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Fitting the Gompertz equation to asymmetric breakthrough curves
Journal Pre-proof Fitting the Gompertz equation to asymmetric breakthrough curves Khim Hoong Chu
PII:
S2213-3437(20)30061-0
DOI:
https://doi.org/10.1016/j.jece.2020.103713
Reference:
JECE 103713
To appear in:
Journal of Environmental Chemical Engineering
Received Date:
8 November 2019
Revised Date:
14 January 2020
Accepted Date:
20 January 2020
Please cite this article as: Chu KH, Fitting the Gompertz equation to asymmetric breakthrough curves, Journal of Environmental Chemical Engineering (2020), doi: https://doi.org/10.1016/j.jece.2020.103713
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Fitting the Gompertz equation to asymmetric breakthrough curves
Khim Hoong Chu* Honeychem, Nanjing Chemical Industry Park, Nanjing 210047, China *
Corresponding author.
E-mail address: [email protected] (K.H. Chu).
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Abstract
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Graphical abstract
Fixed bed adsorption studies often report asymmetric breakthrough curves which exhibit a tailing phenomenon as the effluent approaches the influent concentration. Evaluations of models capable of describing such curves are lacking in the literature. This paper examines the ability of the Gompertz equation to correlate asymmetric breakthrough data collated from reports published in the environmental adsorption literature. It is shown that the Gompertz
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equation, which has received little attention in this field of research, is able to track asymmetric breakthrough curves displaying a moderate degree of tailing. The logistic equation, which is mathematically analogous to the Bohart-Adams, Thomas, and YoonNelson models, cannot effectively describe such asymmetric data. The Gompertz equation provides only an approximate representation of breakthrough data exhibiting a pronounced degree of tailing. To fit such data, this paper presents two modified forms of the Gompertz equation, which are shown to be highly accurate (R2 > 0.996). The Gompertz equation and the
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two modified versions are useful additions to the toolbox of breakthrough curve modeling which has long been filled with the Bohart-Adams, Thomas, and Yoon-Nelson models. These
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popular logistic-based equations are confined to fitting symmetric breakthrough curves.
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Keywords: Adsorption; Breakthrough curve; Fixed bed; Gompertz; Logistic
Roman letters
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Nomenclature
parameter of the logistic equation
b
parameter of the logistic equation
C
outlet adsorbate concentration at time t
Co
inlet adsorbate concentration
J
number of competing models
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a
kBA
Bohart-Adams rate coefficient
kT
Thomas rate coefficient
kYN
Yoon-Nelson rate coefficient
L
bed depth
m
number of observations
2
mass of adsorbent
n
fitting parameter of Eq. (19)
No
adsorption capacity per unit volume of fixed bed
p
number of fitting parameters
qo
adsorption capacity per unit mass of adsorbent
Q
volumetric flow rate
R2
coefficient of determination
t
time or number of pore volumes
tI
time value at inflection point
t*
unit time in the chosen system of units
u
superficial velocity
wi
Akaike weight for model i
Y
response variable of the Gompertz equation
YI
response variable value at inflection point
Ymax
maximum Y value
z mod j
model prediction for observation j
z exp
mean of experimental data
z exp j
experimental data for observation j
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M
Greek letters
parameter of the Gompertz equation
parameter of the Gompertz equation
i
difference in AICc between two models
time required for 50% breakthrough
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Initialisms AICc
corrected Akaike information criterion 3
AICc,i
AICc value for model i
AICc,min
lowest AICc value
SSE
sum of squared errors
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1. Introduction Fixed bed adsorption is an important method for removing organic and inorganic contaminants from drinking water sources and wastewater streams. Several textbooks offer
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extensive theoretical descriptions of the breakthrough dynamics of fixed bed adsorption columns. They are a valuable source of much information on a wide variety of fixed bed
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models based on sophisticated mass transfer principles as well as simple chemical rate laws
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[1-4]. In environmental adsorption research simple models such as the Bohart-Adams, Thomas, and Yoon-Nelson equations are frequently used by investigators to correlate fixed bed breakthrough data. The commonly used linear forms of the three models are written as
C k qM ln o 1 T o kT Co t Q C
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Thomas :
C k N L ln o 1 BA o k BACo t u C
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Bohart - Adams :
Yoon - Nelson :
(1)
(2)
C ln o 1 k YN k YN t C
(3)
where the symbols used are defined at the end of this paper. The three models should not be treated separately (as they commonly are) because mathematically they are equivalent to one another. Each model can be expressed in terms of the logistic equation formulated for adsorption [5-11]:
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C 1 Co 1 exp a bt
(4)
where the general parameters a and b are given by
Bohart - Adams :
Thomas :
a
a
k BA N o L ; u
kT qo M ; Q
Yoon - Nelson :
b k BACo
(5)
b k T Co
a k YN ;
(6)
b k YN
(7)
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Clearly, the three models are exactly analogous. This in turn means that they can be represented by the logistic equation in data fitting, as will be done here. It should however be noted that this mathematical equivalence is sheer algebraic coincidence because there is no
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intrinsic relation between the three models and the logistic equation.
The logistic function was first propounded by Pierre-François Verhulst in 1838 [12] as
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a means of describing dynamic population growth in the presence of restricted resources. It
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has since then been applied to a large number of phenomena that exhibit sigmoidal or Sshaped patterns. However, a major drawback to the use of the logistic equation lies in the fact
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that it predicts symmetric sigmoidal curves. It follows that the Bohart-Adams, Thomas, and Yoon-Nelson models all suffer from the same limitation. Thus, it is no surprise that some papers have reported experimental breakthrough curves that were not amenable to treatment by these logistic-based models [13-20]. Such curves were found to display an asymmetric,
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tailing behavior, i.e., a slow approach of C/Co toward unity. Relatively little progress has been made in the development of models that are capable
of describing asymmetry. Some investigators have proposed the use of neural network or hybrid Thomas-neural network models in lieu of the Thomas model to fit asymmetric breakthrough curves [16-20]. However, these neural network models are coded in specialized
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software; they are not available in the form of simple equations that allow testing by other researchers. This paper reports the use of a simple mathematical expression, known as the Gompertz equation [21], to describe the asymmetric characteristics of fixed bed breakthrough curves. Although the Gompertz equation is in common use across a wide range of disciplines, it has thus far attracted little attention as a modeling tool in environmental adsorption research. Like the logistic equation of Verhulst, Gompertz’s equation has been used extensively in the
inflection, and deceleration phases of growth over time.
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studies of growth processes. It produces sigmoidal curves that describe the acceleration,
In the domain of environmental technology research, several modified forms of the
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Gompertz model have been used to describe microbial growth, substrate consumption, and biogas production rates in waste treatment processes based on anaerobic fermentation [22].
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The Gompertz equation has also been used to correlate the breakthrough behavior of fixed
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bed columns packed with nanomaterials [23-25]. These fixed bed columns were not designed to remove contaminants by adsorption; they were used to disinfect drinking water contaminated by harmful microorganisms.
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There are only limited applications of the Gompertz equation in the realm of environmental adsorption research. Yu et al. [26] used a modified Gompertz equation to model variations of a Freundlich isotherm parameter and a surface diffusion coefficient with
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time. Another modified version of the Gompertz equation was used by Çelekli et al. [27] to model the kinetics of copper adsorption in a batch adsorber. Smith et al. [28] used the Gompertz equation to describe the breakthrough behavior of a rapid small-scale column test apparatus but only one set of breakthrough data was presented in their work. Hu et al. [29] used a double exponential equation, which is similar to the Gompertz equation, to fit two sets of breakthrough data displaying slight asymmetry. The curve-fitting ability of the Gompertz
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equation was not sufficiently tested in these two prior studies on fixed bed adsorption [28,29]. Accordingly, the main aim of this research is to determine how well breakthrough curve asymmetry could be described by the Gompertz equation. The skewed breakthrough data examined here were collated from reports published in the environmental adsorption literature.
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2. Theory The British mathematician, Benjamin Gompertz, put forward a model in 1825 [21] as a means to explain human mortality curves. Modern studies have successfully exploited
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Gompertz’s model to describe growth curves across a wide range of disciplines, such as biology, crop science, medical science, engineering, and economics, to name but a few. A
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commonly used form of the Gompertz equation for growth [30] reads
(8)
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Y Ymax exp exp t
where Y denotes the response variable at time equal to t, Ymax represents the maximum Y value
and .
