Growth Rate Model Research Articles

Static layer melt crystallization: Effects of impurities on the growth behaviors of crystal layers

• Impurity concentration polarization occurred at solid–liquid interface. • Mass transfer coefficient of impurity in melt boundary layer was derived. • Growth rate model correlating growth rate with temperature gradient of crystal layer was proposed. • Latent heat of crystallization decreases during crystallization due to impurity concentration polarization. In this work, the effects of impurities on the growth behaviors of crystal layer during static layer melt crystallization (LMC) was studied. The crystal layer structure, growth rates, melt temperature curves and separation efficiency were experimentally investigated at different impurity concentrations and temperatures. With the experimental results, the mass transfer at the solid–liquid interface was discussed through: a) calculating the supercooling of the melt over the crystallization and b) deducing the mass transfer coefficient of impurity in the melt boundary layer. It was verified that the melt kept supercooled during the crystallization and concentration polarization occurred at the solid–liquid interface. A power law model was proposed to correlate the growth rate with the temperature gradient of crystal layer, which indicates that the concentration polarization results in the decrease of the latent heat of crystallization, and the mixture with lower impurity concentration and crystallization temperature would exhibit more serious concentration polarization.

Growth Equations in Forest Research: Mathematical Basis and Model Similarities

Purpose of ReviewGrowth equations have been widely used in forest research, commonly to assess ecosystem-level behavior and forest management. Nevertheless, the large number of growth equations has obscured the growth-rate behavior of each of these equations and several different terms for referring to common phenomena. This review presents a unified mathematical treatment of growth-rates besides several well-known growth equations by giving their mathematical basis and representing their behavior using tree growth data as an example.Recent FindingsWe highlight the mathematical differences among several growth equations that can be better understood by using their differential equations forms rather than their integrated forms. Moreover, the assumed-and-claimed biological basis of these growth-rate models has been taken too seriously in forest research. The focus should be on using a plausible equation for the organism being modelled. We point out that more attention should be drawn to parameter estimation strategies and behavior analysis of the proposed models. Thus, it is difficult for a single model to capture all possible shapes and rates that such a complex biological process as tree growth can depict in nature.SummaryWe pointed out misleading concepts attributed to some growth equations; however, the differences come from their mathematical properties rather than pure biological reasoning. Using the tree growth data, we depict those differences. Thus, comparisons of some functional forms (at least simple ones) must be carried out before selecting a function for drawing scientific findings.

Modeling of zircon nucleation and growth rates using crystal size distributions in a cooling magmatic intrusion

With a rapid increase in the use of zircons for geochronology and in situ isotopic and chemical analyses it is important to quantify growth rates and nucleation rates of zircons in natural magmatic systems. Here we present a mathematical model of the nucleation and growth of a population of zircon crystals in a cooling magmatic intrusion and apply nucleation laws to derive zircon crystal size distributions (CSD) and particle number density (n per cm3 of melt) from first principles. The model is based on a numerical solution of the one-dimensional diffusion equation that accounts for the nucleation and growth of individual crystals and the dependence of the equilibrium concentration of Zr on temperature. As experimental studies of zircon nucleation at natural concentration levels of Zr are nearly impossible, we rely on measured CSDs of age-invariant (within error of the method <±5ky) zircon populations in rapidly quenched eruptive products of known crystallization duration to calibrate our model. The CSDs of zircon crystals are calculated at different rates of cooling of the magma chamber and are compared to measurements. The combination of the zircon growth software with measurements of zircon CSDs and particle number density permits direct estimation of the zircon nucleation rates and their evolution with time in cooling magmatic systems. We observe a delayed zircon nucleation of 500–2000 years after the melt becomes zircon saturated. Zr supersaturations that drive grown in simulations range from 1 to ∼30ppm and vary non-monotonically as crystallization proceeds. Initially, the nucleation rates increase exponentially from 10−3 to ∼1crystals/cm3/yr but then decrease by an order of magnitude as the distance between zircons decreases. Growth rates stay in 10−15cm/s range for the most duration of zircon growth. The ratio of growth to nucleation rates in our model also varies insignificantly and is used to estimate crystallization time based on the CSD slope, which decreases with the increased duration of cooling. Flatter slopes correspond to longer crystallization durations as is also noted for major phases. Due to non-monotonic variations in growth and nucleation rates with time, the observed deficiency of small zircons resulting in concave up CSD shapes can be reproduced from first principles assuming monotonic cooling of the magmatic system. Variations in the mode of cooling from linear to monotonic during conductive heat loss of a cooling intrusion, and minor coeval precipitation (<10vol%) of major phases has little impact on zircon CSD and its evolution. This study has implication to individual zircons IDTIMS dates and demonstrates that most zircons crystallize much later than magma becomes saturated with Zr, despite sometimes having nominally long tail crystallization rate in the inherited cores. For natural CSDs from eight different eruptions randomly selected zircons predate eruption age only by ∼300–1000years.

