Semenov Number Research Articles

Predicting optimal temperature profiles in single-stage fixed-bed reactors for CO2-methanation

The catalytic conversion of carbon dioxide into methane, known as Sabatier process, is a promising option for chemical storage of excess renewable energy and greenhouse gas emission control. Typically externally cooled fixed-bed reactors (FBR) using supported nickel or ruthenium catalyst are applied. The Sabatier process, however, is strongly exothermic and leads to substantial hot spots within the reactor at stoichiometric feed ratios. Although high temperatures increase the reaction rate in general, they thermodynamically limit the achievable methane-yield in the Sabatier process. Here, we present an easy-to-use method based on a Semenov number optimization (SNO) to compute optimal axial temperature profiles in single-stage fixed-bed reactors that account for kinetic and thermodynamic limitations simultaneously, and thus result in maximized yield for a fixed reactor length. In a case study on CO2-methanation, these temperature profiles result in a twofold improvement of the methane-yield compared to isothermal and adiabatic operation, and thus demonstrate the high potential of thermal optimization that lies in the Sabatier process. The SNO-method provides a valuable tool to compute optimal temperature profiles, and allows intuitive insight into the key parameters for thermal process intensification. Further, it can readily be transferred to other processes that suffer from the dilemma between kinetic and thermodynamic limitations. Our findings illustrate the attractiveness of the SNO-method to compute optimal temperature profiles in fixed-bed reactors, and the need for catalyst supports with enhanced and tailorable heat transport properties.

Explosion Characteristics of a Volatile Explosive

Thermal explosion of volatile energetic materials has been investigated based upon the thermal explosion theory of a gaseous mixture. The critical and transition condition expressions for the thermal explosion of these materials are obtained. The analytical solutions not only deal with the steady-state solution but also cover the unsteady state solution. The different cases and different parameters which affect the critical conditions are investigated in details. Also the numerical solution of the energy equation is offered to give the different families of temperature - time histories. The relation between the critical parameters as ignition timing, ignition temperature, Semenov number and others with different boundary conditions is offered. In this work, the critical and transition conditions of the explosion of the volatile energetic materials through the analytical solution to the problem was introduced. The numerical solution of the unsteady state equation to the problem was offered. The relation between the critical parameters was also presented. The effect of evaporation of the volatile materials as compared with the non-volatile materials on the critical conditions was shown.

Investigation on Thermal Runaway in Batch Reactors by Parametric Sensitivity Analysis

A batch system is an inherently parametric sensitivity system where small changes of input parameters can induce large changes of output variables. In the present work, temperature and temperature sensitivity with respect to the initial temperature were investigated by parametric sensitivity analysis using a dimensionless batch reactor model. The influence of the Semenov number, the heat of reaction parameter and the Arrhenius-type number on reactor temperature and temperature sensitivity were studied. It is demonstrated that batch reactors can exhibit high sensitivity when small changes in input parameters lead to large changes in temperature and temperature sensitivity trajectories. A criterion is established for thermal runaway by analyzing temperature sensitivity trajectories and applied to some experimental examples. This criterion allows runaway and safe conditions to be identified, the results being in agreement with the experimental data. The proposed criterion can satisfactorily predict the safety limits of the operating conditions.

Modelling of critical phenomena for ignition of metal particles

The singularly perturbed system of differential equations describing metal particle ignition is analyzed. Two general types of combustion reactions are recognized, namely subcritical and supercritical. It is shown that there exists a so-called critical regime, which separates domains of subcritical and supercritical regimes. The critical regime is modelled by the special phase trajectory of the system, which includes the repulsive part with slow variation of the temperature of a metal particle and the thickness of the oxide film. The asymptotic formulae for the calculation of critical values of the modified Semenov number are obtained in the cases of parabolic and cubic laws of oxidation kinetics.

