EAS-230 Spring 2020 Programming Project (PP) Due Dates: Sections A, B, and E: Friday 4/17 Score: ——-/150 Instructions: 1. This project is to be done in groups of maximum 2 students. You must choose your partner from any of your instructor sections. Name your Group as 55555555_66666666 The 55555555 and 66666666 are the person numbers for partners 1 and 2, respectively. 2. Each group must fill in and submit the file Group_Info.xlsx to UBlearns no later than 3/28/20 at 11:59 pm. 3. Each partner individually must submit a peer review at the beginning of the project report

EAS-230 Spring 2020 Programming Project (PP) Due Dates: Sections A, B, and E: Friday 4/17 Score: ——-/150 Instructions: 1. This project is to be done in groups of maximum 2 students. You must choose your partner from any of your instructor sections. Name your Group as 55555555_66666666 The 55555555 and 66666666 are the person numbers for partners 1 and 2, respectively. 2. Each group must fill in and submit the file Group_Info.xlsx to UBlearns no later than 3/28/20 at 11:59 pm. 3. Each partner individually must submit a peer review at the beginning of the project report that shows the contribution to every section of the project in percent. However, each partner is fully responsible for every part of the project. Partners may be randomly selected to be tested in person on their understanding of each section of the project. 4. Every group must submit only one zipped file per group named as EAS230S20_PP_GroupName.zip that contains all m-files, txt files and a pdf of your report. See the submission instructions at the end of this document. The zip file must be uploaded to UBlearns by any of the partners before 11:59 PM on the due date shown above (Do Not include in your zip file any files ending in .m~, .sav or .mat.) 5. You must save your m-files in addition to any data file with the exact names as in the text of this assignment in your EAS230S20_PP_GroupName folder before zipping. 6. Your project final report must have all sections shown in the template on UBlearns. Be sure to write the group name, the partners names, the person number and the lab section of each partner on the cover page of the final report. 7. You must write your own code and follow all instructions to get full credit. You are not allowed to use codes or scripts found on the internet or any other references. 8. You must use good programming practices, including indentation, commenting your functions and scripts and choosing meaningful variable names to make your programs self-documenting. 9. It is your responsibility to make sure that your functions/scripts work properly and are free from errors by utilizing the resources at your disposal. Files not running may get a grade of zero. NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 2 of 16 Numerical Analysis of Annular Radial Fins of Uniform Thickness Fig. 1: Annular radial fin of uniform thickness. Background: Figure 1 shows an annular radial fin used to increase the surface area of a circular tube for the purpose of increasing the rate of heat transfer to/from the surface. The rate of heat transfer, __________, from a surface at temperature ____ to the surrounding environment at a temperature ___ is determined from Newtons law of cooling [1] seen in equation 1 where ____ is the surface area and _ is the convective heat transfer coefficient. __________ = ___��(____ _ ___) (equation 1) The surface temperature along the fin length, ____ in the previous equation, is not constant but varies along the fin from its base to its tip. Engineers are usually interested in determining the temperature distribution along the fin and accordingly the rate of heat transfer through the fin, ________. To achieve this goal, engineers can use numerical methods and/or analytical methods. Numerical methods: Fig. 2: Schematic for the nodes used for the numerical analysis. NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 3 of 16 One numerical method is known as the finite difference method where the fin length is divided into a finite number of equally spaced nodes, or points along the fin length, as seen in Figure 2. The fin is divided into __ sections of equal lengths (___) with __ + 1 nodes. Nodes 2, 3, , __ are called internal nodes, while nodes 1 and __ + 1 are boundary nodes at the boundaries of the fin. The temperature at each node or point is then determined by an equation derived from the energy balance at this node as ( ________ ____ _______ ____________________ ____ _____ ________ ________ ____ _____ ________ ) + ( ________ ____ _______ ____________________ ____ _____ _________ ________ ____ _____ ________ ) + ( ________ ____ _______ ____________________ ____ _____ ________ ) = ( ________ ____ ___________ ____ _____ ____________ ______________ ____ _____ ________ ) Since each node has its own equation, this produces a system of __ + 1 linear equations that can be solved using the linear algebra techniques taught in class. In practice, the more nodes, the more