A new heat transfer correlation for supercritical fluids

Abstract

A new method of heat transfer prediction in supercritical fluids is presented. Emphasis is put on the simplicity of the correlation structure and its explicit coupling with physical phenomena. Assessment of qualitative behaviour of heat transfer is conducted based on existing test data and experience gathered from open literature. Based on phenomenological analysis and test data evaluation, a single dimensionless number, the acceleration number, is introduced to correct the deviation of heat transfer from its conventional behaviour, which is predicted by the Dittus-Boelter equation. The new correlation structure excludes direct dependence of heat transfer coefficient on wall surface temperature and eliminates possible numerical convergence. The uncertainty analysis of test data provides information about the sources and the levels of uncertainties of various parameters and is highly required for the selection of both the dimensionless parameters implemented into the heat transfer correlation and the test data for the development and validation of new correlations. Comparison of various heat transfer correlations with the selected test data shows that the new correlation agrees better with the test data than other correlations selected from the open literature.

Keywords

super critical fluids / heat transfer / circular tubes / prediction method

Introduction

The heat transfer of supercritical fluids shows abnormal behavior compared with that of conventional fluids due to the strong variation of thermal-physical properties in the vicinity of the pseudocritical point. In spite of extensive studies in the past five decades and a large number of prediction models, empirical approaches are mainly used in the prediction of heat transfer of SC fluids. There exist a large number of empirical correlations derived based on experimental data with limited parameter ranges [1-5].

As reviewed and summarized by Cheng and Schulenberg [6], most of the empirical correlations have the general form of a modified Bittus-Boelter equation:

(1)N{u}_{\mathrm{B}}=C\mathit{R}{\mathit{e}}_{\mathrm{B}}^n\mathit{P}{\mathit{r}}_{\mathrm{B}}^{\mathit{m}}F.

The correction factor F takes into account the effects resulting from property variation between the bulk temperature and the wall surface temperature. For most of the correlations proposed earlier up to the end of last century, the correction factor is mainly dependent on the specific heat ratio and the density ratio, i.e.,

(2)F=f\left({\displaystyle \frac{{\rho }_{\mathrm{W}}}{{\rho }_{\mathrm{B}}}},{\displaystyle \frac{C_{\mathrm{P},\mathrm{A}}}{C_{\mathrm{P},\mathrm{B}}}}\right),

where the effective specific heat C_P,A is defined as

(3)C_{\mathrm{P},\mathrm{A}}={\displaystyle \frac{h_{\mathrm{W}}-h_{\mathrm{B}}}{T_{\mathrm{W}}-T_{\mathrm{B}}}}.

It was pointed out in Refs. [6,7] that most of the correlations are applicable only to cases without heat transfer deterioration (HTD) and given a maximum heat transfer coefficient at a bulk temperature close to the pseudocritical temperature.

Recently, in the frame of the development of supercritical water cooled reactors (SCWR), further efforts have been made to develop prediction methods of heat transfer in supercritical fluids [8,9]. The approaches applied by most authors are similar. More and more parameters were taken into consideration in order to improve the prediction accuracy, e.g., the correlations proposed by Jackson [8]

(4)F=f_{\mathrm{1}}\left({\displaystyle \frac{{\rho }_{\mathrm{W}}}{{\rho }_{\mathrm{B}}}},{\displaystyle \frac{C_{\mathrm{P},\mathrm{A}}}{C_{\mathrm{P},\mathrm{B}}}}\right)f_{\mathrm{2}}\left({\displaystyle \frac{{\rho }_{\mathrm{W}}}{{\rho }_{\mathrm{B}}}},{\displaystyle \frac{{\mu }_{\mathrm{W}}}{{\mu }_{\mathrm{B}}}},\mathit{R}{\mathit{e}}_{\mathrm{B}},\mathit{P}{\mathit{r}}_{\mathrm{B}},{\displaystyle \frac{q{\beta }_{\mathrm{B}}D}{{\lambda }_{\mathrm{B}}}}\right),

and by Kuang [9]

