role.
Fewster and Jackson (Fewster and Jackson, 2004; Fewster, 1976) conducted heat transfer
experiments for turbulent flow of carbon dioxide inside vertical tubes at supercritical pressures (Figs.
2–5). The objective of these experiments was to investigate various regimes of heat transfer at
supercritical conditions. They found that, in general, three modes of heat transfer at supercritical
pressures exist: (1) normal heat transfer (Fig. 2), (2) improved heat transfer, characterized by
higherthan- expected HTC values than in the normal heat transfer regime (Fig. 3) and (3) deteriorated
heat transfer, characterized by lower-than-expected HTC values than in the normal heat transfer
regime (Figs. 3–5). In general, these findings correspond to those found in SCW (Pioro and Duffey,
2003a). Deteriorated heat transfer may appear at high heat fluxes (Fig. 5) and in any place along the
heated length (Figs. 3–5).
3.1
Normal heat transfer
3.2
Enhanced heat transfer
Silin (1973) investigated heat transfer in forced convection of supercritical carbon dioxide in vertical
and horizontal tubes. He found that at Tb ≤Tpc and Tw ≥Tpc a region with improved heat transfer
existed. During experiments with a 4mm ID tube acoustic effects, such as various noises or whistles,
were observed in the improved heat transfer regime.
All references with primary experimental data are listed in Table 4. Shiralkar and Griffith (1968)
found that the twisted tape installed inside a bare tube improved the heat transfer (Fig. 11).
3.3
Heat transfer deterioration
Impairment of heat transfer, i.e., heat transfer deterioration can be induced by the combined effects
of heat flux and mass velocity in the vertically upward flow. In case of high heat flux, turbulence is
reduced as a result of the thermal acceleration due to heating and the consequent density reduction
of the fluid. Turbulent diffusivity is reduced when the low-density wall layer becomes thick enough to
reduce the shear stress brought about by flow acceleration due to heating. In case of low mass flux,
buoyancy force accelerates the flow velocity near the wall. This makes the flow velocity distribution
to be flat and turbulence energy generation is reduced.
Griffith and Shiralkar propose a mechanism for the deterioration in heat transfer which depends
essentially on physical property variations across the pipe. Thus when the pipe wall passes through
the critical temperature there appears at the wall a low conductivity “gaslike” layer, the core
remaining in a “liquidlike” state moving with a relatively low velocity; the heat transfer coefficient is
therefore reduced. As a greater proportion of the fluid is heated through the critical temperature,
the flow velocity increases and the heat transfer coefficient is thereby restored to something like its
initial value.
While the above mechanism may be valid in the absence of buoyancy effects (i.e., at low values of
Gr/Re1.8), it is radically modified when these effects are large. A mechanism suggested for the effect
of buoyancy is that the shear stress distribution across the pipe, and hence the turbulence
production, is drastically modified. With upward flow the shear stress is rapidly reduced to zero in
the core as the wall passes through the critical temperature and is then reversed, thus re-establishing
turbulence production; the heat transfer coefficient thus passes through a minimum and then
increases. For downward flow the effect of the buoyancy forces is always to increase the shear stress
in the core of the flow and thus to improve heat transfer. The effect described by Shiralkar and
Griffith may also be present, but at the higher values of Gr/Re1.8, it appears to be completely
dominated by the buoyancy effect.
As mentioned before, a strong reduction in heat transfer coefficient can occur, when heat flux is high
and mass flux is low. However, the increase in the wall temperature under heat transfer
deterioration condition is much milder than that at the onset of departure from nucleate boiling [38].
