Liquid Desiccant System Research Articles

Study on the heat transfer characteristics of CO2 and liquid desiccant CaCl2 aqueous solution in the gas cooler and evaporator

The heat exchangers are essential components in the CO2 heat pump-driven liquid desiccant dehumidification system since the heat transfer performance of heat exchangers directly affects the energy efficiency of the system. However, the heat transfer characteristics of CO2 and liquid desiccant in the heat exchanger have yet to be fully revealed, resulting in a lack of appropriate strategies to improve the heat transfer performance. To address this issue, numerical gas cooler and evaporator models are developed and verified. The effects of parameters such as the weight concentration of the CaCl2 aqueous solution, the CO2 inlet pressure, mass flow rates, and the length of the heat exchanger on the heat transfer performance are investigated respectively under various operating conditions. Results show that the heat exchange load of the gas cooler can be significantly impacted by various inlet parameters of both fluids. In particular, the increase in the weight concentration from 36 to 54 wt.% deteriorates the heat exchange load by over 15 %. Nevertheless, the heat exchange load of the evaporator is hardly influenced by the weight concentration. This is attributed to a great gap in the two fluids’ heat capacity rates, causing the evaporator to be in the maximum heat transfer state. The sign of the maximum heat transfer state of the heat exchanger is whether the thermal effectiveness is close to 1, which is related to the heat transfer area. Therefore, modifying the heat exchanger length could avoid the thermal effectiveness approaching 1. Results indicate that the appropriate length of the evaporator is 3–4 m under the designed operating conditions.

Exergy analysis on the low flow rate of solution in the atomization-based liquid desiccant system

The purpose of this study is to analyse the proposed liquid desiccant absorber with low solution flow rate compared with the conventional packbed-type absorber. The total exergy destruction and exergy efficiency were estimated to assess the system performance. To determine the total exergy destruction and exergy efficiency, it was predicted that the specific thermal and chemical exergy in the inlet and outlet both air and solution side of the absorber. The results indicated that the average total thermal and chemical exergy destruction is 0.054 kW and 0.080 kW in the proposed system while it is 0.292 kW and 0.093 kW in the conventional absorber. The exergy efficiency is 0.573 and 0.114 on average in the proposed and conventional absorber, respectively.

Study on the optimal solution recirculation ratio of liquid desiccant dehumidification system based on matching rate evaluation

The recirculation of dehumidification and regeneration solutions has shown promising moisture removal rates and energy-saving effects in liquid desiccant dehumidification systems. The different combinations of dehumidification solution recirculation ratio and regeneration solution recirculation ratio under varying operating conditions can have different impacts on the system. Nevertheless, the system performance under full operating conditions has not been comprehensively studied. To address this research gap, this study proposes a liquid desiccant dehumidification system with a solution recirculation structure, defines a new evaluation index — matching rate, and conducts a parameterized study and optimization of the steady-state performance of the liquid desiccant dehumidification system with a solution recirculation structure within the range of solution recirculation ratios under all operating conditions, by integrating the system cooling capacity and energy effectiveness ratio. The performance of the system is studied by varying seven key parameters: dehumidification solution recirculation ratio, regeneration solution recirculation ratio, total solution mass flow rate, dehumidification solution inlet temperature, regeneration solution inlet temperature, and ambient air temperature-humidity ratio. The results show that under design conditions, the solution recirculation ratio within the full operating range can process air to a range of 17.8℃～18.5℃ and 12.5 g/kg～17.4 g/kg. The maximum system cooling capacity, energy effectiveness ratio and matching rate values are 21.5 kW, 0.21 and 0.42 respectively. The recommended (dehumidification solution recirculation ratio, regeneration solution recirculation ratio) schemes for high, medium and low demand of system cooling capacity are (0.7～0.9, 0～0.5), (0.4～0.6, 0～0.7) and (0～0.3, 0.4～0.7) respectively, based on the requirements of system cooling capacity.

