Empirically tuned model for a precooled MGJT cryoprobe

Cryogenics 52 (2012) 590–603

Contents lists available at SciVerse ScienceDirect

Cryogenics journal homepage: www.elsevier.com/locate/cryogenics

Empirically tuned model for a precooled MGJT cryoprobe H.M. Skye ⇑, K.L. Passow, G.F. Nellis 1, S.A. Klein 1500 Engineering Drive, University of Wisconsin – Madison, WI 53706, USA

a r t i c l e

i n f o

Article history: Received 18 December 2011 Received in revised form 1 August 2012 Accepted 7 August 2012 Available online 30 August 2012 Keywords: Joule–Thomson Mixed gas Cryosurgery Recuperator Experimental

a b s t r a c t Cryosurgery is a medical technique that uses a freezing process to destroy undesirable tissues such as cancerous tumors. The handheld portion of the cryoprobe must be compact and powerful in order to serve as an effective surgical instrument; the next generation of cryoprobes utilizes precooled Mixed Gas Joule–Thomson (pMGJT) cycles to meet these design criteria. The increased refrigeration power available with this more complex cycle improves probe effectiveness by reducing the number of probes and the time required to treat large tissue masses. Selecting mixtures and precooling cycle parameters to meet a cryogenic cooling load in a size-limited application is a challenging design problem. Modeling the precooler and recuperator performance is critical for cycle design, yet existing techniques in the literature typically use highly idealized models of the heat exchangers that neglect pressure drop and assume infinite conductance. These assumptions are questionable for cycles that are required to use compact components. The focus of this research project is to understand how the cycle performance is impacted by transport processes in the heat exchangers and to integrate these findings into an empirically tuned model that can be used for mixture optimization. This effort is carried out through a series of modeling, experimental, and optimization studies. While these results have been applied to the design of a cryosurgical probe, they are also more generally useful in understanding the operation of other compact MGJT systems. A commercially available pMGJT cryoprobe system has been modified in order to integrate a suite of measurement instrumentation that can completely characterize the performance of the individual components as well as the overall system. Measurements include sufficient temperature and pressure sensors to resolve thermodynamic states, as well as flow meters in order to compute the heat and work transfer rates. Temperature sensors are also integrated within the recuperator in order to capture the spatially resolved heat transfer performance; these data are used to overcome the lack of correlations for heat transfer of the multi-phase mixture in the helically wound finned-tube heat exchanger. Test conditions were varied to achieve a range of temperatures, pressures, and thermodynamic qualities with mixtures of argon, R14 and R23. Recuperator and precooler conductance and pressure drop data for these test conditions are presented and fit to simple physics-based correlations; these correlations are integrated with an optimization model of the precooled MGJT cryoprobe that has been described in a previous paper. The predictive capabilities and optimal mixture selections of the model are compared with those of other models available in the literature, including the isothermal enthalpy difference model. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Cryosurgery is a medical procedure designed to destroy targeted tissue with a rapid cooling process. The cold tip of a handheld cryoprobe is placed in contact with the tissue for several minutes, forming a small cryolesion (typically about 1–3 cm) as described in Rubinsky [1]. A comprehensive history of refrigeration ⇑ Corresponding author. Present address: 100 Bureau Drive, MS 8631, Gaithersburg, MD 20899 8631, USA. Tel.: +1 301 975 5871; fax: +1 301 975 8973. E-mail addresses: [email protected] (H.M. Skye), kendrapassow@gmail. com (K.L. Passow), [email protected] (G.F. Nellis), [email protected] (S.A. Klein). 1 Tel.: +1 608 265 6626; fax: +1 608 265 2316. 0011-2275/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.cryogenics.2012.08.004

technology used for cryoprobes is given in Skye et al. [2]. Many modern cryoprobes leverage the high cooling density available from Mixed-Gas Joule–Thomson (MGJT) cycles to provide for a powerful and compact design as shown by Radebaugh [3], Boiarski [4], Marquardt et al. [5], and Dobak et al. [6]. Skye et al. [2] and Alexeev et al. [7] discuss how precooling the gas mixture before it enters the recuperator enables selection of mixtures that achieve even greater cooling from the same cryoprobe size envelope. As part of this project, a thermodynamic modeling tool was created for a pMGJT cryoprobe system that utilizes a conventional vaporcompression (VC) cycle to provide precooling; the model is described in Skye et al. [2] and the system is shown in Figs. 1 and 2. The model was used to investigate cycle design issues related

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

591

Fig. 1. Geometric schematic of a precooled MGJT cryoprobe. The inlet/outlet tubes shown on the right hand side connect to remotely located compressors via flexible tubing. The PRTs embedded in the G10 sheath are used to characterize the spatially resolved conductance.

Fig. 2. Schematic of experimental test facility including (a) measurement instrumentation integrated with the pMGJT cryoprobe system and (b) the recuperator thermodynamic states and section indices.

to proper selection of precooling temperature and mixture compositions. In this paper, experimental data for heat transfer and pressure drop in the recuperator/precooler are used to tune the model to give a more realistic estimate of system performance. The design challenge for MGJT cryoprobes is to provide the maximum amount of cooling within the size constraints imposed by the requirement for an ergonomic surgical instrument. This design task primarily involves a non-trivial process of selecting the refrigerant blend, whose optimal composition depends on the operating pressure [8], cryoprobe tip temperature [9,10], precooling temperature [2] and refrigerant charging mass/pressure [11,26–30]. Mixture optimization is typically carried out by maximizing the ‘‘isothermal enthalpy difference’’ [12,3,13] or some other thermodynamic quantity, such as COP or exergetic efficiency [14–17], of a highly idealized cycle where the heat exchangers are

modeled as having infinite conductance and no pressure drop. Designs based on the thermodynamic optimization targets of the idealized cycle often result in a large recuperator, which is not appropriate for a compact system. Minimizing recuperator size is especially important in this application as the recuperator is part of the handheld cryoprobe as shown in Fig. 1. Keppler et al. [13] developed a thermodynamic model that considered a finite recuperator size in the optimization, where the size of the heat exchanger was assumed to be directly related to the conductance. Fredrickson et al. [9,10] demonstrated how the finite-conductance optimization model could be integrated with a cryoprobe iceball formation model to select the best mixture for a MGJT cryoprobe system. However, none of these models consider the effect of transport phenomena on the cycle performance and optimization. The

592

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

experimental test data described here show that the actual performance of the compact cycle is far from ideal, and subsequently mixtures selected by the idealized models do not perform well in the system. For example, Fig. 3 compares the actual and ideal cycle performance for a nominal test case from the experiment presented here; the refrigeration effect (h7  h6) is much smaller for the actual test case. Part of the difference is related to the finite heat exchanger conductance, where limited surface area and low heat transfer coefficients restrict the recuperator exergy recovery from the cold stream returning from the load. This is especially problematic for compact systems where increasing the heat exchanger size in an effort to approach the idealized performance may not be an option because of size limitations. Design for higher heat transfer coefficients is difficult because of the lack of heat transfer data for mixtures in the helically wound finned tube precooler/recuperator (Giaque-Hampson style heat exchangers) at conditions encountered in the cryogenic MGJT cycle; the existing studies either do not provide sufficient detail to be of general use [18,19] or are for relatively simple geometries [20–22,24]. Exergetic efficiency is also reduced by pressure drop in the precooler/recuperator. This pressure drop reduces the temperature difference between the hot and cold streams, effectively reducing the heat transfer between the streams [25]. The focus of this research was to develop a physics-based empirically tuned mixture optimization model that includes the transport performance of mixtures in the cycle, where particular attention is given to the precooler/recuperator as they largely govern the operation of the entire cycle. The lack of heat transfer or pressure drop data for the mixtures and particular heat exchange geometry is addressed through measurements of recuperator/precooler performance in an operating cycle; these data are reduced to form basic hydrodynamic relations that can be integrated with the system model developed in [2] to select optimal mixtures. An experimental test facility was developed to measure the detailed thermodynamic and heat transfer performance of a commercially available pMGJT system. The test facility provides an in situ measurement of local recuperator conductance that provides, to our knowledge, an unprecedented level of information about the recuperator performance. It should be noted that transport correlations are not universally applicable because: the fidelity of the transport measurements was limited by the integration with a commercial system, the heat exchanger geometry was limited to a single configuration, and the applicability of the correlations is not known for mixtures with