(5)F=f\left(Gr,{\displaystyle \frac{q{\beta }_{\mathrm{B}}}{C_{\mathrm{P},\mathrm{B}}G}},{\displaystyle \frac{{\rho }_{\mathrm{W}}}{{\rho }_{\mathrm{B}}}},{\displaystyle \frac{C_{\mathrm{P},\mathrm{A}}}{C_{\mathrm{P},\mathrm{B}}}},{\displaystyle \frac{{\mu }_{\mathrm{W}}}{{\mu }_{\mathrm{B}}}},{\displaystyle \frac{{\lambda }_{\mathrm{W}}}{{\lambda }_{\mathrm{B}}}}\right).

However, the complex structure of the correlations does not always lead to a significant achievement in the prediction accuracy.

A new approach to derive prediction correlation of heat transfer in supercritical fluids is presented in this paper. Emphasis is put on the simplicity of the correlation structure and on the explicit connection with the physical phenomena. Based on phenomenological assessment of heat transfer behaviour, a new structure of heat transfer correlation is proposed, which contains one single dimensionless number to correlate the correction factor and excludes the direct dependence of heat transfer coefficient on the surface wall temperature. To validate the new heat transfer correlation, the test data of Herkenrath et al. [10] were used. Criteria for the selection of test data are presented. The new heat transfer correlation is compared with the selected test data.

Heat transfer prediction

General features of heat transfer

One of the main features of heat transfer of supercritical fluids is its strong dependence on heat flux, especially as the bulk temperature gets closer to the pseudocritical value. Figure 1 shows schematically the heat transfer coefficient (HTC) versus fluid bulk temperature in dependence on heat flux. At low heat fluxes, e.g., approaching zero, the heat transfer coefficient can be well predicted by the conventional Dittus-Boelter (D-B) equation. Heat transfer behaviour shows its maximum at the pseudocritical point. As heat flux increases, this peak shifts to lower values of bulk temperature; meanwhile, the peak amplitude decreases. As shown in Fig. 1(b), the ratio of heat transfer coefficient to the reference value at zero heat flux increases from 1.0, with increasing bulk temperature, and reaches its maximum at when bulk temperature is still far from the pseudocritical point. After that, it decreases when bulk temperature is approaching the pseudocritical point. It may show a minimum value when bulk temperature is around the pseudocritical point. It approaches to unity again when bulk temperature is far beyond the pseudocritical point. In open literature, the region with the HTC ratio larger than the unity is referred to as heat transfer enhancement, whereas the region with the HTC ratio much smaller than the unity, e.g., 0.3, is referred to as heat transfer deterioration (HTD). Several widely applied correlations were reviewed and assessed by Cheng and Schulenberg [6], and Pioro and Duffey [7]. It was concluded that in spite of the large number of heat transfer correlations available, the deviation between them is great, especially for the case with high heat fluxes.

Heat transfer deterioration is a special issue that could occur in SC fluids at high heat fluxes and low mass fluxes. At HTD, a strong reduction in heat transfer coefficient or a strong increase in heated wall temperature occurs, which is however much milder than that at the onset of DNB (departure from nucleate boiling) at subcritical conditions. HTD is considered to occur only in cases that the bulk temperature is below the pseudocritical value, and the wall temperature exceeds the pseudocritical temperature. Due to a relatively smooth behaviour of the wall temperature at HTD, there is no unique definition for the onset of HTD. This is one of the reasons for the large deviation between different correlations predicting the onset of HTD. In open literature, different definitions were used, most of which are based on the ratio of the heat transfer coefficient to a reference value, i.e., \alpha /{\alpha }_{\mathrm{0}}. Usually, HTC at zero heat flux is taken as the reference value{\alpha }_{\mathrm{0}}. For the onset of HTD, the ratio \alpha /{\alpha }_{\mathrm{0}}=\mathrm{03} is used [4,11].

The validity of most of the existing heat transfer correlations in literature is limited to normal heat transfer phenomena without heat transfer deterioration. However, it is well agreed that the design of SCWR does allow the occurrence of HTD. One of the design criteria is the upper limit of the cladding surface temperature. Therefore, it is now highly required that heat transfer coefficient be predicted also in the range of HTD.

Correlation structure of the correction factor

In this paper, the following criteria were taken for deriving the final structure of the correction factor:

1) The correlation should be based on dimensionless numbers, so that it can be extended and applied to various supercritical fluids.

2) The correlation should contain as less parameters as possible.

3) The correlation will cover both normal and HTD conditions.

The correlation should not contain wall temperature or parameters depending on wall temperature, to avoid numerical instability problems..

The first requirement was well fulfilled by the general structure in Eq. (1). For the second requirement, a simple correlation is to be established, which on one hand considers the main mechanisms of heat transfer and, on the other hand, is easy to apply and enhances phenomenological understanding. The third requirement is crucial for the application to SCWR conditions. As mentioned above, there is no clear separation between normal heat transfer and heat transfer deterioration. In the SCWR design, the target is not to avoid the occurrence of HTD but to predict the heat transfer coefficient and the resulted wall surface temperature. The fourth requirement differs obviously from the structure of most of the existing correlations. As shown in previous sections, the heat transfer of SC fluids depends strongly on wall surface temperature, which leads to a large number of correlations containing parameters, which are directly or indirectly dependent on wall temperature, e.g., Eqs. (4) and (5). However, correlations containing wall temperature requires an iteration procedure, which might not only lead to a convergence problem but also to numerical instability, especially near the pseudocritical point. The following heat transfer correlation is assumed in which the heat transfer coefficient depends on wall temperature:

(6)\alpha =f(P,G,D,q,\cdots ,T_{\mathrm{B}},T_{\mathrm{W}}).

Furthermore, wall temperature and heat transfer coefficient has to fulfil the relationship of

(7)\alpha ={\displaystyle \frac{q}{T_{\mathrm{W}}-T_{\mathrm{B}}}}.

Combining both equations, the following equation can be obtained

(8){\displaystyle \frac{q}{T_{\mathrm{W}}-T_{\mathrm{B}}}}=f(P,G,D,q,\cdots ,T_{\mathrm{B}},T_{\mathrm{W}}).

Depending on function f and the conditions considered, Eq. (8) could have a single solution or more than one solution, as schematically shown in Fig. 2. Therefore, correlations containing wall temperature could result in serious problems of convergence. The case with multisolutions would also lead to numerical instability.

Therefore, it was decided that the specific heat ratio should be eliminated. The effect of wall temperature and, subsequently, the specific heat ratio on heat transfer will be taken into account indirectly with other parameters. Furthermore, both the acceleration parameter and the buoyancy parameter were tightly related to each other. It is thus desirable to eliminate one of them for simplicity. To achieve this, a systematic evaluation of the effect of both parameters on heat transfer was conducted. Figure 3 shows the two examples based on the test data in Ref, [10]. It was seen that the effect of the acceleration parameter could be reasonably described by a single curve, whereas much more complicated dependence of heat transfer on the buoyancy parameter was observed. Based on this evidence, it was decided that the buoyancy parameter should be eliminated, and the final structure of the correction factor was proposed as follows:

(9)F={\displaystyle \frac{Nu}{N{u}_{\mathrm{0}}}}={\displaystyle \frac{Nu}{\mathrm{0.023}\mathit{R}{\mathit{e}}^{\mathrm{0.8}}\mathit{P}{\mathit{r}}^{\mathrm{1}/\mathrm{3}}}}=f({\pi }_{\mathrm{a}}).

Experimental data base

The next step in deriving a new heat transfer correlation is to propose a function with empirical constants, which can be determined based on experimental data base. In addition, the determination of the function requires an extensive analysis of test data and understanding of experimental evidence. For the present study, the experiments in Ref. [10] were selected due to the large number of test data points and well documented experimental procedure.