Artificial neural network-based modeling for the prediction of heat and mass transfer coefficient of the adiabatic liquid desiccant system

This work, based on the data obtained from the literature reported by (Varela et al., 2018), aims to use the artificial neural network approach to predict the heat and mass transfer in a dehumidifier system, using lithium chloride as a liquid desiccant. A neural network model was developed in MATLAB environment based on multilayer perceptron that included an input, hidden, and output layer. The network input parameters are air velocity, air temperature, air humidity ratio, liquid desiccant temperature, liquid flow rate, and liquid desiccant concentration. The network output includes two variables which are the heat transfer coefficient (Kh) and mass transfer coefficient (Km). The performance of the ANN model was evaluated using the statistical parameters between the prediction results and experimental values. The performance regression yields R2 and MSE values of 0.9344 and 9.0032, respectively, for the test data set of heat transfer coefficient (Kh). Moreover, for the mass transfer coefficient (Km), the regression parameter R2 and MSE values for the ANN tests were found to be 0.9657 and 2.0414, respectively. In addition, air velocity, air temperature, solution mass flow rate, and solution concentration are the most influential parameters on the heat and mass transfer between the air and liquid desiccant.

Membrane-based packed-sheet liquid desiccant dehumidification system

Humidity control plays a vital role in both buildings and greenhouses. The thermally-driven liquid desiccant dehumidification systems are getting an increasing traction because of their high moisture removal capacity and the possibility of their integration with waste-heat or renewable energy sources. In this work, a novel design concept for liquid desiccant dehumidification systems is proposed to overcome the practical challenges of the conventional systems, such as solution carryover, crystallization, and corrosion. The presented system utilizes a compact “packed-sheet” sorption bed that houses spherical membrane-based micro-absorbers with LiBr as a liquid desiccant. A bench-scale system is designed and tested under the typical dehumidification working conditions. The experimental results showed that the proposed design has up to two-fold higher moisture removal rate per volume (MRR = 75 g/s-m3) than a conventional LiBr liquid desiccant dehumidification system (MRR = 35 g/s-m3). In addition, a one-dimensional coupled heat and mass transfer mathematical model is developed to simulate the transient behavior of the proposed system. An optimization study, using the validated model, suggested that the performance of proposed design can be maximized to realize moisture removal rates of up to 135 g/s-m3 (270% higher than the conventional liquid desiccant systems) with a coefficient of performance of 0.25.

Performance characteristics of a compact self-circulating liquid desiccant air dehumidification system coupled with a heat pump: A comprehensive parametric study

Integration of a heat pump with a liquid desiccant system for air dehumidification has received growing attention in recent years in view of its more compact size and lower electricity consumption. This paper presents a compact combined heat pump and self-circulating liquid desiccant air dehumidification system targeted at small cooling load applications. The compactness lies in the integration of the dehumidifier and evaporator as well as the integration of the regenerator and condenser. Besides, the design of solution self-circulation (SSC) is intended for energy savings and the auxiliaries are employed to help achieve energy matching between the heat pump and the air dehumidification system. First, a comparative study is performed to identify the more energy efficient one of the two strategies for energy matching. Based on this, a thermodynamic model-based parametric study is conducted to investigate the effects of various influencing factors on the energy matching relationship and performance of the proposed system, as well as the energy-saving potential relative to the basic system without SSC. The results show that there exists an optimum solution self-circulation ratio (SCR) to achieve the maximum COP under a given condition. Basically, the proposed system with a proper SCR could yield a significant energy-saving effect regardless of the performance variations under studied operation conditions. It is found that a reduction of 29.68 % in energy consumption and an improvement of 42.20 % in COP are achieved under the ambient condition of 30 ℃ temperature and 20 g/kg humidity ratio. Furthermore, a correlation analysis is used to reveal the association strength and direction between any two main operation and performance parameters. Finally, the heat recovery is proved to effectively reduce the cooling coils load. This paper provides an in-depth understanding of the energy matching relationship and demonstrates the energy-saving performance benefiting from SSC for future practical applications of the proposed system.

Study on the optimal circulation structure of liquid desiccant dehumidification system based on composite model