Ignition And Transition Conditions In The Theory Of Combustion

In branched-chain thermal explosion theory, it is usually impossible analytically and sometimes a substantial computation to calculate the temperature, activation energy and modiued Semenov's number at transition from discontinuous to continuous behaviour. Invariably, it is possible to reduce the governing equation to accommodate other important mathematical model in the theory of combustion. With this aspect of the equation, transition temperature, activation energy and Semenov's number can be estimated analytically. Previously obtained results are special cases of ours. IFE Journal of Science Vol. 9 (1) 2007 pp. 51-54

EFFECT OF DEGREE OF REACTION ON CRITICAL CONDITIONS AND TIMES TO IGNITION OF A GAS MIXTURE EXPLOSION

The effect of degree of reaction on critical conditions and times to ignition of a gas-mixture explosion is theoretically examined in detail. The Arrhenius form of the chemical kinetic rate and the Frank-Kamenetskii approximation for that form is considered. The study offers analytical and numerical solutions of the problem in pressure-time, temperature-time, and temperature-pressure planes. The analytical solution presented analytical detailed expressions, which relate between the critical parameters at critical point. The fractional order of reaction (n < 1) and order of reaction (n > 1) are studied. The effect of different parameters such as ambient temperature (Θa), exothermicity of reaction (B), modified Semenov number (ψ) on critical conditions for explosion are investigated. A direct numerical solution of the governing equations of the problem is given in different planes. Different trajectories of subcritical (non-ignition), supercritical (ignition), and critical states are illustrated. The relationships between the critical parameters from the numerical solutions are confirmed with those have been obtained from the analytical one. The effect of order of reaction on the critical parameters and so on the ignition times is also investigated.

Effect of self-sustained oscillations on critical fluctuations

As is known [1, 2], the space of the parameters of an exothermic flow reactor includes a bistability region, where either a low- or high-temperature stationary regime is realized, depending on the initial conditions. The boundary of this region has a cusp, where the difference between regimes disappears. The properties of the cusp are similar to those of a critical point on the phase diagram of a substance with the first-order phase transition between the phases with the same symmetry [3]. In particular, it is the endpoint of the equilibrium line between a liquid and vapor (boiling curve) on the pressure‐temperature plane, where the bistability domain is bounded by a spinodal. Fluctuations of the reaction rate must be large near the cusp [4] the same as density fluctuations increase near the critical point. At the same time, this similarity is not universal: the stationary regimes of a reactor lose stability for certain parameter values and become periodic [5‐7]. In [4], an increase in fluctuations was calculated in the linear approximation and disregarding the oscillation mode. In this work, critical fluctuations are analyzed by numerical simulation for the conditions when they cannot be treated as small, as well as upon the transition from the stationary regime to the periodic one at the cusp. 1. The reactor state is determined by three parameters [8] such as the Damkohler number D , which is the ratio of the substance residence time in the reactor to the characteristic reaction time τ R ; the Semenov number, which is the ratio of the heat-exchange time of the reactor with the environment to τ R ; and the Zel’dovich number

The effect of infiltration and reactant gas pressure on spin combustion in a gas-solid system

A mathematical model of surface spin combustion in a binary gas-solid system under the approximation of quasi-homogeneity of the reactant porous media (i.e., low Semenov number, Se >1) and infiltration (Pe∞ 1).

Calculation of the stability criterion in the reaction of energetic materials

The stability in the energetic materials' thermal decomposition reaction is developed from both transient mass and heat transfer equations simultaneously. The derived equation is the sufficient and necessary condition for the reaction system's stability, which was expressed as:φ<(βθ+1)2(1−x)nnϕ(1−x)n−1+exp−θβθ+1 In this stability equation, a modified Semenov number φ is the expressed function of three kinetic parameters, i.e., n, ϕ and β and two variables, i.e., x and θ, respectively. A reaction system is stable if its evaluated modified Semenov number satisfies this stability criterion. In the case of the minimum modified Semenov number φm, we can deduce the stability equation at critical condition asβ2θ2c+2β1+ηβn(1−x)θc+1+ηn(1−x)(2β−1)=0 The criteria of stability for the explosives TNT, RDX, PETN and HMX were evaluated using this formula, which was expressed θc as function of x at given n, η and various β as parameters.

The critical condition and stability of exothermic chemical reaction in a non-isothermal reactor

Abstract Theoretical equations of critical thermal explosion and runaway in a non-isothermal reactor without chemical consumption were derived. The critical ignition, extinction and transition temperatures, modified Semenov's number and the stable criteria of temperature in reaction system can be estimated employing these equations. In an unsteady state reaction system the equations of critical modified Semenov's number and runaway temperature equations were also derived. The transient conversion and temperature profiles in the case of chemical consumption are the function of reaction time, which can be calculated numerically from two heterogeneous first order ordinary differential equations. Four parameters, i.e. n , β, ϕ, and φ, affect those calculated results.

