Opportunities and Challenges in Passive Thermal-Fluid and Energy Systems

Shabgard, Hamidreza; Li, Xianglin; Faghri, Amir

doi:10.1115/1.4055342

Abstract

This article focuses on passive systems that are used in energy and thermal-fluid applications. These passive systems do not have moving parts and are reliable and cost-effective. Fluid motion in these passive devices could be driven by capillary force, gravity, osmotic pressure, and/or concentration gradient. The fundamental mechanisms and limitations of transport phenomena for passive systems are highlighted, followed by their applications in heat pipes, fuel cells, thermal energy storage, and desalination systems. The capabilities of the passive systems are limited by the balance between the driving force and transport resistance. Based on the fundamental understanding of fluid flow and phase change in passive systems, this study proposes associated transport phenomena and quantitative criteria to determine the maximum heat transfer rate, the transport distance, and minimum pore size of wick structures (when relevant) in these passive devices. This article concludes with the discussion of challenges and future opportunities of passive systems.

1 Introduction

Access to clean, affordable, and reliable energy systems has been a crucial factor for prosperity and economic growth since the beginning of the industrial revolution. Energy systems are entering a period of unprecedented challenges: new technologies, new requirements, and new vulnerabilities in addition to environmental impact and efficiency concerns. Passive thermofluidic and energy systems can play a unique and significant role in the transition to a sustainable and environmentally friendly energy ecosystem.

Passive energy terminology is a term initially used in reference to passive solar and natural energy systems with emphasis in passive solar buildings since the 1970s energy crisis [1]. Passivity is also used as a property of engineering systems, most commonly encountered in analog electronics and control systems [2]. However passive technologies are recognized in general as systems with no major moving parts/components such as fan, pump, blower, compressor, turbine, etc. This is the context that we are interested to promote in this paper since this type of passive system has a profound impact on a wide range of energy and thermal systems while simultaneously addressing other challenges, including environmental impact, mass and volume limitations, and low maintenance.

The drive for miniaturization of thermal and energy systems is also of interest among many technologies for different applications due to the benefits of usability it can bring, but there are obviously challenges too. In many applications, the miniaturization is only possible by the integration of the concept of passive methodology. The main advantages of such passive concepts/devices are that they are environmentally safe, portable, cost-effective, and have obviously smaller dimensions. Passive systems are in general devices that are developed with no moving parts. In particular, passive thermal energy systems are defined as

Continuous fluid flow with the transport of mass (and momentum/flow) and heat without moving parts
No moving parts as power sources such as fuel cells, heat engines, heat pumps
No moving parts such as energy transfer or storage systems such as heat pipes, heat exchangers, thermal energy storage systems

Passive systems are typically characterized by small sizes, low weight, portable, green, and low maintenance with applications in energy systems, electronics cooling devices, power sources, medical devices, energy storage systems, and security and emergency power.

The principal mechanism or driving forces for passive thermal and energy systems are

Capillary force (porous media)
Evaporation and condensation
Gravity
Concentration gradient
Osmotic pressure difference

Several critical passive utilization systems are discussed in terms of their principles of operations and benefits as well as their advances and opportunities. These passive thermal and energy technologies are heat pipes, passive direct methanol fuel cells, thermal energy storage systems, and passive desalination processes. These passive technologies share similar basic operating principles and technical challenges and opportunities.

Based on our understanding, no previous archival publication addresses the above observations and technologies from a common perspective. The main objectives of this paper are (a) evaluations of several important passive energy and thermal technologies, (b) discussions of their common fundamental mechanisms, including associated transport phenomena and governing equations, and (c) sharing our perspectives on limitations, challenges, and opportunities of passive systems.

2 Description of Passive Transport Phenomena

Passive transport phenomena include both diffusion and advection. Diffusive transport is driven by spatial gradients in one or more potential functions. For example, electrical potential, pressure, temperature, and mass concentration gradients lead to the transfer of electrical charge, momentum, thermal energy, and chemical species, respectively. Advective transport is induced by inertia or applying a force on the matter [3,4]. Considering a general extensive property

Φ

moving with absolute velocity

V

⁠, the integral form of the transport equation for

Φ

can be written as

{\frac{d Φ}{d t}|}_{system} = \frac{\partial}{\partial t} \int_{V} ρ φ d V + \int_{A} ρ (V_{rel} \cdot n) φ d A

(1)

where $φ$ is the intensive form of $Φ$ (⁠ $φ = Φ / m$ ⁠). The reference frame moves with velocity $V_{ref}$ ⁠, and $V_{rel}$ is the relative velocity, $V_{rel} = V - V_{ref}$ ⁠. The control volume moves with the reference frame at a constant velocity $V_{ref}$ ⁠. A coordinate system is attached to and moves with the reference frame, and n is the control surface unit normal vector. The term on the left-hand side of the equation is obtained from physical laws governing the property being transported. For example, in the case of transport of mass, where $Φ = m$ ⁠, the conservation of mass implies that the mass of the system remains constant, that is ${\frac{d m}{d t}|}_{system} = 0$ ⁠.

For momentum transport, $Φ = m V_{rel}$ and the physical law governing the rate of change of the momentum is the Newton's second law of motion that asserts ${\frac{d m V_{rel}}{d t}|}_{system} = \sum F$ ⁠, where $\sum F$ denotes all the external forces acting on the system.

For energy transport, $Φ = E = U + m V_{rel}^{2} / 2 + m g h$ ⁠, and the physical law governing the rate of change of the energy of the system is the first law of thermodynamics: ${\frac{d E}{d t}|}_{system} = {(\frac{δ Q}{d t})}_{net} - {(\frac{δ W}{d t})}_{net}$ ⁠, where Q and W are the heat and work transfer across the system boundary.

Finally, for conservation of mass for i^th species, $Φ = m_{i}$ and ${\frac{d m_{i}}{d t}|}_{system} = \int_{V} {\dot{m}}_{i}^{‴} d V$ ⁠, where ${\dot{m}}_{i}^{‴}$ is the production rate of species i due to chemical reaction (⁠ ${\dot{m}}_{i}^{‴} = 0$ when there is no chemical reaction). When Eq. (1) is written for transport of species i, the relative advective velocity on the right-hand side of the equation must be the velocity of species i, $V_{i, rel}$ ⁠.

From the above presentation, it is deduced that what drives the transport of momentum, energy and species is one or a combination of the following factors: the forces acting on the system, the heat or work crossing the system boundary, and concentration gradients or generation or consumption of species due to chemical reaction within the system. These driving factors can be induced actively by applying work, e.g., mechanical work to pump a fluid, or can be created passively with no need for external work. Passive transport utilizes naturally occurring physical phenomena that create the driving potential differences with no external work. Passive transport systems may or may not involve the transfer of matter. This work does not cover passive transport processes that do not involve the transportation of mass, either through bulk fluid motion or transport of chemical species. Examples of such passive systems are heat conduction, where thermal energy is transferred passively with no driving work, and thermoelectric devices, where thermal gradients are utilized to generate electricity using the Peltier effect. As such, the focus of this effort is on passive systems involving at least one form of fluid flow and mass transfer, and systems only involving heat conduction are excluded. Furthermore, only passive systems with engineering applications are considered, and those without distinctive applications are excluded.

The most notable examples of passive systems considered in this work involve capillary-driven flows, phase change-induced flows, and osmotic flows. In these systems, fluid flow and mass transfer are driven passively by the pressure differential created in the pores of a porous material, saturation pressure differentials in a vapor exposed to hot and cold surfaces, or by the osmotic pressure difference stemming from the concentration gradients.

