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The role of surfactant adsorption rate in heat and mass transfer enhancement in absorption heat pumps
International Journal of Refrigeration 26 (2003) 129–139 www.elsevier.com/locate/ijrefrig
The role of surfactant adsorption rate in heat and mass transfer enhancement in absorption heat pumps Michael S. Koenig, Gershon Grossman*, Khaled Gommed Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, 32000 Israel Received 23 February 2001; received in revised form 21 December 2001; accepted 4 February 2002
Abstract The importance of heat and mass transfer additives in absorption chillers and heat pumps has been recognized for over three decades. However, a universally accepted model for the mechanisms responsible for enhanced absorption rates has yet to be proposed. The Marangoni effect—an instability arising from gradients in surface tension at the liquid-vapor interface—is generally accepted as the cause of the convective flows that enhance transfer rates. Certain surfactant additives can significantly improve absorption rates and thus reduce the overall transfer area required by a given machine. Any means available that can increase the efficiency and acceptability of absorption machines is to be welcomed, as this technology provides an alternative to vapor compression systems which is both environmentally friendly and more versatile with regards to energy sources. This study investigates the rate at which a surfactant additive adsorbs at a liquid-vapor interface. The residence time of the falling liquid solution in an absorber is quite short. An effective additive must not only reduce the surface tension of the solution; it must do so quickly enough to cause the Marangoni instability within the short absorption process time. The entrance region of an absorber features a freshly exposed interface at which no surfactant has adsorbed. A numerical model is used to analyze surfactant relaxation rates in a static film of additive-laced solution. Kinetic parameters for the combination of the working pair LiBr-H2O and the additive 2-ethyl-1-hexanol are derived from data in the literature for static and dynamic surface tension measurements. Bulk, interfacial and boundary parameters influencing relaxation rates are discussed for surfactant adsorption occurring in the absence of absorption, as well as for concurrent adsorption and stable vapor absorption. Initial solution conditions and absorption driving force are shown to impact the potential for instability in the effect they have on the rate of interfacial additive adsorption. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Heat pump; Adsorption system; Absorber; Surfactant; Mass transfer; Heat transfer
Le roˆle des agents tensio-actifs sur la vitesse d’adsorption et l’ame´lioration des transferts de chaleur et de masse dans les pompes a` chaleur Mots cle´s : Pompe a` chaleur ; Syste`me a` absorption ; Absorbeur ; Agent tensioactif ; Transfert de masse ; Transfert de chaleur
* Corresponding author. Tel.: +972-4-829-2074; fax: +972-4-832-4533. E-mail address: [email protected] (G. Grossman). 0140-7007/03/$20.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(02)00012-9
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Nomenclature B c cA d D Ha k ka, kd KL P Pr Sc t* t T y y Z1, Z2 2EHX
adsorption rate barrier, Eq. (3) (dimensionless) absorbate concentration (kmol/m3) additive (bulk) concentration (kmol/m3) film thickness (m) mass diffusion coefficient (m2/s) heat of absorption (J/kmol) thermal conductivity (W/m C) rate coefficients: ka—adsorption, kd— desorption (m/s, s1) Langmuir isotherm constant, ka/kd1 (m3/kmol) pressure (kPa) Prandtl number, / (dimensionless) Schmidt number, /D (dimensionless) time (s) normalized time, t/(d2/) (dimensionless) temperature ( C) vertical coordinate in film (m) normalized vertical coordinate, y/d (dimensionless) interdiffusion parameters, Eq. (17) (dimensionless) 2-ethyl-1-hexanol
1. Introduction Heat and mass transfer additives have been used in certain absorption chillers and heat pumps for over three decades. The maintenance of quite low concentrations of such additives in machines using closed cycles of lithium bromide–water (LiBr–H2O) as a working pair, affords significantly improved performance of the absorber, and thus of the entire machine, as compared to operation in the absence of additives. Various higher alcohols are known to be good additives for cycles operating in moderate temperature ranges, the most prominent among them being 2-ethyl-1-hexanol (2EHX), an isomer of n-octanol, which is itself also a relatively good additive. However, not all absorption machines benefit from the effects of additives. Ammonia–water absorbers do not have convincing counterparts to the alcohols used in LiBr–H2O absorbers. Likewise, the development of advanced absorption cycles with higher operating temperatures requires a search for additives with greater chemical stability. The search for effective additives for a wide variety of applications suffered—until the latter part of the 1980s—from a lack of understanding of the mechanisms responsible for the absorption rate enhancement. Soon
Greek symbols thermal diffusivity (m2/s) normalized concentration: (dimensionless) Absorbate=(cco)/(ceco) Additive=cA/cA1 Additive surface excess (interfacial) concentration (kmol/m2) solution temperature=(TTo)/(TeTo) (dimensionless) l normalized heat of absorption, HaD(ceco)/ k(TeTo) (dimensionless) kinematic viscosity (m2/s) surface tension (N/m) Subscripts a adsorption A additive d desorption L Langmuir 1 reference or maximum value o initial value NA no additive s subsurface v vapor wall absorber cooling wall LiBr lithium bromide e, eq equilibrium
after the accidental discovery of these additives in the mid 1960s [1]—a welcome phenomenon experienced while searching for antifoaming agents—it was widely and correctly assumed that improved absorption rates were attributable to the surface active nature of the alcohols in question. The additives were understood to cause the Marangoni effect, a hydrodynamic instability at the liquid–vapor interface, brought on by gradients in surface tension. The result of this instability can be an intense mixing of an otherwise smooth laminar liquid film and greatly improved heat and mass transfer rates. Early research focused mainly on a trial and error search for effective additives without much attention being paid to the fundamental aspects of Marangoni convection. The work of Kashiwagi and his group [2] set off a steady international research trend devoted to a more complete and quantitative understanding of the mechanisms involved in the coupled convective heat and mass transfer processes. Such work continues today, as the absorption community is still not convinced that a comprehensive model for absorption enhancement exists, and indeed because there is still a need to identify new additives for applications other than moderate temperature LiBr–H2O machines.
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One aspect of additive research that has begun to receive more attention focuses clearly on the role of additive transport both in the bulk phases of the absorbing system, as well as between the bulk and the interface. Experimental studies [3] clearly have shown that the lowering of static surface tension of absorbent solution is not, in and of itself, a sufficient condition for the occurrence of the Marangoni instability. The surface tension must be lowered, but the time required for this to happen is critical in determining the ultimate success of an additive in enhancing absorption rates. The liquid film of absorbent solution entering the absorber exposes a fresh interface with the vapor phase. The surfactant species—a dual hydrophobic/ hydrophilic molecule present in either or both of the bulk phases—starts a process of adsorption aimed at achieving equilibrium. In doing so, the concentration of surfactant at the interface increases, thus decreasing the surface tension. While static surface tension data—for which significant discrepancies exist in the literature—provide information regarding the ultimate equilibrium surface tension a given system will reach, one cannot assume that this adsorption equilibrium is achieved instantaneously at the entrance region of the absorber. Rather, one must attempt to quantify the rate at which the surfactant adsorbs to the interface to form what is known as the Gibbs layer, or the surface excess concentration. This rate of dynamic surface tension relaxation is a relatively complex function of several parameters of the system, and in effect determines the rate at which the system is capable of experiencing the Marangoni instability. The time constant of the surface tension relaxation must clearly be much smaller than that of the residence time of the solution flowing in the absorber for a given surfactant to be a successful additive, no matter how low the static surface tension is expected to drop due to its presence. Various models exist to predict the transport of surfactant from the bulk to the interface. Early work in this field assumed a necessary equilibrium between the instantaneous concentrations of surfactant at the interface, , and at the bulk directly adjacent to the interface—the so-called subsurface bulk concentration cAs. This approach is referred to as the diffusion model of adsorption since the diffusivity of the surfactant species in the bulk essentially determines how quickly the additive can migrate from the bulk to the interface. Later approaches attempt to apply a kinetic model of simultaneous adsorption and desorption of surfactant to and from the interface. At equilibrium, the net adsorption rate will be zero and the surface excess layer will remain constant. The purely kinetic model assumes that diffusion in the bulk is much quicker than interfacial kinetics, and bulk concentration is taken to be constant. The mixed model considers kinetic transfer
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from bulk to interface and bulk transport in a coupled problem. The current study quantitatively describes the adsorption process in a system consisting of a static film of LiBr–H2O exposed to water vapor. The fact that the static surface tension of this solution is a function of temperature, salt concentration and surfactant concentration, indicates that the kinetic parameters determining adsorption and desorption are functions of all three of these parameters. Cases of both non-absorbing (isothermal, constant salt concentration) as well as absorbing films are considered in order to illustrate the effects of transient system parameters on the rate of surface tension relaxation. Characteristic values for adsorption rate parameters for the additive 2EHX are derived from best fits by the theoretical rate models to scant data available in the literature for static and dynamic surface tension of this system.
