Lumped parameter model heat transfer

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Of more concern is the effect of the storage of thermal energy in the drum metalwork and the transfer of heat between the drum metal and the drum contents. Using a lumped parameter model of the drum metal, an energy balance yields ... 

C. T. Hsu, P. Cheng, and K. W. Wong, A Lumped Parameter Model for Stagnant Thermal Conductivity of Spatially Periodic Porous Media, ASMEJ. Heat Transfer, (117) 264-269,1995. 

WILLIAMS, W.R., ANDERSON, J.C., Estimation of time to rupture in a fire using 6FIRE, a lumped parameter UFg cylinder transient heat transfer/stress analysis model ,... 

FIGURE 1.6 A lumped parameter model of the infant-incut>alor dynamics used by Simon and Reddy (1994) to simulate the effect of various control modes in a convectively heated infant incubator. Infants core, and skin are modeled as two separate compartments. Hie incubator air space, the incubator wall, and the mattress are treat as three compartments. Heat interactions occur between the core (infant s lungs) and the incubator air space through breathing. Skin-core heat interactions are predominantly due to blood flow to the skin. Heat transfer between the infant s skin and the incubator air are due to conduction and convection. Heat transfer from the skin to the mattress is via conduction, and heat transfer to the wall is via radiation from skin and convection from the air. 

With an effective thermal model of the cells, modules and overall system, an analysis of the performance under different situations and load conditions can be evaluated. This proves to be a very useful tool in the development of the pack as these thermal models can be input into computational fluid dynamic (CFD) models to determine how the cells will heat during operation. A good CFD model can be used to determine flow rates, turbulence, and heat transfer within a pack. In addition, it is possible to use a lumped parameter model to develop a simplified model where the external parameters are basically ignored and the model is designed using fully adjustable parameters to do high-level evaluations of the thermal effectiveness of a system. 

Simultaneous heat and mass transfer, the subject of this chapter, is a complicated process. Analyzing this process to And simple but useful results depends on making effective approximations. The approximations exploit both the similar mathematics used for the processes and the similar numerical values of the transport coefficients. This can be true for both distributed and lumped-parameter models. More specifically, for gases, D and a are nearly equal, and k and h/pCp are very similar. 

Bunimovich et al. (1995) lumped the melt and solid phases of the catalyst but still distinguished between this lumped solid phase and the gas. Accumulation of mass and heat in the gas were neglected as were dispersion and conduction in the catalyst bed. This results in the model given in Table V with the radial heat transfer, conduction, and gas phase heat accumulation terms removed. The boundary conditions are different and become identical to those given in Table IX, expanded to provide for inversion of the melt concentrations when the flow direction switches. A dimensionless form of the model is given in Table XI. Parameters used in the model will be found in Bunimovich s paper. 

The application of CFD to packed bed reactor modeling has usually involved the replacement of the actual packing structure with an effective continuum (Kvamsdal et al., 1999 Pedernera et al., 2003). Transport processes are then represented by lumped parameters for dispersion and heat transfer (Jakobsen... 

De Wasch and Froment (1971) and Hoiberg et. al. (1971) published the first two-dimensional packed bed reactor models that distinguished between conditions in the fluid and on the solid. The basic emphasis of the work by De Wasch and Froment (1971) was the comparison of simple homogeneous and heterogeneous models and the relationships between lumped heat transfer parameters (wall heat transfer coefficient and thermal conductivity) and the effective parameters in the gas and solid phases. Hoiberg et al. (1971)... 

Many working groups have modeled the performance of diesel particulate traps during the past few decades. Concentrated parameter models (CSTR assumption) have been applied for the evaluation of formal kinetic models and model parameters. The formal kinetic parameters lump the heat and mass transfer effects with the reaction kinetics of the complicated reaction network of diesel soot combustion. Those models and model parameters were used for the characterization of the performance of different filter geometries and filter materials, as well as of the performance of a variety of catalytically active coatings and fuel additives [58],... 

Considering the models in Table I, it follows that the response of model III-T will be more close to reality due to (i) the correct way the transfer phenomena in and between phases is set up, and (ii) radial gradients are taken into account. Therefore, the responses of the different models will be compared to that one. It is obvious that the different models can be derived from model III-T under certain assumptions. If the mass and heat transfer interfacial resistances are negligible, model I-T will be obtained and its response will be correct under these conditions. If the radial heat transfer is lumped into the fluid phase, model II-T will be obtained. This introduces an error in the set up of the heat balances, and the deviations of type II models responses will become larger when the radial heat flux across the solid phase becomes more important. On the other hand, the one-dimensional models are obtained from the integration on a cross section of the respective two-dimensional versions. In order to adequately compare the different models, the transfer parameters of the simplified models must be calculated from the basic transfer... 

As shown in the preceding parts, kinetic parameters cannot be directly calculated when internal heat transfer limits pyrolysis. A model taking into account both kinetic scheme and heat- mass transfers becomes necessary, A one-dimension model has already been implemented and solved. It features a classical set of equations for heat and mass transfers in porous media, i.c. heat transfer through convection, conduction, radiation and mass transfer due to pressure gradient (Darcy s law) and binary diffusion. Different kinetic schemes from e literature arc and will be tested mass-loss as lumped first order reaction, formation of volatiles, tars and char from decomposition of cellulose, hcmicellulose and lignin [26], the Broido-Shafi2adeh model [30] and the one proposed by Di Blasi [31]. None of them can describe the composition of the volatiles and supplementary schemes have to be found. 

Under normal physiological conditions, the following heat transfer terms should be incorporated in a lumped or distributed parameter model metabolic heat generation conduction and convection within the body heat exchange with the environment by radiation, conduction and convection heat loss from the skin by evaporation of sweat and water diffused across the skin and respiratory heat loss. Expressions for each of these terms, along with the parameter values, are given elsewhere (Cooney, 1976). 

The estimated model parameters are given in table 1. Note that the estimated model parameters cannot be considered to represent intrinsic kinetic constants. They represent lumped parameters which can be disguised by possible heat and mass transfer effects which are not accounted for in the model. 

The general heat balance equation corresponds to the formalization of the general calorimetric model by means of the set of equations with lumped parameters. To consider the thermal properties of a calorimeter, the detailed form of this equation has to be derived. It is necessary to define in it the number and configuration of the distinguished domains and the centers which separate these domains and where heat transfer takes place. 

Due to both common knowledge and related model calculations, at a technical scale that is characterized by L/2Rp > 200, axial dispersion as compared with convection contributes little to overall mass and heat transfer. Neglecting both steps in the calculation of breakthrough curves causes deviations within the order of magnitude of numerical errors for the solution of the resulting equations for convection. Further simplification is possible by lumping parameters. A comparison of mass transport resistances in a pellet leads to the following overall resistance [101] ... 

In this paper, only the latter will be considered for the purpose of showing the influence of algebraic equations on open loop stability of process systems using illustrative examples of a continuous fermentation process model and a countercurrent heat exchanger. Special emphasis is put into the effect of different mechanisms, such as convection, transfer and reaction, occurring in lumped parameter process systems on local stability. 

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