An Adsorption-Desorption Heat Engine for Power Generation from Waste Heat

Mikhail Granovskiy ^*

1. Professional Engineers of Ontario, Canada

* Correspondence: Mikhail Granovskiy

Academic Editor: Kiari Goni Boulama

Special Issue: Applied Thermodynamics and Energy Conversion

Received: October 17, 2023 | Accepted: November 09, 2023 | Published: November 14, 2023

Journal of Energy and Power Technology 2023, Volume 5, Issue 4, doi:10.21926/jept.2304034

Recommended citation: Granovskiy M. An Adsorption-Desorption Heat Engine for Power Generation from Waste Heat. Journal of Energy and Power Technology 2023; 5(4): 034; doi:10.21926/jept.2304034.

© 2023 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.

1. Introduction

Organic Rankine Cycle (ORC) heat engines, which use organic liquids instead of water as their working fluids, are the most cited in processes of low-potential waste heat (≤250°C) utilization [1]. These engines employ indirect heat transfer to evaporate (at higher pressure) and condense (at lower pressure) organic liquids. The highest pressure in the cycle is produced by a pump forcing liquid to flow from condenser to evaporator. The vapor obtained in the evaporator expands to the lowest pressure in a turbine that generates mechanical work for the subsequent conversion into power via electric generator.

The working fluid of the cycle accepts and releases heat indirectly, “through the wall” in the evaporator and condenser, respectively. The latent heat transfer is relatively fast, as evaporation and condensation are characterized with the highest heat transfer coefficients. Organic liquids, due to their low boiling points compared to water, evaporate at lower temperatures to accept low-potential, usually sensible heat from external gases and liquids.

ORC cycles are not robust for small-scale applications up to 40 kWe, because the rotating speed of their turbines significantly increases with decreasing turbine output power [1,2], while their manufacturing costs per kW of produced energy increases dramatically [2,3].

Conventional Stirling engines also employ indirect heat transfer to gaseous working fluids but permit low-kilowatt power generation. Pistons force gas to move between hot and cold zones within one or two connected cylinders. The kinematic mechanism generates extra mechanical work because gas makes greater mechanical work in its heating and expansion than it consumes in its cooling and contraction. The intensity and efficiency of heat transfer is inferior to that of ORC heat engines. Therefore, Stirling engines require a substantial temperature gradient between heat source and sink [4]. As the literature [5] indicates, they can operate efficiently only if the thermal energy source is at temperatures indicatively higher than 300°C, challenging high temperature sealing requirements. Presently, Stirling engines are associated with low on-site operational efficiencies and high manufacturing costs [6].

Recently, the adsorption phenomenon has received increasing attention for employment in sorption heat pumps utilizing low-temperature waste heat. This waste heat is used to desorb refrigerant vapor from adsorbent to allow its adsorption at the next stage, yielding a cooling effect caused by refrigerant evaporation [7,8]. Sorption heat pumps do not require compressors and electricity to run; there is no vibration of moving parts with an associated tendency of their breakage. Allowing a continuous operation, semi-batch units consist of two identical reactors (adsorber and desorber) and heat exchangers (evaporator and condenser), switching their functionalities during the process.

The possibility of reversible chemical and physical processes to produce electricity from low temperature heat was studied in a few publications and rather overlooked. Nomura et al. [9] proposed a chemical engine that used the reversible reaction of hydrogen with metals to form metal hydrides. Hydrogen released from metal hydride (with an uptake of heat) at high pressure and temperature was used to turn the work generating mechanism. At the exit, hydrogen was absorbed by the metal (with a discharge of heat) to allow repeating the process again. The proposed semi-batch unit (with a constant heat supply and withdrawal) required strict synchronization of direct and reverse reaction kinetics with the rotation speed of the work generating mechanism.

Bao et al. [10] numerically investigated for the first time the novel combination of the chemisorption cycle and the scroll expander for refrigeration and power cogeneration. However, because of the mutual constraint between the chemisorption and the expansion when they link in series, the power output of the cogeneration mode was only around one third of the original expectation.

Muller and Schulze-Makuch [11] presented the idea of a sorption heat engine. They described a batch adsorber where adsorbant gas was firstly desorbed (with a heat uptake), expanded to generate work, and readsorbed back (with a heat discharge) to repeat the process again. Periodic heat supply and withdrawal should be synchronized with the corresponding stage. The described process was not thermodynamically analyzed to obtain its performance indicators and, subsequently, define its efficiency.

