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New correlation to predict the heat transfer coefficient during in-tube cooling of turbulent supercritical CO2
International Journal of Refrigeration 25 (2002) 887–895 www.elsevier.com/locate/ijrefrig
New correlation to predict the heat transfer coefficient during in-tube cooling of turbulent supercritical CO2 Srinivas S. Pitlaa, Eckhard A. Grollb,*, Satish Ramadhyanib a
b
Visteon Climate Control Systems, Plymouth, MI 48170, USA Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA
Received 20 December 2000; received in revised form 15 July 2001; accepted 1 October 2001
Abstract The Nusselt number variations of supercritical carbon dioxide during in-tube cooling are presented and discussed. Using data presented in this paper as well as prior publications, a new correlation to predict the heat transfer coefficient of supercritical carbon dioxide during in-tube cooling has been developed. The new correlation is presented in this paper. It is based on mean Nusselt numbers that are calculated using the thermophysical properties at the wall and the bulk temperatures, respectively. It is seen that the majority of the numerical and experimental values are within 20% of the values predicted by the new correlation. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating circuit; CO2; Transcritical cycle; Gas; Cooling; Turbulent flow; Heat transfer coefficient; Modelling
Nouvelle corre´lation pour pre´voir le coefficient de transfert thermique pendant le refroidissement du gaz CO2 en re´gime turbulent supercritique Mots cle´s : Circuit frigorifique ; CO2 ; Cycle transcritique ; Gaz ; Refroidisseur ; E´coulement turbulent ; Coefficient de transfert de chaleur ; Mode´lisation
1. Introduction With the growing awareness of the dual threats of global warming and ozone depletion, the interest in the transcritical carbon dioxide cycle has greatly increased over the last decade. In particular, the transcritical carbon dioxide cycle has recently been investigated as a potential alternative technology for certain applications, such as automobile air conditioners, heat pump water * Corresponding author. Tel.: +1-765-494-9157; fax: +1765-494-0787. E-mail address: [email protected] (E.A. Groll).
heaters, and environmental control units. A comparison of the transcritical carbon dioxide cycle to conventional vapor compression technology shows that the heat rejection process from the supercritical carbon dioxide is of particular interest. This process occurs along an isobar that is close to the critical isobar. The heat exchanger in which the process occurs is called a ‘‘gas cooler,’’ instead of a condenser, because the process occurs at a supercritical pressure and a phase change does not take place. However, the thermophysical properties of the fluid change drastically during the process. While a great deal of research has been performed to determine the heat transfer characteristics during condensation of
0140-7007/02/$22.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(01)00098-6
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and the accuracy of the available correlation is tested against available experimental and numerical data.
Nomenclature A cp D h k m1, m2 . m n1, n2 Nu Pr Re U
Cross-sectional area of the tube (m2) Specific heat (kJ/kg K) Diameter of the tube (mm) Heat transfer coefficient (W/m2 K) Thermal conductivity (W/m K) dimensionless coefficients () Mass flow rate (kg/s) dimensionless coefficients () Nusselt number () Prandtl number () Reynolds number () velocity (m/s) Density (kg/m3) Friction coefficient ()
Subscripts avg average value bulk evaluated at the bulk temperature and pressure PKP Petukhov–Kirilov–Popov (textbook single-phase correlation) wall evaluated at the wall temperature and pressure CFC, HCFC, and HFC refrigerants, comparatively few investigations have been conducted on the in-tube heat transfer of a supercritical fluid. The investigation presented in this paper was concerned with developing a suitable heat transfer correlation for carbon dioxide flow in the supercritical region during intube cooling. The correlation was developed based on the numerical and experimental work carried out by the authors and reported in prior publications. A detailed description of the numerical modeling of the problem and the experimental apparatus and data reduction can be found in Pitla et al. [1,2]. A detailed literature review of heat transfer in the supercritical region can be found in Pitla et al. [3]. In this paper, the general features of the Nusselt number variations are first reviewed. The development of the proposed new correlation is then described,
2. Numerical model The geometry under consideration is a tube-in-tube heat exchanger, as shown in Fig. 1. Water is flowing in the annulus, while carbon dioxide flows in the inner tube in counterflow. The inlet conditions of the carbon dioxide (State 1) are well in the supercritical region. The water entering the heat exchanger is at a temperature below the critical temperature of carbon dioxide (State 2). For the given geometry, a detailed numerical model based on the governing equations of mass, momentum and energy was developed. The numerical model of the complex turbulent flow was solved by two independent methods: Favre-averaging, also referred to as densityweighted averaging, and time averaging. Favre-averaging has an advantage over time averaging the governing equations, as there are fewer assumptions in the former, to achieve closure. The two different methods of achieving closure were studied using various turbulence models. In particular, the time averaged equations were investigated using the Bellmore and Reid [4] mixing length model, the Lee and Howell [5] modification to Bellmore and Reid’s mixing length model, and Nikuradse’s mixing length model with damping function [6]. The Favre averaged equations were studied using Nikuradse’s mixing length model and the k-equation turbulence model that was first proposed by Kolmogorov in 1943 and Prandtl in 1945 [6]. Model validation revealed that the Favre-averaging technique using the kequation turbulence model produced the most accurate results and thus, was used for further analysis and comparison to experimental results [2]. In addition, a grid independence study was conducted to make certain that the solution was grid independent. The study showed that a compressed grid near the nearwall region was necessary to investigate the flow. The size of the grid in the axial direction of the grid is not as critical as in the radial direction. By studying the flow in the near-wall region it was observed that the superficial profiles of the velocity and the temperature look very
Fig. 1. Schematic of the tube-in-tube counter flow heat exchanger. Fig. 1. Schematic of the tube-in-tube counter flow heat exchanger.
