Improved fixed bed models for correlating asymmetric adsorption breakthrough curves

Breakthrough curves of water contaminants are usually analyzed using simple fixed bed models such as the Bohart-Adams, Thomas, and Yoon-Nelson equations, which are by design symmetric. Because breakthrough data often follow an asymmetric pattern, the use of models that do not account for asymmetry could lead to poor fits, consequently resulting in erroneous estimates of breakthrough and exhaustion times. To address this issue, the Bohart-Adams, Thomas, and Yoon-Nelson models were modified by a logarithmic transformation to enhance their data fitting ability. The three modified models were found capable of providing robust fits to seven separate sets of previously reported asymmetric breakthrough data of water contaminants (fluoride, methylene blue, salicylic acid, lead, mercury, nickel, and arsenic), with reported residual root mean square error (RRMSE) values ranging from 0.019 to 0.046. In consequence, the new models were found capable of providing reliable estimates of breakthrough and exhaustion times corresponding to any predetermined concentration level. By contrast, the three original models were found to perform poorly, reporting inferior RRMSE values ranging from 0.038 to 0.086 for data fits and providing grossly inaccurate estimates of breakthrough and exhaustion times. The new models contain only parameters that appear in the original models, and are highly flexible, being able to assume virtually all monotonically increasing sigmoid shapes. They represent a far more accurate alternative to the original models.


Introduction
Adsorption is considered a highly efficient process technology for drinking water and wastewater purification. Because practical applications of adsorption are most often realized in fixed bed mode, considerable attention has been devoted to measuring contaminant breakthrough data using laboratory-scale adsorption columns. Simple fixed bed models such as the Bohart-Adams [1], Thomas [2], and Yoon-Nelson [3] equations are frequently employed for breakthrough curve analysis. Fixed bed models based on Fick's law of diffusion for the solid phase are more realistic [4,5], but they generally require the use of specialized software and programming code. For descriptions of some of the commonly used fixed bed models, the reader is referred to a handful of review articles that have been published in the literature of water contaminant adsorption over the last 30 years [6][7][8][9]. 2 The Bohart-Adams, Thomas, and Yoon-Nelson models are often used in a "correlative" modeling approach, where the model equations are fitted to breakthrough data for parameter estimation.
For design purposes, it is necessary to establish empirical relationships between model parameters and system and operational conditions such as bed length, flow rate, and initial contaminant concentration.
Such a correlative design approach is best exemplified by the bed-depth-service-time or BDST method, which is based on the Bohart-Adams model. The well-known book by Cooney [10] provides practical guidance on the use of the Bohart-Adams, Thomas, and BDST models for sizing activated carbon columns. The recent introductory book by Tien [11] affords a comprehensive overview of these models, which have a long history of use in breakthrough curve modeling.
A prolific research literature on the use of the Bohart-Adams, Thomas, and Yoon-Nelson models in breakthrough curve analysis exists. As examples one may cite two very recent studies [12,13] published in this Journal. Their ubiquitous applicability is attributable to several reasons. One reason is that the availability of linearized forms of the three models allows straightforward data fitting by linear regression. Another reason is that application of the Bohart-Adams and Thomas models needs little input information, for instance, they do not require knowledge of bed void fraction and adsorbent density. At the far end of the scale, not even flow rate or bed length is required by the Yoon-Nelson model, which is devoid of typical system and operational variables.
In recent years, the three popular models have encountered criticisms on two fronts. First, they have been shown to be mathematically equivalent to one another [14][15][16]. As such, they should not be treated separately (as they commonly are). Second, they cannot describe asymmetric breakthrough data to a significant degree of precision because their functional form is intrinsically symmetric. It is the second issue that is the subject of this article.
It is of note that the bulk of reported experimental breakthrough curves are more or less asymmetric, which is largely a result of fronting or tailing. Fronting, also known as leakage, refers to early appearance of contaminant in the effluent while tailing describes a phenomenon where the effluent exhibits a slow approach toward the influent concentration near column saturation. Indeed, it is rare to observe perfectly symmetric breakthrough curves. The inability of the three models to track asymmetric breakthrough data could lead to inaccurate estimates of breakthrough and exhaustion times.
Breakthrough times are fundamental to the evaluation of adsorption column performance while exhaustion times are needed to gauge the performance of series-column systems where certain adsorbers are operated to complete saturation prior to regeneration or replacement.
There have been several previous efforts to boost the data fitting ability of the three models.
One of the earlier attempts was made by Clark and colleagues [17][18][19], who developed a modified logistic equation that incorporates the exponent of the Freundlich isotherm, based on the fact that the Bohart-Adams model is analogous to the logistic equation. Yan et al. [20] incorporated some parameters of the Thomas model in a dose response model containing an empirical exponent. Blagojev et al. [21] enhanced the fitting ability of the Yan model by extending it to a five-parameter equation. Hu et al. [22] converted the Bohart-Adams, Thomas, and Yoon-Nelson models to power law forms by introducing empirical exponents to the original models.
So far, most of the model improvement efforts have adopted the same strategy: addition of one or more empirical shape parameters. This approach works well because extra parameters tend to enhance the data fitting ability of a model. The present paper presents a new way to modify the Bohart-Adams, Thomas, and Yoon-Nelson equations. The modification method is a new departure from the work of earlier authors as it does not rely on extra shape parameters. The three modified models contain only parameters appearing in the original models and can be linearized. Previously reported breakthrough data of water contaminants exhibiting varying degrees of curve asymmetry are used to test the new models. Specific cases examined are the fixed bed adsorption of fluoride, methylene blue, salicylic acid, lead, mercury, nickel, and arsenic by adsorbents of different types.

