Energy Efficiency and Performance Limiting Effects in Thermo-Osmotic

Article Cite This: Environ. Sci. Technol. 2017, 51, 12925-12937

pubs.acs.org/est

Energy Efficiency and Performance Limiting Effects in ThermoOsmotic Energy Conversion from Low-Grade Heat Anthony P. Straub and Menachem Elimelech* Department of Chemical and Environmental Engineering, Yale University, P.O. Box 208286, New Haven, Connecticut 06520-8286, United States S Supporting Information *

ABSTRACT: Low-grade heat energy from sources below 100 °C is available in massive quantities around the world, but cannot be converted to electricity effectively using existing technologies due to variability in the heat output and the small temperature difference between the source and environment. The recently developed thermo-osmotic energy conversion (TOEC) process has the potential to harvest energy from low-grade heat sources by using a temperature difference to create a pressurized liquid flux across a membrane, which can be converted to mechanical work via a turbine. In this study, we perform the first analysis of energy efficiency and the expected performance of the TOEC technology, focusing on systems utilizing hydrophobic porous vapor-gap membranes and water as a working fluid. We begin by developing a framework to analyze realistic mass and heat transport in the process, probing the impact of various membrane parameters and system operating conditions. Our analysis reveals that an optimized system can achieve heat-to-electricity energy conversion efficiencies up to 4.1% (34% of the Carnot efficiency) with hot and cold working temperatures of 60 and 20 °C, respectively, and an operating pressure of 5 MPa (50 bar). Lower energy efficiencies, however, will occur in systems operating with high power densities (>5 W/m2) and with finite-sized heat exchangers. We identify that the most important membrane properties for achieving high performance are an asymmetric pore structure, high pressure resistance, a high porosity, and a thickness of 30 to 100 μm. We also quantify the benefits in performance from utilizing deaerated water streams, strong hydrodynamic mixing in the membrane module, and high heat exchanger efficiencies. Overall, our study demonstrates the promise of full-scale TOEC systems to extract energy from low-grade heat and identifies key factors for performance optimization moving forward.

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INTRODUCTION The massive quantities of low-grade heat available globally have the potential to supply a considerable portion of energy needs if effectively harvested.1 Waste heat discharged from industry and power plants amounts to more than 8000 TWh per year in the United States alone, with industrial facilities discharging about a third of energy consumed during operation.2,3 Geothermal reservoirs at low temperatures (less than 150 °C) are also geospatially abundant and can be accessed using relatively shallow wells, making them a promising potential base load energy source.4,5 To extract the large amounts of energy available from lowgrade heat, technological advancements are needed. In particular, current systems are limited in their ability to extract low-grade heat energy because of the small temperature difference between the source and the environment and, in the case of waste heat, temporal variability in heat output. Binary cycle systems, which are the most technologically mature energy conversion systems for lower temperature heat sources, typically require heat sources with temperatures greater than 100 °C.6,7 Additionally, these sources utilize a working fluid with a fixed boiling point, resulting in a low tolerance for fluctuations in the heat source temperature. Solid state thermoelectric systems have targeted low temperature © 2017 American Chemical Society

ranges but are expensive and have achieved limited efficiencies (less than 12% of the Carnot efficiency).8,9 Other emerging technologies are being developed that utilize thermoelectrochemical phenomena, such as metal complexation reactions and temperature-dependent electrochemical redox potentials,10−12 or rely on creating concentration gradients with a thermal distillation process and recapturing the salinity gradient energy.13−15 However, these emerging systems have mostly shown relatively low efficiencies and have been limited to smallscale laboratory studies. We recently introduced a new thermo-osmotic energy conversion (TOEC) process to effectively harvest energy from low-grade heat sources.16 The TOEC process relies on thermoosmosis or the transport of fluid through a membrane driven by a temperature gradient.17 To convert thermal energy to mechanical work, fluid is driven by thermo-osmosis from a reservoir at ambient pressure to a reservoir at a higher hydrostatic pressure. The pressurized flow of fluid generated across the membrane is Received: Revised: Accepted: Published: 12925