Passive systems may operate in a continuous or noncontinuous manner. In general, for a passive system to work continuously, it needs to transfer heat. Isolated passive systems will eventually reach an equilibrium state and the transport phenomenon will cease. It is noted that in engineering applications, the passive systems incorporate heat transfer to fulfill the desired functionalities, and noncontinuous passive transport processes do not possess significant practical applications. An example of a continuously operating passive system is an evaporating meniscus in a capillary tube. In this system, a liquid comes in contact with the heated wall of a capillary tube with a contact angle smaller than 90 deg. The liquid starts to rise in the tube, and simultaneously evaporation occurs at the meniscus. Eventually, a steady-state is reached, where the height of the liquid column inside the tube does not change, and evaporation from the meniscus continues at a constant rate [5]. The evaporation induces a steady liquid flow from the reservoir to the meniscus. In the same system, if the liquid in the capillary tube is in thermal equilibrium with its surrounding, the liquid would first rise inside the capillary tube and eventually would reach an equilibrium height [6]. At this point, the flow would cease, and the system would come to an equilibrium state, where the surface energy of the system is minimum.

3 Fundamental Mechanism and Limitations

This section discusses the fundamental mechanisms and limitations of the main types of driving forces in passive system, namely, the capillary, evaporation-condensation, concentration gradient, and osmotic force.

3.1 Capillary Induced Flow.

Stable working fluid circulation in capillary-driven passive systems is achieved through the capillary pressure head developed by the wick (porous) structure. The optimum condition attainable in such passive devices is achieved under conditions where the capillary pressure head is greater than or equal to the sum of pressure losses along the fluid path. Heat pipes are an example of passive devices in which the fluid flow is driven by capillary pressure. In order for a heat pipe to operate properly, the following pressure balance must be satisfied [7]

Δ p_{cap, \max} \geq Δ p_{l} + Δ p_{v} + Δ p_{e, δ} + Δ p_{c, δ} + Δ p_{g}

(2)

The maximum capillary head can be calculated by

Δ p_{cap, \max} = \frac{2 σ}{r_{eff}}

(3)

where r_eff is the effective pore radius.

Δ p_{l}

is the pressure drop of the liquid flow in a wick structure due to the frictional drag. This liquid pressure drop is one of the major factors which causes the capillary limit. The vapor pressure drop along the flow path, Δp_v, should be interpreted as the absolute vapor pressure drop along the flow path.

Δ p_{e, δ}

and

Δ p_{c, δ}

are the pressure drops due to the evaporation and condensation at the liquid–vapor interface, respectively, and can be neglected if there are no phase changes. Δp_g is the pressure drop in the liquid due to the effect of the gravitational force in the direction of the flow path, and can be expressed as

Δ p_{g} = ρ_{l} g L_{t} s in ϕ

(4)

where ϕ is the inclination angle of the flow path from horizontal.

Since a macroscopic approach is the most feasible way to model transport, there needs to be a way to model the bulk resistance to the flow caused by the porous zone. In 1856, Henry Darcy experimentally measured the resistance to a steady, one-dimensional, gravitationally-driven flow through an unconsolidated, uniform, rigid, and isotropic solid matrix. He came up with a relationship for the pressure gradients/resistance as a function of the dynamic viscosity, μ_l, the superficial velocity,

V_{l}

⁠, and the permeability, K, now known as Darcy's Law

ρ g - \frac{\nabla (ε p)}{ε} = \frac{μ_{l}}{K} V_{l}

(5)

where the permeability, K, is inversely proportional to the resistance against the flow, which is analogous to the thermal conductivity in the Fourier's Law of heat conduction. It has a unit of meters squared. Darcy's Law can be expanded into vectorial form for an anisotropic solid matrix

ρ g - \frac{\nabla (ε p)}{ε} = \frac{μ_{l}}{K} V_{l}

(6)

where the permeability, K, becomes a second-order symmetric tensor. When the porous media is isotropic (having equal resistance in all directions), the permeability reduces to a scalar (single value), K. The permeability of the porous wick structure can be simplified using the Kozeny–Carman equation [7]. Darcy's Law is valid for Re < 1, or in the creeping flow regime, where the viscous forces dominate. The Reynolds number of the porous media is defined by the superficial velocity

Re = \frac{ρ_{l} V_{l} d}{μ_{l}}

(7)

where d is the average characteristic length scale of the voids. These principles and governing equations generalize multiphase mass transfer in porous media. The balance between the driving force, usually the capillary force, and resistance governs the mass transfer limitations of capillary-driven passive systems and the associated transport phenomena.

3.2 Flow Driven by Evaporation and Condensation.

Evaporation and condensation of a liquid within a system in thermal communication with a heat source and a heat sink can induce a passive continuous flow with no work requirement. The working fluid in such passive systems may circulate in a closed cycle or cross the system boundaries. Thermosyphons [7] are an example of liquid–vapor phase change-driven passive systems in which the working fluid circulates in a closed cycle inside a sealed shell. A wickless closed two-phase thermosyphon is a method of passive heat exchange, based on gravity and phase change, which circulates a fluid without the necessity of a mechanical pump. The operation principle of a thermosyphon is very similar to heat pipes and the only difference is that thermosyphons rely solely on gravity to return the condensed working fluid back to the evaporator section. The dependence of the operation on gravity necessitates that the evaporator section of the thermosyphon is always positioned at a lower level than the condenser. Solar stills [8] are an example of liquid–vapor phase change-driven passive systems in which the working fluid crosses the system boundary.

3.3 Mass transfer Driven by Concentration Gradient.

Concentration gradients in a mixture induce species transport described by the general conservation equations presented in the following [4]:

\frac{\partial ρ_{i}}{\partial t} + \nabla \cdot ρ_{i} V = - \nabla \cdot J_{i} + {\dot{m}}_{i}^{‴}, i = 1, 2, \dots, N

(8)

where

ρ_{i}

and

J_{i}

are the partial density and the diffusive mass flux of species i, respectively, and

{\dot{m}}_{i}^{‴}

is the volumetric mass generation rate of species i. Using the definition of mass fraction

ω_{i} = ρ_{i} / ρ

⁠, Eq. can be rewritten as follows:

ρ \frac{D ω_{i}}{D t} = - \nabla \cdot J_{i} + {\dot{m}}_{i}^{‴}

(9)

where

ρ

is the mixture density. The diffusive mass flux for a binary system of species A and B is obtained from Fick's Law. Using Fick's Law, Eq. can be written as

ρ \frac{D ω_{A}}{D t} = - ρ \nabla \cdot (D_{AB} \nabla ω_{A}) + {\dot{m}}_{A}^{‴}

(10)

The species transport can also be described using the molar concentration of species, c_i, instead of mass concentration

ρ_{i}

\frac{\partial c_{i}}{\partial t} + \nabla \cdot c_{i} V^{*} = - \nabla \cdot J_{i}^{*} + {\dot{n}}_{i}^{‴}

(11)

where ${\dot{n}}_{i}^{‴}$ and $V^{*}$ are the volumetric molar generation rate and mixture's molar-averaged velocity, respectively, and $J_{i}^{*}$ is the diffusive molar flux of species i.

For a binary system of species A and B at constant pressure, the Fick's Law can be used for the species molar diffusion

\frac{\partial c_{A}}{\partial t} + \nabla \cdot c_{A} V^{*} = c \nabla \cdot (D_{AB} \nabla x_{A}) + {\dot{n}}_{A}^{‴}

(12)

where x is the mole fraction. It is noted that often times there are more than two species present in the system and appropriate equations must be employed to relate the diffusive fluxes to the species concentrations.