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or upper asymptote, and and are constants. The three parameters to be fitted are Ymax, ,
The Gompertz equation produces sigmoidal curves of asymmetric shape. A typical feature of these sigmoidal curves is the existence of an inflection point at which the growth
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rate is maximized. The inflection point and maximum growth rate of a Gompertz curve may be determined from the first and second derivatives of Eq. (8) which are given by Eqs. (9) and (10), respectively.
dY Ymax exp t exp exp t dt
(9)
d 2Y Ymax 2 exp t 1exp t exp exp t 2 dt
(10)
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By setting the left-hand side of Eq. (10) to zero, we obtain the expression for the time value, tI, at the inflection point:
tI
(11) Substitution of the preceding equation in Eq. (8) yields the expression for the response
variable value, YI, at the inflection point:
YI
Ymax e
(12)
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where e is the base of the natural logarithm.
Substitution of Eq. (11) in Eq. (9) yields the maximum growth rate at the inflection point:
(13)
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Y dY max e dt tI
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3. Methods
To assess the broad applicability of the Gompertz equation, three asymmetric
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breakthrough curves and one symmetric breakthrough curve were selected from the adsorption literature for testing in this work. For comparison purposes the logistic equation was also included in the curve-fitting process. The two models were fitted to the four data sets by nonlinear regression. This resulted in estimates for the parameters , , a, and b. The
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linearized versions of the two models, expressed by Eqs. (17) and (18), were fitted to one of the data sets by linear regression. For each fit, we calculated the following statistical indicators:
z
2
m
R2
j 1
mod j
2
z exp z mod z exp j j m
j 1
z j 1
(14)
2
m
mod j
z
exp
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p2 SSE AICc m ln 2 p 11 m m p2
(15)
where the symbols and initialisms used are defined at the end of this paper. The coefficient of determination R2 is used for testing the goodness-of-fit of a model and its value ranges from 0 to 1. A model with a higher value of R2 is considered to be the best. The corrected Akaike information criterion AICc provides an implementation of Occam's razor, in which parsimony or simplicity is balanced against goodness-of-fit [31]. Given a data
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set, multiple models may be ranked according to their AICc scores, with the one producing the lowest value being the best.
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4. Results and discussion 4.1. Gompertz’s equation as a model of fixed bed adsorption
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A similarity exists between the pattern of biological growth dynamics and the rise of a fixed bed adsorber’s exit concentration. Both phenomena result in sigmoidal curves with
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respect to time. Also, the rise of the effluent concentration that is restricted by some upper limit to that rise is analogous to a growth process in the presence of restricted resources.
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Accordingly, the response variable Y in Eq. (8) can be replaced by the effluent concentration C, which “grows” over time. The maximum value or upper limit of the effluent concentration is known, which is equal to the feed concentration Co. This allows us to swap the unknown
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parameter Ymax in Eq. (8) with the known parameter Co, reducing the model’s unknown parameters from three to two. These modifications transform the three-parameter Gompertz equation to an empirical model of fixed bed adsorption with two unknown parameters: C exp exp t Co
(16)
The shape of Gompertz and logistic curves is largely dictated by the point of inflection. The inflection point for the logistic model is at the midpoint, i.e., C/Co = 0.5. This property
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restricts the logistic model to producing symmetric sigmoidal curves. That is, the convex and concave curves on either side of the inflection point have the same curvature. In contrast, the Gompertz model provides a relaxation of this restriction by placing its reflection point at YI/Ymax = C/Co = 1/e = 0.368 (see Eq. (12)). The Gompertz curve therefore differs from the logistic in not being symmetric about the point of inflection. This attribute suggests the possibility that the Gompertz equation may be able to correlate asymmetric breakthrough
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curves. This will be investigated in the ensuing sections using published breakthrough data.
4.2. Example 1: adsorption of chromium(VI) by coconut coir pith
Suksabye et al. [13] used a fixed bed apparatus packed with coconut coir pith to treat
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chromium(VI)-contaminated wastewater produced by an electroplating factory. Several breakthrough curves were recorded by varying the adsorption system’s flow rate and bed
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depth. A set of their breakthrough data examined here is depicted in Fig. 1. As can be seen in
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Fig. 1 the fixed bed column took more than 40 hours to reach saturation. The long loading duration produced a relatively smooth sigmoidal curve. Fig. 1 shows the logistic and Gompertz curves calculated using Eqs. (4) and (16) with the nonlinear regression-generated
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parameters given in Table 1, which also gives the regression statistics calculated using Eqs. (14) and (15). The logistic equation cannot adequately fit the S-shaped experimental profile, overpredicting dimensionless effluent concentrations for the initial (C/Co < 20%) and later
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(C/Co > 80%) stages. The logistic curve indicates that the column achieved breakthrough and became saturated at earlier times than those observed experimentally. It seems clear from this result that the logistic model would underestimate the amount of chromium(VI) loaded onto the bed. In contrast to the logistic fit, the Gompertz fit closely matches the form of the entire experimental profile. Indeed, the R2 and AICc values in Table 1 agree with the visual
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observations. This suggests that the Gompertz equation can easily outperform the BohartAdams, Thomas, and Yoon-Nelson models in correlating the Fig. 1 data. The inability of the logistic equation to track the shape of the Fig. 1 data suggests that the experimental curve is asymmetric. One can see that the experimental profile exhibits some tailing; the time span of the later part (C/Co > 0.8) is noticeably longer than that of the initial stage (C/Co < 0.2). The remarkably close correspondence between the Gompertz fit and Fig. 1
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data confirms that the Gompertz equation is able to handle breakthrough curve asymmetry.
Fig. 1. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with nonlinear regression generated-
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parameters given in Table 1. Data of Suksabye et al. [13].
The overestimation of the low concentration region by the logistic equation means that
the model will underpredict the service time for a given breakthrough concentration, leading to underutilization of the capacity of the fixed bed adsorber. From the Fig. 1 data, we note that the experimental breakthrough time for a 5% breakthrough concentration is approximately 657 minutes. For this particular breakthrough concentration, the breakthrough times predicted
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by the logistic and Gompertz equations are respectively 433 and 630 minutes. It is evident that the prediction of the logistic equation is rather poor while that of the Gompertz equation closely matches the observed service time. The Gompertz and logistic equations may be linearized to allow parameter estimation by linear regression. Linearization of the two models gives Gompertz :
(17)
C ln o 1 a bt C
(18)
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Logistic :
C ln ln o t C
The preceding equations suggest that a plot of the left-hand side of Eq. (17) or (18) versus t should be linear with slope ( or b) and intercept ( or a). Since many investigators
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in this field of research are accustomed to using linearized versions of the Bohart-Adams,
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Thomas, and Yoon-Nelson models (Eqs. (1)-(3)) in data fitting, the availability of Eq. (17) could perhaps help promote the Gompertz equationwhich has received very little attention
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in the pastas a practical tool for breakthrough curve modeling.
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Fig. 2. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with linear regression generated-parameters given in Table 1. Data of Suksabye et al. [13].
To assess their data fitting ability, Eqs. (17) and (18) were fitted to the Fig. 1 data by linear regression. The resulting best-fit parameters and R2 and AICc scores are presented in Table 1. From Fig. 2, it is clear that the two model fits are visually worse than those of Fig. 1.
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Comparison of the two sets of R2 and AICc values in Table 1 supports this visual evidence. Fig. 2 shows that the Gompertz equation cannot adequately fit the entire experimental profile, overestimating low dimensionless effluent concentrations (C/Co < 0.3). The trend of the
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logistic fit seems to be the reverse of the trend seen in Fig. 1. The logistic fit lies close to both the low and high concentration regions of the breakthrough curve (C/Co < 0.1, C/Co >0.9) but
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it deviates markedly at intermediate effluent concentrations. It seems that the linear regression
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procedure assigned greatest weight to the points at low and high effluent concentrations.
