# Mathematical Modeling of Urea Reaction with Sulfuric Acid and Phosphoric Acid to Produce Ammonium Sulfate and Ammonium Dihydrogen Phosphate Respectively

^{*}

## Abstract

**:**

_{2}or MgO for the safety of the process. Numerical and simulation analyses were performed by studying the effect of the surrounding temperature, the ratio of acid reagent to urea concentration, the rate of addition, and the reaction rate to estimate the required time to achieve an optimum value of urea conversion into ammonium dihydrogen phosphate or ammonium sulfate as potential technological opportunities for by-product valorization. Full conversion of urea was achieved in about 10 h for reaction rates in the order of 1 × 10

^{−5}s

^{−1}when the ratio of H

_{2}SO

_{4}to CH

_{4}N

_{2}O was 1.5. When increasing the ratio to 10, the time required for full conversion was considerably reduced to 3 h.

## 1. Introduction

_{4}

^{+}) is liberated during amino acid catabolism and protonates the conjugate base of a stronger acid (HCO

_{3}

^{−}) in coupled reactions to produce urea, a less toxic compound which is subsequently transported by the blood to the kidneys and excreted in the urine [1].

_{3}, NH

_{4}

^{+}, and H

^{+}exist in equilibrium, which strongly depends on temperature, humidity, NH

_{4}

^{+}concentration, and pH. By decreasing the pH, the equilibrium is shifted to the left, which decreases the concentration of un-ionized ammonia in the liquid, attenuating the emission of NH

_{3}. According to de Oliveira Vilela et al. [8], at pH lower than 7, the H

^{+}ions increase the amount of nonvolatile NH

_{4}

^{+}while decreasing the quantity of volatile NH

_{3}. This is the basic principle behind acidification of waste slurries to diminish emissions of NH

_{3}during storage. Accordingly, increasing the temperature affects the dissociation constant (K

_{d}) and the fraction of un-ionized ammonia increases. Equation (7) represents the estimation of K

_{d}for dilute aqueous solutions at 25 °C [9]:

^{−1}MgO was necessary to achieve saturation and to increase the pH to a value of 10 and to inhibit the enzyme-catalyzed hydrolytic degradation of urea [10]. It was suggested to avoid higher temperatures, because this could lead to the reactivation of urease, as more than 40 °C decreased the saturation pH to <10.

_{2}SO

_{4}and H

_{2}O, respectively. They reported values of the rate constant between 2 × 10

^{−7}s

^{−1}(60 °C) and 4.15 × 10

^{−5}s

^{−1}(100 °C).

_{3}volatilization, Nitrogen dioxide accumulation and phytotoxicity problems have been associated with adverse effects on seed germination and seedling growth in the presence of NH

_{3}[13]. Some of the studies concerning the production of urea derivatives with application as fertilizers include research by Biskupsi et al. [14] who reported a continuous process for the manufacturing of urea-superphosphate and phosphate fertilizers based on the decomposition of phosphate raw materials with urea solutions (1.5–4 mol) in acid media, and using 90% to 100% with respect of the stoichiometric requirement to achieve complete decomposition of the rock phosphate. They stressed the importance of the possibility of occurrence of exothermic reactions, which could represent a threat to the safety of the process and leading eventually to injury, loss of life, or damage to instruments and property. The main reactions that gave rise to a safety threat were the exothermic urea hydrolysis reaction, including neutralization of released ammonia in the presence of mineral acids (H

_{2}SO

_{4}and H

_{3}PO

_{4}) and exothermic reactions that lead to the formation of amide compounds. Another study reported the reaction of H

_{3}PO

_{4}with urea and ammonia to obtain solid and liquid concentrated fertilizers based on ammonium and urea polyphosphates containing N, P, and K through ammoniation and condensation of urea phosphate, which favored the separation of the impurities such as iron, aluminum, calcium, magnesia, and fluorine that were present in the liquid phase. Accordingly, the products obtained could be utilized after dissolution in water, by spraying, injection, sprinkling, or dilution as other common and commercial fertilizers [15]. Additional works have reported the manufacturing of stable products containing mono and diurea sulfates free of sulfamic acid and ammonium sulfamate produced from the reaction between concentrated urea and sulfuric acid used in stoichiometric quantities [16].