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flow rate, the heat transfer coefficient increases during cooling until a maximum is reached. The region of the maximum in heat transfer coefficient is called the pseudocritical region and coincides with the region where the specific heat has a maximum [3]. The heat transfer coefficient then drops suddenly as the fluid enters the liquid regime. The oscillation in the heat transfer coefficient seen in each of the three curves near their peaks is a numerical artifact produced by the extremely sharp variation in the thermophysical properties with temperature in the pseudocritical region.
Fig. 2. The effect of mass flow rate on the heat transfer coefficient (Tin=395 K, Pin=10 MPa, Twall=303 K).
similar to constant property turbulent flows. However, upon close examination it was seen that the velocity and temperature law of the wall that can be derived for constant property flows are not valid here [1].
3. Effect of mass flow rate on heat transfer coefficient The numerical model was used to study the general features of the Nusselt number distribution and the effect of increasing the flow rate, i.e. increasing the Reynolds number. In this study, the inlet temperature and pressure of CO2 were maintained at 395 K and 10 MPa, respectively, and a constant wall temperature of 303 K (below the critical temperature of CO2) was used. It is seen from Fig. 2 that there is an increase in the heat transfer coefficient, under identical conditions, with just an increase in the flow rate, as would be the case in constant property flow. In addition, for a single mass
4. Effect of pressure on heat transfer coefficient With an increase in pressure, the temperature range of the pseudocritical region increases. The numerical model was also used to study the effect of variations in inlet pressure on the heat transfer coefficient. In this study, the inlet temperature and mass flow rate of CO2 were maintained at 390 K and 0.04 kg/s, respectively, and a constant wall temperature of 310 K (above the critical temperature of CO2) was used. Fig. 3 shows the effect of pressure on the heat transfer coefficient. It is seen that the ‘‘peak’’ in the heat transfer coefficient is shifted to a higher temperature as the pressure increases. This coincides with the shift in the pseudocritical region to higher temperatures with an increase of pressure. At higher pressures the variation in the heat transfer coefficient with temperature is smaller than at pressures near the critical point, as the variation in the thermophysical properties is maximum near the critical point and decreases as the pressure is increased from the critical pressure (7.353 MPa). The peak in the heat transfer coefficient is more pronounced at pressures closer to the critical pressure. This can be attributed to the fact that the thermophysical property variations are more pronounced at pressures closer to the critical point. In particular, the specific heat attains extremely high values as the critical point is approached.
5. Experimental test facility
Fig. 3. The effect of pressure on the heat ransfer coefficient (Tin=390 K, mass flow rate=0.04 kg/s, Twall=310 K).
Fig. 4 shows a schematic diagram of the experimental test facility. Since the operating pressure of the apparatus is between 8.0 and 12.0 MPa, special safety procedures were observed, such as thick-walled stainless steel tubes. The test apparatus consists of a closed carbon dioxide loop, an open water loop, and a closed water–glycol loop. The carbon dioxide loop consists of four main components: a sub-cooler/receiver assembly, a magnetically coupled, sealed, gear pump, a heater, and the test-section. In addition, a mass flow meter, pressure transducers, probe thermocouples, and a pressure relief valve are installed in the loop.
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Fig. 4. Schematic diagram of the test facility. Fig. 1. Schematic diagram of the test facility.
Carbon dioxide enters the test section in the supercritical state at about 120 C. In passing through the test section, the CO2 is cooled by water to approximately 25 C, which is below the thermodynamic critical temperature of carbon dioxide. The fluid enters the subcooler/receiver assembly where it is further cooled to increase its density. The fluid leaves this assembly at about 5 C and is then pumped by the variable speed internal gear pump. The function of the pump is to maintain a certain mass flow rate and overcome the pressure drop throughout the CO2 piping. The cool liquid carbon dioxide then enters the heater, where it is heated to the supercritical inlet conditions of the test section. The pump is a custom-designed, variable speed internal gear pump, which is magnetically coupled to a 1hp motor. It is designed to work at pressures up to 136 bar. Flow into the test section is controlled by the variable speed drive of the motor. The mass-flow meter measures the flow and the drive can be adjusted accordingly. All the tubes are connected with compression fittings. The test section consists of eight subsections. Each subsection is a tube-in-tube counter-flow heat exchanger. Carbon dioxide flows in the center and water flows in the annulus. Five subsections are 1.8 m long and three subsections are 1.3 m long. The tubing of the carbon dioxide loop is made of stainless steel with a nominal OD of 6.35 mm and a wall thickness of 0.815 mm. To minimize heat transfer to the environment, the test sections were heavily insulated.
6. Comparison of numerical and experimental results Numerical simulations were conducted corresponding to the conditions of 12 experimental test runs that are
described in detail by Pitla et al. [7]. In each case, the measured carbon dioxide inlet temperature and the measured water outlet temperature were provided as inputs to the numerical model. The experimental data and the numerical predictions of the heat transfer coefficient and the temperature along the tube length were compared for each test run. An uncertainty analysis was also conducted which showed that the heat transfer coefficients were measured following a specified data reduction procedure [2] with an uncertainty of 17%, while the water temperatures were measured directly using thermistors with an uncertainty of 0.07 C. For three test runs, detailed comparisons of the heat transfer coefficient and temperature along the tube length are provided here. The experimental test conditions of the three runs are summarized in Table 1. The selected test runs represent three different supercritical pressures as well as three different mass flow rates. The selected pressures span the expected pressure range seen during gas cooler operation. The selected mass flow rates are representative of cooling capacities from 3 to 7 kW.
