Conservation equations dimensional analysis

Initial shock-wave overpressure can be determined from a one-dimensional technique. It consists of using conservation equations for discontinuities through the shock and isentropic flow equations through the rarefaction waves, then matching pressure and flow velocity at the contact surface. This procedure is outlined in Liepmatm and Roshko (1967) for the case of a bursting membrane contained in a shock tube. From this analysis, the initial overpressure at the shock front can be calculated with Eq. (6.3.22). This pressure is not only coupled to the pressure in the sphere, but is also related to the speed of sound and the ratio of specific heats. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

A critique and more thorough review of the time-lag ideas has been published [7]. The principal value of time-lag concepts lies in the wide range of problems to which they can be applied with relative ease. Their principal deficiency lies in the difficulty of relating Xi to the fundamental processes occurring. If the one-dimensional, time-dependent conservation equations are linearized about any one of the steady-state solutions of Chapter 7 and p is calculated from a perturbation analysis, then it is found... 

In the theoretical analysis of shock instability, shock waves that are not too strong are presumed to propagate axially back and forth in a cylindrical chamber, bouncing off a planar combustion zone at one end and a short choked nozzle at the other [101], [102]. The one-dimensional, time-dependent conservation equations for an inviscid ideal gas with constant heat capacities are expanded about a uniform state having constant pressure p and constant velocity v in the axial (z) direction. Since nonlinear effects are addressed, the expansion is carried to second order in a small parameter e that measures the shock strength discontinuities are permitted across the normal shock, but the shock remains isentropic to this order of approximation. Boundary conditions at the propellant surface (z = 0) and at the... 

Limiting the analysis to a stationary fully developed one dimensional flow, the conservation equations become ... 

For a vertical condensing liquid film, scale analysis of the 2-dimensional constant-property conservation equations of mass, momentum and energy, with appropriate boundary conditions, shows that Rafii is a measure of the slenderness ratio L/d of the film (18). Here L is taken as the streamwise length of the condensation film and d an average film thickness. This interpretation is valid in the limit where the momentum equation is a balance of viscous and gravitational forces. Under such conditions, a velocity scale is... 

In stable and unstable conditions the velocity profiles of the atmospheric surface layer deviate from the logarithmic law (16.57). In this section we will outline briefly the forms of velocity profiles in these conditions. Since the stratified-boundary-layer conservation equations cannot be solved (because of the closure problem), we must resort to empirical profiles, based largely on dimensional analysis. 

By and large we can describe the results of the analysis of distributed parameter systems (i.e., flow reactors other than CSTRs) in terms of the gradients or profiles of concentration and temperature they generate. To a large extent, the analysis we shall pursue for the rest of this chapter is based on the one-dimensional axial dispersion model as used to describe both concentration and temperature fields within the nonideal reactor. The mass and energy conservation equations are coupled to each other through their mutual concern about the rate of reaction and, in fact, we can use this to simplify the mathematical formulation somewhat. Consider the adiabatic axial dispersion model in the steady state. 

To quantify the balances in a CPFR, one may begin from the analogy between the space profile in a CPFR and the c/t profile in a DCSTR (Fig. 3.30). This implies a reformulation of Equ. 2.3d so that the time and space coordinates are interchanged. Dimensional analysis shows that the term representing length dependence must be multiplied with a velocity. Thus, for an ideal CPFR, with z is the length coordinate, the conservation equation is... 

The answer to this question is the subject oiscaling and dimensional analysis. In general, scaling involves the nondimensionalization of the conservation equations where the characteristic variables used for nondimensionalization are selected as their maximum values, e.g., the maximum values of velocity, temperature, length, and the like, in a particular problem. Let s see specifically how this method works and why it can often lead to a simplification of partial differential equations. 

The question arises how do we identify the variables for a Dimensional Analysis study The best way to identify the variables for use in a Dimensional Analysis is to write the conservation laws and constitutive equations underpinning the process being studied. Constitutive equations describe a specific response of a given variable to an external force. The most familiar constitutive equations are Newton s law of viscosity, Fourier s law of heat conduction, and Pick s law of diffusion. 

For interpretation and extrapolation of the experimental results the central role of a computer code as vital analytical tool has been emphasized. In Ref. 7 an attempt is made to identify areas of particular interest. For these great care should be exercised, when scaled experimental results are interpreted and extrapolated with the aid of dimensional analysis of a fypical set of conservation equations normally used in the lumped parameter codes. Two basic assumptions have been made in the course of the dimensional analysis. These are the geometrical similarity of the test fecilify and the prototype plant and performing the experiments with the same fluid compon ts, which are expected to dominate within the containment atmosphere of the prototype plant. Both of the two basic assumptions are valid in the VICTORIA test fecilify. 

The analysis presented in Chapter 8 was solely in terms of the conservation equations for the particle phase of a fluidized suspension. However, the full one-dimensional description is in terms of the coupled mass and momentum conservation equations for both the particle and fluid phases eqns (8.21)-(8.24). These equations correspond to those derived in Chapter 7, except for the inclusion of the particle-phase elasticity term on the extreme right of eqn (8.22). 

In the linear approach, the steady-state parameters are required as initial conditions for the evaluation of system stability, and they are calculated by the plant analysis code developed for the Super LWR, considering the effects of water rods on the coolant channels. The fuel channel is axially discretized into meshes of equal length, and the fuel and coolant properties are assumed uniform within each mesh. The one-dimensional single-channel single-phase conservation equations are solved for each axial mesh in the fuel channel and in each water rod, starting from the core inlet to the core outlet. The core inlet coolant temperature and mass... 

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 analysis of equations (1.1) begins by establishing that the conservation principle holds exactly as in the previous chapters and so allows for the reduction of (1.1) to a two-dimensional system. Letting E = S-l-jic, -I-X2 —1 and adding the equations (1.1) yields the periodic linear equation... 

The complete description of a flame requires the specification of the pressure, the mass flow rate or burning velocity, the initial gas composition, and the appropriate transport coefficients and thermodynamic data. The remaining information is contained in a set of one-dimensional profiles of composition, temperature, and gas velocity as a function of distance (Fig. 2). Other independent variables than distance could have been used, e.g., temperature or time, but distance is common in experimental studies. Not all of these profiles are independent since there are a number of relations between the variables such as the equation of state, conservation of mass, etc. As an example, gas velocity can be obtained both by direct measurement and from temperature measurements using geometrical and continuity considerations. In the example given the indirect determinations of velocity are the more reliable and were used in the analysis. It is general practice to measure as many variables as convenient because the redundant profiles provide a check on the reliability of the measurements. 

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