Modeling of a membrane-based absorption heat pump

Journal of Membrane Science 337 (2009) 113–124

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Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci

Modeling of a membrane-based absorption heat pump Jason Woods a,∗ , John Pellegrino a , Eric Kozubal b , Steve Slayzak b , Jay Burch b a b

Department of Mechanical Engineering, University of Colorado Boulder, 427 UCB, Boulder, CO 80309-0427, USA National Renewable Energy Laboratory, Golden, CO, USA

a r t i c l e

i n f o

Article history: Received 6 October 2008 Received in revised form 2 March 2009 Accepted 24 March 2009 Available online 1 April 2009 Keywords: Heat pump Energy storage Heat and mass transfer modeling Desiccant Membrane distillation

a b s t r a c t In this paper, a membrane heat pump is proposed and analyzed. Fundamentally, the proposed heat pump consists of an aqueous CaCl2 solution flow separated from a water flow by a vapor-permeable membrane. The low activity of the solution results in a net flux of water vapor across the membrane, which heats the solution stream and cools the water stream. This mechanism upgrades water-side low-temperature heat to solution-side high-temperature heat, creating a “temperature lift.” The modeling results show that using two membranes and an air gap instead of a single membrane increases the temperature lift by 185%. The model predicts temperature lifts for the air-gap design of 24, 16, and 6 ◦ C for inlet temperatures of 55, 35, and 15 ◦ C, respectively. Membranes with lower thermal conductivities and higher porosities improve the performance of single-membrane designs while thinner membranes improve the performance of air-gap designs. This device can be used with a solar heating system which already uses concentrated salt solutions for liquid-desiccant cooling. © 2009 Elsevier B.V. All rights reserved.

1. Introduction As with other renewable technologies, solar thermal energy for residential water and space heating is plagued by the imbalance between the time the resource is available and the time the resource is needed. This occurs over many timescales, but in the extreme, this can be illustrated by the low-load, high-resource summers and high-load, low-resource winters, as shown in Fig. 1. Thermal energy storage can alleviate this imbalance by storing excess solar heat for later use. The amount of sensible energy stored depends on the heat capacity of the storage medium, the maximum temperature limit (e.g., 100 ◦ C for water) and the minimum temperature required to meet the heating load. For the application considered here, the required minimum temperature for space heating and domestic hot water is around 60 ◦ C. Although water has a high specific heat capacity, large storage tanks are still required to meet a large portion of the load. One way to reduce tank size is to use a heat pump during low-resource times to upgrade energy with a temperature too low to be useful to high-temperature energy. Vapor-compression and absorption heat pumps can be used for this purpose. In this paper, we propose a heat pump that uses water and a low-activity salt solution (often referred to as a liquid desiccant). The advantage of this type of heat pump is that the salt solution can be concentrated using solar energy and stored with no (or minimal) losses. The low-activity of the solution then attracts water

vapor, and its corresponding latent heat, and provides a “temperature lift” by upgrading low-temperature heat to high-temperature usable heat. In this way, the energy required for heat pumping is also supplied by stored solar energy. Water and a solution of lithium chloride are used in a previously developed absorption heat pump [2]. This device is a closed-system, using evacuated chambers for the heat pumping process. The proposed heat pump in our study is an open system using microporous membranes to separate the water and the low-activity salt solution. This eliminates the complexity of a vacuum system and also allows for separate storage of the solution, facilitating optimal design of both the capacity (kWh) of the storage and power (kW) of the heat pump device. In addition, with the open system, the salt solution can be used in the summer for air dehumidification. The purpose of this paper is to determine the feasibility of using a membrane exchanger as a heat pump. In our study we evaluated the performance based on calcium chloride (CaCl2 ) as the lowactivity solution. The paper addresses the concept of a membrane exchanger heat pump, the ideal maximum performance, and the predicted deviation from this ideal case when using a membrane exchanger. The performance is predicted with a numerical model. The theoretical effects of membrane properties are also discussed. 2. A membrane exchanger heat pump 2.1. The heat pump process using membranes

∗ Corresponding author. Tel.: +1 317 435 1207. E-mail addresses: [email protected], [email protected] (J. Woods). 0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.03.039

The heat pump process proposed here is similar to membrane distillation (MD) and osmotic distillation (OD) processes [3–6]. In these processes, a water vapor pressure difference across a

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J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

Fig. 1. A representation of the residential heating load (the required energy for space heating and domestic hot water) and the solar resource throughout the year in kWh/month. Recreated from [1].

membrane is used to transfer water as a vapor and to either purify or concentrate a solution. In MD, this vapor pressure difference is created by using two liquids with different temperatures, whereas OD uses liquids of different water activities (i.e., concentrations). The liquids are separated by a microporous hydrophobic membrane. The hydrophobic nature of the membranes prevents the liquids from entering the pores and establishes a liquid–vapor interface at the ends of each membrane pore. There is a net flux

of water vapor through the pores as water evaporates from the stream associated with the higher vapor pressure and condenses into the stream associated with the lower vapor pressure. This evaporation and condensation also carries with it a large amount of energy in the enthalpy of vaporization and imposes a temperature difference across the membrane. In MD and OD, this temperature difference can be suppressed by supplying heat to the evaporating side in order to increase the flux of water vapor. In the membrane heat pump proposed here, this latent energy transfer is used as a means to upgrade low-temperature heat to high-temperature useable heat. A water stream serves as the evaporation source while a low-activity solution can be used as the heated fluid, attracting the water vapor and the corresponding latent heat. The counter-flow membrane exchanger heat pump concept is illustrated in Fig. 2. Each fluid flow is separated by two membranes with a gap in between. As will be discussed, the air gap between the membranes helps reduce the conductive heat transfer between the two flows. A co-current design was also modeled but is inferior to the counter-flow design. In the co-current design the outlet solution is strongly coupled thermally to the outlet water, which due to evaporation is lower in temperature than the inlet water. 2.2. Theoretical maximum performance Before analyzing the counter-flow heat pump exchanger, we will look at the theoretical maximum performance. In this study, an

Fig. 2. Membrane exchanger heat pump concept; s represents the low-activity salt solution while w represents the water. (a) Alternating water-filled and solution-filled channels allow for a net flux of water vapor, Jv , from the water flows to the solution flows. (b) Hypothetical temperature profiles for the water and low-activity solution along the membrane exchanger. (c) Schematic vapor pressure and temperature profiles across the membranes. The air gap reduces thermal interaction between the two streams.