These fundamental equations are the primary means to describe the mass fluxes in systems involving passive transport due to concentration gradients, such as natural draft cooling towers and other evaporative cooling systems. The concentration gradient-induced transport is often accompanied by other passive or active transport processes such as heat transfer and passively or actively induced flow. Analysis of the full system in such cases requires the consideration of the coupled conservation equations for multiphase mass, momentum, energy, and species transport.

3.4 Mass transfer Driven by Osmotic Pressure.

Osmatic mass transfer is induced by osmotic pressure, which is the minimum pressure needed to prevent the flow of a pure solvent into a solution through a semipermeable membrane. The mass flux created by the osmotic pressure is a function of the difference between the bulk osmotic pressures of the liquids on different sides of the membrane. Referring to the less concentrated solution as draw and the more concentrated solution as feed, the osmotic mass flux is described as

J_{w} = A (π_{D, b} - π_{F, b})

(13)

where

π_{D, b}

and

π_{F, b}

are the bulk osmotic pressures of the draw and feed, respectively. This equation assumes that only the solvent passes through the membrane. It also assumes that both the feed and draw solutions are homogeneous, and the external concentration polarization (ECP) is negligible. It is noted that ECP refers to the local changes of the concentration adjacent to the membrane due to the transport of species through the membrane. As such, Eq. may be valid only if the permeate flux is very low. For higher flux rates, Eq. must be modified to include both the concentrative and dilutive ECP [9]

J_{w} = A [π_{D, b} \exp (- \frac{J_{w}}{k_{D}}) - π_{F, b} \exp (\frac{J_{w}}{k_{F}})]

(14)

Note that the dilutive effect is indicated by the negative exponential term modifying the draw solution osmotic pressure. The individual mass transfer coefficients on the feed, k_F, and permeate, k_D, sides of the membrane must also be accounted for. The application of Eq. (16) is limited to dense symmetric membranes. For practical cases where the membrane is asymmetric, internal concentration polarization (ICP) effects are significant and must be taken into consideration [10].

4 Applications of Passive Systems

This section presents operation principles, transport phenomena, and limitations of some of the most important passive devices.

4.1 Heat Pipes.

Of the many different types of systems that transport heat, the heat pipe [7] is one of the most efficient systems known today. The advantage of using a heat pipe over other conventional methods is that large quantities of heat can be transported through a small cross-sectional area over a considerable distance with no additional power input to the system. Furthermore, design and manufacturing simplicity, small end-to-end temperature drops, and the ability to control and transport high heat rates at various temperature levels are all unique features of heat pipes [11]. These features have led to widespread use of heat pipes in a wide range of applications including electronics cooling and heat engines [12], thermal energy storage [13], and gas turbine vane cooling [14].

The operation of a heat pipe is easily understood by using a cylindrical geometry, as shown in Fig. 1. However, heat pipes can be of any size or shape. The components of a heat pipe are a sealed container (pipe wall and end caps), a wick structure, and a small amount of working fluid which is in equilibrium with its own vapor. Different types of working fluids such as water, acetone, methanol, ammonia, or sodium can be used in heat pipes based on the required operating temperature. The length of a heat pipe is divided into three parts: the evaporator section, adiabatic (transport) section, and condenser section. A heat pipe may have multiple heat sources or sinks with or without adiabatic sections depending on specific applications and design. Heat applied externally to the evaporator section is conducted through the pipe wall and wick structure, where it vaporizes the working fluid. The resulting vapor pressure drives the vapor through the adiabatic section to the condenser, where the vapor condenses, releasing its latent heat of vaporization to the provided heat sink. The capillary pressure created by the menisci in the wick pumps the condensed fluid back to the evaporator section. Therefore, the heat pipe can continuously transport the latent heat of vaporization from the evaporator to the condenser section. This process will continue as long as there is a sufficient capillary pressure to drive the condensate back to the evaporator.

Fig. 1

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Schematic of a conventional heat pipe showing the principle of operation and circulation of the working fluid [7]

The vapor pressure changes along the heat pipe are due to friction, inertia and blowing (evaporation), and suction (condensation) effects, while the liquid pressure changes mainly as a result of friction. The liquid–vapor interface is flat near the condenser end cap corresponding to a zero local pressure gradient at very low vapor flow rates. A typical axial variation of the shape of the liquid–vapor interface and the liquid and vapor pressures for low vapor flow rates are shown in Figs. 2(a) and 2(b), respectively.

Fig. 2

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Axial variation of the liquid–vapor interface, and the vapor and liquid pressures along the heat pipe at low vapor flow rates [7]

The maximum local pressure difference occurs near the evaporator end cap. This maximum local capillary pressure should be equal to the sum of the pressure drops in the vapor and the liquid across the heat pipe in the absence of body forces. When body forces are present, such as an adverse gravitational force, the liquid pressure drop is greater, indicating that the capillary pressure must be higher in order to return the liquid to the evaporator for a given heat input.

Heat pipe fluids can be evaluated for maximum capillary heat transport capability by the dimensional Merit number if vapor pressure drops and gravity can be neglected.

Merit Number = \frac{ρ_{l} σ h_{fg}}{μ_{l}}

(15)

where $σ$ and $h_{fg}$ are the surface tension and the latent heat of vaporization of the working fluid, respectively. The wick structure within the heat pipe is present to return condensate to the evaporator section. While small pores are needed at the liquid–vapor interface to develop high capillary pressures, large pores are preferred within the wick so that the movement of the liquid is not restricted too greatly. For this reason, many different types of wick structures have been developed in order to optimize the performance of the capillary heat pipe. There are three properties of wicks that are important in heat pipe design:

Minimum capillary radius: This parameter should be small if a large capillary pressure difference is required, such as in terrestrial operation for a long heat pipe with the evaporator above the condenser, or in cases where a high heat transport capability is needed.
Permeability: Permeability is a measure of the wick resistance to axial liquid flow. This parameter should be large in order to have a small liquid pressure drop and a resulting higher heat transport capability.
Effective thermal conductivity: A large value for this parameter gives a small temperature drop across the wick, which is a favorable condition in heat pipe design.

A high thermal conductivity and permeability and a low minimum capillary radius are somewhat contradictory properties in most wick designs. For example, a homogeneous wick may have a small minimum capillary radius and a large effective thermal conductivity but have a small permeability. Therefore, the designer must always make tradeoffs between these competing factors to obtain an optimal wick design.

For most heat pipes, the wick structures are very thin, so the liquid flow in the wick can be simplified to one-dimensional axial flow. Also, the liquid velocity and its gradient in the axial direction are very small. Since

{\dot{m}}_{l} = w_{l} A_{w} ρ_{l}

⁠, where A_w is the wick cross-sectional area, Eq. can be simplified to

\frac{d ρ_{l}}{d z} = - \frac{μ_{l} {\dot{m}}_{l}}{ρ_{l} A_{w} K}

(16)

One of the major difficulties in dealing with the liquid flow in a porous medium is determining the permeability. In many cases, a specific model is needed to calculate permeability for a specific type of wick structure, and often experimental measurements must be made if an accurate analytical model is not available. For this reason, Eq. (16) is preferred and often used to determine permeability through analytical modeling or experimental measurements.

