Research on the Design of Auxiliary Generator for Enthalpy Reduction and Steady Speed Scroll Expander

Abstract

To help the reverse Brayton cycle cool the refrigerant from 100 K to 50 K, an auxiliary generator, with a housed stator, is studied and optimized, and the influences of weights in the cost- function on the results are discussed. The power demand and adiabatic characteristics of reverse Brayton cycle expansion are analyzed, after which the optimization indexes, including output rated power, efficiency, the air gap between rotor and stator, loss, and volume, are decided. The initial model of the auxiliary generator is then constructed and the parameters to be optimized are also determined. Taking the low loss and sinusoidal back-EMF as the evaluation indexes, the single parameter optimizations of the auxiliary generator are carried out. The co-simulation of the generator and its corresponding driving circuit is investigated, with which the power generation efficiency is calculated. The global optimizations of the generator parameters are carried out using a genetic algorithm. A suitable analytic hierarchy process (AHP) model is proposed, with which a three-order judgment matrix is constructed, and the effects of different weight combinations, in the cost-function, on generator performance are compared. The experimental results show that the output back-EMF amplitude is 28.2 V, which is about 10% smaller than the simulation results; the output power of the auxiliary generator under load is about 3.7 W, meeting the rated demand.

1. Introduction

The micro-cryogenic refrigerator based on the reverse Brayton cycle has unique competitive advantages in space developments and infrared detections due to its small size, and lightweight and compact structure [1,2]. Ideally, in the refrigeration cycle, after the high-pressure medium is transformed into the low-pressure medium by isentropic expansion process in the scroll expander. The enthalpy will be reduced, and the energy will be released to achieve the purpose of refrigeration [3,4]. As one of the core components of the refrigeration cycle, scroll expanders have been studied by many scholars, but there are still many problems to be solved and optimized in their structures and controls.

In practical uses, the expander often deviates from the rated speed due to the influence of inlet pressure fluctuation and operation friction resistance. This makes it difficult for the scroll expander to operate stably at a set intake pressure, expansion ratio, and mass flow rate to complete isentropic expansion of the working medium.

Focusing on the unstable problem of such expanders, a set of DC motor speed regulating devices is proposed in the literature [5] to realize the steady speed of the scroll expander. It analyzes the principle of DC motor speed regulation of the scroll expander and the advantages of the control circuit. In the literature [6], the scroll expander is connected to the generator, and a closed-loop control strategy controls the engine. Simulation and experimental results show that the scheme can improve the operational stability of the scroll expander effectively. In the literature [7], a scroll expander model is established according to the changes in chamber volume, gas volume, and pressure. Control strategies such as valves and buck chopper controllers are used to control the scroll expander. The results show that the optimization method can improve the energy conversion efficiency of the system.

In the above studies, scholars use different control systems and strategies to improve the stability and efficiency of the scroll expander. However, the scroll expanders and generators are usually separately designed and mounted. There are few concerns about the compatibility between the electro-magnetic performance of the generator and the operating conditions of the scroll expander.

The micro and integrated auxiliary generator in this paper helps to realize reverse Brayton cycle expansion refrigeration from 100 K to 50 K with a cooling power of 3.5 W. It is unique with its stator wrapped by the thermal insulation cover in the refrigeration system. The parameters to be optimized, the corresponding objectives, and the strategies adopted, especially the weights of objectives in the cost-function, for the global optimization are investigated and discussed. Our research can make the scroll expander work stably to realize refrigeration and will not let the heat of the auxiliary generator affect the refrigeration. The specific contents of the article are as follows.

The first section describes the components of the reverse Brayton cycle refrigeration and its operating principles. Then the type, initial model, and overall optimization goal of the auxiliary generator are determined according to the working condition requirements. In the second section, taking a low loss and sinusoidal back-EMF as the evaluation indexes, the single parameter optimizations of the auxiliary generator are carried out, and then electro-magnetic joint simulation is conducted to calculate the power output efficiency. The third section uses genetic algorithms to optimize the auxiliary generator globally. In multi-objective optimization, an analytic hierarchy process model is established, and a three-order judgment matrix is constructed. Then, the differences in generator structures and performances with different weight combinations are compared, and the best generator topology is determined. Finally, the prototype, including the control circuit, is made, and the corresponding performance of the auxiliary generator is tested and analyzed.

