Mathematical Biosciences 265 (2015) 40–46
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A sigmoidal model for biosorption of heavy metal cations from aqueous media Rümeysa Özen1, Nihat Alpagu Sayar1,∗, Selcen Durmaz-Sam, Ahmet Alp Sayar Department of Bioengineering, Marmara University, 34722, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 8 April 2014 Revised 10 April 2015 Accepted 15 April 2015 Available online 24 April 2015 Keywords: Modeling Biosorption Ni(II) Fission yeast Validation Optimization
a b s t r a c t A novel multi-input single output (MISO) black-box sigmoid model is developed to simulate the biosorption of heavy metal cations by the fission yeast from aqueous medium. Validation and verification of the model is done through statistical chi-squared hypothesis tests and the model is evaluated by uncertainty and sensitivity analyses. The simulated results are in agreement with the data of the studied system in which Schizosaccharomyces pombe biosorbs Ni(II) cations at various process conditions. Experimental data is obtained originally for this work using dead cells of an adapted variant of S. Pombe and represented by Freundlich isotherms. A process optimization scheme is proposed using the present model to build a novel application of a cost-merit objective function which would be useful to predict optimal operation conditions. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The use of biomass as biosorption agent for heavy metals from aqueous solutions frequently offer easier, cleaner and cheaper routes for certain water remediation and biohydrometallurgical purposes. They are now being investigated in many applications [1]. Among a great number of available biosorption agents especially yeasts show promising performances for removal and bioseparation of heavy metals from aqueous media [2]. The fission yeast Schizosaccharomyces pombe has received increasing interest for heavy metal cation removal and recovery [3]. Often, the experimental load for biosorbent development and process design is a major bottleneck for the development of such bio-applications in industrial processes. The time required to develop a new process is critical. One potential solution is to use mathematical modeling and computer simulations to aid the experimental procedures. Significant benefits could be gained in overall economics and performance when mathematical modeling and simulation techniques are applied to processes especially when combined with a systems type approach, namely the integrated design, control, optimization and operation of processes. This allows prioritization of process improvements through a detailed understanding of the cost-benefit relationships of potential process changes.
∗
Corresponding author: Tel.: +90 216 3480292, fax: +90 216 3450126. E-mail addresses: [email protected] (N.A. Sayar), selcen.durmaz@ marmara.edu.tr (S. Durmaz-Sam), [email protected] (A.A. Sayar). 1 Equally contributed authors. http://dx.doi.org/10.1016/j.mbs.2015.04.007 0025-5564/© 2015 Elsevier Inc. All rights reserved.
Mathematical modeling in process development is used to bring a measure of order to observation and strengthen prediction [4–6]. Important process parameters that are used to define equipment specifications and operational conditions can be identified. Parameter estimation methods are applicable to find the process parameters that fit the specific mathematical model of the system in question [7]. Some of the research described in the scientific literature focuses on using models to simulate the effects of key operating conditions on a particular performance metric [8]. In addition, another section of the scientific literature studies the application of empirical models [9]. To date, the main body of mathematical modeling work within biological sciences focuses on microbial growth kinetics [10], operation kinetics [11], and also biosorption processes [12]. Reactor modeling [13], modeling of other unit operations [14] and biosorption optimization are also common [15]. On the other hand, a major section of the modeling work on biosorption focuses on the microscopic physicochemical and thermodynamic properties of this phenomenon. For example, Plazinski’s [16] review lists models that relate metal uptake to binding site concentration using gas adsorption isotherms. Some other models estimate metal uptake as a function of pH involving the competition phenomenon between metal cations and protons and electrostatic effects. Another concept studied in various modeling works is the relationship between ionic strength, binding mechanism and metal uptake via Donnan volume. Among these models the semi-empirical NICA (non-ideal competitive adsorption) model can be used to predict metal charge in relation to pH change [17–19]. A modified version of this model, NICCA incorporates thermodynamic consistency by considering heterogeneity of the biomass used [20,21].
R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
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Fig. 1. Residuals Plot for Freundlich, Langmuir and Combined Langmuir–Freundlich (Sips) isotherms.
In comparison to the abovementioned models focusing on the specifics of the mechanism of biosorption, the model proposed in the current study aims to relate macroscopic, process related metrics to critical input variables. Metrics are a universal way of looking at the performance and assessing the economic viability of a process [22]. Metrics can be used to relate concepts such as the effort put into experimentation, the timescales of process development and biosorbent specifications to the cost and financial return of the process. In this work, the process studied is the biosorption of Ni (II) cations from aqueous solutions at various independent process conditions (variables) such as Ce , residual Ni (II) concentration at equilibrium (mg L−1 ); pH and T °C. The resulting metric is qe , Ni (II) uptake at equilibrium (mg g−1 ).The experimental data used to validate and verify the proposed model are obtained originally for this work. The current paper offers two novel applications for the modeling and preliminary design approach of the present process. The first is the development of a multi input-single output [23] sigmoid function to model the biosorption operation. The second is the proposal of an optimization scheme depending on a cost-benefit function [24] adapted for this process. Conventional process analysis often requires experimental determination of a set of process responses for a given array of forcing variables. For complex processes such as biosorption with multivariate input and output variables, the necessary experimentation is costly and time consuming. The availability of a simpler black-box model based on the evaluation of a very limited number of experiments allows the simulation of the required process outputs for a systematically chosen set of inputs [25]. 2. Materials and methods 2.1. Biosorption experiments The biosorption data of Ni(II) onto dead wild type S. pombe (972 h−1 ) through a batch equilibrium process have been collected originally for this work at various T and pH conditions of 25, 30, 35,
50 °C and 4, 5 and 6, respectively. The equilibrium values of Ni (II) uptake by the biomass, qe , mg (g dry weight)−1 were calculated using
qe =
(Ci − Ce )V m
(1)
where Ci , Ce , V and m denote, initial Ni(II) concentration of the aqueous phase (mg L−1 ), equilibrium Ni(II) concentration of the aqueous phase (mg L−1 ), volume (L) of the solution contacted by the sorbent and the amount of the sorbent expressed as dry mass (g), respectively. The wild-type (972 h−1 ) S. pombe strain used in this work was obtained from the Department of Molecular Biology and Genetics, University of Istanbul. This strain was grown in aqueous medium containing 30 g L−1 glucose and 5 g L−1 yeast extract, both acquired from Merck. Biosorption experiments were performed following a recently published procedure [26]. Each experiment was repeated three times and arithmetic averages were taken. Filtered samples were analyzed in triplicate with Perkin Elmer AAS 400 atomic absorption spectrophotometer to measure the concentration of Ni (II) species remaining in aqueous medium. 2.2. Model development A universal sigmoid function which is a simple but versatile mathematical function [23,24] can be used to represent the input–output relationship between the abovementioned process variables. The sigmoid relates the independent forcing variables to the process response with a certain quality of fit. The advantage of this function comes from its simple and robust applicability to Jacobian and Hessian operations thus making the development of optimization schemes mathematically more manageable. The raw biosorption data were regressed through Freundlich, Langmuir and combined Langmuir–Freundlich (Sips) isotherms. Sips and Freundlich approaches yielded statistically better fits whereas Langmuir exhibited poorer results. The comparison of goodness of fit for three isotherms was done plotting residuals for each approach in Fig. 1.
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For modeling considerations, Sips isotherm has been chosen to reproduce data because of its physico-chemically consistent parameters. The numerical results for three different isotherms were given in Supplementary Material 1. The universal sigmoid function with Ce , T and pH as process variables and the resulting output % qe is:
%qe = 100 − [a1 expa2 P +a6 expa7 P ]
(2)
where
P = (Ce )a3 (T )a4 (pH)a5
(3)
(qe )act (qe )max
(4)
%qe =
where (qe )act is the simulated value for the given input and (qe )max is the highest value in the whole simulated data matrix. (qe )max for each separate case is obtained by correlating qe versus Ce using Sips isotherms. The dimensionless normalized form of %qe , a percentage expression, is appropriate to be used in validating the model through goodness of fit chi-square test. ai = 1–7 are the sigmoid model parameters to be optimized. Matlab based non-linear curve fitting functions have been used to estimate the model parameters evaluating an input–output array of 64 elements in which Ce varies from 0.48 to 50 mg L−1 , T from 25 to 50 °C and pH from 4 to 6. The proposed sigmoid model structure can also incorporate Ci as one of the input variables instead of Ce since Ce is a function of Ci . The version of the model with Ci as an input variable is given in Supplementary Material 3 with the corresponding model parameter values and optimization statistics.
as a uniform distribution of each input parameter with 10% variability. The Latin-Hypercube sampling method was utilized to create 500 sets of input parameters within the identified input parameter space. 500 dynamic simulations (each using one set of input parameters) were run to propagate the samples. Standard regression coefficient sensitivity analysis was conducted by constructing linear regression models on the outputs obtained from the Monte Carlo procedure [29]. The standardized regression coefficients method was implemented to determine the coefficient values for 7 model parameters. Scalar matrix sy for each output for 500 samples of the Latin hypercube can be composed as:
syi,%qe = d0,%qe +
7 j=1
dj,%qe θi,j + εi,%qe
(6)
θ i , εi , d0 and dj denote probabilistic parameter sampling for i = 500 elements Latin Hypercube, probabilistic regression error for i = 500 elements of the Latin Hypercube, linear regression coefficient and regression coefficients, respectively. The indices i (1–500) and j (1–7) stand for total number of Latin-Hypercube samples and total number of model parameters, respectively. For each sy a multi-independent variable linear equation set was configured using their corresponding mean M and standard deviations, σ :
syi,%qe − Msy%qe
σsy%qe
=
7 j=1
dj,%qe
θi,j − Mθ j + εi,%qe σθ j
(7)
Using multivariable linear regression techniques numerical values R were found for each d coefficient using Matlab . 3. Results and discussion 3.1. Biosorption experiments
2.3. Model validation 2.3.1. Chi-square test A two-sided chi-square, χ 2, test can be used to assess the statistical acceptability of the simulation results [23,27,28]. The test requires a null hypothesis which in this case presumes zero difference between the simulation results and experimental data. The degrees of freedom for this system can be calculated as:
Degrees of freedom = number of input − output sets − number of parameters − 1
(5)
2.3.2. Uncertainty and sensitivity analyses Uncertainty and sensitivity analyses are essential parts of good modeling practice (GMoP) [29]. The quality of the model can be assessed applying uncertainty and sensitivity analyses. Uncertainty analysis indicates how uncertainty in parameter optimization is propagated throughout the simulated model responses. The uncertainty may arise from the imprecision in parameters and the defects of the structure of the model in capturing the right mechanism. The analysis has been conducted setting minimum and maximum values at the 10th and 90th percentiles. The uncertainty analysis results lead to determine which improvements could be implemented to modify the model structure. Its application may indicate which section in the system is a good target for improvement. The evaluation of the model may be extended to include a sensitivity analysis to assist identifying the most influential parameters, for which further collection of data may be in order to improve model accuracy. A Monte Carlo procedure was used to estimate the output uncertainty in the metric of the model. Subsequently, standardized regression coefficient (SRC) sensitivity analysis was conducted on the input parameters. The Monte Carlo procedure for the analysis of uncertainty involves the specification of input uncertainty, sampling, and propagation of uncertainty. The input parameter space was created
Among the adsorption models tested Sips was found to be the best to correlate the experimental results. The related isotherm parameters representing the observed data are given in Table 1. The raw data is not shown in the text. It is available as Supplementary Material 1. It has to be noted that the experimental results concerning the metal uptake exhibit a slight discrepancy with the previously published data [26], possibly due to random mutagenesis of the available S. pombe strain generated from the original source. 3.2. Model parameters and simulation results The model parameters were optimized using experimentally obtained data applying the Matlab based nonlinear curve fitting functions. Numerical values and related regression statistics are given in Table 2. The simulated results are given in the supplementary data section. 3.3. Model verification and validation A test statistics χ 2 with 64 value sets for estimating 7 parameters, corresponding to 56 degrees of freedom, is calculated to be 65.53. The projected upper-tail critical value of (χ 2cv )56 upper with a chance of 10% (α 0.1 56 ) of rejecting the null hypothesis is 69.92. The projected lower-tail critical value of (χ 2 cv )56 lower with a chance of 10% (α 0.1 56 ) of rejecting the null hypothesis is 42.94. Similarly, (α 0.05 54 )upper , (α 0.05 54 )lower are 74.47 and 39.80 respectively. For double-sided validation, test statistics is lesser than the upper critical value and greater than the lower critical value which asserts that null hypothesis is not rejected for both tail tests under the α 0.1 54 and α 0.05 54 conditions [27,28]. For a model with three independent inputs estimating one dependent output, a correlation factor of 0.941 can be found satisfactory to assume the model results are in good agreement with the experimentally generated data.
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Table 1 Langmuir–Freundlich (SIPS) isotherm parameters correlating experimental data with their 95% confidence intervals. β
qe = KS Ce /1 + (α Ce )β KS
α
25 ° C pH = 4 0.27 –0.002 0.26 –0.02 027 0.02 pH = 5 0.52 0.013 0.34 –0.34 0.69 0.37 pH = 6 0.41 0.003 0.41 0.002 0.41 0.003
β
KS
α
β
30 ° C
KS
α
β
35 ° C
KS
α
B
50 ° C
0.38 0.36 0.04
0.24 0.24 0.25
0.015 –0.02 0.05
0.38 0.35 0.41
0.17 0.17 0.18
–0.003 –0.02 0.01
0.45 0.42 0.48
0.05 0.048 0.060
–0.0005 –0.03 0.01
0.68 0.59 0.77
0.51 0.18 0.83
0.46 0.44 0.48
–0.001 –0.04 0.04
0.50 0.46 0.54
0.34 0.29 0.39
–0.008 –0.15 0.14
0.53 0.37 0.69
0.29 0.10 0.49
0.22 –0.58 1.02
0.32 0.01 0.62
0.59 0.59 0.60
0.38 0.38 0.39
–0.0001 –0.002 0.002
0.56 0.56 0.57
0.31 0.31 0.32
–0.003 –0.01 0.004
0.50 0.48 0.51
0.27 0.22 0.33
0.07 –0.18 0.31
0.18 0.12 0.25
Fig. 2. Verification of model performance. Comparison of modeled and observed results via diagonal test.
Table 2 The estimated numerical values with their standard errors (SE) for model parameters and computed statistics. Estimated values for model parameters a1 a2 a3 a4 a5 a6 a7
= 76.14 (1.58) = –1.71 (3.91) = 2.19 (0.33) = –7.16 (1.18) = 8.81 (1.30) = 17.37 (1.61) = –1094.9 (2799)
Regression statistics
R2 = 0.941 χ 2 = 65.53
In the present assessment of the model fit, the test statistics and the fit index have been found satisfactory but insufficient. Because, the nonlinear multivariate regression results with promising RMSE, R2 and χ 2 values may seldom exhibit some disagreement between the predicted and observed results. For the verification, the experimental and the modeled results have been compared through a graphical
diagonal test given in Fig. 2. The test indicates that such disagreement does not occur in this work because the distribution of comparison points along the diagonal remains in an acceptably narrow band. Moreover, the random pattern of points at each side of the diagonal implies that the model is almost free of systematic errors. 3.3.1. Uncertainty and sensitivity analyses The uncertainty results illustrating the three selected cases with varying Ce at T = 37.6 °C and pH = 5; varying T at Ce = 25.5 and pH = 5 and varying pH at Ce = 25.5 and T = 37.6 °C are represented in Figs 3, 4 and 5, respectively. The uncertainty in the outputs is represented using 10th and 90th percentile and mean of the distribution of each output. The gray area indicates the propagated uncertainty range. It is to be noticed that the larger the spread of gray region the higher the uncertainty is. The studies were done for arbitrary simulation cases selected from the middle region of the range of input variables (Ce = 25.5, varying between 0 and 50; T = 37.6 °C, varying between 25 and 50; pH = 5, varying between 4 and 6).
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R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
Fig. 3. Uncertainty propagation with varying Ce ; T and pH constant.
Fig. 4. Uncertainty propagation with varying T; Ce and pH constant.
In cases where Ce and pH are varying (Figs. 3 and 5, respectively), the uncertainty remains in a tolerable band for lower to middle region values and exhibits a tolerable spread area (between the 10th and 90th percentiles). However, for higher input values of the two variables the uncertainty in the model is increased. In the case where T is varying (Fig. 4), the uncertainty propagation exhibits a wider spectrum at lower temperature values but with increasing T values uncertainty in the model decreases. Higher values of uncertainty indicate the regions where experimental coverage should be enlarged. This leads to assume that additional experiments at lower Ce and pH and at higher T values would narrow the uncertainty spread of the model. The sensitivity analysis results are done for the same cases as the uncertainty study. The SRCs were computed using Eqs (6) and (7). The coefficients obtained for each model parameter were ranked for
model output. The analysis was done for each independent input by varying this input in the range of investigation while the others were being held constant at selected case values. The results and related statistics are summarized in Table 3. The values of the coefficients are between –1 and 1. A higher absolute magnitude indicates a higher sensitivity of the model to the particular parameter. The adjusted-R2 values reflecting the linearization degree of the regression profiles are also given. All the adjusted-R2 values were found to be greater than the lower acceptable value of 0.7 [30]. In all of the three cases a4 was found to be the most influential model parameter while a1 , a3 and a5 all have lesser effect with similar magnitudes. The analysis indicates that the simulations are less sensitive to the parameters a2 , a6 and a7 . According to these results it can be assumed that a4 must be optimized with experiments at various temperatures as accurately as possible. Similar suggestions may be made for Ce and pH. This leads
R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
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Fig. 5. Uncertainty propagation with varying pH; Ce and T constant. Table 3 Sensitivity analysis.
Varying pH
The basic linear cost function of the present biosorption operation may be expressed as the summation of the corresponding terms of three independent inputs [18] where Ce is replaced with Ci due to the controllable nature of the latter:
0.906
Fcos t = cp1 Ci + cp2 pH + cp3 T
%qe Varying Ci
Varying T
Adj- R2
0.897
0.925
Rank
θ
SCR
θ
SCR
θ
SCR
a4 a1 a5 a3 a2 a7 a6
0.717 –0.449 0.424 0.216 –0.050 –0.029 –0.0004
a4 a5 a1 a3 a2 a7 a6
0.764 0.450 –0.373 0.227 –0.047 –0.022 0.008
a4 a5 a1 a3 a2 a7 a6
0.726 0.439 –0.430 0.219 –0.050 –0.027 0.007
1 2 3 4 5 6 7
to conclude that the number of experiments can be increased with a focus on the abovementioned variable ranges to reach a lower degree of uncertainty and higher accuracy in simulated model results. As expected, the relative insensitivity of the model to the parameters a6 and a7 is due to highest ranking of their adjacent parameters a1 and a2 in the model equation constituted by two combinatorial terms. According to this, one may question the necessity of the second combinatorial term. The optimization trials with a truncated model with only 6 parameters yielded much less successful simulation results with lower R-squared values of around 0.920 compared to 0.941 of the proposed model (data not shown). It may be concluded that the mathematical structure of the sigmoid needs the inclusion of the second combinatorial term to exhibit sufficient versatility to represent the studied phenomena with higher adequacy. The sensitivity rankings which are almost similar in all three cases may refer to the intrinsic robustness of SCR method when applied to a black box model with multi input–single output. 3.4. Benefits of the sigmoid model The principle task of process optimization is to develop an objective function. The objective function may essentially be composed of two main parts: merit and cost. For the present biosorption process the merit function can be established in terms of %qe :
Fmerit = 100 − [a1 expa2 P + a6 expa7 P ]
(8)
(9)
where cp1 , cp2 and cp3 denote techno-economical parameters relating the input parameter to their effects toward the cost of the process. They are related to the costs of the raw materials entering the process and keeping the solution at desirable pH and T. Let the techno-economical parameters of the chosen inputs be cp1 , cp2 and cp3 for Ci , pH and T, respectively. These coefficients reflect the unit costs of Ci , acid or base used to obtain a unit change in pH and utility consumed per degree of temperature to maintain the solution at desired temperature. For process optimization purposes Ci is considered as a controllable input variable instead of Ce , which is a more conventional variable for process analysis purposes. The magnitude of the unit cost of one of the input variables may be taken as unity while the two other magnitudes are quantified as its multiples.
J(Ci , pH, T ) = Fmerit (Ci , pH, T ) − Fcos t (Ci , pH, T )
(10)
J is then maximized in order to estimate the set of optimal values for key inputs. The maxima of J occur at points where the slope is zero, which is at points where gradient of J, J satisfies
∇J =
∂J ∂J ∂J , , ∂ Ci T pH
T =0
(11)
This proposed optimization scheme can be used for preliminary process design once the relevant techno-economical parameters are identified. The optimization scheme has been implemented for a hypothetical case study. Table 4 summarizes the corresponding sets of optima related to various fictive cost parameter combinations. 4. Conclusion An experimentation based multi-input single-output sigmoid model has been developed for a biosorption process, evaluating the biosorption of Ni (II) cations by S. pombe as a case study. The performance of the model has been validated and verified through goodness
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R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46 Table 4 Optimization of biosorption operation of key inputs.
Ci pH T
cp1 = 0.1 cp2 = 1 cp3 = 3
cp1 = 0.5 cp2 = 2 cp3 = 1
cp1 = 1 cp2 = 1 cp3 = 1
21.84 7.77 25
17.44 8 25
2.01 7.15 25
of fit statistics and the model is evaluated by sensitivity and uncertainty analyses. The proposed model successfully correlates process input variables pH, T and Ce (or Ci ) to the output variable qe exhibiting R-squared values above 0.94. The model uncertainty has been shown to decrease at higher T values and lower pH and Ce values. The model output is found to be most sensitive to the model parameter a4 followed by a1 , a3 and a5 . The remaining parameters a2 , a6 and a7 show relatively less effect on the sensitivity analysis. The suggested modeling approach which provides reliable estimates of operational metric(s) has been shown to be potentially useful and dependable to assist in biosorption development to facilitate knowledge-based decision making. It is assumed that it may reduce the investment of research and analysis efforts eliminating least promising circumstances especially at early stages of bioprocess development. There are several approaches to the modeling of biosorption phenomena focusing on the characterization of the physico-chemical aspects of the process such as those that incorporate the Donnan volume concept. The current model differs both in scope and purpose from these models. The current model aims to empirically correlate process related variables such as pH, T and Ce (or Ci ) to the output metric qe without emphasizing the microscopic properties of biosorption, using a versatile universal sigmoid function which can subsequently be used in a simple techno-economic process optimization scheme. An example of the process optimization scheme is given using dummy variables. For future works, the modeling approach has been proposed as an engineering tool in optimizing cost-benefit trade-off, balancing variable process inputs such as raw material and utility costs with process profitability expressed in process metrics. An optimization scheme was developed to focus on attractive and promising process options which can be taken into consideration as a starting point for a more elaborate techno-economic analysis. Acknowledgment Funding by Marmara University Scientific Research Committee (Project FEN-C-YLP-040712-0280) is gratefully acknowledged. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mbs.2015.04.007. References [1] K. Chojnacka, Biosorption and bioaccumulation- prospects for practical applications, Environ. Int. 36 (2010) 299–307. [2] J. Wang, C. Chen, Biosorbents for heavy metals removal and their future, Biotechnol. Adv. 27 (2009) 195–226.
