Thermodynamic analysis of packed bed thermal energy storage system Huan Guo 1 , Yujie Xu 1,2,* , Cong Guo 1 , Haisheng Chen 1,2 , Yifei Wang 1 , Zheng Yang 1 , Ye Huang 3 ，Binlin Dou 4 1 Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China; 2 University of Chinese Academy of Sciences, Beijing 100049, China; 3 School of the Built Environment, University of Ulster, Co. Antrim, BT37 0QB, UK; 4 School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, 200093, China. * Correspondence: [email protected]; Tel.: +86 10 82543149. Abstract: A packed-bed thermal energy storage (PBTES) device, which is simultaneously restricted by thermal storage capacity and outlet temperatures of both cold and hot heat transfer fluids, is characterized by an unstable operation condition, and its calculation is complicated. To solve this problem, a steady thermodynamics model of PBTES with fixed temperatures on both ends is built. By using this model, the exergy destruction, thermocline thickness, thermal storage capacity, thermal storage time, and other key parameters can be calculated in a simple way. In addition, the model explains the internal reason for the change of thermocline thickness during thermal storage and release processes. Furthermore, the stable operation of the PBTES device is analyzed, and it is found that higher inlet temperature of hot air, and lower temperature difference between cold and hot air can produce less exergy destruction and achieve a larger cycle number of stable operation. The work can be employed as the basis of the design and engineering application of PBTES. Keywords: packed bed thermal energy storage; thermocline; steady; thermodynamic analysis; stable operation 1. Introduction A thermal energy storage (TES) device, which combines thermal storage and heat transfer, aims to overcome the contradiction between heat generation and usage as well as to solve the problem of using intermittent heat sources. For example, in a solar thermal power generation system, the discontinuous and unstable solar thermal energy is stored in TES when sunlight is sufficient. Then, the stored thermal energy is released continuously and stably if needed. In this way, the solar energy can be used efficiently [1, 2]. The heat is stored in thermal storage materials, such as molten salt, microcapsules of phase-change material, pebble, carbon steel, ceramics, aluminum, and stainless steel [3-5]. TES technology can be sorted based on the change in the chemical and physical state of thermal storage materials during thermal storage and release process [6-8] as sensible heat storage (SHS) that stores and releases heat by temperature change, 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1
Transcript
Thermodynamic analysis of packed bed thermal energy storage system
1 Institute of Engineering Thermophysics, Chinese Academy of Sciences, Beijing 100190, China; 2 University of Chinese Academy of Sciences, Beijing 100049, China; 3 School of the Built Environment, University of Ulster, Co. Antrim, BT37 0QB, UK;4 School of Energy and Power Engineering, University of Shanghai for Science and Technology, Shanghai, 200093,
Abstract: A packed-bed thermal energy storage (PBTES) device, which is simultaneously restricted by thermal storage capacity and outlet temperatures of both cold and hot heat transfer fluids, is characterized by an unstable operation condition, and its calculation is complicated. To solve this problem, a steady thermodynamics model of PBTES with fixed temperatures on both ends is built. By using this model, the exergy destruction, thermocline thickness, thermal storage capacity, thermal storage time, and other key parameters can be calculated in a simple way. In addition, the model explains the internal reason for the change of thermocline thickness during thermal storage and release processes. Furthermore, the stable operation of the PBTES device is analyzed, and it is found that higher inlet temperature of hot air, and lower temperature difference between cold and hot air can produce less exergy destruction and achieve a larger cycle number of stable operation. The work can be employed as the basis of the design and engineering application of PBTES.
Keywords: packed bed thermal energy storage; thermocline; steady; thermodynamic analysis; stable operation
1. IntroductionA thermal energy storage (TES) device, which combines thermal storage and heat transfer, aims to
overcome the contradiction between heat generation and usage as well as to solve the problem of using intermittent heat sources. For example, in a solar thermal power generation system, the discontinuous and unstable solar thermal energy is stored in TES when sunlight is sufficient. Then, the stored thermal energy is released continuously and stably if needed. In this way, the solar energy can be used efficiently [1, 2].
The heat is stored in thermal storage materials, such as molten salt, microcapsules of phase-change material, pebble, carbon steel, ceramics, aluminum, and stainless steel [3-5]. TES technology can be sorted based on the change in the chemical and physical state of thermal storage materials during thermal storage and release process [6-8] as sensible heat storage (SHS) that stores and releases heat by temperature change, latent heat storage that stores and releases heat by latent heat, thermochemical heat storage that stores and releases heat during chemical reaction and sorption/desorption. SHS is widely used in large industrial equipment for its strong temperature adaptability, simplicity, and low cost. Thus, SHS is applied in heat pumps, solar thermal power plants, thermal energy storage systems, and novel compressed air energy storage systems[9-12].
