1. Introduction:
Chemical vapor deposition (CVD) is a widely used materials-processing technology. The majority of its applications involve applying solid thin-film coatings to surfaces, but it is also used to produce high-purity bulk materials and powders, as well as fabricating composite materials via infiltration techniques. Chemical vapor deposition is a complex operation with key processes occurring on widely separated length and time scales. At each of the length scale regimes important in CVD, a particular set of modeling techniques and assumptions is appropriate. Frequently though, processes of different scales interact, and thus many problems of interest cannot be cast within the context of a single scale.
Chemical vapor deposition (CVD) is a widely used materials-processing technology. The majority of its applications involve applying solid thin-film coatings to surfaces, but it is also used to produce high-purity bulk materials and powders, as well as fabricating composite materials via infiltration techniques. Chemical vapor deposition is a complex operation with key processes occurring on widely separated length and time scales. At each of the length scale regimes important in CVD, a particular set of modeling techniques and assumptions is appropriate. Frequently though, processes of different scales interact, and thus many problems of interest cannot be cast within the context of a single scale.
2. Objective:
In this project, my objective is to simulate the infiltration of a chemical vapor into the material block after T=100 seconds. The governing equations is given by the following equation.
My first goal is to consider a 1-D problem and try to solve for the concentration at the mid plane using two methods i.e., using analytical method as well as the three point finite difference method and compare the error between the two methods.
My second goal is to repeat the process using the finite difference scheme for 3 different node conditions and plot the concentration profile along the z direction for all the 3 cases.
My second goal is to repeat the process using the finite difference scheme for 3 different node conditions and plot the concentration profile along the z direction for all the 3 cases.
My third goal it to set up a discretized formula for a 3-D version and estimate the no. of time steps to solve the problem and figure the array size that need to be updated and solved for. Finally, we need to estimate the time taken for the entire simulation to take place.
3. MATLAB Analysis:
a. Comparison of the concentration at the midplane (z=0.005) using the Analytical method and Three-Point Finite Difference method for different time intervals:
It can be observed that the concentration at the midplane for different time intervals increases with increase in time. Also, the results are almost the same using the Analytical method and the Three-Point Finite Difference method. At time t=0, the concentration at the midplane is 0. At time t=50 sec, the concentration at the midplane increases to 0.08217. At time t=100 sec, the concentration at the midplane further increases to 0.1505. The increase in the concentration at the midplane is linear.
b. Error Estimation Plot at the mid-plane using the Analytical and 3PFD method at different time intervals:
It can be observed that the error at the midplane using the Analytical method and Three-point Difference method is almost negligible, of the order of 10^(-7). At time t=0, the error value is found out to be 1.8×10^(−7) at the midplane. At time t=100 sec, the error between the two method decreases to 1.473×10^(−7) .It can be observed that with the increase in time the error value keeps on reducing linearly.
c. Combined Plot of Concentration Profile along z-direction for all different node cases:
It can be observed that, the concentration of the chemical vapor keeps reducing from the top face to bottom face. Also, it can be observed that as the number of nodes keeps increasing, the quality of the concentration profile along the z-direction improves and the plots become much smoother. This is because as we increase the no. of nodes, the functions become more continuous and they provide more accurate results. With finer mesh size, we get better simulation results at the cost of simulation time.
