Arbitrary Biot Number Research Articles

Development and validation of an asymptotic solution for a two-phase Stefan problem in a droplet subjected to convective boundary condition

Droplet solidification is governed by classical Stefan problems which have been commonly treated as a single-phase problem by the majority of the studies in the literature. This approach, however, is unable to capture the initial temperature and the start of freezing time correctly. The treatment of two-phase Stefan problem in spherical coordinates is limited. No known exact solution exists, albeit numerical solutions and asymptotics have proven to be useful. We present a singular perturbation solution in the limit of low Stefan number and arbitrary Biot number for the two-phase Stefan problem in a finite spherical domain. An asymptotic solution is developed for a droplet at a non-freezing initial temperature subjected to a convective boundary condition at the surface. The solution is developed for both long-time and short-time scales. The results from asymptotic expansion method are validated with the experimental results in the literature and are further verified by a numerical model of a freezing droplet using enthalpy–porosity method. The sensitivity of the asymptotic solution to the droplet initial temperature, Biot number, and Stefan number has also been studied. The results indicate that the solution from perturbation series and enthalpy–porosity method agrees to within 1%–10% for temperature profile and overall freezing times over a wide range of practical values for initial temperature, Stefan and Biot numbers for the application of spray freezing. Our perturbation series solution is also able to capture the effect of initial temperature on the overall freezing time of the droplet.

Effect of Biot number on unsteady reaction-diffusion phenomena and analytical solutions of coupled governing equations in porous particles with various shapes

Analytical solutions of transient concentration profile inside particles were calculated by solving rection-diffusion equation in spherical, cylindrical, and slab-type porous particles, assuming first order irreversible reaction. Time-dependent concentration in bulk fluid phase was assumed as exponential decay for each particle, and convective boundary conditions were taken using arbitrary Biot number to obtain a general solution by eigenfunction expansion or Laplace transform method. Factors affecting average transient concentration inside particles were studied by adjusting Biot number, reaction time, and Thiele modulus as well as the position inside particles. To predict transient bulk concentration in batch photocatalytic reactor containing porous particles, coupled differential equations were solved by Laplace transform to obtain analytical solutions of bulk concentration as well as average concentration inside porous particles as a function of reaction time. The factors affecting transient concentrations were investigated by adjusting the concentration and porosity of the catalytic particles, morphology of the particles, and Thiele modulus in batch-mode reactor to study reduction speed of reactant concentration during photocatalytic reaction. The solutions from coupled differential equations were useful for the prediction of transient behavior in batch-type photocatalytic reactor and were compared with the results from CSTR containing slab-type photocatalytic particles with various space time.

Conservative bounds on Rayleigh-Bénard convection with mixed thermal boundary conditions

Using the background field variational method developed by Doering and Constantin, we obtain upper bounds on heat transport in Rayleigh-Benard convection assuming mixed (Robin) thermal conditions of arbitrary Biot number η at the fluid boundaries, ranging from the fixed temperature (perfectly conducting, η = 0) to the fixed flux (perfectly insulating, η = ∞) extremes. Solving the associated Euler-Lagrange equations, we numerically find optimal bounds on the averaged convective heat transport, measured by the Nusselt number \(\mathit{Nu}\), over a restricted one-parameter class of piecewise linear background temperature profiles, and compare these to conservative analytical bounds derived using elementary functional estimates. We find that analytical estimates fully capture the scaling behaviour of the semi-optimal numerical bounds, including a clear transition from fixed temperature to fixed flux behaviour observed for any small nonzero η as the usual Rayleigh number \(\mathit{Ra}\) increases, suggesting that in the strong driving limit, all imperfectly conducting boundaries effectively act as insulators. The overall bounds, optimized over piecewise linear backgrounds, are \(\mathit{Nu}\)≤ 0.045 \(\mathit{Ra}^{1/2}\) in the fixed temperature case η = 0, and \(\mathit{Nu}\)≤ 0.078 \(\mathit{Ra}^{1/2}\) in the large-\(\mathit{Ra}\) limit in all other cases, 0 <η≤∞.

