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Midterm f09 solution
- 1. ENSC283 INTRODUCTION TO FLUID MECHANICS 27 February 2009 Midterm Examination M. Bahrami This is a 2-1/2 hours, closed-book and notes examination. You are permitted to use one 8.5 in.× 11 in. crib sheet (double-sided) and the property tables. There are 5 questions to be answered. Read the questions very carefully. Clearly state all assumptions. It is your responsibility to write clearly and legibly. Problem 1: (20 marks) A stream of water of diameter ݀ ൌ 0.1 ݉ flows steadily from a tank of diameter ܦ ൌ 1 ݉ as shown in the figure. Determine the flow rate, ܳ, needed from the inflow pipe if the water depth remains constant, (݄ ൌ 2 ݉). Problem 1: (Solution) ݄ ൌ 2 ݉ ݀ ൌ 0.1 ݉ ܳ ܦ ൌ 1 ݉ For steady, incompressible flow, the Bernoulli equation applied between points 1 and 2 ଵ 1 2 ଶ ߛݖଵ ൌ ଶ ߩܸଵ 1 2 ଶ ߛݖଶ ߩܸଶ ENSC283 Mid-term W2009 1
- 2. ݄ ൌ 2 ݉ ݀ ൌ 0.1 ݉ ܳ ܦ ൌ 1 ݉ 1 2 With the assumptions that ଵ ൌ ଶ ൌ 0, ݖଵ ൌ ݄ and ݖଶ ൌ 0, ଵ ଶ ܸଵ ଶ ݄݃ ൌ ଵ ଶ (I) ଶ ܸଶ Although the water level remains constant, there is an average velocity, ܸଵ, across section 1 because of the flow from the tank. For steady incompressible flow, conservation of mass requires ܳଵ ൌ ܳଶ, where ܳ ൌ ܣܸ. Thus, ܣଵܸଵ ൌ ܣଶܸଶ or ܦଶܸଵ ൌ ݀ଶܸଶ Hence, ܸଵ ൌ ቀௗ ଶ ܸଶ (II) ቁ Combining equations (I) and (II) ܸଶ ൌ ඩ 2݄݃ 1 െ ቀ݀ܦ ସ ൌ ඪ ቁ 2 ቀ9.81 ቂ݉ ݏଶቃቁ ሺ2 ሾ݉ሿሻ 1 െ ൬0.1ሾ݉ሿ 1 ሾ݉ሿ ൰ ݉ ݏ ସ ൌ 6.26 ቂ ቃ Thus, ENSC283 Mid-term W2009 2
- 3. ܳ ൌ ܣଵܸଵ ൌ ߨሺ0.1 ሾ݉ሿሻଶ 4 ݉ ݏ ൈ 6.26 ቂ ቃ ൌ 0.0492 ቈ ݉ଷ ݏ Problem 2: (20 marks) A uniform block of steel (SG = 7.85) will float at a mercury-water interface as shown in the figure. What is the ratio of the distance ܽ and ܾ. Problem 2: (Solution) water steel block mercury: SG = 13.36 a b Let ܮ be the block length into the paper, ܹ is the block width and let ߛ be the water specific weight. Then the vertical force balance on the block is 7.85ߛሺܽ ܾሻܮܹ ൌ 1ߛܽܮܹ 13.56ߛܾܮܹ Canceling ܹ, ܮ and ߛ from both sides of this equation and rearranging, 7.85ܽ 7.85ܾ ൌ ܽ 13.56ܾ ՜ ܽ ܾ ൌ 13.56 െ 7.85 7.85 െ 1 ൌ 0.834 ENSC283 Mid-term W2009 3
- 4. Problem 3: (20 marks) Kerosene at 20Ԩ flows through the pump shown in figure at 0.065 ݉ଷ/ݏ. Head losses between 1 and 2 are 2.4 ݉, and the pump delivers 6 ܹ݇ to the flow. What should the mercury-manometer reading ݄ be? (ߛ௦ ൌ 8016.2 ܰ/݉ଷ and ߛ௨௬ ൌ 133210 ܰ/݉ଷ) 1.5݉ ܸଵ ܦଵ ൌ 7.5 ܿ݉ mercury ݄ ൌ? Problem 3: (Solution) First establish two velocities, i.e. ܸଵ and ܸଶ, ܸଵ ൌ ܳ ܣଵ ൌ pump 0.065 ݉ଷ ݏ ൨ ܦଶ ൌ 15 ܿ݉ ቀߨ4ቁ ሺ7.5 ൈ 0.01ሾ݉ሿሻଶ ܸଶ ൌ 14.7 ቂ ݉ ݏ ቃ ܸଶ ܸଵ ൌ ܣଵ ܣଶ ൌ ൬ ܦଵ ܦଶ ൰ ଶ ՜ ܸଶ ൌ 1 4 ݉ ݏ ܸଵ ൌ 3.675 ቂ ቃ To apply a manometer analysis to determine the pressure difference between points 1 and 2, ଶ െ ଵ ൌ ሺߛ െ ߛሻ݄ െ ߛΔݖ ൌ ቀ133210 ቂ ே యቃ െ 8016.2 ቂ ே యቃቁ ݄ െ ቀ8016.2 ቂ ே యቃቁ ൈ 1.5ሾ݉ሿ ൌ 125193.8݄ െ 12024.3 ቂ ே మቃ Now apply the steady flow energy equation between points 1 and 2 ENSC283 Mid-term W2009 4
