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/m3, specific heat (cp) equal to 500 J/kg, and conductivity equal to 80 w/m2/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:

Ti6n(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/m3, 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(Twall-Tfluid), where the heat transfer coefficient is based upon laminar flow, and Twall 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.

• 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.
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