2. Literature Modeling the specific absorption fluid-surfactant systems of interest (LiBr–H2O and 2EHX or 1-octanol, for example) in an investigation of heat and mass transfer enhancement cannot begin without solid data for both static and dynamic surface tension of these systems. Unfortunately, dynamic data is very scarce and quite new and therefore unvalidated [4,5], whereas static data, while more plentiful [6–8], is rather inconsistent, apparently owing to the different methods used to obtain them. Data for surface tension relaxation with water and various higher alcohols, however, is more available, along with various theoretical models of adsorption. An analytical diffusion model of adsorption, which assumes instantaneous transfer from the subsurface layer to the interface, was put forward by Ward and Tordai [9]. Borwankar and Wasan [10] developed a model based on the coupling of penetration theory and a Frumkin isotherm and compared its numerically computed solution to previously published data for various aqueous surfactant systems. Lin et al. [11] considered existing experimental dynamic surface tension data of systems containing heptanol, octanol and decanol in water exposed to air. They fitted these data to several different adsorption models, including a phase transition model, which views the surfactant at the interface as undergoing a transition from a gaseous to a liquid phase as the surface excess concentration is increased. Their comparative study yielded rate constants for the various models and systems. Chang and Franses [12] analyzed data for various surfactants using a modified Langmuir-Hinshelwood equation for adsorption-desorption rate. Of note is the work of MacLeod and Radke [13], as they considered adsorption of decanol at a gas-liquid
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(water) interface where the surfactant is allowed to adsorb exclusively from either the liquid phase or the vapor phase. Vapor side adsorption may be important in light of recent research by the group led by Herold [14–16], which puts forth the Vapor Surfactant Theory and promotes it as the dominant mechanism causing the Marangoni effect in absorption machines. As for models of adsorption of alcohol surfactants in absorption fluids, Kim and Janule [4] produced dynamic surface tension data for LiBr–H2O with 2EHX. However, they assumed a diffusion adsorption model, and apparently under-predicted the diffusivity of the additive in the liquid phase. Kren et al. [5] applied both diffusion and kinetic models to fit data for 2EHX as well as 1-octanol in LiBr–H2O. They clearly show that the diffusion model alone is not sufficient to model the relaxation process. A large body of work exists that deals with various aspects of absorption additives research, but without explicit consideration of additive adsorption mechanisms. The reader is referred to surveys of this literature available in various sources, for example, Ziegler and Grossman [17].
3. Adsorption model An appropriate starting point for a discussion of the static surface tension of a liquid–vapor interface is the Gibbs equation [18] ¼
ð@=@cA Þ cA RT
ð1Þ
Derived solely from thermodynamic considerations of minimum free energy, Eq. (1) predicts the relationship between the bulk cA and surface excess concentrations of surfactant for a system at equilibrium. At the molecular level, equilibrium is characterized not by a lack of activity, but rather by a balance between the rate at which surfactant molecules in the bulk adsorb to the interface, and the rate at which they go from interface to bulk. There is widespread agreement in the literature [10– 13] that, for an isothermal system in which the surfactant is the only transporting species, the rate of adsorption depends on both the bulk concentration as well as the surface excess, while the rate of desorption is a function only of the surface excess @ ¼ k c 1 kd a A @t 1
ð2Þ
The coefficients ka and kd are not necessarily constants, but rather may depend on the surface excess concentration in the form of an exponential adsorption
barrier, as seen in a modified form of the Langmuir– Hinshelwood equation (2) @ ¼ k c 1 exp B k a A d @t 1 1
ð3Þ
The adsorption barrier parameter B is understood to be a manifestation of the interaction among surfactant molecules at the interface due to considerations of molecular structure (e.g. branching) and intermolecular forces. 1 is the maximum value of surface excess, determined primarily by considerations of molecular shape, bonding and the like. At equilibrium, the net rate of adsorption vanishes, giving the following relationship between surface excess and bulk concentrations of surfactant, known as the Langmuir isotherm: cA ¼
1 KL 1
ð4Þ
Other formulae for isotherms are found in the literature, and are derived by assuming different behavior of the coefficients in (2) as a function of . Combining a given isotherm with the Gibbs equation (1) results in a prediction of static surface tension as a function of surface excess or bulk concentration. The Langmuir isotherm (4) yields the following formula ð5Þ ¼ NA þ 1 RTln 1 1 or, alternately, ¼ NA 1 RTlnð1 þ KL cA Þ
ð6Þ
Data of static surface tension at varying surfactant concentrations cA can be fitted with a given isotherm to determine the parameters 1 and KL. However, such curve fits can only yield the ratio of the adsorption and desorption rate constants, not their absolute values. In order to derive the actual values of ka and kd, one must perform a dynamic simulation of the relaxation process, and compare this to data for dynamic surface tension. In this study, simulations of the dynamic relaxation of surface tension of a given additive are performed for two different models of adsorption: the diffusion (or equilibrium) model, and a mixed diffusion-kinetic model. The diffusion model assumes that the rate of adsorption is much greater than the rate at which the surfactant diffuses within the bulk. If this is so, a state of equilibrium can be assumed to exist at any given time during the relaxation, between the surface excess concentration and the value of bulk concentration directly adjacent to the interface, or the so-called sub-surface bulk concentration cAs
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ðtÞ ¼
ð@=@cA Þ cAs ðtÞ RT
ð7Þ
The buildup of the Gibbs layer proceeds as fast as the bulk is able to supply the sub-surface with surfactant species. This supply rate is governed by the (nondimensionalized) 1-D transport equation for a species following Fick’s law of diffusion @A 1 @2 A ¼ @t ScA @y2
ð8Þ
A no-flux boundary condition is applied at the bottom wall of the static film @A ¼ 0 at y ¼ 0 @y
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out the surface tension-driven instability originating from the adsorption and absorption processes at the interface (i.e., in the case of absorption in the absence of additives), the system in question represents the combined cases of heating from above and salting from below. In terms of Rayleigh (buoyancy) instability, this is a stable scenario. It is the reversal of the behavior of surface tension as a function of T and c that makes the system potentially unstable. For additive-free brine, s is reduced at increasing temperatures and water concentrations @ @ < 0; < 0: @TNA @c NA
ð12Þ
ð9Þ
The transient values of cAs and are determined iteratively by comparing the adsorption flux represented by the difference between two equilibrium values of at successive time steps, and the flux to the subsurface from the bulk given by the diffusion equation. Initial conditions are uniform bulk additive concentration and a fresh interface, with no initial Gibbs layer. A ¼ 1 at t ¼ 0 for all y
ð10Þ
¼ 0 at t ¼ 0 1
ð11Þ
A wholly kinetic model would assume that the diffusion in the bulk is much greater than the adsorption rate from the rate equation (3), and would thus assume a constant value of bulk concentration during the process. This model is not considered explicitly in this study. Rather, a mixed diffusion-kinetic model allows for the combined variation of parameters of interfacial kinetics and bulk diffusion, and will show which mechanism is the controlling feature. The mixed model couples the kinetic rate equation (3) with the diffusion equation. The value of cAs will vary as a result of the flux from the bulk, as well as that to the interface, and will not necessarily be in equilibrium with the value of —as given in the Gibbs equation (1)— except at the end of the relaxation process.
