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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

06/05/2023

_____________________________________________________________________________________________________This report may only be used wordily or entirely for publication purposes. Texts or publications where this report is issued have to be approved by the authors before submission. 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.

_____________________________________________________________________________________________________This report may only be used wordily or entirely for publication purposes. Texts or publications where this report is issued have to be approved by the authors before submission.

http://www.orcnext.be/

ContentFrame..................................................................................................................... 2Content.................................................................................................................. 3Chapter 1 Introduction...........................................................................................4

1. Supercritical state......................................................................................52. Thermophysical fluid properties................................................................6

Chapter 2 Forced convection heat transfer in supercritical fluids........................101. Introduction.............................................................................................102. Literature review.....................................................................................103. Review of a selected group of experimental studies...............................134. Data presentation [4]..............................................................................16

4.1 Description in terms of local conditions only........................................174.2 Presentation in terms of a heat transfer coefficient..............................184.3 Presentation in terms of dimensionless groups....................................20

5. General characteristics for supercritical heat transfer – Heat transfer regimes.............................................................................................................20

5.1 Heat transfer enhancement..................................................................225.2 Heat transfer deterioration...................................................................235.3 Influence of the heat flux......................................................................285.4 Influence of the mass flux....................................................................315.5 Influence of the direction of flow..........................................................325.6 Influence of the diameter of the pipe...................................................335.7 Influence of buoyancy..........................................................................33

6. Summary and future experimental work.................................................35Chapter 3 Correlations for forced convection supercritical heat transfer.............36

1. Introduction.............................................................................................362. Correlations.............................................................................................363. Conclusion...............................................................................................48

References...........................................................................................................53


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Chapter 1IntroductionInvestigation 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. 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, 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 thermo-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



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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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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 (Figure3). 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



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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 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 regimes.



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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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Chapter 2Forced convection heat transfer in supercritical fluids1. 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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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) G (Mg/m²s) Q (MW/m²) D (mm) L (mm) L/D TB (°C) ΔT (°C) Remarks Subject

Dickinson (1958) [11] 25,0-32,1 2,1-3,4 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] 22,0-26,0 0,6-5,1 0,58-4,5 2,0; 4,0 1075; 1233 - =<450 - - Heat transfer,

oscillationBishop (1962, 1965) [13] 22,6-27,5 0,68-3,6 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 (1970) [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 and

horizontal

Heat transfer, Heat transfer deterioration

Griem (1999) [17] 22,0-27,0 0,3-2,5 0,20-0,70 10-24 - - - - - Heat transfer

Sabersky (1967) [18] 7.24-7.58-8.27 - 0,437 - - - 24.9- 25.6-

40.5 - Horizontal

Visualisation, turbulence



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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 MPao G: 0.1 – 5.1 Mg/m²s



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o Q: 0.0 – 4.5 MW/m²o D: 2.0 – 32.0 mmo 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:

T B≤T pc≤T w

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 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



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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: Tpc = 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 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) [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 heated vertical tube with a diameter of 1.9cm (Evans et al. PhD thesis [29]. The



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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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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 Figure9 it can be seen that is not possible.


Bulk temperature

(●) 19°C

(+) 22°C

(∆) 25°C

(x) 28°C

(□) 31°C


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Figure 10: Heat transfer coefficient versus wall temperature for various bulk temperatures [29].

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 ℜn Prs [Eq. 1]

with c, n and s constants.

Goldman, however, suggested collecting all the temperature dependent terms in the dimensionless groups.

q0d1−n

(ρu )n=f (T 0 ,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.


Bulk temperature

(●) 19°C

(+) 22°C

(∆) 25°C

(x) 28°C

(□) 31°C


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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 ( ℜm )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.

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.



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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.

