Heat transfer mathematical models

Zhou ZY, Hou QP, Yu AB Particle scale simulation of heat transfer in fluid bed reactors. In Belmiloudi A, editor Heat transfer—mathematical modelling, numerical methods and information technology, Rijeka, Croatia, 2011a, InTech, pp 383—408. 

In general, the desorptive behavior of contaminated soils and soHds is so variable that the requited thermal treatment conditions are difficult to specify without experimental measurements. Experiments are most easily performed in bench- and pilot-scale faciUties. Full-scale behavior can then be predicted using mathematical models of heat transfer, mass transfer, and chemical kinetics. 

A mathematical model of the operating characteristics of a modem HLW storage tank has been developed (60). This model correlates experimental data for the rate of radiolytic destmction of nitric acid, the rate of hydrogen generation owing to radiolysis of water, and cooling coil heat transfer. These are all functions of nitric acid concentration and air-lift circulator operation. 

Heat transfer in the furnace is mainly by radiation, from the incandescent particles in the flame and from hot radiating gases such as carbon dioxide and water vapor. The detailed theoretical prediction of overall radiation exchange is complicated by a number of factors such as carbon particle and dust distributions, and temperature variations in three-dimensional mixing. This is overcome by the use of simplified mathematical models or empirical relationships in various fields of application. 

Kom GA, Korn TM (1968) Mathematical handbook. McGraw-Hill, Boston Landau LD, Lifshitz EM (1959) Fluid mechanics, 2nd edn. Pergamon, London Morijama K, Inoue A (1992) The thermodynamic characteristics of two-phase flow in extremely narrow channels (the frictional pressure drop and heat transfer of boiling two-phase flow, analytical model). Heat Transfer Jpn Res 21 838-856 Ngan CD, Dussan EBV (1982) On the nature of the dynamic contact angle an experimental study. JEluidMech 118 27- 0... 

The mathematical model of a single die hole consists of equations which describe the polymer flow and heat transfer. In deriving this model the following assumptions are made ... 

The final physical properties of thermoset polymers depend primarily on the network structure that is developed during cure. Development of improved thermosets has been hampered by the lack of quantitative relationships between polymer variables and final physical properties. The development of a mathematical relationship between formulation and final cure properties is a formidable task requiring detailed characterization of the polymer components, an understanding of the cure chemistry and a model of the cure kinetics, determination of cure process variables (air temperature, heat transfer etc.), a relationship between cure chemistry and network structure, and the existence of a network structure parameter that correlates with physical properties. The lack of availability of easy-to-use network structure models which are applicable to the complex crosslinking systems typical of "real-world" thermosets makes it difficult to develop such correlations. 

In order to understand the effect of each process variable, a fundamental understanding of the heat transfer and polymer curing kinetics is needed. A systematic experimental approach to optimize the process would be expensive and time consuming. This motivated the authors to use a mathematical model of the filament winding process to optimize processing conditions. 

Ultrasound can thus be used to enhance kinetics, flow, and mass and heat transfer. The overall results are that organic synthetic reactions show increased rate (sometimes even from hours to minutes, up to 25 times faster), and/or increased yield (tens of percentages, sometimes even starting from 0% yield in nonsonicated conditions). In multiphase systems, gas-liquid and solid-liquid mass transfer has been observed to increase by 5- and 20-fold, respectively [35]. Membrane fluxes have been enhanced by up to a factor of 8 [56]. Despite these results, use of acoustics, and ultrasound in particular, in chemical industry is mainly limited to the fields of cleaning and decontamination [55]. One of the main barriers to industrial application of sonochemical processes is control and scale-up of ultrasound concepts into operable processes. Therefore, a better understanding is required of the relation between a cavitation coUapse and chemical reactivity, as weU as a better understanding and reproducibility of the influence of various design and operational parameters on the cavitation process. Also, rehable mathematical models and scale-up procedures need to be developed [35, 54, 55]. 

The mathematical model describing the two-phase dynamic system consists of modeling of the flow and description of its boundary conditions. The description of the flow is based on the conservation equations as well as constitutive laws. The latter define the properties of the system with a certain degree of idealization, simplification, or empiricism, such as equation of state, steam table, friction, and heat transfer correlations (see Sec. 3.4). A typical set of six conservation equations is discussed by Boure (1975), together with the number and nature of the necessary constitutive laws. With only a few general assumptions, these equations can be written, for a one-dimensional (z) flow of constant cross section, without injection or suction at the wall, as follows. 

The mathematical model is based on the superstructure shown in Fig. 11.2. The heat transfer fluid in heat storage remains in the storage vessel during heat transfer with only the process fluid pumped around. The superstructure also shows that each unit is capable of receiving external heating or cooling in addition to direct and indirect heat integration. 

Alternatively, one might satisfy convection near a boundary by invoking Il6 and Ilg where the heat transfer coefficient is taken from an appropriate correlation involving Re (e.g. Equation (12.38)). Radiation can still be a problem because re-radiation, n7, and flame (or smoke) radiation, II3, are not preserved. Thus, we have the art of scaling. Terms can be neglected when their effect is small. The proof is in the scaled resultant verification. An advantage of scale modeling is that it will still follow nature, and mathematical attempts to simulate turbulence or soot radiation are unnecessary. 

A mathematical model is described [138] in which the self-heating of material layers under industrial conditions is simulated. The model takes into account oxygen (or gas) diffusion and consumption, reactant conversion, heat conduction in, and heat transfer to and from the layer. Scale-up experiments were performed which showed the model can be successfully applied to predict the self-heating phenomenon in the layers. 

If a more complex mathematical model is employed to represent the evaporation process, you must shift from analytic to numerical methods. The material and enthalpy balances become complicated functions of temperature (and pressure). Usually all of the system parameters are specified except for the heat transfer areas in each effect (n unknown variables) and the vapor temperatures in each effect excluding the last one (n — 1 unknown variables). The model introduces n independent equations that serve as constraints, many of which are nonlinear, plus nonlinear relations among the temperatures, concentrations, and physical properties such as the enthalpy and the heat transfer coefficient. 

Trapaga and Szekely 515 conducted a mathematical modeling study of the isothermal impingement of liquid droplets in spray processes using a commercial CFD code called FLOW-3D. Their model is similar to that of Harlow and Shannon 397 except that viscosity and surface tension were included and wetting was simulated with a contact angle of 10°. In a subsequent study, 371 heat transfer and solidification phenomena were also addressed. These studies provided detailed... 

The dynamic relationships discussed thus far in this book were determined from mathematical models of the process. Mathematical equations, based on fundamental physical and chemical laws, were developed to describe the time-dependent behavior of the system. We assumed that the values of all parameters, such as holdups, reaction rates, heat transfer coeflicients, etc., were known. Thus the dynamic behavior was predicted on essentially a theoretical basis. 

DISCUSSION AND CONCLUSIONS In this study a general applicable model has been developed which can predict mass and heat transfer fluxes through a vapour/gas-liquid interface in case a chemical reaction occurs in the liquid phase. In this model the Maxwell-Stefan theory has been used to describe the transport of mass and heat. A film model has been adopted which postulates the existence of a well-mixed bulk and stagnant zones where the principal mass and heat transfer resistances are situated. Due to the mathematical complexity the equations have been solved numerically by a finite-difference technique. In this paper (Part I) the Maxwell-Stefan theory has been compared with the classical theory due to Pick for isothermal absorption of a pure gas A in a solvent containing component B. Component A is allowed to react by a unimolecular chemical reaction or by a bimolecular chemical reaction with... 

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