Analytical Model for Predicting Induction Times in Reverse Osmosis Systems with and without Antiscalants

Abstract

A simple predictive analytical model for induction times in reverse osmosis (RO), both with and without an antiscalant (AS), has been developed based on the fundamental principles of mass and momentum balance. The simplicity of the model arises from the very low Reynolds number in the vicinity of the cluster surface, enabling the use and derivation of exact equations. The main assumption of the induction time without AS, t_0A, is that the net growth of the cluster size results from the difference between adhesion and shear forces. With AS, the induction time, t_A, is extended due to the competition between the AS and the scaling molecules on the cluster’s surface ligands. The model was validated by fitting it to six independent datasets from experiments conducted with spiral-wound and tubular RO membranes under various operational conditions, resulting in an average difference of 8.0% (t_0A) and 8.7% (t_A) between predicted and experimental induction times. It was found that t_0A is governed by three dimensionless parameters: supersaturation ratio (S_a), shear (K_u), and scalant saturation (κ). t_A increases with t_0A and the AS concentration.

1. Introduction

The issue of mineral scaling, as part of fouling in reverse osmosis (RO) desalination, has prompted extensive experimental and theoretical efforts for decades to minimize its economic damage. Research and development efforts were focused on understanding the scaling mechanism and parameters controlling it. When supersaturation is reached, nucleation begins, leading to the formation of small clusters. These clusters then continue to grow through coagulation until they reach a critical size, at which point they become stable and are more likely to grow further than dissolve. Once this critical size is achieved, further growth becomes spontaneous, a process often referred to as scaling [1].

The term induction time can be defined as the period preceding the onset of quantifiable crystallization within a supersaturated solution [2]. It is also described as the time interval between achieving a state of supersaturation and the juncture at which clusters within the solution attain a critical size [3]. The induction time serves as a valuable indicator for the control of scaling. Maximizing the induction time involves optimizing the operational conditions of RO systems. Establishing a relation between operational conditions and the induction time is the objective of experimental and theoretical studies.

When measuring the induction time (t_ind), the objective is to identify the initiation of crystalline growth or to detect crystal growth after its inception [2]. Simple induction time measurements involving static jar tests provide data regarding the bulk precipitation of a scaling species from its supersaturated solution over time. In these tests, the induction time denotes the period before the first observable change in the solution properties, such as the pH, turbidity, conductivity, and concentration of target scaling ions. The ability to measure the initial stages of crystal development depends on the precision of the analytical approach utilized [4]. More sophisticated evaluation methods involve dynamic tests aiming to simulate field conditions, such as reaction kinetics, wall shear stress, fluid velocity, water chemistry, temperature, and pressures. These methods include dynamic tube-blocking tests, rotating cylinder systems, scale coupons, and membrane tests [5]. In the latter, scaling is detected based on a rapid drop in the membrane permeability (flat sheet, tubular, or spiral-wound), a significant increase in feed water turbidity due to bulk precipitation of the scaling species [6,7,8,9], and by real-time imaging of the membrane surface [10,11,12]. A comprehensive review of the characterization of incipient scaling on RO membranes can be found in [13,14,15,16]. It can be concluded that the precision and reliability of the induction time determination depend on the type and sensitivity of the measurements.

A theoretical model for predicting induction was based on classical nucleation theory developed by [17]. In this model, the t_ind is regarded as the sum of three distinct crystallization times: the relaxation time (t_r) required to achieve quasi-steady-state distribution of clusters, nucleation time (t_n) responsible for critical size formation, and the time to crystal growth (t_g) from the critical size to a detectable size. In solutions with low viscosity, it is assumed that t_r << t_n leads to t_ind ~ t_n + t_g. The model takes into consideration three scenarios, t_n >> t_g, t_n ~ t_g, and t_n << t_g, each involving different possible nucleation and growth mechanisms. Another model, based on classical nucleation theory, which considers two distinct phases of crystallization, namely nucleation and crystal growth, was employed to calculate the induction time without the dosage of AS [18]. It was postulated that the AS influences both the molar volume and Gibbs free energy associated with nucleus formation. These effects were quantified to be proportional to the AS concentration.