Table 1 Run matrix of the experimental test runs used in the comparative study Test run
Inlet pressure (MPa)
Inlet to outlet temperatures ( C)
Mass flow rate (kg/s)
1 2 3
10.8 9.4 13.4
124–27 121–34 101–20
0.029 0.020 0.039
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which is placed after the first two heat exchangers. Thus, the ‘‘entrance effect’’ is felt in the third heat exchanger. This is observed from the plots by the heat transfer rapidly diminishing for a small axial distance along the tube. Experimental measurements give an average value of the heat transfer coefficient over a heat exchanger. The average heat transfer predicted by the numerical program was calculated and compared with that of the experimental measured value. The results are shown in Table 2. The average experimentally measured heat transfer coefficient agrees with the numerical prediction within 16%. Fig. 6(a)–(c) shows a comparison of the numerically predicted and the experimentally measured water temperature. The experimental measurements agree within 3% of the numerical prediction.
7. New correlation for the heat transfer coefficient Several investigators attempted to correlate the heat transfer of supercritical carbon dioxide during in-tube heating. Relatively fewer attempts were made to correlate the heat transfer for in-tube cooling of supercritical carbon dioxide. In particular, Krasnoshechekov et al. [8], Baskov et al. [9], and Petrov and Popov [10] attempted to correlate the heat transfer of supercritical carbon dioxide during in-tube cooling. The correlations were complex and were not very accurate, especially in the pseudocritical region. Lee and Howell [5] stated that
Table 2 Comparison between the experimentally measured and the numerically predicted heat transfer coefficient
Fig. 5. Variations of the heat transfer coefficient along the length of the tube: (a) Run 1; (b) Run 2; (c) Run 3.
Fig. 5(a)–(c) shows the variations of the heat transfer coefficient along the tube length. Experimental data was used to compute the average heat transfer coefficient of each subsection. These values can be seen as the horizontal lines in the plots. As shown earlier through numerical simulation, it can also experimentally be observed that the heat transfer coefficient is not constant, but exhibits a peak in the pseudocritical region. Fig. 5(b), and more prominently Fig. 5(c), display a discontinuity. This is accounted for by the water heater,
Heat transfer coefficient times length (W/mK) (numerical prediction) Run 1 13568 23031 27689 23228 Run 2 6684 9726 12653 22923 25255 31346 Run 10 18619 25280 25285
Heat transfer Difference coefficient times (%) length (W/mK) (experimental measurement) 13991 22694 25702 19910
3.0 1.0 7.0 14.0
6852 10080 14625 22696 25004 30696
2.5 3.5 15.5 1.0 1.0 2.0
20383 25418 25371
8.0 0.5 0.5
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Nuwall þ Nubulk kwall Nu ¼ 2 kbulk
ð1Þ
where Nuwall and Nubulk are Nusselt numbers that are evaluated based on the thermophysical properties at the wall and bulk temperatures, respectively. In each case, the Gnielinski correlation [11], as shown in Eq. (2), is used to calculate the respective Nusselt number. Nu ¼
=8ðRe 1000ÞPr pffiffiffiffiffiffiffi 12:7 =8ðPr2=3 1Þ þ 1:07
ð2Þ
Using Eq. (2) requires knowledge of the wall and bulk Reynolds numbers. Several variables were considered to develop the correlation shown in Eq. (1). It was found that the best fit was obtained by using the inlet velocity to compute the Reynolds number at the wall (irrespective of location, i.e. over the whole length of the test section) and by using the local mean velocity to compute the bulk Reynolds number. To obtain the local mean velocity, the length of the test section was divided into finite lengths (finite sections) and the mean velocity was evaluated in each finite section using Eq. (3). Uavg ¼
: m Abulk
ð3Þ
In addition, Eq. (2) requires the knowledge of the friction coefficient, . Appropriate results were obtained by using Filonenko’s correlation as shown in Eq. (4), in which the friction factor is only a function of Reynolds number and which was also used in the correlation proposed by Krasnochekov et al. [8]. ¼ ð0:79lnðReÞ 1:64Þ2
ð4Þ
Once the mean Nusselt number has been obtained, the heat transfer coefficient can be computed as shown in Eq. (5): Fig. 6. The variations of the temperature along the length of the tube: (a) Run 1; (b) Run 2; (c) Run 3.
h¼
it was not possible to develop a generalized correlation for the heat transfer coefficient in the pseudocritical region and a detailed numerical analysis is necessary to predict the heat transfer in this region. Based on the numerical predictions of the heat exchanger problem defined in Fig. 1, it was attempted to develop a new correlation for the Nusselt number in terms of other dimensionless parameters. For this purpose, the numerically predicted Nusselt numbers were correlated. The final outcome was a new correlation that is based on the ‘‘mean Nusselt number’’ and is defined as shown in Eq. (1):
In the following section, the numerical computed values and the experimental data of the heat transfer coefficient are compared with the ones obtained with the proposed correlation. Fig. 7(a)–(c) compares the accuracy of the proposed correlation with the numerical prediction for Runs 1, 2 and 3, respectively. In each figure, the heat transfer coefficient computed using the proposed correlation is plotted together with the numerical prediction (Nu) and the experimental data versus the tube length. To observe the accuracy of the new correlation, two help lines have been plotted. The first line, 1.2 Nu, denotes a +20% value of the numerically predicted heat transfer coefficient, and the second
Nu kbulk D
ð5Þ
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Fig. 7. Comparisons of the proposed correlation with the numerical prediction and experimental data along the length of
Fig. 8. Comparisons of the proposed correlation with the numerical prediction versus the temperature: (a) Run 1; (b) Run 2; (c) Run 3.
