Pressure in a Spherical Tank. Calculate the pressure in psia and kN/m² at the bottom of a spherical tank filled with oil having a diameter of 8.0 ft. The top of the tank is vented to the atmosphere having a pressure of 14.72 psia. The density of the oil is 0.922 g/cm³.

Pressure with Two Liquids, Hg and Water. An open test tube at 293 K is filled at the bottom with 12.1 cm of Hg, and 5.6 cm of water is placed above the Hg. Calculate the pressure at the bottom of the test tube if the atmospheric pressure is 756 mm Hg. Use a density of 13.55 g/cm³ for Hg and 0.998 g/cm³ for water. Give the answer in terms of dyn/cm², psia, and kN/m². See Appendix A.1 for conversion factors.

Head of a Fluid of Jet Fuel and Pressure. The pressure at the top of a tank of jet fuel is 180.6 kN/m². The depth of liquid in the tank is 6.4 m. The density of the fuel is 825 kg/m³. Calculate the head of the liquid in m that corresponds to the absolute pressure at the bottom of the tank.

Measurement of Pressure. An open U-tube manometer similar to Fig. 2.2-4a is being used to measure the absolute pressure p_a in a vessel containing air. The pressure p_b is atmospheric pressure, which is 754 mm Hg. The liquid in the manometer is water having a density of 1000 kg/m³. Assume that the density ρ_B is 1.30 kg/m³ and that the distance Z is very small. The reading R is 0.415 m. Calculate p_a in psia and kPa.

Measurement of Small Pressure Differences. The two-fluid U-tube manometer is being used to measure the difference in pressure at two points in a line containing air at 1 atm abs pressure. The value of R₀ = 0 for equal pressures. The lighter fluid is a hydrocarbon with a density of 812 kg/m³ and the heavier water has a density of 998 kg/m³. The inside diameters of the U tube and reservoir are 3.2 mm and 54.2 mm, respectively. The reading R of the manometer is 117.2 mm. Calculate the pressure difference in mm Hg and pascal.

Pressure in a Sea Lab. A sea lab 5.0 m high is to be designed to withstand submersion to 150 m, measured from sea level to the top of the sea lab. Calculate the pressure on top of the sea lab and also the pressure variation on the side of the container measured as the distance x in m from the top of the sea lab downward. The density of seawater is 1020 kg/m³.

Measurement of Pressure Difference in Vessels. In Fig. 2.2-5b the differential manometer is used to measure the pressure difference between two vessels. Derive the equation for the pressure difference p_A − p_B in terms of the liquid heights and densities.

Design of Settler and Separator for Immiscible Liquids. A vertical cylindrical settler–separator is to be designed for separating a mixture flowing at 20.0 m³/h and containing equal volumes of a light petroleum liquid (ρ_B = 875 kg/m³) and a dilute solution of wash water (ρ_A = 1050 kg/m³). Laboratory experiments indicate a settling time of 15 min is needed to adequately separate the two phases. For design purposes use a 25-min settling time and calculate the size of the vessel needed, the liquid levels of the light and heavy liquids in the vessel, and the height h_A2 of the heavy-liquid overflow. Assume that the ends of the vessel are approximately flat, that the vessel diameter equals its height, and that one-third of the volume is vapor space vented to the atmosphere. Use the nomenclature given in Fig. 2.2-6.

Molecular Transport of a Property with a Variable Diffusivity. A property is being transported through a fluid at steady state through a constant cross-sectional area. At point 1 the concentration Г₁ is 2.78 × 10⁻² amount of property/m³ and 1.50 × 10⁻² at point 2 at a distance of 2.0 m away. The diffusivity depends on concentration Г as follows:

Integration of General-Property Equation for Steady State. Integrate the general-property equation (2.3-11) for steady state and no generation between the points Г₁ at z₁ and Г₂ at z₂. The final equation should relate Г to z.

Shear Stress in Soybean Oil. Using Fig. 2.4-1, the distance between the two parallel plates is 0.00914 m, and the lower plate is being pulled at a relative velocity of 0.366 m/s greater than the top plate. The fluid used is soybean oil with viscosity 4 × 10⁻² Pa · s at 303 K (Appendix A.4).