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Table 1 Characteristics and properties of the module, salt solution, and the hypothetical membrane assumed in this work. Module dimensions are adjusted to achieve different values of NTU. This parameter is explained below. Module dimensions at NTU = 1

No air gap

1-mm air gap

Number of channels Length of channel (m) Width of channel (m) Height of channel (m)

15 0.5 0.2 0.001

40 0.5 0.3 0.001

Assumed membrane propertiesa Porosity, ε Pore diameter, dpore (␮m) Thickness, ımem (␮m) Thermal conductivity, kpolymer (W m−1 K−1 ) Tortuosity, 

0.7 0.2 100 0.2 2

Fluid properties at 35 ◦ C Fig. 3. Equilibrium temperature difference between water and an aqueous solution of CaCl2 for different mass fractions of salt in the solution.

aqueous solution of CaCl2 was used for the low-activity solution. The properties of other salt solutions such as LiCl are better suited for this application, but with the requirement for large amounts of storage, we chose CaCl2 because of its lower cost. The maximum temperature difference is achieved when the vapor pressures, pv , above the water and the aqueous salt solution are equal: Tlift,max = Tsat,s (pv , ω) − Tsat,w (pv )

(1)

where Tsat,s is the saturation temperature at a specified vapor pressure and salt mass fraction, ω, and Tsat,w is the saturation temperature for pure water at the same vapor pressure. The vapor pressures above aqueous solutions of CaCl2 , along with all other CaCl2 properties, were calculated with correlations from Conde [7]. As seen in Fig. 3, the maximum temperature difference is a strong function of the CaCl2 concentration, with mass fractions less than 0.25 providing a temperature difference of only 5 ◦ C compared to nearly 35 ◦ C at a mass fraction of 0.55. The temperature has a lesser effect, but as will be shown, the temperatures are still important due to the reduced solubility at lower temperatures. The temperature lift, defined as the difference between the outlet and inlet temperatures of the heat pump, will be less than this equilibrium temperature difference, Tlift,max , for two reasons. First, as the CaCl2 solution absorbs water vapor along its path, its concentration will decrease, which will reduce the equilibrium difference. This effect is negligible at low concentrations, but reduces the temperature difference by 5% at high concentrations. Second, heat is transferred back to the evaporating water side as the solution temperature increases, limiting the temperature difference. A model was developed to simulate the simultaneous heat and mass transfer that occurs along the exchanger and is outlined in the next section. 3. Modeling The numerical model used to predict the performance of the membrane heat pump is outlined in this section. The model consists of a set of finite-difference equations representing the heat and mass transfer across a flat-plate, counter-flow membrane exchanger at steady-state. With this model, we examined the effect of the air-gap width, the inlet concentration and temperatures, the flow rates, and overall membrane area. In our current analysis, we neglect the heat loss from the module to the ambient and parasitic fluid pumping power. The module is sized to deliver heat to the salt solution at a rate of 4–6 kW, a value that depends on the inlet conditions. The size of the module is not fixed in our model; we vary the number and width of the channels to determine the effect the membrane area

−3

Density,  (kg m ) Specific heat, cp (J kg−1 K−1 ) Viscosity,  (kg m−1 s−1 ) Thermal conductivity, k (W m−1 K−1 ) Vapor pressure, pv (kPa) Enthalpy of mixing, Hmix (J kg−1 ) Diffusion coefficient, Dsw (m2 /s)

Water

CaCl2 solution (ω = 0.5)

994 4183 0.000655 0.609 5.58 – –

1490 2211 0.0177 0.55 1.26 3 × 105 4.17 × 10−10

a The membrane is assumed to have a monodisperse pore size distribution with cylindrical pores and deviations from this ideal are “lumped” into the tortuosity parameter.

has on the heat pump performance. Dimensions for a particular case are shown in Table 1, along with properties for the salt solution at a particular temperature and concentration. For the correlations used for water and CaCl2 solution properties, see Conde [7]. Also shown in Table 1 are the assumed membrane properties, which we selected based on membranes used in MD and OD experiments [4–6,8–11]. The tortuosity () through the membrane thickness is used to take into account the fact that the pores are not straight and a value of 2 is used here, as is often done in the literature [8,9,12]. The water vapor mass flux through a single membrane can be expressed as [8,13–15]: m Jv = K(pm v,w − pv,s )

(2)

m where pm v,w and pv,s are the vapor pressures at the liquid–membrane interfaces for the water and the salt solution, respectively, and K is the membrane mass transfer coefficient. The heat flux equations are obtained by performing an energy balance at each of the liquid–membrane interfaces. For the water side: m b m m qw = hw,BL (Tw − Tw ) = −Jv (Hv + cp,w Tw ) + hmem (Tsm − Tw )

(3)

and for the solution side: m m + Hmix )−hmem (Tsm − Tw ) qs = hs,BL (Tsm − Tsb ) = Jv (Hv + cp,w Tw

(4) where hw,BL , hmem , and hs,BL are the heat transfer coefficients for the water boundary layer, the membrane, and the solution boundary m − T b , T m − T m , and T m − T b are the temlayer, respectively, Tw w s w s s perature differences across the water boundary layer, across the membrane, and across the solution boundary layer. The heat of m term account for the total vaporization, Hv , along with the cp,w Tw enthalpy of the water vapor, where cp,w is the specific heat of liquid m is the water temperature at the membrane surface. water and Tw The heat flux equations were modified slightly from those found in the OD and MD literature [8,9,15,16] by considering the enthalpy of mixing of the CaCl2 solutions, Hmix , which can be significant at high concentrations. In the convention used in this work, Hv and Hmix are always written as positive. The enthalpy of mixing, an exothermic process for CaCl2 solutions, takes into account the energy of both ion-dipole and hydrogen bonds that are broken