A different approach for predicting the induction time is based on Smoluchowski’s coagulation theory, which describes the temporal evolution of the cluster size distribution [3]. Their model relies on the assumption that the collision of two clusters results in a larger cluster containing the combined entities of each, and the breakup of a large cluster leads to the formation of two smaller ones. Experimental validation of the model, based on data from a crystallizer with a magnetic stirrer [19], revealed that the rate of association between two clusters is independent of their size.

A mathematical model of the coagulation process for dispersed colloidal particles in a stirred flow was developed to predict the kinetics of shear-induced coagulation [20]. The model combines the discrete element and final volume methods. It calculates particle trajectories and rotation rates based on the forces, torques, and flow fields within the domain. The model demonstrates that the ultimate size of the produced clusters depends on the applied shear rate and the surface energy, which is a measure of particle adhesion. The cluster size is determined based on the equilibrium between the hydrodynamic force acting on the cluster and the maximum adhesion force among the primary particles. A new mechanistic model for mineral scale crystallization and inhibition has recently been developed. This model accurately predicts the barite crystallization induction time, both in the absence and presence of ten commonly used scale inhibitors [21].

A simple model that correlates the induction time with the saturation level prevailing on the membrane surface was introduced for RO systems [9,22]. The model relies on the assumption that the induction time is inversely proportional to the nucleation rate. Determination of the induction time involves measuring the slope of the linear relationship between ln(t_ind) with 1/(lnS_w)², where S_w is the supersaturation ratio on the membrane surface. Oh et al. [23] developed a model that integrates both surface and bulk crystallization into a resistance in-series framework, which was modified using concentration polarization theory and crystallization kinetics. The model aimed to analyze the scaling of RO membranes across a wide range of conditions, including crossflow velocity, transmembrane pressure, permeate recovery, and operation mode. The model’s outcomes suggest that both surface and bulk crystallization play a significant role in crossflow filtration and that RO scaling due to bulk crystallization tends to increase with a higher crossflow velocity and permeate recovery. Simulations were developed to predict supersaturation along reverse osmosis modules in seawater desalination. These simulations are based on the application of conservation principles and chemical equilibria equations along RO modules. Supersaturation has been used as a key parameter to assess scaling [23].

The objective of the present work was to develop an analytical model for predicting induction times in RO systems, both with and without an antiscalant, and to validate it using independent datasets. Following the validation of the model, the impact of its variables and parameters on induction times was analyzed.

2. Materials and Methods

Two independent datasets were used to validate the developed model. The experiments in the study of Hasson et al. [24] were conducted using a spiral-wound TFC-ULP 2514 membrane from Fluid Systems Corporation (now Koch Membrane Systems), San Diego, CA, USA. The experiments were carried out in a full recycle mode, where the concentrate and permeate were recycled back to the feed tank. The transmembrane pressure was 15 bar, with a flow rate of 475 L/h (corresponding to feed velocity of 0.72 m/s) and a temperature of 28–30 °C. The feed solution contained Ca²⁺ (27–36 mM), Mg²⁺ (12–16 mM), SO₄²⁻ (12–16 mM), and Cl⁻ (55–73 mM), with CaSO₄ supersaturation ratios (S_a) ranging from 1.6 to 2.0 (corresponding to 27–36 mM of CaSO₄, both without and with a dosage of 6 mg/L of a phosphonic acid sodium salt scale-inhibitor). The induction time was determined through measurements of the permeate flow rate, frictional pressure drop along the membrane, feed turbidity, and Ca²⁺ and SO₄²⁻ concentrations. The authors reported that the four different criteria provided similar indications of the onset of scaling. The linear correlation between the induction time and saturation level on the membrane wall, shown in Figure 7 of [24], was used to validate the model.

In the study of Li et al. [25], two different membrane systems were used: a tubular system and a spiral-wound system. In the former, two 30 cm-long tubular AFC99 reverse osmosis membranes (CPI membranes, Wilton, UK) with a diameter of 12.5 mm were connected by a U-tube. Experiments were conducted in a once-through mode, in which both the permeate and the concentrate were withdrawn from the system. The saturation index of CaSO₄ was 3.3 (corresponding to 49 mM of CaSO₄). Flow rates of 0.4–7.0 L/min (equivalent to feed velocities of 0.16–2.64 m/s) were applied, at a pressure of 30 bar and a temperature of 22–24 °C. The spiral-wound module was equipped with a Dow Filmtec SW30-2521 element (membrane area of 1.19 m²). The experiments were conducted in a full recycle mode at a transmembrane pressure of 30 bar, with a flow rate ranging from 246 to 648 L/h (equivalent to feed velocities of 0.07–0.20 m/s) and a temperature of 20–22 °C. The feed solution contained Ca²⁺, Na⁺, SO₄²⁻, and Cl⁻ with a saturation index of 2.54 with respect to gypsum (equivalent to 40 mM of CaSO₄). The experiments were carried out both with and without a dosage of 2 mg/L phosphinocarboxylic acid AS. In this study, the induction time was determined by identifying the onset of the membrane permeability decline. The results of the induction time as a function of the Re number for both the tubular and spiral-wound systems were used to validate the model. These results are presented in Figure 6 of [25].

3. Model Development

Several factors influence the precipitation of sparingly soluble minerals (scalants), with supersaturation playing a dominant role as the driving force behind nucleation and growth phenomena [26]. Nucleation initiates when a solution reaches a saturation level at a concentration C_e. When the feed concentration C_f exceeds the saturation level at C_e, nucleation initiates and continues to grow into small metastable clusters until they reach a critical size n_c, at which they undergo spontaneous growth and precipitate. Introduction of antiscalants into the feed solution initiates a competition between the scalant and the AS ions for occupancy of the bonding sites within the clusters [24]. This competition retards the spontaneous growth of the clusters and, therefore, prolongs the induction time. A standalone analytical model was developed to describe cluster growth without AS and predict the induction time, denoted as t_0A. This model was then extended to describe the contribution of AS to extending the t_0A to t_A.

3.1. Prediction of the Induction Time without AS (t_0A)

At supersaturation, the critical number of molecules in the clusters is determined by Equation (1) using free energy arguments [1,19].

n_c = 32πV_m² γ³/3(kTlnS_a)³

(1)

where V_m is the molar volume of a target scalant and γ is the cluster surface energy density. A value of γ = 9.25 mJ/m² was used in this study [24]; k is the Boltzmann constant, T is the absolute temperature, and S_a is the supersaturation ratio, defined in Equation (2).

S_a = C_f/C_e

(2)

where C_f is the target scalant feed concentration (mM) and C_e is the saturation level (mM).

In this model, the induction time t_0A refers to the period during which a cluster grows until it reaches the critical size [27,28]. Theoretical derivation of the t_0A can commence with the consideration of the rate of cluster growth. A diffusion-reaction approach was employed to account for the cluster’s growth, involving two distinct steps: a diffusion process, wherein solute molecules are transported to the surface of the cluster, and a first-order reaction, where the solute molecules arrange themselves within the lattice of the cluster [29,30]. This approach is presented as follows:

dm/dt = k_cA(C_f − C_e)^z

(3)

where dm/dt is the rate of mass m deposited over time t, k_c is the cluster growth coefficient [kg/(s m² mM^z)], A is the cluster surface area, and z is the order of the overall cluster growth process. When z = 1, the deposition process is primarily controlled by diffusion, whereas at z > 1, the process is dominated by the surface reaction.