April 29, 2017 September 15, 2017 October 12, 2017 October 12, 2017 DOI: 10.1021/acs.est.7b02213 Environ. Sci. Technol. 2017, 51, 12925−12937

Article

Environmental Science & Technology

Figure 1. (A) Schematic diagram of water vapor flux, Jw, and heat flux, q, across a hydrophobic, porous membrane from a hot reservoir to a cold reservoir at increased hydraulic pressure, Ph. The membrane thickness, δ, and pore radius, r, are indicated. Color intensity indicates the relative temperature, with a reduced temperature difference at the membrane air−liquid interface due to temperature polarization. (B) Vapor flux achievable with varying membrane permeability coefficient, Bw, and thermal conductivity divided by thickness, Km/δ. Curves are shown for a membrane with a uniform small pore diameter (30 nm), an asymmetric membrane with a larger pore diameter (500 nm) throughout most of the thickness, and an asymmetric membrane operating in a deaerated system. All three membrane types assume a porosity of 0.8, a tortuosity of 1, and a thermal conductivity of 0.04 W m−1 K−1. Curves are generated by varying the thickness of the membrane. (C) Thermal efficiency of the membrane as a function of the membrane vapor permeability, Bwδ. Different curves are representative of various membrane thermal conductivities, where a thermal conductivity of 0.02 W m−1 K−1 corresponds to that of water vapor, 0.04 m−1 K−1 is representative of a high porosity membrane, and 0.2 W m−1 K−1 corresponds to a dense polymer. In (B) and (C), the heat source is 60 °C, the heat sink is 20 °C, the operating pressure is 5 MPa (50 bar), and the heat transfer coefficient on both sides of the membrane is 5000 Wm−2 K−1.

then depressurized through a turbine to generate electricity. To demonstrate the concept experimentally, we used hydrophobic nanoporous membranes that trap air within their pores when submerged in water.16 When a temperature difference is applied across the membrane, the resulting difference in partial vapor pressure across the air interface results in a net vapor flux from the hot to the cold side of the membrane. The hydrophobic membranes generated fluid flows at pressures up to 1.3 MPa (13 bar) when utilizing low-temperature (less than 60 °C) heat sources, allowing for power densities comparable to other membrane-based power generation technologies such as pressure-retarded osmosis or reverse electrodialysis.18,19 Since the driving force for this system is a partial vapor pressure difference between air−liquid interfaces on either side of the membrane, a wide range of source temperatures could also be used. The process is thus able to operate effectively using lowand variable-temperature sources. Despite the initial promise of the TOEC system, it remains in the very early stages of development. Critical process performance indicators, such as the expected power output, membrane requirements, and energy conversion efficiency of a full-scale system, have not been established since previous testing has only been conducted with small, laboratory-scale membrane coupons. Accurate determination of these metrics is critical for understanding the commercial feasibility and anticipated market for such a technology. Additionally, the optimal design of a TOEC system and its components are not well-understood. An understanding of design criteria is particularly important to aid the fabrication of membranes tailored for the process, which will be requisite to achieving efficient operation. In this work, we systematically investigate the performancelimiting phenomena in thermo-osmotic energy conversion of low-grade heat sources, focusing on the energy efficiency and power density of the process. We first examine the mass and heat transfer in the system, studying the effect of different vapor transport regimes on performance. A framework for analyzing full-scale systems is then presented. The optimal operating flow rates and hydraulic pressure are identified before examining the impact of various system parameters on performance. The importance of membrane propertiesspecifically, the vapor

permeability, thermal conductivity, and membrane thicknessis examined together with hydrodynamic conditions in the system. Heat recovery and alternative configurations of the process are also analyzed and discussed. Overall, the comprehensive analysis accurately describes the expected performance of thermoosmotic power generation and identifies critical system parameters that must be optimized to successfully implement the TOEC system.

