Design and Performance of an Innovative Four-Bed, Three-Stage Adsorption Cycle

The design of a four-bed three-stage adsorption cycle has been proposed to reduce the volume of the six-bed three-stage adsorption cycle. A simulation model for the proposed innovative cycle was developed to analyse the influence of cycle time on the system performance identifying the specific cooling power ( SCP ) and coefficient of performance ( COP ). A particle swarm optimization (PSO) technique was used to optimize the cycle time enabling us to maximize the SCP . PSO results showed that the optimal cycle time was decreased with heat source temperature and SCP value was proportional to heat source temperature. It was found that the proposed cycle could be driven by waste heat as low as 40 °C, along with coolant at 30 °C. Comparative study of optimized result indicated that the proposed cycle increased the performance significantly over a whole range of temperatures from 40 to 70 °C and reduced two adsorbent beds, compared to the six-bed three-stage cycle.


A
heat transfer area (m 2 ) C p specific heat (J kg −1 K −1 ) D so surface specific heat (m 2 s −1 ) E a activation energy (J kg −1 ) ksap overall mass transfer coefficient L latent heat of vaporization (J kg −1 ) m  mass flow rate (kg s −1 ) P s saturated vapour pressure (Pa) q fraction of water content that can be adsorbed by the adsorbent (kg kg −1 ) q* fraction of water content that can be adsorbed by the adsorbent under saturation condition (kg kg −1 ) Q st isosteric heat of adsorption (J kg −1 ) R gas constant (J kg −

Introduction
Utilization of waste heat and renewable energy can lead to significant contributions in the reduction of the consumption of fossil fuels.One of the key technologies that allow the utilization of relatively low-grade heat sources, typically below 100 C, is adsorption heat pump system.It has been established that adsorption heat pumps can produce cooling power utilizing a heat source consuming temperature as low as 40 °C, along with a coolant at 30 °C if a three-stage regenerative scheme is employed [1].Low-temperature waste heat is commonly found as part of any process in industry or in renewable heat sources, such as solar thermal energy, even in moderate solar radiation regions around the World.Moreover, in most cases, adsorption systems neither use ozone layer depleting gases such as CFCs/HCFCs, nor fossil fuel and electricity as their driving source.Therefore, it is widely considered that adsorption heat pump systems are environmentally-friendly and have energy-saving potential.Moreover, they can not only serve the needs for air-conditioning, refrigeration and ice making purposes, but also can meet the demand pure water through adsorption desalination processes [2][3][4][5][6][7].
A basic conventional single-stage adsorption system comprises of two adsorbent beds, one condenser and an evaporator [8,9].Many researchers have studied advanced single-stage adsorption systems such as two-bed mass recovery, three-bed, three-bed mass recovery, and two-bed heat recovery adsorption cycles to improve the performance of the single-stage adsorption cooling system [10,11].However, the required driving temperature for advanced single-stage adsorption systems is 80 °C or higher.
Saha et al. [12] proposed a design of a four-bed two-stage adsorption chiller to reduce the temperature of the heat source.They showed numerically that the chiller can utilize solar/waste heat of temperatures below 70 °C.Alam et al. [13] and Hamamoto et al. [14] identified the influence of design parameters and operating conditions on the performance of the chiller proposed by Saha et al. [12].Moreover, the modification of the chiller in terms of operating conditions such as re-heat schemes was investigated by Khan et al. [15][16][17], Alam et al. [18] and Farid et al. [19].These investigations identified that it can work with a heat source temperature as low as 45 °C [20].Recently, experimental study on a two-stage adsorption freezing machine driven by a low temperature heat source was done by Wang et al. [21].However, the coefficient of performance (COP) of a two-stage adsorption cycle was found to be lower than that of the advanced single-stage chillers [22,23].
To increase the performance while keeping the heat source temperature around 45 °C, Saha et al. [24] proposed a design of advanced three-stage chiller and then investigated the effect of different parameters (e.g., cycle time, heat exchanger flow rate and temperature) on system performance [1].Khan et al. [25] improved system performance by adding a re-heat scheme.It was shown that the COP of a three-stage adsorption chiller reaches up to 0.3 at 70 °C heat source temperature with a cycle time of 4400 s and longer, the heat source temperature can be reduced to as low as 40 °C along with a coolant at 30 C.
The three-stage chiller has six beds (i.e., a pair of beds in each stage) and these beds occupy a large part of the system.Accordingly, the footprint of the three-stage machine is larger than that of the single-and two-stage adsorption machines for the same cooling capacity.This is the drawback of multi-stage systems, therefore, any reduction of the number of adsorbent beds directly contributes to the reduction of system volume.
In the present study, first a design of a four-beds three-stage adsorption system was proposed, instead of a six-beds three-stage adsorption system.Second, cycle time was optimized to maximize the performance of the proposed and former six-bed system to identify the best performance.The particle swarm optimization method was used to obtain the maximum cooling output under standard working conditions.Finally, the performance of the proposed cycle was compared with the optimal performance of former six-bed three-stage adsorption systems.

Description of Three-Stage Adsorption Cycle
Advance single-stage adsorption system is used to achieve better performance, and two-and three-stage cycles were proposed to utilize lower heat source temperature.A single-stage adsorption cycle is not operational when the heat source temperature is below 60 °C along with a coolant of temperature 30 °C or higher.For practical utilization of these temperatures in adsorption system operation, higher staged regeneration is necessary.In this regard, Saha et al. [26] first proposed a six-bed three-stage adsorption cycle.The schematic diagram of the system is shown in Figure 1a and its operation mode is given in Table 1a.In this case, three pairs of adsorber/desorber heat exchanger beds i.e., Bed 1-Bed 2, Bed 3-Bed 4, and Bed 5-Bed 6 were used in the lower cycle, intermediate cycle and upper cycle, respectively.In the proposed four-bed three-stage cycle, in order to reduce the physical geometry of the system, four adsorption/desorption beds in the intermediate and upper cycles were replace by two adsorption/desorption beds.To make the adsorption cycle competitive in the market, it is necessary to consider economical issues as well as the total size of system.A schematic of the proposed cycle with modifications (indicated by arrow marks) is shown in Figure 1b.In the proposed case Bed 3 and Bed 4 can work instead of Bed 3-Bed 4 and Bed 5-Bed 6 in the former cycle, respectively, whereas Bed 1 and Bed 2 were kept as in the former system.The operation mode of the proposed system is given in Table 1b.Operation strategy has taken into account the relatively longer adsorption time compared to the desorption time, whereas in the former cycle adsorption and desorption time were chosen equal.In the first half cycle of the proposed system, Bed 3 adsorbs vapour from Bed 1 and then it transfers to the condenser via Bed 4. Similarly, in the second half cycle, it adsorbs vapour from Bed 2 and then transfers in to the condenser in similar way.As a result, Bed 3 and Bed 4 are completing two full cycles, while Bed 1 and Bed 2 are completing only one full cycle operation at the same time.
The proposed three-stage adsorption cycle consists of four adsorber/desorber heat exchangers, namely Bed 1, Bed 2, Bed 3 and Bed 4, one evaporator and one condenser (Figure 1b).Silica-gel and water have been chosen as adsorbent-refrigerant pair because the regenerative temperature of silica gel is lower than that of other adsorbents and water has large latent heat of vaporization.The main advantage of the three-stage cycle is the possible utilization of lower grade heat sources than possible with single-stage and two-stage adsorption cycles.For example, a temperature of 50 °C is enough to drive the three-stage cycle.However, such low temperature is not enough to pressurize the vapour from the evaporator up to the condenser pressure in a single stage system.The vapour needs to be pressurized three times in the cycle through three-stage adsorption-desorption processes.The first stage pulls up the vapour from evaporator pressure to a certain pressure level, the second stage pulls up an intermediate pressure and then the third stage pushes up the vapour from the intermediate pressure to the condenser pressure.
