Total ordinary differential equation

It is an ordinary differential equation (time dependenee only) where the zeroth moment (/io) is the total erystal number (per unit volume of suspension), the first (/ii) the total erystal length (lined up end to end), the seeond (7 2) is related to total surfaee area, the third (7 3) total volume (and lienee mass) and so on. 

Particle Size Development. Now that a general total property balance equation has been developed (equation (II-9)), one can use it to obtain ordinary differential equations (ode s) which will describe particle size development. What is needed with equation (II-9) is an expression for dp(t,t)/dt, where p denotes a specific property of the system (e.g. particle size). Such an expression can be written for the rate of change of polymer volume in a particle of a certain class. The analysis, which is general and described in Appendix III, will finally result in a set of ode s for Np(t), Dp(t), Ap(t) and Vp(t). 

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

This complex system would be difficult to solve directly. However, the problem is separable by taking advantage of the widely different time scales of conversion and deactivation. For example, typical catalyst contact times for the conversion processes are on the order of seconds, whereas the time on stream for deactivation is on the order of days. [Note Catalyst contact time is defined as the volume of catalyst divided by the total volumetric flow in the reactor at unit conditions, PV/FRT. Catalyst volume here includes the voids and is defined as WJpp — e)]. Therefore, in the scale of catalyst contact time, a is constant and Eq. (1) becomes an ordinary differential equation ... 

The first type, which includes, for example, the problem of strong explosion or propagation of heat in a medium with nonlinear thermal conductivity [3], is characterized by the fact that the exponents are found from physical considerations, from the conservation laws and their dimensionality. In addition, the exponents turn out to be rational numbers. The task of the calculation is to find the dimensionless functions by integration of ordinary differential equations. After this the problem is completely solved, since the numerical constants are determined by normalizing the solution to the conserved quantity (the total energy released in these examples). 

St is the total sorbed concentration (M/M), a, is the first-order mass-transfer rate coefficient for compartment i (1/T), / is the mass fraction of the solute sorbed in each site at equilibrium (assumed to be equal for all compartments), Kp is the distribution coefficient (L3 /M), C is the aqueous solute concentration (M/L3), St is the mass sorbed in compartment i with respect to the total mass of the sorbent (M/M), 0 is the volumetric flow rate through the reactor (L3/T), C, is the influent concentration of solute (M/L3), M, is the mass of sorbent in the reactor (M), and V is the aqueous reactor volume (L3). Using the T-PDF, discrete values for the mass-transfer rate coefficients were generated for the NK compartments. The median value of the mass-transfer rate coefficient within each compartment was chosen as the representative value. The resulting system of ordinary differential equations was solved numerically using a 4th-order Runge-Kutta integration technique. 

Within the rotational sudden approximation we assume that the interaction time is much smaller than the rotational period of the fragment molecule so that the diatom BC does not appreciably rotate from its original position while the two fragments separate. In terms of energies, this requires the rotational energy, Erot, to be much smaller than the total available energy. If that is true, the operator for the rotational motion of BC, hrot, can be neglected in (3.16). The partial differential equation thus becomes an ordinary differential equation,... 

Thus 1 is the total heat energy required to bring a solid from an initial temperature Tso to Tm and to melt it at that temperature. Sundstrom and Young (33) solved this set of equations numerically after converting the partial differential equations into ordinary differential equations by similarity techniques. Pearson (35) used the same technique to obtain a number of useful solutions to simplified cases. He also used dimensionless variables, which aid in the physical interpretation of the results, as shown below ... 

The above discussion completes our description of the single-nephron model. In total we have six coupled ordinary differential equations, each representing an essential physiological relation. Because of the need to numerically evaluate Ce in each integration step, the model cannot be brought onto an explicit form. The applied parameters in the single-nephron model are specified in Table 12.1. They have all been adopted from the experimental literature, and their specific origin is discussed in Jensen et al. [13]. 

The governing equations - that is, mainly the component and the total mass balances in the anode channels - are provided here in dimensionless form. The five ordinary differential equations (ODE) with respect to the spatial coordinate describe the development of the five unknowns in one single anode channel, namely the mole fractions, with i = CH4, H2O, H2, CO2, as well as the molar flow density inside the anode channel, y. Here, the Damkohler numbers, Da/, are the dimensionless reaction rate constant of the reforming and the oxidation reaction, respectively, the rj are the corresponding dimensionless reaction rates, and the v, j are the stoichiometric coefficients ... 

For reactor design calculations it is necessary to know the total devolatilization rate as well as the species production rates. Therefore, one needs to include in the reactor model all the reaction rates that are available for the devolatilization of the particular coal. Kayihan and Reklaitis (8) show that the kinetic data provided by Howard, et al. (5,6) can be easily incorporated in the design calculations for fluidized beds where the coal residence times are long. However, if the residence time of pulverized coal in the reactor is short as it is in entrained bed reactors, then the handling of ordinary differential equations arising from the reaction kinetics require excessive machine computation time. This is due to the stiffness of the differential equations. It is found that the model equations cannot be solved... 