constituents not tested here. Nevertheless, the correlation development is useful to show how transport phenomena affect cycle performance, how this suite of measurements can be used to tune a mixture optimization model, and the general shift in optimal mixture composition when transport phenomena are considered. The measurements are carried out with high precision and in a thermally isolated vacuum environment, so the data can be used to assess the accuracy of various mixture property databases. The predictive capabilities of two commercially available mixture database packages (REFPROP – Lemmon et al. [38] and NIST4 – Ely et al. [39]) developed by the National Institute of Standards and Technology (NIST) are evaluated in order to assess their applicability to the mixtures and operating conditions encountered in a cryogenic cycle. The purpose of the remainder of this paper is to: (1) discuss the design of the experimental test facility, (2) show how the data are reduced to quantify cycle and component performance where heat exchanger pressure drop and conductance data are fit to correlations, and (3) compare the predictive performance and optimal mixture selection of the empirically tuned optimization model with existing models in the literature. Note that a thesis has been prepared for this project and contains an expanded discussion of the material presented here as well as the raw test data: http://sel.me.wisc.edu/publications-theses.shtml [40]. Thesis page numbers with additional details are given for convenience. 2. Methods 2.1. Experimental test facility The experimental test facility shown in Fig. 2 has been constructed by modifying a commercially available cryoprobe system to integrate measurement instrumentation sufficient to capture the thermodynamic and heat transfer performance of the cycle. The modifications to the cryoprobe and instrumentation techniques are discussed in greater detail in [41,40] (pp. 80–146). The 1st stage vapor-compression cycle operates with R410a and the 2nd stage JT cycle uses blends of R14, R23 and argon for the tests reported here. The composition of the circulating gas differs significantly from the charging composition, as much as 20% in [40] (pp. 173–175), and is therefore measured for each data set using a Gas Chromatograph (GC). The measurements are shown in Fig. 2 where the numbers correspond to thermodynamic state number. The measurements include:

Fig. 3. Comparison of the actual and idealized pMGJT cycle performance for a nominal test case. The property data are computed using REFPROP.

593

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

1. temperatures measured with Platinum Resistance Thermometers (PRT) and ThermoCouples (TC), including measurements within the low pressure stream of the recuperator labeled ‘‘PRTc1–PRTc5’’ shown in Figs. 1 and 2, 2. pressures (P), measured with membrane type capacitance transducers, _ 1st and m _ 2nd ), 3. flow rates in the 1st and 2nd stage cycles (m measured with calorimetric flowmeters, and 4. cryoprobe tip thermal load, applied using a Nichrome wire heater where power is computed using voltage (Vload) and current (Iload) measurements. Note that hardware limitations prohibited direct measurement at states 4, 11, and H1–5 (shown in dashed boxes in Fig. 2) so the states were inferred using an energy balance. The cold components are enclosed in a vacuum chamber and wrapped with Multi-Layer radiation Insulation (MLI) to insulate them from thermal parasitics; the total parasitic heat input is estimated at 0.8 W in [40] (pp. 122–125). An interchangeable jewel orifice (0.017500 diameter for the tests here) and a bypass valve are used to regulate the pressure ratio and mass flow applied to the JT cycle. Table 1 lists the instrumentation details including the sensor uncertainties that are used to determine the uncertainties in the performance metric calculations. The overall measurement uncertainty accounts for the sensor calibration, the data acquisition system, and in some cases observed sensor drift (notably in the pressure transducers). Note that we recommend that future studies use a Coriolis meter to eliminate uncertainty associated with correcting the calorimetric flow meter for the thermal capacitance of the mixture and to avoid measurement error and damage to the instrument caused by compressor oil entrained in the gas mixture. The measurements are used to compute the thermodynamic states shown in Figs. 2 and 3 according to the methods described in [41,40] (pp. 159–172). Notable assumptions include: (1) constant evaporation temperature for the R410a in the precooler as determined by the saturation pressure measured at state 8, (2) equal pressure drop between each ‘‘PRTc’’ measurement in the recuperator (note that there are two diametrically opposed sensors at each ‘‘PRTc1-PRTc4’’ location as shown in Fig. 1), and (3) considering the tube lengths and fluid density between states 3,4 and 5, the pressure at state 4 can be estimated by the average between states 3 and 5. The precooler/recuperator have the same tube cross section and helix configuration as shown in Fig. 5, and the tube length between state 4 and 5 is about two times greater than between 3 and 4. However, the warmer and less dense gas between states 3 and 4 will have a larger pressure drop for a given length of tube. (The mixture is often in a two-phase state between 4 and 5 and in a vapor phase between 3 and 4.) 2.2. Experimental test matrix The data collected using the experimental test facility are divided into four different sets based on the 2nd stage mixture

compositions as shown in Table 2. Sets 1 and 2 include tests with pure synthetic refrigerants (R14 and R23), where the R23 data were collected with the precooling cycle deactivated. These tests were used to debug and verify the test facility. For example, the data are used to check component energy balances and to show the agreement between the measured and predicted Joule–Thomson effect across the jewel orifice in the 2nd stage. Verification in this manner can be carried out with higher confidence using pure components with well-defined property data as opposed to mixtures, whose properties must be predicted using complex mixing rules that introduce a larger uncertainty. A bottle containing a precisely formed mixture of argon, R14 and R23 was used to charge the system for the third set of tests. These tests were used to demonstrate the composition shift of the circulating mixture in the cycle and also provided initial data where the recuperator operated in the two phase regime. However, this mixture yielded relatively high thermodynamic quality (above 0.8) in the recuperator. In order to study the heat transfer across the entire vapor dome, the fourth set of mixture compositions eliminated the argon and increased the R23 (which has a relatively high saturation temperature). The majority of the two-phase heat transfer and pressure drop data used to develop the empirical correlations discussed in Sections 3.1–3.3 came from this fourth set. 2.3. Performance metric calculations 2.3.1. Mixture thermodynamic properties Accurate thermodynamic property data for the multi-component mixtures at cryogenic temperatures are critical for cycle analysis. The precise measurements presented here are ideal for evaluating the capability of the computed property data to predict thermodynamic phenomena occurring in the cycle. Specifically, the isenthalpic expansion process across the jewel orifice in the pMGJT cycle (from state 5 to state 6 in Fig. 2) can be characterized by the Joule–Thomson effect temperature change (DTJT). Comparisons of the predicted and measured data showed that REFPROP was able to predict the experimental DTJT better than NIST4 near the edges of the vapor dome [40] (pp. 175–178). This observation is related to the somewhat higher dewpoint and bubble point lines predicted by NIST4 compared to REFPROP (approximately 2–6 K higher). The difference in vapor domes also causes a significant difference in the heat capacity of the mixture near the dewpoint/bubble point, resulting in different computed recuperator temperature profiles. The NIST4 data often predicted a pinch point violation (i.e., a situation in which the hot stream has a local temperature that is lower than the cold stream) [40] (pp. 179–181). Therefore, the REFPROP database was used to analyze the cycle data presented here. Note that the local mixture concentrations entrained in each cycle component may differ from the circulating composition and may play a significant role in the transport phenomena. For example, the cold end of the recuperator may collect non-circulating condensate that will participate in the heat transfer process. This effect is not considered in the present research, rather, the analysis uses the circulating composition for transport properties.

