1. Introduction
Membrane transport processes belong to the group of basic phenomena occurring at the level of organization of physicochemical systems, in which the membrane constitutes a selective barrier separating the interior of the system from its surroundings [
1,
2,
3]. The driving forces of these transport phenomena are a consequence of the occurrence of various types of physical fields, such as concentration, pressure, temperature or electric potential fields, participating in shaping the field constitution of nature [
4]. The flows resulting from the action of these forces, such as diffusion or osmosis, modify the physical fields, an example of which in the case of the concentration field is concentration polarization [
5,
6,
7,
8]. This modification consists in minimizing the concentration gradients, which results in minimizing, inter alia, the osmotic and diffusion fluxes of dissolved substances and the membrane potentials [
8,
9]. Under certain conditions depending on the composition of solutions and the orientation of the membrane with respect to the gravity vector, concentration gradients can be reconstructed by gravitational convection [
8,
9]. In the case of a biological cell, the membrane plays the role of a receiver and regulator of environmental signals [
10].
Certain laboratory features of biological membranes are used in membrane technologies used in various fields of science, technology and medicine, as well as in various industries [
11,
12]. Therefore, the aim of the research is, on the one hand, to understand the mechanisms of membrane transport, and on the other, to develop membrane technologies and techniques useful in biomedicine (hemodialyzer, controlled drug release) and industrial technologies (bioreactors, biorefineries, membrane modules for food processing and water treatment) or sewage treatment) [
1,
11]. Most of the film-forming materials are polymers characterized by high stability and mechanical strength (e.g., polybenzimidazole, polyamide, polytriazole, cellulose acetate or cellulose triacetate) and biodegradable (poly/lactic acid, cellulose, bacterial cellulose or chitosan) [
13]. They are mainly used as materials for membrane systems based on osmosis and diffusion [
14,
15].
Membrane transport mechanisms are based on five thermodynamic forces (four gradients: mechanical pressure, concentration, temperature, electric potential and chemical affinity) and interconnected with them, five thermodynamic fluxes (hydraulic, diffusion, thermal energy, electric charge and reactants). The cause-effect relationships of these forces and fluxes result from simple membrane processes such as osmosis or diffusion, and cross processes such as thermo-osmosis, electrodiffusion or flow potential [
1,
16]. Explaining the mechanisms of membrane transport is based on the methods and laws of non-equilibrium thermodynamics [
17], network thermodynamics [
1,
18] and statistical physics [
19]. Examples include the known laws of Fick, Fourier or Ohm [
1] and the Kedem-Katchalsky [
17], Peusner [
18], Nernst-Planck [
20,
21,
22] and Stefan-Maxwell [
20] mathematical equations. In practice, it uses two groups of membrane techniques, created on the basis of the criterion of the type of driving force of the membrane process (e.g., ultrafiltration, reverse osmosis, pervaporation, dialysis, membrane distillation or electrodialysis) and the criterion of the size of the separated particles (nanofiltration, reverse osmosis and microfiltration) [
12].
In thermodynamic systems, including membrane systems, internal energy can be converted into free energy and dissipated energy. The energy dissipated is the product of absolute temperature (
T) and
S-entropy (
S). The rate of entropy changes of the system (
) is the sum of the rate of entropy exchanged between the system and the environment (
) and the rate of entropy formation inside the system (
) [
1,
15]. The rate of formation or production of entropy inside the system is determined by the expression
, where
≥ 0—denotes the source of entropy that is the rate of
S-entropy formation in the volume unit (
V) of the tested system, (
> 0—in an irreversible process, and
= 0—in a reversible process) [
3]. Moreover, the source of entropy (
) satisfies the relation
. This relation shows that the set of thermodynamic force (
) causes irreversible flows conjugated with them and opposite to them, which are measured by the
fluxes, reducing the value of
and leading the system to the state of thermodynamic equilibrium [
1,
3].
For a membrane system where a Δ
x thick membrane separates two homogeneous electrolyte solutions of different concentrations, the entropy source of the membrane itself is
[
17]. If the solutions contain a solvent and
k solutes, then the global source of entropy is described by the following equation:
where
—global entropy source for the conditions of the homogeneous concentration field of solutions;
,
and
—the
S-entropy produced by
,
and
I, respectively;
and
—fluxes, respectively, volume solution and
k-th solute for the conditions of homogeneity of solutions,
I—electric current, Δ
P and Δ
πk =
RTΔ
Ck—differences of hydrostatic and osmotic pressures, respectively (
RT—the product of the gas constant and temperature, Δ
Ck—difference of the concentrations of the solutions),
—the average concentration of solutes in the membrane (M). Equation (1) is reduced to the written expression for nonelectrolyte solutions when
I = 0 and
E = 0 [
17].