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Fig. 4. Mass transfer (a) and heat transfer (b) resistances for the air-gap design. The mass transfer resistances in the boundary layers (BL) represent an equivalent mass transfer resistance that takes into account both thermal and concentration boundary layers.

and formed as water vapor dilutes the solution and depends on the temperature and concentration of the solution. As a benchmark for CaCl2 , Hmix ≈ 0.18 Hv at a mass fraction of 0.55 and a temperature of 60 ◦ C and decreases as either temperature or mass fraction is reduced (cf. [7]). In addition to the thermal boundary layers taken into account in Eqs. (3) and (4), a concentration boundary layer will form as condensation dilutes the solution near the membrane. According to film theory, the ratio of the mass fraction at the liquid–membrane interface (ωsm ) to the mass fraction in the bulk flow (ωsb ) is found from: ωsm ωsb



= exp



1 Kmem

 Uloss =

+

1 hmem

1 Kgap +

+ 1

hgap

Kmem +

1

(10)

3.1. Mass transfer

−1

1

i = 1 to 15

(7)

where Jv is the water vapor mass flux, s is the density of the solution, and km,BL is the convective mass transfer coefficient in the boundary layer. To model the air-gap design, the membrane mass transfer (K) and heat transfer (hmem ) coefficients in Eq. (2) and Eqs. (3) and (4), respectively, are replaced with: K=

ms,i−1 + Jv,i dAi

(6)

(5)

km,BL s

(ωsb )i−1 ms,i−1

for the heat and mass transfer on the solution side. The subscript i is ˙ w are the ˙ s and m the node along the exchanger. In these equations, m mass flow rates of the solution and the water flow, cp,s is the specific heat of the solution, and qs ,i , qw,i , and Jv,i are determined from Eqs. (2)–(5) for each node. The area available for heat and mass transfer at each node, dAi , is equal to the total exchanger area divided by the number of nodes along the exchanger. Note that the solution flow enters at i = 1, while the water flow enters at i = 15. We used the Engineering Equation Solver (EES) program [17] to solve Eqs. (2)–(5) and (8)–(10), along with the temperature and concentration dependent properties at each node along the exchanger. The module size, flow rates, and inlet conditions are specified, while guesses are made for all other variables. Using a variant of Newton’s method, the guesses are improved at each iteration until the residual, or error, of all equations is less than 1 × 10−5 . Equations for the coefficients and resistances used in the above equations are described in the next section. These equations are solved during the iterative process along with those mentioned above.



−Jv

(ωsb )i =

−1

hmem

(8)

3.1.1. Mass transfer through the membrane The dusty-gas model is often used to model the transport of a vapor or gas through porous media [18,19], and has been used extensively in OD and MD modeling [5,8,9,20]. The dusty-gas model assumes transport through the pores by molecular (ordinary) diffusion, Knudsen diffusion, surface diffusion, and viscous flow and is modeled as shown in Fig. 5. Surface diffusion is assumed negligible as long as the pore area is much larger than the membrane surface area [5], which is true for the pore diameters considered in this work. Since the solubility of air in water is low, the air is assumed to be a stagnant film and the viscous flow is also neglected [5]. The membrane resistance is therefore modeled as a Knudsen diffusion resistance and a molecular diffusion resistance in series. The relative importance of Knudsen and molecular diffusion depends on the Knudsen number, defined as

(9)

Kn =

where Uloss is referred to as the overall heat loss coefficient, Kmem and Kgap are the mass transfer coefficients for the membranes and for the air gap, respectively, and hmem and hgap are the heat transfer coefficients for the membranes and for the air gap, respectively. For the single membrane design, the overall heat loss coefficient is just hmem , but will be referred to as Uloss in subsequent sections. Eqs. (2)–(5) determine the heat and mass fluxes between the flows and can be represented by heat and mass transfer resistance networks, as shown in Fig. 4. Each resistance in this figure will be explained later. The model is split up into fifteen nodes along the exchanger, so that Eqs. (2)–(5) are each solved fifteen times. Mass and energy balances between the nodes along the flow determine how the temperature and concentration change along each flow: b ˙ w cp,w Tw ˙ w cp,w Tw )i−1 (m )i + qw,i dAi = (m

i = 1 to 15

for the heat transfer on the water side, and: ˙ s cp,s Tsb )i + qs,i dAi = (m ˙ s cp,s Tsb )i−1 (m

i = 1 to 15

 dpore

(11)

J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

117

width, and g , g , and ˛g , are the average density, dynamic viscosity, and thermal diffusivity of the vapor–air mixture in the gap. For the operating conditions considered for this application, and for air gaps of 3 mm or less, the Rayleigh number is less than 100. Based on this calculated Rayleigh number, and the model validation performed by Alklaibi and Lior, convection effects are assumed negligible and the mass transfer coefficient for the air gap is expressed as Kgap =

Mw Dva p RT ıgap pa,lm

(16)

where ıgap is the gap width and here T is the average temperature in the gap. 3.2. Heat transfer Fig. 5. Membrane resistance to mass transfer (1/Kmem ). Surface and viscous resistances assumed negligible in this study.

where  is the mean free path of the transported molecules, and dpore is the membrane pore diameter. For Kn > 10, molecule-wall collisions dominate and Knudsen diffusion is the primary mechanism for mass transfer. For Kn < 0.01, molecule–molecule collisions dominate and molecular diffusion is the primary mechanism for mass transfer [19]. In this case, the mean free path of the water molecules is on the same order of magnitude as the pore diameter, and both molecular and Knudsen diffusion are considered. With the molecular and Knudsen resistances in series (Fig. 5), the mass transfer coefficient for the membrane is expressed as [3]: Kmem =