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MASS AND HEAT TRANSFER IN THERMO-OSMOTIC SYSTEMS Selection of Membrane and Working Fluid. The choice of the membrane and working fluid utilized in the thermoosmotic energy conversion (TOEC) system is of critical importance to the overall performance.16,20 Two types of membranes have been experimentally investigated for thermoosmosis. The first membranes investigated were made from dense polymer materials.17 Both charged and uncharged dense polymeric membranes have shown measurable solvent fluxes driven by a temperature gradient, where transport occurs as the solvent partitions into and diffuses through the polymer material.21−23 More recently, a second type of thermo-osmotic membrane has been developed that utilizes hydrophobic, porous materials that create an air gap between two liquid solutions (Figure 1a).24−26 When a temperature difference is imposed across the air gap, the higher partial vapor pressure on the hot side of the membrane will induce a net flux of vapor to the cold side of the membrane. Water therefore evaporates from the hot side of the membrane, travels through the membrane pore in the gas phase, and condenses on the cold side of the membrane. The driving force available from vapor-gap membranes and dense polymeric membranes is fundamentally different. The hydraulic pressure that can be generated in thermo-osmosis with a given temperature difference can be estimated from ΔPh =

T ⎞ Q* ⎛ ⎜1 − C ⎟ VM ⎝ TH ⎠

(1)

where ΔPh is the hydraulic pressure that can be generated, TC is the temperature on the cold side of the membrane, TH is the 12926

DOI: 10.1021/acs.est.7b02213 Environ. Sci. Technol. 2017, 51, 12925−12937

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Environmental Science & Technology temperature on the hot side of the membrane, VM is the molar volume of the liquid, and Q* is the heat transferred per mole of fluid permeating across the membrane. This equation can be derived by examining the chemical potential across a dense polymeric membrane21 or by utilizing the Antoine and Kelvin equations for the partial vapor pressure of water.27,28 For vaporgap membranes, the value of Q* is equal to the enthalpy of vaporization (41 kJ/mol for water). In the case of dense polymer membranes like cellulose acetate, Q* is estimated to be less than 2 kJ/mol when using water as a working fluid.21,22 Since the heat transferred across vapor-gap membranes is much higher than that transferred across polymeric membranes, the associated pressure generation is also much greater in vapor-gap systems. For example, with only a 1 °C temperature difference and a reference temperature of 20 °C, vapor-gap systems can generate pressures of 7600 kPa (76 bar), whereas dense polymers can only generate pressures less than 400 kPa (4 bar). Experimental studies of thermo-osmosis have shown that vapor-gap membranes have thermo-osmotic water fluxes more than an order of magnitude greater than those of dense polymer membranes with the same temperature difference applied.29,23,30 Additionally, the air gap inside the hydrophobic porous membranes allows them to be far more insulating than a pure polymer material, with the thermal conductivity of air (∼0.025 W m−1 K−1) nearly one-tenth that of a polymeric material (∼0.2 W m−1 K−1).31,32 The insulating properties of vapor-gap membranes aid in preventing unwanted heat loss in the system. However, vapor-gap membranes are also vulnerable to wetting of the membrane pores under hydraulic pressure, as we will discuss later. Because of their favorable experimental performance and improved insulating properties, we focus on vapor-gap membranes in the rest of our analysis. Water is selected as the working fluid since it has been well-characterized, is available at negligible cost, and has a high surface tension to prevent wetting of the pores. However, we note that the vapor-gap membrane system using water as a working fluid is only one of the possible configurations of the thermo-osmotic energy conversion system. Determining Water and Heat Flux Across the Membrane. Accurate determination of water flux across the membrane and the corresponding heat transfer (illustrated in Figure 1) is paramount to understanding the overall power generation performance of the TOEC system. The vapor flux across the membrane, Jw, can be calculated as the product of the vapor permeability coefficient, Bw, and the partial vapor pressure difference between the water interfaces on either side of the membrane:29 Jw = Bw [Pv(TH,m , 0) − Pv(TC,m , Ph)]

Convective heat transfer will occur as the transported water flux, Jw, carries the enthalpy (latent heat) of vaporization, hvap, across the membrane. Conductive heat transfer occurs as heat diffuses through the membrane-vapor matrix; the quantity of conductive heat transfer depends on the thermal conductivity of the membrane, Kc, and the membrane thickness, δ. The thermal conductivity of the membrane can be estimated from the thermal conductivity of the air in the pores, the polymer material, and the porosity.31 We note that convective heat transfer is necessary for operation of the system, while conductive heat transfer represents an energetic loss. The water flux and heat flux across the membrane are both dependent on the temperature at the liquid/vapor interface on either side of the membrane.33 The temperature difference at the interface is lower than that of the bulk fluid due to temperature polarization in the thermal boundary layers on either side of the membrane. The impact of temperature polarization is quantified using the heat transfer coefficients on the hot and cold sides of the membrane (hH and hC, respectively), which can be increased by enhancing mixing and turbulence at the membrane surface: q TH,m = TH,b − hH (4) TC,m = TC,b +

Kc (TH,m − TC,m) δ

(5)

Here, TH,b and TC,b are the bulk temperatures on the hot and cold sides of the membrane, respectively, and TH,m and TC,m are the corresponding temperatures at the membrane surface. The heat transfer coefficients are related to the hydrodynamic conditions in the membrane module, where a large heat transfer coefficient corresponds to a higher Reynolds number. Convective flow through the membrane due to permeating water was found to have a negligible impact on temperature polarization and was therefore not included in our analysis. Vapor Transport Resistances from Molecular Diffusion and Knudsen Effects. Combining eqs 2−5, the mass and heat fluxes across the membrane can be calculated for a given temperature difference and hydraulic pressure. These fluxes will be dependent on certain key membrane parameters, mainly the vapor permeability coefficient and the thermal conductivity of the membrane. While the thermal conductivity can be estimated using the membrane material and porosity, determination of the vapor permeability coefficient is more complex since this value is dependent on the pore size and structure of the membrane. To understand the range of feasible values for the membrane vapor permeability, models must be able to relate performance to membrane structure and properties. Vapor transport through porous membranes is described using two transport regimes: molecular and Knudsen diffusion. These regimes are defined by the dominant resistance mechanism for vapor molecules as they move through the membrane. In the molecular diffusion regime, the pore size is much larger than the mean free path of the vapor (130−145 nm for water vapor between 20 and 60 °C at ambient pressure).29,34 Resistance is therefore dominated by the vapor molecules colliding with air molecules in the pore, and transport resistances due to interactions with the pore walls are minimal. The vapor permeability coefficient of a membrane in the molecular diffusion regime is described by

(2)

The partial vapor pressure, Pv, is a function of the temperature (TH,m and TC,m for hot and cold sides of the membrane surface, respectively) and the hydraulic pressure, which is equal to Ph on the permeate side of the membrane and assumed zero on the ambient pressure feed side. The partial vapor pressure dependence on temperature and hydraulic pressure is determined using the Antione and Kelvin equations, respectively.27,28 As water is transported through the membrane, heat will also be carried across. The heat flux across the membrane, q, is the sum of convective and conductive terms: q = Jw hvap +

q hC

BwMD =

(3) 12927

εPDw M w RTPaτδ

(6) DOI: 10.1021/acs.est.7b02213 Environ. Sci. Technol. 2017, 51, 12925−12937

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Environmental Science & Technology where Dw is the diffusion coefficient of water vapor, P is the total pressure in the pore, Pa is the pressure of air in the pore, ε is the membrane porosity, τ is the tortuosity, δ is the thickness, T is the absolute temperature in the pore, R is the ideal gas constant, and Mw is the molecular weight of water. Note that since pore wall collisions are negligible in terms of the overall membrane resistance in the molecular diffusion regime, the vapor permeability coefficient is independent of the pore radius. If a membrane contains pores that are significantly smaller than the mean free path of water vapor, the system operates in the Knudsen regime.24,29,34 Here, vapor transport resistances are dominated by molecule collisions with the pore wall. The vapor permeability coefficient in the Knudsen regime can be calculated using24,29,34 BwK =

2 εrM w 3RT τδ

8RT πM w

much higher permeability than a membrane with small, uniformly sized pores. In the case of a support layer with a pore size in between 200 and 500 nm, transport occurs through a combination of Knudsen and molecular diffusion, where the permeability coefficient for an asymmetric membrane, BAsym w , can be determined by adding the two resistances in series:29 BwAsym

(9)

