ENG3103 Engineering Problem Solving Computations Semester 2, 2013
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1 Assessment: Assignment 2 Due: 16 September 2013 Marks: 100 Value: 10 % Question 1 (70 marks) Introduction You are designing a pipe network system that transfers water from the upper pipe to the lower pipe. Note that Figure 1 is a plan view and the elevation is constant across all pipes. The static pressure difference between points A and D is designed to be P A P D = 3 atm (1 atm = 1 standard atmospheric pressure = kpa). It is necessary to ensure that the speed of the flow through every pipe is at least 2 m/s so that there is no sediment build-up. Determine if this is the case. A 1 2 B C 4 3 D Figure 1: Plan view (looking from above) of pipe network. Theory The change in pressure between two points along a streamline (a flow path) is modelled by the Bernoulli equation P V gh P V gh P (1) A 2 A A B 2 B B where P is the static pressure, the density, V the speed, g gravitational acceleration, h the elevation and P loss is the reduction in pressure due to any losses in the system. The most important loss (and the only one to be accounted for here) is caused by friction: 2 LV Ploss f (2) D 2 where f is the Darcy Weisbach friction factor. Fluid flow is governed by the continuity equation (which is conservation of mass); for incompressible (constant-density) flow, this is: Q VA const (3) where Q is the volume flow rate (m 3 /s) and A is the cross-sectional area. Incompressible flow is a good assumption for liquids. A consequence of Eq. (3) for Eq. (1) is that if the cross-sectional area is constant for a given pipe, the flow speed at the start is equal to the speed at the end and can be defined based on the pipe ID rather than an end-point ID. Pipe networks can be considered to be equivalent to electrical circuits in series and parallel, with pressure change equivalent to potential difference and volume flow rate equivalent to current. However, the resistance cannot be treated as constant: it is a non-linear function of the flow speed. The rules of potential difference and current are still maintained however: loss Page 1 of 6
2 a) The pressure change (potential difference) across multiple branches in parallel is equal (e.g. P B P C is the same regardless of whether pipe 2 or 3 is taken). b) The sum of the volume flow rates (currents) entering a junction is equal to the sum of the volume flow rates exiting a junction (e.g. Q 1 = Q 2 + Q 3 ). Because of the non-linear nature of the system [flow rate is squared in Eq. (1) and f is a nonlinear function of Q], iteration is required to determine the flow speeds in each pipe section. To do this, certain constraints can be applied based on the rules of potential difference and current: 1. The pressure change between points A and D is known and is equal to the sum of the pressure changes along all pipes in series that connect points A and D. This constraint should be used only once. 2. The pressure change across pipes in parallel is the same for each pipe. 3. The sum of volume flow rates entering junction B is equal to the sum of volume flow rates exiting B; similarly for C. These constraints provide a set of equations that can be solved for the speeds in each pipe. To calculate the friction factor, the common formula that is used is the Colebrook formula (Colebrook ): 1 ed log (4) f 3.7 Re f where e is the absolute roughness of the pipe wall and the Reynolds number is Re VD (5) with (the Greek letter nu ) the kinematic viscosity. Equation (4) is only valid for turbulent pipe flow (Re > 2300), otherwise f = 64/Re. The Moody diagram (Figure 2) is a standard method of determining f (the left axis) for hand calculations; the range of possible values for f given the different blue lines that represent the range of plausible pipe roughnesses can be seen. Figure 2: Friction factor for fully developed flow in circular pipes (Moody 1944). Page 2 of 6
3 Network Specifications The pipes are made of drawn tubing. The water temperature is 20 C. The pipe dimensions are: Pipe Number Length (m) Diameter (cm) Hints You should iterate until the values of V do not change significantly from one iteration to another. To linearise the system of equations, it is possible to treat any term including V 2 as V old V, where V old is the value from the previous iteration. Your system of equations then only has V, so becomes a set of linear algebraic equations. This is valid: see the discussion below Eq. (6). (You have to use the explicit method use V old to calculate f anyway.) The system of equations is only conditionally stable. To ensure that the solution converges, you can use an under-relaxation factor 0 < < 1: new V V 1 V (6) where V new is the value that is computed from the set of equations. Note that the same solution is obtained irrespective of the value of because when convergence is obtained V = V new = V old. Generally, an under-relaxation factor slows the rate of convergence (more iterations are required), but this is better than not obtaining a solution at all! Requirements For this assessment item, you must produce MATLAB code which: 1. Iterates until convergence is reached (the previous guess for the velocities is not significantly different from the current guess). 