# Revisiting the Solid Flux Theory

^{1}

^{2}

^{*}

## Abstract

**:**

_{R}, to be applied to the modified return flow formula rather than to the limiting solid flux as in the past, a significant improvement in the comparison between the results by theory and by experiments can be obtained.

## 1. Introduction

_{R}, is introduced. The paper is concluded in Section 5.

## 2. Theory

_{L}(kg m

^{−2}h

^{−1}), is commonly evaluated according to the well-known solids flux theory [15], as applied by many researchers [22,27,28,29]. The total solids flux to the final clarifier, G (kg m

^{−2}h

^{−1}), is given by the following:

_{v}(kg m

^{−2}h

^{−1}) and G

_{u}(kg m

^{−2}h

^{−1}) are the solids flux contributions due to gravity and the activated sludge extraction (underflow flux), respectively, which in turn depend on the zone-settling velocity of the activated sludge v (m h

^{−1}) and on the recycle velocity, u (m h

^{−1}), respectively, and on the suspended solids (SS) concentration, x (kg m

^{−3}), also denoted as biomass concentration.

_{0}(m h

^{−1}) and k (m

^{3}kg

^{−1}) are empirical coefficients expressing the settling velocity under zero suspended solids concentration (scale factor) and the exponential decay constant (shape factor), respectively.

_{L}(kg m

^{−2}h

^{−1}), requires putting the first derivative of Equation (3) with respect to x equal to zero:

_{L}(kg m

^{−3}):

_{L}. Contrarily to what will be shown here, x

_{L}has usually been obtained numerically [17], as observed in Section 1. The G value corresponding to limiting solids flux, G

_{L}, is obtained by imposing x = x

_{L}into Equation (1):

_{L}can also be expressed as follows:

_{L}by u (m h

^{−1}) provides the SS concentration of the recycle flow rate, x

_{r}(kg m

^{−3}):

_{0}= 17.12 m h

^{−1}and k = 0.452 m

^{3}kg

^{−1}, and by assuming u = 0.5 m h

^{−1}, Figure 1a shows the solids flux contributions due to gravity, G

_{v}, and due to the activated sludge extraction, G

_{u}, and their sum G (Equation (1)), which admits the minimum G

_{L}= 6.8 kg m

^{−2}h

^{−1}for x = x

_{L}= 10.82 kg m

^{−3}(Equation (6)). The corresponding SS concentration of the recycle flow rate x

_{r}, which lays on the straight-line G

_{L}= u x

_{r}(Equation (9)), is also indicated. The equation of the dashed straight line is as follows:

_{v}curve at the point x = x

_{L}, showing that at increasing G

_{L}, the recycle flow rate must increase too, but the corresponding SS concentration will reduce (the sludge will be more diluted).

_{max}associated with the minimum biomass concentration x

_{min}, which, together with x

_{L}, delimits the biomass concentration domain (x

_{min}≤ x ≤ x

_{L}), which can be analyzed by varying u, as will be shown in Section 2.5.

_{L}, the clarifier surface area, A (m

^{2}), is derived by the clarifier mass balance, yielding the following:

_{0}(kg m

^{−3}) is the influent SS concentration to the final clarifier; Q (m

^{3}h

^{−1}) and Q

_{r}(m

^{3}h

^{−1}) are the wastewater and the recycle flow discharges, respectively; and R = Q

_{r}/Q is the return ratio.

_{y}, which is useful for the design purpose, as it is known:

#### 2.1. Introducing the Lambert W Function and Dimensionless Groups

_{0}, k, and u), although it does not allow a generalization to any input parameters (v

_{0}, k, and u); moreover, Equation (6) is in an implicit form. To achieve the aim of generalizing the SFT, dimensionless groups can be suitably introduced. In this section, it is shown that introducing the Lambert W function helps address this issue and suggests consolidating input and output parameters in a compacted design procedure. Indeed, dimensionless groups are useful for scaling arguments; for consolidating experimental, analytical, and numerical results into a compact form; and to delimit the parameters domain of interest.

