Separable Equation

Suppose M is a function of t only, and Af is a function of p only, then Equation 2.41 becomes 

Whenever a first-order differential equation can be written in either of the forms Equation 2.42 or Equation 2.43, the equation is said to be separable. If we reconsider Equation 2.38  

this differential equation is separable and results in 

In order to solve Equation 2.44, the left-hand side must first be simplified. Consider now the fi action 

As is evidenced in this later illustration, the potential difficulty in applying this separation of variable technique lies in one s ability to carry out the resulting integration that may arise, such as in Equation 2.44 above. 

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In the few two- and three-dimensional cases that pemiit exact solution of the Schroedinger equation, the complete equation is separated into one equation in each dimension and the energy of the system is obtained by solving the separated equations and summing the eigenvalues. The wave function of the system is the product of the wave functions obtained for the separated equations. 

We have said that the Schroedinger equation for molecules cannot be solved exactly. This is because the exact equation is usually not separable into uncoupled equations involving only one space variable. One strategy for circumventing the problem is to make assumptions that pemiit us to write approximate forms of the Schroedinger equation for molecules that are separable. There is then a choice as to how to solve the separated equations. The Huckel method is one possibility. The self-consistent field method (Chapter 8) is another. 

A separate equation is obtained for each component i Consider the important case of steady-state flow of a single fluid through the pipe section of Figure 21b, where the flow is taken to be perpendicular to the cross sections and Making use of the concept of average velocity ... 

Operation m the Simple Mode if there is no concentration gradient within the liquid pool and if there is no coalescence within the rising foam, then the operation shown by the sohd hnes of Fig. 22-42 is truly in the simple mode, i.e., a single theoretical stage of separation. Equations (22-45) and (22-46) will then apply to the steady-flow operation. 

For many purposes, it is convenient to have a two-point relation between the vapor pressures (P Pi) at two different temperatures (T2, Ti). Such a relation is obtained by first writing separate equations at each temperature ... 

The concentrations are contained in separate terms, and the equation (17) for the whole reaction can be divided into a number of separate equations expressing the equilibrium conditions for various possible partial reactions. 

This integrated version of Equation (3.15) requires viscosity to be constant as well as density, but this assumption is not strictly necessary. See Problem 3.15. Write separate equations for the pressure drop in the large and small reactors and take their ratio. The physical properties cancel to give the following, general relationship ... 

In the general case of a piston flow reactor, one must solve a fairly small set of simultaneous, ordinary differential equations. The minimum set (of one) arises for a single, isothermal reaction. In principle, one extra equation must be added for each additional reaction. In practice, numerical solutions are somewhat easier to implement if a separate equation is written for each reactive component. This ensures that the stoichiometry is correct and keeps the physics and chemistry of the problem rather more transparent than when the reaction coordinate method is used to obtain the smallest possible set of differential... 

The error of approximation of a loccJly one-dimensional scheme. Upon raising the question regarding the error of approximation provided by one or another LOS it is straightforward to verify that every separate equation (21) with the number a does not approximate equation (15) in spite of the fact that the sum of the residuals i/ = + Vb +... 

Equation (1.47) can be rearranged to predict the nuaber of theoretical plates required to give a certain separation, equation (1.48). 

When the object is to perform fast separations equation (1.57) indicates that we should not demand an overly large value for the... 

The equation fully complies with all observations made. The copper ion precipitates as red copper on the zinc strip, and the color of the solution fades on account of this. Zinc metal dissolves and enters the solution as zinc ions hence, the surface of the zinc metal shows pitting. Because a zinc ion solution is colorless, the increase in zinc ion concentration is physically observed as color fading. Energy is liberated, as indicated by the temperature of the solution rising by several degrees. With electrons positioned in their correct places, one can write, for the two processes, separate equations as shown below ... 

Figure 10.22. Determination of J50 from the desired particle separation (Equation 10.3, Zanker, 1977)...

Solution If the gas is assumed to be well mixed, then on the high-pressure (feed-side) side of the membrane, the mole fraction is that of the retentate leaving the membrane. Assuming a binary separation, Equation 10.24 can be written for Component A as ... 

Although elastic strain and plastic deformation are expressed as numbers and have the same units (length/length), since they are physically different entities, they cannot be mixed in arithmetic operations. That is, mixtures of them cannot be added, subtracted, multiplied, or divided. Therefore, separated equations should describe them. Constitutive equations that combine them into a single equation are physically meaningless. A consequence is that elastic... 

