• Fire Science and Engineering
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• v.21 no.4
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• pp.32-37
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• 2007

An accurate knowledge of the flash point is important in developing appropriate preventive and control measures in industrial fire protection. The lower flash points for the 2-propanol+acetic acid and 2-propanol+-n-propionic acid systems were measured by Cleveland open cup apparatus. The experimental data were compared with the values calculated by the Raoult's law, the Wilson equation and the NRTL(non random two liquids) equation. The calculated values based on the Wilson and NRTL equations were found to be better than those based on the Raoult's law. And the predictive curve of the flash point prediction model, based on NRTL equation described the experimentally-derived data were more effective than the case of the Wilson equation.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.785-802
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• 2024

Given M a monoid with a neutral element e. We show that the solutions of d'Alembert's functional equation for n × n matrices Φ(pr, qs) + Φ(sp, rq) = 2Φ(r, s)Φ(p, q), p, q, r, s ∈ M are abelian. Furthermore, we prove under additional assumption that the solutions of the n-dimensional mixed vector-matrix Wilson's functional equation $$\begin{cases}f(pr, qs) + f(sp, rq) = 2\phi(r, s)f(p, q),\\Φ(p, q) = \phi(q, p),{\quad}p, q, r, s {\in} M\end{cases}$$ are abelian. As an application we solve the first functional equation on groups for the particular case of n = 3.

• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1157-1169
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• 2016

We find the distributional solutions of the Wilson's functional equations $$u{\circ}T+u{\circ}T^{\sigma}-2u{\otimes}v=0,\\u{\circ}T+u{\circ}T^{\sigma}-2v{\otimes}u=0,$$ where $u,v{\in}{\mathcal{D}}^{\prime}({\mathbb{R}}^n)$, the space of Schwartz distributions, T(x, y) = x + y, $T^{\sigma}(x,y)=x+{\sigma}y$, $x,y{\in}{\mathbb{R}}^n$, ${\sigma}$ an involution, and ${\circ}$, ${\otimes}$ are pullback and tensor product of distributions, respectively. As a consequence, we solve the $Erd{\ddot{o}}s$' problem for the Wilson's functional equations in the class of locally integrable functions. We also consider the Ulam-Hyers stability of the classical Wilson's functional equations $$f(x+y)+f(x+{\sigma}y)=2f(x)g(y),\\f(x+y)+f(x+{\sigma}y)=2g(x)f(y)$$ in the class of Lebesgue measurable functions.

• Journal of the Korean Society of Safety
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• v.22 no.2 s.80
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• pp.47-52
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• 2007

The flash points and the fire points for the m-xylene+n-propionic acid and n-butanol+n-pentanol systems were measured by using Tag open-cup apparatus(AS1M D 1310-86). The experimental flash points of two binary systems were compared with the values calculated by the Raoult's law, Van Laar equation and Wilson equation. The calculated values based on the Raoult's law on m-xylene+n-propionic acid system were found to be better than those based on Van Laar and Wilson equations. The calculated values based on Van Laar equation on n-butanol+n-pentanol system were found to be better than those based on the Raoult's law and Wilson equation. The the fire points for the m-xylene+n-propionic acid system were about $7{\sim}8^{\circ}C$ above the flash points. In the case of n-butanol+n-pentanol system, the flash points and the fire points had been found to be identical.

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.711-722
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• 2020

Given a semigroup S generated by its squares equipped with an involutive automorphism 𝝈 and a multiplicative function 𝜇 : S → ℂ such that 𝜇(x𝜎(x)) = 1 for all x ∈ S, we determine the complex-valued solutions of the following functional equations f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(x)g(y), x, y ∈ S and f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(y)g(x), x, y ∈ S.

• Communications of the Korean Mathematical Society
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• v.38 no.4
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• pp.1063-1074
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• 2023

Let S be a semigroup. We determine the complex-valued solutions of the following functional equation f(xy) + 𝜇(y)f(𝜎(y)x) = 2f(x)g(y), x, y ∈ S, where 𝜎 : S → S is an automorphism, and 𝜇 : S → ℂ is a multiplicative function such that 𝜇(x𝜎(x)) = 1 for all x ∈ S.

• International Journal of Safety
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• v.10 no.1
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• pp.5-9
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• 2011

Two popular activity coefficient models, Wilson and NRTL equations have been used to correlate the published flash point data on the n-propanol + propionic acid and n-butanol + propionic acid systems through the optimization method. The results of these correlation were compared with the results calculated by Raoult's law. The optimization method were found to be better than those based on the Raoult's law. The optimization method based on the Wilson equation described the published data more effectively than was the case when the optimization method was based upon the NRTL equation.

• Fire Science and Engineering
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• v.25 no.3
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• pp.113-118
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• 2011

The flash point is an important indicator of the flammability of a chemical. The minimum flash point behaviour (MFPB) is exhibited when the flash point of a mixture is below the flash points of the individual components. The identification of this behaviour is critical, because a hazardous situation results from taking the lowest component flash point value as the mixture flash point. In this study, the flash points for the n-butanol + n-decane and n-octane + n-propanol systems which exhibit MFPB, were measured by Tag open-cup apparatus. The experimental data were compared with the alues calculated by the Raoult's law, the van Laar equation and the Wilson equation. The calculated values based on the van Laar and Wilson equations were found to be better than those based on the Raoult's law. It was concluded that the van Laar and Wilson equations were more effective than the Raoult' law at describing the activity coefficients for non-ideal solution such as the n-butanol + n-decane and n-octane + n-propanol systems. The predictive curve of the flash point prediction model based on the Wilson equation described the experimentally-derived data more effectively than was the case when the prediction model was based upon the van Laar equation.

• Journal of the Korean Institute of Gas
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• v.14 no.5
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• pp.19-25
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• 2010

The flash points for the systems, o-xylene+n-pentanol and m-xylene+n-hexanol, were measured by using Tag open-cup tester(ASTM D1310-86). The experimental data were compared with the values calculated by the Raoult's law and the optimization method using van Laar and Wilson equations. The calculated values based on the optimization method were found to be better than those based on the Raoult's law. The predictive curve of the flash point prediction model based on the van Laar equation described the experimentally-derived data more effectively than was the case when the prediction model was based upon the Wilson equation.

• Journal of Korean Society of Transportation
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• v.9 no.2
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• pp.121-126
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• 1991

Wilson made an importent contribution to develop a trip distribution model with the general form of gravity model which is an entropy maximization program. Also Wilson suggested a technique which is called the "iterative balancing method" for soving the model. This te-chnique however is not stable to find solution because it is a heuristic method and sometimes does not converge to the correct solution. In this paper a new solution method using a numerical method for solving the non-linear simultaneous equation system is developed and evaluated in both computers VAX 8700 and PC/AT 286 The result of this method and Wilson's method are compared with each other. Wilson's method resulted in inferior solutions measured by the final norm of residuals.