SHS technology mainly includes packed-bed thermal energy storage (PBTES), indirect thermal energy storage of two tanks, and so on [13-15]. Compared with indirect thermal energy storage of two tanks, PBTES has a simpler structure and stronger temperature adaptation especially in relatively low temperature [1, 14, 16]. In PBTES, the cold and hot fluids exchange heat with the thermal storage material respectively at different times. In this way, the heat of hot fluid is transferred to cold fluid [17]. However, the unsteady heat transfer of the PBTES can lead to unsteady operation of the thermodynamic system when it is strict to the outlet temperature of cold and hot fluids and thermal storage capacity (the maximum heat stored in one cycle). The reason is that the temperature inside PBTES varies with time and space. For example, when the initial temperature of the thermal storage material is unified, the thermocline is thin at the beginning of thermal storage or releasing process; then, the thermocline moves to the cold/hot side and thickens during the
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storage/releasing process. If the thermal storage capacity of the PBTES is kept constant, the outlet temperature of cold or hot fluid fluctuates in a large range [13, 18-20]. If the outlet temperatures of cold and hot fluids are both kept in a small range, the thermal storage capacity becomes smaller with the storage/releasing process, and after several cycles, the thermal storage capacity drops to the lowest when the PBTES operates in a stable state because the temperature change of the thermal storage material becomes smallest; meanwhile, the thickness of thermocline becomes the largest [13, 14]. Thus, conventional PBTES could not meet the requirement of stable outlet temperatures of cold and hot fluids as well as large thermal storage density simultaneously in a thermodynamic system.
Full CFD simulation of the unsteady process of the PBTES is complex, and the computation consumes a significant amount of computing resource when the time and space step is too small for calculation precision [21, 22]. Unsteady 1D models may consume less computing resource if efficiently coded [19, 23], but they are also inconvenient to calculate when applied in large system optimization and system dynamic simulation, where the PBTES is a part of a thermodynamic system for multiple calculation.
Thus, according to the corresponding characteristics of thermal charging and discharging processes, a steady heat transfer model for the PBTES is built, which is suitable for calculating PBTES when it works unsteadily with fixed outlet temperatures of both cold and hot fluids. No research on establishing steady heat transfer model was found in literature at present. Through calculating energy loss and exergy destruction, the steady model can solve the issue of complex calculation and optimization of unsteady process of the PBTES in a thermodynamic system efficiently. It also provides a theoretical support for analysis on the PBTES with fixed outlet temperatures of cold and hot fluids. Then the reason of the variation of the thermocline thickness can be obtained. On this basis, a stable operating PBTES model is proposed to reduce large fluctuation of outlet temperatures of cold and hot fluids and keep constant large thermal storage density. The operating condition is analyzed and the results can provide a theoretical basis for the design and operation of the proposed PBTES system.
2. Working principle and classification of PBTES
Figure 1. A conventional PBTES device
A conventional PBTES device, which is filled with thermal storage materials to store energy, is shown in Fig. 1. In this device, the inlet temperatures of hot and cold fluids are constant. In the thermal storage process, the hot fluid with temperature T1 flows into the PBTES to exchange heat with the thermal storage material. As the temperature of the thermal storage material increases, the outlet temperature of hot fluid becomes T2. The thermal storage process ends in t hours. In the thermal release process, the cold fluid with temperature T3
flows into the PBTES to exchange heat with the thermal storage material, the temperature of thermal storage material decreases, and the outlet temperature of cold fluid becomes T4. The thermal release process ends when the thermal energy stored in the PBTES returns to the original value.
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Figure 2. Four types of PBTES device (T, temperature；t, time)
From the application perspective, the PBTES can be classified into two groups on the basis of whether the outlet temperatures of hot and cold fluids are restricted. The first group has only one restricted outlet temperature, i.e., either the outlet temperature of hot fluid or that of cold fluid (cold-side type is shown in Fig. 2a, in which the outlet temperature of hot fluid is restricted; hot-side type is shown in Fig. 2b, in which the outlet temperature of cold fluid is restricted). The second group has two restricted outlet temperatures, i.e., the outlet temperatures of hot fluid and cold fluid are both restricted to a certain range or a certain value (two-side variable-temperature type is shown in Fig 2.c, in which the outlet temperatures of hot and cold fluids are both restricted in a certain range; two-side constant-temperature type in Fig 2.d, in which the outlet temperatures of hot and cold fluids are both restricted in a certain value).