The Bed 1 and Bed 2 are working at low temperature and pressure (close to the evaporator pressure), Bed 3 is working at intermediate temperature and pressure and Bed 4 is working at higher temperature and pressure (close to the condenser pressure).The conceptual Dühring diagram is shown in Figure 2. The cyclic diagrams a-b-c-d-a, e-f-g-h-e and i-j-k-l-i indicate the lower, intermediate and upper cycle, respectively.All beds are completing four thermodynamic stages, i.e., pre-heating, desorption, pre-cooling and adsorption stage, which are accordingly followed by the sides a- b, b-c, c-d  and d-a, and e-f, f-g, g-h and h-e for the lower and intermediate cycle, respectively.Similarly, the sides i-j, j-k, k-l and l-i indicate the modes pre-heating, desorption, pre-cooling and adsorption stage of Bed 4, respectively.To complete full cycle operation, each bed is to complete eight operational modes, namely A, B, C, D, E, F, G and H, as shown in Table 1b, whereas the former cycle needed four operation modes.The number of operation mode was increased to choose a longer adsorption time compared to desorption time.In Mode A, Bed 1 and Bed 2 are in a cooling state when cooling water is passed through both beds and are connected with the evaporator to adsorb refrigerant vapour from it.This process is called adsorption mode.At the same time the evaporator's lower temperature and pressure is maintained so that water could easily vaporize by seizing heat from chilled water.Bed 2 remains in the adsorption stage up to end of mode E. On the other hand Bed 3 and Bed 4 are connected each other and Bed 3 is heated with hot water to released it own refrigerant vapour to Bed 4. At the same time Bed 4 is cooled with cooling water passing through it to release the adsorption heat.
In mode B, Bed 1 is disconnected from the evaporator and the other two beds (i.e., Bed 3 and Bed 4) are disconnected each other by closing the connecting valve "V3".Then Bed 1 and Bed 4 are heated by hot water to increase their pressure, which is called pre-heating mode and Bed 3 is cooled down by cooling water to reduce its pressure, which is called pre-cooling mode.When the pressures of Bed 1 and Bed 3 become nearly equal, then both are connected each other by opening the connecting valve (i.e., V2), which is starting point of mode C. In this mode, Bed 1 is heated by hot water and Bed 3 is cooled by cooling water, which is called desorption mode for Bed 1 and adsorption mode for Bed 3. Bed 1 is desorbing its adsorbed vapour to the Bed 3, and Bed 3 is adsorbing vapour from Bed 1 during this mode.On the other hand when the pressure of Bed 4 becomes equal to the condenser pressure then it connected with the condenser by opening the connecting valve (V4) to release its own refrigerant vapour to condenser and it is called the desorption of Bed 4 (i.e., mode C).In this mode, adsorbed refrigerant is removed from the bed and goes to the condenser.Then refrigerant vapour is condensed inside condenser and condensation heat is removed by cooling water.Finally, condensed refrigerant goes to the evaporator via a U shape tube.
Mode D is called pre-cooling for Bed 1 and Bed 4, and pre-heating mode for Bed 3, while beds are isolated from other parts of the cycle.In this case Bed 1 is cooled by cooling water to reduce its pressure to the evaporator pressure and then it is connected with the evaporator by opening the connecting valve (V1), which is called adsorption process and starting point for mode E. This process will continue to the end of mode H.Meanwhile, Bed 3 is heated by hot water to increase its pressure and Bed 4 is cooled with cooling water passing through it.When the pressure of both beds becomes equal then both are connected with each other by opening connecting valve V3.This is the starting point of Mode E, which is similar to mode A. Through these modes the first half cycle (from mode A to D) of Bed 3 and Bed 4 are completed and the second half cycle (from mode E to H) is started.The mechanism of modes E, F, G and H of Bed 3 is similar to the modes A, B, C and D, but during these modes, Bed 3 is connected with Bed 2 instead of Bed 1 (i.e., mode G) and vapour is transferred among Bed 2 and Bed 3. In the second half cycle Bed 4 acts exactly the same as in the first half cycle.