equations, the better approximation to the exact/analytical solution. The equation, in dimensionless form, for each internal node (__ goes from 2 to N) can be derived from equation 2 by plugging in the corresponding __ value. ( 1 (___)2 _ 1 ____(2___) )_____1 _ ( 2 (___)2 + __2)____ + ( 1 (___)2 + 1 ____(2___) )____+1 = 0 (equation 2) The distance to each node, ____, can be generated using equation 3. ____ = __ + (__ _ 1)___, ___ = 1___ __ (equation 3) Note that the boundary nodes, node (__ = 1) and node (__ = N+1) require two more equations that depend on the assumptions made at each boundary, known as the boundary conditions. At the fin base, Node 1, the temperature is assumed the same as that of the tubular surface where the fin is mounted on. This is represented by equation 4. __1 = 1 (equation 4) At the fin tip, Node N+1, One of three different assumptions can be made, resulting in 3 different equations (equations 5A-C). Only one of these equations can be used to complete the system of equations. Table 1: Boundary Conditions at the fin tip Boundary condition assumption Equation No heat flow from the fin tip (perfect insulation) _____1 _ 4____ + 3____+1 = 0 (equation 5A) NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 4 of 16 Finite heat flow from the fin tip by convection _____1 _ 4____ + (3 + __2__(2___))____+1 = 0 (equation 5B) Infinitely too long fin ____+1 = 0 (equation 5C) Variables in equations 1 to 5: _ __ is the dimensionless temperature and represents the temperature at each node (these are the unknowns in the system of equations that you need to solve for.) _ __ + 1 is the number of nodes and __ indicates a specific node, __ = 1,2,3,,__,__ + 1 _ __ is the ratio between the inner and outer radii of the fin (__ = __1 __2 ) _ __ represents the dimensionless distance of each node to the center of the tube. __ is the ratio between the distance to each node and the outer radius of the fin (Ri = ____ __2 ). You must notice that R is a vector of (N+1) elements where __1 = __1 __2 = __ and ____+1 = __2 __2 = 1. _ __2 is known as the enlarged Biot number and represents how quickly heat transfers through the fin (__ = __ ____ __2 2) _ __ represents the dimensionless thickness of the fin. It is defined as the ratio between the actual thickness and the outer radius of the fin (__ = __ __2 ). For optimum design consideration, __ is typically in the range of 0.01 to 0.1. Problem PP_PartA (25 pts): Solving the radial fin problem with numerical methods: Deliverable : Function RadFin_Numerical 1. Using equations 2, 4, and 5A, write the system of equations for 6 nodes (__ = 5) using the provided variables: a. __ = 5 resulting in __ = 1, 2, 3, 4, 5, 6 b. __ = 0.2 c. __ = _10 2. Solve your system of equations using MATLAB to determine the temperature distribution. Compare your results with the provided results below in format long: __1 = 1.000000000000000 __2 = 0.486132897434888 __3 = 0.260949304501633 NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 5 of 16 __4 = 0.153710622042675 __5 = 0.104256326792430 __6 = 0.087771561709016 3. Generalize your calculations to solve the system of equations for any number __ + 1 of nodes. a. Generate your R vector using equation 3 for all __ values. b. Pre-allocate your __ matrix and b vector using the zeros function. c. Fill in the middle rows of __ and b (rows 2 to N) using for loops according to equation 2. d. Fill in row 1 of __ and b according to equation 4. e. Fill in row __ + 1 of __ and b according to equations 5A-C depending on the provided boundary condition using a branching structure. f. Once __ and b are filled, solve for theta using left division. 4. Lastly, write your final set of calculations as a function named RadFin_Numerical with the following function definition line. [R, T] = RadFin_Numerical(N, C, Tau, Gamma, BC) _ R is the vector of radii calculated for each node using equation 3 _ T is the vector of dimensionless temperatures __ at each node calculated using the numerical method. _ N, C, Tau (__), and Gamma (__) are defined before. _ BC is a variable for the boundary condition at the tip of the fin that may have a value of the integers 1, 2, or 3 only. If BC = 1, use equation 5A, if BC = 2, use equation 5B, and if BC = 3, use equation 5C in your system of equations. Note: you may use any additional variables that you need and you are free to name your variables any valid names. Analytical methods: The numerical method gives us an approximation of the analytical/exact solution. The analytical solution of the radial fin heat problem can be found in equations 6A-C depending on the boundary condition. The analytical solution for no heat flow from the tip of the fin (perfect insulation/negligible heat loss from the tip) is defined in equation 6A. NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 6 of 16 __ = __1(__)__0(____) + __1(__)__0(____) __1(__)__0(____) + __1(__)__0(____) (equation 6A) The analytical solution for finite heat flow from the fin tip by convection is defined in