The key function of cold-side type is to provide a steady supply of cooled hot fluid. Similarly, the key function of hot-side type is to provide a steady supply of heated cold fluid. By contrast, the key function of two-side variable-temperature type and two-side constant-temperature type can provide both hot and cold fluids for an efficient use, in which, the two-side variable-temperature type may function stably and periodically [14, 24], i.e., the PBTES may undergo the same process in all cycles; and for the two-side constant-temperature type , the outlet temperature of the hot fluid is the same approximately as the inlet temperature of cold fluid for the hot side, and the same situation is for the cold side.
Because of more request on the temperatures, the two-side constant-temperature type and two-side variable-temperature type require more caution in design and operation compared with cold-side type and hot-side type, and the two-side constant-temperature type requires the most. Compared with two-side variable-temperature type, the temperature change in the thermal storage material of two-side constant-temperature type is larger in steady operating conditions, and its thermal storage capacity is greater. The steady operating condition is noted to be an important factor in system design and practical operation. Thus, in this study, the PBTES of two-side constant-temperature type is analyzed and the influence factors of unstable operation of the PBTES are explored; on this basis, a steady operating model is proposed.
3. Steady heat transfer model
3.1 Simplified steady heat transfer model
The proposed PBTES model of two-side constant-temperature type is assumed to be one-dimensional and have no pressure drop, that is because heat exchanged by radiation is small, while in packed bed the pressure drop is always small as well [25, 26]. The heat transferred with surroundings is assumed to be zero due to the reason that an insulating layer is always equipped in the outside of packed bed to keep very low heat dissipation rate. The heat capacity of fluid and thermal storage material is assumed constant. The mass flow
rates of hot and cold fluids are the same as . Air is chosen as heat transfer fluid, which is considered as an ideal gas, and pebble is chosen as the thermal storage material.
The operation process of the PBTES is shown in Fig. 3. The initial temperature of pebbles is Tlow. Thermal storage starts at time 0. At this moment, the hot air with temperature Thigh flows into the PBTES, and the temperature decreases drastically to Tlow. Thus, a steep thermocline emerges. The thermocline moves to the cold side during the thermal storage process. The thermal storage ends at t0. Thermal release also starts at time
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0, which is the same time when the cold air with temperature Tlow and the same mass flow rate as that during the thermal storage process flows into the PBTES and is heated by the hot pebbles. A thermocline also emerges and moves to the hot side during the thermal release process. The thermal release ends at t0. The time duration of thermal storage and release processes are the same because the mass flow rate during thermal storage and thermal release process are the same, and the total air mass is constant.
According to the PBTES work principle, during the thermal storage process, the inlet and outlet temperatures of hot air are approximately the same as the local temperature of pebbles. The reason is that the temperature difference between hot air and pebbles is approximately zero at the two ends of the thermocline. However, the temperature difference between them exits in the inner thermocline. Similarly, during the thermal release process, the temperature difference between cold air and pebbles is approximately zero at the two ends of the thermocline, but the temperature difference between them exits in the inner thermocline [13, 27].
As both the hot and cold air exchange heat with pebbles, it can be assumed that the hot air and cold air exchange heat directly with each other through a simple heat transfer model, in which the thermal storage material, pebbles, is ignored. The steady heat transfer of hot air and cold air occurs at the same time (such as heat transfer in a counterflow heat exchanger). But actually, the heat transfer of hot air and cold air doesn’t occur at the same time in PBTES. To solve the problem, time of the thermal storage and thermal release can be corresponded: time, 0 (beginning of thermal storage) of the thermal storage process corresponds to time, t0
(end of thermal release) of the thermal release process, meanwhile, time, t0 (end of thermal storage) of the thermal storage process corresponds to time, 0 (initial of thermal release) of the thermal release process. Then time t1 of the thermal storage process corresponds to time, t01 of the thermal release process, and the time, t2 of the thermal storage process corresponds to time, t02 of the thermal release process, where
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The direct heat transfer model of hot air and cold air at corresponding times, (t1, t01) and (t2, t02), can be obtained, and the heat exchange process is shown in Fig. 3.
Figure 3. Simplified heat transfer process between cold and hot air under steady operation in PBTES device
As mentioned, the heat transfer in the PBTES can be considered as the counterflow heat transfer between hot and cold air at a corresponding time with temperature difference δT based on the energy balance:
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where cp,coldair, cp,hotair represents the heat capacity of cold air and hot air, respectively; and are the mass flow rate of cold and hot air, respectively; and ∆Tcold and ∆Thot are the local temperature change of cold and hot air, respectively.