For circular passages, such as a circular artery or tunnel wick with radius r, the Hagen-Poiseuille solution for laminar tube flow is applicable

\frac{d p}{d z} = - 2 τ_{w} / r

(17)

By using appropriate equations for

τ_{w}

and dp/dz, the permeability K can be obtained as

K = \frac{2 r^{2} ε}{f {Re}_{l}}

(18)

where

{Re}_{l} = 2 r V_{p, l} / V_{l}

is the axial liquid Reynolds number, f is a nondimensional friction coefficient, and

ε

is the wick porosity. The pore velocity

V_{p, l}

used in the definition of

{Re}_{l}

is related to the volume-averaged velocity

V_{l}

by

V_{l} / V_{p, l} = ε

⁠. For circular arteries,

ε

= 1 and

f {Re}_{l} =

16, therefore

K = r^{2} / 8

(19)

For noncircular and circular annular arteries, the radius in Eq. can be used by introducing a hydraulic diameter and radius and an appropriate equation for

f {Re}_{l}

[15]. By neglecting the pressure drops due to the evaporation and condensation at the liquid–vapor interface in Eq. , the general expression for the capillary limit for heat pipe is

\frac{2 σ}{r_{eff}} \geq Δ p_{l} + Δ p_{v} + ρ_{l} g L_{t} sin ϕ

(20)

where L_t is the total heat pipe length.

Since at steady-state

{\dot{m}}_{l} = | {\dot{m}}_{v} | = Q (z) / h_{fg}

⁠, and if a heat pipe has a uniform heating distribution along its evaporator and condenser sections, the axial mass flux distribution will be described by Eq. . In addition, if both A_w and K are constant along the heat pipe length, the liquid pressure drop in terms of the axial heat flow is

\int_{0}^{(L_{e} + L_{a})} \frac{u_{l} {\dot{m}}_{l}}{ρ_{l} A_{w} K} d z = \frac{u_{l}}{ρ_{l} A_{w} K h_{fg}} \int_{0}^{(L_{e} + L_{a})} Q (z) d z

(21)

A heat transport factor (QL)_cap,max is defined as

{(Q L)}_{cap, \max} = \int_{0}^{(L_{e} + L_{a})} Q (z) d z = (0.5 L_{e} + L_{a}) Q_{e}

(22)

The heat transport factor (QL)_cap,max in this case is

Q_{e} = \frac{{(Q L)}_{cap, \max}}{0.5 L_{e} + L_{a} + 0.5 L_{c}} = \frac{{(Q L)}_{cap, \max}}{L_{eff}}

(23)

where L_eff= 0.5 L_e+ L_a+ 0.5 L_c.

The maximum heat transfer rate due to the capillary limit can be estimated by the following expression if the vapor pressure drop can be neglected in comparison with the liquid pressure drop.

Q_{cap, \max} = \frac{ρ_{l} σ_{l} h_{fg}}{μ_{l}} (\frac{A_{w} K}{L_{eff}}) (\frac{2}{r_{eff}} - \frac{ρ_{l} g L_{t}}{σ_{l}} cos ϕ)

(24)

There are other heat transport limitations for heat pipes such as boiling and sonic limits that should be considered in addition to the capillary limit [7].

4.2 Direct methanol Fuel Cells.

Passive direct methanol fuel cell (DMFC) technology uses various capillary approaches to manage methanol and water transport without the need for a complex microfluidic subsystem [16–19]. At the core of this new technology is a unique passive system that uses the heat pipe concept for fuel delivery (Fig. 3). Furthermore, the fuel cell is designed for both passive water management and effective carbon dioxide removal. The passive components that are critical to the fuel cell design are the fuel delivery, air-breathing, and water recirculation systems. The passive fuel delivery system stores pure methanol, which can be mixed with water in situ without the use of a pumping system, and can be passively supplied to the fuel cell anode at an optimal concentration. Since water is needed in the anode for the methanol reaction to occur, the water created in the cathode can be passively supplied to the anode. This water recirculation, in conjunction with passive methanol fuel delivery, can dramatically extend the operation time of the fuel cell per refueling.

Fig. 3

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Passive direct methanol fuel cell using heat pipe concept [20]

The passive mass transfer concept (wick structure) developed in heat pipe technology is an effective approach for mass transfer management in various fuel cell technologies [20]. The proposed DMFC technologies developed were operated passively, without moving parts, which resulted in a highly reliable system. Due to their significantly longer charging life, passive miniature DMFC systems are being seriously considered for replacing the battery for applications such as cell phones, digital cameras, and laptops [21].

During operation, the liquid fuel (methanol-water solution) transfers from the fuel chamber through the gas diffusion layer (GDL) to the anode catalyst layer (CL). The anode methanol oxidation reaction (MOR) oxidizes methanol and water and generates carbon dioxide, while the generated carbon dioxide gas moves in the opposite direction from the anode CL through the GDL to the fuel chamber and is discharged to the environment. The mass transfer limitation of the fuel delivery system in fuel cells is governed by the balance among capillary pressure, pressure drops of the vapor, pressure drop of the liquid, the gravitational pressure drop, and the acceleration pressure drop. Considering the small thicknesses of fuel cell components, L_t (in the order of mm or less), the pressure drop due to the gravitational force is negligible in fuel cells. When the system is under equilibrium, there is no acceleration pressure drop, and the pressure balance equation is reduced to $Δ p_{cap} = Δ p_{v} + Δ p_{l}$ ⁠. The capillary pressure plays a critical role in the fuel delivery, water management, and electrochemical performance of the passive DMFCs.

Fuel Delivery.

In order to deliver fuel from the fuel reservoir to the fuel cell, the capillary pressure needs to overcome the liquid pressure drop. When the flow is laminar, the liquid pressure drop when the fuel flows through the porous wick structure, p_l, can be calculated by Darcy's equation (Eq. ). If the gravitational force is negligible, and the porosity is uniform, Darcy's equation can be simplified to

\frac{Δ p_{l}}{L} = \frac{μ_{l}}{K} \frac{{\dot{Q}}_{l}}{A}

(25)

where μ_l is the viscosity of the liquid, K is the permeability of the porous media (Eq. ),

{\dot{Q}}_{l}

is the volumetric liquid flow rate, and A is the cross-sectional area of the wick structure. Based on the mass balance, the volumetric flow rate of liquid (methanol and water) is proportional to the operating current density of the fuel cell I

{\dot{Q}}_{l} = \frac{I \times (M_{H 2 O} + M_{MeOH})}{6 F} / ρ_{l}

(26)

Given the small pore size of the wick structure (in the order of μm), the Renumber of the liquid flow is expected to be much less than 1 (creeping flow). Darcy's Law is valid for the fuel flow in the wick structure. As a result, if the wick structure is fully saturated by liquid, the capillary pressure required to deliver the fuel to support the electrochemical reaction in the fuel cell is

Δ p_{cap} = \frac{μ_{l}}{K} \cdot \frac{I \times (M_{H 2 O} + M_{MeOH})}{6 F ρ_{l}} \cdot \frac{L}{A} = \frac{μ_{l}}{\frac{1}{150} \frac{ε^{3}}{{(1 - ε)}^{2}} d^{2}} \cdot \frac{I \times (M_{H 2 O} + M_{MeOH})}{6 F ρ_{l}} \cdot \frac{L}{A}

(27)

The capillary pressure required is determined by the operating current, properties of the liquid, and geometry of the wick structure: length L, cross-sectional area A, and pore size d. For a wick with pore diameter of d and contact angle of $θ$ ⁠, the capillary pressure is $4 σ cos θ / d$ ⁠. The following table includes geometrical parameters of a typical wick structure.