4. Semi-coupled adsorption and absorption It is clear from surface tension data and from an understanding of the Marangoni instability that the surface tension in an absorbing solution is a function not only of the surfactant concentration, but also of the temperature and concentration of the brine. This is in fact what causes the system to become unstable. With-
However, for a given value of bulk additive concentration, the static surface tension is increased as the temperature and absorbate concentrations are increased [6,8]. @ @ > 0; >0 @TA @c A
ð13Þ
This can be explained by taking into account the effect of these properties on the balance of additive concentrations between the interface and the bulk. In cold solutions of strong salt concentration, the less soluble additive will have a higher surface excess concentration, thus giving a lower surface tension than in a warmer, more dilute solution. It is of interest, then, to establish the relationship between the adsorption rate parameter ratio (KL) and the temperature and concentration of the solution, which change during the course of the absorption process. The rate of relaxation is expected to be affected by these properties. If this influence manifests itself within the time frame relevant to the flow in an absorber, then the variation of parameters governing the absorption process will have a direct impact on the potential for instability in the system. Relaxation simulations are performed concurrent to stable absorption in the static film. The values of the adsorption rate constants are allowed to vary with the interfacial temperature and absorbate concentrations. The initial conditions are for a liquid film with a vapor pressure lower than that of the absorbate vapor to which it is exposed. The absorption model assumes an equilibrium state at all times at the interface ðy ¼ 1Þ ¼ eq ð ðy ¼ 1Þ; Pv Þ
ð14Þ
The solution is modeled as a linear absorbent where temperature is a linear function of absorbate concentration at a given vapor pressure
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þ ¼1
ð15Þ
This assumption has been used extensively in the literature, and is quite acceptable for the relatively narrow range of values of temperature and concentration over which the absorber works [19–21]. The concurrent modeling of absorption and adsorption is referred to in this study as ‘‘semi-coupled’’, since the influence of the two processes on one another is oneway. The current work does not allow for the development of Marangoni convection due to the adsorption process, but the absorption is allowed to affect the rate of adsorption. In this way, the model is really an indicator of the potential for instability, and does not attempt to model that instability. The equations (non-dimensionalized) governing the absorption process are for one-dimensional heat and mass transfer in the bulk @ 1 @2 ¼ @t Sc @y2 " # @ 1 @2 1 @ 2 @ @ þ þZ2 ¼ Z1 @t Pr @y2 Sc @y @y @y
ð16Þ
ð17Þ
where and are the dimensionless liquid temperature and absorbate concentration, respectively and Z1 and Z2 are parameters of interdiffusion, or heat of mixing, associated with the brine, which is explicitly accounted for in the model. A more detailed discussion of the development of the absorption model, including the interdiffusion terms, can be found in [19]. Non-dimensional terms are defined in the Nomenclature section of this paper. The brine is at an initial uniform temperature and concentration ðt ¼ 0Þ ¼ ðt ¼ 0Þ ¼ 0
ð18Þ
No-flux and constant temperature conditions are applied at the bottom wall ð y ¼ 0Þ ¼ wall
ð19Þ
@ ¼0 @y y¼0
ð20Þ
At the interface ( y ¼ 1), an energy balance yields proportional heat and mass fluxes into the liquid layer @ @ ¼ l ð21Þ @yy¼1 @y y¼1
where l is the dimensionless heat of absorption, assumed to be transferred entirely to the liquid phase. The vapor pressure is kept constant, and the liquid layer heats up at the interface. The wall temperature, together with the film depth and Prandtl number, determines the rate of cooling for the film.
5. Solution method The state variables (, and A) are solved for with a forward-time, central-spaced finite difference scheme on a uniform Cartesian grid. The coupling of interfacial conditions for temperature and concentration requires an iterative solution for each time step. Iteration is also required to solve for additive bulk and surface concentrations in the diffusion model. The solution for and is completely independent of the solution for A, while the solution for A depends on the instantaneous values of and at the interface, as these determine the values of the adsorption rate parameters ka and kd.
6. Results and discussion Data for additive-free surface tension of LiBr–H2O as published in [8] were fitted with polynomial curves in order to facilitate their use in formulae [such as Eq. (6)] for additive-influenced surface tension. Fig. 1 shows the correlation coefficients, and is included here for the use it may serve others modeling this system. In order to establish an approximate correlation for the equilibrium adsorption rate constant (KL) as a function of temperature and solution concentration, each of five sets of data [6] were fitted with a Langmuir isotherm. The value of the maximum surface excess concentration was taken as the average of those found for independent fits of the data sets and was set to 1=7.2E-09 kmol/m2. The sets were then refitted (Fig. 2) with the same value of 1 to yield values of KL. These values are tabulated in Table 1. Static data alone can give only the ratio between adsorption and desorption rate constants (KL ¼ ka =kd 1 ). Dynamic data is needed to reveal absolute values for ka and kd. Fig. 3 shows the data of [5] for dynamic surface tension of LiBr–H2O with 80 ppm of 2EHX at a published solution state of 53% and ‘‘room temperature’’. Runs of kinetic-diffusion model simulations of -relaxation with varying values of kd as inputs were compared to the data. A value of 5 for the adsorption barrier B was found, and is in agreement with that of [5]. Values of ka were then computed as per the value of KL corresponding to the static data fit (Table 1) at the solution conditions used in the experiment. It can be seen that a figure of about 0.1 (s1) is an approximate value for kd, while it is understood that
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Fig. 1. Polynomial curve fit of surface tension of additive-free LiBr–H2O. From data of [8].
this is only a rough estimate, based on the scant data available. In any case, a precise determination of this parameter is not the goal of this study, and a reasonable value for the system at hand can be inferred from the data. 6.1. Relaxation Simulations without Absorption An abbreviated presentation of the results of nonabsorbing relaxation simulations is in order, as this has been treated in similar systems by others (see ‘‘Literature’’ section). Fig. 4 illustrates the fundamental difference between the diffusion and mixed diffusion-kinetic models. One can see that for the diffusion model, the requirement for additive equilibrium between the subsurface (cAs) and interfacial () layers creates the situation in which the subsurface layer is initially starved of additive by the interface, which is additive-free at time equals zero. The diffusion process in the bulk requires some time to replenish the subsurface. Ultimately, this replenishment and concurrent buildup of the Gibbs layer occur rapidly. The time at which this occurs depends directly on the bulk diffusivity of the additive. The mixed model, on the other hand, allows for a non-equilibrium buildup of the surface excess layer as dictated by the Langmuir-Hinshelwood rate equation (3). Until the additive adsorption ‘‘takes off’’, the subsurface layer is unaffected. When begins to grow, one can see a moderate dip in cAs indicative of a partial depletion of bulk additive during the period of high bulk-to-interface flux. From Fig. 4 one can see that the magnitude of this depletion depends in part on the relative rate of adsorption as determined by the rate barrier B. The subsurface layer recovers and tends to a value slightly lower than the initial bulk concentration, due to the fact that the interface has robbed the bulk of some of its initial inventory. In the purely kinetic model, the subsurface layer would not be affected at all, and the
Fig. 2. Fits of static surface tension data for LiBr–H2O with 2EHX, using Langmuir isotherms. Top: effect of brine concentration on the Langmuir constant. Bottom: effect of solution temperature. Data of [6].
Table 1 Values of the Langmuir constant KL [1/ppm] for different solution temperatures and concentrations T\CLIBR
40%
50%
60%
24 C 48 C
0.02728
0.08039 0.03399
0.26127 0.06475
Derived from data of [6].
entire bulk would remain at a uniform additive concentration. Fig. 5 illustrates adsorption simulated with the kineticdiffusion model with different initial bulk concentrations cAo and diffusivities DA. For two consecutive orders of magnitude (DA ffi 2:0E-9 m2/s at ScA ¼ 104 and 2.0E-10 at ScA ¼ 105 ) a lower additive diffusivity is seen to decrease the relaxation rate very slightly, as the bulk transport is slower in replenishing the subsurface layer. Fig. 5(b) shows lower dips in this concentration for the lower diffusivity case. For all values of cAo considered, the different values of static surface tension are reached at about the same time. So it can be seen that the mixed
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Fig. 3. Fit of dynamic surface tension data of LiBr–H2O with 2EHX [5] with mixed diffusion-kinetic model using LangmuirHinshelwood rate equation. Adsorption barrier B=5. Langmuir constant KL=0.11 [1/ppm].
Fig. 4. Profiles of additive concentration: (1) subsurface bulk cAs/cA1, (2) surface excess /1. Diffusion and mixed diffusionkinetic models with different adsorption rate barriers.
model is closer to a purely kinetic model than a diffusion model, with a slight effect due to the bulk transport properties of the additive. Indeed, only the mixed model with an adsorption rate barrier is seen to behave in accordance with the data. 6.2. Relaxation simulations with absorption A number of runs were performed simulating concurrent processes of surfactant adsorption and vapor absorption. The Langmuir–Hinshelwood rate equation employed to model the flux of additive from the subsurface to the interface used parameters which were updated each time step to account for changing temperature and absorbate concentration at the interface. When initial brine concentration was allowed to vary, it was found that the change in relaxation profile was negligible. The equilibrium condition for the brine at a given temperature and vapor pressure immediately yields nearly identical interfacial absorbate concentrations, despite the different bulk concentrations (Fig. 6a). This can be understood by considering that bulk temperature, as opposed to concentration, is the dominant property in determining the interfacial equilibrium state of the brine, due to the fact that energy diffusion is much quicker than mass diffusion in LiBr–H2O. The concentration sustains a sharp gradient from the interface into the film, as opposed to temperature, which communicates its bulk value much more quickly to the interface. In terms of the characteristic parameters of the absorption system, this is seen in the high Schmidt number and low Prandtl number. The Lewis number (Le=Pr/Sc) for LiBr–H2O at the states in question is on the order of 0.025. Indeed, it is this higher concentration gradient across the film that causes the stronger solution to absorb vapor more quickly. However, the very similar
Fig. 5. Relaxation profiles of surface tension (top) and subsurface additive concentration (bottom). Effect of initial bulk concentration and additive diffusivity.
interfacial states at the early stages of the process for the different initial concentrations gave identical surface tension behavior during the time in question, and therefore the potential for instability is not affected by initial solution concentration.
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Fig. 7. Surface tension relaxation profiles for different initial solution temperatures (at 53% LiBr).
Fig. 6. Profiles of the interfacial brine state for varying (a) initial concentration (at 30 C) and (b) temperature (at 53% LiBr).
