# A Dynamic Optimization Model for Designing Open-Channel Raceway Ponds for Batch Production of Algal Biomass

^{*}

## Abstract

**:**

## 1. Introduction

^{−2}·day

^{−1}. Hill and Lincoln developed a mathematical model for continuous flow mass culture of microalgae in multi rectangular open channels. The model predicted algal production, dissolved oxygen, pH, and concentrations of nutrients [7]. Grobbelaar et al. developed a deterministic mathematical model to describe the production of the green microalgae Scenedesmus. obliquus (S. obliquus) and Coelastrum. sphaericum (C. sphaericum) in outdoor raceways [8]. Assuming that CO

_{2}and other nutrients were supplied in excess during cultivation, the dynamic model incorporated sixteen months of irradiance and temperature measurements to evaluate the productivity. The productivity varied between 1.7 and 16.92 g·m

^{−2}·day

^{−1}. Sukenik et al. developed a deterministic simulation model to estimate productivity of the marine algae Isochrysis. Galbana (I. galbana) in an outdoor raceway pond [9]. The effects of pond depth and chlorophyll concentration on production rate were evaluated in various seasons. The model predicted an annual average productivity of 9.7 g of carbon uptake per square meter per day (gC·m

^{−2}·day

^{−1}). A new regulatory model was developed by Geider et al. [10] to describe the acclimation of phytoplankton growth rate to irradiance, temperature and nutrient availability. The model used Poisson function to model photosynthesis, Arrhenius function to model temperature and Michaelis-Menten kinetics to model nutrient uptake. James et al. [11] simulated hydrodynamics coupled with growth kinetics of Phaeodactylum. tricornutum (P. tricornutum) in open-channel raceways using modified versions of Environmental Fluid Dynamics Codes (EFDC) [12] and U.S. Army Corp of Engineers’ water quality code (CE-QUAL) [13]. System parameters such as flow velocity and raceway depth were varied to improve growth rate. Their results show that the depth of the pond should vary in response to the atmospheric temperature to maximize algae yield. They also reported that increasing the flow rate above 6.25 L/s does not further improve the algae growth rate.

^{2}. Talent et al. [17] developed a protocol for harvesting microalgae massive cultures from open ponds. According to them, a minimum of 10% of the pond volume per day should be harvested and replaced with new feedstock water.

## 2. Problem Statement

_{2}diffusers, harvesting, flocculation, centrifugation, water and nutrient supply system, waste treatment system, buildings and structures, electrical supply and distribution, instrumentation and machinery, land, and engineering and contingency costs. Operating costs include electricity, water, nutrients, flocculants, waste disposal, maintenance and repairs, operating supplies, depreciation, taxes and insurance, financing, labor and overhead costs, and general expenses.

## 3. Algal Production Model for Raceway Ponds

^{3}) is the volume of the raceway pond and BC (g·m

^{−3}) and X

_{DA}(g) represent the biomass concentration and the amount of dry algae in the pond at any given time.

_{avg}(t) (µE·m

^{−2}·s

^{−1}), experienced by a single cell moving inside the culture medium (via Equation (2)) based on the position of the sun relative to the pond and Beer-Lambert relationship [21].

_{o}(t) (µE·m

^{−2}·s

^{−1}) is the incident light on the surface which is a function of daily maximum and minimum temperatures as explained by Hargreaves [22]. It is shown via Equation (24) of the Supplementary Materials provided with this manuscript. CD

_{eq}(t) is the length of light path from the surface to any point inside the pond. The length of light path depends on the depth of the pond, CD (m), and the position of the sun relative to the earth’s surface called solar zenith angle as shown in Equation (3). The greater the pond depth, the longer is the length of the light path. The lower the sun’s position from the Earth’s surface, during sunrise and sunset, the longer is the length of the light path in the pond. However, when the sun’s position is high in the sky, the length of the light path is shorter.

_{k}(μE·m

^{−2}·s

^{−1}), and the availability of light inside the pond, I

_{avg}(t) as shown in Equation (4). The equation does not take into account photoinhibition. Although it is well documented, it has often been disregarded. Studies suggest that growth models that express µ in terms of the average irradiance raised to some power greater than unity better fit experimental observations [21,23]. Equation (4) is a light-limited growth equation which was developed and tested by Grima, Camacho, Pérez, Sevilla, Fernández, and Gómez [19]. Based on the study, material balance performed in a chemostat where specific growth rate is expressed as a function of irradiance, it was observed that the algae growth rate predictions of this model exhibit a good agreement with the measured ones. It models the effect of light attenuation on the observed algae growth rate in the cultivation unit. It defines a hyperbolic relationship between the specific growth rate of algae and average solar irradiance inside the culture.

_{max}, (day

^{−1}) of algae species depend strongly on the temperature of the pond and the species itself [24]. Several models describe the influence of temperature on growth rate of algae, we have chosen the model presented in [25] where eight species of marine phytoplankton were grown at different temperatures ranging from 10 to 25 °C. They experimentally modeled the general response of growth rate to temperature as ${\mathsf{\mu}}_{max}=\mathsf{\alpha}.exp\left(\mathsf{\beta}.T\right)$ where α and β are species dependent constants and T is temperature of medium. In our work, we have used their data to model the growth response of the selected species. We used the same relationship in our model via Equation (5).