Based on the preceding assumption, cp,hotair is equal to cp,coldair. and are the same as . Thus, the local temperature changes of cold air and hot air are the same, and the temperature difference, δT, does not change along the axial direction (or thermocline direction) during the heat transfer process at corresponding time. However, according to the PBTES work principle, the temperature difference at the ends of the thermocline is approximately 0. It can be explained that, during the thermal storage process, the heat sink is the cold energy stored in pebbles, which cools the hot air after its temperature decreases to Tlow+δT, and the hot air is cooled to Tlow. During the thermal release process, the heat source is the thermal energy stored in
pebbles, which heats the cold air after its temperature increases to Thigh- δT, and the cold air is heated to Tlow. Meanwhile, the cold/thermal energy stored in pebbles, which is supplied for this corresponding time, can be seen as energy coming from other corresponding times substantially.
In fact, the supplement of the cold/thermal energy from pebbles to air occurs not only at the ends of the thermocline but throughout the entire thermocline. Thus, δT is not constant along the thermocline, and it changes at different corresponding times [13, 27]. Given a stable working condition, δT is an average value during several cycles and can be regarded as a constant. The effect of all unstable factors on the PBTES is reflected on δT in this model. Meanwhile, it is assumed that the supplement of the cold/thermal energy from pebbles to air only occurred at the ends of the thermocline in this model.
3.2 Steady thermodynamic model of PBTES
3.2.1 Balance of thermal energy and exergy
The thermocline of pebbles in the PBTES device thickens with the process of thermal storage and thermal release. The shape of the thermoclines of pebbles at the corresponding time, (t1， t01), is shown in Fig. 4, where ADCEF is the shape of the thermocline at t1 during thermal storage, and ABCGF is the shape of the thermocline at corresponding time t01 during thermal release in the same cycle.
From t1 in thermal storage to t01 in thermal release process, according to the balance of energy,
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where △Usto represents the change of thermal energy stored in pebbles in this process. Hhotair is the input of thermal energy from hot air. Hcoldair is the thermal energy transferred from pebbles to cold air. cp, air represents the heat capacity of cold and hot air, which is constant. In the following equation, △T is the temperature change of cold and hot air, which is also the temperature change of pebbles:
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where Thigh represents the inlet temperature of hot air. Tlow represents the inlet temperature of cold air. When equation (1) is substituted to equation (3), the thermal energy stored in the pebbles does not change from t1 in the thermal storage to its corresponding time, t01, in the thermal release process.
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Thus, in the aforementioned period of time, the thermal energy stored in region 1 (ABCDA) is transferred equivalently to region 2 (CEFGC), as shown in Fig. 4. The distribution of pebbles along the axial direction is assumed to be uniform, and the abscissa value in Fig. 4 represents the mass of pebbles, while the ordinate value represents the temperature. Based on equation (6), the area of region 1 (AreaⅠ) represents the released thermal energy of pebbles when the line ADC changes to ABC because of a thicker thermocline. In the same way, the region 2 area (AreaⅡ) represents the absorbed thermal energy of the pebbles when line CEF changes to CGF. AreaⅠis equal to AreaⅡ because the thermal energy stored in region 1 (ABCDA) is transferred equivalently to region 2 as follows:
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where Qtransfer represents the released/absorbed thermal energy of pebbles when line ADC/CEF changes to ABC/CGF because of a thicker thermocline. cp,sto represents the constant heat capacity of the pebbles. δTsto
represents the temperature change of the pebbles. m represents the mass of the pebbles.
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Figure 4. Shape of thermocline at corresponding time
From t1 of the thermal storage to its corresponding time, t01, in the thermal release process, the exergy balance of the thermal storage and thermal release can be respectively shown in the following equations (7) and (8):
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()
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where ∆Es,sto , ∆En,sto is the change of exergy stored in the pebbles during thermal storage and that during the thermal release process, respectively. Ehotair is the exergy change of hot air during the thermal storage process. Ecoldair represents the exergy change of cold air during the thermal release process. Is and In respectively represent the exergy destruction because of the temperature difference between air and pebbles during the thermal storage and the thermal release process. ein,hotair, eout,hotair is the inlet exergy and outlet exergy of hot air, respectively. ein,coldair, eout,coldair is the inlet exergy and outlet exergy of cold air, respectively.