Normally, it is a slow and smooth behaviour. Therefore, it is difficult to define the onset point of heat
transfer deterioration. In the literature, different definitions were used, most of which are based on
the ratio of the heat transfer coefficient to a reference value:
𝑐=
𝛼
𝛼0
Yamagata (1972) [16] and Koshizuka et al. [39] used the heat transfer coefficient at zero heat flux (or
approaching zero) as the reference value α0. The ratio 0.3 is defined as criterion for the onset of heat
transfer deterioration. It is well agreed that the higher the mass flux is, the higher is the critical heat
flux at which heat transfer deterioration occurs. Based on experimental data in a 10 mm circular
tube, Yamagata (1972) [16] proposed the following equation for detecting the onset of heat transfer
deterioration:
𝑞 = 200. 𝐺 1.2
Based on test data obtained in a 22 mm circular tube, Styrikovich [40] proposed the following
equation for the onset of heat transfer deterioration:
𝑞 = 580. 𝐺
According to the studies available in the literature, heat transfer deterioration is caused mainly by
buoyancy effect and by the acceleration effect resulted by a sharp variation of density near the
pseudo-critical line. Based on a simple analysis of the effect of buoyancy on the shear stress, Jackson
et al. [8] derived the following equation for the onset of heat transfer deterioration:
𝑞𝑤 𝜕𝜌
𝜇𝑤 𝜌𝑤 −0.5 1
( ) ( )( )
≥𝐶
𝜌𝑏 𝐺 𝜕ℎ 𝑝,𝑏 𝜇𝑏 𝜌𝑏
𝑅𝑒𝑏0.7
The constant C should be determined by using test data. By taking a 5% reduction in the shear stress
at the location y+ = 20 as a criterion for the onset of heat transfer deterioration, the coefficient C is
set to be 2.2 ⋅10−6.
Taking into account the acceleration effect on the heat transfer behaviour, Ogata (1972) [41] derived
the following equation for the onset of heat transfer deterioration in cryogens (He, H2 and N2):
𝑓 𝑐𝑝
𝑞 = 0.034√ ( ) 𝐺
8 𝛽 𝑝𝑐
Based on the same mechanism, Petukhov [42] derived a similar theoretical model for the onset of
heat transfer deterioration:
𝑐𝑝
𝑞 = 0.187 𝑓 ( ) 𝐺
𝛽 𝑝𝑐
Figure 23 shows the critical heat flux calculated according to different equations for a pressure of 25
MPa and a tube diameter of 4 mm. Large deviation between different correlations is obtained. Both
empirical correlations of Yamagata (1972) [16] and of Styrikovich [40] give much smaller critical heat
flux than other three semi-empirical correlations.
Figure 23: Critical heat flux according to different correlations [3].
Relating to the heat transfer deterioration, some comments are made by Cheng X. et al. [3]:


Heat transfer deterioration is considered to occur only in case that the bulk temperature is
below the pseudo critical value and the wall temperature exceeds the pseudo-critical
temperature. Most of the correlations do not take this limitation into consideration.
Due to a relatively smooth behaviour of the wall temperature, there is no unique definition
of the onset of heat transfer deterioration. This is one of the reasons for the large deviation
between different correlations.
The increase in the heated wall temperature at the onset of heat transfer deterioration is limited and
does normally not lead to an excessive high temperature of the heated wall. Therefore, in some
design proposals of e.g. a supercritical light water reactor heat transfer deterioration is not taken as a
design criterion. Efforts should be made to predict heat transfer coefficient after the onset of heat
transfer deterioration
Kondrat’ev (1971) and Protopopov and Silin (1973) proposed non-dimensional correlations to
estimate the starting point of the deteriorated heat transfer, but these correlations have not been
checked independently in SCW and carbon dioxide.
Bourke and Pulling (1971a,b) investigated the deteriorated heat transfer in supercritical carbon
dioxide. They found that in the upstream part of a tube there was a reduction in the turbulence level,
which caused a local deterioration in heat transfer. Further downstream the turbulence increased,
which lead to improved heat transfer.
Tanaka et al. (1971) conducted experiments with supercritical carbon dioxide flowing in vertical
smooth and rough tubes. In general, they investigated the deterioration of heat transfer near the
pseudocritical temperature. They showed that surface roughness has some effect on heat transfer at
supercritical pressures, i.e.,with increase in tube surface roughness from 0.2 to 14m the heat transfer
also increased.