_{−1}of the Lambert W function, also denoted as the omega function or product logarithm, Equation (6), which is usually solved numerically [17], can also be expressed in the following form:

_{*}denotes the u value normalized with respect to v

_{0}:

^{y}, where y is any complex number and e

^{y}is the exponential function, and it has been applied in different contexts [38,39]. When dealing with real numbers, as in the considered case, only the two brunches W

_{0}and W

_{−1}can be considered, and the general equation y e

^{y}= x can be solved for y, if x ≥ −1/e. In particular, for x > 0, y = W

_{0}(x), and for −1/e ≤ x < 0, the two values W

_{0}and W

_{−1}occur. In this case, imposing the condition x ≥ −1/e provides:

_{0}values, including v

_{0}= 7.4 m h

^{−1}(i.e., u

_{th}= 1 m h

^{−1}) and 17.12 m h

^{−1}(Figure 1a), Table 1 reports the corresponding u

_{th}= v

_{0}e

^{2}values.

_{*}), a graphical illustration of the u

_{*}domain for which Equation (13) provides real solutions can be performed. Figure 2 plots the two branches of the Lambert W function, W

_{0}and W

_{−}

_{1}, versus −e u

_{*}, together with the threshold value −e u

_{*th}(Table 1) and the vertical dashed line that delimits the physical circumstance that u

_{*}has to be positive (downward), depicting the x

_{L}real solutions domain. Figure 2 also illustrates the W value corresponding to u

_{*}= 0.0292 (−e u

_{*}= −3.892) that refers to the application of Figure 1 and to the applications that will be performed later, which lay on the W

_{−1}brunch.

_{0}brunch allow for calculating the minimum value of the biomass concentration x

_{min}, associated with G

_{max}(Figure 1a), if replacing W

_{−}

_{1}by W

_{0}into Equation (13), as will be recalled in Section 2.5.

_{0}= 17.12 m h

^{−1}and k = 0.452 m

^{3}kg

^{−1}); for u = 1, 1.5; and for the threshold u

_{th}= 2.32 m h

^{−1}(Table 1), for which the minimum of Equation (3) does not occur, the effect of the recycle velocity is shown in Figure 1b–d, where the solid fluxes vs. the biomass concentration are plotted.

_{0}and the SS concentration of the recycle flow rate x

_{r}decrease. For u = u

_{th}, the minimum does not occur, but a horizontal point of inflection occurs where the curvature of the G function changes sign, thus producing a flat tangent line as the double derivate will equal to zero at its coordinates. This means that for u > u

_{th}, x

_{0}itself determines the limiting concentration of the incoming biomass to the final clarifier. For u = u

_{th}, the underflow flux achieves the solid flux contribution due to gravity (G

_{u}= G

_{v}, Figure 1d). The occurrence u = u

_{th}is illustrated by the red dot in the Lambert W function (Figure 2).

_{L}in their dimensionless product, we can write:

_{0}(kg m

^{−2}h

^{−1}) a “virtual” solids flux corresponding to the settling velocity under zero SS concentration [40,41], we can write the following:

_{L}with respect to G

_{0}provides the dimensionless G

_{*}and G

_{*L}relationships, respectively:

_{L}, derived by the simplified SFT [16]. This occurrence was ascribed to the hydrodynamics of the final clarifier, which behaves differently to what the simplified 1D SFT is able to describe [42].

_{*L}is plotted in Figure 3 (red lines) versus the normalized recycle velocity, u

_{*}, in a log-log scale (Figure 3a) and in a linear scale (Figure 3b), together with the k x

_{L}parameter (Equation (18)) versus u

_{*}. The key parameter k x

_{L}is useful to derive since it allows for determining any k and u

_{*}values, the limiting SS concentration, x

_{L}, and thus G

_{*L}. In dimensionless terms, besides u

_{*}, no parameters are required since they are arranged in the dimensionless groups; thus, Figure 3 covers all the possible combinations of the design parameters.

