Algebra simultaneous equations

The gas volumes in the body of the turbine are very small, so that the establishment of flow in the turbine is very fast, with a response time measured in tenths of a second. Since these time constants are an order of magnitude less than the rotor time constant associated with turbine speed, it is permissible to treat the equations of fluid flow as algebraic, simultaneous equations. This simplification receives even greater justification when the model of the turbine and the machine it drives are interfaced with a model of the rest of the process plant, which will be invariably much slower still. It is only when the turbine is being considered in detail on its own, for example in designing a speed-control system, that some of the larger turbine pressure constants, such as those associated with the turbine inlet manifolds, will need to be considered. 

The summations in Eqs. (2-73) are over all i. Equations (2-73) are called the normal regression equations. With the experimental observations of 3, as a function of the Xij, the summations are carried out, and the resulting simultaneous equations are solved for the parameters. This is usually done by matrix algebra. Define these matrices ... 

Equations (4.1) or (4.2) are a set of N simultaneous equations in iV+1 unknowns, the unknowns being the N outlet concentrations aout,bout, , and the one volumetric flow rate Qout- Note that Qom is evaluated at the conditions within the reactor. If the mass density of the fluid is constant, as is approximately true for liquid systems, then Qout=Qm- This allows Equations (4.1) to be solved for the outlet compositions. If Qout is unknown, then the component balances must be supplemented by an equation of state for the system. Perhaps surprisingly, the algebraic equations governing the steady-state performance of a CSTR are usually more difficult to solve than the sets of simultaneous, first-order ODEs encountered in Chapters 2 and 3. We start with an example that is easy but important. 

The formal, algebraic, method. The presence of recycle implies that some of the mass balance equations will have to be solved simultaneously. The equations are set up with the recycle flows as unknowns and solved using standard methods for the solution of simultaneous equations. 

An alternative method of solving the equations is to solve them as simultaneous equations. In that case, one can specify the design variables and the desired specifications and let the computer figure out the process parameters that will achieve those objectives. It is possible to overspecify the system or to give impossible conditions. However, the biggest drawback to this method of simulation is that large sets (tens of thousands) of nonlinear algebraic equations must be solved simultaneously. As computers become faster, this is less of an impediment, provided efficient software is available. 

Thus, considering equation 4-2, we note that the matrix expression looks like a simple algebraic expression relating the product of two variables to a third variable, even though in this case the variables in question are entire matrices. In equation 4-2, the matrix /f represents the unknown quantities in the original simultaneous equations. If equation 4-2 were a simple algebraic equation, clearly the solution would be to divide both sides of this equation by A, which would result in the equation B = C/A. Since A and C both represent known quantities, a simple calculation would give the solution for the unknown B. 

DERIVATION OF MORE COMPLICATED RATE EQUATIONS. So far, the rate equations that describe one-substrate enzyme systems have been fairly simple, and the usual algebraic manipulations of substitution and/or addition of simultaneous equations have permitted us to obtain the pertinent rate law. When the number of steps increases and especially when there are branched pathways involved, these manual methods become cumbersome, and more systematic procedures are required. The next two sections should allow the reader to develop a working knowledge of effective methods for obtaining multisubstrate enzyme rate expressions. 

Accdg to von Stein Alster (Ref 41), accurate determination of isochoric adiabatic flame temp of an expl often involves a series of tedious calcns of the equilibrium of compn of the expln products at several temps. Calcg the expln product compn at equilibrium is a tedious process for it.requires the soln of a number of non-linear simultaneous equations by a laborious iterative procedure. Damkoehler Edse (Ref 11) developed a graphical procedure and Wintemitz (Ref 26a) improved it. by transforming it into its algebraic equivalent. Unfortunately both.methods proved less useful with.hetorogeneous equilibria which. contain solid carbon... 

Here is where we stand Equations 16-9 and 16-10 are both statements of algebraic truth. But neither one alone allows us to find E+, because we do not know exactly what tiny concentrations of Fe2+ and Ce4+ are present. It is possible to solve the four simultaneous equations 16-7 through (6-10 by first adding Equations 16-9 and 16-10 ... 

Finally, by algebraic solution of the two simultaneous equations it follows that... 

We leave the algebra to you as an exercise. Solving the two simultaneous equations you will find that the most probable thickness of the critical nucleus is (Equation 10-25) ... 

The degree-of freedom analysis tells us that there are five unknowns and that we have five equations to solve for them [three mole balances, the density relationship between V2 (= 225 Uh) and hj, and the fractional condensation], hence zero degrees of freedom. Hie problem is therefore solvable in principle. We may now lay out the solution—still before proceeding to any algebraic or numerical calculations—by writing out the equations in an efficient solution order (equations involving only one unknown first, then simultaneous pairs of equations, etc.) and circling the variables for which we would solve each equation or set of simultaneous equations. In this... 