Liquid desiccant dehumidification systems have demonstrated good energy savings and dehumidification effects in extremely hot and humid areas. However, to improve system performance, the matching problem between the desiccant-regeneration solution recirculation ratio and system performance has not been thoroughly investigated. Therefore, in this study, we constructed a liquid desiccant dehumidification system with solution recirculation and set four circulation modes: NS mode without solution recirculation, CS mode with only desiccant solution recirculation, RS mode with only regeneration solution recirculation, and CRS mode with both desiccant and regeneration solution recirculation. The orthogonal test method was used to determine the influence of solution recirculation on the evaluation indicators, and multi-objective optimization of the major parameters under the four circulation modes was performed based on the NSGA-II and TOPSIS algorithms. The results showed a clear conflict between the moisture removal rate (MRR) and energy consumption (Q). In extremely hot and humid areas, a high air flow rate is suitable for use in the regenerator to process small-flow-rate solutions or set larger sizes. Under typical environmental conditions, it is recommended to set up a liquid desiccant dehumidification system with a solution recirculation structure: when MRR demand is high, the CS mode should be used, where MRR is 6.72 g/s; when Q demand is high, the RS mode should be used where Q is 35.34 kW; considering both MRR and Q effects, the CRS mode should be used where unit energy consumption moisture removal rate is 0.1196 g/(s·kW).

Numerical investigation and comparison on energy performance and exergy destruction of cascading deep dehumidification systems with dehumidification amount exceeding 95%

Liquid desiccant and desiccant wheel systems are feasible approaches for the dehumidification process in modern buildings. Some specialized buildings and industrial fields have the demand for deep dehumidification process to produce supply air with low dew point temperature below 0 °C. In this study, the numerical investigation has been conducted to evaluate the energy and exergy performance of multiple cascading deep dehumidification systems assisted with liquid desiccant and desiccant wheel under a great dehumidification gradient. To process the air from 21 g/kg to 1 g/kg, the results indicate that the cascading liquid desiccant and desiccant wheel deep dehumidification system (LD-DW system) driven by two stages of heat pump exhibits the lowest demand of power consumption compared with the conventional dehumidification systems. It also shows less exergy loss and significantly higher exergy efficiency than other systems. The effects of influencing parameters on the dehumidification performance are also evaluated. The LD-DW system can effectively match the air state with the proper treatment dehumidification process, thus achieving a superior coefficient of system performance (COPsys) of 3.70, and it is an energy and exergy efficient method for advancing the deep dehumidification process.

Performance evaluation of an innovative liquid desiccant air-conditioning system directly integrating a heat pump with two membrane energy exchangers

For high moisture removal applications with improved indoor air quality, the use of liquid-to-air membrane energy exchangers (LAMEEs) in liquid desiccant (LD) air-conditioning systems has been suggested in the literature. This study introduces a novel configuration for a heat pump LD system to improve its dehumidification and energy use. The evaporator and condenser coils of the heat pump (HP) are embedded in the LD channels of two LAMEEs; one is an internally cooled dehumidifier (ICD), and the other is an internally heated regenerator (IHR). These LAMEEs are designed to directly use the HP condenser coils for the IHR's regeneration process while also effectively controlling the local temperature rise of the LD solution inside the ICD. An integrated CFD model for both the ICD and the IHR is simultaneously solved, considering the coupling of the HP components and the closed loop of the LD solution. This model, which counts all existing balances of energy, mass, and salt content in various components, determines the condensing temperature employed in the regeneration process at each operational condition. This system-level model, which requires few inputs, is novel and very reliable for evaluating the overall performance of the proposed system in terms of specific dehumidification rates and COP. Case studies were carried out using the annual dehumidification design conditions of five Middle Eastern cities known for their hot and humid climates. For the cities of Sharm El Sheikh and Dammam, respectively, the system achieved promising results with specific dehumidification rates of 130.3 and 172.2 g/(m2.h) and system COPs of 6.38 and 5.67, when the temperature and specific humidity of outside fresh air of (30 °C, 19 g/kg) and (32.8 °C, 23.2 g/kg) were cooled and dehumidified to (20.18 °C, 6.52 g/kg) and (21.0 °C, 6.61 g/kg). Using the regeneration air flow rate as high as possible boosts the performance of the proposed HPLD system, concluding that the system is preferable for all-fresh-air handling units.

