Steady state, Eulerian

When the unsteady-state equations are being integrated forward in time either towards the Eulerian steady state or towards the Lagrangian steady propagation rate, a criterion is required for deciding when the steady situation is achieved. In both cases this is provided by the space integral reaction rates qiA dy, which for sufficiently small finite difference intervals may be replaced by the summations x y)n or an equivalent quantity, where A dy) is the volume of the nth element and A a weighted mean area within the element. For the quasi-one-dimensional steady-state flame we have from Eq. (2.12)... 

Equation (4.43) arises as a result of expressing the conservation conditions in finite-difference form, there being one equation for each species i and one for the enthalpy. However, the closure of the set of equations at grid point / requires three further relations. Two of these are provided by the auxiliary equations (2.25) and (2.26). In certain computations directed towards an Eulerian steady state these relations may be used as they stand, as will be indicated in Section 4.4(b). Otherwise, both equations are expanded to first order in the dependent variables to... 

In the Eulerian steady-state, stationary flame the time derivatives in Eqs. (4.12) and (4.13) become zero, and the computational problem becomes one of determining convective fluxes and flame profiles that will allow this to occur everywhere. 

The Eulerian steady state solution to these equations may be obtained by a parallel approach to that outlined in Sections 4 and 7.2(a), with the velocities constrained by Eqs. (7.2b) and (7.26) as above. Equation (7.26) must be integrated on the assumption of linear variation of / with a> between nodes so as to evaluate the reduced velocities V at the cell boundaries. 

We will now describe the basic hydrodynamic relationships applicable in the case of steady-state flow in which the Eulerian velocity field is time-independent and written as v(r). Here the rate of strain elements are given by [1]... 

This flow field can be maintained in a steady state, at least in the Eulerian sense, either by use of a four-roll mill [18] as in Figure 2.8.4(a) or by means of opposed jet flow as in Figure 2.8.4(b). However, it is important to note that the flow is still transient in the Lagrangian sense. That is, pure planar extension is confined to the central stagnation... 

In the present study, two-dimensional Two-Fluid Eulerian model was used to describe the steady state, dilute phase flow of a wet dispersed phase (wet solid particles) in a continuous gas phase through a pneumatic dryer. The predictions of the numerical solutions were compared successfully with the results of other one-dimensional numerical solutions and experimental data of Baeyens et al. [5] and Rocha [13], Axial and the radial distributions of the characteristic properties were examined. 

At steady state, the local derivative, d Cydt) , of the Eulerian derivative is zero. Also, as the wastewater enters the reactor, its dissolved oxygen content is practically zero thus, [C<, ] is equal to zero. [Ce], as it goes out of the secondary basin must also be equal to zero. Thus,... 

At steady state, the local derivative is equal to zero. Thus, from the Reynolds transport theorem, (Lagrangian derivative = to the Eulerian derivative) ... 

For steady-state conditions the species mass balance equation is a first-order ODE with the z-coordinate as the only independent variable. The Eulerian species mass balance (6.1) becomes ... 

It defines the transformed Eulerian mean (TEM) or residual circulation. When these definitions are applied to the simplified form (3.63) of the thermodynamic equation, the resulting circulation is called the diabatic circulation and the meridional and vertical velocities will be noted (vd,wd) rather than (v, w ). In this case, the simplified thermodynamic equation and the steady state approximation for the continuity equation become... 

Three-dimensional steady-state calculations of drying process in vertical pneumatic dryer were performed by [21], The theoretical model is based on two-phase Eulerian-Lagrangian approach and incorporates advanced drying kinetics for wet particles. The model was utilized for simulation of the drying process of wet PVC and silica particles in a large-scale vertical pneumatic dryer. 

For steady-state flows (here we consider the steady-state in the Eulerian viewpoint. In a steady Eulerian velocity field (which is not necessarily Lagrangian steady), 0M,/0f = 0. For the analysis of polymer melt flow occurring at a very low Reynolds number, the inertia term puj dujldxj will usually be neglected. The body force effect is usually negligible too. This reduces the equations of motion to... 

The time-dependent behavior of the flow in the reaction zone of detonating PBX-9404 was investigated using one-dimensional Lagrangian and three-dimensional Eulerian numerical hydrodynamics. The steady-state solution was stable and perturbations were found to decay. This is in contrast to the time-dependent, unstable, periodic reaction zones reported for liquid TNT and nitromethane. 

In this type of apparatus, the tensile stress is measured through the deflection of a spring associated with one of the rotating wheels. In determining with the methods described in this section, one may have to wait a long time for the stress to build up to the level where a steady-state (in both the Lagrangian and Eulerian senses) is attained because the specimen used is strain free before the test begins. 

A few additional comments about when and under what conditions one must use a nonlinear viscoelastic constitutive equation are discussed here. At this time it seems that whenever the flow is unsteady in either a Lagrangian DvIDt 0) or a Eulerian (9v/9r 0) sense, then viscoelastic effects become important. In the former case one finds flows of this nature whenever inhomogeneous shear-free flows arise (e.g., flow through a contraction) and in the latter case in the startup of flows. However, even in simple flows, such as in capillaries or slit dies, viscoelastic effects can be important, especially if the residence time of the fluid in the die is less than the longest relaxation time of the fluid. Then factors such as stress overshoot could lead to an apparent viscosity that is higher than the steady-state viscosity. In line with these ideas one defines a dimensionless group referred to as the Deborah number ... 

The continuity equations of Sections 2.1 to 2.4 have been derived for a reference frame which is stationary in space. This is known as an Eulerian frame of reference, and in it at the steady state the time derivatives vanish, as in... 

In our experience, on the final approach to the steady state in an Eulerian type of time-dependent reactive flow calculation (see Section 4), when the properties at a point in the flow field vary slowly with time and fresh calculations of the transport coefficients are only required at infrequent intervals, use of the detailed transport model presents little problem. A sound basis then exists for studies of the effect of variations in basic transport parameters on the steady-state flow. Unsteady flows, on the other hand, are most economically treated by less demanding, though more approximate, methods. Some approaches along these lines are discussed below and in Section 3.4. 

To set against the drawback of possible uncertainty about effects of the numerical diffusion associated with the convective terms, the Eulerian approach has very considerable operational advantages if the computational objective is solely the determination of steady-state properties. Because the properties at each grid point are time-invariant in the steady state, Eulerian time steps are in themselves successive approximations to the end result. They... 

The solutions discussed so far, both Eulerian and Lagrangian, are known as block implicit solutions. Eulerian integrations into the steady state may, however, be further simplified by uncoupling the separate equations, so that each dependent variable is considered separately during each time step, under... 

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