We can quantify the capillary pressure and flow resistance using parameters of the wick structure and relevant physical properties of liquid methanol under room temperature, including the surface tension of 22.5 mN/m and the liquid viscosity of 0.5435 mPa·s. The capillary pressure is calculated to be 5,785 Pa, and the liquid pressure drop when the current is 1 Ampere (which is a relatively high current) is 4.445 Pa. The capillary pressure of the wick structure is three orders of magnitude higher than the pressure drop of the liquid moving through the wick structure. If the pore size, d, decreases, the capillary pressure increases with d⁻¹, while the pressure drop of liquid increases with d⁻². Unless the pore size of the wick structure is of the order of 10 nm or smaller, the capillary pressure of the wick structure is sufficient to supply sufficient fuel to the fuel cell.

Liquid Pressure Drop.

Similarly, the flow pressure drop of liquid flow through porous media, Δp_l, can be calculated by

\frac{Δ p_{l}}{δ} = - \frac{150 μ}{d_{avg, l}^{2}} \frac{{(1 - ε)}^{2}}{ε^{3}} V_{s, l}

(28)

where d_avg,l is the average diameter of pores smaller than d (which are filled with liquid). The superficial velocity is calculated by the volumetric liquid flow rate, Q_l, and the cross-sectional area that have gas flow

V_{s, l} = \frac{Q_{l}}{A \times CDF (d)}

(29)

The volumetric flow rate of liquid (methanol and water) per projected area of the MEA is proportional to the operating current density i

\frac{Q_{l}}{A} = \frac{i \times (M_{H 2 O} + M_{MeOH})}{6 F} / ρ_{l}

(30)

Therefore, the liquid pressure drop through the thickness direction of the fuel cell component can be calculated by

Δ p_{l} = - \frac{150 μ}{d_{avg}^{2}} \frac{{(1 - ε)}^{2}}{ε^{3}} \times \frac{i \times (M_{H 2 O} + M_{MeOH})}{6 F ρ_{l}} \times \frac{δ}{CDF (d)}

(31)

Capillary Pressure.

The capillary pressure is the pressure difference between the liquid and gas on two sides of a meniscus formed in a capillary conduit. Since the liquid pressure drop (Eq. ) is much smaller than the gas pressure drop, assume the liquid pressure, p_l, is around the atmospheric pressure, p₀. The gas pressure, p_g, that is required to break through pores and maintain the flow rate to match the generation rate of CO₂ is

p_{g} = \frac{4 σ cos θ}{d} + \frac{150 μ}{d_{avg, g}^{2}} \frac{{(1 - ε)}^{2}}{ε^{3}} \times \frac{i}{6 F} \times \frac{R_{u} T}{p_{g}} \times \frac{δ}{(1 - CDF (d))}

(32)

We can quantify the pressure change along the gas and liquid flow in fuel cell by applying the previous equations in porous fuel cell components with uniform or distributed pore sizes. As an example, we assume log-normal distributions of pore sizes and summarize the pore size distributions (PSDs) of major components in the MEA in Table 2. Based on the PSDs, the gas pressure in different components when the current density is 0.1 A/cm² is illustrated in Fig. 4. Pressure distributions at other current densities follow a similar trend.

Fig. 4

Gas pressure through different fuel cell components at the current density of 0.1 A/cm2

View large Download slide

Gas pressure through different fuel cell components at the current density of 0.1 A/cm²

Table 2

Properties of MEA components in this model [22,23]

	Thickness δ (μm)	Mean pore size μ (μm)	Shape Factor S (/)	Porosity ε (/)	Contact angle θ (°)	Permeability K (m²)
GDL	200	10	1.0	0.8	50 [24]	3 × 10⁻¹²
MPL	50	0.1	1.0	0.3	70	7 × 10⁻¹³
CL	20	0.1	1.0	0.2	30	3 × 10⁻¹⁴
PEM	50	0.01	1.0	0.2	25 [25]	7.13 × 10⁻²⁰ [26]

	Thickness δ (μm)	Mean pore size μ (μm)	Shape Factor S (/)	Porosity ε (/)	Contact angle θ (°)	Permeability K (m²)
GDL	200	10	1.0	0.8	50 [24]	3 × 10⁻¹²
MPL	50	0.1	1.0	0.3	70	7 × 10⁻¹³
CL	20	0.1	1.0	0.2	30	3 × 10⁻¹⁴
PEM	50	0.01	1.0	0.2	25 [25]	7.13 × 10⁻²⁰ [26]

Water Recirculation.

When concentrated methanol or even pure methanol is supplied as fuel, recirculation of water generated in the oxygen reduction reaction is critical to provide water for the MOR. Considering the small size of portable DMFCs and the small amount of water generation, actively collecting condensed water from the cathode is often impossible. Therefore, passive water recirculation by capillary pressure is critical for the success of portable DMFCs. Principles of water recirculation driven by capillary pressure are similar to those for fuel delivery. To recirculate water from the cathode to the anode, the capillary pressure needs to overcome the liquid pressure drop

\frac{4 σ cos θ}{d} \geq \frac{μ_{l}}{\frac{1}{150} \frac{ε^{3}}{{(1 - ε)}^{2}} d^{2}} \cdot \frac{I \times (M_{H 2 O})}{2 F ρ_{l}} \cdot \frac{L}{A}

(33)

The capillary pressure and liquid pressure drop can be calculated based on the geometrical parameters of a typical wick structure in Table 1 and the properties of liquid water under room temperature, including the surface tension of 72 mN/m and the liquid viscosity of 1.002 mPa·s. The capillary pressure is calculated to be 18,512 Pa, and the liquid water pressure drop when the current is 1 A is 1.98 Pa. The capillary pressure of the wick structure is four orders of magnitude higher than the pressure drop of the liquid moving through the wick structure. If the pore size, d, decreases, the capillary pressure increases with d⁻¹, while the pressure drop of liquid increases with d⁻². Unless the pore size of the wick structure is of the order of 1 nm or smaller, the capillary pressure of the wick structure is sufficient to recirculate liquid water to the anode of the fuel cell.

Table 1

Properties of wick structure

	Length (mm)	Cross-sectional area (mm²)	Mean pore size d_avg (μm)	Porosity ε (/)	Contact angle θ_H2O (°)	Contact angle θ_MeOH (^o)
Wick	10	10	10	0.8	50	50

4.3 Thermosyphon-Assisted Passive Thermal Energy Storage Systems.

Innovative passive thermal energy storage (TES) systems can be designed to capture heat or cold from the ambient or any other reservoir with fluctuating temperatures. Applications include diurnal cold storage in a desert where the temperature swings between the day and night are large and seasonal cold storage to supply the air conditioning needs during summer. For such TES to be effective, it should enable unidirectional heat transfer allowing only heat absorption (release) from (to) the media to be cooled (heated). Otherwise, the cold or heat captured from the reservoir will be returned to it once the temperature difference between the two reservoirs changes direction. In other words, the TES must be able to capture the heat or cold when the reservoir temperature is favorable and lock it inside to supply the demand when needed. The unidirectional heat transfer feature of thermosyphons can be utilized for this purpose. An embodiment of a passive thermal energy storage (PTES) system integrated with thermosyphons is shown in Fig. 5. The PTES is comprised of a tank containing a phase change material (PCM) such as organic PCM, salt-hydrate, or inorganic salt mixtures, and two sets of gravity-assisted thermosyphons. The thermosyphons penetrate the storage tank from top and bottom, with their opposite ends extending out of the tank (Fig. 5).

