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Thiele Modulus
The Thiele modulus was developed by Ernest Thiele in his paper 'Relation between catalytic activity and size of particle' in 1939.Thiele, E.W. Relation between catalytic activity and size of particle. Industrial and Engineering Chemistry, 31 (1939), pp. 916–920 Thiele reasoned that a large enough particle has a reaction rate so rapid that diffusion forces can only carry the product away from the surface of the catalyst particle. Therefore, only the surface of the catalyst would experience any reaction. The Thiele Modulus was developed to describe the relationship between diffusion Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ... and reaction rates in porous catalyst pellets with no mass transfer limitations. This value is generally used to measure the effectiveness fac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ernest Thiele
Ernest W. Thiele (pronounced ; 1895–1993) was an influential chemical engineering researcher at Standard Oil (then Amoco, now BP) and professor of chemical engineering at the University of Notre Dame. He is known for his highly impactful work in chemical reaction engineering, complex reacting systems, and separations, including distillation theory. Early life and education Ernest Thiele, born on December 8, 1895, grew up in Chicago, Illinois. In 1916, he earned an A.B. degree from Loyola University in Chicago. He was stationed at the University of Illinois at Urbana–Champaign with the U.S. Army, where he completed a bachelor's degree in chemical engineering in 1919. In the fall of 1922, matriculated to MIT where he began graduate studies under the direction of Professor Robert T. Haslam; he earned his M.S. degree in 1923 and a doctoral degree in 1925 with a thesis on steam-carbon reactions. Career Before starting his graduate degree at MIT, Thiele worked for six months fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Mass Diffusivity
Diffusivity, mass diffusivity or diffusion coefficient is usually written as the proportionality constant between the molar flux due to molecular diffusion and the negative value of the gradient in the concentration of the species. More accurately, the diffusion coefficient times the local concentration is the proportionality constant between the negative value of the mole fraction gradient and the molar flux. This distinction is especially significant in gaseous systems with strong temperature gradients. Diffusivity derives its definition from Fick's law and plays a role in numerous other equations of physical chemistry. The diffusivity is generally prescribed for a given pair of species and pairwise for a multi-species system. The higher the diffusivity (of one substance with respect to another), the faster they diffuse into each other. Typically, a compound's diffusion coefficient is ~10,000× as great in air as in water. Carbon dioxide in air has a diffusion coefficient of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Rate Equation
In chemistry, the rate equation (also known as the rate law or empirical differential rate equation) is an Empirical relationship, empirical Differential equation, differential Expression (mathematics), mathematical expression for the reaction rate of a given reaction in terms of concentrations of chemical species and constant parameters (normally rate coefficients and partial orders of reaction) only. For many reactions, the initial rate is given by a power law such as :v_0\; =\; k[\mathrm]^x[\mathrm]^y where and are the molar concentrations of the species and usually in Mole (unit), moles per liter (molarity, ). The exponents and are the partial ''orders of reaction'' for and , respectively, and the ''overall'' reaction order is the sum of the exponents. These are often positive integers, but they may also be zero, fractional, or negative. The order of reaction is a number which quantifies the degree to which the rate of a chemical reaction depends on concentrations of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Term (logic)
In mathematical logic, a term denotes a mathematical object while a formula denotes a mathematical fact. In particular, terms appear as components of a formula. This is analogous to natural language, where a noun phrase refers to an object and a whole sentence refers to a fact. A first-order term is recursively constructed from constant symbols, variable symbols, and function symbols. An expression formed by applying a predicate symbol to an appropriate number of terms is called an atomic formula, which evaluates to true or false in bivalent logics, given an interpretation. For example, is a term built from the constant 1, the variable , and the binary function symbols and ; it is part of the atomic formula which evaluates to true for each real-numbered value of . Besides in logic, terms play important roles in universal algebra, and rewriting systems. Definition Given a set ''V'' of variable symbols, a set ''C'' of constant symbols and sets ''F''''n'' of ''n''-ary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Volumetric Flow Rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes \dot V). Its SI unit is cubic metres per second (m3/s). It contrasts with '' mass flow rate'', which is the other main type of fluid flow rate. In most contexts a mention of "rate of fluid flow" is likely to refer to the volumetric rate. In hydrometry, the volumetric flow rate is known as '' discharge''. The volumetric flow rate across a unit area is called '' volumetric flux'', as defined by Darcy's law and represented by the symbol . Conversely, the integration of a volumetric flux over a given area gives the volumetric flow rate. Units The SI unit is cubic metres per second (m3/s). Another unit used is standard cubic centimetres per minute (SCCM). In US customary units and imperial units, volumetric flow rate is often expressed as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Molar Concentration
