Solved: Under what conditions is the ideal-gas assumption

Problem 106P The specific gravities of solids and carrier fluids of a slurry are usually known, but the specific gravity of the slurry depends on the concentration of the solid particles. Show that the specific gravity of a water-based slurry can be expressed in terms of the specific gravity of the solid SGs. and the mass concentration of the suspended solid particlesCsmass as

Problem 16P What is vapor pressure? How is it related to saturation pressure?

Problem 17P Dues water boil at higher temperatures at higher pressures? Explain.

Problem 18P If the pressure of a substance is increased during a boiling process, will the temperature also increase or will it remain constant? Why?

Problem 20P In a piping system, the water temperature remains under 35°C. Determine the minimum pressure allowed in the system to avoid cavitation.

Problem 19P What is cavitation? What causes it?

Problem 21P The analysis of a propeller that operates in water at 20°C shows that the pressure at the tips of the propeller drops to2 kPa at high speeds. Determine if there is a danger of cavitation for this propeller.

Problem 23P What is the difference between the macroscopic and microscopic forms of energy?

Problem 56P Nitrogen enters a steady-flow heat exchanger at 150 kPa, 10°C, and 100 m/s, and it receives heat in the amount of 120 kJ/kg as it flows through it. Nitrogen leaves the heat exchanger at 100 kPa with a velocity of 200 m/s. Determine thejvlach number of the nitrogen at the inlet and the exit of the heat exchanger.

Problem 24P What is total energy? Identify the different forms of energy that constitute the total energy.

Problem 22P A pump is used to transport water to a higher reservoir. If the water temperature is 30°C, determine the lowest pressure that can exist in the pump without cavitation.

Problem 55P Carbon dioxide enters an adiabatic nozzle at 1200 K with a velocity of 50 m/s and leaves at 400 K. Assuming constant specific heats at room temperature, determine the Mach number (a) at the inlet and (b) at the exit of the nozzle. Assess the accuracy of the constant specific heat assumption.

Problem 57P Assuming ideal gas behavior, determine the speed of sound in refrigerant-134a at 0.1 MPa and 60°C.

Problem 58P The Airbus A-340 passenger plane has a maximum takeoff weight of about 260,000 kg, a length of 64 m, a wing span of 60 m, a maximum cruising speed of 945 km/h, a seating capacity of 271 passengers, maximum cruising altitude of 14,000 m, and a maximum range of 12,000 km. The air temperature at the crusing altitude is about ?60°C. Determine the Mach number of this plane for the stated limiting conditions.

Problem 62P Air expands isentropically from 1.5 MPa and 60°C to 0.4 MPa. Calculate the ratio of the initial to final speed of sound.

Problem 63P Repeat Prob. 9-20 for helium gas. PROBLEM: Air expands isentropically from 1.5 MPa and 60°C to 0.4 MPa. Calculate the ratio of the initial to final speed of sound.

Problem 64P What is viscosity? What is the cause of it in liquids and in gases? Do liquids or gases have higher dynamic viscosities?

Problem 65P What is a Newtonian fluid? Is water a Newtonian fluid?

Problem 97P Nutrients dissolved in water are carried to upper parts of plants by tiny tubes partly because of the capillary effect. Determine how high the water solution will rise in a tree in a 0.002-mm-diameter tube as a result of the capillary effect. Treat the solution as water at 20°C with a contact angle of 15°. FIGURE P2-103

Problem 98P The surface tension of a liquid is to be measured using a liquid film suspended on a U-shaped wire frame with an 8-cm-long movable side. If the force needed to move the wire is 0.024 N, determine the surface tension of this liquid in air.

Problem 99P Contrary to what you might expect, a solid steel ball can float on water due to the surface tension effect. Deter mine the maximum diameter of a steel ball that would float on water at 20°C. What would your answer be for an aluminum ball? Take the densities of steel and aluminum balls to be 7800 kg/m3 and 2700 kg/m3, respectively.