Here, the author extends the above-mentioned idea to an adsorption-desorption semi-batch unit to generate power from a low-grade waste heat. A distinctive feature of the unit is the absence of mass transfer (gas flow) between desorber and adsorber. A detailed thermodynamic analysis is carried out to compare its performance with Stirling, Carnot, ORC, and Rankine cycles.

2. An Adsorption-Desorption Heat Engine

2.1 A Configuration of an Adsorption-Desorption Heat Engine

A principal schematic of an adsorption-desorption heat engine is presented in Figure 1. Adsorber and desorber are identical devices accommodating two coiled heat exchangers. Adsorbent tubes and one heat exchanger are submerged into a low-boiling point heat transfer liquid (for instance, n-pentane, which is frequently used in ORC cycles [12]).

Figure 1 A principle schematic of an adsorption-desorption heat engine.

Adsorber and desorber are physically linked through two vessels with an inert high-boiling point thermal oil (for instance, Duratherm LT [13], with a boiling point higher than 589 K) and a power generation mechanism incorporating one or more power generation tubes. A power generation tube consists of a permanent piston-magnet sliding along a cylindrical tube, stretching one spring and compressing another in response to a pressure difference in two vessels at open Valves 7 and 8 and closed Valve 9.

2.2 The Principles of Electricity Production and Heat Regeneration

In desorber, a hot gas (waste heat carrier) enters the bottom heat exchanger (Valve 2 is open, Valve 1 is closed) and vaporizes n-pentane. This vapor condenses on outer surfaces of adsorbent tubes with a heat release causing adsorbant gas to desorb. With Valves 5, 6, and 7 closed (Figure 1), the pressure of the gas in Vessel_1 increases.

In adsorber, n-pentane is vaporized near adsorbent tubes where exothermic (heat releasing) adsorption takes place. A cooling water or air enters the top heat exchanger (cooler) (Valve 3 is open, Valve 4 is closed) to condense vapor and maintain a temperature to advance adsorption. With Valves 5, 6, and 8 closed Figure 1), the pressure of the gas in Vessel_2 decreases.

When desorption and adsorption approach their equilibriums in desorber and adsorber, Valves 7 and 8 get opened, while Valve 9 is closed. Due to higher gas pressure in Vessel_1, thermal oil pushes a piston-magnet to the right, stretching and compressing the left and right springs, respectively (Figure 2).

Figure 2 A principle of power generation in the adsorption-desorption heat engine.

Due to an increase in gas volume, the pressure in desorber and Vessel_1 decreases. This promotes further gas desorption. A continuous heat uptake by adsorbent is provided by a corresponding heat supply that maintains near-isothermal conditions.

Due to a respective decrease in gas volume, the pressure in adsorber and Vessel_2 increases. This promotes further gas adsorption and heat release. A continuous heat removal from adsorbent is provided by corresponding condensation of saturated n-pentane vapor via cooling air or water that also maintains near-isothermal conditions. The temperature in adsorber is lower than in desorber.

At a designed pressure difference in vessels, Valves 7 and 8 get closed and Valve 9 gets opened (Figure 2). Continuous conversion between potential and kinetic energy of springs induces back and forth movements (oscillations) of a piston-magnet. These oscillations generate a variable magnetic field and alternate electric current in a wire spiraled around the tube.

This current can be directed to and collected by an energy storage device (battery or capacitor). For instance, in Figure 3, a well-known diode bridge converts alternate current into a direct one, and the latter charges a capacitor or battery coupled with the load. Amplitudes of piston-magnet oscillations as well as current (voltage) attenuate during this process. When piston-magnet oscillations are practically over, Valve 9 gets closed.

Figure 3 A principal scheme of an energy storage device (capacitor) charging.