Mw RTımem



1 Kn Dw

+

pa,lm

−1

eff pDva

(12)

where the effective diffusivity of water vapor in air filled pores is Dveff a

ε = Dva 

(13)

and the Knudsen diffusivity is given by dpore ε Dv = 3



Kn

8 RT Mw

(14)

In these equations, Mw is the molecular mass of water, R is the universal gas constant, p and pa,lm are the total pressure and the logarithmic-mean pressure of air, respectively, and ımem , ε, and  are the thickness, porosity, and tortuosity of the membrane, respectively. 3.1.2. Mass transfer across the air gap Air-gap membrane distillation (AGMD) is a membrane separation process consisting of a feed stream separated from a falling film of condensate by a membrane and an air gap (cf. [11,21]). Although AGMD is different from the process here since it uses only one membrane and a falling film instead of a second pressure driven flow, it can be looked at for some guidance on modeling the air gap. Alklaibi and Lior [21] assumed negligible convection in the air-gap region when modeling the AGMD process. They modeled air gaps of up to 5 mm and validated their model with experimental work done by Banat [22]. In general, the transport of a gas across an air gap is assumed to occur by diffusion alone if the Rayleigh number is less than a critical value, usually on the order of 103 [23]. The Rayleigh number is defined as Ra =

gg ı3gap g g ˛g

(15)

where g is the acceleration due to gravity, g is the difference in density of the vapor–air mixture across the gap, ıgap is the gap

3.2.1. Heat transfer through the membrane Heat is transferred between the two streams through the membrane itself and through the pores in the membrane. If these two paths are assumed to operate independent of one another (i.e., the parallel-path or isostrain model), the heat transfer coefficient across the membranes can be expressed as hmem =

(1 − ε)kpolymer + εkg

(17)

ımem

where kpolymer is the thermal conductivity of the membrane material and kg is the thermal conductivity of the vapor–air mixture in the pores. Radiation through the membrane pores is neglected. When modeling membrane distillation, Schofield et al. [24] found that Eq. (17) predicted the actual membrane conductivity of their membranes within 10%. 3.2.2. Heat transfer across the air gap Since convection is assumed negligible (Ra < 103 ), heat transfer across the air gap is modeled with conduction and radiation. The heat transfer coefficient between the membranes is hgap =

kg SB + ıgap 1/εrad,s + 1/εrad,w − 1 2 2 × (Tmem,w + Tmem,s )(Tmem,w + Tmem,s )

(18)

where the first term takes into account conduction and the second term takes into account radiation. In this equation,  SB is the Stefan–Boltzmann constant, and εrad and Tmem are the emissivities and inner surface temperatures of the membranes, respectively. 3.3. Thermal and concentration boundary layers The hydrodynamic and thermal entrance lengths for both flows were found to be less than 1 cm and therefore these two boundary layers were assumed to be fully developed. The average Nusselt number for fully developed laminar flow [4,25] can be expressed as Nud,avg = 0.13 Re0.64 Pr 0.38 d

Red =

Vdh

l

Pr =

l ˛l

(19)

where Red is the Reynolds number, Pr is the Prandtl number, V is the velocity of the liquid, l and ˛l are the dynamic viscosity and thermal diffusivity of the liquids, respectively, and dh is the hydrodynamic diameter of the channel. Phattaranawik et al. [26] compared different correlations with their experimental values and found Eq. (19) to be one of two correlations that was most suitable for MD during laminar flow. The average Nusselt number is calculated for each flow and the heat transfer coefficient is found from the definition of the Nusselt

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J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

number: hw,BL =

Nud,avg,w kl,w

hs,BL =

dh

Nud,avg,s kl,s

(20)

dh

where kl,w and kl,s are the thermal conductivity of the water and solution, respectively. Since the thermal boundary layer is fully developed, this leads to a relatively constant Nusselt number and heat transfer coefficient along the exchanger and this average value is used at each node. The concentration boundary layer would take one meter to become fully developed, so the boundary layer is still developing along the entire length of the exchanger. Therefore, the Sherwood number correlation for a developing boundary layer, as opposed to a fully developed boundary layer, was used to find the mass transfer coefficient. A correlation for the average Sherwood number commonly used for OD processes is [3,4,15]: 1/3

Shx,avg = 1.62 Gzx

d Gzx = h Red Sc x

Sc = l Dws

(21)

where Sc is the Schmidt number, Gzx is the Graetz number, x is the distance from the entrance of the channel, and Dws is the water binary diffusion coefficient for CaCl2 solutions. The average Sherwood number in Eq. (21) is used in the OD literature with the x replaced with L, the length of the tube. However, the mass transfer coefficient will change as the boundary layer develops and the local Sherwood number needs to be used at each node. From the definition of the average Sherwood number, the local Sherwood number is found to be: Shx =

2 1/3 Shx,avg = 1.08 Gzx 3

(22)

The boundary layer convective mass transfer coefficient, used in Eq. (5), is then found for each node along the exchanger from the definition of the Sherwood number: km,BL =

Shx Dws dh

(23)

3.4. Analysis methods The transport properties of the heat pump can be simplified to just two parameters. The first is the overall mass transfer coefficient, which characterizes how much vapor, and therefore latent energy, is transferred from the water to the solution:



Kov =

1 Kw,BL

+

1 Kmem

+

1 Kgap

+

1 Kmem

+

1 Ks,BL

−1

(24)

The boundary layer coefficients, which take into account both the concentration and thermal boundary layers, are based on the difference in vapor pressures between the liquid–membrane interface and the bulk liquid flows: Kw,BL =

