2. Heat transfer to fluids at supercritical pressure

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2. Heat transfer to fluids at supercritical pressure

2.1 Physical properties of fluids at critical and supercritical regions

At supercritical pressure, the heat transfer is influenced by the variations of the thermo-physical properties. The most significant variation is near the critical and pseudocritical points.

At the critical point the distinction between the liquid and gas phases disappears: both phases have the same temperature, pressure and volume. Hence at this point all pure substances have a unique value. Moreover, it is:

( )

( )

Over the critical temperature the fluid cannot be liquefied by simple compression. In the supercritical region, the fluid is neither liquid nor gas, but is approximately a perfect gas at high temperature (Pioro, Duffey, 2007). Dense gases cannot be treated as ideal gas, but only the water vapor at very low pressures.

The thermo physical properties of water at different pressures can be calculated by NIST ¹ (2002) software. Water is a working fluid frequently used for supercritical steam generators. All thermo physical properties of water undergo significant variations at the pseudocritical temperature and even more at the critical temperature. In this work water was addressed for making the simulation and especially we have considered supercritical pressure. Table 1 shows the critical values of water and the pseudocritical ones at some operating pressures of interest.

Critical values Operating pressures and related pseudocritical values

P [MPa] 22.06 23.5 25 25.5

ρ [kg/m] 347.17 315.47 317.02 317.69

T [°C] 373.94 379.35 384.89 386.71

Table 1: Critical values of pressure, density and temperature for water

1

The NIST software (2002) calculates the thermophysical properties of 82 fluids. In this software, the fundamental equation for the Helmholtz energy per unit mass (kg) as a function of temperature and density is used. This equation was combined with a function for the ideal gas Helmoholtz energy to define a complete Helmoholtz energy surface.

All other thermodynamic properties are stated to be obtained by differentiation on this surface.

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Fig.1 represents the trends of density and dynamic viscosity. It is possible to see a significant drop, almost vertical for pressures very close to the critical pressure, in a very narrow range of temperature. The density decreases by a factor seven between 350 °C and 450 °C while the dynamic viscosity decreases about three times in the same range.

a) b)

Fig.1: a) Water density and b) dynamic viscosity trends at some supercritical pressures

Fig. 2 shows the trend of specific enthalpy and kinematic viscosity. Their behavior is opposite with respect to density and dynamic viscosity, showing a sharp increase with the temperature.

a) b)

Fig. 2: a) Water specific enthalpy trend, b) kinematic viscosity trend at some supercritical pressures

Fig.3 and Fig. 4 show the trend of the thermal conductivity and the the molecular Prandtl

number, repectively, and they illustrate peaks near the critical and the supercritical pressures.

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The value of the pseudocritical temperature is defined considering the maximum value of the specific heat. The Prandtl number, evaluated as the ratio of the kinematic viscosity and the thermal diffusivity is very high at the pseudocritical temperature. The thickness of the thermal boundary layer thus is smaller than thickness of the velocity boundary layer.

a) b)

Fig.3: a) Water thermal conductivity trend, b) Prandtl number trend at some supercritical pressures

a) b)

Fig. 4: a) Water volume expansivity trend, b) Specific heat trend at some supercritical pressures

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2.2. Mechanism of the heat transfer deterioration

Understanding the behaviour of supercritical fluids and, specifically heat transfer, is very important to define the safety margins of the SCWR power plants. In this section it is presented the mechanisms that influence heat transfer to supercritical fluids.

Many experiments showed the deterioration of heat transfer when the pseudocritical temperature is greater than the bulk temperature but smaller than the wall temperature. At supercritical pressures three different heat transfer modes are recognized. The first is the normal heat transfer:

it occurs when the fluid properties are constant or change very little; the enhancement of heat transfer happens when the heat transfer coefficient (HTC) has a local increase and the wall temperature stabilise to an almost constant value. However, the enhancement of heat transfer is reduced with the increase of heat flux and/or pressure. In fact, above a certain threshold of the ratio of heat flux to mass velocity, heat transfer effectiveness can experience a severe degradation. This lead the deteriorated heat transfer is observed when the HTC decreases in some parts of the pipe, whereas the wall temperature increases.

The criteria to identify deterioration involve the consideration of property variation, buoyancy and acceleration.

Variation of thermodynamic and thermophysical properties

The variation of the properties of fluids influences the convective heat transfer. It is possible to estimate this effect through the Dittus-Boelter equation:

where:

⁄

⁄

Hence the convective heat transfer coefficient is directly dependent on specific heat and the

thermal conductivity and depends inversely on the dynamic viscosity. Considering Fig. 4 b) it is

possible to note that the specific heat has the largest value at the pseudocritical temperature,

where its behaviour is generally predominant. Fig.5 shows the trend of HTC for water at

different pressures for assuming a mass flow rate equal to 0.01579 kg/s and a diameter of 6.28

mm, and considering the properties of water.