4.3. Example 2: adsorption of copper by biosorbent
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Copper uptake by a biosorbent in fixed bed column mode was investigated by Izquierdo et al. [14]. Five sets of breakthrough data obtained at different feed concentrations are available in their paper. One of the data sets exhibiting tailing behavior is reproduced in
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Fig. 3 as discrete points. The logistic and Gompertz equations were fitted to the Fig. 3 data by nonlinear regression (the variable t in this case represents the number of pore volumes instead of time). The figure includes results for another model and these will be discussed later. Both the model fits in Fig. 3 and R2 and AICc scores in Table 1 reveal that the Gompertz equation describes the experimental data better than the logistic. However, compared with the excellent Gompertz fit of Fig. 1, the Gompertz fit of Fig. 3 is less satisfactory. The deviations seen at
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high effluent concentrations (C/Co > 0.8) show that the Gompertz equation provides only an approximate representation of the dynamic behavior. It is clear that the experimental breakthrough curve is substantially asymmetric, with an apparent broadening close to saturation conditions which cannot be matched by the Gompertz equation. Nonetheless, the early rising portion of the breakthrough curve, which is most important from a practical viewpoint, is reasonably well represented by the Gompertz model. The logistic fit of Fig. 3 is similar to that of Fig. 1: it provides a reasonable representation of dimensionless effluent
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concentrations in the 0.2-0.8 range but overestimates those for both the early and later stages.
Fig. 3. Adsorption of copper by a biosorbent showing comparison of experimental data and Gompertz and logistic curves calculated with parameters given in Table 1. Data of Izquierdo
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et al. [14]. Also shown is the theoretical prediction reported by Izquierdo et al. [14].
The distribution of residuals (i.e., the difference between the experimental and
computed values of C/Co) provides useful information on the suitability of the Gompertz and logistic equations for describing the Fig. 3 data. A plot was constructed of residual against the number of pore volumes for each of the two equations, as depicted in Fig. 4. The figure
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includes results for another model and these will be discussed later. If the correct model has been chosen, the residuals should be scattered in a random fashion. For the Gompertz and logistic equations, the residuals lie blockwise under and over the horizontal axis, indicating that both equations are inappropriate. It is apparent that the Gompertz equation must be modified if it is to be an accurate description of the experimental results. We will discuss two
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modified forms of the Gompertz equation in Section 4.6.
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Fig. 4. Residual plots for the Fig. 3 data fitted to the Gompertz equation, logistic equation, and mechanistic model of Izquierdo et al. [14].
The Fig. 3 modeling results demonstrate the difficulty in describing asymmetry using
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the simple Gompertz equation, which can be viewed as a convenient mathematical relationship for empirically fitting asymmetric data. Application of the Gompertz model in the present study has implicitly assumed the parameters and to be of no physical significance. One concern that must be addressed is whether the lack-of-fit errors are due to the empirical nature of the Gompertz equation. Would a mechanistic model provide a more accurate prediction? An attempt was made by Izquierdo et al. [14] to model the asymmetric data of Fig.
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3 by using a mechanistic model that accounts for axial dispersion and intraparticle diffusion expressed in the form of a linear driving force rate expression. The model was solved numerically and fitted to the Fig. 3 data by adjusting the linear driving force rate coefficient to ensure that model prediction would agree well with data at the onset of breakthrough. The resulting best-fit curve is shown as broken line in Fig. 3. It is evident that there is good agreement between the model prediction and low dimensionless effluent concentrations up to about C/Co = 0.2. For the remainder of the experimental profile, the correspondence between
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prediction and experiment is rather poor, indicating that the mechanistic model is not adequate enough. Fig. 4 shows a plot of the residuals for this model fit. The majority of the points lie blockwise under the horizontal axis, revealing a misfit between the model and the data.
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Clearly, further refinement of the mechanistic model is necessary. The large discrepancy is presumably due to the failure of the mechanistic model to consider factors responsible for the
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tailing behavior.
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Various reasons have been put forward in the literature to explain the tailing phenomenon observed in fixed bed adsorbers including flow non-uniformity, nonspecific adsorption, enhanced adsorption under continuous flow conditions, heterogeneous binding
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mechanism with fast and slow binding sites, heterogeneous particle or pore size distribution effects, etc. Rigorous mechanistic models accounting for these factors generally require inordinate effort to generate substantial data for model validation and parameter estimation. In
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reality, the tailing phenomenon is much complicated and the exact mechanisms leading to tailing have been identified only in very limited circumstances. In fact, different mechanisms may be responsible for the breakthrough tailing in individual cases. With these considerations in mind, empirical models such as the Gompertz equation would seem preferable as they are more convenient to use, computationally simpler, and could have immediate practical benefits.
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Admittedly, there is no mechanistic background behind the Gompertz equation when it is applied to fixed bed adsorption. However, it must be borne in mind that the commonly used Bohart-Adams and Thomas models can also be viewed as mere curve-fitting tools. The reason for this is that these models are based on chemical reaction rate laws which contradict the actual diffusion mechanisms present in most adsorption systems of interest. In addition, their parameters are empirical in the sense that they vary with operational and system variables such as flow rate, initial adsorbate concentration, and bed length. The validity of these
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parameters is limited to the range of conditions used during experimental tests conducted to generate breakthrough data for model calibration. In other words, the fitted parameters cannot be used to provide a priori breakthrough predictions over a wide range of operating conditions.
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When viewed in this way, it becomes obvious that the empirical nature of the Gompertz
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equation is no different from those of the Bohart-Adams and Thomas models.
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4.4. Example 3: adsorption of methylene blue by activated carbon The ability of the Gompertz equation to describe asymmetry was further challenged using a set of breakthrough data reported by Li et al. [15] who investigated dye adsorption in
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fixed bed adsorbers packed with activated carbon. Fig. 5 presents the Gompertz and logistic fitting results graphically. For both models, Fig. 5 shows that large lack-of-fit errors are evident, indicating the presence of a pronounced asymmetry in the experimental data. It can
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be seen that the entire breakthrough pattern is skewed by the tailing behavior. As a result, the early rising and final tailing portions of the breakthrough curve are not predicted exactly by the Gompertz equation. Dimensionless effluent concentrations in the intermediate region are also poorly represented by the Gompertz equation. The logistic fit seems worse than the Gompertz fit. This visual observation is supported by the statistical evaluation presented in Table 1.
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Although the Gompertz model is inherently asymmetric, the results of Examples 2 and 3 reveal that it is not accurate enough in describing the asymmetric characteristics of these two data sets. The reason for this unsatisfactory performance could be attributed to the fact that the Gompertz equation has a fixed inflection point which is located at C/Co = 1/e. With this fixed inflection point, which is below the midpoint, the Gompertz equation is able to fit breakthrough curves with a moderate degree of tailing but is unable to handle those with a marked degree of tailing. To fit such curves, the inflection point would have to be located
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further away from the midpoint. This would require the use of models with a variable
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inflection point. The analysis of such models is beyond the scope of the present study.
Fig. 5. Adsorption of methylene blue by activated carbon showing comparison of
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experimental data and model curves calculated with parameters given in Table 1. Data of Li et al. [15].
4.5. Example 4: adsorption of methylene blue by activated carbon It is of interest to compare the relative performance of the Gompertz and logistic equations in fitting symmetric data. The two equations were fitted to a set of breakthrough
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data reported by Foo and Hameed [32] who investigated methylene blue adsorption in an activated carbon column. Fig. 6 depicts the selected experimental curve along with its model counterparts. The logistic equation is shown to capture the experimental observations very well, as expected. For the Gompertz fit, the agreement is not as close. It is apparent that low and high dimensionless effluent concentrations are much less well predicted. Because the Gompertz equation has no point of symmetry, the modest levels of lack-of-fit error can be considered tolerable. Note that the R2 score for the logistic fit is 0.9994, suggesting that this
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data set is highly symmetric with slight to no tailing. It is evident that the logistic equation is a more appropriate mathematical representation of this data set. In view of the results reported in Examples 3 and 4, one can conclude that the Gompertz equation is not a versatile modeling
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tool because it specializes in fitting mildly asymmetric breakthrough curves with a moderate
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degree of tailing.
Fig. 6. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Foo and Hameed [32].
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4.6. Modified Gompertz equations To reduce the lack-of-fit error of a model, an approach that is often taken is to increase the number of free parameters. That is, adding an adjustable parameter to a model will almost always improve fit to some degree. Accordingly, the Gompertz equation can be extended by adding an exponent to the time variable t. The equation so modified is given by
C exp exp t n Co
(19)
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where n is a dimensionless fitting parameter. In this work the preceding equation is known as the power law Gompertz. Note that Eq. (19) is called the Sloboda equation in the growth curve modeling literature [33].
