Ternary Equations Research Articles

Separation of an azeotropic mixture through the use of organic solvent nanofiltration

Experimental data, along with theoretical modelling, are combined to explore the potential for the separation of an azeotropic polar/non-polar solvent mixture through an OSN membrane. A mixture of toluene/butanol was used for the model development, and then applied to toluene/ethanol, and toluene/heptane mixtures. Experimental data was obtained at various testing conditions to validate simulation results from a non-ideal solution diffusion transport model incorporating the Flory–Huggins ternary equations. Accounting for the effects of non-ideality in the feed and permeate solutions generated a better fit to the experimental data. The system of equations was solved using parameters obtained from mono component membrane data only. The model developed can be used as a pre screening check for the separation of a potential solvent mixture that an OSN membrane can offer prior to embarking upon an extensive experimental program.

OBSERVATIONS ON THE INTEGRAL SOLUTIONS OF TERNARY QUADRATIC EQUATION X2 + Y2 =Z2 - 8

This paper illustrates the process of obtaining different sets of non-zero distinct integer solutions to the non-homogeneous ternary quadratic Diophantine equations given by X2 + Y2 =Z2 - 8

Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

Abstract: The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

Inter-influence of Temperature-time-reflectivity in the Coal Metamorphism Thermodynamics Equation (CMTE)

The influence and control of time and temperature on coalification are comprehensive. In order to quantitatively analyze the thermal kinetics of coal metamorphosis process by temperature and time, the qualitative analysis of coal metamorphosis thermodynamic equation (CMTE) based on Wu Chonglong was further verified from the mathematical point of view. Wu Chong Long's Coal Metamorphism Thermodynamics Equation (CMTE) is a ternary equation containing coal forming period or metamorphic age (time), metamorphic temperature (temperature) and metamorphic degree (reflectivity). The temperature calculated by fixing time and reflectivity is the lowest theoretical metamorphic temperature. The reflectivity calculated by fixing time and temperature is the maximum theoretical metamorphic degree. The “time-temperature ratio” is defined as the equivalent of extending million years to increasing 1°C temperature regarding coal metamorphism. This ratio is used to compare the impact of extending time or increasing temperature on improving the metamorphic degree and its significance. The results showed that in the same coal forming period, the ratio is decreased with the reflectivity increasing; at the same reflectivity, the ratio decreased with younger coal forming period. At R_o=6.1%, the calculated ratio is 3.67, which means, at R_o=6.1%, increasing 1°C is equaled extending 3.67 million.

A New Numerical Algorithm of Three Parameter Generalized Extreme Value Distribution of Wind Speed

Maximum likelihood estimation method is used to solve the problem of parameter estimation of three-parameter generalized extreme value distribution. Based on the theory of order reducing，a new numerical algorithm is presented to resolve the problem of maximum likelihood estimation of three-parameter generalized extreme value distribution．Firstly，the shape parameter is assumed to be known and ternary likelihood equations can be transferred into binary ones that are solved with the dichotomy．And then，scale and location parameters are the functions of shape parameter. Further，the maximum likelihood function is described as a unitary function of shape parameter. The optimal estimation of shape parameters can be obtained by applying dichotomy again.

On Some Ternary Diophantine Equations of Signature (p, p, k)

In this paper, we study some ternary Diophantine equations of the form $$Aa^n+Bb^n=Cc^m$$ , where $$m \in \{2,3\} $$ , and $$n\geqslant 7 $$ is a prime. In fact, we completely solve some particular cases: $$A=5^{\alpha }, ~B=64,~ C=3$$ , when $$m=2$$ ; $$A=2^{\alpha },~ B=27, ~C \in \{7,13\}$$ , when $$m=3$$ .

Study on Residual Stress of Welded Hoop Structure

Residual stress is inevitable during welding, which will greatly affect the reliability of the structure. The purpose of this paper was to study the residual stress of the hoop structure caused by the cooling shrinkage of the weld when the outer cylinder was wrapped and welded under the condition of the existing inner cylinder. In this paper, the “method of killing activating elements” of ANSYS was used to simulate the three-dimensional finite element of the hoop structure. In the case of applying interlayer friction, the welding-forming process and welding circumferential residual stress of the hoop structure were analyzed. The blind hole method was used to test the residual stress distribution of the hoop structure, and the test results were compared with the finite element simulation results to verify the reliability of the simulation calculation method and the reliability of the calculation results. Then, the influence factors of the maximum welding residual stress of the hoop structure were studied. The results show that the maximum residual stress of the outer plate surface of the hoop structure decreases with the increase of the welding energy, the thickness of the laminate, the width of the weld seam, the welding speed, and the radius of the container. Based on the results of numerical simulation, the ternary first-order equations of the maximum residual stress of the hoop structure with respect to the welding speed, the thickness of the laminate, and the width of the weld seam were established. Then, the optimal welding parameters were obtained by optimizing the equations, which provided an important basis for the safe use and optimal design of the welding hoop structure.

