Backwards differencing

Example 8.12 Use the backward differencing method to solve the heat transfer problem of Example 8.3. Select A-t = 0.25 and A = 0.0625. 

The results are sensitive to the differencing technique used to approximate dF(t)/dt. The two most common methods are backward differencing ... 

Differences between equations 19.3-19 and 19.3-20 are most significant if samples are collected infrequently. Ultimately, if they lead to substantially different estimates of t and of, it is necessary to verify the results using an appropriate mixing model. Example 19-2 illustrates the method for evaluating t and of from a step response, using both central and backward differencing. 

The results for backward differencing are given in Table 19.2 in the form of a spreadsheet analysis, analogous to that in Example 19-1. Column (4) gives Fft) calculated from... 

Table 19.2 Spreadsheet analysis for Example 19-2 using backward differencing...

Using the data from Example 19-2, estimate values of N based on both central and backward differencing, and determine which differencing technique best describes the tracer outflow concentrations. 

The exit-age distribution function E(t) is approximately obtained by following the experimentally often used method of backward differencing ... 

The method of lines is called an explicit method because the new value T(r, z + Az) is given as an explicit function of the old values T(r, z),T(r — Ar, z),. See, for example, Equation (8.57). This explicit scheme is obtained by using a first-order, forward difference approximation for the axial derivative. See, for example, Equation (8.16). Other approximations for dTjdz are given in Appendix 8.2. These usually give rise to implicit methods where T(r,z Az) is not found directly but is given as one member of a set of simultaneous algebraic equations. The simplest implicit scheme is known as backward differencing and is based on a first-order, backward difference approximation for dT/dz. Instead of Equation (8.57), we obtain... 

To illustrate the validity of the models presented in the previous section, results of validation experiments using lab-scale BSR modules are taken from Ref. 7. For those experiments, the selective catalytic reduction (SCR) of nitric oxide with excess ammonia served as the test reaction, using a BSR filled with strings of a commercial deNO catalyst shaped as hollow extrudates (particle diameter 1.6 or 3.2 mm). The lab-scale BSR modules had square cross sections of 35 or 70 mm. The kinetics of the model reaction had been studied separately in a recycle reactor. All parameters in the BSR models were based on theory or independent experiments on pressure drop, mass transfer, or kinetics none of the models was later fitted to the validation experiments. The PDFs of the various models were solved using a finite-difference method, with centered differencing discretization in the lateral direction and backward differencing in the axial direction the ODEs were solved mostly with a Runge-Kutta method [16]. The numerical error of the solutions was... 

In forward differencing we let 5V, = V, — Vi i in backward differencing we let 5Vi- = Vi — Vi. In central differencing we say that a particular difference is associated with the interval between two of the original values, and that therefore <5V 1/2 K — K i While this is less convenient for computation (how many programming languages allow for half-integer valued subscripts ), it reflects the semantics of the sequence better, and we will use it as our convention. 

Assuming that the tube-side fluid is liquid, we may apply backward differencing for the x-variable to the liquid-flow equation (20.14) so that the ordinary time differential of temperature at the end of each section... 

In order to generate (pnit) using backward differencing, we have... 

Numerical analysis. Equation 21-76 is conveniently solved, again using finite difference time-marching schemes. We always central difference our first and second-order space derivatives, while backward differencing in time, with respect to the nodal point (ri,tn). Furthermore, we will evaluate all nonlinear saturation-dependent coefficients at their previous values in time. This leads to... 

These formulations were solved using second-order accurate implicit schemes in the work just presented that is, our approach was implicit pressure, implicit saturation. This is in contrast to the popular implicit pressure, explicit saturation codes used in the industry, which are only conditionally stable. (The von Neumann stability of both implicit and explicit schemes was considered in Chapter 20.) This so-called IMPES scheme, in addition to its stability problems, yields undesirable saturation oscillations and overshoots that are often fixed by upstream (that is, backward) differencing of spatial derivatives. But this... 

The backward differencing method requires the solution of 7+1 simultaneous equations to find the radial temperature profile. It is semi-implicit since the solution is still marched-ahead in the axial direction. Fully implicit schemes exist where (7-I- l)(7-l-1) equations are solved simultaneously, one for each grid point in the total system. Fully implicit schemes may be used for problems where axial diffusion or conduction is important so that second derivatives in the axial direction, or 9 r/9z, must be retained in the partial differential equa-... 

One important conclusion can be drawn from this simple example. The forward difference technique has severe problems with differential equations that have differing time constants. Also as shown by diese examples, the accuracy does not approach that of the trapezoidal or backwards differencing rule. Thus the forward differencing algorithm will be eliminated from further consideration as a general purpose technique for the numerical solution of differential equations. 

For the time derivative there are several possible approximations. Three of these have been discussed in detail in Chapter 10 and are known as flie explicit forward differencing (FD) method, flie imphcit backwards differencing (BD) method and the trapezoidal rule (TP) which averages flie time derivative between two successive time points. From the discussion of fliese methods in Section 10.1, one would expect different long term stability results for each of fliese methods and this is certainly the case for partial differential equations as well as single variable differential equations. The forward and backwards time differencing methods leads to the set of equations ... 

If the time derivative is approximated by the backwards differencing method for some time interval h = At then the resulting equation becomes ... 

The other extreme is to evaluate the remaining terms at time n + 1)((50 the fully implicit or backward differencing approach. It leads to a set of algebraic equations from which the dependent variables at time (n -h 1)( 0 can be calculated. This approach is unconditionally stable (Richtmyer and Morton, 1967), and is the approach used here. We may of course also use other schemes in which intermediate weights are given to the forward and backward differences. These partially implicit schemes lead to improved accuracy. However, if attempts are made to use them on systems of stiff equations, the latter must be treated by asymptotic techniques. In chemical situations such techniques are equivalent to the use of the chemical quasi-steady-state or partial equilibrium assumptions at long times. They will be considered again in Section 9. 

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