On the structure of a distinguished-limit quasi-isothermal deflagration for the generalized reaction-rate model

The structure of the quasi-isothermal deflagration is examined by means of an asymptotic analysis of the physical-plane boundary-value problem, with Lewis–Semenov number unity, in the limit of the activation-temperature ratio,β=Ta/Tb, greater than order unity, for the generalized reaction-rate-model case of: (1) the heat-addition-temperature ratio,α=(Tb−Tu)/Tu, of orderβ−1/2, less than order unity [whereTa,Tb, andTuare the activation, adiabatic-flame (and/or burned-gas), and unburned-gas temperatures, respectively]; and (2) the exponent,a, which characterizes the pre-exponential thermal dependence of the reaction-rate term, unity. The examination indicates that, as in the order-unity heat-addition case, this deflagration has a four-region structure: the upstream diffusion-convection and downstream diffusion-reaction regions, and the far-upstream (or cold-boundary) and the far-downstream (or hot-boundary) regions.

Parametric sensitivity and self-similarity in thermal explosion theory

Abstract We study the relations between thermal runaway (also called parametric sensitivity) and self-similarity, an interesting property of the sensitivity functions that has been numerically verified in many explosion and combustion systems. Both concepts are sensitivity related but independent of the particular parameter being perturbed. This independence is emphasized by proposing a new generalized condition for parametric sensitivity. Critically is defined as the point in the parameter space where the nominal trajectory exhibits maximum sensitivity to arbitrary, unstructured perturbations applied at the maximum temperature. The condition for critically reduces to the analysis of the eigenvalues of the Jacobian matrix. In addition to its conceptual generality, the new condition shows that in certain cases there exists no critical Semenov number. The sensitivity functions are shown to satisfy self-similarity relations if and only if the system exhibits critical or supercritical behavior. The onset of self-similarity is explained in terms of two properties of explosion systems, both related to parametric sensitivity. First, the temperature is a dominant variable, and any perturbation in the system affects the conversion mainly through the changes induced in the temperature. This strong coupling of the variables is shown by decomposing the sensitivity functions into direct and indirect terms. Second, the sensitivity equations are pseudo-homogeneous in a characteristic time window, in which the system becomes relatively insensitive to parameter perturbations applied within the same interval. The two properties are shown to imply self-similarity of the sensitivity functions. Relations to earlier parametric sensitivity and self-similarity conditions are discussed.

Accounting for reactant consumption in the thermal explosion problem. Part I: Mathematical foundation

New definitions for the criticality conditions of the thermal explosion problem are founded on the mathematical behavior of the governing equations. The paper deals with uniform temperature and concentration (Semenov problem). It is well known that the results can be applied to the distributed temperature and concentration case by the use of correction factors. It is shown that criticality can be defined in the temperature-time plane as accepted by most authors. However, using our definitions of criticality in the temperature-concentration plane confirms the previous findings of Adler and Enig. They showed that the classically defined critical state in the temperature-time plane is always a subcritical state in the temperature-concentration plane. At the same time, the critical state in the temperature-concentration plane is always a supercritical state in the temperature-time plane. However, the critical state in the temperature-concentration plane is in agreement with that in the Semenov number (Ψ)-temperature plane. It is shown that the critical states in all planes coincide only when n = 0 or B = ∞ and agree with the well known results neglecting reactant consumption. The difference between the critical and ignition temperatures is discussed. It is shown that as B approaches infinity, the solution for Φ as a function of τ for any value of n approaches the solution for n = 0. Hence for B = ∞, substituting for n = 0 in Alder and Enig results produces the classical Semenov result. This resolves the objections expressed against these results before. At the same time the locus of the critical states for n = 0, with finite B is determined. The effect of the degree of reaction on the induction time for adiabatic systems is demonstrated. The conditions required for ignition of subcritical systems is demonstrated. The conditions required for ignition of subcritical systems are discussed as well as the effect of initial conditions on criticality.