_{pond}is temperature of the pond. The species specific constants α and β are taken from our previous findings [26]. The experimental data modeled the response of growth rates of eight species of marine phytoplankton to changes in temperature under continuous light [27].

_{pond}(t) (K), is calculated via an overall energy balance around the pond (Equation (17)). According to [28], conduction was of low importance in the total heat balance for shallow algae ponds. Thus conductive heat flux is neglected for this model since we assume no heat transfer between the pond bottom and the soil beneath. We also ignored the changes in pond temperature caused due to rainfall, and hence, removed the rain heat flux term from the model. Therefore, temperature of the outdoor pond is determined by six heat fluxes: (1) Heat flux due to pond radiation; (2) heat flux due to solar radiation; (3) heat flux due to air radiation; (4) Evaporation; (5) Convection; and (6) Inflow heat flux.

_{p}(t) is estimated by Stefan-Boltzmann’s fourth power law as shown in Equation (7) [28]. Since heat transfer takes place from pond surface to the atmosphere, this heat flux is negative. The higher the radiation from the pond, the higher heat transfer from pond to the atmosphere becomes, which lead to lower pond temperature.

_{w}= 0.97 represents the emissivity of water [29], σ = 5.67 × 10

^{−8}represents Stefan-Boltzmann constant (W·m

^{−2}·K

^{−4}) and SA is the surface area of the pond (m

^{2}). (2) Heat flux due to solar radiation: Radiation received by the pond from the sun Q

_{s}(t) is calculated from incident irradiance on the surface in Equation (8) [28]. The higher the radiation from the sun to the pond, the higher would be the temperature of the pond.

_{a}is the fraction of sunlight converted by algae into chemical energy during photosynthesis. It is assumed to be constant and equal to 2.5%. (3) Heat flux due to air radiation: Radiation from air, Q

_{a}(t), is given by Stefan-Boltzmann’s fourth power law as shown in Equation (9) [28]. T

_{surr}(t) (K) is calculated by the method outlined by Woodhead [30]. The higher the radiation from the air to the pond, the higher would be the pond temperature.

_{a}= 0.8 represents the emissivity of air. (4) Evaporation: Evaporative heat transfer, Q

_{ev}(t), is related to rate of evaporation, me(t) (kg·m

^{−2}·s

^{−1}), and it is expressed in Equation (10) [28].

_{w}= 2.45 × 10

^{6}J·kg

^{−1}represents the latent heat of water. Theoretically, applying Buckingham theorem to evaporation at the pond surface yields three dimensionless numbers namely, the Sherwood number Sh

_{L}, the Schmidt number Sch

_{L}, and the Reynolds number Re

_{L}. The rate of evaporation is shown to be dependent on the three dimensionless numbers [31]. For mass transfer in a horizontal surface, the three dimensionless groups are correlated as follows:

^{−1}) determined via Equation (11), D

_{h}is the hydraulic diameter of the pond (m), D

_{w,a}= 2.4 × 10

^{5}m

^{2}·s

^{−1}represents the mass diffusion coefficient of water vapor in air, ϑ

_{a}= 1.5 × 10

^{5}m

^{2}·s

^{−1}denotes the air kinematic viscosity, and WV(t) represents the wind velocity (m·s

^{−1}). Equation (11) has been derived with an assumption that wind is constant with respect to height. The rate of evaporation is calculated by Equation (12).

_{H2O}is the molecular weight of water (kg·mole

^{−1}), R = 8.314 Pa·m

^{3}mol

^{−1}·K

^{−1}is the ideal gas constant, and RH(t) represents the relative humidity of the air over the pond surface. Relative humidity is important with regard to both evaporation losses and pond cooling. When humidity is low, high rates of evaporation occur, particularly during windy periods. When humidity is high and there is little or no wind, and when sunlight is abundant, the water in shallow cultures may heat up. In Equation (12), P

_{a}and P

_{w}are saturated vapor pressures (Pa) at T

_{surr}and T

_{pond}, respectively, and can be determined using Equation (13) [32].

_{cv}(t), is expressed similar to evaporative heat transfer as heat transfer and mass transfer obey the same laws. Applying Buckingham theorem to convection at the pond surface yields three dimensionless numbers namely, the Nusselt number, the Prandtl number, and the Reynolds number. Convection coefficient is calculated using the correlation between the three dimensionless numbers (Equation (14)) [31].

_{cv}(t) (W·m

^{−2}·K

^{−1}) is the convection coefficient, λ

_{a}= 2.6 × 10

^{2}W·m

^{−1}·K

^{−1}represents air thermal conductivity, and α

_{a}= 2.2 × 10

^{5}m

^{2}·s

^{−1}represents air thermal diffusivity. The convective heat flux Q

_{cv}(t) is given by Equation (15).

_{i}(t) can be expressed as a steady flow thermal energy equation shown in Equation (16). Water is added to compensate for evaporation from the pond surface, and it is assumed to be at the surrounding temperature T

_{surr}(t) when added to the pond.

