Force on a Cylinder in a Wind Tunnel. Air at 101.3 kPa absolute and 25°C is flowing at a velocity of 10 m/s in a wind tunnel. A long cylinder having a diameter of 90 mm is placed in the tunnel and the axis of the cylinder is held perpendicular to the air flow. What is the force on the cylinder per meter length?

Wind Force on a Steam Boiler Stack. A cylindrical steam boiler stack has a diameter of 1.0 m and is 30.0 m high. It is exposed to a wind at 25°C having a velocity of 50 miles/h. Calculate the force exerted on the boiler stack.

Effect of Velocity on Force on a Sphere and Stokes' Law. A sphere having a diameter of 0.042 m is held in a small wind tunnel, where air at 37.8°C and 1 atm abs and various velocities is forced past it.

Drag Force on Bridge Pier in River. A cylindrical bridge pier 1.0 m in diameter is submerged to a depth of 10 m. Water in the river at 20°C is flowing past at a velocity of 1.2 m/s. Calculate the force on the pier.

Surface Area in a Packed Bed. A packed bed is composed of cubes 0.020 m on a side and the bulk density of the packed bed is 980 kg/m³. The density of the solid cubes is 1500 kg/m³.

Derivation for Number of Particles in a Bed of Cylinders. For a packed bed containing cylinders, where the diameter D of the cylinders is equal to the length h, do as follows for a bed having a void fraction ε:

Derivation of Dimensionless Equation for Packed Bed. Starting with Eq. (3.1-20), derive the dimensionless equation (3.1-21). Show all steps in the derivation.

Flow and Pressure Drop of Gases in Packed Bed. Air at 394.3 K flows through a packed bed of cylinders having a diameter of 0.0127 m and length the same as the diameter. The bed void fraction is 0.40 and the length of the packed bed is 3.66 m. The air enters the bed at 2.20 atm abs at the rate of 2.45 kg/m² · s based on the empty cross section of the bed. Calculate the pressure drop of air in the bed.

Flow of Water in a Filter Bed. Water at 24°C is flowing by gravity through a filter bed of small particles having an equivalent diameter of 0.0060 m. The void fraction of the bed is measured as 0.42. The packed bed has a depth of 1.50 m. The liquid level of water above the bed is held constant at 0.40 m. What is the water velocity ν' based on the empty cross section of the bed?

Mean Diameter of Particles in Packed Bed. A mixture of particles in a packed bed contains the following volume percent of particles and sizes: 15%, 10 mm; 25%, 20 mm; 40%, 40 mm; 20%, 70 mm. Calculate the effective mean diameter, D_pm, if the shape factor is 0.74.

Permeability and Darcy's Law. A sample core of porous rock obtained from an oil reservoir is 8 cm long and has a diameter of 2.0 cm. It is placed in a core holder. With a pressure drop of 1.0 atm, the water flow at 20.2°C through the core is measured as 2.60 cm³/s. What is the permeability in darcy?

Minimum Fluidization and Expansion of Fluid Bed. Particles having a size of 0.10 mm, a shape factor of 0.86, and a density of 1200 kg/m³ are to be fluidized using air at 25°C and 202.65 kPa abs pressure. The void fraction at minimum fluidizing conditions is 0.43. The bed diameter is 0.60 m and the bed contains 350 kg of solids.

Minimum Fluidization Velocity Using a Liquid. A tower having a diameter of 0.1524 m is being fluidized with water at 20.2°C. The uniform spherical beads in the tower bed have a diameter of 4.42 mm and a density of 1603 kg/m³. Estimate the minimum fluidizing velocity and compare with the experimental value of 0.02307 m/s found by Wilhelm and Kwauk (W5).

Fluidization of a Sand-Bed Filter. To clean a sand-bed filter, it is fluidized at minimum conditions using water at 24°C. The round sand particles have a density of 2550 kg/m³ and an average size of 0.40 mm. The sand has the properties given in Table 3.1-2.