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Contents lists available at ScienceDirect
Mathematical Biosciences journal homepage: www.elsevier.com/locate/mbs
A sigmoidal model for biosorption of heavy metal cations from aqueous media Rümeysa Özen1, Nihat Alpagu Sayar1,∗, Selcen Durmaz-Sam, Ahmet Alp Sayar Department of Bioengineering, Marmara University, 34722, Istanbul, Turkey
a r t i c l e
i n f o
Article history: Received 8 April 2014 Revised 10 April 2015 Accepted 15 April 2015 Available online 24 April 2015 Keywords: Modeling Biosorption Ni(II) Fission yeast Validation Optimization
a b s t r a c t A novel multi-input single output (MISO) black-box sigmoid model is developed to simulate the biosorption of heavy metal cations by the fission yeast from aqueous medium. Validation and verification of the model is done through statistical chi-squared hypothesis tests and the model is evaluated by uncertainty and sensitivity analyses. The simulated results are in agreement with the data of the studied system in which Schizosaccharomyces pombe biosorbs Ni(II) cations at various process conditions. Experimental data is obtained originally for this work using dead cells of an adapted variant of S. Pombe and represented by Freundlich isotherms. A process optimization scheme is proposed using the present model to build a novel application of a cost-merit objective function which would be useful to predict optimal operation conditions. © 2015 Elsevier Inc. All rights reserved.
1. Introduction The use of biomass as biosorption agent for heavy metals from aqueous solutions frequently offer easier, cleaner and cheaper routes for certain water remediation and biohydrometallurgical purposes. They are now being investigated in many applications [1]. Among a great number of available biosorption agents especially yeasts show promising performances for removal and bioseparation of heavy metals from aqueous media [2]. The fission yeast Schizosaccharomyces pombe has received increasing interest for heavy metal cation removal and recovery [3]. Often, the experimental load for biosorbent development and process design is a major bottleneck for the development of such bio-applications in industrial processes. The time required to develop a new process is critical. One potential solution is to use mathematical modeling and computer simulations to aid the experimental procedures. Significant benefits could be gained in overall economics and performance when mathematical modeling and simulation techniques are applied to processes especially when combined with a systems type approach, namely the integrated design, control, optimization and operation of processes. This allows prioritization of process improvements through a detailed understanding of the cost-benefit relationships of potential process changes.
∗
Corresponding author: Tel.: +90 216 3480292, fax: +90 216 3450126. E-mail addresses: [email protected] (N.A. Sayar), selcen.durmaz@ marmara.edu.tr (S. Durmaz-Sam), [email protected] (A.A. Sayar). 1 Equally contributed authors. http://dx.doi.org/10.1016/j.mbs.2015.04.007 0025-5564/© 2015 Elsevier Inc. All rights reserved.
Mathematical modeling in process development is used to bring a measure of order to observation and strengthen prediction [4–6]. Important process parameters that are used to define equipment specifications and operational conditions can be identified. Parameter estimation methods are applicable to find the process parameters that fit the specific mathematical model of the system in question [7]. Some of the research described in the scientific literature focuses on using models to simulate the effects of key operating conditions on a particular performance metric [8]. In addition, another section of the scientific literature studies the application of empirical models [9]. To date, the main body of mathematical modeling work within biological sciences focuses on microbial growth kinetics [10], operation kinetics [11], and also biosorption processes [12]. Reactor modeling [13], modeling of other unit operations [14] and biosorption optimization are also common [15]. On the other hand, a major section of the modeling work on biosorption focuses on the microscopic physicochemical and thermodynamic properties of this phenomenon. For example, Plazinski’s [16] review lists models that relate metal uptake to binding site concentration using gas adsorption isotherms. Some other models estimate metal uptake as a function of pH involving the competition phenomenon between metal cations and protons and electrostatic effects. Another concept studied in various modeling works is the relationship between ionic strength, binding mechanism and metal uptake via Donnan volume. Among these models the semi-empirical NICA (non-ideal competitive adsorption) model can be used to predict metal charge in relation to pH change [17–19]. A modified version of this model, NICCA incorporates thermodynamic consistency by considering heterogeneity of the biomass used [20,21].
R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
41
Fig. 1. Residuals Plot for Freundlich, Langmuir and Combined Langmuir–Freundlich (Sips) isotherms.
In comparison to the abovementioned models focusing on the specifics of the mechanism of biosorption, the model proposed in the current study aims to relate macroscopic, process related metrics to critical input variables. Metrics are a universal way of looking at the performance and assessing the economic viability of a process [22]. Metrics can be used to relate concepts such as the effort put into experimentation, the timescales of process development and biosorbent specifications to the cost and financial return of the process. In this work, the process studied is the biosorption of Ni (II) cations from aqueous solutions at various independent process conditions (variables) such as Ce , residual Ni (II) concentration at equilibrium (mg L−1 ); pH and T °C. The resulting metric is qe , Ni (II) uptake at equilibrium (mg g−1 ).The experimental data used to validate and verify the proposed model are obtained originally for this work. The current paper offers two novel applications for the modeling and preliminary design approach of the present process. The first is the development of a multi input-single output [23] sigmoid function to model the biosorption operation. The second is the proposal of an optimization scheme depending on a cost-benefit function [24] adapted for this process. Conventional process analysis often requires experimental determination of a set of process responses for a given array of forcing variables. For complex processes such as biosorption with multivariate input and output variables, the necessary experimentation is costly and time consuming. The availability of a simpler black-box model based on the evaluation of a very limited number of experiments allows the simulation of the required process outputs for a systematically chosen set of inputs [25]. 2. Materials and methods 2.1. Biosorption experiments The biosorption data of Ni(II) onto dead wild type S. pombe (972 h−1 ) through a batch equilibrium process have been collected originally for this work at various T and pH conditions of 25, 30, 35,
50 °C and 4, 5 and 6, respectively. The equilibrium values of Ni (II) uptake by the biomass, qe , mg (g dry weight)−1 were calculated using
qe =
(Ci − Ce )V m
(1)
where Ci , Ce , V and m denote, initial Ni(II) concentration of the aqueous phase (mg L−1 ), equilibrium Ni(II) concentration of the aqueous phase (mg L−1 ), volume (L) of the solution contacted by the sorbent and the amount of the sorbent expressed as dry mass (g), respectively. The wild-type (972 h−1 ) S. pombe strain used in this work was obtained from the Department of Molecular Biology and Genetics, University of Istanbul. This strain was grown in aqueous medium containing 30 g L−1 glucose and 5 g L−1 yeast extract, both acquired from Merck. Biosorption experiments were performed following a recently published procedure [26]. Each experiment was repeated three times and arithmetic averages were taken. Filtered samples were analyzed in triplicate with Perkin Elmer AAS 400 atomic absorption spectrophotometer to measure the concentration of Ni (II) species remaining in aqueous medium. 2.2. Model development A universal sigmoid function which is a simple but versatile mathematical function [23,24] can be used to represent the input–output relationship between the abovementioned process variables. The sigmoid relates the independent forcing variables to the process response with a certain quality of fit. The advantage of this function comes from its simple and robust applicability to Jacobian and Hessian operations thus making the development of optimization schemes mathematically more manageable. The raw biosorption data were regressed through Freundlich, Langmuir and combined Langmuir–Freundlich (Sips) isotherms. Sips and Freundlich approaches yielded statistically better fits whereas Langmuir exhibited poorer results. The comparison of goodness of fit for three isotherms was done plotting residuals for each approach in Fig. 1.
