Industrial Mathematics: Case Studies in the Diffusion of by Glenn R. Fulford, Philip Broadbridge

By Glenn R. Fulford, Philip Broadbridge

The focal point during this textual content is on mathematical modelling inspired by means of modern business difficulties related to warmth conduction and mass diffusion. those contain non-stop steel casting, laser drilling, spontaneous combustion of business waste, water filtration and crop irrigation. the economic difficulties turn out to be a superb surroundings for the advent and reinforcement of modelling abilities, equation fixing thoughts, qualitative figuring out of partial differential equations and their dynamical houses. Mathematical issues contain developing partial differential equations and boundary stipulations, dimensional research, scaling, perturbation expansions, boundary valuer difficulties, Fourier sequence, symmetry savings, Stefan difficulties and bifurcations.

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2 ∂x ∂y ∂z Dividing by ρc gives ✗ ∂u = α∇2 u, ∂t ✔ where α = k . ρc (11) ✖ ✕ This is the three-dimensional generalisation of the linear heat equation. 7 Boundary conditions 21 Note that for one-dimensional problems, where u is a function of only x and t then equation (11) reduces to the one dimensional heat equation. Just as we generalised the 1-D heat equation to account for nonlinear conduction, heat sources, advection and heterogeneity, we can do the same for the 3-D heat equation. The generalised version is ✔ ✗ ρc ∂u + ∇ · (vu) ∂t = ∇ · (k(u)∇u) + Q, (12) ✖ ✕ where v is the bulk velocity of the material, k(u) is the temperature dependent conductivity and Q is the rate of production of heat per unit time per unit volume.

1-D model for the solidification and heat transport processes in the molten steel and water cooled drum. The temperature of the molten metal is very close to the solidification 1400 ◦C for steel. We will neglect radiative heat temperature, uf transfer† from the liquid-air interface, assuming that all the heat flows towards the cooled drum. Thus we can assume that the temperature in the liquid phase is a constant, uf . We take the temperature in the † Neglecting radiative transfer, for this problem, can be justified by a simple model of one-dimensional heat conduction with radiative transfer from the liquid-air interface and ignoring solidification.

The main skill lies in deciding which variables are important in a problem, the rest is routine. Given a functional relationship of the form φ = f (x1 , x2 , . . , xn ) first take the dimensions of both sides. This yields something of the form [φ] = [x1 ]a1 [x2 ]a2 . . [xn ]an . By reducing each of the quantities [xi ], i = 1 . . n to fundamental dimensions L, M, T, and Θ and equating with the LHS, a linear system of equations should be obtained for the unknowns a1 , . . , an . Solving the linear system yields several possibilities.

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