line, 0.8 Nu, represents a 20% value of the numerically predicted heat transfer coefficient. It is seen from Fig. 7(a)–(c) that the proposed correlation agrees within the range of 20% for most of the tube length of the three test runs, except for two deviations. In Run 1 [Fig. 7(a)], the new correlation overpredicts the heat transfer coefficient after the pseudocritical region. In Run 2 [Fig. 7(b)], the new correlation underpredicts the heat transfer coefficient far out in the vapor region. Fig. 8(a)–(c) shows the variation of the mean Nusselt number computed using the proposed correlation together with the numerically predicted Nusselt number (Nu) versus the bulk temperature for Runs 1, 2 and 3, respectively. As in Figs. 7 (a)–(c), the 20% curves (1.2
Nu and 0.8 Nu, respectively) are also plotted in the figures. It is seen that the Nusselt numbers calculated using the proposed correlation agree to within 20% for all test runs, except for Run 1 [Fig. 8(a)] at lower temperatures (after the pseudocritical temperature), where there is a sharp change in the thermophysical properties. From this comparison it can be seen that the new correlation is most accurate for the data obtained during Run 3 [Fig. 8(c)], where the pressure is the highest (13.4 MPa). As the pressure is increased from the critical point, the variation in the thermophysical properties is less severe. It is thus possible to correlate the heat transfer coefficient of supercritical carbon dioxide more accurately at pressures away from the critical point as would
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(6) and (7), respectively, and to predictions obtained using the Gnielinski correlation [Eq. (2)] evaluated at the bulk temperature. Nuwall ¼ NuPKP;wall
wall bulk
Nuwall ¼ NuPKP;wall
n1
cp;avg cp;wall
cp;avg cp;wall
m1
m2 bulk n2 wall
ð6Þ
ð7Þ
The values of the exponents m1, m2, n1 and n2 are to be determined from graphical (Eq. (6)) and tabular (Eq. 7)) data as provided in the references. Fig. 9(a)–(c) show the heat transfer coefficient computed with the new proposed correlation and the other three correlations together with the numerically predicted heat transfer coefficient versus the bulk temperature for Runs 1, 2 and 3, respectively. It is seen from Figs. 9(a)–(c) that the correlations by Krasnoshchekov et al. [8] and Baskov et al. [9] overpredict the numerical prediction, especially in the pseudocritical region, while the Gnielinski correlation underpredicts the numerical prediction throughout. Since the predictions obtained using the Gnielinski correlation are computed at the bulk temperature there is no effect of the heat transfer improvement near the wall which is below the critical temperature in the pseudocritical region. In comparison to the other three correlations, the new proposed correlation matches the numerical predictions the closest.
9. Conclusions
Fig. 9. Comparisons of the heat transfer coefficient predicted by various correlations: (a) Run 1; (b) Run 2; (c) Run 3.
be the case with any single-phase textbook correlation for the heat transfer coefficient.
8. Comparison to literature correlations The proposed correlation was also compared to predictions obtained with two existing correlations for the supercritical heat transfer coefficient by Krasnoshchekov et al. [8] and by Baskov et al. [9], which are given in Eqs.
The heat transfer coefficient in supercritical fluids is not constant and varies as both the wall temperatures and the bulk temperatures of the fluid vary. It was seen that there is a spike in the heat transfer coefficient in the pseudocritical region. A new correlation to predict the heat transfer coefficient has been developed based on curve fits of the numerically predicted and experimentally obtained data and is presented in this paper as Eq. (1). The new correlation predicts an average Nusselt number based on wall and bulk Nusselt numbers. The Nusselt numbers of the wall and bulk are calculated using the Gnielinski correlation evaluated based on the thermophysical properties at the wall and the bulk temperatures, respectively. It was seen that 85% of the heat transfer coefficient values predicted by the new correlation were accurate to within 20%. A comparison of the heat transfer coefficient calculated using the new correlation and three existing correlations found in the literature showed that the accuracy in predicting the heat transfer coefficient with the new correlation was greatly increased, especially in the pseudocritical region.
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Acknowledgements The study presented in this paper was funded by ASHRAE 913-RP. The authors would like to thank ASHRAE for the sponsorship.
References [1] Pitla SS, Ramadhyani S, Groll EA. Convective heat transfer from in-tube flow of turbulent supercritical carbon dioxide: part 1—numerical analysis. Int J HVAC&R Research, Vol. 7, No. 4, October 2001, pp. 345–66. [2] Pitla SS, Groll EA, Ramadhyani S. Convective heat transfer from in-tube cooling of turbulent supercritical carbon dioxide: part 2—experimental data and numerical predictions. Int J HVAC&R Research, Vol. 7, No. 4, October 2001, pp. 367–82. [3] Pitla SS, Robinson DM, Groll EA, Ramadhyani S. Heat transfer from supercritical carbon dioxide in tube Flow: A Critical Review. Int’l J HVAC&R Research 1998;4(3): 281–301. [4] Bellmore CP, Reid RL. Numerical prediction of wall temperatures for near-critical para-hydrogen in turbulent upflow inside vertical tubes. Transactions of the ASME 1983;105:536–41.
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[5] Lee SH, Howell JR. Turbulent developing convective heat transfer in a tube for fluids nar the critical point. International Journal Heat Mass Transfer 1997;41(10): 1205–18. [6] Schlichting H. McGraw Hill: Boundary layer theory, 1960. [7] Pitla SS, Robinson DM, Zingerli A, Groll EA. Ramadhyani S. ‘‘Heat Transfer and Pressure Drop Characteristics During In-Tube Gas Cooling of Supercritical Carbon Dioxide. Final Report ASHRAE 913-RP, American Society of Heating, Refrigerating, and Air Conditioning Engineers, Inc., Atlanta, GA, August 2000. [8] Krasnoshechekov EA, Kuraeva IV, Protopopov VS. Local heat transfer of carbon dioxide at supercritical pressure under cooling conditions. Teplofizika Vysokikh Temperatur 1970;7(5):922–30. [9] Baskov VL, Kuraeva IV, Protopopov VS. Heat transfer with the turbulent flow of a liquid at Supercritical Pressure in tubes under cooling conditions. Teplofizika Vysokikh Temoperatur 1997;15(1):96–102. [10] Petrov NE, Popov VN. Heat transfer and resistance of carbon dioxide being cooled in the supercritical region. Thermal Engineering 1985;32(3):131–4. [11] Gnielinski V. New equation for heat and mass transfer in turbulent pipe and channel flow. Int Chemical Engineering 1976;16:359–68.
