DIONYSIUS C. GROENEVELD, ANALYTICAL AND ...users.ugent.be/~mvbelleg/literatuur SCHX - Stijn...

DIONYSIUS C. GROENEVELD, ANALYTICAL AND EXPERIMENTAL PROGRAM OF SUPERCRITICAL HEAT TRANSFER RESEARCH AT THE UNIVERSITY OF OTTAWA, NUCLEAR ENGINEERING AND TECHNOLOGY, VOL.40 NO.2 SPECIAL ISSUE ON THE 3RD INTERNATIONAL SYMPOSIUM ON SCWR, 2007

Fluid-to-fluid modelling of SCHTSuccessful fluid-to-fluid modelling or scaling of SCHT requires the use of appropriate similarity relationships. It is proposed to apply fluid-to-fluid modelling of SCHT using the following dimensionless groups:

P/Pc and T/Tc For the subcritical region, the saturation lines of CO2, water and R-134a nearly coincide on a P/Pc vs. T/Tc (absolute temperatures) diagram, as can be seen in Figure 2.

For SC conditions, we hypothesized that the dependence of the pseudocritical temperature Tpc on pressure might be similar to the dependence of the saturation temperature on pressure, because the enthalpy gradient dh/dT reaches a maximum at both temperatures. This was confirmed in Figure 2 where a remarkable degree of similarity in the SC behaviours of these three fluids is noted. On a P/Pc vs. T/Tc plot, the pseudocritical lines for all three fluids nearly coincide and the pseudocritical line appears to be an extension of the saturation line.

Reynolds number and Nusselt number At SC conditions that are well beyond the critical or pseudo critical points, single-phase-like flow characteristics prevail and the conventional Nu = f (Re, Pr) relationship is applicable for predicting the heat transfer. The heat transfer mode at these SC conditions is labeled as

“normal” (Pioro and Duffey [14]). Thus, the product Re.Pr0.5 can be used as a first approximation to determine equivalent mass flux conditions, especially when the Prandtl number is not far from unity.

Not surprisingly, for near-pseudocritical conditions, at which the fluid properties change drastically, the heat transfer also displays an atypical behaviour, as shown in Figure 1.

Because the atypical behaviour appears to be similar for both CO2 and water, when normalizing Nu by NuDittus-Boelter (Figure 1), this methodology will be used also in our preliminary fluid-to-fluid modelling approach. We will therefore scale test conditions for all three fluids by maintaining the same value of Re.Pr0.5. We also expect that the similarity would apply equally to heat transfer in the deteriorated heat transfer region, which, compared to the normal heat transfer mode, is characterized by lower values of the heat transfer coefficient (see again Figure 1) and hence higher values of wall temperature within parts of a test section at high heat fluxes and low mass fluxes.

Mechanisms responsible for this deterioration in heat transfer have been described by various authors (e.g., Jackson and Hall [3]). This anomalous behaviour has been observed in various fluids operating at SC conditions. Additional confirmation that the fractional decrease in heat transfer (or the ratio Nuexp /NuDB) is the same in all three fluids of interest is required for a wider range of fluids, and values of P/Pc and Re within the entire ranges of interest.

Heat flux Jackson and Hall [3] examined the governing SC heat transfer equations and suggested that the values of the heat flux parameter q.D/(k.Tb) should be kept the same in the prototype and modelling fluid; in this expression, D is the tube inside diameter, k is the thermal conductivity and Tb is the absolute bulk coolant temperature. Yang and Khartabil [15] found that the heat flux parameter q/(G.Hb), when included in a Nu = f(Re, Pr, Tb/Tc, P/Pc) type correlation, provided an improved prediction for the deteriorated heat transfer region for the AECL SC CO2 data and the SC water data by Yamagata et al. [16]. We will therefore explore both heat flux parameters for SC fluid-to-fluid modelling.

S.K. Yang and H.F. Khartabil, “Normal and deteriorated heat transfer correlations for supercritical fluids”, Trans. ANS Meeting, 95, Washington DC, USA (2005).

Geometry Both Re and Nu contain an equivalent diameter, which should, in principle, account for differences in geometries in scaled tests. However, in previous CHF and film boiling modelling studies (e.g., Groeneveld et al. [11]), it was found that the accuracy of fluid-to-fluid modelling could be adversely affected by significant geometrical differences. To remove this uncertainty, SC fluid-to-fluid modelling experiments at the University of Ottawa will be based on identical test section geometries.

Experimental studies The ranges of similarity parameters used to scale the tests are listed in Table 3.

Reference heat transfer and pressure drop measurements: Surface temperature and pressure drop of CO2 flow in a 2 m long, 8 mm ID, vertical tube will be measured for the conditions listed in Table 4.

These measurements will serve as reference and will be compared to CO2 measurements from the literature. Heating will be applied such as to generate conditions for both normal and deteriorated heat transfer.

Yang and Khartabil [15] proposed a criterion for the onset of deteriorated heat transfer for CO2 in 8 mm ID tubes as q > 0.27 G0.94, where q is the heat flux in kW/m2 and G is the mass flux in kg/m2s. This is analogous to the condition q > 0.20 G1.2, suggested by Yamagata et al. [16] for water in 10 mm tubes.

Cheng and Schulenberg [44] have reviewed additional criteria for deteriorated heat transfer and demonstrated that they provide vastly different estimates. This issue will be examined in detail in the future. For planning purposes, the range of heat fluxes for the present tests was estimated to extend from one order of magnitude lower to one order of magnitude higher than the value given by the criterion of Yang and Khartabil.

Effect of fluid type: Tests similar to those in CO2 will be performed in Freon R-134a to facilitate the development and validation of fluid-to-fluid scaling laws for SC heat transfer and pressure drop.

Effect of orientation: The proposed Canadian Generation IV reactor design uses horizontal fuel channels. Although some SCHT tests have been performed in horizontal channels (see Table 2), no systematic investigation of the orientation effect has yet been performed. It is proposed to conduct heat transfer and pressure drop measurements in horizontal tubes over the complete range of conditions of interest and in both fluids. These results will be compared to corresponding measurements in vertical tubes for an assessment of the orientation effect.

Effect of flow geometry: Upon the completion of the circular-tube tests, rod-bundle subassemblies will be tested as part of a systematic study of the effects of equivalent diameter, heater curvature, inter-element gap size, and rod-wall gap size. A three-rod subassembly is already available for these tests.

Effect of flow obstructions: Nuclear fuel bundles require spacers between fuel rods and between fuel rods and pressure tubes or containment channels. Spacers affect both pressure drop and heat transfer significantly (Yao et al. [45], Groeneveld et al. [11]), depending on the flow blockage ratio, their shape and their axial pitch. Spacer effects will be investigated initially by inserting simple obstructions in a heated tube and will be extended later to include more realistic obstructions in the rodbundle subassembly

Measurements of mean and turbulent velocity and temperature: Traverses of Pitot-tubes and micro thermocouples will be made across different test sections to measure the average velocity and temperature profiles. In addition, cold-wire/hot-wire probe combinations will be used to measure simultaneously the velocity and temperature fluctuations at selected locations, including narrow gaps. These results will be valuable in understanding SCHT phenomena, for developing phenomenological models and for validating SC subchannel analysis codes and CFD studies.

X. Cheng, Y.H. Yang, S.F. Huang, A simplified method for heat transfer prediction of supercritical fluids in circular tubes, Annals of Nuclear Energy 36 (2009) 1120–1128

Abstract New method for heat transfer prediction via simple correlation structure Explicit coupling with physical phenomena Introduction of 1 dimensionless number, the acceleration number’, to correct the deviation

of the supercritical fluids to that of conventional fluids New correlation excludes direct dependence of the HTC on T_wall and eliminates possible

numerical instability It gives a reasonable prediction accuracy over a wide parameter range + capable of

predicting HT behaviour in the HTD region

Heat transfer prediction

General features of heat transfer Strong dependence on heat flux especially as T_b approaches T_pc.

o At low heat fluxes (approaching ‘0’): HTC well predicted by Dittus-Boelter equationo For increasing heat flux: peak shifts to lower T_b and also decreases

Ratio of HTC to HTC at zero fluxo Starts at ‘1’o Then reaches a maximum at T_b still far below T_pco After maximum it decreases as T_b approaches T_pco Minimum at T_b around T_pco At T_b >> T_pc HTC ration approaches ‘1’ again

Region where ration >1 = HT ENHANCEMENT Region where ration <<1 = HT DETERIORATION

Selection of dimensionless parameters

The new correlation is of the same type: o The goal is to develop a correlation for the factor ‘F’ which accounts for the deviation

of heat transfer from the Dittus-Boelter correlation.

The deviation is mainly due to 3 issues:o Property variationo Buoyancy effecto Acceleration effect

1. Property variation Small heat flux HTC prediction is good with Dittus-Boelter correlation Increasing heat flux deviation is more significant

o Due to strong dependence of thermo-physical properties on temperature, especially Cp

o The near wall sub-layer properties deviate from the ones in the bulk region deviation increases as q↑

o Many researchers accepted the use of the effective specific heat Cp,a: to account for the effect of property variation

o : ration of effective cp to cp at T_bulko The behaviour of cp-ratio is similar to behaviour of HTC-ratio

abnormal behaviour could be caused by cp-variation also without any significant change in flow structure!

cp-ratio plays an dominant roll amongst all thermo-physical properties2. Acceleration effect

The acceleration effect can be characterized by the density gradient in the main flow

direction:

o introduction of dimensionless parameter: (called here: ‘acceleration number’)

3. Buoyancy effect

Characterized by density gradient in the radial direction:

Simplification: property parameters based on T_bulk (called here: ‘buoyancy number’)

Correlation structure of correction factor Criteria for deriving structure of correlation

o Based on dimensionless numbers so to extend to other SC fluidso As few as parameterso Cover normal and HTD conditionso No T_wall’s or parameters depending of T_w to avoid numerical instability

correlations containing the wall temperature require an iterative solution procedure, which might not only lead to convergence problems, but also to numerical instability, especially near the pseudo-critical point!

