The transport of through a given pipe is governed by the following equation for steady-state convection and diffusion.
x
V ndS =
ndS +
q dV
(1)
S
S
V
(a) Using the central differencing scheme, calculate the distribution of (x) for the following three cases. GRAPH. Assume constant velocity along pipe.
Case 1:
5 control volumes and V
= 0.1 m/s,
Case 2:
5 control volume and V
= 2.5 m/s, and
Case 3:
20 control volumes and
V = 2.5 m/s.
(b) For all three cases, compare your numerical solution to the following analytical solution. GRAPH.
−
L
=
exp( V x / ) −1
R
−
L
exp( V L / ) −1
(c) Calculate the average error for each of the three cases using the following formula.
iexact − i
= i
N
(d) Are the numerical results that you obtained what you expected? Why or why not?
Given
Pipe length = 1.0 m
= 1.0 kg/m3 (constant) = 0.1 kg-s/m (constant) Q =0.0
Dirichlet boundary conditions, L = 100, R = 50
(2)
(3)