On the other hand, variation of initial solution temperature causes an initial variation in surface tension before any adsorption has even taken place (Figs. 6b and 7). The increased initial interfacial temperature dictated by a higher initial bulk temperature gives a lower initial water concentration in order to satisfy the brinevapor equilibrium requirement. While surface tension is expected to decrease with increasing temperature for the additive-free interface, the lowered absorbate concentration (i.e. high salt content) actually causes an initial positive jump in the pre-adsorption value of (Fig. 7). Eventually, by about half a second, is primarily influenced by the growing additive surface excess concentration, since the interfacial states of the brine tend to converge at about this time to the same wall temperature for the different runs. Here, it can be inferred that the potential for instability is increased somewhat for a solution having a higher initial temperature, even though this solution is less hygroscopic than colder ones, and is expected to absorb more slowly in the stable regime. Turning attention to other factors—external to the bulk solution—that influence the driving force for absorption, one can consider the effects of wall temperature (i.e. cooling flux) and evaporator temperature
Fig. 8. Effect of cooling wall temperature on rates of vapor absorption and additive adsorption. Temperatures and water concentrations are interfacial values. Initial parameters: LiBr conc.=58%, solution temp.=30 C, absorbate vapor pressure=1.228 kPa, additive (2EHX) conc.=80 ppm.
(i.e. vapor phase pressure) on the -relaxation profile. Fig. 8 shows the influence of wall temperature on the rates of absorption and on the behavior of the interface, in terms of brine state and surfactant adsorption. One can see that in each case, the solution approaches the constant wall temperature. The solution on the colder wall will approach a higher water concentration at its cooler interface. The lower excess surfactant concentration (the surfactant is more soluble in a subsurface solution with lower salt concentration) in this case gives a higher equilibrium value of . However, the adsorption equilibrium will be reached more quickly due to the higher heat flux of the cooler wall. Granted, for the
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In this sense, the potential for instability is increased with the driving force, as was the case with the wall temperature study.
7. Conclusions
Fig. 9. Effect of vapor phase pressure on rates of vapor absorption and additive adsorption. Pressure correspond to evaporator temperatures of 20, 25 and 30 C. Temperatures and water concentrations are interfacial values. Initial parameters: LiBr conc.=53%, solution temp.=30 C, Cooling wall temperature=20 C, Additive (2EHX) conc.=80 ppm.
temperature range of 20 to 30 C for the cooling wall, which is representative for water from a cooling tower, the difference in surface tension at a given time for the different runs is not great—and it cannot be stated here whether this will increase the rate or intensity of instability in the system. However, this result is qualitatively in line with the results of Kulankara et al. [14]. They conducted water condensation experiments on a cold tube with and without additives. It was found that for colder tubes, the mass transfer coefficient was increased, and they took this to be a sign of enhancement due to the presence of the additive in the vapor phase. Assuming vapor additive transport to be similar to adsorption from the liquid bulk, then this agreement makes sense. As far as is known, no reliable data is available for dynamic surface tension of absorption fluids with the surfactant originating from the vapor side. However, the work of MacLeod and Radke [13] with aqueous decanol solutions showed similar profiles and relaxation times for liquid- and vapor-side adsorption experiments. The driving force for absorption can also be increased by increasing the vapor phase pressure. Fig. 9 shows the influence of this parameter on -relaxation. It can be seen that the higher the vapor pressure, the larger the initial negative jump to the value of s corresponding to the immediate (pre-adsorption) equilibrium state of the brine at the interface. Even if this jump is not ‘‘immediate’’, it is still evident that the relaxation of surface tension is quicker for the higher vapor pressure.
Simulations of surface tension relaxation were performed in cases of both non-absorbing and absorbing static films of LiBr–H2O in the presence of surfactant additive, with adsorption properties characteristic of 2-ethyl-1-hexanol being used. Langmuir isotherm constants (KL) expressing the ratio of adsorption to desorption parameters in the Langmuir–Hinshelwood rate equation (3) were derived from static surface tension data taken from [6]. This equilibrium is a function of temperature and brine concentration, and reflects the variation of additive solubility with the solution state. Representative absolute values of ka and kd were derived from dynamic surface tension data of [5]. It is evident that a mixed kinetic-diffusion model with an adsorption barrier of about 5 fits experimental data of both static and dynamic surface tension of the system in question. A diffusion model alone gives a relaxation profile which is too steep and occurs too early for reasonable diffusion coefficients. Kim and Janule [4] derived a value of DA on the order of 3.0E-11 m2/s by fitting their dynamic data with a diffusion model. This value is an order of magnitude smaller than estimates based on a mixed model [5] and used in this study. The effects of initial additive concentration in the bulk, additive diffusivity and kinetic parameters on adsorption rate were illustrated. A semi-coupled model of concurrent additive adsorption and stable vapor absorption was used to simulate the effects of the absorption process on the surface tension relaxation rate. The behavior of the liquid–vapor interface was seen to be influenced by four of the factors that determine the driving force for absorption: initial solution concentration and temperature, wall temperature (cooling side heat flux) and vapor phase pressure. Variations in brine concentration in the range of 50– 58% LiBr did not affect the relaxation profiles. Temperature variations in the range of 20–30 C yielded slightly quicker relaxation times for warmer—or less hygroscopic—solutions. For both external factors of wall temperature and vapor phase pressure, relaxation rates were seen to increase for the cases of higher absorption driving force. In terms of wall temperature, this was so for the colder wall, or the higher heat flux case. In terms of vapor pressure, it was so for the higher vapor pressure, or for the case of a higher temperature evaporator. This result provides cursory corroboration to the experimental water condensation data of [14] in which mass transfer coefficients increased with higher
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condensation heat fluxes in the presence of vapor-side additives. The manifestation of the -relaxation in the form of the Marangoni instability is not modeled in this study. It would be of interest to further the model of Marangoni convection in a static absorber [19] using the results of the current study as a basis for predicting the dynamic surface tension of the system. The previous model used constant Marangoni numbers, in effect making invariant the gradients of surface tension against temperature, absorbate concentration and additive concentration. The merging of these two works may allow for a simulation of the entrance region of an absorber. It is understood that data [4–8] used to derive adsorption parameters such as those in Table 1 of this paper must be taken in their proper context, based on the limited information available as to the experimental methods used to obtain them. The additive concentrations reported for these data are proposed to be liquid concentrations, and are thus viewed in the context of this paper. However, they more accurately represent system concentration, since it seems that no efforts were made to isolate the additive in the liquid phase. Vapor additive concentrations were seemingly not accounted for in obtaining the data. There is, therefore, a need for dynamic surface tension experimentation for absorption fluids laced with additives in the vapor phase. Such data can allow for the calculation of rate constants for this manner of adsorption, and may allow for further enlightenment regarding the vapor surfactant theory put forth by [14]. The vapor surfactant model of Herold’s group is recognized as a promising theory explaining the origin of the additive causing the Marangoni instability in LiBr–H2O systems, and further data is warranted to model the behavior of the additive in its transport between the vapor phase and the absorbing liquid phase.
References [1] Bourne JR, Eisberg KV. Maintaining the effectiveness of an additive in absorption refrigeration systems. US Patent No. 3276217, 1966. [2] Kashiwagi T, Kurosaki Y, Shishido H. Enhancement of vapor absorption into a solution using the marangoni effect. Nihon Kikai Gakkai Ronbunshu, 463B, 1985 [Translated for ORNL]. [3] Beutler A, Greiter I, Wagner A, Hoffmann L, Schreier S, Alefeld G. Surfactants and fluid properties. International Journal of Refrigeration 1996;19:342–6. [4] Kim KJ, Janule VP. Dynamic surface tension of aqueous lithium bromide with 2-ethyl-1-hexanol. International Communications in Heat and Mass Transfer 1994;23:907–15. [5] Kren Ch, Hellmann H-M, Ziegler F. Dynamic surface
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tension of libr-solutions with higher alcohols as heat transfer additives, Proceedings of the International Sorption Heat Pump Conference, Munich; 1999. pp. 375–80. Kim KJ, Berman NS, Wood BD. Surface tension of aqueous lithium bromide plus 2-ethyl-1-hexanol. Journal of Chemical and Engineering Data 1994;39:122–4. Ishida K, Mori YH. Surface tension of aqueous lithium bromide solutions containing 1-octanol as a heat transfer additive. International Communications in Heat and Mass Transfer 1996;23:907–15. Yao W, Bjurstrom H, Setterwall F. Surface tension of lithium bromide solutions with heat-transfer additives. Journal of Chemical and Engineering Data 1991;36:96–8. Ward AFH, Tordai L. Time dependence of boundary tensions of solutions. I. The role of diffusion in timeeffects. Journal of Chemical Physics 1946;14:453–61. Borwankar RP, Wasan DT. The kinetics of adsorption of surface active agents at gas-liquid surfaces. Chemical Engineering Science 1983;38:1637–49. Lin SY, McKeigue K, Maldarelli C. Diffusion-limited interpretation of the induction period in the relaxation in surface tension due to the adsorption of straight chain, small polar group surfactants: theory and experiment. Langmuir 1991;7:1055–66. Chang CH, Franses EI. Modified Langmuir–Hinshelwood kinetics for dynamic adsorption of surfactants at the air/water interface. Colloids and Surfaces 1992;69: 189–201. MacLeod CA, Radke CJ. Surfactant exchange kinetics at the air/water interface for dynamic tension of growing liquid drops. Journal of Colloid and Interface Science 1994;166:73–88. Kulankara S, Verma S, Herold KE. Theory of heat/mass transfer additives in absorption chillers. Proceedings of the ASME Advanced Energy Systems Division, 1999;39:199– 206. Yuan Z, Herold KE. Surface tension of aqueous lithium bromide with controlled vapor concentration of 2-ethylhexanol. ASHRAE AT-01–4-2. Kulankara S. Effect of enhancement additives on the absorption of water vapor by aqueous lithium bromide. PhD d#issertation, Department of Mechanical Engineering, University of Maryland, 1999. Ziegler F, Grossman G. Review Paper: Heat-transfer Enhancement by Additives. International Journal of Refrigeration 1996;19:301–9. Davies, Rideal. Interfacial phenomena. Academic Press, 1961. Koenig MS, Grossman G, Gommed K. Additive-induced enhancement of heat and mass transfer in a static absorber: a numerical study Proceedings of the International Sorption Heat Pump Conference, Munich; 1999. pp. 359– 66. Grossman G. Simultaneous heat and mass transfer in film absorption under laminar flow. International Journal of Heat and Mass Transfer 1983;26:357–71. Ji W, Setterwall F. Effect of heat transfer additives on the instabilities of an absorbing falling film. Chemical Engineering Science 1995;50:3077–97.