Assume a HT correlation which depends on T_w:

Also the following relation has tob e fulfilled:

This equation could have 1 solution or more than 1 (see figure)

convergence problems! The case with more than 1 solution would lead to numerical instability

eliminate cp-ratio The effect of T_w and cp ratio on heat transfer will be taken into account indirectly with

other parameters: e.g. acceleration parameter (mathematically contains the heat flux and affects T_w and thus the property variation)

Acceleration and buoyancy parameter are tightly related to each other:

o eliminate one for simplicity and a systematic evaluation of the effect of both parameters showed that the selction of the acceleration parameter would be more favourable

PROPOSED FINAL CORRELATION STRUCTURE of ‘F’:

Experimental database The correlation was validates with the experimental data of Herkenrath et al. [1] (1967). Error analysis of test data:

New heat transfer correlation Extensive analysis is done for the effects of the various parameters on the correction factor

o Acceleration factor has a strong influence and is unique

Relation between F and acceleration factor 2 regions

o Region 1: Small values of acc. factor F↑ as acc. factor↑ relationship described with ONE single curve for different experimental

conditionso Region 2: large acc. factor F↓ as acc. factor↑

different curves required for different parameters combinations For each combination of pressure, mass flux and heat flux, the acc. factor

depends on T_fluid Ratio thermal expansion coefficient and cp maximum at T_pc gives maximum

acceleration number at T_pc

Correlation

o Region 1:

o Region 2: o Constants determined based on the criterion that the error parameter

has its minimum value

o

Assessment of the new HT correlation For larger values of the acc. factor large scatter 70% of data points: deviation between calculated and measured Nu fall into 20% error band

Comparison correlation and measurements for the HTC-ratio

o As T_b approaches T_pc: ratio ↓ to values lower dan 0.2 = HTDo The correlation predicts well the behaviour of HTD, but quantitatively it needs more

accuracy Comparison low heat flux

o No HTD, maximum of HTC at T_pc after first a slow decreaseo Correlation of Griem shows a good agreemento New correlation: stronger increase, but peak value well predicted

Comparison high heat flux

o Besides the new correlation and the one from Griem, the rest fails to predict correctly the test data near T_pc

Other test data for comparison

o Average value:

o Standard deviation: o The new correlation shows the best agreement with all the selected data (except the

test data of Xu)

ONSET OF HTD Smoother behaviour of T_w at HTD compared to a much sharper increase in T_W at voiling

crisis at subcritical pressure conditions no unique definition for the onset of HTD Criterion if Koshizuka et al (1995)

o Koshizuka, S., Takano, N., Oka, Y., 1995. Numerical analysis of deterioration phenomena in heat transfer to supercritical water. Int. J. Heat Mass Transfer 38 (16), 3077–3084.

o

o onset

According to , the correction factor has a minimum value as T_b approaches T_pc

o criterion for HTD onset at a given parameter combination of pressure, heat flux an

mass flux: o Relationship between heat flux and mass flux at onset of HTD

o The heat flux at onset of HTD Increases with increasing pressure

V.A. Kurganov, Yu.A. Zeigarnik, I.V. Maslakova, Heat transfer and hydraulic resistance of supercritical-pressure coolants. Part I: Specifics of thermophysical properties of supercritical pressure fluids and turbulent heat transfer under heating conditions in round tubes (state of the art), International Journal of Heat and Mass Transfer 55 (2012) 3061–3075

Abstract Objective: present a systematized picture of the main results of studies of HT regularities and

the specifics of the hydrodynamics of SCP fluid flows under heating in channels of a standard form (tubes). These studies were conducted at different scientific centers and the results constitute the basis of the current knowledge on the heat transfer mechanism at SCP. We assume that a compact representation of these data, as well as the main results of their application in the thermohydraulic design, will be useful for competent planning and comprehensive arranging of new-generation studies for new fields of SCP coolant application.

PART I: Specifics of thermophysical properties of supercritical pressure fluids and turbulent heat transfer under heating conditions in round tubes

PART II: Results of hydraulic and flow-sounding studies PART III: Discussion of practical problems methods of calculating normal and

deteriorated heat transfer using new standards for fluid heat conductivity, assessing the “boundaries” of the normal heat transfer region, and enhancing HT to prevent its deterioration.

Specific features of typical heat transfer modes are pointed out: normal, deteriorated and improved HT

Discussion of the existing concept of HTD Proposal of a simple classification of the heat transfer regimes under high heat loads

makes it possible to determine the reasons for and assess the degree of danger of HTD

Introduction Very few studies have been conducted within the range of parameters and geometric

characteristics of cooling channels typical of future SCP reactors same as for ORCs using waste heat!!!

Many “old” studies carried out with relatively poor developed and inadequate old-fashioned measurements devices compared with the present state, computational base, that ensures high-quality of experiments

During the entire second half of 20th century, intense investigations and refinement of the thermophysical properties of substances in the near critical range of parameters took place. At the end of the century, even for such thoroughly studied SCP coolants as water and carbon dioxide, the necessity of considerably correcting standardized tabular data on transfer properties, viscosity, and especially conductivity, was revealed

o A.A. Aleksandrov, A.I. Ivanov, A.B. Matveev, Dynamic viscosity of water and steam within a wide range of pressures and temperatures, Therm. Eng. 22 (4) (1975) 77–83

o A.A. Aleksandrov, International tables and equations for the thermal conductivity of water and steam, Therm. Eng. 27 (4) (1980) 235–240

o V. Vesonic, W.A. Wakeham, G.A. Olchowy, J.V. Sengers, J.T.R. Watson, L. Millat, The transport properties of carbon dioxide, J. Phys. Chem. Ref. Data 19 (3) (1990) 763–808

Meanwhile, the old experimental and calculated data and empiric correlations for heat transfer under SCP that constitute almost 100% of the available reference material are based on the old standards of properties. With the transition to the new standard IAPWS-97 for water properties, the question arises, whether many widely used empiric correlations will retain their workability!!!

Also a large volume of experimental, calculation and theoretical studies of heat transfer at SCP were caused by difficulties and failures that occurred in the first phase as SCP apparatuses were introduced in power engineering, rocket building and other advanced technology fields.

o As revealed post facto, they stemmed from insufficient scientific exploration of the new problem and imperfection in the scientific and methodological concepts of the problem at the time.

o Back then, Powell’s experimental data were already known [10], obtained in the heating of SCP oxygen under parameters typical of rocket technologies. These data showed the possibility of an extremely deep drop in heat transfer coefficient values in the vicinity of tb = tm under large t values. It is presently known that this is determined by the effect of thermal acceleration of the flow. However, at that time, such a drop in the heat transfer coefficient was explained as a consequence of the peculiar value and specific behavior of the

complex of oxygen, as compared to those of water and CO2 [11].

Powell’s work did not provoke any anxiety concerning the possibility of heat transfer deterioration while operating with water and CO2. Persuasive arguments were later obtained for the fact that the main role in these phenomena is played by the change in fluid density over the tube cross section and along its length, rather than the specific behavior of or even Cp.

Thus very often it can be said even as a rule that the all the existing knowledge appears insufficient for introducing new technical ideas without certain problems. Therefore, the long list of works on heat transfer of SCP coolants should not make us overconfident that all of the main problems in this field have been solved to the proper extent and that there is no acute need to regenerate large-scale experimental and theoretical studies in advance to create nuclear reactors with SCP fluid cooling.

Specifics of the behaviour of thermophysical properties of coolants in the single-phase near-critical region: effect of gas admixtures on the properties of SCP CO2 and water

Typical behaviour of thermophysical properties of SCP fluids with changes in T and enthalpy (In SCP fluid flow and heat transfer, changes in pressure are most often small as compared to the absolute value and do not considerably affect the properties of a fluid. Therefore, as a rule, pressure is considered only as a parameter of the temperature (enthalpy) dependence of fluid properties)

o Presently: generally accepted to consider SCP fluids as single- phase media with variable physical properties and correlate the specifics of SCP fluid heat transfer just with peculiarities in the behavior of the thermophysical properties of such a fluid [16].B.S. Petukhov, Heat transfer in a single-phase medium under supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709

o The attempts of some authors to consider the region of maximum specific heat capacity at SCP as a special kind of phase transition zone have not been sufficiently substantiated [17]. A.M. Sirota, On supercritical transitions in single-component systems, Teploenergetika 19 (8) (1972) 73–78 (in Russian)

Nevertheless, this zone is very often called the ‘‘pseudophase transition region’’; this has certain sense, because on both sides of this zone, the dependences of thermophysical properties on temperature and pressure are quite different.

o FIG 1-4 Left of critical isotherm dependence of thermophysical properties

remains qualitatively the same as that for dropletlike fluids at subcritical pressures (liquid phase) the specific volume, as well as specific heat and heat conductivity, change slightly with temperature and almost do not depend on pressure, while viscosity and, correspondingly, the Prandtl number considerably decrease with temperature

Right of T_crit at a certain distance from it : pattern of changes in SCP fluid properties qualitatively correspond to that of gases with variable properties when an increase in the and values with temperature and constancy or a slight growth in Cp are observed, while the density is nearly proportional to the pressure and inverse temperature 1/T. The Prandtl number is on the order of unity and slightly depends on temperature and pressure.