equation 6B where __1 = (1 + __)__. __ = __1(__1)__0(__1__) + __1(__1)__0(__1__) __1(__1)__0(__1__) + __1(__1)__0(__1__) (equation 6B) The analytical solution for infinitely too long fin is defined in equation 6C. __ = __0(__)__0(____) _ __0(__)__0(____) __0(__)__0(____) _ __0(__)__0(____) (equation 6C) In equations 6A-C, ____(__) denote the modified Bessel functions of the first kind of order __ and ____(__) denote the modified Bessel functions of the second kind of order __ and can be calculated using the besselk and besseli functions in MATLAB. Note that R and __ are both vectors of __ + 1 elements containing the radii and the dimensionless temperature, respectively. Problem PP_PartB (15 pts): Solving the radial fin problem with analytical methods: Deliverable : Function RadFin_Analytical 1. Generate your R vector using equation 3 for all __ values. 2. Perform the analytical calculation based on the boundary condition defined in equations 6A-C using a branching statement. Use the besseli and besselk functions to determine ____(__) and ____(__) respectively. Hint: besseli(nu,z) computes the modified Bessel function of the first kind, ____(__), for each element of the array z and besselk(nu,z) computes the modified Bessel function of the second kind, ____(__), for each element of the array z. 5. Generalize your final set of calculations to a function named RadFin_Analytical with the following function definition line. [R, T] = RadFin_Analytical(N, C, Tau, Gamma, BC) _ R is the vector of radii calculated for each node using equation 3 _ T is the vector of temperatures at each node using the analytical method NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 7 of 16 _ N, C, Tau (__), and Gamma (__) are defined before _ BC is a variable for the boundary condition at the tip of the fin. If ____ = 1, use equation 6A, if ____ = 2, use equation 6B, and if ____ = 3, use equation 6C in your system of equations. Note: you may use any additional variables that you need and are free to name your variables any valid names. Fin efficiency: The fin efficiency, ________, is a measure of the fin performance. It represents how much heat is actually transferred by the fin out of the maximum heat that can be transferred through the ideal fin. 0 _ ________ _ 100%. The fin efficiency can be calculated using equation 7 where the limits on the finite integral indicate the start of the fin (__) to the end of the fin (1). ________ = 2_ ________ 1 __ (1 _ __2) (equation 7) This equation requires the calculation of a definite integral. Most of the time in computations (programming), calculating an integral requires numerical integration methods where the integral is estimated through summation of function values. For this project, we will use either numerical integral method 1 or numerical integral method 2 or a combination of method 1 and method 2. Method 1 can be used to calculate the integral for an even number of subintervals __ and is expressed by equation 8 and illustrated graphically in Figure 3. __(__) = _ ______ __ __ _ _ 3 [__(__) + 4 _ __(____) __ __=2,4,6,.. + 2 _ __(____) ___1 __=3,5,7,.. + __(__)] (equation 8) NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 8 of 16 Figure 3 Numerical Integral Method 1 Method 2 can be used to calculate the integral for a number of subintervals M divisible by 3, i.e. 3, 6, 9, 12, etc, and is expressed by equation 9 and illustrated graphically in Figure 4. __(__) = _ ______ __ __ _ 3_ 8 [__(__) + 3 _ [__(____) + __(____+1)] ___1 __=2,5,8,.. + 2 _ __(____) ___2 __=4,7,10,.. + __(__)] (equation 9) Figure 4 Numerical Integral Method 2 For both equations, M is the number of subintervals of the numerical integral which is less than the number of function values __ by 1. f is nothing but the elements of __ times __ (____) NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 9 of 16 vector. _ = _____ __ , where a and b are the limits on the definite integral. A combination of the two methods can be used to calculate the full integral. Example: 1. If the ____ vector has 25 elements (25 nodes), the number of subintervals is __ = 25 _ 1 = 24 accordingly M is even > 2, accordingly method 1 can be used for the entire interval to estimate the integral _ ________ 1 __ . Where __ = __, __ = 1, _ = ___, and __ is the same as __. 2. If the ____ vector has 28 elements (28 nodes), the number of subintervals is __ = 28 _ 1 = 27 accordingly M is divisible by 3 and > 3, accordingly method 2 can be used for the entire interval to estimate the integral _ ________ 1 __ . Where __ = __, __ = 1, _ = ___, and __ is the same as __. 