Table 1 Instrumentation type and accuracy used in the experiment. Measurement

Sensor

Manufacturer

Model

Uncertainty

Temperature Temperature Refrigerant flow Pressure (P3, P5, P9) Pressure (P1, P7, P8) Heater power Gas composition

Platinum resistance thermometer Type E thermocouple Calorimetric flow meter Capacitance-type transducer Capacitance-type transducer Digital multimeter Gas Chromatograph

Lakeshore Lakeshore Omega Setra Setra Nat’l Instruments Hewlett Packard

PRT-111 36 AWG FMA1742 206 (500 psig) 206 (100 psig) SCXI 1100 5890 Series II

0.5 K 1K 3% FS 3 psi 1.5 psi 1 lW 3%

594

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

Table 2 Summary of test parameters for the collected data. Set 1

Set 2

Set 3

Set 4

2nd stage Mole fraction argon Mole fraction R14 Mole fraction R23 High pressure Low pressure Tip temperature (T7) Tip thermal load (Q_

– 100% – 185–350 psig 10–40 psig 163–255 K 0.5–17.5 W

– – 100% 240–350 psig 11–25 psig 240–292 K 7–24 W

10–15% 55–65% 25–35% 190–255 psig 10–25 psig 170–215 K 0.5–7.5 W

– 20–50% 50–80% 160–290 psig 14–100 psig 175–260 K 0.3–43 W

_ 2nd ) Mass flow (m

0.8–1.6 g/s

0.65–0.9 g/s

0.8–1.0 g/s

0.7–1.7 g/s

1st stage Working fluid Evaporator temperature _ 1st ) Mass flow (m

R410a 237–240 K 1.4–2 g/s

N/A N/A N/A

R410a 240–242 K 2–2.2 g/s

R410a 235–242 K 1.2–1.4 g/s

load )

2.3.2. Heat exchanger conductance 2.3.2.1. Precooler. The 1st stage evaporator temperature, as well as the 2nd stage (JT cycle) thermodynamic states and temperature profiles shown in Fig. 3, are used to compute the precooler and recuperator conductances. Discrete models divide the heat exchangers into sections with small temperature changes over which the specific heat of the mixture is nearly constant and therefore the effectiveness-NTU relationship can be accurately applied. The governing equations are solved using the Engineering Equation Solver (EES – Klein et al. [42]), where property data are computed using the REFPROP [39] mixture database that has been interfaced with the EES program [43]. Fig. 4 shows how the discrete model divides the precooler into Npc sections, each exchanging an equal amount of heat. Fifteen sections were used based on a numerical sensitivity analysis. The overall energy exchange is computed according to:

_ 2nd ðh3  h4 Þ Q_ pc ¼ m

ð1Þ

where h3 and h4 are the enthalpies at state 3 and 4 in the 2nd stage, _ 2nd is the 2nd stage mass flow. The intermediate 2nd stage and m enthalpies at each node are computed using an energy balance on each of the sections:

(a)

h2nd ;pc;i ¼ h2nd;pc;i1 

Q_ pc 1 _ 2nd Npc m

i ¼ 1 . . . Npc

ð2Þ

h2nd;pc;0 ¼ h3

ð3Þ

h2nd;pc;Npc ¼ h4

ð4Þ

where i is the node index and h2nd,pc,i is the enthalpy of the mixture in the 2nd stage at each node. The model also assumes a linear pressure drop on the 2nd stage side so that the pressure drop is assumed to be equal in each of the sections:

P2nd ;pc;i ¼ P3 

ðP3 —P 4 Þ ðiÞ i ¼ 0 . . . Npc Npc

ð5Þ

where P2nd,pc,i is the pressure in the 2nd stage at each node, and P3 and P4 are the pressures at states 3 and 4. Nodal temperatures can then be computed using the mixture property data relations in REFPROP using the enthalpy and pressure:

  T 2nd ;pc;i ¼ temperature P2nd ;pc;i ; h2nd ;pc;i i ¼ 0 . . . Npc

ð6Þ

Finally, the 1st stage (VC cycle) refrigerant in the precooler is assumed to evaporate at a constant pressure and therefore at a

(b)

Fig. 4. (a) Precooling heat exchanger divided into Npc sections and (Npc + 1) nodes. (b) First differential heat exchanger element.

595

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

Fig. 5. Cross sectional view of (a) the recuperator and (b) the precooler.

constant temperature, computed as the saturation temperature for the pressure measured at state 8. Using the 1st stage temperatures, as well as the 2nd stage temperatures and enthalpies, the effectiveness-NTU equations (given in more detail in Skye et al. [2], Incropera and DeWitt [44], or Nellis and Klein [45]) are applied to compute the conductance of each section.

UApc;i

  ln ¼ min C_ 2nd;pc;i ; C_ 1st;pc;i



epc;i 1 epc;i C r;pc;i 1

C r;pc;i1

 i ¼ 1 . . . Npc

ð7Þ

where C_ 2nd;p;c;i and C_ 1st;p;c;i are the thermal capacities of the 1st and 2nd stage fluids in each section of the precooler, Cr,pc,i is the capacity ratio, and epc,i is the section effectiveness. The overall precooler conductance (UApc) is then represented by the sum of the conductances in each section. Finally, the conductance is normalized by the tube length:

PNpc ðUA=LÞpc ¼

i¼1 UApc;i

ð8Þ

Lf ;pc

where the coiled, finned-tube length in the precooler (Lf,pc) is 0.55 m, as listed in Table 3. Note that only the finned portion of the tube is considered; short sections of smooth tube that do not significantly contribute to heat transfer are ignored. 2.3.2.2. Recuperator. The temperature measurements within the low pressure stream of the recuperator allow for more resolved

Table 3 Summary of important cryoprobe heat exchanger assembly dimensions shown in Fig. 5. Measurement description

Value

Mandrel radius Finned tube helix radius G10 sheath inner radius Precooler SS sheath inner radius Fin height (above tube) Fin thickness Number of fins per revolution Tube ID Tube OD Frontal flow area in the low pressure (cold) side of the recuperator/precooler Flow area inside the precooler/recuperator tube (hot side) Tube helix pitch (tube center-to-center spacing) Length of tube for single helix revolution Length of finned precooler tube (wrapped) Length of finned recuperator tube

6.33 mm 7.78 mm 9.32 mm 9.32 mm 0.737 mm 0.18 mm 63 1.04 mm 1.42 mm 6.136  105 m2 8.518  107 m2 3.15 mm 4.90 cm 0.55 m 1.17 m

conductance measurements than those in the precooler. Fig. 2b shows the recuperator divided into six sections (labeled sec 0–5) between the 7 cold stream measurement locations including states 1, C1–C5, and 7. The hot stream states (4, H1–H5) that are computed using energy balances on the recuperator sections are also shown. Note that Section 5 represents a short, unfinned section between PRTc5 and PRT 7 and is not considered in the heat exchanger conductance data presented here. The conductance in each section is computed using a discrete model that further divides the sections into Nsub sub-sections (where Nsub was chosen to be 10 based on a numerical sensitivity analysis) that each exchange an equal portion of the energy that is transferred in each section [2,40] (pp. 162–171). The effectiveness-NTU equations are applied in each sub-section following the method used for the precooler, where the sub-section conductances (UArec,sub,j) are summed to find the conductance of each section (UArec,sec,i). Furthermore, the conductances in each section are normalized by the finned tube length in each section (Lf,rec,sec,i, which ranges from 17.8 cm to 25.6 cm; the overall tube length is 1.17 m as listed in Table 3) in order to facilitate a comparison of heat exchanger performance between sections containing different tube lengths.