The deteriorated heat transfer usually appears at higher heat fluxes and lower mass fluxes. This
phenomenon can be suppressed or significantly delayed by increasing the turbulence level with flow
obstructions and other heat transfer enhancing devices.
Deteriorated HT in vertical tubes
One of the distinctive heat transfer characteristics at a supercritical condition is that the heat transfer
from the tube wall can be deteriorated when the fluid temperature in the near-wall region
approaches the pseudocritical temperature even though the buoyancy is not so strong. This is
because the flow is accelerated in the near-wall region due to the abrupt decrease in the fluid
density at the pseudocritical temperature. The heat transfer of turbulent flow in the tubes is affected
by heat conduction through the viscous sublayer, and then energy diffusion from the rim of viscous
layer to the core of the turbulent flow. The energy diffusion to the core region is more effective heat
transfer mechanism which is proportional to the turbulence production, i.e., velocity gradient
between the core and viscous sublayer (Aicher and Martin, 1997). Therefore, the flow acceleration in
the near-wall region reduces the velocity gradient, and hence the turbulence production, which
results in the heat transfer deterioration. When the buoyancy is getting stronger after the
deterioration, the velocity near the viscous sublayer becomes faster than that in the core region,
which produces negative shear stress in the near-wall region. Hence, the turbulence production
starts to increase, and as a result, the heat transfer to the core region is recovered. Therefore, a local
peak in the wall temperature distribution appears due to the heat transfer deterioration caused by
the buoyancy effect.
Chapter
Correlations for forced
supercritical heat transfer
4
convection
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.
Table 9 gives an overview of existing correlations for supercritical heat transfer.
Table 9: Summary of the correlations for supercritical fluids.
Fluid
Bringer and Smith
(1957) [43]
Correlation
Water
𝑁𝑢𝑋
= 0.0266𝑅𝑒𝑋0.77 𝑃𝑟𝑤 0.55
CO2
𝑁𝑢𝑋
= 0.0375𝑅𝑒𝑋0.77 𝑃𝑟𝑤 0.55
Dickinson (1958)
Water
[11]
Miropolsky and
Shitsman (1959, Water
1963) [28]
𝑇𝑝𝑐 − 𝑇𝑏
<0
𝑇𝑤 − 𝑇𝑏
𝑇𝑝𝑐 − 𝑇𝑏
𝑇𝑥 = 𝑇𝑝𝑐 𝑖𝑓 0 ≤
≤1