Jv pbv,w

− pm v,w

Ks,BL =

Jv pm v,s

− pbv,s

(25)

where pbv,w and pbv,s are the vapor pressures in the bulk water and solution flows, respectively. Although a vapor pressure in a bulk liquid has little meaning, it is used here to represent the vapor pressure that would be seen at the liquid–membrane interface if there were no thermal or concentration boundary layers. The second parameter, defined previously as the overall heat loss coefficient (Uloss ), characterizes the amount of heat that is lost from the solution back to the water flow through conduction across the membrane. This coefficient does not include the boundary layer coefficients since the heat transfer across the boundary layers should be maximized in order to reduce the difference between the bulk and liquid–membrane interface’s vapor pressures. The design

objective is thus simplified to maximizing the overall mass transfer coefficient and minimizing the overall heat loss coefficient. For characterizing the performance of the heat pump, we use two metrics: the effectiveness, εlift , and the efficiency, . The effectiveness represents the ratio of the achieved temperature lift to the maximum temperature lift. The efficiency represents the fraction of the total heat transfer from vapor transport that results in increasing the temperature of the solution, as opposed to being conducted back across the membrane to the water flow. These two metrics are defined in more detail below. Note that the efficiency is concerned with the concentration of the solution, and how efficiently that potential for vapor transport is being used, as opposed to being concerned with thermal energy. The thermal energy conducted back to the water side is not lost, but returns to the water storage tank. 3.4.1. Effectiveness-NTU A modified version of the effectiveness-NTU method, commonly used to characterize heat exchangers (cf. [27]), was used to analyze the heat pump performance. The number of transfer units (NTU) of a heat exchanger represents the ratio of the heat capacity rate provided by the exchanger to the heat capacity of the flow: NTUthermal =

UA ˙ p mc

(26)

where the numerator is the overall heat transfer coefficient (U) times the transfer area (A) and the denominator is the mass flow ˙ times the specific heat of the flow (cp ). The NTU of a heat rate (m) exchanger, and the NTU for the membrane exchanger heat pump, are representations of the overall size, and thus cost, for a particular application. For the membrane heat pump, the NTU is analogously defined to be the ratio of the total energy transfer (due to vapor transport) provided by the exchanger to the total energy required by the flow to reach the maximum temperature lift: NTU =

Kov Apv,lm (Hv + cp,w Tw + Hmix ) ˙ s cp,s Tlift,max m

(27)

where pv,lm is the log-mean difference in bulk-liquid vapor pressures between the two streams at each end of the exchanger, and A is the total area available for heat and mass transfer. An NTU of one is obtained when the energy terms in the numerator and denominator of this equation are equal. This means that if we ignore conduction losses back through the membrane, the boundary layer effects, and the effect of the concentration change in the solution, the maximum temperature lift results when the NTU equals one. The NTU was varied in the simulations by varying the number and width of the channels. The effectiveness of a heat exchanger is defined as the ratio of the actual amount of heat transfer to the maximum possible amount of heat transfer. For the heat pump, this leads to: εlift =

Tlift,actual Tlift,max

(28)

which can be seen as the fraction of the maximum temperature lift that is achieved. 3.4.2. Heat capacity rate ratio In heat exchanger analysis, the heat capacity rate ratio dictates how the temperature of each stream changes. From conservation of energy, the ratio of the heat capacity rates of the two flows equals the reciprocal of the change in temperatures. Although this relationship will not be exact for the heat pump application due to the enthalpy of mixing, Hmix , the general relationship is still approximately true. Here the heat capacity rate ratio (Rc ) is defined as the ratio of the solution flow heat capacity rate to the water flow heat

J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

Fig. 6. Comparison of model-predicted membrane mass transfer coefficients, Kmem , to previous experimental work. Table 1 shows the properties of the membranes used in the experimental studies.

capacity rate: Rc =

˙ p )s (mc ˙ p )w (mc

≈

Tw Ts

(29)

3.4.3. Efficiency The usefulness of the salt solution decreases after passing through the membrane exchanger heat pump since the concentration, and therefore the vapor pressure depression, is reduced. Since the reduction in concentration is directly proportional to how much water vapor is transferred, an efficiency can be defined as the fraction of the total energy transfer due to vapor transport that results in increasing the temperature of the solution: =

˙ s cp,s Tlift,actual m Jv A(Hv + cp,w Tw + Hmix )

(30)

The efficiency is a measure of how efficiently the concentrated solution is being used. The efficiency would be 100% if there were no conduction losses from the solution back to the water. 4. Results and discussion This section presents results from our numerical model for several air-gap widths as well as a design with no air gap. The intention here is not to validate the complete model. Rather, we aim to estimate the performance of the heat pump and determine how key parameters affect this performance. No experimental work was found on a dual-membrane exchanger with an air gap. Since the largest uncertainty in our model is from the membrane mass transfer modeling, we compare the single-membrane model with available experimental data from the literature for membrane mass transfer coefficients (Table 2). The model predicts the membrane mass transfer coefficients within 30% of the experimental values,

119

Fig. 7. Effect of heat capacity rate ratio on water and solution outlet temperatures for the case when Tw,in = Ts,in = 45 ◦ C and ωs,in = 0.55. Results shown are for an air-gap design with ıgap = 1 mm.

as seen in Fig. 6, and the membrane modeling was considered adequate for this analysis. The following results are for conditions where the inlet temperature of the solution is the same as the inlet temperature of the water. This is not required for the heat pump to operate, but it is a way to simplify the analysis and observe overall trends. 4.1. Heat capacity rate ratio The outlet temperatures of both the water and solution from the heat pump for various heat capacity rate ratios are shown in Fig. 7. As Rc increases, the magnitude of the temperature change in the water flow increases. This has two effects. The first is to reduce the vapor pressure on the water side, which reduces the driving potential for water vapor transfer and thus reduces the possible temperature lift on the solution side. The second effect of the lower water temperatures is to increase the driving potential for conductive and radiative heat transfer from the solution back to the water, which increases losses and reduces the efficiency. For these two reasons, the highest temperature lifts and the highest efficiencies are achieved when operating at low values of Rc , which means the water mass flow rate should be much larger than the solution mass flow rate. However, increasing the water flow rate increases the parasitic pumping power, implying an optimum ratio could be calculated if parasitic power were considered. Subsequent results are for Rc = 0.1. 4.2. Mass transfer to heat transfer ratio: effect of the air gap Since the NTU is defined as the ratio of the actual latent heat transfer to the maximum heat transfer possible, an NTU value of one would theoretically give the maximum temperature lift (i.e., an effectiveness of 100%). This theoretical limit is shown in Fig. 8 along with the predicted performance of several cases. The first case