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Another method sometimes used to improve fit operates by converting independent variables to logarithmic terms. First, the Gompertz equation expressed by Eq. (16) is rewritten
(20)
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C t exp exp t * * Co t
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as
where t* is the unit time in the chosen system of units (e.g., 1 s or 1 min). Thus (t/t*) is a
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dimensionless quantity numerically equal to t while (t*) is also a dimensionless quantity numerically equal to . It is necessary to introduce the dimensioned quantity t* because we cannot take the logarithm of t which is of course a dimensioned quantity. Next, we modify Eq. (20) empirically by taking the logarithm of (t/t*):
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C t exp exp t * ln * Co t
(21)
For convenience, the preceding equation is rewritten as C exp exp ln t Co
(22)
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We stress that the t and terms in the preceding equation do not have their usual meanings; t represents pure numbers numerically equal to the actual t while represents pure numbers numerically equal to the actual . Here, Eq. (22) is called the log-Gompertz equation. Compared to the power law Gompertz which has three fitting parameters, the log-Gompertz remains a two-parameter equation. We note in passing that the log-Gompertz equation can be linearized to facilitate parameter estimation by linear regression.
C ln ln o ln t C
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(23)
A plot of the left-hand side of Eq. (23) versus lnt yields a slope and an intercept from which
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and are obtained.
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Fig. 7. Comparison of the Fig. 5 data and model curves calculated with parameters given in Table 2.
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Fig. 8. Comparison of the Fig. 5 data and model curves calculated with parameters given in
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Table 2.
To evaluate their data fitting ability, Eqs. (19) and (22) were fitted to the highly
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asymmetric data of Fig. 5 by nonlinear regression. The resulting parameters are summarized
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in Table 2, and comparisons between the experimental data and the model fits are shown in Figs. 7 and 8. For comparison purposes both figures also show the original Gompertz fit given
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by Eq. (16). It can be seen that the two modifications provide fits that are visually much better, eliminating most of the lack-of-fit errors present in the original Gompertz fit. Table 2 shows the R2 and AICc scores for the model fits in Figs. 7 and 8, which confirm that asymmetry can be effectively modeled by the three-parameter power law
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Gompertz and the two-parameter log-Gompertz. The R2 and AICc scores for these two modified Gompertz equations are fairly similar. Ranking rival models with different numbers of fitting parameters in terms of the R2 test is a simplistic approach because the R2 indicator does not compensate for possible bias to parameter-rich models. In contrast to the R2 test, the AICc test addresses the tradeoff between gain in fit and addition of new parameters [31]. It is a quantitative way to rank competing models with different numbers of adjustable parameters
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and identify the model that is most justified by the data at hand. In Eq. (15) the first term measures fit, while the second term penalizes complex models, i.e., models with more adjustable parameters. An effective way to interpret AICc values calculated for a set of competing models is via a set of positive Akaike weights [34,35]. The Akaike weight for model i among a cohort of J competing models is given by exp 0.5 i
(24)
exp 0.5 J
j
j 1
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wi
where
i AICc,i AICc,min
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In the preceding equation, wi is the Akaike weight for model i, i is the difference in the AICc
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values for model i (AICc,i) and the model with the lowest AICc score (AICc,min). The Akaike weight can be interpreted as the probability that model i is the best model given the data at
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hand and the chosen set of candidate models.
The wi values calculated from Eq. (24) for fitting to the Gompertz, log-Gompertz, and
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power law Gompertz equations are presented in Table 2. The wi value for the Gompertz fit is practically zero, indicating that it receives no support from the Fig. 5 data. This result is not surprising given that its AICc score is substantially greater (i.e., less negative) than those of the power law Gompartz and log-Gompertz. The three-parameter power law Gompertz is also
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not supported by the Fig. 5 data even though its AICc value is only slightly inferior to that of the log-Gompertz. It is apparent that the conclusion from the AICc test is to use the logGompertz equation to fit this data set; it is the best one in this group of three competing models.
5. Conclusions
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The Gompertz and the logistic equations have been applied to the analysis of published breakthrough data on contaminant adsorption in fixed bed adsorbers. Of these two equations the Gompertz has attracted little attention as a breakthrough curve model while the logistic is mathematically analogous to the widely used Bohart-Adams, Thomas, and YoonNelson models. The logistic equation cannot effectively fit breakthrough curves of asymmetric shape. This in turn means that the Bohart-Adams, Thomas, and Yoon-Nelson models are generally unable to handle much of the asymmetric breakthrough data commonly
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observed in environmental adsorption studies. The Gompertz equation provides a partial solution to the problem of describing breakthrough tailing close to column saturation. Mildly asymmetric breakthrough curves exhibiting a moderate degree of tailing can be adequately
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represented by the Gompertz equation. Its performance is less satisfactory in describing breakthrough data with a marked degree of tailing. Two modified forms of the Gompertz
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equation, known as the power law Gompertz and the log-Gompertz, permit strong asymmetry
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to be effectively modeled. The log-Gompertz equation has two fitting parameters while the power law Gompertz has three. The former also has a linear form. Therefore, the
purposes.
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mathematically simpler log-Gompertz equation may be an adequate choice for practical
Credit Author Statement
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This is a single-author manuscript.
Khim Chu
Declaration of competing interests
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The author declares that he has no known competing financial interests or personal
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relationships that could have appeared to influence the work reported in this paper.
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[30] K.M.C. Tjørve, E. Tjørve, 2017. The use of Gompertz models in growth analyses, and new Gompertz-model approach: an addition to the Unified-Richards family. PLoS One. 12, e0178691. [31] H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19 (1974) 716–723. [32] K.Y. Foo, B.H. Hameed, Dynamic adsorption behavior of methylene blue onto oil palm shell granular activated carbon prepared by microwave heating, Chem. Eng. J. 203
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Figure captions
Fig. 1. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with nonlinear regression generatedparameters given in Table 1. Data of Suksabye et al. [13].
Fig. 2. Adsorption of chromium(VI) by coconut coir pith showing comparison of
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experimental data and model curves calculated with linear regression generated-parameters given in Table 1. Data of Suksabye et al. [13].
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Fig. 3. Adsorption of copper by a biosorbent showing comparison of experimental data and Gompertz and logistic curves calculated with parameters given in Table 1. Data of Izquierdo
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et al. [14]. Also shown is the theoretical prediction reported by Izquierdo et al. [14].
Fig. 4. Residual plots for the Fig. 3 data fitted to the Gompertz equation, logistic equation,
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and mechanistic model of Izquierdo et al. [14].
Fig. 5. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Li et
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al. [15].
Fig. 6. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Foo and Hameed [32].
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Fig. 7. Comparison of the Fig. 5 data and model curves calculated with parameters given in Table 2.
Fig. 8. Comparison of the Fig. 5 data and model curves calculated with parameters given in
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Table 2.
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Table 1 Best-fit parameters and regression statistics for the Gompertz and logistic fits in Figs. 1-3, 5, and 6. Example Regression no.
Logistic model
method
/ min-1
R2
AICc a
b / min-1
R2
AICc
Nonlinear
2.671
0.0025
0.9988
-404
4.505
0.0036
0.9933
-324
Linear
2.229
0.0022
0.9933
-327
4.805
0.0034
0.9611
-240
2
Nonlinear
7.562
0.0117a
0.9818
-118
11.832
0.0174b
0.9626
-103
3
Nonlinear
1.620
0.0128
0.9827
-235
2.870
0.0179
0.9564
-200
4
Nonlinear
7.504
0.0270
0.9939
-129
11.616
0.0393
0.9994
-176
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1
/ (number of pore volumes)-1 ; b b / (number of pore volumes)-1
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a
Gompertz model
Table 2
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Best-fit parameters and regression statistics for the Gompertz, log-Gompertz, and power law Gompertz fits in Figs. 7 and 8. Model
Number of
/ min-1
n
R2
AICc
wi
parameters 2
1.620
0.0128
0.9827
-235
0.00000
Log-Gompertz
2
9.487
1.985
0.9979
-316
0.99993
Power law Gompertz
3
12.748
6.244
0.9968
-297
0.00007
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Gompertz
32
0.15
PII:
S2213-3437(20)30061-0
DOI:
https://doi.org/10.1016/j.jece.2020.103713
Reference:
JECE 103713
To appear in:
Journal of Environmental Chemical Engineering
Received Date:
8 November 2019
Revised Date:
14 January 2020
Accepted Date:
20 January 2020
Please cite this article as: Chu KH, Fitting the Gompertz equation to asymmetric breakthrough curves, Journal of Environmental Chemical Engineering (2020), doi: https://doi.org/10.1016/j.jece.2020.103713
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Fitting the Gompertz equation to asymmetric breakthrough curves
Khim Hoong Chu* Honeychem, Nanjing Chemical Industry Park, Nanjing 210047, China *
Corresponding author.