If the number of unknowns equals the number of equations relating them (i.e., if the system has zero degrees of freedom), write the equations in an efficient order (minh mizing simultaneous equations) and circle the variables for which you will solve (as in Example 4.3-4). Start with equations that only involve one unknown variable, then pairs of simultaneous equations containing two unknown variables, and so on. Do no algebra or arithmetic in this step. 

One algebraic method involves rates of reactions when the reagent concentration is large compared with [A]o -I- [B]q. Two simultaneous equations are solved at two times of observation, one near the optimum (Section 21-3) and the other when the reaction is nearly complete. This method is less restricted than graphical extrapolation with respect to both [A]q/[B]o and kjk, because Ata[A], need not be negligible compared with A b[B], to make analytically useful observations feasible. With this method the error in analysis increases when the ratio kjk approaches unity and when the second observation is made at a time prior to complete reaction. 

This is a system of three simultaneous equations in three unknowns. Each unknown age is represented by a variable. Each time a particular variable appears in an equation, it stands for the same quantity. In order to see how the concept of a matrix enters, rewrite the above equations, using the standard rules of algebra, as ... 

Thus, the required %v/v propylene glycol is (27.1/ 46.5) X 100 = 58.3. Alternatively, the %v/v of the new cosolvent can be solved using an algebraic method involving the solution of simultaneous equations however, aligation is a simpler method when more than one cosolvent is to be included in the formulation. When a vehicle is to be formulated for the first time, it is necessary to experimentally determine the concentration of some cosolvent necessary to maintain the required concentration of drug in solution. This value can then be used to calculate the ADR and the final vehicle calculated as illustrated previously. 

Under the first assumption, each electron moves as an independent particle and is described by a one-electron orbital similar to those of the hydrogen atom. The wave function for the atom then becomes a product of these one-electron orbitals, which we denote P (r,). For example, the wave function for lithium (Li) has the form i/ atom = Pa ri) Pp r2) Py r3). This product form is called the orbital approximation for atoms. The second and third assumptions in effect convert the exact Schrodinger equation for the atom into a set of simultaneous equations for the unknown effective field and the unknown one-electron orbitals. These equations must be solved by iteration until a self-consistent solution is obtained. (In spirit, this approach is identical to the solution of complicated algebraic equations by the method of iteration described in Appendix C.) Like any other method for solving the Schrodinger equation, Hartree s method produces two principal results energy levels and orbitals. 

Many real-world applications of chemistry and biochemistry involve fairly complex sets of reactions occurring in sequence and/or in parallel. Each of these individual reactions is governed by its own equilibrium constant. How do we describe the overall progress of the entire coupled set of reactions We write all the involved equilibrium expressions and treat them as a set of simultaneous algebraic equations, because the concentrations of various chemical species appear in several expressions in the set. Examination of relative values of equilibrium constants shows that some reactions dominate the overall coupled set of reactions, and this chemical insight enables mathematical simplifications in the simultaneous equations. We study coupled equilibria in considerable detail in Chapter 15 on acid-base equilibrium. Here, we provide a brief introduction to this topic in the context of an important biochemical reaction. 

Regardless of which tools were used in the indexing of the powder diffraction pattern, the most reliable solution should result in the minimum discrepancies in the series of simultaneous equations (Eq. 5.5 or its equivalent for a different crystal system) constructed with the observed 20 substituted into the left hand side and the assigned index triplets and refined unit cell dimensions substituted into the right hand side of each equation. While the minimum combined discrepancy is easily established algebraically, e.g. as the sum of the squared differences... 

Even conceptually simple reactions - such as the textbook anti-Markovnikov addition of HBr to isobutene, catalysed by dibenzoyl peroxide, can lead to sets of simultaneous equations which may need computer algebra such as Mathe-matica to solve. 

You are free, of course, to pick three characteristic wavelengths, and to solve the resulting three simultaneous equations. The added noise will then (rather strongly) affect your results, but you will have no way of knowing by how much. The method illustrated below is not only much less sensitive to noise, but also provides error estimates and, most importandy, is much easier to implement. Of course it uses matrix algebra, just as you did in section 6.2, but that will be completely invisible to you, the user. The entire analysis comes prepackaged with the spreadsheet. 

In this section we will briefly review the most salient aspects of matrix algebra, insofar as these are used in solving sets of simultaneous equations with linear coefficients. We already encountered the power and convenience of this method in section 6.2, and we will use matrices again in section 10.7, where we will see how they form the backbone of least squares analysis. Here we merely provide a short review. If you are not already somewhat familiar with matrices, the discussion to follow is most likely too short, and you may have to consult a mathematics book for a more detailed explanation. For the sake of simplicity, we will restrict ourselves here to two-dimensional matrices. 

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