_{*}condition, u

_{*th}, does come back to k x

_{L}as well as G

_{*L}, which are denoted as (k x

_{L})

_{th}and (G

_{*L})

_{th}, respectively, as indicated in Figure 3, laying in the vertical dashed line, u

_{*}= u

_{*th}(Table 1). In Figure 3, the pairs (u

_{*}, k x

_{L}) and (u

_{*}, G

_{*L}) corresponding to the applications of Figure 1 are also indicated.

_{L}(Equation (21)) also allows for inspection of the behavior of the SS concentration of the recycle flow rate, Q

_{r}, corresponding to the limiting condition x

_{r}(kg m

^{−3}), the return rate R, and the hydraulic loading rate C

_{h}.

_{r}, substituting Equations (5) and (8) into Equation (9) provides the following:

_{L}as follows:

_{*}, confirming the dominant role of k x

_{L}in the final clarifier behavior.

_{0}, differs from the limiting SS concentration, x

_{L}, and it can be shown that x

_{0}is related to x

_{L}and to the return ratio, R, even by a dimensionless relationship. To show this, the mass balance needs to be invoked, as described in the next section, in dimensionless terms.

#### 2.2. The Return Ratio by Dimensionless Groups

_{0}(kg m

^{−3}) is the influent SS concentration to the final clarifier. Substituting Equation (25) into Equation (27) yields the following:

_{0}and k x

_{L}:

_{0}≤ 2 and k x

_{0}≥ 2, by varying k x

_{L}, Equation (29) is plotted in Figure 4a and Figure 4b, respectively. The figures show that k x

_{L}decreases at increasing R, as could be expected. For a fixed R, at increasing k x

_{0}, k x

_{L}increases for both k x

_{0}≤ 2 and k x

_{0}≥ 2, with a greater influence on the latter that could require unsuitable high R > 1 values. Contrarily, it can also be observed that for k x

_{0}≤ 2, R is less than one.

_{0}≤ 2 and for any R ≤ 1, (k x

_{L})

_{th}equals 2, whereas for k x

_{0}≥ 2 and for R ≥ 1, since the x

_{L}minimum does not occur, x

_{0}itself provides the limiting SS concentration. Therefore, imposing x

_{L}= x

_{0}into Equation (29) yields the following:

_{c}denotes the corresponding limiting return ratio. Figure 4a also indicates the dot corresponding to the application performed later.

_{*}= u

_{*th}, it can also be verified that substituting (k x

_{L})

_{th}= 2 into Equation (30) provides R

_{c}= 1, as indicated in Figure 4a,b. In conclusion, the following limiting conditions occur:

_{L}and Equation (29) makes it possible to determine, for any R and k x

_{0}, the limiting solid flux, G

_{*L}, and the SS concentration of the recycle flow rate, k x

_{r}, by using Equations (24) and (26), respectively.

#### 2.3. Normalized Hydraulic Loading Rate

_{0}, the dimensionless hydraulic loading rate C

_{*h}can be expressed as:

_{*h}can also be rewritten as follows:

_{L}in C

_{*h}. Equation (34) shows the dependence of C

_{*h}on ρ, R, k x

_{L}, and k x

_{0}. The normalized influent SS concentration k x

_{0}can be also expressed by Equation (29):

_{L}, and thus R (Equation (29)), for k x

_{0}≤ 2 and k x

_{0}≥ 2, Equation (36) is plotted in Figure 5a and Figure 5b, respectively. The figures show that C

_{*h}increases at increasing R, and that for a fixed R at increasing k x

_{0}, C

_{*h}increases for k x

_{0}≤ 2 whereas it decreases for k x

_{0}≥ 2.

_{0}≥ 2 correspond low C

_{*h}values that of course make this occurrence not recommended. It can also be observed that for k x

_{0}≤ 2, R is less than the unity, whereas for the not recommended condition k x

_{0}≥ 2, R can also be higher than the unity.

_{*h}condition was also plotted. Similarly to k x

_{L}, two cases can be distinguished: (i) k x

_{0}≤ (k x

_{L})

_{th}(R

_{c}≤ 1) and (ii) k x

_{0}≥ (k x

_{L})

_{th}(R

_{c}≥ 1).