Molar concentration (also called molarity, amount concentration or substance concentration) is the number of moles of solute per liter of solution. Specifically, It is a measure of the concentration of a chemical species, in particular, of a solute in a solution, in terms of amount of substance per unit volume of solution. In chemistry, the most commonly used unit for molarity is the number of moles per liter, having the unit symbol mol/L or mol/ dm3 (1000 mol/ m3) in SI units. A solution with a concentration of 1 mol/L is said to be 1 molar, commonly designated as 1 M or 1 M. Molarity is often depicted with square brackets around the substance of interest; for example, the molarity of the hydrogen ion is depicted as + Definition Molar concentration or molarity is most commonly expressed in units of moles of solute per litre of solution. For use in broader applications, it is defined as amount of substance of solute per unit volume of solution, or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Boundary Value Problem
In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them. Problems involving the wave equation, such as the determination of normal modes, are often stated as boundary value problems. A large class of important boundary value problems are the Sturm–Liouville problems. The analysis of these problems, in the linear case, involves the eigenfunctions of a differential operator. To be useful in applications, a boundary value problem should be well posed. This means that given the input to the problem there exists a unique solution, which depends continuously on the input. Much theoretical work in the field of partial differential equations is devot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Limit Of A Function
Although the function is not defined at zero, as becomes closer and closer to zero, becomes arbitrarily close to 1. In other words, the limit of as approaches zero, equals 1. In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input which may or may not be in the domain of the function. Formal definitions, first devised in the early 19th century, are given below. Informally, a function assigns an output to every input . We say that the function has a limit at an input , if gets closer and closer to as moves closer and closer to . More specifically, the output value can be made ''arbitrarily'' close to if the input to is taken ''sufficiently'' close to . On the other hand, if some inputs very close to are taken to outputs that stay a fixed distance apart, then we say the limit ''does not exist''. The notion of a limit has many applications in modern calc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Differential Of A Function
In calculus, the differential represents the principal part of the change in a function y = f(x) with respect to changes in the independent variable. The differential dy is defined by dy = f'(x)\,dx, where f'(x) is the derivative of with respect to x, and dx is an additional real variable (so that dy is a function of x and dx). The notation is such that the equation dy = \frac\, dx holds, where the derivative is represented in the Leibniz notation dy/dx, and this is consistent with regarding the derivative as the quotient of the differentials. One also writes df(x) = f'(x)\,dx. The precise meaning of the variables dy and dx depends on the context of the application and the required level of mathematical rigor. The domain of these variables may take on a particular geometrical significance if the differential is regarded as a particular differential form, or analytical significance if the differential is regarded as a linear approximation to the increment of a function. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Reaction Rate Constant
In chemical kinetics, a reaction rate constant or reaction rate coefficient () is a proportionality constant which quantifies the rate and direction of a chemical reaction by relating it with the concentration of reactants. For a reaction between reactants A and B to form a product C, where :A and B are reactants :C is a product :''a'', ''b'', and ''c'' are stoichiometric coefficients, the reaction rate is often found to have the form: r = k mathrmm mathrm Here is the reaction rate constant that depends on temperature, and and are the molar concentrations of substances A and B in moles per unit volume of solution, assuming the reaction is taking place throughout the volume of the solution. (For a reaction taking place at a boundary, one would use moles of A or B per unit area instead.) The exponents ''m'' and ''n'' are called partial orders of reaction and are ''not'' generally equal to the stoichiometric coefficients ''a'' and ''b''. Instead they depend on the reacti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Porosity
Porosity or void fraction is a measure of the void (i.e. "empty") spaces in a material, and is a fraction of the volume of voids over the total volume, between 0 and 1, or as a percentage between 0% and 100%. Strictly speaking, some tests measure the "accessible void", the total amount of void space accessible from the surface (cf. closed-cell foam). There are many ways to test porosity in a substance or part, such as industrial CT scanning. The term porosity is used in multiple fields including pharmaceutics, ceramics, metallurgy, materials, manufacturing, petrophysics, hydrology, earth sciences, soil mechanics, rock mechanics, and engineering. Void fraction in two-phase flow In gas-liquid two-phase flow, the void fraction is defined as the fraction of the flow-channel volume that is occupied by the gas phase or, alternatively, as the fraction of the cross-sectional area of the channel that is occupied by the gas phase. Void fraction usually varies from location to l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Catalytic Activity
Catalysis () is the increase in rate of a chemical reaction due to an added substance known as a catalyst (). Catalysts are not consumed by the reaction and remain unchanged after it. If the reaction is rapid and the catalyst recycles quickly, very small amounts of catalyst often suffice; mixing, surface area, and temperature are important factors in reaction rate. Catalysts generally react with one or more reactants to form intermediates that subsequently give the final reaction product, in the process of regenerating the catalyst. The rate increase occurs because the catalyst allows the reaction to occur by an alternative mechanism which may be much faster than the noncatalyzed mechanism. However the noncatalyzed mechanism does remain possible, so that the total rate (catalyzed plus noncatalyzed) can only increase in the presence of the catalyst and never decrease. Catalysis may be classified as either homogeneous, whose components are dispersed in the same phase (usuall ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |