Questions and Answers of Introduction To Chemical Engineering Fluid Mechanics

A flow in which the velocity changes somewhere from subsonic (Ma < 1) to supersonic (Ma > 1), or the reverse, is called transonic. Equation (12.5-9) shows that this cannot occur in a
Sounds are the result of small changes in pressure. To predict the speed at which such disturbances travel, assume that P, ρ, v, and T in a fluid are each perturbed slightly from their constant
Fatalities have resulted from the careless handling of compressed-gas cylinders, both during transportation and during use. The objective of this problem is to illustrate the consequences of a large,
Assume that an air tank at T1 = 27 °C and P1 = 2 atm discharges into the atmosphere through a steel pipe of diameter D = 2.5 cm. The tank is large enough that T1 and P1 remain nearly constant. The
Suppose that the pressure in a car tire is discovered to be 20 psig, which is well below the 30 psig recommended for that type of tire. A nearby service station provides compressed air at P1 = 50
A rigid steel tank of finite volume V is filled with compressed air and at t = 0 a valve is to be opened to allow the tank to discharge into the atmosphere. It is desired to predict the resulting
A large balloon is to be inflated from radius a1 to a2 using a piston pump that delivers air at a constant mass flow rate w. If the wall tension is proportional to how much the radius a exceeds a0,
Figure P12.13 shows a conical diffuser, a tube segment that tapers outward linearly in the direction of flow. Its local diameter is D(z) = D1 + (D2 − D1)(z/L). The rate of diameter increase
A tank is to be added to a refinery to increase the storage capacity for a product such as home heating oil. The oil level in the new tank may be as much as a height H above that in the tank leading
All of the liquid in a tank is to be transferred to another tank of the same size. As shown in Fig. P12.11, both are of radius R and at first they are slightly less than half full, with a liquid
To boost production in a chemical plant without enlarging any of the tubular reactors, several might be arranged in parallel, as shown in Fig. P12.10. Each might be an open tube of diameter DR and
In distribution manifolds it is often desired to have the same flow rate QB in each branch. If the branches are identical, the inlet pressure at each would need to be the same. There are at least two
The setup in Fig. P12.8 was used to help characterize porous membranes employed in diffusion studies (Epstein, 1979). The membrane hydraulic permeability km obtained using this apparatus was used to
It is desired to infer the value of the loss coefficient of the valve–spigot combination in a household bathtub. As shown in Fig. P12.7, leading from the underground water main to the house is pipe
When it is necessary to reverse the flow in the system in Fig. P12.5, the pump is bypassed by a short section of pipe. The valves involved have negligible resistance when open. If the downward flow
Suppose that water is to be pumped upward at a volume flow rate Q from one reservoir to another, as in Fig. P12.5. The difference in the water levels is nearly constant at H. The connection is a
For the siphon in Fig. P11.13, suppose that d = 0.10 m, HA = 0.50 m, HB = 1.25 m, HC = 1.00 m, and the liquid is water at 20 °C. Assume that the tube is smooth and has a total length L = 5.0 m.(a)
A nozzle increases the fluid velocity by tapering inward, whereas a diffuser decreases it by tapering outward. Figure P12.3 shows a rounded nozzle followed by a diffuser whose diameter varies from D1
It is desired to estimate the entrance length LE for laminar flow at large Re in a parallel-plate channel with wall spacing 2H and mean velocity U. In such a channel the velocity profile develops
Consider air flow at a mean velocity of 5.0 m/s through a smooth pipe of diameter 0.10 m and length 10 m, at 27 °C. The air comes from a tank at atmospheric pressure.(a) If there is a sudden
An underground plastic pipe is to be used to carry water away from a building. As shown in Fig. P11.15, the pipe of diameter D will run downward and have both ends open to the air (pressure P0). The
A nominal 1⁄2 hp electric sump pump for household use comes with the following data:where GPM is gallons per minute. A typical installation is shown in Fig. P11.14. The average water depth at the
A siphon may be used to draw a liquid above its level in an open tank or channel and discharge it below that level. Once initiated, the flow is sustained by the height difference. The simplest