Thirdly, when we separated equation 43-51 into two terms, we only worked with the first term. The second term, which we presented in equation 43-52B, was neglected. Is it possible that the nonlinear effects observed for equation 43-52A will also operate on equation 43-52B The answer is yes, it will, but... And the but..is this AEs is a random variable, just as AEr is. Furthermore, it is uncorrelated with AEx. Therefore, in order to evaluate the integral representing the variation of both AEs and AEr, it would be necessary to perform a double integration over both variables. Now, for each value of AEs, the nonlinearity caused by the presence of AEr in the denominator would apply. However, AEs is symmetrically distributed around zero, therefore for every positive value of AEs there is an equal but negative value that is subject to exactly the same nonlinear effect. The net result is that these pairs always form equal and opposite contributions to the integral, which therefore cancel, leaving no effect due to AEs. 

In the low-noise case we were able to justify separating equation 53-5 into two terms and setting T equal to EJ(EX + AEr). Here we cannot do that for several reasons ... 

In E-Z Solve, a separate equation entry is required for each data pair. The software then determines the values of m and b which minimize the sum of squared residuals (SSR). The E-Z Solve syntax is shown below. 

If an error probability of 0.1 is considered, then 12.03 > 9.24 (from statistical tables) and one may say that inconsistency is important at this error probability level. After sequential processing of the measurements, as is shown in Table 10, feed temperature and reboiler and condenser duties are suspected to contain gross errors. Since feed temperature and reboiler duty do not appear in separate equations, it is difficult to isolate the gross error when it happens in one or both of duties. In this case, we need... 

Equations 11 and 12 are not written for constant molality, and can not be easily used with the Gibbs-Duhem equation to obtain an analytical expression for the activity of water in the ternary solution. However, it is possible to propose a separate equation for the activity coefficient of water that is consistent with the proposed model of concentrated solutions. 

One of the most successful methods for calculating the number of plates necessary for a given separation is due to Lewis and Matheson(38). This is based on the Lewis-Sorel method, described previously for binary mixtures. If the composition of the liquid on any plate is known, then the composition of the vapour in equilibrium is calculated from a knowledge of the vapour pressures or relative volatilities of the individual components. The composition of the liquid on the plate above is then found by using an operating equation, as for binary mixtures, although in this case there will be a separate equation for each component. 

Another key point of differentiation is the fact that nearly all PSA separations are bulk separations and any investigator interested in a high fidelity description of the problem of adsorption must solve a mass balance equation such as Eq. (9.9), the bulk separation equation, together with the uptake rate model and a set of thermal balance equations of similar form. In addition to the more complicated pde and its attendant boundary and initial conditions the investigator must also solve some approximate form of a momentum balance on the fluid flow as a whole. 

Equation (2.18) can be split into two separate equations as foUows ... 

Experiments carried out with accelerating horizontal bubble flow, that is, with variable quality, were correlated by similar but separate equations for f-in., -in., and -in. diameter pipe. For the -in. pipe,... 

The critical gas velocity, Vocb, at the start of entrainment is expressed by three separate equations, for low, medium and high Reynolds numbers. 

Many word problems lend themselves to more than one equation with more than one variable. It s easier to write two separate equations, but it takes more work to solve them for the unknowns. And, in order for there to be a solution at all, you have to have at least as many equations as variables. 

Separating Equation A-10 into two equations helps to clarify the procedure needed to quickly calculate the q1 quantile, Xq, of pdf(x) when x is a lognormally distributed random variable. Since q and Zq are related by Equation A-2, Equation A-ll can be used to solve for Xq given q or for q given Xq. 

The right-hand side is constant in r, so the left-hand side must also be constant in r. Contrariwise, both sides are constant in 3 and (p. In other words, the variables are separated into different terms, a happy accident that we can exploit. We started with one differential equation involving three variables, and ended up with two separate equations, one involving one variable, and one involving two variables. Thus we have reduced the problem (finding solutions to the original equation) to two simpler problems. Of course, this simplification works only if our supposition (that there are solutions of the given form) turns out to be true. 

We place no upper bound on j since we assume that the chain is long enough to neglect the end effect, which manifests itself in separate equations for dPNjdt and dPs i dt. 

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