New correlation to predict the heat transfer coefficient during in-tube cooling of turbulent supercritical CO2 Srinivas S. Pitlaa, Eckhard A. Grollb,*, Satish Ramadhyanib a
b
Visteon Climate Control Systems, Plymouth, MI 48170, USA Ray W. Herrick Laboratories, Purdue University, West Lafayette, IN 47907, USA
Received 20 December 2000; received in revised form 15 July 2001; accepted 1 October 2001
Abstract The Nusselt number variations of supercritical carbon dioxide during in-tube cooling are presented and discussed. Using data presented in this paper as well as prior publications, a new correlation to predict the heat transfer coefficient of supercritical carbon dioxide during in-tube cooling has been developed. The new correlation is presented in this paper. It is based on mean Nusselt numbers that are calculated using the thermophysical properties at the wall and the bulk temperatures, respectively. It is seen that the majority of the numerical and experimental values are within 20% of the values predicted by the new correlation. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Refrigerating circuit; CO2; Transcritical cycle; Gas; Cooling; Turbulent flow; Heat transfer coefficient; Modelling
Nouvelle corre´lation pour pre´voir le coefficient de transfert thermique pendant le refroidissement du gaz CO2 en re´gime turbulent supercritique Mots cle´s : Circuit frigorifique ; CO2 ; Cycle transcritique ; Gaz ; Refroidisseur ; E´coulement turbulent ; Coefficient de transfert de chaleur ; Mode´lisation
1. Introduction With the growing awareness of the dual threats of global warming and ozone depletion, the interest in the transcritical carbon dioxide cycle has greatly increased over the last decade. In particular, the transcritical carbon dioxide cycle has recently been investigated as a potential alternative technology for certain applications, such as automobile air conditioners, heat pump water * Corresponding author. Tel.: +1-765-494-9157; fax: +1765-494-0787. E-mail address: [email protected] (E.A. Groll).
heaters, and environmental control units. A comparison of the transcritical carbon dioxide cycle to conventional vapor compression technology shows that the heat rejection process from the supercritical carbon dioxide is of particular interest. This process occurs along an isobar that is close to the critical isobar. The heat exchanger in which the process occurs is called a ‘‘gas cooler,’’ instead of a condenser, because the process occurs at a supercritical pressure and a phase change does not take place. However, the thermophysical properties of the fluid change drastically during the process. While a great deal of research has been performed to determine the heat transfer characteristics during condensation of
0140-7007/02/$22.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(01)00098-6
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S.S. Pitla et al. / International Journal of Refrigeration 25 (2002) 887–895
and the accuracy of the available correlation is tested against available experimental and numerical data.
Nomenclature A cp D h k m1, m2 . m n1, n2 Nu Pr Re U
Cross-sectional area of the tube (m2) Specific heat (kJ/kg K) Diameter of the tube (mm) Heat transfer coefficient (W/m2 K) Thermal conductivity (W/m K) dimensionless coefficients () Mass flow rate (kg/s) dimensionless coefficients () Nusselt number () Prandtl number () Reynolds number () velocity (m/s) Density (kg/m3) Friction coefficient ()
Subscripts avg average value bulk evaluated at the bulk temperature and pressure PKP Petukhov–Kirilov–Popov (textbook single-phase correlation) wall evaluated at the wall temperature and pressure CFC, HCFC, and HFC refrigerants, comparatively few investigations have been conducted on the in-tube heat transfer of a supercritical fluid. The investigation presented in this paper was concerned with developing a suitable heat transfer correlation for carbon dioxide flow in the supercritical region during intube cooling. The correlation was developed based on the numerical and experimental work carried out by the authors and reported in prior publications. A detailed description of the numerical modeling of the problem and the experimental apparatus and data reduction can be found in Pitla et al. [1,2]. A detailed literature review of heat transfer in the supercritical region can be found in Pitla et al. [3]. In this paper, the general features of the Nusselt number variations are first reviewed. The development of the proposed new correlation is then described,
2. Numerical model The geometry under consideration is a tube-in-tube heat exchanger, as shown in Fig. 1. Water is flowing in the annulus, while carbon dioxide flows in the inner tube in counterflow. The inlet conditions of the carbon dioxide (State 1) are well in the supercritical region. The water entering the heat exchanger is at a temperature below the critical temperature of carbon dioxide (State 2). For the given geometry, a detailed numerical model based on the governing equations of mass, momentum and energy was developed. The numerical model of the complex turbulent flow was solved by two independent methods: Favre-averaging, also referred to as densityweighted averaging, and time averaging. Favre-averaging has an advantage over time averaging the governing equations, as there are fewer assumptions in the former, to achieve closure. The two different methods of achieving closure were studied using various turbulence models. In particular, the time averaged equations were investigated using the Bellmore and Reid [4] mixing length model, the Lee and Howell [5] modification to Bellmore and Reid’s mixing length model, and Nikuradse’s mixing length model with damping function [6]. The Favre averaged equations were studied using Nikuradse’s mixing length model and the k-equation turbulence model that was first proposed by Kolmogorov in 1943 and Prandtl in 1945 [6]. Model validation revealed that the Favre-averaging technique using the kequation turbulence model produced the most accurate results and thus, was used for further analysis and comparison to experimental results [2]. In addition, a grid independence study was conducted to make certain that the solution was grid independent. The study showed that a compressed grid near the nearwall region was necessary to investigate the flow. The size of the grid in the axial direction of the grid is not as critical as in the radial direction. By studying the flow in the near-wall region it was observed that the superficial profiles of the velocity and the temperature look very
Fig. 1. Schematic of the tube-in-tube counter flow heat exchanger. Fig. 1. Schematic of the tube-in-tube counter flow heat exchanger.
