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applied fluid mechanics
Questions and Answers of
Applied Fluid Mechanics
A slurry containing $40 %$ solids by volume is delivered to a rotary drum filter, which is $4 \mathrm{ft}$ in diameter and $6 \mathrm{ft}$ long and operates at a vacuum of $25 \mathrm{in}$.
A slurry is to be filtered with a rotary drum filter that is $5 \mathrm{ft}$ in diameter, $8 \mathrm{ft}$ long, rotates once every $10 \mathrm{~s}$ and has $20 %$ of its surface immersed in the
A rotary drum filter is to be installed in your plant. You run a test in the lab on the slurry to be filtered using a $0.1 \mathrm{ft}^{2}$ sample of the filter medium at a constant pressure drop of
A slurry of $\mathrm{CaCO}_{3}$ in water at $25^{\circ} \mathrm{C}$ containing $20 %$ solids by weight is to be filtered in a plateand-frame filter. The slurry and filter medium are tested in a
An algal sludge is to be clarified by filtering. A lab test is run on the sludge using an area A of the filter medium. At a constant pressure drop of $40 \mathrm{kN} / \mathrm{m}^{2}$, a plot of the
A slurry containing $0.2 \mathrm{~kg}$ of solids per $\mathrm{kg}$ water is filtered through a rotary drum filter, operating at a pressure difference of $65 \mathrm{kN} / \mathrm{m}^{2}$. The drum is
You want to filter an aqueous slurry using a rotary drum filter, at a total rate (of filtrate) of $10,000 \mathrm{gal} / \mathrm{day}$. The drum rotates at a rate of $0.2 \mathrm{rpm}$, with $25 %$
You want to use a plate-and-frame filter to filter an aqueous slurry at a rate of $1.8 \mathrm{~m}^{3} / 8 \mathrm{~h}$ day. The filter frames are square, with a length on each side of $0.45
An aqueous slurry is filtered in a plate-and-frame filter, which operates at a constant $\Delta P$ of $100 \mathrm{psi}$. The filter consists of 20 frames, each of which have a projected area per
You must transport a sludge product from an open storage tank to a separations unit at $1 \mathrm{~atm}$, through a 4 in. sch 40 steel pipeline that is $2000 \mathrm{ft}$ long, at a rate of $250
Consider a dilute aqueous slurry containing solid particles with diameters from 0.1 to $1000 \mu \mathrm{m}$ and a density of $2.7 \mathrm{~g} / \mathrm{cc}$, flowing at a rate of $500
Calculate the flow rate of water (in gpm) required to fluidize a bed of 1/16 in. diameter lead shot $(S G=11.3)$. The bed is $1 \mathrm{ft}$ in diameter, $1 \mathrm{ft}$ deep, and has a porosity of
Calculate the range of water velocities that will fluidize a bed of glass spheres $(S G=2.1)$ if the sphere diameter is (a) $2 \mathrm{~mm}$, (b) $1 \mathrm{~mm}$, and (c) $0.1 \mathrm{~mm}$.
A coal gasification reactor operates with particles of $500 \mu \mathrm{m}$ diameter and a density of $1.4 \mathrm{~g} / \mathrm{cm}^{3}$. The gas may be assumed to have properties of air at
A bed of coal particles, $2 \mathrm{ft}$ in diameter and $6 \mathrm{ft}$ deep, is fluidized using a hydrocarbon liquid with a viscosity $15 \mathrm{cP}$ and a density of $0.9 \mathrm{~g} /
A catalyst having spherical particles with $d_{p}=50 \mu \mathrm{m}$ and $ho_{s}=1.65 \mathrm{~g} / \mathrm{cm}^{3}$ is used to contact a hydrocarbon vapor in a fluidized reactor at $900^{\circ}
A fluid bed reactor contains catalyst particles with a mean diameter of $500 \mu \mathrm{m}$ and a density of $2.5 \mathrm{~g} / \mathrm{cm}^{3}$. The reactor feed has properties equivalent to
Water is pumped upward through a bed of $1 \mathrm{~mm}$ diameter iron oxide particles $(S G=5.3)$. If the bed porosity is 0.45 , over what range of superficial water velocity will the bed be
A fluidized bed combustor is $2 \mathrm{~m}$ in diameter and is fed with air at $250^{\circ} \mathrm{F}, 10 \mathrm{psig}$, at a rate of $2000 \mathrm{scfm}$. The coal has a density of $1.6
A fluid bed incinerator, $3 \mathrm{~m}$ in diameter and $0.56 \mathrm{~m}$ high, operates at $850^{\circ} \mathrm{C}$ using a sand bed. The sand density is $2.5 \mathrm{~g} / \mathrm{cm}^{3}$, and
Determine the range of flow rates (in gpm) that will fluidize a bed of $1 \mathrm{~mm}$ cubic silica particles $(S G=2.5)$ with water. The bed is 10 in. in diameter, 15 in. deep.
