Modeling and Optimization of Algae Growth - Industrial & Engineering

Article pubs.acs.org/IECR

Modeling and Optimization of Algae Growth Suresh K. Jayaraman and R. Russell Rhinehart* School of Chemical Engineering, Oklahoma State University, 423 EN, Stillwater, Oklahoma 74078-5201, United States S Supporting Information *

ABSTRACT: Microalgae is a promising source of renewable biofuels, and optimization and control of the biomass growth stage can make techno-economic improvements. This work explores the development of an algae growth model from first-principles, which includes the impact of natural vagaries of weather (and such) associated with production in an open system. Consequently, the process simulation is stochastic as well as fundamental; it returns a distribution of results representing the day-to-day realizations from natural variability. It also expresses day-to-night variation in the cycling solar energy. The simulation is then used to optimize pond design and management (the growth time, raceway depth, pH control, etc.) to improve profitability. Since the simulation is stochastic, nonlinear, and with multiple optima, a multiplayer direct search optimization technique with steady state convergence criteria is used for optimization. Conclusions are that (1) accounting for natural variation in the optimization leads to noticeable improvement in profitability, (2) sensitivity analysis of the model reveals where fundamental science research is needed to underpin critical techno-economic phenomena, and (3) the stochastic optimization approach has wide ranging applicability.

1. INTRODUCTION Algae are ubiquitous photosynthetic organisms in aqueous habitats. They vary from small, unicellular forms to multicellular colonies; and their simple structure allows them to adapt to prevailing environmental conditions.1 The algae consume CO2 as part of their growth process and are widely used as fertilizers, animal feed, pest controls, or in sewage treatment. Oil content of some microalgae may be as high as 90% of dry biomass weight under extreme growth conditions,2 and this potential high oil content and dense biomass makes algae an interesting renewable fuel source. Algae, as renewable resource, provide a means to reuse CO2 in large volume, thereby reducing greenhouse gas emission. In addition, microalgae are also used to produce methane by anaerobic digestion of algal biomass.3 The growth of algae also requires nitrate and phosphates which can be supplied from industrial wastewater.4 This would both treat the effluents, and decrease the annual expense on nutrients for algae production. Biofuels, derived from algae, have the potential to replace fossil fuel, providing a domestic and carbon-neutral fuel.5 Major challenges in algal biofuel are related to optimization of the growth conditions and to control actions to achieve high oil content while maintaining exponential or high growth of organism. These issues can be addressed by developing mathematical models to accurately capture the algal growth and lipid production. Research work has been published on algal growth development and its validation using small-scalelaboratory data; justified due to the lack of peer reviewed, published, scalable growth data.6 The microalgae biomass growth models reported in the literature generally include the estimation of light attenuation within the culture and predict the rate of biomass growth based on incident or absorbed light.7 But, commercial scale modeling has other issues. Developing a mathematical model to determine the optimum growth method and conditions will maximize algal oil production.8 © XXXX American Chemical Society

This work explores a way to develop an algae growth model from first-principles and fits the model to experimental data. A first-principles approach is a mechanistic, phenomenological way of modeling the truth about nature but seeks an adequate, yet simple embodiment, as opposed to a fully rigorous attempt at modeling. It balances utility with perfection and seeks to create a model that is sufficient for techno-economic design. The algal growth model along with other nutrient consumption models is regressed against experimental data to obtain various coefficient values. The simulation uses the model but also includes the impact of natural vagaries of weather, temperature, and nutrient availability associated with algae production in open raceway ponds. Consequently, the simulation is stochastic as well as fundamental, which returns a distribution of results mimicking the day-to-day vagaries of nature. The simulation is then used for model based optimization of harvest time and pond design (pond depth) to maximize profitability. Since the simulation objective function (OF) is nonlinear and stochastic with multiple optima, a recently developed heuristic optimization technique called Leapfrogging is extensively used in this work. Bootstrapping analysis is also done in order to determine the impact of experimental uncertainty in the model. Sensitivity analysis of model inputs like nutrient initial concentrations and system conditions is done to see the effect of model inputs on growth model. The model results are applied to perform an economic analysis using net present value as the profitability index of the algae production process over a 20-year period. This work focuses on modeling of the pond and maximizing harvest value; not on the consequence of the downstream processes. Received: May 5, 2015 Revised: August 4, 2015 Accepted: August 5, 2015