Population-balance study of protein crystal growth from solution using a hyperbolic rate law

Kinetic data for crystal size, recently reported for bulk protein crystal growth from solution, is shown to obey a hyperbolic rate law beginning at some critical time after the commencement of the growth run. Given this rate law, the linear growth rate is established and mass-balance methods are then used to derive analytic time dependent expressions for the nucleation rate, the supersaturation and the distribution function for crystal sizes. Our population-balance approach involves a truncated Fokker–Planck equation along with an integro-differential equation for mass conservation. This model is used to describe three cases of protein crystal growth from solution for which kinetic data has been reported: lysozyme, β–lactoglobulin and insulin. Additionally, we show that this empirical rate law for crystal size is approximately related to the Burton-Cabrera-Frank (BCF) and Kossel-Stranski (KS) growth rates. Further, insight into the so called two-step, process is obtained. For two of the growth cases studied here, two-step activity was experimentally confirmed. The hyperbolic tangent rate law most accurately describes growth beginning at some critical time when two-step activity begins to wane and the bulk crystal phase becomes the more dominate form in the solution. Therefore, there seems to be a transition between growth modes at the critical time. We propose here that two-step activity creates a diffusion limited growth situation in the solution for times less than the critical time. The kinetics of crystal size are estimated, during the early diffusion limited time period, using a known model for diffusion limited growth leading to the crystal radius being directly proportional to the square root of time. At the critical time, the two models meet and the kinetics of the crystal radius can thereafter be described using the hyperbolic rate law. Additionally, features of the hyperbolic growth rate model are used to develop a lever rule that can be used to predict the critical time when the transition between growth modes occurs.

Numerical modeling and simulation of fatigue crack growth rate due to cyclic loading on doubler structure fuselage skin station number 360-380 stringer 6L-7L Boeing 737-900 extended range aircraft

Untuk mengetahui kekuatan struktur doubler dalam menahan terjadinya kegagalan akibat retakan yang disebabkan oleh cyclic loading , diperlukan suatu analisis tersendiri yang tidak terdapat pada panduan Structural Repair Manual (SRM). Cyclic loading dapat mengurangi kekuatan skin doubler secara bertahap ( fatigue ) hingga mencapai kegagalan ( fracture ). Pada penelitian ini akan dilakukan analisis crack growth rate dan jumlah siklus pembebanan yang dibutuhkan untuk memunculkan retakan dengan panjang tertentu ( fatigue cycle ) terhadap struktur doubler fuselage skin pesawat Boeing 737-900 Extended Range dengan station number 360-380 di antara stringer 6L-7L pada setiap panjang retakan dan ketinggian terbang simulasi. Analisis dilakukan menggunakan pendekatan numerik dengan metode Modified Virtual Crack Closure Integral (MVCCI) untuk mendapatkan nilai Stress Intensity Factor (SIF) melalui software berbasis metode elemen hingga. Melalui nilai yang diperoleh dapat diketahui bahwa nilai crack growth rate dan fatigue cycle berbanding lurus terhadap variasi panjang retakan dan ketinggian terbang simulasi yang diberikan. Nilai crack growth rate paling rendah terjadi pada struktur doubler dengan panjang retakan 8,5 mm dan kondisi terbang 5000 feet yaitu sebesar 2,964 mm/ cycle , dan nilai tertinggi sebesar 5,471 mm/ cycle terjadi pada struktur doubler dengan panjang retakan 51 mm dan kondisi terbang 40000 feet . Sedangkan, nilai fatigue cycle paling rendah terjadi pada struktur doubler dengan panjang retakan 8,5 mm dan kondisi terbang 40000 feet yaitu sebesar 2,540 cycle , dan nilai tertinggi sebesar 5,470 cycle terjadi pada struktur doubler dengan panjang retakan 51 mm dan kondisi terbang 5000 feet .