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R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
For modeling considerations, Sips isotherm has been chosen to reproduce data because of its physico-chemically consistent parameters. The numerical results for three different isotherms were given in Supplementary Material 1. The universal sigmoid function with Ce , T and pH as process variables and the resulting output % qe is:
%qe = 100 − [a1 expa2 P +a6 expa7 P ]
(2)
where
P = (Ce )a3 (T )a4 (pH)a5
(3)
(qe )act (qe )max
(4)
%qe =
where (qe )act is the simulated value for the given input and (qe )max is the highest value in the whole simulated data matrix. (qe )max for each separate case is obtained by correlating qe versus Ce using Sips isotherms. The dimensionless normalized form of %qe , a percentage expression, is appropriate to be used in validating the model through goodness of fit chi-square test. ai = 1–7 are the sigmoid model parameters to be optimized. Matlab based non-linear curve fitting functions have been used to estimate the model parameters evaluating an input–output array of 64 elements in which Ce varies from 0.48 to 50 mg L−1 , T from 25 to 50 °C and pH from 4 to 6. The proposed sigmoid model structure can also incorporate Ci as one of the input variables instead of Ce since Ce is a function of Ci . The version of the model with Ci as an input variable is given in Supplementary Material 3 with the corresponding model parameter values and optimization statistics.
as a uniform distribution of each input parameter with 10% variability. The Latin-Hypercube sampling method was utilized to create 500 sets of input parameters within the identified input parameter space. 500 dynamic simulations (each using one set of input parameters) were run to propagate the samples. Standard regression coefficient sensitivity analysis was conducted by constructing linear regression models on the outputs obtained from the Monte Carlo procedure [29]. The standardized regression coefficients method was implemented to determine the coefficient values for 7 model parameters. Scalar matrix sy for each output for 500 samples of the Latin hypercube can be composed as:
syi,%qe = d0,%qe +
7 j=1
dj,%qe θi,j + εi,%qe
(6)
θ i , εi , d0 and dj denote probabilistic parameter sampling for i = 500 elements Latin Hypercube, probabilistic regression error for i = 500 elements of the Latin Hypercube, linear regression coefficient and regression coefficients, respectively. The indices i (1–500) and j (1–7) stand for total number of Latin-Hypercube samples and total number of model parameters, respectively. For each sy a multi-independent variable linear equation set was configured using their corresponding mean M and standard deviations, σ :
syi,%qe − Msy%qe
σsy%qe
=
7 j=1
dj,%qe
θi,j − Mθ j + εi,%qe σθ j
(7)
Using multivariable linear regression techniques numerical values R were found for each d coefficient using Matlab . 3. Results and discussion 3.1. Biosorption experiments
2.3. Model validation 2.3.1. Chi-square test A two-sided chi-square, χ 2, test can be used to assess the statistical acceptability of the simulation results [23,27,28]. The test requires a null hypothesis which in this case presumes zero difference between the simulation results and experimental data. The degrees of freedom for this system can be calculated as:
Degrees of freedom = number of input − output sets − number of parameters − 1
(5)
2.3.2. Uncertainty and sensitivity analyses Uncertainty and sensitivity analyses are essential parts of good modeling practice (GMoP) [29]. The quality of the model can be assessed applying uncertainty and sensitivity analyses. Uncertainty analysis indicates how uncertainty in parameter optimization is propagated throughout the simulated model responses. The uncertainty may arise from the imprecision in parameters and the defects of the structure of the model in capturing the right mechanism. The analysis has been conducted setting minimum and maximum values at the 10th and 90th percentiles. The uncertainty analysis results lead to determine which improvements could be implemented to modify the model structure. Its application may indicate which section in the system is a good target for improvement. The evaluation of the model may be extended to include a sensitivity analysis to assist identifying the most influential parameters, for which further collection of data may be in order to improve model accuracy. A Monte Carlo procedure was used to estimate the output uncertainty in the metric of the model. Subsequently, standardized regression coefficient (SRC) sensitivity analysis was conducted on the input parameters. The Monte Carlo procedure for the analysis of uncertainty involves the specification of input uncertainty, sampling, and propagation of uncertainty. The input parameter space was created
Among the adsorption models tested Sips was found to be the best to correlate the experimental results. The related isotherm parameters representing the observed data are given in Table 1. The raw data is not shown in the text. It is available as Supplementary Material 1. It has to be noted that the experimental results concerning the metal uptake exhibit a slight discrepancy with the previously published data [26], possibly due to random mutagenesis of the available S. pombe strain generated from the original source. 3.2. Model parameters and simulation results The model parameters were optimized using experimentally obtained data applying the Matlab based nonlinear curve fitting functions. Numerical values and related regression statistics are given in Table 2. The simulated results are given in the supplementary data section. 3.3. Model verification and validation A test statistics χ 2 with 64 value sets for estimating 7 parameters, corresponding to 56 degrees of freedom, is calculated to be 65.53. The projected upper-tail critical value of (χ 2cv )56 upper with a chance of 10% (α 0.1 56 ) of rejecting the null hypothesis is 69.92. The projected lower-tail critical value of (χ 2 cv )56 lower with a chance of 10% (α 0.1 56 ) of rejecting the null hypothesis is 42.94. Similarly, (α 0.05 54 )upper , (α 0.05 54 )lower are 74.47 and 39.80 respectively. For double-sided validation, test statistics is lesser than the upper critical value and greater than the lower critical value which asserts that null hypothesis is not rejected for both tail tests under the α 0.1 54 and α 0.05 54 conditions [27,28]. For a model with three independent inputs estimating one dependent output, a correlation factor of 0.941 can be found satisfactory to assume the model results are in good agreement with the experimentally generated data.
