Questions and Answers of Fundamentals Of Chemical Engineering Thermodynamics

The following rates were observed for a first-order, irreversible reaction, carried out on a spherical catalyst:\[\begin{array}{lr}d_{p 1}=0.625 \mathrm{~cm} & r_{o b s, 1}=0.09 \mathrm{~mol} /
Consider a catalyst pellet in the shape of a Raschig ring (a hollow cylinder of inner radius, \(r_{i}\), and outer radius, \(r_{o}\) ). A first-order reaction occurs inside the catalyst.\[a
A first-order heterogeneous irreversible reaction is taking place within a spherical catalyst pellet \(200 \mu \mathrm{m}\) in diameter.\[a \xrightarrow{k^{\prime \prime}} b\]The reactant
An inventor has the bright idea of running a gas-solid catalytic reaction with an ultrasonic assist. The concept is to induce acoustic waves inside the pores of the solid catalyst and therefore
Consider a chemical reaction, \(a \rightarrow b\), taking place in a bed of spherical catalyst pellets. The true reaction rate is \(\mathrm{n}^{\text {th }}\)-order in the concentration of \(a\) and
An elementary reaction is taking place inside a cylindrical catalyst pore. The process contains a small amount of a catalyst poison, \(P\), and you would like to know how this affects the
A first-order isomerization reaction is taking place in a non-isothermal, spherical catalyst pellet. Determine the effectiveness factor and the maximum temperature rise in the pellet for the
If an equation obeys Laplace's equation, then at what angle must lines of constant temperature cross lines of constant flux?
In a two-dimensional, Cartesian system, if we have irrotational flow of a fluid, what is the definition of the stream function?
How are the velocity and vorticity related in a 2-D Cartesian system?
What is the definition of circulation?
Fick's Law states that mass flows in the direction of decreasing concentration. What is it about the activator/inhibitor system that allows Fick's Law to be circumvented?
Suppose we are interested in the steady-state temperature distribution inside a piece of crystalline quartz. Quartz is well-known for having material properties that are different for different
Consider the fin of section 9.2.1 and its solution given by equation (9.21).a. Develop expressions for the effectiveness and efficiency of such a fin.b. Estimate the effectiveness for the following
A miniature heat pipe is a heat transfer device that operates via an evaporation/condensation mechanism. The heat source evaporates fluid at one end of the device. The fluid travels the length of the
One of the body's ways of increasing heat transfer with the environment is the use of goose bumps. The small raised regions on the skin offer increased surface area for transport. We would like to
A thin slab of metal is formed in the shape of a square. The \(x-y\) vertices of the square are \((0,0),(1,0),(0,1)\), and \((1,1)\). The faces of the slab in the \(z\)-direction are insulated so
We are interested in determining the temperature distribution inside a concrete slab at a stoplight. The slab is exposed to cars' catalytic converters whose surface temperatures are \(300^{\circ}
Consider one three-dimensional, time-dependent problem to illustrate the power of a Green's function solution. We are interested in doping a semi-infinite boule of initially pure, single crystal
Consider the lines of constant concentration and streamlines in Figure P9.13. Based on the relationship to one another and the boundaries of the system, describe what is going on and what the
If the ellipse:\[a\left(x^{2}-y^{2}\right)+2 b x y-\frac{1}{2} \omega_{o}\left(x^{2}+y^{2}\right)+c=0\]is full of liquid and is rotated about the origin with an angular velocity, \(\omega_{o}\), the
Find the temperature profile in the semi-infinite solid defined by the Figure P9.15. The side at \(x=0\) is held at a temperature \(T=0\). The side defined by \(y=0\) is insulated for \(01\). Use the
Show that the stream function for flow in the corner of Figure P9.16 is given by:\[\psi(r, \theta)=A r^{4} \sin (4 \theta)\]Use the transformation \(z=w^{4}\) where \(z=x+i y\) and \(w=\phi+i \psi\).