A

DOI: 10.1021/acs.iecr.5b01635 Ind. Eng. Chem. Res. XXXX, XXX, XXX−XXX

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Industrial & Engineering Chemistry Research

Where B (g/L) is biomass concentration, t (days) is time, k1 (1/days) is a growth rate constant, k2 (1/days) is a death rate constant, P is a biomass production attenuation coefficient10 which is a product of functions for light intensity, f(I), system temperature, f(T), and the availability of nutrients like phosphate, f(P), nitrate, f(N), and CO2, f(C). Due to insufficient data, death rate is assumed to be zero in this work. All these functions except for light intensity would be spatially uniform because of the mixing due to thermal and wind-driven currents. So, uniform chemical and temperature composition is assumed throughout the development of the model. 2.2.1. Light Dependence Modeling. Light is the primary energy source for algae growth. Light intensity varies with depth of water, and seasonally, and between day-and-night.11 The light intensity decreases with pond depth due to attenuation by both water, suspended particles, and biomass. The day-and-night cycle of algae growth has been modeled as shown in eq 5,

There are many models that could be chosen to represent the natural vagaries of any one particular geographic location, or of the time forecast of economic prices and costs. Further, there are several forms of a profitability index that combine capital and expenses, which may be preferred under different economic systems. There are many algae species that could be used as the lab-scale prototype to generate data that would shape the model coefficients in regression. This work does not claim to provide the global best values of application-dependent pond depth or harvest time but to reveal the way of determining those values.

2. METHOD 2.1. Overview of Modeling Equations. Models and numerical simulations are relatively inexpensive, making them a powerful tool that can be used to enhance economic competitiveness: to minimize resource consumption and maximize profitability at the same time. The governing equations and parameters of an algae growth model are explained in following sections. The main objective is to develop a robust model that captures the various stages of algae growth along with nutrient consumption. The kinetic model is developed in a generalized way to adapt both open and closed pond systems; however, natural vagaries and economics are different for the two. This explores open pond or raceway processes. The most influencing factors of algae growth are light intensity, photosynthetic rate, temperature, nutrient availability, and pH.9 2.2. Algae Growth. Figure 1 shows the pictorial representation of an algae pond with the necessary nutrients

⎧ ⎛ (t − 7)2π ⎞⎫ Iβ = max⎨0, Io sin⎜ ⎟⎬ ⎝ ⎠⎭ 24 ⎩

(5)

Io (W/m2) is incident light at its maximum and t (h) is time. The total light extinction12 in a pond system is calculated as a linear function of nonalgal turbidity and algal turbidity as shown in eq 6 α = k n + kaB

(6)

α (1/m) is total extinction coefficient where kn (1/m) accounts for nonalgal turbidity and ka (L/g m) accounts for algal turbidity. The range of kn is 2.22−7.13 and the constant ka is 0.014 ± 0.003.12 The light intensity is calculated using a simple Beer−Lambert extinction model as shown in eq 7.

I = Iβe−αD

(7)

2

I (W/m ) is light intensity and D (m) is depth of pond. Since the pond is assumed to be well mixed, the biomass will spend time feeling the light intensity at all depths, so the model uses an average light intensity to depict the effect of light source on growth of algae as shown in eq 8. ⎛ Iβ ⎞ Iavg = ⎜ ⎟(1 − e−αD) ⎝ αD ⎠

Figure 1. Schematic representation of algae pond.

Iavg is the average light intensity calculated along the depth of the pond. 2.2.2. Photosynthetic Rate Modeling. Quinn et al. calculated biomass growth based on an energy balance incorporating photosynthetic, respiration, and energy required for nutrient uptake.13 Photosynthetic light-response (P−I) curves are also used in modeling algal productivity.14 Data from a classic P−I response curve in Figure 2 is used to model photosynthetic rate.14 Here, the average intensity is used in the photosynthetic rate modeling to calculate the photosynthetic rate f(I) as shown in eq 9.

for growth at monitored temperature and pH. Estimates of the time scale for either diffusion or thermal mixing indicate that the volume will be a completely mixed batch (of uniform concentration) but with spatially dependent light intensity. The time development of both live (BL) and dead (BD) algae biomass are represented by conventional kinetic models, accounting for both algae growth and death rate. The sum of live and dead algae accounts for total biomass concentration as shown in eq 3

dB L = k1PBL − k 2BL dt

(1)

dB D = k 2BL dt

(2)

B = B L + BD

(3)

P = f (I )f (T )f (P )f (N )f (C )

(4)

(8)

f (I ) = 9.34(1 − e(−.0044Iavg)) − 1.60

(9)

f(I) represents the influence of light intensity on algae growth; and f(I) decreases as the depth increases because of the lack of sun light attenuation and changes in time due to the cyclic sun position. B


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and f(P) represent the algae growth dependence on nitrogen and phosphates, respectively. When nitrogen is present in the medium in the form of nitrate, uptake is an energy-linked process and happens mostly during daylight.9 Ammonia and diammonium phosphate (DAP) are used as the source of nitrogen and phosphate in this model for economic analysis. The fact that CO2 affects the pH of the system is used to infer the CO2 concentration of the pond by pH monitoring. For simplicity of the model, and because of inadequate literature on the metabolic byproducts, it is assumed that the pH is solely determined by the presence of CO2. The consumption of CO2 by the algae leads to the reduction of CO2 concentration in the pond, which in turn increases pH of the pond. It is known that algae growth is optimal only in the certain range of pH,21 so it is a must to maintain the pond pH. This model is regressed with growth data of Galdieria sulphuraria, which grows at the low range of pH.22 So, 2.5 is used as optimal pH in the model. If CO2 concentration is insufficient, it would increase the pH and become unfavorable for algae growth. Since pH of the system can be easily measured, this could be used as a controller strategy to check on CO2 concentration. Following literature-reported models, algae growth rate dependence on CO2 is modeled using the probit function, which monotonically decreases with pH as CO2 concentration increases.

Figure 2. Photosynthetic rate modeling.

2.2.3. Temperature Dependence Modeling. The model accounts for the effect of temperature on algal growth as a radial basis function of pond temperature, centered on the optimal temperature for a given species, as shown below15 2

f (T ) = e(−k t(Tr − Topt) )

(10)

Tr (K) is the pond temperature, Topt (K) is the optimal growth temperature, and K t (1/K 2 ) is the temperature-effect coefficient. Most of the strains used in biofuel production grow best between 20 and 30 °C.16 However, the model is regressed with growth data of Galdieria sulphuraria, which grows best at a relatively higher temperature. The optimum temperature of 323 K, pond temperature of 313 K, and temperature-effect coefficient of 0.0001 are chosen to match the growth rate data of Galdieria sulphuraria.17 f(T) represents the influence of pond water temperature on algae growth, and f(T) increases with temperature up to an optimal temperature and then decreases gradually. 2.2.4. Nutrient Dependence Modeling. Nutrient uptake has been modeled in various ways in the literature. Bonachela et al. modeled nutrient uptake of phytoplankton using the Michaelis−Menten model.18 Lemesle and Mailleret modeled nutrient limitation using the Droop model, assuming that algal growth rate is dependent on intracellular nitrogen concentration.19 The growth model of this work assumes that microalgae growth is limited by both nitrogen and phosphate availability. The consumption of nutrients is modeled using a kinetic model as shown in eq 11 and f(Cx) is modeled using the Monod equation20

dCx = −kxrxPBL dt f (Cx ) =

f (CO2 ) =

+ Cx

(13)

3. MODEL REGRESSION Values for model parameters are obtained by regression of the model using the experimental data obtained from the algal research group at New Mexico State University (NMSU). They used thermo-tolerant and acidophilic algal strain (5572) of Galdieria sulphuraria, to generate algal growth and nutrient consumption data over the span of 10 days.17,22 Fitting a model to data requires an optimizer, for which there are many good choices. Leapfrogging23 is a recently developed heuristic, multiplayer optimization technique which relocates the worst trial solution (player), by leaping the worst over the best into a random spot in the reflected decision variable hypervolume. Usually an optimizer algorithm moves the best player toward the minimum, rather than relocating the worst. However, leaping the worst player over the best has two advantages. It cuts the average distance in half at each leap over, leading to end-game convergence, but expands the cluster during a downhill search. Leapfrogging has been tested with many functions including the nonlinear constrained horizon predictive control, regression modeling of viscoelastic tissues, and chemical process modeling and found to be robust with fewer function evaluations and higher probability of finding the global optimum compared to other optimizers. Both the algae growth and nutrient (ammoniacal nitrogen and phosphate) consumption data were used to regress the model in order to calculate eight different species-dependent model parameters such as growth rate, half saturation constant for N and P source, rate constant of N and P source, and CO2 model, and nutrient uptake rates of the species. Leapfrogging is used to determine model coefficient values that minimize the sum of squared deviation (SSD) between the modeled and the measured values. Since the three measured

(11)

Cx kxh

1 (1 + e λ(pH − pHopt))

(12)

Cx (mg/L) is the nutrient concentration, kx is the rate constant, rx is the nutrient consumption rate, and khx is the half-saturation constant for species x. In laboratory experiments on algal growth, the nutrients are normally added in excess initially. Later, after consumption, algae become nutrient deprived, and the value of half-saturation constant plays a more vital role in the algae growth.10 By contrast the nutrient dependence in this simulation of a commercial production is done in such a way that whenever nutrient concentration decreases to one-third of initial concentration, half of the initial concentration is added to the pond. So, the simulation indicates nutrient concentration is monitored in order to avoid nutrient depletion. The terms f(N) C


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Figure 3. Comparison of measured and simulated (a) biomass concentration of algae strain 5572, (b) N source concentration, and (c) P source concentration.

variables have disparate units and values, the SSD is a weighted sum as shown below SSD =

λ12 ∑ (Be − Bm )2 λ 2 ∑ (Ne − Nm)2 + 2 n1 n2 +

λ32 ∑ (Pe − Pm)2 n3

(14)

Where, λ is the Lagrange multiplier, which determines the relative importance of each term. B, N, and P represent the concentration of biomass, nitrogen, and phosphates, respectively. The subscripts e and m stand for experimental and model; whereas, n represents the number of data points used for regression. The value of λ1, λ2, and λ3 are 300, 0.5, and 3.0, respectively. The model is adjusted to match the experimental setup in the preadaptation period and during the light−dark cycles. The modeled temperature, initial concentrations, and pH are the same as the experimental setup.17 The comparison of measured and modeled values of biomass, N source, and P source concentration are shown in Figure 3. The modeled biomass concentration matches the experimental data better than the other two variables because of the values of Lagrange multipliers chosen here to represent data certainty (effectively the reciprocal of the square of the standard deviation of the experimental data). The model parameter values obtained from strain 5572 are used to predict the algae growth experimental data obtained from a different strain (5587.1) of the same species. The model fairly well matches the experimental data as shown in Figure 4. So, the model from a strain also predicts the growth data of a different strain of the same species which supports the robustness of the model. The oscillations in the simulated curve of Figures 3 and 4 are due to the day−night induced growth intervals. Once the model coefficient values have been

Figure 4. Comparison of measured and simulated biomass concentration of algae strain 5587.1.

identified, the model can serve in the further business optimization studies.

4. ECONOMIC OPTIMIZATION The data-validated model and Leapfrogging is used to optimize the harvest time and depth of the pond, so that the process is profitable. If batch growth time is too short then there is the expense of many harvests each year, but with little total production. Contrasting, if growth time is too long, then the nearly completed batch waits and there is less harvested production each year. Similarly there is an optimum depth. If depth is too shallow, that decrease the biomass concentration for the given area, but if it is too deep, algae would lack sunlight because of excessive accumulation of algae. Thus, both the harvest time and depth must be optimized to maximize the biomass concentration. Net present value (NPV) is a common profitability index and is calculated using the following equation D


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Industrial & Engineering Chemistry Research N

NPV(i , N ) =

∑ t=0

Rt (1 + i)t

4.2. Optimization of BiomassDeterministic Process. The algae growth model along with the economics is used to optimize harvest time (t) and pond depth (d) that would maximize process NPV as shown in Figure 5. The objective of the optimization statement in eq 16 is to maximize NPV of the process by finding the optimum t and d.