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Table 1 Langmuir–Freundlich (SIPS) isotherm parameters correlating experimental data with their 95% confidence intervals. β
qe = KS Ce /1 + (α Ce )β KS
α
25 ° C pH = 4 0.27 –0.002 0.26 –0.02 027 0.02 pH = 5 0.52 0.013 0.34 –0.34 0.69 0.37 pH = 6 0.41 0.003 0.41 0.002 0.41 0.003
β
KS
α
β
30 ° C
KS
α
β
35 ° C
KS
α
B
50 ° C
0.38 0.36 0.04
0.24 0.24 0.25
0.015 –0.02 0.05
0.38 0.35 0.41
0.17 0.17 0.18
–0.003 –0.02 0.01
0.45 0.42 0.48
0.05 0.048 0.060
–0.0005 –0.03 0.01
0.68 0.59 0.77
0.51 0.18 0.83
0.46 0.44 0.48
–0.001 –0.04 0.04
0.50 0.46 0.54
0.34 0.29 0.39
–0.008 –0.15 0.14
0.53 0.37 0.69
0.29 0.10 0.49
0.22 –0.58 1.02
0.32 0.01 0.62
0.59 0.59 0.60
0.38 0.38 0.39
–0.0001 –0.002 0.002
0.56 0.56 0.57
0.31 0.31 0.32
–0.003 –0.01 0.004
0.50 0.48 0.51
0.27 0.22 0.33
0.07 –0.18 0.31
0.18 0.12 0.25
Fig. 2. Verification of model performance. Comparison of modeled and observed results via diagonal test.
Table 2 The estimated numerical values with their standard errors (SE) for model parameters and computed statistics. Estimated values for model parameters a1 a2 a3 a4 a5 a6 a7
= 76.14 (1.58) = –1.71 (3.91) = 2.19 (0.33) = –7.16 (1.18) = 8.81 (1.30) = 17.37 (1.61) = –1094.9 (2799)
Regression statistics
R2 = 0.941 χ 2 = 65.53
In the present assessment of the model fit, the test statistics and the fit index have been found satisfactory but insufficient. Because, the nonlinear multivariate regression results with promising RMSE, R2 and χ 2 values may seldom exhibit some disagreement between the predicted and observed results. For the verification, the experimental and the modeled results have been compared through a graphical
diagonal test given in Fig. 2. The test indicates that such disagreement does not occur in this work because the distribution of comparison points along the diagonal remains in an acceptably narrow band. Moreover, the random pattern of points at each side of the diagonal implies that the model is almost free of systematic errors. 3.3.1. Uncertainty and sensitivity analyses The uncertainty results illustrating the three selected cases with varying Ce at T = 37.6 °C and pH = 5; varying T at Ce = 25.5 and pH = 5 and varying pH at Ce = 25.5 and T = 37.6 °C are represented in Figs 3, 4 and 5, respectively. The uncertainty in the outputs is represented using 10th and 90th percentile and mean of the distribution of each output. The gray area indicates the propagated uncertainty range. It is to be noticed that the larger the spread of gray region the higher the uncertainty is. The studies were done for arbitrary simulation cases selected from the middle region of the range of input variables (Ce = 25.5, varying between 0 and 50; T = 37.6 °C, varying between 25 and 50; pH = 5, varying between 4 and 6).
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R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
Fig. 3. Uncertainty propagation with varying Ce ; T and pH constant.
Fig. 4. Uncertainty propagation with varying T; Ce and pH constant.
In cases where Ce and pH are varying (Figs. 3 and 5, respectively), the uncertainty remains in a tolerable band for lower to middle region values and exhibits a tolerable spread area (between the 10th and 90th percentiles). However, for higher input values of the two variables the uncertainty in the model is increased. In the case where T is varying (Fig. 4), the uncertainty propagation exhibits a wider spectrum at lower temperature values but with increasing T values uncertainty in the model decreases. Higher values of uncertainty indicate the regions where experimental coverage should be enlarged. This leads to assume that additional experiments at lower Ce and pH and at higher T values would narrow the uncertainty spread of the model. The sensitivity analysis results are done for the same cases as the uncertainty study. The SRCs were computed using Eqs (6) and (7). The coefficients obtained for each model parameter were ranked for
model output. The analysis was done for each independent input by varying this input in the range of investigation while the others were being held constant at selected case values. The results and related statistics are summarized in Table 3. The values of the coefficients are between –1 and 1. A higher absolute magnitude indicates a higher sensitivity of the model to the particular parameter. The adjusted-R2 values reflecting the linearization degree of the regression profiles are also given. All the adjusted-R2 values were found to be greater than the lower acceptable value of 0.7 [30]. In all of the three cases a4 was found to be the most influential model parameter while a1 , a3 and a5 all have lesser effect with similar magnitudes. The analysis indicates that the simulations are less sensitive to the parameters a2 , a6 and a7 . According to these results it can be assumed that a4 must be optimized with experiments at various temperatures as accurately as possible. Similar suggestions may be made for Ce and pH. This leads
R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46
45
Fig. 5. Uncertainty propagation with varying pH; Ce and T constant. Table 3 Sensitivity analysis.