,
and
I fluxes can be described by the appropriate Kedem-Katchalsky equations for the homogeneity conditions of electrolyte solutions [
17]:
where
Lp,
and
—hydraulic permeability, reflection and solute permeability coefficients,
(1 ≤
≤ 2)—stands for the Vant Hoff coefficient,
β—electroosmotic coefficient,
i—represent electric current through the membrane,
—transference number of ions,
—valence of ions,
F—Faraday number, κ—conductance coefficient, Δ
E—electromotive force difference. Equations (2)–(4) reduces to the expression for nonelectrolyte when
I = 0. Due to the lack of accumulation or depletion of ions in the electroneutral membrane and due to the electroneutrality of the solution, it can be concluded that
=
=
(
k = 1 or 2).
Under real conditions, the homogeneity of the solution concentration field may be disturbed by concentration polarization. As a result, concentration boundary layers are spontaneously formed on both sides of the membrane. For the conditions of concentration polarization, and for
I = 0 or
E = 0, Equation (1) takes the form:
where
—global entropy source for the conditions of concentration polarization,
is the
S-entropy produced by
,
is the
S-entropy produced by
,
and
—the volume and
k-th solute fluxes, respectively, for the concentration polarization conditions of the solutions,
r = A or B means the configuration of the membrane system. The Kedem-Katchalsky equations for the fluxes
and
and for
I = 0 can be written as:
where
,
,
and
are the hydraulic, osmotic, diffusive and adjective concentration polarization coefficients, respectively [
23]. As in the previous case, due to the lack of accumulation or depletion of ions in the electroneutral membrane and the electroneutrality of the solutions, it can be assumed that
=
=
(
k = 1 or 2). For this reason, in the vicinity of the electroneutral membrane, there only a phenomenon of concentration polarization of the membrane having an important influence on substances “1” and/or “2” transport through the membrane. Due to the electroneutrality of the concentrated electrolyte solutions, the electric current through the membrane (electroneutral membrane without bounded ions) during the measurement is negligible (
I = 0) [
17,
24].
In [
4,
8] it was shown that
and
depend on the transport properties of the membrane, the configuration of the membrane system as well as the physicochemical properties and composition of solutions separated by the membrane. The value of these fluxes is greater under convective than in non-convective conditions. In the case of ternary solutions (consisting of water and two dissolved substances, one of which causes an increase in density and the other a decrease in density as their concentration increases), the
and
fluxes are non-linear functions of the concentration difference. Due to Equation (2), the global source of entropy for the conditions of concentration polarization (
), is a non-linear function of
and
[
23,
25].
The aim of the present study was to determine , , and in a single-membrane system, in which the hemodialyzer biomembrane Nephrophan® (Orwo VEB Filmfabrik, Wolfen, Germany) situated in the horizontal plane separates water and a ternary solution consisting of water, ammonia and/or HCl. In order to achieve this goal, the influence of the concentration of individual components of the solutions and the configuration of the membrane system on the value of , , and fluxes under the conditions of concentration polarization, respectively, and under the conditions of homogeneity of solutions were investigated. Based on the results of the and tests, the sources of entropy (, ), the diffusion-convective effects ( = ) and the convective effects ( = ) in the global entropy source (k = 1, 2 represents the component number of the solution and r = A, B—configuration of the membrane system). The experiments were performed under the conditions of E = 0 and I = 0.
2. Model of the Electrochemical Membrane Cell
The subject of considerations, as well as several of our previous works, is transport in a membrane system illustrated schematically in
Figure 1 [
4,
26]. This figure shows a model of a membrane system in which the membrane (M), situated in the horizontal plane, separates two solutions with the initial concentrations
Chk and
Clk (
Chk >
Clk,
k = 1, 2). In configuration A, in the compartment above the membrane there is a solution with a concentration of
Clk, and in the compartment under the membrane—a solution with a concentration of
Chk. In configuration B—solutions with the concentration of
Clk, and
Chk are changed places. If we assume that the driving force for osmotic flows is the difference in concentrations between the solutions filling the upper and lower compartments, then Δ
Ck for configuration A has a negative sign, and for configuration B—positive.
According to the laws of diffusion, water and substances dissolved in it, penetrating through the membrane, causing the phenomenon of concentration polarization, form, on both its sides, concentration boundary layers and (r = A, B) with thicknesses respectively and . The consequence of the formation of these layers is the reduction of the concentration difference from the value of Chk—Clk to the value of , where > , Chk > and > Clk.