(15)

Where, t (years) is the time of the cash flow, i (%) is the rate of return, Rt ($) is the annual net cash flow, and N is the number of years for business analysis. If the NPV is greater than zero, it means that the investment would be more profitable than investing in an interest earning account of i%/y. NPV analysis of the batch reaction is carried out for the time period of N = 20 years at the annual rate of return i = 5%. Detailed economic analysis is done in order to obtain the annual profit which is discussed in the next section. 4.1. Economic Analysis for Biomass Production. Data for the economic analysis are collected through literature review from a plant that produces 4500 gal algae oil/y and the lipid content of dried algae is assumed to be 50%.24 The initial land cost and working capital sums up to fixed capital investment. Cost data are updated to year 2014 using the annual cost index from The Chemical Engineering periodical. The input data for the economic analysis are tabulated in Tables 1−3. It is assumed that the rate of inflation of labor, energy, and land price are identical, so there is no change in the tax rate.

max J = NPV

Figure 6 shows the contour of NPV of the process with harvest time and pond depth scaled as decision variables (DV) on a 0−10 basis using the following equation DVs = min + DVu(max − min)/10

facility assumptions

input data 1500 20 1000 0.05 547.00 87 460.98 100 000.00

Table 2. Variable ExpensesNutrient and Power variable expenses

input data

price of CO2 ($/ton) price of DAP ($/ton) price of ammonia ($/ton) avg price of electricity ($/ton BM) price wet algae ($/ton)

487.65 580.00 720.00 29.46 490.00

Table 3. Fixed ExpensesLabor Cost labor cost ($/y) position

no. required

project manager operation manager admin field operations lab manager

1 1 2 5 1

2014 salary

2014 cost

154,772.05 73,701.18 37,903.40 42,114.65 58,960.73 salary total labor burden (20%) total

154,772.05 73,701.18 75,806.81 210,573.25 58,960.73 573,814.01 114,762.80 688,576.81

(17)

DVs and DVu are scaled and unscaled decision variables respectively, and max and min are the maximum and minimum value of a decision variable are chosen to focus on the region of the global optimum. The maximum and minimum values considered for harvest time are 10 and 4 days respectively, and the maximum and minimum values considered for pond depth are 1.5 and 0.1 m, respectively. Leapfrogging is used to optimize the DVs to maximize the NPV of the process. The optimum is the spot near the center within the oval. The contours represent how NPV changes with harvest time and depth, and they are irregular because of the light dark cycles. Just off the left and right are local optima for harvesting the day prior and day after. The optimum harvest time is found to be 4.78 days and depth to be 0.36 m. This shows that the harvest should be done at the end of the fifth day, just after evening, (when the algae growth is almost zero) so that the process yields maximum profit. Optimized parameters are listed in Table 4. 4.3. Optimization of Biomasswith Uncertainty. Algae grown in an open system is prone to various environmental uncertainties like temperature fluctuation, weather change, consumption of the nutrients by other microorganisms in the pond, and stray pathogens. The optimization procedure of biomass with stochastic simulator is shown in Figure 7. These natural vagaries and uncertainties are included in the simulation by adjusting model coefficient values with a classic first-order autoregressive function about a nominal value of unity, as shown in eq 18. This work only explores some of the external vagaries and not any of the internal vagaries like biological variability, however other vagaries can be included in a similar way. This work also uses a Gaussian driven first-order autoregressive trend in the disturbances. For a particular situation, different disturbance models may be more appropriate.

Table 1. Input Data for Facility Assumptions facility land size (acres) pond width (m) pond length (m) biomass loss in harvest % of harvest volume price of land $/acre water well construction ($) office construction ($)

(16)

{d , t }

Ji = β(1 + 0.1Yi ) + (1 − β )Ji − 1

(18)

Yi =

(19)

−2log(U1) sin(2πU2)

J is the uncertainty factor involved with the model value, Y is calculated for each simulation Δt interval for every variable, using the Box−Muller method, which follows a Gaussian distribution mimicking the nature’s perturbations. U1 and U2 are random numbers uniformly distributed between 0 and 1, and the first-order β parameter is calculated using the following equation

The expenses were categorized as variable expenses and fixed expenses. The cost incurred by algae production accounts for variable expenses whereas labor cost and field expenses accounts for fixed expenses. The 50% of harvested algae is used to extract algae oil, and the remainder is used to produce other useful bioproducts.

