Determining the differential equations

The strategy for determining the differential equations for biochemical reactants, pH, and binding ions is to express the equations for reactants based on the stoichiometry of the reference reactions and to determine the kinetics of pH and binding ions based on mass balance. 

For example, if the ATP hydrolysis flux (for the reaction of Equation (6.1)) were denoted. /ATPase, then the differential equations for concentrations of reactants ATP, ADP, and PI would follow from the stoichiometry of the reference species in the reference reaction  

However, the differential equations for the pH, [Mg2+], and [K+] (and concentrations of any other binding ions) are not as straightforward to determine as the equations for the biochemical reactants. 

The kinetics of pH are governed by proton binding and unbinding as well as the consumption and generation of protons via chemical reactions. For a general system of Nr reactants, and considering [H+], [Mg2+], and [K+] binding, the total 

If the system is closed, then the rate of change of free [H+] can be calculated based on mass conservation. The total proton concentration in the system is given by 

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Figure 2.21 shows a passive eleetrieal network. Determine the differential equation relating v t) and viit). 

The methods previously discussed in this chapter can be used to determine the differential equations, solutions and parameters for a number of mechanical models using a variety of combinations of springs and damper elements. Table 5.1 is a tabulation of the differential equation, parameter inequalities, creep compliances and relaxation moduli for frequently discussed basic models. Note that the equations are given in terms of the pj and qj coefficients of the appropriate differential equation in standard format. The reader is encouraged to verify the validity of the equations given and is also referred to Flugge (1974) for a more complete tabulation. 

The orientation of particles of a liquid is characterized by one unit vector 71 The rheological constants p, t, tij, 112, and X are usually experimentally determined. The differential equation characterizes the change in the orientation of the particles of the liquid caused by flow. The viscosity tensor of a simple Ericksen liquid is... 

The focus herein is a survey of contemporary experimental approaches to determining the form of equation 3 and quantifying the parameters. In general, the differential equation could be very compHcated, eg, the concentrations maybe functions of spatial coordinates as well as time. Experimental measurements are arranged to ensure that simplified equations apply. 

Method of Variation of Parameters This method is apphcable to any linear equation. The technique is developed for a second-order equation but immediately extends to higher order. Let the equation be y" + a x)y + h x)y = R x) and let the solution of the homogeneous equation, found by some method, he y = c f x) + Cofoix). It is now assumed that a particular integral of the differential equation is of the form P x) = uf + vfo where u, v are functions of x to be determined by two equations. One equation results from the requirement that uf + vfo satisfy the differential equation, and the other is a degree of freedom open to the analyst. The best choice proves to be... 

Therefore, the slope of the linear plot Cg versus gives the ratio kj/kj. Knowing kj -i- kj and kj/kj, the values of kj and kj ean be determined as shown in Figure 3-10. Coneentration profiles of eom-ponents A, B, and C in a bateh system using the differential Equations 3-95, 3-96, 3-97 and the Runge-Kutta fourth order numerieal method for the ease when Cgg =Cco = 0 nd kj > kj are reviewed in Chapter 5. 

Cbo = Cco = 0- For known values of kj and kj, simulate die eoneen-tradons of A, B, and C for 10 minutes at a dme interval of t = 0.5 min. A eomputer program has been developed using die Runge-Kutta fourdi order mediod to determine die eoneentrations of A, B, and C. The differential Equations 5-64, 5-65, and 5-66 are expressed, respeetively, in die form of X-arrays and funetions in die eomputer program as... 

Tables 13-2 and 13-3 elueidate how the eommon dimensionless groups are derived. The boundary eonditions governing the differential equations eombined with the relative size of the system should be eonsidered when determining dimensionless parameters. Using Table 13-2 to determine the dimensionless groups for any of the three equations, divide one set of the dimensions into all the others ineluding the boundary eonditions.

The restrictions on engineering constants can also be used in the solution of practical engineering analysis problems. For example, consider a differential equation that has several solutions depending on the relative values of the coefficients in the differential equation. Those coefficients in a physical problem of deformation of a body involve the elastic constants. The restrictions on elastic constants can then be used to determine which solution to the differential equation is applicable. 

One can say, thus, that the differential equation merely prescribes the pattern of closed trajectories, but not the individual (physical) trajectories themselves the latter are determined uniquely by the initial conditions. 