Assuming a pore size of 500 nm in the membrane support, the vapor permeability coefficient of a 100 μm thick membrane would equal 1.30 × 10−6 kg Pa−1 s−1 m−2, which is almost an order of magnitude higher than the permeability of a similar membrane with a uniform pore size of 30 nm. Transport resistance in the membrane can be further reduced by decreasing the pressure of air inside the membrane pores. In a closed system, the removal of air in the pores can be performed by deaerating the water or by placing the entire system under vacuum.26,36,37 Either method will decrease the pressure inside the pores to around the partial vapor pressure of water, and molecule−molecule collisions will be reduced (i.e., the mean free path increases). Therefore, in a deaerated system, even membranes with relatively large pore sizes will operate in the Knudsen regime due to the reduced frequency of molecule− molecule collisions, but the resistance will be very low due to the longer path length between each collision. For the 500 nm pore size asymmetric membrane, the vapor permeability with deaeration is equal to 6.07 × 10−6 kg Pa−1 s−1 m−2. We note that models of vapor transport under vacuum also include a relatively minor contribution from viscous flow, which is further discussed in the Supporting Information. Analysis of Coupon-Scale Water Flux and Heat Transport. The first step in understanding TOEC performance is examining the mass and heat transport that occurs in a smallscale membrane coupon, which can be calculated by solving eqs 2−5. These values represent the local water and heat fluxes of a differential area element along a membrane module and will later be integrated with respect to the membrane area to determine full-scale performance. The coupon-scale water flux achievable with a given membrane permeability coefficient, Bw, and thermal conductivity coefficient, Km/δ, is shown in Figure 1B. Both of these parameters (Bw and Km/δ) include thickness and are thus specific to a given membrane, rather than intrinsic material properties. It is straightforward that increasing the vapor permeability and decreasing the thermal conductivity both result in higher water flux, but this improvement diminishes as the water flux increases due to temperature polarization. The permeabilities and thermal conductivities achievable in thermo-osmotic energy conversion will be limited by the structure and properties of the membrane as described in the previous subsection. Figure 1B includes curves that are representative of the performance achievable using membranes with uniform small-sized pores (30 nm diameter), membranes that have an asymmetric structure with larger pores (500 nm diameter) underlying a thin upper layer, and a deaerated system utilizing asymmetric membranes. Each curve is generated by varying the thickness of the membrane while holding the other membrane parameters constant, where a thinner membrane has not only an increased vapor permeability but also a higher thermal conductivity. Optimized membrane structure and deaeration can result in markedly improved performance, with 100 μm thick membranes demonstrating fluxes of 4.37 × 10−3,

(7)

where r is the pore radius. In the Knudsen regime, there is a strong dependence of the vapor permeability on the pore radius. The molecule-wall collisions in small pore size membranes operating in the Knudsen regime will always result in more transport resistances than larger pore size membranes operating under molecular diffusion. Thus, if the only goal of a system is to maximize the vapor permeability, it would be advisible to use a membrane with very large pores. However, to maintain the air gap in the hydrophobic membrane, the pore size must be sufficiently small for capillary forces to prevent water from displacing air in the pores. The wetting (or liquid entry) pressure difference for given membrane, ΔPwet, is described by the Young−Laplace equation35 ΔPwet = −2βγl cos θ /rmax

−1 ⎛ 1 1 ⎞ ⎜ ⎟ = + K MD Bw ⎠ ⎝ Bw

(8)