[12] M.P. Vukalovich, S.L. Rivkin, A.A. Aleksandrov, Tables of Thermophysical Properties of Water and Steam, Izd. Standartov, Moscow, 1969 (in Russian).

[13] S.L. Rivkin, Thermophysical Properties of Water in Critical Region, Izd. Standartov, Moscow, 1970. p.635 (in Russian).

[14] A.A. Aleksandrov, B.A. Grigor’ev, Tables of Thermophysical Properties of Waterand Steam, second ed., MEI Publishing House, Moscow, 2006 (in Russian).

[15] V.V. Altunin, Thermophysical Properties of Carbon Dioxide, Izd. Standartov, Moscow, 1975. p. 551 (in Russian).

When the passage through the region of the specific-heat maximum occurs, we observe a considerable change in such an important parameter as the relative work of expansion Eq = (pdV/ dq)p = p/(Cp), which is performed by a substance in the course of thermal expansion against external pressure forces (see Figs. 2a and 3a).

o At t<<tm, the values of Eq are small and have an order of magnitude of 10 -2, which is typical of dropletlike liquids.

o the right of tm, Eq increases to rather high values of 0.2–0.4, which are characteristic of gases. In this connection, we propose using Eq to determine the boundaries of the pseudophase transition region.

o Beyond these boundaries, we can consider an SCP fluid as a certain analog of a dropletlike liquid or as a gas with variable physical properties

o It is clear from Figs. 2a and 3a that it is expedient to make such a definition using the dependence of Eq on enthalpy h. In so doing, the region of the pseudoliquid state (I) can be determined from the condition Eq ≤ 0.02–0.03. We designate the lower boundary of the pseudophase transition region (II), which corresponds to this condition, as hm0. Then, the values of hm0 will be as follows: for water, hm0 ≈ 1500 kJ/kg, and for CO2 ≈ 500 kJ/kg.

o It is expedient to designate the upper boundary of the pseudophase transition region, from which SCP fluid can be considered as a gas (region III), as hm1, the value of enthalpy at which Eq reaches the level E0

q = R=C0p typical of a particular

substance in an ideal gas state. For water, hm1 = 2800–3000 kJ/kg (2900 kJ/kg, on average), and that for CO2 = 750–800 kJ/kg (780 kJ/kg). Note that the difference h m1 - hm0, which is ≈1400 kJ/kg, for water and ≈280 kJ/kg for CO2, corresponds to the heat of evaporation of water at p ≈ 8.5 MPa and that of CO2 at p ≈ 1.97 MPa (in both cases p/ pcr ≈ 0.27). This gives additional grounds to call the range hm0 < h < hm1 the pseudophase transition region and to apply the customary thermal-engineering terminology to characterize the stages of SCP-flow heating, i.e., economizer-type heating (hb < hm0), steam generation (hm0 ≤ hb ≤ hm1), and steam superheating (hb > hm1).

In cases when the temperature parameters of the heat transfer process (tb and tw) at SCP do not fall beyond the boundaries of the regions of the liquid (I) and gaseous (III) states, we should anticipate that the hydraulic-resistance and heat-transfer characteristics will satisfy the regularities obtained for viscous liquids or gases with variable physical properties. This has been confirmed by numerous experimental data. The specific features of SCP heat transfer manifest themselves in cases when the temperature range within which the process takes place fully or partially falls in the pseudophase transition region (II).

While studying hydrodynamic and heat transfer processes of SCP coolants experimentally, especially if we are dealing with the pseudophase transition region, it is of great importance to be able to quite accurately determine the thermophysical properties of the studied fluid. This primarily concerns the thermodynamic parameters of a state. Such a possibility depends on the availability of sufficiently accurate and detailed tables of the properties of a substance, as well as on properly arranging of primary measurements, which account for the specifics of the near-critical region.

Presently, there are few substances, for which rather detailed and reliable data on thermophysical properties in the near-critical region are available. Water and carbon dioxide have been investigated better than other fluids. Recently, the near-critical region of helium was rather intensely explored. For water and pure carbon dioxide (99.9% CO2 or more) mutually consistent tables of the thermodynamic and calorific parameters of state have been elaborated [14,15]. These tables are based on the vast experimental material on p-V-T data, as well as those on enthalpy and specific heat.

Transport properties ( and ) in the near-critical region remain insufficiently studied, even for water and CO2. Recently, the data on viscosity and heat conductivity of water and steam were considerably refined (see [7,8]). Since 1997, the new standard on the thermophysical properties of water IAWPS-97 (briefly IF-97) has been in force. All calculations for industrial equipment must use IF-97 data [14]. The new tables on the heat conductivity of water have constructed with allowance for the existence of peak k values in the pseudophase transition region, which were recognized in many experiments (Fig. 2). The region of elevated values encompasses a rather large interval of pressures and enthalpies. As is clear from Fig. 2, the presence of peaks considerably changes the value of the Prandtl number in the vicinity of tm. In the range of p/pcr from 1.02 to 1.12, the decrease in the Pr(tm) value is by a factor of ~2.5–1.9; in the range of p/pcr from 1.2 to 1.35, by ~1.5–1.35.

It is easy to obtain, with the use of the known McAdams formula [18], for example, that such changes in the Prandtl number lead to a considerable increase in the calculated heat transfer coefficients, in the vicinity of tm, by a factor of ~1.75–1.18. The consequences of these changes in the standard tabular thermophysical properties on heat transfer calculations will be discussed in detail in the third part of the paper.[18] W.H. MacAdams, Heat Transmission, McGraw-Hill, New York, 1942

Analysis of the experimental data on the heat conductivity of CO2 conducted in [15] also shows the presence of peaks in the pseudocritical region. For CO2, however, the discrepancy in the experimental values in these peaks is very large. Because of this, the interpolating equation for heat conductivity of CO2, which was used in constructing the tables [15], does not allow for the presence of peaks in the near-critical region. Proceeding from the well-acknowledged experimental data, the authors of [9], which was published later than [15], consider the presence of peaks on (t) isobars doubtless and propose coordinating

correlations to introduce proper corrections to the canonic heat conductivity values of SCP CO2. Fig. 4a shows the (t) values calculated with the use of correlations from [9] and [15] at p = 7.7 and 9.0 MPa (p/pcr = 1.05 and 1.23, respectively). It is clear that allowance for the excesses in k at pressures close to critical also considerably changes the commonly used Prandtl number values in the vicinity of tm, Fig. 4b. For example, with p/pcr = 1.05, the Prandtl number decreases by a factor of ~1.8, and with p/pcr = 1.23, by ~15%.

When the hydraulic resistance coefficients and velocity fields are determined experimentally by measuring the distribution of static and dynamic pressures in the flow, it is necessary to exactly know the fluid density under the conditions of the experiment. In this connection, we have always paid special attention to the problem of determining the actual state and density of an SCP fluid at temperatures corresponding to the pseudophase transition region. We consider this problem as it relates to carbon dioxide, which was used in our experiments as the working substance. Naturally, the main conclusions are applicable to the other SCP fluids.

Experimental data on regularities of turbulent heat transfer in tubes under SCP conditions

Large amount of experimental works on HT to near-critical pressure coolants in tubes and channels Reviews:

o V.S. Protopopov, Study of heat transfer under turbulent flow of supercriticalpressure carbon dioxide. Part 1. Heat transfer and hydraulic resistance under turbulent flow of a supercritical-pressure fluid in tubes (analysis of the stateof-the-art), Report B376948, MEI, Moscow, 1975

o I.L. Pioro, R.B. Duffey, Heat Transfer and Hydraulic Resistance at Supercritical Pressures in Power Engineering Applications, ASME Press, New York, 2006. p. 334

o B.S. Petukhov, Heat transfer in a single-phase medium under supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709

o B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Advances in Heat Transfer, Vol. 6, Academic Press, New York, 1970. pp. 503–564.

o R.C. Hendricks, R.J. Simoneau, R.V. Smith, Survey of heat transfer to nearcritical fluids, Adv. Cryogenic Eng., vol. 15, Plenum Press, USA, 1970. pp. 197–237.

o G.V. Alekseev, A.M. Smirnov, Heat transfer in turbulent flow of liquids at supercritical pressures in channels, FEI, Obninsk, 1973, p. 83 (in Russian).

o W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int. Heat Trans fer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392.

o J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical pressure, in: S. Kakacˇ, D.B. Spalding (Eds.), Turbulent Forced Convection in Channels and Bundles, Vol. 2, Hemisphere, Washington, 1979, pp. 563–612.

o A.F. Polyakov, Heat transfer under supercritical pressures, Adv. Heat Transfer 21 (1991) 1–53.

o V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical pressure of the coolant. Part I: specifics of thermophysical properties, hydrodynamics, and heat transfer of the liquid. Regimes of normal heat transfer, Therm. Eng. 45 (3) (1998) 177–185.

o V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical pressure of the coolant. Part II: Heat transfer and friction at high heat fluxes. The influence of additional factors. Enhancement of deteriorated heat transfer, Therm. Eng. 45 (4) (1998) 301–310.

o S. Yoshida, H. Mori, Heat Transfer to Supercritical Fluids Flowing in Tubes. SCR–2000, University of Tokyo, Japan, 2000.

