Mitscherlich Model Research Articles

Model-averaging as an accurate approach for ex-post economic optimum nitrogen rate estimation

AbstractFinding economic optimum fertilizer rate with good accuracy is essential for optimal crop yield, efficient resource utilization, and environmental well-being. However, the prevailing incomplete understanding of input-output relationships leads to imprecise crop yield response functions, such as those for winter wheat, and potentially biased fertilizer choices. From a statistical point of view, there is uncertainity with regards to which model is most suitable to estimate the economic optimum fertilizer rate. This complexity is amplified when considering site-specific nitrogen fertilization, which factors into elements like soil attributes, topography, and crop variations within a field, as opposed to uniform application. This study undertakes a comparative analysis to evaluate biases, variance, mean squared errors and confidence intervals in Economic Optimum Nitrogen Rate (EONR) estimations across different functional forms. The goal is to uncover performance discrepancies among these forms and explore potential advantages of adopting model averaging for optimizing nitrogen use in crop cultivation. The results of simulations reveal noteworthy biases when comparing diverse yield functions with the averaged model, particularly evident in the Linear-Plateau and Mitscherlich models. Moreover, analysis of empirical data indicates that confidence intervals for the averaged model overlap with the projected ranges of all functions. This implies that the averaged model could be suitable for determining EONR and effectively address the problem of model specification without focusing on one specific functional form. The effectiveness of model averaging hinges on incorporating models that well approximate the true model. However, even if the true model is not known, the average model can provide reasonable information for determining the EONR, provided that similar model specifications are considered. This has implications for modelling of yield response for various applications and can contribute to unbiased estimations of yield response.

Characterizing the role of hydraulic retention time on nitrate removal indices in denitrifying bioreactors by nonlinear models

Denitrifying bioreactors (DNBRs) are a sustainable and cost-effective practice commonly used at the edge of fields to reduce nitrate from agricultural runoff. The hydraulic retention time (HRT) is a crucial variable that affects nitrate removal rate (NRR, g N m−3 d−1), nitrate removal efficiency (NRE, %), and nitrate concentration reduction per length (Nrd, mg N L−1 m−1). In this study, two nonlinear models, the developed Michaelis-Menten (MM) model and the Mitscherlich (MT) model, were developed to characterize the relationship between nitrate removal indices (NRR, NRE, and Nrd) and HRT. This study first utilizes nonlinear models to quantitatively understand the relationship between NRR, NRE, Nrd, and HRT. To verify the models, eight experiments were conducted under different conditions, including different scales (laboratory and field), media (woodchip, woodchip+biochar, woodchip+silage leachate, woodchip+biochar+silage leachate), and influent nitrate concentrations (6.8–70 mg N L−1). The results showed that the MT model outperformed the MM model and MT could accurately characterize the nitrate removal changes with HRT and provide the optimal HRT (HRTO). Overall, the model could be beneficial for designers and practitioners to optimize nitrate removal.

Optimizing ruminant nutrition: Insights from a comprehensive analysis of silage composition and in vitro gas production dynamics using nonlinear models

Achieving sustainable livestock management necessitates optimizing animal production while minimizing environmental impact. To achieve this, feed efficiency must be enhanced, and nutrition blueprints must be understood. In ruminant nutrition, this is of paramount importance, as it exposes degradation kinetics and nutritional benchmarks, allowing feed management and formulations to be more ecologically balanced. Previous research efforts have focused on exploring the relationship between a restricted set of nutrient parameters and the in vitro gas production dynamics. In the current study, an extensive dataset derived from freeze-dried kefir culture treated white clover silage was used to examine intricate relationships between eight nonlinear models and diverse variables. This dataset contains in vitro gas production data along with nutritional composition, microbial populations, fermentation quality, digestibility, mineral concentration, and fatty acid profiles. Through rigorous application of mathematical models, the performance in capturing gas production dynamics was critically assessed. Among these, the Michaelis‒Menten (MM) and Mitscherlich (MIT) models fit the data well and offer superior predictions of gas production dynamics. Asymptotic gas volume was negatively correlated with crude protein content, emphasizing the influence of protein on gas production. Fiber composition plays a significant role in fermentation kinetics, as evidenced by significant correlations between degradation rate constant and crude protein concentrations. The degradation rate constant of insoluble fraction exhibited significant positive correlations with crude protein and neutral detergent fiber contents. Moreover, mineral content had significant effects on gas production dynamics. Zinc content showed a strong and significant positive correlation with the gas production rate coefficient, underscoring its crucial role in enhancing microbial activity. Conversely, calcium content displayed a significant but weak negative correlation with the final asymptotic gas volume, indicating its potential to modulate gas production. In summury, this study provides detailed insights into the intricate relationship between mathematical models and various variables in rumen fermentation. The MM and MIT models have proven to be robust tools, offering nuanced perspectives on gas production dynamics. These findings pave the path for improving sustainable ruminant nutritional practices and refining feed management strategies.