Shear Stress and Shear Rate in Fluids. Using Fig. 2.4-1, the lower plate is being pulled at a relative velocity of 0.40 m/s greater than the top plate. The fluid used is water at 24°C.

Reynolds Number for Milk Flow. Whole milk at 293 K having a density of 1030 kg/m³ and viscosity of 2.12 cp is flowing at the rate of 0.605 kg/s in a glass pipe having a diameter of 63.5 mm.

Pipe Diameter and Reynolds Number. An oil is being pumped inside a 10.0-mm-diameter pipe at a Reynolds number of 2100. The oil density is 855 kg/m³ and the viscosity is 2.1 × 10⁻² Pa · s.

Average Velocity for Mass Balance in Flow Down Vertical Plate. For a layer of liquid flowing in laminar flow in the z direction down a vertical plate or surface, the velocity profile is

Flow of Liquid in a Pipe and Mass Balance. A hydrocarbon liquid enters a simple flow system shown in Fig. 2.6-1 at an average velocity of 1.282 m/s, where A₁ = 4.33 × 10⁻³ m² and ρ₁ = 902 kg/m³. The liquid is heated in the process and the exit density is 875 kg/m³. The cross-sectional area at point 2 is 5.26 × 10⁻³ m². The process is steady state.

Average Velocity for Mass Balance in Turbulent Flow. For turbulent flow in a smooth, circular tube with a radius R, the velocity profile varies according to the following expression at a Reynolds number of about 10⁵:

Bulk Velocity for Flow Between Parallel Plates. A fluid flowing in laminar flow in the x direction between two parallel plates has a velocity profile given by the following:

Overall Mass Balance for Dilution Process. A well-stirred storage vessel contains 10 000 kg of solution of a dilute methanol solution (w_A = 0.05 mass fraction alcohol). A constant flow of 500 kg/min of pure water is suddenly introduced into the tank and a constant rate of withdrawal of 500 kg/min of solution is started. These two flows are continued and remain constant. Assuming that the densities of the solutions are the same and that the total contents of the tank remain constant at 10000 kg of solution, calculate the time for the alcohol content to drop to 1.0 wt%.

Overall Mass Balance for Unsteady-State Process. A storage vessel is well stirred and contains 500 kg of total solution with a concentration of 5.0% salt. A constant flow rate of 900 kg/h of salt solution containing 16.67% salt is suddenly introduced into the tank and a constant withdrawal rate of 600 kg/h is also started. These two flows remain constant thereafter. Derive an equation relating the outlet withdrawal concentration as a function of time. Also, calculate the concentration after 2.0 h.

Mass Balance for Flow of Sucrose Solution. A 20 wt % sucrose (sugar) solution having a density of 1074 kg/m³ is flowing through the same piping system as in Example 2.6-1 (Fig. 2.6-2). The flow rate entering pipe 1 is 1.892 m³/h. The flow divides equally in each of pipes 3. Calculate the following:

Kinetic-Energy Velocity Correction Factor for Turbulent Flow. Derive the equation to determine the value of α, the kinetic-energy velocity correction factor, for turbulent flow. Use Eq. (2.7-20) to approximate the velocity profile and substitute this into Eq. (2.7-15) to obtain (ν³)_av. Then use Eqs. (2.7-20), (2.6-17), and (2.7-14) to obtain α.

Flow Between Parallel Plates and Kinetic-Energy Correction Factor. The equation for the velocity profile of a fluid flowing in laminar flow between two parallel plates is given in Problem 2.6-4. Derive the equation to determine the value of the kinetic-energy velocity correction factor α. [Hint: First derive an equation relating ν to ν_av. Then derive the equation for (ν³)_av and, finally, relate these results to α.]

Temperature Drop in Throttling Valve and Energy Balance. Steam is flowing through an adiabatic throttling valve (no heat loss or external work). Steam enters point 1 upstream of the valve at 689 kPa abs and 171.1°C and leaves the valve (point 2) at 359 kPa. Calculate the temperature t₂ at the outlet. [Hint: Use Eq. (2.7-21) for the energy balance and neglect the kinetic-energy and potential-energy terms as shown in Example 2.7-1. Obtain the enthalpy H₁ from Appendix A.2, steam tables. For H₂, linear interpolation of the values in the table will have to be done to obtain t₂.] Use SI units.