The cluster size is determined based on a balance between the adhesive forces and hydrodynamic or shear forces that act at the solution–cluster interface. The shear forces are induced by the velocity difference u_d = u_f − u_c (between the feed u_f and cluster u_c velocities) at the cluster surface, the viscosity of the solution, and the cluster’s surface roughness. Removal from the cluster’s surface occurs through mechanisms, such as dissolution, erosion, and spalling [20,29]. Spacer-filled RO channels create local shear stresses resulting in the detachment of deposited particles on a cluster [15].

The present analytical model is based on the assumption that the overall shear forces F_s, acting on a cluster surface, slow down the rate of the cluster growth. This assumption leads to the modification of Equations (3) and (4).

dm/dt = k_cA(C_f − C_e )^z − k_sF_s

(4)

where k_s is a shear rate constant (s/m), which depends on the cluster’s surface shape and roughness, the adhesion strength of its molecules, the local velocity profile in the feed solution, and the concentrations C_f and C_e (M). The local velocity profile may be affected by factors, such as the presence or absence of membrane spacers or the types of spacers used [31].

The flow around a cluster surface is influenced by the velocity difference u_d. Due to the small magnitudes of u_d (<u_f < 1 m/s [32]) and the cluster size (<10⁻⁸ m [33]), the Reynolds number is very low (Re << 1). In this case, the Stokes analytical solution of the velocity field around a sphere [34] is used to derive the shear stress (τ) of the flow on the cluster surface. Integration of τ over the cluster surface area A yields the resulting total shear force (F_s) acting on a cluster’s surface:

$${\mathrm{F}}_{\mathrm{s}}={\int }_0^{\mathsf{\pi }}\mathsf{\tau }\mathrm{dA}-3/2{\mathsf{\pi }}^2\mathsf{\eta }\mathrm{R}{\mathrm{u}}_{\mathrm{d}}$$

(5)

where R is the cluster radius and η is the dynamic viscosity of the solution. The velocity difference u_d can be derived from the force balance between Stokes’ drag force and the inertial force resulting from the change in the cluster mass over time (Equation (6)).

6πηRu_d = (u_f − u_d)dm/dt

(6)

where u_f is the flow velocity of the feed solution.

It can be assumed that clusters flowing in the feed solution result in u_d << u_f. This assumption can be verified by rearranging Equation (6) to obtain Equation (7):

u_d/u_f = f/1 + f where f = (dm/dt)/6πηR

(7)

Equation (4) shows that dm/dt is proportional to R² through A, and f in Equation (7) is proportional to R with a proportionality coefficient α of the order 10 1/m. As the cluster size is less than 1 μm [20], the resulting value of f is much smaller than 1. Therefore, Equation (7) yields u_d/u_f << 1, which justifies the neglecting of u_d in Equation (6). By neglecting u_d relative to u_f in Equation (6) and combining it with Equation (5), Equation (8) is obtained.

dm/dt = 4F_s/πu_f

(8)

The shear force F_s is derived from Equations (4) and (8).

F_s = πu_f k_cA(C_f − C_e)^z/(4 + πu_fk_s)

(9)

The rate of mass growth of the cluster, dm/dt, is obtained from Equations (4) and (9).

dm/dt = 4k_cA(C_f − C_e)^z/(4 + πu_fk_s)

(10)

The variables m and A on both sides of Equation (10) are time dependent as they both depend on the number n(t) ≥ 1 of molecules in a growing cluster with size and time. The mass of the cluster composed of n molecules is given by the following:

m(n) = (M_w/N_A)n

(11)

The cluster’s volume V(n) is approximated by considering n hard spheres randomly packed with a bulk density of 0.635 [35].

V(n) = nV_m/0.635N_A

(12)

where V_m is the molar volume. Although n is a discrete variable, the cluster’s packed volume V(n) in Equation (12) is considered a continuous volume. This assumption can be justified by analyzing a single average cluster that represents the growth process of the cluster population in the feed at a specific time t. Using Equation (12) the cluster surface area is calculated by the following:

A(n) = 4π(3V(n)/4π)^2⁄3

(13)

The cluster surface area A in Equation (13) is time-dependent through the cluster size n(t), represented as A = A(n(t)) = A(t). It is specified using Equations (12) and (13).