_{0}≤ (k x

_{L})

_{th}(R

_{c}≤ 1), by putting k x

_{L}= (k x

_{L})

_{th}= 2 into Equation (29), the limiting condition can be written as a function of k x

_{0}:

_{L})

_{th}= 2 and substituting Equation (37) into Equation (36), the corresponding limiting C

_{*h}, (C

_{*h})

_{th}, can be derived (Figure 5a):

_{0}≥ (k x

_{L})

_{th}, since the x

_{L}minimum does not occur, x

_{0}itself provides the limiting SS concentration, as previously observed. Putting x

_{L}= x

_{0}into Equation (35), and considering that R

_{c}can be expressed by Equation (30), the limiting C

_{*h}condition equals the following (Figure 5b):

_{c}= 1, both Equations (38) and (39) yield (C

_{*h})

_{th}= e

^{−2}(0.1353), matching u

_{*th}, as reported in Table 1.

_{c}(x

_{0}= x

_{L}), C

_{*h}does not depend on R (Figure 5a,b). The latter can be derived by substituting Equation (32) into Equation (36), showing that C

_{*h}matches the normalized settling velocity corresponding to x

_{0}(ρ e

^{−k x}

_{0}, Equation (2) with ρ = 1).

_{0}, but during the operating conditions, e.g., after heavy rainfalls (storm water influent flow), loading variability can be expected. This issue is addressed in the next section.

#### 2.4. Varying the Influent SS Concentration

_{0}, does not figure into Equation (36) but of course affects C

_{*h}, since x

_{L}depends on u (Equation (13)), which in turn depends on R (u = RQ/A) and thus x

_{0}(Equation (29)). To express C

_{*h}as a function of k x

_{0}, which could be useful in practice, a dimensionless relationship between k x

_{L}and k x

_{0}needs to be determined. From Equation (29), we can write the following:

_{0}(1 + R) equals the rate of the amount of volumetric solids to the final clarifier, Q

_{s}(m

^{3}h

^{−1}), normalized with respect to the influent discharge to WWTP, Q (m

^{3}h

^{−1}).

_{L}. The corresponding discriminant, Δ, is as follows:

_{L}> 2 (Equation (19)), only the solution corresponding to the plus sign needs to be taken into account:

_{0}, k, and R, Equations (36) and (44) make it possible to determine the normalized hydraulic loading rate, C

_{*h}, as a function of k x

_{0}, and thus, for any Q, the surface area clarifier area, A, which would be necessary to cope with the inflow SS loadings’ variability (k x

_{0}).

_{L}(Equation (44)) and C

_{*h}(Equation (36)) as a function of k x

_{0}. The limiting conditions (k x

_{L})

_{th}= 2 and Equation (38) are also illustrated. As expected, for any selected SS sample (v

_{0}and k), an increase in the influent SS concentration (k x

_{0}) determines a k x

_{L}increase but also requires decreasing C

_{*h}—i.e., decreasing Q—or increasing the clarifier surface area (for the design purpose) or the return ratio. Figure 3, Figure 4 and Figure 6 strongly evidence its general validity for all the design/verification parameters’ combinations.

#### 2.5. The Domains of the SS Concentration and of the Solid Flux

_{min}delimits the biomass concentration domain (x

_{min}≤ x ≤ x

_{L}), which can even be analyzed in dimensionless terms by varying u

_{*}. The possible domain of the biomass concentration, Δx

_{*}, and the corresponding solid flux domain, ΔG

_{*}, are expressed by the following:

_{min}by replacing W

_{−1}with W

_{0}in Equation (21), using Equation (24) to calculate G

_{*max}and G

_{*L}, and substituting into Equation (45) and Equation (46) provides Δx

_{*}and ΔG

_{*}relationships that only depend on W

_{−1}and W

_{0}:

_{*}. As expected, both Δx

_{*}and ΔG

_{*}, in between which the pairs (k x

_{0}, G

_{*}) could lay, decrease at increasing u

_{*}, meaning that for high u

_{*}, WWTP misoperation could be expected, depending on the occurring variable influent SS concentration. Of course, for u

_{*}= u

_{*th}= 0.1353 (Table 1), Δx

_{*}and ΔG

_{*}are equal to zero. In Figure 7a, the black dots correspond to the parameters Δx

_{*}and ΔG

_{*}of the application performed in the next section (Figure 7b).

