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Energy Efficiency and Performance Limiting Effects in Thermo-Osmotic Energy Conversion from Low-Grade Heat Anthony P. Straub, and Menachem Elimelech Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.7b02213 • Publication Date (Web): 12 Oct 2017 Downloaded from http://pubs.acs.org on October 16, 2017

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Environmental Science & Technology

Energy Efficiency and Performance Limiting Effects in Thermo-Osmotic Energy Conversion from Low-Grade Heat

Anthony P. Straub and Menachem Elimelech*

Department of Chemical and Environmental Engineering, Yale University, New Haven, Connecticut 06520-8286, United States

* Corresponding author; Address: P.O. Box 208286, Yale University, New Haven, CT 06520;

Phone: +1 (203) 432-2789; Fax: +1 (203) 432-2881; email: [email protected]

1 ACS Paragon Plus Environment


ABSTRACT 1

Low-grade heat energy from sources below 100 °C is available in massive quantities around the

2

world, but cannot be converted to electricity effectively using existing technologies due to

3

variability in the heat output and the small temperature difference between the source and

4

environment. The recently developed thermo-osmotic energy conversion (TOEC) process has the

5

potential to harvest energy from low-grade heat sources by using a temperature difference to create

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a pressurized liquid flux across a membrane, which can be converted to mechanical work via a

7

turbine. In this study, we perform the first analysis of energy efficiency and the expected

8

performance of the TOEC technology, focusing on systems utilizing hydrophobic porous vapor-

9

gap membranes and water as a working fluid. We begin by developing a framework to analyze

10

realistic mass and heat transport in the process, probing the impact of various membrane

11

parameters and system operating conditions. Our analysis reveals that an optimized system can

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achieve heat-to-electricity energy conversion efficiencies up to 4.1% (34% of the Carnot

13

efficiency) with hot and cold working temperatures of 60 and 20 °C, respectively, and an operating

14

pressure of 5 MPa (50 bar). Lower energy efficiencies, however, will occur in systems operating

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with high power densities (>5 W/m2) and with finite-sized heat exchangers. We identify that the

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most important membrane properties for achieving high performance are an asymmetric pore

17

structure, high pressure resistance, a high porosity, and a thickness of 30 to 100 µm. We also

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quantify the benefits in performance from utilizing deaerated water streams, strong hydrodynamic

19

mixing in the membrane module, and high heat exchanger efficiencies. Overall, our study

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demonstrates the promise of full-scale TOEC systems to extract energy from low-grade heat and

21

identifies key factors for performance optimization moving forward.


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INTRODUCTION

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The massive quantities of low-grade heat available globally have the potential to supply a

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considerable portion of energy needs if effectively harvested.1 Waste heat discharged from

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industry and power plants amounts to more than 8000 TWh per year in the United States alone,

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with industrial facilities discharging about a third of energy consumed during operation.2,3

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Geothermal reservoirs at low-temperatures (less than 150 °C) are also geospatially abundant and

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can be accessed using relatively shallow wells, making them a promising potential base load

29

energy source.4,5

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To extract the large amounts of energy available from low-grade heat, technological

31

advancements are needed. In particular, current systems are limited in their ability to extract low-

32

grade heat energy because of the small temperature difference between the source and the

33

environment and, in the case of waste heat, temporal variability in heat output. Binary cycle

34

systems, which are the most technologically mature energy conversion systems for lower

35

temperature heat sources, typically require heat sources with temperatures greater than 100 °C.6,7

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Additionally, these sources utilize a working fluid with a fixed boiling point, resulting in a low

37

tolerance for fluctuations in the heat source temperature. Solid state thermoelectric systems have

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targeted low temperature ranges, but are expensive and have achieved limited efficiencies (less

39

than 12% of the Carnot efficiency).8,9 Other emerging technologies are being developed that utilize

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thermo-electrochemical phenomena, such as metal complexation reactions and temperature-

41

dependent electrochemical redox potentials,10–12 or rely on creating concentration gradients with a

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thermal distillation process and recapturing the salinity gradient energy.13–15 However, these

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emerging systems have mostly shown relatively low efficiencies and have been limited to small-

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scale laboratory studies.