ANÁLISE DA TENDÊNCIA E CRESCIMENTO DAS EXPORTAÇÕES CHINESA E SEUS IMPACTOS NO COMPLEXO SOJA SUL AMERICANO: 1990 A 2020

The study proposes an analysis of the trends and export profiles of products from the soy complex in Argentina, Brazil, Paraguay and China in the period from 1990 to 2020. The econometric model of trend and log-linear growth rate was used as a methodology. Among the results, it was found that china has resized its international trade in the soy complex. Through industrial policy, it constituted a competition and a large industrial park. Based on trade policy, it strengthened the industry, through tariff and non-tariff barriers. On the other hand, in the South American continent, it was observed that as a promotion policy in China, it stimulates the growth of soy production in Brazil. In contrast, Argentina and Paraguay's policy was effective in maintaining and expanding activities related to soy processing.

Fatigue crack initiation and propagation behavior of nickel-based single crystal DD6 under different drilling processes

The fatigue crack initiation and propagation behavior of film cooling holes (FCHs) under different drilling processes are of great significance to key structural parts. In this paper, it mainly comprehensively and systematically summarized the difference in surface condition caused by different drilling processes of nickel-based superalloys combining several characterization devices and described whole process from crack initiation to fracture in detail. The in-situ fatigue crack growth test at room temperature on the plate specimens with a different drilling FCHs (EDM and LDM) made of [001] nickel-based single crystal was carried out, and proposed a comprehensive evaluation method for the crack initiation mechanism from the three dimensions of geometry, metallurgy and mechanics. The four main octahedral slip systems are proved by finite element simulation. Then, we clarified the difference of fracture surface, crack path and fracture morphology of different drillings, and the competition mechanism between crack propagation path and slip surface was investigated. Finally, a simple unified model of fatigue crack growth rate was proposed, which provides a useful crack propagation law of different drilling hole.

The association of opening K–12 schools with the spread of COVID-19 in the United States: County-level panel data analysis

This paper empirically examines how the opening of K-12 schools is associated with the spread of COVID-19 using county-level panel data in the United States. As preliminary evidence, our event-study analysis indicates that cases and deaths in counties with in-person or hybrid opening relative to those with remote opening substantially increased after the school opening date, especially for counties without any mask mandate for staff. Our main analysis uses a dynamic panel data model for case and death growth rates, where we control for dynamically evolving mitigation policies, past infection levels, and additive county-level and state-week "fixed" effects. This analysis shows that an increase in visits to both K-12 schools and colleges is associated with a subsequent increase in case and death growth rates. The estimates indicate that fully opening K-12 schools with in-person learning is associated with a 5 (SE = 2) percentage points increase in the growth rate of cases. We also find that the association of K-12 school visits or in-person school openings with case growth is stronger for counties that do not require staff to wear masks at schools. These findings support policies that promote masking and other precautionary measures at schools and giving vaccine priority to education workers.

Loss of density dependence underpins decoupling of livestock population and plant biomass in intensive grazing systems.

Across the world, social-ecological rangeland systems have been transformed from a preindustrial extensive status to intensive exploitation, often leading to long-term livestock population booms, overgrazing, and rangeland degradation. To understand the regulatory mechanisms involved in such historical social-ecological transformations, we collected population data on the native sheep of the last nomadic county in the Inner Mongolia Autonomous Region (1961-2005). We detected changes in internal feedbacks (e.g., density-dependent effects) and external disturbance (e.g., winter harshness, rainfall, harvest) between the extensive and intensive management periods using regression models of sheep population growth rate and counterfactual analyses. We found that, in the extensive period, sheep populations were regulated during harsh winters by climate, while they were regulated during mild winters by negative density dependence. In the intensive period, the negative feedback of density dependence was removed through the provision of additional forage and shelter, and only winter climate and growing season rainfall regulated sheep populations. Counterfactual analyses also confirmed the irreplaceable role of density-dependence in maintaining a sustainable rangeland ecosystem. Although herders attempted to adapt to the removal of negative feedbacks by improving livestock harvest, overgrazing and grassland degradation remain a challenge in this system. We conclude that internal feedbacks within social-ecological systems should be carefully anticipated and accounted for when managing rangelands for sustainability.