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flow rate, the heat transfer coefficient increases during cooling until a maximum is reached. The region of the maximum in heat transfer coefficient is called the pseudocritical region and coincides with the region where the specific heat has a maximum [3]. The heat transfer coefficient then drops suddenly as the fluid enters the liquid regime. The oscillation in the heat transfer coefficient seen in each of the three curves near their peaks is a numerical artifact produced by the extremely sharp variation in the thermophysical properties with temperature in the pseudocritical region.
Fig. 2. The effect of mass flow rate on the heat transfer coefficient (Tin=395 K, Pin=10 MPa, Twall=303 K).
similar to constant property turbulent flows. However, upon close examination it was seen that the velocity and temperature law of the wall that can be derived for constant property flows are not valid here [1].
3. Effect of mass flow rate on heat transfer coefficient The numerical model was used to study the general features of the Nusselt number distribution and the effect of increasing the flow rate, i.e. increasing the Reynolds number. In this study, the inlet temperature and pressure of CO2 were maintained at 395 K and 10 MPa, respectively, and a constant wall temperature of 303 K (below the critical temperature of CO2) was used. It is seen from Fig. 2 that there is an increase in the heat transfer coefficient, under identical conditions, with just an increase in the flow rate, as would be the case in constant property flow. In addition, for a single mass
4. Effect of pressure on heat transfer coefficient With an increase in pressure, the temperature range of the pseudocritical region increases. The numerical model was also used to study the effect of variations in inlet pressure on the heat transfer coefficient. In this study, the inlet temperature and mass flow rate of CO2 were maintained at 390 K and 0.04 kg/s, respectively, and a constant wall temperature of 310 K (above the critical temperature of CO2) was used. Fig. 3 shows the effect of pressure on the heat transfer coefficient. It is seen that the ‘‘peak’’ in the heat transfer coefficient is shifted to a higher temperature as the pressure increases. This coincides with the shift in the pseudocritical region to higher temperatures with an increase of pressure. At higher pressures the variation in the heat transfer coefficient with temperature is smaller than at pressures near the critical point, as the variation in the thermophysical properties is maximum near the critical point and decreases as the pressure is increased from the critical pressure (7.353 MPa). The peak in the heat transfer coefficient is more pronounced at pressures closer to the critical pressure. This can be attributed to the fact that the thermophysical property variations are more pronounced at pressures closer to the critical point. In particular, the specific heat attains extremely high values as the critical point is approached.
5. Experimental test facility
Fig. 3. The effect of pressure on the heat ransfer coefficient (Tin=390 K, mass flow rate=0.04 kg/s, Twall=310 K).
Fig. 4 shows a schematic diagram of the experimental test facility. Since the operating pressure of the apparatus is between 8.0 and 12.0 MPa, special safety procedures were observed, such as thick-walled stainless steel tubes. The test apparatus consists of a closed carbon dioxide loop, an open water loop, and a closed water–glycol loop. The carbon dioxide loop consists of four main components: a sub-cooler/receiver assembly, a magnetically coupled, sealed, gear pump, a heater, and the test-section. In addition, a mass flow meter, pressure transducers, probe thermocouples, and a pressure relief valve are installed in the loop.
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Fig. 4. Schematic diagram of the test facility. Fig. 1. Schematic diagram of the test facility.
Carbon dioxide enters the test section in the supercritical state at about 120 C. In passing through the test section, the CO2 is cooled by water to approximately 25 C, which is below the thermodynamic critical temperature of carbon dioxide. The fluid enters the subcooler/receiver assembly where it is further cooled to increase its density. The fluid leaves this assembly at about 5 C and is then pumped by the variable speed internal gear pump. The function of the pump is to maintain a certain mass flow rate and overcome the pressure drop throughout the CO2 piping. The cool liquid carbon dioxide then enters the heater, where it is heated to the supercritical inlet conditions of the test section. The pump is a custom-designed, variable speed internal gear pump, which is magnetically coupled to a 1hp motor. It is designed to work at pressures up to 136 bar. Flow into the test section is controlled by the variable speed drive of the motor. The mass-flow meter measures the flow and the drive can be adjusted accordingly. All the tubes are connected with compression fittings. The test section consists of eight subsections. Each subsection is a tube-in-tube counter-flow heat exchanger. Carbon dioxide flows in the center and water flows in the annulus. Five subsections are 1.8 m long and three subsections are 1.3 m long. The tubing of the carbon dioxide loop is made of stainless steel with a nominal OD of 6.35 mm and a wall thickness of 0.815 mm. To minimize heat transfer to the environment, the test sections were heavily insulated.
6. Comparison of numerical and experimental results Numerical simulations were conducted corresponding to the conditions of 12 experimental test runs that are
described in detail by Pitla et al. [7]. In each case, the measured carbon dioxide inlet temperature and the measured water outlet temperature were provided as inputs to the numerical model. The experimental data and the numerical predictions of the heat transfer coefficient and the temperature along the tube length were compared for each test run. An uncertainty analysis was also conducted which showed that the heat transfer coefficients were measured following a specified data reduction procedure [2] with an uncertainty of 17%, while the water temperatures were measured directly using thermistors with an uncertainty of 0.07 C. For three test runs, detailed comparisons of the heat transfer coefficient and temperature along the tube length are provided here. The experimental test conditions of the three runs are summarized in Table 1. The selected test runs represent three different supercritical pressures as well as three different mass flow rates. The selected pressures span the expected pressure range seen during gas cooler operation. The selected mass flow rates are representative of cooling capacities from 3 to 7 kW.