## 3. Example of Application

^{3}h

^{−1}and operates with a return ratio R = 40%, so that the return sludge discharge Q

_{r}equals 21.6 m

^{3}h

^{−1}. The parameters k and v

_{0}of the Vesilind model are equal to 8 m h

^{−1}and 0.375 m

^{3}kg

^{−1}, respectively, and the clarifier surface area A is 60.16 m

^{2}. By assuming ρ = 1 for an influent SS concentration to the final clarifier of x

_{0}= 4.27 kg m

^{−3}(k x

_{0}= 1.6), we want to establish whether the WWTP operates properly—i.e., the solids do not escape in the supernatant—and how the WWTP operates under R or x

_{0}variability.

_{*}= R Q/(v

_{0}A), equals 0.045, which for e u

_{*}= 0.122 (W

_{−1}= −3.3, Figure 2) allows calculating k x

_{L}by Equation (21) (Figure 3), resulting to 4.297. Thus, k x

_{L}is properly higher than (k x

_{L})

_{th}= 2. Knowing u

_{*}makes it possible to calculate G

_{*L}= 0.251 and k x

_{r}= 5.6, which are also indicated in Figure 3. The goodness of the WWTP design can also be observed in Figure 4a, where for k x

_{0}= 1.6, the pair (R = 0.4, k x

_{L}= 4.297) is illustrated, together with the limiting condition.

_{0}and R variation can be observed in Figure 4a and Figure 5a, and in Figure 6, where the considered parameters, k x

_{0}= 1.6 and R = 0.4, match those of this application, and the corresponding dots are indicated. In Figure 5a, it can be observed that at increasing R, C

_{*h}, which for x = x

_{0}equals 0.112, increases too, meaning that for the fixed clarifier surface area, A = 60.16 m

^{2}, a higher influent discharge, Q, to the WWTP should be required. Meanwhile, for a fixed R = 0.4 at increasing k x

_{0}, Figure 5a shows that C

_{*h}decreases, and so should Q.

_{0}= k x

_{L}), the normalized settling velocity, v/v

_{0}, highly decreases (from 0.202 to 0.014), and for the imposed Q, the clarifier surface area should be 161.55 m

^{2}—thus 2.68 times greater than for x = x

_{0}(Table 2).

_{*v}and G

_{*u}and their sum G

_{*}(Equation (1)) versus the normalized biomass concentration, k x, as graphed in Figure 7b. Figure 7b also shows the normalized solid fluxes corresponding to both k x

_{0}= 1.6 and k x

_{L}= 4.3, in green and red dots, respectively. The minimum value of the biomass concentration x

_{min}, associated with G

_{max}, and the domains Δx

_{*}and ΔG

_{*}are also displayed in Figure 7b.

_{0}= v

_{0}/k (Equation (22)), which is equal to 21.35 kg m

^{−2}h

^{−1}. In dimensional terms, for the imposed influent discharge Q = 54 m

^{3}h

^{−1}, Figure 8 shows the expected decreasing trends versus u

_{*}of x

_{L}, x

_{r}, and G

_{u}/G

_{v}in the secondary axis for x = x

_{L}and x = x

_{0}, A(x

_{L}) and A(x

_{0}), respectively, and the increasing trend of G

_{L}.

_{h}= 3 h (i.e., clarifier volume V = Q t

_{h}= 162 m

^{3}), the height of the final clarifier, H, versus u

_{*}is also graphed. For any u

_{*}value, the clarifier surface area corresponding to x

_{0}= 4.27 kg m

^{−3}, A(x

_{0}), provides lower values than A(x

_{L}). The parameters corresponding to the application of Figure 7b are indicated by black dots.