Determine the range of velocities over which a bed of granite particles $\left(S G=3.5, \mathrm{a}_{\mathrm{s}}=0.012 \mu \mathrm{m}^{-1}\right.$, $\psi=0.8, \mathrm{~d}=0.6 /
Calculate the velocity of water that would be required to fluidize spherical particles with $S G=1.6$ and a diameter of $1.5 \mathrm{~mm}$, in a tube with a diameter of $10 \mathrm{~mm}$. Also,
You want to fluidize a bed of solid particles using water. The particles are cubical, with a length on each side of $1 / 8$ in., and a SG of 1.2 .(a) What is the sphericity factor for these
Solid particles with a density of $1.4 \mathrm{~g} / \mathrm{cm}^{3}$ and a diameter of $0.01 \mathrm{~cm}$ are fed from a hopper into a line where they are mixed with water, which is draining by
A sludge is clarified in a thickener, which is $50 \mathrm{ft}$ in diameter. The sludge contains $35 %$ solids by volume $(S G=1.8)$ in water, with an average particle size of $25 \mu \mathrm{m}$.
In a batch thickener, an aqueous sludge containing $35 %$ by volume of solids $(S G=1.6)$ with an average particle size of $50 \mu \mathrm{m}$ is allowed to settle. The sludge is fed to the settler
Ground coal is slurried with water in a pit, and the slurry is pumped out of the pit at a rate of $500 \mathrm{gpm}$ with a centrifugal pump and into a classifier. The classifier inlet is $50
You want to concentrate a slurry from $5 %$ (by vol.) solids to $30 %$ (by vol.) in a thickener. The solids density is $200 \mathrm{lb}_{\mathrm{m}} / \mathrm{ft}^{3}$ and that of the liquid is $62.4
You must determine the maximum feed rate that a thickener can handle to concentrate a waste suspension from $5 %$ solids by volume to $40 %$ solids by volume. The thickener has a diameter of $40
Determine the weight of $1 \mathrm{~g}$ mass at sea level in units of:(a) dynes(b) $\mathrm{lb}_{\mathrm{f}}$(c) $\mathrm{g}_{\mathrm{f}}$(d) poundals
One cubic foot of water weighs $62.4 \mathrm{lb}_{\mathrm{f}}$ under conditions of standard gravity.(a) What is its weight in dynes, poundals, and $\mathrm{g}_{\mathrm{f}}$ ?(b) What is its density
The acceleration due to gravity on the moon is about $5.4 \mathrm{ft} / \mathrm{s}^{2}$. If your weight is $150 \mathrm{lb}_{\mathrm{f}}$ on the earth:(a) What is your mass on the moon, in slugs?(b)
You weigh a body with a mass $m$ on an electronic scale, which is calibrated with a known mass.(a) What does the scale actually measure, and what are its dimensions?(b) If the scale is calibrated in
Explain why the gravitational "constant" $(g)$ is different at Reykjavik, Iceland, than it is at Quito, Ecuador. At which location is it greatest, and why? If you could measure the value of $g$ at
You have purchased a $5 \mathrm{oz}$ bar of gold ( $100 %$ pure), at a cost of $\$ 400 / \mathrm{oz}$. Because the bar was weighed in air, you conclude that you got a bargain, because its true mass
You purchased $5 \mathrm{oz}$ of gold in Quito, Ecuador $\left(g=977.110 \mathrm{~cm} / \mathrm{s}^{2}\right)$, for $\$ 400 / \mathrm{oz}$. You then took the gold and the same spring scale on which
Calculate the pressure at a depth of 2 miles below the surface of the ocean. Explain and justify any assumptions you make. The physical principle that applies to this problem can be described by the
