APPLIED MATHEMATICS 5 (AMT580S)
Chapter 2: Finite Difference Equations
Tutorial Problems
1. The governing equation for 1-dimensional heat transfer, through a stationery rectangular
solid, was shown to be:
𝜕𝑇
= 𝛼∇2 𝑇
𝜕𝑡
(a) Develop a finite-difference equation for the partial differential equation.
(b) Perform a von Neumann stability analysis and determine under which conditions is the
difference equation stable.
(c) Solve the difference equation for Dirichlet boundary conditions, for 10 internal nodes of
the spatial coordinate.
2. Acetone can be removed from acetone-air mixtures using simple counter-current cascades,
by adsorption onto charcoal (Foust et al., 1980). We wish to find the required number of
equilibrium stages to reduce a gas stream carrying 0.222 kg acetone per kg air to a value of
0.0202 kg acetone per kg air. Clean charcoal (X0 = 0) enters the system at 2.5 kg/s, and the air
rate is constant at 3.5 kg/s. Equilibrium between the solid and gas can be taken to obey the
Langmuir-type relationship:
𝑌𝑛 =
𝐾𝑋𝑛
; 𝐾 = 0.5
1 + 𝐾𝑋𝑛
where
Yn = kg acetone/kg air
Xn = kg acetone/kg charcoal
(a) Show that a finite difference representation of the above cascade system gives rise to
the Riccati equation
𝑋𝑛 𝑋𝑛+1 + 𝐴𝑋𝑛+1 + 𝐵𝑋𝑛 + 𝐶 = 0
(b) Calculate the number of equilibrium stages required.
Dr B. Godongwana
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