Euler time integration

The concentration of each chemical species, as a function of time, during cure can be calculated numerically from Equations 3-6 using the Euler-Romberg Integration method if the initial concentrations of blocked isocyanate and hydroxyl functionality are known. It is a self-starting technique and is generally well behaved under a wide variety of conditions. Details of this numerical procedure are given by McCalla (12). 

Each trial curve generated by the Euler-Romberg integration method is similarly normalized by dividing each of its points by a value corresponding to the time at which the experimental curve reached its maximum. 

Treating variations of f and f as independent, because they lead to equivalent Euler-Lagrange equations, and integrating the time integral by parts as in Euler s theory, the resulting variational expression is... 

The system of equations is discretized in space by a finite voJume approach, while for the time integration an implicit Euler method is used. Particle and flow model are solved consecutively, which implies that conditions in the bed change slowly compared to the integration step. To reduce the required computation time the flow model is solved here only for one dimension even if the software library TOSCA provides also classes for a two dimensional approach. 

The volume fraction variables were assumed to be time dependent in the semi-implicit discretization scheme used to solve the continuity equations. Considering the last two terms in the above relation, the transient term was then approximated by an explicit Euler time discretization scheme followed by volume integration in which the transient term was kept constant. If we thereafter multiply the resulting relation by Atjp, the relation can be rewritten as ... 

If a single-stage Euler explicit time-integration scheme is used, the updated moment set can be written as... 

Eq. (B.l). Thus, as a first step, we need to consider the volume-average form of Eq. (B.l) or, equivalently, the volume-average forms of the individual terms in Eqs. (B.2)-(B.5). Using a single-stage Euler explicit time-integration scheme (Leveque, 2002), the finite-volume expression for the updated NDF has the form ... 

Let us now discuss in detail the question of moment conservation during time integration. Consistently with Chapter 8, the source terms due to phase-space processes are set to zero so that only transport terms in real space are considered in this discussion. When Eq. (D.23) is integrated using an explicit Euler scheme, the volume-average moment of order k in the cell centered at X at time (n + l)Af is directly calculated from the volume-average moment of order k at time n Af from the following equation ... 

For the spatial solution of the nonlinear coupled multi-field problem given in Sect 1.3, the Finite Element Method (FEM) is applied. The equations for the three fields are solved with a Newton-Raphson algorithm, and the time integration is performed with the implicit Euler backwards scheme. 

Economics in process control, 3, 10-11, 15, 26, 532-34 Environmental regulations, 3 Equal-percentage valve, 254, 255 Equations of state, 57 Equilibria, 56, 78 chemical, 56 phase, 56-57, 71, 75, 78 Error criteria (see Time integral criteria) Euler s identities, 131-32, 149 Experimental modeling, 45, 656 frequency response techniques, 668 process identification, 657-62 time constant determination, 228, 232 Exponential function, 130 approximations, 215-16 Laplace transform, 130 z-transform, 592... 

SO dh(ntank)=(f(ntank- I)-f(ntank))/area c integrate a la Euler time=time+delta do 60 ntank=l,2... 

There were no dynamic simulation issues experienced in this system. The default Implicit Euler numerical integration algorithm worked well, giving quite short simulation times (1 min of real time to simulate lOh of process time). 

Forward-Euler discrete integrator (FEDI) A discrete-time integrator based on the forward-Euler numerical integration scheme. 

The relaxation time was obtained from the time integral of stretched exponential which is expressed with Euler T function,... 

To discretize the continuity, the differential equation is first integrated over the scalar grid cell volume. This is the same grid cell volume as was used for the generalized scalar quantity in Sect. 12.8. By use of the first order implicit Euler time discretization scheme and the basic theorem of integration for the convective fluxes, a discrete form of (12.195) can be expressed as ... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

Starting with an initial value of and knowing c t), Eq. (8-4) can be solved for c t + At). Once c t + At) is known, the solution process can be repeated to calciilate c t + 2At), and so on. This approach is called the Euler integration method while it is simple, it is not necessarily the best approach to numerically integrating nonlinear differential equations. To achieve accurate solutions with an Eiiler approach, one often needs to take small steps in time. At. A number of more sophisticated approaches are available that allow much larger step sizes to be taken but require additional calculations. One widely used approach is the fourth-order Bunge Kutta method, which involves the following calculations ... 

V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler s method, V is determined at each time step... 

If Euler s method is used for integration, the time step is limited by... 

Much better results are obtained by using the parallel surface method (Section III.F.3), because the integral methods are used to determine (K). Nevertheless, PSM gives only approximate estimation of the Euler characteristic and is extremely time-consuming in comparison to the methods described below. 

Conservation equations are written for all reactive species initiators, monomer, polymer carbon radicals and DTC radicals. They are integrated forward in time using the forward Euler technique, and the results can be presented as functions of either time or conversion. The results for these simulations are given in the following section. 

These step sizes scale directly with the time constant t. If t were 10, we could take steps that were 10 times bigger. So the maximum stable step size for the Euler integration is twice the time constant. 

Then to step to the next point in time, using Euler integration with a step size DELTA,... 

Therefore we have an additional ODE that comes from the controller. This must be solved at the same time as the three ODBs describing the process. Using Euler with a DELTA integration step gives... 

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