β = 1 − e−Δt/ τ

(20)

Δt is the time step in days and τ is persistence of the perturbing event in a day. The uncertainties in the weather change, E


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Figure 5. Optimization procedure with deterministic simulator.

the objective function to a stochastic value. Replicating depth and harvest time choice will not result in exactly the same NPV. Figure 8 shows the contour of the NPV of the process with harvest time and pond depth as scaled DVs. The disturbed

Figure 6. NPV contour−deterministic process.

Table 4. Optimized ParametersDeterministic Process harvest time (days) pond depth (m)

4.78 0.36

Figure 8. NPV contourprocess with uncertainties.

temperature fluctuation, and nutrient scarcity are accounted by multiplying the J-value from eq 18 to eqs 9, 10, and 12. The average value of Y is zero, so the structure of eqs 18 and 19 keep J about its average value of 1.0. The variance on random variable Y is 1.0, and the factor 0.1 in eq 18 is chosen to make the variance on J have a value of 0.1(β/(2 − β))1/2, which appears to provide a reasonable match to natural vagaries. The random perturbations on each factor, representing daily changes in conditions mean that every run of the simulation will give different results, and consequently the NPV of any 20year simulation will be somewhat different from the NPV of a seemingly replicate simulation. The inclusions of the natural vagaries take an otherwise deterministic simulation and convert

pattern in the contour reveals the impact of the natural vagaries influencing the process. Since the model has stochastic effects, leapfrogging with replicates25 is used to optimize the DVs to maximize NPV of the process. Leapfrogging with replicates (five replicates) rechecks whether the prior best player remains the best by recalculating the objective function. It is assigned the worst of the replicate values, if it remains the best, the worst leaps over it. If not, the worst leaps over the remaining best. Convergence is defined when worst player OF value approached a steady state. Since uncertainty is associated with the NPV calculation of the process, NPV is calculated for 100 trials and 90% of the cumulative distribution function (NPV) is taken as the objective function for every set of

Figure 7. Optimization procedure with stochastic simulator. F


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decides to present it. The bootstrapping procedure with deterministic simulator is shown in Figure 9. The deterministic model is used in the bootstrapping procedure, in order to match the experimental setup. From the experimental data set of biomass and nutrient concentrations, nine data from biomass concentration, four data from N source concentration, and two data from P source concentration are randomly sampled to be regressed with model data in order to determine eight species dependent model parameter values. This process is repeated for 100 number of trials to get 100 different sets of eight model parameters. These 100 different sets of model parameters are substituted back into the NPV optimizer to estimate the optimum harvest time and pond depth. The result reveals how the uncertainty in the experimental data affects the model which affects the optimum DV. The bootstrapping analysis affirms that the NPV is maximum when the harvest is done at the end of the fourth day as shown in Figure 10. Additionally, the range on the optimum harvest

decision variables. The optimum harvest time is found to be 4.80 days and depth to be 0.38 m as shown in Table 5. Table 5. Optimized ParametersProcess with Uncertainties harvest time (days) depth (m)

4.80 0.38

As shown in Table 6, when the simulator is deterministic, the optimal DV values are 4.78 days and 0.36 m with an NPV of Table 6. NPV and Optimum Results Using Deterministic and Stochastic Simulations applied to optimized on

deterministic

stochastic

deterministic stochastic

$308 M (4.78, 0.36)

$273 M (4.78, 0.36) $295 M (4.80, 0.38)

$308 M. When the simulator is stochastic, the optimum DV values are 4.80 days and 0.38 m and NPV is $295 M. The $295 M represents a more likely expectation than $308 M, since it accounts for time variations in uncontrolled environmental conditions. If the optimum values from the deterministic optimization were used in the stochastic simulation, which accounts for natural vagaries, the return is only $273 M. This reveals that the result from stochastic simulation is better for optimizing the expected situation, the false representation of a deterministic simulator generates suboptimal choices for a process subject to vagaries. The values for the stochastic simulations in the far right column of Table 6 are the average of five 20-year trials and have a 95% uncertainty of about $3 M. They are significantly different.

Figure 10. Cumulative distribution function of optimized harvest time in days.