Problem 61P The isentropic process for an ideal gas is expressed as PVk = constant. Using this process equation and the definition of the speed of sound (Eq. 9-1), obtain the expression for the speed of sound for an ideal gas (Eq. 9-3). eq.9-1 eq. 9-3

Problem 101P The combustion in a gasoline engine may be approximated by a constant volume heat addition process, and the contents of the combustion chamber both before and after figure p2-107 combustion as air. The conditions are 1.8 MPa and 450°C before the combustion and 1300°C alter it. Determine the pressure at the end of the combustion process.

Problem 102P A rigid tank contains an ideal gas at 300 kPa and 600 K. Half of the gas is withdrawn from the lank and the gas is at 100 kPa at the end of the process. Determine (a) the final temperature of the gas and (b) the final pressure if no mass were withdrawn from the tank and the same final temperature were reached at the end of the process.

Problem 103P The absolute pressure of an automobile tire is measured to be 290 kPa before a trip and 310 kPa after the trip. Assuming the volume of the tire remains constant at 0.022 m3, determine the percent increase in the absolute temperature of the air in the tire.

A \(10-\mathrm{m}^{3}\) tank contains nitrogen at \(25^{\circ} \mathrm{C}\) and \(800 \mathrm{kPa}\). Some nitrogen is allowed to escape until the pressure in the tank drops to \(600 \mathrm{kPa}\). If the temperature at this point is \(20^{\circ} \mathrm{C}\), determine the amount of nitrogen that has escaped.

Problem 105P The composition of a liquid with suspended solid particles is generally characterized by the fraction of solid particles either by weight or massCsmass= ms/mm or by volume,Csvol = ?s/ ?m wherem is mass and ? is volume. The subscriptss and m indicate solid and mixture, respectively. Develop an expression for the specific gravity of a water- based suspension in terms ofCsmass and Cs vol.

Problem 25P List the forms of energy that contribute to the internal energy of a system.

Problem 26P How are heat, internal energy, and thermal energy related to each other?

Problem 27P What is flow energy? Do fluids at rest possess any flow energy?

Problem 66P Consider two identical small glass balls dropped into two identical containers, one filled with water and the other with oil. Which ball will reach the bottom of the container first? Why?

Problem 67P How does the dynamic viscosity of (a) liquids and (b) gases vary with temperature?

Problem 68P How does the kinematic viscosity of (a) liquids and (b) gases vary with temperature?

Problem 107P A closed tank is partially filled with water at 80°C. If the air above the water is completely evacuated, determine the absolute pressure in the evacuated space. Assume the temperature to remain constant.

Problem 108P The variation of the dynamic viscosity of water with absolute temperature is given as T, K ?, Pa • s 273.15 1.787 × 10?3 278.15 1.519 ×10?3 283.15 1.307 × 10?3 293.15 1.002 × 10?3 303.15 7.975 × 10?4 313.15 6.529 × 10?4 333.15 4.665 × 10?4 353.15 3.547 × 10?4 373.15 2.828 × 10?4 Using tabulated data, develop a relation for viscosity in the form of ? = ? (T) = A + BT + CT2 + DT3 + ET4 Using the relation developed, predict the dynamic viscosity of water at 50°C at which the reported value is 5.468 × 10?4 Pa • s. Compare your result with the results of Andrade’s equation, which is given in the form of ? = D • eB/T, where D and B are constants whose values are to be determined using the viscosity data given.

Problem 109P Consider laminar flow of a Newtonian fluid of viscosity ? between two parallel plates. The flow is one-dimensional, and the velocity profile is given as u(y) = 4umax[y/h ? (y/h)2], where y is the vertical coordinate from the bottom surface, h is the distance between the two plates, and umax is the maximum flow velocity that occurs at midplane. Develop a relation for the drag force exerted on both. by the fluid in the flow direction per unit area of the plates.

Problem 28P How do the energies of a flowing fluid and a fluid at rest compare? Name the specific forms of energy associated with each case.

Problem 30P Using average specific heats, explain how enthalpy changes of ideal gases and incompressible substances can be determined

Problem 29P Using average specific heats, explain how internal energy changes of ideal gases and incompressible substances can be determined.

Problem 69P For flow over a plate, the variation of velocity with vertical distance y from the plate is given as u(y) = ay – by2where a and b are constants. Obtain a relation for the wall shear stress in terms ofa, b, andµ.