Energy Balance on a Heat Exchanger and a Pump. Water at 93.3°C is being pumped from a large storage tank at 1 atm abs at a rate of 0.189 m³/min by a pump. The motor that drives the pump supplies energy to the pump at the rate of 1.49 kW. The water is pumped through a heat exchanger, where it gives up 704 kW of heat and is then delivered to a large open storage tank at an elevation of 15.24 m above the first tank. What is the final temperature of the water to the second tank? Also, what is the gain in enthalpy of the water due to the work input? (Hint: Be sure to use the steam tables for the enthalpy of the water. Neglect any kinetic-energy changes but not potential-energy changes.)

Steam Boiler and Overall Energy Balance. Liquid water under pressure at 150 kPa enters a boiler at 24°C through a pipe at an average velocity of 3.5 m/s in turbulent flow. The exit steam leaves at a height of 25 m above the liquid inlet at 150°C and 150 kPa absolute, and the velocity in the outlet line is 12.5 m/s in turbulent flow. The process is steady state. How much heat must be added per kg of steam?

Energy Balance on a Flow System with a Pump and Heat Exchanger. Water stored in a large, well-insulated storage tank at 21.0°C and atmospheric pressure is being pumped at steady state from this tank by a pump at the rate of 40 m³/h. The motor driving the pump supplies energy at the rate of 8.5 kW. The water is used as a cooling medium and passes through a heat exchanger, where 255 kW of heat is added to the water. The heated water then flows to a second large, vented tank, which is 25 m above the first tank. Determine the final temperature of the water delivered to the second tank.

Mechanical-Energy Balance in Pumping Soybean Oil. Soybean oil is being pumped through a uniform-diameter pipe at a steady mass-flow rate. A pump supplies 209.2 J/kg mass of fluid flowing. The entrance abs pressure in the inlet pipe to the pump is 103.4 kN/m². The exit section of the pipe downstream from the pump is 3.35 m above the entrance and the exit pressure is 172.4 kN/m². Exit and entrance pipes are the same diameter. The fluid is in turbulent flow. Calculate the friction loss in the system. See Appendix A.4 for the physical properties of soybean oil. The temperature is 303 K.

Pump Horsepower in Brine System. A pump pumps 0.200 ft³/s of brine solution having a density of 1.15 g/cm³ from an open feed tank having a large cross-sectional area. The suction line has an inside diameter of 3.548 in. and the discharge line from the pump a diameter of 2.067 in. The discharge flow goes to an open overhead tank, and the open end of this line is 75 ft above the liquid level in the feed tank. If the friction losses in the piping system are 18.0 ft lb_f/lb_m, what pressure must the pump develop and what is the horsepower of the pump if the efficiency is 70%? The flow is turbulent.

Pressure Measurements from Flows. Water having a density of 998 kg/m³ is flowing at the rate of 1.676 m/s in a 3.068-in.-diameter horizontal pipe at a pressure p₁ of 68.9 kPa abs. It then passes to a pipe having an inside diameter of 2.067 in.

Draining Cottonseed Oil from a Tank. A cylindrical tank 1.52 m in diameter and 7.62 m high contains cottonseed oil having a density of 917 kg/m³. The tank is open to the atmosphere. A discharge nozzle of inside diameter 15.8 mm and cross-sectional area A₂ is located near the bottom of the tank. The surface of the liquid is located at H = 6.1 m above the center line of the nozzle. The discharge nozzle is opened, draining the liquid level from H = 6.1 m to H = 4.57 m. Calculate the time in seconds to do this. [Hint: The velocity on the surface of the reservoir is small and can be neglected. The velocity ν₂ m/s in the nozzle can be calculated for a given H by Eq. (2.7-36). However, H, and hence ν₂, are varying. Set up an unsteady-state mass balance as follows: The volumetric flow rate in the tank is (A_t dH)/dt, where A_t is the tank cross section in m² and A_t dH is the m³ liquid flowing in dt s. This rate must equal the negative of the volumetric rate in the nozzle, or −A₂ν₂ m³/s. The negative sign is present since dH is the negative of ν₂. Rearrange this equation and integrate between H = 6.1 m at t = 0 and H = 4.57 m at t = t_F.]