A = A(n) = A(1)n^2/3, where A(1) = 6.55(V_m/N_A)^2/3

(14)

In the analyzed datasets, the measured induction time of RO membranes, t_0A, results from the initial flux decline due to partial-scale coverage of the membrane surface. Since for n > n_c, the cluster size grows spontaneously [1] and t_c = t(n_c), it is assumed that in RO systems, (t_0A − t_c)/t_0A << 1. This means that practically t_c ~ t_0A. By combining Equations (10), (11) and (13) and with n = n_c, and t = t_0A, t_0A is obtained as follows:

$${\mathrm{t}}_{0\mathrm{A}}=\frac{\left(4+\mathsf{\pi }{\mathrm{k}}_{\mathrm{s}}{\mathrm{u}}_{\mathrm{f}}\right){\mathrm{M}}_{\mathrm{w}}}{4{\mathrm{k}}_{\mathrm{c}}{\mathrm{N}}_{\mathrm{A}}\mathrm{A}\left(1\right){({\mathrm{C}}_{\mathrm{f}}-{\mathrm{C}}_{\mathrm{e}})}^{\mathrm{z}}}3{\mathrm{n}}_{\mathrm{c}}^{1/3}$$

(15)

t_0A in Equation (15) predicts the induction time with the shear rate constant k_s as a single adjustable parameter of RO systems without AS.

3.2. Derivation of Induction Time with AS (t_A)

The presence of AS in the solution hinders nucleation because it competes with scalants for bonding sites on the clusters [36,37]. Molecules of the antiscalant attached to the cluster’s surface create a surface coverage fraction denoted as θ, leaving a surface fraction of 1 − θ for further cluster growth. The growth of θ is governed by the reaction equation [38]:

dθ/dt = k_aC_a(1 − θ) − k_dθ

(16)

where k_a is the adsorption rate constant of AS molecules on the cluster surface, C_a is the AS concentration, and k_d is the desorption rate constant of the AS from the cluster surface. The solution of Equation (16) with θ(0) = 0 yields

θ(t) = (a/b)(1 − e^−bt); a = k_aC_a, b = a + k_d

(17)

At time t, the surface coverage fraction θ, representing the cluster’s occupancy by the AS, leaves a free surface area of (1 − θ)A available for further cluster growth. In this case, the surface area A in Equation (10) is substituted with (1 − θ)A to obtain the following:

dm/dt = (4k_cA(1 − θ)(C_f − C_e)^z)/(4 + πu_fk_s)

(18)

Integration of Equation (18) yields the required induction time t_A, accounting for the impact of AS on increasing t_A over t_0A.

$${\mathrm{t}}_{\mathrm{A}}=\frac{1}{\mathrm{b}-\mathrm{a}}(\mathrm{b}{\mathrm{t}}_{0\mathrm{A}}-\frac{\mathrm{a}}{\mathrm{b}}\left(1-{\mathrm{e}}^{-\mathrm{b}{\mathrm{t}}_{\mathrm{A}}}\right))$$

(19)

According to Equation (19), the induction time t_A is approximately proportional to the concentration of the AS, C_a. When C_a = 0, t_A = t_0A, as derived in Equations (17) and (19). Equations (15) and (19) predict the induction times t_0A and t_A respectively, based on the determination of the parameters k_s in Equation (15) and k_a and k_d in Equation (19) for specific datasets.

It is worth noting that the model is applicable when the feed velocity (u_f) is constant, >> than the permeate velocity, as in the case of RO desalination. Otherwise, where both permeate velocity and Stokes gravitational velocity become significant, these factors were not taken into account in the present model and are irrelevant to RO systems. Consequently, the analytical model is not valid for such low bulk velocities.