^{−1}·K

^{−1}). The overall energy balance for the pond is given in Equation (17).

_{p}(t) is the rate of heat flow from the pond surface (W), Q

_{s}(t) is the rate of heat flow from sun to the pond (W), Q

_{a}(t) is the rate of heat flow from air to the pond (W), Q

_{ev}(t) is the rate of heat flow by evaporation (W), Q

_{cv}(t) is the rate of heat flow by convection at the surface of the pond (W), and Q

_{i}(t) is the rate of heat flow associated with water inflow (W). The mass flow rate, $\dot{m}$(t), can be calculated via Equation (18) using the concepts of fluid dynamics where mass flow rate is density times volumetric flow rate.

_{avg}(t) is the average daily velocity of the pond, CW is the channel width, and CD is the channel depth.

## 4. Dynamic Model

#### 4.1. Subscripts

#### 4.2. Parameters

_{PW}); (2) empirical constant for interior regions (EmpA) and coastal regions (EmpB) used for calculating incident irradiance at a location; (3) solar constant (μE·m

^{−2}·s

^{−1}) (Sc); (4) initial biomass concentration (g·m

^{−3}) at seeding of the pond (BCini); (5) algae biomass (g) demand (demand); (6) number of days between harvests is harvest period (harvP); (7) number of harvest periods in a month is harvest cycle (harvC); and (8) density (g·m

^{−3}) of water (ρ).

_{s}); (2) light absorption coefficient (m

^{2}·g

^{−1}) of biomass (Ka

_{s}); and (3) a species dependent constant (μE·m

^{−2}·s

^{−1}) used in the calculation of growth rates (I

_{k,s}). Similarly, the relevant parameters for a geographical location are: (1) latitude of location l (latitude

_{l}); (2) longitude of location l (longitude

_{l}); (3) time zone of location l (TZ

_{l}); (4) average maximum and minimum temperatures (°C) of the day (T

_{maxl,d}and T

_{minl,d}); (5) surrounding temperature (°C) (T

_{surrl,d}(t)); (6) total capital cost coefficient ($·m

^{−2}) at location l (TCI

_{l}); (7) total operating cost coefficient ($·m

^{−2}) at location l (TPC

_{l}); (8) electricity costs ($·kWh

^{−1}) at location l (ElCost

_{l}); (9) water costs ($·m

^{−3}) at location l (WtCost

_{l}); (10) availability of sunlight at location l (μE·m

^{−2}·s

^{−1}) (I

_{ol}(t)); (11) relative humidity (%) for location l (RH

_{l,d}(t)); (12) wind velocity (m·s

^{−1}) at location l (WV

_{l,d}(t)); (13) sunrise time for location l (sunrise

_{l,d}); (14) sunset time for location l (sunset

_{l,d}); and (15) zenith angle for location l (θ

_{l,d}(t)).

#### 4.3. Decision Variables

^{−3}) (BC(t)); (2) biomass concentration (g·m

^{−3}) on harvest day hd (BCharv

_{hd,}(t=sunset

_{l})); (3) biomass specific growth rate (h

^{−1}) (µ(t)); (4) biomass maximum specific growth rate (h

^{−1}) (µ

_{max}(t)); (5) biomass volumetric productivity (g·m

^{−3}·day

^{−1}) (PrV).

^{−1}) of the pond (U

_{avg}(t)); (2) average irradiance (μE·m

^{−2}·s

^{−1}) inside the pond (I

_{avg}(t)); (3) length (m) of light path from the surface to any point in the pond (CD

_{eq}(t)); (4) mass (g) of component c (X

_{c}(t)); and (5) accumulation (g) of component c until harvest day hd (Xharv

_{c,hd}(t=sunset

_{l})) (g); (6) annual production (g) of component c (D

_{c,y}); (7) mass (g) of products accumulated (m(t)); (8) Areal productivity (g·m

^{−2}·day

^{−1}) (PrA); (9) Reynolds number of flowing stream (Re

_{d}); (10) pond temperature (°C) (T

_{pond}(t)); (11) heat transfer (W) due to pond radiation (Q

_{r}(t)); (12) heat transfer (W) due to solar radiation (Q

_{s}(t)); (13) heat transfer (W) due to air radiation (Q

_{a}(t)); (14) heat transfer (W) due to evaporation (Q

_{ev}(t)); (15) heat transfer (W) due to convection (Q

_{cv}(t)); (16) heat transfer (W) due to inflow water (Q

_{i}(t)); (17) head loss (m) due to friction (h

_{F}); (18) kinetic head loss (m) (h

_{K}); (19) total head loss (m) (h

_{T}); (20) power needed to drive the paddle wheel (W) (PP(t)); (21) annual energy (kWh) requirements of paddle wheel (kW) (EPP); (22) annual water (m

^{−3}) requirements (Awater); (23) power needed to supply the water lost during harvesting, evaporation, and recycling (W) (PW(t)); (24) annual energy (kWh) requirements of pumping lost water (kW) (EPW); (25) Industrial water (m

^{3}) added into the pond (Indwater).