where β is a geometric pore coefficient, γl is the liquid surface tension, θ is the contact angle, and rmax is the maximum pore radius. For the sizable pressure differences across the membrane required in thermo-osmotic energy conversion, nanoscale pore sizes will be necessary to prevent pore wetting. For example, a system operating at 5 MPa (50 bar) will require a maximum pore diameter of 30 nm assuming a contact angle, θ, of 120° and cylindrical pores (β = 1). Asymmetric Pore Structure and Operation Under Vacuum. The membrane structure and system conditions will govern the vapor transport resistances that occur and thus dictate the vapor permeability coefficient of the membrane, Bw. In the simplest scenario, the membrane would have pores with a uniform diameter throughout the entire thickness of the membrane, as is schematically illustrated in Figure 1A. Since the nanoscale pore sizes required for thermo-osmotic energy conversion are far below the mean free path of water vapor, the membrane will operate in the Knudsen diffusion regime with high transport resistances. For a membrane with a 30 nm pore diameter, a 100 μm thickness, and a porosity of 0.8, the vapor permeability coefficient, Bw, is 3.36 × 10−7 kg Pa−1 s−1 m−2. The transport resistances of the membrane can be reduced by using an asymmetric structure with a thin, small pore size upper layer on top of a thick and larger pore size support. The small pore size layer aids in preventing pore wetting on the permeate side of the membrane, but if the thickness is sufficiently small (less than a hundredth of the total membrane thickness), the contribution of this layer to the overall transport resistances will be negligible ( 17000 W m−2 K−1).31 In most of our calculations, we assume a heat transfer coefficient of 5000 W m−2 K−1, which corresponds to transient flow between the laminar and turbulent regimes. Increasing the heat transfer coefficient beyond this value leads to relatively small performance improvements; quadrupling the value to 20000 W m−2 K−1 will only increase the achievable maximum energy conversion efficiency by 27%. This increase will come at a substantial energetic cost, as any increase in the heat transfer coefficient will result in a corresponding increase in pressure losses across the channel as water is being pumped at a higher flow rate. Conversely, reducing pumping energy by operating the system in laminar flow (h = 1000 W m−2 K−1) will result in a 48% decrease in the achievable maximum energy conversion efficiency. The selection of an appropriate heat transfer coefficient will therefore rely on balancing the energetic cost of increasing pumping with the improvement in both power density and energy efficiency.

ηHX =

HX HX TC,f − TC,0 HX HX TH,0 − TC,0

(14)

The overall system energy conversion efficiency as a function of the heat exchanger efficiency and membrane area is shown in Figure 5. Small decreases in the heat exchanger efficiency result in

Figure 5. Energy conversion efficiency as a function of the heat exchanger efficiency and normalized membrane area. The heat source temperature is 60 °C, and the heat sink temperature is 20 °C. The hydraulic pressure difference between the two streams is 5 MPa (50 bar), and equal flow rates are flow rates used at any point in the membrane module. The membrane permeability coefficient, Bw, is 1 × 10−6 kg m−2 s−1 Pa−1; the thermal conductivity of the membrane, Km, is 0.04 W m−1 K−1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m−2 K−1; and the thickness is 100 μm.

substantial system efficiency losses; a heat exchanger efficiency greater than 90% is required to achieve approximately half of the maximum possible system energy conversion efficiency. The substantial impact of the heat exchanger arises because large quantities of heat are transferred across the membrane module in the enthalpy of vaporization or via conductive heat transfer. Without proper heat recovery, the heat transferred across the membrane from the feed stream must be entirely recuperated from the heat source. With heat recovery using a heat exchanger, the system will partially reheat the feed stream, decreasing the load on the heat source and improving the efficiency. The heat exchanger efficiency also impacts the optimal membrane area in the system, where decreases in the heat exchanger efficiency correspond to lower normalized membrane areas. The requirement for a higher heat exchanger efficiency can be partially relaxed by operating the system with a higher hydraulic pressure difference. Increasing the hydraulic pressure difference allows more power to be generated with every molecule of water that is transported across the membrane, even though nearly the 12933

DOI: 10.1021/acs.est.7b02213 Environ. Sci. Technol. 2017, 51, 12925−12937

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energy efficiency increases from 16% of Carnot with a 40 °C heat source and 20 °C heat sink to 28% with an 80 °C heat source and a 20 °C heat sink. The associated overall energy conversion efficiencies are 1.0% and 4.7%, respectively. The optimal membrane area increases at higher source temperature differences, however, meaning that larger membrane areas will be required to realize the achievable gains in energy efficiency. With water as a working fluid, the system is also limited to operating with a source temperature below the boiling point. Overall, energy conversion efficiency results show the promise of TOEC in handling fluctuation temperatures, although they also reveal that system designs must be tailored to target a given source temperature.