_{u}/G

_{v}≥ 1 which matches the ratio u/v

_{L}is interesting to consider since it represents how much higher the involved recycling flow rate, related to the energy required for recycling, is than the gravitational settling velocity. Of course, for u = u

_{th}, the ratio G

_{u}/G

_{v}≡ u/v

_{v}is equal to the unity. This occurrence is seldom achieved in practice since it is associated with low x

_{L}values and high values of the settling velocity. Moreover, for u > u

_{th}, the energy required for the sludge extraction would be less than that corresponding to the gravitational hindered settling, but high u values make the wastewater treatment not efficient from an energetic point of view. In practice, high u values also need to be checked according to the required cellular residence time, depending on the wastewater characteristics.

## 4. Comparison with Experimental Data and the New Hydrodynamic Factor

_{0}= 1.8, 2.3, 3.0, and 3.8 kg m

^{−3}, which correspond to k x

_{0}= 0.864, 1.104, 1.440, and 1.840, respectively, for the k = 0.48 m

^{3}kg

^{−1}Vesilind parameter that we considered.

_{*L,m,i}, and those calculated by Equation (24) was determined:

_{*}< u

_{*th}. The SEE values are plotted in Figure 9a, showing that for ρ = 0.836, the minimum SSE, SSE

_{min}= 0.0856, was obtained (Table 4).

_{*L}values is plotted in Figure 9b, with k x

_{0}as a parameter. The obtained ρ values almost agree with the results found by other researchers such as Ekama and Marais [42] and Gohle et al. [43], who suggested ρ = 0.8. Watts et al. [32] also found ρ values in the range of 0.55–0.94, with an average of ρ = 0.73.

_{*h}versus R is graphed (Equation (36)) together with the corresponding experimental values derived by d’Antonio and Carbone (1987) for ρ = 1 (Figure 10a, no correction) and for ρ = 0.836 (Figure 10b), with k x

_{0}as a parameter.

_{h}measurements carried out for u

_{*}< u

_{*tk}and the 18 measurements carried out for u

_{*}> u

_{*th}(Figure 11a), yielding ρ = 0.809.

_{*L}only partially corrects the simplified SFT, another attempt was made to improve the accuracy of the predictions derived by the SFT because according to Equation (24), k x

_{L}, which affects the other design variables, certainly also requires correction.

_{*L}on k x

_{L}is not linear; applying a correction to R by introducing a new reduction hydrodynamic factor, ρ

_{R}, into Equation (29) would overcome this issue:

_{R}= 0.796 and Equation (36) with ρ = 1 strongly improved the fitting of the experimental measurements to the new dimensionless SFT (especially for k x

_{0}= 0.864), as can be observed in Figure 11b. The corresponding SEE values seem to validate the new ρ

_{R}hydrodynamic factor to be applied to Equation (50) since for this scenario, the lowest SEE = 0.0378 was obtained (Table 4).

## 5. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Solids flux contributions due to gravity, G

_{v}, and the activated sludge extraction, G

_{u}, and their sum G (Equation (1)) versus the biomass concentration, x, for (

**a**) u = 0.5 m h

^{−1}, (

**b**) u = 1 m h

^{−1}, (

**c**) u = 1.5 m h

^{−1}, and (

**d**) u = u

_{th}= 2.32 m h

^{−1}. The limiting SS concentration, x

_{L}; the corresponding solid flux, G

_{L}; and the SS concentration of the recycle flow rate, x

_{r}, are also indicated.

**Figure 2.**The two branches of the Lambert W function, W

_{0}and W

_{−}

_{1}. In the branch W

_{−}

_{1}, the Lambert W function values corresponding to the application performed later are indicated. The domain of the real x

_{L}solutions − e u

_{*th}≤ − e u

_{*}≤ 0 and the dots corresponding to the applications are also reported.