Flow Measurement Using a Pilot Tube. A pitot tube is used to measure the flow rate of water at 20°C in the center of a pipe having an inside diameter of 102.3 mm. The manometer reading is 78 mm of carbon tetrachloride at 20°C. The pitot tube coefficient is 0.98.

Gas Flow Rate Using a Pitot Tube. The flow rate of air at 37.8°C is being measured at the center of a duct having a diameter of 800 mm by a pitot tube. The pressure-difference reading on the manometer is 12.4 mm of water. At the pitot-tube position, the static-pressure reading is 275 mm of water above 1 atm abs. The pitot-tube coefficient is 0.97. Calculate the velocity at the center and the volumetric flow rate of the air.

Pitot-Tube Traverse for Flow-Rate Measurement. In a pitot-tube traverse of a pipe having an inside diameter of 155.4 mm in which water at 20°C is flowing, the following data were obtained:

Metering Flow by a Venturi. A venturi meter having a throat diameter of 38.9 mm is installed in a line having an inside diameter of 102.3 mm. It meters water having a density of 999 kg/m³. The measured pressure drop across the venturi is 156.9 kPa. The venturi coefficient C_ν is 0.98. Calculate the gal/min and m³/s flow rate.

Use of a Venturi to Meter Water Flow. Water at 20°C is flowing in a 2-in. schedule 40 steel pipe. Its flow rate is measured by a venturi meter having a throat diameter of 20 mm. The manometer reading is 214 mm of mercury. The venturi coefficient is 0.98. Calculate the flow rate.

Metering of Oil Flow by an Orifice. A heavy oil at 20°C having a density of 900 kg/m³ and a viscosity of 6 cp is flowing in a 4-in. schedule 40 steel pipe. When the flow rate is 0.0174 m³/s, it is desired to have a pressure-drop reading across the manometer equivalent to 0.93 × 10⁵ Pa. What size orifice should be used if the orifice coefficient is assumed as 0.61? What is the permanent pressure loss?

Water Flow Rate in an Irrigation Ditch. Water is flowing in an open channel in an irrigation ditch. A rectangular weir having a crest length L = 1.75 ft is used. The weir head is measured as h₀ = 0.47 ft. Calculate the flow rate in ft³/s and m³/s.

Sizing a Flow Nozzle. A flow nozzle is to be sized for use in a pipe having an internal diameter of 1.25 in. to meter 0.60 ft³/min of water at 25°C. A pressure drop of 10.0 in. of water is to be used. Calculate the size of the flow nozzle and the permanent power loss in hp. Assume the coefficient C_n = 0.96.

Brake Horsepower of Centrifugal Pump. Using Fig. 3.3-2 and a flow rate of 60 gal/min, do as follows:

kW Power of a Fan. A centrifugal fan is to be used to take a flue gas at rest (zero velocity) and at a temperature of 352.6 K and a pressure of 749.3 mm Hg and discharge this gas at a pressure of 800.1 mm Hg and a velocity of 38.1 m/s. The volume flow rate of gas is 56.6 std m³/min of gas (at 294.3 K and 760 mm Hg). Calculate the brake kW of the fan if its efficiency is 65% and the gas has a molecular weight of 30.7. Assume incompressible flow.

Adiabatic Compression of Air. A compressor operating adiabatically is to compress 2.83 m³/min of air at 29.4°C and 102.7 kN/m² to 311.6 kN/m². Calculate the power required if the efficiency of the compressor is 75%. Also, calculate the outlet temperature.

(NPSH)_R forFeed to Distillation Tower. A feed rate of 200 gpm of a hydrocarbon mixture at 70°C is being pumped from a tank at 1 atm abs pressure to a distillation tower. The density of the feed is 46.8 lb_m/ft³ and its vapor pressure is 8.45 psia. The velocity in the inlet line to the pump is 3 ft/s and the friction loss between the tank and pump is 3.5 ft of fluid. The net positive suction head required is 6 ft.

