Homework Solution

Homework #1-solution 1. Define incompressible flow and incompressible fluid. Must the flow of a compressible fluid necessarily be treated as compressible?

2. A 150-lbm astronaut took his bathroom scale (a spring scale) and a beam scale (compares masses) to the moon where the local gravity is g = 5.48 ft/s2. Determine how much he will weigh (a) on the spring scale and (b) on the beam scale.

3. The pressure in an automobile tire depends on the temperature of the air in the tire. When the air temperature is 25°C, the pressure gage reads 210 kPa. If the volume of the tire is 0.025 m3, determine the pressure rise in the tire when the air temperature in the tire rises to 50°C. Also, determine the amount of air that must be bled off to restore pressure to its original value at this temperature. Assume the atmospheric pressure to be 100 kPa.

4. A 50-cm ⅹ 30-cm ⅹ 20-cm block weighing 150 N is to be moved at a constant velocity of 0.8 m/s on an inclined surface with a friction coefficient of 0.27. (a) Determine the force F that needs to be applied in the horizontal direction. (b) If a 0.4-mm-thick oil film with a dynamic viscosity of 0.012 Pa∙s is applied between the block and inclined surface, determine the percent reduction in the required force.

5. A 0.03-in-diameter glass tube is inserted into kerosene at 20°C. The contact angle of kerosene with a glass surface is 26°. Determine the capillary rise of kerosene in the tube.

Homework #2-solution 1. A gas is contained in a vertical, frictionless piston–cylinder device. The piston has a mass of 4 kg and a cross-sectional area of 35 cm2. A compressed spring above the piston exerts a force of 60 N on the piston. If the atmospheric pressure is 95 kPa, determine the pressure inside the cylinder.

2. The water in a tank is pressurized by air, and the pressure is measured by a multifluid manometer as shown in figure below. Determine the gage pressure of air in the tank if h1 = 0.2 m, h2 = 0.3 m, and h3 = 0.46 m. Take the densities of water, oil, and mercury to be 1000 kg/m3, 850 kg/m3, and 13,600 kg/m3, respectively.

3. Freshwater and seawater flowing in parallel horizontal pipelines are connected to each other by a double U-tube manometer, as shown in figure below. Determine the pressure difference between the two pipelines. Take the density of seawater at that location to be ρ = 1035 kg/m3. Can the air column be ignored in the analysis?

4. Consider a large cubic ice block floating in seawater. The specific gravities of ice and seawater are 0.92 and 1.025, respectively. If a 10-cm-high portion of the ice block extends above the surface of the water, determine the height of the ice block below the surface.

Homework #3-solution 1. Consider steady, incompressible, two-dimensional flow through a converging duct (Fig. 1). A simple approximate velocity field for this flow is

   V  (u , v )  (U 0  bx ) i  byj

where U0 is the horizontal speed at x = 0. Note that this equation ignores viscous effects along the walls but is a reasonable approximation throughout the majority of the flow field. Calculate the material acceleration for fluid particles passing through this duct. Give your answer in two  ways: (1) as acceleration components ax and ay and (2) as acceleration vector a .

Figure 1

2. The water is sprayed out to the 5 N-weighted flat plate through the orifice by the water level of h1 from the water tank A without friction. This plate is attached to the water tank B with the water level of h2 as shown in Fig. 2. The friction coefficient between the plate and the orifice of the tank B is µ. The cross-sectional area of the orifice A and B is a. When the h2 is given, what will be h1 to hold the plate?

Figure 2

3. A desktop computer is to be cooled by a fan whose flow rate is 0.34 m3/min. Determine the mass flow rate of air through the fan at an elevation of 3400 m where the air density is 0.7 kg/m3. Also, if the average velocity of air is not to exceed 110 m/min, determine the diameter of the casing of the fan.

Figure 4

4. A siphon pumps water from a large reservoir to a lower tank that is initially empty. The tank also has a rounded orifice 6 m below the reservoir surface where the water leaves the tank. Both the siphon and the orifice diameters are 5 cm. Ignoring frictional losses, determine to what height the water will rise in the tank at equilibrium.

5. A piezometer and a Pitot tube are tapped into a 3-cm diameter horizontal water pipe, and the height of the water columns are measured to be 20 cm in the piezometer and 35 cm in the Pitot tube (both measured from the top surface of the pipe). Determine the velocity at the center of the pipe.

6. Air at 110 kPa and 50°C flows upward through a 6-cm-diameter inclined duct at a rate of 45 L/s. The duct diameter is then reduced to 4 cm through a reducer. The pressure change across the reducer is measured by a water manometer. The elevation difference between the two points on the pipe where the two arms of the manometer are attached is 0.20 m. Determine the differential height between the fluid levels of the two arms of the manometer.