𝑇𝑤 − 𝑇𝑏
𝑇𝑝𝑐 − 𝑇𝑏
𝑇 𝑖𝑓
>1
{ 𝑤
𝑇𝑤 − 𝑇𝑏
p (MPa)
G (Mg/m²s)
Q (MW/m²)
D
L (mm)
(mm)
L/D TB (°C)
T
Remarks
(°C)
<34.5
-
-
-
-
-
-
-
-
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
-
-
-
-
-
-
-
-
-
-
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) [44] CO2
[45]
Valid within:
2𝑥104 < 𝑅𝑒𝑏 < 8.6𝑥105
̅̅̅𝑏 < 65
0.85 < 𝑃𝑟
0.90 <
𝜇𝑏
< 3.60
𝜇𝑤
1.00 <
𝑘𝑏
< 6.00
𝑘𝑤
0.07 <
𝑐̅𝑝
< 4.50
𝑐𝑝,𝑏
-
𝑁𝑢𝑏 = 0.1𝑅𝑒𝑏0.66 𝑃𝑟𝑏1.2 𝑓𝑜𝑟 𝑇𝑤 ≥ 350°𝐶
𝜇𝑤
𝑁𝑢𝑏 = 0.036𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 ( ) 𝑓𝑜𝑟 𝑇𝑤 = 250 − 350°𝐶
𝜇𝑏
𝜌𝑤 0.43
2.4 𝐷
0.90 ̅̅̅̅̅̅0.66
𝑁𝑢𝑏,𝑥 = 0.0069𝑅𝑒𝑏,𝑥
𝑃𝑟𝑏,𝑥
( )
(1 +
)
𝜌𝑏 𝑥
𝑥
𝑥 = 𝑎𝑥𝑖𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 𝑎𝑙𝑜𝑛𝑔 𝑡ℎ𝑒 ℎ𝑒𝑎𝑡𝑒𝑑 𝑡𝑢𝑏𝑒
𝑐̅ 𝜇
ℎ − ℎ𝑏
̅̅̅𝑏 = 𝑝 𝑏 𝑎𝑛𝑑 𝑐̅𝑝 = 𝑤
𝑃𝑟
𝜆𝑏
𝑇𝑤 − 𝑇𝑏
Domin (1963) [12] Water
Bishop
(1962,
Water
1965) [13]
Kutateladze
Leontiev
(1964)
and
[46] -
2
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 [2⁄√𝜌𝑤 ⁄𝜌𝑏 + 1]
𝜌𝑤 0.231
( )
𝜌𝑏
ℎ𝑤 − ℎ𝑏
𝑎𝑛𝑑 𝑐̅𝑝 =
𝑇𝑤 − 𝑇𝑏
0.923 𝑃𝑟
̅̅̅𝑤
𝑁𝑢𝑤 = 0.00459𝑅𝑒𝑤
Swenson
[14]
(1965)
Touba
McFadden
(1966)
Sabersky
[17]
Water
̅̅̅
𝑃𝑟𝑤 =
and
[47] Water
(1967)
CO2
𝑐̅𝑝 𝜇𝑤
𝜆𝑤
22,0-26,0 0,6-5,1
0,58-4,5
2,0;
4,0
1075;
1233
-
22,6-27,5 0,68-3,6
0,31-3,5
2,55,1
-
30294-525
565
Upward
16inside tube
216
and annulus
-
-
-
-
-
-
-
22,7-41,3 0,2-2,0
0,2-2,0
9,4
1830
-
70-575
6,0285
-
-
-
-
-
-
-
-
0,437
-
-
-
24.925.640.5
-
Horizontal
-
𝑁𝑢𝑏 = 0.0068𝑅𝑒𝑏0.80 ̅̅̅
𝑃𝑟 𝑒 [2.19(ℎ𝑏⁄ℎ𝑝𝑐−0.801)]
-
7.247.58-8.27
-
𝑁𝑢𝑏 = 0.020𝑅𝑒𝑏0.80
Valid within the range of:
104 < 𝑅𝑒 < 4𝑥105 𝑎𝑛𝑑 𝑇𝑏 = 130 − 600°𝐶.
12.02
25.2-32.0 -
-
Water
-
7.62
-
-
105-537
-
220-545
al.
Water
Yamagata (1972)
Water
[16]
𝜌𝑤
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
𝑛1
𝐹𝐶 = 0.67𝑃𝑟𝑚−0.05 (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 ) 𝑓𝑜𝑟 0 ≤ 𝐸 ≤ 1
𝐹𝐶 = (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 )
𝑛2
-
𝑓𝑜𝑟 𝐸 < 0
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
0,116-0,930
7,5;
10,0
15002000
-
230-540
-
Vertical and
horizontal
0.3
Ornatsky et
(1970) [49]
Horizontal
tubes
Vertical
tubes
260-560
9.73;
6.35
≤24.3
Ackermann
(1970) [15]
-
0.613
22.8-30.4
Kondrat’ev (1969)
Water
[48]
=<450
-
22,6-29,4 0,31-1,83
5
𝑇𝑝𝑐 − 𝑇𝑏
𝑇𝑤 − 𝑇𝑏
𝑛1 = −0.77(1 + 1⁄𝑃𝑟𝑝𝑐 ) + 1.49
𝑛2 = 1.44(1 + 1⁄𝑃𝑟𝑝𝑐 ) − 0.53
2
𝑁𝑢
𝑁𝑢
al.
= [1 − 0.2
𝛽(𝑇𝑤 − 𝑇𝑏 )]
Helium
𝑁𝑢0
𝑁𝑢0
Where 𝑁𝑢0 is calculated with the Dittus-Boelter equation.