As the inlet/outlet specific exergy of hot air is equal to the outlet/inlet specific exergy of cold air, equation (9) can be obtained from equation (7) and (8) as follows:
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where ∆Esto is the change in exergy stored in the pebbles during one cycle. Isn represents the exergy destruction in the heat transfer process between cold air and hot air under the temperature difference, δT.
Therefore, greater exergy decrease is observed in the pebbles during the thermal release process compared with the increased exergy of the pebbles during the thermal storage process to keep the fixed outlet temperatures of cold and hot air. The exergy stored in pebbles decreases as a result of the heat transfer of cold and hot air. The change of exergy stored in pebbles indicates the change of the temperature distribution in the pebbles. The heat is transferred from the hot side to the cold side of the pebbles during t1 to t01, as shown in Fig. 4, and the exergy stored in the pebbles decreases with heat transfer.
As analyzed, in Fig. 4, the equality of AreaⅠand AreaⅡis due to energy balance, and the size of AreaⅠ(Ⅱ) is decided by exergy destruction.
In general, in the steady thermodynamic model of the PBTES, given the supplement of cold and thermal energy from pebbles to air at the ends of the thermocline, the heat transfer process can be considered as a superficial reversible process because of the zero temperature difference at two ends of the thermocline. The supplement of cold and thermal energy in the pebbles leads to the heat transfer from the hot side to the cold side of the pebbles. Then, the thermocline thickens with the thermal storage/release processes as shown in
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Fig. 4. Thus, the exergy destruction of heat transfer between cold air and hot air under temperature difference, δT, is reflected on the exergy destruction of the pebbles.
The exergy analysis for PBTES can also be found in some literature covering sensible and latent heat storage[25, 28, 29], but those are all based on the numerical results of temperature profiles. There is no research on establishing exergy destruction model for PBTES at present. For example, Bindra et al.[25] developed a robust system-scale heat transfer CFD model for the packed bed, which accounts for wall heat transfer and intra-particle diffusion effects. Based upon the temperature field modeling, recovered and lost exergy were calculated. The way to decrease exergy destruction was proposed. For sensible thermal storage, higher energy density results in higher exergy recovery, but for latent thermal storage, the result is opposite.
3.2.2 Key performance parameters
The thermal storage capacity of the PBTES decreased as the thermocline thickened, as shown in Fig. 5. The area of OPQRO and OTQSO can both represent the storage capacity of the PBTES. As the process of thermal storage and release proceeds, the thermocline thickens, and the area of the curve decreases as shown in equation (10). Then, the thermal storage capacity decreases until it reaches 0 when the thermocline has crossed the entire PBTES,
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where SOPQRO, SOTQSO represent the areas of curve OPQRO and curve OTQSO, respectively.
Figure 5. Change of thermal storage capacity of PBTES device during operation of several cycles
To keep the thermal storage capacity constant during several operating cycles, some cold or thermal energy can be reserved in the two sides of the PBTES to supply local cold or thermal energy, which is needed during the thermal storage or release process. The PBTES work principle with reserved part is shown in Fig. 6, where the thermocline is assumed to be a straight line that is for briefly analysis in the following part.
Figure 6. Thermal storage capacity of PBTES device with reserved section
In the beginning, A-B is the reserved part with temperature Thigh. The temperature of other parts is Tlow, which is not in a full load of the PBTES to store thermal energy in the first thermal storage process, and the stored thermal energy is only equal to that in part B-F. C-D is the reserved part accordingly. The PBTES with reserved parts can maintain a constant thermal storage capacity during several cycles. For example, as shown in Fig. 6, the area of HIKJ represents the thermal storage capacity after the PBTES has gone through several cycles, which is equal to the area of BECF, the initial thermal storage capacity.
It should be noted that, the above analysis about the decrease of thermal storage capacity is under the
condition that the outlet temperatures of the PBTES are fixed. If the temperatures at the outlets of the PBTES
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change, the thermal storage capacity can be maintained. Al-Azawii et al. [28] conducted an experiment to
study the cyclic charge-discharge behavior of PBTES using alumina beads as packing material. The control
parameter is time, and the outlet temperature is changing, thus the thermal storage capacity is maintained.
Johnson et al.[30] studied PBTES with supercritical carbon dioxide as the working fluid and α- alumina as the
storage material. An axisymmetric model produces temperature profiles in the bed, insulation, and pressure
vessel in the axial and radial directions over time. During thermal charge period, the outlet temperature can be
changed, while the outlet temperature during thermal discharging varies in a small range, then on that
condition the thermal storage capacity can be also maintained.(1) Thermal storage capacity and thermal storage/release time
The constant thermal storage capacity during several cycles is
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where msto represents the total mass of pebbles from point B to point F. LB-F is the length of point B to point F. ρm,sto is the mass of pebbles contained per unit length.