(a) Use the principle in Problem 8 to calculate the pressure at a depth of $1000 \mathrm{ft}$ below the surface of the ocean (in psi, $\mathrm{Pa}$, and atm). Assume that the ocean water density is
The following formula for the pressure drop through a valve was found in a design manual:\[h_{L}=\frac{522 K q^{2}}{d^{4}}\]where$h_{L}$ is the "head loss" in feet of fluid flowing through the
When the energy balance on the fluid in a stream tube is written in the following form, it is known as the Bernoulli
Determine the value of the gas constant, $R$, in units of $\mathrm{ft}^{3} \mathrm{~atm} / \mathrm{lb} \mathrm{mol}{ }^{\circ} \mathrm{R}$ ), starting with the value of the standard molar volume of a
Calculate the value of the Reynolds number for sodium flowing at a rate of $50 \mathrm{gpm}$ through a tube with a $1 / 2$ in. ID, at $400^{\circ} \mathrm{F}$.
The conditions at two different positions along a pipeline (at points 1 and 2) are related by the Bernoulli equation (see Problem 11). For flow in a pipe,\[e_{f}=\left(\frac{4 f
The Peclet number $\left(N_{P e}\right)$ is defined as\[N_{P e}=N_{R e} N_{P r}=\left(\frac{D V ho}{\mu}\right)\left(\frac{c_{p} \mu}{k}\right)=\frac{D G c_{p}}{\mu},\]where$D$ is the pipe
The heat transfer coefficient $(h)$ for a vapor bubble rising through a boiling liquid is given by\[h=A\left(\frac{k V ho c_{p}}{d}\right)^{1 / 2} \quad \text { where } V=\left(\frac{\Delta ho g
Determine the value of the Reynolds number for water flowing at a rate of $0.5 \mathrm{gpm}$ through a $1 \mathrm{in}$. ID pipe. If the diameter of the pipe is doubled at the same flow rate, how much
The pressure drop for a fluid with a viscosity of $5 \mathrm{cP}$ and a density of $0.8 \mathrm{~g} / \mathrm{cm}^{3}$ flowing at a rate of $30 \mathrm{~g} / \mathrm{s}$ in a $50 \mathrm{ft}$ long $1
In the steady flow of a Newtonian fluid through a long uniform circular tube, if $N_{R e}
Perform a dimensional analysis to determine the groups relating the variables that are important in determining the settling rate of very small solid particles falling in a liquid. Note that the
A simple pendulum consists of a small, heavy ball of mass $m$ at the end of a long string of length $L$. The period of the pendulum should depend on these factors, as well as gravity, which is the
An ethylene storage tank in your plant explodes. The distance that the blast wave travels from the blast site $(R)$ depends upon the energy released in the blast $(E)$, the density of the air $(ho)$,
It is known that the power required to drive a fan depends upon the impeller diameter $(D)$, the impeller rotational speed $(\omega)$, the fluid density $(ho)$, and the volume flow rate $(Q)$. (Note
A centrifugal pump with an 8 in. diameter impeller operating at a rotational speed of $1150 \mathrm{rpm}$ requires $1.5 \mathrm{hp}$ to deliver water at a rate of $100 \mathrm{gpm}$ and a pressure of
A gas bubble of diameter $D$ rises with a velocity $V$ in a liquid of density $ho$ and viscosity $\mu$.(a) Determine the dimensionless groups that include the effects of all of the significant
You must predict the performance of a large industrial mixer under various operating conditions. To obtain the necessary data, you decide to run a laboratory test on a small-scale model of the unit.