Modeling description of interface shear deformation: A theoretical study on damage statistical distributions

The shear constitutive model is important for the analysis of the interface shear deformation mechanism. In this study, five statistical distributions, including four that have never been applied in interface shear deformation, are introduced to describe the damage evolution of the interface. The corresponding statistical damage constitutive models are developed, and they have been validated using a series of experimental data (both laboratory and in-situ tests) at the rock-concrete interface. The comparison results with laboratory data show that the Mitscherlich model has the lowest prediction accuracy. By comparing with in-situ data, the Weibull model shows the best-predicted performance. Based on the Akaike information criterion metric, the Morgan-Mercer-Flodin (MMF) model performs much better than the other models in the laboratory tests, which indicates the MMF model can be introduced for a comprehensive analysis of interface shear deformation. In addition, the similarities between the MMF model and the Weibull model are found through the canonical transformation of the MMF model. And the analysis of the two unknown parameters of the MMF model shows that they are related to the yield characteristics and strength characteristics of the interface, respectively. Based on the damage variable evolution, the shear deformation of the interface can be divided into three phases, i.e., damage initiation phase, damage acceleration phase and damage slowing phase. And the parameters of the MMF model have an important influence on the damage variable curves.

Bayesian inference of biomass growth characteristics for sugi (C. japonica) and hinoki (C. obtusa) forests in self-thinned and managed stands

BackgroundForests are an important sink for atmospheric carbon and could release that carbon upon deforestation and degradation. Knowing stand biomass dynamic of evergreen forests has become necessary to improve current biomass production models. The different growth processes of managed forests compared to self-managed forests imply an adaptation of biomass prediction models.MethodsIn this paper we model through three models the biomass growth of two tree species (Japanese cedar, Japanese cypress) at stand level whether they are managed or not (self-thinning). One of them is named self-thinned model which uses a specific self-thinning parameter α and adapted to self-managed forests and an other model is named thinned model adapted to managed forests. The latter is compared to a Mitscherlich model. The self-thinned model takes into account the light competition between trees relying on easily observable parameters (e.g. stand density). A Bayesian inference was carried out to determine parameters values according to a large database collected.ResultsIn managed forest, Bayesian inference results showed obviously a lack of identifiability of Mitscherlich model parameters and a strong evidence for the thinned model in comparison to Mitscherlich model. In self-thinning forest, the results of Bayesian inference are in accordance with the self-thinning 3/2 rule (α=1.4). Structural dependence between stand density and stand yield in self-thinned model allows to qualifying the expression of biological time as a function of physical time and better qualify growth and mortality rate. Relative mortality rate is 2.5 times more important than relative growth rate after about 40 years old. Stand density and stand yield can be expressed as function of biological time, showing that yield is independent of initial density.ConclusionsThis paper addressed stand biomass dynamic models of evergreen forests in order to improve biomass growth dynamic assessment at regional scale relying on easily observable parameters. These models can be used to dynamically estimate forest biomass and more generally estimate the carbon balance and could contribute to a better understanding of climate change factors.