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].

q (W /c m2) m /A (gm /s cm ²) d (cm) Flow directiona Shitsman [19] 34 43 0.8 vertical upwardb Shitsman [19] 28.5 43 0.8 vertical upwardc Shitsman [19] 28.0 43 0.8 vertical upwardd Domin [12] 72.5 68.6 0.2 horizontale Domin [12] 72.5 72.4 0.2 horizontal

Table 3: Experimental conditions for supercritical water at p = 1.15 pcrit [32].

q (W /c m2) m /A (gm /s cm ²) d (cm) Flow direction

a Vikrev and Lokshin [33] 69.9 100 0.8 horizontalb Vikrev and Lokshin [33] 69.9 40 0.8 horizontalc Schmidt [34] 58 61 0.5 horizontald Schmidt [34] 82 61 0.5 horizontal


p = 1.05 pcrit p = 1.15 pcrit


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e Domin [12] 91 101 0.2 horizontalf Shitsman [19] 39.6 44.9 0.8 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 T Bulk≤T crit ≤T wall, 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 T Bulk≤T crit ≤T wall. 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 T Bulk is slightly below the pseudo-critical temperature T pc and when the wall temperature TWall is slightly above T pc. 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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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.

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].

m /A (gm /s cm ²) q (W /c m2) Flow direction1 382 27 Vertical upward


(1) Theory (∆) Experimental: m = 140±4.4 kg/h; q = 1.44 W/cm²

(2) Theory(x) Experimental: m = 140±3.1 kg/h; q = 2.73 W/cm²

(3) Theory (○) Experimental: m = 280±5.6 kg/h; q = 3.32 W/cm²

(4 Theory (●) Experimental: m = 280±7.8 kg/h; q = 5.20 W/cm²


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2 382 37 Vertical upward3 400 45 Vertical upward4 375 52 Vertical upward5 400 27 Vertical downward6 400 36 Vertical downward7 393 43 Vertical downward8 381 50 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>T pc, while the deteriorations in water from Shitsman [36], occurred below T pc as well as above T pc. 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.

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.



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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 (Figure16). 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.


(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


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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): m /A=60gm /s cm ² and q=52W /c m2 [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 CO2 flowing up- and downwards in a vertical pipe [4].

Reference d (cm) q(W /cm2)

ℜ Gr q .d(W /cm)

p¿

Shiralkar and Griffith [38] 0.635 15.8 1.0 1 10.0 75.8Jackson and Evans-Lutterodt [37] 1.905 5.67 1.24 27 10.8 75.8Bourke et al. [39] 2.285 5.1 0.82 46.5 11.6 74.5

Figure 17 shows the wall temperature as a function of the bulk enthalpy for a downward and upward flow.


(1) Horizontal pipe – upper surface

(2) Horizontal pipe – lower surface

(3) Vertical pipe – upward flow

(4) Bulk fluid temperature


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Figure 17: Comparison of the data of Shiralkar and Griffith [38], Jackson and Evans-Lutterodt [37] and Bourke et al. [39] 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 [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.


Legend:

_.__._: Shiralkar and Griffith [38]

_____: Jackson and Evans-Lutterodt [37]

_ _ _ _ : Bourke et al. [39]


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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=cteℜ0.8Prn [Eq. 4]

Where:

Nu, the Nusselt Number (¿h L/ λ) ¿; ℜ, the Reynolds Number (¿ m L /μ) ¿; Pr, the Prandtl Number (¿ μc p/ λ) ¿; h, the heat transfer coefficient W /m2K ; 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; m, 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 m0.8λ0.6 cp

0.4

L0.2μ0.4[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 (α 0), presented as the ratio by α /α 0 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 q /m 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 .10−3 /m ≈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])

5.5 Influence of the direction of flow



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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 CO2 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.

5.6 Influence of the diameter of the pipe



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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.

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



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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 q /m (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

Grbℜb

2.7 <10−5 [Eq. 6]



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where Grb is the Grashof number (¿ g ( ρb−ρ )D3

νb2 ) [Eq. 7], in which ρ is the

integrated mean density and the subscript b indicates physical properties evaluated at the local bulk temperature.

The buoyancy parameter Bo, defined as Bo=8 x104 Grℜ3.425Pr0.8

[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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Chapter 3Correlations for forced convection supercritical heat transfer1. IntroductionDue 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. CorrelationsTable 6 gives an overview of existing correlations for supercritical heat transfer (this table cannot be considered complete).



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Table 6: Summary of the correlations for supercritical fluids.