Table 1 Run matrix of the experimental test runs used in the comparative study Test run
Inlet pressure (MPa)
Inlet to outlet temperatures ( C)
Mass flow rate (kg/s)
1 2 3
10.8 9.4 13.4
124–27 121–34 101–20
0.029 0.020 0.039
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which is placed after the first two heat exchangers. Thus, the ‘‘entrance effect’’ is felt in the third heat exchanger. This is observed from the plots by the heat transfer rapidly diminishing for a small axial distance along the tube. Experimental measurements give an average value of the heat transfer coefficient over a heat exchanger. The average heat transfer predicted by the numerical program was calculated and compared with that of the experimental measured value. The results are shown in Table 2. The average experimentally measured heat transfer coefficient agrees with the numerical prediction within 16%. Fig. 6(a)–(c) shows a comparison of the numerically predicted and the experimentally measured water temperature. The experimental measurements agree within 3% of the numerical prediction.
7. New correlation for the heat transfer coefficient Several investigators attempted to correlate the heat transfer of supercritical carbon dioxide during in-tube heating. Relatively fewer attempts were made to correlate the heat transfer for in-tube cooling of supercritical carbon dioxide. In particular, Krasnoshechekov et al. [8], Baskov et al. [9], and Petrov and Popov [10] attempted to correlate the heat transfer of supercritical carbon dioxide during in-tube cooling. The correlations were complex and were not very accurate, especially in the pseudocritical region. Lee and Howell [5] stated that
Table 2 Comparison between the experimentally measured and the numerically predicted heat transfer coefficient
Fig. 5. Variations of the heat transfer coefficient along the length of the tube: (a) Run 1; (b) Run 2; (c) Run 3.
Fig. 5(a)–(c) shows the variations of the heat transfer coefficient along the tube length. Experimental data was used to compute the average heat transfer coefficient of each subsection. These values can be seen as the horizontal lines in the plots. As shown earlier through numerical simulation, it can also experimentally be observed that the heat transfer coefficient is not constant, but exhibits a peak in the pseudocritical region. Fig. 5(b), and more prominently Fig. 5(c), display a discontinuity. This is accounted for by the water heater,
Heat transfer coefficient times length (W/mK) (numerical prediction) Run 1 13568 23031 27689 23228 Run 2 6684 9726 12653 22923 25255 31346 Run 10 18619 25280 25285
Heat transfer Difference coefficient times (%) length (W/mK) (experimental measurement) 13991 22694 25702 19910
3.0 1.0 7.0 14.0
6852 10080 14625 22696 25004 30696
2.5 3.5 15.5 1.0 1.0 2.0
20383 25418 25371
8.0 0.5 0.5
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Nuwall þ Nubulk kwall Nu ¼ 2 kbulk
ð1Þ
where Nuwall and Nubulk are Nusselt numbers that are evaluated based on the thermophysical properties at the wall and bulk temperatures, respectively. In each case, the Gnielinski correlation [11], as shown in Eq. (2), is used to calculate the respective Nusselt number. Nu ¼
=8ðRe 1000ÞPr pffiffiffiffiffiffiffi 12:7 =8ðPr2=3 1Þ þ 1:07
ð2Þ
Using Eq. (2) requires knowledge of the wall and bulk Reynolds numbers. Several variables were considered to develop the correlation shown in Eq. (1). It was found that the best fit was obtained by using the inlet velocity to compute the Reynolds number at the wall (irrespective of location, i.e. over the whole length of the test section) and by using the local mean velocity to compute the bulk Reynolds number. To obtain the local mean velocity, the length of the test section was divided into finite lengths (finite sections) and the mean velocity was evaluated in each finite section using Eq. (3). Uavg ¼
: m Abulk
ð3Þ
In addition, Eq. (2) requires the knowledge of the friction coefficient, . Appropriate results were obtained by using Filonenko’s correlation as shown in Eq. (4), in which the friction factor is only a function of Reynolds number and which was also used in the correlation proposed by Krasnochekov et al. [8]. ¼ ð0:79lnðReÞ 1:64Þ2
ð4Þ
Once the mean Nusselt number has been obtained, the heat transfer coefficient can be computed as shown in Eq. (5): Fig. 6. The variations of the temperature along the length of the tube: (a) Run 1; (b) Run 2; (c) Run 3.
h¼
it was not possible to develop a generalized correlation for the heat transfer coefficient in the pseudocritical region and a detailed numerical analysis is necessary to predict the heat transfer in this region. Based on the numerical predictions of the heat exchanger problem defined in Fig. 1, it was attempted to develop a new correlation for the Nusselt number in terms of other dimensionless parameters. For this purpose, the numerically predicted Nusselt numbers were correlated. The final outcome was a new correlation that is based on the ‘‘mean Nusselt number’’ and is defined as shown in Eq. (1):
In the following section, the numerical computed values and the experimental data of the heat transfer coefficient are compared with the ones obtained with the proposed correlation. Fig. 7(a)–(c) compares the accuracy of the proposed correlation with the numerical prediction for Runs 1, 2 and 3, respectively. In each figure, the heat transfer coefficient computed using the proposed correlation is plotted together with the numerical prediction (Nu) and the experimental data versus the tube length. To observe the accuracy of the new correlation, two help lines have been plotted. The first line, 1.2 Nu, denotes a +20% value of the numerically predicted heat transfer coefficient, and the second
Nu kbulk D
ð5Þ
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Fig. 7. Comparisons of the proposed correlation with the numerical prediction and experimental data along the length of
Fig. 8. Comparisons of the proposed correlation with the numerical prediction versus the temperature: (a) Run 1; (b) Run 2; (c) Run 3.