Continuous color model as a tool to improve speleothem age model development

Because they can archive a variety of geochemical proxies and be precisely and accurately dated with the U-Th decay series chronometer, stalagmites are widely used for paleoclimate reconstructions. However, limitations in the use of this chronometer arise because U-Th dating is analytically time consuming, expensive, and requires a relatively large sample size. These limitations restrict the number of absolute dates usually obtained, which can result in significant uncertainties in the age model and inhibit the ability to archive high resolution records of environmental variability, particularly in those stalagmites where there are variations in growth rate not constrained by U-Th dates. Here, we explore the relationship between stalagmite color and growth rate. Consequently, we evaluate the use of a simple, practically non-destructive approach to model the age-depth relationship of stalagmites using the sample color to provide a continuous record of growth rate. The method was developed by comparing high-resolution color images with pre-determined U-Th dates along the growth axes of seven stalagmites. The obtained results suggest that prior to dating, a color-derived, continuous growth rate model may be used to identify important changes in growth rate which may aid in the determination of the most efficacious locations for U-Th dating. Further, continuous color-derived interpolations between U-Th derived dates may be superior to traditional linear interpolation methods. Such an approach has the potential to greatly improve a researcher’s ability to efficiently choose sampling locations for more precise, albeit laborious and costly, U-Th dating.

Impact of Technological Change on Growth and Agricultural Sector in Gujarat State of India: A Time-Series Data Study

This study assessed the growth rate of commercial and food-grain crops due to technological change in Gujarat. Growth rate model was employed to examine the growth rate of area sown, production and yield of crops. Subsequently, impact of technological change, and other inputs on yield of individual crop was estimated using a Cobb-Douglas production function model. Time trend factor was used as a proxy variable to capture the impact of technological change, and other inputs (i.e., area sown, irrigated area, application of fertilizer, agricultural labors, rural literate population and annual actual rainfall) on yield of crops. Growth rate of cropped area, production and yield of cotton, sugarcane, castor, potato, rice, arhar, maize, gram and wheat crops were seemed positive in Gujarat. Yield of cotton, sugarcane, castor, rice, arhar, maize, bajra, gram, wheat, jowar ragi, potato, groundnut, sesamum, rapeseed&amp;mustard and soyabeans crops was positively associated with time trend factors. Furthermore, the regression coefficient of time trend factor with yield of cotton, tobacco, potato, groundnut, sesamum, rapeseed &amp; mustard, rice, arhar, maize, bajra, gram, wheat, jowar and ragi was reported positive and statistically significant. Hence, the estimates shows that yield of aforesaid crops were improved due to application of technological change in agricultural sector in Gujarat. Several practical policy suggestions are given to increase the use of technology in agricultural sector to improve the growth of major food-grain and commercial crops.

Environmental Control of Plant Primary Metabolism: Exploitation of Plant Plasticity in Perennial and Tree Crops

Perennial and tree crops are interwoven with environmental challenges in multiple ways, as anthropogenic global changes are a fundamental component in a variety of pressures that have negative consequences for farming. Climate controls have a wide range of detrimental effects on the land and crops. Rainfall, temperature, heat waves, pests or bacteria, CO2 or ozone levels, and marine flows are a few examples of environmental controls life. These alterations have a negative influence on the metabolisms of primary and secondary in plants, but they make use of the adaptability of plants also, which is referred to as plasticity.&#x0D; Biological and metabolic characteristics, as well as plant genome mutations for greater adaptability, play an important impact on growth patterns. Pathogens and herbivores, for example, are important climatic regulators that induce unique plasticity within the plant system. The incredible adaptability is that the plants thrive under extreme conditions. Furthermore, more research and investigations are needed to determine how and to what extent plasticity can aid endurance. Because of the influence of various other factors, the results of previous studies have been inconsistent. They sense the stressor in the environment, become engaged, and then trigger the appropriate physiological responses. According to the GDB theory, the metabolic exchange is responsible for plant elasticity including the processes of growth and differentiation.&#x0D; The genetic trade-off in plant life development is caused by the biological impact on growth and genetic alterations, as well as herbivory and plant-plant competition. In a traditional growth rate model, researchers separate the biological and evolutionary components to characterize the impact of competition in the development of this flexibility. Plant breeding is unquestionably important in the application of plasticity to stressful controls. In the current circumstances, larger yields under harsh environmental conditions are required to meet food demand.