Natural convection effects on thermal ignition in a porous medium. I. Semenov model

A one-dimensional diffusion-convection-reaction model is formulated to account for natural convection effects on thermal ignition in an open system consisting of a porous medium. Various limiting cases of the model are considered. A detailed analysis of the Semenov (lumped) model is presented. Explicit relations are derived for the dependence of the critical Semenov number ( ψ c ) on the Rayleigh number ( R *). It is shown that for R * → 0, ψ c approaches the classical (conduction) limit e -1 , while for R * ≫ 1, the ignition locus is given by the convection asymptote ψ c / R * = 4 e -2 . Inclusion of reactant consumption shows that the conduction asymptote disappears at B = 4 while the convection asymptote ceases to exist for B Ls &lt; 3 + 2√2, where Ls is a modified Lewis number and B is the heat of reaction parameter. It is shown that the Semenov model has five different types of bifurcation diagrams of temperature against Rayleigh number (particle size), (single-valued, inverse S , isola, inverse S + isola and mushroom). This behaviour is found to be qualitatively identical to that of the forced convection problem investigated by Zeldovich &amp; Zysin.

Natural convection effects on thermal ignition in a porous medium. II. Lumped thermal model-I

In this paper, we examine the disappearance of criticality, ignition locus and bifurcation diagrams of temperature against Rayleigh number of a one-dimensional diffusion-convection-reaction model with the assumption of infinite thermal conductivity and zero species diffusivity. The predictions of this model are compared with those of the Semenov model to determine the impact of the species diffusion term. It is shown that for large values of the Rayleigh number (R* ≫ 1), the ignition locus may be expressed in a parametric form B Ls = t /ln t + t /( t - 1) (1 &lt; t ≼ 3.4955), ψ / R * = ( B Ls ) 2 (( t - 1)/ t ) exp{ - B Ls + B Ls / t } ln t , where B is the heat of reaction parameter, ψ is the Semenov number and Ls is a (modified) Lewis number. Criticality is found to disappear at B Ls = 4.194. When these results are compared with those of the Semenov model, it is found that neglecting the species diffusion term gives conservative approximations to the ignition locus, and criticality boundary. It is found that the lumped thermal model-I has five different types of bifurcation diagrams of temperature against Rayleigh number (single­-valued, isola, inverse S , mushroom, inverse S + isola). These diagrams are qualitatively identical to the bifurcation diagrams of temperature against flow rate for the forced convection problem under the assumption of infinite thermal conductivity and zero species diffusivity.

Spontaneous ignition and stability in closed and open systems. Part 1.—Foundations: first-order reactions in the adiabatic case

When a first-order, exothermic reaction is carried out in a closed vessel, thermal explosion may result. For well stirred systems of high exothermicity, reactant consumption can be ignored and the value of a single parameter (the Semenov number) determines the course of events, and calculation of its critical value permits the prediction of the pressure–temperature boundary between quiescent and explosive behaviour. When the same reaction is studied in an open system so contrived as to operate adiabatically there is no such (Pcr, Tcr) boundary. There is, however, another single parameter (the Zel'dovich number) which determines whether slow reaction or fast reaction or jumps from one state to another occur. Calculation of this number allows the prediction of critical flow rates for ignition and extinction. This case is presented here as the foundation of a broader study of unstable behaviour in open systems.

A Criticality Criterion for Thermal Explosions in Well-stirred Systems with Reactant Consumption

The autonomous differential equations for the temperature and reactant consumption in a first-order well-stirred exothermic reaction are considered. An examination of the phase-plane solutions allows the qualitative behaviour of the Semenov number ψ as a function of maximum temperature rise Θ* to be established. In the limit of infinite adiabatic temperature rise (B→∞) and zero activation energy parameter (ɛ = 0), the relationship between ψ and stationary temperature Θs is known to be ψeΘ1 = Θs. Criticality arises at the maximum of ψ(Θs) and leads to the critical Semenov values (Θs)cr = 1, ψcr = e−1. For sufficiently large B, it is shown that the ψ(Θ*) curve has a bifurcation at Θ* = 1, with the upper branch monotonically increasing and the lower branch monotonically decreasing for Θ* > 1. In the limit B→ ∞ these become respectively the straight line ψ = e−1, Θs ≥ 1 and the unstable branch of ψ = Θse−1, Θs ≥ 1 and the unstable branch of ψ = Θs e−Θ. Criticality for finite B is defined as occurring at the bifurcation, namely Θ*cr = 1, with ψcr(B) the value of ψ at this point. Values of these Semenoy numbers are obtainable from the numerical calculations of Boddington et al. [Proc. R. Soc. Lond. (1983), 390, 13–30]. The new criterion is applied to an approximate phase-plane solution. The corresponding critical parameter is found to be ψcr = e−1[1+B−½(2−e−1)½ +O(B−1)].