𝑛
𝜌𝑤 0.3 𝑐̅𝑝
𝑁𝑢𝑏 = 𝑁𝑢0,𝑏 ( ) (
)
𝜌𝑏
𝑐𝑝,𝑏
𝐸=
Yaskin
et
(1977) [50]
𝑁𝑢0,𝑏 =
Water
-
-
-
-
-
-
-
-
-
22,5-26,5 0,7-3,6
≤602 G
1.620
-
-
-
-
-
(1.061.33)xpcrit
4.6x104 < q
4.1
< 2.6
2000
-
-
-
ReB=8 104-5
105
(x/D)≥15
𝑓𝑏
𝑅𝑒𝑏 ̅̅̅̅
𝑃𝑟
8
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
𝑇𝑤
𝑛 = 0.4 + 0.2 (
− 1) 𝑓𝑜𝑟 𝑇𝑏 ≤ 𝑇𝑝𝑐 ≤ 𝑇𝑤
𝑇𝑝𝑐
Petukhov,
Krasnoshchekov
and
Protopopov(1966)
(1979) [22]
𝑇𝑤
𝑇𝑏
𝑛 = 0.4 + 0.2 (
− 1) (1 − 5 (
− 1))
𝑇𝑝𝑐
𝑇𝑝𝑐
𝑓𝑜𝑟 𝑇𝑝𝑐 ≤ 𝑇𝑏 ≤ 1.2𝑇𝑝𝑐 𝑎𝑛𝑑 𝑇𝑏 < 𝑇𝑤
Valid within:
CO2
8𝑥104
< 𝑅𝑒𝑏 <
5𝑥105
0.85 < ̅̅̅
𝑃𝑟 < 65
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
2002) [7]
(1979,
-
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
𝑁𝑢𝑏 = 0.023𝑅𝑒𝑏0.8 𝑃𝑟𝑏 0.4 [2⁄√(0.8𝜓 + 0.2) + 1] 𝐹
Yeroshenko and
Yaskin (1981) [51]
0,28
𝐹 = (𝑐̅𝑝 ⁄𝑐𝑝,𝑏 )
𝑎𝑡 𝑐̅𝑝 > 𝑐𝑝,𝑏
𝐹 = 1 𝑎𝑡 𝑐̅𝑝 ≤ 𝑐𝑝,𝑏
𝜓 = 1 + 𝛽𝑏 (𝑇𝑤 − 𝑇𝑏 )
Where 𝑁𝑢0 is calculated with the Dittus-Boelter equation and
𝛽𝑏 , the volumetric thermal expansion coefficient.
̅̅̅̅𝑏 0.295
̅̅̅̅𝑏
3000𝐺𝑟
𝐺𝑟
]
𝑓𝑜𝑟
< 10−4
2.7 ̅̅̅ 0.5
2.7 ̅̅̅ 0.5
𝑅𝑒𝑏 𝑃𝑟𝑏
𝑅𝑒𝑏 𝑃𝑟𝑏
̅̅̅̅𝑏 0.295
̅̅̅̅𝑏
7000𝐺𝑟
𝐺𝑟
𝑁𝑢 = 𝑁𝑢𝑣𝑎𝑟 𝑝 [1 − 2.7 0.5 ]
𝑓𝑜𝑟
> 10−4 ̅̅̅𝑏
̅̅̅𝑏0.5
𝑅𝑒𝑏 𝑃𝑟
𝑅𝑒𝑏2.7 𝑃𝑟
With:
̅̅̅̅𝑏
𝜌𝑤 0.35
𝐺𝑟
𝑁𝑢𝑣𝑎𝑟 𝑝 = 0.021𝑅𝑒𝑏0.8 ̅̅̅
𝑃𝑟𝑏0.55 ( )
𝑓𝑜𝑟
< 10−4
̅̅̅𝑏0.5
𝜌𝑏
𝑅𝑒𝑏2.7 𝑃𝑟
𝑁𝑢 = 𝑁𝑢𝑣𝑎𝑟 𝑝 [1 −
Watts (1982) [52]
Water
0.35
𝑐̅𝑝
𝑁𝑢
=(
)
𝑁𝑢0
𝑐𝑝,𝑏
Bogachev et al.