The Pitot tube, invented by the French engineer Henri Pitot (1695–1771) to measure fluid velocities, has various forms. Pitot tubes are used to monitor the airspeed of commercial jets, as
Puget Power Plant No. 1 at Snoqualmie Falls, WA is a hydroelectric facility in which water is directed straight down through a pair of steel penstocks located upstream, passes through a set of
When the velocity field in a conduit is completely known, Ev can be calculated directly from Eqs. (11.4-3) and (11.4-4). Confirm that, for fully developed laminar flow of an incompressible fluid in a
Suppose that an upward jet of water strikes an unsupported, horizontal plate of mass mp, as shown in Fig. P11.9. The diameter of the turbulent jet is D(z) and its velocity is v(z). The diameter and
Syringe pumps such as in Problem 2.4 are limited by the force that the linear motor can apply to the plunger. To guide the design of a new line of pumps, it is desired to predict the forces for
A laboratory water tank designed to study wave motion is equipped with an end wall that can be moved inward at a desired constant velocity. As shown in Fig. P11.7(a), a wall velocity U creates a wave
A jet ejector is a device without moving parts that can be used as a pump. As shown in Fig. P11.6, on the upstream side are concentric tubes of diameters D and λD, where λ < 1. A liquid at a high
The objective is to estimate the drag on a cylinder from the velocity reduction measured in its wake. The cylinder of diameter d and length L (≫ d) is perpendicular to the approaching flow. Figure
The drag on an object can be inferred from the velocity reduction in its wake. Figure P11.4 depicts the boundary layer and wake on one side of a flat plate. The surfaces S1 through S4 enclose a
Figure P11.3 shows two nozzles, one straight and the other curved, in which the diameter is reduced from D1 to D2. Suppose that water enters each at a mean velocity U and pressure P1 and exits to the
Water clocks were used in Egypt and other parts of the ancient world to monitor the daily passage of time. In the version shown in Fig. P11.2 an open container of varying radius R(z) drains through a
Suppose that the tank in Fig. 11.3 empties through a hole of diameter do near its bottom, instead of draining through a long pipe. The opening is smooth and rounded on the inside.Figure 11.3(a) If
Axisymmetric, submerged turbulent jets are conical on average (Example 10.6- 2). As shown in Fig. P10.12, let β be the angle at which the velocity at any z is half-maximal. It has been found
Derive OM estimates of the half-thickness (δ), maximum velocity (vm), and volume flow rate per unit width (q) of a turbulent jet that passes from a slit into a large volume of the same fluid. Assume
As discussed in Section 2.5, at very high Re the friction factor in a tube becomes independent of Re and is affected only by the relative roughness of the wall. In this “fully rough” regimewhere
A weakness of both the power-law and logarithmic velocity profiles is that they giveat y = R. Because symmetry requires that the shear stress vanish at the centerline, this implies that ε → 0 as y
Consider a parallel-plate channel with wall spacing 2H. If the logarithmic velocity profile given by Eq. (10.5-16) is assumed to apply throughout the channel and if Re is based on the hydraulic
When applied to tube flow, the 1⁄7-power velocity profile can be written aswhere C is a constant. This is analogous to Eq. (10.6-4) for a turbulent boundary layer. Show that this expression implies
As shown in Fig. 10.5, the velocity profile in a tube at Re = 4.1 × 104 is represented quite well by a 1⁄7-power relationship. The objective is to see what this implies about the magnitude of the
The velocity profile near a wall is sometimes represented as(a) Plot this piecewise function and, based on the data in Fig. 10.4, comment on its accuracy.(b) Derive and plot the function ε(y+)⁄ν
In addition to the time-smoothed axial velocities for tube flow in Fig. 10.5, Laufer (1954) measured the fluctuations in all three velocity components. Figure 10.5Some of those data are shown in
Figure P.10.3 provides an instantaneous view of a submerged, circular jet of water. The water coming from the nozzle at the top was visualized using laserinduced fluorescence. In the lower part of
It has been suggested that, when turbulence is intense enough to create eddies that are comparable to or smaller than the size of a cell, the fluctuating shear stresses may cause cellular damage. For
Calculate the velocity, length, and time scales of the largest and smallest eddies in 20 °C water that is flowing at a mean velocity of 1.0 m/s in a smooth pipe with a diameter of 10 cm.