Fluid Correlation p (MPa) G (Mg/m²s) Q (MW/m²)

D (mm) L (mm) TB (°C) Remarks

Bringer and Smith (1957) [45]

Water NuX=0.0266ℜX0.77 Prw

0.55

T x={ T b ifT pc−T bT w−T b

<0

T pc if 0≤T pc−T bT w−Tb

≤1

Tw ifT pc−T bT w−T b

>1

<34.5 - - - - - -CO2 NuX=0.0375ℜX

0.77Prw0.55

Dickinson (1958) [11] Water - 25,0-32,1 2,1-3,4 0,88-1,8 7,6 1600 - -Miropolsky and Shitsman (1959, 1963) [31]

Water Nub=0.023ℜb0.8Prmin

0.8

where Prmin isthe lesser of Prb∧Prw22,0-25,0 0,3-1,5 <1,16 8 1500 =<450 -

Petukhov, Krasnoshchekov and Protopopov (1959, 1961) [46] [47]

Water andCO2

Nub=Nu0 , b( c pcp ,b )0.35

( λbλw )−0.33

( μbμw )0.11

Nu0 ,b=(f b8

ℜb Pr

12.7( f b8 )0.5(Pr

23−1)+1.07 )

f=(1.82 log10 (ℜb )−1.64 )−2

Valid within:

2 x104<ℜb<8.6 x 105

0.85<Prb<65

- - - - - - -



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0.90<μbμw

<3.60

1.00<kbkw

<6.00

0.07<c pc p , b

<4.50

Domin (1963) [12] Water

Nub=0.1ℜb0.66Prb

1.2 for T w≥350 °C22,0-26,0 0,6-5,1 0,58-4,5 2,0;

4,01075; 1233 =<450 Horizontal

tubesNub=0.036ℜb0.8Prb

0.4 ( μwμb ) for T w=250−350 °C

Bishop (1962, 1965) [13] Water

Nub , x=0.0069ℜb , x0.90 Prb , x

0.66( ρwρb )x0.43

(1+ 2.4Dx )x=axial lengthalong the heated tube

Prb=c pμbλb

∧c p=hw−hbT w−Tb

22,6-27,5 0,68-3,6 0,31-3,5 2,5-5,1 - 294-525

Upward inside tube and annulus

Kutateladze and Leontiev [48] (1964)

- Nub=0.023ℜb0.8Prb

0.4 [2 /√ ρw / ρb+1 ]2 - - - - - - -

Swenson (1965) [14] Water

Nuw=0.00459ℜw0.923Prw

0.613( ρwρb )0.231

Prw=c pμwλw

∧c p=hw−hbTw−Tb

22,7-41,3 0,2-2,0 0,2-2,0 9,4 1830 70-575 -

Touba and McFadden [49] (1966)

Water Nub=0.0068ℜb0.80Pr e[2.19 (hb /h pc−0.801)] - - - - - - -

Sabersky (1967) [18]

CO2 - 7.24-7.58-8.27

- 0,437 - - 24.9- 25.6-

Horizontal



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40.5

Kondrat’ev (1969) [50] Water

Nub=0.020ℜb0.80

Valid within the range of: 104<ℜ<4 x105∧T b=130−600 ° C.

22.8-30.4

- -

12.02

-

260-560 Vertical tubes

25.2-32.0 7.62 105-537 Horizontal tubes

≤24.3 9.73; 6.35 220-545

Vertical annular channel

Ackermann (1970) [15] Water - 22,7-44,1 0,135-2,17 0,12-1,7 9,4-

24,4 - 77-482 -

Ornatsky et al. (1970) [51] Water Nub=0.023ℜb

0.8Prmin0.8( ρwρb )

0.3

where Prmin isthe lesser of Prb∧Prw

- - - - - -Inside 5 parallel tubes

Yamagata (1972) [16] Water

Nub=0.0135ℜb0.85Pr0.8FC

FC=1.0 for E>1FC=0.67 Prm

−0.05 (c p/c p ,b )n1 for 0≤ E≤1FC=(c p/c p , b )n2 for E<0

E=T pc−T bT w−T b

n1=−0.77 (1+1/Prpc )+1.49n2=1.44 (1+1/Pr pc )−0.53

22,6-29,4 0,31-1,83 0,116-0,930

7,5; 10,0 1500-2000 230-540 Vertical and

horizontal

Yaskin et al. (1977) [52]

Helium

NuNu0

=[1−0.2 NuNu0 β (T w−T b )]2

Where Nu0 is calculated with the Dittus-Boelter equation.