Problem 70P A 50-cm × 30-cm × 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.

Problem 71P Consider the flow of a fluid with viscosity ? through a circular pipe. The velocity profile in the pipe is given as u(r) = umax(l ? rn/Rn), where umax is the maximum flow velocity, which occurs at the centerline; r is the radial distance from the centerline; and u(r) is the flow velocity at any position r. Develop a relation for the drag force exerted on the pipe wall by the fluid in the flow direction per unit length of the pipe.

Problem 110P Some non-Newtonian fluids behave as a Bingham plastic . For which shear stress can be expressed as ? = ?y+ ? (duldr) for laminar flow of a Bingham plastic in a horizontal pipe radius R, the velocity profile is given as u(r) = (?P/4?L)(r2 ? R2) + (?y/?)(r ? R), where ?P/L is the constant pressure drop along the pipe per unit length, fi is the dynamic viscosity, r is the radial distance from the centerline, and ?y is the yield stress of Bingham plastic. Determine (a) the shear stress at the pipe wall and (b) the drag force acting on a pipe section of length L.

Problem 111P In some damping systems, a circular disk immersed in oil is used as a damper, as shown in Fig. P9-56. Show that the damping torque is proportional to angular speed in accordance with the relation Tdamping = C ? where C = 0,5??(l/a +1/b)R4 Assume linear velocity profiles on both sides of the disk and neglect the tip effects.

Problem 113P Derive a relation for the capillary rise of a liquid between two large parallel plates a distance t apart inserted into the liquid vertically. Take the contact angle to be ?.

Problem 114P Consider a 40-cm-long journal bearing that is lubricated with oil whose viscosity is 0.1 kg/m ? s at 20°C at the beginning of operation and 0.008 kg/m ? s at the anticipated steady operating temperature of 80°C. The diameter of the shaft is 8 cm, and the average gap between the shaft and the journal is 0.08 cm. Determine the torque needed to overcome the bearing friction initially and during steady operation when the shaft is rotated at 800 rpm.

Problem 115P A 10-cm diameter cylindrical shaft rotates inside a 40-cm long 10.3-cm diameter bearing. The space between the shaft and the bearing is completely filed with oil whose viscosity at anticipated operating temperature is 0.300 N-s/m2. Determine the power required to overcome friction when the shaft rotates at a speed of (a) 600 rpm and (b) 1200 rpm.

Problem 116P The diameter of one arm of a U-tube is 5 mm while the other arm is large. If the U-tube contains some water, and both surface are exposed to atmospheric pressure, determine the difference between the water levels in the two arms.

Problem 31P What does the coefficient of compressibility of a fluid represent? How does it differ from isothermal compressibility?

Problem 32P What does the coefficient of volume expansion of a fluid represent? How does it differ from the coefficient of compressibility?

Problem 33P Can the coefficient of compressibility of a fluid be negative? How about the coefficient of volume expansion?

Problem 72P A thin 40-cm× 40-cm flat plate is pulled at 1 m/s horizontally through a 3.6-mm-thick oil layer sandwiched between two plates, one stationary and the other moving at a constant velocity of 0.3 m/s, as shown in Fig. P2-77. The dynamic viscosity of oil is 0.027 Pa ? s. Assuming the velocity in each oil layer to vary linearly, (a) plot the velocity profile and find the location where the oil velocity is zero and (b) determine the force that needs to be applied on the plate to maintain this motion. FIGURE P2-77

Problem 73P A frustum-shaped body is rotating at a constant angular speed of 200 rad/s in a container filled with SAE 10W oil at 20°C (? = 0.1 Pa • s), as shown in Fig. P9-31. If the thickness of the oil film on all sides is 1.2 mm, determine the power required to maintain this motion. Also the determine reduction in the required power input when the oil the reduction in the required power input when the oil temperature rises to 80°C(? = 0.0078 Pa • s)

The clutch system shown in Fig. P2-79 is used to transmit torque through a 2-mm-thick oil film with µ = 0.38 N ? s/m2 between two identical 30-cm-diameter disks. When the driving shaft rotates at a speed of 1450 rpm, the driven shaft is observed to rotate at 1398 rpm. Assuming a linear velocity profile for the oil film, determine the transmitted torque. FIGURE P2-79

Problem 117P Some rocks or bricks contain small air pockets in them and have a spongy structure;. Assuming the air spaces form columns of an average diameter of 0.004 mm, determine how high water can rise in such a material. Take the surface tension of the air-water interface in that material to be0.085 N/m.