The Bohart-Adams model
This model [1], originally developed for gas phase adsorption, was reparameterized and presented by Eckenfelder [23] and Eckenfelder and Ford [24] as a design tool for sizing liquid phase activated carbon columns based on pilot test breakthrough data. A simplified version of the Bohart-Adams model is written as

The Thomas model
This model was originally developed to describe liquid phase ion-exchange in fixed bed columns [2]. A simplified form of the original Thomas model [25], which was reparameterized and presented by Reynolds [26] as a model for sizing fixed bed adsorbers, reads where kT ≡ the Thomas

The Yoon-Nelson model
This model, formulated for describing the breakthrough characteristics of adsorption cartridges for respiratory protection [3], may be expressed as where kYN ≡ the Yoon-Nelson rate coefficient and τ ≡ the time required for 50% breakthrough. Eq. (5) can be rearranged to give the following linear form:

Mathematical equivalence
As mentioned previously, the Bohart-Adams, Thomas, and Yoon-Nelson models are mathematically analogous to one another [14][15][16]. A simple way to demonstrate this mathematical equivalence is to express each model in terms of the logistic equation with two general parameters: where the parameters a and b are given by Fitting the logistic equation given by Eq. (7) to breakthrough data will yield a and b from which all the unknown parameters of the three models may be computed using Eqs. (9)- (11). Accordingly, it is not necessary to fit the three models one by one to the same set of breakthrough data in order to estimate their parameters.

Modified models
To enhance the fitting ability of the three models, we propose that certain groups of parameters and variables in the three models be replaced with logarithmic terms. Since one can take the logarithm only of pure, dimensionless numbers, the proposed logarithmic transformation is best illustrated by an example.
Step 2: We define a new set of variables/parameters in the chosen system of units whose numerical values are equal to unity, e.g., * Step 3: The new variables/parameters are introduced into the Bohart-Adams model: Note that the preceding equation is numerically identical to the original Bohart-Adams model.
Step 4: Upon rearrangement of the four bracketed terms within the exponential term in Eq. (12), Step 5: We define two new parameters: Step 6: Substitution of Eq. (14) into Eq. (13) yields All the four bracketed terms within the exponential term in Eq. (15) are dimensionless.
Step 7: Eq. (15) is empirically modified by taking the logarithm of the second and fourth bracketed terms within the exponential term: Step 8: Because α and β are numerically equal to unity, for convenience, these two parameters are omitted from Eq. (16): The preceding equation is the new form of the Bohart-Adams model proposed in this work. It is referred to as the log-Bohart-Adams model.
In this work, Eqs. (18) and (19) are known as the log-Thomas model and the log-Yoon-Nelson model, respectively. Just like the original models on which they are based, the log-Bohart-Adams, log-Thomas, and log-Yoon-Nelson models are equivalent to one another. Note that each of the three modified models remains a two-parameter model. Each expression contains no new parameters, but uses existing parameters and variables in new combinations. The three modified equations can be rearranged to give the following linear forms: A plot of the left-hand side of the three preceding equations versus ln(Cot) or ln(t) should result in a straight line if the data follow the linearized model.

