Mass momentum and energy balance equation

To describe the flow in a horizontal heated capillary we use the mass, momentum and energy balance equations. At moderate velocity, the effects due to compressibility of liquid and vapor, as well as energy dissipation in gaseous and liquid phases are negligible. Assuming that thermal conductivity and viscosity of the vapor and the liquid are independent of temperature and pressure, we arrive at the following system of equations ... 

Estimates (9.12) and (9.14) effectively reduce the problem of flow in a heated micro-channel to solving a system of one-dimensional mass, momentum and energy balance equations. They have the following form ... 

Based on the above-mentioned assumptions, the mass, momentum and energy balance equations for the gas and the dispersed phases in two-dimensional, two-phase flow were developed [14], In order to solve the mass, momentum and energy balance equations, several complimentary equations, definitions and empirical correlations were required. These were presented by [14], In order to obtain the water vapor distribution the gas phase the water vapor diffusion equation was added. During the second drying period, the model assumed that the particle consists of a dry crust surrounding a wet core. Hence, the particle is characterized by two temperatures i.e., the particle s crust and core temperatures. Furthermore, it was assumed that the heat transfer from the particle s cmst to the gas phase is equal to that transferred from the wet core to the gas phase i.e., heat and mass cannot be accumulated in the particle cmst, since all the heat and the mass is transferred by diffusion through the cmst from the wet core to the surrounding gas. Based on this assumption, additional heat balance equation was written. 

When the contribution of the particles to the heat capacity of the gas stream is important (i.e., the high particle loading case), the mass, momentum and energy balance equations in (7.77) must be solved simultaneously. Typically, the details of streams 1 and 2 are known, and we need to calculate the outlet velocity and temperature (i.e., stream 3). Using the mass balance, we can calculate. With Wg, we... 

The third chapter covers convective heat and mass transfer. The derivation of the mass, momentum and energy balance equations for pure fluids and multi-component mixtures are treated first, before the material laws are introduced and the partial differential equations for the velocity, temperature and concentration fields are derived. As typical applications we consider heat and mass transfer in flow over bodies and through channels, in packed and fluidised beds as well as free convection and the superposition of free and forced convection. Finally an introduction to heat transfer in compressible fluids is presented. 

Unlike the aforementioned models, Fyhr and Rasmuson [41,42] and Cartaxo and Rocha [43] used an Eulerian-Lagrangian approach, in which the gas phase is assumed as the continuous phase and the solids particles are occupying discrete points in the computational domain. As a consequence, mass, momentum, and energy balance equations were solved for each particle within the computational domain. 

In extrusion, as well as in many other processes, one deals with the transport of mass, momentum, and energy. Balance equations are used to describe the transport of these quantities. They are universal physical laws that apply to all media (solids and fluids). Matter is considered as a continuum. Thus, the volume over which the balance equation is formulated must be large enough to avoid discontinuities. 

In this section, we discuss the role of numerical simulations in studying the response of materials and structures to large deformation or shock loading. The methods we consider here are based on solving discrete approximations to the continuum equations of mass, momentum, and energy balance. Such computational techniques have found widespread use for research and engineering applications in government, industry, and academia. 

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

Boundary and interface conditions must be known if solutions to the conservation equations are to be obtained. Since these conditions depend strongly on the model of the particular system under study, it is difficult to give general rules for stating them for example, they may require consideration of surface equilibria (discussed in Appendix A) or of surface rate processes (discussed in Appendix B). However, simple mass, momentum, and energy balances at an interface often are of importance. For this reason, interface conditions are derived through introduction of integral forms of the conservation equations in Section 1.4. 

In this sense, differential equations appear more tractable since they do not require particle tracking. Indeed, the solution of the coupled equations of mass, momentum and energy balance including the material equation, properly described on a suitable finite element mesh, theoretically provides the material lines. Nevertheless, the correct description of the basic experiments often requires the use of strong nonlinear terms. Such improvements may be unsatisfying from the numerical point of view since they can lead to stiff systems of nonlinear equations and to many convergence related problems. 

B Derive the differential equations that govern convection on the basis of mass, momentum, and energy balances, and solve these equations for some simple cases such as laminar flow over a flat plate,... 

For a flat plate, Ihe characteristic length is the distance x from the leading edge. The Reynolds number at which the flow becomes turbulent is called the critical Reynolds number. For flow over a flat plate, its value is taken to be Re = V.V ,/v = 5 X 10 The continuity, momentum, and energy equations for steady Iwo-diruensional incompressible flow with constant properties are determined from mass, momentum, and energy balances to be... 

The steady-state fluid mechanics problem is solved using the Fluent Euler-Euler multiphase model in the fluid domains. Mass, momentum and energy balances, the general forms of which are given by eqn. (4), (5), and (6), are solved for both the liquid and the gas phases. In solid zones the energy equation reduces to the simple heat conduction problem with heat source. By convention, / =1 designates the H2S04 continuous liquid phase whereas H2 bubbles constitute the dispersed phase 0 =2). 

Bird RB (1957) The equations of change and the macroscopic mass, momentum, and energy balances. Chem Eng Sci 6 123-181... 

The differential PEM fuel cell reactor is motivated by considering a small element in the serpentine flow channel fuel cell as shown in Figure 3.1 [12]. Mathematical models of fuel cells use differential mass, momentum and energy balances around the differential element as the defining equations for modeling larger and more complex flow fields [12]. In the differential element the only compositional variations are transverse to the membrane. The key element of a differential fuel cell is that the compositions in the gas phases in the flow channels at the anode and cathode are uniform. 

The heat transfer and flow inside the extruder are calculated by solving the classical continuum mechanics equations (mass, momentum and energy balances), according to the local geometry and boundary conditions. 

The temporal and spatial evolutions of the above five fields are determined by so-called balance equations (abbr. balances) for mass, momentum and energy. These equations represent axioms and read ... 

The model equations for mass, momentum, and energy balance and the EoS that will be useful in the following discussions are summarized in the following paragraphs. Any number of textbooks have detailed derivations and discussions (eg, Todreas and Kazimi, 1990 Collier and Thome, 1996). An area-averaged, transient, one-dimensional formulation for compressible fluids will be sufficient for the applications considered in this chapter. Some aspects of accounting for separate speeds for the vapor and liquid phases in a two-phase mixture are also included. 

A microscopic description assumes that a process acts as a continuum and that the mass, momentum, and energy balances can be written in the form of phenomenological equations. This is the usual level of transport phenomena where detailed molecular... 

The basic equations describing the steady-state and time-dependent behavior of a molten spinline made up of a single flber of circular cross section were first developed by Kase and Matsuo [55]. It was assumed that the filament cross-sectional area and polymer velocity and temperature varied with axial position down the spinline but that there were no radial variations. The model, therefore, is a one-dimensional model that is vahd provided that the fiber curvature is small. To derive the equations, we consider the differential control volume shown in Figure 15.26 and carry out a simultaneous mass, momentum, and energy balance. 

The conservation of mass gives comparatively Httle useful information until it is combined with the results of the momentum and energy balances. Conservation of Momentum. The general equation for the conservation of momentum is... 

To calculate the flow parameters under the conditions when the meniscus position and the liquid velocity at the inlet are unknown a priori. The mass, momentum and energy equations are used for both phases, as well as the balance conditions at the interface. The integral condition, which connects flow parameters at the inlet and the outlet cross-sections is derived. 

As for the mass and energy balance equations, steady-state conditions are obtained when the rate of change of momentum in the system is zero and... 

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