Power for Liquid Agitation. It is desired to agitate a liquid having a viscosity of 1.5 × 10⁻³ Pa · s and a density of 969 kg/m³ in a tank having a diameter of 0.91 m. The agitator will be a six-blade open turbine having a diameter of 0.305 m operating at 180 rpm. The tank has four vertical baffles, each with a width J of 0.076 m. Also, W = 0.0381 m. Calculate the required kW. Use curve 2, Fig. 3.4-4.

Power for Agitation and Scale-Up. A turbine agitator having six flat blades and a disk has a diameter of 0.203 m and is used in a tank having a diameter of 0.61 m and height of 0.61 m. The width W = 0.0405 m. Four baffles are used having a width of 0.051 m. The turbine operates at 275 rpm in a liquid having a density of 909 kg/m³ and viscosity of 0.020 Pa · s.

Scale-down of Process Agitation System. An existing agitation process operates using the same agitation system and fluid as described in Example 3.4-1a. It is desired to design a small pilot unit with a vessel volume of 2.0 liters so that effects of various process variables on the system can be studied in the laboratory. The rates of mass transfer appear to be important in this system, so the scale-down should be on this basis. Design the new system specifying sizes, rpm, and kW power.

Anchor Agitation System. An anchor-type agitator similar to that described for Eq. (3.4-3) is to be used to agitate a fluid having a viscosity of 100 Pa · s and a density of 980 kg/m³. The vessel size is D_t = 0.90 m and H = 0.90 m. The rpm is 50. Calculate the power required.

Design of Agitation System. An agitation system is to be designed for a fluid having a density of 950 kg/m³ and viscosity of 0.005 Pa · s. The vessel volume is 1.50 m³ and a standard six-blade open turbine with blades at 45°C (curve 3, Fig. 3.4-4) is to be used with D_a/W = 8 and D_a/D_t = 0.35. For the preliminary design a power of 0.5 kW/m³ volume is to be used. Calculate the dimensions of the agitation system, rpm, and kW power.

Scale-Up of Mixing Times for a Turbine. For scaling up a turbine-agitated system, do as follows:

Mixing Time in a Turbine-Agitated System. Do as follows:

Effect of Viscosity on Mixing Time. Using the same conditions for the turbine mixer as in Example 3.4-4, part (a), except for a viscous fluid with a viscosity of 100 Pa · s (100 000 cp), calculate the mixing time. Compare this mixing time with that for the viscosity of 10 cp. Also calculate the power per unit volume.

Mixing Time in a Helical Mixer. A helical mixer with an agitator pitch/tank diameter = 1.0 and with D_t = 1.83 m and D_a/D_t = 0.95 is being used to agitate a viscous fluid having a viscosity of 200 000 cp and a density of 950 kg/m³. The value of N = 0.3 rev/s. Calculate the mixing time and the power per unit volume.

Pressure Drop of Power-Law Fluid, Banana Purée. A power-law biological fluid, banana purée, is flowing at 23.9°C, with a velocity of 1.018 m/s, through a smooth tube 6.10 m long having an inside diameter of 0.01267 m. The flow properties of the fluid are K = 6.00 N · s^0.454/m² and n = 0.454. The density of the fluid is 976 kg/m³.

Pressure Drop of Pseudoplastic Fluid. A pseudoplastic power-law fluid having a density of 63.2 lb_m/ft³ is flowing through 100 ft of pipe having an inside diameter of 2.067 in. at an average velocity of 0.500 ft/s. The flow properties of the fluid are K = 0.280 lb_f · sⁿ/ft² and n = 0.50. Calculate the generalized Reynolds number and also the pressure drop, using the friction factor method.

Turbulent Flow of Non-Newtonian Fluid, Applesauce. Applesauce having the flow properties given in Table 3.5-1 is flowing in a smooth tube having an inside diameter of 50.8 mm and a length of 3.05 m at a velocity of 4.57 m/s.

Agitation of a Non-Newtonian Liquid. A pseudoplastic liquid having the properties n = 0.53, K = 26.49 N · s^n'/m², and ρ = 975 kg/m³ is being agitated in a system such as in Fig. 3.5-4 where D_t = 0.304 m, D_a = 0.151 m, and N = 5 rev/s. Calculate μ_a, , and the kW power for this system.