Figure 6

7. Water enters a hydraulic turbine through a 30-cm diameter pipe at a rate of 0.6 m3/s and exits through a 25-cm diameter pipe. The pressure drop in the turbine is measured by a mercury manometer to be 1.2 m. For a combined turbine– generator efficiency of 83 percent, determine the net electric power output. Disregard the effect of the kinetic energy correction factors.

Figure 7

Homework #4-solution 1. A horizontal water jet of constant velocity V impinges normally on a vertical flat plate and splashes off the sides in the vertical plane. The plate is moving toward the oncoming water jet with velocity 1 V. If a force F is required to maintain the plate stationary, how much force is 2

required to move the plate toward the water jet?

Figure 1

2. A 3 m3/s water jet is moving in the positive x-direction at 6 m/s. The stream hits a stationary splitter, such that half of the flow is diverted upward at 45° and the other half is directed downward, and both streams have a final speed of 6 m/s. Disregarding gravitational effects, determine the x- and z-components of the force required to hold the splitter in place against the water force.

Figure 2

3. Firefighters are holding a nozzle at the end of a hose while trying to extinguish a fire. If the nozzle exit diameter is 6 cm and the water flow rate is 5 m3/min, determine (a) the average water exit velocity and (b) the horizontal resistance force required of the firefighters to hold the nozzle.

Figure 3

4. Water enters a centrifugal pump axially at atmospheric pressure at a rate of 0.12 m3/s and at a velocity of 7 m/s, and leaves in the normal direction along the pump casing, as shown in Fig. 4. Determine the force acting on the shaft (which is also the force acting on the bearing of the shaft) in the axial direction.

Figure 4

5. Pelton wheel turbines are commonly used in hydroelectric power plants to generate electric power. In these turbines, a high-speed jet at a velocity of Vj impinges on buckets, forcing the wheel to rotate. The buckets reverse the direction of the jet, and the jet leaves the bucket making an angle b with the direction of the jet, as shown in Fig. P6–51. Show that the power produced by a Pelton wheel of radius r rotating steadily at an angular velocity of  is W   rV (V   r )(1  cos  ) , where  is the density and V is the volume flow rate shaft

j

of the fluid. Obtain the numerical value for =150 rpm,



=160°, and



= 1000 kg/m3,

V j =50 m/s.

Figure 5

r = 2 m,

V

= 10 m3/s,

n

6. Water is flowing through a 12-cm-diameter pipe that consists of a 3-m-long vertical and 2m-long horizontal section with a 90° elbow at the exit to force the water to be discharged downward, as shown in Fig. 6, in the vertical direction. Water discharges to atmospheric air at a velocity of 4 m/s, and the mass of the pipe section when filled with water is 15 kg per meter length. Determine the moment acting at the intersection of the vertical and horizontal sections of the pipe (point A). What would your answer be if the flow were discharged upward instead of downward?

Figure 6

Homework #5-solution 1. An important application of fluid mechanics is the study of room ventilation. In particular, suppose there is a source S (mass per unit time) of air pollution in a room of volume V (Fig. 1). Examples include carbon monoxide from cigarette smoke or an unvented kerosene heater, gases like ammonia from household cleaning products, and vapors given off by evaporation of volatile organic compounds (VOCs) from an open container. We let c represent the mass concentration (mass of contaminant per unit volume of air). V is the volume flow rate of fresh air entering the room. If the room air is well mixed so that the mass concentration c is uniform throughout the room, but varies with time, the differential equation for mass concentration in the room as a function of time is

V

dc   cA k  S  Vc s w dt

where kw is an adsorption coefficient and As is the surface area of walls, floors, furniture, etc., that adsorb some of the contaminant. Write the primary dimensions of the first three additive terms in the equation, and verify that those terms are dimensionally homogeneous. Then determine the dimensions of kw. Show all your work.

Figure 1

2. A liquid of density ρ and viscosity µ flows by gravity through a hole of diameter d in the bottom of a tank of diameter D (Fig. 2). At the start of the experiment, the liquid surface is at height h above the bottom of the tank, as sketched. The liquid exits the tank as a jet with average velocity V straight down as also sketched. Using dimensional analysis, generate a dimensionless relationship for V as a function of the other parameters in the problem. Identify any established nondimensional parameters that appear in your result. (Hint: There are three length scales in this problem. For consistency, choose h as your length scale.)

Figure 2

3. Water at 15°C (ρ = 999.1 kg/m3 and µ = 1.138 × 10-3 kg/m · s) is flowing steadily in a 30m-long and 4-cm-diameter horizontal pipe made of stainless steel at a rate of 8 L/s. Determine (a) the pressure drop, (b) the head loss, and (c) the pumping power requirement to overcome this pressure drop.

Figure 3

4. A highly viscous liquid discharges from a large container through a small-diameter tube in laminar flow. Disregarding entrance effects and velocity heads, obtain a relation for the variation of fluid depth in the tank with time.

Figure 4