Figure 1A shows the schematic diagram of the waste heat-driven ejection cooling cycle, which consists of two parts: collector cycle and ejector refrigeration cycle. The collector cycle mainly composes water tank 1 (low temperature), water tank 2 (high temperature), a series (four in the case of this paper) of dust collectors, and two water pumps. For the ERC, it consists of three heat exchangers (a generator, a condenser, and an evaporator), an ejector, a circulation pump, and an expansion valve. Figure 1(b) shows the ejection cycle in p–h chart.

In the basis of ERC, the working fluid is compressed to a high-pressure state by the pump and then absorbs heat in the generator. The heat is recovered from the collector and stored in water tank 2. The primary fluid vaporised in the generator enters the ejector nozzle, so as to entrain the low-pressure secondary vapor coming from the evaporator (Akbari and Sorin, 2020). The two flows then undergo mixing and introduce shock trains in the mixing chamber, followed by pressure recovery in the diffuser of the ejector (Chen et al., 2015). The mixed middle-pressure stream enters the condenser, releasing heat into the environment. After then, the stream flows to two parts: one passes through the expansion valve and then enters the evaporator, while the other is compressed by the pump and returns to the generator to complete the cycle.

The collector cycle is boxed by green-dashed line in Figure 1. Water tank 1, where low-temperature water with a temperature of 80°C is stored, is connected to several parallelly arranged dust collectors, passing valve 1 and pump 1. Water tank 2 receives and stores high-temperature water (about 90°C) from collectors. It is connected to pump 2, which is further connected to the generator.

The temperature of the collector is controlled by water from water tank 1, and the reaction heat is simultaneously stored in water tank 2. Then, with this heat, the ERC further cools the collectors. Therefore, the cooling process of the collector system consists of two main steps.

In the first step, air is slowly charged into the dust collector, which is vacuum during the production process. It takes no shorter than 2 h to avoid violent chemical reaction (2SiO + O₂ = 2SiO₂) in the collector that may release a large amount of heat energy and damage collector components. To remove the heat, valve 1 and valve 2 are both open, and the system is directly cooled by water from water tank 1. The cooling water flows through the tube-fin heat exchanger at the bottom of the collector (Figure 2), maintaining the temperature in the collector around 100°C. In other words, the waste reaction heat is delivered and stored into water tank 2, carried by water.

The second step (step 2) starts when air pressure in the collector reaches one atmospheric pressure after enough air is fed into the collector (the reaction almost ends). In this step, valve 1 and valve 2 are closed, while valve 3, valve 4, and valve 5 are all open. Driven by the low-temperature heat energy source from water tank 2, ERC works with a high efficiency. The collectors are cooled down (expected in 15 min) much faster than conventional ones (3–8 h) by water from the evaporator Figure 3until. When the reaction product temperature is lower than 20°C, step 2 ends, and the collector is ready for cleaning operation.

A dust collector with typical 330L/S processing capacity is chosen for this study, which can reach 99.9% dust collection efficiency. The Q235-made collector body is 1012 mm in diameter and 2323 mm in height. The upper temperature limit is 260°C. With some of the reaction heat leaking from the collector wall, the total heat released in step 1 is tested as 4.3875MJ, which is determined by Eq. 1 as water is used to remove the heat. Q = m h t c p Δ T t , (1) where m _ht is the mass of cooling water, c _p is water isobaric specific heat, and ΔT _t is temperature change of water. Since step 1 is finished in 2 hours, heat dissipating capacity can be found as 609W. Thus, the fin-tube heat exchanger is designed with tube diameter being 4 mm (0.15 mm wall thickness), tube pitch (c _t) being 10mm, aluminum fin thickness (δ _f) being 0.13mm, and fin space (c _f) being 2.2 mm. The total length of heat exchange tube and aluminum fin can be determined by L = 2 ∑ n = 1 n = r / c 2 ( r 2 − ( r − n c ) 2 ) . (2)