Freezing of a supercooled spherical droplet with mixed boundary conditions

The freezing of a supercooled droplet occurs in two steps: recalescence, that is, a rapid return to thermodynamic equilibrium at the freezing temperature leading to a liquid–solid mixture and a longer stage of complete freezing. The second freezing step can be modelled by the one-phase Stefan problem for an inward solidification of a sphere, assuming the droplet to be spherical. A convective heat transfer with the ambient immiscible fluid is modelled by a mixed boundary condition on the outer surface of the droplet. This condition depends on the Biot number (ratio of the heat transfer resistances inside the droplet and at its surface). A novel asymptotic solution is developed for a small Stefan number and an arbitrary Biot number. Applying the method of matched asymptotic expansions, uniformly valid solutions are obtained for the temperature profile and freezing front evolution in the whole stage of complete freezing. For an infinite Biot number, that is, for a fixed temperature at the droplet outer boundary, known solutions are recovered. In parallel, numerical results are obtained for an arbitrary Stefan number using a finite-difference scheme based on the enthalpy method. The asymptotic and numerical solutions are in good agreement.

A model for the extracted heat and the phase front position in solidification with boundary condition of the third kind

In this paper an approximate analytical solution is given for the extracted heat and the phase front position as a function of time for a finite slab of phase change material with a surface heat transfer coefficient and a fixed melting point. It will be shown how, for an arbitrary Biot number, an approximate solution can be obtained from the solutions in the case Bi → 0 (homogeneous temperature in phase change material) and Bi → ∞ (constant temperature boundary condition). The accuracy of the solution will be assessed by comparison with a numerical finite difference solution obtained by the enthalpy method.

Multiplicity of steady state solutions for combustion in a solid sphere with arbitrary Biot number

The Frank-Kamenetskii model for steady state thermal regimes in a spherical region has gross multiplicity in the case of infinite Biot number. This paper uses phase plane techniques to show that similar behaviour extends to the case of arbitrary Biot number, with the analytical condition for infinite multiplicity given as σ ∞=2 exp(− 2 Bi ). Criticality is seen to be represented by a tangency condition in the phase plane.

Thermal explosion and times to ignition in systems with distributed temperatures. IV. Behaviour at points of criticality and transition

The conductive theory of thermal explosion in systems with distributed temperatures expresses conditions for the breakdown of stability as the critical value of a single dimensionless group δ , Frank-Kamenetskii’s parameter. For values of δ above critical, there is thermal runaway; for values of δ less than critical, the system is quiescent. Analytic solutions for excess temperatures as a function of position and time are not generally available for these systems, but can be found when δ is close to the critical value, by using the techniques of asymptotic analysis. In this paper we consider two related problems. First, the approach of temperature to the critical, steady-state profile is found when δ takes precisely its critical value. The second concerns systems where criticality is just being lost through the coalescence of two critical points. It establishes how the new ‘transitional’ steady-state profile is then approached. In both circumstances we neglect reactant consumption. We allow for an arbitrary dependence of reaction rate on temperature, and an arbitrary Biot number in the boundary conditions. The solutions are found in terms of leading-order descriptions of the behaviour of the reduced temperature-excess as it moves towards the steady state profile.

Thermal explosions, criticality and the disappearance of criticality in systems with distributed temperatures. ।. Arbitrary Biot number and general reaction-rate laws