2. Scroll Expander Operating Conditions and Auxiliary Generator Analysis

2.1. Adiabatic Working Environment for Scroll Expanders

The refrigeration system described in the patent [8] is shown in Figure 1. It is mainly divided into two subsystems: the circulating refrigeration system and the auxiliary control system. It is composed of a heat exchanger, a scroll expander, a generator, and a control circuit. The relevant parameters of the scroll expander are shown in

Table 1. Relevant parameters of scroll expander.

Scroll Expander Parameters	Inlet Gas Parameters	Pressure 0.75 MPa Mass Flow 0.055 kg/s
	Radius of base circle	2.4 mm
	Involute starting angle	42.3°
	Number of vortices	2.38
	Vortex wall thickness	1.5 mm
	Vortex height	4 mm

. The working gas of the system is helium.

Ideally, after the isentropic expansion process, the 100 K gas entering the scroll expander will be cooled to 80 K. The temperature of the output gas is 80 K, which will cool 100 K inlet gas to 90 K in the heat exchange tube. Therefore, the gas for a new round of isentropic expansion starts from 90 K and cools to 70 K. After each refrigeration cycle, the temperature of the gas output from the scroll expander will decrease until it reaches the refrigeration limit of 50 K.

From the perspective of energy, when the high-pressure gas is converted into a low-pressure gas, energy is released. This part of energy will work on the scroll expander and convert it into the kinetic energy of the scroll expander. The speed of the scroll expander will increase continuously and deviate from its rated speed. The scroll expander is guaranteed to complete the cooling stably only when this part of the energy is exactly consumed.

Ideally, the isentropic expansion process is mainly completed in a closed system consisting of two parts, a scroll expander, and an auxiliary generator. Its schematic diagram is shown in Figure 2. Since the ball bearings connected to the moving scroll disc have voids, the two parts of the scroll expander and the auxiliary generator are connected. The cold air obtained through the isentropic expansion process will come into contact with the auxiliary generator. The heat generated by the auxiliary generator stator during operation will reduce the cooling effect.

In order to prevent the heat generated by the generator from mixing with the cold gas, the stator part of the auxiliary generator is sealed with an insulation sealing cover, and together with the base, a fully enclosed stator sealing chamber is formed. At the same time, in order to reduce the heat generation of the stator winding, the electrical energy generated by the auxiliary generator is output to the load at the external ambient temperature by designing a circuit. The sealed stator structure places specific requirements on the loss and output efficiency of the auxiliary generator.

2.2. Auxiliary Generator Optimization Analysis

2.2.1. Analysis of Optimization Objectives

In the above system composed of the scroll expander and the auxiliary generator, due to the existence of stator seal housing, the performance requirements of high efficiency and atmospheric clearance are put forward for the auxiliary generator. In order to effectively cool the scroll expander, the auxiliary generator also needs to output sufficient power. In order to reduce the volume of the whole system, the volume of the auxiliary generator should be minimized.

2.2.2. Initial Model of the Generator

Permanent magnet electric machines have the advantages of small size, high efficiency, and ample starting torque, and the control scheme is relatively mature, which is very suitable for auxiliary enthalpy reduction and stable speed scroll expander operation. Therefore, the auxiliary generator takes the form of a permanent magnet electric machine, and its performance requirements and initial structural parameters are shown in

Table 2. Initial structural parameters and performance requirements of the auxiliary generator.

Parameters	Values	Parameters	Values
Rated power P [W]	3.5	Power output efficiency η	90%
Rotate speed v [rpm]	2500	Motor thickness depth [mm]	9
Stator outer Radius r1 [mm]	26	Stator inner Radius r2 [mm]	13
Airgap air [mm]	1.3	Stator split ratio τ	0.5
Stator tooth thickness Ht [mm]	0.8	Magnet thickness Mt [mm]	2
Stator cogging ratio Wt	0.6	Polar arc coefficient α	0.4
Rotor pole number N1	10	Stator slot number N2	12

. Define the ratio of the stator inner diameter to the stator outer diameter as the stator split ratio τ, and the ratio of stator tooth width to groove width as the stator tooth groove ratio W_t. The initial model of the auxiliary generator is shown in Figure 3.