PNsub ðUA=LÞrec;sec;i ¼

j¼1

UArec;sub;j

Lf ;rec;sec;i

ð9Þ

2.3.3. Heat exchanger Reynolds numbers Measuring heat exchanger conductance within an operating cycle (rather than in a dedicated heat exchanger test facility where the thermal inputs could be more precisely controlled) provides valuable information about the cycle dynamics related to component interactions. However, the measurements are restricted in that they cannot be used to distinguish the individual thermal resistances of the recuperator, including conduction through the finned-tube wall, and the heat transfer coefficients on either side of the tube. Conductance measurements are presented in this paper with relation to the independent variables (e.g. Reynolds number, thermodynamic quality) that likely govern the variations in heat transfer performance observed in this research. The precooler and recuperator conductances are presented in Sections 3.2 and 3.3 as a function of the mixture Reynolds numbers for test conditions where the mixture is in a gas phase throughout the precooler or recuperator. The Reynolds number for the low pressure side (rather than the high pressure side) of the recuperator was chosen because the overall conductance for this type of heat exchanger is usually limited by the lower density, lower velocity fluid flow on

596

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

the low pressure side [46]. Conversely, for the precooler, the Reynolds number in the high pressure side (i.e. 2nd stage gas mixture side) is used because the flow conditions in the precooling cycle do not vary significantly for the tests presented here. The heat exchanger geometry is nearly identical for the precooler and the recuperator: a small tube (1.42 mm outer diameter) with annular fins is wrapped around a mandrel and enclosed in an outer sheath (see Fig. 1). The high pressure, warmer fluid flows inside the tube and the low pressure, cool fluid flows over the finned tube in the annulus formed by the mandrel and an outer sheath. Fig. 5a shows the cross-sectional view of the recuperator including the finned tube and the low pressure return annulus which is bound on the outside by the G10 sheath with the embedded PRTs. The precooler is shown in Fig. 5b where there is no G10 sheath or embedded PRTs, and the Stainless Steel (SS) sheath inner diameter narrowly clears the tube fins. The hot stream frontal flow area (FFA) is computed using the tube ID, and the cold stream FFA is computed using the area inside the G10 sheath (for the recuperator) or the SS sheath (for the precooler) debited by the FFA of the mandrel and finned-tube. Table 3summarizes the heat exchanger dimensions including the FFAs [40] (pp. 110–113). Reynolds numbers for the low pressure stream of the recuperator are computed using the averaged property data in each recuperator section (defined in Fig. 2) and the low pressure annular FFA. The Reynolds number of the high pressure mixture in the precooler was computed based on the FFA inside of the finned tube as listed in Table 3 and the property data at state 3. Correlating the precooler conductance with the Reynolds number at state 4 and averaged between states 3 and 4 was also considered, but this alternative did not significantly change or improve the correlation [40] (pp. 209–211); therefore, for simplicity, the data are presented as a function of the Reynolds number calculated at state 3.

ever, most of the pressure drop will occur in the warmer sections of the heat exchanger where density is low and therefore the velocities are high (leading to large frictional losses). The first order approximation made here is to assume that the pressure drop is governed by flow parameters computed at the warm end of the heat exchanger where the flow is in or near a vapor state. Existing single phase pressure drop correlations for tubes typically follow the form:

DP ¼ f ðRe; e; DÞ

q v2 L 2 D

ð10Þ

3. Discussion

where DP is the pressure drop, f is the friction factor (which is a function of Reynolds number Re, tube roughness e, and diameter D), q is the density, v is the velocity, and L is the flow passage length. The geometry is fixed so e, D and L are constant, and the Reynolds number does not vary significantly in these experiments. Therefore, the pressure drop in the recuperator can be approximately correlated with the dynamic pressure value (1/2qm2) computed at the warm (high pressure) stream inlet and cold (low pressure) stream outlet. Fig. 6 shows the measured recuperator cold (a) and hot (b) side pressure drop data (with error bars representing the uncertainty in the measurements) presented as a function of the dynamic pressure. Linear curve fits have been applied to the data and the resulting curve fits are displayed in the figures (198 data points and R2 = 0.61 for both hot and cold side, RMS error hot side = 50 kPa, RMS error cold side = 4.8 kPa). It is expected that the velocity and pressure drop terms would go to zero together; however, the linear fit yields an offset of 106 kPa and 32 kPa, respectively, for the hot and cold streams. Nevertheless, the curve fits track the data adequately for the results presented here. As discussed in Section 2.1, the pressure drop on the hot sides of the precooler and recuperator are assumed to be equal, so the correlation in Fig. 6b is also used to compute the precooler hot side pressure drop.

3.1. Pressure drop data and correlations

3.2. Recuperator conductance data and correlations

The pressure drop in the recuperator is computed directly in the cold stream using measurements at state 7 and 1. The hot side pressure drop is computed using the measurement at state 5 and the estimated pressure at state 4 (taken to be the average between 3 and 5, as discussed in Section 2.1). A rigorous presentation of the two-phase pressure drop should include a detailed model that integrates the local pressure gradient, which is governed by liquid/vapor densities, superficial velocities and viscosities (e.g. the Muller-Steinhagen and Heck correlation as shown in [47]). How-

Conductance data for each of the recuperator sections are presented in this section and examined according to flow regime. In order to make these data more general, the conductance values are normalized by the length (Lf,rec,sec,i) of the finned tube in their sections, where the data from all the sections are combined to form a correlation that governs all sections (an expanded analysis of the data presented here, including data where the conductance in each section is distinguished, can be found in pp. 184–196 and 200–219 of [40]). The correlation shown in Fig. 7 spans the three phase

Fig. 6. Recuperator pressure drop empirical correlations for the (a) cold, and (b) hot streams of the recuperator.

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

597

Fig. 7. Recuperator conductance data and correlation for (a) two-phase and (b) vapor phase.

regimes where the equation for each phase is highlighted by framed text in the figure. Note that only data from the finned tubes in Sections 1–4 are included for the correlation. Section 0 has been removed as a significant outlier; this section contains a significant length of unfinned tube and includes the space between the precooler shell and the outer SS sheath (as shown in p. 99 of [40]) which introduces an unknown heating/cooling effect on the stream. Section 5 has also been excluded as it represents a section of unfinned tube only. 3.2.1. Vapor-phase conductance Fig. 7b shows the vapor-phase recuperator conductance data normalized by finned tube length. The data used for the correlation have been selected so that they have an uncertainty level of 40% or less, which reduces the scatter substantially while still including most of the data (415 total points, pp. 201–202 of [40]). Several different methods were explored to fit the data: both linear and constant value fits were used, and the data were normalized by the mixture thermal conductivity to form a Nusselt number-like metric [40] (pp. 188–189). The constant value fit (UA/L = 3.0 W/K m with RMS error 1.57 W/K m) provided a reasonable estimate of performance and was selected for the model as the other, more complicated methods did not significantly improve the fit. 3.2.2. Two-phase conductance The thermal resistance network in the recuperator includes convection/condensation in the hot stream, conduction through the finned tube wall, and convection/evaporation in the cold stream. Condensation and evaporation heat transfer coefficients are both relatively large so it is not obvious which of these will represent the dominant resistance. The cold stream (low pressure) vapor quality was chosen because it is likely the limiting factor for the conductance; the cold stream exits the vapor dome at a somewhat lower temperature than the hot stream for the mixtures and operating conditions presented here and thus will have lower heat transfer due to two-phase dry out. Additionally, as discussed in Section 2.3.3, the low pressure side fluid has low density and velocity and therefore is more likely to have the smaller heat transfer coefficient. The two-phase conductance data are presented as a function of cold stream vapor quality in Fig. 7a. Fig. 7 shows the data exhibit somewhat similar trends to those shown in previous research [21–23] where local heat transfer coefficients for mixtures at cryogenic temperature in small tubes were measured across the entire range of thermodynamic quality. In the present research the heat transfer coefficient for a liquid and vapor are relatively small, and the heat transfer is enhanced by as much as an order of magnitude inside the vapor dome (the peak occurs at

a vapor quality between 0.6 and 0.8). A few of the liquid rich values showed relatively high conductances; we are not sure if these data are outliers or are related to some repeatable physical mechanism. The conductance data normalized by the low pressure stream mass flux (not shown here) did not significantly change the shape of the trend [40] (p. 193), indicating that nucleate boiling and/or condensation, rather than convection, dominate the heat transfer in the low and high pressure streams [48]. The calculated uncertainty in the data presented in Fig. 8a is relatively large (data included have uncertainty of 80% or less), although the grouping of the data suggest the actual uncertainty may be smaller. This observation may be attributed to the simultaneous calibration of the recuperator low pressure stream thermometers (‘‘PRTc’’ in Figs. 1 and 2); the differential temperature measurement used to compute the section heat transfer may be more accurate than the estimate based on summing the influence of individual thermometer uncertainties in quadrature. A 3rd order curve fit (rather than a 2nd order fit) shown in Fig. 7a is used to capture the asymmetry in the parabolic-like trend. This asymmetry is in agreement with the data presented in [21–23], which features a sharp increase in heat transfer between a quality of 1 and 0.8, and a slow progression downward from a quality of 0.3–0. The curve fit (16—62:5xc þ 224:6x2c  176:1x3c ) is forced to pass through the constant vapor (UA/L) value (2.99 W/K m) at a quality of 1 so that the conductance-quality relationship is continuous on the dewpoint line. Note that the two-phase data with quality of 0.95–1 trend downward very close to this value, so the enforcement of the vapor value does not cause a noticeable distortion in the 3rd order fit. The curve fit was applied to the data with varying levels of maximum uncertainty criteria where the fit statistics are presented in p. 205 of [40]; the trend does not change significantly between the 40% uncertainty, 80% uncertainty, and ‘‘all data’’ selections. Therefore, the 80% data were selected to include a larger set of data and yet eliminate a few of the significant outliers. The statistics of the fit include an RMS error of 3.1 W/K m and an R2 value of 0.75 as reported on p. 205 of [40]. The data and the 3rd order fit exhibit an upward inflection in the low quality regions, which disagrees with the downward (and roughly linear) trend with decreasing quality for this region observed in [21,23]. Furthermore, the measurements in [21,23] show liquid heat transfer coefficients that nearly match the vapor values. The heat transfer measurements from the [21,23] study are much more controlled and precise than the measurements presented here, so the correlation for the present data was forced to follow a downward, linear trend in the low quality region. A linear projection (5 + 25xc) of the 3rd order fit at a quality slightly higher than the inflection point (0.3) is drawn in Fig. 7a,