According to the energy balance, the mass of air flowed through the PBTES during the thermal storage or release process, which is expressed as follows:
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The thermal storage/release time in one cycle is
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(2) Exergy destruction of heat transfer per unit of time
At the corresponding time, the temperature difference, δT, between cold and hot air leads to exergy destruction, which is approximately equal to the exergy destruction of a counterflow heat transfer between hot air (the temperatures for the two ends are Thigh and Tlow+δT, respectively) and cold air (the temperatures for the two ends are Tlow and Thigh-δT, respectively).
The entropy change of hot air per unit of time is
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The entropy change of cold air per unit of time is
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The exergy destruction per unit of time is
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where represents the entropy generation in the heat transfer process between cold and hot air per unit of time. T0 represents the environment temperature.
(3) Exergy destruction rate
Exergy destruction rate is the ratio of the total exergy destruction in one cycle to the maximum amount of stored exergy when the thermocline is straight. In Fig. 6, zone CEBF contains the maximum amount of stored exergy, which is the ideal working condition during the operation of the PBTES. From equations (3), (11), (13), and (16), exergy destruction rate, ηgen, can be expressed as follows:
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Based on equation (17), ηgen is only related to the temperatures of two ends, the temperature difference, and the environment temperature.
(4) Thickness of thermocline
As shown in Fig. 6, the shape of the thermocline after one cycle is H-I. In other words, the thermal energy of zone 1 is transferred to zone 2 during one cycle. The change of entropy in zone 1 is
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where mB-H is the mass of pebbles from point B to H, and Tmid is the temperature of point M.Since the areas of zones 1 and 2 are the same, LB-H is equal to LE-I. The mass of pebbles from point E to
point I is also mB-H. The change of entropy in zone 2 is
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The total exergy destruction in one cycle is
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where Sgen,sto is the entropy generation caused by the change of entropy in zones 1 and 2.Based on equation (9), the exergy destruction caused by the heat transfer between cold and hot air is equal
to the exergy destruction caused by heat transfer from the hot to the cold side of the pebbles, which is
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Based on equation (13), (16), (20), and (21), the half thickness of thermocline, LB-H, after one cycle is
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The change in the thermocline thickness is related to the inlet temperature of cold/hot air, the length of the PBTES, and the temperature difference of heat transfer.
A comparison between the present model and a standard numerical model[13] is carried out for verification. The numerical model has been verified with experiment data and they matched well[13]. The basic model parameters of the present model are shown in Table 1, and the comparison result between the present model and the CFD model is shown in Fig.7, in which the temperature profiles of the two models after one cycle are indicated. It can be seen that the temperature profiles match with each other well. The mean relative error in the thermocline zone is 3.47%. Meanwhile, relatively larger deviation of the temperature profiles exists in two ends of the thermocline, the reason of which is that the shape of thermocline is assumed to be straight and the heat dissipation to environment is not considered in present model. The temperature of intersection point of initial profile and profile after one cycle for CFD model is slightly higher than that for present model because of heat dissipation to environment.
Table 1. The parameters for verification
Parameters ValueAxial length of PBTES (m) 10
Diameter of PBTES (m) 2Porosity of PBTES 0.4
Thigh (K) 600Tlow (K) 298
Air pressure (MPa) 7Air mass flow-rate (kg/s) 5
Thermal storage/release time (s) 3600
Figure 7. The verification of present model
4. Stable operation of PBTES As mentioned, the thermal storage capacity of the PBTES decreases during the thermal storage and release
process because of the exergy destruction. When the PBTES has two reserved parts (hot and cold), the thermal storage capacity can be kept constant in several cycles, while the outlet temperatures of cold air and
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hot air remain the same. However, as the thermocline thickens to a certain value, the outlet temperatures of cold air and hot air change. Thus, the shape of the thermocline should be modified to become thin in time to guarantee the thermal storage capacity and air outlet temperature.
In this paper, it is assumed that minimum exergy is needed for restoring the thermocline, in other words, the process of restoring the thermocline is assumed to be a reversed process. Then the transfer process of exergy destruction of the PBTES is divided into two parts. First, the exergy destruction of heat transfer of hot air and cold air is transferred to be the exergy destruction of equivalent heat transfer of pebbles. Second, a exergy supplement is made for offsetting the exergy destruction in the pebbles, as shown in Fig. 8b, which means that the exergy destruction in pebbles is reflected on the exergy supplement by outside. The supplement embodies the principle of entropy compensation in the second law of thermodynamics, that is, for changing the result of an irreversible process (unordered) to that of a reversible process (ordered), the supplement of entropy production (decrease of exergy) of surroundings is needed.