Problem 34P The volume of an ideal gas is to be reduced by half by compressing it isothermally. Determine the required change in pressure.

Problem 36P Using the definition of the coefficient of volume expansion and the expression ?ideal gas = 1/T, show that the percent increase in the specific volume of an ideal gas during isobaric expansion is equal to the percent increase in absolute temperature.

Problem 37P Water at 1 atm pressure is compressed to 550 atm pressure isothermally. Determine the increase in the density of water. Take the isothermal compressibility of water to be 4.80 × 10–5 atm–1.

Problem 75P Reconsider Prob. 2–79. Using EES (or other) software, investigate the effect of oil film thickness on the torque transmitted. Let the film thickness vary from 0.1 mm to 10 mm. Plot your results, and state your conclusions.

Problem 76P The viscosity of some fluids changes when a strong electric field is applied on them. This phenomenon is known as the electrorheological (ER) effect, and fluids that exhibit such behavior are known as ER fluids. The Bingham plastic model for shear stress, which is expressed as ? = ?y + ?(duldy) is widely used to describe ER fluid behavior because of its simplicity. One of the most promising applications of ER-clutch A typical multidisk ER clutch consists of several equally spaced steel disks of:inner radius R1 and outer radius R2, N of them attached to the input shaft. The gap h between the parallel disks is filled with a viscous fluid, (a) Find a relationship for the torque generated by the clutch when the output shaft is stationary and (b) calculate the torque for an ER clutch with N = 11 for R1 = 50 mm, R2 = 200 mm, and rpm if the fluid is 10 with ? = 0.1 Pa • s, ?y = 2.5 kPa, and h = 1.2 mm.

Problem 77P The viscosity of some fluids, called magnetorheologi- cal (MR) fluids, changes when a magnetic field is applied. Such fluids involve micron-sized magnetizable particles suspended in an appropriate carrier liquid, and are suitable for use in controllable hydraulic clutches. See Fig. P2-49. The MR fluids can have much higher viscosities than the ER fluids, and they often exhibit shear-thinning behavior in which the viscosity of the fluid decreases as the applied shear force increases. This behavior is also known as pseudoplastic behavior, and can be successfully represented by Herschel-Bulkley constitutive model expressed as ? =?y + K(du/dy)m. Here ? is the shear stress applied, ?y is the yield stress,K is the consistency index, and m is the power index. For a Herschel-Bulkley fluid with ?y, = 900 Pa,K = 58 Pa ? sm, and m = 0.82, (a) find a relationship for the torque transmitted by an MR clutch for N plates attached to the input shaft when the input shaft is rotating at an angular speed of ? while the output shaft is stationary and (b) calculate the torque transmitted by such a clutch with N = 11 plates for R1 = 50 mm,R2 = 200 mm,? = 3000 rpm, and h = 1.5 mm.

Water at 15°C and 1 atm pressure is heated to 100°C at constant pressure. Using coefficient of volume expansion data, determine the change in the density of water.

Problem 39P Saturated refrigerant-134a liquid at 10°C is cooled to 0°C at constant pressure. Using coefficient of volume expansion data, determine the change in the density of the refrigerant.

Problem 40P A water tank is completely filled with liquid water at 20°C. The tank material is such that it can withstand tension caused by a volume expansion of 1.2 percent. Determine the maximum temperature rise allowed without jeopardizing safety. For simplicity, assume ? = constant = ? at 40°C.