Source of test data

Herkenrath et al. [10] conducted experimental investigation on heat transfer in vertical circular tubes at a high pressure of up to 25.0 MPa. The heat transfer at supercritical pressures was performed at steady state conditions. Two circular tubes whose inner diameters are 10 mm and 20 mm were used and were electrically directly heated. The tube wall thickness is 2.0 mm and 4.0 mm, respectively. An analysis of the uncertainty of the test parameters is given in Ref. [10]. The test data are given in the form of Figures with the wall inner temperature versus the fluid enthalpy. The digitalization of the Figures results in additional uncertainties, which are estimated to be less than 1.2 °C in wall temperature and 5 kJ/kg in enthalpy.

Thermocouples are axially uniformly distributed over the heated section. For test section No.1, with the tube inner diameter of 10 mm, the heated length of 5.1 m, 34 thermocouples were distributed over the entire heated length with an interval of 150 mm. The first thermocouple locates at 75 mm from the beginning of the heated section. Test section No.2 has an inner and outer diameter of 20 mm and 28 mm, respectively. The total heated length is 8500 mm. Forty-six thermocouples were uniformly distributed over the heated length with an interval of 185 mm. Test section No.3 has an inner and outer diameter of 10 mm and 14 mm, respectively. The total heated length is 3000 mm. Test section No. 3 was to check the reproducibility of the test data obtained in test section No.1. Twenty thermocouples were uniformly distributed over the heated length with an interval of 150 mm. Test sections No. 4 and No. 5 had identical geometrical parameters with test sections No.1 and No.2. Test sections No. 4 and No. 5 were to check the reproducibility of the experiments. About 4600 data points were collected with test parameter ranges, as shown in Table 1.

Selection of test data

The selection of test data is mainly based on the above uncertainty analysis. The following criteria are applied for the data selection:

Uncertainty of heat transfer coefficient　　　<20%

Uncertainty of Reynolds number　　 　　 　<20%

Uncertainty of Prandtl number　　　 　　　<50%

Uncertainty of acceleration parameter　　 　 <20%

Uncertainty of Nusselt number　　　 　　　<30%

Furthermore, test data with L/D<50 were also excluded to avoid possible entrance effect. Based on the above criteria, a total 2152 data points are selected for the present studies.

New heat transfer correlation

Based on the selected data points, an extensive analysis was conducted of the effect of various parameters on the correction factor derived from D-B correlation. It was found that the relationship between the correction factor and the acceleration parameter can be divided into two regions, as illustrated in Fig. 4. In the region of small values of the acceleration parameter, the correction factor increases with an increasing acceleration parameter, whereas in the region of large acceleration parameter, a decrease in the correction factor was obtained. The relationship in the first region is well described with a single curve for different experimental conditions, whereas in the second region, various curves are required for different parameter combinations, which can be characterized by the maximum value of the acceleration parameter. For each combination of pressure, mass flux, and heat flux, the acceleration parameter {\pi }_{\mathrm{a}} depends on the temperature. As shown in Fig. 5, the ratio of the thermal expansion coefficient to specific heat shows its maximum value at the pseudocritical temperature. This gives the maximum acceleration number at the pseudocritical temperature.

The following relationships of the correction factor are proposed:

In the first region,

(10)F=a_{\mathrm{1}}+a_{\mathrm{2}}{({\pi }_{\mathrm{a}})}^{a_{\mathrm{3}}},

and in the second region,

(11)F={\displaystyle \frac{b_{\mathrm{1}}}{{\pi }_{\mathrm{a},\mathrm{p}\mathrm{c}}^{b_{\mathrm{2}}}}}+b_{\mathrm{3}}\left(\mathrm{1}-{\displaystyle \frac{{\pi }_{\mathrm{a}}}{{\pi }_{\mathrm{a},\mathrm{p}\mathrm{c}}}}\right).