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Fig.5: Water heat transfer coefficient trend at some supercritical pressures

Buoyancy effects

Heat transfer at supercritical pressure is sometimes different from normal heat transfer due to the strong variation of thermal and physical properties near the pseudocritical temperature. The strong variation of the density leads to buoyancy effects at low flow rates and acceleration effects at high flow rates.

The buoyancy effects cause shear stress and radial velocity redistributions leading to heat transfer deterioration and recovery: this is called “external” effect. Moreover the buoyancy influences the turbulent kinetic energy, which is called “structural” effect (Zhang et al, 2012).

Upward flow

In supercritical pressure fluid flows when the heat flux to mass velocity is low, heat transfer is enhanced near the pseudocritical region. Under certain conditions, however, heat transfer may be deteriorated, especially in upward flow. To avoid heat transfer deterioration in practice, it is important to predict the onset conditions of the deterioration as accurately as possible. The apparent sign of heat transfer deterioration is a large increase in the wall temperature. The buoyancy and the thermal acceleration effects are considered to be the main causes of the heat transfer degradation.

This effect may cause the velocity profile to be flattened (Fig.6). The shear stress distribution

varies accordingly and the turbulence production reduces. This leads to laminarization of the

flow and hence deterioration in heat transfer.

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Numerical studies indicate that the transition from turbulence to laminar flow may occur under the effect of buoyancy (Zhang et al, 2012). This happens because near the wall, the axial component of velocity increases where the fluid is lighter and an M-shaped profile appears. This effect causes a decrease of both of the production of turbulence and accordingly the turbulent kinetic energy decreases.

Fig.6: Radial distribution of axial velocity and turbulent kinetic energy. Mass flux, G=39 kg/m

s, heat flux [W/m

]:

A)100, B) 450, C)550, D)2000 (Koshizuka et al.,1995))

The fluid near the pseudocritical temperature, in the upward flow, exhibits a large thermal expansion coefficient Fig. 4a) and a relatively small kinetic viscosity Fig. 2 b). Hence the Grashof number becomes large:

It is possible evaluating the Grashof number through the wall heat flux and the thermal conductivity as in the following equation:

The criteria for the onset of deterioration is defined by dimensionless buoyancy parameters (McEligot, et al., 2004; Jackson, 2009):

17 Downward flow

In these conditions there is generally an enhancement of heat transfer, because of the increase in the production of turbulence and then in turbulent kinetic energy. This is due to an increase of the shear stress.

Acceleration effect

Large variations of density near wall lead to a strong buoyancy force at low flow rate and to flow acceleration at high flow rate. When the wall temperature is above the pseudocritical point, the density decreases in the boundary layer with respect to the values in the bulk region; hence, the velocity increases near the wall. This effect leads to an increase of both the velocity gradient and the viscous shear stress close to the wall decreases.

For this reason the turbulent production becomes smaller and there is a shift to a less turbulent regime.

2.3. Experimental data

Data of Ornatskii et al. (1971)

Ornatskii and his group performed experimental tests for supercritical water flow in circular tubes whose pipe diameter is 3 mm. The value of pressure adopted in the experiments is 25.5 MPa; With this pressure the pseudocritical temperature is nearly 660 K. An inlet unheated section of 0.5 m was used to stabilize the flow and the heated length was 1.6 m long. In these experiments, the fluid flow was only in the upward direction. Since it was not possible to find the original paper (the experimental data were performed long time ago) the operating conditions, reported in

Table 2, were obtained from a secondary source (Shams, 2010). The operating pressure is relatively far from the critical value; moreover, the high mass flux G, makes the effects of buoyancy relatively unimportant.

T _in [°C] G [kg/m ² s] q’’ [kW/m ² ] D _in [mm]

26.85 1500

1320

3

Upward

1810 Upward

Table 2. Boundary conditions for the data of Ornatskii et al .

18 Data of Pis’menny et al. (2005)

In this paper the results for heat transfer to supercritical water above and under the critical temperature are presented.

In

Fig. 7 a general schematic of the experimental set-up is shown (Pis’menny et al. 2005). The stainless steel loop operated at pressures up 23.5 MPa and at temperatures up to 600 °C. Water was heated by means of alternate or a direct electrical current passing through the tube wall from the inlet to the outlet power terminals.

Current and voltage were measured for calculating the heating power. Chromel-Alumel thermocouples of 0.2 mm diameter measured the temperature. The thermocouples are inserted into the fluid stream inside the mixing chamber. They measured also the temperature at several pipe cross sections corresponding to locations on the external wall. The mixing chamber is used to minimize the distribution of the non-uniformity temperature in the cross sectional and for decreasing the pressure pulsations with the test section.

Thirteen wall thermocouples were used for the pipe diameter equal to 9.5 mm and sixteen thermocouples for the diameter equal to 6.28 mm. The ratio between is used to estimate the effect of free convection. Usually, in a range of 0.01-0.6 free convection decreases the heat transfer coefficient (HTC). The deterioration heat-transfer regime can start earlier at the upward flow compared to that at the downward flow due to the effect of free convection.