H2O, CO2, O2, N2, H, NH3, He and different kind of refrigerants To simulate HT in the pseudoliquid region and in the vicinity of T_pc refrigerants can be

applied! (CO2 not because at t=20°C h is already > hmo) Investigations revealed the extraordinary complexity of the regularities of SCP HT!

o Sorting of HT regimes depending on heat load and thermodynamic state of a fluid: normal, deteriorated and improved HT

o B.S. Petukhov, Heat transfer in a single-phase medium under supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709

o B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Advances in Heat Transfer, Vol. 6, Academic Press, New York, 1970. pp. 503–564

NORMAL HT o corresponds qualitatively with the existing ideas on turbulent HT with constant or

slightly variable physical propertieso For boundary conditions (q_w = cte, like in most experiments), for any h_in, T_w

changes MONOTONIC along the heated tube (common for linear HT problems)o At a small distance from the inlet (x/d>20-40) stabilization of the HT intensity

Depends only on the local parameters and hardly on the temperature (enthalpy) at the inlet and the geometric characteristics of the inlet

For identical q_w, d and ρu stabilized values of T_w are described by a single monotonic curve IIRESPECTIVE of h_in

See FIG 8 for q_w < 300 kW/m² See FIG 9 curve 1

[36] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow in tubes at supercritical pressure, Heat Transfer-V, in: Proceedings of the fifth All- Union conference on heat mass transfer, Nauka i Tekhnika, Minsk, Belarus’, 1976, 1(1) pp. 261–269 (in Russian).

[37] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under water flow at supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–2349.

[38] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89.

o Effect of the starting section remains qualitatively the same as for constant fluid properties within 15-20 gages of the tube there is a relative decrease in HT rate caused by the formation of the temperature field in the liquid flow.

B.S. Petukhov, V.S. Protopopov, V.A. Silin, Experimental investigation of worsened heat transfer conditions with the turbulent flow of carbon dioxide at supercritical pressure, High Temp. 10 (2) (1972) 304–310

o The influence of the type of boundary condition (e.g. a change in q_w value along the tube) and of the wall roughness remains in within the limits of typical developed turbulent flows

o Effect of variable properties on the local characteristics of normal HT is significant HTC and Nu numbers differ with a factor 1.5 to 0.2 compared to the correlations for constant properties

The different between Nu_b and Nu_b,0 (ref cte properties) are very high in the pseudophase transition region (< or > 1)!

Nu_b / Nu_b,0 > 1 for T_b < T_pc and T_w≈T_pc: condition of increased heat capacity in the wall layer compared to the flow core.

Nu_b / Nu_b,0 << 1 for T_b ≈ T_pc and T_w>>T_pc: decrease in density and c_p in the wall layer flow

NORMAL HT regimes meets requirements for reliable and safe operation of SCP HXo BUT, this regime is limited to relative LOW HEAT LOADS q / ρuo At HIGH heat loads more complex HT unfavourable phenomena e.g. HTD =

sharp reduction in HTC at certain limited sections of the tube! For boundary condition q_w = cte peaks in T_w appear (FIG 8 and 9)

superheating of the wall can occur dangerous to wall strength A significant HTD due to a small increase in heat load (1-10%) is

UNFAVOURABLE

M.E. Shitsman, Impairment of heat transmission at supercritical pressures, High Temp. 1 (2) (1963) 237–244

o THE DETERMINATION OF THE REASONS AND CONDITIONS AT WHICH THE TRANSITION O THE REGIME OF HTD AND THE OOCURS ARE THE MOST ACUTE PROBLEMS!!!

HEATTRANSFER DETERIORATIONSo Numerous studies: depending on d, mass flow rate and h_in non-monotonic T_w-

distribution under heating of tube according to the q_w=cte law can originate near the tube inlet (“inlet-peaks in T_w) + in the range of flow enthalpies h_b which correspond to the pseudophase transition

o Inlet peaks in T_w for different dia (5.7-32.2mm) for upward flow for moderate mass flow rates (200-1000 kg/m²s) for water and CO2

S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under water flow at supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–2349.

[43] M.E. Shitsman, Peculiarities of a temperature regime in tubes under supercritical pressures, Teploenergetika 15 (5) (1968) 57–61 (in Russian).

[44] Yu.V. Vikhrev, Yu.D. Barulin, A.S. Kon’kov, Study of heat transfer in vertical tubes under supercritical pressures, Therm. Eng. 14 (9) (1967) 116–119.

[45] I.S. Alferov, R.A. Rybin, B.F. Balunov, Heat transfer under turbulent flow of water in vertical tubes with significant effect of free convection, Teploenergetika 16 (12) (1969) 66–70 [in Russian]..

[46] P.J. Bourke, D.J. Pulling, L.E. Gill, W.H. Denton, Forced convective heat transfer to turbulent CO2 in the supercritical region, Int. J. Heat Mass Transfer 13 (8) (1970) 1339–1348.

[47] I.I. Belyakov, L.Yu. Krasyakova, A.V. Zhukovskii, N.D. Fefelova, Heat transfer in vertical and horizontal tubes under supercritical pressure, Teploenergetika 18 (11) (1971) 39–43 (in Russian).

[48] J.N. Ackerman, Heat transfer during pseudoboiling of water in supercritical region in smooth and finned tubes, Trans. ASME J. Heat Transfer 3 (1970) 490–498.

[49] N.P. Ikryannikov, B.S. Petukhov, V.S. Protopopov, Calculation of heat transfer in single phase near-critical region under viscous-inertial-gravitational flow, High Temp. 11 (5) (1973) 949–955.

[50] D.J. Brassington, D.N.H. Cairns, Measurements of forced convective heat transfer to supercritical helium, Int. J. Heat Mass Transfer 20 (8) (1977) 207– 214.

[51] V.A. Bogachev, V.M. Eroshenko, L.A. Yaskin, Heat transfer in upward flow of supercritical-pressure helium in a transition-flow regime in a round tube, High Temp. 21 (3) (1983) 611–619.

[52] M.J. Watts, C.T. Chou, Mixed convective heat transfer to supercritical pressure water, Proc. 7th Int. Heat Transfer Conf. Munchen 3 (1982) 495–500.

[53] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in Russian].

[54] V.A. Kurganov, Heat transfer of gases in turbulent flow in tubes, Teploenergetika (Thermal Engineering) 39 (5) (1992) 2–9 (in Russian).

o These peaks are more typical for tubes with larger diameter (16-32.2mm) Located at inlet section of 0≤x/d≤30-50 (initial thermal section)

o Inlet peaks recognized for different values of liquid enthalpy at inlet h_in up to h_m and more


o The T_w in the inlet peaks can be higher or lower than T_pco As a rule: DOWNSTREAM of the inlet maximum T_w the section of normal or

increased HT was observed (especially for h_in<<h_m)o Very long tube with h_b>h_m0 secondary HTD occurs increase in h_in 2

regions of HTD (2 maxima in T_w or even more) DOWNWARD AND HORIZONTAL FLOW (under same conditions as UPWARD flow)

o No inlet peaks of T_w! inlet peaks in T_w originate as a result of the considerable effect of buoyancy forces on turbulent flow in the region of the initial thermal section!

Evolution of T_w at the initial section of large diameter tubes with ↑ heat flow rate q (rest of conditions equal) complex nature depending of values of ρu and h_in

o Typical if q_w ↑: T_w peak shifts to inlet of the tube For small mass flows (200 kg/m²s) the inlet peak can degrades as q_w↑

T_w distribution monotonic again (like at small heat flow rates) [49] N.P. Ikryannikov, B.S. Petukhov, V.S. Protopopov, Calculation

of heat transfer in single phase near-critical region under viscous-inertial-gravitational flow, High Temp. 11 (5) (1973) 949–955.


HTD for h_b in the pseudophase transition is most typicalo This KIND of HTD (for h_b in pseudophase transition) can occur in tubes of different

DIAMETER! S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow

in tubes at supercritical pressure, Heat Transfer-V, in: Proceedings of the fifth All-Union conference on heat mass transfer, Nauka i Tekhnika, Minsk, Belarus’, 1976, 1(1) pp. 261–269 (in Russian).

B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89.

V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in Russian].

Yu.D. Barulin, Yu.V. Vikhrev, B.V. Dyadyakin, et al., Heat transfer in turbulent flow of supercritical-state water in vertical and horizontal tubes, Inzh. Fiz. Zhurnal (Eng. Phys. Journal) 20 (5) (1971) 929–930 (in Russian).

REVIEWS V.S. Protopopov, Study of heat transfer under turbulent flow of

supercriticalpressure carbon dioxide. Part 1. Heat transfer and hydraulic resistance under turbulent flow of a supercritical-pressure fluid in tubes (analysis of the stateof-the-art), Report B376948, MEI, Moscow, 1975

I.L. Pioro, R.B. Duffey, Heat Transfer and Hydraulic Resistance at Supercritical Pressures in Power Engineering Applications, ASME Press, New York, 2006. p. 334

B.S. Petukhov, Heat transfer in a single-phase medium under supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709

B.S. Petukhov, Heat Transfer and Friction in Turbulent Pipe Flow with Variable Physical Properties, Advances in Heat Transfer, Vol. 6, Academic Press, New York, 1970. pp. 503–564.

R.C. Hendricks, R.J. Simoneau, R.V. Smith, Survey of heat transfer to nearcritical fluids, Adv. Cryogenic Eng., vol. 15, Plenum Press, USA, 1970. pp. 197–237.

G.V. Alekseev, A.M. Smirnov, Heat transfer in turbulent flow of liquids at supercritical pressures in channels, FEI, Obninsk, 1973, p. 83 (in Russian).

W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int. Heat Trans fer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392.