Fig. 5

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Schematic view of the passive thermosyphon-integrated PCM cold storage [30]

The heat transfer through thermosyphons is enabled by evaporation and condensation of a small amount of a working fluid recirculating in a sealed solid shell. As shown in Fig. 5, the thermosyphon working fluid accumulates at the bottom end of the device. Upon supplying heat to the bottom portion of the thermosyphon, referred to as evaporator section, the working fluid boils and the vapor travels to the opposite end, where the thermosyphon is exposed to a lower temperature medium. The saturated vapor inside the thermosyphon condenses as it comes in contact with the colder condenser wall. The condensed working fluid returns to the liquid pool due to the gravity.

The primary heat transfer limits in the operation of a thermosyphon are flooding, dryout, and boiling limit. Flooding limit [27] is due to the interactions between the counterflowing vapor and liquid at the interface of the vapor and liquid film in a thermosyphon. As the relative velocity between the liquid and vapor increases, the shear interfacial forces retard the return of liquid from the condenser section to the evaporator. The flooding limit is reached when the lack of liquid delivery to the evaporator section causes unstable and unsteady thermal fluid condition. When the flooding limit is exceeded, the interfacial viscous force overcomes the surface tension force and thus liquid droplets are detached from the liquid film and are entrained into the vapor region of the condenser section [7]. Faghri et al. [27] presented the following equation for the prediction of the flooding limit in two-phase closed thermosyphons

Q_{\max} = K h_{fg} A {[ρ σ (ρ_{l} - ρ_{v})]}^{\frac{1}{4}} {(ρ_{v}^{- 1 / 4} + ρ_{l}^{- 1 / 4})}^{- 2}

(34)

where A is the inner cross-sectional area of the thermosyphon and K is Kutateladze number [7].

Dryout limit also occurs in two-phase closed thermosyphons at the bottom of the evaporator when the fill volume of the working fluid is insufficient for the input heat. In this situation, the liquid pool is completely vaporized, and the liquid film vanishes due to evaporation before reaching the bottom of the evaporator. The temperature of the lower region of the evaporator section increases since there is no liquid to absorb the input heat upon evaporation. The dry zone temperature increases steadily, and the dry region progresses upward. Dryout limit can be predicted using correlations such as the one proposed by Shiraishi et al. [28].

Boiling limit is also seen in thermosyphons with large fill volumes and high radial heat fluxes in the evaporator section. When the evaporator heat flux reaches a critical value, the vapor bubbles coalesce near the pipe wall, which essentially blocks the liquid from touching the wall. At this point, the wall temperature increases rapidly. In thermosyphons, the film boiling leads to a significant pressure increase which may lead to a rupture of the container. Gorbis and Savchenkov [29] proposed the following correlation for the prediction of the boiling limit

\frac{q_{\max}}{q_{\max, \infty}} = C^{2} {[0.4 + 0.012 R \sqrt{\frac{g (ρ_{l} - ρ_{v})}{σ}}]}^{2}

(35)

where

q_{\max, \infty}

is the critical heat flux for pool boiling

q_{\max, \infty} = 0.142 \sqrt{ρ_{v}} {[g σ (ρ_{l} - ρ_{v})]}^{1 / 4}

(36)

The coefficient C in Eq. is

C = R {(\frac{D}{L_{c}})}^{- 0.44} {(\frac{D}{L_{e}})}^{0.55} V_{r}^{n}

(37)

where V_r is the fill ratio defined as the liquid volume to the total inner volume of the thermosyphon and L_e and L_c are the heated and cooled lengths, respectively. The coefficient R and power n are related to the fill ratio as follows:

\begin{matrix} R = 0.538 and n = 0.13 for V_{r} \leq 0.35 \\ R = 3.54 and n = - 0.37 for V_{r} > 0. 35 \end{matrix}

The above basic analysis establishes the primary limiting mechanisms of passive transport phenomena driven by evaporation and condensation in two-phase closed thermosyphons as one of the most fundamental passive devices in this category. The operation limit of other passive devices driven by evaporation and condensation must be analyzed considering their unique characteristics with similar methodology.

Since the liquid pool is always at the lower end of the thermosyphon, heating the upper end of the thermosyphon will not create any boiling, and heat transfer will be limited to negligible conduction through the solid wall. Due to this unique feature, thermosyphons are also referred to as “thermal diodes”. By strategically positioning the heat source and heat sink (ambient air) below and above the PCM, respectively, the PCM can never transfer heat to the heat transfer fluid (HTF), however, the HTF releases heat to the PCM when the PCM is relatively colder. Likewise, the only possible heat transfer between the ambient air and PCM is from PCM to relatively colder air. The melting point of the PCM is selected to be slightly lower than the desired HTF temperature. This allows for the PCM to regulate the HTF temperature by absorbing the excess heat once the HTF temperature rises above the threshold value. The thermosyphon-PCM cool storage does not need any power input for operation and runs completely passively when the favorable temperature gradients are established (when the atmospheric air temperature falls below the PCM melting point). To further improve the heat transfer from the thermosyphon-integrated PCM cold storage to the air, the condenser section of the output thermosyphons can be finned. Fins can be also applied to the evaporator section of the input thermosyphons.

The thermosyphon arrays not only enable the HTF-to-PCM and PCM-to-air heat transfer, but they also serve to circumvent the thermal resistances within the PCM by delivering the heat transfer rates deep through the PCM mass. Further heat transfer improvements can be achieved by embedding the PCM in a solid matrix, such as a metal foam.

4.4 Passive Desalination Systems.

A solar still is a passive device that uses solar thermal energy to produce distilled water from input saline water. A conventional single-slope single-basin solar still integrated with latent heat thermal energy storage (LTES) is shown in Fig. 6. It incorporates a basin that is covered by a transparent top cover exposed to solar insolation. The solar radiation is absorbed and converted to heat by an absorber plate placed at the bottom of the basin. The temperature increases inside the still, due to the greenhouse effect, and facilitates the evaporation of the water. The water vapor in the saturated air inside the still condenses on the inner surface of the top cover by releasing heat to the outside air or another cooling medium. The condensate drips down the inclined surface of the top cover and is gathered in a separate basin. Conventional solar stills are simple and passive, however, they are often characterized by poor efficiency and productivity [31]. Several approaches can be taken to improve the performance of a solar still, such as the inclusion of reflectors, solar collectors, increased surface area, recovery of the latent heat of condensation for increased evaporation, and integration with thermal energy storage [8,32].

Fig. 6

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Schematic diagram of a conventional single-slope single-basin solar still integrated with LTES, (1) makeup water, (2) glass cover, (3) storage medium (PCM), (4) mixture of heated air and steam, (5) basin water, (6) basin liner (absorber), (7) output distilled water, and (8) thermal insulation [8]

The application of the LTES in thermally-driven desalination systems is conceptually simple and straightforward. As shown in Fig. 6, the most common method for integrating the LTES with solar stills is embedding the PCM underneath the absorber plate at the bottom of the saline water basin. The LTES is charged whenever there is an excess amount of thermal energy and then releases the stored energy to run the desalination system when the primary energy source is not able to supply the demand. As such, LTES can be potentially a good fit for integration with solar thermal desalination systems due to the periodic nature of the energy source.