3. Single Parameter Optimizations and Efficiency Estimation

3.1. Single Parameter Optimizations

According to the above analysis, the rated power and power output efficiency are important output characteristics of the auxiliary generator. Power is related to the magnitude of the back-EMF, while efficiency is related to copper loss and iron loss. Therefore, the main objectives of single parameter optimizations are to make the back-EMF waveform more sinusoidal while reducing the loss of the generator.

The size of the auxiliary generator is small, and the area of each slot is about 50 mm². In addition, due to the inner rotor structure, it is more difficult to fill the winding. It is determined empirically that the filling coefficient of the groove δ is 0.3. The current density of the wire remains constant. On this basis, Figure 4 shows the effect of the air gap, polar arc coefficient, stator split ratio, and stator cogging ratio on the performance of the auxiliary generator [9,10,11,12,13,14,15,16,17,18,19,20,21,22].

It can be seen from Figure 4 that the four optimization variables have great influence on the amplitude of back-EMF of the auxiliary generator. After comprehensively considering the magnitude of back-EMF and its harmonic ratio, the values of various optimization parameters are determined, shown in

Table 3. The result of single parameter optimizations.

Parameters	Initial Values	Single Parameter Optimized Values
air	1.3	1.6
α	0.4	0.6
τ	0.45	0.53
Wt	0.6	0.75

. The back-EMF amplitude of the optimized auxiliary generator is 38.2 V, and the THD is about 10.5%.

3.2. Electrical-Magnetic Simulation of the Auxiliary Generator

After calculation and simulation, the resistance r of each phase winding of the auxiliary generator is 18 Ohm, and the inductance L is 13 mH. At the rated speed, the total impedance of each phase winding is about 35 Ohm. When the generator generates electricity, the impedance on the generator windings is regarded as the internal resistance of the power supply. According to the maximum power output theorem, to achieve an output efficiency of 90%, the ratio of the internal resistance of the power supply to the external resistance needs to reach 1:9.

According to reference [23], the external load resistance R is connected to each phase output of the generator. Compared with the circuit with a rectifier bridge, this circuit causes less torque fluctuation and will not affect the stability of the generator. In the circuit shown in Figure 5, L represents the inductance of each phase winding, and r represents the resistance of each phase winding. The resistance value of external load resistance R is 315 Ohm.

With a constant speed of 2500 r/min for the auxiliary generator, ignoring the wind friction loss, the line voltage V_L and line current I_L in the electrical-magnetic simulation are shown in Figure 6. The amplitude of V_L is about 50 V, and that of I_L is about 0.22 A. After the Fourier transforms, the phase difference θ between the line voltage and the line current is 70.2 deg. The output power P_out can be calculated according to Formula (1).

$$P_{out}=\sqrt{3}{V}_L{I}_L\cos \theta =5.85\mathrm{W}$$

(1)

Due to the large air gap of the auxiliary generator, the loss of the stator core P_Fe is only 15 mW, and the loss of copper P_Cu on the winding is 0.7 W. The power output efficiency of the auxiliary generator is.

$$\eta =\frac{P_{out}}{P_{out}+P_{Cu}+P_{Fe}}=\frac{5.85}{5.85+0.7+0.015}\approx 89\%$$

(2)

4. Global Optimizations

The output power of the auxiliary generator after the single parameter optimizations has exceeded the rated demand. In order to reduce the volume of the entire system and the temperature accumulation in the sealed cover, a part of the power can be sacrificed to make the auxiliary generator obtain a smaller motor thickness and less copper consumption.

For this purpose, multi-parameter and multi-objective global optimization is needed. Genetic algorithm is an optimization algorithm widely used in the process of motor optimization, which has the characteristics of strong global search ability [24,25,26,27,28,29,30,31].

