Lumped-differential Formulations Research Articles

An improved lumped model for freezing of a freely suspended supercooled water droplet in air stream

This work deals with the mathematical modeling of the transient freezing process of a supercooled water droplet in a cold air stream. The aim is to develop a simple yet accurate lumped-differential model for the energy balance for a freely suspended water droplet undergoing solidification, that allows for cost effective computations of the temperatures and freezing front evolution along the whole process. The complete freezing process was described by four distinct stages, namely, supercooling, recalescence, solidification, and cooling. At each stage, the Coupled Integral Equations Approach (CIEA) is employed, which reduces the partial differential equation for the temperature distribution within the spherical droplet into coupled ordinary differential equations for dimensionless boundary temperatures and the moving interface position. The resulting lumped-differential model is expected to offer improved accuracy with respect to the classical lumped system analysis, since boundary conditions are accounted for in the averaging process through Hermite approximations for integrals. The results of the CIEA were verified using a recently advanced accurate hybrid numerical-analytical solution through the Generalized Integral Transform Technique (GITT), for the full partial differential formulation, and comparisons with numerical and experimental results from the literature. After verification and validation of the proposed model, a parametric analysis is implemented, for different conditions of airflow velocity and droplet radius, which lead to variations in the Biot numbers that allow to inspect for their influence on the accuracy of the improved lumped-differential formulation.

On the Thermal Performance of a Microparallel Channels Heat Exchanger

Abstract This paper presents the experimental and theoretical analysis of a micro heat exchanger designed for the waste heat recovery from a high concentration photovoltaic (HCPV) system. A test bench was built to analyze the thermal behavior of a heat exchanger targeted to work in a similar condition of an existing HCPV panel. A high power heater was encapsulated inside a copper cartridge, covered by thermal insulation, leading to dissipated heat fluxes around 0.6 MW/m2, representative of the heat flux over the solar cell within the HCPV module. The experimental campaign employed water as the coolant fluid and was performed for three different mass flow rates. An infrared camera was used to nonintrusively measure the temperature field over the micro heat exchanger external surface, while thermocouples were placed at the contact between the heat exchanger and the heater, and at the water inlet and outlet ports. In the theoretical analysis, a hybrid numerical–analytical treatment is implemented, combining the numerical simulation through the comsolmultiphysics finite elements code for the micro heat exchanger, and the analytical solution of a lumped-differential formulation for the electrical heater cartridge, offering a substantial computational cost reduction. Such computational simulations of the three-dimensional conjugated heat transfer problem were critically compared to the experimental results and also permitted to inspect the adequacy of a theoretical correlation based on a simplified prescribed heat flux model without conjugation effects. It has been concluded that the conjugated heat transfer problem modeling should be adopted in future design and optimization tasks. The analysis demonstrates the enhanced heat transfer achieved by the microthermal system and confirms the potential in reusing the recovered heat from HCPV systems in a secondary process.

Theoretical–experimental analysis of heat transfer in nonhomogeneous solids via improved lumped formulation, integral transforms and infrared thermography

Theoretical and experimental methodologies for the identification of spatially variable thermophysical properties and for simulating multidimensional heat transfer in heterogeneous materials are illustrated by using plate samples with aluminum oxide nanoparticles dispersed in a polymeric matrix. First, the heterogeneous nanocomposite plate is thermally characterized by means of a fairly simple experimental setup which can be modeled by a one-dimensional heat conduction formulation with space variable properties. Non-intrusive temperature measurements are obtained via infrared thermography, while the direct problem is handled by an error-controlled integral transform solution with an improved lumped-differential formulation, and the inverse analysis is undertaken via Bayesian inference, making use of the Markov Chain Monte Carlo method. Then, in order to illustrate the application of the methodologies here presented, an experimental multidimensional demonstration is provided consisting of a small electrical resistance attached to the plate, simulating a heat generating electronic device installed on the nanocomposite substrate, which in such situation works as a heat spreader modeled by an improved lumped-differential two-dimensional heat conduction formulation. The integral transform solution of the lumped-differential two-dimensional problem is then critically compared against the infrared thermography experimental results.