Nonlinear least-squares regression
The modified models and their original counterparts are nonlinear functions with two fitting parameters. Nonlinear least-squares regression was applied to estimate their parameters. The overall fit of a model was measured with the coefficient of determination R 2 . Although it should be well known that R 2 (adjusted or unadjusted) is not valid for nonlinear regression, it is still being used routinely within the environmental adsorption literature to assess the goodness of nonlinear fits. The most general definition of R 2 is given by where N ≡ the number of data points; ˆj z ≡ model values; zj ≡ experimental values; and z ≡ the mean of experimental values. Given that R 2 can be misleading and inappropriate for discrimination of nonlinear models [27][28][29], it is supplemented here with a more general goodness-of-fit statistic known as the residual root mean square error (RRMSE) [30,31]: where p ≡ the number of regression parameters and (Np) ≡ the number of degrees of freedom of the regression. The RRMSE metric can be interpreted as the typical deviation of the data points from the fitted curve, and is valid for both linear and nonlinear regression models. Note that the RRMSE measure is also known as the standard error of the residuals [32]. A lower RRMSE value indicates a better fit.

Results and discussion
Three sets of skewed breakthrough data, selected from reports published in the literature of environmental adsorption, are used to test the three modified models. Because the modified models are analogous, it is not necessary to fit all three to the same set of breakthrough data. Accordingly, for a given set of breakthrough data, one of the three modified models will be selected for evaluation. The Additional modeling results for another four sets of breakthrough data are provided in supplementary material, and these will be discussed later.

Case 1: adsorption of fluoride by bone char
This case reports the breakthrough characteristics of fluoride adsorption on two types of commercial bone char [33]. Fixed bed experiments were performed in a glass column with an internal diameter of 2.5 cm and a total length of 13 cm. The column was packed with 27.69 g of bone char, resulting in a packed length of 7.5 cm. For each bone char adsorbent, six breakthrough curves were measured using different flow rates and feed concentrations of fluoride. Tovar-Gómez et al. [33] found noticeable discrepancy between breakthrough data exhibiting tailing and prediction of the linearized Thomas model given by Eq. (4). To obtain better fits, they proposed the use of a hybrid neural network-Thomas modeling approach.
Here, we have the option of choosing one out of the three modified models to correlate a set of breakthrough data taken from the work of Tovar-Gómez et al. [33]. We can select the log-Bohart-Adams model since the length of packed bed (L) is available. With the mass of adsorbent (M) given, the log-Thomas model is also a good choice. It is of course possible to pick the log-Yoon-Nelson model since it is free of standard system and operational variables. The log-Bohart-Adams model given by Eq.
(17) was selected for this case study.
