Differential with constant

The molar entropy and the molar enthalpy, also with constants of integration, can be obtained, either by differentiating equation (A2.1.56) or by integrating equation (A2.T42) or equation (A2.1.50) ... 

The procedure we followed in the previous section was to take a pair of coupled equations, Eqs. (5-6) or (5-17) and express their solutions as a sum and difference, that is, as linear combinations. (Don t forget that the sum or difference of solutions of a linear homogeneous differential equation with constant coefficients is also a solution of the equation.) This recasts the original equations in the foiin of uncoupled equations. To show this, take the sum and difference of Eqs. (5-21),... 

Neither of these equations tells us which spin is on which electron. They merely say that there are two spins and the probability that the 1, 2 spin combination is ot, p is equal to the probability that the 2, 1 spin combination is ot, p. The two linear combinations i i(l,2) v /(2,1) are perfectly legitimate wave functions (sums and differences of solutions of linear differential equations with constant coefficients are also solutions), but neither implies that we know which electron has the label ot or p. 

Differentiate with respect to T, assuming the temperature dependence of the concentrations is negligible compared to that of the rate constants ... 

Equation 163, written as = G- /-RT, clearly shows that In ( ) " is a partial molar property with respect to G /KT. MultipHcation of equation 175 by n and differentiation with respect to at constant T, P, and in accord with equation 116 yields, after reduction, equation 179 (constant T,x), where is the partial molar compressibiUty factor. This equation is the partial-property analogue of equation 178. 

All applications are for closed systems with constant mass. If a process is reversible and only p-V work is done, the first law and differentials can be expressed as follows. 

Linear Differential Equations with Constant Coeffieients and Ri ht-Hand Member Zero (Homogeneous) The solution of y" + ay + by = 0 depends upon the nature of the roots of the characteristic equation nr + am + b = 0 obtained by substituting the trial solution y = in the equation. 

All these results generalize to homogeneous linear differential equations with constant coefficients of order higher than 2. These equations (especially of order 2) have been much used because of the ease of solution. Oscillations, electric circuits, diffusion processes, and heat-flow problems are a few examples for which such equations are useful. 

Cauchy Momentum and Navier-Stokes Equations The differential equations for conservation of momentum are called the Cauchy momentum equations. These may be found in general form in most fliiid mechanics texts (e.g., Slatteiy [ibid.] Denu Whitaker and Schlichting). For the important special case of an incompressible Newtonian fluid with constant viscosity, substitution of Eqs. (6-22) and (6-24) lead to the Navier-Stokes equations, whose three Cartesian components are... 

The number of independent rate equations is the same as the number of independent stoichiometric relations. In the present example. Reactions (1) and (2) are reversible reactions and are not independent. Accordingly, C,. and C, for example, can be eliminated from the equations for and which then become an integrable system. Usually only systems of linear differential equations with constant coefficients are solvable analytically. 

The unsteady material balances of tracer tests are represented by linear differential equations with constant coefficients that relate an input function Cj t) to a response function of the form... 

Equations (2.9), (2.10) and (2.11) are linear differential equations with constant coefficients. Note that the order of the differential equation is the order of the highest derivative. Systems described by such equations are called linear systems of the same order as the differential equation. For example, equation (2.9) describes a first-order linear system, equation (2.10) a second-order linear system and equation (2.11) a third-order linear system. 

Linear differential equations with constant coefficients can be solved by a mathematical technique called the Laplace transformation . Systems of zero-order or first-order reactions give rise to differential rate equations of this type, and the Laplaee transformation often provides a simple solution. 

Systems of reversible first-order reactions lead to sets of simultaneous linear differential equations with constant coefficients. A solution may be obtained by means of a matrix formulation that is widely used in quantum mechanics and vibrational... 

The usual Raleigh Equation form [130] is for the conditions of a binary simple differential distillation (no trays or packing), no reflux, but with constant boilup. 

Batch with Constant Reflux Ratio, 48 Batch with Variable Reflux Rate Rectification, 50 Example 8-14 Batch Distillation, Constant Reflux Following the Procedure of Block, 51 Example 8-15 Vapor Boil-up Rate for Fixed Trays, 53 Example 8-16 Binary Batch Differential Distillation, 54 Example 8-17 Multicomponent Batch Distillation, 55 Steam Distillation, 57 Example 8-18 Multicomponent Steam Flash, 59 Example 8-18 Continuous Steam Flash Separation Process — Separation of Non-Volatile Component from Organics, 61 Example 8-20 Open Steam Stripping of Heavy Absorber Rich Oil of Light Hydrocarbon Content, 62 Distillation with Heat Balance,... 

The preceding two equations are examples of linear differential equations with constant coefficients and their solutions are often found most simply by the use of Laplace transforms [1]. 

Differential Systems.—For differential systems with constant coeffidents the following theorem holds ... 

Suppose we have a simple differential equation with constant coefficients... 

For both of these cases, Eqs. (13)—(15) constitute a system of two linear ordinary differential equations of second order with constant coefficients. The boundary conditions are similar to those used by Miyauchi and Vermeulen, which are identical to those proposed by Danckwerts (Dl). The equations may be transformed to a dimensionless form and solved analytically. The solutions may be recorded in dimensionless diagrams similar to those constructed by Miyauchi and Vermeulen. The analytical solutions in the present case are, however, considerably more involved algebraically. 

The value of P2 can be obtained by differentiation with respect to m, since m is the moles of solute per constant number of moles of solvent (n — l/M, where M is the molecular weight of the solvent in kg-mol-1). Differentiating equation (5.28) gives... 

For constant upstream conditions, the maximum flow through the pipe is found by differentiating with respect to v2 and putting (dG/dn2) equal to zero. The maximum flow is thus shown to occur when the velocity at the downstream end of the pipe is the sonic velocity y/yP2v2 (equation 4.37). 

The dAc/dz term is usually zero since tubular reactors with constant diameter are by far the most important application of Equation (3.7). For the exceptional case, we suppose that Afz) is known, say from the design drawings of the reactor. It must be a smooth (meaning differentiable) and slowly varying function of z or else the assumption of piston flow will run into hydrodynamic as well as mathematical difficulties. Abrupt changes in A. will create secondary flows that invalidate the assumptions of piston flow. 

This is an inhomogeneous linear differential equation of second order with constant coefficient a, where g is its right hand side. The parameter a is very small, and it is approximately... 

We have obtained an inhomogeneous differential equation with constant coefficients. As follows from the theory of linear equations, its solution is a sum... 

In this section, we will outline only those properties of the Laplace transform that are directly relevant to the solution of systems of linear differential equations with constant coefficients. A more extensive coverage can be found, for example, in the text book by Franklin [6]. 

Compartmental analysis is the most widely used method of analysis for systems that can be modeled by means of linear differential equations with constant coefficients. The assumption of linearity can be tested in pharmaeokinetic studies, for example by comparing the plasma concentration curves obtained at different dose levels. If the curves are found to be reasonably parallel, then the assumption of linearity holds over the dose range that has been studied. The advantage of linear... 

The linearisation of the non-linear component and energy balance equations, based on the use of Taylor s expansion theorem, leads to two, simultaneous, first-order, linear differential equations with constant coefficients of the form... 

Fig. 3.79. Dead-stop end-point titration, i.e. measuring the current across two Pt-IE s with constant potential difference AE (differential amperometric titration), curves being obtained from Fig. 3.78.

What are some of the mathematical tools that we use In classical control, our analysis is based on linear ordinary differential equations with constant coefficients—what is called linear time invariant (LTI). Our models are also called lumped-parameter models, meaning that variations in space or location are not considered. Time is the only independent variable. 

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