Conservation equations inviscid flows

In order to illustrate how these integral equations are derived, attention will be given to two-dimensional, constant fluid property flow. First, consider conservation of momentum. It is assumed that the flow consists of a boundary layer and an outer inviscid flow and that, because the boundary layer is thin, the pressure is constant across the boundary layer. The boundary layer is assumed to have a distinct edge in the present analysis. This is shown in Fig. 2.20. 

The governing inviscid flow equations of continuity, energy, and momentum (in conservation form) in gas region are Euler Equations (9)-(l 1) ... 

Boundary Layer Concept. The transfer of heat between a solid body and a liquid or gas flow is a problem whose consideration involves the science of fluid motion. On the physical motion of the fluid there is superimposed a flow of heat, and the two fields interact. In order to determine the temperature distribution and then the heat transfer coefficient (Eq. 1.14) it is necessary to combine the equations of motion with the energy conservation equation. However, a complete solution for the flow of a viscous fluid about a body poses considerable mathematical difficulty for all but the most simple flow geometries. A great practical breakthrough was made when Prandtl discovered that for most applications the influence of viscosity is confined to an extremely thin region very close to the body and that the remainder of the flow field could to a good approximation be treated as inviscid, i.e., could be calculated by the method of potential flow theory. 

When Ugo > af o, it is necessary to insert a discontinuity in the flow in order to obtain a solution to the inviscid conservation equations. Across the discontinuity, all interphase transfer processes are effectively frozen and hence no changes occur in droplet radius, temperature and velocity. The vapour phase conditions donwstream of the discontinuity are easily calculated by a standard Rankine-Hugoniot analysis and these provide the initial conditions for the numerical integration procedure. 

Fluid mechanics is the study of the motion of flowing or stirred fluids, usually liquids or gases. In electrochemical technology it has two different applications (1) to describe the movement of electrolyte solutions in a cell, since this will be a principal driving force for mass transport to the electrodes and (2) to ensure the proper design of the pipes, valves and junctions which join the cell to the rest of the plant. Quantitative fluid mechanics is based upon the continuity equations which state that at all points in space, charge, mass, momentum and, for inviscid flow (i.e. fluids where no viscous forces operate) energy must be conserved. This section will deal mainly with the qualitative concepts because of the very complex nature of flow in most electrolysers. 

The physical aspects of any fluid flow are governed by three principles mass is conserved, Newton s second law is fulfilled (also referred as momentum equation) and energy is conserved these principles are expressed in integral equations or partial differential equations (continuity, momentum and energy equations), being the most common form the Navier-Stokes equations for viscous flows and the Euler equations for inviscid flows. 

Anderson and McFadden [8] listed a representative (but relatively simple) model that have been considered by a number of authors. The hydrodynamic equations governing inviscid, compressible flow of a single component fluid near its critical point are described by conservation equations for mass, momentum, and entropy (energy). 

As remarked in Section 2.2, most flames are essentially constant pressure systems, and if the flow is inviscid it is then not necessary to consider the momentum equation at all. Inviscid flow will be assumed for the purpose of the following discussion. Since the flame lies in the radial direction (Fig. 11), the gradients of all dependent variables except the radial velocity are zero in this direction, and the radial diffusive fluxes also vanish. By use of Eqs. (2.1) and (2.2b), conservation of mass in the steady state axisymmetric flow gives... 

The governing equations for the flow are obtained by one-dimensional approximations of conservation laws for mass, momentum, and energy. Often the additional assumptions of inviscid, adiabatic flow are invoked, and as a consequence, the flow is regarded as isentropic throughout. From the conservation laws, one may derive an important relation between the cross-sectional area, the velocity, and the local Mach number ... 

The steady two-dimensional diabatic flow is described by the equations for mass, momentum and energy in conservation form (Schnerr and Dohrmann [7], Dohrmann [8]). Real gas effects are not yet included and inviscid fluids are assumed. Here the classical nucleation theory of Volmer [9] is used which gives a good qualitative representation of the behavior of condensing in the supersaturated state (Wegener [iO]). Oswatitsch [11] introduced this theory into the calculation of flow processes, a summary of all basic relationships for compressible flows with heat addition is given by Zierep [12]. To compute the nucleation rate J per unit time and volume, we take... 

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