The chosen set of breakthrough data is shown in Fig. 1. The relevant experimental conditions are as follows: Co = 9 × 10 -3 mg cm -3 , u = 0.67 cm min -1 , and L = 7.5 cm. The data points in the initial portion of the breakthrough curve indicate that fluoride breakthrough occurred almost immediately. A sharp rise of the dimensionless effluent concentration to about 70% was followed by a relatively slow approach to saturation, a characteristic of the tailing phenomenon which has been attributed to a variety of factors, including nonspecific adsorption, flow non-uniformity, heterogeneous particle size distribution, and so on. Accounting for these possible causes of tailing mechanistically requires the use of complex models.   1) and (17) respectively using the best-fit parameters obtained from nonlinear regression analysis. Table 1 presents a summary of the parameter estimates and goodness-of-fit statistics for the two data fits.  Table 1. It is apparent that the log-Bohart-Adams model is highly effective in correlating this set of asymmetric data.  (1), which reduces to the following form when t = 0: This undesired attribute is due to the fact that the Bohart-Adams model is not valid for t < εL/u [34] where ε is the bed voidage. It follows that the Bohart-Adams equation is undefined when t = 0.
The log-Bohart-Adams model is discernibly undefined when t = 0 because the time variable t has been replaced by the logarithmic term ln(t). Both the Bohart-Adams and log-Bohart-Adams models have the undesirable property of not being constrained to go through the 0,0 origin. However, the latter has the ability to predict very low effluent concentrations as t approaches zero. Since both the Bohart-Adams and log-Bohart-Adams models have linear forms, it is of interest to know their relative performance. To answer this question the linear versions of the two models given by Eqs. (2) and (20) were fitted to the Fig. 1 data by linear regression, the details of which are not shown here for brevity (regression plots and parameter estimates). The Bohart-Adams linear fit with an R 2 score of 79.5% is very poor. In the case of the linearized log-Bohart-Adams the fit quality is reasonable (R 2 = 98.1%). Fig. 2 plots the model curves computed from Eqs. (1) and (17) Given C/Co values, Eqs. (26) and (27) Table 2 includes results for other cases and these will be discussed later. The physical significance of the fitted parameters given in Table 1 requires some discussion.
The fitted kBA value of the log-Bohart-Adams model is much bigger than that of the Bohart-Adams.
This difference is due to the effect of the logarithmic transformation. As is well known, the Bohart-Adams rate formulation is based on a chemical reaction kinetics theory. Its single rate coefficient kBA does not reflect the actual rate mechanism present in a typical liquid phase adsorption system, which is invariably controlled by mass transport processes. The parameter kBA lacks physical significance and can thus be viewed as a purely empirical parameter which is adjusted to fit the shape of a given breakthrough curve. This consideration applies to both the Bohart-Adams and the log-Bohart-Adams models. The numerical value of kBA does not in any way reflect the kinetics of the assumed chemical reaction.
In contrast to kBA, No is a physically significant parameter. This is because the Bohart-Adams where te ≡ a time value at which C/Co reaches unity or the breakthrough curve has leveled off. In Eq.
where Vb ≡ the bed volume.
According to Tovar-Gómez et al. [33], the void fraction ranged from 25% to 30% in all fixed As can be seen in Table 1, the fitted No value of the log-Bohart-Adams model is 1.62 mg cm -3 , which is 33% smaller than the Wa/Vb quantity. Although the log-Bohart-Adams data fit is excellent, the logarithmic transformation distorted the fitted No value, making it unsuitable as a bed capacity indicator.
Nevertheless, the main objective of the modification is to enhance the ability of the original Bohart-Adams model to fit asymmetric breakthrough curves, and to provide reliable estimates of breakthrough and exhaustion times. Accordingly, the fact that a less accurate No value has been obtained from the log-Bohart-Adams model is considered to be of secondary importance for our present concerns.