_{3}PO

_{4}and H

_{2}SO

_{4}to promote the formation of a coagulated solid phase that can be used as a valuable organic nitrogen fertilizer.

#### 1.1. Role of Sulfur in Plants

^{−1}); however, in order to be absorbed through the root system it is required to be in the form of sulfate. The reaction of urea with sulfuric acid leads to the generation of ammonium sulfate, a compound that is generally used as fertilizer containing 21% of N and 24% of S, which is normally incorporated within the irrigation water by the drip system and also through direct soil application. It is quickly absorbed by the plant and can be used alone or in a mixture combined with other fertilizers. Once absorbed by the roots, it allows sulfur to be incorporated into the structure of o-acetylserine, which eventually contributes to the formation of L-cysteine by a complex variety of isoforms that are present in the cytosol, chloroplasts, and mitochondria [22]. The most significant sources of S are ammonium sulfate, single superphosphate, and potassium sulfate. Sulfur concentration in plant tissues varies between 0.1 and 0.5% [23]. For most common crops, SO

_{4}

^{−2}concentrations of 3–5 ppm are sufficient, except for rapeseed/canola, alfalfa, and broccoli, which require higher concentration. Deficiency of sulfur in the soil has been associated with reduced plant growth, occurrence of uniform yellowing of leaf tissue due to a lack of chlorophyll on younger leaves, reduction in N and P fixation by affecting nodule development and function, and accumulation of nonprotein N as NH

_{2}and NO

_{3}

^{−}in leaves. This affects the optimum N:S ratio needed for effective N use by rumen microorganisms and reduces food quality [24].

_{4}

^{+}to NO

_{3}

^{−}liberates H

^{+}and reduces the alkalinity of the soil. This can be explained thorough Equations (8)–(10). It can be seen that ammonium sulfate generates double the amount of H

^{+}in comparison to urea or ammonium nitrate and promotes acidification. As a result, this creates a favorable environment to keep other elements in solution, such as P, Fe, Zn, B, Cu, and Mn, especially in cases of alkaline soils, and increases its availability and further absorption.

#### 1.2. Importance of Phosphorous in Plant

## 2. Materials and Methods

#### 2.1. Stabilization of Urea with Sulfuric Acid

_{4})

_{2}SO

_{4}by following the conversion of urea. Regarding the ammonium sulfate concentration, the rate of formation is slow at lower times. It is possible to observe a change in the slope of the conversion curves as the ratio between the amount of sulfuric acid and initial concentration of urea increases. At lower values of the rate constant, the conversion continues to increase significantly as time continues.

#### 2.2. Estimation of Adiabatic Heat Difference

^{3}of wastewater containing diluted urine to which 217 kg of H

_{2}SO

_{4}was added. The concentration of diluted acid $\left({\mathrm{a}}_{{\mathrm{H}}_{2}{\mathrm{SO}}_{4}}\right)$ was estimated according to Equation (20) where ${\mathrm{m}}_{{\mathrm{H}}_{2}{\mathrm{SO}}_{4}}$ and ${\mathrm{m}}_{{\mathrm{H}}_{2}{}_{\mathrm{O}}}$. corresponds to the initial mass of H

_{2}SO

_{4}and wastewater respectively, and corresponds to 0.7%:

_{2}SO

_{4}$\left(972\mathrm{kJ}\xb7{\mathrm{kg}}^{-1}\right)$,$\mathrm{C}\mathrm{p}$ is the specific heat of water $\left(4.18\mathrm{kJ}\xb7{\mathrm{kg}}^{-1}\xb7{\mathrm{K}}^{-1}\right)$. This leads to a value of $\Delta \mathrm{t}=1.63\xb0\mathrm{C}$.

