Integrate numerical

E. Hairer and Ch. Lubich. The life-span of backward error analysis for numerical integrators. Numer. Math. 76 (1997) 441-462... 

At constant T Eq. (7) may be integrated numerically to yield the temperature as a function of the number of moles... 

Put n o = 6, substitute Eq. (1) into the flow reactor equation, and integrate numerically. 

The conversion achieved in the vessel is obtained by the solution of the differential equation at the exit of the vessel where the hfe expectation is t = 0. The starting point for the integration is tj). When integrating numerically, however, the RTD becomes essentially 0 by the time becomes 3 or 4, and the value of the integral beyond that point becomes nil. Accordingly, the integration interval is from (f, t,<2> or 4) to (/effluent, = 0)-jl IS found from Eq. (23-46). 

This can be readily integrated numerically as long as we use the appropriate nonequilibrium equivalent specific volume in the integration. A reasonably simple form for has been suggested by Chisholm (1983), which makes use of estabhshed correlations for the slip velocity K, which depends on the Lockhart-Martinelh parameter X. Integrating Eq. (26-118) gives ... 

Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

In the work that follows, the experimental data were fitted by minimizing the sum of least squares and the differential equations were integrated numerically. 

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... 

A more complete discussion of the kinetics of isocyanate crosslinking in the presence of humidity along with experimental verification of the rate equations used above has been given by van der Ven, et al. (18). Equations 38 and 39 can be integrated numerically for different values of isocyanate to hydroxy ratio and H. Plots of the consumption of hydroxy and isocyanate functionality are shown in Figure 2 for H-0 and H-0.66 assuming equal Initial Isocyanate and hydroxy levels. High humidity increases the rate of consumption of isocyanate and reduces the consumption of hydroxy. 

The preceding equations form a set of algebraic and ordinary differential equations which were integrated numerically using the Gear algorithm (21) because of their nonlinearity and stiffness. The computation time on the CRAY X-MP supercomputer for a typical case in this paper was about 5 min. Further details on the numerical implementation of the algorithm are provided in (Richards, J. R. et al. J. ApdI. Polv. Sci.. in press). 

The concentration-time curve can be integrated numerically and yields the so-called area under the curve (AUC) ... 

These data can be integrated numerically between times 0 and 120 minutes. The remaining part between 120 minutes and infinity must be extrapolated from the downslope of the curve (P-phase) which can be modelled by means of the exponential function ... 

Numeric-symbolic approaches are particularly important in process applications because the time series of data is by far the dominant form of input data, and they are the methods of choice if annotated data exist to develop the interpretation system. With complete dependence on the annotated data to develop the feature mapping step, numeric-symbolic mappers can be used to assign labels directly. However, as the amount and coverage of available annotated data diminishes for the given label of interest, there is a need to integrate numeric-symbolic approaches with... 

Equation (3-124) has been integrated numerically, and results can be presented. Figure 3.41 is a plot of liquid holdup EL versus W+ for various values of the parameter R+. 

A Simple Model for The Reaction. Several models have be en developed for the self-condensation of A. CSMP (Continuous System Modeling Program) programming has been used to integrate numerically the set of differential equations associated with each model and to calculate methylol and methylene ether group concentrations as a function of time. 

If the functional form of this relation is known (e.g., if one is dealing with a gaseous system that behaves ideally), this relationship can be combined with equation 3.0.1 and the appropriate rate function to obtain a differential equation, which can then be integrated numerically or in explicit form. If we consider a generalized rate... 

Fields of Application. In SAXS a calibration to absolute intensity is required if extrapolated or integrated numerical values must be compared on an absolute scale. Examples are the determination of density fluctuations or the density difference between matrix and domains as a function of materials composition. 

Since we have seen that the integral cannot be evaluated analytically, there are several alternatives to analytic integration of equation 43-63 we can perform the integration numerically, we can investigate the behavior of equation 43-63 using a Monte-Carlo simulation, or we can expand equation 43-63 into a power series. 

The E-Z Solve software can be used to integrate numerically the differential equation resulting from the combination of the material-balance equation (15.2-17) and the rate law [equation (A)]—see file exl5-3.msp. The same results are obtained for V and t. 

If simplifying conditions such as adiabatic behavior and temperature-independence of (—AHra) and CP are not valid, the material and energy balances may have to be integrated numerically as a system of two coupled ordinary differential equations. 

Again, equations (A) to (C) are integrated numerically (file exl8-6.msp) for each stage, taking into account the appropriate initial conditions. The results at the outlet of the third stage are ... 

Ca2+ homeostasis that results from graded Ca2+ influx through voltage-dependent Ca2+ channels (Fleischmann et al 1994) and spontaneous intracellular Ca2+ release through ryanodine receptors. Thus the myocyte integrates numerous Ca2+ inputs and the concentration of cytosolic free Ca2+ ([Ca2+]j) at any given time is the net result of the activity of these Ca2+ channels, as well as that of Ca2+ pumps and exchangers that remove Ca2+ ions from the cytosol. 

The mol fractions are given by Eqs (1) and (2). Find V for each value of nc by solving the vdW equation numerically. Substitute into Eq (3) and integrate numerically. The values are tabulated. In this case, kt = 0.8531... 

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