J.D. Jackson, W.B. Hall, Forced convection heat transfer to fluids at supercritical pressure, in: S. Kakacˇ, D.B. Spalding (Eds.), Turbulent Forced Convection in Channels and Bundles, Vol. 2, Hemisphere, Washington, 1979, pp. 563–612.

A.F. Polyakov, Heat transfer under supercritical pressures, Adv. Heat Transfer 21 (1991) 1–53.

V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical pressure of the coolant. Part I: specifics of thermophysical properties, hydrodynamics, and heat transfer of the liquid. Regimes of normal heat transfer, Therm. Eng. 45 (3) (1998) 177–185.

V.A. Kurganov, Heat transfer and pressure drop in tubes under supercritical pressure of the coolant. Part II: Heat transfer and friction at high heat fluxes. The influence of additional factors. Enhancement of deteriorated heat transfer, Therm. Eng. 45 (4) (1998) 301–310.

S. Yoshida, H. Mori, Heat Transfer to Supercritical Fluids Flowing in Tubes. SCR–2000, University of Tokyo, Japan, 2000.

o For higher MASS FLOW RATES (1500 kg/m²s or more for CO2 and H2O) in SMALL DIAMETER (up to ~10mm) an INCREASE in HEAT FLOW RATE q_w HTD regardless of FLOW DIRECTION!! (FIG 9)

Here: ARCHIMEDES EFFECT is SECONDARY, but it can causes a considerable difference in local HT under upward and downward flows as well as a large temp maldistribution over the circumference in a horizontal tube, as well as a change in the location, height and configuration of the T_wall peak under equal heating conditions!!!

THIS KIND of HTD occurs at h_in < and > than h_m (FIG 9) Hydraulic measurements (SEE PART II) in HTD regimes like in THIS CASE:

pressure drop p_i due to flow acceleration is considerably higher than the friction resistance p_

the flow is significantly gradient in nature in contrast to the normal HT regimes (FIG 12)

B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89

MODERATE MASS FLOW RATES (200<ρu<1000 kg/m²s for CO2 and H2O) + h_b in PSEUDOPHASE TRANSITION

o HT depends on FLOW ORIENTATION in gravity field (FIG 8) As q_w ↑ greatest danger for HTD is for UPWARD FLOW Same conditions:

Downward: strong T_w peaks Upward: T_w distributions remains monotonically

o local HTC differ a lot!o LOW values of h_in<<h_m in LONG tubes + rather LARGE diameter + UPWARD FLOW

TWO T_w peaks = inlet peak + peak in region of PSEUDOPHASE TRANSITION

[44] Yu.V. Vikhrev, Yu.D. Barulin, A.S. Kon’kov, Study of heat transfer in vertical tubes under supercritical pressures, Therm. Eng. 14 (9) (1967) 116–119.

[45] I.S. Alferov, R.A. Rybin, B.F. Balunov, Heat transfer under turbulent flow of water in vertical tubes with significant effect of free convection, Teploenergetika 16 (12) (1969) 66–70 [in Russian]..

[46] P.J. Bourke, D.J. Pulling, L.E. Gill, W.H. Denton, Forced convective heat transfer to turbulent CO2 in the supercritical region, Int. J. Heat Mass Transfer 13 (8) (1970) 1339–1348.

[47] I.I. Belyakov, L.Yu. Krasyakova, A.V. Zhukovskii, N.D. Fefelova, Heat transfer in vertical and horizontal tubes under supercritical pressure, Teploenergetika 18 (11) (1971) 39–43 (in Russian).

[48] J.N. Ackerman, Heat transfer during pseudoboiling of water in supercritical region in smooth and finned tubes, Trans. ASME J. Heat Transfer 3 (1970) 490–498.

As h_in increases TWO region come together T_w distribution show two peaks (or more) each following directly one another

= TYPICAL for CO2, because h_in<<h_m0!!! M.E. Shitsman, Peculiarities of a temperature regime in tubes under

supercritical pressures, Teploenergetika 15 (5) (1968) 57–61 (in Russian).

V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in Russian].

A. Watson, The influence of axial wall conduction in variable property convection-with particular reference to supercritical pressure fluids, Int. J. Heat Mass Transfer 20 (1) (1977) 65–71

B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int. Heat Transfer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392.

HORIZONTAL AND INCLINED tubes with MODERATE MASS FLOW RATESo In region of HIGH HEAT LOADS q_w/ρu HT has some specific features!

Due to presence of SECONDARY FREE-CONVECTIVE FLOWS + STEADY DENSITY DISTRIBUTION NEAR THE UPPER GENERATRIX

[59] M.E. Shitsman, The effect of natural convection on temperature conditions in horizontal tubes at supercritical pressures, Therm. Eng. 13 (7) (1966) 69–75.

[60] A.V. Zhukovskii, L.Yu. Krasyakova, I.I. Belyakov, N.D. Fefelova, Heat transfer in a horizontal tube at SCP, Energomashinostroenie 2 (1971) 23–26 (in Russian).

[61] V.M. Solomonov, V.A. Lokshin, Temperature conditions and heat transfer in horizontal and inclined tubes of steam generators

at supercritical pressure under conditions of joint free and forced convection, Therm. Eng. 22 (7) (1975) 74–77.

[62] J.A. Adebiyi, W.B. Hall, Experimental investigation of heat transfer to supercritical pressure carbon dioxide in a horizontal tube, Int. J. Heat Mass Transfer 19 (8) (1976) 715–720.

Region of HTD localized NEAR the UPPE GENERATRIX of the tube, while NEAR the LOWER GENERATRIX HIGH level of HT!!!

Temperature difference between upper and lower generatrix can reach 200K for CO2 and H2O!!!

One of the main difficulties on generalizing the experimental data on HTD is the fact that all T_w(h_b) curves corresponding to different h_in (the rest of the conditions equal) have an individual pattern which coincide with one another only at h_b ≈ h_m1!

o BUT, for water and possible refrigerants, a certain REGULARITY exists in the HTD in certain enthalpy regions within h_m0<h_b<h_m, regardless of h_in, if a fluid is heated from the tube section with h_in<<h_m0 (the tube has an economizer section of considerable length)

H. Komita, S. Morooka, S. Yoshida, H. Mori, Study on the heat transfer to the supercritical water cooled power reactor development, NURETH-10, Seoul, Korea, October 5–9, 2003

o EXPLANATION REGULARITY: in these cases, a flow enters the “vapour-generation” section with ± appr the same hydrodynamic structure, because t_w<t_pc (in the heating zone) and the scale of variation of density and other physical properties over the tube cross section and length is relatively small.

STUDIES about HTD:o THE ULTIMATE HEAT LOAD VALUE, where the criteria of normal HT and qualitative

description of the HTD process are maintained, depend strongly on ADDITIONAL FACTORS that do NOT affect Normal HT!!

Conditions at the tube inlet Wall roughness

The data in [65] also showed that wall roughness DELAYED the transition to HTD regime possible to increase the allowable heat load level by 15-20%!!

[65] H. Tanaka, N. Nishiwaki, M. Hirata, A. Tsuge, Forced convection heat transfer to fluid near critical point flowing in circular tube, Int. J. Heat Mass Transfer 14 (6) (1971) 739–750

BUT, as q_w increased, eventually HTD occurred and the increase in T_W was even more sharply and to a greater value! The used test section with uncontrolled roughness of commercial-grade smooth tubes COULD BE the source of VARIATION between the experimental data on HTD

FEW CASES where hydraulic resistance was preliminarily measured under adiabatic conditions considerable influence of wall roughness on hydraulic resistance was recognized in the same rang of Re numbers, within which the abnormal HT data were than later obtained.

E.g. Water: salt deposits and fouling on heated walls affect HT deterioration.

o [67] G.V. Alekseev, V.A. Silin, A.M. Smirnov, V.I. Subbotin, Study of thermal conditions on the wall of a pipe during the removal of heat by water at asupercritical pressure, High Temp. 14 (4) (1976) 683–687.

o [68] M.E. Shitsman, L.S. Midler, A.V. Firsov, Temperature regime of tubes and critical phenomena in solutions, Heat Transfer-IV, in: Proceedings of the fourth All-Union conference on heat mass transfer, Nauka i Tekhnika, Minsk, Belarus’, 1972, 2(1) pp. 30–36. (in Russian).

o [69] O.K. Smirnov, Yu.P. Michurov, Study of thermohydraulic characteristics of a steam generating tube with formation of a solid phase of admixtures in supercritical parameters medium, Teploenergetika (Therm. Eng.) 22 (7) (1975) 83–86 (in Russian).

Also water chemistry of the coolant (admixtures) can change the conditions and the pattern of HTD.

o E.g. dissolved gases in the SCP fluid convert the solution in a subcritical state the gas admixture can provoke a SHARP HTD!! (compared to HTD in the SC region) resembles the HT CRISIS under subcooled liquid boiling.

o E.g. CO2-N2 mixture (0.5 to 4.0 mol% N2 concentration) B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, A.G.

Kaptil’nyi, Dissolved gas effect on heat transfer under supercritical-pressure carbon dioxide turbulent flow in a tube, High Temp. 23 (4) (1985) 742–747.