Development of Rigorous and Reduced Heterogeneous Dynamic Models for Fixed Bed Catalytic Reactor and Three-Phase Catalytic Slurry Reactor

In this work, the dynamic behavior of two chemical reactors (Fixed Bed Catalytic Reactor and the Three-Phase Reactor), in which different reactions of great industrial interest take place, is presented. Rigorous and reduced models were developed and their capabilities to adequately predict the dynamic behavior of these reactors were compared, indicating which model is more suitable for a specific application (design, optimization and control). Different approaches [Classic, Hermite, Finlayson, Pirkle, Dixion, Generic and Orthogonal collocation (one point)] were used to obtain the reduced models. With these techniques, a mathematical order reduction was carried out, which eliminates the spatial co-ordinate (of the catalyst particle or of the reactor) and promotes radially lumped-differential formulations. The proposed rigorous models were able to predict the main characteristics of the dynamic behavior of the fixed bed catalytic reactor as well as of the three-phase reactor, including the inverse response phenomena and the hot spot present in the former. This knowledge is essential to design and control these reactors. The computational time demanded for the solution of the rigorous models is high in comparison to the reduced ones, which restricts the use of the rigorous models to cases where time is not a limiting factor. Otherwise, when on-line applications are required, the reduced models are more adequate. The models based on reduction techniques overcame computational burden with a faster and easier numerical solution, as well as other difficulties found in rigorous heterogeneous models, especially related to the large number of parameters and sophisticated numerical procedures required by the solution.

Improved lumped model for thermal analysis of high burn-up nuclear fuel rods

High burn-up nuclear fuel elements have been intensively studied for prolonged lifetime of existing reactors and for next-generation advanced reactors. This paper presents an improved lumped-differential formulation for one-dimensional transient heat conduction in a heat generating cylinder with temperature-dependent thermo-physical properties typical of high burn-up nuclear fuel rods. Two-points Hermite approximations for integrals ( H 1,1/ H 1,1) were used to obtain the average temperature and heat flux in the radial direction. As a testing case, transient heat conduction in a nuclear fuel rod was computed with the thermo-physical properties represented by the MATPRO correlations. The problem was formulated and solved by using the symbolic/numerical computation software system MATHEMATICA. The solutions of the proposed improved lumped model were validated by comparing with numerical solutions of the original distributed parameter formulation of the transient heat conduction problem, as well as by comparing with simulation results of a cold leg SBLOCA with RELAP5/MOD3.

Improved lumped-differential formulations and hybrid solution methods for drying in porous media

The present paper illustrates the construction of hybrid tools for the problem formulation and solution methodology towards the simulation of heat and mass transfer during drying in capillary porous media. First, a problem reformulation strategy is discussed, known as the coupled integral equations approach (CIEA), which offers improved lumped-differential formulations in different classes of problems, in comparison against classical lumping schemes, allowing for a reduction on the number of independent variables to be considered in specific formulations. Second, the generalized integral transform technique (GITT) is employed, as a hybrid numerical–analytical solution methodology for convection–diffusion problems. An example is provided related to drying in capillary porous cylindrical media as formulated by the two-dimensional Luikov's system of equations.

Contaminant transport in finite fractured porous medium: integral transforms and lumped-differential formulations : Cotta, R. M. et al. Annals of Nuclear Energy, 2003, 30, (3), 261–285

This work presents a thermoeconomic analysis of a cogeneration system using the exhaust gas from a natural gas-fueled diesel power generator as heat source for an ammonia-water absorption refrigeration system. The purpose of the analysis is to obtain both unit exergetic and exergoeconomic costs of the cogeneration system at different load conditions and replacement rates of diesel oil by natural gas. A thermodynamic model of the absorption chiller was developed using the Engineering Equation Solver (EES) software to simulate the exergetic and exergoeconomic cogeneration costs. The data entry for the simulation model included available experimental data from a dual-fuel diesel power generator operating with replacement rates of diesel oil by natural gas of 25%, 50% and 75%, and varying engine load from 10 kW to 30 kW. Other required data was calculations using the GateCycle software, from the available experimental data. The results show that, in general, the cogeneration cold unit exergetic and exergoeconomic costs increases with increasing engine load and decreases with increasing replacement rate of diesel oil by natural gas under the conditions investigated. Operating with 3/4 of the rated engine power and replacing 50% of diesel oil by natural gas, the exergoeconomic cost of the produced power is increased by 75%, and the exergoeconomic cost of the produced cold is decreased by 17%. The electric power unit exergetic and exergoeconomic costs indicate that the replacement of diesel oil by natural gas is feasible in the present considerations for engine operation at medium and high loads.

Contaminant transport in finite fractured porous medium: integral transforms and lumped-differential formulations

Mass transfer within saturated porous media with discrete finite fractures is examined, by simultaneously solving the convection–diffusion equation for the contaminant transport along the fracture and the two-dimensional diffusion equation for contamination within the porous matrix. A lumped-differential formulation based on Hermite integration is proposed for the porous matrix thus eliminating the dependence on the transverse direction. The resulting coupled partial differential equations are then handled through the generalized integral transform technique (GITT), which yields analytical expressions for the space dependence and numerical estimates for the concentration fields as a function of time. Different analytical filtering strategies are proposed and analyzed in terms of convergence rates. An illustrative example is considered for both the constant and time-variable contamination physical situations.