Mathematical models describing disappearance of Lucerne hay in the rumen using the nylon bag technique

It is essential to study the dynamics of rumen degradation of feeds before their potential use in formulating diets for ruminants. Various mathematical models have been developed to describe this degradation. The non-lagged exponential model (Model I), the lagged exponential model (Model II), the Gompertz model (Model III), and the generalized Mitscherlich model (Model IV) were examined using two alternative software (SAS and MATLAB) to determine their efficacy in accounting for variation in ruminal disappearance of dry matter (DM) and crude protein (CP) of lucerne hay from three cuttings. All models described DM degradability well (R2 >0.98). Only Models I and II converged when fitted to CP degradability data (R2 >0.98). It was concluded that any of these models could be used to describe the degradation of DM, whereas only Models I and II could be used to describe the degradation of CP from three cuttings of Lucerne hay. All the models that were fitted to the DM degradation data performed reasonably well, with only minor differences in goodness of fit. However, these models differed in values of the parameter estimates. Additionally, SAS failed to converge in the analyses of CP with Models III and IV, and MATLAB converged to nonsensical values with Model III. Model I might be recommended because it fitted the data well and required estimates of the fewest parameters
 Keywords: alfalfa hay, in situ digestion, model selection, nonlinear regression

Site-specific machine learning predictive fertilization models for potato crops in Eastern Canada.

Statistical modeling is commonly used to relate the performance of potato (Solanum tuberosum L.) to fertilizer requirements. Prescribing optimal nutrient doses is challenging because of the involvement of many variables including weather, soils, land management, genotypes, and severity of pests and diseases. Where sufficient data are available, machine learning algorithms can be used to predict crop performance. The objective of this study was to determine an optimal model predicting nitrogen, phosphorus and potassium requirements for high tuber yield and quality (size and specific gravity) as impacted by weather, soils and land management variables. We exploited a data set of 273 field experiments conducted from 1979 to 2017 in Quebec (Canada). We developed, evaluated and compared predictions from a hierarchical Mitscherlich model, k-nearest neighbors, random forest, neural networks and Gaussian processes. Machine learning models returned R2 values of 0.49-0.59 for tuber marketable yield prediction, which were higher than the Mitscherlich model R2 (0.37). The models were more likely to predict medium-size tubers (R2 = 0.60-0.69) and tuber specific gravity (R2 = 0.58-0.67) than large-size tubers (R2 = 0.55-0.64) and marketable yield. Response surfaces from the Mitscherlich model, neural networks and Gaussian processes returned smooth responses that agreed more with actual evidence than discontinuous curves derived from k-nearest neighbors and random forest models. When conditioned to obtain optimal dosages from dose-response surfaces given constant weather, soil and land management conditions, some disagreements occurred between models. Due to their built-in ability to develop recommendations within a probabilistic risk-assessment framework, Gaussian processes stood out as the most promising algorithm to support decisions that minimize economic or agronomic risks.

Determining the optimum range of soil Olsen P for high P use efficiency, crop yield, and soil fertility in three typical cropland soils

Soil Olsen P level has a major influence on crop yield, efficient P utilization, and soil fertility. In this study, the optimum Olsen P range was determined from long-term (1990–2012) field experiments in three typical soil types of China under single cropping of maize or double cropping of maize and wheat. The critical soil Olsen P value for crop yield was evaluated using three different models, and the relationships among P use efficiency (PUE), Olsen P, and total P were analyzed. The agronomic critical soil Olsen P values obtained from the three models for the neutral soil of Gongzhuling and the calcareous soil of Zhengzhou were similar; however, the values from the linear-linear and linear-plateau models for both maize and wheat were substantially lower than those from the Mitscherlich model for the acidic soil of Qiyang. The PUE response change rates (linear equation slopes) under different soil Olsen P levels were small, indicating slight or no changes in the PUE as the soil Olsen P increased in all three soils. A comparison of the Olsen P levels that achieved the maximal PUE with the agronomic critical values derived from the three models indicated that the linear-plateau model exhibited the best performance. The regression equation coefficients of Olsen P response to total P decreased as follows: Zhengzhou (73 mg g−1) > Qiyang (65 mg g−1) > Gongzhuling (55 mg g−1). The Olsen P level increased as the total P increased, which may result in a decrease in PUE. To achieve a relatively high crop yield, PUE, and soil fertility, the optimum Olsen P range should be 13–40, 10–40, and 29–40 mg kg−1 at Gongzhuling, Zhengzhou, and Qiyang, respectively.