4. Results

4.1. Experimental Validation of the Induction Time Prediction

Analytical models are often considered validated even when supported by limited experimental datasets [39,40]. The presented analytical model was validated using datasets from two independent studies [24,25], each employing different membranes, operating conditions, and types of antiscalants, as described in detail in the Experimental Section. It should be mentioned that in both studies, the RO systems were operated at a full recycle mode, where both permeate and concentrate are recycled to the feed vessel. In these tests, the time required for the onset of scaling was measured allowing for the evaluation of the lower scaling threshold limit. This limit is defined as the point at which precipitation is prevented or at least delayed for a sufficiently long period. The results presented by Hasson et al. [24] showed a correlation between the induction time data and the supersaturation level derived from nucleation theory. Due to the prevention of scaling onset in commercial RO systems, induction time data for such systems were not found in the literature.

While the prediction of t_0A is a standalone model, the t_A prediction depends on t_0A as described in Equation (19). In other words, the prediction of t_A needs experiments without AS to determine k_s in Equation (15) for a specific RO system. Subsequently, further experiments employing AS based on the same RO system become imperative to determine the values of k_a and k_d, as defined in Equation (19). It is worth recalling that k_s depends on the local shear stress exerted on each cluster. Cluster size decreases with k_su_d (Equations (4) and (5)). The shear rate constant k_s is determined by fitting the model to the data, and u_d is determined from the balance equations (Equations (4)–(6)). Therefore, k_su_d necessarily includes the resultant u_d of all shear velocity differences at various points in the channels and channel types, such as narrow, spacer-filled, or tubular ones.

Different RO systems exhibit different local velocities, resulting in various shear stresses. Consequently, k_s inherently exhibits system-dependent characteristics. However, k_a and k_d are reaction rate constants that depend on each specific pair of scalant and AS molecules, independent of any particular desalination system. Consequently, k_a and k_d may be determined in advance for each pair of scalant and AS, applicable across all desalination systems. This leaves k_s as the sole fitting parameter for predicting t_0A and t_A using Equations (15) and (19), respectively, for a given RO system.

The experimental data used to validate the model were for calcium sulfate scaling, hence the exponent z in Equation (4) is 2 [41], indicating that the calcium sulfate cluster growth is surface reaction controlled. A cluster growth coefficient of k_c = 5.31 × 10⁻⁸ kg/(s m² mM²) was calculated from Table 4 of [42].

Fit results of the model in Equations (15) and (19) to the experimental data of [24] are illustrated in Figure 1. Each of the experimental/prediction pair symbols is related to an individual experiment. The model’s adjustable parameters used were k_s = 1.73 × 10⁴ s/m for t_0A and k_a = 3.17 × 10⁻⁴ L/mg s and k_d = 4.00 × 10⁻⁴ 1/s for t_A. As seen in

Table 1. Fit results of the model parameters k_s, k_a, and k_d.

Membrane System	ks (s/m)	ka (L/mg s)	kd (1/s)	* Δavrg − t0A (%)	* Δavrg − tA (%)	Ref.
Spiral-wound	1.73 × 104	3.17 × 10−4	4.0 × 10−4	10.8 (3.5–16.7)	10.8 (4.9–18.9)	[24]
Spiral-wound	2.43 × 105	1.00 × 10−3	6.5 × 10−5	10.7 (5.3–20.0)	5.2 (0.4–9.6)	[25]
Tubular	7.37 × 104	1.00 × 10−3	6.5 × 10−5	2.4 (1.5–3.2)	10.0 (6.2–15.3)	[25]

* Average prediction errors Δ_avrg of the induction times t_0A and t_A (lowest Δ–highest Δ), where Δ = 100 × |measured-predicted|/measured (t_0A or t_A).

and Figure 1, a good fit has been obtained between the predicted and experimental induction times, with an average prediction error of 10.8% for both datasets, both with and without AS. Individual errors for both datasets range between 3.5% and 18.9%.