Enthalpy balances for the dry layers and the wet layer can be formulated along with a pertinent drying rate equation. Formulation by Beckwith and Beard results in three ordinary differential equations that describe the dry fabric temperature, the wet layer temperature and most importantly, the moisture content of the total fabric as a function of timeQJ. By predetermining the fabric speed through the dryer, residence time can be converted to dryer length. 

In equilibrium NVE Molecular Dynamics simulation new molecular positions are obtained solving by Newton s equation of motion numerically. To solve Equation Al we use Equation A3 and also specify the initial and boundary conditions of our d dimensional system. This results in a set of d x TV coupled second-order ordinary differential equations and a total of d x TV degrees of freedom. This set of equations are discretized and new positions and velocities for each atom is found numerically by integrating forward in time. Below we give the MD recipe ... 

We now insert rate laws written in terms of molar flow rates [e.g., Equation (3-45)] into the mole balances (Table 6-1). After performing this operation for each species we arrive at a coupled set of first-order ordinary differential equations to be solved for the molar flow rates as a function of reactor volume (i.e., distance along the length of the reactor). In liquid-phase reactions, incorporating and solving for total molar flow rate is not necessary at each step along the solution pathway because there is no volume change with reaction. 

The method of zonation was applied to the energy and material conservation equations. Based on centered finite difTerence approximations, this method can transform three partial differential equations in radial distance and time to ordinary differential equations in time only. Following this, the ordinary differential equations were solved by using Crank-Nicholson algorithm. On the basis of this, the volumetric fluxes of those tar-phase and total volatile phase components were integrated with time by using in roved Euler method to evaluate overall pyrolysis product yields, and afterwards the gas yield can be deduced. 

Simultaneous to the graph creation, kinetic properties in each vRxn are used to create the appropriate reaction rate equations (ordinary differential equations, ODE). These properties include rate constants (e.g., Michaelis constant, Km, and maximum velocity, Vmax, for enzyme-catalyzed reactions, and k for nonenzymatic reactions), inhibitor constants, A) and modes of inhibition or allosterism. The total set of rate equations and specified initial conditions forms an initial value problem that is solved by a stiff ODE equation solver for the concentrations of all species as a function of time. The constituent transforms for the each virtual enzyme are compiled by carefully culling the literature for data on enzymes known to act on the chemicals and chemical metabolites of interest. 

Newton s second law relates the motion of a particle to the force acting on it. If we have N molecules, then Newton s second law yields 3N second-order, ordinary differential equations. For an isolated system of particles with position vectors r, and momenrnm p, the total energy of the system (potential energy plus kinetic energy) will be conserved. The total energy of the system will take the form ... 

As in the gravity sedimentation problem, characteristic solutions also exist for nondiffusive sedimentation in an ultracentrifuge. We recall the definition of a characteristic as a line in the distance-time plane, here the r-t plane, on which an ordinary differential equation may be written. Such an equation must be expressed as a relation connecting total differentials in which partial derivatives do not appear. Since we wish to obtain relations involving total differentials, we write... 

A dynamic model of (his process contains two ordinary differential equations, which arise from the total mass balance on each of the tanks. We assume constant density. 

No fewer than 11 variables appear in the reaction scheme (5.5). These are the concentrations of the metabolites and of the free and complexed forms of the receptor and adenylate cyclase. Because of the existence of conservation relations for the total amount of adenylate cyclase and of receptor, the number of independent variables can be reduced to nine. In the conditions of spatial homogeneity corresponding to the experiments in well-stirred cellular suspensions, the time evolution of the system is governed by the system of nine ordinary differential equations (Martiel Goldbeter, 1987a see Appendix, p. 234) ... 

This problem requires an analysis of coupled thermal energy and mass transport in a differential tubular reactor. In other words, the mass and energy balances should be expressed as coupled ordinary differential equations (ODEs). Since 3 mol of reactants produces 1 mol of product, the total number of moles is not conserved. Hence, this problem corresponds to a variable-volume gas-phase flow reactor and it is important to use reactor volume as the independent variable. Don t introduce average residence time because the gas-phase volumetric flow rate is not constant. If heat transfer across the wall of the reactor is neglected in the thermal energy balance for adiabatic operation, it... 

Hence, one must solve the following ordinary differential equation for the time dependence of total pressure ... 

The number of boundary or initial conditions required to solve an ordinary differential equation, as we saw in Chapter 2, corresponded to the number of arbitrary constants of integration generated in the course of analysis. Thus, we showed that nth order equations generated n arbitrary constants this also implies the application of boundary or initial conditions totalling n conditions. In partial differential equations, at least two independent variables exist, so, for example, an equation describing transient temperature distribution in a long cylindrical rod of metal that is,... 

The set of ordinary differential equations generated from Equation 4.48 can be solved for a given set of concentrations corresponding to different characteristic reaction times, and assuming a series of mixing times, At the total consumption of acid, the final concentrations of iodine and tri-iodine are obtained. 

---