In some cases there also occur semistable limit cycles (in this discussion the single term cycle is used wherever it is unambiguous or if no confusion is to be feared) characterized by stability on one side and instability on the other side. Figure 6-5(a), (b), and (c) illustrate these definitions. Physically, only stable cycles are of interest the unstable cycles play the role of separating the zones of attraction of stable cycles in the case when there are several cycles. It is seen from this definition that, instead of an infinity of closed trajectories, we have now only one such trajectory determined by the differential equation itself and the initial conditions do not play any part. In fact, the term initial conditions means just one point (x0,y0) of the phase plane as a spiral trajectory O passes through that point and ultimately winds itself onto the cycle 0, it is clear that the initial conditions have nothing to do with this ultimate closed trajectory C—the stable [Pg.329]

This theory is adequate to explain practically all oscillatory phenomena in relaxation-oscillation schemes (e.g., multivibrators, etc.) and, very often, to predict the cases in which the initial analytical oscillation becomes of a piece-wise analytic type if a certain parameter is changed. In fact, after the differential equations are formed, the critical lines T(xc,ye) = 0 are determined as well as the direction of Mandelstam s jumps. Thus the whole picture of the trajectories becomes manifest and one can form a general view of the whole situation. The reader can find numerous examples of these diagrams in Andronov and Chaikin s book4 as well as in Reference 6 (pp. 618-647). 

Since X + In X is a transcendental function, Eq. (2-67) cannot be solved for [A], Two methods are usually used. The method of initial rates is the more common one, since it converts the differential equation into an algebraic one. Values of v(, determined as a function of [A]o, are fit to the equation given for v. This application to enzyme-catalyzed reactions will be taken up in Chapter 4. The other method regularly used relies on numerical integration these techniques are given in Chapter 5. 

These three equations (11), (12), and (13) contain three unknown variables, ApJt kn and sr The rest are known quantities, provided the potential-dependent photocurrent (/ph) and the potential-dependent photoinduced microwave conductivity are measured simultaneously. The problem, which these equations describe, is therefore fully determined. This means that the interfacial rate constants kr and sr are accessible to combined photocurrent-photoinduced microwave conductivity measurements. The precondition, however is that an analytical function for the potential-dependent microwave conductivity (12) can be found. This is a challenge since the mathematical solution of the differential equations dominating charge carrier behavior in semiconductor interfaces is quite complex, but it could be obtained,9 17 as will be outlined below. In this way an important expectation with respect to microwave (photo)electro-chemistry, obtaining more insight into photoelectrochemical processes... 

Before giving further motivations, we would like to recall the basic aspects concerned with the elementary problem of determining eigenfunctions and eigenvalues for the differential equation... 

The ripple is stipulated by the fact that difference harmonics reveal the dispersion, that is, determination of a harmonic velocity depends on its number, whereas for the differential equation all harmonics have the same velocity a. In order to improve the quality of a scheme, one needs to minimize the dispersion. Among various schemes (44) with weights the scheme relating to ... 

In general the rate of a reaction is determined by monitoring its progress over time. Hence, we need expressions that relate the concentrations of reacting molecules to time, as opposed to the differential equations in the preceding section, which relate the rate of reaction to the concentrations of the participating molecules. 

Equation (4.3.37) can be used to determine the function = T1(c1), which is the adsorption isotherm for the given surface-active substance. Substitution for c1 in the Gibbs adsorption isotherm and integration of the differential equation obtained yields the equation of state for a monomole-cular film = T jt). 

The principles of conservation of mass and momentum must be applied to each phase to determine the pressure drop and holdup in two phase systems. The differential equations used to model these principles have been solved only for laminar flows of incompressible, Newtonian fluids, with constant holdups. For this case, the momentum equations become... 

For many safety studies the maximum flow rate of vapor through the hole is required. This is determined by differentiating Equation 4-48 with respect to PIP0 and setting the derivative equal to zero. The result is solved for the pressure ratio resulting in the maximum flow ... 

Example 14-7 can also be solved using the E-Z Solve software (file exl4-7.msp). In this simulation, the problem is solved using design equation 2.3-3, which includes the transient (accumulation) term in a CSTR. Thus, it is possible to explore the effect of cAo on transient behavior, and on the ultimate steady-state solution. To examine the stability of each steady-state, solution of the differential equation may be attempted using each of the three steady-state conditions determined above. Normally, if the unsteady-state design equation is used, only stable steady-states can be identified, and unstable... 

Equation (2.19), which concerns a situation without processes in the biofilm, can be extended to include transformation of a substrate, an electron donor (organic matter) or an electron acceptor, e.g., dissolved oxygen. If the reaction rate is limited by j ust one substrate and under steady state conditions, i.e., a fixed concentration profile, the differential equation for the combined transport and substrate utilization following Monod kinetics is shown in Equation (2.20) and is illustrated in Figure 2.8. Equation (2.20) expresses that under steady state conditions, the molecular diffusion determined by Fick s second law is equal to the bacterial uptake of the substrate. 

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