The fit between the predicted values (Equations (15) and (19)) and experimental data from [25] is displayed in Figure 2 and Figure 3. A very good fit was obtained between the experimental and predicated induction times, with an average difference of 10.7% and 2.4% for t_0A in the spiral-wound and tubular systems, respectively, ranging from 1.5% to 20.0%. The average difference for t_A for the spiral-wound and tubular systems is 5.2% and 10.0%, respectively, with a range between 0.4% and 15.3% (Table 1). As expected, the same k_a = 1.0 × 10⁻³ L/mg s and k_d = 6.5 × 10⁻⁵ 1/s were obtained for predicting t_A in different RO membrane systems with the same pair of scalant and AS. This result supports the assumption that k_a and k_d for a specific pair of scalant and AS are independent of a particular RO system. The analysis for both studies validated the analytical model. It should be noted that generally, a tolerance of up to 30% for measurement system variability is considered acceptable [43].

4.2. Impact of the Model’s Parameters on t_0A and t_A

4.2.1. Induction Time without AS (t_0A)

Validating the model enables an analysis of the impact of its components on its magnitude. Such an analysis has the potential to optimize the efficiency of RO systems with a crossflow velocity significantly greater than the permeate velocity. For this purpose, a feed velocity of V_f = 0.189 m/s was selected. Any other velocity within the performance envelope of RO membranes, such as the 8-inch spiral-wound brackish water membrane [44], is also valid for analysis. The shear rate coefficient k_s = 2.43 × 10⁵ s/m (Table 1) and standard surface energy of 9.25 mJ/m² [24] were used in the analysis.

Since dimensionless numbers reduce the number of variables and increase the physical interpretability [45], the induction time in Equation (15) was normalized by the following:

τ_0A = t_0A/t_h

(20)

where t_h is any time magnitude at the same unit as t_0A in Equation (15). Since t_0A is calculated or measured in seconds, it is convenient to normalize it using t_h as a unit of either 60 s or 3600 s, which provides dimensionless values τ_0A numerically identical to minutes or hours, respectively.

Next, a dimensionless shear rate K_u is set as follows: K_u = k_su_f. The dimensionless parameter κ (Equation (21)) representing the denominator in Equation (15), is introduced as the target dimensionless concentration at saturation.

κ = (t_h/M_w)k_cN_AA(1)c_e^z

(21)

where A(1) is defined in Equation (13) for n = 1. For calcium sulfate, the exponent in Equation (21) z = 2 [41]. Finally, the model can be expressed in a dimensionless form:

$${\mathsf{\tau }}_{0\mathrm{A}}=\frac{3(4+\mathsf{\pi }{\mathrm{K}}_{\mathrm{u}}{)\mathrm{n}}_{\mathrm{c}}^{1/3}}{4\mathsf{\kappa }{({\mathrm{S}}_{\mathrm{a}}-1)}^{\mathrm{z}}}$$

(22)

The normalized induction time (τ_0A) in Equation (22) depends on three dimensionless parameters S_a, K_u, and κ specific to the scalant in question. Notice that, according to Equation (1), the critical cluster size (n_c) depends on S_a. The analyzed experimental data used in the present study show that K_u >> 4/π. Therefore, practically, τ_0A is directly proportional to K_u. Since S_a appears in both Equations (1) and (22), its impact on τ_0A is illustrated graphically in Figure 4. As seen in the Figure, the most influential parameter affecting τ_0A is S_a. τ_0A exhibits a monotonic decrease with S_a. For S_a < 1.6, τ_0A decreases sharply as S_a increases, while for S_a > 1.6, τ_0A decreases asymptotically with S_a. At a low S_a (<1.6), the impact of K_u and κ on τ_0A is considerable. Its impact decreases asymptotically for a high S_a (>1.6) until it becomes negligible for S_a > 2. This dimensionless analysis simplifies the understanding of the impact of all variables and constants on τ_0A (Equations (1)–(15)) by reducing it to only three variables: S_a, K_u, and κ. This dimensionless analysis provides the following quantitative conclusion: longer induction times are achieved at a high feed velocity, high local velocity in the vicinity of the cluster surface (resulting from spacers, for example), and low scalant species concentrations, and vice versa.