line, 0.8 Nu, represents a 20% value of the numerically predicted heat transfer coefficient. It is seen from Fig. 7(a)–(c) that the proposed correlation agrees within the range of 20% for most of the tube length of the three test runs, except for two deviations. In Run 1 [Fig. 7(a)], the new correlation overpredicts the heat transfer coefficient after the pseudocritical region. In Run 2 [Fig. 7(b)], the new correlation underpredicts the heat transfer coefficient far out in the vapor region. Fig. 8(a)–(c) shows the variation of the mean Nusselt number computed using the proposed correlation together with the numerically predicted Nusselt number (Nu) versus the bulk temperature for Runs 1, 2 and 3, respectively. As in Figs. 7 (a)–(c), the 20% curves (1.2
Nu and 0.8 Nu, respectively) are also plotted in the figures. It is seen that the Nusselt numbers calculated using the proposed correlation agree to within 20% for all test runs, except for Run 1 [Fig. 8(a)] at lower temperatures (after the pseudocritical temperature), where there is a sharp change in the thermophysical properties. From this comparison it can be seen that the new correlation is most accurate for the data obtained during Run 3 [Fig. 8(c)], where the pressure is the highest (13.4 MPa). As the pressure is increased from the critical point, the variation in the thermophysical properties is less severe. It is thus possible to correlate the heat transfer coefficient of supercritical carbon dioxide more accurately at pressures away from the critical point as would
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(6) and (7), respectively, and to predictions obtained using the Gnielinski correlation [Eq. (2)] evaluated at the bulk temperature. Nuwall ¼ NuPKP;wall
wall bulk
Nuwall ¼ NuPKP;wall
n1
cp;avg cp;wall
cp;avg cp;wall
m1
m2 bulk n2 wall
ð6Þ
ð7Þ
The values of the exponents m1, m2, n1 and n2 are to be determined from graphical (Eq. (6)) and tabular (Eq. 7)) data as provided in the references. Fig. 9(a)–(c) show the heat transfer coefficient computed with the new proposed correlation and the other three correlations together with the numerically predicted heat transfer coefficient versus the bulk temperature for Runs 1, 2 and 3, respectively. It is seen from Figs. 9(a)–(c) that the correlations by Krasnoshchekov et al. [8] and Baskov et al. [9] overpredict the numerical prediction, especially in the pseudocritical region, while the Gnielinski correlation underpredicts the numerical prediction throughout. Since the predictions obtained using the Gnielinski correlation are computed at the bulk temperature there is no effect of the heat transfer improvement near the wall which is below the critical temperature in the pseudocritical region. In comparison to the other three correlations, the new proposed correlation matches the numerical predictions the closest.
9. Conclusions
Fig. 9. Comparisons of the heat transfer coefficient predicted by various correlations: (a) Run 1; (b) Run 2; (c) Run 3.
be the case with any single-phase textbook correlation for the heat transfer coefficient.
8. Comparison to literature correlations The proposed correlation was also compared to predictions obtained with two existing correlations for the supercritical heat transfer coefficient by Krasnoshchekov et al. [8] and by Baskov et al. [9], which are given in Eqs.
The heat transfer coefficient in supercritical fluids is not constant and varies as both the wall temperatures and the bulk temperatures of the fluid vary. It was seen that there is a spike in the heat transfer coefficient in the pseudocritical region. A new correlation to predict the heat transfer coefficient has been developed based on curve fits of the numerically predicted and experimentally obtained data and is presented in this paper as Eq. (1). The new correlation predicts an average Nusselt number based on wall and bulk Nusselt numbers. The Nusselt numbers of the wall and bulk are calculated using the Gnielinski correlation evaluated based on the thermophysical properties at the wall and the bulk temperatures, respectively. It was seen that 85% of the heat transfer coefficient values predicted by the new correlation were accurate to within 20%. A comparison of the heat transfer coefficient calculated using the new correlation and three existing correlations found in the literature showed that the accuracy in predicting the heat transfer coefficient with the new correlation was greatly increased, especially in the pseudocritical region.
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Acknowledgements The study presented in this paper was funded by ASHRAE 913-RP. The authors would like to thank ASHRAE for the sponsorship.
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[5] Lee SH, Howell JR. Turbulent developing convective heat transfer in a tube for fluids nar the critical point. International Journal Heat Mass Transfer 1997;41(10): 1205–18. [6] Schlichting H. McGraw Hill: Boundary layer theory, 1960. [7] Pitla SS, Robinson DM, Zingerli A, Groll EA. Ramadhyani S. ‘‘Heat Transfer and Pressure Drop Characteristics During In-Tube Gas Cooling of Supercritical Carbon Dioxide. Final Report ASHRAE 913-RP, American Society of Heating, Refrigerating, and Air Conditioning Engineers, Inc., Atlanta, GA, August 2000. [8] Krasnoshechekov EA, Kuraeva IV, Protopopov VS. Local heat transfer of carbon dioxide at supercritical pressure under cooling conditions. Teplofizika Vysokikh Temperatur 1970;7(5):922–30. [9] Baskov VL, Kuraeva IV, Protopopov VS. Heat transfer with the turbulent flow of a liquid at Supercritical Pressure in tubes under cooling conditions. Teplofizika Vysokikh Temoperatur 1997;15(1):96–102. [10] Petrov NE, Popov VN. Heat transfer and resistance of carbon dioxide being cooled in the supercritical region. Thermal Engineering 1985;32(3):131–4. [11] Gnielinski V. New equation for heat and mass transfer in turbulent pipe and channel flow. Int Chemical Engineering 1976;16:359–68.