Problem 78P The viscosity of a fluid is to be measured by a viscometer constructed of two 75-cm-long concentric cylinders. The outer diameter of the inner cylinder is 15 cm, and the gap between the two cylinders is 1 mm. The inner cylinder is rotated at 300 rpm, and the torque is measured to be 0.8 . N ? m. Determine the viscosity of the fluid. FIGURE P2-83

Problem 79P In regions far from the entrance, fluid flow through a circular pipe is one-dimensional, and the velocity profile for laminar flow is given byu(r) =umax( 1 –r2/R2), whereR is the radius of the pipe,r is the radial distance from the center of the pipe, and umax is the maximum flow velocity, which occurs at the center. Obtain(a) a relation for the drag force applied by the fluid on a section of the pipe of lengthL and (b) the value of the drag force for water flow at 20°C with R = 0.08 m, L = 30 m, umax = 3 m/s, and µ = 0.0010 kg/m ? s. FIGURE P2-85

Problem 80P Repeat Prob. 2–85 for umax = 7 m/s.

Problem 1P What is the state postulate?

Problem 2P For a substance, what is the difference between mass and molar mass? How are these two related?

Problem 3P What is the difference between intensive and extensive properties?

Problem 41P Repeat Prob. 2–43 for a volume expansion of 1.5 percent for water.

Problem 45P Prove that the coefficient of volume expansion for an ideal gas is ?deal gas = 1/T

Problem 42P The density of seawater at a free surface where the pressure is 98 kPa is approximately 1030 kg/m3. Taking the bulk modulus of elasticity of seawater to be 2.34 × 109 N/m2and expressing variation of pressure with depthz asdP=?g dz determine the density and pressure at a depjh of 2500 m. Disregard the effect of temperature.

A thin plate moves between two parallel, horizontal, stationary flat surfaces at a constant velocity of 5 m/s. The two stationary surfaces are spaced 4 cm apart, and the medium between them is filled with oil whose viscosity is \(0.9\mathrm{\ N}\cdot\mathrm{s}/\mathrm{m}^2\). The part of the plate immersed in oil at any given time is 2-m long and 0.5-m wide. If the plate moves through the mid-plane between the surfaces, determine the force required to maintain this motion. What would your response be if the plate was 1 cm from the bottom surface \(\left(h_{2}\right)\) and 3 cm from the top surface \(\left(h_{1}\right)\)? FIGURE P2-87

Problem 82P Reconsider Prob. 2–87. If the viscosity of the oil above the moving plate is 4 times that of the oil below the plate, determine the distance of the plate from the bottom surface (h2) that will minimize the force needed to pull the plate between the two oils at constant velocity.

A rotating viscometer consists of two concentric cylinders - an inner cylinder of radius \(R_{i}\) rotating at angular velocity (rotation rate) \(\omega_{i}\), and a stationary outer cylinder of inside radius \(R_{o}\). In the tiny gap between the two cylinders is the fluid of viscosity \(\mu\). The length of the cylinders (into the page in Fig. P2-83) is \(L. L\) is large such that end effects are negligible (we can treat this as a two-dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed. (a) Showing all of your work and algebra, generate an approximate expression for \(T\) as a function of the other variables. (b) Explain why your solution is only an approximation. In particular, do you expect the velocity profile in the gap to remain linear as the gap becomes larger and larger (i.e., if the outer radius \(R_{o}\) were to increase, all else staying the same)?

Problem 4P What is specific gravity? How is it related to density?

Problem 5P Under what conditions is the ideal-gas assumption suitable for real gases?

Problem 6P What is the difference between R and Ru? Hdw are these two related?

Problem 46P When modeling fluid flows with small changes in temperature and pressure, the Boussinesq approximation is often used in which the fluid density is assumed to vary linearly with changes in temperature. The Boussinesq approximation is ? = ?()[1 –?(T– To)], where ? is assumed to be constant over the given temperature range; ? is evaluated at reference temperature T0, taken as some average or mid-value temperature in the flow; and ?0 is a reference density, also evaluated at T0. The Boussinesq approximation is used to model a flow of air at nearly constant pressure,P = 95.0 kPa, but the temperature varies between 20°C and 60°C. Using the mid-way point (40°C) as the reference temperature, calculate the density at the two temperature extremes using the Boussinesq approximation, and compare with the actual density at these two temperatures obtained from the ideal gas law. In particular, for both temperatures calculate the percentage error caused by the Boussinesq approximation.

