Simulation and Evaluation on The Dynamic Performance of a Cryogenic Turbo-based Reverse Brayton Refrigerator

In this study, a cryogenic turbo-based refrigerator, with high-speed foil gas bearing turboexpander and plate-fin heat exchanger, was presented. For the cooling process of a cryogenic refrigerator is long and complex, the efficient simulation of the dynamic performance is vital for its evaluation and operation. It is difficult to achieve this purpose. Because the cryogenic refrigerator will go through great temperature changes, resulting in dynamic performances of components going to complexity. In this article, the turboexpander was simulated using CFX, and the matching models with expansion works of consumption and utilization were unified and modified. With numerical calculation of heat exchanger, the dynamic simulation model of the refrigerator was improved via MATLAB code. The calculation method of the required input energy of the refrigerator in cooling processes was proposed based on the simulation and integration. The energy efficiency and economy cost of the refrigerator were evaluated under different operation modes. The refrigerator with a motor was more stable than the one with a blower. The refrigerator was tested under different conditions, the results of which agreed well with simulation data. The simulation method is efficient to serve practical applications, improve energy efficiency and reduce costs.


Introduction
The cryogenic turbo-based refrigerator, which consists of the reverse Brayton cycle, Claude cycle, and Kapitza cycle etc., is extensively applied in many areas, such as the controllable nuclear fusion [1], air separation and liquefaction [2], Zero Boil Off liquid hydrogen storage [3], low temperature environmental simulator [4], superconducting [5,6], food industry [7,8], and heat pump [9].Because of the low operational temperature, the thermal capacity of system components, and so on, the cooling process of a cryogenic turbo-based refrigerator is quite long and complex [4,7].Moreover, the cryogenic refrigerator usually operates under varying conditions, which makes the dynamic performance more complex and unpredictable.Therefore, an efficient simulation and evaluation method is quite necessary for the turbo-based refrigerators, which can help evaluating, and optimizing, the cryogenic refrigerator; and improving the energy efficiency and energy economy.The turbo-based reverse Brayton refrigerator is a basic and important form of cryogenic turbo-based systems [10], it was chosen as the objective to research the dynamic cooling characteristics.
Many experiments and simulations have been conducted to research the dynamic performances of turbo-based systems.Spence et al. [11] tested an air refrigeration system employing a turboexpander compressor for road transport under several conditions.Zhao et al. [12] experimentally studied the dynamic performance of a gas bearing turboexpander for conditioner in aircraft, which achieved high efficiency.Hirai et al. [5,13] manufactured a neon reverse Brayton refrigerator for superconducting applications.They investigated the dynamic cooling characteristics under different conditions, but further analysis was relatively rare.Yang et al. studied dynamic characteristics of the turboexpander with brake blower [4] and turboexpander compressor [14] respectively.Then they proposed a transient cooling model of the reverse Brayton refrigerator and analyzed the refrigerator cooling performance under several typical operational modes [15].The matching analyses were separate, and the system efficiency and economy were not clarified.
Some dynamic simulations of the turbo-based refrigerator were also done.The refrigerator mainly includes many components, which interact with each other mutually.Many working processes and dynamic characteristics of components are simplified for easy of calculation.In some models, the steady-state calculation of heat exchanger was adopted for dynamic simulation, resulting in great errors [16,17].Ahmadi et al. [18] conducted thermal optimization of a cryogenic system using the genetic algorithm.In the research, the heat exchanger was researched in detail, but the performance of expander was simplified.Fazlollahi et al. carried out costs and exergy analysis for cryogenic refrigerators [19,20].In their research, the turboexpander efficiencies were set as constant.Through thermodynamic simplification, Michelsen et al. [21] proposed a dynamic system model for the control of the TEALARC process.Zhan et al. [22] carried out a simulation of a cryogenic N2 expansion cycle.The calculation of heat exchanger was illustrated detailly, but the matching performance of turboexpander was simplified.
In these works, the matching characteristics of turboexpander were usually simplified for easy simulation or specific to a certain type (e.g., a brake blower).Moreover, the dynamic calculation of energy efficiency and economic cost of different modes were rare, especially the quantitative evaluation method.The turboexpander is the main device that influences the refrigerator greatly.It is quite an efficient equipment to gain low temperature, and is one of the main advantages for turbobased refrigerators.Thus, it is necessary to establish a dynamic simulation and evaluation method, which could evaluate the system energy efficiency accurately and consider the turboexpander matching characteristics completely.
In this study, the dynamic characteristics of turboexpander were resolved into the energy consumption and energy recovery.The expansion process was simulated using Ansys CFX.The matching characteristics of turboexpander with a motor were analyzed in detail, which was compared with the characteristics of energy consumption.Through analysis, different matching modes were unified, and the turboexpander matching model was modified and improved.Then based on the transient model, a dynamic model of different operation modes was modified via MATLAB code.The quantitative evaluation method of required power in the cooling process was given.The energy efficiency and economic costs under varied modes were clarified.The starting characteristics of turboexpander together with the cooling performance of the refrigerator were simulated and analyzed.The changing input power of the refrigerator was calculated and evaluated under different rotating speeds of turboexpander.The dynamic simulation conclusion was applied in the experiment and simulated results were compared with experimental data.Test apparatus of the refrigerator were improved with a separate brake cycle.The aims of this work were to simulate and evaluate the dynamic matching and refrigerating performance of the cryogenic turbo-based reverse Brayton refrigerator, provide a basis for the evaluation and operation of refrigerators, and allow for improving the energy and economy efficiencies.