In the case when a solution with a lower density is placed in the compartment under the membrane, and a solution with a higher density in the compartment above the membrane, the system
/M/
loses hydrodynamic stability and, consequently, gravitational convection may occur in the concentration boundary layers region [
27,
28,
29,
30,
31,
32]. It appears when the thickness of the boundary concentration layers (
,
) exceeds the critical value (
) and/or the concentration polarization coefficients (
) exceeds the critical value (
) and when the concentration Rayleigh number (
RCk) that control the process of the appearance of gravitational convection, will exceed their critical values [
28,
33,
34]. The concentration Rayleigh number for membrane transport processes of ternary solutions can be represented by the expressions [
35,
36]:
where
—concentration Rayleigh Number,
—mass density,
—kinematic viscosity of solution,
RT—product of the gas constant and temperature,
—solute permeability coefficient, g—gravitational acceleration,
—variation of density with concentration,
—concentration polarization coefficient,
—diffusion coefficient, (
k = 1, 2). It is worth noting that Equation (8) does not contain the concentration thickness of the boundary layer (
). To get
it is enough to change the index “1” to “2”.
Over time, the destructive effect of gravitational convection limits the growth of
and
and accelerates the diffusion of substances beyond the layers, which extends the effect of convection to the entire volume of the solution. Under certain conditions, even liquid structuring may occur, which is manifested in the appearance of “plum structures” [
37,
38].
The process of creating concentration boundary layers is accompanied by a decrease in the volume osmotic fluxes from
to
and the solute fluxes from
to
[
7]. Using Equations (1) and (5), the global source of entropy for ternary solutions can be represented as:
To calculate the sources of entropy and , it is enough to experimentally determine the concentration dependences of the fluxes , , and .
3. Methodology for Measuring the Volume Osmotic and Solute Fluxes
The study of volume osmotic transport and transport of dissolved substances was carried out using the measuring set described in a previous paper [
29]. The set consisted of two cylindrical measuring vessels with a volume of 200 cm
3 each. One of the vessels contained the tested binary solution (aqueous HCl or NH
3·H
2O solution) or ternary (aqueous solution of HCl and NH
3·H
2O). In turn, the second vessel in all experiments contained an aqueous solution of HCl and/or NH
3·H
2O (NH
4OH) with a constant concentration
Cl1 =
Cl2 = 1 mol m
−3. The solutions in the vessels were separated by the Nephrophan
® (Orwo VEB Filmfabrik, Wolfen, Germany) biomembrane, set in a horizontal plane, with an area of
A = 3.36 cm
2 and transport properties determined by the following factors: hydraulic permeability (
Lp), reflection (
σ) and diffusion permeability (
ω). The values of these coefficients for HCl (index 1) and NH
3·H
2O (index 2), determined in a series of independent experiments carried out according to the procedure described in paper [
16], were:
Lp = 5 × 10
−12 m
3N
−1s
−1,
σ1 = 0.06,
σ2 = 0.01,
ω11 = 1.24 × 10
−9 mol N
−1s
−1,
ω12 = 1.4 × 10
−12 mol N
−1s
−1,
ω22 = 2.68 × 10
−9 mol N
−1s
−1 and
ω21 = 2.5 × 10
−12 mol N
−1s
−1. Nephrophan
® (Orwo VEB Filmfabrik, Wolfen, Germany) is a microporous, highly hydrophilic and electroneutral membrane made of regenerated cellulose [
39].
A graduated (every 0.5 mm
3) pipette set in a plane parallel to the plane of the membrane was connected to the vessel containing the higher concentration. The change in volume (Δ
Vr) of the solution in this vessel of the plumbing system was measured with this pipette. In turn, the second vessel was connected to a reservoir of an aqueous solution of HCl and/or NH
4OH (NH
3·H
2O) with a concentration of
Cl1 =
Cl2 = 1 mol m
−3, with adjustable height relative to the pipette. This made it possible to compensate for the hydrostatic pressure (Δ
P = 0) present in the measurement set. The measurements were performed according to the procedure described in [
8], which consisted of two stages. In the first stage, the increases of Δ
Vr were measured under the conditions of intensive mechanical stirring of the solutions with an angular speed of 500 rpm. The second stage started as soon as steady-state flows were achieved, and the stirring of the solutions was turned off. In this step, the increases of Δ
Vr were also measured until the steady state of the flows was obtained. Each experiment was performed for configurations A and B of the membrane system. In configuration A, the test solution was filled into the vessel under the membrane, and in configuration B—the vessel over the membrane. It should be noted that the volume flows took place from the vessel with a lower concentration of solutions to the vessel with a higher concentration of solutions, and the flows of dissolved substances in the opposite direction. Therefore, it was assumed that in the configuration A the fluxes
,
,
and
and the concentration differences Δ
Ck (
k = 1, 2) are negative (
,
—directed vertically downwards,
and
—vertically upwards), and in configuration B—positive (
,
—vertically upwards,
,
—vertically downwards).