The conductive theory of thermal explosion in its original form (Frank-Kamenetskii, Acta phys. -chim . URSS (1939)) expresses the balance between heat generation and heat conduction in terms of the dimensionless parameter δ = [QEA a 2 0 c n 0 exp ( - E/RT a )]/ K RT 2 a Stability is lost when δ exceeds a critical value which, in this approximation, depends only on the geometry of the system. Matters are usually more complicated than this. First, heat transfer is often impeded both in the interior (by conduction) and at the surface; the relative importance of these impedances is expressed by the Biot number B i = X a o / K - Second, the temperature dependence of reaction rate may not be well enough represented by the 'exponential approximation’ (which simply implies a doubling of rate every so many degrees). The natural and convenient dimensionless measure here is the parameter ϵ = RT a /E. In the present paper, critical values for the parameter δ and for the dimensionless central-temperature excess Ɵ o have been evaluated for the whole range of Biot number from the uniform case (Semenov extreme, B i → 0) to the Frank-Kamenetskii extreme (B i →∞). The procedures can handle any temperature-dependence of rate and are illustrated here for the Arrhenius and 'bimolecular’ forms for which, k∝ exp ( — E /RT ) and k ∝T ½ exp ( - E /R T ) respectively. When E /RT a is not large, criticality is lost at ϵ = ϵ tr ≤ ¼. Such transitional values for the reduced ambient temperature ϵ, for the critical value of δ, and for the dimensionless central temperature excess Ɵ 0 have also been obtained. They are represented both graphically and numerically. The present results are also compared with earlier work.

Surface temperature gradients and rates of internal heat evolution at criticality for generalized heat transfer (arbitrary Biot number)

The dependence on the heat-transfer conditions (Biot number, Bi) of the temperature gradient at the surface, Γ, of an exothermic reactant mass at criticality is presented. Bodies of slab, cylindrical and spherical geometry have the same values of Γcr at the Frank-Kameneskii extreme (Bi→∞, Γcr= 2) and also at the Semenov extreme (Bi= 0, Γcr= 0). For general boundary conditions the critical surface temperature gradient depends on the Biot number and shape.

The disappearance of criticality for small activation energy with arbitrary biot number

For exothermic reactions obeying the Arrhenius equation the critical conditions for thermal ignition are known to exist for activation energy E large enough satisfying ϵ ≡ RT a E < ϵ tr . This transition value ϵ tr of ϵ has values less than 0.25 in all geometries, (Boddington et al. [1]) and exactly 0.25 for the uniform temperature reactor (the Semenov case). The precise values of α tr and the corresponding values of the maximum dimensionless temperature θ and the Frank-Kamenetskii parameter δ (the transition values) were calculated for a slab by Boddington et al. [2] and Kordylewski [3] for infinite Biot number (Frank-Kamenetskii conditions). Kordylewski [3] gave a precise criterion for the appearance of these transition conditions and calculated the transition values for the other class A geometries of an infinite circular cylinder and a sphere, also for infinite Biot number. This criterion is modified here to the case of finite Biot number to obtain the transition values for a range of Biot numbers. The transition values are shown to match on to the values of the Semenov case for small values of the Biot number α as ϵ tr= 0.25 − O(α 2), ϵ tr= 4(j + 1)α/e 2) +O(α 2), θ tr= 4 + O(α), where j = 0, 1, 2 for slab, cylinder, and sphere, respectively.

Temperature distributions, critical conditions and scaling for exothermic materials under different boundary conditions. Part 2.—Intermediate boundary conditions (arbitrary biot number) and bodies of low symmetry

An approximate method enabling rapid calculation of critical size as a function of the ambient temperature for bodies of arbitrary shape under generalized boundary conditions is given. These boundary conditions apply when there is resistance to heat transfer both within and at the surface of the reactant. The approach uses the same size and shape factors R0 and j as before and invokes two lengths, lin and lex, which are fixed by the ambient temperature, reactant composition and heat-transfer characteristics. Our results allow small-scale experiments to be used to give safe predictions for larger systems. This is important as conventional methods may err on the dangerous side.