3. If the ____ vector has 30 elements (30 nodes), the number of subintervals is __ = 30 _ 1 = 29 which is not even nor divisible by 3. __ = 29 can be broken into 2 subintervals __1 = 3 (divisible by 3) and __2 = 26 (even number). Method 2 can be used for the first 3 + 1 = 4 values of ____ where __1 = __, __1 = __4, _ = ___, and __1 = 3. Method 1 can be used for the rest 26 subintervals where __2 = __4, __2 = 1, _ = ___, and __2 = 26. The integral _ ________ 1 __ is estimated as _ ________ 1 __ = _ ________ __4 __ + _ ________ 1 __4 . Problem PP_PartC (30 pts): Determining the radial fin efficiency using numerical integration: Deliverables: Functions Integral_Numerical_1, Integral_Numerical_2 and FinEfficiency 1. Create a function that performs numerical integration using method 1 (equation 8) with the following function definition line: I = Integral_Numerical_1(f, M, a, b) _ I is the calculated integral _ f if the vector of function values (____) _ M is the number of subintervals, M = length(f)-1, M must be an even number _ a is the lower limit of the definite integral _ b is the upper limit of the definite integral Note: each summation requires a separate for loop NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 10 of 16 2. Create a second function that performs numerical integration using method 2 (equation 9) with the following function definition line: I = Integral_Numerical_2(f, M, a, b) _ I is the calculated integral _ f if the vector of function values (____) _ M is the number of subintervals, M = length(f)-1, M must be divisible by 3. _ a is the lower limit of the definite integral _ b is the upper limit of the definite integral Note: each summation requires a separate for loop 3. Create a third function that calculates fin efficiency (________) using your Integral_Numerical_1 and Integral_Numerical_2 functions with the following function definition line: nfin = FinEfficiency(R, T) _ nfin is the calculated fin efficiency _ R is the vector of radii _ T is the temperature distribution calculated using the numerical or analytical solutions a. Calculate the ____ vector using the two inputs. b. Check the number of subinterval __ and select the proper numerical integration method to use: i. If __ is an even number, use Integral_Numerical_1 to calculate the integral. ii. If __ is divisible by 3, use Integral_Numerical_2 to calculate the integral. iii. If __ is not even or divisible by 3, divide the integral into two separate integrals, I1 and I2, where I1 is calculated with the first four values in ____ with Integral_Numerical_2 and the I2 is calculated with the last __ _ 3 values in ____ with Integral_Numerical_1. The total integral is determined as the sum of both integrals, I = I1 + I2. c. Calculate the fin efficiency (________) using equation 7. Note __ is the first value in the R vector. NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 11 of 16 Problem PP_PartD (25 pts): Comparison between the numerical and analytical solutions for radial fin: Deliverable: Script PP_PartD.m 1. Create a new script file and name it PP_PartD.m with appropriate comments at the top. 2. In your script file, prompt the user for the values specified in Table 2. For each value entered by the user, use a while loop to validate each value according to Table 2. You may name your variables whatever you want. Table 2: Variables to be entered by the user Order Variable Valid values 1st Number of subintervals (__) N > 1 2nd The ratio between the distance to the first node and the last node (__) 0 < c="">< 1=""> 3rd The thickness of the fin (__) 0 <>__ < 1="" 4th="" the="" parameter="">__) __ > 0 5th The type of boundary conditions at the tip (BC) BC = 1, 2 or 3 3. Use the user inputs to call your RadFin_Numerical and RadFin_Analytical functions to determine the temperature distribution from your numerical calculations and your analytical calculations, respectively. 4. Store your results as N+1 x 1 vectors named R_num and T_num to store your numerical results and R_ana and T_ana to store your analytical results. 5. Calculate the fin efficiency (________) for both your numerical and analytical results. Store your results in nfin_num and nfin_ana, respectively. 6. Plot the results, __ versus R, for both the numerical results and analytical results in the same figure. Use different line styles and widths to distinguish between the two results. Fully annotate your plot with a title, axis labels, etc. Your legend should contain text in the following format where X.XXXXXX indicates your results from calculating ________. Fin Efficiency from Numerical Solution = XX.XXX % Fin Efficiency from Analytical Solution = XX.XXX % 7. Use your script with the following data: First test your input validation loops by running the following case NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 12 of 16 __ = _1 ______ _______ 10,__ = _1 ______ _______ 0.2, __ = _1 ______ _______ 0.15, __ = _1 ______ _______ 4, ____ = _1 ______ _______ 2 Then use your program with the inputs in Table 3 and save your plots: Table 3: Cases to be run with script PP_PartD Case N C __ __ Type of boundary condition i 29 0.15 0.1 3 1 ii 29 0.15 0.1 3 2 iii 29 0.15 0.1 3 3 iv 59 0.1 0.05 5 1 v 59 0.2 0.05 5 1 vi 59 0.5 0.05 5 1 NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 13 of 16 Problem PP_PartE (15 pts): Developing the diagram of radial fin efficiency Deliverable: Script PP_PartE.m 1. Create a new script file and name it PP_PartE.m with appropriate comments at the top. 2. In your script file, use the following values. You may name your variables whatever you want. a. __ = 40 b. __ = 0.05 c. ____ = 2 assuming a convective boundary condition d. __ = a vector from 0.1 to 0.9 with 5 values e. __ = a vector from 0.0 to 3 with 16 values 3. Calculate the fin efficiency (________) for every (C, __) combination using your RadFin_Numerical and FinEfficiency functions. a. Initialize a matrix named nfin_partE as a 16 x 5 array of zeros. b. Use nested for loops (one for __ and one for __ ) to go through all possible combinations of __ and __. Store your results in nfin_partE where each column contains the fin efficiencies for every __ for a single __ value. 4. In one figure, plot your results as __ versus ________ resulting in one curve per every __ value, i.e. your plot will be 5 lines in one figure. Use different line/marker colors distinguish between results. Fully annotate your plot with a title, axis labels, legends, etc. Problem PP_PartF (20 pts): Calculate results using data from a file Deliverable: Script PP_PartF.m 1. Create a new script file and name it PP_PartF.m with appropriate comments at the top. 2. Read in the data from the RadFin.txt file. The data in the file is arranged as shown in the Table 4 below. Each row represents a different radial fin. Table 4: The text file RadFin.txt __1 [m] __2 [m] __ [m] _ [W/m2K] __ [W/mK] ____ [C] ___ [C] 0.05 0.5 0.005 30 50 75 20 NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 14 of 16 0.06 0.5 0.005 35 55 80 21 0.07 0.5 0.005 40 60 85 22 _ _ _ _ _ _ _ 3. Use the following values to calculate the fin efficiency (________) for each row of data. Store your results in a 21 x 1 vector named nfin_partF. All other variables can be named whatever you want. a. __ = 37 b. ____ = 2 assuming a convective boundary condition at the tip c. __ = __ __2 d. __ = __1 __2 e. __ = __ ____ __2 2 4. Calculate the overall rate of heat transfer (________) for the fin in each row using the following equation. Store your results in a 21 x 1 vector named qfin. ________ = ________[2__(__2 2 _ __1 2)_(__�� _ ___)] 5. Combine the original data with your results for __, C, __, ________, and ________ so that your final matrix is a 21 x 12 where the columns represent the values according to the table header below. Write out your data to a new text file named RadFin_Results.txt. You can do that by using the fprintf feature for writing to a file. __1 [m] __2 [m] __ [m] _ [W/m2K] __ [W/mK] ____ [C] ___ [C] __ C __ ________ ________ Problem PP_PartG (20 pts): Project final report Deliverable: File PP_Report.pdf 1. Download the programming project report template from UBlearns. 2. Write your introduction and description of code sections as described in the document. 3. Insert the following results. NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 15 of 16 a. Six figures comparing the numerical versus analytical results with the parameters listed in Table 3 before. One plot should be generated per row. Use your PP_PartD.m file to generate each plot. b. The figure comparing the fin efficiencys generated across different C and __ values generated from your PP_PartE.m. c. A table showing the data calculated in the RadFin_Results.txt file from your PP_PartF.m script file. Your table must include all the data with an appropriate header row. 4. Write your conclusion section as described in the document. 5. Copy and paste all of your .m files to the appendix section of your report. 6. Finish up by completing your summary/abstract section, list of tables and figures (8 figures and 1 table at minimum from your results), and table of contents. 7. Remove any remaining red text and complete the title page. 8. Save your report as a PDF and name it PP_Report.pdf. Submission Process Zip your PP_GroupName folder to a PP_GroupName.zip file that contains all your .m files, text file, and your PDF report. Your zip file should contain the following files: Table 5: The files to be submitted in your zipped folder Filename Type RadFin_Numerical.m Function file RadFin_Analytical.m Function file Integral_Numerical_1.m Function file Integral_Numerical_2.m Function file FinEfficiency.m Function file PP_PartD.m Script file PP_PartE.m Script file PP_PartF.m Script file RadFin_Results.txt Text file PP_Report.pdf PDF file containing your report NAME: LAB SECTION: EAS 230 Spring 2020 PP Page 16 of 16 Submit your zip file to the Programming Project assignment on UBlearns. References: [1] A. Compo and S. Chikh, Reproduction of the Fin Efficiency Diagram for Annular Fins of Uniform Thickness by Solving Systems of up to Four Algebraic Equations, International Journal of Mechanical Engineering Education, Volume: 34-1, pp 85-92, 2006. [2] Y. Cengel, Heat Transfer A Practical Approach, 2nd edition, McGraw-Hill. [3] F.P. Incropera and D.P. DeWitt (2002). Fundamentals of heat and mass transfer. New York: J. Wiley.

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