Simulate transient spatial and temporal dependence of a cylinder when exposed to a sudden change in environment. This task was accomplished through simulation via theconstruction of a MATLAB program. Temperature profiles for Biot numbers (Bi=0.01,0.1,1,10,100) were created for Fourier numbers (Fo=0.01,0.1,0.2,0.33,1,2) resulting in a total of 30 graphs available for analysis.
Analysis of the results for the varying Biot numbers led to the general conclusion that the Lumped Capacitance Method (LCM) assumption was valid for Bi<0.1. For these small Biot values, the rate of convection between the cylinder’s surface and the surrounding fluid was determined to be negligible when compared to the internal rate of conduction within the object. These isothermal results can be visualized by the flat, planar temperature profiles obtained for Bi<0.1. Additionally, an increasing rate of cooling was found for increasing Bi values.
Similarly, the first term approximation was proved to be valid for Fo>0.2. Analyses of the results illustrate that for these large Fourier numbers, the discrepancy between the first term approximation and exact solution were non-existent. This can be attributed to two simple facts: the first of which deals with the direct proportionality between the time and Fourier number, while the second is concerned with the relationship between the temperature profile obtained from the approximation and the midpoint temperature. For large Fo values corresponding to longer time intervals, the temperature of the centerpoint was determined to be an accurate representation of the entire cylinder.