ON THE TERNARY QUADRATIC DIOPHANTINE EQUATION 6z2 = 6x2 – 5y2

The ternary homogeneous quadratic equation given by 6z2 = 6x2 -5y2 representing a cone is analyzed for its non-zero distinct integer solutions. A few interesting relations between the solutions and special polygonal and pyramided numbers are presented. Also, given a solution, formulas for generating a sequence of solutions based on the given solutions are presented.

Linear equations over multiplicative groups in positive characteristic II

Not long ago it was observed that the equation x+y−z=1 has an unexpected “doubly infinite family” of solutions, parametrized by two independent Frobenius actions, over the group G generated by t and 1−t in Fp(t). In an earlier paper we already put that observation into context by finding all such solutions. Here we extend our work to the full (thus not necessarily “prime-to-p”) radical or division group of G in the algebraic closure of Fp(t). In particular we find new families now parametrized by elements of the algebraic closure of Fp.This seems to be the first example of the complete solution of a ternary equation in the context of Mordell–Lang and Manin–Mumford.

Simplification of Ternary Function using Variable Entered Mapping Technique

This paper describes the steps for minimization of ternary function i.e. three level logic using variable entered mapping technique. Here we applied VEM technique successfully to the ternary function and result is verified using truth table of given ternary equations. We also implemented simplified expression using decoder and ternary gates as well as using ternary multiplexer. It will be more difficult as radix of the number system increases to get more difficult in minimization and to design circuit. In this paper we successfully applied VEM technique to ternary system, the proposed technique is developed for the ternary logic function simplification. It incorporates all designed rules for ternary logic system design and gives the output in the form of Sum-of-Product (SOP) terms. Generally VEM technique is used for the minimization of binary function so the rule used here are different than binary logic. Output equation of ternary digital system is in the form of F = F2 + 1. (F1) Where, F2 = 2‟s minterms and F1 = 1‟s minterms.

A discrete two-level model for charge pricing of electric vehicles based on evolution game theory

According to the present vehicle-to-grid situation and existing pricing strategy in China, this paper proposed a two-level evolution game model for pricing analysis problem. The model contains two level game models: the up level game operated between monopoly electricity sale enterprise and small electricity sale enterprise; the second level is game between electricity sale enterprise and EV user; a pay-off matrix in ternary nonlinear equation form is built to descries the evolution process under different strategies. After the Linearization of nonlinear equations, the evolution stable strategy (ESS) is deduced for evolution game model. Analysis work is carried out depending on the ESS. Finally, the numerical results are given by computer simulation. Several suggestions of pricing strategies in China’s electricity market are obtained.

The ternary exponential Diophantine equation concerning Pythagorean triplets

Let r be a positive integer with r > 1, and let m be a positive even integer. Further let a=|V(m, r)|, b=|U (m, r)| and c=m2+1, where V (m, r) + U (m, r)√−1=(m+√−1)r . In this paper, using the Gel′ fond-Baker method, we prove that if 2 | r and m>max(1015, 2r3), then the equation ax+by=cz has only the positive integer solution (x, y, z)=(2, 2, r).

Shifted powers in binary recurrence sequences

AbstractLet {uk} be a Lucas sequence. A standard technique for determining the perfect powers in the sequence {uk} combines bounds coming from linear forms in logarithms with local information obtained via Frey curves and modularity. The key to this approach is the fact that the equation uk = xn can be translated into a ternary equation of the form ay2 = bx2n + c (with a, b, c ∈ ℤ) for which Frey curves are available. In this paper we consider shifted powers in Lucas sequences, and consequently equations of the form uk = xn+c which do not typically correspond to ternary equations with rational unknowns. However, they do, under certain hypotheses, lead to ternary equations with unknowns in totally real fields, allowing us to employ Frey curves over those fields instead of Frey curves defined over ℚ. We illustrate this approach by showing that the quaternary Diophantine equation x2n±6xn + 1 = 8y2 has no solutions in positive integers x, y, n with x, n > 1.

