Final Project: Combined Flow and Conduction
First, I expect a single detailed writeup as your final product,
written so that another graduate student who had not taken this course
could understand
everything that you did and why you did it.
You are modeling flow in a circular tube with heated walls. The tube is
2 meters long with an internal diameter of 0.004m. Thickness of
the tube wall is
0.001 meters (1 mm). A power of 200 watts is delivered uniformly
through the volume of the tube wall by electrical heating. The wall
metal has a density of 8000 kg/m^{3}, specific heat (c_{p})
equal to 500 J/kg, and conductivity equal to 80 w/m^{2}/K. The
outer wall of the tube can be
assumed to be perfectly insulated. Use a zero curvature boundary
condition for the flow outlet. Ignore the effects of gravity.
Initially water flows into the channel with an area averaged
velocity of 0.1 m/s. Take the cross-channel profile for the velocity to
be that of fully developed laminar flow. Inlet water temperature is
300K
and outlet pressure is 2 MPa. Calculate the steady state associated
with these boundary conditions. Next, using
the steady state for
initial
conditions, let the inlet temperature change with time
according
to the following expression:
T_{i6n}(t)
= 300 + 75*t**2 - 25*t**3 for
transient time from 0 through 2 seconds
= 400
for
transient time greater than 2 seconds
Calculate the
transient behavior for 30 seconds.
You may model the flow in this system in one of two ways. The
first
option is to assume that the density is constant at 1000 kg/m^{3},
and velocity profile does not vary down the channel. You must then
solve a time dependent 2-D (r-z) advection/diffusion equation for the
energy
transport within the water, and 2-D conduction equation for
energy transport in the tube wall.
The second acceptable fluid analysis is 1-D, allowing for the
temperature dependence of the liquid density. Solution of the flow
equations should be at the level of detail of my 1-D flow example. Adaptation
of that example to the requirements of this project is acceptable. The
heat source to the
energy equation should be taken in the form h(T_{wall}-T_{fluid}),
where the heat transfer coefficient is based upon laminar flow, and T_{wall}
is obtained from a 2-D (r-z) conduction solution in the pipe wall.
Fluid properties not specified above, must be taken from EosM.f90.
You are free to choose your numerical methods, with the
restrictions
that they must be at least second order accurate in space for interior
cells and must use
either a fully implicit or Crank-Nicholson time level method. The
fluid/metal boundary must have at least second order flux matching.
You
may obtain the steady state either through execution of a sufficiently
long transient or solution of a true steady state set of equations. In
the latter case be sure that the spatial difference method matches the
one used in your transient calculation.
Provide a copy of your final
program
in the drop box (source code only).
For the 1-D flow option you may use any sparse (direct or iterative)
solution
package to solve the linearized equations associated with the Newton
iteration. For the 2-D flow option you must implement either a
preconditioned Krylov package or a multi-grid method.
Your project writeup must include:
- A description of all equations used (differential and
difference)
and all terms used in these equations;
- Reasons why you selected specific numerical methods;
- A Taylor Series based error analysis for interior points in
the
wall conduction and
fluid difference
equations, and for the fluid/wall surface flux matching;
- Statement of stability limits for your chosen combination
of
methods, backed by your own stability analysis (analysis of interior
points is adequate, and analysis not required for fluid portion of
Option 2);
- Description of simplified analytic solutions and/or
Method
of Manufactured solutions,
cleanly testing all terms in your difference equations, and
plots showing comparison
of
your calculated results against the simplified solutions (for each
result clearly indicate which term or terms you are testing);
- Description of a mesh and time step sensitivity study based
on
Richardson Extrapolation, and using a sequence of at least three mesh
sizes and three time step sizes.
Use this analysis to show that the maximum Grid Convergence Index for
the fluid centerline temperatures at 10 seconds is less than 0.2%
for your finest mesh (use a Factor of Safety equal to 1.25).
Perform a similar analysis to demonstrate errors associated with time
step size are less than 0.2% at on the tube centerline at 10 seconds.
Calculate a mean order of accuracy "p" in x, y, and t, associated with
these centerline numbers, and compare to
the results of your Taylor series
truncation error analysis.
- Plots of temperature vs. distance along the tube (use
centerline for 2-D) at 5, 10, 15, 20, 25, and 30 seconds (all on the
same graph);
- Plots of temperature vs. time at the outer tube wall's
insulated surface
for locations at the tube entrance, tube exit, and halfway down
the tube.
Deadlines:
- Thursday April 10, Hand in a description of your chosen
model
equations, and difference equations.
- Project Stability analysis, due Wednesday April 16
- Project Taylor Series Error analysis, due Monday
April 21
- Initial Steady State, due Thursday April 24
- Final Project, due May 5 (or April 28 to skip the next exam)
Grading on the April 10, 16, 21, and 24 submissions will be at
three levels:
- Complete, you will receive a perfect score on this
section of the project when you hand in the full report;
- Needs work, deficiencies will be noted for you and
your
revised section graded after the full project is due;
- No effort, half of the maximum points for that
section
will be deducted from your final grade, regardless of your final
submission on that section.