Process Analysis
The flow process and T-s diagram of the cryogenic refrigerator are presented in Figure 1.According to whether the expansion work is used, the refrigerator could be classified as two types: Energy consumption and energy recovery, as shown in Figure 1a.Considering the operational temperature ranges, the refrigerator could be divided into two parts: The normal temperature part and the low temperature part.As shown in Figure 1a, the low temperature part mainly consists of a heat exchanger, a cold load, and an expander (cold side of turboexpander), which are the same in refrigerators of energy recovery and consumption.The other components belong to the normal temperature part.There are somewhat different in two types.As shown in Figure 1b, in the refrigerator with energy consumption, a brake blower or an eddy current brake absorbs and just consumes the expansion work.As shown in Figure 1c, in the refrigerator with energy recovery, a booster compressor or a motor absorbs the expansion work, and the output energy is recycled.Under the same energy input of the front compressor, the recycled energies were different in two operation modes.Also, there are big differences between the dynamic matching characteristics of turboexpander, resulting in complex variations of cooling performance.

Test Apparatus
As shown in Figure 2, the schematic view of the test rig is presented.The system used air as the working fluid and run in an open cycle.The normal temperature part is divided as the air supply section and the brake section.The screw compressor can provide 260 Nm 3 h -1 compressed air at 1.00 MPa.The cooling and filtering devices (a filter, a freezing dryer and a molecular sieve) are applied to cool down the high temperature gas and remove the water and oil in the system.The cold box has a vacuum degree of 0.01 Pa, consisting of a cryogenic plate-fin heat exchanger, a cold load, and pipes.The expander was the same as the one shown in Reference [15].It has a diameter of 24.00 mm, and was supported by foil air bearings.The expander inlet operates in the range of 0.300-0.600MPa, and the outlet is slightly higher than environmental pressure.The design isentropic expander efficiency is 0.578.The heat exchanger is a plate-fin regenerator and the design effectiveness is 0.930.The low temperature part and the air supply section are similar to those in References [4,15].While, the brake section is improved and quite different from the previous one in References [4,15].In the present study, the brake section is an independent and complete circuit.As shown in Figure 2, a centrifugal blower is applied to make the pressure in circulation can meet the demand of tests.When the required brake pressure is high, the valve is switched and the centrifugal blower runs to raise the pressure.When the brake pressure is low, the centrifugal blower stops working.The brake gas is not supplied by the air supply section any more [4].It is a benefit for the stability of the system and accurate calculation of energy efficiency.The blower has a diameter of 27.00 mm.Based on the NI data acquisition board and the LabVIEW program, a real-time data acquisition, calculation, and display system was developed for the effectively dynamic control of the refrigerator.
In the experiment, the temperatures were measured by platinum resistance thermometers and the rotating speed was measured by an eddy current sensor.The measure range and accuracy of tested parameters were listed in Table 1.

