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a boat propelled by a jet of water. The boat must be capable of exerting a 200-lbf pull on a line which holds it motionless. The intake to the boat is inclined at an angle of 85" and is a pipe with a 6-in. inside diameter. The motor and pump are capable of delivering 150 gaVmin through the horizontal outlet. a. Calculate the outlet pipe size which gives the required thrust. b. Calculate the minimum horsepower output required of the motor if its efficiency is 100 percent. I ' 6 i n diometer 4.3 Water is flowing out of the frictionless nozzle shown in the following illustration, where p1 = 120 psig and the nozzle discharges to the atmosphere. a. Show a control volume and all control surfaces with all forces labeled. b. Find the resultant force in the bolts at (1) which is necessary to hold the nozzle on the pipe. c. If mercury, sp. gr. = 13.6, were flowing through the nozzle instead of water at the same conditions, what would be the new resultant force on the bolts? 204 Chapter 4: The Momentum Balances 4.4 Consider the straight piece of pipe - Flow 1 pi = 82 psig Q = 300 gaVmin p = 62.4 lbdft3; 0.1337 ft3/gal D = 4 in. a. Draw the control volume and label the control surfaces. b. Calculate the force on the bolts at point 1. Are the bolts in tension or compression? 4.5 Water is flowing out of a frictionless nozzle discharging to the atmosphere. The inlet diameter and nozzle diameter are shown on the sketch. The pressure at p2 = 80 bar. a. Sketch a control volume and label it. b. What is the resultant force on the bolts holding the nozzle to the pipe? c. If a fluid of specific gravity = 10 flows through the nozzle what is the resulting force on the bolts? Chapter 4: The Momentum Balances 205 150 rnrn - f l o w 4.6 Consider the sketch below where p1 = 300 psig; p2 = 270 psig; Q = 400 gpm D = 5 in, r = 62.4 Ibm/ft3 a. Draw a control volume and label it b. Calculate the force on the bolts c. Are the bolts in compression or tension? I - Flow I n 4.7 Water (p = 62.4 lbm/ft3) is flowing through the 180- elbow represented by the sketch. The inlet gage pressure is 15 lbf/in2. The diameter of the pipe at 1 is 2 inches. The diameter of the opening to the atmosphere at 2 is 1 inch v1 = 10 ft/s. a. Write down the macroscopic total-momentum balance and simplify it for this problem (state your assumptions). b. What is the horizontal component of force required to hold the elbow in place? 4.8 Water is flowing through a 60° reducing elbow as in the sketch. The volume of the elbow is 28.3 in3. Calculate the resultant force on the elbow. 206 Chapter 4: The Momentum Balances Flow p1 = 30 psig; p2 = 28 psig AI = 12.56 in2; A2 = 3.14 in2 w = 75 lbdsec 4.9 Consider steady, constant-temperature flow of water through the 45" reducing elbow shown below. The volume of the elbow is 28.3 in3. Flow p1 = 26 pig; w = 85 lbdsec AI = 12.56 in2; A2 = 3.14 in2 Draw a control volume and label all control surfaces and forces on the control volume. Calculate the total resultant force acting on the elbow. 4.10 Two streams of water join together at a reducing tee where the upstream pressure of both is 40 pig. Stream 1 is flowing in a 1-in. pipe at 50 lbdmin, stream 2 in a 2-in. pipe at 100 lbm/min, and stream 3 exits from the tee in a 3- Chapter 4: The Momentum Balances 207 in. pipe. The straight run flow is from stream 2 to stream 3. Calculate the force of the pipe threads on the tee if the downstream pressure is 35 psig. Use positive coordinates in the direction of flow as shown in the following figure. 4.11 Consider the horizontal lawn sprinkler shown. Observe that if the sprinkler is split in half as shown, the two halves are identical and thus only one half need be considered. For one-half of the sprinkler. a. Use a mass balance to obtain an expression for vo in terms of v1. b. Use the frictionless mechanical energy balance to find vl . c. Find the value of FR that will just prevent the sprinkler from rotating. 4.12 A stand is to be built to hold a rocket ship stationary while a lateral or side thrust engine is being fired. Twenty l b d s of fuel is consumed and ejected only out of the side engine at a velocity of 4,000 ft/s. The direction of flow is as shown in the following diagram. Find the x and y components of the restraining force required of the stand. 208 Chapter 4: The Momentum Balances " I 4.13 The tank shown below is secured to a concrete slab by bolts and receives an organic liquid (sp. gr. 0.72) from a filler line which enters the tank 1 ft above the tank floor. The liquid enters at a rate of 4,500 gal/min. What are the forces exerted on the restraining bolts? Assume that the filler line transmits no force to the tank (due to expansion joints). Over f tow pipe J 1 4500 goJ/min 4.14 A cylindrical storage tank is connected to a nozzle by a short length of horizontal pipe as shown in the following diagram. At time equal to zero, the height of liquid in the tank is H ft above the axis of the discharge pipe. The liquid discharges through the nozzle into the atmosphere as the liquid level h(t) changes. a. Using a mass balance, show that the liquid level h(t) is related to the nozzle discharge velocity v2 by dh dt Chapter 4: The Momentum Balances 209 b. Using a mechanical energy balance on the discharge pipe and the results of part a, show that the nozzle discharge velocity v2 and the gage pressure at the pipe entrance (pl - p2) are given as functions of time by where b = @2/D1)2 c. Using a momentum balance on the nozzle and the results of part b, show that the force exerted on the threads of the nozzle FT, is given as a function of time by Assume there is no pressure drop across the short section of pipe between the tank and the nozzle. -.. -..-- ,-I .. -A .. - .- -.. - -..- This page intentionally left blank 5 APPLICATION OF DIMENSIONAL ANALYSIS 5.1 Systems of Measurement A dimension or unit is a way of assigning a numerical value to a property, For example, one could assign the number 66 to the property height as measured in the dimension or unit of inches. The appropriate number if the property were measured in cm would be 167.6; if the property were measured in feet, 5.5. To have physical meaning, equations must be dimensional ly homogeneous: in an equation of the form each of the terms f l , f2, f3, ... fm must have the same dimension for the equation to be meaningful. For example, suppose that I have two bank accounts, one in the United States of America and one in England, and am asked, \u201cHOW much money do you have?\u201d Suppose further that my account in the United States has a balance of $1000 and that my account in England has a balance of 500E. I do not get a meaningful application of tbe equation total money = money in US + money in England if I perform the calculation as (5.1-2) total money = $1OOO+ 500& = 1500? 212 Chapter 5: Application of Dimensional Analysis The answer is meaningless, because I have added two numbers that result from measurements using two different scales. The calculation is meaningful if (assuming the current exchange rate to be $1 S O = 1&) I convert the terms to the same units, either as $1 5 (5.1-4) totalmoney = $1O00+ 5OOf f = $1750 totalmoney = $1OOOf:+5OOE = 1167& $1.5 (5.1-5) Notice that one can formally multiply and divide dimensions (multiplied by the appropriate scale factor) like algebraic quantities (even though they are not). In effect, this amounts to multiplication by unity - in the above example (5.1-6) In this example we were concerned with measuring only one variable - money. In the usual engineering