Establishing reference values for soil microbial biomass-C in agroecosystems in the Atlantic Forest Biome in Southern Brazil

Abstract Soil microbial biomass plays a key role in the biogeochemical cycles, degradation of organic pollutants and soil aggregation, and its estimation through soil microbial biomass-C (MB-C) is often referred as a potential indicator of soil functioning. This study aimed to identify reference values of MB-C that would relate to crop yield. Soybean and maize grain yields and microbial biomass-C (MB-C) data were gathered from studies listed in “Web of Science” performed in agroecosystems located in the Atlantic Forest Biome in Southern Brazil. Database was composed by 78 datasets, where soil samplings for evaluation of MB-C were performed at the 0–10 cm layer at flowering stage, and evaluated by the fumigation-extraction method. A meta-regression using the Mitscherlich model was applied to determine the MB-C reference value corresponding to 80% of maximum yield. The effects of soil management systems and crop systems were verified by comparing the confidence intervals. Soil tillage decreased MB-C and yield, while crop rotation increased yield but not MB-C. It was not possible to find a reference value for soybean, because of the variability not explainable by the MB-C. The reference value related to high maize yield (>6,752 kg ha−1) was of 170 μg C g−1 soil, but further results must be included to decrease the variability.

Optimizing Soil and Fertilizer Phosphorus Management According to the Yield Response and Phosphorus Use Efficiency of Sugarcane in Southern China

The objective of this study was to determine the suitable range of available phosphorus (P) and fertilizer application rate in a sugarcane-growing region with red soil in southern China. Thirty-five monitoring sites were selected to investigate the soil available P in relation to crop yield, P-use efficiency (PUE) and soil total P under four different P rates. Compared with the no-P treatment, all three treatments involving fertilizer P presented sugarcane yields increased to varying extents, with an average increase of 17.6–25.8%. The yield-increasing effects and PUE in each P treatment decreased with increasing soil available P. The agronomic critical values of available P for sugarcane according to the linear-linear, linear-plateau and Mitscherlich models were 27.7, 41.5 and 17.0 mg kg−1, respectively. When the response of the yield-increasing effects of P fertilizer to soil available P and the agronomic critical values of the three models were combined, the linear-linear model exhibited the best fit. An observable change occurred in the available P response to total P. After the change point, the available P increased dramatically with increasing total P, possibly leading to decreased PUE or even triggering water pollution. Therefore, 28–50 mg kg−1 is the suggested range of suitable soil available P for the sugarcane-growing region. In addition, when the soil P budget and current relatively high level of available P are considered, applications of P fertilizer at 120 kg P2O5 ha−1 (1/2 the conventional P fertilizer rate) are recommended in this region.

Critical Value of Soil Potassium for Potato Crops in Volcanic Soils

The critical value of soil exchangeable potassium (K) (mg K kg−1) is a valuable tool for an efficient K fertilization management in potato production systems. The aim of the present study was to determine the critical value of soil exchangeable K for potato crops grown in different environments and differing availabilities of initial exchangeable K. The data used in this study were taken from 34 experiments on K fertilization conducted in southern Chile from 1977 to 2017 by the National Potato Center at the Instituto de Investigaciones Agropecuarias (INIA). Yield responses to K fertilization rates were assessed with linear-plateau and Mitscherlich models in order to determine the critical soil exchangeable K across environments. Across experiments, the average fresh tuber yield ranged between 18 and 91 Mg ha−1. Fresh tuber yields decreased by up to 70% when no K fertilization was applied. Yield reduction was concentrated around 25% in response to K deficiency. Relative yields were described as significant by both linear-plateau and Mitscherlich models. However, relative yields were better described by the linear-plateau model (p < 0.01, R2 = 0.62; SEE = 0.11). The critical soil exchangeable K value identified with the linear-plateau model was 224 mg K kg−1. This critical K value will be useful for potato farmers that produce their crops in volcanic soils, such as Andisols. Moreover, this critical K value will help to increase K use efficiency through better K fertilization management strategies aimed at building and/or maintaining K levels in soils.

