Zero-gradient condition

At the solid walls, the boundary conditions state that the velocity is zero (i.e. no slip). Also at the walls, the temperature is either fixed or a zero-gradient condition is applied. At the surface of the spinning disk the gas moves with the disk velocity and it has the disk temperature, which is constant. The inlet fiow is considered a plug fiow of fixed temperature, and the outlet is modeled by a zero gradient condition on all dependent variables, except pressure, which is determined from the solution. 

Alternatively, note that one could take advantage of symmetry, applying a zero-gradient condition at z = L/2 and solve the problem on half the domain 0 < z < L/2.) In addition to the boundary conditions, the velocity distribution must be constrained to deliver a certain mass-flow rate m that must be independent of r,... 

Eq. (8.103) allows the dimensionless pressure, Pi, to be determined. Once this has been found, Eq. (8.92) can be used to find the C/, j values and Eq. (8.97) applied sequentially outward from j = 2 as discussed above allows the Vjj values to be found. The boundary conditions give = 0 and since to first order of accuracy the zero.gradient condition on the center line gives ... 

The same boimdaiy conditions are applied as in the single-phase model (section 3). Again, symmetry boimdary conditions are applied in the y and the z directions, thereby redueing the size of the computational domain and computational costs. In the x direction, zero flux conditions are applied at all interfaces except for the flow channels. At the inlets of the gas-flow charmels, the incoming velocity is calculated as a function of the desired current density and stoichiometric flow ratio, as described in section 3.7. The gas streams entering the cell are fully humidified, but no liquid water is contained in the gas stream. At the outlets, the pressure is prescribed for the momentum equation and a zero gradient conditions are imposed for all scalar equations. 

As a result, there is a jump discontinuity in the temperature at Z=0. The condition is analogous to the Danckwerts boimdary condition for the inlet of an axially dispersed plug-flow reactor. At the exit of the honeycomb, the usual zero gradient is imposed, i.e. 

The zero flux condition at the closed outlet requires a zero gradient, thus... 

Referring to Fig. 4.15, it is seen that the concentration and the concentration gradient are unknown at Z=0. The above boundary condition relation indicates that if one is known, the other can be calculated. The condition of zero gradient at the outlet (Z=L) does not help to start the integration at Z=0, because, as Fig. 

Knowledge of the derivative at the centre of the rod requires a solution involving repeated estimates of the temperature gradient at the rod end. The integration proceeds from the end to the rod centre, defined by the condition TFIN = L/2 = 0.25, and a check made for the zero gradient. The initial estimate is revised at X = L/2, accordingly to... 

As the electric field always points in the direction of the electrode, the densities of the electrons and negative ions are set equal to zero at the electrode. It is assumed that the ion flux at the electrodes has only a drift component, i.e., the density gradient is set equal to zero. The conditions in the sheath, which depend on pressure, voltage drop, and sheath thickness, are generally such that secondary electrons (created at the electrodes as a result of ion impact) will ionize at most a few molecules, so no ionization avalanches will occur. Therefore, secondary electrons can be neglected. 

Instead of a formal development of conditions that define a local optimum, we present a more intuitive kinematic illustration. Consider the contour plot of the objective function fix), given in Fig. 3-54, as a smooth valley in space of the variables X and x2. For the contour plot of this unconstrained problem Min/(x), consider a ball rolling in this valley to the lowest point offix), denoted by x. This point is at least a local minimum and is defined by a point with a zero gradient and at least nonnegative curvature in all (nonzero) directions p. We use the first-derivative (gradient) vector Vf(x) and second-derivative (Hessian) matrix V /(x) to state the necessary first- and second-order conditions for unconstrained optimality ... 

For the boundary condition of particle concentration at the wall, a zero normal gradient condition is frequently adopted that is... 

Another special area is the insulating plane outside the disk, defined by Z = 0, R > 1. Here, the boundary condition is usually given as in the set (12.18), zero gradient with respect to Z. This is expressed as a four-point first derivative, as... 

Wagner enhancement factor — describes usually the relationships between the classical - diffusion coefficient (- self-diffusion coefficient) of charged species i and the ambipolar - diffusion coefficient. The latter quantity is the proportionality coefficient between the - concentration gradient and the - steady-state flux of these species under zero-current conditions, when the - charge transfer is compensated by the fluxes of other species (- electrons or other sort(s) of -> ions). The enhancement factors show an increasing diffusion rate with respect to that expected from a mechanistic use of -> Ficks laws, due to an internal -> electrical field accelerating transfer of less mobile species [i, ii]. 

It is clear from Fig. 2.5 that a surface, arbitrarily drawn through a charge distribution, will be crossed by gradient vectors of p and will not satisfy the zero-flux condition. Any surface including a nuclear position coordinate is also excluded as p is not defined there and again the zero-flux condition, eqn (2.9), will be violated. 

A gradient path of V has a simple physical interpretation. It is a line of force—the path traversed by a test charge moving under the influence of the potential F(r X). At a critical point other than a (3, — 3) critical point, the force vanishes. Thus a critical point in the field V(r X) denotes a point of electrostatic balance between the attractors of the system. Since trajectories defining the surface which separates neighbouring basins satisfy the zero-flux condition... 

In order to solve mass and energy balance equations, initial and boundary conditions should be given. Initially, the biomass material is in a quiescent environment at atnbient conditions and thus is specified as uniform tenqrerature and solid compositions. For 1>0, the spatial conditions at the centerline (i=0) are specified by symmetry, the two sides (r=R) are exposed to radiation by a constant temperature of heater. At these two sides, radiant heat transfer from the surface takes places. Further conditions of constant ambient pressure and zero gradient of tar and gas mass at centerline are also used. 

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