E-mail address: [email protected] (K.H. Chu).
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Abstract
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Graphical abstract
Fixed bed adsorption studies often report asymmetric breakthrough curves which exhibit a tailing phenomenon as the effluent approaches the influent concentration. Evaluations of models capable of describing such curves are lacking in the literature. This paper examines the ability of the Gompertz equation to correlate asymmetric breakthrough data collated from reports published in the environmental adsorption literature. It is shown that the Gompertz
1
equation, which has received little attention in this field of research, is able to track asymmetric breakthrough curves displaying a moderate degree of tailing. The logistic equation, which is mathematically analogous to the Bohart-Adams, Thomas, and YoonNelson models, cannot effectively describe such asymmetric data. The Gompertz equation provides only an approximate representation of breakthrough data exhibiting a pronounced degree of tailing. To fit such data, this paper presents two modified forms of the Gompertz equation, which are shown to be highly accurate (R2 > 0.996). The Gompertz equation and the
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two modified versions are useful additions to the toolbox of breakthrough curve modeling which has long been filled with the Bohart-Adams, Thomas, and Yoon-Nelson models. These
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popular logistic-based equations are confined to fitting symmetric breakthrough curves.
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Keywords: Adsorption; Breakthrough curve; Fixed bed; Gompertz; Logistic
Roman letters
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Nomenclature
parameter of the logistic equation
b
parameter of the logistic equation
C
outlet adsorbate concentration at time t
Co
inlet adsorbate concentration
J
number of competing models
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a
kBA
Bohart-Adams rate coefficient
kT
Thomas rate coefficient
kYN
Yoon-Nelson rate coefficient
L
bed depth
m
number of observations
2
mass of adsorbent
n
fitting parameter of Eq. (19)
No
adsorption capacity per unit volume of fixed bed
p
number of fitting parameters
qo
adsorption capacity per unit mass of adsorbent
Q
volumetric flow rate
R2
coefficient of determination
t
time or number of pore volumes
tI
time value at inflection point
t*
unit time in the chosen system of units
u
superficial velocity
wi
Akaike weight for model i
Y
response variable of the Gompertz equation
YI
response variable value at inflection point
Ymax
maximum Y value
z mod j
model prediction for observation j
z exp
mean of experimental data
z exp j
experimental data for observation j
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M
Greek letters
parameter of the Gompertz equation
parameter of the Gompertz equation
i
difference in AICc between two models
time required for 50% breakthrough
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Initialisms AICc
corrected Akaike information criterion 3
AICc,i
AICc value for model i
AICc,min
lowest AICc value
SSE
sum of squared errors
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1. Introduction Fixed bed adsorption is an important method for removing organic and inorganic contaminants from drinking water sources and wastewater streams. Several textbooks offer
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extensive theoretical descriptions of the breakthrough dynamics of fixed bed adsorption columns. They are a valuable source of much information on a wide variety of fixed bed
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models based on sophisticated mass transfer principles as well as simple chemical rate laws
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[1-4]. In environmental adsorption research simple models such as the Bohart-Adams, Thomas, and Yoon-Nelson equations are frequently used by investigators to correlate fixed bed breakthrough data. The commonly used linear forms of the three models are written as
C k qM ln o 1 T o kT Co t Q C
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Thomas :
C k N L ln o 1 BA o k BACo t u C
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Bohart - Adams :
Yoon - Nelson :
(1)
(2)
C ln o 1 k YN k YN t C
(3)
where the symbols used are defined at the end of this paper. The three models should not be treated separately (as they commonly are) because mathematically they are equivalent to one another. Each model can be expressed in terms of the logistic equation formulated for adsorption [5-11]:
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C 1 Co 1 exp a bt
(4)
where the general parameters a and b are given by
Bohart - Adams :
Thomas :
a
a
k BA N o L ; u
kT qo M ; Q
Yoon - Nelson :
b k BACo
(5)
b k T Co
a k YN ;
(6)
b k YN
(7)
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Clearly, the three models are exactly analogous. This in turn means that they can be represented by the logistic equation in data fitting, as will be done here. It should however be noted that this mathematical equivalence is sheer algebraic coincidence because there is no
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intrinsic relation between the three models and the logistic equation.
The logistic function was first propounded by Pierre-François Verhulst in 1838 [12] as
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a means of describing dynamic population growth in the presence of restricted resources. It
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has since then been applied to a large number of phenomena that exhibit sigmoidal or Sshaped patterns. However, a major drawback to the use of the logistic equation lies in the fact
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that it predicts symmetric sigmoidal curves. It follows that the Bohart-Adams, Thomas, and Yoon-Nelson models all suffer from the same limitation. Thus, it is no surprise that some papers have reported experimental breakthrough curves that were not amenable to treatment by these logistic-based models [13-20]. Such curves were found to display an asymmetric,
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tailing behavior, i.e., a slow approach of C/Co toward unity. Relatively little progress has been made in the development of models that are capable
of describing asymmetry. Some investigators have proposed the use of neural network or hybrid Thomas-neural network models in lieu of the Thomas model to fit asymmetric breakthrough curves [16-20]. However, these neural network models are coded in specialized
5
software; they are not available in the form of simple equations that allow testing by other researchers. This paper reports the use of a simple mathematical expression, known as the Gompertz equation [21], to describe the asymmetric characteristics of fixed bed breakthrough curves. Although the Gompertz equation is in common use across a wide range of disciplines, it has thus far attracted little attention as a modeling tool in environmental adsorption research. Like the logistic equation of Verhulst, Gompertz’s equation has been used extensively in the
inflection, and deceleration phases of growth over time.
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studies of growth processes. It produces sigmoidal curves that describe the acceleration,
In the domain of environmental technology research, several modified forms of the
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Gompertz model have been used to describe microbial growth, substrate consumption, and biogas production rates in waste treatment processes based on anaerobic fermentation [22].
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The Gompertz equation has also been used to correlate the breakthrough behavior of fixed
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bed columns packed with nanomaterials [23-25]. These fixed bed columns were not designed to remove contaminants by adsorption; they were used to disinfect drinking water contaminated by harmful microorganisms.
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There are only limited applications of the Gompertz equation in the realm of environmental adsorption research. Yu et al. [26] used a modified Gompertz equation to model variations of a Freundlich isotherm parameter and a surface diffusion coefficient with
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time. Another modified version of the Gompertz equation was used by Çelekli et al. [27] to model the kinetics of copper adsorption in a batch adsorber. Smith et al. [28] used the Gompertz equation to describe the breakthrough behavior of a rapid small-scale column test apparatus but only one set of breakthrough data was presented in their work. Hu et al. [29] used a double exponential equation, which is similar to the Gompertz equation, to fit two sets of breakthrough data displaying slight asymmetry. The curve-fitting ability of the Gompertz
6
equation was not sufficiently tested in these two prior studies on fixed bed adsorption [28,29]. Accordingly, the main aim of this research is to determine how well breakthrough curve asymmetry could be described by the Gompertz equation. The skewed breakthrough data examined here were collated from reports published in the environmental adsorption literature.
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2. Theory The British mathematician, Benjamin Gompertz, put forward a model in 1825 [21] as a means to explain human mortality curves. Modern studies have successfully exploited
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Gompertz’s model to describe growth curves across a wide range of disciplines, such as biology, crop science, medical science, engineering, and economics, to name but a few. A
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commonly used form of the Gompertz equation for growth [30] reads
(8)
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Y Ymax exp exp t
where Y denotes the response variable at time equal to t, Ymax represents the maximum Y value
and .