Velocity Profile of a Bingham Plastic Fluid. For the conditions of Example 3.5-3, do as follows:

Pressure Drop for Bingham Plastic Fluid. A Bingham plastic fluid has a value of τ₀ = 1.2 N/m² and a viscosity μ = 0.4 Pa · s. The fluid is flowing at 5.70 × 10⁻⁵ m³/s in a pipe 2.5 m long with an internal diameter of 3.0 cm. Calculate the pressure drop (p₀ − p_L) in N/m² and r₀. (Hint: This is a trial-and-error solution. As a first trial, assume τ₀ = 0.)

Flow Properties of a Non-Newtonian Fluid from Rotational Viscometer Data. Following are data obtained on a fluid using a Brookfield rotational viscometer:

Equation of Continuity in a Cylinder. Fluid having a constant density ρ is flowing in the z direction through a circular pipe with axial symmetry. The radial direction is designated by r.

Change of Coordinates for Continuity Equation. Using the general equation of continuity given in rectangular coordinates, convert it to Eq. (3.6-27), which is the equation of continuity in cylindrical coordinates. Use the relationships in Eq. (3.6-26) to do this.

Combining Equations of Continuity and Motion. Using the continuity equation and the equations of motion for the x, y, and z components, derive Eq. (3.7-13).

Average Velocity in a Circular Tube. Using Eq. (3.8-17) for the velocity in a circular tube as a function of radius r,

Equation 3.8-17

Equation 3.8-19

Laminar Flow in a Cylindrical Annulus. Derive all the equations given in Example 3.8-4 showing all the steps. Also, derive the equation for the average velocity ν_{z av}. Finally, integrate to obtain the pressure drop from z = 0 for p = p₀ to z = L for p = p_L.

Velocity Profile in Wetted-Wall Tower. In a vertical wetted-wall tower, the fluid flows down the inside as a thin film δ m thick in laminar flow in the vertical z direction. Derive the equation for the velocity profile ν_z, as a function of x, the distance from the liquid surface toward the wall. The fluid is at a large distance from the entrance. Also, derive expressions for ν_{z av} and ν_{z max}. (Hint: At x = δ, which is at the wall, ν_z = 0. At x = 0, the surface of the flowing liquid, ν_z = v_{z max}.) Show all steps.

Velocity Profile in Falling Film and Differential Momentum Balance. A Newtonian liquid is flowing as a falling film on an inclined flat surface. The surface makes an angle β with the vertical. Assume that in this case the section being considered is sufficiently far from both ends that there are no end effects on the velocity profile. The thickness of the film is δ. The apparatus is similar to Fig. 2.9-3 but is not vertical. Do as follows:

Velocity Profiles for Flow Between Parallel Plates. In Example 3.8-2 a fluid is flowing between vertical parallel plates with one plate moving. Do as follows:

Conversion of Shear Stresses in Terms of Fluid Motion. Starting with the x component of motion, Eq. (3.7-10), which is in terms of shear stresses, convert it to the equation of motion, Eq. (3.7-36), in terms of velocity gradients, for a Newtonian fluid with constant ρ and μ. Note that (∇·ν) = 0 in this case. Also, use of Eqs. (3.7-14)-(3.7-20) should be considered.

Derivation of Equation of Continuity in Cylindrical Coordinates. By means of a mass balance over a stationary element whose volume is r Δr Δθ Δz, derive the equation of continuity in cylindrical coordinates.

Flow between Two Rotating Coaxial Cylinders. The geometry of two coaxial cylinders is the same as in Example 3.8-5. In this case, however, both cylinders are rotating, the inner rotating with an annular velocity of ω₁ and the outer at ω₂. Determine the velocity and the shear-stress distributions using the differential equation of momentum.

Potential Function. The potential function ϕ for a given flow situation is ϕ = C(x² - y²), where C is a constant. Check to see if it satisfies Laplace's equation. Determine the velocity components ν_x and ν_y.

Determining the Velocities from the Potential Function. The potential function for flow is given as ϕ = Ax + By, where A and B are constants. Determine the velocities ν_x and ν_y.