(a) Using the integral method with profile (3) of Table 9.2, determine the boundary-layer thickness for flow past a 90° wedge, where u(x) = ax1/3.(b) With the wall shear stress written asa numerical
(a) Using the integral method with profile (3) of Table 9.2, determine the boundary-layer thickness for planar stagnation flow, where u(x) = ax.(b) With the wall shear stress written asa numerical
Bubbles at large Re are typically shaped as spherical caps, as shown in Fig. P9.14. The trailing surface is more irregular than what is depicted, but the leading surface is almost a perfect spherical
Rotating-disk electrodes are widely used to study electrochemical kinetics. As depicted in Fig. P9.12, a metal (e.g., platinum) electrode is embedded in the end of an insulating rod of radius R,
Applying suction on at least part of the surface of an object has been used to reduce drag. This can delay or prevent flow separation and also reduce the wall shear stress. This problem focuses on
The fluid moving along a flat plate gradually slows, as shown in Example 9.3-1. That deceleration creates a positive vy at the outer edge of the boundary layer. It is desired to compare the average
It is desired to determine the general features of boundary layers in fluids that obey Eq. (6.5-17). The velocity scale is U and the object dimension is L.(a) Show that if δ⁄L ≪ 1, the momentum
Analogous to the planar jet in Example 9.3-2 is one that emerges from a circular opening. The boundary-layer momentum equation for an axisymmetric flow isand the kinematic momentum is defined now
In any long pipe or other conduit of constant cross-section the velocity profile eventually becomes fully developed. The distance from the inlet at which this occurs is the entrance length, LE.
Suppose that a tube is immersed in a liquid and that air flow through the tube creates a bubble at its end, as shown in Fig. P9.7. Beginning at t = 0, when the bubble radius is R0, the volume flow
A uniform coating of liquid on a flat surface can be created by rapidly spinning the solid substrate after some liquid is applied. It is found that such films level quickly and then gradually become
An object moving through a static fluid sets part of the fluid in motion, and if it accelerates, the neighboring fluid must accelerate as well. The acceleration of the fluid tends to slow the
Circular jets directed at one another have been used to study gas-phase combustion. Typically, one jet contains a fuel such as methane and the other contains oxygen. Upon ignition, a stationary flame
An axisymmetric stagnation flow results from directing a circular stream perpendicular to a planar surface, which deflects the fluid outward radially. Suppose that the surface is at z = 0, and the
Consider irrotational flow at velocity U past a half-cylinder of radius R and length L, as in Fig. P9.2. A numerical solution for the velocity potential would be needed to obtain precise results for
Consider irrotational flow at velocity U past a stationary sphere of radius R, with coordinates as in Fig. 8.7.(a) As in Example 9.2-1, the velocity potential is a separable function. Show thatand
Walker and Beebe (2002) demonstrated a way to pump liquid through small tubes without the need for moving parts, and their idea has been applied to pointof- care diagnostics and other microfluidic
Injection molding is a manufacturing process in which a mold is filled with a liquid that solidifies after cooling. The mold is then opened and the product removed. Such molds have openings for
Liquid uptake by porous media is typically driven by surface tension. Suppose that a long, horizontal pore of radius a is initially filled with air, and that at t = 0 one end is immersed in a liquid.
The interstitium of a body tissue, which is the space outside blood vessels and cells, contains fibers of collagen, glycosaminoglycans, and other biopolymers that are surrounded by water. Fluid
Consider creeping flow at velocity U past a long cylinder of radius R and length L that is perpendicular to the approaching fluid, as in Fig. P6.6. Unlike the case of flow past a sphere, there are no
Dropping-mercury electrodes have been used extensively in analytical chemistry and electrochemical research. As shown in Fig. P8.11, a small mercury drop of radius a is suspended from a tube in an
The arrangement in Fig. P8.10 is well suited for determining the viscosity of small liquid samples. An inverted cone of radius R and small angle β is brought into contact with a pool of the liquid
Suppose that an extremely viscous liquid fills a space of thickness H between two disks of radius R, as shown in Fig. P8.9. The upper disk rotates at a constant angular velocity ω and the lower one
(a) Use the velocity and pressure in Problem 8.7 to calculate the drag coefficient for a bubble in creeping flow. How do the pressure and viscous contributions compare?(b) Derive an expression for
Consider creeping flow of a liquid at velocity U relative to a spherical bubble of radius R, with coordinates as in Fig. 8.7. Show that the velocity and pressure in the surrounding liquid arewhere μ
Suppose that a small sphere of radius R rotates at angular velocity ω in a large container of otherwise static fluid. The radius and velocity are such that With the sphere rotating about the z
In blade coating, which is widely used for sheet materials such as paper, a substrate passes through an opening of fixed height as it is pulled from a pool of the coating liquid. The opening can have
A manufacturer is planning to coat candy centers with thin films of chocolate. This is to be done by flowing molten chocolate onto a solid center by gravity, and then cooling quickly to solidify the
Randall and Doyle (2005) discovered that diffusion of water through the poly(dimethylsiloxane) (PDMS) that formed the top of a microfluidic channel produced an observable flow. Their device is shown
Consider a small, permeable tube of radius R and length L that is closed at one end, as in Fig. P8.2. Suppose that a high external pressure creates a constant inward velocity vw at the wall, but with
It is desired to predict the effects of imperfections that might occur in fabricating a parallel-plate channel. One possibility is a sinusoidal variation in the halfheight,as depicted in Fig. P5.2.