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We recently introduced a new thermo-osmotic energy conversion (TOEC) process to effectively

46

harvest energy from low-grade heat sources.16 The TOEC process relies on thermo-osmosis, or the

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transport of fluid through a membrane driven by a temperature gradient.17 To convert thermal

48

energy to mechanical work, fluid is driven by thermo-osmosis from a reservoir at ambient pressure

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to a reservoir at a higher hydrostatic pressure. The pressurized flow of fluid generated across the

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membrane is then depressurized through a turbine to generate electricity. To demonstrate the

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concept experimentally, we used hydrophobic nanoporous membranes that trap air within their 3 ACS Paragon Plus Environment


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pores when submerged in water.16 When a temperature difference is applied across the membrane,

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the resulting difference in partial vapor pressure across the air interface results in a net vapor flux

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from the hot to the cold side of the membrane. The hydrophobic membranes generated fluid flows

55

at pressures up to 1.3 MPa (13 bar) when utilizing low-temperature (less than 60 °C) heat sources,

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allowing for power densities comparable to other membrane-based power generation technologies

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such as pressure-retarded osmosis or reverse electrodialysis.18,19 Since the driving force for this

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system is a partial vapor pressure difference between air-liquid interfaces on either side of the

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membrane, a wide range of source temperatures could also be used. The process is thus able to

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operate effectively using low- and variable-temperature sources

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Despite the initial promise of the TOEC system, it remains in the very early stages of

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development. Critical process performance indicators, such as the expected power output,

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membrane requirements, and energy conversion efficiency of a full-scale system have not been

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established since previous testing has only been conducted with small, laboratory-scale membrane

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coupons. Accurate determination of these metrics is critical for understanding the commercial

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feasibility and anticipated market for such a technology. Additionally, the optimal design of a

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TOEC system and its components are not well-understood. An understanding of design criteria is

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particularly important to aid the fabrication of membranes tailored for the process, which will be

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requisite to achieving efficient operation.

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In this work, we systematically investigate the performance-limiting phenomena in thermo-

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osmotic energy conversion of low-grade heat sources, focusing on the energy efficiency and power

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density of the process. We first examine the mass and heat transfer in the system, studying the

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effect of different vapor transport regimes on performance. A framework for analyzing full-scale

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systems is then presented. The optimal operating flow rates and hydraulic pressure are identified

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before examining the impact of various system parameters on performance. The importance of

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membrane properties—specifically, the vapor permeability, thermal conductivity, and membrane

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thickness—is examined together with hydrodynamic conditions in the system. Heat recovery and

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alternative configurations of the process are also analyzed and discussed. Overall, the

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comprehensive analysis accurately describes the expected performance of thermo-osmotic power

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generation and identifies critical system parameters that must be optimized to successfully

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implement the TOEC system.


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MASS AND HEAT TRANSFER IN THERMO-OSMOTIC SYSTEMS

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Selection of Membrane and Working Fluid. The choice of the membrane and working fluid

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utilized in the thermo-osmotic energy conversion (TOEC) system is of critical importance to the

85

overall performance.16,20 Two types of membranes have been experimentally investigated for

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thermo-osmosis. The first membranes investigated were made from dense polymer materials.17

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Both charged and uncharged dense polymeric membranes have shown measurable solvent fluxes

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driven by a temperature gradient, where transport occurs as the solvent partitions into and diffuses

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through the polymer material.21–23 More recently, a second type of thermo-osmotic membrane has

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been developed that utilizes hydrophobic, porous materials that create an air gap between two

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liquid solutions (Figure 1b).24–26 When a temperature difference is imposed across the air gap, the

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higher partial vapor pressure on the hot side of the membrane will induce a net flux of vapor to the

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cold side of the membrane. Water therefore evaporates from the hot side of the membrane, travels

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through the membrane pore in the gas phase, and condenses on the cold side of the membrane.