Using the transformation \(w=\sin (z)\), determine the equation for the stream function for flow inside the semi-infinite region \(y \geq 0,-\pi / 2 \leq x \leq \pi / 2\) shown in Figure P9.17. -/2
Consider the flow formed by placing a source of strength, \(q_{o}\), a distance, \(d\), from an infinitely long wall as shown in Figure P9.18. The velocity potential for this incompressible and
A stream function is given by:\[\psi=\sin \left(\frac{x}{L}\right) \sinh \left(\frac{y}{K}\right)\]where \(L\) and \(K\) are constants, \(0 \leq x \leq \pi L\) and \(y \geq 0\).a. Does \(\psi\)
Previously, we solved Laplace's equation using separation of variables for the velocity potential about a cylinder with circulation. The solution was of the form:\[\phi=c_{1}
The method of images uses precisely oriented collections of sources, sinks, and vortices to establish artificial walls and so simulate more complicated flow fields. One such flow field is formed by
Another example of the method uses flow past a cylinder of radius, \(R\), and its image to simulate the flow of fluid past a cylinder that lies close to a plane wall. The situation is shown in Figure
What would the stream function and velocity potential look like for the combination of plane flow and a tornado? Sketch the flow field. Is it realistic? Comment on the event horizon and escape
A semitransparent slab of photoresist is being exposed to ultraviolet radiation. When exposed, the photoresist generates heat in direct proportion to the intensity of the radiation. The radiation
Reconsider the solenoid problem of section 9.4.2 shown in Figure P9.25 but with convective heat transfer from the surface at \(r=r_{o}\). The convection is characterized by a heat transfer
The surface of a catalytic slab is exposed to a concentration distribution \(c_{a o}(x)\). As material diffuses into the slab it reacts according to a first-order chemical reaction \((a \rightarrow
The cylindrical surface of a catalytic pellet is exposed to a concentration distribution \(c_{a o}(z)\). The ends of the pellet are impermeable as shown in Figure P9.27. As material diffuses into the
Consider a thin circular sheet of water of radius, \(r_{o}\), and depth, \(h\), that is being excited by steady, periodic oscillations of its outer rim at a frequency, \(\omega\). We want to catalog
Consider the diffusion situation described in section 9.5. Instead of having pure diffusion, we put a catalyst into the fluid that forces a first-order chemical reaction \(a \rightarrow b\) to occur.
In section 9.4.3 we discussed the diffusion of a compound from a skin patch and its transport and reaction once it entered the skin. We did a very simple example of this, not considering what
The mechanical energy equation applies only to isothermal systems.
Irrotational flows can have vorticity associated with them.
Mole numbers are conserved in the species continuity equation.
The Navier-Stokes equations are a statement of Newton's Law \((F=m a)\) per unit volume of fluid.
Viscous dissipation is always a negative quantity.
At steady state, a flow must always be incompressible.
An incompressible fluid flows in a two-dimensional space. It has the following velocity component.\[v_{x}=x^{2} y+2 x y\]a. Determine an expression for \(v_{y}\).b. Is the flow irrotational?c. What
Evaluate the stream function for a flow with the following velocity components.\[\begin{aligned}& v_{x}=2 x+e^{x} \cos (y) \\& v_{y}=-3 x^{2}-2 y-e^{x} \sin (y)\end{aligned}\]Find the stream function
Using Table 10.5, describe under what conditions, if any, the aspherical objects might exhibit the same or lower drag than a sphere in Stoke's flow. For systems that have parallel and perpendicular
Given the following velocity profile for steady flow around a sphere, determine if the fluid is incompressible. Is it also irrotational?\(v_{r}=v_{\infty} \cos
A fluid flows through a circular pipe of radius, \(r_{o}\). The flow is fully developed. The pipe is oriented vertically with respect to gravity (z-direction) and also has a constant pressure
The surface of a rotating cylinder of fluid takes on a parabolic shape. Engineers have used that fact to produce large telescope mirrors using mercury as the liquid. You propose to make a 3-meter
We have a rotating tank of diameter, \(0.5 \mathrm{~m}\) and height, \(1.5 \mathrm{~m}\). The tank is filled with water at \(300 \mathrm{~K}\) to a depth of \(1 \mathrm{~m}\).a. How fast would we
A journal bearing of the form shown in Figure 10.8 is being designed to carry a load of two metric tons. The liquid being used is conventional lubricating oil with a viscosity of \(1 \mathrm{~Pa}
Consider the spin-coating process used to coat silicon wafers with photoresist, television picture tubes with phosphorescent layers, etc. (Figure P10.15). In all cases, the process is designed to
Consider the system shown in Figure P10.16 of two concentric rotating cylinders. The two cylinders each rotate at a constant but different angular velocity.a. Determine the velocity profile
An infinite rigid rod of radius, \(r_{o}\), rotates in an infinite, incompressible, Newtonian fluid of viscosity, \(\mu\), and density, \(ho\), with a constant rotational velocity, \(\omega\).