#### 2.3. Neutralization Using Ca(OH)_{2}

_{2}SO

_{4}need to be neutralized with Ca(OH)

_{2}, we proceed to estimate the adiabatic neutralization with Ca(OH)

_{2}, which is described in the following reaction:

_{2}corresponds to ${\left(\Delta \mathrm{H}\right)}_{\mathrm{r}}=-214.22\times {10}^{3}\mathrm{kJ}\xb7{\mathrm{kmol}}^{-1}$. Considering the total amount of ${\mathrm{H}}_{2}{\mathrm{SO}}_{4}$, and the total mass of wastewater, the estimated temperature difference $\left(\Delta \mathrm{t}\right)$ is thus equal to 3.7 °C. As a result, if we consider the heat of neutralization and the heat of dilution of H

_{2}SO

_{4}, the value of temperature difference would correspond to 5.33 °C. This value is considered safe for practical applications.

#### 2.4. Neutralization Using MgO

_{2}

^{+}and Mg

^{+2}trade places to produce the ionic compound MgSO

_{4}, and water. The combined reaction enthalpy of all three reactions $\left({\Delta}_{\mathrm{H}}\right)$ corresponds to $-233.251\mathrm{kJ}\xb7{\mathrm{mol}}^{-1}$. The energy balance can be calculated following Equation (28), where $\dot{\mathrm{n}}$ is the molar flow rate of ${\mathrm{H}}_{2}{\mathrm{SO}}_{4}$ ($\mathrm{mol}\xb7{\mathrm{s}}^{-1}$), $\mathrm{m}$ the total amount of the solution(mol), $\mathrm{k}$ the heat transfer coefficient $\left(\mathrm{kJ}/\left(\mathrm{s}\xb7{\mathrm{m}}^{2}\xb7\mathrm{K}\right)\right)$, $\mathrm{S}$ the surface of the tank $\left({\mathrm{m}}^{2}\right)$, ${\mathrm{t}}_{0}$ the initial temperature (°C), $\mathrm{t}$ the final temperature (°C), $\mathrm{Cp}$ the specific heat capacity of the blend $\mathrm{kJ}\xb7{\left(\mathrm{kmol}\xb7\mathrm{K}\right)}^{-1}$, and $\tau $ the time of the reaction (s).

^{−1}, $\mathrm{S}:4.7{\mathrm{m}}^{2}$, $\mathrm{m}:46.16\mathrm{kmol}$, $\mathrm{k}:0.01\mathrm{kJ}\xb7{\left(\mathrm{s}\xb7{\mathrm{m}}^{2}\xb7\mathrm{K}\right)}^{-1}$, ${\mathrm{t}}_{0}:10\xb0\mathrm{C}$, $\mathrm{Cp}:134.5\mathrm{kJ}\xb7{\left(\mathrm{kmol}\xb7\mathrm{K}\right)}^{-1}$, which were obtained considering a 20% solution of MgSO

_{4}. The surrounding temperature of 10 °C simulates the case of performing the experiment during winter time, whereas hotter temperatures of about 30 °C represent average summer temperatures.

_{2}SO

_{4}on time and temperature was also studied for a range of 0 to 10 and is presented in Figure 4. This represents how fast the sulfuric acid is added and ultimately also affects the time the reaction will finish. The faster the H

_{2}SO

_{4}is added, the sooner the reaction will take place.

^{−1}, when the addition of H

_{2}SO

_{4}is 1x (5 kg·min

^{−1}), using 60 kg of ${\mathrm{H}}_{2}{\mathrm{SO}}_{4}$ ($0.61\mathrm{kmol})$ since $\dot{\mathrm{n}}=8.5\times {10}^{-4}\mathrm{kmol}\xb7{\mathrm{s}}^{-1}$, the amount of time required corresponds to $0.61\mathrm{kmol}/8.5\times {10}^{-4}\mathrm{kmol}\xb7{\mathrm{s}}^{-1}=717.6\mathrm{s}=11.96\mathrm{min}$. The required time is substituted in Equation (31), where constants A and B are determined as follows:

#### 2.5. Stabilization of Urea with Phosphoric Acid

^{−1}). Besides its use as fertilizer, ammonium dihydrogen phosphate also finds application during the fermentation of wine grapes and for the production of wine vinegar using apple cider to complement nutritional requirements for acetic acid bacteria that produce vinegar with up to 15% acetic acid [26]. We consider that the following reaction takes place:

_{3}PO

_{4}ratios. When the amount of H

_{3}PO

_{4}is 1.5 times higher, there is no full urea conversion. This shows that complete conversion of urea can be achieved using concentrations of H

_{3}PO

_{4}which are at least 2.5 times higher than the initial concentration of urea. However, this is achieved at longer times (>30 h). Therefore, fast conversion can be obtained with higher concentrations of the acid. To exemplify this, numerical analysis was performed considering concentrations which are 5, 8, and 10 times higher than the initial concentration of urea. This is represented in Figure 5b).

_{3}PO

_{4}and urea was developed using High Performance Liquid Chromatography. Experiments were realized between phosphoric acid and urea in a molar ratio using $0.66{\mathrm{mol}\mathrm{H}}_{3}{\mathrm{PO}}_{4}/\mathrm{mol}\mathrm{urea}$, $1.22{\mathrm{mol}\mathrm{H}}_{3}{\mathrm{PO}}_{4}/\mathrm{mol}\mathrm{urea}$ and $1.5{\mathrm{mol}\mathrm{H}}_{3}{\mathrm{PO}}_{4}/\mathrm{mol}\mathrm{urea}$. A method was developed for reliable separation and identification of both compounds. The main challenge was that both compounds eluted at a similar retention time, which compliated their quantification. We used a Shim-Pack VP-ODS (250 mm × 4.6 mm) column. After several approaches, the best method that separated both compounds was using a temperature of 30 °C in isocratic mode with a mixture of Acetonitrile: H

_{2}O (5:95 v/v) as mobile phase, with a flow of 0.4 mL·min

^{−1}using a UV and RI detector. A calibration curve was prepared between 0.25% and 2% for both detectors. Urea (in solid form) and H

_{3}PO

_{4}were of analytical grade and obtained from PENTA s.r.o. company. Analytical measurements using HPLC showed decrement in H

_{3}PO

_{4}concentration but not in urea concentration, as presented in Figure 6. This indicates that the value of rate constant for the reaction condition is of lower magnitude (<1 × 10

^{−5}).

#### 2.6. Sedimentation Experiments

^{3}of wastewater per day, and the average value of COD is 6000 mg·L

^{−1}. The main task is to design a settling system that would allow the desired separation of the solid phase so that the separated (upper) layer decreases in chemical oxygen demand (COD). After calculation of the dimensionless criteria of Lyashenko (Ly) and Reynolds (Re), we found parameters for estimating the settling rate. To perform the experiment, 100 mL of rinsing water from a dairy factory with pH 7, dry matter content of 0.9%, and COD value of 2816 mg·L

^{−1}were mixed with 2 g MgO for 60 min and then poured into a 100 mL graduated measuring cylinder (26 mm diameter) to reach a pH of 11, and the rate of sedimentation was monitored at different times. The solution was then neutralized with 0.52 g of H

_{3}PO

_{4}topH 7 and the rate of sedimentation of the clarified liquid was measured again. The average rate of settlement using MgO was $2.41\times {10}^{-5}\mathrm{m}\xb7{\mathrm{s}}^{-1}$ and $7.3\times {10}^{-5}\mathrm{m}\xb7{\mathrm{s}}^{-1}$ using H

_{3}PO

_{4}. The final amount of dry matter of the clarified layer was 0.7% and 2112 mg·L

^{−1}of COD. The sediment settling rate was significantly higher after neutralization. In a second settlement experiment, 100 mL of rinsing water was stirred with 1 g Ca(OH)

_{2}for 15 min (pH 11) and then neutralized with 2 g of H

_{3}PO

_{4}. Sediment deposition in the measuring cylinder was monitored. Dry matter of the upper layer after neutralization was 1.0%, the COD value of the upper layer was 1440 mg ·L

^{−1}, and the nitrogen content in the upper layer was 0.055%. The settling rate of the experiments performed is presented in Figure 7. These experiments showed the potential of using a sedimentation system (i.e., Ca(OH)

_{2}-H

_{2}SO

_{4}) to reduce up to 50% of the initial COD consumption. This has the additional benefit of contributing to a potential reuse of water, particularly in operations such as floor cleaning and sanitization.