Effect of Stress Magnitude on Pit Growth Rate of 304 Austenitic Stainless Steel in Chloride Environments

In this study, a corrosion pit test using notched bar specimens was conducted to investigate the effect of stress magnitude on the pit growth rate. To produce the notched bar specimens, 304 austenitic stainless steel was used, which is a material used for spent nuclear fuel canisters. Furthermore, three levels of stresses were generated using different notch radii. The corrosion pits were quantitatively measured through scanning electron microscopy and analyzed by finite element analysis. Based on experimental data, the pit growth rate model is suggested in terms of the stress and exposure time.

A preliminary evaluation of the environmental toxicology of road salts on plant root dynamics

Although critical for the deicing of roads and bridges throughout the winter season in North America, road salts continue to have yearlong negative impacts on both anthropogenic and biological systems. Our objective was to measure the impact of four commonly used road salt ingredients on the dynamics of plant growth and development. Using onions as a model system, we hypothesized that exposing plants to different concentrations of sodium chloride, calcium chloride, potassium chloride, and magnesium chloride treatments would allow us to calculate the environmental concentration at which root development is halted by each road salt. We found that root growth decreased with sodium chloride, calcium chloride, and potassium chloride treatments, but increased with magnesium chloride. We used a linear model of root growth rates and concentrations to calculate the level at which each road salt becomes toxic to plants in the surrounding environment. We also apply the ecological theory of the principle of allocation to discuss how the chemical induction of root growth by magnesium chloride road salts may result in a life history trade-off that reduces investment toward plant-defense. The predicted trade-off in plant defense suggests that crop species in urban environments would be more susceptible to herbivory and as such would rely more on agricultural pesticides to sustain food security. We propose further studies to evaluate applications of plant defense theory in an urban context.

Selective area growth rates of III-V nanowires

Selective area growth (SAG) of semiconductors is a scalable method for fabricating gate-controlled quantum platforms. This letter reports on the adatom diffusion, incorporation, and desorption mechanisms that govern the growth rates of SAG nanowire (NW) arrays. We propose a model for the crystal growth rates that considers two parameter groups: the crystal growth control parameters and the design parameters. Using GaAs and InGaAs SAG NWs as a platform, we show how the design parameters such as NW pitch, width, and orientation have an impact on the growth rates. We demonstrate that by varying the control parameters (i.e., substrate temperature and beam fluxes) source, balance, and sink growth modes may exist in the SAG selectivity window. Using this model, we show that inhomogeneous growth rates can be compensated by tuning the design parameters.

A minimal mathematical model of neutrophil pseudopodium formation during chemotaxis

The directed movement of neutrophils is provided by the rapid polymerization of actin with the formation of a protrusion growing forward. In our previous work we observed impaired neutrophil movement for patients with Wiskott-Aldrich syndrome (WAS) compared to healthy donors.In this work, we set out to explain the impairment of neutrophil chemotaxis in patients by observation and computer modeling of the linear growth rates of the anterior pseudopodia. The neutrophil chemotaxis was observed by means of low-angle fluorescent microscopy in parallel-plate flow chambers. The computational model was constructed as a network-like 2D stochastic polymerization of actin guided by the proximity of cell membrane with branching governed by Arp2/3 and WASP proteins.The observed linear velocity of neutrophil pseudopodium formation was 0.22 ± 0.04 μm/s for healthy donors and 0.23 ± 0.08 μm/s for WAS patients. The model described the velocity of the pseudopodium formation for healthy donors well. For the description of WAS patients data, a variation of branching velocity (governed by WASP) by an order of magnitude was applied, which did not significantly alter the linear protrusion growth velocity.We conclude that the proposed mathematical model of neutrophil pseudopodium formation could describe the experimental data well, but the data on overall neutrophil movement could not be explained by the velocities of the pseudopodium growth.