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The driving force available from vapor-gap membranes and dense polymeric membranes is

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fundamentally different. The hydraulic pressure that can be generated in thermo-osmosis with a

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given temperature difference can be estimated from ∆ Ph =

Q* VM

 TC  1 −   TH 

(1)

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where ∆Ph is the hydraulic pressure that can be generated, TC is the temperature on the cold side

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of the membrane, TH is the temperature on the hot side of the membrane, VM is the molar volume

100

of the liquid, and Q* is the heat transferred per mole of fluid permeating across the membrane.

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This equation can be derived by examining the chemical potential across a dense polymeric

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membrane21 or by utilizing the Antoine and Kelvin equations for the partial vapor pressure of

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water.27,28 For vapor-gap membranes, the value of Q* is equal to the enthalpy of vaporization (41

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kJ/mol for water). In the case of dense polymer membranes like cellulose acetate, Q* is estimated

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to be less than 2 kJ/mol when using water as a working fluid.21,22 Since the heat transferred across

106

vapor-gap membranes is much higher than that transferred across polymeric membranes, the

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associated pressure generation is also much greater in vapor-gap systems. For example, with only

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a 1 °C temperature difference and a reference temperature of 20 °C, vapor-gap systems can



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generate pressures of 7600 kPa (76 bar), whereas dense polymers can only generate pressures less

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than 400 kPa (4 bar).

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Experimental studies of thermo-osmosis have shown that vapor-gap membranes have thermo-

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osmotic water fluxes more than an order of magnitude greater than those of dense polymer

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membranes with the same temperature difference applied.29,23,30 Additionally, the air gap inside

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the hydrophobic porous membranes allows them to be far more insulating than a pure polymer

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material, with the thermal conductivity of air (~0.025 W m-1K-1) nearly one-tenth that of a

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polymeric material (~0.2 W m-1K-1).31,32 The insulating properties of vapor-gap membranes aide

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in preventing unwanted heat loss in the system. However, vapor-gap membranes are also

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vulnerable to wetting of the membrane pores under hydraulic pressure, as we will discuss later.

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Because of their favorable experimental performance and improved insulating properties, we

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focus on vapor-gap membranes in the rest of our analysis. Water is selected as the working fluid

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since it has been well-characterized, is available at negligible cost, and has a high surface tension

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to prevent wetting of the pores. However, we note that the vapor-gap membrane system using

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water as a working fluid is only one of the possible configurations of the thermo-osmotic energy

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conversion system.

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Determining Water and Heat Flux Across the Membrane. Accurate determination of

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water flux across the membrane and the corresponding heat transfer (illustrated in Figure 1) is

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paramount to understanding the overall power generation performance of the TOEC system. The

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vapor flux across the membrane, Jw, can be calculated as the product of the vapor permeability

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coefficient, Bw, and the partial vapor pressure difference between the water interfaces on either

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side of the membrane:29

J w = Bw[ Pv (TH ,m ,0) − Pv (TC ,m , Ph )]

(2)

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The partial vapor pressure, Pv, is a function of the temperature (TH,m and TC,m for hot and cold sides

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of the membrane surface, respectively) and the hydraulic pressure, which is equal to Ph on the

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permeate side of the membrane and assumed zero on the ambient pressure feed side. The partial

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vapor pressure dependence on temperature and hydraulic pressure is determined using the Antione

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and Kelvin equations, respectively.27,28

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FIGURE 1 6 ACS Paragon Plus Environment

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137 138


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 = J whvap +

Kc

δ

(TH ,m − TC ,m )

(3)

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Convective heat transfer will occur as the transported water flux, Jw, carries the enthalpy (latent

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heat) of vaporization, hvap, across the membrane. Conductive heat transfer occurs as heat diffuses

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through the membrane-vapor matrix; the quantity of conductive heat transfer depends on the

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thermal conductivity of the membrane, Kc, and the membrane thickness, ߜ. The thermal

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conductivity of the membrane can be estimated from the thermal conductivity of the air in the

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pores, the polymer material, and the porosity.31 We note that convective heat transfer is necessary

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for operation of the system while conductive heat transfer represents an energetic loss.