Thermal explosions with extensive reactant consumption: a new criterion for criticality

In classical treatments of thermal runaway, reactant consumption is commonly ignored. Criticality is readily identified as the disappearance of steady states. The distinction between subcritical and supercritical systems is sharp. For explosive behaviour, infinite excess temperatures ( θ → ∞) are reached in a finite time; non-explosive reaction is characterized by a low stationary state of self-heating( θ ≼ 1). These are divided discontinuously by a critical value for the Semenov number ψ : ψ = QVAc m 0 e - E / RT a / X S ( RT 2 a / E ) ; ψ cr = e -1 ; θ cr = 1. When reactant consumption occurs, this sharpness disappears. Each temperature-time history evolves to a maximum temperature excess ∆ T * or θ * before decaying back to ambient. A new criterion is presented here for thermal ignition in systems with extensive reactant consumption. It is based on the dependence of θ * on the initial reaction rate. We identify criticality with the maximum sensitivity of θ * to ψ , and look for the point of inflexion in the function θ *( ψ ). Such a definition is closely related to that used implicitly by an experimentalist. The critical value of the Semenov number is derived as an integral equation. This is solved numerically to yield ψ cr as a function of the reaction exothermicity B . The calculations lead to values for up to 2 or 3 times greater than the classical value found when reactant consump­tion is ignored. There is a smooth transition between new and old values close to the classical limit (at very large B ). Our results coincide with the predictions from asymptotic analysis for B greater than ca . 50. They peel away significantly as B diminishes through the range 50 &gt; B &gt; 4, a range that can be met in practice for dilute gases or solid masses of low reactivity.

Thermal explosion, times to ignition and near-critical behaviour in uniform-temperature systems. Part 2.—Generalized dependences of rates on temperature and concentration

This paper extends the range of earlier but narrower treatments of the time evolution of reactant temperature in exothermic, homogeneous systems near to criticality. It considers (i) generalized temperature dependences of the reaction rate coefficient, ƒ(θ), and (ii) generalized concentration dependences of the reaction rate, g(w). Here θ is the dimensionless temperature excess and text-decoration:overlinew=(c/c0) is the fractional concentration.When reactant consumption can be neglected, the temperature dependence of the rate of reaction is represented by k∝ exp (–E/RTa)ƒ(θ). The Arrhenius form implies ƒ(θ)= exp [θ/(1 +Iµθ)], and the case of Iµ→ 0 corresponds to the exponential approximation. Time to ignition has the form [graphic omitted], where M′={2π2/[ƒ(θ0)ƒθθ(θ0)]}1/2. This correctly generates the value M′= [graphic omitted]2π/e for the exponential approximation.When transition is reached, our results show that M′→∞; therefore tign lengthens without limit even for a fixed degree of supercriticality.A similar systematic approach allows the influence of reactant consumption on the critical value of the Semenov number to be assessed for a very wide variety of reaction rate expressions (and not merely first or nth order). We find the general form: ψcr=ψ0[1 + 2.946(ƒ0/ƒθθ)1/3(gw/B)2/3]. (For an nth-order deceleratory reaction gw=n.)

Flame structure in initially unmixed reactants under one-step kinetics

The burning of initially unmixed reactants at Lewis—Semenov number unity for a direct one-step reversible reaction of arbitrary order is examined under Arrhenius kinetics. The study concentrates on the stagnation region formed by the blowing of a hot stream of oxidant at an adiabatically vaporising planar wall of pure nongaseous fuel. Asymptotic expansion are developed to describe the chemical behavior for limiting values of the first Damkohler number. i.e. for nearly frozen and nearly equilibrium flow. Because of the nonlinearities owing to the law of mass action, numerical integration is required for intermediate values of the Damkohler number (moderate rates of reaction). Two specific results are sought. First, the relative thickness of the flame zone and departure of maximum temperature from adiabatic flame temperature in the near-equilibrium limit are furnished, with emphasis on variance with the order of the reaction. Further, the asymptotic methods are continued to higher order to indicate the first effects owing to convection in the reaction region. Second the additional insight asymptotic methods give into previously noted possibilities of nonuniqueness of solution to the steady-state combustion problem is developed.