Simulation Methods and Materials
In the analysis, it is assumed that the temperature, pressure and refrigerant concentration throughout the adsorbent bed are uniform.For the specific heats of metals, adsorbent and adsorbate inside the heat exchanger and for adsorption heat, average values are adopted throughout the entire temperature range.Moreover, both adsorption and desorption occur at constant pressure, thus specific heat is chosen as C p .Refrigerant mass flows between evaporator and adsorber, condenser and desorber, adsorber and desorber are taken as equivalent.Total work of the system is considered as zero and total heat is mainly taken into account.Under the above assumptions, the energy balance of the system is explained in following sections.

Adsorption and Desorption Energy Balance
The heat transfer equation for the adsorption/desorption bed can be written as: where heat transfer fluid temperature The energy balance equation for the adsorption/desorption bed is given below: The same equation is used for adsorption and desorption purposes.During adsorption both hot col in T / and hot col out T / represent the cooling water temperature and during desorption they represent the hot water temperature.The left-hand side of the equation indicates the amount of sensible heat required to cool or heat the silica-gel (s), water (w) as well as metallic parts of the heat exchanger (hex) during adsorption or desorption.The first term of the right hand side stands for the total amount of heat released to the cooling water upon adsorption or provided by the hot water upon desorption, the second term for the adsorption heat release or desorption heat input and last term for the sensible heat of the adsorbed vapour.α is either 0 or 1 depending on whether the bed is working as a desorber or adsorber.The heat balance equation [Equation (2)] does not consider the external heat losses to the environment.

Evaporator Energy Balance
The heat transfer equation of evaporator can be expressed as: where ) (t W e w is the mass of water inside the evaporator that is changing with cycle time due to adsorption and desorption process and L w is the latent heat of vaporization.The left hand side of the equation indicates the sensible heat required by the liquid refrigerant (water) and the metal of heat exchanger tube of evaporator.The first term of the right hand side represents the latent heat for the adsorbed refrigerant; the second term represents the sensible heat required to cool down the incoming condensed refrigerant from the condensation temperature; and the last term represents the total amount of heat loss by the chilled water.

Condenser Energy Balance
The heat transfer equation of the condenser can be written as: ) exp where c hex A and c hex U is the heat transfer area and heat transfer coefficient of condenser, respectively.
The energy balance equation of the condenser is as follows: The left hand side of the equation represents the sensible heat required by the liquid refrigerant and the metallic parts of heat exchanger tubes due to the temperature variations in the condenser.The first part of the right hand side represents the latent heat of vaporization (L w ) for the desorbed refrigerant; the second term represents the sensible heat due to vapour transfer from desorber to the condenser; and the last term takes into account the total amount of heat released to the cooling water.

Adsorption and Desorption Rate
The adsorption/desorption rate can be express as [27]: des ads des ads q q ksap dt dq    (7) where the overall mass transfer coefficient (ksap) for adsorption is given by: 2 ) /( 15 Here R p denotes the average radius of a silica gel particle.The adsorption rate is controlled by surface diffusion inside the gel particle and surface diffusivity (D S ), which can be expressed by Arrhenius equation as a function of temperature as: where D so is a pre-exponential term whose value is taken as 2.54 × 10 −4 m 2 s −1 [28].E a denotes activation energy; R denotes gas constant and T stands for temperature.* q is the adsorption uptake at equilibrium state that can be expressed as [29]:

Mass Balance
Total mass of refrigerant in the system was assumed constant.The rate of change of the mass of liquid refrigerant can be expressed through the following equation: where b s W is the mass of silica gel packed in each bed and e w W is the mass of liquid refrigerant in the evaporator.Superscripts "des-con" and "ads-evp" stand for the vapor flow from desorber to condenser and evaporator to adsorber, respectively.In this case, vapour transfer of Bed 3 was not taken into account.It was assumed that it desorbed all refrigerant vapour it adsorbed.

System Performance
The performance of three-stage adsorption cycle is mainly characterized by specific cooling power (SCP), which is the cooling capacity per unit mass of adsorbent and coefficient of performance (COP), which is the ratio between heat released by the evaporator and heat source input to the bed, these can be calculated by using the following equations: Specific cooling power (SCP):

Optimization Procedure
A complete simulation program was developed using MATLAB (MathWorks, Natick, MA, USA) software to solve the equations mentioned above.Adsorption/desorption rate equation, energy balance equation of beds, condenser and evaporator, and mass balance equation have been solved simultaneously using MATLAB ode45 solver.All input parameters such as adsorbent-refrigerant properties, flow rates of heat transfer fluids and heat exchangers specifications were initially given as those for which the system cyclic operation can be continued.
Particle swarm optimization (PSO) methodology was first introduced in 1995 [30] to optimize the nonlinear function based on a swarm of particles (e.g., birds and fishes).These particles are moved around in the search-space according to formulae.The movements of the particles are guided by their own best known position (i.e., personal best) as well as the entire swarm's best known position (i.e., group best).When better positions are being discovered these will then come to guide the movements of the swarm.The formula to update a particle's velocity and position can be written as: where x i and v i are the position and velocity of the ith particle.p i is the personal best position that the ith particle had reached and p g , is the group best position that all the particles had reached.t represents the iteration number.r 1 and r 2 are two random numbers between 0 and 1. C 1 and C 2 are two constant numbers, which are often called the acceleration coefficients.In this study, PSO was applied to optimize the cycle time based on maximum value of SCP, whereas SCP was chosen as the objective function and cycle time (i.e., τ 1 and τ 2 ) was chosen as the variable.In PSO, a particle holds the values of variables and updates the values toward the optimal solution.After a number of iterations, all the particles hold the same value and the objective function value is maximized.In this case, the number of particles and the number of iterations were considered as 24 and 1500, respectively.It was observed that all particles reached their best position around 500 iterations.