The tests were carried out in isobaric-isothermal conditions for
T = 295 K and Δ
P = 0. The volume flow was calculated on the basis of the volume changes (Δ
Vr) in the pipette over time ∆
through the membrane surface
S, using the formula
= (
)
S−1(Δ
t)
−1 (
r = A, B). Flows of dissolved substances were calculated on the basis of the formula
=
S−1(Δ
t)
−1 (
k = 1, 2;
r = A, B),
—volume of the measuring vessel,
—global concentration exchanes in the solutions studied was performer by the standard physico-chemical method [
40,
41]. In this expression, due to the lack of accumulation or depletion of ions inside the electroneutral membrane and in its surroundings (electroneutral solutions), we assume that
=
=
),
=
=
),
=
=
(
) and
=
=
(
).
The study of volume flows and flows of dissolved substances in both configurations consisted in determining the characteristics = f(t), = f(t), = f(t) and = f(t),
(k = 1, 2; r = A, B) for different concentrations of solutions. Each measurement series was repeated 3 times. The relative error in determining , , and was not greater than 5%. Based on the characteristics = f(t), = f(t), = f(t) and = f(t) for the steady state, the characteristics , = constant), , = constant) , = constant), , = constant), = constant), = constant), = constant) and = constant). Based on these characteristics, the concentration source of entropy was calculated: = constant), = constant), = constant), = constant), = constant) and = constant).
5. Conclusions
In the paper, the authors present the results of research on the effects of the concentration and orientation of aqueous HCl and/or ammonia solutions in relation to a horizontally oriented membrane, under Earth gravity conditions, on the value of osmotic volume fluxes ( and dissolved substances ). It has been shown that for the polarization conditions of the concentration and of aqueous HCl or ammonia solutions, and are linear, and for aqueous HCl and ammonia solutions, non-linear functions of solution concentration differences. Moreover, it has been shown that the values of and depend on the alignment of the solutions with respect to the horizontally oriented membrane. In the case of mechanically stirred solutions, and are independent of the orientation of the solutions in relation to the horizontally oriented membrane and are a linear function of the difference in concentrations of the solutions of both aqueous HCl or ammonia solutions and aqueous HCl and ammonia solutions. For the investigated fluxes, the following relations are satisfied: > and > .
A common feature of the and concentration relationships for aqueous HCl and/or ammonia solutions is the change in the nature of transport from osmotic-diffusion to osmotic-diffusion-convective or the other way around. This means that under the Earth’s gravitational field conditions and concentration field dependency on the density of the solutions separated by the membrane, gravitational convection appears or disappears. The measure of the effect of gravitational convection is the coefficient , which can take positive or negative values. A positive value of this coefficient indicates that the convective movements that destroy CBLs are vertically downward, and negative—vertically upward. The transition from non-convective to convective or the other way has the characteristics of a pseudo-phase transition. All the above-mentioned features have a global source of entropy (), which for solutions containing a solvent and two dissolved substances is the sum of three partial sources of entropy, the global source of entropy is the sum of three components , and , (k = 1, 2). It is similar in the case of homogeneous solutions: the global source of entropy is the sum of , and (k = 1, 2). There are relations between the above-mentioned quantities > , > , > and > , (k = 1, 2). The largest share in are the components and and in the case of and .
It has been shown that the coefficient
can be related to the concentration number Rayleigh
, i.e., with the parameter controlling the transition from the non-convective (diffusive) state to the convective state. The article uses an innovative approach consisting in replacing the expression
with a Katchalsky number (
Ka):
This number acts as a switch between the two states of the concentration field: convective (with a higher entropy source value) and non-convective (with a lower entropy source value). The operation of this switch indicates the regulatory role of Earth’s gravity in relation to membrane transport.
This number acts as a switch between two states of the concentration field: convective (with a higher entropy source value) and convection-less (with a lower entropy source value). The operation of this switch indicates the regulatory role of Earth’s gravity in relation to membrane transport.