Case 2: adsorption of methylene blue by activated carbon
This example deals with dye removal by activated carbon synthesized from paper mill sewage sludge [35]. The carbon adsorbent was packed in a fixed bed column which was used to remove the dyes methylene blue and reactive red 24 from aqueous solutions. The column had an internal diameter of 1.2 cm and a total length of 20 cm. The effects of influent dye concentration, particle size, adsorbent mass, and flow rate on the breakthrough characteristics of the two dyes were determined. Several sets of breakthrough curves are available in the article of Li et al. [35], many of which are asymmetric. Fig.   3 shows a set of methylene blue breakthrough data taken from their paper. In contrast to case 1, Fig. 3 shows that there is no immediate contaminant breakthrough in this case. The early portion of the experimental breakthrough curve is much shorter than the later stage, suggesting the presence of tailing.

Fig. 3. Thomas and log-Thomas data fits compared to observed methylene blue breakthrough data.
The bed length parameter, L, for the selected fixed bed test is not available, ruling out the use of the Bohart-Adams and log-Bohart-Adams models to fit the Fig. 3 data. Since the mass of adsorbent is known, we can select the Thomas and log-Thomas models given by Eqs.  Table 3. The Thomas data fit is rather poor, showing significant deviations at both the initial and later stages of the experimental curve, and displaying a fairly large effluent concentration at t = 0. By contrast, it can be seen that the log-Thomas data fit is closely aligned with the entire experimental profile. The difference in the fit quality of the two models is also observable in the R 2 and RRMSE scores presented in Table   3. The log-Thomas model seems to work extremely well for this asymmetric data set. The log-Thomas model is undoubtedly the clear winner here.
The fitted kT value of the log-Thomas model differs greatly from that of the Thomas model, as may be seen in Table 3. The logarithmic transformation turned kT into a powerful shape parameter, producing the excellent log-Thomas data fit by optimizing the numerical value of kT. The Thomas rate coefficient (kT), just like the Bohart-Adams rate coefficient (kBA), can be viewed as a purely empirical parameter. This is because the Thomas model is based on an unrealistic surface reaction mechanism. In most cases mass transport processes control the kinetics of adsorption in porous activated carbon particles. . This condition is however ignored in many studies that use the simplified Thomas model in data fitting. The prevailing practice is to extract qo from breakthrough data without regard to the type of equilibrium relationship of the adsorbate-adsorbent system under study. Table 3 shows that the qo value of the Thomas model obtained by this modeling approach is 100.3 mg g -1 .
As mentioned previously, the area above the breakthrough curve gives the amount of contaminant adsorbed. Since the void fraction is not reported, we will use Eq. (28) in lieu of Eq. (29) to estimate the total amount of methylene blue retained by the fixed bed at saturation. The values of Co and Q required by Eq. (28) are given above. The final data point in Fig. 3  There is good agreement between the fitted qo value of the Thomas model (100.3 mg g -1 ) and the W/M quantity even though the data fit is not very accurate. As discussed in case 1, the good agreement is probably fortuitous given the rather poor Thomas data fit. The fitted qo value of the log-Thomas model (93.32 mg g -1 ) is surprisingly quite comparable to the W/M quantity of 110.2 mg g -1 . It seems that in this case the distortion effect of the logarithmic transformation on qo was not significant.

Case 3: adsorption of salicylic acid by imprinted adsorbent
This third case describes breakthrough results for salicylic acid adsorption on a wollastonitebased imprinted adsorbent [36]. A glass column with an internal diameter of 1 cm and a length of 30 cm (packed length = 0.9-4.6 cm) was used to produce a series of breakthrough curves as functions of initial adsorbate concentration, flow rate, bed depth, and temperature. A set of breakthrough data taken from the work of Meng et al. [36] is presented in Fig. 4. The experimental conditions for this fixed bed test are as follows: Co = 0.01 mg cm -3 , Q = 1 cm 3 min -1 , and L = 0.9 cm. Fig. 4 shows that salicylic acid emerged fairly quickly at the column outlet. Since the original and modified Bohart-Adams and Thomas models have been tested in the previous two cases, for this example the only ones left to be tested are the Yoon-Nelson model given by Eq. (5) and the log-Yoon-Nelson given by Eq. (19). As noted previously, use of the Yoon-Nelson model and its logarithmic variant in data fitting is straightforward because they lack typical system and operational variables. Table 4 presents the regression results while Fig. 4 plots the Yoon-Nelson and log-Yoon-Nelson data fits. The Yoon-Nelson data fit is somewhat decent, especially around the saturation portion of the breakthrough curve. There is however noticeable deviation in the early stage of salicylic acid breakthrough. In addition, there is an obvious nonzero effluent concentration at t = 0 even though the observed breakthrough curve is not highly asymmetric. From the Fig. 4 plot, one can see that the log-Yoon-Nelson model is very effective, tracing the majority of the data points. However, the fact that the log-Yoon-Nelson model does not contain important fixed bed variables such as bed length makes it unsuitable as a design tool for sizing fixed bed adsorption columns.
The fourth column of Table 2

Additional cases
The Bohart-Adams and log-Bohart-Adams models were tested against another four sets of breakthrough data taken from reports published in the Journal of Water Process Engineering. The modeling results of these four cases, designated by S1-S4, are presented in supplementary material.
Case S1 deals with the fixed bed adsorption of lead by an adsorbent based on waterworks sludge [37].
Case S2 illustrates the breakthrough behavior of mercury adsorption on a nano-adsorbent [38]. Case S3 reports breakthrough results for nickel sequestration by a seaweed biosorbent [39]. Case S4 describes the arsenic removal by a nano-adsorbent in fixed bed columns [40]. The results of all four cases, presented in Figs. S1-S4 and Tables S1-S4, indicate that the better fitting model is the log-Bohart-Adams. These results, taken together with those of cases 1-3, confirm that the modified models are very flexible and much more effective than the original versions in correlating asymmetric breakthrough data.