4.2.2. Induction Time with AS _tA

The induction time with AS, t_A, depends on two independent variables: the induction time without AS, t_0A, and the AS concentration, C_a as derived in Equation (19) for t_A. However, since Equation (19) is an implicit function of the three variables, t_0A, C_a, and t_A, it is useful to analyze the impact of t_0A and C_a on t_A through the partial derivatives via Equation (19). The derivation results are presented in Equations (23) and (24) and show that t_A increases monotonously with the increase in both t_0A and C_a.

The main parameter that contributes to the increment of t_A is k_a in both Equations (23) and (24). This means that an AS with a better adsorption rate constant yields a further increased rate of t_A with both t_0A and C_a. As expected, k_d, the desorption rate constant, has the opposite effect to that of k_a in both equations.

$$\frac{\partial {\mathrm{t}}_{\mathrm{A}}}{\partial {\mathrm{t}}_{0\mathrm{A}}}=\frac{\mathrm{b}}{{\mathrm{k}}_{\mathrm{d}}+\mathrm{a}{\mathrm{e}}^{-\mathrm{b}{\mathrm{t}}_{\mathrm{A}}}}>0$$

(23)

$$\frac{\partial t_A}{\partial C_a}=\frac{k_a\left[b^2{t}_{0A}-k_d+\left(k_d-ab{t}_A\right)e^{-b{t}_A}\right]}{b^2\left(k_d+a{e}^{-b{t}_A}\right)}>0$$

(24)

An explicit relationship between t_A and t_0A may be approximated in Equation (19) when e^−btA << 1. In this case, t_A increases linearly with t_0A, as shown in Equation (25). As a numerical example, the conditions e^−btA << 1, calculated using Equation (25) for t_A with b = 2.65 × 10⁻⁴ 1/s derived from the data of [25], resulted in t_A > 10⁴ s.

t_A ≈ (1/(b − a))(bt_0A − a/b)

(25)

5. Conclusions

When small clusters of sparingly soluble salt move at a constant velocity in the feed solution of RO systems, a small relative velocity, u_d, flowing over their surface is generated, resulting in very low Reynolds numbers (Re = u_dR/ν << 1) close to the cluster surface. This characteristic has been utilized to develop a simple analytical model for predicting the induction times in both AS-free and AS-present RO systems. The predictive model is based on principles of mass and momentum balance. The very low Reynolds number allows for the derivation of precise equations for drag and shear forces.

The main assumption for predicting t_0A is that the net growth of the cluster size arises from the interplay between adhesion and shear forces, yielding an exact equation for the induction time t_0A. For t_A, the primary assumption is the competition between AS and the scalant molecules on the surface ligands of clusters, which results in a prolonged induction time.

The model was validated by fitting it to six independent datasets generated from experiments conducted using spiral-wound and tubular membranes under various operational conditions. Average differences of 2.4% to 10.8% were obtained for t_0A, with 5.2% to 10% for t_A, between the experimental and predicted induction times, with a range from 1.5% to 20%.

The predicted induction time, t_0A, was normalized (τ_0A), yielding three independent and dimensionless parameters that exert opposite effects on τ_0A. The shear variable K_u extends τ_0A, while the supersaturation ratio S_a and the saturation parameter κ shorten it. Specifically, for S_a < 1.6, τ_0A sharply decreases as S_a increases. At higher S_a values, τ_0A asymptotically decreases with S_a, tending towards zero, rendering the impact of all three parameters negligible. An analysis of the t_A behavior indicates a monotonic increase with t_0A and C_a.

The analytical model provides simple quantitative insight into the induction times of RO systems. To facilitate the model’s prediction of the induction time for pairs of scalants and antiscalants, the parameters k_a and k_d need to be a priori determined experimentally. These parameters are common to all sizes of desalination systems.