Forced pressure oscillations (flow rate) due to pump operationArtificial damping of pressure oscillations downstream of the pump under water heating in a rising tube sharp change in temperature regime of the wall tube HTD!!K. Yamagata, K. Nishikawa, S. Hasegava, T. Fujii, S. Yoshida, Forced convective heat transfer to supercritical water flowing in tubes, Int. J. Heat Mass Transfer 15 (12) (1972) 2575–2593

For HT of SC CO2 in a smooth tube (d=6mm and ρu = 1180-2350 kg/m²s) [65]

For a gear-type pump pressure oscillations generated an increase in heat flow rate in the deteriorated HT regimes (T_w same level as in [38,66]) + shift of the T_w maximum to the region h_b>h_m


[66] B.S. Shiralkar, P. Griffith, Deterioration in heat transfer to fluids at supercritical pressure and high heat fluxes, Trans. ASME J. Heat Transfer 91 (1) (1969) 27–36.

o ALL these factors should be taken into account when comparing the QUANTITATIVE DATA on HTD in literature!!!

During transition to HTD regime excitation of a self-oscillating process is often observed!o Acoustic instability on the flow is also a feature of heating turbulent SCP fluids in

tubes especially of small diameter with h_b<h_m0 leads often to resonance thermoacoustic oscillations (TAOs) of pressure in tubes at a natural frequency and overtones

o SPAN of pressure oscillations in the antinodes of standing waves can be very high (up to 50% of the absolute pressure) DANGEROUS FOR THE TUBE STRENGTH!

E. Stewart, P. Stewart, A. Watson, Thermo-acoustic oscillations in forced convection heat transfer to supercritical pressure water Int. Journal of Heat and Mass Transfer Vol. 16, pp. 257-270. 16 (2) (1973) 257–270.

o TAOs contribute to HT ENHANCEMENT! V.V. Sevastyanov, A.T. Sinitsin, F.L. Yakaitis, Study of heat exchange

process in supercritical region of parameters under conditions of high-frequency pressure oscillations, High Temp. 18 (3) (1980) 433–439

also called regimes of enhanced HT [16] or HT with pseudoboiling [16] B.S. Petukhov, Heat transfer in a single-phase medium under

supercritical conditions (survey), High Temp. 6 (4) (1968) 696–709o Conditions for TAOs development:

Existence of turbulent mode of the flow Considerable subcooling of the fluid relative to T_pc A level of heat flow rates measuring T_w>T_pc

an ultimate change in density and other properties of the fluid in the wall layer

o LOWER and UPPER boundaries of the acoustic instability as function of the heat flow rate V.I. Vetrov, V.A. Gerliga, V.G. Razumovskii, Experimental study of thermoacoustic oscillations in heated channels under supercritical pressures of water, Voprosy Atomnoi nauki i Tekhniki, Ser. Dinamika Yadernykh Energeticheskikh Ustanovok (Problems of Nuclear Sci. Techn., Ser. Dynamics of Nuclear Power Plants), 2 (12) (1977) pp. 51–57 (in Russian).

o Attemps for theoretical models for this phenomenon [72] V.V. Sevastyanov, A.T. Sinitsin, F.L. Yakaitis, Study of heat exchange

process in supercritical region of parameters under conditions of high-frequency pressure oscillations, High Temp. 18 (3) (1980) 433–439.

[74] V.I. Vetrov, Mechanism of thermoacoustic instability under forced flow of a coolant, Izv. AN SSSR, ser. Energetika i Transport (J. USSR Acad. of Sciences, ser. Energy and Transport), No. 1, (1987), pp. 119–127 (in Russian).

o E.g. TAO development in tests on HTD in steam-generating tubes Also in V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance

and heat transfer in vertical heated tubes under supercritical pressure of a

coolant, in: A.F. Polyakov (Ed.), Turbulent heat transfer under mixed convection in vertical tubes, IVTAN, Moscow, 1989, pp. 95–160 [in Russian].

For CO2 (p = 7.7 and 9 MPa, tin = 20°C) 8 mm diameter tube TAO at ρu=1000 and 1350 kg/m²s with the transition to HTD regime

Freq. of 50 an 350 Hz – amplitude = 0.5MPa!! distribution of static pressure

[38] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89

As h_out approached h_m1 TAO stopped No significant effect of these oscillations in HT were recognized, only

in a 22.7mm tube TAO improved the HT in the inlet peak region of T_w!

The nature and classification of deteriorated heat transfer regimes GOLDMAN (1954) and SHLYKOV (1971)

o Reason for turbulent HTD: localization of the T_pc region in certain zones of the flow causes difficulties in HT through the zone with higher values of and c_p.

o Proposed generalized variables, but local analysis of the problem based on the this concept appeared UNSUCCESSFUL

POLYAKOV (1973-1975)o In non-isothermal fluid flow with variable properties turbulent HT changes due to

the action of:

Mass forces induced by thermal acceleration of the flow and Archimedes forces originated due to a non-uniform density distribution over

the flow cross sectiono The assumed that these forces act immediately on the intensity of turbulent HT.o The correlation ρ ' u ' in the equation of the turbulent energy balance (in the term

ρ ' u ' [±g+u( ∂u∂ χ )]) was considered an agent of this effect which can

considerably changer the local parameters of turbulent transfer, fluid flow and heat transfer. This happens PRIOR to the distortion of the averaged flow according to the equation of motion.

o On this basis, Polyakov did estimations for the boundaries of the effect of mass forces in friction and heat transfer in tubes and expressions were presented for specific criteria of the thermal acceleration and the effects of Archimedes forces (SEE PART III)

A.F. Polyakov, Mechanism and limits on the formation of conditions for impaired heat transfer at a supercritical coolant pressure, High Temp. 13 (6) (1975) 1119–1126

At the same time (1970-1971) calculations and experimental works that demonstrated that in DHT regimes COMPLETE REARRANGEMENT of the AVERAGED FLOW takes place velocity profile becoming M-SHAPED!

o HTD develops at the phase of velocity profile rearrangement when the zone with

very small values of the product τ t( ∂u∂ y ) appears, which is responsible for the

gradient generation of turbulent energy = BARRIER LAYER formed in the flowo [76] C.A. Bankston, D.M. McEligot, Turbulent and laminar heat transfer to gases

with varying properties in the entry region of circular ducts, Int. J. Heat Mass Transfer 13 (2) (1970) 319–344.

o [77] P.J. Bourke, D.J. Pulling, Experimental explanation of deterioration in heat transfer to supercritical carbon dioxide, Paper ASME, No. 71-HT-24, 1971, p. 7

ESTIMATE describing the above ideas of POLYAKOV

o Prandtl and Boussinesq hypotheses correlation Expression for turbulent momentum transfer coefficient:

o a decrease in shear stresses in the wall region of the flow reduction in turbulent transfer and heat transfer

THE FACTORS THAT DETERMINE THESE TRANDS are (HALL AND JACKSON):1. THERMAL ACCELARATION (for SCP fluid and gas)2. LIFTING ARCHIMEDES FORCES (BUOYANCY FORCES) (for vertical tubes)W.B. Hall, J.D. Jackson, Heat transfer near the critical point, Proc. VI Int. Heat Transfer Conf, Vol. 6, Hemisphere, New York, 1978. pp. 377–392

the pressure gradient that provides fluid motion is determined NOT ONLY by FRICTION RESISTANCE (as in case of constant properties fluid), but also

ADDITIONAL DYNAMIC FACTORS that appear due to DENSITY CHANGES over the tube cross section and along its length:

With ‘+’ in gravitational term = upward flow and ‘-‘ = downward flow

With:

With: = coefficient of inertial flow resistance

In , in the inertial resistance of non-isothermal SCP flows the momentum coefficient S_b and its changes along the tube length play an ACTIVE ROLE! actual ξ i-values can considerably differ from mass-averaged estimates (SEE PART II)

The solution to the equation of motion (

) of an axisymmetric VERTICAL flow, in the wall region (R1), where convective terms

can be neglected

Where ‘C’ = averaged value of the ratio in the interval from R to 1

o Roughest estimate C≈0.5 asymptotic formula:

o The smaller K_g/K_i, the closer to realityo Analysis of the results of hydraulic and sounding

measurements (SEE PART II) showed that in NORMAL HT regimes, the shear-stress profile satisfied the following

condition: CONCLUSION: from the correlations it is clear that for K_ig>>1

and Eq. and

radical changes in the shear-stress profiles and in the values of the turbulent transfer coefficient which can lead to HTD!!!!

The experimental data on HT in VERTICAL heated tubes in literature showed that the general behaviour of the T_w at boundary condition q_w≈cte + rather high heat loads q_w/ρu (thus

beyond the limits of NHT), DEPEND on the PARAMETER: this parameter specifies the potential scale of BUOYANCY EFFECT on the DYNAMICS of a fluid IN THE WALL LAYER.

CLASSIFICATION OF DHT REGIMES UNDER MIXED CONVECTION IN VERTICAL HEATED TUBED OF SCP FLUIDS 6 typical groups

o Schematic presentation of heat transfer in vertical tubes at q_w≈const and high heat loads depending on the scale of Archimedes forces

o Supposes tubes that are sufficiently long L/d~100 or more!