Use of nonlinear mixed models for describing testicular volume growth curve in Nellore bulls

This study aimed to describe longitudinal testicular volume (TV) data of Nellore bulls by using nonlinear mixed models. Dataset consisted of 2,294 TV measurements from 505 bulls with ages ranging from 563 to 4,307 days. Nine nonlinear models were evaluated: Brody, Gompertz, Hill, Logistic I and II, Meloun, Michaelis-Menten, Mitscherlich and von Bertalanffy. Goodness of fit was evaluated by Akaike's information criterion (AIC), Bayesian information criterion (BIC), adjusted R2, percentage of convergence, error sum of squares (ESS), mean absolute deviation (MAD), and average prediction error (APE). These criteria were used to select the best model, then the absolute growth rate (AGR) for TV was estimated by the first derivate of the adjusted model related to time (∂Y/∂t). The values of adjusted R2, ESS, MAD and APE were similar among models. Percentage of convergence was higher for the Logistic I (76.8%), Logistic II (75.5%) and Mitscherlich (78.6%) models, but Logistic I and II showed the lowest values of AIC and BIC, indicating a better fit, so the Logistic I model was chosen for subsequent analyses. The TV growth occurred at a high rate until the inflection point, which was estimated at approximately 22 months of age; it stabilized and reached a plateau at approximately 2,500 days of age. This may suggest that TV is more related to sexual maturity than precocity. Additionally, the Logistic I model was used to estimate the growth curve and the AGR of the testicular length and width. As a result, testicular length increased at a higher rate than testicular width until approximately 1,600 days of age, indicating that the testes become longer with increasing age.

Hydrodynamic Modelling, Thermodynamic and Textural Variations during Common Beans Soaking

Hydrodynamic characteristics and its associated thermodynamic and textural variation of three common Malawian beans varieties (Boma, Sugar and Mandondo) during soaking were evaluated at four temperature regimes (25°C, 35°C, 45°C and 55°C). The equilibrium water uptake of 127% ± 5% was reached in 10, 6, and 4 hours respectively, for 25°C, 35°C and 45°C. Not much variation was observed between 45°C and 55°C except for sugar beans where equilibrium water uptake was reached within two hours of soaking at 55°C. Three models namely Peleg, two-parameter Mitscherlich model and viscoelastic model were used to evaluate the comparative predicting capabilities of the bean hydrodynamic characteristics. All models predicted the water absorption accurately (R2 > 0.903, RMSE activation kinetic parameters to be between 25 - 65 kJ/mol. Sugar beans were found to be the least hard. At room temperature, its hardness reduced by 58% within 2 hours of soaking. At higher temperature (55°C) hardness values were reduced to 12.5%, 11.1% and 15.0% within the first hour for Boma, Sugar and Mandondo beans, respectively.

Contrasting yield responses to phosphorus applications on mineral and organic soils from extensively managed grasslands: Implications for P management in high ecological status catchments

AbstractPhosphorus (P) loss from grassland production is one of the main causes affecting high status water bodies in Europe. Soils with a high content in organic matter (OM), even if extensively managed, are particularly vulnerable to P losses due to their poor P sorption capacities, and can affect the water quality of high status catchments if the dynamics of applied P in these soils is not well understood. The aim of this study was to assess dry matter yield, herbage P content, and P use efficiency in six soils deficient in P and ranging in OM content from 8.7% to 76.4% in a pot experiment under increasing P applications using the Mitscherlich equation. Of the six soils investigated, there was a better response in dry matter yield and greater P use efficiency in the soils with greater OM content than the mineral soils. The Mitscherlich model described grass response precisely in organic soils due to the higher plant availability of applied P as a consequence of the poor P sorption capacities of these soils. Despite the higher availability of applied P for plants in organic soils, the P requirements to meet the threshold herbage P content for dietary P supply to ruminants were still very high, which may pose a risk of P loss to the environment if P fertiliser is applied based on recommendations obtained from plant analysis. These results indicate that P fertilisation of organic soils in sensitive catchments poses a potentially high risk of P transfer to water bodies.

Responses to Potassium Application and Economic Optimum K Rate of Maize under Different Soil Indigenous K Supply

Potassium (K) is an essential macronutrient for maintaining crop productivity, but the economic benefit of K fertilizer often has been neglected. We collected a database from 60 maize field experimental sites in Northeast China between 2005 and 2012 to study the impacts of potassium (K) application rates on yield, K concentrations in grain and straw, plant K uptake, and to evaluate the economic optimum K rate (EOKR) for maize under different levels of soil indigenous K supply (IKS). The results showed that the average maize yield in Krec treatment (the recommended K rate) was highest and was 32.1% higher than that in K0 treatment (no K fertilization). Compared to K0, the application of K did not significantly increase grain K concentration, whereas it significantly increased K concentration in the straw. Plant K uptake in K150 treatment (150% of the recommended K rate) was higher than that in Krec treatment, but grain yield was lower than that in Krec treatment. Thus, the result indicates that luxury K absorption occurred in K150% treatment due to excessive K application. Thus, the application rate of K should be further optimized. Based on the modified Mitscherlich model, the average economic optimum K rate (EONR) was 135.24, 124.27, and 96.54 kg K2O/ha for the low, medium, and high levels of soil indigenous K supply (IKS), respectively. The average economic optimum yield (EOY) at a high IKS level was similar to the average yield in Krec treatment, whereas the EOKR could reduce by a K rate of 37.9–63.7 kg K2O/ha compared to Krec treatment. This study highlights the importance of K application to improve grain yield and provides a promising fertilizer recommendation method for minimizing fertilizer inputs and optimizing maize production.

Carbon stocks and tree allometries in the savannahs of the Plateau Batéké, central Africa

Tree allometric equations that are used to predict the biomass of a tree from its dendrometrical characteristics are key elements for the estimation of forest carbon stocks. Few equations compared to the diversity of forest ecosystems are currently available for the tropical forests of the Congo Basin. Pantropical equations are available but a pending question is whether they remain valid for all types of forest ecosystems in central Africa. The Plateau Batéké that covers a large area across Congo, Gabon and the Democratic Republic of Congo is dominated by savannahs that contrast with the dense rain forest often found under this tropical wet climate. Four 1-ha plots were inventoried in a savannah of the Plateau Batéké in Congo. Ninety trees from the five most abundant species were destructively sampled for aboveground biomass, while belowground biomass was measured for a sub-sample of 36 of these trees. The relationship between height (H in m) and diameter at breast height (dbh, D in cm) was modelled by a Mitscherlich model: H = 6.755–5.570 exp(−0.0946 D). The relationship between biomass (B in kg) and D, H and wood density (ρ in g cm−3) was modelled as: B = 0.6229 × (ρD2H)0.7031. This locally fitted allometric model had a bias of −2%, while a commonly used pantropical equation had a bias of −38%. Nevertheless, the bias of the pantropical equation at the Batéké site was of the same order as the bias of this pantropical equation at other sites used to establish it. The shoot-root ratio was 0.272. The savannah of the Plateau Batéké had a basal area of 6.32 m2 ha−1 and an aboveground woody biomass of 11.7 Mg ha−1 while belowground woody biomass accounted for an additional 3.4 Mg ha−1. The estimate of the aboveground woody biomass dropped by 50% when using the pantropical allometric equation. Insofar as site differences in biomass allometry are negligible in the pantropical equation, so is the difference between the pantropical equation and the locally fitted biomass equation. Nevertheless, the difference between the estimate of aboveground biomass using the locally fitted equation and the pantropical equation is large enough to call for a further examination of site differences in pantropical approaches.

Soil Phosphorus Management Based on the Agronomic Critical Value of Olsen P

ABSTRACTFertilizer phosphorus (P) is generally added to agricultural soils to meet the needs of crop production. In this study, the crop yield and soil Olsen P were measured every year (5–18 years) at 16 winter wheat (Triticum aestivum L.) –maize (Zea mays L.) crop rotation sites in cinnamon soil (Luvisols in FAO system). The mean agronomic critical value of Olsen P for maize was 14.2 mg kg−1 and for winter wheat was 14.4 mg kg−1 when using the Liner-plateau and Mitscherlich models. The change in soil Olsen P was positively linearly correlated with the P budget (P < 0.01), and an increase of 4.70 mg kg−1 in soil Olsen P for each 100 kg ha−1 of P budget in the 0–20 cm soil layer. A model of P fertilizer recommendation rate that integrated values of the change in soil Olsen P in response to P budget and the agronomic critical value of Olsen P was used, in order to adjust current levels of soil Olsen P to the agronomic critical value at the experimental sites over the next 5 years, P fertilizer application rate should be in the range of 0–87.5 kg P ha−1.

A comparison of soil tests for available phosphorus in long‐term field experiments in Europe

SummaryMost soil tests for available phosphorus (P) perform rather poorly in predicting crop response. This study was set up to compare different established soil tests in their capacity to predict crop response across contrasting types of soil. Soil samples from long‐term field experiments, the oldest &gt;100 years old, were collected in five European countries. The total number of soil samples (n = 218), which differed in cropping and P treatment, and originated from 11 different soil types, were analysed with five tests: ammonium oxalate (Ox), ammonium lactate (AL), Olsen P, 0.01 m CaCl2 and the diffusive gradient in thin film (DGT). The first three tests denote available P quantity (Q), whereas the last two indicate P intensity (I) of the soil solution. All five tests were positively related to the crop yield data (n = 317). The Q‐tests generally outperformed I‐tests when evaluated with goodness of fit in Mitscherlich models, but critical P values of the I‐tests varied the least among different types of soil. No test was clearly superior to the others, except for the oxalate extraction, which was generally poor. The combination of Q‐ and I‐tests performed slightly better for predicting crop yield than any single soil P test. This Q + I analysis explains why recent successes with I‐tests (e.g. DGT) were found for soils with larger P sorption than for those in the present study. This systematic evaluation of soil tests using a unique compilation of established field trials provides critical soil P values that are valid across Europe.Highlights We compared soil P tests for predicting crop response across contrasting soil types. No test was clearly superior to the others except for the oxalate extraction, which was generally poor. This study suggests that intensity tests do not perform markedly better than quantity tests. The evaluation of soil P tests on this unique dataset provided critical soil P values across Europe.

Modeling the hydration process of bean grains coated with carnauba wax

Edible waxes are widely used to maintain foodstuff until they are consumed. However, some products may be subjected to industrial procedures, such as hydration, prior to their consumption. Hydration of a material is a complex process, which aims to reconstitute the original characteristics of a product when in contact with a liquid phase. An important agricultural product that requires this procedure is beans. Thus, the purpose of this work is to study the hydration process of beans (cultivar BRSMG Majestoso) in different temperatures and concentrations of carnauba wax, which is applied on the product surface. Beans with initial moisture content of 0.2015, 0.1972 and 0.1745 (d.b.) corresponding to treatments 0 (witness), 1 (wax diluted in water in the ratio 1:1), and 2 (carnauba wax, without dilution) were used. Later, these samples were imbibed in distilled water at temperatures of 20, 30 and 40 ºC, for 15 h. The temperature and the carnauba wax influenced the water absorption rate. The Peleg model described satisfactory experimental data and the Mitscherlich model presented biased residual distribution. The constants C1 and C2 of the Peleg model exhibited opposite behaviors with increasing temperatures in the hydration process.

Modeling the hydration process of bean grains coated with carnauba wax

Edible waxes are widely used to maintain foodstuff until they are consumed. However, some products may be subjected to industrial procedures, such as hydration, prior to their consumption. Hydration of a material is a complex process, which aims to reconstitute the original characteristics of a product when in contact with a liquid phase. An important agricultural product that requires this procedure is beans. Thus, the purpose of this work is to study the hydration process of beans (cultivar BRSMG Majestoso) in different temperatures and concentrations of carnauba wax, which is applied on the product surface. Beans with initial moisture content of 0.2015, 0.1972 and 0.1745 (d.b.) corresponding to treatments 0 (witness), 1 (wax diluted in water in the ratio 1:1), and 2 (carnauba wax, without dilution) were used. Later, these samples were imbibed in distilled water at temperatures of 20, 30 and 40 ºC, for 15 h. The temperature and the carnauba wax influenced the water absorption rate. The Peleg model described satisfactory experimental data and the Mitscherlich model presented biased residual distribution. The constants C1 and C2 of the Peleg model exhibited opposite behaviors with increasing temperatures in the hydration process.