2. Calculates the friction factor for the current guess of velocities. 3. Reports the flow velocity in each pipe to the Command Window in addition to confirming that the speeds meet the required level 4. Validates the code by checking that the volume flow rates in the branches are consistent. 5. Verifies the code by ensuring that the friction factor calculation is valid. 6. Displays to the Command Window a brief discussion (fewer than 5 lines) stating the value you selected for the under-relaxation factor and why you selected that value. 7. Has appropriate comments throughout. An important component of quality assurance is to test (verify) that each function works correctly: 8. Write a test program that supplies block of code with a known input and confirm (verify) that the output is correct. (Writing blocks of code as functions makes the code more transparent, more portable and easier to test!) Assessment Criteria Your code will be assessed using the following scheme. Does the code run without error? Does the code have an appropriate header and comments? 10 marks Correct calculation of pipe velocities plus quality of methodology and implementation? 2 Confirmation of minimum pipe speed? Quality of validation of volume flow rates? Quality of verification of friction factor calculation? Quality of discussion for selection of under-relaxation factor? Quality of testing of code blocks? 10 marks old Page 3 of 6
4 References Colebrook, CF , Turbulent Flow in Pipes, with Particular Reference to the Transition Region between the Smooth and Rough Pipe Laws, Journal of the Institution of Civil Engineers, London, vol. 11, pp Moody, LF 1944, Friction Factors for Pipe Flow, Transactions of the ASME, vol. 8, pp Page 4 of 6
5 Question 2 (2) A square-wave pulse is shown in Figure 3(a) with its corresponding Fourier transform in Figure 3(b). Figure 3: Plot of square-wave pulse (a) and its Fourier transform (b). The analytical result for the Fourier transform is F 2sin a (7) If a particular signal has the property a = 5, use a numerical method to determine a from the signal F(). (This sounds perverse, but pretend you have data for a signal and you don t know the value: at least you know whether your code is any good!) To attempt this, you should calculate a value of where either is a known function of a or F() is a known function of a. The main numerical method you use should not rely on complex MATLAB functions (i.e. it should take more than 5 lines of code!). You should validate your code using another numerical method (which could use a complex MATLAB function) and verify your code by using the same numerical method but solving by hand for 3 iterations. (Note that many exam questions require you to conduct the same verification exercise, but actually solving the problem with a lower level of accuracy, since you won t actually run a code. When verifying a code, you just need to confirm that the computer produces values on each iteration that you expect.) Be careful that you obey one example of l Hôpital s rule (computers are not aware of this mathematical relationship ): sin x 1 (8) x x0 Requirements For this assessment item, you must produce MATLAB code which: 1. Finds the value of a using a numerical method that has been taught in the content [satisfying Course Objective 6 (in the Course Specs)] and reports the result to the Command Window. 2. Validates the numerical method using any other numerical method available to you and reports the result to the Command Window. 3. Validates both numerical methods by computing the relative error compared to the analytical solution and reports the result to the Command Window. 4. Verifies the primary method by solving it by hand for 3 iterations. Submit this as a pdf file. Page 5 of 6
6 Assessment Criteria Your code will be assessed using the following scheme. Does the code have an appropriate header and comments? 4 marks Correct calculation of a plus quality of methodology and implementation? 10 marks Validation of a using another numerical method 3 marks Reporting of results to Command Window 3 marks Verification of numerical method Question 3 () After submissions have closed for Assignment 1, you will be provided with a dummy submission for Assignment 1 containing a number of errors. Fill in the table in the file Corrections to Assignment 1 dummy submission.doc with how the code should be amended. Do not submit corrections to your own Assignment 1 submission. Submission Submit your assignment by the due date to the StudyDesk. Note that: you should upload all of your files individually (not zipped together). (If a code is comprised of multiple files, it cannot be run directly from the archive: it must be unpacked first.) you will NOT receive any confirmation of receipt. If you can see that the file(s) has uploaded, then you have successfully submitted your assignment. There is no need to click a send for marking button. Page 6 of 6
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