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Bean, E.S.; Atkinson, D.E. Regulation of the Rate of Urea Synthesis in Liver by Extracellular pH. J. Biol. Chem.
**1984**, 259, 1552–1559. [Google Scholar] [CrossRef] - Bristow, A.W.; Whitehead, D.C.; Cockburn, J.E. Nitrogenous constituents in the urine of cattle, sheep and goats. J. Sci. Food Agric.
**1992**, 59, 387–394. [Google Scholar] [CrossRef] - Gratzfeld, D.; Heitkämper, J.; Debailleul, J.; Olzmann, M. On the influence of water on urea condensation reactions: A theoretical study. Z. Phys. Chem.
**2020**, 234, 1311–1327. [Google Scholar] [CrossRef] - Mikkelsen, R. Biuret in urea fertilizer. Fertil. Res.
**1990**, 26, 311–318. [Google Scholar] [CrossRef] - Seifan, M.; Sarabadani, Z.; Berenjian, A. Development of an Innovative Urease-Aided Self-Healing Dental Composite. Catalysts
**2020**, 10, 84. [Google Scholar] [CrossRef] [Green Version] - Sigurdarson, J.J.; Svane, S.; Karring, H. The molecular processes of urea hydrolysis in relation to ammonia emissions from agriculture. Rev. Environ. Sci. Biotechnol.
**2018**, 17, 241–258. [Google Scholar] [CrossRef] [Green Version] - Randall, D.G.; Krähenbühl, M.; Köpping, I.; Larsen, T.A.; Udert, K.M. A novel approach for stabilizing fresh urine by calcium hydroxide addition. Water Res.
**2016**, 95, 361–369. [Google Scholar] [CrossRef] [Green Version] - Vilela, M.O.; Gates, R.S.; Souza, C.F.; Teles Junior, C.G.S.; Sousa, F.C. Nitrogen transformation stages into ammonia in broiler production: Sources, deposition, transformation, and emission into the environment. Dyna
**2020**, 87, 221–228. [Google Scholar] [CrossRef] - Simha, P.; Friedrich, C.; Randall, D.G.; Vinnerås, B. Alkaline Dehydration of Human Urine Collected in Source-Separated Sanitation Systems Using Magnesium Oxide. Front. Environ. Sci.
**2021**, 8, 619901. [Google Scholar] [CrossRef] - Krajewska, B.; Ureases, I. Functional, catalytic and kinetic properties: A review. J. Mol. Catal. B Enzym.
**2009**, 59, 9–21. [Google Scholar] [CrossRef] - Shaw, W.H.R.; Bordeaux, J.J. The decomposition of Urea in aqueous media. J. Am. Chem. Soc.
**1955**, 77, 4729–4733. [Google Scholar] [CrossRef] - Hocking, M.B. Ammonia, Nitric Acid and Their Derivatives. In Handbook of Chemical Technology and Pollution Control; Academic Press: Cambridge, MA, USA, 2006; pp. 321–364. [Google Scholar]
- Bremner, J.M. Recent research on problems in the use of urea as a nitrogen fertilizer. Fertil. Res.
**1995**, 42, 321–329. [Google Scholar] [CrossRef] - Biskupski, A.; Mieczyslaw, B.; Dawidowicz, M.; Igras, J.; Kowalski, Z.; Kruk, J.; Malinowski, P.; Mozenski, C.; Myka, A.; Rusek, P.; et al. Method and Plant for Continuous Manufacture of Granular USP Nitrogen and Phosphate Type Fertilizers and Products on Their Basis. EP Patent No. 2774907-A2, 10 September 2014. [Google Scholar]
- Gittenait, M. Reaction of Phosphoric Acid, Urea, and Ammonia. U.S. Patent No. 3713802A, 30 January 1973. [Google Scholar]