598

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

Fig. 8. Precooler conductance data and correlation for (a) two-phase and (b) vapor phase.

where the projection intersects the bubble point (xc = 0) at a conductance of 5.2 W/K m. Few data were collected in the liquid regime, so a rigorous correlation for the liquid conductance is not presented here. Rather, the observation of the nominally constant conductance for the vapor regime is extended to the liquid regime, and a constant value equal to 5.2 W/K m is assumed. Inaccuracies introduced by this assumption do not significantly impact the model verification here, as the validation data include very few points where the mixture is a subcooled liquid. 3.3. Precooler conductance correlation The flow conditions of the R410a evaporating in the precooler (referred to as the ‘‘cold stream’’) did not significantly change for the tests presented here. Therefore, the heat transfer characteristics of the high pressure 2nd stage working fluid (hot stream) were used to correlate the precooling conductance data. The 2nd stage fluid existed in both a vapor and a two phase state, so the conductance data are compared with the hot stream Reynolds number and the local thermodynamic quality (as was done with the recuperator data). These data are not as resolved as the recuperator conductance measurements because only the cold stream exit temperature/pressure and hot stream inlet temperature/pressure are directly measured. Assumptions about pressure drop through the precooler/recuperator tubes and an energy balance on the recuperator are used to compute the hot exit temperature at state 4 (as discussed in Section 2.1), and the pressure drop of the R410a on the shell side of the precooler is neglected. These measurements represent a total heat transfer over the whole length of the precooler and the results are normalized by the length of the finned tube. 3.3.1. Vapor-phase conductance Fig. 8b shows precooler vapor conductance data normalized by the overall precooler tube length as a function of the Reynolds number at the 2nd stage cycle precooler inlet (i.e., state 3). The data were filtered based on maximum uncertainty criteria for UApc including <20%, <50%, and <70%. A linear best fit was applied to the data for each level of uncertainty, and the final curve fit, shown in Fig. 8b, was selected to include data with <50% uncertainty to encompass as many points as possible while reducing the scatter. The data were correlated with both the hot stream inlet Reynolds number (Re3) as well as the Reynolds number averaged between the hot inlet and outlet (Re3 + Re4)/2. Furthermore, a curve fit where the conductance was normalized by the average thermal conductivity was created to produce a Nusselt number-like quantity. However, neither normalizing by conductivity nor using the

average Reynolds number significantly improved the fit (pp. 209–211 of [40]), so to simplify the analysis the final correlation (1.13 + 5.48E5 Re3) shown in Fig. 8b relates the conductance (UApc/Lpc) to the Re3. 3.3.2. Two-phase conductance Computing the conductance for two-phase flow is more difficult in the precooler than in the recuperator. There are no internal temperature measurements that can be used to compute the variation of the thermodynamic quality of the mixture within the heat exchanger. Rather, the numerical precooler model (presented in Section 2.3.2, more details are available in pp. 211–217 of [40]) is used to infer the thermodynamic quality. The model is used to determine the UA needed for condensation to begin, and the conductance is translated into a tube length using the empirical vapor conductance correlation shown in Fig. 8b. The length of tube containing two-phase flow is subsequently computed by debiting this length from the overall precooler tube length (Lpc) and is used to compute the normalized precooler conductance (UA/L)pc. Fig. 8a shows the normalized conductance values correlated against the quality (of the 2nd stage fluid) averaged between the condensation point (i.e. when x2nd,pc is 1) and the precooler hot exit (state 4). There are relatively few data where the 2nd stage refrigerant exited the precooler in a two-phase state so the results shown in Fig. 8a represent a limited range of thermodynamic quality. Furthermore, the tube length required to cool the 2nd stage refrigerant to the dewpoint line, as predicted using the vapor conductance model, exceeded the actual precooler tube length for some of the data. The length-normalized conductance for these data could not be inferred. It was not possible to create a highly resolved curve fit for the limited precooler two-phase data (six points), so the fit is accomplished using a few assumptions that are based on observations from the recuperator conductance data. A second order polynomial (5 þ 290:1x2nd;pc  291:6x22nd;pc , shown bracketed by the ‘‘two-phase’’ label in Fig. 8a) is used to ensure that the curve exhibits a peak with respect to quality. The saturated vapor value for the two-phase correlation is specified as 3.5 W/K m to roughly intersect the average vapor values from Fig. 8b, and the saturated liquid value (quality = 0) is specified somewhat higher at 5 W/K m. The liquid value does not reflect a physical phenomenon but rather is chosen to be about 50% higher than the vapor value; this percentage is based on the ratio of the single phase liquid/vapor conductance values for the recuperator. Note that the significantly enhanced conductance in the two-phase region is attributed to condensation; the sensitivity of the precooler conductance to the condensation suggests that the 2nd stage vapor flow (rather than 1st stage refrigerant flow boiling) dominates the thermal resistance in the precooler.

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

The vapor conductance is a function of Reynolds number, so the transition between vapor and two-phase is more involved than with the recuperator. An exponentially decaying term (Eterm) is applied to transition between the ‘‘two-phase’’ and ‘‘vapor’’ terms between a quality of 0.96 and 1 as shown in Fig. 8. 3.4. Model evaluation 3.4.1. Empirically tuned model 3.4.1.1. Precooler/recuperator model. The empirical correlations for conductance and pressure drop in the precooler/recuperator were integrated with the component- and system-level models from [2] to evaluate the ability of the model to predict heat exchanger performance and overall refrigeration power. Including the empirical correlations allows for specification of the heat exchanger lengths, which better represents the physical constraints of the system compared to the pinch point temperatures specified in [2]. The model is evaluated using the experimental data set that was used to create the correlations. Ideally, this model would be validated against a different set of data (with different pressure ratios, flow rates, and mixture constituents); however, such an analysis is beyond the scope of this paper. The numerical precooler/recuperator models presented in [2] divide the heat exchangers into sections of equal energy exchange, and compute the temperature distribution and conductance given the temperature difference at one end (DTrec,hot = T4  T1 and DTpc,cold = T4  T11). These conductance values combined with the conductance correlations presented in Section 3.2 are used to compute the heat exchanger tube length; an iteration scheme is then used to adjust the temperature differences end (DTrec,hot and DTpc,cold) until the computed tube lengths match the actual tube lengths in the cryoprobe (Table 3). The tube lengths in the recuperator and precooler are computed respectively, according to:

Ltube;f ;rec ¼

N rec X UArec;i UArec ðxc;i Þ i¼1 Lrec

ð11Þ

599

relatively basic in order to simulate a design environment where very few measurements are available for the system. Inputs to the model from experimental data include the 2nd stage circulating mixture composition, mass flow, compressor suction and discharge pressures (P1 and P3), load temperature (T7), and high pressure inlet temperature (T3, which is nominally ambient temperature), as well as the 1st stage evaporator (precooler) saturation temperature (T8 = T11). Given this information the model can compute the cycle performance including all the thermodynamic state points in the 2nd stage, the precooler and recuperator conductances and pressure drops, and the refrigeration load. Fig. 9 compares the refrigeration power measured during the tests with the values predicted using the empirical model. The prediction for tests where the experimental heat input is less than 15 W is excellent – these include the tests with the pure refrigerants, and a number of tests with mixtures. As the heat input increases (e.g. points with 30–40 W), the agreement tends to become worse; these points include many of the tests that used the R14–R23 mixtures and resulted in performance that was better than the manufacturer’s original mixture as shown on pp. 173 of [40]. Differences between the experimentally measured and predicted refrigeration values can be attributed to component level modeling errors in the recuperator, precooler, and the jewel orifice. The differences between measured and predicted 2nd stage cycle operation were compared using P–h (pressure-enthalpy) diagrams for several different test cases [40] (pp. 219–225). The majority of the under-prediction in the refrigeration capacity observed in Fig. 9 for the larger loads (>15 W) can be attributed to an overly-conservative estimate of the precooler effectiveness. The next largest source of error is related to the under-prediction of recuperator effectiveness at low temperatures; this condition occurs when less heat is applied to the cycle so the recuperator spans most of the vapor dome. Refrigeration prediction errors that are caused only by inaccurate property data, as quantified by deviation from an isenthalpic expansion process from state 5 to 6, are the smallest at about 20–30%.

and:

Ltube;f ;pc ¼

N pc X UApc i¼1 Lpc

UApc;i ðRe3 x2nd;pc Þ

ð12Þ

where Nrec (60) and Npc (15) are the number of discrete heat exchanger sections, UAi and UApc,i are the conductances in each of the heat UA rec exchanger sections, and UA ðxc;i Þ and Lpcpc ðRe3 ; x2nd;pc Þ are the conLrec ductance correlations presented in Figs. 7 and 8. Furthermore, the pressure drop correlations are integrated into the heat exchanger models by applying an equal fraction of the pressure drop into each of the sections. The accuracy of the precooler/recuperator conductance models was evaluated by comparing the measured and predicted heat exchanger effectiveness. Effectiveness values were predicted generally within 10% for the recuperator, and 10–20% for the precooler [40] (pp. 207–209, 217–219). Inclusion of the pressure drop model (rather than using all measured values of pressure) did not significantly change the effectiveness predictions, indicating the pressure drop model is sufficient for the purpose of computing precooler/recuperator conductances. 3.4.1.2. System model. The empirically tuned precooler/recuperator models were included in the system-level model [2] to evaluate the ability of the model to predict refrigeration power at a specified load temperature (T7). The prediction is compared with the precisely controlled experimental thermal load applied between states 6 and 7, where T7 nominally represents temperature applied to the tissue for surgery. The model inputs are selected to be

3.4.2. Model comparisons The refrigeration performance for the cycle was also computed using two simpler models, including the isothermal enthalpy difference model and the pinch point model. These simple models described in the current MGJT literature do not account for transport phenomena (i.e. heat transfer and pressure drop) in the heat exchangers. The first model includes evaluating the minimum isothermal enthalpy difference (DhJT) over the operating temperature span of the recuperator (T4 to T7) as described in [40] (pp. 5–16). Pressure drop in the heat exchangers is not considered so the compressor suction and discharge pressures represent the high and low cycle pressures. The refrigeration effect is computed as: _ 2nd Q_ load ¼ m  min

h     i 2nd  h P high;2nd ;T; y 2nd for T ¼ T 4 to T 7 h Plow;2nd ;T; y ð13Þ

_ 2nd is the 2nd stage mass flow, T is the temperature, y 2nd is where m the 2nd stage mixture composition, Phigh,2nd and Plow,2nd are the discharge and suction pressures, and T4 and T7 respectively are the temperatures at the hot and cold inlets of the recuperator. The temperature at state 4 is computed by assuming a precooler cold end temperature difference (DTpc,cold = T4–T11). The precooler cold end temperature difference is specified as 2 K; this difference does not reflect any physical phenomena, but rather is based on the general observation that the experimental cold end temperature difference is typically about 1–2 K. The other inputs to the model from the

600

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

100

model empirical pinch point minimum hJT

90

Modeled refrigeration [W]

80

70

60 +20% +10%

50

exact -10%

40

-20%

The single component 2nd stage working fluid tests (distinguished from tests with mixtures in [40] (p. 222)) referred to as ‘‘Set 1’’ and ‘‘Set 2’’ in Table 2 had relatively low heat input power, about 15 W or less. For these tests the pinch point and DhJT models over-predict the refrigeration by 30–40%. Conversely, the empirically tuned model provided a prediction within 10–20% for the same data. In the same 0–15 W experimental heat input range, the simpler models grossly over-predict the refrigeration capacity by as much as 500%, whereas the empirically tuned model appropriately penalizes these mixtures for poor heat transfer performance. The data points with higher refrigeration values (20– 50 W) exclusively used mixtures in the 2nd stage (set 4 in Table 2), and the advantage of the empirically tuned model is somewhat diminished. The simpler models provide over-optimistic refrigeration predictions, but here the empirical model tends to under-predict the refrigeration, albeit to a lesser degree than the deviations with the simpler models.

30

3.5. Using the model as a mixture selection tool 20

10

0 0

10

20

30

40

50

Measured refrigeration [W] Fig. 9. Measured and predicted refrigeration power for the empirical model, the pinch point model, and the isothermal enthalpy difference model.

experimental data are the same as those presented for the empirical ‘‘system model’’ in the 1st paragraph of the ‘‘System Model’’ part of Section 3.4.1. The pinch point model presented in [2] is also compared to the empirical model. Similar to the isothermal enthalpy difference model, the pressure drop in the heat exchangers is neglected so all the 2nd stage cycle pressures before the jewel orifice are specified at the compressor discharge pressure (Phigh,2nd), and pressures after the orifice are specified at the suction pressure (Plow,2nd). The precooler cold end temperature difference (DTpc,cold) was specified as 2 K (the pinch point temperature changes the refrigeration power but not the optimal mixture, as discussed in [40] (p. 235). The recuperator heat transfer performance is specified by selecting a pinch point temperature difference (DTpp,rec); this value is necessarily a guess in this design environment where the detailed heat transfer performance of the heat exchanger is unknown. Specifying heat exchanger performance using the pinch point temperatures is a significant weakness of the pinch point model as this specification does not represent any physical parameters related to the working fluid or heat exchanger geometry. In addition to the thermodynamic state points, the pinch point model also computes the heat exchanger conductances (UArec and UApc), which can be used as a metric to minimize the cryoprobe size by maximizing Q_ load =UAtotal as described in [2]. The other inputs to this model from the experimental data are the same as those listed for the empirical model in Section 3.4.1. Fig. 9 compares the cryoprobe tip refrigeration measured in the experiment with the values predicted by each of the empirical, isothermal enthalpy difference, and pinch point models. The general trend observed in the comparison is that both the pinch point and DhJT models (referred to as the ‘‘simpler models’’) tend to over-predict the refrigeration. Comparatively, the empirically tuned model provides a more realistic but sometimes over-conservative estimate.