It was recognized that the boundary values of for different groups of regimes, which were specified from the experimental data, form a straight line with equidistant nodes in logarithmic coordinates allows to determine the number of the group Ngr into which the regime under consideration falls using the simple formula:

It is advised to perform calculations with an accuracy of 0.1 If Ngr ↑ effect of Archimedes forces ↓and the influence of thermal acceleration ↑ In very long tubes (thermodynamic state can change considerably along the tube) the HT

regime can pass into another group in accordance with the local values of parameter

Also possible to introduce the buoyancy effect parameter F_g (= the value which will give an idea on the relative role of this effect on the HT pattern)

o Consider between 3rd and 4th group an equal effect of buoyancy and thermal acceleration + consider in groups 1 and 2: F_g 1, while in regimes of 5th and 6th

groups (where thermal acceleration rules): F_g 0

simple expression for buoyancy effect parameter F_g: o THIS EXPRESSION CAN BE USED IN THE DEVELOPMENT OF CORRELATIONS FOR

FORECASTING HTD

Conclusions New recent data on specifics of the behaviour of thermophysical properties

o Advised to DIVIDE the entire region of possible states in 3 typical zones Pseudoliquid state Pseudophase transition: with in here T_pc Pseudogas state (superheated vapour)

o Determine boundaries of these zones via parameter (characterises the specific work of fluid expansion per unit of heat added

o The new standards contain sharp changes in (T) and Pr(T) must be taken into account when using the known correlations and constructing new formulas

o Effect that small admixtures of dissolved gases have on (T) and T(h) dependences and other properties of SCP water and CO1 are considered

The reason for HTD is WEAKENING of turbulent transfer in the wall region caused by a DECREASE in SHEAR STRESS due to the effect of THERMAL ACCELERATION and ARCHIMEDES FORCED (buoyancy – in vertical tubes) this concept is in agreement with Hall and Jackson’s ideas

Results of experimental studies of HT regularities for q_w≈cte considered existing data

base on DHT was classified (for high heat loads q_w/ρu) using the PARAMETER which characterises the scale of the effect of ARCHIMEDES FORCES on turbulent flow and heat transfer. 6 representative groups were distinguished with this parameter + corresponding typical modes of T_w behaviour within these groups were described.

V.A. Kurganov, Yu.A. Zeigarnik, I.V. Maslakova, Heat transfer and hydraulic resistance of supercritical-pressure coolants. Part II: Experimental data on hydraulic resistance and averaged turbulent flow structure of supercritical pressure fluids during heating in round tubes under normal and deteriorated heat transfer conditions, International Journal of Heat and Mass Transfer 58 (2013) 152–167

Abstract Experimental results on hydraulic and friction resistances in round tubes under adiabatic and

heating conditions Experimental technique + results sounding studies under normal and deteriorated HT Possible errors in hydraulic resistance calculations using 1-D flow model are pointed out Analysis between HTD and significant changes in the averaged flow structure (due to fluid

thermal acceleration and Archimedes forces effects)

Introduction PART I:

o Divide range of SCP fluid states in 3 regions: pseudoliquid (h<h_m0) pseudophase transition h_m0<h<h_m1 pseudogas (superheated vapour) (h>h_m1)

o the reference enthalpy values (h_m0, h_m and h_m1) determined with E_q=p./(c_p.)

In the range of p/p_crit ≈1.05 to 1.30 values remain unchanged for water and CO2

o Classification of HT regimes in vertical round tubes at the boundary condition of q_w ≈cte and high heat loads q_w/ρu.

Classification depends on buoyancy force and thermal acceleration effects. 6 typical groups that differs from the T_wall behaviour and its dependence

on the flow direction in the gravity field

Group number via (for L<100-150dia) Where: = the

potential parameter of the effect of Archimedes forces For very long tubes additionally account the local value of K_rx

parameter determines the regime number far from the tube inlet.

Buoyancy effect parameter value gives an understanding on the relative contribution of the buoyancy and thermal acceleration effects in building-up the HT pattern typical of the given group.

Group1 3: 1≥F_g≥0.6 the flow pattern is primarily determined by the buoyancy effects

Group 4: Archimedes forces retain their triggering role in HTD in an upward flow and damp this process in a downward flow

Regimes that fall in this group during HTD thermal acceleration effect increases a lot and provokes unfavourable T_w changes!!

Group 5 and 6: F_g 0 HTD is initiated by the thermal acceleration, regardless the flow direction. Buoyancy effects are of the second order of significance and can either intensify or weaken the deterioration rate!!

COMPLEX THERMOHYDRAULIC INVESTIGATIONS = HT studies + measurements of hydrodynamic parameters of the process (hydraulic resistance + components)

o very important for understanding the nature of SCP fluid turbulent HT specifics + developing methods for its forecasting and calculating!

EXPERIMENTAL DATA on SCP TURBULENT FLOW INTERNAL STRUCTURE is needed by SOUNDING MEASUREMENTS!

Hydraulic resistance in tubes under SCP heating VERY LIMITED experimental data on SCP fluids hydraulic resistance in tubes under heating

conditions!!! COSTING Experimental technique = measuring p along the tube length separated into components

caused by friction, flow thermal acceleration and hydrostatic head. ‘integral equation of motion’ = main correlation for this.

o Within the frames of the usual boundary layer theory assumptions for an axially symmetric flow in a rather long straight round vertical tubes equation

o ‘+’ = upward flowo ‘-‘ = downward flow

o Dividing the terms by the local conditional dynamic head correlation

between the resistance coefficients: o Take a tube segment of length L (first cross section of this segment has the origin of

the x-coordinate)

Friction resistance:

The inertial resistance:

Hydraulic resistance due to fluid flow:

Effective hydrostatic head: Flow momentum and its inertial resistance using the momentum coefficient

(Boussinesq coefficient):

the momentum coefficient S_b plays a significant role in the dynamics of non-isothermal SCP flow, because the inertial resistance depends on both its absolute value and its changing along the tube length!! (see later)

o To separate p in components important to have DETAILES INFORMATION on the AVERAGED FLOW STRUCTURE and its TEMPERATURE FIELD in different cross sections of the tube

Early works no such info estimations of p_i and p_g correspond to the 1D homogeneous flow model (flow with uniform distribution of temperature T_b over the tube cross section and constant momentum coefficient value S_b = S_0 = cte)

With: In turbulent flow S_0 = S_0t = 1.02-1.03 ≈1 In developed laminar flow with Poiseuille velocity profile S_0l=1.33 1D Darcy expression for local and averaged resistance coefficients:

MOST EXPERIMENTAL STUDIES MEAN HYDRAULIC RESISTANCE in the heating zone of round tubes measured

o Main regime parameter in existing studies:

o When processing these result 1D correlations have been used (similar to above Darcy expressions)

o The inner structure of the flows could also differ! E.g. tube of 12Kh1MF steel resistance=f(T) q_w = var Horizontal tubes difference in q_w values at upper and lower part (up to

30-50%) Tube roughness has also an important affect

due to this the results from these studies are for qualitative assessment and were instructive for further studies

In the existing experiments + special measurement NO specifics in the friction resistance in the region of T_pc!!

o [5] N.V. Tarasova, A.I. Leont’ev, Hydraulic resistance during flow of water in heated pipes at supercritical pressure, High Temp. 6 (4) (1968) 721–722.

o [8] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and friction in water flow in tubes at supercritical pressure, heat transfer – V, in: Proc. Fifth All-Union Conf. on Heat Mass Transfer, Nauka i Tekhnika, Minsk, Belarus’, vol. 1, issue no. 1, 1976, pp. 261–269 (in Russian).

o [9] S. Ishigai, M. Kadji, M. Nakamoto, Heat transfer and pressure drop under water flow at supercritical pressure, JSME J. Ser. B 47 (424) (1981) 2333–2349.

o [10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic resistance and heat transfer of smooth channels with turbulent flow of water of supercritical pressure, Thermal Eng. 31 (2) (1984) 109–113.

o [14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental study of heat transfer and hydraulic resistance in turbulent flow of CO2 of supercritical pressure, Report B914102, IVTAN, Moscow, 1979 (in Russian).

o [15] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, V.S. Grigor’ev, Experimental investigation of drag and heat transfer in a turbulent flow of fluid at supercritical pressure, High Temp. 18 (1) (1980) 90–99.

o [16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New experimental data on pressure drop and heat transfer in a round tube under heating carbon dioxide of near-critical parameters of the state, in: Convective Heat Transfer. Method and Results from the Studies, IVTAN, Moscow, 1982, pp. 29–68 (in Russian).

o [17] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Heat transfer and flow resistance in the turbulent pipe flow of a fluid with near-critical state parameters, High Temp. 21 (1) (1983) 81–89.

o [18] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Total flow resistance and fluid friction associated with ascending and descending supercritical fluid flow in heated pipes, High Temp. 27 (1) (1989) 87–94.

o [19] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent Heat Transfer under Mixed Convection in Vertical Tubes, IVTAN, Moscow, 1989, pp. 95–160 (in Russian).

Data from [5.839] show a DECREASE in under heating conditions

o Measurements in short test sections [10,11] at h_in ≥ h_m0 (typical CO2) show that 1D analysis of the measured pressure drops at high heat loads under HTD gives

Due to actual p_i values are much larger than p_io.d. under the conditions

of [10,11] [10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic resistance

and heat transfer of smooth channels with turbulent flow of water of supercritical pressure, Thermal Eng. 31 (2) (1984) 109–113.

[11] I.V. Kuraeva, V.S. Protopopov, Mean friction coefficients for turbulent flow of a liquid at a supercritical pressure in horizontal circular tubes, High Temp. 12 (1) (1974) 194–196.

Data from Table 1 generalized by empirical correlationso Tarasova and Leont’ev

With:

_w and _b = the mean integral viscosity values along the measurement section

= The friction coefficient at constant physical properties, calculated via Filinenko formula

oo [24] G.K. Filonenko, Hydraulic resistance in pipelines,

Teploenergetika (Thermal Engineering) 1 (4) (1954) 40–44 (in Russian).

[5] N.V. Tarasova, A.I. Leont’ev, Hydraulic resistance during flow of water in heated pipes at supercritical pressure, High Temp. 6 (4) (1968) 721–722.

At Re~105 correlates well with Lafay’s and Razumovskii experimental data

[25] J. Lafay, Mesure du coefficient de frottement avec transfert de chaleur en convection forcee dans un canal circulaire, Centre d’Etudes nucleairs de Grenoble, 1970, Report CEA-R-3896, pp. 52.