Friction Loss in Turbine Water Power System. Water is stored in an elevated reservoir. To generate power, water flows from this reservoir down through a large conduit to a turbine and then through a similar-sized conduit. At a point in the conduit 89.5 m above the turbine, the pressure is 172.4 kPa, and at a level 5 m below the turbine, the pressure is 89.6 kPa. The water flow rate is 0.800 m³/s. The output of the shaft of the turbine is 658 kW. The water density is 1000 kg/m³. If the efficiency of the turbine in converting the mechanical energy given up by the fluid to the turbine shaft is 89% (η_t = 0.89), calculate the friction loss in the turbine in J/kg. Note that in the mechanical-energy-balance equation, the W_s is equal to the output of the shaft of the turbine over η_t.

Pipeline Pumping of Oil. A pipeline laid cross-country carries oil at the rate of 795 m³/d. The pressure of the oil is 1793 kPa gage leaving pumping station 1. The pressure is 862 kPa gage at the inlet to the next pumping station, 2. The second station is 17.4 m higher than the first station. Calculate the lost work ( ∑F friction loss) in J/kg mass oil. The oil density is 769 kg/m³.

Test of Centrifugal Pump and Mechanical-Energy Balance. A centrifugal pump is being tested for performance, and during the test the pressure reading in the 0.305-m-diameter suction line just adjacent to the pump casing is −20.7 kPa (vacuum below atmospheric pressure). In the discharge line with a diameter of 0.254 m at a point 2.53 m above the suction line, the pressure is 289.6 kPa gage. The flow of water from the pump is measured as 0.1133 m³/s. (The density can be assumed as 1000 kg/m³.) Calculate the kW input of the pump.

Friction Loss in Pump and Flow System. Water at 20°C is pumped from the bottom of a large storage tank where the pressure is 310.3 kPa gage to a nozzle which is 15.25 m above the tank bottom and discharges to the atmosphere with a velocity in the nozzle of 19.81 m/s. The water flow rate is 45.4 kg/s. The efficiency of the pump is 80% and 7.5 kW are furnished to the pump shaft. Calculate the following:

Power for Pumping in Flow System. Water is being pumped from an open water reservoir at the rate of 2.0 kg/s at 10°C to an open storage tank 1500 m away. The pipe used is schedule 40 3in. pipe and the frictional losses in the system are 625 J/kg. The surface of the water reservoir is 20 m above the level of the storage tank. The pump has an efficiency of 75%.

Momentum Balance in a Reducing Bend. Water is flowing at steady state through the reducing bend in Fig. 2.8-3. The angle α₂ = 90° (a right-angle bend). The pressure at point 2 is 1.0 atm abs. The flow rate is 0.020 m³/s and the diameters at points 1 and 2 are 0.050 m and 0.030 m, respectively. Neglect frictional and gravitational forces. Calculate the resultant forces on the bend in newtons and lb force. Use ρ = 1000 kg/m³.

Forces on Reducing Bend. Water is flowing at steady state and 363 K at a rate of 0.0566 m³/s through a 60° reducing bend (α₂ = 60°) in Fig. 2.8-3. The inlet pipe diameter is 0.1016 m and the outlet 0.0762 m. The friction loss in the pipe bend can be estimated as Neglect gravity forces. The exit pressure p₂ = 111.5 kN/m² gage. Calculate the forces on the bend in newtons.

Force of Water Stream on a Wall. Water at 298 K discharges from a nozzle and travels horizontally, hitting a flat, vertical wall. The nozzle has a diameter of 12 mm and the water leaves the nozzle with a flat velocity profile at a velocity of 6.0 m/s. Neglecting frictional resistance of the air on the jet, calculate the force in newtons on the wall.

Flow Through an Expanding Bend. Water at a steady-state rate of 0.050 m³/s is flowing through an expanding bend that changes direction by 120°. The upstream diameter is 0.0762 m and the downstream is 0.2112 m. The upstream pressure is 68.94 kPa gage. Neglect energy losses within the elbow and calculate the downstream pressure at 298 K. Also calculate R_x and R_y.