- Young, D.C. Method of Producing Urea-Sulfuric Acid Reaction Products. U.S. Patent No. 4397675A, 5 November 1981. [Google Scholar]
- Tischer, S.; Börnhorst, M.; Amsler, J.; Schoch, G.; Deutschmann, O. Thermodynamics and reaction mechanism of urea decomposition. Phys. Chem. Chem. Phys.
**2019**, 21, 16785–16797. [Google Scholar] [CrossRef] [Green Version] - Schaber, P.M.; Colson, J.; Higgins, S.; Thielen, D.; Anspach, B.; Brauer, J. Thermal decomposition (pyrolysis) of urea in an open reaction vessel. Thermochim. Acta
**2004**, 424, 131–142. [Google Scholar] [CrossRef] - Wang, D.; Dong, N.; Niu, Y.; Hui, S. A Review of Urea Pyrolysis to ProduceNH3 Used for NOx Removal. J. Chem.
**2019**, 2019, 6853638. [Google Scholar] [CrossRef] [Green Version] - Tempelman, C.; Warning, N.; van Geel, J.; van Bommel, F.; Lamers, K.; Hashish, M.; Schippers, J.; Gundlach, M.; Luijendijk, E. An Infrared and thermal decomposition study on solid deposits originating from heavy-duty diesel SCR urea injection fluids. Reactions
**2020**, 1, 72–88. [Google Scholar] [CrossRef] - Chen, J.P.; Isa, K. Thermal Decomposition of Urea and Urea Derivatives by Simultaneous TG/(DTA)/MS. J. Mass Spectrom. Soc. Jpn.
**1998**, 46, 299–303. [Google Scholar] [CrossRef] - Alvarez, C.; Calo, L.; Romero, L.C.; García, I.; Gotor, C. An O-acetylserine(thiol)lyase homolog with L-cysteine desulfhydrase activity regulates cysteine homeostasis in Arabidopsis. Plant Physiol.
**2010**, 152, 656–669. [Google Scholar] [CrossRef] [Green Version] - Lucheta, A.R.; Lambais, M.R. Sulfur in agriculture. Rev. Bras. Ciência Solo
**2012**, 36, 5. [Google Scholar] [CrossRef] [Green Version] - Havlin, J.L.; Tisdale, S.L.; Nelson, W.L.; Beaton, J.D. Soil Fertility and Fertilizers, 8th ed.; Pearson: London, UK, 2017. [Google Scholar]
- Fotyma, M.; Hammond, L.; Kesik, K. Suitability of North Carolina natural phosphate to Polish agriculture, in Fertilizers and Environment. Dev. Plant Sail Sci.
**1996**, 66, 151–154. [Google Scholar] - Ebner, H.; Wilsberg, S.S. Vinegar, Acetic Acid Production. In Encyclopedia of Bioprocess Technology: Fermentation, Biocatalysis and Bioseparation; Wiley-Interscience: Hoboken, NJ, USA, 1999; p. 2798. [Google Scholar]
- Adam, N.; Mitchell, J.L.; Pickering, K.D. Development of low-toxicity urine stabilization for spacecraft water recovery systems. In Proceedings of the International Conference on Environmental Systems (ICES), Vail, CO, USA, 14–18 July 2013. [Google Scholar]
- Kolomazník, K.; Langmaier, F.; Mládek, M.; Janáčová, D.; Taylor, M.M. Experience in Industrial Practice of Enzymatic Dechromation of Chrome Shavings. J. Am. Leather Chem. Assoc.
**2000**, 95, 55–63. [Google Scholar]