Effectiveness of temperature and humidity treatment processes: Unified expression and thermodynamic analysis

Temperature control and humidity control are the two main objectives of the heating, ventilation, and air conditioning (HVAC) system. Previous studies have mainly focused on analyzing one or more existing configurations with regular indoor conditions of public and residential buildings. In this study, a unified expression is derived to evaluate the thermodynamic effectiveness from the ideal temperature and humidity treatment processes to the actual cooling and dehumidification systems, and it is modified by progressively rejecting the “ideality” assumptions. It is indicated that the thermodynamic effectiveness of the ideal temperature and humidity treatment process only depends on the initial state, the final state, and the flow rate of the air streams, and the process with the highest COPth for dehumidification is coupled with the cooling process. Based on the illustrative non-ideality characteristics in the actual cooling and dehumidification systems, the exergy performances of multiple system configurations are investigated. The results also elucidate that the liquid desiccant systems attain less irreversible loss and has a relatively high thermodynamic effectiveness, but the desiccant wheel system also shows great potential for deep dehumidification applications. This research provides a theoretical guide and illustrates the potential paths for the future design of exergy efficient cooling and dehumidification systems.

Optimization of a multistage liquid desiccant dehumidifier by neural networks & particle swarm techniques

To overcome the inconsistent and limited dehumidification by single stage dehumidification system, a novel multistage, reciprocating dynamic liquid desiccant dehumidification system has been proposed to dehumidify larger space consistently. Experimental investigation on the proposed unit has been conducted by varying air velocity, cam shaft speed, and inlet relative humidity (RH). System's performance in terms of moisture removal rate (MRR), dehumidification efficiency (DE) and energy consumed has been compared with that of a single stage system. It is observed that multistage system performed better than the single stage system in terms higher MRR and efficiency. To optimise the system performance, neural network-based Particle Swarm optimization (PSO) has been employed. Inertia weight distribution for PSO has been obtained by the statistical Taguchi method. PSO optimized results for randomly distributed weights have been compared with the Taguchi given weights. It is found that the performance of the PSO with this method is better than the former in terms of optimized paramters. PSO predicted optimum input variables are found as air velocity of 8.2 m/s, inlet RH of 90% and cam speed of 15.16 rpm. The maximum performance parameters obtained are MRR of 7.549 g/s, Dehumidification efficiency of 0.762 and energy consumption of 2165.4 Watts. Confirmation experiments indicated that PSO predicted results are in well agreement with the experimental results with a maximum deviation of 1.8%. Current work predicts and optimizes the dehumidification performance which minimizes experimentation time and also the cost involved.

Response surface modeling and optimization scheme of an internally cooled liquid desiccant air conditioning system

Applying an internally cooled liquid desiccant air conditioning system (LDAC) is a promising energy-saving and emission-reduction scheme for hot and humid areas. This research investigates the application of an internally cooled LDAC for performance enhancement in hot and humid areas like Hong Kong. The combined system integrates liquid desiccant dehumidification (LDD) and regenerative indirect evaporative cooling (RIEC) without a power-intensive compressor. The internally cooled LDD removes latent heat from the hot and humid air before the RIEC cools it. To ensure efficient energy utilization, the LDD captures the exhaust air to assist in the dehumidification and initial cooling of the fresh air, which alleviates the efficiency deterioration of the desiccant. An all-fresh air system is used for better indoor air quality. To optimize the performance of a system with many influencing parameters, the response surface method (RSM) and multi-objective optimization are used to optimize and assess the potential and performance of the system. The system cooling capacity (C), latent heat removal rate (Qd), and dehumidification efficiency (ηd) are used as the optimization objectives. The response surface model and desirability function approach optimize six critical parameters, achieving a 7.6% improvement in dehumidification performance with low airspeed (1.5 m/s) and high desiccant concentration (40%) during high-humidity months. Increasing the extraction ratio of the RIEC by 20% in warmer months enhances the peak cooling capacity by 23.6%. This research contributes to implementing internally cooled LDAC systems and provides insights into optimizing monthly operation patterns in hot and humid regions.

Study on the regeneration process and overall performance of a microencapsulated phase change material slurry dehumidification system

Building on previous findings that showed the efficacy of microencapsulated phase change materials (MicroPCM) in augmenting liquid desiccant dehumidification performance, this study endeavors to further explore the extent of the impact of MicroPCM in the regeneration and the entire dehumidification system. The results of the regeneration experiment on the microencapsulated phase change material slurry (MPCMS) reveal that the addition of MicroPCM impedes regeneration. Specifically, as the concentration of MicroPCM increases, both the regeneration rate and the MPCMS concentration after regeneration decrease, while the regeneration effectiveness exhibits an initial decline followed by an increase. An empirical correlation was proposed to predict the regeneration rate of MPCMS for their future application. The results showed that the addition of MicroPCM led to a decrease in the regeneration rate by as much as 48.1%, when compared to a pure LiCl solution. However, a simulation analysis of the entire system revealed that within a certain concentration range of MicroPCM addition, the coefficient of performance (COP) of the system could still be increased by up to 4.28%. This finding suggests that if a more suitable non-heating regeneration method were used, the addition of MicroPCM could further enhance the overall performance of the liquid desiccant dehumidification system.