**Figure 3.**Relationship between the dimensionless parameters k x

_{L}(Equation (21)), G

_{*L}(Equation (24)), and k x

_{r}(Equation (26)) versus the normalized velocity recycle flow rate u

_{*}(

**a**) in a log-log scale and (

**b**) in a linear scale. Dots correspond to the applications performed later. For u

_{*}= u

_{*th}= 0.1353, the threshold values (k x

_{L})

_{th}, (G

_{*L})

_{th}, and (k x

_{r})

_{th}(Table 1) are also indicated.

**Figure 4.**Dimensionless parameters k x

_{L}(Equation (29)) versus the return ratio, R, (

**a**) for k x

_{0}≤ 2 and (

**b**) for k x

_{0}≥ 2. The limiting k x

_{L}conditions (Equations (31) and (32)) are also indicated. In (

**a**), the dot corresponds to the applications performed later.

**Figure 5.**Hydraulic loading rate normalized with respect to v

_{0}, C

_{*h}(Equation (36)) versus the return ratio, R, for (

**a**) k x

_{0}≤ 2 and (

**b**) k x

_{0}≥ 2. The limiting C

_{*h}conditions (Equation (38) and Equation (39), respectively) are also indicated. In (

**a**), the dot corresponds to the application performed later.

**Figure 6.**Relationship between k x

_{L}and C

_{*h}versus k x

_{0}for different return ratios R. The limiting k x

_{L}(k x

_{L}= 2), and C

_{*h}conditions (Equations (38)) are also indicated.

**Figure 7.**(

**a**) Biomass concentration and solid flux domains, Δx

_{*}and ΔG

_{*}(secondary axis), versus u

_{*}. Dots refer to the application of (

**b**). (

**b**) For the parameters of the example application (Table 2 and Table 3), G

_{*v}, G

_{*u}, and their sum G

_{*}versus the normalized biomass concentration k x. The normalized minimum value of the biomass concentration k x

_{min}, associated with G

_{*max}, and the biomass concentration and solid flux domains are also indicated.

**Figure 8.**For k = 0.375 m

^{3}kg

^{−1}and t

_{h}= 3 h, relationship between the dimensional parameters: limiting SS concentration, x

_{L}; the corresponding solid flux G

_{L}; the SS concentration of the recycle flow rate, x

_{r}; the G

_{u}/G

_{v}ratio; the clarifier height, H; and the clarifier surface area, A (secondary axis), for x = x

_{L}and x = x

_{0}, A(x

_{L}) and A(x

_{0}), respectively, versus u

_{*}. The dots refer to the example application of Figure 7b. The dashed line delimits the real x

_{L}solutions domain (u

_{*th}= 0.1353).

**Figure 9.**(

**a**) Standard error on the estimate, SEE, between the experimental limiting normalized solid flux, G

_{*}, obtained by data of d’Antonio and Carbone [34], and the corresponding theoretical values (Equation (24)) versus the limiting solid flux reduction factor ρ. (

**b**) Comparison between measured and calculated G

_{*L}for the minimum SEE value, SEE

_{min}, obtained for ρ = 0.836, for different k x

_{0}values.

**Figure 10.**Relationship between the normalized hydraulic loading rate, C

_{*h}(Equation (36)), versus R (

**a**) for ρ = 1 and (

**b**) for ρ = 0.836 for the k x

_{0}values (k x

_{0}≤ 2), corresponding to the experimental measurements carried out by d’Antonio and Carbone [34]. The corresponding values derived by d’Antonio and Carbone [34] are also plotted.

**Figure 11.**Relationship between the normalized hydraulic loading rate, C

_{*h}(Equation (36)), versus R (

**a**) for ρ = 0.809 and (

**b**) for ρ = 1 and ρ

_{R}= 0.796 for the k x

_{0}values (k x

_{0}≤ 2), corresponding to the experimental measurements carried out by d’Antonio and Carbone [34]. The corresponding values derived by d’Antonio and Carbone [34] are also plotted.