- - - - - - -



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Petukhov, Krasnoshchekov and Protopopov(1966) (1979) [21]

Water

Nub=Nu0 , b( ρwρb )0.3

( c pc p ,b )n

Nu0 ,b=(f b8

ℜb Pr

12.7( f b8 )0.5(Pr

23−1)+1.07 )

f=(1.82 log10 (ℜb )−1.64 )−2

n=0.4 for Tb≤T w≤T pc∧1.2T pc≤T b≤T w

n=0.4+0.2( T wT pc−1) for T b≤T pc≤Tw

n=0.4+0.2( T wT pc−1)(1−5( T bT pc−1))

for T pc≤T b≤1.2T pc∧T b<T w

Valid within:

8 x104<ℜb<5 x 105

0.85<Pr<65

0.09<ρbρw

<1.0

22,5-26,5 0,7-3,6 ≤602 G 1.6-20 - - -

CO2 (1.06-1.33)xpcrit

- 4.6x104 < q < 2.6

4.1 2000 - ReB=8 104-5 105

(x/D)≥15



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0.02<c pc p ,b

<4.0

Jackson (1979, 2002) [7]

CO2

Nub=0.0183ℜb0.82Prb

0.5( ρwρb )0.3

( c pcp , b )n

f=(1.82 log10 (ℜb )−1.64 )−2

n=0.4 for Tb≤T w≤T pc∧1.2T pc≤T b≤T w

n=0.4+0.2( T wT pc−1) for T b≤T pc≤Tw

n=0.4+0.2( T wT pc−1)(1−5( T bT pc−1))

for T pc≤T b≤1.2T pc∧T b<T w

Simplified form (Jackson and Fewster):

Nub=0.0183ℜb0.82Prb

0.5( ρwρb )0.3

- - - - - - -

Water

Yeroshenko and Yaskin (1981) [53]

- Nub=0.023ℜb0.8Prb

0.4 [2 /√ (0.8ψ+0.2 )+1 ]2FF=(c p/c p , b )0,28at c p>c p , b

F=1at c p≤cp , bψ=1+βb(T w−T b)

- - - - - - -



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Where Nu0 is calculated with the Dittus-Boelter equation and βb, the volumetric thermal expansion coefficient.

Watts (1982) [54] Water

Nu=Nuvar p [1−3000Grbℜb2.7 Prb

0.5 ]0.295

forGrb

ℜb2.7Prb

0.5 <10−4

Nu=Nuvar p [1−7000Grbℜb2.7 Prb

0.5 ]0.295

forGrb

ℜb2.7Prb

0.5 >10−4

With:

Nuvar p=0.021ℜb0.8 Prb

0.55( ρwρb )0.35

forGrb

ℜb2.7 Prb

0.5<10−4

- - - - - - Buoyancy effect

Bogachev et al. [55] (1983) - Nu

Nu0=( cpc p ,b )

0.35

0.23-0.3 (0.190.26)x10-3 kg/s

(0.1-1,85) x10-3 1.8 0.4

(Ltot=0.51)Tin=4.21-4.24 < Tpc

Local values of Re: (36–90)×103

and (Gr/Re2) <10−2

Gorban’ et al. (1990) [56]

Water Nub=0.0059ℜb0.9Prb

−0.12 for T >T crit - - - - - - -R-12 Nub=0.0094ℜb

0.86Prb−0.15 for T>Tcrit

Griem (1995, 1999) [17] Water

Nu=0.0169ℜ0.8356Pr0.432( ρwρb )0.231

ω

λ=0.5 (λb+ λw )μ=μbc p=

13

¿

ω=min { 1.0

max { 0.820.82+9.10−7 (h−1.54 .106 )