Based on the criterion that the error parameter

(12)\mathrm{\Delta }={\displaystyle \frac{\mathrm{1}}{N}}{{\displaystyle \sum _{i=\mathrm{1}}^N\left|{\displaystyle \frac{N{u}_{\mathrm{C}}}{N{u}_{\mathrm{m}}}}-\mathrm{1}\right|}}_i

has its minimum value, the constants in Eqs. (10) and (11) are determined. The derived new correlation is as follows:

(13)F=\min (F_{\mathrm{1}},F_{\mathrm{2}}),

(14)F_{\mathrm{1}}=\mathrm{0.85}+\mathrm{0.776}{({\pi }_{\mathrm{a}})}^{\mathrm{2.4}},

(15)F={\displaystyle \frac{\mathrm{0.48}}{{\pi }_{\mathrm{a},\mathrm{p}\mathrm{c}}^{1.55}}}+\mathrm{1.21}\left(\mathrm{1}-{\displaystyle \frac{{\pi }_{\mathrm{a}}}{{\pi }_{\mathrm{a},\mathrm{p}\mathrm{c}}}}\right).

It has to be pointed out that according to Eqs. (13)-(15), the heat transfer coefficient at zero acceleration parameter conditions is, on the average, about 15% lower than that predicted by the conventional Dittus-Boelter equation. This is due to the test data used for determining the empirical constants in Eqs. (10) and (11). Future work is thus required to optimize the proposed correlation based on test the data base of a wide range of parameters.

Figure 6 shows the ratio of the measured Nusselt number to the values calculated with the new derived heat transfer correlation versus various parameters. For nearly all data points, the ratio varies between 0.5 and 1.5. There is no systematic deviation related to various parameters, i.e., Reynolds number, Prandtl number, and Acceleration number. However, a large scattering can be observed in the range of large acceleration number. For all the 2152 data points, the average value of the ratio is 0.995, and the standard deviation is 19.5%.

Figure 7 presents the distribution density function and the accumulated probability of the ratio. The ratio is well symmetrically distributed with an average value of about 1. For about 70% of the data points, the deviation between the measured and the calculated Nusselt number falls into an error band of 20%. It has to be pointed out that if all the 4599 data points of Ref. [10] are taken for comparison, an average value and a standard deviation of 1.068 and 40.9% can be obtained, respectively. For more than 10% of the data points, the correlation gives an overprediction of more than 60%. This indicates the importance of a thorough evaluation of test data points.

Table 2 summarizes the comparison of the results of the present correlation and some other existing correlations, i.e., Dittus-Boelter equation, correlations of Refs. [1,2,3,5,12]. Compared with the Dittus-Boelter equation, all correlations show significant improvement. The present new correlation, Eqs. (13)-(15), gives the best results. Among the existing correlations, the correlation of Ref. [5] has obviously the best capability to reproduce the test data of Ref. [10].

Summary

Due to the great variation of thermal-physical properties in the vicinity of the pseudocritical point, the heat transfer of supercritical fluids shows abnormal behavior compared with that of conventional fluids. Empirical approaches are mainly used in the prediction of the heat transfer of SC fluids. Extensive studies of the last five decades lead to more complicated structure of heat transfer correlations, whereas the improvement in the prediction accuracy is still limited.

A new approach is proposed in this study to derive prediction correlation of heat transfer in supercritical fluids with an emphasis on the simplicity of the correlation structure and on the explicit connection with the physical phenomena. A new heat transfer correlation is derived based on phenomenological assessment of the heat transfer behaviour and a thorough evaluation of the test data base. A single dimensionless number, the acceleration number, is introduced to correct the deviation of the heat transfer from its conventional behaviour. Furthermore, the new correlation structure excludes direct dependence of the heat transfer coefficient on the wall surface temperature and eliminates possible numerical instability. This correlation can be applied to both normal and HTD conditions.

An extensive uncertainty analysis is proven to be highly required for the selection of both the dimensionless parameters implemented into the heat transfer correlation and the test data for the development and validation of new correlations. In addition, it provides information about uncertainty sources, which could be taken into account in future experimental studies.

Comparison of various heat transfer correlations with the selected test data shows that the new correlation agrees better with the test data than other correlations selected from open literature. Further improvement of the new heat transfer correlation is required with respect to an enlarged, well assessed test data base, and for various supercritical fluids.

References

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[13] Jackson J D. HTFS design report No. 34 – Heat transfer to supercritical pressure fluids, Part 1 – Summary of design recommendation and equations. <patent>AERE-R8157</patent> 1975

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