_{h}); (7) pond volume (m

^{3}) (V); and (8) pond surface area (m

^{2}) (SA).

#### 4.4. The Model

_{Cap-cost}and operating costs: Z

_{Electric-mixing}, Z

_{Electric-pumping}, Z

_{Water}, and Z

_{Product-cost}. The Minimum Acceptable Rate of Return (MARR) is 15%. Equation (20) assumes that the capital cost changes linearly with the pond surface area. Equations (21) and (22) calculate the cost of electricity required for mixing using paddle wheel and cost of electricity required for pumping additional water that is lost during evaporation. Equation (23) estimates the cost of water. Equation (24) calculates the total production cost. The estimation of cost coefficients for capital costs (TCI

_{l}) and operating costs (TPI

_{l}) are detailed in Section 6.1.

^{−2}·day

^{−1}[35]. The cost coefficients in the objective function, TCI

_{l}and TPC

_{l}, were taken assuming the average daily areal productivity of 60 g·m

^{−2}·day

^{−1}.

^{−1}have been shown to prevent thermal stratification and sedimentation. Improper mixing may also result in poor CO

_{2}mass transfer rates causing low biomass productivity [37]. Channel flow velocity is typically between 0.15 and 0.30 m·s

^{−1}, and Equation (49) enforces these bounds [38]. Equation (50) constrains biomass concentration to be below 10,000 g·m

^{−3}in order to inhibit light attenuation due to dense cultures and to maintain the Newtonian behavior of the growth medium while keeping the fluid properties similar to that of water [39]. Equations (51) and (52) capture the dynamic changes in biomass concentration that occurs due to the production of algae biomass. These equations are analogues to the differential equations–Equations (1) and (6).

## 5. Solution Approach

#### Approach to Convert Annual Production to Hourly Production

_{maxl,d}), minimum temperatures (T

_{minl,d}), hourly wind velocity (WV

_{l,d}(t)), and relative humidity (RH

_{l,d}(t)) for a year are collected from Wolfram Mathematica 8 Mathematica Weather data database [40]. Monthly minimum and maximum of temperatures are calculated, and assigned to their respective representative day of the month. The averages for the wind velocity and relative humidity is calculated for each hour of the day within a month, and they are assigned to their corresponding representative day. Zenith angle is the position of the sun with respect to earth’s surface. It not only helps in determining the incident irradiance at a location, I

_{o}(t), but also the sunrise and sunset times of the location. Hourly weighted averages of zenith angle were computed for all days in a month, and their monthly means were specified to their representative day of each month. By doing so, we have successfully created a day that represents the average changes in the relevant parameters within a day for each month.

_{hd}(t = sunset) is the biomass concentration at the time of harvest, BC(t = sunrise–1) and BC(t = sunset) are the biomass concentrations at sunrise and sunset of the representative day, and harvP is the harvest period.

_{c,hd}(t), from the first day of harvest period to the last day of harvest period, to mass produced at sunrise and sunset, X

_{c}(t), and harvest period, harvP. Here, X

_{c}(t), (g) is the hourly production of the component c produced in a raceway pond on representative day.

_{c,hd}(t), from the sunrise of the first day to the sunset of the last day of harvest period times the number of harvest cycles, harvC, in a month as shown in Equation (55). This equation replaces Equations (27) and (28) in our model.

^{3}), to run a raceway pond and to compensate for the evaporated water is calculated via Equation (58).

^{7}. CONOPT option Rtmaxv = 1 × 10

^{30}was used as the upper limit on the variables. Maximum number of domain errors was set to 1000 and maximum number of solver iterations was set to 5000. Because CONOPT is a local optimization solver, we used a multiple-start approach, where the model variables were initialized to different values generated using Latin Hypercube Sampling technique, in an effort to find the global solution.

## 6. Case Study

_{l}) and total production cost coefficient (TPC

_{l}) is shown in Section 6.1. In this work, all the cost coefficients have been inflated to the year 2012. The annual algae biomass demand in Equation (25) is five tons per annum. The harvest period and harvest cycle in each month were taken to be six days and five cycles, respectively. It was assumed that the raceway is seeded just before sunrise after each harvest to yield a biomass concentration of 0.6 g·m

^{−3}[11].

#### 6.1. Economic Assumptions

- ■
- Estimation of capital investment cost (TCI
_{l})- □
- Direct costs: They include site preparation, grading, compaction; pond levees, geotextiles; mixing (paddle wheels); CO
_{2}diffusers; harvesting; flocculation; centrifugation; water and nutrient supply system; waste treatment (blow down); building and structures; electrical supply and distribution; instrumentation and machinery; and land costs [35]. The total direct costs add to $63,700/ha - □
- Indirect costs: Engineering and contingency = 15% of direct costs excluding land costs = $9,255/ha
- □
- Fixed-capital investment = direct costs + indirect costs = $72,955/ha
- □
- Working capital = 15% of total capital investment
- □
- Total capital investment = fixed-capital investment + working capital = $85,829/ha
- □
- 10% rate of return
- □
- 10 years plant life
- □
- Straight line depreciation
- □
- Cost index for 2012