The role of surfactant adsorption rate in heat and mass transfer enhancement in absorption heat pumps Michael S. Koenig, Gershon Grossman*, Khaled Gommed Faculty of Mechanical Engineering, Technion, Israel Institute of Technology, Haifa, 32000 Israel Received 23 February 2001; received in revised form 21 December 2001; accepted 4 February 2002
Abstract The importance of heat and mass transfer additives in absorption chillers and heat pumps has been recognized for over three decades. However, a universally accepted model for the mechanisms responsible for enhanced absorption rates has yet to be proposed. The Marangoni effect—an instability arising from gradients in surface tension at the liquid-vapor interface—is generally accepted as the cause of the convective flows that enhance transfer rates. Certain surfactant additives can significantly improve absorption rates and thus reduce the overall transfer area required by a given machine. Any means available that can increase the efficiency and acceptability of absorption machines is to be welcomed, as this technology provides an alternative to vapor compression systems which is both environmentally friendly and more versatile with regards to energy sources. This study investigates the rate at which a surfactant additive adsorbs at a liquid-vapor interface. The residence time of the falling liquid solution in an absorber is quite short. An effective additive must not only reduce the surface tension of the solution; it must do so quickly enough to cause the Marangoni instability within the short absorption process time. The entrance region of an absorber features a freshly exposed interface at which no surfactant has adsorbed. A numerical model is used to analyze surfactant relaxation rates in a static film of additive-laced solution. Kinetic parameters for the combination of the working pair LiBr-H2O and the additive 2-ethyl-1-hexanol are derived from data in the literature for static and dynamic surface tension measurements. Bulk, interfacial and boundary parameters influencing relaxation rates are discussed for surfactant adsorption occurring in the absence of absorption, as well as for concurrent adsorption and stable vapor absorption. Initial solution conditions and absorption driving force are shown to impact the potential for instability in the effect they have on the rate of interfacial additive adsorption. # 2002 Elsevier Science Ltd and IIR. All rights reserved. Keywords: Heat pump; Adsorption system; Absorber; Surfactant; Mass transfer; Heat transfer
Le roˆle des agents tensio-actifs sur la vitesse d’adsorption et l’ame´lioration des transferts de chaleur et de masse dans les pompes a` chaleur Mots cle´s : Pompe a` chaleur ; Syste`me a` absorption ; Absorbeur ; Agent tensioactif ; Transfert de masse ; Transfert de chaleur
* Corresponding author. Tel.: +972-4-829-2074; fax: +972-4-832-4533. E-mail address: [email protected] (G. Grossman). 0140-7007/03/$20.00 # 2002 Elsevier Science Ltd and IIR. All rights reserved. PII: S0140-7007(02)00012-9
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Nomenclature B c cA d D Ha k ka, kd KL P Pr Sc t* t T y y Z1, Z2 2EHX
adsorption rate barrier, Eq. (3) (dimensionless) absorbate concentration (kmol/m3) additive (bulk) concentration (kmol/m3) film thickness (m) mass diffusion coefficient (m2/s) heat of absorption (J/kmol) thermal conductivity (W/m C) rate coefficients: ka—adsorption, kd— desorption (m/s, s1) Langmuir isotherm constant, ka/kd1 (m3/kmol) pressure (kPa) Prandtl number, / (dimensionless) Schmidt number, /D (dimensionless) time (s) normalized time, t/(d2/) (dimensionless) temperature ( C) vertical coordinate in film (m) normalized vertical coordinate, y/d (dimensionless) interdiffusion parameters, Eq. (17) (dimensionless) 2-ethyl-1-hexanol
1. Introduction Heat and mass transfer additives have been used in certain absorption chillers and heat pumps for over three decades. The maintenance of quite low concentrations of such additives in machines using closed cycles of lithium bromide–water (LiBr–H2O) as a working pair, affords significantly improved performance of the absorber, and thus of the entire machine, as compared to operation in the absence of additives. Various higher alcohols are known to be good additives for cycles operating in moderate temperature ranges, the most prominent among them being 2-ethyl-1-hexanol (2EHX), an isomer of n-octanol, which is itself also a relatively good additive. However, not all absorption machines benefit from the effects of additives. Ammonia–water absorbers do not have convincing counterparts to the alcohols used in LiBr–H2O absorbers. Likewise, the development of advanced absorption cycles with higher operating temperatures requires a search for additives with greater chemical stability. The search for effective additives for a wide variety of applications suffered—until the latter part of the 1980s—from a lack of understanding of the mechanisms responsible for the absorption rate enhancement. Soon
Greek symbols thermal diffusivity (m2/s) normalized concentration: (dimensionless) Absorbate=(cco)/(ceco) Additive=cA/cA1 Additive surface excess (interfacial) concentration (kmol/m2) solution temperature=(TTo)/(TeTo) (dimensionless) l normalized heat of absorption, HaD(ceco)/ k(TeTo) (dimensionless) kinematic viscosity (m2/s) surface tension (N/m) Subscripts a adsorption A additive d desorption L Langmuir 1 reference or maximum value o initial value NA no additive s subsurface v vapor wall absorber cooling wall LiBr lithium bromide e, eq equilibrium
after the accidental discovery of these additives in the mid 1960s [1]—a welcome phenomenon experienced while searching for antifoaming agents—it was widely and correctly assumed that improved absorption rates were attributable to the surface active nature of the alcohols in question. The additives were understood to cause the Marangoni effect, a hydrodynamic instability at the liquid–vapor interface, brought on by gradients in surface tension. The result of this instability can be an intense mixing of an otherwise smooth laminar liquid film and greatly improved heat and mass transfer rates. Early research focused mainly on a trial and error search for effective additives without much attention being paid to the fundamental aspects of Marangoni convection. The work of Kashiwagi and his group [2] set off a steady international research trend devoted to a more complete and quantitative understanding of the mechanisms involved in the coupled convective heat and mass transfer processes. Such work continues today, as the absorption community is still not convinced that a comprehensive model for absorption enhancement exists, and indeed because there is still a need to identify new additives for applications other than moderate temperature LiBr–H2O machines.