Figure 8. Restoration of thermocline by exergy supplement
As shown in Fig. 9, the original shape of the thermocline is B-E (A-B is the reserved part of hot side, cold side has a reserved part of the same length too), and after several cycles, this shape changes to its ultimate state, A-L. Based on the calculation of equation (20), the total exergy destruction generated (i.e., the supplement exergy needed from surroundings) during the time when the shape of thermocline, B-E, changes to A-L is
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The thermal storage/release time of the stable operation of the PBTES is as follows:
2424\* MERGEFORMAT ()
The cycle numbers of operation can be calculated as follows:
11
348349350351352353354355356357358359
360
361
362363364365366
367
368
369
370
371
2525\* MERGEFORMAT
()
Since the outlet temperature will change when the shape of thermocline changes to A-L, the PBTES needs an exergy supplement after n cycles. Based on equation (25), n is related to the length distribution of the PBTES and the temperatures. n is irrelevant to the mass flow rate of the air and the total mass of pebbles.
To illustrate the influence of key parameters on the performance of the PBTES, the device is analyzed under typical and variable working conditions. The present model is calculated in the software of Matlab.
Figure 9. Limited thickness of thermocline after operation of several cycles
(1) Typical working condition
As shown in Fig. 6, the total length of the PBTES is 15 m; the lengths of reserved parts, the part from point A to point B, and the part from point C to point D are all 5 m. The length of the part from point B to point F is also 5 m. The total mass of a pebble is 6×104 kg. The mass flow rate of the cold and hot air is 3 kg/s. The inlet temperature of hot air, Thigh, is 600 K, and the inlet temperature of cold air, Tlow, is 298 K. The environment temperature, T0, is 298 K. The heat transfer temperature difference between the cold and hot air, δT, is 10 K (i.e., the average value in a stable operating cycle). The heat capacity of air, cp,air, is 1.08kJ/(kg•K), and the heat capacity of pebble, cp,sto, is 0.780 kJ/(kg•K). The calculated results are shown in Table 2.
Table 2. Performance parameters of PBTES device in a typical operation condition
Parameters ValueQ/kJ 4.71×106
ηgen/% 5.17tcycle/h 1.33
t/h 6.68n 5
It can be seen that the exergy destruction rate of the PBTES is 5.17%. The operation time of one cycle is 1.33 h, and it can operate 6.68 h continuously in the condition that thermal storage capacity and outlet temperatures keep unchanged. The number of steady operation cycle is 5, that means after 5 cycles, the PBTES needs an exergy compensation. If the reserve parts are larger than 5 m, the number of steady operation cycle will increase.
(2) Variable working conditions
The effect of Thigh and δT on the performance of the PBTES is analyzed in this section, keeping other basic parameters unchanged. In practice, the mass flow rate of air and working pressure affects the temperature
12
372
373
374375376377378379
380
381
382
383384385386387388389
390
391392393394395
396
397398
difference, δT. In this paper, the influence of all unstable factors, which has an effect on δT, on the performance of the PBTES results in the influence of δT on the performance of the PBTES.
Based on equations (4) and (11), the thermal storage capacity, Q, has a linear relationship with Thigh, but it is irrelevant to δT. Based on equations (12) and (13), the thermal storage/release time of one cycle, tcycle, is irrelevant to Thigh and δT. Based on equations (16), (23), (24), and (25), the variation tendencies of the stable operation time of thermal storage/release, t, and cycle number, n, with δT and Thigh are consistent. The exergy destruction rate ηgen and cycle number n with Thigh and δT are shown in Figs. 10 and 11 by calculating equations (17) and (25).
As shown in Fig. 10, the exergy destruction rate, ηgen, increases with δT and decreases with Thigh. Meanwhile, ηgen is more affected by Thigh and tcycle when Thigh is lower. The reason is that the maximum stored exergy is invariable when Thigh is constant, whereas the exergy destruction of heat transfer between cold and
hot air per unit of time, , increases with δT. Meanwhile, the thermal storage/release time of one cycle, tcycle, is constant. Based on equation (17), the exergy destruction rate, ηgen, increases with δT. When δT is
invariable, and the maximum stored exergy both decrease with Thigh, but the former is more pronounced. Thus, based on equation (17), ηgen decreases with Thigh.
