Courant number

All the methods have a limit to the time step that is set by the convection term. Essentially, the time step should not be so big as to take the material farther than it can go at its velocity. This is usually expressed as a Courant number limitation. 

In a backwards-in-distance solution for advective transport in the absence of dispersion or diffusion, the Courant criterion limits the time step. In one dimension, the grid Courant number is the number of nodal blocks the fluid traverses over a time step. By the Courant criterion, the Courant number Co must not exceed one, or... 

Numerical dispersion can be minimized in several ways. The nodal block spacing Ax can be set small by dividing the domain into as many blocks as practical. The value specified for Dl can be reduced to account for the anticipated numerical dispersion. And a time step can be chosen to give a grid Courant number Co as close to one as allowed by the stability criterion (Eqn. 20.32). 

See Pope (2000) for an alternative estimate of the time-step scaling based on the Courant number. The overall conclusion, however, remains the same DNS is computationally prohibitive for high Reynolds numbers. 

The breakthrough curve for different values of the Courant number is given in Figure E7.3.1. A lower Courant number, less than 1, adds more numerical diffusion to the solution. If the Courant number is greater than 1, the solution is unstable. This Cou > 1 solution is not shown in Figure E7.3.1 because it dwarfs the actual solution. Thus, for a purely convective problem, the Courant number needs to be close to 1, but not greater than 1, for an accurate solution. In addition to the value of the Courant number, the amount of numerical diffusion depends on the value of the term UAz, which is the topic discussed next. 

Figure E7.3.1. Computational solutions to the purely convective filter transport problem with varying Courant number.

Let US return to the discussion of computational transport routines, where each computational cell is the equivalent of a complete mix reactor. If we are putting together a computational mass transport routine, we could simply specify the size of the cells to match the diffusion/dispersion in the system. The number of well-mixed cells in an estuary or river, for example, could be calculated from equation (6.44), assuming a small Courant number. Then, the equivalent longitudinal dispersion coefficient for the system would be calculated from equation (6.44), as well, for a small At (At was infinitely small in equation 6.44) ... 

The analytical solution of Equation (E7.4.7) is compared with the computational solution with a grid Courant number of 0.1 (and no diffusion coefficient) in Figure E7.4.1. It is seen that there is not much difference between the two. 

Figure E7.4.1. Solution to convective transport problem at Co = 0.1 and comparison with equation (E7.4.7) with D = U zl2. Cou, Courant number.

The problem of Example 7.3 will again be solved with explicit and implicit exponential differences, and compared with the analytical solution, equation (E7.4.7). This solution is given in Figure E7.5.1. Note that the explicit solution is close to the analytical solution, but at a Courant number of 0.5, whereas the implicit solution could solve the problem with less accuracy at a Courant number of 5. In addition, the diffusion number of the explicit solution was 0.4, below the limit of Di < 0.5. The implicit solution does not need to meet this criteria and had Di = 4. 

Figure E7.5.1. Comparison of explicit, implicit, and analytical solutions for the filter problem. Fe = 1.25 for explicit and implicit solutions. Cou, Courant number.

Numeric dispersion can be eliminated largely by a high-resolution discretisation. The Grid-Peclet number helps for the definition of the cell size. Pinder and Gray (1977) recommend the Pe to be < 2. The high resolution discretisation, however, leads to extremely long computing times. Additionally the stability of the numeric finite-differences method is influenced by the discretisation of time. The Courant number (Eq. 104) is a criterion, so that the transport of a particle is calculated within at least one time interval per cell. 

Numerical dispersion or oscillation effects can occur as accidental source of error when using finite differences and finite element methods while modeling mass transport. Utilizing the criteria of numerical stability (Grid-Peclet number or Courant number) or the random walk procedure, these errors can be either reduced or even eliminated. 

Conditions 10.86 to 10.88 are frequently called the Courant-Friedrichs- Lewy (CFL) convergence conditions [58] and a is called the Courant number. 

In these equations, Uzfi is the migration velocity of the solute at infinite dilution given by Eq. 10.85, a is the Courant number, and the order of the error made in the three models is the first order, 0 h + r). 

The value of the former integration parameter is positive only if H decreases with increasing fcg, and the value of the latter is positive only if H decreases rapidly enough with increasing fcg to compensate for the opposite variation of the term 1 + fcg If we introduce the values of t and H given by Eq. 11.13a,b into the definition of the Courant number, we obtain... 

A further condition applies to the value of the Courant number, which must exceed 1 with the forward-backward scheme. The requirements that both t and h are positive and that fli and 2 are larger than unity are often difficult to satisfy. Furthermore, if T and h are small, the computation time may become excessive. Similar results are obtained with the other schemes. 

Accordingly, it is preferable to accept the introduction of an additional error and to choose the value of the Courant number because of its importance for the numerical stability of the calculation. Then, we can choose the value of H and, accordingly, the value of Da derived from the truncation error, for one single component. For aU the other components, H and Da are fixed and there is no reason that these values are those observed in the practical chromatographic problem studied. These values may not even be realistic [6-8]. Even in linear chromatography, where Eqs. 11.9a to 11.9c are exact, if one simulates the band profiles of a series of compoimds using the multicomponent scheme (although there would not be any... 

For example, let us consider a two-component problem and use the forward-backward scheme to calculate numerical solutions, with a Courant number equal to fli for the first component. In linear chromatography, the contribution of the numerical dispersion for this first component would be exactly equivalent to the true HETP, Hi, if the space and time increments are chosen according to Eqs. 10.86 and 10.96... 

For these values of the time and space increments, the value of the Courant number for the second component is given by... 

Thus, in the calculation of the individual band profiles in the case of multicomponent mixtures, there is a third source of errors, besides the two classical error sources observed with finite difference methods, which we have discussed in the study of the single-component problem. Obviously, these two sources are also found in the calculations of solutions of the multicomponent problems. As can be seen from Eqs. 11.17 to 11.19, this new error increases with the difference between the retention factors of the two components, and it decreases with decreasing Courant number. The error would disappear with the second and third schemes (Eqs. 11.18 and 11.19) and the numerical dispersion for the two solutes would become equal and correspond to the proper value of H if a was dose enough to 0. This observation is important because, for these two schemes (Eqs. 10.87 and 10.88), we can always select low values of the Courant raunber if needed by combining a large space increment h and a suitably small time increment t (Eq. 11.10). 

Figure 11.6 Comparison of profiles calculated with OCFE and a finite difference method. N = 1000 plates. Line 1, forward-backward finite difference algorithm, Courant number = 2 line 2, OCFE, fp = 5 s. (a) 1 9 mixture, a = 1.5. (b) 1 1 mixture, a. = 1.5. (c) 5 1 mixture, a = 1.2.

Courant number, A number characterizing the risk of instability in the numerical integration of partial differential equations (see Eqs. 10.86 to 10.88). 

Implicit time integration schemes are not as efficient as the corresponding explicit schemes due to the computational time required on the iterative process. With larger time steps the accuracy of implicit schemes decrease rapidly. The widespread use of the implicit schemes with Courant numbers ten- or even hundredfold the magnitude of what is used in an explicit method, is not justifiable in the presence of gradients or steps in the convected variable. 

Courant number, used in the Courant-Friedrichs-Lewy necessary stability condition for hyperbolic equations... 

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