Simulation and Evaluation Method
In Reference [15], a transient cooling model of revers Brayton refrigerator was built, which used the method of dual non-steady time steps to connect dynamic characteristics of the turboexpander and cryogenic heat exchanger.Similar to the reverse Brayton refrigerator, the normal temperature part of the cryogenic turbo-based system operates under relatively stable boundary conditions, and the dynamic performance of the system is mainly affected by the low temperature part.In a cryogenic turbo-based system, with related component model inserted in the transient model, the dynamic performance of the system could be simulated and predicted.
As a critical component of the turbo-based refrigerator, matching performances of the turboexpander with a blower were researched and the mathematical model was built in References [4,15].The working power and rotating speed are two equilibrium parameters between the expander and blower.The relations between the expander and blower could be presented as the following [4,15]: As shown in the above equations, M is nearly constant for a certain refrigerator, and T at this time.Also, the constant brake means that the expansion work is certain.The expansion work is as below: Therefore, with a motor to recycle the output energy, the essential of the matching characteristics is also to obtain the performance curve.ANSYS CFX was performed to simulate the expansion performance, and the Shear Stress Transport model was applied to resolve the turbulence term [15].The working fluid was the real air.The inlet boundary conditions were the mass flow rate and static temperature, and the outlet one was the static pressure.The temperature contours from CFX simulations of the expander under the design point are presented in Figure 3.The gas was first expanded in the nozzles and then further expanded and cooled in the expansion wheel.In the diffuser, the temperature increased a bit, which was due to the diffusion effect.As shown in the temperature contours, the variations of temperature in the expansion process were uniform.According to the analysis and the CFX simulation, the unified turboexpander dynamic model could be established, as shown in Figure 4.The flow chart of the simulation method is shown in Figure 5.With a brake of energy consumption, the simulation method is the same as the one in Reference [15].In the dynamic cooling process, the total computation time is tt, and the 1 st time step is t .In the 1 st time step,   is divided as the 2 nd time step.In t , inlet temperature of the heat exchanger at the cold side is assumed to be equal to the expander outlet temperature, and they are considered as constant.Then the transient convection heat transfer in the heat exchanger is calculated by   in t .At the end of t , the outlet temperature at the hot side of the heat exchanger is obtained through simulation.That is, the expander inlet temperature is got.Then, the expander outlet temperature could be calculated according to the matching model and the transient state in next t could be simulated later.The heat exchanger is calculated using the method of numerical heat transfer and computational fluid dynamics by considering both the heat conduction and cold loss [15,24].
While with a brake of energy recovery, the inlet pressure will also vary during the cooling process.Therefor the cooling model is modified, as shown in Figure 5. Through the dynamic matching model, the required expander inlet pressure was calculated and then input as the boundary parameter of the heat exchanger.
With a brake of energy consumption, the purpose is to obtain better refrigerating performance.Under certain brake power or expander efficiency, the characteristics of the expander and refrigerator could be simulated.With a brake of energy recovery, both the refrigerating and energy performances recycle should be considered.When ensuring the cooling effect, the energy input should be reduced as possible to improve the energy and economy efficiency.In a cryogenic refrigerator, the required power is changing during the cooling process, which could be calculated through the simulation model.Then the total power could be obtained by integration.For easy evaluation, the mass flow rate is specified.The work per unit of cold production could be calculated using the following function: where t is the cooling time.

Simulation of Energy Consumption
In the cryogenic system, such as the controllable nuclear fusion [1], the expansion work of the turboexpander was usually consumed.Because the operation temperature is quite low, the total energy input of the system is much bigger than the expansion work.In these systems, reliability and adjustability needs to be guaranteed firstly.With the energy consumption, the flow process of the turboexpander becomes simple and the turboexpander is easy to regulate.There are several kinds of energy consumption methods, consisting of the oil brake, eddy brake and brake of a blower.The specific brake characteristics of these methods were somewhat different, but the approaches and goals are similar.The rotating speed of the turboexpander was regulated through changing the brake power, resulting in high expansion efficiency.Because the blower is easy to adjust, different brake characteristics could almost be simulated by a blower.
In a cryogenic refrigerator, as the turboexpander with the brake of a blower, there are two commonly used methods: Certain brake pressure and certain expander efficiency.The method of certain brake pressure has advantages of reliability, stability, simple process and low economy cost.Whereas, with this method, the turboexpander is unable to adjust and can only adapt to change with the operation parameters of the cold end.With certain expander efficiency, additional devices are required, and the economy cost increases.For example, a brake circuit, which is composed of a buffer, control valve, pipes and coolers, is necessary, as shown in Figure 2.But the advantages are obvious, it means the cooling capacity and energy efficiency can be greater under varied conditions.Both the two conditions were simulated, and the cooling performance and energy efficiency are presented and compared.The simulations were conducted with heat exchanger inlet pressure of 0.400 MPa, atmospheric pressure and temperature of 0.100 MPa and 300.0 K.
As shown in Figure 6, the dynamic cooling characteristics of the refrigerator under different operation modes are presented.In the beginning, the cooling speed of the refrigerator with optimal expander efficiency is significantly faster than those under certain brake pressure.As the cooling goes on, the cooling speeds of different conditions became similar, as shown in Figure 6b.The expander efficiency under brake pressure of 0.100 MPa increases first and then decreases, and it reaches the optimal value at the 18th min.The expander efficiency under brake pressure of 0.118 MPa reaches the optimal value at the 36th min.In most of the cooling period, the expander efficiency under brake pressure of 0.118 MPa is higher than the one under brake pressure of 0.100 MPa, and is closer to the optimal one.As shown in Figure 6d, the refrigerator with this brake pressure also has higher effectiveness of heat exchanger.
More working conditions were simulated to evaluate the operation modes, as shown in Figure 7.The lowest ultimate refrigerating temperature is achieved under optimal expander efficiency, and the value is 109.7 K.With the increase of brake pressure, the ultimate temperature decreases first and then increases.It comes close to the lowest one in the brake pressure range of 0.112-0.124MPa, as shown in Figure 7a.The ultimate temperature is 111.1 K at the brake pressure of 0.118 MPa.The highest ultimate temperature of the refrigerator is 113.0 k, which is obtained at the brake pressure of 0.100 MPa.The required times for the refrigerator to achieve this value were compared under different conditions, as shown in Figure 7b.When pB = 0.100 MPa, it takes 148 min for the refrigerator to reach 113.0 K.Under optimal expander efficiency, the required time is 54 min shorter than the longest one.The shorter cooling time means the shorter energy and economy cost.When pB is around 0.118 MPa, the cooling time is quite near the shortest one and the deviation is about 10 min.Form Figures 6 and 7, some conclusions could be conducted.First, the cryogenic refrigerator will achieve the best dynamic and ultimate cooling performance under the method of optimal expander efficiency, which could be achieved through the feedback control [4].Second, when adopting a proper brake pressure, the refrigerator might still achieve a satisfactory cooling effect under control of constant brake pressure, which is very close to the best result of running under optimal expander efficiency.Although the refrigerating ability is less capable than the optimal one, it has advantages of low cost, stability and easy operation.In the actual application, a separate brake circuit is required, and the proper value under different conditions could be gained through the simulation.

Simulation of Energy Recovery
To recycle the energy, the expansion work could be absorbed by a motor or a compressor.The common advantage of the two modes is that the output energy is recycled, and the energy efficiency is raised.While, the dynamic characteristics of the turboexpander and refrigerator are quite different under the two operation modes.As a motor is employed, the rotating speed of turboexpander is fixed, which means the expansion work is also fixed.With a compressor applied, the expansion work is absorbed by the coupled compressor, which raises the pressure of gas before flowing into the expander [14].The rotating speed and other coupling parameters might vary during the cooling process, which was presented in detail in Reference [14].As a result, the energy and economic efficiencies under two modes were also different.The energy and economy efficiency of the energy recycle method with a motor will be studied in detail in the present work.
As shown in Figure 8, the dynamic characteristics of the refrigerator under different rotating speeds are presented, both of which operated at the expansion work of 800 W. As shown in Figures 8a and 8b, the cooling curves are much smoother than those of the refrigerator with certain brake pressure, which means the cooling process is more stable.With fixed expansion work and rotating speed, the variations of expansion parameters might be more controllable, resulting in the slow change of the refrigerator.That is, the more constrains, the less uncertainty.It also can be seen that, with a lower rotating speed, the temperature declines faster first and then slower.As shown in Figure 8c, the heat exchanger effectiveness was nearly the same at the beginning of the cooling process, and increased with the rotating speed later.Although a higher rotating speed was related to higher effectiveness, the refrigerator has poor performance.This shows that the expander efficiency played a more important role in this refrigerator.As shown in Figure 8d, the ultimate temperature of the refrigerator decreases with the increase of rotating speed.The same expansion work represents the same energy recycled.Whereas, the refrigerating temperature and cooling capacity were different.At this expansion work, the greater rotating speed, which is corresponding to higher effectiveness of heat exchanger, results in relatively weaker cooling performance finally.The changing trend is a comprehensive reflect of dynamic characteristics of both the heat exchanger and turboexpander.
As shown in Figure 9, the expansion performances are illustrated.Under rotating speed of 170 krpm, the expansion ratio is much higher at the beginning, and it is corresponding to a lower expansion efficiency.With the cooling of the refrigerator, the expansion ratio decreases first and then increases.The expansion efficiency increases until to the optimal value, and then decreases.But during the most period, the expansion efficiency is relatively high.That's why at this speed, the refrigerator reveals a better cooling performance.As the rotating speed decreases, the variation tendencies of the expansion ratio and efficiency begin to change obviously.The starting pressure of expander becomes smaller and the turning point gradually disappears.The expansion ratio increases and efficiency declines, which leads to less satisfactory performance.Although the refrigerator may achieve good performance under lower rotating speed, the starting pressure is quite high and changes dramatically in the beginning.Those were unfavorable to the stability and reliability of the cryogenic refrigerator.Both the cooling effect and operational performance should be considered in the actual application.The required compression work of the front-end compressor was also analyzed.The expansion ratio nearly equals the lowest pressure ratio that the front-end compressor (e.g., screw compressor in Figure 2) must provide.The required input energy of the cryogenic refrigerator could be calculated through the integration of compression work using Equation (5).Assuming the compressor operates under =0.60 C  and the mass flow rate of 0.03 kg s -1 , the input compression works under varied rotating speed were calculated and presented in Figure 10.The curves of compression work are similar to those of the compression ratio (expansion ratio in Figure 9), but the change trends vary less dramatically.Because there is an exponential relationship with them and the index is less than 1.0, as shown in Equation ( 4).In the figure of compression work with time, the fill area under the curve represents the integration of compression work with time, which is the total energy input in the cooling process, as shown in Figures 10b, 10c and 10d.The total input energies are 71861.1,75110.3, and 78265.3kJ at the rotating speed of 170, 190, 210 krpm respectively.Also, the differences are not as large as those of the expansion ratios.