- 5. ଵ ߛ ܸଵ ଶ 2݃ ݖଵ ൌ ଶ ߛ ܸଶ ଶ 2݃ ݖଶ ݄ െ ݄ where ݄ ൌ ܲ ߛܳ ൌ 6000ሾܹሿ ݉ଷቃ ൈ 0.068 ݉ଷ 8016.2 ቂ ܰ ݏ ൨ ൌ 11 ሾ݉ሿ Thus, ଵ 8016.2 ቂ ܰ ݉ଷቃ ቀ14.7 ቂ݉ݏ ቃቁ ଶ 2ሺ9.81ሻ 0 ൌ ଶ 8016.2 ቂ ܰ ݉ଷቃ ቀ3.675 ቂ݉ݏ ቃቁ ଶ 2ሺ9.81ሻ 1.5 ሾ݉ሿ 2.4ሾ݉ሿ െ 11ሾ݉ሿ ՜ ଶ െ ଵ ൌ 139685 ቂ ே మቃ ൌ 125193.8݄ െ 12024.3 ቂ ே మቃ or ݄ ൌ 1.211 ሾ݉ሿ Problem 4: (20 marks) Circular-arc gate ABC pivots about point O. For the position shown, determine (a) the hydrostatic force on the gate (per meter of width into the paper); and (b) its line of action. water Problem 4: (Solution) ܴ O ܴ ߙ ߙ ݈ଵ ܮ ܪ ݈ଶ ൌ ENSC283 Mid-term W2009 5 2ܮ 3 ܣ ൌ ܮܪ 2 ݈ଵ ൌ 2ܴݏ݅݊ߙ 3ߙ ܣ ൌ ߙܴଶ ܥܩ ܥܩ ܪ Hint: A B C
- 6. The horizontal hydrostatic force is based on the vertical projection: ܨு ൌ ߛ௪݄ீܣ௩ ൌ ߛ௪ܪଶ 2 and the point of action for the horizontal force is 2H/3 below point C. The vertical force is upward and equal to the weight of the missing water in the segment ABC shown hatched in the following figure. ܴ ߙ ߙ ܥ ܪ/2 ܪ/2 ܣ ܤ ݈ The segment ABC area can be calculated from ܴ ܣ௦ ൌ ߙܴଶ െ ܱ ܴܪܿݏߙ 2 This area gives the volume of water in segment ABC per unit width into the paper. Hence the water weight is, ܨ ൌ ߛ௪ܣ௦ ൌ ߛ௪ ൬ߙܴଶ െ ܴܪܿݏߙ 2 ൰ To calculate the point of action, center of weight for segment ABC should be calculated. To do so, momentum balance around point O should be applied, െ ܪܴܿݏߙ 2 ൈ 2ܴܿݏߙ 3 ߙܴଶ ൈ 2ܴݏ݅݊ߙ 3ߙ ൌ ൬ߙܴଶ െ ܴܪܿݏߙ 2 ൰ ݈ ՜ ݈ ൌ 2ܴ 3 ቆ 2ܴݏ݅݊ߙ െ ܪܿݏଶߙ 2ߙܴ െ ܪܿݏߙ ቇ The net force is thus ଶ ܨு ܨ ൌ ටܨ ଶ ENSC283 Mid-term W2009 6
- 7. Per meter of width. This force is acting upward to the right at an angle of β where ߚ ൌ ݐܽ݊ିଵ ቆ ܪଶ 2ߙܴଶ െ ܪܴܿݏߙ ቇ Problem 5: (20 mark) A rigid tank of volume ܸ ൌ 1݉ଷ is initially filled with air at 20Ԩ and ൌ 100 ݇ܽ. At time ݐ ൌ 0, a vacuum pump is turned on and evacuates air at a constant volume flow rate of ܳ ൌ 80 ܮ/݉݅݊ (regardless of pressure). Assume an ideal gas and isothermal process. (a) Set up a differential equation for this flow. (b) Solve this equation for t as a function of ሺܸ, ܳ, , ሻ. (c) Compute the time in minutes to pump the tank down to ൌ 20 ݇ܽ. Problem 5: (Solution) ݐ ൌ 0 ൌ 100 ݇ܽ ܶ ൌ 20Ԩ ܳ ൌ 80 ܮ/݉݅݊ The control volume encloses the tank, as shown is selected. ݐ ൌ 0 CV ൌ 100 ݇ܽ ܶ ൌ 20Ԩ Continuity equation for the control volume becomes ݀ ݀ݐ ܳ ൌ 80 ܮ/݉݅݊ ൬න ߩ݀ݒ൰ ݉ሶ௨௧ െ ݉ሶ ൌ 0 Since no mass enters the control volume Σ mሶ୧୬ ൌ 0. Assuming that the gas density is uniform throughout the tank ρdv ൌ ρv. Hence ENSC283 Mid-term W2009 7
- 8. ݒ ݀ߩ ݀ݐ ߩܳ ൌ 0 or ௗఘ ఘ ൌ െ ொ ௩ ݀ݐ (part a) This differential equation can be easily solved yielding ݈݊ ൬ ߩ ߩ ൰ ൌ െ ܳݐ ݒ where ρ is the gas initial density. For an isothermal ideal gas, ρ/ρ ൌ p/p, therefore ݐ ൌ െ ௩ ொ ݈݊ ቀ బ ቁ (part b) For given values of Q ൌ 80L/min ൌ 0.08 mଷ/min, the time to pump a 1 mଷ tank down from 100 to 20 kpa is ݐ ൌ െ 1ሾ݉ଷሿ 0.08 ሾ ݉ଷ ݉݅݊ሿ ݈݊ ൬ 20 100 ൰ ൌ 20.1 ሾ݉݅݊ሿ ENSC283 Mid-term W2009 8