Valves 7 and 8 get closed after charging the power generation mechanism; then, the latter becomes separated from adsorber and desorber, and switching desorber and adsorber functionalities occur (Valves 1 and 4 are open; Valves 2 and 3 are closed; see Figure 1). This switching starts with the heat regeneration. If desorber is on the left (as in Figure 1), Valve 5 gets opened; if it is on the right, Valve 6 gets opened. A high pressure and high temperature vapor flows from desorber to adsorber. It condenses within a relatively cold liquid in adsorber and heats it and adsorbent up, causing an inception of gas desorption there and preparing adsorber to be desorber at this stage. A decrease in vapor pressure in desorber induces spontaneous liquid vaporization and declination in liquid and adsorbent temperatures, causing an inception of gas adsorption there and preparing desorber to be adsorber at this stage. When both devices reach the same vapor pressure, Valve 5 (or 6) gets closed, and the system starts a new power generation cycle.

3. Thermodynamic Analysis of Adsorption-Desorption Heat Engine: A Case Study

3.1 A Source of Waste Heat: Air at 500 K (227°C)

The temperature of 500 K is within a usual temperature range for flue gases from a typical boiler. The adsorption-desorption heat engine (as any heat engine) generates work from the heat that is supplied at higher temperatures and withdrawn at lower temperatures. If n-pentane vapor pressure in adsorber is kept 1 Bar, it boils and absorbs heat at 309 K [14]. The produced vapor is condensed in the top heat exchanger (cooler) to control the vapor pressure and release heat to ambient air or cooling water. An ambience (ambient air) is considered a universal heat sink, and its temperature is often taken equal to T₀ = 298 K [15].

The waste heat is transferred to adsorbent in desorber. This transfer includes boiling liquid n-pentane and condensing its vapor on adsorbent tubes. The following analysis allows finding an optimal boiling temperature for n-pentane in desorber.

When hot air cools down from T_in = 500 K to an output temperature T_out it transfer heat Q_Air as follows:

\[ \mathrm{Q_{air}=C_{p}^{air}(T_{in}-T_{out})} \tag{1} \]

where $\mathrm{C_{p}^{air}}$ is the isobaric heat capacity of air (29.2 J/mol·K).

The exergy of Q_air is defined as the maximum work that can be obtained in the ideal Carnot engine with a heat sink temperature T₀. This exergy E_air is determined as follows:

\[ \mathrm{E}_{\mathrm{air}} \ = \ \mathrm{Q}_{\mathrm{air}}\big(1-{}^{\mathrm{T}_0}\big/_{\mathrm{T}^*}\big) \tag{2} \]

where T^* is an average thermodynamic temperature reflecting air cooling from T_in to T_out as follows:

\[ T^*=\frac{\int_{T_{in}}^{T_{out}}dH_{air}}{\int_{T_{in}}^{T_{out}}dS_{air}}=\frac{\int_{T_{out}}^{T_{in}}C_{p}^{air}dT}{\int_{T_{out}}^{T_{in}}\frac{C_{p}^{air}}{T}dT}=\frac{T_{in}-T_{out}}{\ln\frac{T_{in}}{T_{out}}} \tag{3} \]

where dH_air and dS_air are differential changes in air enthalpy and entropy along with the transfer of sensible heat. Figure 4 shows that with decreasing T_out the difference between energy (heat) and exergy (ability to generate work) increases.

Figure 4 The exergy and heat transferred from air (T_in = 500 K) as a function of output temperature T_out.

If n-pentane boils at temperature T_out it absorbs sensible heat of air Q_air (see Eq. (1)) in the range of T_in = 500 K to T_out to produce saturated vapor at T_out. The exergy of saturated vapor that condenses and releases heat at T_out on adsorbent tubes is expressed as follows:

\[ \mathrm{E_{sv}=Q_{air}\left(1-\frac{T_{0}}{T_{out}}\right)} \tag{4} \]

Exergy of saturated vapour E_sv in equation (4) is a product of two terms. The first term Q_air increases with decreasing T_out as shown in Figure 4 and expressed by equation (1). The second term $(\mathrm{1-\frac{T_{0}}{T_{out}}})$ decreases with decreasing T_out (to T₀). The behavior of two terms points out a product function with an extreme. Exergy of saturated vapour E_sv as a function of T_out (Figure 5) has a maximum of T_out = 390 K. This exergy is transferred to adsorbent. To maintain this temperature, the corresponding n-pentane pressure in desorber should be 8.5 Bar [13]. The average thermodynamic temperature of air that cools down from 500 K to 390 K is T^* = 443 K [Eq. (3)].

Figure 5 Exergy of saturated vapour E_sv as a function of outlet air temperature T_out.