Experimental Study
Experiments were carried out to investigate the dynamic performance and verify the simulation method.The cooling characteristics of the refrigerator with energy consumption were studied experimentally.The operating parameters of the tests and simulations are listed in Table 2.In two tests, the refrigerator run under the same expansion ratio and boundary conditions.With optimal expansion efficiency, the expander operated under optimal characteristic ratio through the feedback control [4].With certain brake pressure, the blower inlet pressure was fixed in the cooling process.Each experimental data was the mean of three replicates.11, the simulated results agreed well with the experimental data under two operation modes.The dynamic simulation method and analysis in Section 3 were verified.The refrigerator in test 1 had a faster cooling speed than that in test 2. The refrigerator ultimate temperature stabilized at 115.3 K in test 1 finally while it stabilized at 117.2 K in test 2. The ultimate temperatures in simulations 1 and 2 were 110.8 and 112.1 K, respectively.The relative deviations of the ultimate temperature in simulations 1 and 2 were 3.9% and 4.5%.The refrigerator achieved a better cooling effect under optimal expansion efficiency.But with a proper constant brake pressure, the refrigerator could still obtain fairly good results.It verified the conclusions in Section 4.1 and showed that the simulation model can be applied in actual applications.
There were errors between the simulated and tested results.That is because the models of expander and heat exchanger both adopted some simplifications and assumptions.To simplify the calculation, the heat exchanger was simulated using a one-dimensional method, which caused the deviation.Also, the expansion process was considered as quasi-steady during the cooling process.These resulted in the limitations of the simulation method.The simulation method is more suitable to be applied in the working condition without dramatic change.The simulated results will be satisfactory for a long running turbo-based refrigerator.

Conclusions
This paper performed dynamic analysis of a cryogenic reverse Brayton refrigerator.The turboexpander was simulated using CFX, and the matching models with expansion works of consumption and utilization were unified and modified.The dynamic simulation model of the turbobased refrigerator was improved via MATLAB code.The calculation method of total input energy of the refrigerator in the cooling process was proposed based on the simulation and integration.
When adopting a proper brake pressure, the refrigerator might still achieve a satisfactory cooling effect and economic cost under the control of constant brake pressure, which is very close to the best result of running under optimal expander efficiency.The ultimate temperature at pB = 0.118 MPa is about 1.4 K above the lowest one achieved under feedback control.
The cooling process of the refrigerator with a motor is more stable than the one with a brake blower.With a motor as a brake, the variations of expansion parameters might be more controllable, resulting in the slow change of the refrigerator.The refrigerator may achieve better refrigerating performance and less energy consuming under lower rotating speed, but the starting pressure is quite high and changes dramatically in the beginning.Both the cooling effect and operational stability should be considered.
The experiments show that the simulation and evaluation method can be applied to the actual application.The simulation method is more suitable to be applied in the working condition without dramatic changes.It can be used to predict and evaluate the operation and control scheme of cryogenic turbo-based systems, and help to promote energy efficiency and reduce economic costs.

Figure 1 .
Figure 1.Flow process and T-s diagram of a cryogenic reverse Brayton refrigerator: (a) Flow process; (b) T-s diagram of energy consumption; (c) T-s diagram of energy recovery.

Figure 2 .
Figure 2. Schematic view of the turbo-based air refrigerator.

Figure 3 .
Figure 3. Temperature Contours of the expansion process under E  =4.53, 1 / s u c =0.671: (a) Temperature contour in the nozzles and wheel; (b) temperature contour in the diffuser.

Figure 4 .
Figure 4.The unified matching model of turboexpander.

Figure 6 .
Figure 6.Comparisons between different conditions: (a) Cooling characteristics; (b) cooling characteristics in the later period; (c) variation of expander efficiency; (d) variation of heat exchanger effectiveness.

Figure 7 .
Figure 7. Comparisons between different conditions: (a) Ultimate refrigerating temperature; (b) the cooling time to 113.0 K under different operation modes.

Figure 8 .
Figure 8. Dynamic cooling characteristics with a motor: (a) Refrigerator cooling characteristics; (b) cooling characteristics at the beginning; (c) effectiveness of the heat exchanger; (d) ultimate refrigerating temperature.

Table 2 .
The operating parameters of test and simulation.