Stream Function and Velocity Vector. Flow of a fluid in two dimensions is given by the stream function ψ = Bxy, where B = 50s^-1 and the units of x and y are in cm. Determine the value of v_x, v_y, and the velocity vector at x = 1 cm and y = 1 cm.

Stream Function and Potential Function. A liquid is flowing parallel to the x axis. The flow is uniform and is represented by ν_x = U and v_y = 0.

Velocity Components and Stream Function. A liquid is flowing in a uniform manner at an angle of β with respect to the x axis. Its velocity components are ν_x = U cos β and ν_y = U sin β. Find the stream function and the potential function.

Flow Field with Concentric Streamlines. The flow of a fluid that has concentric streamlines has a stream function represented by ψ = 1/(x² + y²). Find the components of velocity ν_x and ν_y. Also, determine if the flow is rotational and, if so, determine the vorticity, 2ω_z.

Potential Function and Velocity Field. In Example 3.9-2 the velocity components were given. Show if a velocity potential exists and, if so, also determine ϕ.

Euler's Equation of Motion for an Ideal Fluid. Using the Euler equations (3.9-2)–(3.9-4) for ideal fluids with constant density and zero viscosity, obtain the following equation:

Plot of Streamlines. For Ex. (3.9-3) plot the streamlines for ψ = 0 and ψ = 2 when x > 0.

Stream Function for Two-Dimensional Flow. Find the stream function of the two-dimensional flow with constant density where ν_x = U[(y/L)² − (y/L)] and ν_y = 0. The flow is between two parallel plates spaced L distance apart. Also plot the velocity profile of ν_x versus y.

Stream and Potential Functions and Plots of These Functions. Determine the stream function ψ when the velocities are ν_x = 2x and ν_y = -2y. Also determine the potential function ϕ. Plot the stream function for ψ = 1 and ψ = 2. Also plot the equal potential lines for ϕ = 1 and ϕ = 4.

Velocity Field from Stream Function. Given the stream function ψ = 3x² + 2y², calculate ν_x and ν_y and draw the streamlines for ψ = 1 and ψ = 2.

Streamline from Velocities. The velocity v_x = x² and v_y = -2xy. Determine the stream function ψ.

Stream Function from Velocity Potential. Find the stream function ψ from the velocity potential ϕ = UL[(x/L)³ − (3xy²)/L³], where U and L are constants. Also, find ν_x and ν_y.

Laminar Boundary Layer on Flat Plate. Water at 20°C is flowing past a flat plate at 0.914 m/s. The plate is 0.305 m wide.

Air Flow Past a Plate. Air at 294.3 K and 101.3 kPa is flowing past a flat plate at 6.1 m/s. Calculate the thickness of the boundary layer at a distance of 0.3 m from the leading edge and the total drag for a 0.3-m-wide plate.

Boundary-Layer Flow Past a Plate. Water at 293 K is flowing past a flat plate at 0.5 m/s. Do as follows:

Transition Point to Turbulent Boundary Layer. Air at 101.3 kPa and 293 K is flowing past a smooth, flat plate at 100 ft/s. The turbulence in the air stream is such that the transition from a laminar to a turbulent boundary layer occurs at N_Re,L = 5 × 10⁵.

Dimensional Analysis for Flow Past a Body. A fluid is flowing external to a solid body. The force F exerted on the body is a function of the fluid velocity ν, fluid density ρ, fluid viscosity μ, and a dimension of the body L. By dimensional analysis, obtain the dimensionless groups formed from the variables given. (Note: Use the M, L, t system of units. The units of F are ML/t². Select ν, ρ, and L as the core variables.)

Dimensional Analysis for Bubble Formation. Dimensional analysis is to be used to correlate data on bubble size with the properties of the liquid when gas bubbles are formed by a gas issuing from a small orifice below the liquid surface. Assume that the significant variables are bubble diameter D, orifice diameter d, liquid density ρ, surface tension σ in N/m, liquid viscosity μ, and g. Select d, ρ, and g as the core variables.