In Eq. 2, for tube, c _t = 10mm, and for fin, c _f = 2.6 mm. The total length is obtained as 80 and 309.1 m for tube and fin, respectively. The heat exchange area of fin can be calculated by Eq. 3. A f = 2 ( L f ⋅ c t − π r 2 ) . (3)

The heat exchanger area of tube is given by A t = π d L t π ( c f − δ f ) / c f . (4)

Consequently, heat exchange area A _L of one tube layer can be obtained as 4.8986 m², with 0.68 fin efficiency. A _f is calculated as 5.7936 m², and A _t is calculated as 0.959 m², using Eq. 5. A L = A f ⋅ η f + A t . (5)

With designed average temperature difference (ΔT) of 15 °C and overall estimated heat transfer coefficient (h) of 5 W/(m²K), the needed heat transfer area A _n can be obtained as 8.12 m² from Eq. 6. Q = A n h Δ T . (6)

Therefore, choosing heat exchanger layers to be 2, the designed area A _tot is now 9.7972 m². It has a 20.7% design margin, showing the design may be suitable.

Based on the above heat exchanger, a mathematical model of the cooling system including the heat exchanger is proposed, aiming at the verification of the feasibility of the cooling system and investigation of system performance. The theoretical model is established with main equations shown in Table 1, based on the following assumptions:

1) The flow inside the ejector is steady and one-dimensional.

Taking real gas model and constant pressure mixing assumption into account, the ejector model of this work is obtained, which is the same as the study by Xu et al. (2021b). In that work, the ejection model was employed in an ejection–compression cycle and validated by comparing against two experimental datasets. The relative average and minimum deviations with variations in condensing temperature were 13.6 and 3.0%, respectively. And, those with variations in evaporating temperature were 6.8 and 14.9%, respectively, indicating the model performance was acceptable. In step 2, dust temperature decreases as cooling system works. The relation between dust temperature and cooling temperature can be simplified using lumped parameter method, as shown in Eq. 7. θ / θ 0 = e x p ( − h A t o t t / ( m ⋅ c p ) ) , (7) where θ is the temperature difference between dust and coolant, as shown in Eq. 8. θ = T d u s t − T c o o l . (8)

T _cool is simply calculated as the average of temperature of cooling water entering and exiting the heat exchanger, as the temperature increase is not large. In Eq. 7, subscript 0 indicates the parameter gives initial value of the beginning of step 2. c _p is the equivalent specific heat of the cooled substance. From the experiment test, about 10% of the reaction occurs in step 2, as well as the cooling process of dust occurs from 100 to 20°C. Based on testing data, it can be calculated using Eq. 1 that total heat of step 2 is 3270.7 kJ. Assuming that the heat is uniformly released with time, equivalent specific heat can be estimated as 20.97 kJ/kg/K. With constant heat transfer coefficient h and designed area A _tot , the time (t) needed for step 2 can be calculated. The working condition used in modeling and analysis is listed in Table 2.

Figure 4 illustrates that the total cooling energy can be provided by the ejection refrigeration cycle, which is driven by the waste heat collected in step 1. Since the temperature of heat source (waste heat in water tank) is constant, superheating degree in generator decreases with increasing generating temperature (Figure 4). With constant ΔT _con of 2°C, as generating temperature (T _g) increases (superheating degree decreases), total cooling energy firstly increases to 3255 kJ at T _g = 73°C and then decreases. This is because when generating temperature is higher than 73°C, the ejector runs under the critical mode, whose entrainment ratio decreases as generating pressure increases. With an increasing condenser temperature difference, the maximum cooling energy reduces, while the corresponding optimization generating temperature raises. For the curve of condenser temperature difference higher than 4°C, its optimization generating temperature is beyond the studied range. Therefore, to meet the requirement of cooling load, a generating temperature not too high combining with a small condenser temperature difference is suitable for system operation.