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The water flux and heat flux across the membrane are both dependent on the temperature at the

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liquid/vapor interface on either side of the membrane.33 The temperature difference at the interface

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is lower than that of the bulk fluid due to temperature polarization in the thermal boundary layers

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on either side of the membrane. The impact of temperature polarization is quantified using the heat

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transfer coefficients on the hot and cold side of the membrane (hH and hC, respectively), which can

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be increased by enhancing mixing and turbulence at the membrane surface:

TH ,m = TH ,b −

q hH

(4)

TC,m = TC,b +

q hC

(5)

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Here, TH,b and TC,b are the bulk temperatures on the hot and cold side of the membrane,

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respectively, and TH,m and TC,m are the corresponding temperatures at the membrane surface. The

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heat transfer coefficients are related to the hydrodynamic conditions in the membrane module,

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where a large heat transfer coefficient corresponds to a higher Reynolds number. Convective flow

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through the membrane due to permeating water was found to have a negligible impact on

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temperature polarization and was therefore not included in our analysis.

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Vapor Transport Resistances from Molecular Diffusion and Knudsen Effects.

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Combining equations 2-5, the mass and heat fluxes across the membrane can be calculated for a 7 ACS Paragon Plus Environment


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given temperature difference and hydraulic pressure. These fluxes will be dependent on certain

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key membrane parameters, mainly the vapor permeability coefficient and the thermal conductivity

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of the membrane. While the thermal conductivity can be estimated using the membrane material

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and porosity, determination of the vapor permeability coefficient is more complex since this value

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is dependent on the pore size and structure of the membrane. To understand the range of feasible

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values for the membrane vapor permeability, models must be able to relate performance to

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membrane structure and properties.

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Vapor transport through porous membranes is described using two transport regimes: molecular

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and Knudsen diffusion. These regimes are defined by the dominant resistance mechanism for

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vapor molecules as they move through the membrane. In the molecular diffusion regime, the pore

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size is much larger than the mean free path of the vapor (130-145 nm for water vapor between

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20 °C and 60 °C at ambient pressure).29,34 Resistance is therefore dominated by the vapor

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molecules colliding with air molecules in the pore, and transport resistances due to interactions

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with the pore walls are minimal. The vapor permeability coefficient of a membrane in the

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molecular diffusion regime is described by: BwMD =

ε PD w M W RTPaτδ

(6)

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Where Dw is the diffusion coefficient of water vapor, P is the total pressure in the pore, Pa is the

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pressure of air in the pore, ε is the membrane porosity, τ is the tortuosity, δ is the thickness, T is

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the absolute temperature in the pore, R is the ideal gas constant, and Mw is the molecular weight

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of water. Note that since pore wall collisions are negligible in terms of the overall membrane

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resistance in the molecular diffusion regime, the vapor permeability coefficient is independent of

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the pore radius.

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If a membrane contains pores that are significantly smaller than the mean free path of water

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vapor, the system operates in the Knudsen regime.24,29,34 Here, vapor transport resistances are

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dominated by molecule collisions with the pore wall. The vapor permeability coefficient in the

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Knudsen regime can be calculated using24,29,34 BwK =

2 ε rM W 3 RT τδ

8 RT π MW


(7)

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where r is the pore radius. In the Knudsen regime, there is a strong dependence of the vapor

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permeability on the pore radius.

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The molecule-wall collisions in small pore size membranes operating in the Knudsen regime

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will always result in more transport resistances than larger pore size membranes operating under

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molecular diffusion. Thus, if the only goal of a system is to maximize the vapor permeability, it

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would be advisable to use a membrane with very large pores. However, to maintain the air gap in

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the hydrophobic membrane, the pore size must be sufficiently small for capillary forces to prevent

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water from displacing air in the pores. The wetting (or liquid entry) pressure difference for given

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membrane, ∆Pwet, is described by the Young-Laplace equation:35

∆Pwet = −2βγ l cosθ rmax

(8)

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where β is a geometric pore coefficient, γ l is the liquid surface tension, θ is the contact angle, and

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rmax is the maximum pore radius. For the sizable pressure differences across the membrane required

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in thermo-osmotic energy conversion, nanoscale pore sizes will be necessary to prevent pore

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wetting. For example, a system operating at 5 MPa (50 bar) will require a maximum pore diameter

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of 30 nm assuming a contact angle, θ, of 120° and cylindrical pores (β = 1).