What's in a name?
The ability of the log-Bohart-Adams, log-Thomas, and log-Yoon-Nelson models to correlate asymmetric breakthrough data is mainly due to their superior functional form. To dig deeper into their functional form, it is necessary to rearrange the model equations defined by Eqs. (17)- (19). Using the log-Bohart-Adams as example, Eq. (17) The general form of the preceding equation is given in Eq. (32). This equation can be derived by applying the logarithmic transformation method to the logistic model defined by Eq. (7). Referring to Eq. (7), if we replace a by ln(a) and t by ln(t) (assuming t has been made dimensionless), then we obtain which may be rewritten as Comparison of this last equation and Eq. (32) confirms that they are equivalent, with different constants.
Additionally, a growth curve model attributed to Hossfeld [42] is a direct analog of Eq. (32).
This model, known as the Hossfeld IV equation, has been devised for the description of tree growth: where y ≡ the response variable; ymax ≡ the maximum value of y; and m3, n3 ≡ constants. Evidently, the Hossfeld IV model, originally proposed in 1822 and therefore much older than the Hill equation, is functionally identical with Eq. (32).
The Sips adsorption isotherm [43], also known as the Langmuir-Freundlich isotherm, is well known for its ability to model a wide variety of equilibrium data. It bears an obvious resemblance to When n4 < 1, the Sips isotherm yields a portion of a rectangular hyperbola, and when n4 > 1 it produces a sigmoid curve. When n4 = 1, the Sips isotherm reduces to the Langmuir isotherm.
Interestingly, in 1909 Hill [44] put forward what later became known as Langmuir's adsorption isotherm. Langmuir [45]seemingly without knowledge of the prior work of Hillpublished his namesake isotherm equation a few years later. In some publications, the eponym "Hill-Langmuir" is used to denote Langmuir's hyperbolic relation.
We see from the foregoing discussion Eq. (32)

Inflection points
where n > 1 and n = kBA, kT, kYN (these rate coefficients are treated as dimensionless quantities, see Eqs. (16) and (17)). The inflection point is a function of n; it is greater than zero and the upper limit is 0.5.
The inflection points for the breakthrough curves predicted by the three modified models for cases 1-3, computed from Eq. (45) with the fitted values of kBA, kT, and kYN, are presented in Table 5.
Mathematically, the three modified models possess two distinct characteristicsa floating inflection point and a flexible functional formthat together enhance their ability to accurately correlate the asymmetric breakthrough data of cases 1-3, and to afford reliable estimates of breakthrough and exhaustion times (see Table 2). wayit is the biggest, and so the case 1 data set was the most difficult to fit, implying that case 1 must have exhibited the strongest asymmetry. It should be recalled that the three modified models are analogous to one another; so are the three original versions. The comparison of the three cases in Fig.   5 is therefore valid although each case was analyzed using a particular modified model and its original form.

Conclusions
The three modified models have been shown to successfully represent the asymmetric breakthrough curves for the adsorption of seven different water contaminants in fixed bed columns packed with adsorbents of different types. Consequently, they were found capable of providing reliable estimates of breakthrough and exhaustion times corresponding to any predetermined concentration level. Thus, their practical value is evident. In all the cases studied, the three original models were found to perform poorly, resulting in inaccurate estimates of breakthrough and exhaustion times. However, the fitted capacity parameter of the modified Bohart-Adams model was found somewhat inaccurate.
This shortcoming is of secondary importance because the practical utility of the modified Bohart-Adams model is estimation of breakthrough and exhaustion times.
A unique feature of the modified models is that they contain no new parameters, which is one of the key points of this paper. Existing models capable of describing asymmetric breakthrough curves invariably rely on extra shape parameters. Although the modified models are highly flexible, being capable of assuming a great variety of different sigmoid shapes, it could be argued that their general applicability cannot be inferred from the seven cases tested here. It is likely that contaminant adsorption on synthetic and natural adsorbents of different types will produce a myriad of breakthrough curve shapes. Additional testing using experimental breakthrough data of other water contaminants such as micropollutants will be required to give confidence that the modified models may be universally applied.

Funding
This work was not in any way directly or indirectly supported, funded, or sponsored by any organization or entity.
Case S1: Adsorption of lead by waterworks sludge reported by Faisal et al. [S1] Table S1. Parameter estimates and goodness of fit statistics for the Bohart-Adams and log-Bohart-Adams data fits shown in Fig. S1. Model input parameters: Co = 0.01 mg cm -3 , L = 10 cm, and u = 0.41 cm min -1 .