Transient boundary conditions

Example 4.2. Heat Conduction with Transient Boundary Conditions... 

Chan et al. 1998) or rock mechanical modelling based on complex 3D geological structural models (Hansson et al. 1995). In this study time-dependent ice mechanical load and water pressure at the ice/bedrock interface, calculated by the ice-sheet/drainage model, were applied as transient boundary conditions for the coupled H-M models. 

Ghoreishy, M. H. R. and Nassehi, V., 1997. Modelling the transient flow of rubber compounds in the dispersive section of an internal mixer with slip-stick boundary conditions. Adv. Poly. Tech. 16, 45-68. 

For the determination of downdraft risk in the winter case, three-dimensional and transient CFD computauons were performed using the TASC flow code. Boundary conditions were defined from the results of the thermal modeling. 

In the SMB operation, the countercurrent motion of fluid and solid is simulated with a discrete jump of injection and collection points in the same direction of the fluid phase. The SMB system is then a set of identical fixed-bed columns, connected in series. The transient SMB model equations are summarized below, with initial and boundary conditions, and the necessary mass balances at the nodes between each column. 

Fig. 2.9 The state transition graph Gc, computed for a Tdim lattice consisting of iV = 4 points (with periodic boundary conditions), and totalistic rule T2 . The vertices labeled ti represent transient configurations those labeled cd represent cyclic states, and give rise to the formal cycle cum decomposition C[ j =[3, lj-f[2,2].

Numerical Observations Figure 3.42 shows a schematic plot of H versus A for A = 8 Af = 5 two dimensional CA. The lattice size is 64 x 64 with periodic boundary conditions. In the figure, the evolution of the single-site entropy is traced for four different transition events. In each case, for a given A, a rule table consistent with that A is randomly chosen and the system is made to evolve for 500 steps to allow transients to die out before H is measured. 

Consider the case of transient diffusion at constant potential (constant surface concentration). The first boundary condition, (11.2), is preserved and the second boundary condition can be written (for any time t) as... 

With these boundary conditions, the differential transient-diffusion equation (11.1) has the solution... 

Associated with the pole of the S-matrix is a Seigert state, I-Ves, which has purely outgoing boundary conditions and satisfies (with some caveats) the equation, // I res = z les,H being the system Hamiltonian.44 If a square integrable approximation to I res is constructed, then its time evolution, k . (/,), wiH exhibit pure exponential decay after a transient induction period. Of course any L2 state will show quadratic, and hence non-exponential, decay at short times since... 

The high temperatures and pressures created during transient cavitation are difficult both to calculate and to determine experimentally. The simplest models of collapse, which neglect heat transport and the effects of condensable vapor, predict maximum temperatures and pressures as high as 10,000 K and 10,000 atmospheres. More realistic estimates from increasingly sophisticated hydrodynamic models yield estimates of 5000 K and 1000 atmospheres with effective residence times of <100 nseconds, but the models are very sensitive to initial assumptions of the boundary conditions (30-32). 

In both experimental and theoretical investigations on particle deposition steady-state conditions were assumed. The solution of the non-stationary transport equation is of more recent vintage [102, 103], The calculations of the transient deposition of particles onto a rotating disk under the perfect sink boundary conditions revealed that the relaxation time was of the order of seconds for colloidal sized particles. However, the transition time becomes large (102 104 s) when an energy barrier is present and an external force acts towards the collector. 

In subcooled impact, the initial droplet temperature is lower than the saturated temperature of the liquid of the droplet, thus the transient heat transfer inside the droplet needs to be considered. Since the thickness of the vapor layer may be comparable with the mean free path of the gas molecules in the subcooled impact, the kinetic slip treatment of the boundary condition needs to be applied at the liquid-vapor and vapor-solid interface to modify the continuum system. 

Solving sets of (partial) differential equations inherently requires the specification of boundary conditions and, in case of transient simulations, also initial conditions. This is not as simple as it looks like, especially for turbulent flows in complex process equipment. 

Applying Immersed or Embedded Boundary Methods (Mittal and Iaccarino, 2005) circumvents the whole issue of the friction between the more or less steady overall flow in the bulk of the vessel and the strongly transient character of the flow in the zone of the impeller. These methods are introduced below. In the context of a LES, Derksen and Van den Akker (1999) introduced a forcing technique for both the stationary vessel wall and the revolving impeller. They imposed no-slip boundary conditions at the revolving impeller and at the stationary tank wall (including baffles). To this purpose, they developed a specific control algorithm. 