Problem 47P What is sound? How is it generated? How does it travel? Can sound waves travel in a vacuum?

Problem 48P Is it realistic to assume that the propagation of sound waves is an isentropic process? Explain.

Problem 86P A rotating viscometer consists of two concentric cylinders—a stationary inner cyliner of radiusRi and an outer cylinder of inside radiusR0 rotating at angular velocity (rotation rate) w0. In the tiny gap between the two cylinders is the fluid whose viscosity (µ) is to be measured. The length of the cylinders (into the page in Fig. P2-92) is L. L is large such that end effects are negligible (we can treat this as a two- dimensional problem). Torque (T) is required to rotate the inner cylinder at constant speed. Showing all your work and algebra, generate an approximate expression of T as a function of the other varilables. FIGURE P2-92

Problem 89P Consider a soap bubble. Is the pressure inside the bubble higher or lower than the pressure outside?

Problem 49P Is the sonic velocity in a specified medium a fixed quantity, or does it change as the properties of the medium change? Explain.

Problem 50P In which medium does a sound wave travel faster: in cool air or in warm air?

Problem 51P In which medium will sound travel fastest for a given temperature: air, helium, or argon?

Problem 7P A 100-L container is filled with 1 kg of air at a temperature of 27°C. What is the pressure in the container?

Problem 90P What is the capillary effect? What is it caused by? How is it affected by the contact angle?

Problem 9P A fluid that occupies a volume of 32 L weighs 280 N at a location where the gravitational acceleration is 9.80 m/s2. Determine the mass of this fluid and its density.

Problem 91P A small-diameter tube is inserted into a liquid whose contact angle is 110°. Will the level of liquid in the tube be higher or lower than the level of the rest of the liquid? Explain.

Problem 92P Is the capillary rise greater in small- or large-diameter tubes?

Problem 10P A spherical balloon with a diameter of 9 m is filled with helium at 20°C and 200 kPa. Determine the mole number and the mass of the helium in the balloon.

Problem 11P Reconsider Prob. 2–12. Using EES (or other) fiH software, investigate the effect of the balloon diameter on the mass of helium contained in the balloon for the pressures of (a) 100 kPa and (b) 200 kPa. Let the diameter vary from 5 m to 15 m. Plot the mass of helium against the diameter for both cases.

Problem 12P The 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. FIGURE P2-14

Problem 15P The density of atmospheric air varies with elevation, decreasing with increasing altitude, (a) Using the data given in the table, obtain a relation for the variation of density with elevation, and calculate the density at an elevation of 7000 m. (b) Calculate the mass of the atmosphere using the correlation you obtained. Assume the earth to be a perfect sphere with a radius of 6377 km, and take the thickness of the atmosphere to be 25 km. r, km ?, kg/m3 6377 1.225 6378 1.112 6379 1.007 6380 0.9093 6381 0.8194 6382 0.7364 6383 0.6601 6385 0.5258 6387 0.4135 6392 0.1948 6397 0.08891 6402 0.04008 Vapor Pressure and Cavitation

Problem 52P In which medium does a sound wave travel faster: in air at 20°C and 1 atm or in air at 20°C and 5 atm?

Problem 53P Does the Mach number of a gas flowing at a constant velocity remain constant? Explain.

Problem 54P Determine the speed of sound in air at (a) 300 K and (b) 1000 K. Also determine the Mach number of an aircraft moving in air at a velocity of 240 m/s for both cases.

Problem 93P Consider a 0.2-mm diameter air bubble in a liquid. Determine the pressure difference between the inside and outside of the air bubble if the surface tension at the air-liquid interface is (a) 0.08 N/m and (b) 0.12 N/m.

Problem 95P A 1.2-mm-diameter tube is inserted into an unknown liquid whose density is 960 kg/m3, and it is observed that the liquid rises 5 mm in the tube, making a contact angle of 15°. Determine the surface tension of the liquid.

Problem 96P Determine the gage pressure inside a soap bubble of diameter (a) 0.2 cm and (b) 5 cm at 20°C.

Problem 88P What is surface tension? What is it caused by? Why is the surface tension also called surface energy?