- ■
- Estimation of total production cost (TPC
_{l})- □
- Manufacturing costs: They include utilities–power required for harvesting, processing, lighting up buildings; nutrients; flocculants; waste disposal; maintenance and repairs = 5% of fixed-capital investment; operating supplies = 15% of maintenance and repairs; depreciation = fixed-capital investment/number of recovery years; local taxes and insurance = 4% of fixed-capital investment; financing = 5% of total capital investment; and labor and overhead costs. The total manufacturing costs add to $28,600.08/ha/year
- □
- General expenses = 20% of total production cost = $7,150.02/ha/year
- □
- Total production cost = manufacturing costs + general expenses = $35,750.1/ha/year

_{l}) was calculated as $13.144 m

^{−2}and total production cost (TPC

_{l}) was calculated as $5.475 m

^{−2}·year

^{−1}. Here, cost index for 1996 and 2012 were taken as 382 and 585, respectively. Cost of electricity and water are taken from literature [35] as ElCost

_{l}: 0.07 $ (kW·h)

^{−1}and WtCost

_{l}: 3.18 $ (1000 gal)

^{−1}, respectively. It should be noted that these economic parameters are taken same for all locations to understand the sole influence of algae species, geographical location, and raceway pond geometry on one another.

#### 6.2. Problem Statistics and Solution Approach

## 7. Results and Discussion

_{avg}), channel velocity (U

_{avg}), and pond temperature (T

_{pond}); average annual culture properties such as biomass concentration (BC) and specific growth rate (µ); and average daily productivities such as volumetric productivity (PrV) and areal productivity (PrA). Average annual culture and physical properties are highlighted here to give an idea of how much the average values would be for a given year. However, the culture properties and physical properties fluctuate on a daily basis based on the climatic conditions. These fluctuations are shown via Figure 3 through 8.

^{−1}. It can be observed that I. galbana species require shallower and wider ponds and Hyderabad, India location requires shallower and narrower ponds for growing algae.

_{k}. This enables higher light absorption inside the pond which triggers higher productivity and lower costs. This behavior can be observed in Table 1. Cases 5 through 8 where I. galbana species is cultured requires smaller pond volumes when compared to Cases 1 through 4 where P. tricornutum species is cultured, except for Case 7. We hypothesize that the solution of Case 7 is a bad local minima.

^{−2}·s

^{−1}). It is evident that as the day progresses, sunlight keeps increasing until mid-day and later decreases as it comes close to nightfall. It can also be observed that as summer approaches, solar irradiance tends to build up until the month of March and later declines as winter approaches. Figure 4 shows how average light intensity (I

_{avg}) inside the pond varies with day light time of the representative day for I. galbana species cultured in Hyderabad, India. The x-axis defines the times in a 24 h format for each representative day of the month and y-axis is the average light intensity (μE·m

^{−2}·day

^{−1}). It follows a trend similar to that of solar irradiance (I

_{o}). It can be observed that in a representative day, the average light intensity inside the pond gradually increases from the time of sunrise until mid-day and later decreases as the sun goes down. This trend continues for all representative days of the year. However, on an overall scale, the light intensity is maximum inside the pond during the months of February, March, April, and May. Figure 5 shows the biomass concentration profile (g·m

^{−3}) in the same format. On each representative day just before sunrise, biomass concentration is initiated to 0.6 g·m

^{−3}. During the course of the day with sunlight, the concentration in the pond builds up continuously until it reaches sunset. The concentration has rapidly raised from sunrise to sunset in the months of summer when compared to the winter months. Figure 6 shows the trend in biomass production (X

_{f}and X

_{g}). The x-axis shows day light time (sunrise through sunset) in a given day while y-axis shows the biomass production. It follows the same trend as biomass concentration. It is clear that as production rises, concentration of biomass inside the pond rises. In Figure 7, x-axis is the clock times in 24 h format for each representative day of the month and y-axis is the growth rate (μ). For a given day, growth rate of algae biomass tends to increase in the first few hours until mid-day and later decreases. This behavior of deteriorating growth rate is mainly due to the decrease in the pond temperature in the afternoon hours. Algal growth rate is rapid in the summer months of March, April, and May because temperature of the pond oscillated around the optimal growth temperature of I. galbana (~27 °C).

_{surr}) and raceway pond temperature (T

_{pond}) with time. Surrounding temperature depends on maximum and minimum temperature of the day, longitude, and time zone. It rises from sunrise until mid-day and then after goes down as the sun goes down. Temperature of the pond is driven by surrounding temperature and thermal gains/losses in the pond due to solar radiation, air radiation, pond radiation, evaporative heat transfer, convective heat transfer, and heat transfer due to inflow water. It is also dependent on the growth rate of algae as shown in Equations (4) and (5). The temperature of the pond rises as the day progresses and reaches a maximum after which it decreases. The decrease in pond temperature is mostly due to the cooling effects of evaporation. It can be observed that the surrounding temperature and pond temperature are high during the warmer months of March, April, and May.