The empirical model was used to computationally investigate the optimization of a binary mixture of R14 and R23 applied to the cryoprobe system studied here; the optimization could be readily extended to include mixtures with more than two constituents to increase the probe capacity [8] by utilizing the optimization algorithm described in [2] rather than the parametric study presented here. The optimal refrigeration capacity is evaluated at various load temperatures for a set of nominal operating conditions that represent the typical operating conditions of the cycle. The operating conditions specified in Table 4 include the 2nd stage mass flow, suction and discharge pressures, and 1st stage evaporator temperature. Based on information from the cryoprobe manufacturer, the iceball size, rather than probe refrigeration, is used to judge the medical effectiveness of the instrument. Developing a model to select gas mixtures that optimize the refrigeration performance of the MGJT cycle at a given load temperature is a major component of the design process to maximize the iceball size produced by the cryosurgical probe. However, this model does not represent a complete design tool as the optimal tip temperature must still be determined. The mixture optimization model must be combined with a heat transfer analysis of the cryolesion (iceball) formation that considers: (1) the geometry of the active portion of the cryoprobe tip in contact with the tissue, (2) the thermal storage, phase change and heat transport properties of the tissue, and (3) the heat transfer from biological processes related to metabolic generation and blood perfusion [9,10]. This heat transfer analysis establishes important design benchmarks including the optimal cryoprobe tip temperature and the refrigeration capacity required to achieve a certain cryolesion size. Optimization of the cryoprobe tip temperature is beyond the scope of this paper, but examples can found in [9,10,40] (pp. 237–244). Fig. 10a shows the variation in refrigeration power at various load temperatures predicted by the empirically tuned model as

Table 4 Specified system operating conditions for the optimal binary mixture selection for the cryoprobe system studied here. Parameter

Value

Load temperature (T7) 2nd stage compressor discharge pressure (P3) 2nd stage compressor suction pressure (P1) _ 2nd ) 2nd stage mass flow (m 1st stage evaporation temperature (T8) Mixture constituents

170–210 K 289.5 psig 31.7 psig 0.0012 kg/s 241.5 K R14 and R23

601

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

Fig. 10. Mixture optimization using the empirical model compared to (a) the minimum DhJT model, and (b) the pinch point model (right scale).

Table 5 Optimal R14 compositions selected by the various models for the fixed geometry cryoprobe. Cryoprobe system model

DTpp DP

Empirical

Minimum DhJT

Pinch point

Pinch point

Pinch point

Pinch point

N/A Correlation

0 0

5 0

2 0

10 0

5 Correlation

50% 42% 38% 32% 14%

50% 44% 40% 32% 14%

50% 44% 40% 32% 14%

52% 44% 40% 32% 14%

78% 40% 38% 32% 22%

Load temperature (Tload) 170 K 82% 180 K 32% 190 K 28% 200 K 24% 210 K 20%

well as the simpler minimum isothermal enthalpy difference (DhJT) model. As shown before in Fig. 9, the DhJT model over-predicts the refrigeration capacity, so a design using this model would result in an underpowered probe. The optimal mixture compositions predicted by the models differ by 5–30% depending on the load temperature, where mixtures selected by the DhJT model do not perform well in the actual cycle (assuming the empirical model represents a more realistic estimate of performance). For example, the DhJT model selects a 32% R14 mixture as the optimum for a 200 K load temperature. A 32% R14 mixture is predicted by the empirical model to produce about 20 W of refrigeration at 200 K; this is a significant reduction from the optimal mixture of 22% R14 selected by the empirical model that yields 60 W. This observation, along with the over-prediction of capacity, highlights the importance of incorporating the heat transfer and pressure drop behavior in the mixture optimization model. Table 5 summarizes the optimal compositions selected by these two models as well as the variations of the pinch point model [40] (pp. 231–237). The load curves from the empirical model were also compared with the load curves from the pinch point model described in Section 3.4.2 and [2]. Mixture selection using the pinch point model is carried out by maximizing the cryoprobe compactness target (Q_ load =UAtotal ) rather than the refrigeration capacity, as the UAtotal is not known a priori with the pinch point model. The precooler and recuperator performances are defined by pinch point temperature differences, respectively specified as 2 K and 5 K. Specifying the pinch point temperature differences effectively defines the lengths of the heat exchangers, where the lengths are different than the actual tube lengths in the cryoprobe. The pinch point model is therefore not strictly comparable to the empirical model; however, comparing these methods is still useful for illustrating

the differences in optimal mixtures selected by the two models. Fig. 10b compares the refrigeration capacity predicted using the empirical model with the Q_ load =UAtotal predicted by the pinch point model as a function of R14 fraction. The pinch point model predicts the optimal mixture will have less R14 for the 170 K and 210 K tip temperatures, whereas the empirical model predicts the optimal mixture will have less R14 for the 180 K, 190 K and 200 K tip temperatures. Variations to the pinch point model were explored using recuperator pinch point temperatures of 2 and 10 K, as well as the inclusion of the pressure drop model discussed in Section 3.1. The optimal mixtures for each of these variations are also displayed in Table 4. The optimal mixture composition did not significantly change with pinch point temperature, but including the pressure drop model yielded optimal mixtures much closer to those selected by the empirical model. It is very time-consuming to perform the experimental testing described here, so there would be a considerable advantage to using the pinch point model (rather than the empirical model) with a pressure drop model to select the mixtures. Evaluation of this method is beyond the scope of this paper. Note that a more detailed discussion of the pinch point model analysis is found in pp. 233–237 of [40]. 4. Conclusion The test facility presented here was used to study the performance of multi-phase mixtures in the components of a compact cryocooler applied to cryosurgery. The measurements show a significant deviation from the idealized performance (predicted by the isothermal enthalpy difference model) related to pressure drop and finite recuperator/precooler conductance as shown in Fig. 3. The experimental measurements were used to develop empirical

602

H.M. Skye et al. / Cryogenics 52 (2012) 590–603

but physics-based correlations to predict pressure drop and conductance in the heat exchangers; these correlations were integrated with the system model developed in [2] so that transport phenomena can be included in the mixture optimization. The predictive capability of empirical model was evaluated using experimental data and compared with predictions from the isothermal enthalpy difference and pinch point models (i.e. the ‘‘simpler models’’). The empirical model was shown to appropriately penalize mixtures with very low refrigeration (<15 W) related to poor heat transfer or high pressure drop, where the simpler models did not. For mixtures with larger refrigeration power (15–50 W), the empirical/simpler models respectively under-predict/over-predict refrigeration power, where the empirical model had somewhat better accuracy. The three models were used to optimize a binary mixture of R23/R14 for the cryoprobe system studied here. A comparison of the models shows that the transport processes, considered in the empirical model, cause the optimal mixture to be relatively rich in the high-boiling component (R23 boils at 191.2 K, compared to R14 at 145.2 K; see pp. 251–252 of [40] for expanded discussion of how each constituent contributes to the heat transfer, JT effect, and ultimately the overall performance of the mixture). A higher R23 concentration is less favorable in terms of thermodynamic properties, as the isothermal enthalpy difference over the cycle temperature span is lower (and subsequently the idealized refrigeration power is lower). However, the higher R23 concentration raises the dew point of the mixture so larger portions of the heat exchangers are occupied by two-phase flow with lower velocities (leading to lower pressure drop) and high heat transfer coefficients [21–23]. The fraction of the recuperator dedicated to cooling the mixture to the dewpoint decreases and leaves more room for the two-phase section where most of the total energy exchange occurs; this increased recuperator performance reduces the system performance bottleneck related to poor heat transfer in the recuperator. The refrigeration power will subsequently be closer to the ideal performance predicted by the isothermal enthalpy difference of the mixture. The research carried out here is both costly and time consuming and may not be practical for a commercial environment. An alternative approach may be to use the isothermal enthalpy difference model to select an initial mixture, and then incrementally increase the high- and mid-boiling components to remove any performance bottleneck related to poor heat transfer caused by dry-out. Acknowledgements The authors would like to thank ASHRAE for funding this research project (Project 1472-RP). Additional support provided by American Medical Systems and the Wisconsin Space Grant Consortium is gratefully acknowledged. Technical assistance by Mike Perkins significantly contributed to the construction and troubleshooting of the experimental test facility. Many undergraduate students contributed to the progress of this project, grateful acknowledgements are given to Jeremiah Osborn, Jake Kilbane, Greg Marsicek, Joe Jaeckels, Ben Cox, and Branden Krause. References [1] Rubinsky B. Cryosurgery. Annu Rev Biomed Eng 2000;2:157–87. [2] Skye HM, Klein SA, Nellis GF. Modeling and optimization of a two-stage mixed gas Joule–Thomson cryoprobe system (RP-1472). ASHRAE transactions paper TRNS-00196-2008; 2008. [3] Radebaugh R. Recent development in cryocoolers. 19th international congress of refrigeration; 1995. p. 973–90. [4] Boiarski. Retrospective of mixed-refrigerant technology and modern status of cryocoolers based on one-stage, oil-lubricated compressors. Adv Cryog Eng 1998;43:1701–8.