Performance analysis and optimization of a coupled open and close liquid desiccant sorption system using LiCl solution

The efficient utilization of low-grade heat for air conditioning has been a topic of world concern, especially for low-grade heat below 80℃. In this study, a novel coupled open and close liquid desiccant sorption system driven by 70 ∼ 80℃ low-grade heat is proposed. This system employs a generator to provide strong solution for both the dehumidifier and absorber, achieving the air temperature and humidity independent control to reduce driving temperature and enhance system performance. A mathematical model is developed to simulate the proposed new system, and a comparative analysis is conducted with a traditional combined absorption refrigeration with liquid desiccant dehumidification system using LiCl solution. It is found that the COP of the new system is 0.76, which is higher than 0.65 of the traditional system when the hot water temperature is 75℃, the reason is that the new system avoids the sensible heat consumption of regenerated air which accounts for 17% of the total heat consumption in the traditional system. Furthermore, it is observed that the outlet concentration difference between the absorber and dehumidifier exists in the new system. The cycle is optimized to improve the concentration difference, leading to a 5.5% increase in the COP. After that, economic performance analysis shows a payback period of 2.65 years for a 150 kW cooling capacity system, with a reduction of 86.6 tons of CO2 emissions per year due to the decrease in power consumption.

A cleaner and more efficient energy system achieving a sustainable future for road transport

A novel integrated cooling system is developed for future medium to large electric vehicles by integrating the fuel cell, battery, metal-hydride, heat pump, and liquid desiccant dehumidification system to reduce power consumption and extend vehicle's driving range. The system benefits from the reuse of the normally wasted energy in the form of pressure difference and wasted thermal energy, from the hydrogen vessel, the fuel cell stack, and the battery pack. A numerical model for the proposed system and a finite element model for the dehumidifier and regenerator are developed and validated by experimental results from published data. A comprehensive evaluation of the impacts of the ambient air temperature and humidity, fuel cell current output, battery discharging C rate, and air mass flow rate on the Coefficient of Performance (COP), outlet air temperature, and cooling capacity is conducted. Two operating modes, namely non-compressive mode and heat pump supplemental mode are investigated and a detailed comparison between these two modes is undertaken. Furthermore, the proposed system under heat pump supplemental mode has been compared to other published cooling systems and dehumidification systems. Under non-compressive mode, results indicate that the proposed system can provide sufficient cooling capacity without the need of the compressor when the supply air mass flow rate is lower than 0.03 kg/s, under the specific operating situation. Under the heat pump supplemental mode, the proposed system can operate at 36 °C with a COP greater than 4, which is 56% higher than the cited published results, although the COP of the proposed system also considers battery cooling. Heat pump supplemental mode drastically reduces the 12-s insufficient cooling period that occurs at the beginning of the charging to discharging transition between the two metal hydrides to 2 s compared to non-compressed mode. Overall, this study provides a potential solution for future zero-emission vehicles by utilizing the heat and electric co-generation characteristic of the fuel cell, the isothermal characteristic of the metal hydride, and dehumidification and cooling characteristics of the liquid desiccant dehumidification system to extend the driving range of the electric vehicles and reduce energy consumption for cooling. Moreover, the proposed system can also provide domestic cooling loads and power by integrating the system into residential buildings.