**Figure 1.**(

**a**) Time evolution of urea conversion according to the corresponding values of reaction rate. (+) $1.5\times {10}^{-5}{s}^{-1}$; (∘) $2\times {10}^{-5}{\mathrm{s}}^{-1};$ (∗) $3\times {10}^{-5}{\mathrm{s}}^{-1};$ (●) $4.5\times {10}^{-5}{\mathrm{s}}^{-1}$; when the initial concentration of H

_{2}SO

_{4}is 1.5 times higher than the initial concentration of urea. (

**b**) Numerical simulation of urea conversion at different reaction rates values when the initial concentration of H

_{2}SO

_{4}is 3 times higher than the initial concentration of urea. (

**c**) Numerical simulation of urea conversion considering different possible values of reaction rates when the initial concentration of H

_{2}SO

_{4}is half the initial concentration of urea. (

**d**) Numerical simulation of urea conversion at different values of ${\mathrm{C}}_{{\left({\mathrm{H}}_{2}{\mathrm{SO}}_{4}\right)}_{0}}/{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}$ ratio with a rate constant vale equal to $4.5\times {10}^{-5}$ 0.001 s

^{−1}.

**Figure 2.**Variation of the reaction temperature with surrounding temperature (10 °C < t

_{0}< 30 °C) during the neutralization step using MgO.

**Figure 3.**(

**a**) Comparison of the reaction blend temperature considering the influence of input flow addition (O) and constant volume (◊), (

**b**) error variation presented as percentage, $0\le \tau \le 60\mathrm{min}$.

**Figure 4.**(

**a**) Influence of rate of H

_{2}SO

_{4}addition on temperature and reaction time; (

**b**) selected values of addition rate: (□) rate: 5x (25 kg·min

^{−1}); (O) rate: 3x (15 kg·min

^{−1}); (+) rate: 2x (10 kg·min

^{−1}); (●) rate: 1x (5 kg·min

^{−1}). $0\le \mathrm{\tau}\le 800\mathrm{s}$.

**Figure 5.**(

**a**) Plot of amount of urea reacted vs. time considering values of phosphoric acid concentration between 1 and 2.5. (

**O**) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=2.5,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$; (

**◊**) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=1.5,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$; (□) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=1,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$. (

**b**) Plot of ${\mathrm{X}}_{{\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}}$ vs. time considering values of phosphoric acid concentration between 5 and 10. (Δ) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=5,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$; (

**×**) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=8,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$; (●) ${\mathrm{C}}_{{\left({\mathrm{H}}_{3}{\mathrm{PO}}_{4}\right)}_{0}}=10,{\mathrm{C}}_{{\left({\mathrm{CH}}_{4}{\mathrm{N}}_{2}\mathrm{O}\right)}_{0}}=1,\mathrm{k}=1\times {10}^{-5}$.

**Figure 6.**Variation of H

_{3}PO

_{4}(

**a**) and urea concentration (

**b**) in experiments performed using a molar ratio of 0.66 mol H

_{3}PO

_{4}/mol urea (♦), 1.22 mol H

_{3}PO

_{4}/mol urea (▪), 1.5 mol H

_{3}PO

_{4}/mol urea (●).

**Figure 7.**Average settling rate of suspended particles after sedimentation using MgO, MgO-H

_{3}PO

_{4}, and Ca(OH)

_{2}-H

_{2}SO

_{4}.

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Beltrán-Prieto, J.C.; Kolomazník, K.
Mathematical Modeling of Urea Reaction with Sulfuric Acid and Phosphoric Acid to Produce Ammonium Sulfate and Ammonium Dihydrogen Phosphate Respectively. *Energies* **2021**, *14*, 8004.
https://doi.org/10.3390/en14238004

**AMA Style**

Beltrán-Prieto JC, Kolomazník K.
Mathematical Modeling of Urea Reaction with Sulfuric Acid and Phosphoric Acid to Produce Ammonium Sulfate and Ammonium Dihydrogen Phosphate Respectively. *Energies*. 2021; 14(23):8004.
https://doi.org/10.3390/en14238004

**Chicago/Turabian Style**

Beltrán-Prieto, Juan Carlos, and Karel Kolomazník.
2021. "Mathematical Modeling of Urea Reaction with Sulfuric Acid and Phosphoric Acid to Produce Ammonium Sulfate and Ammonium Dihydrogen Phosphate Respectively" *Energies* 14, no. 23: 8004.
https://doi.org/10.3390/en14238004