First-order ordinary differential

The remaining terms in equation set (4.125) are identical to their counterparts derived for the steady-state case (given as Equations (4.55) to (4.60)). By application of the 9 time-stepping method, described in Chapter 2, Section 2.5, to the set of first-order ordinary differential equations (4.125) the working equations of the solution scheme are obtained. The general form of tliese equations will be identical to Equation (2.111) in Chapter 2,... 

In Chapter 3, the analytieal method of solving kinetie sehemes in a bateh system was eonsidered. Generally, industrial realistie sehemes are eomplex and obtaining analytieal solutions ean be very diffieult. Beeause this is often the ease for sueh systems as isothermal, eonstant volume bateh reaetors and semibateh systems, the designer must review an alternative to the analytieal teehnique, namely a numerieal method, to obtain a solution. Eor systems sueh as the bateh, semibateh, and plug flow reaetors, sets of simultaneous, first order ordinary differential equations are often neeessary to obtain die required solutions. Transient situations often arise in die ease of eontinuous flow stirred tank reaetors, and die use of numerieal teehniques is die most eonvenient and appropriate mediod. 

The solution of boundary value problems depends to a great degree on the ability to solve initial value problems.) Any n -order initial value problem can be represented as a system of n coupled first-order ordinary differential equations, each with an initial condition. In general... 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

These equations are a set of nonlinear first-order ordinary differential equations that describe the evolution of the n species as a function of time starting from a set of initial conditions... 

This results In a set of first-order ordinary differential equations for the dynamics of the moments. However, the population balance Is still required In the model to determine the three Integrals and no state space representation can be formed. Only for simple MSMPR (Mixed Suspension Mixed Product Removal) crystallizers with simple crystal growth behaviour, the population balance Is redundant In the model. For MSMPR crystallizers, Q =0 and hp L)=l, thus ... 

These relationships define the equations of motion of the atoms, which can be written as a system of 6N first-order ordinary differential equations ... 

Now we see that we appear to have three coupled first-order ordinary differential equations with Cc apparently coupled in the second and third equations. However, we can eliminate... 

These are first-order ordinary differential equations that have two initial conditions at the inlet to the reactor,... 

The first situation involves two algebraic equations, the second involves an algebraic equation (the mixed phase) and a first-order ordinary differential equation (the unmixed phase), and the third situation involves two coupled differential equations. Countercurrent flow is in fact more compHcated than cocurrent flow because it involves a two-point boundary-value problem, which we will not consider here. 

This is a first-order ordinary differential equation and its solution is lCf[A]o-/Cb[B]o,... 

In simplifying the packed bed reactor model, it is advantageous for control system design if the equations can be reduced to lit into the framework of modern multivariable control theory, which usually requires a model expressed as a set of linear first-order ordinary differential equations in the so-called state-space form ... 

The group structure under a change of A forces Y(A) to change according to a first order ordinary differential equation. To show this we write... 

We now have to solve the following system of two nonlinear coupled first-order ordinary differential equations for the given initial conditions ... 

Process Transfer Function Models In continuous time, the dynamic behaviour of an ideal continuous flow stirred-tank reactor can be modelled (after linearization of any nonlinear kinetic expressions about a steady-state) by a first order ordinary differential equation of the form... 

Equating these two equations for the radial flux yields a first-order ordinary differential equation in c, which may be integrated to yield c(eo) — c(a +... 

Solve the following system of first-order ordinary differential equations and prepare a state-space plot where x is plotted against y using the solution. 

The accuracy of a solution is affected by two properties of the difference equations and by the rounding error involved in the calculation. The first property of the difference equations is the truncation error, that is, the discrepancy between the differential equation written for some point and the difference equation that is supposed to correspond to it. The second property is the order of the difference equation with respect to the axial step. These effects can be illustrated by the way they appear in the solution of a first order ordinary differential equation. 

Since all quantities in equations (77) and (78) except a, q>, and x are constants, these equations comprise two first-order ordinary differential equations describing the flow the boundary conditions are tr = = 1 at x = 0. 

When only the feed side and permeate side mass balance equations are considered under the isothermal condition, the resulting equations arc a set of first-order ordinary differential equations. Furthermore, a co-current purge stream renders the set of equations an initial value problem and well established procedures such as the... 

Combining the mass eonservation (Eq. (1)), momentum eonservation equation (Eq. (2)), mass balanee and energy eonservation, (Eq. (3)) and pressure balance (Eq. (5)) along with Eq. (6) for the eurvature for the miero-region results in a set of three nonlinear first-order ordinary differential Eqs. (7), (8) and (9), as derived in [15],... 

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. 

During the growth phase we could also relate the rate of formation of product, tp, to the cell growth rate, r. The coupled first-order ordinary differential equations above ean be solved by a variety of numerical techniques. 

Using Equation (14-3) to substitute for Ci, we obtain the first-order ordinary differential equation... 

In 50 the author presents a further investigation of the frequency evaluation techniques which are recently proposed by Ixaru et al. for exponentially fitted multistep algorithms for the solution of first-order ordinary differential equations (ODEs). These studies have a scope which is to maximize the benefits of the exponentially-fitted methods via the evaluation of the frequency of the problem. The proposed by Ixaru and co-workers method for frequency... 

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