[53] (1983)
Gorban’ et
(1990) [64]
𝑁𝑢𝑏 = 0.0059𝑅𝑒𝑏0.9 𝑃𝑟𝑏 −0.12 𝑓𝑜𝑟 𝑇 > 𝑇𝑐𝑟𝑖𝑡
𝑁𝑢𝑏 = 0.0094𝑅𝑒𝑏0.86 𝑃𝑟𝑏 −0.15 𝑓𝑜𝑟 𝑇 > 𝑇𝑐𝑟𝑖𝑡
𝜌 0.231
̅̅̅̅ = 0.0169𝑅𝑒 0.8356 𝑃𝑟 0.432 ( 𝑤 )
𝑁𝑢
𝜔
𝜌𝑏
𝜆 = 0.5(𝜆𝑏 + 𝜆𝑤 )
𝜇 = 𝜇𝑏
al. Water
R-12
Water
Komita
[54]
Water
(2003)
-
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 )
𝑁𝑢 = 0.015𝑅𝑒 0.85 𝑃𝑟 𝑚
8100
𝑚 = 0.69 −
+ 𝑓𝑐 𝑞
𝑞𝑑ℎ𝑡
𝑞𝑑ℎ𝑡 = 200𝐺1.2
0.11
𝑘𝐽
2.9 𝑥 10−8 +
𝑓𝑜𝑟 0 ≤ 𝐻𝑏 ≤ 1500
𝑞𝑑ℎ𝑡
𝑘𝑔
0.65
𝑘𝐽
𝑓𝑐 = −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
{
-
-
-
-
-
-
Buoyancy
effect
Local values
of Re: (36–
90)×103 and
(Gr/Re2) <
10−2
Tin=4.214.24 <
Tpc
0.23-0.3
(0.190.26)x10-3
kg/s
(0.1-1,85)
x10-3
1.8
0.4
(Ltot=0.51)
-
-
-
-
-
-
-
-
-
22,0-27,0 0,3-2,5
0,20-0,70
1024
-
-
-
-
-
-
100-1750 kg/m²s
0-1.8
MW/m²
-
-
-
Tb: 20550°C
-
-
-
-
-
-
-
-
Deteriorated
heat
5
Griem
(1995,
Water
1999) [19]
Kitoh (1999) [65]
-
-
transfer
+
Buoyancy
effect
Sakurai
[18]
(2000)
CO2
Kim et al [58]
CO2
(2007)
-
For circular tubes :
-
-
-
-
-
-
-
-
-
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 [55] 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) [43] 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) [28], on the other hand, assumed that the
thermal conductivity was a smooth decreasing function of the temperature near the critical ad
pseudo-critical points.
Krasnoshchekov, Protopopov and Petukhov (1959, 1960, 1961) [44] [45] 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], [28]) 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 9.
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) [19], 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 pseudocritical temperature was a gradually decreasing function of temperature.
In 1966, Petukhov, Krasnoshchekov and Protopopov [22] 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 [56] 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.
Shiralkar and Griffith (1968) conducted experiments with supercritical carbon dioxide in circular
tubes over a wide range of flow conditions (Figs. 6–8). They found that deteriorated heat transfer
started at certain ratio of q/G (Fig. 6, q/G= 0.116) and was affected with inlet temperature (Fig. 7)
and direction of flow (Fig. 8). In general, wall temperature excursion within the deteriorated heat
transfer region is more significant in downward flow than in upward flow at similar conditions (Fig.
8). However, in SCW the deteriorated heat transfer appears at q/G> 0.4 (Pioro and Duffey, 2003a).
The very simple correlation proposed by Kondrat’ev (1969) [48] (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) [49] modified the correlation proposed by Miropolsky and Shitsman (1959,
1963) [28], 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 [22] (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 [57] (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 9.