Results and Discussion
The simulation program developed for the proposed four-bed three-stage adsorption cycle was operated by adopting the basic input parameters and standard working conditions as presented in Table 2 [23].The cycle time, operating time of each mode (i.e., τ 1 and τ 2 ), is one of the most important parameters of the system and effectively influences the overall performance.The specific cooling power (SCP) represents the cooling capacity per unit quantity of adsorbent, and therefore it is a useful index for minimizing the size of the system.At first the cycle time was optimized based on the maximization of SCP using the PSO procedure.Cycle time of the system was optimized using various heat source temperatures varying from 40 to 70 °C with an increment of 5 °C by keeping outlet temperature of the chilled water at 9 °C.Accordingly, the chilled water flow rate was optimized simultaneously with cycle time to adjust the chilled water outlet at a constant temperature.Figure 3 shows that optimum cycle time of the proposed system decreases with the increase of heat source temperature.Total optimal cycle time is the sum of the cycle time of all modes (Table 1).The cycle time ratio is defined as the ratio of τ 1 or τ 2 time of different operation modes to the total cycle time.Figure 4 represents the optimal cycle time ratio of the system for different heat source temperatures.It was observed that the cycle time ratios for τ 1 and τ 2 were slightly decreased and increased with heat source temperature, respectively.Similarly, decreasing optimal cycle time with heat source temperature was also observed for the former six-bed three-stage cycle (Figure 5), but the optimal cycle time ratio of the former cycle indicated an opposite trend, i.e., τ 1 and τ 2 were decreased and increased with heat source temperature, respectively (Figure 6).It is noted that both systems are giving shorter pre-cooling/pre-heating (i.e., τ 2 ) times.
The variation of chilled water flow rate with heat source temperature for both cycles is shown in Figure 7.It was observed that average chilled water flow rate increases with the increase of driving heat source temperature.Since, heat removal from the chilled water increased with heat source temperature and to keep a constant average chilled water outlet temperature of 9 °C, chilled water flow rate was increased.It was observed that the chilled water flow rate was similar for both cycles, This is due to both systems having two beds (i.e., Bed 1 and Bed 2) in the bottom cycle with equal masses of silica gel and producing similar effects.

Proposed Former
The optimal performance of the proposed system is shown in Figure 10.It is apparent that the cycle can operate with a low temperature heat source as low as 40 °C and with a coolant at 30 °C.The SCP increased with the increase in heat source temperature, whereas COP increased with heat source temperature up to 55 °C and then it was decreased slightly with heat source temperature.An increase of heat source temperature with a fixed heat sink causes the system to increase the amount of refrigerant adsorption/desorption promptly, resulting in higher SCP at higher heat source temperature.Due to a similar effect, the total optimal cycle time was found to be decreased with heat source temperature, as shown in Figure 11.It is generally observed that the behaviour of COP of adsorption cycle increased with longer cycle time.However at lower temperature (<55 °C), cycle time was found to be increased, whereas COP was found to be decreased because at lower driving heat source temperature longer time is needed for effective desorption.The maximum COP was observed to be 0.218 at 55 °C heat source temperature.
To compare the performance of different adsorption cycles fairly, it is emphasized that the optimized performance of each system should be compared.From this point of view, the optimal performance of the proposed cycle was compared with the optimal performance of the former six-bed three-stage cycle as shown in Figure 10.It can be seen from this figure that both SCP and COP of the proposed cycle were increased significantly over the whole range of studied heat source temperatures, which is the great benefit of the proposed cycle.Comparison of optimal total cycle time of the proposed cycle with the former six-bed three-stage cycle is also shown in Figure 11.It was observed that proposed cycle provided a little bit longer optimal cycle time at each heat source temperature.The higher SCP value of the proposed cycle is due to the effects of the lower total mass of adsorbent (silica-gel) of the system.Because the denominator values as described in Equation ( 13) is lower as the four beds silica-gel mass is smaller (i.e., 64 kg) than that of six beds of silica-gel (i.e., 96 kg).Heat release as described in the numerator is the same for both systems because both cycles have two beds in lower temper and pressure cycle and producing a similar cooling effect to the evaporator.It is expected mathematically that the SCP of four-bed system is larger than that of six-bed system by a factor of 3/2, which was exactly the value achieved in the proposed case.
Although the total cycle time of the proposed system is little bit longer, the proposed cycle provided around 12% to 34% higher COP value compare to the former cycle.This is due to a longer adsorption time as well as a shorter desorption time, which is shown in Table 1.Not only that in proposed case, it was observed that adsorption time is more than twice longer than the desorption time as well as the sum of desorption time, pre-cooling and pre-heating time.Aristov et al. [31] also gave similar practical recommendations to manage cooling cycles with non-equal durations of adsorption and desorption time.Due to the longer adsorption time the total adsorption of the proposed cycle is higher than that of the the former cycle.As a result COP as well as SCP were increased over the whole range of heat source temperatures.
A comparative study between the former six-bed three-stage and the proposed four-bed three-stage adsorption system along with improvements due to modification are given in Table 3.Thus, it can be concluded the proposed cycle can significantly reduce the overall size of the three-stage adsorption system while providing an increase in the overall performance.