T _in [°C] G [kg/m ² s] q’’ [kW/m ² ] D _in [mm]

17 248 113; 181; 277; 370 6.28 Upward

100 248 118; 175; 275; 396; 495 9.5 Upward and Downward

200 249 167; 253; 289 6.28 Upward

300 509 390 6.28 Upward and Downward

Table 3: Data of Pis’menny et al.

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Fig. 7: General schematic of the supercritical water experimental set-up: 1 electro-distillator; 2 ion-exchange filter; 3 accumulator reservoirs; 4 boosting pump; 5 plunger pumps; 6 and 7 regulating valves; 8 damping reservoirs; 9 turbine flowmeter; 10 heat exchanger; 11 electrical preheater; 12 electrical generator(s); 13 test

section; 14 throttling valve; 15 damping reservoir; 16 electro-isolating flanges; 17 main power supply;

18 cooler; 19 throttling valves.

Data of Watt (1980)

Experimental readings were taken using a vertical 25.4 mm bore test section of length 2 m (Fig.

8), first with upward and then with downward flow, at 25 MPa, over the following ranges of independent variables described in Table 4. The relatively large diameter favours the occurrence of buoyancy phenomena. In the follow table, are presented the boundary conditions for the experiments.

T _in [°C] G [kg/m ² s] q’’ [kW/m ² ] D _in [mm]

150 273 250 25.4 Upward

150 367 250 25.4 Upward

150 526 250 25.4 Upward

Table 4 Data of Watt

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Fig. 8: Supercritical pressure water test facility, Simon Laboratories Manchester

2.4. Previous Results

This works is the continuation of previous thesis works concerning the analysis of heat transfer

phenomena for supercritical fluids. All these studies belong to the EU THINS project and were

performed in cooperation with PSI. The experimental data were analysed with CFD models in

previous works (Sharabi, 2008; De Rosa, 2010; Badiali, 2011; Pucciarelli, 2013) where relevant

results were presented. In particular, the capability of various turbulent models implemented in

STAR CCM+ and in the in-house THEMAT code (Pucciarelli, 2013) were analysed, comparing

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the numerical results with experimental data. Under specific conditions, the numerical results reproduced the general trends present in the experimental data, but a completely adequate prediction of the local values was not always achieved. Analyses were made with different models and it was found that there some models tend to overestimate the experimental data of wall temperature, as models, and others tend instead to underestimate the effect of deterioration, as models.

In particular, the turbulence models implemented in CFD commercial codes sometimes, above the supercritical condition, do not reliably describe the damping of turbulent kinetic energy (flow laminarization) occurring due to buoyancy and acceleration or tend to overestimate or underestimate the turbulence production.

The first studies looked at the two-equation models with STAR CCM+ where the best trends were obtained using the Lien model (1996) for supercritical water and for carbon dioxide.

This model, implemented in STAR-CCM+, gave superior results than other available models in the code. It was also noted that the accuracy of the results provided by the Lien model depends on the value of the inlet fluid temperature with respect to the pseudocritical value.

Fig. 9: Wall temperature distribution calculated with Lien model for the case by Pismenny’s with G=248 kg/m

s and q’’=113 kW/m

(Badiali 2011)

In particular, the results obtained for the Pis’menny et al. (2005) data showed better performance

in the case of the Lien model (Fig. 9) when the inlet fluid temperature is low and it became

worse with increasing the inlet temperature.

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Hence it is presently necessary to consider which type of turbulent model could be better applied in simulating some specific data, revealing the need to better understand the phenomena occurring in supercritical fluids.

Pucciarelli (2013) in his work adopted four-equation models making possible to calculate, with two more transport equations, the temperature variance ^̅ and its dissipation. This offers the possibility of using a more advanced relationship for evaluating the turbulent heat flux, in addition to the classical assumption of constant turbulent Prandtl number. Fig 10 shows that this approach may result effective in some specific case.

Fig 10: Wall temperature distribution calculated with Zhang et al. (2010) for the case by Pismenny’s with G=509 kg/m

s and q’’=390 kW/m

(Pucciarelli 2013)

The previous tests analysed the behaviour of buoyancy effect introducing advanced methods for modelling the turbulent heat flux. Based on all these works, we try here to consider the effect of the variation of the turbulent Prandtl number below the unity for four-equation models and to vary the coefficient that influences the production term due to buoyancy. This parameter is introduced in Zhang et al. (2010).

In general, the Zhang et al. (2010) model, in the original form, tends to underestimate the wall

temperature trend, especially far from the pseudo-critical temperature. The model shows anyway

the best performances when working close to the pseudo-critical temperature, e.g. in the case of

water. It had a particularly good performance in the case by Pis’menny described in Fig 10.