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Asymmetric Pore Structure and Operation Under Vacuum. The membrane structure

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and system conditions will govern the vapor transport resistances that occur, and thus dictate the

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vapor permeability coefficient of the membrane, Bw. In the simplest scenario, the membrane would

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have pores with a uniform diameter throughout the entire thickness of the membrane, as is

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schematically illustrated in Figure 1A. Since the nanoscale pore sizes required for thermo-osmotic

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energy conversion are far below the mean free path of water vapor, the membrane will operate in

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the Knudsen diffusion regime with high transport resistances. For a membrane with a 30 nm pore

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diameter, a 100 µm thickness, and a porosity of 0.8, the vapor permeability coefficient, Bw, is 3.36

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× 10-7 kg Pa-1s-1m-2.

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The transport resistances of the membrane can be reduced by using an asymmetric structure with

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a thin, small pore size upper layer on top of a thick and larger pore size support. The small pore

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size layer aids in preventing pore wetting on the permeate side of the membrane, but if the

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thickness is sufficiently small (less than a hundredth of the total membrane thickness), the

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contribution of this layer to the overall transport resistances will be negligible ( 17000 W m-2K-1).31 In most of our calculations, we assume a heat transfer coefficient of

507

5000 W m-2K-1, which corresponds to transient flow between the laminar and turbulent regimes.

508

Increasing the heat transfer coefficient beyond this value leads to relatively small performance

509

improvements; quadrupling the value to 20000 W m-2K-1 will only increase the achievable

510

maximum energy conversion efficiency by 27%. This increase will come at a substantial energetic

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cost, as any increase in the heat transfer coefficient will result in a corresponding increase in

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pressure losses across the channel as water is being pumped at a higher flow rate. Conversely,

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reducing pumping energy by operating the system in laminar flow (h = 1000 W m-2K-1) will result

514

in a 48% decrease in the achievable maximum energy conversion efficiency. The selection of an

515

appropriate heat transfer coefficient will therefore rely on balancing the energetic cost of

516

increasing pumping with the improvement in both power density and energy efficiency.

517

INFLUENCE OF HEAT RECOVERY AND SOURCE TEMPERATURE

518

Impact of Heat Exchanger on Energy Efficiency. The extent of heat recovery that occurs

519

in a heat exchanger can largely dictate the expected performance of a system. The heat exchanger

520

efficiency, ηHX, defines how much of the heat entering the exchanger is transferred to the exiting

521

streams. Since a balanced system will have equal flow rates on either side of the heat exchanger

522

and the specific heat capacity is equal in all streams, we can simply define the heat exchanger


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efficiency based on the initial temperature of the cold stream ( TCHX ,0 ), the initial temperature of the

524

hot stream ( THHX,0 ), and the final temperature of the cold stream ( THHX, f ): η HX =

HX TCHX , f − TC ,0

THHX,0 − TCHX ,0

(14)

525

The overall system energy conversion efficiency as a function of the heat exchanger efficiency

526

and membrane area is shown in Figure 5. Small decreases in the heat exchanger efficiency result

527

in substantial system efficiency losses; a heat exchanger efficiency greater than 90% is required to

528

achieve approximately half of the maximum possible system energy conversion efficiency. The

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substantial impact of the heat exchanger arises because large quantities of heat are transferred

530

across the membrane module in the enthalpy of vaporization or via conductive heat transfer.

531

Without proper heat recovery, the heat transferred across the membrane from the feed stream must

532

be entirely recuperated from the heat source. With heat recovery using a heat exchanger, the

533

system will partially reheat the feed stream, decreasing the load on the heat source and improving

534

the efficiency. The heat exchanger efficiency also impacts the optimal membrane area in the

535

system, where decreases in the heat exchanger efficiency correspond to lower normalized

536

membrane areas.