Varying pH
The basic linear cost function of the present biosorption operation may be expressed as the summation of the corresponding terms of three independent inputs [18] where Ce is replaced with Ci due to the controllable nature of the latter:
0.906
Fcos t = cp1 Ci + cp2 pH + cp3 T
%qe Varying Ci
Varying T
Adj- R2
0.897
0.925
Rank
θ
SCR
θ
SCR
θ
SCR
a4 a1 a5 a3 a2 a7 a6
0.717 –0.449 0.424 0.216 –0.050 –0.029 –0.0004
a4 a5 a1 a3 a2 a7 a6
0.764 0.450 –0.373 0.227 –0.047 –0.022 0.008
a4 a5 a1 a3 a2 a7 a6
0.726 0.439 –0.430 0.219 –0.050 –0.027 0.007
1 2 3 4 5 6 7
to conclude that the number of experiments can be increased with a focus on the abovementioned variable ranges to reach a lower degree of uncertainty and higher accuracy in simulated model results. As expected, the relative insensitivity of the model to the parameters a6 and a7 is due to highest ranking of their adjacent parameters a1 and a2 in the model equation constituted by two combinatorial terms. According to this, one may question the necessity of the second combinatorial term. The optimization trials with a truncated model with only 6 parameters yielded much less successful simulation results with lower R-squared values of around 0.920 compared to 0.941 of the proposed model (data not shown). It may be concluded that the mathematical structure of the sigmoid needs the inclusion of the second combinatorial term to exhibit sufficient versatility to represent the studied phenomena with higher adequacy. The sensitivity rankings which are almost similar in all three cases may refer to the intrinsic robustness of SCR method when applied to a black box model with multi input–single output. 3.4. Benefits of the sigmoid model The principle task of process optimization is to develop an objective function. The objective function may essentially be composed of two main parts: merit and cost. For the present biosorption process the merit function can be established in terms of %qe :
Fmerit = 100 − [a1 expa2 P + a6 expa7 P ]
(8)
(9)
where cp1 , cp2 and cp3 denote techno-economical parameters relating the input parameter to their effects toward the cost of the process. They are related to the costs of the raw materials entering the process and keeping the solution at desirable pH and T. Let the techno-economical parameters of the chosen inputs be cp1 , cp2 and cp3 for Ci , pH and T, respectively. These coefficients reflect the unit costs of Ci , acid or base used to obtain a unit change in pH and utility consumed per degree of temperature to maintain the solution at desired temperature. For process optimization purposes Ci is considered as a controllable input variable instead of Ce , which is a more conventional variable for process analysis purposes. The magnitude of the unit cost of one of the input variables may be taken as unity while the two other magnitudes are quantified as its multiples.
J(Ci , pH, T ) = Fmerit (Ci , pH, T ) − Fcos t (Ci , pH, T )
(10)
J is then maximized in order to estimate the set of optimal values for key inputs. The maxima of J occur at points where the slope is zero, which is at points where gradient of J, J satisfies
∇J =
∂J ∂J ∂J , , ∂ Ci T pH
T =0
(11)
This proposed optimization scheme can be used for preliminary process design once the relevant techno-economical parameters are identified. The optimization scheme has been implemented for a hypothetical case study. Table 4 summarizes the corresponding sets of optima related to various fictive cost parameter combinations. 4. Conclusion An experimentation based multi-input single-output sigmoid model has been developed for a biosorption process, evaluating the biosorption of Ni (II) cations by S. pombe as a case study. The performance of the model has been validated and verified through goodness
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R. Özen et al. / Mathematical Biosciences 265 (2015) 40–46 Table 4 Optimization of biosorption operation of key inputs.
Ci pH T
cp1 = 0.1 cp2 = 1 cp3 = 3
cp1 = 0.5 cp2 = 2 cp3 = 1
cp1 = 1 cp2 = 1 cp3 = 1
21.84 7.77 25
17.44 8 25
2.01 7.15 25
of fit statistics and the model is evaluated by sensitivity and uncertainty analyses. The proposed model successfully correlates process input variables pH, T and Ce (or Ci ) to the output variable qe exhibiting R-squared values above 0.94. The model uncertainty has been shown to decrease at higher T values and lower pH and Ce values. The model output is found to be most sensitive to the model parameter a4 followed by a1 , a3 and a5 . The remaining parameters a2 , a6 and a7 show relatively less effect on the sensitivity analysis. The suggested modeling approach which provides reliable estimates of operational metric(s) has been shown to be potentially useful and dependable to assist in biosorption development to facilitate knowledge-based decision making. It is assumed that it may reduce the investment of research and analysis efforts eliminating least promising circumstances especially at early stages of bioprocess development. There are several approaches to the modeling of biosorption phenomena focusing on the characterization of the physico-chemical aspects of the process such as those that incorporate the Donnan volume concept. The current model differs both in scope and purpose from these models. The current model aims to empirically correlate process related variables such as pH, T and Ce (or Ci ) to the output metric qe without emphasizing the microscopic properties of biosorption, using a versatile universal sigmoid function which can subsequently be used in a simple techno-economic process optimization scheme. An example of the process optimization scheme is given using dummy variables. For future works, the modeling approach has been proposed as an engineering tool in optimizing cost-benefit trade-off, balancing variable process inputs such as raw material and utility costs with process profitability expressed in process metrics. An optimization scheme was developed to focus on attractive and promising process options which can be taken into consideration as a starting point for a more elaborate techno-economic analysis. Acknowledgment Funding by Marmara University Scientific Research Committee (Project FEN-C-YLP-040712-0280) is gratefully acknowledged. Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mbs.2015.04.007. References [1] K. Chojnacka, Biosorption and bioaccumulation- prospects for practical applications, Environ. Int. 36 (2010) 299–307. [2] J. Wang, C. Chen, Biosorbents for heavy metals removal and their future, Biotechnol. Adv. 27 (2009) 195–226.
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