Interval halving

This technique is quite simple and easy to visualize and program. It is not very rapid in converging to the correct solution, but it is rock-bottom stable (it won t blow up on you numerically). It works well in dynamic simulations because the step size can be adjusted to correspond approximately to the rate at which the variable is changing with time during the integration time step. 

We keep moving in the eorreet direction at this fixed step size until there is a change in e sign of the term (P - p ). This means we have erossed over the correct value of T. Then we back up halfway, i.e., we halve the increment AT. With each successive iteration we again halve AT, always moving either up or down in temperature. 

MAIN PROGRAM SETTS DIFFERENT VALUES FOR INITIAL GUESS OF TEMPERATURE AND DIFFERENT INITIAL STEP SIZES SPECIFIC CHEMICAL SYSTEM IS BENZENE/TOLUENE AT 760 MM HG PRESSURE PROGRAM MAIN 

1 FORMATC LOOP IN BUBPT SUBROUTINE ) STOP ENDIF 

Clearly, the number of iterations to eonverge depends on how far the initial guess is from the correct value and the size of the initial step. Table 4.1 gives results for several initial guesses of temperature (TO) and several step sizes PTO). The interval-halving algorithm takes 10 to 20 iterations to converge to the eorrect temperature. 

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Successive linearisation has the advantage of relative simplicity and fast calculation. In addition, it can be modified to choose a step size that minimizes a prespecified penalty function. The step size is chosen by the method of interval halving (Pai and Fisher, 1988). However, variable bounds cannot be handled it may fail to converge to the desired minimum and it might oscillate when multiple minima exist. 

The method involves a simple iteration on only one variable, pH. Simple interval-halving convergence (see Chap. 4) can be used very effectively. The titration curves can be easily converted into simple functions to include in the computer program. For example, straight-line sections can be used to interpolate between data points. 

Figure 4.1 sketches the interval-halving procedure graphically. An initial guess of temperature Tq is made. is calculated from Eq. (4.6). Then P " is compared to P. A fixed increment in temperature AT is added to or subtracted from the temperature guess, depending on whether P is greater or less than P. 

Interval halving can also be used when more than one unknown must be found. For example, suppose there are two unknowns. Two interval-halving loops could be used, one inside the other. With a fixed value of the outside variable, the inside loop is converged first to find the inside variable. Then the outside variable is changed, and the inside loop is reeonverged. This procedure is repeated until both unknown variables are found that satisfy all the required equations. 

This convergence technique is a combination of Newton-Raphson and interval halving. An initial guess Tq is made, and the fimction , is evaluated. A step is taken in the correct direction to a new temperature T, and is evaluated. If... 

Compare convergence times, using interval halving, Newton-Raphson, and false position, for on ideal, four-component, vapor-liquid equilibrium system. The pure component vapor pressures are ... 

C AND PRESSURE FROM KNOWN DENSITY USING C INTERVAL HALVING FLAGP=-1. 

One solution technique is to guess a value for Then Eq. (14.50) for model 4 [(14.52) for model 5] is solved for. Finally the right-hand side of Eq. (14.51) U14.53)] is calculated let us call this If the actual argument of A) is equal to the correct values of Xpi and have been found, Interval-halving can be used to reguess Tpi if A and arc not sufficiently close. 

FORTRAN program with a subroutine ZNT which does this iterative calculation using simple interval halving. 

The golden section search guarantees that each new function evaluation will reduce the uncertainty interval to a length of >. times the previous interval. This is comparable to, but not as good as interval halving in the bisection method of solving a nonlinear equation. You can easily calculate that to attain an error tolerance EP we need... 

We know Qcol, U, Acoil, TR and TCm. Combining the two equations above gives one equation in one unknown, the temperature of the coolant leaving the coil Tc,out. However, the log term precludes an analytic solution, so an iterative interval halving solution method is used. 

Iterative calculation for tcout using log mean temperature difference % using interval halving % Make initial guess of tcout tcout=tr-1 ... 

Either Equation (4.481) or Equation (4.483) can be solved for with % to follow, by searching with a suitable numerical procedure in the interval from 0 to 1. The existence of an extremum within the search interval can be troublesome, though not serious for example, interval halving will provide a solution. However, a more robust equation for numerical solution is obtained by combining the two equations. Subtracting one equation from the other gives... 

Discuss the methods of interval halving, successive substitution, and New-ton-Raphson for solving nonlinear algebraic equations. What are their relative advantages and disadvantages ... 

Calculation of pH titn. curves and end-points. Iterative method with interval halving... 

It is more advantageous to apply methods based on minimalisation of the Gibbs function. In this case we may proceed as follows We determine the equilibrium composition, e.g. by means of the method of Lagrangian multipliers for several temperature values in the vicinity of the expected Tg value. To do this, we must know the dependence of c,- = G]jRT + In P values on temperature. At every temperature, for which we have calculated the equilibrium composition, we determine the values of AH = HE D START- Let AH < 0 apply for the temperature and AH > 0 for the temperature The required temperature Tg will then lie in the interval (Te Furthermore we can apply e.g. the interval halving method or the regula falsi method (see Appendix 3). The c = Cf(T) i = 1, 2,. ..,iV relationship is determined as follows values of — (G — Hp)IT, are tabulated in the literature for various values of T. The standard temperature is usually OK or 298.15 K. The quantity is independent of temperature, and polynomial development to at most the third or fourth degree will usually suffice to elucidate the value of —(Gy — Hy)/T. 

The method of interval halving is a two-step, first-order method. Before it is applied, two approximations must be known, such that x e... 

In this section two iterative schemes for calculating the density given values for the pressure and temperature are described the bisection or interval-halving method, and the Newton-Raphson technique. These methods and others are described in more detail by Burden et al. (1978), who also give algorithms for these procedures. 

For weak adds and bases with double and triple dissociations. Equations 2 4b and 2-4c are used, respectively, in place of Equation 2-4a in Equation 2-4h. Appendix F has a FORTRAN subroutine listing that finds the pH that satisfies the charge balance for i adds and bases with single, double, or triple dissodations by interval halving, The pKj, and pK coeffidents and the solution density should be corrected for process temperature and composition. The effect of changes in the activity coefficients with ionic strength should also be factored in [Ref. 2.6]. The reader is directed to References 2.4 and 2.8 for a more detailed discussion on the effect of ion concentrations on activity coeffidents. The polynomial equation for acid and sodium ion error at die end of the subroutine should be replaced with one that fits the data from the glass manufacturer. 

An interval halving search to find the pH that satisfies the charge balance is the most efficient method to compute pH for a complex mixture. 

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