3.2 Adsorbent-Adsorbate Pair: Activated Carbon and CO₂

3.2.1 An Isotherm of CO₂ Adsorption on Activated Carbon (AC)

A widely used activated carbon (AC) (Maxsorb-III) and CO₂ gas are chosen as an adsorbent-adsorbate pair. The fitting parameters for the Tόth isotherm equation are taken from [16] (see Table 1):

\[ q=\frac{q_0bP}{((1+(bP)^t)^{1/t}} \tag{5} \]

where q denotes the absolute mass of CO₂ adsorbed by unit mass of adsorbent (kg/kg adsorbent at equilibrium state); q₀ is the saturated amount adsorbed (kg/kg); t is the heterogeneity parameter; and b is the adsorption affinity (kPa^-1), given by:

\[ b=b_0\text{exp}\left(\frac Q{RT}\right) \tag{6} \]

where b₀ is the adsorption affinity at infinite temperature (kPa^-1), and Q is the isosteric heat of adsorption (J/mol).

Table 1 Parameters for Tόth isotherm equation.

Based on the Tόth equation, three isobars of adsorption are presented in Figure 6.

Figure 6 An adsorption-desorption thermodynamic cycle (1-2-3-4). The points and arrows denote the thermodynamic trajectory of the cycle.

3.2.2 Adsorption-Desorption Thermodynamic Cycle

The pressure P_eq = 1500 kPa at which gas expansion in desorber equilibrates with its compression in adsorber at open Valves 7 and 8 (Figure 1 and Figure 2) is taken as a starting point for the cycle simulation. Note that the validity of formulas (5) and (6) includes this value.

The adsorption-desorption thermodynamic cycle is presented in Figure 6 and Figure 7 (1-2-3-4) in q-T and P-V coordinates, respectively. As seen in Figure 6 and Figure 7, Points 1 and 3 correspond to adsorption equilibriums at the same pressure P_eq = 1500 kPa but different temperatures: T_ads = 309 K at Point 1 in adsorber and T_ds = 390 K at Point 3 in desorber. At these points, isothermal expansion and desorption and isothermal compression and adsorption attain equal pressures. To continue, desorption and adsorption must switch. Technically, this means that Valves 7 and 8 get closed, and desorber and adsorber exchange their functionalities by starting up heat regeneration and redirection of heating and cooling flows (see Chapters 2.1 and 2.2).

Figure 7 Thermodynamic trajectory of adsorption-desorption (1-2-3-4) and Stirling (6-3-7-5) cycles in P-V coordinates. Thermodynamic trajectory of adsorption-desorption cycle (1-2-3’-4’) is for a specific mechanical load (see the text).

Then, adsorbent in desorber is isochorically heated from T_ads = 309 K at Point 1 to T_ds = 390 K at Point 2, reaching adsorption equilibrium at the elevated pressure. Adsorbent in adsorber is cooled from T_ds = 390 K at Point 3 to T_ads = 309 K at Point 4, reaching adsorption equilibrium at the lower pressure. After reaching isochoric equilibriums, Valves 7, 8 open to proceed with isothermal expansion 2-3 and isothermal compression 4-1 in desorber and adsorber, respectively.

The difference in the gas amounts (desorbed CO₂) corresponding to full desorption and adsorption at Points 3 and 1 is ∆N = 10.6 mol/kg_AC (see Figure 6, where q is the mass of adsorbed CO₂). Another parameter, N_min, is needed for the cycle simulation. It is a molar quantity of CO₂ that permanently remains in gas. For instance, if N_min = 10 mol/kg_AC, the gas phase is comprised of 10 mol of CO₂ after complete adsorption (Point 1) and 20.6 mol of CO₂ after complete desorption (Point 3) per 1 kg of adsorbent, as in adsorber and desorber.