**Table 1.**For different v

_{0}values, threshold u values, u

_{th}, and the corresponding u

_{*th}, −e u

_{*th}, (k x

_{L})

_{th}, (G

_{*L})

_{th}, (k x

_{r})

_{th}, and for R

_{c}= 1, (C

_{*h})

_{th}, for which Equation (21) admits real x

_{L}solutions.

v_{0} (m h^{−1}) | u_{th} (m h^{−1}) | u_{*th} | −e u_{*th} | (k x_{L})_{th} | (G_{*L})_{th} | (k x_{r})_{th} | (C_{*h})_{th} |
---|---|---|---|---|---|---|---|

2.1 | 0.28 | 0.1353 | −0.3679 | 2 | 0.5413 | 4 | 0.1353 (R _{c} = 1) |

3 | 0.41 | ||||||

4 | 0.54 | ||||||

5 | 0.68 | ||||||

6 | 0.81 | ||||||

7 | 0.95 | ||||||

7.4 | 1.00 | ||||||

17.12 | 2.32 |

**Table 2.**Dimensional parameters and their symbols and values, corresponding to the application of Figure 7b.

Dimensional Parameters | Symbol | Value |
---|---|---|

Settling velocity under zero SS concentration (x = 0) | v_{0} (m h^{−1}) | 8.0 |

Exponential decay constant | k (m^{3} kg^{−1}) | 0.375 |

Virtual solids flux under zero SS concentration (x = 0, v = v_{0}) | G_{0} (kg m^{−2} h^{−1}) | 21.35 |

Influent SS concentration to the final clarifier | x_{0} (kg m^{−3}) | 4.27 |

Settling velocity for settling velocity for x = x_{0} | v (x_{0}) (kg m^{−3}) | 1.62 |

Total solid flux for x = x_{0} | G (x_{0}) (kg m^{−2} h^{−1}) | 8.43 |

Influent discharge to the treatment plant | Q (m^{3} h^{−1}) | 54.0 |

Return sludge discharge | Q_{r} (m^{3} h^{−1}) | 21.6 |

SS concentration of the recycle flowrate | x_{r} (kg m^{−3}) | 14.94 |

Recycle velocity | u (m h^{−1}) | 0.359 |

Limiting SS concentration | x_{L} (kg m^{−3}) | 11.47 |

Limiting solids flux | G_{L} (kg m^{−2} h^{−1}) | 5.37 |

Total solid flow to the final clarifier | (Q + Q_{r}) x_{0} (kg h^{−1}) | 322.8 |

Clarifier surface area | A (m^{2}) | 60.16 |

Hydraulic loading rate | C_{h} (m h^{−1}) | 0.90 |

**Table 3.**Dimensionless groups and their symbols and values, corresponding to the application of Figure 7b.

Dimensionless Parameters | Symbol | Value |
---|---|---|

Return ratio | R | 0.40 |

Recycle velocity, u, normalized with respect to v_{0} | u_{*} | 0.045 |

Dimensionless influent SS concentration | k x_{0} | 1.600 |

Dimensionless limiting SS concentration | k x_{L} | 4.297 |

Limiting solids flux normalized with respect to G_{0} | G_{*L} | 0.251 |

Dimensionless SS concentration of the recycle flowrate | k x_{r} | 5.600 |

Hydraulic loading rate normalized with respect to v_{0} | C_{*h} | 0.112 |

**Table 4.**Sum square error, SSE; sample size, N; and standard error of the estimate, SEE, for different values of the reduction hydrodynamic factor.

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**MDPI and ACS Style**

Baiamonte, G.; Baiamonte, C.
Revisiting the Solid Flux Theory. *Soil Syst.* **2022**, *6*, 91.
https://doi.org/10.3390/soilsystems6040091

**AMA Style**

Baiamonte G, Baiamonte C.
Revisiting the Solid Flux Theory. *Soil Systems*. 2022; 6(4):91.
https://doi.org/10.3390/soilsystems6040091

**Chicago/Turabian Style**

Baiamonte, Giorgio, and Cristina Baiamonte.
2022. "Revisiting the Solid Flux Theory" *Soil Systems* 6, no. 4: 91.
https://doi.org/10.3390/soilsystems6040091