537

FIGURE 5

538

The requirement for a higher heat exchanger efficiency can be partially relaxed by operating the

539

system with a higher hydraulic pressure difference. Increasing the hydraulic pressure difference

540

allows more power to be generated with every molecule of water that is transported across the

541

membrane, even though nearly the same amount of heat transfer occurs. The heat exchanger

542

requirements can therefore be eased. Figure S1 shows the effect of heat exchanger efficiency for

543

a system operating at 20 MPa (200 bar), highlighting the decreased impact of an inefficient heat

544

exchanger on performance. For example, a system operating with a heat exchanger efficiency of

545

90% at 20 MPa achieves 63% of the energy conversion efficiency possible with a perfect heat

546

exchanger, whereas the same system would only reach 29% of the possible energy conversion

547

efficiency operating at 5 MPa (50 bar). Additionally, the high-pressure system also needs less

548

membrane area to reach similar efficiencies.



549

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Examining the impact of heat exchanger inefficiencies highlights a fundamental challenge for

550

TOEC using vapor-gap membranes and water as a working fluid.

The high enthalpy of

551

vaporization results in a large amount of heat transfer across the membrane, which quickly

552

diminishes the driving force across the module. In a single pass, less than 7% of the feed flow rate

553

will permeate through the membrane before the driving force available from the temperature

554

difference is expended.16 The rest of the flow must be redirected through the heat exchanger for

555

energy recovery. Thus, the process heavily relies on the heat exchanger to enable a high efficiency.

556

In addition, the theoretical hydraulic pressure that can be generated by the temperature difference

557

(eq 1) is much greater than the operating pressure realistically possible in the system, meaning the

558

system does not operate at its peak efficiency. These fundamental challenges may be mitigated by

559

future refinements in the membrane and working fluid used in the process.

560

Effect of Source Temperature Difference. To utilize low-grade heat sources effectively,

561

the TOEC system must operate with a variety of heat source temperatures. The energy conversion

562

efficiency as a function of source temperature is shown in Figure 6. Since the maximum possible

563

energy efficiency defined by the Carnot limit will increase as the source temperature increases, the

564

results are displayed as a percent of the Carnot efficiency. FIGURE 6

565 566

The system can operate favorably with the realistic range of source temperatures, but higher

567

fractions of the Carnot efficiency can be obtained at higher source temperatures. The achievable

568

energy efficiency increases from 16% of Carnot with a 40 °C heat source and 20 °C heat sink to

569

28% with an 80 °C heat source and a 20 °C heat sink. The associated overall energy conversion

570

efficiencies are 1.0% and 4.7%, respectively. The optimal membrane area increases at higher

571

source temperature differences, however, meaning that larger membrane areas will be required to

572

realize the achievable gains in energy efficiency. With water as a working fluid, the system is also

573

limited to operating with a source temperature below the boiling point. Overall, energy conversion

574

efficiency results show the promise of TOEC in handling fluctuation temperatures, although they

575

also reveal that system designs must be tailored to target a given source temperature.

576

IMPLICATIONS


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In this study, we identify the heat-to-electricity energy conversion efficiency and power output of

578

thermo-osmotic energy conversion (TOEC) in a variety of scenarios. Our analysis indicates that

579

there is a huge potential to enhance energy conversion efficiency by optimizing the membrane

580

design and system operation. We find that improving the pressure resistance of vapor-gap

581

membranes beyond current experimentally demonstrated values (up to 1.3 MPa or 13 bar) will be

582

critical to achieving high efficiencies, with the optimal pressure for a TOEC system exceeding 5

583

MPa (50 bar). Some of the largest gains in performance can also be achieved by improving the

584

vapor permeability of the membrane, both by utilizing asymmetric membrane structure and by

585

operating the system with deaerated water. For high membrane permeabilities to be fully exploited,

586

the system must also reduce the impact of temperature polarization by operating with sufficient

587

hydrodynamic mixing at the membrane surface. Decreasing the thermal conductivity can also

588

improve performance, but relatively small gains are possible from improving this value beyond

589

what is possible with current membranes. To aide in membrane design, our analysis also finds that

590

the ideal membrane thickness to achieve both high energy efficiency and power density is between

591

30 and 100 µm. Thicker membranes will have a very low vapor permeability, while thinner

592

membranes will not be sufficiently insulting.