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One aspect of additive research that has begun to receive more attention focuses clearly on the role of additive transport both in the bulk phases of the absorbing system, as well as between the bulk and the interface. Experimental studies [3] clearly have shown that the lowering of static surface tension of absorbent solution is not, in and of itself, a sufficient condition for the occurrence of the Marangoni instability. The surface tension must be lowered, but the time required for this to happen is critical in determining the ultimate success of an additive in enhancing absorption rates. The liquid film of absorbent solution entering the absorber exposes a fresh interface with the vapor phase. The surfactant species—a dual hydrophobic/ hydrophilic molecule present in either or both of the bulk phases—starts a process of adsorption aimed at achieving equilibrium. In doing so, the concentration of surfactant at the interface increases, thus decreasing the surface tension. While static surface tension data—for which significant discrepancies exist in the literature—provide information regarding the ultimate equilibrium surface tension a given system will reach, one cannot assume that this adsorption equilibrium is achieved instantaneously at the entrance region of the absorber. Rather, one must attempt to quantify the rate at which the surfactant adsorbs to the interface to form what is known as the Gibbs layer, or the surface excess concentration. This rate of dynamic surface tension relaxation is a relatively complex function of several parameters of the system, and in effect determines the rate at which the system is capable of experiencing the Marangoni instability. The time constant of the surface tension relaxation must clearly be much smaller than that of the residence time of the solution flowing in the absorber for a given surfactant to be a successful additive, no matter how low the static surface tension is expected to drop due to its presence. Various models exist to predict the transport of surfactant from the bulk to the interface. Early work in this field assumed a necessary equilibrium between the instantaneous concentrations of surfactant at the interface, , and at the bulk directly adjacent to the interface—the so-called subsurface bulk concentration cAs. This approach is referred to as the diffusion model of adsorption since the diffusivity of the surfactant species in the bulk essentially determines how quickly the additive can migrate from the bulk to the interface. Later approaches attempt to apply a kinetic model of simultaneous adsorption and desorption of surfactant to and from the interface. At equilibrium, the net adsorption rate will be zero and the surface excess layer will remain constant. The purely kinetic model assumes that diffusion in the bulk is much quicker than interfacial kinetics, and bulk concentration is taken to be constant. The mixed model considers kinetic transfer
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from bulk to interface and bulk transport in a coupled problem. The current study quantitatively describes the adsorption process in a system consisting of a static film of LiBr–H2O exposed to water vapor. The fact that the static surface tension of this solution is a function of temperature, salt concentration and surfactant concentration, indicates that the kinetic parameters determining adsorption and desorption are functions of all three of these parameters. Cases of both non-absorbing (isothermal, constant salt concentration) as well as absorbing films are considered in order to illustrate the effects of transient system parameters on the rate of surface tension relaxation. Characteristic values for adsorption rate parameters for the additive 2EHX are derived from best fits by the theoretical rate models to scant data available in the literature for static and dynamic surface tension of this system.
2. Literature Modeling the specific absorption fluid-surfactant systems of interest (LiBr–H2O and 2EHX or 1-octanol, for example) in an investigation of heat and mass transfer enhancement cannot begin without solid data for both static and dynamic surface tension of these systems. Unfortunately, dynamic data is very scarce and quite new and therefore unvalidated [4,5], whereas static data, while more plentiful [6–8], is rather inconsistent, apparently owing to the different methods used to obtain them. Data for surface tension relaxation with water and various higher alcohols, however, is more available, along with various theoretical models of adsorption. An analytical diffusion model of adsorption, which assumes instantaneous transfer from the subsurface layer to the interface, was put forward by Ward and Tordai [9]. Borwankar and Wasan [10] developed a model based on the coupling of penetration theory and a Frumkin isotherm and compared its numerically computed solution to previously published data for various aqueous surfactant systems. Lin et al. [11] considered existing experimental dynamic surface tension data of systems containing heptanol, octanol and decanol in water exposed to air. They fitted these data to several different adsorption models, including a phase transition model, which views the surfactant at the interface as undergoing a transition from a gaseous to a liquid phase as the surface excess concentration is increased. Their comparative study yielded rate constants for the various models and systems. Chang and Franses [12] analyzed data for various surfactants using a modified Langmuir-Hinshelwood equation for adsorption-desorption rate. Of note is the work of MacLeod and Radke [13], as they considered adsorption of decanol at a gas-liquid
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(water) interface where the surfactant is allowed to adsorb exclusively from either the liquid phase or the vapor phase. Vapor side adsorption may be important in light of recent research by the group led by Herold [14–16], which puts forth the Vapor Surfactant Theory and promotes it as the dominant mechanism causing the Marangoni effect in absorption machines. As for models of adsorption of alcohol surfactants in absorption fluids, Kim and Janule [4] produced dynamic surface tension data for LiBr–H2O with 2EHX. However, they assumed a diffusion adsorption model, and apparently under-predicted the diffusivity of the additive in the liquid phase. Kren et al. [5] applied both diffusion and kinetic models to fit data for 2EHX as well as 1-octanol in LiBr–H2O. They clearly show that the diffusion model alone is not sufficient to model the relaxation process. A large body of work exists that deals with various aspects of absorption additives research, but without explicit consideration of additive adsorption mechanisms. The reader is referred to surveys of this literature available in various sources, for example, Ziegler and Grossman [17].
3. Adsorption model An appropriate starting point for a discussion of the static surface tension of a liquid–vapor interface is the Gibbs equation [18] ¼
ð@=@cA Þ cA RT
ð1Þ
Derived solely from thermodynamic considerations of minimum free energy, Eq. (1) predicts the relationship between the bulk cA and surface excess concentrations of surfactant for a system at equilibrium. At the molecular level, equilibrium is characterized not by a lack of activity, but rather by a balance between the rate at which surfactant molecules in the bulk adsorb to the interface, and the rate at which they go from interface to bulk. There is widespread agreement in the literature [10– 13] that, for an isothermal system in which the surfactant is the only transporting species, the rate of adsorption depends on both the bulk concentration as well as the surface excess, while the rate of desorption is a function only of the surface excess @ ¼ k c 1 kd a A @t 1
ð2Þ
The coefficients ka and kd are not necessarily constants, but rather may depend on the surface excess concentration in the form of an exponential adsorption
barrier, as seen in a modified form of the Langmuir– Hinshelwood equation (2) @ ¼ k c 1 exp B k a A d @t 1 1
ð3Þ
The adsorption barrier parameter B is understood to be a manifestation of the interaction among surfactant molecules at the interface due to considerations of molecular structure (e.g. branching) and intermolecular forces. 1 is the maximum value of surface excess, determined primarily by considerations of molecular shape, bonding and the like. At equilibrium, the net rate of adsorption vanishes, giving the following relationship between surface excess and bulk concentrations of surfactant, known as the Langmuir isotherm: cA ¼
1 KL 1
ð4Þ
Other formulae for isotherms are found in the literature, and are derived by assuming different behavior of the coefficients in (2) as a function of . Combining a given isotherm with the Gibbs equation (1) results in a prediction of static surface tension as a function of surface excess or bulk concentration. The Langmuir isotherm (4) yields the following formula ð5Þ ¼ NA þ 1 RTln 1 1 or, alternately, ¼ NA 1 RTlnð1 þ KL cA Þ
ð6Þ
Data of static surface tension at varying surfactant concentrations cA can be fitted with a given isotherm to determine the parameters 1 and KL. However, such curve fits can only yield the ratio of the adsorption and desorption rate constants, not their absolute values. In order to derive the actual values of ka and kd, one must perform a dynamic simulation of the relaxation process, and compare this to data for dynamic surface tension. In this study, simulations of the dynamic relaxation of surface tension of a given additive are performed for two different models of adsorption: the diffusion (or equilibrium) model, and a mixed diffusion-kinetic model. The diffusion model assumes that the rate of adsorption is much greater than the rate at which the surfactant diffuses within the bulk. If this is so, a state of equilibrium can be assumed to exist at any given time during the relaxation, between the surface excess concentration and the value of bulk concentration directly adjacent to the interface, or the so-called sub-surface bulk concentration cAs
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ðtÞ ¼
ð@=@cA Þ cAs ðtÞ RT
ð7Þ
The buildup of the Gibbs layer proceeds as fast as the bulk is able to supply the sub-surface with surfactant species. This supply rate is governed by the (nondimensionalized) 1-D transport equation for a species following Fick’s law of diffusion @A 1 @2 A ¼ @t ScA @y2
ð8Þ
A no-flux boundary condition is applied at the bottom wall of the static film @A ¼ 0 at y ¼ 0 @y
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out the surface tension-driven instability originating from the adsorption and absorption processes at the interface (i.e., in the case of absorption in the absence of additives), the system in question represents the combined cases of heating from above and salting from below. In terms of Rayleigh (buoyancy) instability, this is a stable scenario. It is the reversal of the behavior of surface tension as a function of T and c that makes the system potentially unstable. For additive-free brine, s is reduced at increasing temperatures and water concentrations @ @ < 0; < 0: @TNA @c NA
ð12Þ
ð9Þ
The transient values of cAs and are determined iteratively by comparing the adsorption flux represented by the difference between two equilibrium values of at successive time steps, and the flux to the subsurface from the bulk given by the diffusion equation. Initial conditions are uniform bulk additive concentration and a fresh interface, with no initial Gibbs layer. A ¼ 1 at t ¼ 0 for all y
ð10Þ
¼ 0 at t ¼ 0 1
ð11Þ
A wholly kinetic model would assume that the diffusion in the bulk is much greater than the adsorption rate from the rate equation (3), and would thus assume a constant value of bulk concentration during the process. This model is not considered explicitly in this study. Rather, a mixed diffusion-kinetic model allows for the combined variation of parameters of interfacial kinetics and bulk diffusion, and will show which mechanism is the controlling feature. The mixed model couples the kinetic rate equation (3) with the diffusion equation. The value of cAs will vary as a result of the flux from the bulk, as well as that to the interface, and will not necessarily be in equilibrium with the value of —as given in the Gibbs equation (1)— except at the end of the relaxation process.