The minimum gas volume in adsorber after isothermal compression at Point 1 is defined by the Clapeyron-Mendeleev equation as follows:

\[ \mathrm{N}_{1} \ = \ \mathrm{N}_{\mathrm{min}};\mathrm{P}_{1} \ = \ \mathrm{P}_{\mathrm{eq}} \tag{7} \]

\[ \mathrm{V_1=V_{min}=\frac{N_{min}RT_{ads}}{P_{eq}}=17.1L} \tag{8} \]

Similarly, the maximum gas volume in desorber (after isothermal expansion) at Point 3 is as follows:

\[ N_3=N_{max}=N_{min}+\Delta N;\mathrm{P_3}=\mathrm{P_{eq}} \tag{9} \]

\[ \mathrm{V_3=V_{max}=\frac{N_{max}RT_{ds}}{P_{eq}}=44.5~L} \tag{10} \]

Thermodynamic parameters at Point 2 (pressure P₂ and CO₂ quantity N₂) are obtained using isochoric desorption at constant volume V_min with increasing temperature from T₁ (T₁ = T_ads) to T₂ (T₂ = T_ds) and pressure from P_eq to P₂. The Clapeyron-Mendeleev equation connects P₂ and N₂ as follows:

\[ \mathrm{V_2=V_{min};P_2=\frac{N_2RT_{ds}}{V_{min}}} \tag{11} \]

The thermodynamic desorption equilibrium is described by Formulas (5) and (6). This is another correlation between two unknowns, P₂ and N₂ (through q), to determine their values.

Similarly, thermodynamic parameters at Point 4 (pressure, P₄ and CO₂ quantity, N₄) are obtained using isochoric adsorption at constant volume V_max with decreasing temperature from T₃ (T₃ = T_ds) to T₄ (T₄ = T_ads) and pressure from P_eq to P₄. The Clapeyron-Mendeleev equation connects P₄ and N₄ as follows:

\[ \mathrm{V}_4=\mathrm{V}_{\mathrm{max}};P_4=\frac{N_4RT_{ads}}{V_{max}} \tag{12} \]

The thermodynamic adsorption equilibrium is also described by Formulas (5) and (6). This is another correlation between two unknowns, P₄ and N₄ (through q), to determine their values.

The calculated parameters of the thermodynamic cycle in Figure 6 and Figure 7 are listed in Table 2.

Table 2 Parameters of adsorption-desorption thermodynamic cycle in Figure 6 and Figure 7.

3.2.3 Energy Balance of Adsorption-Desorption Cycle and Its Efficiency

The heat released during adsorption is equal to that consumed during desorption, as follows:

\[ \Delta H_{ds}=-\Delta H_{ads}=Q \tag{13} \]

For desorption, a molar enthalpy difference is associated with heat consumption as follows:

\[ \Delta H_{ds}=H_{CO2}-H_{CO2-AC}=C_v(T-T_0)+P\Delta V-H_{CO2-AC} \tag{14} \]

where H_CO2 is molar enthalpy of CO₂ in the gas; H_CO2-AC is molar enthalpy of CO₂ in adsorbent; C_v is CO₂ isochoric heat capacity (C_v = 29.0 J/mol·K); T_o = 298 K is a standard reference temperature; P is pressure; and ∆V is the volume change caused by desorption of one mol of CO₂.

Considering the adsorbed CO₂ as an incompressible substance referring to liquids or solids, its enthalpy H_CO2-AC is equal to its internal energy I_CO2-AC. Accounting for P∆V = RT, molar H_CO2-AC and I_CO2-AC can be obtained from (14) as follows:

\[ H_{CO2-AC}=I_{CO2-AC}(T)=C_v(T-T_0)+RT-\Delta H_{ds} \tag{15} \]

It is commonly accepted that isosteric heat of adsorption or desorption (calculated using the Clausius-Clapeyron equation) is independent of temperature [17]; in our case, it equals 19,297 J/mol (see Table 1). Based on Formula (15), the internal energies have the following values:

\[ I_{CO2-AC}(T=T_{ds}=390^oK)=-13387J/mol;I_{CO2-AC}(T=T_{ads}=309^oK)=-16409J/mol \tag{16} \]

The heat consumed at isochoric heating along with desorption (Points 1-2) induces an increase in internal energy as follows:

\[ \begin{aligned} Q_{1-2}& =\Delta I_{1-2} \\ &=[n_2I_{CO2-AC}(T_2)+N_2C_v(T_2-T_0)+m_{AC}C_{AC}(T_2-T_0)] \\ &-[n_1I_{CO2-AC}(T_1)+N_1C_{v}(T_1-T_0)+m_{AC}C_{AC}(T_1-T_0)] \end{aligned} \tag{17} \]

where T₁ = T_ads, T₂ = T_ads; n₁, n₂ are amounts of CO₂ (moles) in the solid phase (adsorbed) (see Table 2); N₁, N₂ are amounts of CO₂ (moles) in the gas (desorbed) (see Table 2); and m_AC = 1 kg and C_AC = 900 J/kg·K [18] are the mass and specific heat capacity of activated carbon, respectively.