[5] Marquardt ED, Radebaugh R, Dobak J. A cryogenic catheter for treating heart arrhythmia. Adv Cryog Eng 1998;43:903–10. [6] Dobak J, Xiaou Yu, Kambriz G. A novel closed loop cryosurgical device. Adv Cryog Eng 1998;43:897–902. [7] Alexeev A, Haberstroh Ch, Quack H. Mixed gas J–T cryocooler with precooling stage. Cryocoolers 1999;10:475–9. [8] Maytal B, Nellis GF, Klein SA, Pfotenhauer JM. Elevated-pressure mixedcoolants Joule–Thomson cryocooling. Cryogenics 2006;46:55–67. [9] Fredrikson K. Optimization of cryosurgical probes for cancer treatment. M.S. thesis. Madison, WI USA: University of Wisconsin – Madison, Mechanical Engineering Dept. ; 2004. [10] Fredrikson K, Nellis G, Klein SA. A design method for cryosurgical probes. Int J Refrig 2006;29:700–15. [11] Rozhentsev A. Refrigerating machine operating characteristics under various mixed refrigerant mass charges. Int J Refrig 2008;31:1145–55. [12] Gong MQ, Luo EC, Zhou Y, Liang JT, Zhang L. Optimum composition calculation for multicomponent cryogenic mixture used in Joule–Thomson refrigerators. Adv Cryog Eng 2000;45:283–90. [13] Keppler F, Nellis G, Klein SA. Optimization of the composition of a gas mixture in a Joule–Thomson cycle. HVAC&R Res 2004;10:213–30. [14] Gong MQ, Wu JF, Luo EG. Performances of the mixed-gas Joule–Thomson refrigeration cycles for cooling fixed-temperature heat loads. Cryogenics 2004;44:847–57. [15] Gong M, Ercang L, Zhou Yuan. Research on a mixed-refrigerant J–T refrigerator used for cryosurgery. In: Proceedings of the eighteenth international cryogenic engineering conference (ICEC 18), Mumbai, India; 2000. p. 571–4. [16] Venkatarathnam G. Cryogenic mixed refrigerant processes. In: Timmerhaus KD, editor. International cryogenics monographs series. New York (NY): Springer Publishing; 2008. [17] Nayak HG, Venkatarathnam G. Performance of an auto refrigerant cascade refrigerator operating in liquid refrigerant supply (LRS) mode with different cascade heat exchanger. Cryogenics 2010;50:720–7. [18] Bioarski M, Khatri A, Kovalenko V. Design optimization of the throttle-cycle cooler with mixed refrigerant. Cryocoolers 1999;10:457–65. [19] Alexeev A, Theil A, Haberstoh Ch, Quack H. Study of behavior in the heat exchanger of a mixed gas Joule–Thomson cooler. Adv Cryog Eng 2000;45:307–14. [20] Gong MQ, Wu JF, Luo EC, Qi YF, Hu QG, Zhou Y. Study on the overall heat transfer coefficient for the tube-in-tube heat exchanger used in mixed-gas coolers. Adv Cryog Eng 2002;47:1483–90. [21] Hughes CB, Nellis GF, Pfotenhauer JM. Measurement of heat transfer coefficients for non-azeotropic hydrocarbon mixtures at cryogenic temperatures. In: 2004 ASME international mechanical engineering congress and exposition, IMECE; 2004. p. 415–22. [22] Hughes CB. Experimental measurement of heat transfer coefficients for mixed gas working fluids in Joule–Thomson systems M.S. thesis. Madison, WI USA: University of Wisconsin – Madison, Mechanical Engineering Dept.; 2004 [23] Nellis G, Hughes C, Pfotenhauer J. Heat transfer coefficient measurements for mixed gas working fluids at cryogenic temperatures. Cryogenics 2005;45(8):546–56. [24] Little WA. Heat transfer efficiency of Kleemenko cycle heat exchangers. Adv Cryog Eng 2008;53:606–13. [25] Hwang G, Sangkwon Jeong. Pressure loss effect on recuperative heat exchanger and its thermal performance. Cryogenics 2010;50:13–7. [26] Gong M, Zhou W, Wu J. Composition shift due to different solubility in lubricant oil for multicomponent mixtures. Cryocoolers 2007;14. [27] Gong M, Deng Z, Wu J. Composition shift of a mixed-gas Joule–Thomson refrigerator driven by an oil-free compressor. Cryocoolers 2007;14. [28] Gong MQ, Wu JF, Qi YF, Hu G, Zhou Y. Research on the change of mixture compositions in mixed-refrigerant Joule–Thomson cryocoolers. Adv Cryog Eng 2002;47. [29] Reddy RK, Srinivasa MS, Venkatarathnam G. Relationship between the cooldown characteristics of J–T refrigerators and mixture composition. Cryogenics 2010;50(6–7):421–5. [30] Lakshimi N, Venkatarathnam G. A method for estimating the composition of the mixture to be charged to get the desired composition in circulation in a single stage JT refrigerator operating with mixtures. Cryogenics 2010;50:93–101. [38] Lemmon EW, Huber ML, McLinden MO. NIST reference fluid thermodynamic and transport properties – REFPROP. 2007, 8.0. . [39] Ely JF. Huber ML. NIST thermophysical properties of hydrocarbon mixtures database (SUPERTRAPP). 1992, 3.2. . [40] Skye HM. Modeling, experimentation, and optimization for a mixed gas Joule– Thomson cycle with precooling for cryosurgery. PhD thesis. Madison, WI, USA: University of Wisconsin – Madison, Mechanical Engineering Dept. . [41] Skye HM, Klein SA, Nellis GF. Experimental apparatus for measuring the performance of a precooled mixed gas Joule Thomson cryoprobe. ASHRAE transactions paper TRNS-86020-2011, 2011, submitted to the 2011 ASHRAE Conference in Las Vegas, NV. [42] Klein SA. EES – Engineering equation solver, v8.401, f-Chart Software; 2011. . [43] Klein SA. EES-NIST4 and EES-REFPROP interface routines, f-Chart Software; 2011. .

H.M. Skye et al. / Cryogenics 52 (2012) 590–603 [44] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 4th ed. New York: John Wiley & Sons; 2002. [45] Klein SA, Nellis GF. Heat transfer. 2nd ed. New York (NY): Cambridge University Press; 2009. p. 851. [46] Timmerhaus KD, Schoenhals RJ. Design and selection of cryogenic heat exchangers. Adv Cryog Eng 1974;1:445–62. [47] Passow KL. Empirical model improvements for a mixed gas Joule–Thomson cycle with precooling for cryosurgery. M.S. thesis. Madison, WI, USA: University of Wisconsin – Madison, Mechanical Engineering Dept., p. 55–61. . [48] Wambsganss MW, France DM, Jendrzejczyk JA, Tran TN. Boiling heat transfer in a horizontal small-diameter tube. Trans ASME J Heat Transfer 1993;115(4):963–72.

Further Reading [31] Huber ML, Ely JF. A predictive extended corresponding states model for pure and mixed refrigerants including an equation of state for R134a. Int J Refrig 1994;17(1):18–31.

603

[32] Klein SA, McLinden MO, Lasecke A. An improved extended corresponding states method for estimation of viscosity of pure refrigerants and mixtures. Int J Refrig 1997;20(3):208–17. [33] McLinden MO, Klein SA, Perkins RA. An extended corresponding states model for the thermal conductivity of refrigerants and refrigerant mixtures. Int J Refrig 2000;23:43–63. [34] Khatri A, Boiarski M. Development of JT coolers operating at cryogenic temperatures with nonflammable mixed refrigerants. Adv Cryog Eng 2008;53:3–10. [35] Naer V, Rozhentsev A. Application of hydrocarbon mixtures in small refrigerating and cryogenic machines. Int J Refrig 2002;25:836–47. [36] Podtcherniaev O, Boiarski M, Flynn K. Performance of throttle-cycle coolers operating with mixed refrigerants designed for industrial applications in a temperature range of 110 to 190 K. Adv Cryog Eng 2002;47:863–72. [37] Bioarski M, Khatri A, >Podtcherniaev O, Kovalenko V. Modern trends in designing small-scale throttle-cycle coolers operating with mixed refrigerants. Cryocoolers 2001;11:513–21.