[10] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Hydraulic resistance and heat transfer of smooth channels with turbulent flow of water of supercritical pressure, Thermal Eng. 31 (2) (1984) 109–113

[26] V.G. Razumovskii, A.P. Ornatskii, K.M. Maevskii, Determination of resistance factors under turbulent flow of supercritical-pressure water in smooth channels, Prom. Teplotechn. (Indust. Ther. Eng.) 7 (5) (1985) 24–28 (in Russian).

In vicinity of T_pc at HIGH MASS FLOW RATES and Re numbers: p_io.d. =

several percents of p_o.d. overestimates the friction coefficient even at SMALL HEAT LOADS

In [8,9], the o.d. experimental values for HORIZONTAL tubes (FIG. 3)

acceptable (+-20%) for formula

0.6< <4 HORIZONTAL TUBES NATURAL CONVECTION EFFECT difference

between upper and lower temperature (up to 70°C) At ρu ≅ 1000 kg /m ² s Grρ / ℜb

2≈ (1.3−3.5)10−2

o Using this in the correlations in [11] o.d. values are 1.4-1.5

times HIGHER than via formula o [11] I.V. Kuraeva, V.S. Protopopov, Mean friction

coefficients for turbulent flow of a liquid at a supercritical pressure in horizontal circular tubes, High Temp. 12 (1) (1974) 194–196

In an UPWARD FLOW does NOT describe

the experimental data corresponding to 0.9< <3o The data lays lower!

The o.d./ values for a VERTICAL TUBE at MODERATE HEAT LOADS (q_w/ ρu≤0.6 kJ/kg) (UPWARD AND DOWNWARD) correlations

With as governing enthalpy Most of the experimental results follow this correlation within +-15% HOWEVER: at _b/_w < 2.5 the effect of changes in fluid density

is HIGHER than predicted with the correlation

o More at a level of

At ρu ≅ 1500 kg /m ² s and qwρu

=0.75−0.9kJ /kg (_b/_w ≈3-5)

experimental points have a lot of scatter and the trend is an INCREASING ξo . d . compared to the correlation

CONCLUSION:o Contradictive results from Table 1 experiments insufficient data on the averaged

hydraulic resistance for understanding the reasons and mechanism of HTD!!o Necessary to develop NEW measurements techniques that allows us to analyse the

structure of LOCAL hydraulic resistances Institute for High Temperatures (IVTRAN, Russia) small diameter tubes

unreal to perform sounding measurements proposed the unique 2 pressure drops method (method for high mass flow rate weak Archimedes forces effect)

[14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental study of heat transfer and hydraulic resistance in turbulent flow of CO2 of supercritical pressure, Report B914102, IVTAN, Moscow, 1979 (in Russian).

[15] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, V.S. Grigor’ev, Experimental investigation of drag and heat transfer in a turbulent flow of fluid at supercritical pressure, High Temp. 18 (1) (1980) 90–99.

[16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New experimental data on pressure drop and heat transfer in a round tube under heating carbon dioxide of near-critical parameters of the state, in: Convective Heat Transfer. Method and Results from the Studies, IVTAN, Moscow, 1982, pp. 29–68 (in Russian).

The ‘2 pressure drops method’

o Simultaneous measurement of pressure drops at the heated section of the tube with length L (FULLY DEVELOPED FLOW AT THE SECTION INLET) and unheated (adiabatic) section with length Lad ≈ 50- 60 dia.

o At adiabatic section: velocity and temperature profiles levelling occur = heated fluid flow recovers its usual turbulent flow structure!

o Further, due to INCOMPRESSIBILITY of SCP flow + SMALL DISSIPATIVE HEAT RELEASE the sale mass averaged fluid parameters are retained along the adiabatic section

(= parameters at outlet cross section)

o the and values can be determined from the results of the thermal measurements

o Experimental data showed that the T_w relaxes at <10 dia after heated zone end

(x=L) and with in an UPWARD FLOW, pi/2 in a HORIZONTAL FLOW and pi in a DOWNWARD FLOW.

o MOMENTUM BAMANCE in heated zone and adiabatic section:

o A change in friction coefficient value at the adiabatic section can be described as the

exponential correlation: a = 9 (average)

o At with via (Filonenko) with Re=Re_bL (known from the thermal measurements of the heated tube)

o ESTIMATIONS of

Assuming in (at large fluid flow rates hydrostatic heads are relatively small)

in the heated zone (clos to reality FIG. 5)

Calculation of the integral via the thermal measurement data

Solving system to the unknown and via

and In [16,17] series of p and pad measurements with different L_i (0<L_i≤L)

have been conducted.

Constructed the curves via ITERATIVE LOOP starting from the approximation for the first section 0-L1 (

).

differentiating these curves local values of +

and coefficients Details of the iterative technique for solving

see [14,19] [14] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, Experimental

study of heat transfer and hydraulic resistance in turbulent flow of CO2 of supercritical pressure, Report B914102, IVTAN, Moscow, 1979 (in Russian).

[16] B.S. Petukhov, V.A. Kurganov, V.B. Ankudinov, New experimental data on pressure drop and heat transfer in a round tube under heating carbon dioxide of near-critical parameters of the state, in: Convective Heat Transfer. Method and Results from the Studies, IVTAN, Moscow, 1982, pp. 29–68 (in Russian).


[19] V.A. Kurganov, V.B. Ankudinov, A.G. Kaptil’nyi, Hydraulic resistance and heat transfer in vertical heated tubes under supercritical pressure of a coolant, in: A.F. Polyakov (Ed.), Turbulent Heat Transfer under Mixed Convection in Vertical Tubes, IVTAN, Moscow, 1989, pp. 95–160 (in Russian).

Results

NHT flow direction doesn’t influence values

inertial resistance coefficient does not exceed the friction

resistance coefficient In this regimes, the momentum coefficient S_b ≈ S_0t (FIG. 8) CONCLUSION: in NHT regime, the calculation of the SCP flow hydraulic

characteristics is not a problem and can be performed using the correlation for 1D flow model:

Data on LOCAL HYDRAULIC RESISTANCE structure in DT regimes at HIGH MASS FLOW RATES (= rel. small Archimedes forces effects) in DHT region (located in the vicinity of T_pc regardless of tube orientation)

INERTIAL resistance >> FRICTION resistance CONCLUSION: at high mass flow rates ρu, HTD is first connected with THERMAL ACCELERATION! + exp. Data shows that the 1D-model is NOT suitable for analysing the local inertial resistance in the HTD zone.

See FIG. 7: values of local relative coefficients + 1D analogs in the regimes with peaks in T_w

In SHORT tubes + HIGH HEAT FLOW RATES +T wall-fluid the behaviour of the local and mean (1D) values create an illusion of the friction resistance growth, the actual p_i values are much LARGER ( 1D) due to strong increase in the momentum coefficient S_b (FIG 8) and the actual p_ values are much SMALLER

REASON of large difference between actual and 1D INERTIAL resistance in DHT regimes

Differences connected with behaviour of the momentum coefficient S_b along the tube length see

FIG 7

o For fast INCREASE of S_b (left side FIG 8)

o For a DECREASE of S_b PHYSICAL REASONS of INERTIAL RESISTANCE INSTABILITY calculations

Heating SCP fluid turbulent velocity profile becomes MORE FILLED (flat in the main part of the flow)

o Flow core velocity u_c

With = relative momentum thickness (several %)o Mean flow velocity over the tube cross section =

With = relative displacement thickness (~10-2)

where

o Similar

o Comparing with usual

definition

o (these correlations have also been used in the 2 pressure

drops technique for determining hydrostatic head in vertical tubes)

o Call = “actual” momentum coefficient

For = cte Kurganov simulated this S_00 for water and CO2

using the following exponential functions with

m and n varying from 69 Results of simulation:

In contrast to S_b coefficient (increases to 1.2-1.3 at high h and b/w see FIG 8), the S_00 value changes weakly and as first approximation

assume Sounding measurements validate the correlations

and confirm the conservative nature of the actual momentum coefficient (see later)

On the basis of the correlations and the equations of flow, continuity and energy

In gases (and pseudo-gas) when Eq≈cte inertial resistance coefficient

, regardless of the HT regime

Under stabilized NHT conditions + uniformly heated fluid over the cross section + heat flow rate dissipation described by:

(Petukhov)B.S. Petukhov, L.G. Genin, S.A. Kovalev, Heat Transfer in Nuclear Power Installations, Energoatomizdat, Moscow, 1986. 472 p. (in Russian)

at small h (typical for NHT regime) a thin wall region with elevated Eq values but with small mass flow rate doesn’t significantly contribute to the integral sum

However if + are an

order of a magnitude higher than

possible that esp. in HTD development (q/qw profile

differs from ) a large amount of heat is retained in the wall gas layer

Due to the high expansion work this layer forces back the dense flow core from the wall a kind of gas-nozzle appears in the tube that causes sharply accelerating of the dense flow core, which carries the main part of the flow momentum + fast increasing of

and values along the tube length The latter is accompanied by an INCREASE in

INERTIAL RESISTANCE COEFFICIENT I and in the gradient consequences in eddy diffusivity values and heat transfer intensity

These ideas are confirmed by numerical simulations of

Simulation with formula , power velocity and enthalpy profiles and the following q/qw profile (reflect specifics of HT at stages of HTD and HTE (downstream of

2peaks)) Where -2≤A≤2

Corresponding q/qw curves FIG 9a FIG 9b calculated values of inertial

resistance coefficient

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