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Heat Transfer
Published in Yeong Koo Yeo, Chemical Engineering Computation with MATLAB®, 2020
In the method of lines, only the spatial derivatives are discretized using finite differences, not the time derivatives. For the one-dimensional parabolic equation ∂u/∂t=α∂2u/∂x2, the discretization of only the spatial derivative term yields duidt=α∆x2(ui+1−2ui+ui−1)
Hybrid Numerical–Analytical Solutions
Published in M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta, Finite Difference Methods in Heat Transfer, 2017
M. Necati Özişik, Helcio R.B. Orlande, Marcelo José Colaço, Renato Machado Cotta
Equations (12.18a–g) form an infinite coupled system of nonlinear one-dimensional PDEs for the transformed potentials, T¯ki(x3,t), which is unlikely to be analytically solvable. Nonetheless, reliable algorithms are readily available to numerically handle this PDE system after truncation to a sufficiently large finite order. For instance, the Mathematica system provides the built-in routine NDSolve (Wolfram 2015), which employs the Method of Lines based on finite difference formulae of variable order and step size, and can handle this system under automatic absolute and relative errors control. The Method of Lines is a numerical technique for PDEs that involves the finite difference approximation of all the space coordinates differential operators, and thus transforming the original PDE into an ODE system as an initial value problem in the time variable, either in an actual transient problem or in a pseudo-transient formulation. One interesting aspect of this approach is that reliable and automatic solvers for initial value problems can then be readily employed. In our partial transformation hybrid approach, once the transformed potentials have been numerically computed, the Mathematica routine automatically provides an interpolating function object that approximates the x3 and t variables behavior of the solution in a continuous form. Then, the inversion formula [equation (12.17b)] can be recalled to yield the potential field representation at any desired position r and time t.
Numerical Methods
Published in William F. Ames, George Cain, Y.L. Tong, W. Glenn Steele, Hugh W. Coleman, Richard L. Kautz, Dan M. Frangopol, Paul Norton, Mathematics for Mechanical Engineers, 2022
The method of lines, when used on PDEs in two dimensions, reduces the PDE to a system of ordinary differential equations (ODEs), usually by finite difference or finite element techniques, if the original problem is an initial value (boundary value) problem, then the resulting ODEs form an initial value (boundary value) problem. These ODEs are solved by ODE numerical methods.
Assessment of reduced-order modeling strategies for convective heat transfer
Published in Numerical Heat Transfer, Part A: Applications, 2020
Victor Zucatti, Hugo F. S. Lui, Diogo B. Pitz, William R. Wolf
The equations governing the current convective heat transfer problem (Eqs. (1)–(3)) contain partial derivatives with respect to both spatial coordinates and time. Using the method of lines one can first approximate the spatial derivatives producing a system of ODEs. In the most general notation, for each mesh point, these ODEs would be expressed in the form where G is a nonlinear operator that contains all the spatial derivatives, and is the vector of flow variables.
Computational modelling of multi-material energetic materials and systems
Published in Combustion Theory and Modelling, 2020
Alberto M. Hernández, D. Scott Stewart
To numerically solve Equation (1), the computational domain is discretised in a structured finite difference gird where . Node points are located in the computational grid by i, j and k indices which represent x, y and z-coordinate locations along the physical domain. The method-of-lines (MOL) approach is used to reduce the PDEs of Equation (1) to a system of ordinary differential equations allowing the temporal and spatial problem to be solved independently. Equation (1) can then be re-written as where is the spatial operator on U and is taken to be where S is the source term, , and are the interface fluxes in the x, y and z-coordinate direction, respectively. The spatial operator is calculated component-wise as a 1D directional solve along a given coordinate direction (i.e. to solve the interface flux the j and k indices are held fixed). The higher order Weighted Essential Non-Oscillatory (WENO) interpolation of Jiang and Shu [23] is used to approximate the cell interface flux, along with Lax–Friedrich flux splitting given by Equation (9) where α is the dissipation coefficient defined as . Numerical integration of the system of ODEs is performed using a Total Variation Diminishing (TVD) third-order Runge–Kutta solver presented by Shu and Osher [24] To advance the solution from time n to n + 1 a time step given by Equation (11) is taken where c is the sound speed, dim is the dimension of the problem (for three-dimensional simulations ) and CFL is the Courant–Friedrichs–Lewy condition.
Effects of gas absorption with chemical dissociation reaction on single slurry droplet drying
Published in Drying Technology, 2020
Yehonatan David Pour, Andrew Fominykh, Boris Krasovitov, Avi Levy
The system of partial differential equations (PDEs) given in Equations (1)–(5) was solved using the d03ph function of the NAG toolbox library for MATLAB. The d03ph function integrates a system of nonlinear parabolic PDEs and coupled ordinary differential equations (ODEs). Spatial discretization is performed using finite differences, and the method of lines is used to reduce the PDEs to a system of ODEs. The resulting system can then be solved using a backward differentiation formula method. The mesh points were spaced adaptively using the following formula: where, is the chosen number of mesh points and is an integer. Thus, mesh points cluster near the boundary where the gradients are steep. In our calculations, we assumed = 3. For the numerical solution, we employed 151 mesh points and an error tolerance of 10−5 in the time integration steps. During the numerical solution of Equations (1)–(5), properties such as and were calculated simultaneously at each mesh point for each time step. Formulas with appropriate physical data for calculating these properties were taken from Reid et al.[58] Calculations were stopped when the droplet radius reached the value of (Equation (30)). Thus, in our calculations, we assumed that the initial volume fraction of solid particles in the slurry droplet, was equal to 0.1. The suggested model of gas absorption by an evaporating slurry droplet was applied to the drying of a slurry droplet immersed in a stagnant ternary gaseous mixture. Air is a commonly used spray-drying medium. However, when the drying materials are flammable or sensitive to oxygen, nitrogen is used instead.[60] Notably, the evaporation rates of water droplets in air and in nitrogen are similar. Therefore, the model of gas absorption by an evaporating slurry droplet presented here was applied to the drying of a slurry droplet immersed in a stagnant ternary gaseous mixture composed of nitrogen (N2), soluble ammonia (NH3), and the vapor of the water droplet. In our model, we accounted for the effects of gas absorption with subsequent chemical dissociation reactions in the evaporation of a coal-water slurry droplet.