Figure 10. Variations of exergy loss rate, ηgen with δT under different Thigh
Figure 11. Variations of cycle number, n with δT under different Thigh
As shown in Fig. 11, the cycle number, n, decreases with δT and increases with Thigh. The reason is that when Thigh is invariable, the stable operation time of thermal storage/release, t, decreases with δT, whereas the thermal storage/release time of one cycle, tcycle, is irrelevant to δT, as shown in equations (16), (23), and (24). Based on equation (25), the cycle number, n, decreases with δT. When δT is invariable, the total exergy
13
399400401402403404405406407408409
410411
412413
414
415
416
417
418419420421
destruction of the pebbles, Isto, and the exergy destruction of the heat transfer between cold and hot air per unit
of time, , both decrease with Thigh. Being affected by the two factors, the stable operation time of thermal storage/release, t, increases with Thigh. Thus, the cycle number, n, rises with Thigh from equation (25).
Therefore, in the PBTES operation, when the inlet temperature of hot air, Thigh, is high (Tlow is invariable), the temperature difference, δT, is low and the performance of the PBTES is significant. This condition also produces less exergy destruction and a larger cycle number.
5. ConclusionWhen the outlet temperature of cold and hot fluids, and thermal storage capacity are strictly required by a
thermodynamic system, a conventional PBTES encounters problems, such as unstable working condition, and the calculation of the unstable process becomes complex. To solve these problems, an idea is proposed to convert the unsteady heat transfer process of the PBTES with fixed outlet temperatures of cold and hot fluids to a steady process for a counterflow heat transfer. A thermodynamic PBTES model is developed based on the steady process at a corresponding time, which can be used to calculate the key performance parameters of the PBTES, including thermal storage capacity, thermal storage/release time, exergy destruction of heat transfer per unit of time, exergy destruction rate, and thickness of thermocline.
The exergy destruction occurred in the PBTES can be indicated by the exergy destruction of equivalent amount of heat transfer between cold and hot fluids at a corresponding time based on the analysis of the stable thermodynamic model. The equivalent heat transfer changes the thickness of the thermocline and the thermal storage capacity of the PBTES with fixed outlet temperatures of cold and hot fluids. More exergy destruction leads to a thicker thermocline.
The stable operation of the PBTES, through exergy supplement, is analyzed. It is found that higher inlet temperature of hot air and lower temperature difference can produce less exergy destruction and more cycle numbers of stable operation. For a given inlet temperature of hot air, the decrease of exergy destruction rate and cycle number get slow with the increase of temperature difference. For a given temperature difference, the decrease of exergy destruction also gets slow, while the increase rate of cycle number almost keep unchanged.
Acknowledgment: The authors acknowledge the support provided by National Key R&D plan (No. 2017YFB0903605), National Natural Science Foundation of China (No.51806210), Newton Advanced Fellowship of the Royal Society (No. NA170093) , and International Partnership Program, Bureau of International Cooperation of Chinese Academy of Sciences (No. 182211KYSB20170029).
Nomenclature
t timecp,hotair heat capacity of hot aircp,coldair heat capacity of cold air
mass flow rate of hot air
mass flow rate of cold air△Thot local temperature change of hot air△Tcold local temperature change of cold air△Usto change of thermal energy stored in pebblesmass flow rate of cold and hot air
Hhotair input of thermal energy from hot airHcoldair thermal energy transferred from pebbles to cold aircp, air constant heat capacity of cold and hot air△T temperature change of cold and hot airThigh inlet temperature of hot airTlow inlet temperature of cold aircp,sto constant heat capacity of the pebbles.δTsto temperature change of the pebblesδT temperature difference between cold and hot air△Es,sto change of exergy stored in the pebbles during thermal storage process
△En,sto change of exergy stored in the pebbles during thermal release processEhotair exergy change of hot air during the thermal storage processEcoldair exergy change of cold air during the thermal release processIs exergy destruction during the thermal storage process.In exergy destruction during the thermal release process.△Esto change in exergy stored in the pebbles during one cycle
Isnexergy destruction in the heat transfer process between cold air and hot air under the temperature difference, δT.
Q constant thermal storage capacity (the maximum heat stored in one cycle)L lengthρm,sto mass of pebbles contained per unit lengthMair mass of air flowed through the PBTES during the thermal storage or release processtcycle thermal storage/release time in one cycle
entropy change of hot air per unit of timeentropy change of cold air per unit of time
T0 environment temperatureexergy destruction per unit of time
entropy generation in the heat transfer process per unit of timeηgen exergy destruction rateIsto total exergy destruction in one cycleLB-H half thickness of thermoclinen cycle numbers of operation
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