_{p}), radiation from the air (Q

_{a}), radiation from the sun (Q

_{s}), evaporative heat flux (Q

_{ev}), convective heat flux (Q

_{cv}), and inflow heat flux (Q

_{i}). Radiation from the pond (Q

_{p}) depends on pond temperature (T

_{pond}). Hence, rate of heat flow from the pond, (Figure 9A), follows a similar trend as pond temperature in Figure 8. It can be observed that at the beginning of the day, less radiation is emitted by the pond and as the day progresses, the pond temperature increases and so does the radiation from the pond. Radiation from the air (Q

_{a}) (Figure 9B) follows a similar trend as surrounding temperature (T

_{surr}) (Figure 8), and radiation from the sun (Q

_{s}) (Figure 9C) follows a similar trend as irradiance at the location (I

_{o}) (Figure 3) as they depend on one another. It can be observed that evaporative heat flux (Q

_{ev}) (Figure 9D) is dominant in the month of May when the surrounding temperature is high. The increase in the heat transfer due to evaporation during this month has led to cooling in the pond, and hence, there is a decrease in the pond temperature. Evaporation increases from sunrise to mid-day and later decreases with the course of the day. However, in summer months, at sunrise when the surrounding temperature is higher than the pond, evaporation is high. Rate of heat flow due to convection (Q

_{cv}) (Figure 9E) is higher in May and June similar to that of evaporation heat flux. Inflow heat flux (Q

_{i}) (Figure 9F) is the rate of heat transfer between the water that is added to the pond, that is lost during evaporation, and the pond water itself. It can be observed that during the months of May and June, when the rate of evaporation is high, the pond water is relatively cooler. The water added to the pond is assumed to be at the surrounding temperature. Hence, heat transfer from inflow water to the pond water is high during those months. Among all the heat fluxes, rate of heat transfer due to evaporation plays a major role in determining the temperature of the pond.

#### Sensitivity Analysis

^{−3}. This would violate the model assumptions used in this work. In the current model, we assumed biomass concentrations to be below 10,000 g·m

^{−3}in order to inhibit light attenuation due to dense cultures and keeping the fluid properties similar to that of water. Thus, from our analysis, the next better option to harvest the algae biomass is six days. Since, the length of the pond for a six day harvest period case was found to be the upper bound on pond length, sensitivity analysis was performed by increasing the upper bound of pond length to see its influence on the net present cost.

## 8. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

**Table A1.**Summary of species dependent parameters [41].

Species | Percentage of Dry Algae (%) | Lipid Content (dw) | Light Absorption Coefficient of Biomass (m^{2}·g^{−1}) [42] | Constant (μE·m^{−2}·s^{−1}) |
---|---|---|---|---|

P. tricornutum | 20 | 0.31 | 0.0369 | 114.67 [42] |

I. galbana | 30 | 0.21 | 0.0369 | 170.68 |

Scalar | Value | Unit |
---|---|---|

η_{PW} | 0.17 | |

EmpA | 0.16 [43] | |

EmpB | 0.19 [43] | |

E_{photon} | 225.3 | kJ·mol^{−1} |

Date | Tulsa | Hyderabad | Cape Town | Rio de Janeiro | ||||
---|---|---|---|---|---|---|---|---|

T_{maxl,d} (°C) | T_{minl,d} (°C) | T_{maxl,d} (°C) | T_{minl,d} (°C) | T_{maxl,d} (°C) | T_{minl,d} (°C) | T_{maxl,d} (°C) | T_{minl,d} (°C) | |

January 2012 | 10 | −2 | 32 | 16 | 27 | 16 | 28 | 22 |

February 2012 | 8 | 2 | 34 | 18 | 25 | 15 | 30 | 23 |

March 2012 | 17 | 10 | 38 | 21 | 23 | 14 | 29 | 23 |

April 2012 | 20 | 13 | 39 | 25 | 22 | 14 | 28 | 23 |

May 2012 | 25 | 17 | 42 | 28 | 19 | 11 | 26 | 20 |

June 2012 | 29 | 20 | 36 | 25 | 18 | 11 | 26 | 21 |

July 2012 | 35 | 22 | 31 | 23 | 17 | 10 | 25 | 20 |

August 2012 | 31 | 20 | 31 | 23 | 16 | 10 | 26 | 19 |

September 2012 | 27 | 18 | 32 | 23 | 19 | 12 | 27 | 19 |

15 October 2012 | 16 | 10 | 32 | 20 | 21 | 13 | 29 | 21 |

15 November 2012 | 12 | 3 | 31 | 18 | 22 | 13 | 26 | 21 |

15 December 2012 | 9 | 0 | 32 | 16 | 26 | 15 | 32 | 24 |

**Table A4.**List of average relative humidity and wind velocity parameters for the locations. They are taken the same for all times of the day.