593

Operating with an asymmetric membrane and deaeration, the TOEC system can achieve a peak

594

energy conversion efficiency around 4.1% (34% of the Carnot limit) with a 60 °C heat source and

595

a 20 °C heat sink, a 5 MPa (50 bar) operating pressure, and ideal heat recovery. At this peak energy

596

efficiency, the system will have a relatively low power density (~1 W/m2), and some of the

597

achievable energy efficiency will need to be sacrificed to reach higher power density values. The

598

achievable energy efficiency of the TOEC system compares favorably to other emerging

599

processes. For example, the maximum energy efficiency shown for thermoelectric systems is

600

around 12% of Carnot.8 Electrochemical systems have generally achieved efficiencies less than

601

2% of Carnot,51 with a few recent studies showing efficiencies greater than 10% of Carnot.10–12

602

The power densities estimated for the TOEC system are also comparable to those of other

603

membrane-based power generation systems, such a pressure-retarded osmosis52,53 and reverse

604

electrodialysis.54–56 While the estimated power output in the TOEC process is promising, heat

605

exchanger losses will dramatically reduce the process efficiency, with a 5% decrease in the heat

606

exchanger efficiency reducing the total energy conversion efficiency by around 55%.

607

Additionally, parasitic loads in the system, such as the energy required for pumping, will reduce 23 ACS Paragon Plus Environment


608

the net energy output. Quantification of the energy output with these losses will be critical to more

609

accurately compare the efficiency of the TOEC system to other processes.

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610

Further studies must push forward the development of membranes for the TOEC system with

611

increased pressure resistance and a high-performance structure. Additionally, sizable gains in

612

performance may be achieved by operating the TOEC system with alternative configurations to

613

those discussed here. In the current work, we have revealed that heat exchanger inefficiencies can

614

substantially reduce the achievable energy conversion efficiency. These losses may be partially

615

mitigated by employing innovative heat recovery methods or by utilizing working fluids other than

616

water with a smaller heat of vaporization. By decreasing the enthalpy of vaporization, heat transfer

617

across the membrane may be reduced, and the demands for heat recovery will be relaxed.

618

However, the use of alternative working fluids will require tailored membranes that can maintain

619

and air gap inside their pores. The combined development of new membranes and introduction of

620

innovative process designs will further improve the prospects of TOEC for low-grade heat energy

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harvesting.

NOMENCLATURE ‫ܣ‬௠

membrane area in the module (m2) vapor permeability coefficient of the membrane (kg Pa-1s-1m-2) diffusion coefficient of water (m2 s-1) heat transfer coefficient (W m-2K-1)

‫ܤ‬௪ ‫ܦ‬௪ ℎ ℎ௅ specific enthalpy of liquid water (J kg-1) ℎ௩௔௣ specific enthalpy of water vapor (J kg-1) ‫ܬ‬௪ transmembrane mass flux of water (kg m-2s-1) average transmembrane mass flux of water (kg m-2s-1) ‫ܬ‬തതത ௪ ‫ܭ‬௖ ‫ܯ‬௪

thermal conductivity of the membrane (W m-2K-1) molar weight of water (g mol-1)

ܲ௛

hydraulic pressure (Pa) total pressure of in the membrane pore (Pa) pressure of air in the membrane pore (Pa)

ܲ ܲ௔ ܲ௩ partial vapor pressure (Pa) ܲ௪௘௧ wetting pressure of the membrane (Pa) ܲ‫ ܦ‬power density of the membrane module (W m-2) heat flux across the membrane (W m-2) ‫ݍ‬ 24 ACS Paragon Plus Environment

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ܳ ܳ∗ ܳ௛ ‫ݎ‬ ‫ݎ‬௠௔௫ ܴ ‫ݏ‬ ܶ ܸெ ܹ

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-1mol-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)

Greek Symbols ߚ ߛ௅ ߜ ߝ ߟ

geometric pore coefficient (-) liquid surface tension (N m-1) membrane thickness (-) membrane porosity (-) heat-to-electricity energy conversion efficiency (-) maximum efficiency equal to the Carnot limit (-)