same amount of heat transfer occurs. The heat exchanger requirements can therefore be eased. Figure S1 shows the effect of heat exchanger efficiency for a system operating at 20 MPa (200 bar), highlighting the decreased impact of an inefficient heat exchanger on performance. For example, a system operating with a heat exchanger efficiency of 90% at 20 MPa achieves 63% of the energy conversion efficiency possible with a perfect heat exchanger, whereas the same system would only reach 29% of the possible energy conversion efficiency operating at 5 MPa (50 bar). Additionally, the high-pressure system also needs less membrane area to reach similar efficiencies. Examining the impact of heat exchanger inefficiencies highlights a fundamental challenge for TOEC using vapor-gap membranes and water as a working fluid. The high enthalpy of vaporization results in a large amount of heat transfer across the membrane, which quickly diminishes the driving force across the module. In a single pass, less than 7% of the feed flow rate will permeate through the membrane before the driving force available from the temperature difference is expended.16 The rest of the flow must be redirected through the heat exchanger for energy recovery. Thus, the process heavily relies on the heat exchanger to enable a high efficiency. In addition, the theoretical hydraulic pressure that can be generated by the temperature difference (eq 1) is much greater than the operating pressure realistically possible in the system, meaning the system does not operate at its peak efficiency. These fundamental challenges may be mitigated by future refinements in the membrane and working fluid used in the process. Effect of Source Temperature Difference. To utilize lowgrade heat sources effectively, the TOEC system must operate with a variety of heat source temperatures. The energy conversion efficiency as a function of source temperature is shown in Figure 6. Since the maximum possible energy efficiency defined by the Carnot limit will increase as the source temperature increases, the results are displayed as a percent of the Carnot efficiency. The system can operate favorably with the realistic range of source temperatures, but higher fractions of the Carnot efficiency can be obtained at higher source temperatures. The achievable

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IMPLICATIONS In this study, we identify the heat-to-electricity energy conversion efficiency and power output of thermo-osmotic energy conversion (TOEC) in a variety of scenarios. Our analysis indicates that there is a huge potential to enhance energy conversion efficiency by optimizing the membrane design and system operation. We find that improving the pressure resistance of vapor-gap membranes beyond current experimentally demonstrated values (up to 1.3 MPa or 13 bar) will be critical to achieving high efficiencies, with the optimal pressure for a TOEC system exceeding 5 MPa (50 bar). Some of the largest gains in performance can also be achieved by improving the vapor permeability of the membrane, both by utilizing asymmetric membrane structure and by operating the system with deaerated water. For high membrane permeabilities to be fully exploited, the system must also reduce the impact of temperature polarization by operating with sufficient hydrodynamic mixing at the membrane surface. Decreasing the thermal conductivity can also improve performance, but relatively small gains are possible from improving this value beyond what is possible with current membranes. To aid in membrane design, our analysis also finds that the ideal membrane thickness to achieve both high energy efficiency and power density is between 30 and 100 μm. Thicker membranes will have a very low vapor permeability, while thinner membranes will not be sufficiently insulting. Operating with an asymmetric membrane and deaeration, the TOEC system can achieve a peak energy conversion efficiency around 4.1% (34% of the Carnot limit) with a 60 °C heat source and a 20 °C heat sink, a 5 MPa (50 bar) operating pressure, and ideal heat recovery. At this peak energy efficiency, the system will have a relatively low power density (∼1 W/m2), and some of the achievable energy efficiency will need to be sacrificed to reach higher power density values. The achievable energy efficiency of the TOEC system compares favorably to other emerging processes. For example, the maximum energy efficiency shown for thermoelectric systems is around 12% of Carnot.8 Electrochemical systems have generally achieved efficiencies less than 2% of Carnot,51 with a few recent studies showing efficiencies greater than 10% of Carnot.10−12 The power densities estimated for the TOEC system are also comparable to those of other membrane-based power generation systems, such as pressureretarded osmosis52,53 and reverse electrodialysis.54−56 While the estimated power output in the TOEC process is promising, heat exchanger losses will dramatically reduce the process efficiency, with a 5% decrease in the heat exchanger efficiency reducing the total energy conversion efficiency by around 55%. Additionally, parasitic loads in the system, such as the energy required for pumping, will reduce the net energy output. Quantification of the

