Transport in composite materials: Reduction to a self adjoint formalism

Ramkrishna, D.; Amundson, Neal R.

doi:10.1016/0009-2509(74)80170-3

Cited by 48 publications

(24 citation statements)

References 2 publications

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“…[21], in cases with initial temperature distributions of the steady-state form of Eq. [5] and in cases with the more general dynamic-state initial field. The formulas [A1] through [A3] are valid for the Neumann function Y, as well.…”

Section: Appendixmentioning

confidence: 99%

“…The coefficient matrix in Eq. [5] is determined by the geometry and boundary condition parameters, according to…”

Section: A the Principal Solutionmentioning

confidence: 99%

“…For the material regions at steady state, this can be done easily by using Eq. [5], but for the time-dependent temperature field in layer 1, the matter of satisfying Eq. [22] at each time instant is not straightforward.…”

Section: B Coupling Of the Individual Solutions For The Layersmentioning

confidence: 99%

“…The solution of the heat-conduction equation by various analytical techniques involves a SturmLiouville eigenvalue problem, [4] which in the case of composite media is fairly complicated, due to the discontinuous coefficients of the governing differential equation. Ramkrishna and Amundson [5] have given a convenient for- malism for the solution of the composite media diffusion problem and criteria for self-adjointness of the differential operator involved. Mikhailov et al [6] give a method for computer-aided solution of the eigenvalue problem for this case.…”

Section: Introductionmentioning

confidence: 99%

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Model for temperature profile estimation in the refractory of a metallurgical ladle

Fredman

Saxén

1998

Metall Mater Trans B

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Modeling of the transient thermal state of metallurgical ladles is motivated by the need for estimating the drop in temperature of the liquid metal in the ladle. On-line estimation of the state is required, since the same ladle is used in a number of casting cycles with rapid changes in boundary conditions for the temperature field, and the conditions in the current as well as previous cycles affect the thermal state. Although a large number of methods for the numerical solution of conduction-diffusion partial differential equations have been developed, there are still advantages to keeping thermal field computations at a relatively simple level, in contrast to the situation in the design process of ladles, where two-dimensional modeling may be required. Extensive computations under nonverifiable boundary and initial parameter values are not especially suited for real-time simulation of industrial processes. This article presents a novel approach to the solution of the one-dimensional transient heat conduction problem applied to ladle linings, relying on classical analytical techniques in combination with numerical methods. The performance of the model was validated by a comparison of predictions to thermocouple measurements from the refractory of a steelmaking ladle during a campaign of 26 casting cycles. Reasonable agreement between the measured and simulated variables could be established, which demonstrates the feasibility of the approach.

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Section: Appendixmentioning

confidence: 99%

“…The coefficient matrix in Eq. [5] is determined by the geometry and boundary condition parameters, according to…”

Section: A the Principal Solutionmentioning

confidence: 99%

Section: B Coupling Of the Individual Solutions For The Layersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Model for temperature profile estimation in the refractory of a metallurgical ladle

Fredman

Saxén

1998

Metall Mater Trans B

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“…It has been shown in [12] that this system is self-adjoint in the inner product (when the velocity in the two fluids are equal…”

Section: Plug Flow Regimementioning

confidence: 99%

Comparison of laminar and plug flow-fields on extraction performance in micro-channels

Malengier¹,

Tamalapakula²,

Pushpavanam³

2012

Chemical Engineering Science

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Several applications such as liquid-liquid extraction in micro-fluidic devices are concerned with the flow of two immiscible liquid phases. The stratified flow regime in these micro-channels has the inherent advantage that separation at the exit is efficient. In pressure driven flows in microchannels the flow profile is laminar and parabolic. This induces axial dispersion in the system which is known to reduce the efficiency of the process. In micro-channels electro-osmosis has been employed to overcome this drawback, leading to a flow profile that approaches the classical plug flow behaviour.In this work fundamental features of the velocity profiles in stratified flows are analysed. First the flow between two rectangular infinite flat plates is discussed. The influence of physical properties, in particular viscosity of the two liquids, on the velocity profiles is determined symbolically. The flow-profiles are classified in the parameter space of physical properties (viscosity ratio) and operating conditions (flow-rate ratio). The mass transfer in these systems is compared with that found when the two liquids follow an ideal plug flow behaviour. A symbolic solution based on the method of separation of variables is proposed for concentration when the flow profile is a plug flow. It is found that the plug flow profile gives a poorer performance than the laminar profile. This is attributed to the lower velocity prevailing at the interface in the plug flow regime. To understand this more clearly the effect of changing the height of the interface was also analysed and it was found that under some conditions the ideal plug flow can give a better performance than the laminar reactor. The results of this work will aid the experimentalist establish guidelines to determine if introducing electro-osmosis in a Hagen-Poiseuille flow will be beneficial or detrimental in liquid-liquid extraction.

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Experimental and modeling analysis of diffusive release from single‐shell microcapsules

et al. 2009

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in Wiley InterScience (www.interscience.wiley.com).There is much experimental and mathematical work that describes chemical transport from multilayered films of planar geometries. There is less so, however, for chemical transport from multilayered spheres, a common structure for controlled-release materials. Based on the Sturm-Liouville approach of Ramkrishna and Amundson (1974), explicit analytical solutions for the concentration profiles and release kinetics from spherical capsules are presented. Fluorescent dye-release studies using singleshelled microspheres called nanoparticle-assembled capsules were performed to validate the model for uniformly and nonuniformly sized capsules. The combined experiment-modeling approach allows optical microscopy images and release measurements to be readily analyzed for estimating diffusion coefficients in capsule core and shell walls. V IntroductionThe availability of new types of materials with engineered composition, porosity, internal gradation, and morphology is providing exciting opportunities in encapsulation and delivery applications, like in the biotechnology arena. Unilamellar and multilamellar vesicles, for example, have long been studied as drug and chemical delivery agents, 1,2 in which the NACs were synthesized by first mixing 14 ml of 5 mg/ml of PAH solution with 35 ml of 14.2 mM citrate solution in a 250 ml beaker under gentle magnetic stirring (speed ''4'' of 0 through 10) for 10 s. The resulting suspension turned turbid instantly indicating the formation of polymer-salt aggregates. After aging these aggregates for 20 min, 35 ml of a 1 mg/ml fluorescein sodium salt (''Na-Flu'') dye solution was added under gentle magnetic stirring for 10 s. This mixture was aged for another 10 min and 35 ml of 1.2 wt % To compare our model predictions of the fractional release rate with experiments, the following parameters were required to be estimated from experiments: (i) x 1 , the ratio between the inner and outer radii, (ii) P 12 , partition coefficient between the polymer-salt aggregate and the interior of the shell, (iii) P 2b , the partition coefficient between the outer shell surface

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Transport in composite materials: Reduction to a self adjoint formalism

Cited by 48 publications

References 2 publications

Model for temperature profile estimation in the refractory of a metallurgical ladle

Model for temperature profile estimation in the refractory of a metallurgical ladle

Comparison of laminar and plug flow-fields on extraction performance in micro-channels

Experimental and modeling analysis of diffusive release from single‐shell microcapsules

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