5. BOOTSTRAPPING AND SENSITIVITY ANALYSIS 5.1. Bootstrapping Analysis. Bootstrapping is a technique to estimate the uncertainty in model parameter values and in the model prediction from randomized sampling from the experimental data. This technique is used to determine the impact of uncertainty without creating replicate experimental data sets. The advantage of bootstrapping over conventional propagation of uncertainty is that there is no necessity to estimate the uncertainty on individual elements in the model. Bootstrapping uses the uncertainty in the data, as Nature

time is small revealing high confidence in the 4.80 value. This indicates that normal lab data variability has an insignificant impact on the model-optimized harvest time. Figure 11 suggests that NPV is maximum when the pond depth is about 0.38 m and also shows that the uncertainty in experimental data increases the uncertainty on model coefficient values. When the range of models is used for optimization, the 95% range of pond depth value falls between

Figure 9. Bootstrapping procedure with deterministic simulator. G


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ology to use the models for economic optimization, which includes vagaries of nature, and to propagate uncertainty from lab data on to the economic optimum values is revealed. The exercise indicates that more frequently sampled data, data on alternate strains, etc., are needed to generate models that could be claimed to be definitive. The optimization used NPV (not profit), which includes startup capital value, and the time-value of subsequent cash flow to provide a comprehensive economic profitability analysis. Continued studies should consider an appropriate investment profitability index, such as NPV. This work assumes the day/ night cycle of a typical summer day; however when the same procedure is done for a typical winter day, the optimum harvest time and depth are 4.70 days and 0.15 m, respectively. The economic study reveals the need for economic data. Bootstrapping and sensitivity studies reveal the need for more data related to the death rate portion of the model. Stochastic simulations reveal that the optimum from a deterministic simulation provides a false indication. It is recommended that economic studies use a Monte Carlo simulation to include the impact of vagaries of Nature. The results of this study represent the behavior of one particular algae species in one particular environment. Although the results would be different for other situations, the methodology (of modeling, regression, profitability index, and optimization of a stochastic simulation) would remain the same as demonstrated.

Figure 11. Cumulative distribution function of optimized depth in meters.

0.35 and 0.4 m. This indicates that lab-data variability has an impact on the optimum design, and that more data is needed to get a model with higher certainty, leading to a more precise optimization results. 5.2. Sensitivity Analysis. Sensitivity analysis is a technique to estimate the impact of inputs to the model. A sensitivity analysis is performed on initial biomass, nutrient concentrations and algal system conditions. For simplicity, the deterministic simulation is used for sensitivity analysis. The sensitivity analysis is done by increasing and decreasing each input parameter by 10% and evaluating the NPV at optimum pond depth and harvest time. The sensitivity of initial concentrations and system conditions are presented in tornado plot format as shown in Figure 12, which shows that the system temperature

7. CONCLUSION A phenomenological algae growth model is developed using first-principles, and uncertainties are added to the model to create a simulator to mimic the production vagaries of nature. This work also demonstrated a procedure to optimize the harvest time and the pond depth to make the process economically viable. Bootstrapping analysis is done to estimate the uncertainty associated with model parameters, and also to evaluate the range of optimized harvest time and pond depth. This work also demonstrated Leapfrogging as a viable technique for regression modeling, and for optimizing stochastic processes. The developed algae growth model also predicts the growth pattern of different strains of the same species, which provides evidence of model robustness. The method is demonstrated as functional and robust. The application reveals the need for experimental data related to biomass growth and economic values.

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Figure 12. Sensitivity analysis of model inputs in tornado plot format.

parameter is very sensitive for both ±10% of base case model. The NPV is insensitive to the nutrient and biomass initial concentration, suggesting initial concentrations of ±10% of base case model have insignificant effect on NPV of the process. Though optimum temperature and pH are species dependent, the results of sensitivity analysis shows that ±10% variation from the base case conditions has some signifcant effect on NPV of the process. Results from this sensitivity analysis are important to consider when adapting the growth model to other microalgae species or strains.13

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.iecr.5b01635. Table of notations (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

6. DISCUSSION The study reveals a procedure to generate nonlinear kinetic models. It includes constitutive models on growth rate with respect to light, CO2, pH, T, N, and P. The modeling approach is shown to match lab-scale experimental data. The method-

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ACKNOWLEDGMENTS We would like to acknowledge the algal research group from New Mexico State University (Drs. Thinesh Selvaratnam and H


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Nagamany Nirmalakhandan) for providing the algae growth and nutrient consumption data for this modeling work.

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Modeling and Optimization of Algae Growth - Industrial & Engineering

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