4. Semi-coupled adsorption and absorption It is clear from surface tension data and from an understanding of the Marangoni instability that the surface tension in an absorbing solution is a function not only of the surfactant concentration, but also of the temperature and concentration of the brine. This is in fact what causes the system to become unstable. With-
However, for a given value of bulk additive concentration, the static surface tension is increased as the temperature and absorbate concentrations are increased [6,8]. @ @ > 0; >0 @TA @c A
ð13Þ
This can be explained by taking into account the effect of these properties on the balance of additive concentrations between the interface and the bulk. In cold solutions of strong salt concentration, the less soluble additive will have a higher surface excess concentration, thus giving a lower surface tension than in a warmer, more dilute solution. It is of interest, then, to establish the relationship between the adsorption rate parameter ratio (KL) and the temperature and concentration of the solution, which change during the course of the absorption process. The rate of relaxation is expected to be affected by these properties. If this influence manifests itself within the time frame relevant to the flow in an absorber, then the variation of parameters governing the absorption process will have a direct impact on the potential for instability in the system. Relaxation simulations are performed concurrent to stable absorption in the static film. The values of the adsorption rate constants are allowed to vary with the interfacial temperature and absorbate concentrations. The initial conditions are for a liquid film with a vapor pressure lower than that of the absorbate vapor to which it is exposed. The absorption model assumes an equilibrium state at all times at the interface ðy ¼ 1Þ ¼ eq ð ðy ¼ 1Þ; Pv Þ
ð14Þ
The solution is modeled as a linear absorbent where temperature is a linear function of absorbate concentration at a given vapor pressure
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þ ¼1
ð15Þ
This assumption has been used extensively in the literature, and is quite acceptable for the relatively narrow range of values of temperature and concentration over which the absorber works [19–21]. The concurrent modeling of absorption and adsorption is referred to in this study as ‘‘semi-coupled’’, since the influence of the two processes on one another is oneway. The current work does not allow for the development of Marangoni convection due to the adsorption process, but the absorption is allowed to affect the rate of adsorption. In this way, the model is really an indicator of the potential for instability, and does not attempt to model that instability. The equations (non-dimensionalized) governing the absorption process are for one-dimensional heat and mass transfer in the bulk @ 1 @2 ¼ @t Sc @y2 " # @ 1 @2 1 @ 2 @ @ þ þZ2 ¼ Z1 @t Pr @y2 Sc @y @y @y
ð16Þ
ð17Þ
where and are the dimensionless liquid temperature and absorbate concentration, respectively and Z1 and Z2 are parameters of interdiffusion, or heat of mixing, associated with the brine, which is explicitly accounted for in the model. A more detailed discussion of the development of the absorption model, including the interdiffusion terms, can be found in [19]. Non-dimensional terms are defined in the Nomenclature section of this paper. The brine is at an initial uniform temperature and concentration ðt ¼ 0Þ ¼ ðt ¼ 0Þ ¼ 0
ð18Þ
No-flux and constant temperature conditions are applied at the bottom wall ð y ¼ 0Þ ¼ wall
ð19Þ
@ ¼0 @y y¼0
ð20Þ
At the interface ( y ¼ 1), an energy balance yields proportional heat and mass fluxes into the liquid layer @ @ ¼ l ð21Þ @yy¼1 @y y¼1
where l is the dimensionless heat of absorption, assumed to be transferred entirely to the liquid phase. The vapor pressure is kept constant, and the liquid layer heats up at the interface. The wall temperature, together with the film depth and Prandtl number, determines the rate of cooling for the film.
5. Solution method The state variables (, and A) are solved for with a forward-time, central-spaced finite difference scheme on a uniform Cartesian grid. The coupling of interfacial conditions for temperature and concentration requires an iterative solution for each time step. Iteration is also required to solve for additive bulk and surface concentrations in the diffusion model. The solution for and is completely independent of the solution for A, while the solution for A depends on the instantaneous values of and at the interface, as these determine the values of the adsorption rate parameters ka and kd.
6. Results and discussion Data for additive-free surface tension of LiBr–H2O as published in [8] were fitted with polynomial curves in order to facilitate their use in formulae [such as Eq. (6)] for additive-influenced surface tension. Fig. 1 shows the correlation coefficients, and is included here for the use it may serve others modeling this system. In order to establish an approximate correlation for the equilibrium adsorption rate constant (KL) as a function of temperature and solution concentration, each of five sets of data [6] were fitted with a Langmuir isotherm. The value of the maximum surface excess concentration was taken as the average of those found for independent fits of the data sets and was set to 1=7.2E-09 kmol/m2. The sets were then refitted (Fig. 2) with the same value of 1 to yield values of KL. These values are tabulated in Table 1. Static data alone can give only the ratio between adsorption and desorption rate constants (KL ¼ ka =kd 1 ). Dynamic data is needed to reveal absolute values for ka and kd. Fig. 3 shows the data of [5] for dynamic surface tension of LiBr–H2O with 80 ppm of 2EHX at a published solution state of 53% and ‘‘room temperature’’. Runs of kinetic-diffusion model simulations of -relaxation with varying values of kd as inputs were compared to the data. A value of 5 for the adsorption barrier B was found, and is in agreement with that of [5]. Values of ka were then computed as per the value of KL corresponding to the static data fit (Table 1) at the solution conditions used in the experiment. It can be seen that a figure of about 0.1 (s1) is an approximate value for kd, while it is understood that
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Fig. 1. Polynomial curve fit of surface tension of additive-free LiBr–H2O. From data of [8].
this is only a rough estimate, based on the scant data available. In any case, a precise determination of this parameter is not the goal of this study, and a reasonable value for the system at hand can be inferred from the data. 6.1. Relaxation Simulations without Absorption An abbreviated presentation of the results of nonabsorbing relaxation simulations is in order, as this has been treated in similar systems by others (see ‘‘Literature’’ section). Fig. 4 illustrates the fundamental difference between the diffusion and mixed diffusion-kinetic models. One can see that for the diffusion model, the requirement for additive equilibrium between the subsurface (cAs) and interfacial () layers creates the situation in which the subsurface layer is initially starved of additive by the interface, which is additive-free at time equals zero. The diffusion process in the bulk requires some time to replenish the subsurface. Ultimately, this replenishment and concurrent buildup of the Gibbs layer occur rapidly. The time at which this occurs depends directly on the bulk diffusivity of the additive. The mixed model, on the other hand, allows for a non-equilibrium buildup of the surface excess layer as dictated by the Langmuir-Hinshelwood rate equation (3). Until the additive adsorption ‘‘takes off’’, the subsurface layer is unaffected. When begins to grow, one can see a moderate dip in cAs indicative of a partial depletion of bulk additive during the period of high bulk-to-interface flux. From Fig. 4 one can see that the magnitude of this depletion depends in part on the relative rate of adsorption as determined by the rate barrier B. The subsurface layer recovers and tends to a value slightly lower than the initial bulk concentration, due to the fact that the interface has robbed the bulk of some of its initial inventory. In the purely kinetic model, the subsurface layer would not be affected at all, and the
Fig. 2. Fits of static surface tension data for LiBr–H2O with 2EHX, using Langmuir isotherms. Top: effect of brine concentration on the Langmuir constant. Bottom: effect of solution temperature. Data of [6].
Table 1 Values of the Langmuir constant KL [1/ppm] for different solution temperatures and concentrations T\CLIBR
40%
50%
60%
24 C 48 C
0.02728
0.08039 0.03399
0.26127 0.06475
Derived from data of [6].
entire bulk would remain at a uniform additive concentration. Fig. 5 illustrates adsorption simulated with the kineticdiffusion model with different initial bulk concentrations cAo and diffusivities DA. For two consecutive orders of magnitude (DA ffi 2:0E-9 m2/s at ScA ¼ 104 and 2.0E-10 at ScA ¼ 105 ) a lower additive diffusivity is seen to decrease the relaxation rate very slightly, as the bulk transport is slower in replenishing the subsurface layer. Fig. 5(b) shows lower dips in this concentration for the lower diffusivity case. For all values of cAo considered, the different values of static surface tension are reached at about the same time. So it can be seen that the mixed
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Fig. 3. Fit of dynamic surface tension data of LiBr–H2O with 2EHX [5] with mixed diffusion-kinetic model using LangmuirHinshelwood rate equation. Adsorption barrier B=5. Langmuir constant KL=0.11 [1/ppm].