Since the system is mainly driven by recovered waste heat, which can be considered as totally free, thermal COP is no longer important, and cooling energy provided by ERC is what we focus on. However, in ERC, small quantity of electricity is still consumed by the pump, and the second variable should be carefully investigated is COP_e, which is defined as the ratio of cooling energy of ERC to pump electric energy, as shown in Figure 5. The curves of COP_e also have a optimizing point with low condenser temperature difference. However, comparing with the curves in Figure 4, the best generating temperature for COP_e is smaller than that of cooling energy. For example, at ΔT _con = 2°C, the best generating temperature lies between 72°C and 73°C. The reason is that the pump electricity consumption decreases with decreasing generating temperature, as long as the ejector can still work. Therefore, it is better to choose generating temperature for operation, a little smaller than the best generating temperature for cooling energy.

It can also be found that the curves in Figure 4 and Figure 5 seem similar, which is again related to pump electricity cost shown in Figure 6. The reason is that the decrease in pump electricity costs with the increasing condenser temperature difference is very light (within 1% per Celsius degree), which will not change the interval between curves too much. To explain the behavior of pump electricity cost, we refer to Figure 7. Firstly, as generating temperature decreases, specific generating heat increases, but specific pump work decreases. Since the collected waste heat remains unchanged, the total refrigerant mass decreases, which leads to decrease in pump electricity consumption with the joint effect of the specific pump work. Secondly, as temperature difference in the condenser increases, total refrigerant mass and specific pump work vary in opposite ways this time. The former increases as the specific generating heat decreases, while the latter decreases, resulting in a negligible light decrease in pump electricity cost.

With temperature difference in the evaporator is 7°C, cooling water temperature ranges from 2 to 14°C, and return water temperature ranges from 4 to 16°C, then studied evaporating temperature ranges from −4–8°C. Using the conclusion of the previous subsection, T _g is 73°C. For the same reason as Figure 4, cooling energy is the key parameter, which decreases with a decreased evaporating temperature and increased condenser temperature difference (Figure 8). The tested cooling energy requirement is shown as a horizontal dashed line of 3270kJ, above which the cooling energy is qualified. In the qualified region, the highest condenser temperature difference is 5°C at an evaporating temperature of 8°C and the lowest evaporating temperature is 3°C with a condenser temperature difference of 2°C. During the cooling operation, temperatures need to be controlled in this range. Another important performance parameter is the time taken to cool the dust to the desired temperature. As shown in Figure 9, the lower the evaporating temperature, the faster the cooling speed can be achieved for a larger heat exchange rate. Considering both cooling energy and speed, an evaporating temperature around 6°C may be suitable. Anyway, even with the largest evaporating temperature of 8°C, the longest cooling down time is only 620s, much faster (about 92% faster) than the conventional method (about 2 h). These results indicate the new cooling system can save a lot of time and significantly improve the producing efficiency of single-crystal silicon.

To explain the variation trends in cooling energy, Figure 10 is illustrated. With constant condensing temperature and generating temperature, the ejector entrainment ratio slowly decreases with the decrease in evaporating temperature first. When the evaporating temperature is low enough, the ejector reaches the critical point as shown in Figure 9. From then on, as the evaporating temperature goes on decreasing, the ejector works under the subcritical mode and degradation in the entrainment ratio accelerates. Based on this trend, as total cycled refrigerant mass keeps unchanged (both generating heat and specific generating heat would not change), cooling energy can be decided with specific cooling capacity shown in Figure 11. Because specific cooling capacity presents an almost linear variation with the evaporating temperature, the trends of cooling energy with evaporating temperature are similar to that of the entrainment ratio, showing the same critical point. The decrease in specific cooling capacity with the increasing condenser temperature difference is even smaller (about 1% per Celsius degree), and the trends of cooling energy with condenser temperature difference are very similar to those of the entrainment ratio.

Figure 12 shows the total heat rejected from the condenser. The difference in total condensing heat is relatively small compared to their absolute value. At the critical point, the total condensing heat is 17231kJ, which is just 12.9% smaller than the largest total condensing heat of 19778 kJ at the largest cooling energy point. The result indicates that the size of the condenser does not need to change too much for different designs, neither does its cost. In other words, not much attention needs to be paid on the condenser.