The heat consumed at isothermal expansion along with desorption (Points 2-3) increases in internal energy ∆I_2-3 and generation of mechanical work A_2-3 as follows:

\[ T_{2}=T_{3}=T_{d\mathbf{s}} \tag{18} \]

\[ Q_{2-3}=\Delta I_{2-3}+A_{2-3} \tag{19} \]

\[ \Delta I_{2-3}=[n_3I_{CO2-AC}(T_3)+N_3C_\nu(T_3-T_0)]-[n_2I_{CO2-AC}(T_2)+N_2C_v(T_2-T_0)] \tag{20} \]

\[ A_{2-3}=\int_{V_2}^{V_3}P \ dV\approx\frac{P_2+P_3}2(V_3-V_2) \tag{21} \]

Q_3-4, ∆I_3-4, Q_4-1, ∆I_4-1, A_4-1 are calculated similarly. Note that negative values correspond to the heat release and mechanical work from adsorption and gas compression, respectively. The calculated energy-related parameter are shown in Table 3.

Table 3 Energy balances along with adsorption-desorption cycle in Figure 6 and Figure 7.

The available (useful) work ∆A is obtained as a difference between absolute values of expansion and compression works and can be used to generate power. As shown in Table 3, the energy balance is met because ∆A is equal to the difference between absolute values of consumed and released heats in desorber and adsorber, respectively, as follows:

\[ \Delta A=Q_{ds}-Q_{ads} \tag{22} \]

Thermal efficiency ζ_th of the heat engine is defined as work ∆A per consumed heat Q_ds and is expressed as follows:

\[ \zeta_{th}=\frac{\Delta A}{Q_{ds}}=\frac{31.5}{376.5}=0.084(8.4\%) \tag{23} \]

Note that the value of consumed heat Q_ds is rather conservative without accounting for heat regeneration.

Specific work ∆A_m per weight of working fluid (CO₂ in gaseous phase) is expressed as follows:

\[ \Delta A_{m}=\frac{\Delta A}{M_{CO2}(N_{max}+N_{min})}=\frac{31.5}{44*(10+20.6)*10^{-3}}=23.4\frac{kJ}{kg_{CO_{2}}} \tag{24} \]

where M_CO2 is molecular weight of CO₂ (44 g/mol); N_max + N_min is the sum of maximum and minimum molar amounts of CO₂ in the gas at P_eq in desorber and adsorber, respectively. This specific work indicates an ability of the gas (working fluid) to generate useful work ∆A.

Note that energy balance is carried out without accounting for heat regeneration.

4. Results and Discussion

4.1 Adsorption-Desorption Cycle in Comparison with Stirling, Carnot, ORC, and Rankine Engines

The ideal Stirling cycle also consists of two isochores and two isobars, but it fundamentally differs due to an unchanged amount of gas N_S in the cycle. For a clear presentation on P-V diagram in Figure 7, this value is arbitrarily taken as equal to the maximum amount of gaseous CO₂ in desorber, as follows:

\[ N_s=const=20.6 \ mol \tag{25} \]

Assigning the same temperature range T_min = T_ads = 309 K, T_max = T_ds = 309 K and P_eq = 1500 kPa, unknown pressures ($\mathrm{P_{max}^{s},~P_{min}^{s}}$) and volumes ($V_{max}^{s},V_{min}^{s}$) can be calculated using the Clapeyron-Mendeleev equation. The calculated parameters of the Stirling cycle are presented in Figure 7 (cycle 5-6-3-7).