Conclusions
A modified design of a four-bed, three-stage adsorption cycle which can be powered by low-temperature thermal energy sources as low as 40 C has been introduced.A simulation program has been developed to investigate the performance of the system and cycle time was optimized using a particle swarm optimization method.It was observed that the specific cooling power of the proposed innovative cycle increased with heat source temperature.The maximum coefficient of performance of the system was observed for a regeneration temperature at 55 °C.The optimal cycle time was observed to be largely dependent on the corresponding heat source temperature and it decreased with higher heat source temperature when the chilled water outlet temperature was fixed at 9 °C.With the present innovative design, the overall system footprint can be significantly reduced compared to the six-bed three-stage cycle.Moreover, both SCP and COP were also increased significantly over the whole range of heat source temperatures as compared to that of six-bed three-stage cycle.The proposed cycle could be useful from the economical point of view due to its higher performance and significant reduction in system footprint.

Figure 1 .
Figure 1.Schemata of (a) six-bed, three-stage chiller of Saha et al. [1] and (b) proposed design of four-bed, three-stage adsorption cycle with operation mode A (see Table1b).
Figure 1.Schemata of (a) six-bed, three-stage chiller of Saha et al. [1] and (b) proposed design of four-bed, three-stage adsorption cycle with operation mode A (see Table1b).

Figure 2 .
Figure 2. Conceptual Dühring diagram of the proposed cycle.
cooling water temperature upon adsorption and hot water temperature upon desorption.b T stands for adsorption/desorption bed temperature.Heat transfer is fully depending on heat transfer coefficient b hex U and heat transfer area b hex A of bed.

T
indicate the chilled water inlet and outlet temperature, respectively; T e is the evaporator temperature; and e hex A and e hex U are the heat transfer area and heat transfer coefficient of evaporator, respectively.The energy balance equation of evaporator is given below: indicating the total cycle time and total mass of silica-gel of the system.Coefficient of performance (COP): indicating the hot water flow rate in the desorption bed; and hot in T and hot out T are indicating hot water inlet and outlet temperature, respectively.

1 9 °CFigure 3 .
Figure 3. Optimal cycle time of the proposed cycle for different heat source temperature.

Figure 5 .Figure 6 .Figure 7 .
Figure 5. Optimal cycle time of former six-bed three-stage cycle for different heat source temperature.

Figure 10 .
Figure 10.Comparison of optimal SCP and COP of proposed cycle with the former six-bed three-stage cycle.

Figure 11 .
Figure 11.Comparison of optimal cycle time of proposed cycle with the former six-bed three-stage cycle.

Table 1 .
Operation mode of former six-bed three-stage cycle and proposed four-bed three-stage cycle with optimal cycle time (in second) at 50 °C heat source temperature. (a

Table 2 .
Basic input parameters and standard working conditions.
Optimal cycle time ratio of the proposed cycle for different heat source temperature.

Table 3 .
Comparative study of former six-bed three-stage and proposed four-bed three-stage cycle.