Separation of N2O and CO2 using Room-Temperature Ionic Liquid [bmim][Ac

We have developed a ternary equation of state (EOS) model for the N2O/CO2/1-butyl-3-methylimidazolium acetate ([bmim][Ac]) system in order to understand the separation of N2O and CO2 using room-temperature ionic liquids (RTILs). The present model is based on a generic RK (Redlich-Kwong) EOS, with empirical interaction parameters for each binary system. The interaction parameters have been determined using our measured VLE (vapor-liquid-equilibrium) data for N2O/[bmim][Ac] and literature data for CO2/[bmim][Ac] and N2O/CO2. The binary EOS model for the N2O/[bmim][Ac] system correctly predicted the liquid-liquid phase separation found in VLLE experiments. The validity of the ternary EOS model has been checked by conducting VLE experiments for the N2O/CO2/[bmim][Ac] system over a temperature range from 296 to 313 K. With this EOS model, solubility (VLE) behavior has been calculated for various (T, P, and feed compositions) conditions. Over a range of N2O/CO2 feed ratios, the N2O/CO2 gas selectivity [α N 2 O/CO 2 = (y N 2 O /x N 2 O )/(y CO 2 /x CO 2 )] increases by at least 5 orders of magnitude when adding [bmim][Ac] (α = 1 × 102 to 1 × 107), compared with the absence of the ionic liquid (α = 0.96 to 0.98). The addition of [bmim][Ac] may provide a practical means of separating CO2 and N2O. Supplemental materials are available for this article. Go to the publisher's online edition of Separation Science and Technology to view the free supplemental file.

Binomial Thue equations, ternary equations and power values of polynomials

We explicitly solve the equation Ax n − By n = ±1 and, along the way, we obtain new results for a collection of equations Ax n − By n = z m with m ∈ {3, n}, where x, y, z, A, B, and n are unknown nonzero integers such that n ≥ 3, AB = p α q β with nonnegative integers α and β and with primes 2 ≤ p < q < 30. The proofs depend on a combination of several powerful methods, including the modular approach, recent lower bounds for linear forms in logarithms, somewhat involved local considerations, and computational techniques for solving Thue equations of low degree.

Separation of N2O and CO2Using Room-Temperature Ionic Liquid [bmim][BF4

We have developed a ternary equation of state (EOS) model for the N(2)O/CO(2)/1-butyl-3-methylimidazolium tetrafluoroborate ([bmim][BF(4)]) system in order to understand separation of these gases using room-temperature ionic liquids (RTILs). The present model is based on a generic RK (Redlich-Kwong) EOS, with empirical interaction parameters for each binary system. The interaction parameters have been determined using our measured VLE (vapor-liquid equilibrium) data for N(2)O/[bmim][BF(4)] and CO(2)/[bmim][BF(4)] and literature data for N(2)O/CO(2). The binary EOS models for the N(2)O/[bmim][BF(4)] and CO(2)/[bmim][BF(4)] systems correctly predicted the liquid-liquid phase separation found in VLLE experiments. The validity of the ternary EOS model has been checked by conducting VLE experiments for the N(2)O/CO(2)/[bmim][BF(4)] system over a range in temperature from 296 to 315 K. With this EOS model, solubility (VLE) behavior has been calculated for various (T, P, and feed compositions) conditions. For both large and small N(2)O/CO(2) feed ratios, the N(2)O/CO(2) gas selectivity [α(N(2)O/CO(2)) = (y(N(2)O)/x(N(2)O))/(y(CO(2))/x(CO(2)))] is α = 1.4-1.5, compared with (α = 0.96-0.98) in the absence of ionic liquid. While the concentration of the ionic liquid does not affect the selectivity, the addition of an ionic liquid provides the only practical means of separating CO(2) and N(2)O.

STUDIES OF DYNAMIC VISCOSITY AND GIBBS ENERGY OF ACTIVATION OF BINARY AND TERNARY MIXTURES OF BENZYLCHLORIDE + NITROBENZENE + ISOBUTANOL FROM T = (288.15 TO 313.15) K

Densities and viscosities of ternary mixtures of benzylchloride + nitrobenzene + isobutanol and three corresponding binary mixtures (benzyl chloride + isobutanol, benzyl chloride + nitrobenzene, and nitrobenzene + isobutanol) have been determined over the whole concentration at a temperature range of 288.15 to 313.15 K under atmospheric pressure. Experimental data of resulting mixtures were used to calculate viscosity deviations, Δln η, and excess energies of activation of viscous flow, ΔG*E, of the ternary system. The calculated data for binary mixtures have been fitted by the Redlich-Kister equation to determine the appropriate coefficients. In order to determine the coefficients of ternary data, Cibulka, Singh, and Nakata equations were used. The results have also provided a test for the Grunberg and Nissan equation for correlating the dynamic viscosities of binary and ternary mixtures with mole fractions.

Separation of CO2 and H2S Using Room-Temperature Ionic Liquid [bmim][MeSO4

We have developed a ternary equation of state (EOS) model for the CO 2 /H 2 S/1-butyl-3-methylimidazolium methylsulfate ([bmim][MeSO 4 ]) system to understand separation of these gases using room-temperature ionic liquids (RTILs). The present model is based on a modified RK (Redlich-Kwong) EOS, with empirical interaction parameters for each binary system. The interaction parameters have been determined using our measured VLE (vapor-liquid equilibrium) data for H 2 S/[bmim][MeSO 4 ] and literature data for CO 2 / [bmim][MeSO 4 ] and CO 2 /H 2 S. Due to limited VLE data for H 2 S/[bmim][MCSO 4 ], we have also used VLLE (vapor—liquid—liquid equilibrium) measurements to construct the EOS model. The VLLE for H 2 S/ [bmim][MeSO 4 ] is highly asymmetric with a narrow (mole fraction H 2 S between 0.97 and 0.99) LLE gap which is the first such case reported in the literature and exhibits Type V phase behavior, according to the classification of van Konynenburg and Scott. The validity of the ternary EOS model has been checked by conducting VLE experiments for the CO 2 /H 2 S/[bmim][MeSO 4 ] system. With this EOS model, solubility (VLE) behavior has been calculated for various (T, P, and feed compositions) conditions. For large (9/1) and intermediate (1/1) CO 2 /H 2 S feed ratios, the CO 2 /H 2 S gas selectivity is high (10 to 13, compared with <4.5 in the absence of ionic liquid) and nearly independent of the amount of ionic liquid added. For small CO 2 /H 2 S mole ratios (1/9) at 298.15 K, increasing the ionic liquid concentration increases the CO 2 /H 2 S gas selectivity from about 7.4 to 12.4. For high temperature (313.15 K) and large CO 2 /H 2 S feed ratios, the addition of the ionic liquid provides the only means of separation because no VLE exists for the CO 2 /H 2 S binary system without the ionic liquid.

Separation of CO 2 and H 2S using room-temperature ionic liquid [bmim][PF 6

We have developed a ternary equation of state (EOS) model for the CO 2/H 2S/1-butyl-3-methylimidazolium hexafluorophosphate ([bmim][PF 6]) system in order to understand the separation of these gases using room-temperature ionic liquids (RTILs). The present model is based on a generic RK (Redlich-Kwong) EOS, with empirical interaction parameters for each binary system. These interaction parameters have been determined using our previously measured VLE (vapor–liquid equilibrium) data for CO 2/[bmim][PF 6] and literature data for H 2S/[bmim][PF 6] and CO 2/H 2S. VLLE (vapor–liquid–liquid equilibrium) measurements have been made and validate EOS model predictions which suggest that the CO 2/[bmim][PF 6] and H 2S/[bmim][PF 6] systems are Type V phase behavior, according to the classification of van Konynenburg and Scott. The validity of the ternary EOS model calculations has also been checked by conducting VLE experiments for the CO 2/H 2S/[bmim][PF 6] system. With this EOS model, isothermal ternary phase diagrams and solubility (VLE) behavior have been calculated for various ( T, P, and feed compositions) conditions. The CO 2/H 2S gas selectivity is nearly independent of the amount of ionic liquid addition and ranged from about 3.2 to 4.0. For large CO 2/H 2S mole ratios (9/1) at 298.15 K, the addition of the ionic liquid increases the CO 2/H 2S gas selectivity from about 1.2 to 3.7. For high temperature (333.15 K) and high CO 2/H 2S feed ratios, the addition of the ionic liquid provides the only means of separation because no VLE exists for the CO 2/H 2S binary system without the ionic liquid.

Separation of Carbon Dioxide and Sulfur Dioxide Using Room-Temperature Ionic Liquid [bmim][MeSO4

A ternary equation of state (EOS) model for the CO2/SO2/1-butyl-3-methylimidazolium methyl sulfate ([bmim][MeSO4]) system has been developed in order to gain further our understanding of capturing and enhanced gaseous selectivity of industrial flue gases containing CO2 and SO2 using room-temperature ionic liquids. The present model is based on a generic Redlich−Kwong (RK) EOS. The empirical binary interaction parameters have been determined using our measured vapor−liquid-equilibrium (VLE) data for SO2/[bmim][MeSO4] and literature data for CO2/[bmim][MeSO4] and CO2/SO2. The validity of the present EOS has been checked by conducting ternary VLE experiments for the present system. With this EOS, an isothermal ternary phase diagram and solubility (VLE) behaviors have been calculated for various (T, P, and feed compositions) conditions. The addition of the [bmim][MeSO4] for small and equimolar CO2/SO2 mole ratios significantly increased the selectivity. For large CO2/SO2 mole ratios, the selectivity was high for even a small addition of ionic liquid and in certain cases showed a maximum selectivity due to preferential chemical absorption of SO2. The enhancement in CO2/SO2 selectivity using [bmim][MeSO4] was significantly higher than using 1-hexyl-3-methylimidazolium bis(trifluoromethylsulfonyl)imide ([hmim][Tf2N]) from our previous work and may make the simultaneous capture and separation of these acid gases practical.