When an open tank with a free surface is stirred with an impeller, a vortex will form around the shaft. It is important to prevent this vortex from reaching the impeller, because entrainment of air
The variables involved in the performance of a centrifugal pump include the fluid properties $(\mu$ and $ho$ ), the impeller diameter $(d)$, the casing diameter $(D)$, the impeller rotational speed
The purpose of a centrifugal pump is to increase the pressure of a liquid in order to move it through a piping system. The pump is driven by a motor, which must provide sufficient power to operate
When a ship moves through the water, it causes waves. The energy and momentum in these waves must come from the ship, which is manifested as a "wave drag" force on the ship. It is known that this
You want to find the force exerted on an undersea pipeline by a $10 \mathrm{mph}$ current flowing normal to the axis of the pipe. The pipe is $30 \mathrm{in}$. in diameter, and the density of
You want to determine the thickness of the film when a Newtonian fluid flows uniformly down an inclined plane at an angle $\theta$ with the horizontal at a specified flow rate. To do this, you design
You would like to know the thickness of a syrup film as it drains at a rate of $1 \mathrm{gpm}$ down a flat surface that is $6 \mathrm{in}$. wide and is inclined at an angle of $30^{\circ}$ from the
The size of liquid droplets produced by a spray nozzle depends upon the nozzle diameter, the fluid velocity, and the fluid properties (which may, under some circumstances, include surface
Small solid particles of diameter $d$ and density $ho_{s}$ are carried horizontally by an air stream moving at velocity $V$. The particles are initially at a distance $h$ above the ground, and you
You want to find the wind drag on a new automobile design at various speeds. To do this, you test a $1 / 30$ scale model of the car in the lab. You must design an experiment whereby the drag force
The power required to drive a centrifugal pump and the pressure that the pump will develop depends upon the size (diameter) and speed (angular velocity) of the impeller, the volumetric flow rate
In a distillation column, vapor is bubbled through the liquid to provide good contact between the two phases. The bubbles are formed when the vapor passes upward through a hole (orifice) in a plate
A flag will flutter in the wind at a frequency that depends upon the wind speed, the air density, the size of the flag (length and width), gravity, and the "area density" of the cloth (i.e., the mass
If the viscosity of the liquid is not too high (e.g., less than about $100 \mathrm{cP}$ ), the performance of many centrifugal pumps is not very sensitive to the fluid viscosity. You have a pump with
The pressure developed by a centrifugal pump depends on the fluid density, the diameter of the pump impeller, the rotational speed of the impeller, and the volumetric flow rate through the pump
(a) Using tabulated data for the viscosity of water and SAE 10 lube oil as a function of temperature, plot the data in a form that is consistent with each of the following equations:(i) $\mu=A \exp
The viscosity of a fluid sample is measured in a cup and bob viscometer. The bob is $15 \mathrm{~cm}$ long with a diameter of $9.8 \mathrm{~cm}$, and the cup has a diameter of $10 \mathrm{~cm}$. The
A fluid sample is contained between two parallel plates separated by a distance of $2 \pm 0.1 \mathrm{~mm}$. The area of the plates is $100 \pm 0.01 \mathrm{~cm}^{2}$. The bottom plate is stationary,
The following materials exhibit flow properties that can be described by models that include a yield stress (e.g., Bingham plastic): (a) catsup, (b) toothpaste, (c) paint, (d) coal slurries, and (e)
Consider each of the fluids for which the viscosity is shown in Figure 3.7, all of which exhibit a "structural viscosity" characteristic. Explain how the "structure" of each of these fluids
Starting with the equations for $\tau=f n(\dot{\gamma})$ that define the power law and Bingham plastic fluids, derive the equations for the viscosity functions for these models as a function of shear
A paint sample is tested in a Couette (cup and bob) viscometer that has an outer radius of $5 \mathrm{~cm}$, an inner radius of $4.9 \mathrm{~cm}$, and a bob length of $10 \mathrm{~cm}$. When the