A related technique is the current-step method The current is zero for t < 0, and then a constant current density j is applied for a certain time, and the transient of the overpotential 77(f) is recorded. The correction for the IRq drop is trivial, since I is constant, but the charging of the double layer takes longer than in the potential step method, and is never complete because 77 increases continuously. The superposition of the charge-transfer reaction and double-layer charging creates rather complex boundary conditions for the diffusion equation only for the case of a simple redox reaction and the range of small overpotentials 77 [Pg.177]

The derivation of a steady-state solution requires boundary conditions, but no initial condition. Steady-state can be seen as the asymptotic solution (so never mathematically reached at any finite time [43]) of the transient, which -for practical purposes - can be approached in a reasonably short time. For instance, limiting-flux diffusion of a species with diffusion coefficient Di = 10-9 m2 s 1 towards a spherical organism of radius rQ = 1 jxm is practically attained at t r jDi = 1 ms. 

Although time is not explicitly represented in Figure 8, we can follow the transient evolution towards steady-state on this plot. At t = 0, and Ju are zero and the uptake flux starts from the origin of coordinates then, with increasing (but still short) t, Ju increases. At short t, Jm is extremely large (in comparison with /u), but decreasing until both fluxes (/u and Jm) eventually meet at a t which is easily identified as tm lx (the time when cjy, reaches its maximum value). Indeed, if /u = Jm, the boundary condition equation (27) prescribes dcM (Vo, t)/At - 0, which is the condition for the maximum. Thus, while t < tmax, Jm > Ju, with their difference indicating the rate of accumulation... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

For a rising transient, this equation has been solved using Laplace transformation and appropriate initial and boundary conditions, and Fig. 26 illustrates the expected rising transient so derived. 

Except for stagnant fluids, discussed in Section C.2, there are no general solutions for the case where the transient resistances in both phases are significant. If the external resistance is assumed constant, Eq. (3-56) must be solved with boundary conditions given by Eqs. (3-65), (3-66), (3-69), and... 

The case of a fluid sphere moving at constant velocity and suddenly exposed to a step change in the composition of the continuous phase has been treated by solving Eq. (3-56), with Eqs. (3-40), (3-41), (3-42), and (3-57) as boundary conditions for potential flow (R14). The transient external resistance is given within 3% by... 

The theories of transient processes leading to steady detonation waves have been concerned on the one hand with the prediction of the shape of pressure waves which will initiate, described in Section VI, A of Ref 66, and on the other hand with the pressure leading to the formation of such.an initiating pulse, described in Section VI, B. In Section V it was shown that the time-independent side boundary conditions are important in determining the characteristics of steady, three-dimensional waves. It now becomes necessary to take into consideration time-dependent rear boundary conditions. For one-dimensional waves, the side boundary conditions are not involved... 

The systems considered here are isothermal and at mechanical equilibrium but open to exchanges of matter. Hydrodynamic motion such as convection are not considered. Inside the volume V of Fig. 8, N chemical species may react and diffuse. The exchanges of matter with the environment are controlled through the boundary conditions maintained on the surface S. It should be emphasized that the consideration of a bounded medium is essential. In an unbounded medium, chemical reactions and diffusion are not coupled in the same way and the convergence in time toward a well-defined and asymptotic state is generally not ensured. Conversely, some regimes that exist in an unbounded medium can only be transient in bounded systems. We approximate diffusion by Fick s law, although this simplification is not essential. As a result, the concentration of chemicals Xt (i = 1,2,..., r with r sN) will obey equations of the form... 

Thus the partially reflecting boundary condition reduces the effective encounter distance by a factor of fcact (4nRD + fcact) 1 for both the steady-state and transient terms in the rate coefficient. 

The transient heat equation (Eq. 3.285) often serves as the model for parabolic equations. Here the solution depends on initial conditions, meaning a complete description of T(0, x) for the entire spatial domain at t =0. Furthermore the solution T(t,x) at any spatial position x and time t depends on boundary conditions up to the time t. The shading in Fig. 3.14 indicates the domain of influence for the solution at a point (indicated by the dot). 

The time-rate-of-change of surface species k due to heterogeneous reaction is given by Eq. 11.102. As discussed above, the effects of surface chemistry must be accounted for as boundary conditions on gas-phase species through flux-matching conditions such as Eq. 11.123. For a transient simulation, a differential equation for the site fraction Zk of surface species k can be written... 

Deriving the compressible, transient form of the stagnation-flow equations follows a procudeure that is largely analogous to the steady-state or the constant-pressure situation. Beginning with the full axisymmetric conservation equations, it is conjectured that the solutions are functions of time t and the axial coordinate z in the following form axial velocity u = u(t, z), scaled radial velocity V(t, z) = v/r, temperature T = T(t, z), and mass fractions y = Yk(t,z). Boundary condition, which are applied at extremeties of the z domain, are radially independent. After some manipulation of the momentum equations, it can be shown that... 

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