In Section 10.2.2 we discussed how to find the shape of the surface of a rotating liquid. Consider the situation where the fluid is confined between two counter-rotating cylinders as shown in Figure
The stream function for flow in a \(90^{\circ}\) corner is:\[\psi=\frac{r v_{o}}{\pi^{2}-4}\left(2 \pi \theta \sin \theta+4 \theta \cos \theta-\pi^{2} \sin \theta\right)\]a. What are the velocity
A two-dimensional flow exists between fixed boundaries at \(\theta=\pi / 4\) and \(\theta=-\pi / 4\). The flow is due to a source of strength, \(m\), at \(r=a, \theta=0\), and a sink of equal
If the ellipse:\[a\left(x^{2}-y^{2}\right)+2 b x y-\frac{1}{2} \omega_{o}\left(x^{2}+y^{2}\right)+c=0\]is full of liquid and is rotated about the origin with an angular velocity, \(\omega_{o}\), the
A Newtonian fluid of viscosity, \(\mu\), and density, \(ho\), is contained in between two vertical pipes of diameter, \(d_{o}\) and \(d_{i}\). The situation is shown in Figure P10.22. If left to its
A fluid of viscosity, \(\mu\), flows down a vertical rod of radius, \(r_{o}\). At some point down the rod, the flow reaches a steady condition where the film thickness, \(h\), is constant and the
Skimmers are used to remove viscous fluids, such as oil, from the surface of water. As shown on the diagram in Figure P10.24, a continuous belt moves upward at velocity \(v_{s}\) through the fluid
In Section 10.2.2 we calculated the velocity field for flow past a stationary sphere. Such a sphere experienced a buoyant force but no lift. To generate lift, we need to rotate the sphere. We can
A sliding bearing, modeled as flow between two plates separated by a distance, \(\delta\), is lubricated by a Newtonian fluid of viscosity, \(\mu\), and density, \(ho\). The top plate moves at a
We would like to look at a case of coupled transport: natural convection between horizontal walls. The situation is shown in Figure P10.27. We assume that the length of the device is much larger than
Consider a gas metal arc welding electrode as shown in Figure P10.29 where we are consuming the electrode as we weld ( rod velocity \(\left.=v_{o}\right)\). We are interested in the steady-state
Consider the welding situation of the previous problem. This time let's consider the case where we have convective heat transfer from the rod's surface. For a rod of radius, \(r_{o}\), and a heat
Consider the heat transfer to a falling film problem of Section 10.3. The mass transfer analog is the dissolution of a solid wall into the falling film. Assuming the solid wall is composed of salt,
Vorticity is generated any time a fluid contacts a solid surface. The vorticity diffuses from the solid surface and is convected away by the fluid. Consider the heat transfer to a falling film
An annealing furnace has a belt that moves material in at a specific velocity, \(U\) (Figure P10.34). Only the region inside the oven is heated and the material sees a radiant heat flux of
In Smoluchowski's Theory of Coagulation we focus on an individual sphere and assume that other like particles diffuse toward it. Once they reach the sphere, they collide and form a new spherical
As a biochemical engineer you are evaluating a drug delivery system for an artificial protein to combat Alzheimer's. The protein is very large (200,000 molecular weight) and bulky and is sensitive to
Consider the example in Section 10.4 about mass transfer to a growing bubble. Calculate the mass transfer coefficient for that system. Does it increase or decrease over time? Why should this be so?
Consider the mass transfer example concerning diffusion into a falling film shown in Figure P10.38. A new inventor claims that he can rig the device to operate such that the flux of \(a\) from the
Consider absorption into a falling film, where the film contains a reactive component in great excess. This results in reaction that is pseudo-first-order in absorbate concentration. Dilute solutions
Consider the motion of sodium and chloride ions in a membrane under the influence of an externally applied electric field. The situation is shown in the Figure P10.40. Charge transport in the
The log-mean temperature difference is an average temperature difference between the hot and cold fluids over the length of a heat exchanger.