Fig. 4. Profiles of additive concentration: (1) subsurface bulk cAs/cA1, (2) surface excess /1. Diffusion and mixed diffusionkinetic models with different adsorption rate barriers.
model is closer to a purely kinetic model than a diffusion model, with a slight effect due to the bulk transport properties of the additive. Indeed, only the mixed model with an adsorption rate barrier is seen to behave in accordance with the data. 6.2. Relaxation simulations with absorption A number of runs were performed simulating concurrent processes of surfactant adsorption and vapor absorption. The Langmuir–Hinshelwood rate equation employed to model the flux of additive from the subsurface to the interface used parameters which were updated each time step to account for changing temperature and absorbate concentration at the interface. When initial brine concentration was allowed to vary, it was found that the change in relaxation profile was negligible. The equilibrium condition for the brine at a given temperature and vapor pressure immediately yields nearly identical interfacial absorbate concentrations, despite the different bulk concentrations (Fig. 6a). This can be understood by considering that bulk temperature, as opposed to concentration, is the dominant property in determining the interfacial equilibrium state of the brine, due to the fact that energy diffusion is much quicker than mass diffusion in LiBr–H2O. The concentration sustains a sharp gradient from the interface into the film, as opposed to temperature, which communicates its bulk value much more quickly to the interface. In terms of the characteristic parameters of the absorption system, this is seen in the high Schmidt number and low Prandtl number. The Lewis number (Le=Pr/Sc) for LiBr–H2O at the states in question is on the order of 0.025. Indeed, it is this higher concentration gradient across the film that causes the stronger solution to absorb vapor more quickly. However, the very similar
Fig. 5. Relaxation profiles of surface tension (top) and subsurface additive concentration (bottom). Effect of initial bulk concentration and additive diffusivity.
interfacial states at the early stages of the process for the different initial concentrations gave identical surface tension behavior during the time in question, and therefore the potential for instability is not affected by initial solution concentration.
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Fig. 7. Surface tension relaxation profiles for different initial solution temperatures (at 53% LiBr).
Fig. 6. Profiles of the interfacial brine state for varying (a) initial concentration (at 30 C) and (b) temperature (at 53% LiBr).
On the other hand, variation of initial solution temperature causes an initial variation in surface tension before any adsorption has even taken place (Figs. 6b and 7). The increased initial interfacial temperature dictated by a higher initial bulk temperature gives a lower initial water concentration in order to satisfy the brinevapor equilibrium requirement. While surface tension is expected to decrease with increasing temperature for the additive-free interface, the lowered absorbate concentration (i.e. high salt content) actually causes an initial positive jump in the pre-adsorption value of (Fig. 7). Eventually, by about half a second, is primarily influenced by the growing additive surface excess concentration, since the interfacial states of the brine tend to converge at about this time to the same wall temperature for the different runs. Here, it can be inferred that the potential for instability is increased somewhat for a solution having a higher initial temperature, even though this solution is less hygroscopic than colder ones, and is expected to absorb more slowly in the stable regime. Turning attention to other factors—external to the bulk solution—that influence the driving force for absorption, one can consider the effects of wall temperature (i.e. cooling flux) and evaporator temperature
Fig. 8. Effect of cooling wall temperature on rates of vapor absorption and additive adsorption. Temperatures and water concentrations are interfacial values. Initial parameters: LiBr conc.=58%, solution temp.=30 C, absorbate vapor pressure=1.228 kPa, additive (2EHX) conc.=80 ppm.
(i.e. vapor phase pressure) on the -relaxation profile. Fig. 8 shows the influence of wall temperature on the rates of absorption and on the behavior of the interface, in terms of brine state and surfactant adsorption. One can see that in each case, the solution approaches the constant wall temperature. The solution on the colder wall will approach a higher water concentration at its cooler interface. The lower excess surfactant concentration (the surfactant is more soluble in a subsurface solution with lower salt concentration) in this case gives a higher equilibrium value of . However, the adsorption equilibrium will be reached more quickly due to the higher heat flux of the cooler wall. Granted, for the
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In this sense, the potential for instability is increased with the driving force, as was the case with the wall temperature study.
7. Conclusions
Fig. 9. Effect of vapor phase pressure on rates of vapor absorption and additive adsorption. Pressure correspond to evaporator temperatures of 20, 25 and 30 C. Temperatures and water concentrations are interfacial values. Initial parameters: LiBr conc.=53%, solution temp.=30 C, Cooling wall temperature=20 C, Additive (2EHX) conc.=80 ppm.
temperature range of 20 to 30 C for the cooling wall, which is representative for water from a cooling tower, the difference in surface tension at a given time for the different runs is not great—and it cannot be stated here whether this will increase the rate or intensity of instability in the system. However, this result is qualitatively in line with the results of Kulankara et al. [14]. They conducted water condensation experiments on a cold tube with and without additives. It was found that for colder tubes, the mass transfer coefficient was increased, and they took this to be a sign of enhancement due to the presence of the additive in the vapor phase. Assuming vapor additive transport to be similar to adsorption from the liquid bulk, then this agreement makes sense. As far as is known, no reliable data is available for dynamic surface tension of absorption fluids with the surfactant originating from the vapor side. However, the work of MacLeod and Radke [13] with aqueous decanol solutions showed similar profiles and relaxation times for liquid- and vapor-side adsorption experiments. The driving force for absorption can also be increased by increasing the vapor phase pressure. Fig. 9 shows the influence of this parameter on -relaxation. It can be seen that the higher the vapor pressure, the larger the initial negative jump to the value of s corresponding to the immediate (pre-adsorption) equilibrium state of the brine at the interface. Even if this jump is not ‘‘immediate’’, it is still evident that the relaxation of surface tension is quicker for the higher vapor pressure.
Simulations of surface tension relaxation were performed in cases of both non-absorbing and absorbing static films of LiBr–H2O in the presence of surfactant additive, with adsorption properties characteristic of 2-ethyl-1-hexanol being used. Langmuir isotherm constants (KL) expressing the ratio of adsorption to desorption parameters in the Langmuir–Hinshelwood rate equation (3) were derived from static surface tension data taken from [6]. This equilibrium is a function of temperature and brine concentration, and reflects the variation of additive solubility with the solution state. Representative absolute values of ka and kd were derived from dynamic surface tension data of [5]. It is evident that a mixed kinetic-diffusion model with an adsorption barrier of about 5 fits experimental data of both static and dynamic surface tension of the system in question. A diffusion model alone gives a relaxation profile which is too steep and occurs too early for reasonable diffusion coefficients. Kim and Janule [4] derived a value of DA on the order of 3.0E-11 m2/s by fitting their dynamic data with a diffusion model. This value is an order of magnitude smaller than estimates based on a mixed model [5] and used in this study. The effects of initial additive concentration in the bulk, additive diffusivity and kinetic parameters on adsorption rate were illustrated. A semi-coupled model of concurrent additive adsorption and stable vapor absorption was used to simulate the effects of the absorption process on the surface tension relaxation rate. The behavior of the liquid–vapor interface was seen to be influenced by four of the factors that determine the driving force for absorption: initial solution concentration and temperature, wall temperature (cooling side heat flux) and vapor phase pressure. Variations in brine concentration in the range of 50– 58% LiBr did not affect the relaxation profiles. Temperature variations in the range of 20–30 C yielded slightly quicker relaxation times for warmer—or less hygroscopic—solutions. For both external factors of wall temperature and vapor phase pressure, relaxation rates were seen to increase for the cases of higher absorption driving force. In terms of wall temperature, this was so for the colder wall, or the higher heat flux case. In terms of vapor pressure, it was so for the higher vapor pressure, or for the case of a higher temperature evaporator. This result provides cursory corroboration to the experimental water condensation data of [14] in which mass transfer coefficients increased with higher
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condensation heat fluxes in the presence of vapor-side additives. The manifestation of the -relaxation in the form of the Marangoni instability is not modeled in this study. It would be of interest to further the model of Marangoni convection in a static absorber [19] using the results of the current study as a basis for predicting the dynamic surface tension of the system. The previous model used constant Marangoni numbers, in effect making invariant the gradients of surface tension against temperature, absorbate concentration and additive concentration. The merging of these two works may allow for a simulation of the entrance region of an absorber. It is understood that data [4–8] used to derive adsorption parameters such as those in Table 1 of this paper must be taken in their proper context, based on the limited information available as to the experimental methods used to obtain them. The additive concentrations reported for these data are proposed to be liquid concentrations, and are thus viewed in the context of this paper. However, they more accurately represent system concentration, since it seems that no efforts were made to isolate the additive in the liquid phase. Vapor additive concentrations were seemingly not accounted for in obtaining the data. There is, therefore, a need for dynamic surface tension experimentation for absorption fluids laced with additives in the vapor phase. Such data can allow for the calculation of rate constants for this manner of adsorption, and may allow for further enlightenment regarding the vapor surfactant theory put forth by [14]. The vapor surfactant model of Herold’s group is recognized as a promising theory explaining the origin of the additive causing the Marangoni instability in LiBr–H2O systems, and further data is warranted to model the behavior of the additive in its transport between the vapor phase and the absorbing liquid phase.
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