ߟ௠௔௫ ߟு௑ efficiency of the heat exchanger (-) ߟ௧௛ thermal efficiency of the membrane (-) ߠ ߶

contact angle of liquid water on the membrane surface (°) initial feed flow rate fraction (-)

Superscripts and Subscripts 0 ܾ ‫ܥ‬ ݂ ‫ܨ‬

initial condition bulk cold stream final condition feed stream

‫ ܪ‬hot stream ‫ ܺܪ‬heat exchanger ݉ membrane surface



ܲ

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permeate stream

ACKNOWLEDGMENTS We acknowledge the National Science Foundation Graduate Research Fellowship DGE-1122492 awarded to A.P.S.

SUPPORTING INFORMATION AVAILABLE 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). This material is available free of charge via the Internet at http://pubs.acs.org.

NOTE The authors declare no competing financial interest.


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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-1K-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-1K-1 corresponds to that of water vapor, 0.04 m-1K-1 is representative of a high porosity membrane, and 0.2 W m-1K-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-2K-1.



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Figure 2. (A) Schematic of a closed-loop thermo-osmotic energy conversion system with heat recovery. Streams exit the heat source at a hot temperature, TH,0, and the heat sink is at a cold temperature, TC,0. Mass and heat are transferred across the membrane module into a pressurized zone (yellow area encircled by dotted line). The mass flow is depressurized through a turbine to generate power. (B) Temperature profiles of the feed and permeate streams in the membrane module. The horizontal axis corresponds to the relative position in the membrane module. Three normalized membrane areas, Am/QF,0, are used in calculations as indicated next to each curve. 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). The membrane permeability coefficient, Bw, is 1 × 10-6 kg m-2s-1Pa-1; the thermal conductivity of the membrane, Km, is 0.04 W m-1K-1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m-2K-1; and the membrane thickness is 100 µm. Equal initial feed and permeate flow rates are assumed. 32 ACS Paragon Plus Environment

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Figure 3. (A) Heat-to-electricity energy conversion efficiency as a function of the initial feed flow rate fraction (that is, the initial feed flow rate divided by the total initial flow rate of the feed and permeate). Different curves are shown for various normalized membrane areas, Am/Qtot. Symbols (open circles) indicate the efficiency and flow rate fraction that corresponds to a balanced condition with equal flow rates at any point along the membrane module. The hydraulic pressure difference between the two streams is 5 MPa (50 bar). (B) Energy conversion efficiency as a function of the hydraulic pressure difference across the membrane for TOEC systems operating with a balanced feed flow rate fraction. In both figures, the heat source temperature is 60 °C and the heat sink temperature is 20 °C. The membrane permeability coefficient, Bw, is 1 × 10-6 kg m-2s-1Pa-1; the thermal conductivity of the membrane, Km, is 0.04 W m-1K-1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m-2K-1; and the membrane thickness is 100 µm.



Figure 4. Energy conversion efficiency and power density curves for TOEC systems with varied membrane and hydrodynamic properties. Curves are generated by increasing the membrane area of a given system with higher membrane area from right to left. Results are shown for variations in: (A) membrane thickness, δ, (B) vapor permeability coefficient, Bw, (C) thermal conductivity, Km, and (D) heat transfer coefficient, h. Unless otherwise stated, the vapor permeability of the membrane, Bw, is 1 × 10-6 kg m-2s-1Pa-1; the thermal conductivity of the membrane, Km, is 0.04 W m-1K-1; the thickness is 100 µm; and the heat transfer coefficient, h, on both sides of the membrane is 5000 W m-2K-1. 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), equal flow rates are used at any point in the membrane module, and a perfect heat exchanger efficiency is assumed. 34 ACS Paragon Plus Environment

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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 are used at any point in the membrane module. The membrane permeability coefficient, Bw, is 1 × 10-6 kg m-2s-1Pa-1; the thermal conductivity of the membrane, Km, is 0.04 W m-1K-1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m2

K-1; and the thickness is 100 µm.



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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-2s-1Pa-1; the thermal conductivity of the membrane, Km, is 0.04 W m-1K-1; the heat transfer coefficient, h, on both sides of the membrane is 5000 W m-2K-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.


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