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or upper asymptote, and and are constants. The three parameters to be fitted are Ymax, ,
The Gompertz equation produces sigmoidal curves of asymmetric shape. A typical feature of these sigmoidal curves is the existence of an inflection point at which the growth
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rate is maximized. The inflection point and maximum growth rate of a Gompertz curve may be determined from the first and second derivatives of Eq. (8) which are given by Eqs. (9) and (10), respectively.
dY Ymax exp t exp exp t dt
(9)
d 2Y Ymax 2 exp t 1exp t exp exp t 2 dt
(10)
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By setting the left-hand side of Eq. (10) to zero, we obtain the expression for the time value, tI, at the inflection point:
tI
(11) Substitution of the preceding equation in Eq. (8) yields the expression for the response
variable value, YI, at the inflection point:
YI
Ymax e
(12)
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where e is the base of the natural logarithm.
Substitution of Eq. (11) in Eq. (9) yields the maximum growth rate at the inflection point:
(13)
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Y dY max e dt tI
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3. Methods
To assess the broad applicability of the Gompertz equation, three asymmetric
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breakthrough curves and one symmetric breakthrough curve were selected from the adsorption literature for testing in this work. For comparison purposes the logistic equation was also included in the curve-fitting process. The two models were fitted to the four data sets by nonlinear regression. This resulted in estimates for the parameters , , a, and b. The
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linearized versions of the two models, expressed by Eqs. (17) and (18), were fitted to one of the data sets by linear regression. For each fit, we calculated the following statistical indicators:
z
2
m
R2
j 1
mod j
2
z exp z mod z exp j j m
j 1
z j 1
(14)
2
m
mod j
z
exp
8
p2 SSE AICc m ln 2 p 11 m m p2
(15)
where the symbols and initialisms used are defined at the end of this paper. The coefficient of determination R2 is used for testing the goodness-of-fit of a model and its value ranges from 0 to 1. A model with a higher value of R2 is considered to be the best. The corrected Akaike information criterion AICc provides an implementation of Occam's razor, in which parsimony or simplicity is balanced against goodness-of-fit [31]. Given a data
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set, multiple models may be ranked according to their AICc scores, with the one producing the lowest value being the best.
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4. Results and discussion 4.1. Gompertz’s equation as a model of fixed bed adsorption
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A similarity exists between the pattern of biological growth dynamics and the rise of a fixed bed adsorber’s exit concentration. Both phenomena result in sigmoidal curves with
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respect to time. Also, the rise of the effluent concentration that is restricted by some upper limit to that rise is analogous to a growth process in the presence of restricted resources.
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Accordingly, the response variable Y in Eq. (8) can be replaced by the effluent concentration C, which “grows” over time. The maximum value or upper limit of the effluent concentration is known, which is equal to the feed concentration Co. This allows us to swap the unknown
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parameter Ymax in Eq. (8) with the known parameter Co, reducing the model’s unknown parameters from three to two. These modifications transform the three-parameter Gompertz equation to an empirical model of fixed bed adsorption with two unknown parameters: C exp exp t Co
(16)
The shape of Gompertz and logistic curves is largely dictated by the point of inflection. The inflection point for the logistic model is at the midpoint, i.e., C/Co = 0.5. This property
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restricts the logistic model to producing symmetric sigmoidal curves. That is, the convex and concave curves on either side of the inflection point have the same curvature. In contrast, the Gompertz model provides a relaxation of this restriction by placing its reflection point at YI/Ymax = C/Co = 1/e = 0.368 (see Eq. (12)). The Gompertz curve therefore differs from the logistic in not being symmetric about the point of inflection. This attribute suggests the possibility that the Gompertz equation may be able to correlate asymmetric breakthrough
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curves. This will be investigated in the ensuing sections using published breakthrough data.
4.2. Example 1: adsorption of chromium(VI) by coconut coir pith
Suksabye et al. [13] used a fixed bed apparatus packed with coconut coir pith to treat
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chromium(VI)-contaminated wastewater produced by an electroplating factory. Several breakthrough curves were recorded by varying the adsorption system’s flow rate and bed
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depth. A set of their breakthrough data examined here is depicted in Fig. 1. As can be seen in
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Fig. 1 the fixed bed column took more than 40 hours to reach saturation. The long loading duration produced a relatively smooth sigmoidal curve. Fig. 1 shows the logistic and Gompertz curves calculated using Eqs. (4) and (16) with the nonlinear regression-generated
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parameters given in Table 1, which also gives the regression statistics calculated using Eqs. (14) and (15). The logistic equation cannot adequately fit the S-shaped experimental profile, overpredicting dimensionless effluent concentrations for the initial (C/Co < 20%) and later
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(C/Co > 80%) stages. The logistic curve indicates that the column achieved breakthrough and became saturated at earlier times than those observed experimentally. It seems clear from this result that the logistic model would underestimate the amount of chromium(VI) loaded onto the bed. In contrast to the logistic fit, the Gompertz fit closely matches the form of the entire experimental profile. Indeed, the R2 and AICc values in Table 1 agree with the visual
10
observations. This suggests that the Gompertz equation can easily outperform the BohartAdams, Thomas, and Yoon-Nelson models in correlating the Fig. 1 data. The inability of the logistic equation to track the shape of the Fig. 1 data suggests that the experimental curve is asymmetric. One can see that the experimental profile exhibits some tailing; the time span of the later part (C/Co > 0.8) is noticeably longer than that of the initial stage (C/Co < 0.2). The remarkably close correspondence between the Gompertz fit and Fig. 1
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data confirms that the Gompertz equation is able to handle breakthrough curve asymmetry.
Fig. 1. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with nonlinear regression generated-
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parameters given in Table 1. Data of Suksabye et al. [13].
The overestimation of the low concentration region by the logistic equation means that
the model will underpredict the service time for a given breakthrough concentration, leading to underutilization of the capacity of the fixed bed adsorber. From the Fig. 1 data, we note that the experimental breakthrough time for a 5% breakthrough concentration is approximately 657 minutes. For this particular breakthrough concentration, the breakthrough times predicted
11
by the logistic and Gompertz equations are respectively 433 and 630 minutes. It is evident that the prediction of the logistic equation is rather poor while that of the Gompertz equation closely matches the observed service time. The Gompertz and logistic equations may be linearized to allow parameter estimation by linear regression. Linearization of the two models gives Gompertz :
(17)
C ln o 1 a bt C
(18)
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Logistic :
C ln ln o t C
The preceding equations suggest that a plot of the left-hand side of Eq. (17) or (18) versus t should be linear with slope ( or b) and intercept ( or a). Since many investigators
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in this field of research are accustomed to using linearized versions of the Bohart-Adams,
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Thomas, and Yoon-Nelson models (Eqs. (1)-(3)) in data fitting, the availability of Eq. (17) could perhaps help promote the Gompertz equationwhich has received very little attention
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in the pastas a practical tool for breakthrough curve modeling.
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Fig. 2. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with linear regression generated-parameters given in Table 1. Data of Suksabye et al. [13].
To assess their data fitting ability, Eqs. (17) and (18) were fitted to the Fig. 1 data by linear regression. The resulting best-fit parameters and R2 and AICc scores are presented in Table 1. From Fig. 2, it is clear that the two model fits are visually worse than those of Fig. 1.
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Comparison of the two sets of R2 and AICc values in Table 1 supports this visual evidence. Fig. 2 shows that the Gompertz equation cannot adequately fit the entire experimental profile, overestimating low dimensionless effluent concentrations (C/Co < 0.3). The trend of the
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logistic fit seems to be the reverse of the trend seen in Fig. 1. The logistic fit lies close to both the low and high concentration regions of the breakthrough curve (C/Co < 0.1, C/Co >0.9) but
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it deviates markedly at intermediate effluent concentrations. It seems that the linear regression
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procedure assigned greatest weight to the points at low and high effluent concentrations.
4.3. Example 2: adsorption of copper by biosorbent
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Copper uptake by a biosorbent in fixed bed column mode was investigated by Izquierdo et al. [14]. Five sets of breakthrough data obtained at different feed concentrations are available in their paper. One of the data sets exhibiting tailing behavior is reproduced in
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Fig. 3 as discrete points. The logistic and Gompertz equations were fitted to the Fig. 3 data by nonlinear regression (the variable t in this case represents the number of pore volumes instead of time). The figure includes results for another model and these will be discussed later. Both the model fits in Fig. 3 and R2 and AICc scores in Table 1 reveal that the Gompertz equation describes the experimental data better than the logistic. However, compared with the excellent Gompertz fit of Fig. 1, the Gompertz fit of Fig. 3 is less satisfactory. The deviations seen at
13
high effluent concentrations (C/Co > 0.8) show that the Gompertz equation provides only an approximate representation of the dynamic behavior. It is clear that the experimental breakthrough curve is substantially asymmetric, with an apparent broadening close to saturation conditions which cannot be matched by the Gompertz equation. Nonetheless, the early rising portion of the breakthrough curve, which is most important from a practical viewpoint, is reasonably well represented by the Gompertz model. The logistic fit of Fig. 3 is similar to that of Fig. 1: it provides a reasonable representation of dimensionless effluent
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concentrations in the 0.2-0.8 range but overestimates those for both the early and later stages.
Fig. 3. Adsorption of copper by a biosorbent showing comparison of experimental data and Gompertz and logistic curves calculated with parameters given in Table 1. Data of Izquierdo
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et al. [14]. Also shown is the theoretical prediction reported by Izquierdo et al. [14].
The distribution of residuals (i.e., the difference between the experimental and
computed values of C/Co) provides useful information on the suitability of the Gompertz and logistic equations for describing the Fig. 3 data. A plot was constructed of residual against the number of pore volumes for each of the two equations, as depicted in Fig. 4. The figure
14
includes results for another model and these will be discussed later. If the correct model has been chosen, the residuals should be scattered in a random fashion. For the Gompertz and logistic equations, the residuals lie blockwise under and over the horizontal axis, indicating that both equations are inappropriate. It is apparent that the Gompertz equation must be modified if it is to be an accurate description of the experimental results. We will discuss two
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modified forms of the Gompertz equation in Section 4.6.
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Fig. 4. Residual plots for the Fig. 3 data fitted to the Gompertz equation, logistic equation, and mechanistic model of Izquierdo et al. [14].
The Fig. 3 modeling results demonstrate the difficulty in describing asymmetry using
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the simple Gompertz equation, which can be viewed as a convenient mathematical relationship for empirically fitting asymmetric data. Application of the Gompertz model in the present study has implicitly assumed the parameters and to be of no physical significance. One concern that must be addressed is whether the lack-of-fit errors are due to the empirical nature of the Gompertz equation. Would a mechanistic model provide a more accurate prediction? An attempt was made by Izquierdo et al. [14] to model the asymmetric data of Fig.
15
3 by using a mechanistic model that accounts for axial dispersion and intraparticle diffusion expressed in the form of a linear driving force rate expression. The model was solved numerically and fitted to the Fig. 3 data by adjusting the linear driving force rate coefficient to ensure that model prediction would agree well with data at the onset of breakthrough. The resulting best-fit curve is shown as broken line in Fig. 3. It is evident that there is good agreement between the model prediction and low dimensionless effluent concentrations up to about C/Co = 0.2. For the remainder of the experimental profile, the correspondence between
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prediction and experiment is rather poor, indicating that the mechanistic model is not adequate enough. Fig. 4 shows a plot of the residuals for this model fit. The majority of the points lie blockwise under the horizontal axis, revealing a misfit between the model and the data.
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Clearly, further refinement of the mechanistic model is necessary. The large discrepancy is presumably due to the failure of the mechanistic model to consider factors responsible for the
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tailing behavior.
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Various reasons have been put forward in the literature to explain the tailing phenomenon observed in fixed bed adsorbers including flow non-uniformity, nonspecific adsorption, enhanced adsorption under continuous flow conditions, heterogeneous binding
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mechanism with fast and slow binding sites, heterogeneous particle or pore size distribution effects, etc. Rigorous mechanistic models accounting for these factors generally require inordinate effort to generate substantial data for model validation and parameter estimation. In
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reality, the tailing phenomenon is much complicated and the exact mechanisms leading to tailing have been identified only in very limited circumstances. In fact, different mechanisms may be responsible for the breakthrough tailing in individual cases. With these considerations in mind, empirical models such as the Gompertz equation would seem preferable as they are more convenient to use, computationally simpler, and could have immediate practical benefits.
16
Admittedly, there is no mechanistic background behind the Gompertz equation when it is applied to fixed bed adsorption. However, it must be borne in mind that the commonly used Bohart-Adams and Thomas models can also be viewed as mere curve-fitting tools. The reason for this is that these models are based on chemical reaction rate laws which contradict the actual diffusion mechanisms present in most adsorption systems of interest. In addition, their parameters are empirical in the sense that they vary with operational and system variables such as flow rate, initial adsorbate concentration, and bed length. The validity of these
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parameters is limited to the range of conditions used during experimental tests conducted to generate breakthrough data for model calibration. In other words, the fitted parameters cannot be used to provide a priori breakthrough predictions over a wide range of operating conditions.
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When viewed in this way, it becomes obvious that the empirical nature of the Gompertz
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equation is no different from those of the Bohart-Adams and Thomas models.
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4.4. Example 3: adsorption of methylene blue by activated carbon The ability of the Gompertz equation to describe asymmetry was further challenged using a set of breakthrough data reported by Li et al. [15] who investigated dye adsorption in
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fixed bed adsorbers packed with activated carbon. Fig. 5 presents the Gompertz and logistic fitting results graphically. For both models, Fig. 5 shows that large lack-of-fit errors are evident, indicating the presence of a pronounced asymmetry in the experimental data. It can
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be seen that the entire breakthrough pattern is skewed by the tailing behavior. As a result, the early rising and final tailing portions of the breakthrough curve are not predicted exactly by the Gompertz equation. Dimensionless effluent concentrations in the intermediate region are also poorly represented by the Gompertz equation. The logistic fit seems worse than the Gompertz fit. This visual observation is supported by the statistical evaluation presented in Table 1.
17
Although the Gompertz model is inherently asymmetric, the results of Examples 2 and 3 reveal that it is not accurate enough in describing the asymmetric characteristics of these two data sets. The reason for this unsatisfactory performance could be attributed to the fact that the Gompertz equation has a fixed inflection point which is located at C/Co = 1/e. With this fixed inflection point, which is below the midpoint, the Gompertz equation is able to fit breakthrough curves with a moderate degree of tailing but is unable to handle those with a marked degree of tailing. To fit such curves, the inflection point would have to be located
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further away from the midpoint. This would require the use of models with a variable
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inflection point. The analysis of such models is beyond the scope of the present study.
Fig. 5. Adsorption of methylene blue by activated carbon showing comparison of
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experimental data and model curves calculated with parameters given in Table 1. Data of Li et al. [15].
4.5. Example 4: adsorption of methylene blue by activated carbon It is of interest to compare the relative performance of the Gompertz and logistic equations in fitting symmetric data. The two equations were fitted to a set of breakthrough
18
data reported by Foo and Hameed [32] who investigated methylene blue adsorption in an activated carbon column. Fig. 6 depicts the selected experimental curve along with its model counterparts. The logistic equation is shown to capture the experimental observations very well, as expected. For the Gompertz fit, the agreement is not as close. It is apparent that low and high dimensionless effluent concentrations are much less well predicted. Because the Gompertz equation has no point of symmetry, the modest levels of lack-of-fit error can be considered tolerable. Note that the R2 score for the logistic fit is 0.9994, suggesting that this
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data set is highly symmetric with slight to no tailing. It is evident that the logistic equation is a more appropriate mathematical representation of this data set. In view of the results reported in Examples 3 and 4, one can conclude that the Gompertz equation is not a versatile modeling
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tool because it specializes in fitting mildly asymmetric breakthrough curves with a moderate
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degree of tailing.
Fig. 6. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Foo and Hameed [32].
19
4.6. Modified Gompertz equations To reduce the lack-of-fit error of a model, an approach that is often taken is to increase the number of free parameters. That is, adding an adjustable parameter to a model will almost always improve fit to some degree. Accordingly, the Gompertz equation can be extended by adding an exponent to the time variable t. The equation so modified is given by
C exp exp t n Co
(19)
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where n is a dimensionless fitting parameter. In this work the preceding equation is known as the power law Gompertz. Note that Eq. (19) is called the Sloboda equation in the growth curve modeling literature [33].
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Another method sometimes used to improve fit operates by converting independent variables to logarithmic terms. First, the Gompertz equation expressed by Eq. (16) is rewritten
(20)
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C t exp exp t * * Co t
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as
where t* is the unit time in the chosen system of units (e.g., 1 s or 1 min). Thus (t/t*) is a
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dimensionless quantity numerically equal to t while (t*) is also a dimensionless quantity numerically equal to . It is necessary to introduce the dimensioned quantity t* because we cannot take the logarithm of t which is of course a dimensioned quantity. Next, we modify Eq. (20) empirically by taking the logarithm of (t/t*):
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C t exp exp t * ln * Co t
(21)
For convenience, the preceding equation is rewritten as C exp exp ln t Co
(22)
20
We stress that the t and terms in the preceding equation do not have their usual meanings; t represents pure numbers numerically equal to the actual t while represents pure numbers numerically equal to the actual . Here, Eq. (22) is called the log-Gompertz equation. Compared to the power law Gompertz which has three fitting parameters, the log-Gompertz remains a two-parameter equation. We note in passing that the log-Gompertz equation can be linearized to facilitate parameter estimation by linear regression.
C ln ln o ln t C
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(23)
A plot of the left-hand side of Eq. (23) versus lnt yields a slope and an intercept from which
ur na
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and are obtained.
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Fig. 7. Comparison of the Fig. 5 data and model curves calculated with parameters given in Table 2.
21
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Fig. 8. Comparison of the Fig. 5 data and model curves calculated with parameters given in
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Table 2.
To evaluate their data fitting ability, Eqs. (19) and (22) were fitted to the highly
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asymmetric data of Fig. 5 by nonlinear regression. The resulting parameters are summarized
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in Table 2, and comparisons between the experimental data and the model fits are shown in Figs. 7 and 8. For comparison purposes both figures also show the original Gompertz fit given
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by Eq. (16). It can be seen that the two modifications provide fits that are visually much better, eliminating most of the lack-of-fit errors present in the original Gompertz fit. Table 2 shows the R2 and AICc scores for the model fits in Figs. 7 and 8, which confirm that asymmetry can be effectively modeled by the three-parameter power law
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Gompertz and the two-parameter log-Gompertz. The R2 and AICc scores for these two modified Gompertz equations are fairly similar. Ranking rival models with different numbers of fitting parameters in terms of the R2 test is a simplistic approach because the R2 indicator does not compensate for possible bias to parameter-rich models. In contrast to the R2 test, the AICc test addresses the tradeoff between gain in fit and addition of new parameters [31]. It is a quantitative way to rank competing models with different numbers of adjustable parameters
22
and identify the model that is most justified by the data at hand. In Eq. (15) the first term measures fit, while the second term penalizes complex models, i.e., models with more adjustable parameters. An effective way to interpret AICc values calculated for a set of competing models is via a set of positive Akaike weights [34,35]. The Akaike weight for model i among a cohort of J competing models is given by exp 0.5 i
(24)
exp 0.5 J
j
j 1
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wi
where
i AICc,i AICc,min
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In the preceding equation, wi is the Akaike weight for model i, i is the difference in the AICc
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values for model i (AICc,i) and the model with the lowest AICc score (AICc,min). The Akaike weight can be interpreted as the probability that model i is the best model given the data at
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hand and the chosen set of candidate models.
The wi values calculated from Eq. (24) for fitting to the Gompertz, log-Gompertz, and
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power law Gompertz equations are presented in Table 2. The wi value for the Gompertz fit is practically zero, indicating that it receives no support from the Fig. 5 data. This result is not surprising given that its AICc score is substantially greater (i.e., less negative) than those of the power law Gompartz and log-Gompertz. The three-parameter power law Gompertz is also
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not supported by the Fig. 5 data even though its AICc value is only slightly inferior to that of the log-Gompertz. It is apparent that the conclusion from the AICc test is to use the logGompertz equation to fit this data set; it is the best one in this group of three competing models.
5. Conclusions
23
The Gompertz and the logistic equations have been applied to the analysis of published breakthrough data on contaminant adsorption in fixed bed adsorbers. Of these two equations the Gompertz has attracted little attention as a breakthrough curve model while the logistic is mathematically analogous to the widely used Bohart-Adams, Thomas, and YoonNelson models. The logistic equation cannot effectively fit breakthrough curves of asymmetric shape. This in turn means that the Bohart-Adams, Thomas, and Yoon-Nelson models are generally unable to handle much of the asymmetric breakthrough data commonly
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observed in environmental adsorption studies. The Gompertz equation provides a partial solution to the problem of describing breakthrough tailing close to column saturation. Mildly asymmetric breakthrough curves exhibiting a moderate degree of tailing can be adequately
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represented by the Gompertz equation. Its performance is less satisfactory in describing breakthrough data with a marked degree of tailing. Two modified forms of the Gompertz
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equation, known as the power law Gompertz and the log-Gompertz, permit strong asymmetry
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to be effectively modeled. The log-Gompertz equation has two fitting parameters while the power law Gompertz has three. The former also has a linear form. Therefore, the
purposes.
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mathematically simpler log-Gompertz equation may be an adequate choice for practical
Credit Author Statement
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This is a single-author manuscript.
Khim Chu
Declaration of competing interests
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The author declares that he has no known competing financial interests or personal
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relationships that could have appeared to influence the work reported in this paper.
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[30] K.M.C. Tjørve, E. Tjørve, 2017. The use of Gompertz models in growth analyses, and new Gompertz-model approach: an addition to the Unified-Richards family. PLoS One. 12, e0178691. [31] H. Akaike, A new look at the statistical model identification, IEEE Trans. Autom. Control 19 (1974) 716–723. [32] K.Y. Foo, B.H. Hameed, Dynamic adsorption behavior of methylene blue onto oil palm shell granular activated carbon prepared by microwave heating, Chem. Eng. J. 203
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Figure captions
Fig. 1. Adsorption of chromium(VI) by coconut coir pith showing comparison of experimental data and model curves calculated with nonlinear regression generatedparameters given in Table 1. Data of Suksabye et al. [13].
Fig. 2. Adsorption of chromium(VI) by coconut coir pith showing comparison of
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experimental data and model curves calculated with linear regression generated-parameters given in Table 1. Data of Suksabye et al. [13].
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Fig. 3. Adsorption of copper by a biosorbent showing comparison of experimental data and Gompertz and logistic curves calculated with parameters given in Table 1. Data of Izquierdo
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et al. [14]. Also shown is the theoretical prediction reported by Izquierdo et al. [14].
Fig. 4. Residual plots for the Fig. 3 data fitted to the Gompertz equation, logistic equation,
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and mechanistic model of Izquierdo et al. [14].
Fig. 5. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Li et
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al. [15].
Fig. 6. Adsorption of methylene blue by activated carbon showing comparison of experimental data and model curves calculated with parameters given in Table 1. Data of Foo and Hameed [32].
30
Fig. 7. Comparison of the Fig. 5 data and model curves calculated with parameters given in Table 2.
Fig. 8. Comparison of the Fig. 5 data and model curves calculated with parameters given in
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Table 2.
31
Table 1 Best-fit parameters and regression statistics for the Gompertz and logistic fits in Figs. 1-3, 5, and 6. Example Regression no.
Logistic model
method
/ min-1
R2
AICc a
b / min-1
R2
AICc
Nonlinear
2.671
0.0025
0.9988
-404
4.505
0.0036
0.9933
-324
Linear
2.229
0.0022
0.9933
-327
4.805
0.0034
0.9611
-240
2
Nonlinear
7.562
0.0117a
0.9818
-118
11.832
0.0174b
0.9626
-103
3
Nonlinear
1.620
0.0128
0.9827
-235
2.870
0.0179
0.9564
-200
4
Nonlinear
7.504
0.0270
0.9939
-129
11.616
0.0393
0.9994
-176
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1
/ (number of pore volumes)-1 ; b b / (number of pore volumes)-1
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a
Gompertz model
Table 2
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Best-fit parameters and regression statistics for the Gompertz, log-Gompertz, and power law Gompertz fits in Figs. 7 and 8. Model
Number of
/ min-1
n
R2
AICc
wi
parameters 2
1.620
0.0128
0.9827
-235
0.00000
Log-Gompertz
2
9.487
1.985
0.9979
-316
0.99993
Power law Gompertz
3
12.748
6.244
0.9968
-297
0.00007
Jo
Gompertz
32
0.15