Figure 6. Percent of Carnot efficiency obtainable as a function of the heat source temperature. The heat sink temperature is fixed at 20 °C. Various normalized membrane areas, Am/QF,0, are shown. The membrane permeability coefficient is 1 × 10−6 kg m−2 s−1 Pa−1; the thermal conductivity of the membrane, Km, is 0.04 W m−1 K−1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m−2 K−1; and the membrane thickness is 100 μm. The hydraulic pressure difference between the two streams is 5 MPa (50 bar), equal flow rates at any point in the membrane module are assumed, and perfect heat recovery is utilized. 12934

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energy output with these losses will be critical to more accurately compare the efficiency of the TOEC system to other processes. Further studies must push forward the development of membranes for the TOEC system with increased pressure resistance and a high-performance structure. Additionally, sizable gains in performance may be achieved by operating the TOEC system with alternative configurations to those discussed here. In the current work, we have revealed that heat exchanger inefficiencies can substantially reduce the achievable energy conversion efficiency. These losses may be partially mitigated by employing innovative heat recovery methods or by utilizing working fluids other than water with a smaller heat of vaporization. By decreasing the enthalpy of vaporization, heat transfer across the membrane may be reduced, and the demands for heat recovery will be relaxed. However, the use of alternative working fluids will require tailored membranes that can maintain an air gap inside their pores. The combined development of new membranes and introduction of innovative process designs will further improve the prospects of TOEC for low-grade heat energy harvesting.

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GREEK SYMBOLS β geometric pore coefficient (−) γL liquid surface tension (N m−1) δ membrane thickness (−) ε membrane porosity (−) η heat-to-electricity energy conversion efficiency (−) ηmax maximum efficiency equal to the Carnot limit (−) ηHX efficiency of the heat exchanger (−) ηth thermal efficiency of the membrane (−) θ contact angle of liquid water on the membrane surface (deg) ϕ initial feed flow rate fraction (−)

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.est.7b02213. Discussion of the vapor permeability coefficient in a deaerated system and data on the energy efficiency with an imperfect heat exchanger at elevated hydraulic pressure differences (Figure S1) (PDF)

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heat flux across the membrane (W m−2) mass flow rate of the stream (kg s−1) heat transferred per mole of permeating fluid (J mol−1) heat supplied by the heat source (J) pore radius of the membrane (m) maximum pore radius of the membrane (m) ideal gas constant (J K−1 mol−1) position along the area of the membrane module (m2) absolute temperature (K) molar volume of liquid (m3 mol−1) work output of the system (J)

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SUPERSCRIPTS AND SUBSCRIPTS 0 initial condition b bulk C cold stream f final condition F feed stream H hot stream HX heat exchanger m membrane surface P permeate stream

AUTHOR INFORMATION

Corresponding Author

*Phone: +1 (203) 432-2789. Fax: +1 (203) 432-2881. E-mail: [email protected]. ORCID

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Menachem Elimelech: 0000-0003-4186-1563 Notes

The authors declare no competing financial interest.

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REFERENCES

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ACKNOWLEDGMENTS We acknowledge the National Science Foundation Graduate Research Fellowship DGE-1122492 awarded to A.P.S. NOMENCLATURE Am membrane area in the module (m2) Bw vapor permeability coefficient of the membrane (kg Pa−1 s−1 m−2) Dw diffusion coefficient of water (m2 s−1) h heat transfer coefficient (W m−2 K−1) hL specific enthalpy of liquid water (J kg−1) hvap specific enthalpy of water vapor (J kg−1) Jw transmembrane mass flux of water (kg m−2 s−1) Jw average transmembrane mass flux of water (kg m−2 s−1) Kc thermal conductivity of the membrane (W m−2 K−1) Mw molar weight of water (g mol−1) Ph hydraulic pressure (Pa) P total pressure of the membrane pore (Pa) Pa pressure of air in the membrane pore (Pa) Pv partial vapor pressure (Pa) Pwet wetting pressure of the membrane (Pa) PD power density of the membrane module (W m−2) 12935

DOI: 10.1021/acs.est.7b02213 Environ. Sci. Technol. 2017, 51, 12925−12937

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