Exploratory analysis of Spirulina platensis LB 2340 growth in varied concentrations of anaerobically digested pig effluent (ADPE)

There is a significant interest in novel waste management solutions to treat wastewater from swine operations. Anaerobic digestion is a rising and prominent solution, but this technology still generates highly concentrated effluent that requires further remediation. Therefore, the aim of this study was to explore the feasibility of cultivating the cyanobacterium Spirulina platensis in swine effluent for future applications in biological waste treatment and value-added fermentation. To accomplish this goal, growth of S. platensis was characterized in varying proportions of ideal, synthetic Zarrouk medium and anaerobically digested pig effluent (ADPE) to obtain growth rate models. Results yielded a positive correlation between S. platensis growth rate and Zarrouk medium proportion, with the highest growth rate in 100% Zarrouk media but comparable growth in the 50/50% Zarrouk/ADPE mixture. This study demonstrates the potential for S. platensis to further improve the treatment efficacy of anaerobic digestion systems, and the exploratory analysis also highlights that further testing is required to investigate possible carbon availability, chemical inhibition, and overall nutrient reduction in ADPE. This research contributes important data toward the feasibility of producing value-added cyanobacterial biomass while simultaneously consuming excess nutrients to aid in agricultural wastewater management efforts and generate cost-effective products in a more sustainable manner.

Synergism between fatigue and cyclic‐stress corrosion cracking: Electrochemically active surface area

AbstractAfter over 50 years of research on the topic, researchers continue to consider how time‐dependent environmental effects can be integrated with cycle‐dependent fatigue crack growth rate (CGR) models. Early corrosion‐fatigue (CF) model assumptions were that cycle‐dependent CF and time‐dependent stress corrosion cracking (SCC) contributions are separable, operate in parallel, are non‐interacting. However, research has shown that CF and SCC may interact synergistically to produce CGRs greater than that obtained by simple summation. Proposed here is that CF and SCC mechanisms may interact synergistically as each is dependent on the electrochemically active surface area (E‐ASA) at a crack tip. A phenomenological model is developed for the effects of crack tip strain rate, film‐rupture, and repassivation rates on E‐ASA. Analysis of hold‐time CGR data provides a method of separating cyclic‐ and static‐load hold‐time effects on E‐ASA. Five model parameters, each having physical interpretation, are found to have reasonable magnitudes.

The relationships between economic growth and cropland changes in Bangladesh: An evidence based on annual land cover data

Bangladesh is experiencing rapid economic progress since 1990s, at the same time, facing acute shortage of agricultural land. Using 19 years (2000-2018) time series remote-sensing data and Random Forest machine learning algorithm in Google Earth Engine (GEE), this study classifies the land covers in Bangladesh; determines the trend of land cover changes using least-square growth rate model; and finds the associations between the trends of cropland and growth of socio-economic factors using correlation and regression analysis. The annual average area of cropland decline is 29271 hectare, which is higher than the absolute changes of other major land covers, such as forest (25932 hectare) and built-up area (1649 hectare). With the annual growth of urban population (3.81%) and gross domestic product (GDP) (6.28%), the predicted declining rate of cropland is estimated 0.29%, which is very close to the observed annual declining rate (0.28%). However, the partial effects of urban population and GDP could not be detected. Therefore, the study infers that urbanization and economic growth in Bangladesh are happening simultaneously. As a result, the joint determination of both phenomena can explain the degradation of agricultural land in Bangladesh.

Modeling and Use of Inter-Criteria Decision Analysis for Selecting Growth Rate Models for Batch Cultivation of Yeast Kluyveromyces marxianus var. lactis MC 5

Ten unstructured models of Monod, Mink, Tessier, Moser, Aiba, Andrews, Haldane, Luong, Edward, and Han-Levenspiel are considered in this paper to explain the kinetics of cell growth for batch cultivation of the yeast Kluyweromyces marxianus var. lactis MC 5. For the first time, two independent kinetic models are used to model the process for the two basic substrates—lactose and oxygen. The selection of the most appropriate growth rate models has been made through a new multi-criteria decision-making approach called the Inter-Criteria Decision Analysis (ICDA) method. The application of ICDA to the growth rate of lactose and oxygen alone has shown that there have been many correlations between the studied models. Thus, the models for the growth rate, depending only on lactose, are reduced to one—Monod model and there are two models—Monod and Mink—depending on oxygen only. Separate kinetic process models have been developed for the combination of Monod–Monod and Monod–Mink models. For the first time, in addition to the multiplicative form, the additive form of a specific growth rate has been studied. The comparison of the obtained results has shown that the additive form has shown better results than the multiplicative one. For this reason, the additive form of the Monod–Monod model will be used to model the process.