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Chemical Reactor Modeling pp 1089–1273Cite as

Numerical Solution Methods

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Abstract

In this chapter several numerical methods frequently employed in reactor engineering are introduced. To simulate the important phenomena determining single- and multiphase reactive flows, mathematical equations with different characteristics have to be solved. The relevant equations considered are the governing equations of single phase fluid mechanics, the multi-fluid model equations for multiphase flows, and the population balance equation.

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Notes

  1. 1.

    FIDAP is a general purpose finite element code for simulating two-dimensional, axisymmetric, or full three dimensional equations of viscous incompressible Newtonian or non-Newtonian fluid flow, including the effects of heat transfer.

  2. 2.

    COMSOL Multiphysics/FEMLAB is an interactive FEM-based environment for modeling, implementation and solving scientific and engineering problems involving PDEs. This simulation software offers interface to MATLAB with toolboxes which gives MATLAB the ability to solve 2D PDEs by the Finite Element Method, including meshing, preprocessing and post processing capabilities.

  3. 3.

    A matrix which is \(n \times m\) with \(k\) non-zero entries is sparse if \(k \ll n \times m\).

  4. 4.

    The names elliptic, parabolic, and hyperbolic that denote the different characters of the equation, have arisen by analogy with the conic sections of analytic geometry [215, 257].

  5. 5.

    In the case of solving integro-PDE problems, the operator defined by formula (12.24) must also be used.

  6. 6.

    The (Lanczos) method was named the tau method because Lanczos used the letter \(\tau \) to represent the error.

  7. 7.

    This is an \(L_2\) inner product. The orthogonality condition states that the residual is orthogonal to the space of basis functions.

  8. 8.

    In an attempt to find an exact formula for the integral, we may resort to the mean value theorem of calculus. This theorem states that if the integrand is evaluated at a particular known instant \(t = \tau \) between \(t_n\) and \(t_{n+1}\), the integral is equal to \(f(\tau ,\psi (\tau )) \varDelta t\). However, in the present case the theorem is of little use since the instant \(\tau \) is unknown.

  9. 9.

    In a few textbooks these methods are referred to as two-level methods because they involve the values of the unknown at two time levels [56].

  10. 10.

    The conservation law together with piecewise constant data having a single discontinuity is known as the Riemann problem [134].

  11. 11.

    The early meteorological finite difference studies of long-term numerical time integrations of the equations of fluid motion, which involve non-linear convection terms, revealed the presence of non-linear instabilities due to aliasing errors [6, 171–173, 259]. To avoid the occurrence of these non-linear instabilities, Arakawa [6] was the first to recognize the importance of the use of numerical schemes which conserve kinetic energy.

  12. 12.

    The velocity- and pressure correction equations in IPSA are frequently derived using the SIMPLEC method (i.e., the SIMPLE- Consistent approximation) by van Doormal and Raithby [239].

  13. 13.

    It is noted that the method of presumed distribution functions was originally proposed by Hulburt and Katz [90]. The QMOM may thus be considered an extension of the Hulburt and Katz [90] procedure [230].

  14. 14.

    The matrix \({\mathcal {M}}\) is symmetric if \({\mathcal {A}}={\mathcal {A}}^T\). The matrix is said to be positive definite if the Euclidean inner product \(({\mathbf {x}},{{\mathcal {M}}}{\mathbf {x}})>0\) whenever \({\mathbf {x}}\ne 0\) [205]. The Euclidean inner product between two vectors \({\mathbf {x}}\) and \({\mathbf {y}}\) is defined as \(({\mathbf {x}},{\mathbf {y}})={\mathbf {x}}^T {\mathbf {y}}=\sum _{l=1}^n x_l y_l\).

  15. 15.

    If the matrix \({\mathcal {A}}\) is symmetric, then two vectors \({\mathbf {x}}\) and \({\mathbf {y}}\) are conjugate or A-orthogonal if the A-inner product \(({\mathbf {x}},{\mathbf {y}})_{{\mathcal {A}}} = ({\mathcal {A}} {\mathbf {x}},{\mathbf {y}})=({\mathbf {x}},{\mathcal {A}} {\mathbf {y}}) = 0\) holds [205]. Vectors are orthogonal if \(({\mathbf {x}},{\mathbf {y}})=0\).

  16. 16.

    Several stop criteria can be defined in terms of different norms of the residual [205]. The general \(p\)-norm of a vector is defined as \(||{\mathbf {r}}||_p = (\sum _{i=1}^n |r_i|^p)^{1/p}\). When \(p\) tends to infinity, the vector norm becomes \(||{\mathbf {r}}||_\infty = \max |{\mathbf {r}}|\).

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Jakobsen, H.A. (2014). Numerical Solution Methods. In: Chemical Reactor Modeling. Springer, Cham. https://doi.org/10.1007/978-3-319-05092-8_12

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