The quantities that are measured in a cup and bob viscometer are the rotation rate of the cup (rpm) and the corresponding torque transmitted to the bob. These quantities are then converted to
What is the difference between shear stress and momentum flux? How are they related? Illustrate each one in terms of the angular flow in the gap in a cup and bob viscometer, in which the outer
A sample of coal slurry is tested in a Couette (cup and bob) viscometer. The bob has a diameter of \($10.0\) \mathrm{~cm}$ and a length of \($8.0\) \mathrm{~cm}\($,\) and the cup has a diameter of
You must analyze the viscous properties of blood. Its measured viscosity is $6.49 \mathrm{cP}$ at a shear rate of $10 \mathrm{~s}^{-1}$ and $4.66 \mathrm{cP}$ at a shear rate of $80
The following data were measured for the viscosity of a $500 \mathrm{ppm}$ polyacrylamide solution in distilled water:Shear Rate $\left(\mathbf{s}^{-1}\right)$Viscosity $(\mathbf{c P})$Shear Rate
What viscosity model best represents the following data? Determine the values of the parameters in the model. Show a plot of the data together with the line that represents the model, to show how
You would like to determine the pressure drop in a slurry pipeline. To do this, you need to know the rheological properties of the slurry. To evaluate these properties, you test the slurry by pumping
A film of paint, $3 \mathrm{~mm}$ thick, is applied to a flat surface that is inclined to the horizontal by an angle $\theta$. If the paint is a Bingham plastic, with a yield stress of $150
A thick suspension is tested in a Couette (cup and bob) viscometer that has a cup radius of $2.05 \mathrm{~cm}$, a bob radius of $2.00 \mathrm{~cm}$, and a bob length of $15 \mathrm{~cm}$. The
You have obtained data for a viscous fluid in a cup and bob viscometer that has the following dimensions: cup radius $=2 \mathrm{~cm}$, bob radius $=1.5 \mathrm{~cm}$, bob length $=3 \mathrm{~cm}$.
A sample of a viscous fluid is tested in a cup and bob viscometer that has a cup radius of $2.1 \mathrm{~cm}$, a bob radius of $2.0 \mathrm{~cm}$, and a bob length of $5 \mathrm{~cm}$. When the cup
You have a sample of a sediment that is non-Newtonian. You measure its viscosity in a cup and bob viscometer having a cup radius of $3.0 \mathrm{~cm}$, a bob radius of $2.5 \mathrm{~cm}$, and a
Acrylic latex paint can be described by the Bingham plastic model with a yield stress of $112 \mathrm{dyn} / \mathrm{cm}^{2}$, a limiting viscosity of $80 \mathrm{cP}$, and a density of $0.95
Santa Claus and his loaded sleigh are sitting on your roof, which is covered with snow. The sled's two runners each have a length $L$ and width $W$, and the roof is inclined at an angle $\theta$ to
You must design a piping system to handle a sludge waste product. However, you don't know the properties of the sludge, so you test it in a cup and bob viscometer with a cup diameter of $10
A fluid sample is tested in a cup and bob viscometer that has a cup diameter of 2.25 in., a bob diameter of $2 \mathrm{in}$., and length of $3 \mathrm{in}$. The following data are obtained:Rotation
You test a sample in a cup and bob viscometer to determine the viscosity. The diameter of the cup is $55 \mathrm{~mm}$, that of the bob is $50 \mathrm{~mm}$, and the length is $65 \mathrm{~mm}$. The
Consider each of the fluids for which the viscosity is shown in Figure 3.7, all of which exhibit a typical "structural viscosity" characteristic. Explain why this is a logical consequence of the
You are asked to measure the viscosity of an emulsion, so you use a tube flow viscometer similar to that illustrated in Figure 3.4, with the container open to the atmosphere. The length of the tube
You must determine the horsepower required to pump a coal slurry through an $18 \mathrm{in}$. diameter pipeline, 300 miles long, at a rate of 5 million tons/year. The slurry can be described by the
You want to determine how fast a rock will settle in mud, which behaves like a Bingham plastic. The first step is to perform a dimensional analysis of the system.(a) List the important variables that
A pipeline has been proposed to transport a coal slurry 1200 miles from Wyoming to Texas, at a rate of 50 million tons/year, through a 36 in. diameter pipeline. The coal slurry has the properties of
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