Thermal modelling of potassium formate based cross-flow liquid desiccant dehumidification system

Thermal modelling of potassium formate based cross-flow liquid desiccant dehumidification system

Scalable and high-efficiency lignocellulose sponge-based evaporators for solar-driven desalination and desiccant regeneration

Liquid desiccant air conditioning systems (LDAC) have attracted increasing interest for the high efficiency of temperature and humidity control. Improving the regeneration performance with low-grade energy could make LDAC systems more competitive. In this work, by combining Chinese ink and lignocellulose sponge, we developed an inexpensive, efficient and large-scale evaporator that allows regeneration of desiccant via solar energy. Results show that the evaporator exhibits remarkable salt-resistance capacity, rendering it suitable for effective seawater desalination and regeneration of high-concentration desiccant. Even for the 40 wt% desiccant at about 22 °C, the regeneration rate can reach 0.63 kgm−2 h−1 under 1-sun irradiation. The outdoor experiments further confirmed the high regeneration rate of the evaporator. Additionally, concentrated desiccant presents an alternative means of solar energy storage. Considering the high efficiency, easy preparation, and remarkable salt resistance, the lignocellulose sponge-based evaporators could advance the practical application of solar heat localization for seawater desalination and desiccant regeneration.

Directional vapor transported and water-proof nanofibrous membranes for liquid desiccant dehumidification systems

The membrane-based liquid desiccant dehumidification system is a newly developed method in the field of air dehumidification. In this study, double-layer nanofibrous membranes (DLNMs) with directional vapor transport and water repellency for liquid dehumidification were fabricated by a simple electrospinning process. Specifically, the combination of thermoplastic polyurethane nanofibrous membrane and polyvinylidene fluoride (PVDF) nanofibrous membrane forms a cone-like structure in DLNMs, resulting in directional vapor transportation. The nanoporous structure and rough surface of PVDF nanofibrous membrane provide waterproof performance for DLNMs. Compare with the commercial membranes, the proposed DLNMs have a significantly higher water vapor permeability coefficient, which is as high as 539.67 g·μm m−2·24 h·Pa. This study not only provides a new route to construct a directional vapor transport and waterproof membrane, but also demonstrates the huge application prospect of the nanofibrous membrane formed by electrospinning in the field of solution dehumidification.

Performance enhancement of electrodialysis regeneration of deep eutectic solvent for liquid desiccant air conditioning systems

Despite the promising energy-saving potential of the electrodialysis regeneration technique for liquid desiccant regeneration in air conditioning systems, there are still issues that need to be resolved in electrodialysis separation systems, such as the high initial cost owing to the complexity of the cell gaps, over-limiting current, and electro-osmotic flow effects. This study utilized stereolithography (SLA) 3-D printing technology to manufacture a low-cost scalable model of an electrodialysis system. An electrodialysis system with a cell gap of 5 mm and serpentine flow feed spacers was investigated to regenerate deep eutectic solvent (DES) developed from choline chloride and ethylene glycol as a bio-friendly liquid desiccant for air conditioning systems. Ammonium chloride electrode rinsing solution with about 250 mS/cm electric conductivity was utilized to enhance the conductive properties of the system. The optimum parametric operating conditions for the system are found to be around 0.0058 m/s for the circulation flow rate, 0–4 wt.% for the concentration difference of the diluted and regenerated solutions' cells, and 6–8 V voltage supply energy input. In 60 minutes of operation, the current efficiency value of the constructed model was discovered to be as high as 65.82%. Matching the performance threshold of the commercial electrodialysis equipment described in the literature for regenerating lithium chloride conventional liquid desiccant.

Energy and exergy analysis of a liquid desiccant heat and mass transfer loop with natural convection: The effect of heat sink and heat source temperature

Desiccant systems, as a promising air conditioning technology have opened new aspects as a replacement for vapor compression systems. The liquid desiccant energy exchanger system with natural convection flow is proposed here; the system is a loop with two hollow fiber membrane air-desiccant exchanger of shell-and-tube type that works as the humidifier and regenerator. The LiCl aqueous solution as desiccant directed over the fibers and the air is inside the fibers; the water stream flows outside the modules to provide the required heat exchange. The heat sink and source as merged subparts of the exchangers have a major influence on system performance. The system is examined from the energetic and exergetic viewpoints at different heat sink and heat source temperatures. Keeping the fixed air flow rate and using the standard air properties, the cold and hot water temperatures of 15–25 °C and 50–70 were employed and the effectiveness coefficients including the latent, sensible and total and 2nd law efficiency evaluated. It was shown that the values of NTU and Cr* depend on each other and attenuating effect of NTU reduction is higher than enhancing effect of Cr* increase on effectiveness values. The prominent factor on exergy performance which hinders the effect of the other parameters is the solution flow rate. Also, the highest and lowest values of 2 nd law efficiency are 81% and 5% which are for temperature pairs of (52, 20) and (65,25), respectively.