4.1. Determine the Objective Function and Optimization Variables

For multiple optimization objectives, the expression for its total cost-function cost is:

$$cost={\displaystyle {\sum }_{i=1}^n{w}_i(g_i-G_i)}$$

(3)

where w_i is the weight of the objective i, G_i is the objective value of the objective, g_i is the objective function.

$$g_i=1+(X-X_{\max })\times 9/X_{\min }$$

(4)

where X is the type of parameter to be optimized, and X_max and X_min represent the maximum and minimum values that X can achieve within the range of variables.

The rated output power of the auxiliary generator affects the cooling capacity of the scroll expander, so the rated output power P is the primary optimization objective. The size of the stator slot determines the amount of copper used. Under the condition of keeping the current density unchanged, reducing the amount of copper can reduce the copper loss and inhibit the temperature accumulation in the sealing cover. However, it will reduce the electromagnetic characteristics of the auxiliary generator. So the copper loss P_Cu is also taken as one of the optimization objectives. The motor thickness directly determines the size of the auxiliary generator and the power output. In order to balance the contradiction between the volume and the output power, the motor thickness depth can also be used as an optimization objective. Therefore, the objective function and total cost-function of the three objectives of rated output power P, copper loss P_Cu, and motor thickness depth are:

$$\left\{\begin{array}{l}{g}_1=1+(P-2.7)\times 9/1.9\\ g_2=1+(P_{Cu}-0.1)\times 9/0.3\\ g_3=1+(depth-0.0075)\times 9/0.0015\\ COST=w_1{(g_1-4.8)}^2+w_2{(g_2-1)}^2+w_3{(g_3-1)}^2\end{array}\right.$$

(5)

The optimization objective for P is 3.5 W, P_Cu is 0.1 W, and depth is 7.5 mm. In general, each structural parameter of the motor can be used as an optimization variable, but the more variables in the optimization, the more complex the optimization process is. Changes in the air gap and stator split ratio can cause great changes in many other parameters, increasing the difficulty of optimization. Therefore, according to the above optimization results, the air gap of the auxiliary generator is fixed at 1.6 mm and the stator split ratio τ is 0.53.

In addition, the motor thickness and stator slot type (stator cogging ratio W_t, stator tooth thickness H_t, stator slot width W_s) have a greater impact on the power and copper loss of the auxiliary generator. The changes in these parameters are independent, and the optimization process is simpler. The optimization ranges and discrete steps are shown in

Table 4. Optimization parameters, ranges, and discrete steps.

Parameters	Optimization Ranges	Discrete Steps
depth	7.5–9	0.01
Wt	0.6–0.9	0.01
Ht	0.8–1.4	0.01
Ws	1.2–2.4	0.01

.

4.2. Multi-Objective Optimization

Different parameters have different effects on generator performance, and power is more sensitive to changes in motor thickness than the shape of the stator teeth. Therefore, multi-parameter and multi-objective optimization adopts two schemes: step-by-step optimization and comprehensive optimization.

4.2.1. Step-by-Step Optimization

First, the rated power and volume are minimized with the objective to determine the appropriate motor thickness. Then optimize the shape of the stator groove to meet the rated power and minimal losses. When optimizing power and volume, the optimization results considering multiple-weight combinations are shown in Figure 7:

As can be seen from the figure above, the power is positively correlated with the size of the motor thickness. When the weight of power is relatively small, genetic algorithms tend to optimize a smaller motor thickness, but the power at this time cannot meet the demand. As the weighted share of power increases, the focus of optimization shifts to increasing power. When the weight of power reaches more than 0.5, the power can meet the rated demand. After that, the increase in power causes an increase in volume. The minimum motor thickness to reach the rated power is 7.9 mm.

The optimization results of various weight combinations, aiming at power and loss, are shown in Figure 8.

From the optimization results, it can be seen that the loss gradually increases with the increase of power, accounting for about 10% of the power. When the proportion of the optimized weight of power reaches more than 0.8, the power meets the demand. In order to leave a certain margin for the subsequent sample mechanism, the weight of the power is 0.9, and the results of multi-parameter step-by-step optimizations are shown in

Table 5. The results of multi-parameter step-by-step optimizations.

Parameters	Single Parameter Optimized Values	Multi-Parameter Step-by-Step Optimization of Values
P	5.85	3.81
PCu	0.7	0.37
depth	9	7.9
Ws	-	1.32
Ht	1.1	1.16
Wt	0.8	0.83

.