Force of Stream on a Wall. Repeat Problem 2.8-3 for the same conditions except that the wall is inclined 45° to the vertical. The flow is frictionless. Assume no loss in energy. The amount of fluid splitting in each direction along the plate can be determined by using the continuity equation and a momentum balance. Calculate this flow division and the force on the wall.

Momentum Balance for Free Jet on a Curved, Fixed Vane. A free jet having a velocity of 30.5 m/s and a diameter of 5.08 × 10⁻² m is deflected by a curved, fixed vane as in Fig. 2.8-5a. However, the vane is curved downward at an angle of 60° instead of upward. Calculate the force of the jet on the vane. The density is 1000 kg/m³.

Momentum Balance for Free Jet on a U-Type, Fixed Vane. A free jet having a velocity of 30.5 m/s and a diameter of 1.0 × 10⁻² m is deflected by a smooth, fixed vane as in Fig. 2.8-5a. However, the vane is in the form of a U so that the exit jet travels in a direction exactly opposite to the entering jet. Calculate the force of the jet on the vane. Use ρ = 1000 kg/m³.

Momentum Balance on Reducing Elbow and Friction Losses. Water at 20°C is flowing through a reducing bend, where α₂ (see Fig. 2.8-3) is 120°. The inlet pipe diameter is 1.829 m, the outlet is 1.219 m, and the flow rate is 8.50 m³/s. The exit point z₂ is 3.05 m above the inlet and the inlet pressure is 276 kPa gage. Friction losses are estimated as 0.5 /2 and the mass of water in the elbow is 8500 kg. Calculate the forces R_x and R_y and the resultant force on the control-volume fluid.

Momentum Velocity Correction Factor β for Turbulent Flow. Determine the momentum velocity correction factor β for turbulent flow in a tube. Use Eq. (2.7-20) for the relationship between ν and position.

Film of Water on Wetted-Wall Tower. Pure water at 20°C is flowing down a vertical wetted-wall column at a rate of 0.124 kg/s · m. Calculate the film thickness and the average velocity.

Shell Momentum Balance for Flow Between Parallel Plates. A fluid of constant density is flowing in laminar flow at steady state in the horizontal x direction between two flat and parallel plates. The distance between the two plates in the vertical y direction is 2y₀. Using a shell momentum balance, derive the equation for the velocity profile within this fluid and the maximum velocity for a distance L m in the x direction. [Hint: See the method used in Section 2.9B to derive Eq. (2.9-9). One boundary condition used is dν_x/dy = 0 at y = 0.]

Velocity Profile for Non-Newtonian Fluid. The stress rate of shear for a non-Newtonian fluid is given by

Shell Momentum Balance for Flow Down an Inclined Plane. Consider the case of a Newtonian fluid in steady-state laminar flow down an inclined plane surface that makes an angle θ with the horizontal. Using a shell momentum balance, find the equation for the velocity profile within the liquid layer having a thickness L and the maximum velocity of the free surface. (Hint: The convective-momentum terms cancel for fully developed flow and the pressure-force terms also cancel, because of the presence of a free surface. Note that there is a gravity force on the fluid.)

Viscosity Measurement of a Liquid. One use of the Hagen–Poiseuille equation (2.10-2) is in determining the viscosity of a liquid by measuring the pressure drop and velocity of the liquid in a capillary of known dimensions. The liquid used has a density of 912 kg/m³, and the capillary has a diameter of 2.222 mm and a length of 0.1585 m. The measured flow rate is 5.33 × 10⁻⁷ m³/s of liquid and the pressure drop 131 mm of water (density 996 kg/m³). Neglecting end effects, calculate the viscosity of the liquid in Pa · s.

Frictional Pressure Drop in Flow of Olive Oil. Calculate the frictional pressure drop in pascal for olive oil at 293 K flowing through a commercial pipe having an inside diameter of 0.0525 m and a length of 76.2 m. The velocity of the fluid is 1.22 m/s. Use the friction factor method. Is the flow laminar or turbulent? Use physical data from Appendix A.4.