22,0-27,0 0,3-2,5 0,20-0,70 10-24 - - -

Kitoh (1999) [57] Water Nu=0.015ℜ0.85Prm - 100-1750 kg/m²s

0-1.8 MW/m²

- - Tb: 20-550°C

-



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m=0.69−8100qdht

+ f c q

qdht=200G1.2

f c={ 2.9 x10−8+0.11qdht

for 0≤H b≤1500kJkg

−8.7 x10−8−0.65qdht

for 1500≤Hb≤3300kJkg

−9.7 x10−7+ 1.30qdht

for 3300≤ H b≤4000kJkg

Komita (2003) [58] Water - - - - - - -

Deteriorated heat transfer + Buoyancy effect

Sakurai (2000) [25] CO2 - - - - - - - -

Kim et al [59] (2007) CO2

Nu=NuF ( ξMξF )( c pc p , b )0.6

( ρwρb )n

NuF=0.0243ℜb0.8 Prb

0.4

For circular tubes :

n=0.955−0.0087( qG )+1.30 x10−5( qG )2

- - - - - - -

Mokry et al [60] (2009) Water Nub=0.0061ℜb

0.904 Prb0.684( ρwρb )

0.564

- - - - - - -



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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 pseudo-critical 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.

In 1966, Petukhov, Krasnoshchekov and Protopopov [21] adapted their earlier proposed correlation for supercritical water using a Dittus-Boelter form with



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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<ℜ<4 x105∧T b=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).

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



Report no. 00000

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

(Gr/Re2) or k ¿=(1− ρwρb )Grℜ2 [Eq. 9]. For k*<0.4 or (Gr/Re2) < 0.6, deteriorated 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:

NuNu0

=( cpc p ,b )n

( ρwρb )m

(for k*< 0.01) [Eq. 10]

NuNu0

=( cpc p ,b )n

( ρwρb )m

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* 0.01 0.02 0.04 0.06 0.08 0.1 0.2 0.4φ (k¿) 1 0.88 0.72 0.67 0.65 0.65 0.74 1

The local values of Nu0 for smooth circular tubes under turbulent flow can be calculated as follows:

Nu0=( f /2 ) ℜ Pr

b+4.5 f 1 /2 (Pr2 /3−1 )(Pr=1−2000 and ℜ=(4−5000)103) [Eq. 12]

Where T b is the characteristic temperature, b=1+(900/ℜ ) and f=(1.82 log10 (ℜb )−1.64 )−2.

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.

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



Report no. 00000

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 c p=hb−hwT b−T w

is the integral average specific heat in the range of

(T b−T w ). 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 T b/T pc and T w /T pc, 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 (c p/c pb )≥1 , n=0.7; for (c p /c pb )<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.



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=CℜXn Pr X

m 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.



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 .ℜbn. Prb

m .F

With

F=f 1( ρwρb ,c p , Ac p ,B ) . f 2( ρwρb , μwμb ,ℜb , Prb ,

q .βbDλb )

Kuang et al. [67] (2008) used the following correction factor:

F= f (Gr , ρwρb , cp , Ac p , B,q .βbDc p , BG

,μwμb,λwλb )

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).



Report no. 00000

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.



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) RemarksKurganov (1985, 1993) [70]

CO2, Water, He

NuNun

={ 1 ,~K≤1~K−m,~K>1

~K=[ εu0F +gxGrmℜB2 ] 1εn [1−exp (−ℜf /30000 ) ]

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

ε n=ε 0( ρWρB )0.4

, ε0=[0.55/ log (ℜB/8 ) ]2

Nun=εnℜB PrB

1+ (900/ ℜB )+12.7 (εn/8 )0.5 (Pr B2 /3−1)Nun represents the Nusselt number at normal heat transfer conditions, i.e. without heat transfer deterioration.

ε u=8qW βBGc p , B

,Gru=gd3

vb2 (1− ρWρB )

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:

m=0.55 [1−exp(− xd50 )]

- - - - - ≥40 - Based on a mechanistic analysis.

Application:Circular tubes, downward, upward and horizontal

g Du¿2 ≤0.015

ℜ¿≥2.104

+ no considerable change in wall heat flux over the length



Report no. 00000

F=1−0.8exp [−3( qW βBGcp , Bd /4 xln ( ρ¿/ ρB ) )

2]

Koshizuka (2000) [71]

-

NuB=0.015ℜB0.85Pr B

0.69−81000 /CHF+f cq

{ f c=2.9 .10−8+ 0.11

CHFfor h≤1.5MJ /kg

f c=−8.7 .10−8− 0.65CHF

for 1.5≤h≤3.3MJ /kg

f c=−9.7 .10−7+ 1.30CHF

for 3.3≤h≤4 MJ /kg

CHF=200G1.2

- 1.0-1.75 00-1.8 - - - 20-550 -



Report no. 00000

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