The well-known analytical expressions for available (useful) work ∆A_S and consumed heat Q_S in the Stirling cycle without heat regeneration are as follows [19]:

\[ \mathrm{\Delta A_{s}=\mathrm{R(T_{max}-T_{min})}\ln\frac{V_{max}^{s}}{V_{min}^{s}}=0.156\frac{\mathrm{kJ}}{\mathrm{mol}}} \tag{26} \]

\[ Q_s=C_v(T_{max}-T_{min})+RT_{max}\ln\frac{V_{max}^s}{V_{min}^s}=3.095\frac{kJ}{mol} \tag{27} \]

Respectively, thermal efficiency $\zeta_{th}^{s}$ and specific work are equal, as follows:

\[ \zeta_{th}^{s}=\frac{\Delta A_{s}}{Q_{s}}=5.0\% \tag{28} \]

\[ \Delta A_m^s=\frac{\Delta A_s}{10^{-3}M_{CO2}}=3.55\frac{kJ}{kg_{CO_2}} \tag{29} \]

Note that both thermal efficiency $\zeta_{th}^{s}$ and specific work $A_m^s$ are independent of the total amount of gas in the cycle. A significantly higher thermal efficiency ζ_th and specific work A_m of adsorption-desorption cycle obtained under the same conditions indicates a positive impact of the adsorption-desorption mechanism on the efficiency of the useful work generated.

The Carnot efficiency of an ideal, reversible thermodynamic cycle between T_min = 309 K and T_max = 390 K is 20.8% vs. 8.4% obtained in the absorption-desorption cycle. Recalling that a constant amount of gaseous CO₂ in the cycle (N_min) is always present in adsorber and desorber, it will be of interest to see if reducing its value allows the cycle efficiency to approach the Carnot value.

Table 4 demonstrates increasing efficiency indicators with decreasing N_min in the cycle. At the same P_eq = 1500 kPa, the difference between P_min and P_max increases with a decrease in gas volume. At N_min = $4\frac{\mathrm{mol}}{\mathrm{kg\_AC}}$ in adsorber, the thermal efficiency of the adsorption-desorption cycle exceeds more than twice and specific work more than ten times the efficiency and specific work of Stirling engine, respectively.

Table 4 Parameters and efficiency indicators of adsorption-desorption cycle at decreasing N_min and P_eq = 1500 kPa per one kg of AC in desorber and adsorber.

Tarrad’s [20] thermodynamic analysis of ORCs with different working fluids presented thermal efficiencies (ζ_th) in the range of 7.6-8.5%. Certain ORC enhancements increase their thermal efficiencies to about 12.6% [21]. The proposed adsorption-desorption heat engine retains the same efficiency indicators and could compete with ORCs due to an absence of expensive turbines or expanders.

To compare adsorption-desorption and water-steam Rankine cycle efficiencies, the quality of incoming heat should be considered. In other words, available work should be given per unit of exergy. For the case in the last row of Table 4, the exergy of the hot air sensible heat accepted in desorber is as follows:

\[ \mathrm{E}_{\mathrm{air}}=\mathrm{Q}_{\mathrm{air}}\left(1-^{\mathrm{T}_0}/_{\mathrm{T}^*}\right)=120.2\frac{kJ}{kg_{AC}} \tag{30} \]

where Q_air = Q_ds = 367.3 kJ/kg_AC and an average thermodynamic temperature of air is T^* = 443 K (see Chapter 3.1). The exergy efficiency of the cycle is expressed as follows:

\[ \zeta_{ex}=\frac{\Delta A}{E_{air}}=\frac{41.5}{120.2}=0.345(34.5\%) \tag{31} \]

The most sophisticated Rankine cycles transform only 40% of the fuels’ (coal, natural gas) heating values into work [22], and their lower heating values are very close to their exergies [23]. The obtained exergy efficiency of adsorption-desorption cycle corresponds well to the exergy efficiency of a regular Rankine cycle unit.

4.2 An Available Work to Charge a Spring Mechanism

The power generation mechanism can be envisioned as a set of similar parallel tubes with two springs and a piston-magnet between them, as shown in Figure 1, Figure 2, and Figure 8 (for clarity, the copper coils are not pictured there). Assume that all springs have the same spring constants (K).

Figure 8 A set of “n” consecutively charged power generation tubes (see Figure 2 and Figure 3). For clarity, copper coils are not shown.

Thermal oil pushes a piston-magnet and charges springs consecutively - after displacement Δx is reached in one tube, that tube gets closed, and charging continues in the next tube. The potential energy ΔE of springs is a function of displacement Δx and is expressed as follows:

\[ \Delta E=nK(\Delta x)^2 \tag{32} \]

where Δx is a positive displacement from equilibrium for both springs and n is the number of tubes.