Improved lumped-differential formulations in hyperbolic heat conduction

The present work aims at applying the ideas in the so-called Coupled Integral Equations Approach (CIEA) to the one-dimensional thermal wave propagation problem in a finite solid medium, leading to improved lumped-differential formulations through Hermite-type approximations for integrals. The application of CIEA methodology makes it possible to reduce the partial differential equation governing the hyperbolic heat conduction problem to a simple system consisting of two or three ordinary differential equations for the average and surface temperatures, depending on the choice of the Hermite-type approximation. The Runge-Kuta method, enclosed in the IVPRK routine from IMSL Library [1], is used to solve the system of ordinary differential equations. Results for the average and surface temperatures are computed for different thermal relaxation times.

Enhanced lumped-differential formulations of diffusion problems

This paper is directly related to the task of modelling diffusion problems, prior to the choice of solution strategies. The approach presented is in fact a reformulation tool, aimed at reducing, as much as possible and within prescribed accuracy requirements, the number of dimensions in a certain diffusion formulation. It is shown how appropriate integration strategies can be employed to deduce mathematical formulations of comparable simplicity and improved accuracy in comparison with well-established classical lumping procedures. The approach is demonstrated through representative heat conduction problems, and the enhancement characteristics are examined against the classical lumped system analysis (CLSA) and the exact solutions of the fully differential systems.

IMPROVED APPROXIMATE FORMULATIONS FOR ANISOTROPIC HEAT CONDUCTION

Approximate lumped-differential formulations for heat conduction in anisotropic media are presented, here illustrated for a fin-type example. The proposed formulation is as simple as the classical lumped system analysis, but considerably more accurate, since temperature gradients in the lumped direction are more adequately approximated. Results obtained fin are compared to the exact two-dimensional solution, as well as to the classical lumped system analysis, for a range of the governing parameters, aspect ratio, thermal conductivity ratios and Biot numbers. Illustrative results are reported for a general anisotropic material. © 1997 Elsevier Science Ltd

Lumped- Differential Formulations for Drying in Capillary Porous Media

ABSTRACT Simultaneous heat and mass transfer, during drying of moist capillary porous media, is modeled through different lumped-differential formulations, obtained from spatial integration of the original set of Luikov's equations for temperature and moisture potential. Besides the classical lumped system analysis, improved formulations are proposed, of same mathematicai simplicity, based on approximate evaluations of both temperature and moisture gradients within the medium. The relative merits of such improved zero-dimensional formulations are established, through critical comparisons against analytical solutions for the full partial diflerential system, over a wide range of the governing parameters.

Improved Hybrid Lumped-Differential Formulation for Double-Pipe Heat Exchanger Analysis

Based on recent developments in the so-called Coupled Integral Equation Approach for the approximate solution of heat and mass diffusion problems, the classical lumped-differential formulation for double-pipe heat exchangers is extended and significantly improved in terms of overall accuracy. The energy equation for the outer annular channel is formally integrated and temperature gradients approximately taken into account, in contrast to the conventional lumped system analysis. The resulting partial differential equation for the inner stream, involving a more general type of boundary condition, is analytically handled through the generalized integral transform technique. A critical accuracy analysis is then performed in order to demonstrate the improvement over the plain lumped-differential formulation.

Dynamic analysis of double-pipe heat exchangers subjected to periodic inlet temperature disturbances

A mixed lumped-differential formulation is employed to model the transient energy equations for fully developed laminar-laminar or laminar-turbulent flow situations in concurrent or countercurrent double-pipe heat exchangers. The temperature distribution in the outer annular channel is radially lumped, providing a more general boundary condition for the inner channel differential energy equation, coupled through the interface condition. The case of periodically varying inlet temperatures is more closely considered, and the dynamic response of the exchanger is established in terms of the governing dimensionless parameters, such as heat capacity flow rate ratio, dimensionless inlet temperature oscillation frequency, and relative wall thermal resistence. The ideas in the generalized integral transform technique are extended to yield analytical solutions to the related periodic problem defined in the complex domain, and offer highly accurate numerical results for quantities of practical interest, such as fluids bulk temperatures.

Counterflow double-pipe heat exchanger analysis using a mixed lumped-differential formulation

An analysis is made of double-pipe heat exchangers under a thermally developing countercurrent flow condition. A lumped-differential mixed formulation is employed, by radially lumping the temperature field in the outer channel, which results in a more involved boundary condition for the inner differential system, involving the axially varying outer channel bulk temperature. The generalized integral transform technique is utilized to provide a reliable and straightforward analytical solution to this class of problems. Numerical results for heat transfer quantities are presented in terms of the dimensionless governing parameters along the thermal entry region, allowing for critical comparisons against the concurrent flow situation, limiting solutions and engineering-type correlations.