4.2.2. Comprehensive Optimization

Comprehensive optimization is the optimization of three objectives simultaneously, and when there are more than two optimization objectives, the determination of weight coefficients is more complicated [32,33]. In this paper, the hierarchical analysis method combines the generator performance requirements index with the experience of structural optimization to determine the weight coefficient.

The analytic hierarchy method is a hierarchical weight decision analysis method applied to multi-objective comprehensive evaluation. The main steps are: establishing a hierarchical model, constructing a judgment matrix, testing consistency, and calculating weights.

(1)

establishing a hierarchical model;

Rated output power P, copper loss P_Cu and motor thickness depth are the main optimization objectives of the auxiliary generator and play an essential role in whether the scroll expander can operate efficiently. Using these three parameters as the criterion layer to determine the generator model to establish a hierarchical model for the objective layer [34], shown in Figure 9:

(2)

constructing a judgment matrix;

According to the scale table shown in

Table 6. Scale table.

aij	Scale Meaning
1	i is of equal importance compared to j
3	i is slightly more important than j
5	i is significantly more important than j
7	i is very important compared to j
2, 4, 6	The median of the two adjacent judgments mentioned above
reciprocal	j compared to i aji = 1/aij

, the three criteria are compared to construct a judgment matrix. This approach reduces the difficulty of comparing factors of different natures with each other in order to improve accuracy.

In order to obtain a suitable weight coefficient ratio, consider a variety of situations when constructing the judgment matrix. Matrix A indicates that power has higher importance than volume and loss, with the main purpose of increasing power. Matrix B gives the highest loss of importance to optimize the structure with the lowest loss. Matrix C gives power the same importance as volume, reduces the importance of loss, and aims to find the best structure between increasing power and reducing volume. The judgment matrix is shown in

Table 7. Evaluate the judgment matrix, relative weight and consistency judgment list of the objective function.

A	W1	W2	W3	Wi	B	W1	W2	W3	Wi	C	W1	W2	W3	Wi
W1	1	5	6	−0.96	W1	1	0.5	5	−0.49	W1	1	3	1	−0.69
W2	0.2	1	3	−0.26	W2	2	1	7	−0.87	W2	0.33	1	0.33	−0.23
λ = 3.10					λ = 3.01					λ = 3.00
W3	0.17	0.33	1	−0.12	W3	0.5	0.14	1	−0.11	W3	1	3	1	−0.67
CR = 0.08 < 0.1					CR = 0.01 < 0.1					CR = 0.00 < 0.1

. In order to unify the data, the scores take two significant digits.

(3)

testing consistency;

The indicators for consistency are:

$$\left\{\begin{array}{c}CI=\frac{\lambda -n}{n-1}\\ CR=\frac{CI}{RI}\end{array}\right.$$

(6)

where λ is the maximum eigenvalue of the matrix, n is the order of the matrix, CI is a consistency index, RI is the random consistency index, and when the matrix is third order, RI = 0.58. CR is the consistency ratio, the smaller the value of CR, the higher the consistency. The CR values of the four judgment matrices are listed in Table 7, which meet the requirements.

(4)

calculating weights;

The eigenvectors corresponding to the maximum eigenvalue λ of matrix A are:

W_i = [−0.9584 −0.2601 −0.1177]

(7)

After normalization, it is converted to weights.

W_A = [0.72 0.19 0.19]

(8)

Similarly, it can be calculated that the different weight combinations are:

W_B = [0.33 0.59 0.08]
W_C = [0.43 0.14 0.43]

(9)

In addition, another weight combination is added to balance and optimize the three objectives:

W_D = [0.33 0.34 0.33]

(10)

(5)

The result of the optimizations;

The optimization results of the four weight combinations are shown in Figure 10. From the perspective of power, only the optimization result of weight combination A can achieve the rated power demand, but the winding loss is the highest. Weight combination B has the lowest loss, but at the same time, it limits the power increase. Compared with weight combination B, weight combination C increases the power by optimizing the slot type and reducing the volume. Weight combination D gives the three objectives similar weights, and although it has a small motor thickness and loss, the power does not meet the rated demand.