Date | Tulsa | Hyderabad | Cape Town | Rio de Janeiro | ||||
---|---|---|---|---|---|---|---|---|

RH_{l,d}(t) | WV_{l,d}(t) (m·s^{−1}) | RH_{l,d}(t) | WV_{l,d}(t) (m·s^{−1}) | RH_{l,d}(t) | WV_{l,d}(t) (m·s^{−1}) | RH_{l,d}(t) | WV_{l,d}(t) (m·s^{−1}) | |

January 2012 | 0.597 | 8.705 | 0.520 | 5.866 | 0.755 | 19.63 | 0.813 | 11.06 |

February 2012 | 0.698 | 7.581 | 0.429 | 6.393 | 0.726 | 15.83 | 0.753 | 11.77 |

March 2012 | 0.740 | 8.323 | 0.311 | 6.292 | 0.803 | 17.10 | 0.805 | 12.81 |

April 2012 | 0.765 | 6.847 | 0.446 | 5.088 | 0.728 | 18.72 | 0.829 | 11.74 |

May 2012 | 0.758 | 6.817 | 0.308 | 8.592 | 0.809 | 14.03 | 0.793 | 10.13 |

June 2012 | 0.750 | 6.088 | 0.489 | 15.03 | 0.777 | 12.87 | 0.832 | 10.58 |

July 2012 | 0.575 | 5.789 | 0.687 | 14.74 | 0.744 | 18.25 | 0.779 | 10.31 |

August 2012 | 0.597 | 5.186 | 0.715 | 12.00 | 0.733 | 20.06 | 0.749 | 11.38 |

September 2012 | 0.629 | 6.618 | 0.693 | 10.35 | 0.718 | 16.62 | 0.732 | 12.86 |

15 October 2012 | 0.711 | 6.775 | 0.565 | 6.608 | 0.711 | 23.42 | 0.725 | 12.84 |

15 November 2012 | 0.723 | 5.840 | 0.559 | 6.045 | 0.733 | 19.12 | 0.833 | 12.51 |

15 December 2012 | 0.730 | 7.563 | 0.455 | 6.015 | 0.755 | 22.51 | 0.762 | 12.64 |

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**Figure 1.**Schematic of an open channel raceway pond. Here, length denotes the pond length (m), q the pond width (m), p the channel length (m), and CW the channel width (m). The schematic shows a paddle wheel used for mixing purpose and the direction of flow.

**Figure 2.**Schematic of the variables listed in Equation (53). The figure shows how biomass concentration changes from day-1 to day-15 (until harvest). The figure in the inset shows the accumulation over the period of day-1.

**Figure 6.**Trend that shows the production of algae biomass with time on all representative days of the year.

**Figure 9.**(

**A**) Rate of heat flow from the pond; (

**B**) Rate of heat flow from air; (

**C**) Rate of heat flow from the sun; (

**D**) Rate of heat flow due to evaporation; (

**E**) Rate of heat flow due to convection; (

**F**) Inflow heat flux during the course of the day; (

**G**) Variation in heat fluxes during the course of the day.

Case | Z ($) | CD (m) | CW (m) | Length (m) | SA (m^{2}) | V (m^{3}) | Iavg (μE·m^{−2}·s^{−1}) | Uavg (m·s^{−1}) | BC (g·m^{−3}) | Μ (h^{−1}) | PrV (g·m^{−3}·Day^{−1}) | PrA (g·m^{−2}·Day^{−1}) | T_{pond} (°C) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1(P.tricornutum, Tulsa) | 179,080 | 0.43 | 2.57 | 300 | 1538 | 656 | 1457 | 0.102 | 51 | 0.042 | 8.448 | 3.602 | 15 |

2(P.tricornutum, Hyderabad) | 96,566 | 1.5 | 0.56 | 300 | 337 | 505 | 2060 | 0.117 | 66 | 0.060 | 10.972 | 16.458 | 23 |

3(P.tricornutum, Cape Town) | 493,900 | 0.81 | 4.74 | 300 | 2822 | 2274 | 1797 | 0.100 | 15 | 0.040 | 2.436 | 1.963 | 16 |

4(P.tricornutum, Rio de Janeiro) | 102,970 | 0.79 | 0.997 | 300 | 597 | 468 | 1707 | 0.100 | 71 | 0.062 | 11.829 | 9.274 | 23 |

5(I.galbana, Tulsa) | 131,660 | 0.3 | 2.24 | 300 | 1342 | 403 | 1463 | 0.114 | 122 | 0.035 | 20.313 | 6.094 | 15 |

6(I.galbana, Hyderabad) | 61,109 | 0.3 | 0.95 | 300 | 571 | 171 | 2125 | 0.100 | 286 | 0.070 | 47.74 | 14.323 | 24 |

7(I.galbana, Cape Town) | 2,215,200 | 1.49 | 13.64 | 300 | 8022 | 11954 | 1773 | 0.100 | 4 | 0.024 | 0.684 | 1.019 | 15 |

8(I.galbana, Rio de Janeiro) | 84,363 | 0.42 | 1.19 | 300 | 713 | 297 | 1724 | 0.100 | 165 | 0.063 | 27.506 | 11.474 | 24 |

Case | harvP | Z ($) | CD (m) | CW (m) | Length (m) | V (m^{3}) | SA (m^{2}) | Iavg (μE m^{−2}·s^{−1}) | Uavg (m·s^{−1}) | BC (g·m^{−3}) | μ (h^{−1}) | PrV (g·m^{−3}·Day^{−1}) | PrA (g·m^{−2}·Day^{−1}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