Table 2 Membranes used in previous experiments for comparison to the model-predicted membrane mass transfer coefficient (Fig. 6). Data point in Fig. 6

Membrane supplier

Solutions used in experiment

dpore (␮m)

Porosity ε (%)

ımem (␮m)

Reference

1 2 3 4 5 6 7 8

Microdyn GmbH Microdyn GmbH Microdyn GmbH Millipore Pall Gelman Science Millipore Membrana GmbH Millipore

Water, CaCl2 Water, NaCl Water, NaCl Water, CaCl2 Water, CaCl2 Water, CaCl2 Water, LiCl Water, CaCl2

0.20 0.20 0.20 0.10 0.10 0.22 0.29 0.45

70 70 70 75 55 75 75 75

1500 373 178 125 90 125 178 125

[28] [29] [28] [9] [13] [9] [10] [9]

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Fig. 8. Effect of the air gap on the temperature lift effectiveness. Results shown are for Rc = 0.1, ωin = 0.5 and Tw,in = Ts,in = 45 ◦ C.

is for a membrane exchanger with no losses. The temperature lift is slightly less than the maximum achievable due to the effect of the concentration change and the effect of the thermal and concentration boundary layers in the two flows. Since there are no sensible heat losses, the efficiency for this case is 100%. The other cases shown are for a single-membrane design and for a dual-membrane design with an air gap of different widths. For the single-membrane design, the achievable temperature lift is limited because of the heat conducted back through the membrane. The temperature lift at an NTU value of one is only 22% of the maximum achievable. In addition, the concentrated solution is only being utilized at an efficiency of 29%. There are three important points to be made about the air-gap design. When the NTU is one, an air gap improves the temperature lift 2–3 times over the single-membrane design, depending on the width of the gap. The reason for this is the difference in the heat and mass transfer coefficients per unit thickness for the air gap and for the membrane. Compared to a membrane of equal thickness, an air gap has a 450% higher mass transfer coefficient and a 33% lower heat transfer coefficient. This difference in properties between a membrane and an air gap results in the overall mass transfer coefficients and overall heat loss coefficients shown in Fig. 9. Including an air gap in the membrane exchanger reduces both of these coefficients, but the ratio of mass transfer to heat transfer improves with the inclusion of an air gap and increases as the gap becomes larger. The second point regarding the air-gap design is that the efficiencies are significantly improved, increasing from 29% for the

single-membrane design to 68% for the 1-mm air-gap design. As a result, the air gap will prolong the use of the concentrated solution. The final point regarding the air-gap design is that the amount of membrane area required increases significantly over the singlemembrane design. Besides the fact that there are two membranes instead of one, the overall mass transfer coefficient (Kov ) decreases as the width of the gap increases (Fig. 9). From Eq. (27), it can be seen that for a constant NTU, the exchanger area (A) must increase as Kov decreases. When the NTU is one, the amount of actual membrane area (including the dual-membrane requirement) increases 7, 10, and 15 times over the single-membrane design for gap widths of 0.5, 1, and 2 mm, respectively. The amount each resistance contributes to the overall heat and mass transfer resistances (Fig. 4) are shown in Table 3. When an air gap is used, the overall heat transfer resistance is dominated by the air gap, even for small gap widths. Increasing the width of the air gap significantly increases the overall heat transfer resistance due to lower conductive losses. Radiative heat transfer (not shown individually in Table 3) was found to have a smaller impact, although it becomes more important for larger gap widths since radiation is independent of distance. The percent of the heat losses across the gap from radiation increases from 5% to 30% as the gap increases from 0.25 mm to 2 mm. When considering the mass transfer resistances, the membranes become much more important. This again emphasizes how the air gap improves performance. The design of the membrane exchanger heat pump should have low resistance to mass transfer and high resistance to heat transfer. The membranes amount to 42% of the overall mass transfer resistance, but provide only 8% of the heat transfer resistance for a gap width of 1 mm. The air gap provides a similar amount, 44%, of the overall mass transfer resistance, but 92% of the total heat transfer resistance. Due to the higher mass flow rates and the fact that there is no concentration boundary layer in the water flow, the boundary layer resistance to mass transfer on the water side is less than 1% of the total mass transfer resistance. Conversely, the solution boundary layers contribute a significant amount to the mass transfer resistance since the flow rate is much lower and there is a concentration boundary layer in this flow. The boundary layer mass transfer resistance (Ks,BL ) results from the difference between the vapor pressures in the bulk liquid and at the membrane surface. It was found that roughly 2/3 of this difference was due to the concentration boundary layer while 1/3 was due to the thermal boundary layer. Also, note that the solution boundary layer mass transfer resistances are average values for the exchanger. The Sherwood number and the convective mass transfer coefficient for the boundary layer, Eqs. (21)–(23), change significantly along the exchanger as it develops. For the 1-mm air-gap design, the mass transfer resistance, 1/Ks,BL , increases from 5% of the total mass transfer resistance at the channel inlet to 18% of the total mass transfer resistance at the exit. 4.3. Effect of operating conditions

Fig. 9. Effect of the air gap on the overall mass transfer coefficient (Kov ) and overall heat loss coefficient (Uloss ) for the membrane exchanger. Values shown are for Rc = 0.1, NTU = 1.0, ωin = 0.5, and Tw,in = Ts,in = 45 ◦ C.