In short, the optimization effect of weight combination A is the best. Compared with the results of step-by-step optimization, the results of comprehensive optimization are slightly reduced in power, volume and loss. The final structure of global optimization is shown in

Table 8. The result of global optimization.

Parameters	Single Parameter Optimized Values	Multi-Parameter Step-by-Step Optimization of Values	Global Optimization Values
P	5.85	0.7	3.63
PCu	3.81	0.37	0.36
depth	9	7.9	7.8
Ws	-	1.32	1.2
Ht	1.1	1.16	1.23
Wt	0.8	0.83	0.88

.

5. Experiment and Analysis of Results

Figure 11a shows the actual stator, rotor, house used to fix the generator, and the final installed generator prototype. Due to the processing accuracy, there are slight differences between the actual prototype and the global optimization results.

The no-load experiment platform is shown in Figure 11b. The experiment equipment mainly includes a drive motor, tested generator, coupling, controller, power supply, torque tester, and switching mode power supply. The experiment must ensure that the bearings of the drive mover, torque meter, and the generator under test have a high degree of concentricity to measure the most accurate results. The main technical parameters of the equipment in the experiment are shown in

Table 9. Main technical parameters of experimental equipment.

Equipment	Technical Parameters
Switching Mode Power Supply	Output voltage	24 V(DC)
	Output current	10 A
Controller	Speed regulation range	100–8000 rpm
Drive motor	Rated speed	3000 rpm
	Rated torque	1.27 Nm
Dynamic torque transducer	Accuracy	0.0001 Nm
	Range	0.1 Nm
Tested generator	Rated speed	2500 rpm

.

The drive motor provides a speed of 2500 rpm for the auxiliary generator, and in the no-load state, the back-EMF of the auxiliary generator is shown in Figure 12. It can be seen that the voltage waveform has good sinusoidal, and after Fourier analysis, its total harmonic distortion is about 12%. The amplitude of the back-EMF actually measured is 28.2 V, which is about 10% smaller than the results of the simulation, within the standard error range. This proves that the generator made meets the standards of the experiment and can be carried out in the next step of the experiment.

The connection of the three-phase output of the auxiliary generator is shown in Figure 5, and each phase is connected to an external 315 Ohm resistor. Figure 13 is the line voltage and line current at the output end under the load, and the power output P_out by the auxiliary generator in the load state can be found to be 3.7 W:

With the torque transducer, the input power and torque on the auxiliary generator shaft can be read. As shown in Figure 14, the torque transducers have a torque measurement accuracy of 0.0001 Nm and a measuring range of 0.1 Nm. The input power P_in of the auxiliary generator can be found by using the relationship between torque and speed:

$$P_{in}=0.0257\times 2500/9.55\approx 6.7\mathrm{W}$$

(11)

Then the power output efficiency of the auxiliary generator is:

$$\eta =\frac{P_{out}}{P_{in}}=\frac{3.7}{6.7}\approx 55.2\%$$

(12)

Error analysis: the error of the back-EMF waveform of the auxiliary generator and the result of the finite element simulation during the no-load experiment is within a reasonable range; In the load experiment, the efficiency of the auxiliary generator was only 55.2%. This is because the auxiliary generator, torque meter, and drive motor are not completely concentric, and additional torque will be generated between the bearings during rotation, resulting in greater input power.

6. Conclusions

Based on the scroll expander operating under special conditions, this paper optimizes a permanent magnet synchronous motor to assist its operation. The novelty of this paper is that the optimization of the auxiliary generator is combined with the working condition of the scroll expander in the optimization process. Firstly, through the working condition analysis, it is determined that the optimization direction of the auxiliary generator is high efficiency, small volume, and low loss. Secondly, the weight ratio problem of multi-objective optimization is solved by combining the analytic hierarchy process with the working conditions of the scroll expander.

After single parameter optimization and global optimization, the output power of the auxiliary generator is 3.63 W, and the electric energy output efficiency is 91%. Then, the prototype is made and tested. Under no-load conditions, the amplitude of back-EMF of the auxiliary generator is 28.2 V, which is about 10% less than that of simulation. In the load experiment, the output power of the auxiliary generator is measured to be 3.7 W, which meets the needs of the scroll expander.