The log-mean temperature difference is larger for a co-flow heat exchanger.
The log-mean temperature difference can be a negative quantity if one is boiling or condensing.
Plug flow reactors can be viewed as a sequential train of continuous stirred tank reactors.
In a purely converging nozzle, the fluid can exit the nozzle at a velocity greater than its speed of sound.
The macroscopic continuity equation only applies to incompressible fluids.
The macroscopic momentum equation is used to calculate forces a fluid exerts on a system and the reaction forces necessary to oppose those.
In the mechanical energy balance, we assumed that:\[\overline{v^{3}}=\bar{v}^{3} \quad \text { and } \quad \overline{v^{2}}=\bar{v}^{2}\]This can introduce a lot of error into the final calculations
A water jet pump, Figure P11.9, has a jet area \(A_{j}=0.01 \mathrm{~m}^{2}\) and a jet velocity \(v_{j}=30 \mathrm{~m} / \mathrm{s}\) which entrains a secondary stream of water having a velocity
Fluid is accelerated as it passes through the slipstream of a propeller, Figure P11.10. Show that the average velocity of the fluid as it passes the plane of the propeller blades is the average of
Water flows steadily through a fire hose and nozzle. The hose diameter is \(100 \mathrm{~mm}\) and the nozzle tip is \(30 \mathrm{~mm}\) in diameter. The gauge pressure of the water in the hose is
Water exits a pipe from a series of holes drilled into the side, Figure P11.12. The pressure at the inlet section is \(35 \mathrm{kPa}\) (gauge). Calculate the volumetric flow rate at the inlet
A jet of water issuing from a nozzle at a volumetric flow rate, \(Q\), with a velocity, \(v_{j}\), strikes a series of vanes mounted on a wheel as in Figure P11.13. This configuration is termed a
An ancient device for measuring the passage of time uses an axisymmetric vessel shaped so that the water level falls at a constant rate (see Figure P11.14). The top of the vessel is open to the
A small piece of \(3 \times 5\) card can be held onto a spool of thread by blowing air through the hole in the center as shown in Figure P11.15. The harder one blows, the tighter the card hangs on.
The depth of fluid following a hydraulic jump can be determined from an application of conservation of mass and conservation of momentum as discussed in Chapter 11.This change in height must also
An advertisement from the manufacturer of a low-flow showerhead claims that their unique showerheads inject oxygen into the water to provide greater coverage with less water. They further claim to
Cooling water is pumped from a reservoir to drills at the site of a geothermal power plant. The drills are located \(150 \mathrm{~m}\) above the level of the pump and the distance from the pump to
Find the force components, \(F_{x}\) and \(F_{y}\) required to hold the box in Figure P11.20 stationary. The fluid is oil with a specific gravity of 0.85 . Neglect gravity and assume the pressure
A hole is to be drilled into the side of the tank shown in Figure P11.21 so the liquid will travel the farthest horizontal distance, \(L\). You may assume the liquid height in the tank remains
A plastic tube of \(50 \mathrm{~mm}\) diameter is used to siphon water from the large tank as shown in Figure P11.22. If the pressure on the outside of the tube is \(25 \mathrm{kPa}\) greater than
Consider the case of isentropic flow through a nozzle.a. What is the critical temperature, \(T_{c}\), for the flow?b. Show that the average velocity at any point in the nozzle can be described
If we fail to insulate the nozzle in the gas flow system of Problem 11.23, we allow heat transfer to occur as the gas flows from the reservoir. Show under these conditions that the entropy of the gas
We showed in Section 11.5.1 that for a counterflow heat exchanger, the log-mean temperature difference was:\[\Delta T_{l m, c o u n t e r}=\left[\frac{\left(T_{h i}-T_{c o}\right)-\left(T_{h o}-T_{c
In developing our model for a heat exchanger, we assumed that the overall heat transfer coefficient, \(\mathcal{U}\), was a constant. In most instances, especially if the temperature difference

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