A fundamental energy analysis of a basic LTES-integrated single-slope solar still, shown in Fig. 7 is presented. The system includes a heat exchanger for preheating the makeup water (the saline water input replacing the discharging distilled water and concentrated saline water) by heat recovery from the discharging distillate and concentrated brine streams. The following simplifying assumptions are made: the system is well-insulated and thermal energy leaves the system only through the top cover glass; the heat exchanger used for heat recovery from the discharging streams is ideal and, hence, the outlet temperature of distillate and brine streams are the same as the inlet temperature of the makeup water that is equal to the ambient temperature T_a; the air, vapor and liquid water inside the solar still are all at the same temperature of T_w; the heat transfer coefficient between the absorber plate and the basin water is sufficiently large and, thus, the absorber plate and the basin water are at the same temperature; the thermal mass of the basin water, absorber plate, and the cover glass are negligible compared to the LTES; the sensible heat storage in the PCM is negligible compared to latent heat (Ste → 0); and mass of the water within the still remains unchanged during the operation. The mass balance for the system implies

{\dot{m}}_{m} = {\dot{m}}_{b} + {\dot{m}}_{d}

(38)

where

\dot{m}

shows the mass flow rate and subscripts m, b, and d are related to makeup water, discharged concentrated brine, and discharged distillate, respectively. Considering the above assumptions, the energy balance for the basin water can be written as

α_{w} G_{h} + {\dot{m}}_{m} h_{m} = {\dot{m}}_{b} h_{b} + {\dot{m}}_{d} h_{d} + q_{c} + q_{e} + q_{r} + q_{s}

(39)

where h is the specific enthalpy, G_h is the global solar radiation on a horizontal surface, $α_{w}$ is the fraction of G_h transferred to the water in the basin, $q_{c}, q_{e}, q_{r}$ are heat transfer rates from the basin water to the cover glass due to convection, evaporation, and radiation, respectively, and $q_{s}$ denotes the heat exchange rate between the basin water and the PCM. A value of $α_{w} = 0.8$ is considered to include the effects of partial reflection of the solar radiation by the cover glass, basin water, and the absorber plate, as well as partial absorption of radiation by the cover glass [33].

Fig. 7

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Schematic of the energy exchange of a basic LTES-integrated solar still [8]

It is known that at a fixed temperature, the enthalpy of moderately concentrated salt-water solutions does not change significantly with the salinity [34,35]. Thus, considering a constant h, combining Eqs. and yields

α_{w} G_{h} = q_{c} + q_{e} + q_{r} + q_{s}

(40)

Appropriate equations for the convective, evaporative, and radiative heat transfer rates from the basin water to the cover glass can be found in Dunkle [36].

The conduction-controlled heat transfer rate between the basin water and the PCM can be determined from

q_{s} = \frac{(T_{w} - T_{m})}{\frac{δ_{PCM}}{k_{PCM}} + \frac{δ_{p}}{k_{p}}}

(41)

where

δ

and k show the thickness and thermal conductivity, and subscripts PCM and p are related to the PCM and the absorber plate, respectively. It is noted that

δ_{PCM}

varies with time. Neglecting the thermal mass of the cover glass, the internal glass temperature, T_g,i, can be related to the external glass temperature, T_g,o, using the following equation

q_{c} + q_{e} + q_{r} = \frac{k_{g}}{δ_{g}} (T_{g, i} - T_{g, o})

(42)

where

k_{g}

and

δ_{g}

are the thermal conductivity and thickness of the cover glass, respectively. Finally, the external glass temperature can be coupled to the convective and radiative heat dissipation from the cover glass to the ambient

0.85 σ (T_{g, o}^{4} - T_{sky}^{4}) + 14.5 (T_{g, o} - T_{a}) = α_{g} G_{h} + \frac{k_{g}}{t_{g}} (T_{g, i} - T_{g, o})

(43)

where T_sky is the apparent sky temperature for long-wave radiation exchange and can be related to the ambient temperature, T_a, using T_sky = 0.0552 T_a^1.5 [37], and $α_{g}$ is the absorptivity of the cover glass assumed to be $α_{g} = 0.05$ [33].

The performance of solar stills is quantified by the distilled water production per unit area of the basin. The primary limiting factors in the operation of solar stills are the ability of the system to absorb and reject heat for evaporation and condensation, respectively. The amount of the absorbed solar thermal energy mainly depends on the available solar radiation G_h, absorptivity of the basin liner, and transmissivity of the cover glass. The ability of the system to reject the condensation heat depends mainly on the cooling of the cover glass. In a passively cooled solar still, the cooling capacity is controlled by the temperature difference between the basin water and the ambient. There have been efforts to enhance the solar still performance by increasing both the amount of absorbed solar energy and the cooling rate. The absorbed solar energy can be increased by including external solar collectors in thermal communication with the solar still [38]. The cooling rate is usually increased using active methods such as forced air cooling of the cover glass by a fan [39]. It is noted that there are several other design variations for improving the performance of solar stills including multistage solar stills, cascaded, and pyramid and tubular solar stills among others.

5 Challenges and Opportunities

There are a number of challenges and opportunities to employ passive systems in a broader range of applications with better functionality and efficiency. Several key challenges are listed below to encourage further thoughts, discussions, and creative solutions.

The balance among conductivity, permeability, and capillary pressure: High thermal conductivity, high permeability, and strong capillary action is desired in the wick structures for efficient heat and fluid flow through the wick. However, high thermal conductivity, permeability, and low minimum capillary radius are somewhat contradictory properties in most homogeneous wick designs. For example, a homogeneous porous structure may have a small minimum capillary radius and a large effective thermal conductivity but have a small permeability. Therefore, one always needs to tradeoff between these competing factors to obtain an optimal wick design. Using non-homogeneous porous structures may provide an opportunity to achieve high thermal conductivity, high permeability, and a minimum capillary radius at the same time. Some efforts were made in this regard with limited success using inverted meniscus phenomena [40]. Carefully engineered porous media with the pore size distributions and spatial pore connectivities could provide low flow resistance and high driven force at the same time. More fundamental focused research in this regard is needed.
There has been an increasing amount of research on developing bio-inspired porous materials for enhanced flow and temperature control in engineering systems. The bio-inspired porous materials mimic the biological systems such as leaves [41], wood [42], and lung [43] in macroscopic design, hydrophilicity, and hydrophobicity properties, or the microscopic multiscale pore network structures. In the multiscale pore network structures, the millimeter- or micron-scale pores facilitate the mass transfer while the submicron-scale and nanoscale pores promote the capillary pressure and/or the ion diffusion and electrochemical reaction rates. These bio-inspired porous materials can be employed in a large variety of applications including redox flow batteries [43], flexible heat pipes [44], health monitoring devices [45], solar energy [42], and passive flow control [46] to improve the performance by passively increasing the heat and mass transfer rates. Despite the recent advances in the area of bio-inspired porous materials, there is still plenty of room for fundamental and applied research and innovation to develop engineering systems with a performance approaching their biological counterparts. As an example, a redwood tree (Fig. 8) can passively pump about 500 gallons of water to an astonishing height of about 70 m in a day [47]. Such performance is unparalleled in engineering science [48].
Enabling high flux and long-distance flow transport in porous media: Some passive systems involve multiphase flow and heat transfer in porous structures. The physical phenomena associated with multiphase flow and heat transfer, including phase change in porous structures, are not well understood. In general, porous structures can transport fluid over short distances. More fundamental studies are needed to enable long-distance fluid transport in porous conduits.
Design and fabrication of heterogeneous porous structures with tailored properties: It is common in heat pipes and fuel cells that the local liquid and vapor flow directions are different [7,50]. In the example of a heat pipe, heat transfer at the evaporator and condenser sections occurs mainly in the radius direction. In contrast, the mass transfer of liquid water and steam is mainly along the axis direction. The interactions and flow resistance between the two phases lead to high flow resistance, which often limits the devices' heat and mass transfer capabilities. To this end, designing heterogeneous porous structures to reduce the high flow resistance caused by liquid-vapor two-phase (counter or cross) flow can enhance the performance of the passive system. The advance in additive manufacturing (AM) has enabled the design and fabrication of structures that are impossible to make using traditional manufacturing. AM opens up new horizons in porous structure prototyping and materials fabrication for passive devices specially in small sizes.
Accurate predictions of multiphase transport properties of porous media: In order to design and manufacture novel porous structures, including non-homogeneous structures, it's critical to develop empirical correlations and prediction design tools to estimate permeability and thermal conductivity, considering factors beyond pore size and porosity. The permeability is estimated using the Kozeny-Carman equation. The equation cannot capture the effect of phase connectivity and wettability [51]. The empirical parameter of sphericity can only approximate the pore geometry. Similarly, empirical correlations of effective conductivities are usually limited to simplified geometries, range of porosity, and materials [52]. These predictions will have extremely high uncertainties when applied to multi-phase systems, especially when predicting other geometries and porosity beyond the applicable range. Integrate pore-scale simulations with advanced imaging technologies such as microCT to consider the spatial distributions of different phases [53]. The new simulation capability can predict thermophysical properties more accurately. It provides unique tools to design customized porous media with unprecedented performance and properties.
Development of advanced physical models that are more efficient than traditional CFD models, such as statistical models [52,54], pore-network models, etc. [55,56], and machine learning-accelerated simulations [57,58]. Pore-network models consider the connectivity and provide accurate simulations with high computational efficiency. Statistical models derived based on the thermal resistance model can return thermal conductivities of porous materials in less than a second. It could integrate with pore-scale imaging technologies to predict material properties in situ. Machine learning has several emerging areas that have shown great potential to significantly accelerate fluid dynamics simulations. The success of machine learning relies heavily on high-quality data, which are often generated by CFD simulations. The research community is working toward the integration of machine learning algorithms with CFD models to unlock the full potential of machine learning.

Fig. 8

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The Stagg Tree in California [49]. Trees absorb large amounts of water from the soil and transport it against the gravity passively by using a network of multiscale porous structures from root to the leaf, coupled with transpiration at the leaf surface.

6 Conclusions

Active and passive systems both have advantages and disadvantages. The specific application of the system can determine which type or a hybrid between the two is optimal. In terms of some energy and electronic cooling applications, passive systems are advantageous due to their simplicity and reliability. This can result in a device that is lightweight, durable, and more efficient. This is made possible by eliminating the parasitic losses associated with components of active systems, thereby minimizing weight and eliminating moving parts that could lead to failure. This makes passive system technology optimal for some applications, requiring little to no maintenance. Also, it may lead to a replacement for the conventional battery. More fundamental studies are required to better understand the limitations of passive devices to overcome the challenges for achieving higher performance.

Funding Data

Advanced Research Projects Agency-Energy (ARPA-E) (Grant No. DE-AR0001069; Funder ID: 10.13039/100006133).
U.S. Department of Energy's Office of Energy Efficiency and Renewable Energy (EERE) (Grant No. DE-0008440; Funder ID: 10.13039/100006136).
Kansas NASA EPSCoR Research Infrastructure Development Program (Grant No. #80NSSC19M0042; Funder ID: 10.13039/100000104).

Nomenclature

F =: force
g =: gravitational acceleration
J =: mass flux
$\dot{n}$ =: molar flux
V* =: molar-averaged velocity
V =: velocity

A =: area
Bo =: bond number
c =: molar concentration
c_p =: heat capacity
D =: diameter or mass diffusivity
d =: pore diameter
D_h =: hydraulic diameter
E =: energy
F =: Faraday constant
f =: friction coefficient
G_h =: global solar radiation on a horizontal surface
h =: specific enthalpy or height
h_fg =: latent heat of evaporation
h_sl =: latent heat of fusion
I =: electrical current
i =: current density
K =: permeability or Kutateladze number
k =: conductivity or mass transfer coefficient
L =: length
m =: mass
M =: molecular weight
$\dot{m}$ =: mass flow rate
n =: coefficient
p =: pressure
$q^{″}$ =: heat flux
q =: heat transfer rate
r =: radius
R =: coefficient
Re =: Reynolds number
R_g =: gas constant
R_u =: universal gas constant
S =: shape factor
Ste =: Stefan number
T =: temperature
t =: time
U =: internal energy
u =: velocity on x direction
v =: velocity on y direction
V =: volume
V_s =: superficial velocity
w =: pore velocity in the artery
W =: work
x =: molar fraction

Δ =: property difference
α =: fraction of absorbed solar radiation
δ =: thickness
ε =: porosity
θ =: contact angle
μ =: viscosity
π =: osmotic pressures
ρ =: density
σ =: surface tension
Φ =: a general extensive property
φ =: a general intensive property
ϕ =: inclination angle
ω =: mass fraction

a =: ambient temperature
b =: brine or bulk
c =: condensation or convection
cap =: capillary
D =: draw
d =: distillate
e =: evaporation
eff =: effective
F =: feed
g =: gravitational
g,i =: inner surface of glass
g,o =: outer surface of glass
H₂O =: water
i =: species i
in =: inlet
l =: liquid
lv =: liquid to vapor evaporation
m =: melting temp or makeup water
max =: maximum
MeOH =: methanol
max,∞ =: critical heat flux
net =: net
PCM =: phase change material
r =: radiative
ref =: reference
rel =: relative
sky =: apparent sky temperature
system =: system
v =: vapor
vl =: vapor to liquid condensation
w =: water

CDF =: cumulative distribution function
CL =: catalyst layer
DMFC =: direct methanol fuel cell
ECP =: external concentration polarization
GDL =: gas diffusion layer
HTF =: heat transfer fluid
ICP =: internal concentration polarization
LTES =: latent heat thermal energy storage
MEA =: membrane electrode assembly
MOR =: methanol oxidation reaction
MPL =: micro porous layer
PCM =: phase change material
PEM =: proton exchange membrane
PTES =: passive thermal energy storage
TES =: thermal energy storage

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Opportunities and Challenges in Passive Thermal-Fluid and Energy Systems

Abstract

1 Introduction

2 Description of Passive Transport Phenomena

3 Fundamental Mechanism and Limitations

3.1 Capillary Induced Flow.

3.2 Flow Driven by Evaporation and Condensation.

3.3 Mass transfer Driven by Concentration Gradient.

3.4 Mass transfer Driven by Osmotic Pressure.

4 Applications of Passive Systems

4.1 Heat Pipes.

4.2 Direct methanol Fuel Cells.

Fuel Delivery.

Liquid Pressure Drop.

Capillary Pressure.

Water Recirculation.

4.3 Thermosyphon-Assisted Passive Thermal Energy Storage Systems.

4.4 Passive Desalination Systems.

5 Challenges and Opportunities

6 Conclusions

Funding Data

Nomenclature

References

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Opportunities and Challenges in Passive Thermal-Fluid and Energy Systems

Abstract

1 Introduction

2 Description of Passive Transport Phenomena

3 Fundamental Mechanism and Limitations

3.1 Capillary Induced Flow.

3.2 Flow Driven by Evaporation and Condensation.

3.3 Mass transfer Driven by Concentration Gradient.

3.4 Mass transfer Driven by Osmotic Pressure.

4 Applications of Passive Systems

4.1 Heat Pipes.

4.2 Direct methanol Fuel Cells.

Fuel Delivery.

Liquid Pressure Drop.

Capillary Pressure.

Water Recirculation.

4.3 Thermosyphon-Assisted Passive Thermal Energy Storage Systems.

4.4 Passive Desalination Systems.

5 Challenges and Opportunities

6 Conclusions

Funding Data

Nomenclature

References

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