Frictional Loss in Straight Pipe and Effect of Type of Pipe. A liquid having a density of 801 kg/m³ and a viscosity of 1.49 × 10⁻³ Pa · s is flowing through a horizontal straight pipe at a velocity of 4.57 m/s. The commercial steel pipe is in. nominal pipe size, schedule 40. For a length of pipe of 61 m, do as follows:

Trial-and-Error Solution for Hydraulic Drainage. In a hydraulic project a cast-iron pipe having an inside diameter of 0.156 m and a 305-m length is used to drain wastewater at 293 K. The available head is 4.57 m of water. Neglecting any losses in fittings and joints in the pipe, calculate the flow rate in m³/s. (Hint: Assume the physical properties of pure water. The solution is trial and error, since the velocity appears in N_Re, which is needed to determine the friction factor. As a first trial, assume that ν = 1.7 m/s.)

Mechanical-Energy Balance and Friction Losses. Hot water is being discharged from a storage tank at the rate of 0.223 ft³/s. The process flow diagram and conditions are the same as given in Example 2.10-6, except for different nominal pipe sizes of schedule 40 steel pipe as follows. The 20-ft-long outlet pipe from the storage tank is pipe instead of 4-in. pipe. The other piping, which was 2-in. pipe, is now 2.5-in. pipe. Note that now a sudden expansion occurs after the elbow in the pipe to a pipe.

Friction Losses and Pump Horsepower. Hot water in an open storage tank at 82.2°C is being pumped at the rate of 0.379 m³/min from the tank. The line from the storage tank to the pump suction is 6.1 m of 2-in. schedule 40 steel pipe and it contains three elbows. The discharge line after the pump is 61 m of 2-in. pipe and contains two elbows. The water discharges to the atmosphere at a height of 6.1 m above the water level in the storage tank.

Pressure Drop of a Flowing Gas. Nitrogen gas is flowing through a 4-in. schedule 40 commercial steel pipe at 298 K. The total flow rate is 7.40 × 10⁻² kg/s and the flow can be assumed as isothermal. The pipe is 3000 m long and the inlet pressure is 200 kPa. Calculate the outlet pressure.

Entry Length for Flow in a Pipe. Air at 10°C and 1.0 atm abs pressure is flowing at a velocity of 2.0 m/s inside a tube having a diameter of 0.012 m.

Friction Loss in Pumping Oil to Pressurized Tank. An oil having a density of 833 kg/m³ and a viscosity of 3.3 × 10⁻³ Pa · s is pumped from an open tank to a pressurized tank held at 345 kPa gage. The oil is pumped from an inlet at the side of the open tank through a line of commercial steel pipe having an inside diameter of 0.07792 m at the rate of 3.494 × 10⁻³ m³/s. The length of straight pipe is 122 m, and the pipe contains two elbows (90°) and a globe valve half open. The level of the liquid in the open tank is 20 m above the liquid level in the pressurized tank. The pump efficiency is 65%. Calculate the kW power of the pump.

Flow in an Annulus and Pressure Drop. Water flows in the annulus of a horizontal, concentric-pipe heat exchanger and is being heated from 40°C to 50°C in the exchanger, which has a length of 30 m of equivalent straight pipe. The flow rate of the water is 2.90 × 10⁻³ m³/s. The inner pipe is 1-in. schedule 40 and the outer is 2-in. schedule 40. What is the pressure drop? Use an average temperature of 45°C for bulk physical properties. Assume that the wall temperature is an average of 4°C higher than the average bulk temperature so that a correction can be made for the effect of heat transfer on the friction factor.

Derivation of Maximum Velocity for Isothermal Compressible Flow. Starting with Eq. (2.11-9), derive Eqs. (2.11-11) and (2.11-12) for the maximum velocity in isothermal compressible flow.

Pressure Drop in Compressible Flow. Methane gas is being pumped through a 305-m length of 52.5-mm-ID steel pipe at the rate of 41.0 kg/m² · s. The inlet pressure is p₁ = 345 kPa abs. Assume isothermal flow at 288.8 K.

Pressure Drop in Isothermal Compressible Flow. Air at 288 K and 275 kPa abs enters a pipe and is flowing in isothermal compressible flow in a commercial pipe having an ID of 0.080 m. The length of the pipe is 60 m. The mass velocity at the entrance to the pipe is 165.5 kg/m² · s. Assume 29 for the molecular weight of air. Calculate the pressure at the exit. Also, calculate the maximum allowable velocity that can be attained and compare with the actual.