Yaskin et al. (1977) [50] 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) [51] proposed a correlation by analysing the correlating equations of
Miropol’skii and Shitsman (1957), Krasnoshchekov and Protopopov (1966), Pron’ko et al. [58] (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).
Bogachev et al. [53] (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 (36–90)×103 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.
Gorban’ et al. (1990) [64] proposed a correlation for the forced heat transfer to R-12 and water at
temperatures above the critical temperature.
In 1999, Kitoh [65] 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 24.
Figure 24: 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 pseudocritical 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 pseudocritical 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.
Recently, new type correlations were proposed by considering the effect of buoyancy on the heat
transfer rates. Buoyancy effect driven by the abrupt density difference between near the wall and
centre of the tube is one of the important parameters controlling the heat transfer characteristics at
supercritical pressure. As for the normal heat transfer regime, Watts et al. [52] developed a heat
transfer correlation for the vertically upward flowing supercritical pressure water as follows:
As for the Watts correlation, Komita et al. [54] modified the equation for deteriorated heat transfer
and found their HCFC-22 experimental data to be predicted in ±20% of error range. In the Watts and
the Komita correlations, the criteria term is the buoyancy parameter which was driven by Jackson
and Hall.
Kim et al. [58] (2007) suggested a correlation which included the friction coefficient, because the
friction coefficient is proportional to the wall shear stress.
The velocity gradient between the wall and the core region is smoothed out because of flow
acceleration and buoyancy in the near-wall region. Hence, flow turbulence is suppressed due to the
reduction in the shear stress, which results in a decrease in the heat transfer rate. Thus, the friction
coefficient should be included in the formulation of the heat transfer correlation because the friction
coefficient is proportional to the wall shear stress. The friction coefficient M for the mixed
convection is defined as
where ρb and ub are the bulk fluid density and velocity, respectively, and the wall shear stress τw is
given by
where ρw is the fluid density at the wall temperature, and uT is the friction velocity. The friction
velocity is obtained from the logarithmic-overlap layer at y+ ≈30. It follows that
The friction coefficient F for the forced convection is given by
where Reb is evaluated based on the properties at the local bulk fluid temperature. In addition, the
heat transfer of the supercritical flow can be varied due to the property changes near the
pseudocritical temperature region even in the case when the buoyancy effect is not strong. Thus, it is
necessary to include the wall-to-bulk property ratio to compensate the effect of property variation.
To do so, the following wall-to-bulk property ratio terms are included in the heat transfer correlation:
The best fit of the exponent m is found to be 0.6 from experimental data.
Although these correlations have been found to predict reasonably well for their original
experimental data, large deviations cannot be avoided in estimating the heat transfer rates for the
different operating conditions such as fluids, geometry, and system parameters and so on. These
deviations could be attributed to the complicated heat transfer characteristics of the supercritical
pressure fluids and poor understanding of heat transfer impairment phenomena.
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.
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 10.
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 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.
Table 10: 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
The parameter K~ accounts the effect of buoyancy and
the effect of acceleration induced by the density
variation near the heated wall.
The friction at supercritical condition is computed by
Kurganov
CO2,
(1985,
Water,
1993)
He
[59]
TB (°C)
Remarks
Based
on
mechanistic
analysis.
≥40
a
Application:
Circular
tubes,
downward, upward
and horizontal
Nun represents the Nusselt number at normal heat
transfer conditions, i.e. without heat transfer
deterioration.
In case of a strong effect of buoyancy and acceleration
(K~ ≥1), a correction factor is introduced to account the
heat transfer reduction. The exponent m is dependent
on the heated length and expressed as:
+ no considerable
change in wall heat
flux over the length
Koshizuk
a (2000) [60]
1.0-1.75
00-1.8
20-550
Chapter
Friction and pressure
supercritical fluids
drop
5
in
Chapter
Experimental
reduction
1. Experimental setup
2. Procedure and conditions
3. Data reduction
4. Uncertainty analysis
system
and
6
data
Chapter
Numerical analysis
7
Chapter
Free convection heat
supercritical fluids
transfer
8
in
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