Correction of kinetic data in non-isothermal reactions with non-uniform temperatures: analytical treatments for spherical reactant masses

Exothermic reactions normally proceed non-isothermally, and to treat them quantitatively allowance must be made for variations in internal temperature. Except for the special cases of the infinite slab and infinite cylinder, this has so far had to be done solely by numerical computation. The sphere and other finite, three-dimensional geometries have eluded analytical solution, even in the stationary state. The present paper presents two useful treatments for spherical geometry. They provide families of compact analytical expressions of very satisfactory precision for all the important aspects, errors rarely being greater than 2% right up to criticality. The aspects studied include temperature-position profiles, central temperature-excesses and surface temperature-gradients. (Both stable and unstable subcritical conditions as well as critical cases can be treated.) From these there follow values for rate-constant correction-factors f (and their reciprocals, the effectiveness factors η) and, in turn, the means of correcting errors in Arrhenius activation energies and in reaction orders - again in expressive, simple forms. In addition, explicit equations have been set out for the first time to cover the whole range of boundary conditions (arbitrary Biot number, β ) from Frank-Kamenetskii ( β → ∞) to Semenov ( β ═ 0) extremes. This is a major development, since mere tabulation of numerical solutions has hitherto been a formidable task. Calculating procedures are detailed in an Appendix. Endothermic reactions show self-cooling and self-repression instead of self-stimulation. These aspects are all expressively encapsulated in the change of sign of single coefficients in tidy equations. The two routes - quintic approximation (q. a.) and second-order reversion (r2) - can both be applied most conveniently in parametric form (this is necessary for r2 but not for q. a). The parameters l and x that appear naturally in the two treatments bear a close relation to corresponding integration constants encountered in one- and two-dimensional geometry. Each has striking physical significance, l being most directly related to the effectiveness factor and x to the central temperature excess.

Thermal theory of spontaneous ignition: criticality in bodies of arbitrary shape

For an exothermic reaction to lead to explosion, critical criteria involving reactant geometry, reaction kinetics, heat transfer and temperature have to be satisfied. In favourable cases, the critical conditionstreatments are either confined to idealized geometries, namely, the sphere, infinite cylinder or infinite slab, or require the simplest representations of heat transfer. may be summarized in a single parameter, Frank-Kamenetskii’s δ being the best known, but analytical. In the present paper a general steady-state description is given of the critical conditions for explosion of an exothermic reactant mass of virtually unrestricted geometry in which heat flow is resisted both internally (conductive flow) and at the surface (Newtonian cooling). The description is founded upon the behaviour of stationary-state systems under two extremes of Biot number—that corresponding to Semenov’s case ( B i → 0 ) and that corresponding to Frank-Kamenetskii’s case ( B i → ∞ ) it covers these and intermediate cases. For Semenov’s conditions, the solution is already known, but a fresh interpretation is given in terms of a characteristic dimension—the mean radius R s . A variety of results for criticality is tabulated. For Frank-Kamenetskii’s conditions, the central result is an approximate general solution for the stationary temperature distribution within any body having a centre. Critical conditions follow naturally. They have the simple form: q σ A exp ⁡ ( − E / R T a ) k R T a 2 / E R 0 2 er ≡ δ er ( R 0 ) = 3 F ( j ) , where F ( j ) is close to unity, being a feeble function of shape through a universally defined shape parameter j , and δ cr ( R 0 ) is Frank-Kamenetskii’s δ evaluated in terms of a universally defined characteristic dimension R 0 —a harmonic square mean radius weighted in proportion to solid angle: 1 R 0 2 = 1 4 π ∫ ∫ d ω a 2 . Expressions for the mean radius R 0 have been evaluated and are tabulated for a broad range of geometries. The critical values generated for δ are only about 1 % in error for a great diversity of shapes. No adjustable parameters appear in the solution and there is no requirement of an ad hoc treatment of any particular geometric feature, all bodies being treated identically. Critical sizes are evaluated for many different shapes. For arbitrary shape and arbitrary Biot number (0 &lt; &lt; ∞) an empirical criterion is proposed which predicts critical sizes for a great diversity of cases to within a few parts per cent. Rigorous, closely adjacent upper and lower bounds on critical sizes are derived and compared with our results and with previous investigations, and the status of previous approaches is assessed explicitly. For the most part they lack the generality, precision and ease of application of the present approach.