The performance of the heat pump depends on the temperatures and concentrations at which it is operating. The modeled performance of the air-gap design is shown in Fig. 10 for 45 ◦ C inlet temperatures and various inlet concentrations. The initial concentration has a small impact on both the efficiency and the effectiveness. Since the effectiveness is the percent of the maximum temperature lift, the temperature lift will follow the same trend as seen in Fig. 3, scaled by an appropriate effectiveness value. The improvement in effectiveness levels off as the NTU of the exchanger goes beyond one. Since the purpose of the heat pump is to provide high temperature lifts, a higher NTU value may be desirable. However, there are two tradeoffs that must be considered with this. First, for constant operating conditions, increasing

J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

121

Table 3 Effect of each region (Fig. 4) on the overall heat and mass transfer. Case shown is for Rc = 0.1, NTU = 1.0, ωin = 0.5, and Tw,in = Ts,in = 45 ◦ C. Percent of total heat transfer resistance, 1/Uloss ıgap

1/Uloss (K m2 W−1 )

1/hmem,1

1/hgap

1/hmem,2

No gap 0.25 mm 0.5 mm 1 mm 2 mm

0.0013 0.0111 0.0188 0.0320 0.0523

100% 11% 7% 4% 2%

n/a 77% 87% 92% 95%

n/a 11% 7% 4% 2%

Percent of total mass transfer resistance, 1/Kov 2

−1

ıgap

1/Kov (kPa m h kg

No gap 0.25 mm 0.5 mm 1 mm 2 mm

0.93 1.99 2.40 3.20 4.73

)

1/Kw,BL

1/Kmem,1

1/Kgap

1/Kmem,2

1/Ks,BL

<1% <1% <1% <1% <1%

71% 33% 27% 21% 14%

n/a 18% 29% 44% 60%

n/a 34% 28% 21% 14%

29% 15% 15% 14% 12%

the NTU increases the amount of membrane area required. Secondly, increasing the NTU results in a reduction in the efficiency, which means more concentrated salt solution is required. The optimal design will depend on the importance of the temperature lift, on the cost per area of membrane, and on the importance of the efficiency in using the salt solution (i.e., the cost to concentrate and store the low-activity solution). The results for other inlet temperatures are similar to those shown in Fig. 10. However, the inlet temperature also affects the performance of the heat pump by changing the maximum possible inlet concentration. At lower temperatures, the solution will saturate at lower concentrations (e.g., CaCl2 solution becomes saturated at a concentration of 0.56 at 45 ◦ C and at a concentration of 0.41 at 15 ◦ C (cf. [7])). Taking the effect of saturation into account, the overall heat pump performance is mapped out in Fig. 11 where the temperature lift is plotted for various inlet concentrations and temperatures. The case shown is for a 1-mm air-gap design with Rc = 0.1 and an NTU value of 1.2. For the cases less than 45 ◦ C, the plot ends at the saturation concentration. The efficiencies (Fig. 11) improve as either temperature or concentration increases, although concentration has a smaller impact. Due to the effect of saturation at lower temperatures, the optimal operating range of a heat pump using CaCl2 will likely be in the 35–55 ◦ C range. 4.4. Effect of membrane properties

Fig. 11. Performance of dual-membrane air-gap design with ıgap = 1 mm, Rc = 0.1, and NTU = 1.2. The equilibrium temperature difference shown is for a water temperature of 45 ◦ C. Each temperature line ends at the mass fraction corresponding to saturation. Lower and upper bound on efficiencies are for a mass fraction of 0.2 and the saturated mass fraction, respectively.

that some of these properties have on the achievable temperature lift of the heat pump were estimated by changing the porosity, conductivity, and thickness of the modeled membranes and are shown in Fig. 12. Case (e) shows a near-best-case scenario: a 10-␮m thick

The membrane properties used for this analysis were selected based on common properties found in the literature. The effects

Fig. 10. Efficiency and effectiveness for different inlet concentrations for the air-gap design. Case shown is for Rc = 0.1 and Tw,in = Ts,in = 45 ◦ C.

Fig. 12. Effect of membrane properties on temperature lift. Case (a) is for the assumed properties outlined in the modeling section (ε = 70%, kpolymer = 0.2 W/mK, ımem = 100 ␮m), cases (b–d) show the individual effects of each of the changes to porosity, conductivity, and thickness. Case (e) shows the combined effect of all three changes.

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membrane with 90% porosity and a 10× lower polymer conductivity. Note that the 10× lower polymer conductivity is roughly equal to the conductivity of the vapor–air mixture in the pores. All of these changes together essentially makes the resistances of the membrane negligible, but results in an increase in the temperature lift of only 10%. Further improvement in performance is limited by the air gap since the heat and mass transfer resistances for the two membranes together comprise only 8% and 42% of the overall heat and mass transfer resistances, respectively (Table 3). However, it should be noted that as Kov is reduced, less membrane area is needed for a given NTU. For case (e), the membrane area is reduced by 35%. The most significant impact is made by reducing the membrane thickness. The opposite is true for the single-membrane design, where increasing the membrane thickness improves performance, but minimally, and the porosity and polymer conductivity have a much larger effect. With all of the membrane properties changed, the airgap design provides a 35% increase in the temperature lift over the single-membrane design, compared to the 185% increase for the baseline membranes. Depending on the application, if a membrane with these properties were available, the simplicity of the single-membrane design might outweigh the extra temperature lift provided by the air-gap design. An additional property of hydrophobic membranes relevant to the operation of the membrane exchanger heat pump is the penetration pressure. The penetration pressure is reached when the force from the pressure difference across the membrane exceeds the surface tension forces and liquid can enter and pass through the pores. This allows water transport without evaporation and condensation, and thus eliminates the latent energy transfer. Once the penetration pressure is reached, reducing the pressure in the flow does not remove the water from the pores, and the membrane must be completely dried to restore the liquid–vapor interface at the pores. To avoid these problems, the operation of the membrane exchanger heat pump is limited to conditions where the pressure difference across the membrane is well below the penetration pressure. Increasing the penetration pressure of the membrane could help performance in two ways. Higher operating pressures allow higher flow rates on the water side, which lowers the heat capacity rate ratio and improves the temperature lift and the efficiency (Fig. 7). Secondly, a larger penetration pressure may allow for the use of a partially evacuated air gap between the two membranes, which would increase the diffusivity and the mass flux. The effect on conductivity and the heat flux is negligible for pressures as low as 0.1% of atmospheric pressure, so very low pressures would be required to reduce the heat losses. 4.5. Hollow-fiber membrane configuration As opposed to using flat sheet membranes, the heat pump device can be made from hollow fiber membranes. In this configuration, both streams would flow through hollow fiber lumens and the space between the fibers would act as the air gap. The channels shown in Fig. 2 would be replaced by rows of hollow fiber membranes, alternating between water and solution in each successive plane of tubes. Alternatively, a device similar to that described for contained liquid membranes [30] could be used. One advantage of using hollow fiber membranes is to provide the necessary structural support to maintain an air gap between the two flows. Also, the amount of transfer area per length of exchanger increases by a factor of /2, with a possible reduction in the size of the device. The next step in this research is to model rows of hollow fibers in place of the flat-plate channels and to determine how the cylindrical geometry and the variable air gap affect heat and mass transfer. Experimental work is also planned to verify the modeled results.

5. Conclusions In order to extend the temperature range of a sensible heat storage, a membrane exchanger as a heat pump using water and a liquid desiccant can be used to upgrade low-temperature heat to high-temperature usable heat. A nodal model of a steady-state, flat-plate, counter-flow membrane exchanger was used to estimate its performance. Optimizing the design depends on the application and an economic model, but certain operating and design regions were identified that provide the best performance. There are tradeoffs between the achievable temperature lift, the amount of membrane area required, and how efficiently the concentrated solution is being used. Since the purpose of the heat pump is to increase the temperature of low-grade energy, the temperature lift will likely be most heavily weighted. The predicted temperature lift for a dual-membrane air-gap design increases by 185% over the single-membrane design due to the lower mass transfer resistance and higher heat transfer resistance of an air gap relative to a membrane. Increasing the size of the gap improves performance, but since it also reduces the overall mass flux, it also requires more membrane area. The optimal gap width appears to be near 1 mm. For a given set of operating conditions, a larger NTU results in higher temperature lifts, larger amounts of membrane area, and reduced efficiency. From the modeled results, the optimal design will likely be near the theoretical NTU requirement of one. When the NTU is 1.2, the predicted temperature lifts for a dual-membrane air-gap design were found to be 24, 16, and 6 ◦ C for inlet temperatures of 55, 35, and 15 ◦ C, respectively, for concentrations near saturation. Due to the reduced performance at lower temperatures, a membrane exchanger heat pump using CaCl2 would be suitable for a heat pumping process in the 35–55 ◦ C range. The effects of the membrane properties on performance were also investigated. If membranes with a conductivity on the order of the conductivity of air and porosities approaching 90% were available, the predicted temperature lift for the air-gap design is only 35% higher than that for a single membrane, which could make the single-membrane design more attractive due to its simplicity and reduced size. For the air-gap design, increasing the porosity, reducing the thickness, and/or reducing the conductivity of the membranes improved the performance. Of these properties, the thickness was found to have the largest impact. Acknowledgements The authors would like to acknowledge the support of Robert Hassett and the U.S. Department of Energy (DOE) Solar Heating and Cooling Program, as well as Ed Pollock and the Emerging Technologies Program from the DOE Energy Efficiency and Renewable Energy Buildings Technologies Program.

Nomenclature List of symbols A exchanger area available for heat and/or mass transfer (m2 ) cp specific heat capacity (J kg−1 K−1 ) Dij binary diffusion coefficient of species i in species j (m2 s−1 ) Dveff effective ordinary diffusion coefficient through a a membrane pore (m2 s−1 ) Kn Dv effective Knudsen diffusion coefficient through a membrane pore (m2 s−1 )

J. Woods et al. / Journal of Membrane Science 337 (2009) 113–124

dh dpore g Gzx h i Jv k K km,BL Kn Kov ˙ m Mw NTU Nud p pi Pr q R Rc Ra Red Sc Shx T U Uloss V

hydrodynamic diameter (m) membrane mean pore diameter (m) acceleration due to gravity (m s−1 ) Graetz number based on length x heat transfer coefficient (W m−2 K−1 ) node number along exchanger mass flux of water vapor (kg m−2 s−1 ) conductivity (W m−1 K−1 ) mass transfer coefficient (kg m−2 h−1 kPa−1 ) boundary layer convective mass transfer coefficient (m s−1 ) Knudsen number mass transfer coefficient based on bulk liquid vapor pressures (kg m−2 h−1 kPa−1 ) mass flow rate (kg s−1 ) molecular mass of water, (18.02 kg kmol−1 ) number of transfer units Nusselt number based on hydrodynamic diameter d pressure (kPa) partial pressure of component i (kPa) Prandtl number heat flux (W m−2 ) universal gas constant, (8314 J mol−1 K−1 ) heat capacity rate ratio Rayleigh number Reynolds number based on hydrodynamic diameter d Schmidt number Sherwood number based on length x temperature (K or ◦ C) overall heat transfer coefficient (W m−2 K−1 ) overall heat loss coefficient between the liquid–membrane interfaces (W m−2 K−1 ) average fluid velocity (m s−1 )

Greek symbols ˛ thermal diffusivity, cp /k (m2 s−1 ) ı thickness or width X change or difference in X Hv enthalpy of vaporization of water (J kg−1 ) Hmix enthalpy of mixing of the aqueous solution (J kg−1 ) Tlift difference between the outlet and inlet solution temperatures (K) ε membrane porosity εlift temperature lift effectiveness εrad radiative emissivity efficiency in the use of concentrated solution  mean free path of a molecule

kinematic viscosity (m2 s−1 )  density (kg m−3 )  SB Stefan–Boltzmann constant (5.67 × 10−8 W m−2 K−4 )  membrane tortuosity ω mass fraction Subscripts a air avg average value BL boundary layer g vapor–air mixture in the gap or in the membrane pores gap air gap l liquid phase

lm mem polymer s sat

v w

123

log-mean average or log-mean difference membrane property of the membrane material low-activity, aqueous salt solution flow saturated liquid–vapor condition vapor water flow

Superscripts b bulk m liquid–membrane interface

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