Whole blood consists of a concentrated suspension of cells in plasma. The cells are mainly red blood cells (RBCs), which are flexible, rounded disks about 2 μm thick and 8 μm in diameter. Their
Suppose that temperature variations within the parallel-plate channel of Fig. 7.1 noticeably alter the viscosity. In particular, assume thatwhere α > 0. That is, there is a linear temperature
House paint is formulated so that, if applied properly, it will not run down a vertical surface before it dries. It can be modeled as a Bingham fluid.(a) Suppose that a latex paint with a density
Suppose that an elongated bubble is rising at its terminal velocity U in an open, liquid-filled tube, as in Fig. P7.14. The bubble is shaped as a cylinder of radius a and length L with hemispherical
A method for determining the viscosity of a liquid is to measure the terminal velocity of a cylinder settling in a tube. As shown in Fig. P7.13, the cylinder radius is a, the tube radius is b, and
Suppose that a long and shallow cavity of length L and depth H is filled with liquid, as in Fig. P7.12. If something at the top surface causes the liquid there to move from left to right (i.e., if vx
In the coating process in Fig. P7.11 a liquid is extruded through a slot in a die onto a solid sheet moving horizontally at velocity V. The pressure at the slot exit is P1 and there is a constant gap
In the coating process in Fig. P7.11 a liquid is extruded through a slot in a die onto a solid sheet moving horizontally at velocity V. The pressure at the slot exit is P1 and there is a constant gap
Suppose that a liquid is pumped upward at constant volume flow rate Q through a tube of outer radius R, as shown in Fig. P7.9. The liquid overflows the top and runs down the outside. At a certain
Simultaneous flow of two or more thin layers of liquid is needed to form various composite coatings. Suppose that immiscible liquids 1 and 2 flow down a surface that is inclined at an angle β
Suppose that an open container of radius R is filled with liquid to an initial height h0. It is then rotated at a constant angular velocity ω, as shown in Fig. P7.7. After an initial transient, the
It is desired to predict the Darcy permeability k of a porous material that consists of many cylindrical fibers arranged in parallel. For simplicity, assume that long fibers of radius R are equally
An alternative to the no-slip boundary condition at a stationary solid, proposed by Navier in the 1820s and revived in recent years for modeling liquid flows in extremely small channels, is to
Consider fully developed flow at mean velocity U in a conduit with the elliptical cross-section in Fig. P7.4. The semi-axes are a and b and the length is L. The location of the wall is governed by(a)
An exact solution for the velocity in a conduit is usually obtainable only when each wall is a coordinate surface. An interesting exception is a duct with an equilateral triangular cross-section, as
Consider pressure-driven flow in an annular channel. The inner and outer radii are as in Fig. P7.1, but now both cylinders are stationary and there is a mean fluid velocity U in the z direction. The
In a Couette viscometer the liquid to be studied fills the annular space between two cylinders, as shown in Fig. P7.1. The inner and outer radii are κR and R, respectively. The viscosity is
Use the velocity field in Problem 5.6(b) to calculate ℘(r, z) for flow at small Re between porous and solid disks. You may assume that all inertial terms in the Navier–Stokes equation are
Use the velocity field in Problem 5.4 to show thatfor flow at small Re (creeping flow) past a solid sphere. You may assume that all inertial terms in the Navier–Stokes equation are negligible for
Consider flow at velocity U relative to a solid cylinder of radius R and length L whose axis is perpendicular to the approaching fluid, as in Fig. P6.6.(a) Show how to calculate the drag if ℘(R,
Figure P6.4 depicts pressure-driven flow in a horizontal parallel-plate channel with wall spacing H, length L, and width W (not shown). This is called plane Poiseuille flow. Assuming that the flow is

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