The available work ΔA of the adsorption-desorption cycle is converted into potential energy of springs as follows:

\[ \Delta A=\int_{V_2}^{V_3}PdV-\int_{V_4}^{V_1}PdV\approx\frac{P_2+P_3}2(V_3-V_2)-\frac{P_1+P_4}2(V_4-V_1)=nK(\Delta x)^2 \tag{33} \]

where, as seen in Figure 7:

\[ \Delta\mathrm{V}=(\mathrm{V}_3-\mathrm{V}_2)=(\mathrm{V}_4-\mathrm{V}_1)=27.7\mathrm{L} \tag{34} \]

In the frame of thermodynamic analysis: (i) a relatively small pressure difference between P₃ and P₁ due to different levels of thermal oil in Vessel_1 and Vessel_2 is neglected; and (ii) an indefinite number of tubes with a very small spring constant K is assumed. This allows application of the following equation:

\[ P_{3}=P_{1}=P_{eq} \tag{35} \]

Substituting (35) and (34) into (33), an expression of springs’ potential energy is as follows:

\[ \Delta A=\left(\frac{P_2-P_4}2\right)\Delta V=\left(\frac{P_{max}-P_{min}}2\right)\Delta V=nK(\Delta x)^2 \tag{36} \]

As follows from this equation, increasing P_max and decreasing P_min leads to an increase in available work ΔA and, as seen in Table 4, ΔA grows despite a decrease in ΔV with decreasing N_min.

At finite numbers of tubes P₃ - P₁ ≠ 0 the thermodynamic cycle may look like 1-2-3’-4’ (Figure 7), with the following balance of forces in the last tube:

\[ (P_3^{\prime}-P_1)\frac{\pi\mathrm{d}^2}4=2K\Delta x \tag{37} \]

ΔV’ of the cycle 1-2-3’-4’ is expressed as follows:

\[ \Delta\mathrm{V'}=(\mathrm{V_3}^{\prime}-\mathrm{V_2})=(\mathrm{V_4}^{\prime}-\mathrm{V_1}) \tag{38} \]

Furthermore, ΔV’ should equal the following:

\[ \Delta\mathrm{V'}=n\frac{\pi\mathrm{d}^2}4\Delta x \tag{39} \]

where d is an inner diameter of power generation tubes.

5. Conclusion

A new power generation heat engine based on adsorption-desorption cycle to utilize low-temperature waste heat (<250°C) is proposed. The sensible heat of air wasted at temperature 500 K (227°C) and CO₂ adsorption on activated carbon were selected as a case to conduct thermodynamic analysis. The heat engine accepts heat at a higher temperature in desorber and releases it at a lower temperature in adsorber. For a continuous semi-batch operation, a switch between desorber and adsorber functionalities is required.

Thermodynamic analysis of the heat engine cycle is carried out for the pair adsorbant-adsorbent: CO₂-activated carbon. Its efficiencies are calculated accepting the ideal gas law and an adsorption-desorption equilibrium at the key points of the cycle. The cycle consists of two isochores and two isotherms like that of the Stirling engine, but at the same temperature range and without heat regeneration, its thermal efficiency (work per heat supplied) can reach 11.3% vs. 5.0% and specific work 50.7$\frac{kJ}{kg_{-}CO2}$ vs. 3.55$\frac{kJ}{kg_{-}CO2}$. in the latter. These dramatic increases in efficiencies result from the fact that the expansion work is increased due to CO₂ desorption and the compression work is decreased due to CO₂ adsorption to the adsorbent.

The proposed unit has thermal efficiency in the range of Organic Rankine Cycle units and can be utilized in small-scale applications up to 40 kWe, where manufacturing cost of turbines or expanders for ORCs increases dramatically. Accounting for quality (temperature) of utilized heat, the exergy efficiency of the proposed cycle ζ_ex = 34.5% approaches that of water-steam Rankine cycles utilizing natural gas or coal combustion.

An experimental kinetic study is needed to obtain the heat engine capacity (wattage) in response to the frequency of switching between desorber and adsorber. The selection of an advanced adsorbant-adsorbent pair should be based on the temperature of waste heat and the required capacity.

Nomenclature

Greek Symbols

Subscripts

Superscripts

Abbreviations

Author Contributions

The author conducted all the research work for this study.

Competing Interests

There are no conflicts to declare.