6-SA1 | 1 | 1.2384 × 10^{6} | 1.5 | 7.363 | 300 | 6557 | 4371 | 2061 | 0.114 | 1.25 | 0.059 | 1.247 | 1.871 |

6-SA2 | 2 | 1.1000 × 10^{6} | 1.5 | 6.527 | 300 | 5819 | 3880 | 2061 | 0.117 | 2.81 | 0.059 | 1.405 | 2.108 |

6-SA3 | 3 | 6.8332 × 10^{5} | 1.5 | 4.036 | 300 | 3611 | 2408 | 2061 | 0.117 | 6.79 | 0.059 | 2.264 | 3.396 |

6 | 6 | 6.1109 × 10^{4} | 0.3 | 0.953 | 300 | 171 | 571 | 2125 | 0.100 | 286 | 0.0696 | 47.74 | 14.323 |

6-SA4 | 10 | 9.703 × 10^{3} | 0.042 | 1.45 | 48 | 5.71 | 136 | 2140 | 0.131 | 14329 | 0.0635 | 1433 | 60 |

Case | Upper Bound on Length (m) | Z ($) | CD (m) | CW (m) | Length (m) | V (m^{3}) | SA (m^{2}) | Iavg (μE·m^{−2}·s^{−1}) | Uavg (m·s^{−1}) | BC (g·m^{−3}) | μ (h^{−1}) | PrV (g·m^{−3}·Day^{−1}) | PrA (g·m^{−2}·Day^{−1}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

6 | 300 | 6.1109 × 10^{4} | 0.300 | 0.953 | 300 | 171 | 571 | 2125 | 0.100 | 286 | 0.0696 | 47.74 | 14.323 |

6-SA7 | 500 | 5.0875 × 10^{4} | 0.331 | 0.449 | 500 | 149 | 449 | 2123 | 0.100 | 330 | 0.0709 | 54.950 | 18.216 |

6-SA8 | 1000 | 4.9553 × 10^{4} | 0.372 | 0.299 | 688 | 153 | 411 | 2121 | 0.100 | 321 | 0.0710 | 53.434 | 19.895 |

Case | Lower Bound on CD (m) | Z ($) | CD (m) | CW (m) | Length (m) | V (m^{3}) | SA (m^{2}) | Iavg (μE·m^{−2}·s^{−1}) | Uavg (m·s^{−1}) | BC (g·m^{−3}) | μ (h^{−1}) | PrV (g·m^{−3}·Day^{−1}) | PrA (g·m^{−2}·Day^{−1}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

6 | 0.3 | 6.1109 × 10^{4} | 0.3 | 0.953 | 300 | 171 | 571 | 2125 | 0.100 | 286 | 0.0696 | 47.74 | 14.323 |

6-SA8 | 0.2 | 6.0927 × 10^{4} | 0.278 | 0.979 | 300 | 163 | 586 | 2126 | 0.100 | 301 | 0.0700 | 50.18 | 13.944 |

**Table 5.**Summary of sensitivity analysis results by changing the initial biomass concentration in the pond.

Case | Bcini (g/m^{3}) | Z ($) | CD (m) | CW (m) | Length (m) | V (m^{3}) | SA (m^{2}) | Iavg (μE·m^{−2}·s^{−1}) | Uavg (m·s^{−1}) | BC (g·m^{−3}) | μ (h^{−1}) | PrV (g·m^{−3}·Day^{−1}) | PrA (g·m^{−2}·Day^{−1}) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

6-SA5 | 0.1 | 2.9456 × 10^{5} | 0.306 | 4.611 | 300 | 842 | 2748 | 2139 | 0.100 | 58 | 0.0708 | 9.71 | 2.975 |

6 | 0.6 | 6.1109 × 10^{4} | 0.3 | 0.953 | 300 | 171 | 571 | 2125 | 0.100 | 286 | 0.0696 | 47.74 | 14.323 |

6-SA6 | 0.9 | 4.6405 × 10^{4} | 1.5 | 1.925 | 42 | 240 | 160 | 2024 | 0.117 | 205 | 0.0526 | 34.10 | 51.152 |

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Yadala, S.; Cremaschi, S.
A Dynamic Optimization Model for Designing Open-Channel Raceway Ponds for Batch Production of Algal Biomass. *Processes* **2016**, *4*, 10.
https://doi.org/10.3390/pr4020010

**AMA Style**

Yadala S, Cremaschi S.
A Dynamic Optimization Model for Designing Open-Channel Raceway Ponds for Batch Production of Algal Biomass. *Processes*. 2016; 4(2):10.
https://doi.org/10.3390/pr4020010

**Chicago/Turabian Style**

Yadala, Soumya, and Selen Cremaschi.
2016. "A Dynamic Optimization Model for Designing Open-Channel Raceway Ponds for Batch Production of Algal Biomass" *Processes* 4, no. 2: 10.
https://doi.org/10.3390/pr4020010