Diff EQ Lab II

Diff EQ Lab II - Figure A ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř Figure 7 Figure 8 µ Vs ρ Figure 9 ũ Vs ř Figure 10 Set 1 Linear

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Unformatted text preview: Figure A ũ Vs. ř ũ Vs. ř ũ Vs. ř ũ Vs. ř ũ Vs. ř ũ Vs. ř Figure 7 Figure 8 µ Vs. ρ Figure 9 ũ Vs. ř Figure 10 Set 1: Linear Approximation ũ Vs. ř Set 2: Linear Approximation ũ Vs. ř ũ Vs. ř Set 1: Parabolic Approximation ũ Vs. ř Set 2: Linear Approximation ũ Vs. ř Set 1: Cubic Approximation ũ Vs. ř Set 2: Cubic Approximation µ Vs. ρ ũ Vs. ř Lab 2: Fitting Velocity Profiles APPM 2360 Spring 2008 Michaela Cui SID: 810585604 Theodoros Horikis Recitation 36: Sean Nixon Brandon Parks SID: 810689666 Theodoros Horikis Recitation 34: John Vilavert 1.0 Introduction The goal of this lab is to accurately create a model for the flow of a liquid through a pipe, such as the one in Figure A . Polynomials, as well as a logarithmic function, will be used to approximate two data sets. The method of approximation, coupled with careful linear algebra computations, will be the method of least squares. It was found in this lab that a higher degree polynomial will not always model a set of data more accurately than one of lesser degree, and that a logarithmic polynomial may suit a given set of data even more closely than basic polynomials. The residual for each polynomial equation was calculated in order to more clearly see which polynomial better suited each set of data. While not all models represented their given data sets perfectly, it did prove that the method of least squares is a solid approximation tool, and not a tool for prediction. 2.0 An Analytic Solution When determining a model for the laminar flow of a liquid through a circular pipeline, the following differential equation can be used: = ( ) drdtrdudr Pr 1 In Equation 1 , u is the velocity of the substance, r is the distance from the center of the pipe, and P is a pressure-related constant. Integration of Equation 1 results in the following differential; = + dudr P2r cr ; where c is a constant of integration. Using separation of variables, the equation can be integrated once more to obtain an equation for u in terms of r : = + + ( ) ur P4r2 clnr d 2 In this equation, d is another constant of integration. Given restraints for this equation are u(R) = 0 and u(0) € . Ɽ When substituting in r = 0 , the term containing the natural log in Equation 2 results in infinity for c ≠ 0 , which violates the given boundary. To resolve this problem, we set the first constant of integration, c , equal to zero. Now the first boundary can be tested: = + = uR P4R2 d 0 Through this equation, it is found that the second constant of integration, d , is - P4R2 . The resulting equation is as follows: = - ( ) ur P4r2 P4R2 3 Setting u(0) equal to the value uc , a reference point, we find that = uc- P4R2 , which is equal to the integration constant, d . Equation 3 can be manipulated further to create an alternative equation for laminar flow. Substituting = u ũuc and = r řR into Equation 3 results in the following: = ( ) - ( ) ũuc P4 řR 2 P4R2 4 Manipulation of Equation 4 results in the more manageable equation that is...
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This lab report was uploaded on 04/04/2008 for the course APPM 2360 taught by Professor Williamheuett during the Fall '07 term at Colorado.

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Diff EQ Lab II - Figure A ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř ũ Vs ř Figure 7 Figure 8 µ Vs ρ Figure 9 ũ Vs ř Figure 10 Set 1 Linear

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