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engineering
chemical engineering
Questions and Answers of
Chemical Engineering
Derive Eq. (12.34) leading to the exponential integral solution of the segregated flow model. CA,b(exit) = Da expint(Da/2) + (1 - Da/2)exp(-Da/2) (12.34)
Find the exit concentration and the conversion for a laminar-flow reactor under the following conditions using the segregated model: radius \(1 \mathrm{~cm}\), length \(500 \mathrm{~cm}\), mass flow
Mass transfer can be enhanced by flow oscillation. Your goal in this study is to review the literature and, in particular, examine the following model problem, viz., mass transfer in pipe flow with a
Water is contained between two vertical plates with a gap width of \(2 \mathrm{~cm}\). The temperatures of the plates are 25 and \(75^{\circ} \mathrm{C}\). Find the velocity profile and plot the
Consider again the problem of a highly viscous fluid contained between two parallel plates of gap width \(d\). The top plate is moving with a velocity of \(V\) and this generates heat in the system
Owing to the non-homogeneous term in Eq. (13.24), direct separation of variables is not possible. A partial solution has to be found, and the problem has to be solved by the modified method of
Reexamine the problem of laminar flow with heat generation. Now do the analysis for a constant heat flux at the walls. How should the dimensionless temperature \(\theta\) be defined here? How is the
Extend the analysis to a power-law fluid. Compare the changes in the nature of the profile with varying power-law index. *- (). P (13.4)
The thermodynamic wet-bulb temperature is defined as the temperature at which water evaporates and brings the air to equilibrium conditions. Show that the thermodynamic wet-bulb temperature is the
At a point in a dryer benzene is evaporating from a solid. The air temperature is \(80^{\circ} \mathrm{C}\) and the pressure is \(1 \mathrm{~atm}\). The relative humidity of benzene in air is \(65
A wall subject to intense radiative and convective heating is to be protected by sweat cooling. For this purpose water is injected onto the surface through a porous stainless steel plate at a rate
Water is condensing on a surface at \(310 \mathrm{~K}\). The gas mixture has \(65 \%\) water vapor and is at a temperature of \(370 \mathrm{~K}\). The total pressure is \(1 \mathrm{~atm}\).Calculate
The condensation of a binary mixture A and B in the presence of an inert species \(C\) can be analyzed in the same manner. The liquid side has only \(\mathrm{A}\) and \(\mathrm{B}\), and hence the
Now consider the above problem for a case where a reaction between the condensing vapor A and the relatively non-condensing gas B can take place in the liquid according to\[\mathrm{A}+\mathrm{B}
The effectiveness factors for a porous catalyst in the presence of significant temperature gradients can be larger than one. Explain why.Your goal is to generate such a plot of \(\eta\) as a function
The time of oscillation \(t\) of a simple pendulum is expected to be a function of the mass of the pendulum, its length \(L\), and the gravitational constant. How many dimensionless groups can be
The eminent British fluid dynamicist G. I. Taylor deduced the energy release \(E\) from the first atomic explosion simply by dimensional analysis. This was strictly classified information, but Taylor
In correlating the diameter of drops formed at the orifice, the viscosity is to be included as the core group rather than density as done in the text. Form groups with \(d_{0}, ho\), and \(\mu\) as
A vertical plate of height \(L\) at a temperature of \(T_{\mathrm{s}}\) is immersed in a fluid at a temperature of \(T_{\mathrm{a}}\). Local density variation due to temperature causes a flow, which
It is required to find the cooking time of a large turkey. Experimental data are available for a small turkey of mass \(m_{\mathrm{s}}\), and the time is \(t_{\mathrm{s}}\). Suggest the cooking time,
Apply scaling analysis for gas absorption with first-order reaction based on the film model. Derive a relation for the depth of penetration of the gas for a fast reaction. Using this relation, verify
Gas-absorption systems are usually modeled by assuming a film thickness and steady-state diffusion. In an attempt to modify this picture one can assume a finite film but allow for a transient
How does the power in an agitated vessel depend on the impeller speed and the impeller diameter in (a) the laminar regime and (b) the turbulent regime?
The mass transfer coefficient in agitated gas-liquid systems is often proportional to the power per unit volume of the reactor rather than the total power \(P\) dissipated in the system. On this
For gas-liquid dispersions in agitated vessels, the gas flow rate \(Q_{\mathrm{G}}\) is also important. Show that an additional group, namely the flow number defined as \(Q_{\mathrm{G}} /\left(\Omega
The following data were obtained by Sharma and Danckwerts (1963) for \(\mathrm{CO}_{2}\) absorption in a laminar jet of solution in which the gas underwent a first-order reaction:Interpret the data
For turbulent flow of water in a pipe of diameter \(5 \mathrm{~cm}\) with \(R e=10^{5}\) estimate the magnitude of the length and velocity scales at which viscous dissipation becomes important.
An agitated tank has an impeller diameter of \(10 \mathrm{~cm}\) and operates at a speed of revolution of 10 r.p.s. The tank has a diameter of \(20 \mathrm{~cm}\) and is filled up to a height of \(20
Treating an oil spill by application of a chemical dispersant is a useful means for dispersion of oil. Knowledge of the droplet size formed as a function of the energy dissipation rate is useful to
Consider for illustration the diffusion-reaction problem given by\[\begin{equation*}\frac{d^{2} c}{d x^{2}}=M c^{2} \tag{14.54}\end{equation*}\]with the same boundary conditions as before.Use the
Solve the problem of heat generation in a slab with a variable thermal conductivity. Show that the problem can be represented as\[\frac{d \theta}{d \xi}\left[(1+\beta \theta) \frac{d \theta}{d
Solve the following problem:\[\frac{d^{2} c}{d x^{2}}-P e \frac{d c}{d x}=0\]Use the boundary conditions \(c(0)=1\) and \(c(1)=0\).Solve both for large Péclet number, \(P e\), where this is a
Solve the following singular perturbation problem:\[\epsilon \frac{d^{2} c_{\mathrm{b}}}{d \xi^{2}}=\xi c_{\mathrm{b}}-1\]which arises in a model for a consecutive reaction in a liquid film (Deen,
Your task is to obtain the solution by analytical and numerical methods and compare it with the perturbation solution.Verify the following analytical solution for the concentration distribution and
The problem was solved analytically using complex variables. The perturbation method is also suitable for this problem and provides additional physical insight into the solution.The model equation in
Pulsatile flow at large \(W o\). For large values of \(W o\) the problem is of singular perturbation type. Verify that the inner solution is of a plug-flow type. The pressure is balanced by inertial
Consider again the radial flow between two parallel disks examined. The case of low Reynolds number was examined there, and a solution in which the non-linear terms were ignored was obtained. Here we
Complete the solutions for \(v_{1}\) in Example 14.4 for domain perturbation in the text. What are the boundary conditions for the \(v_{2}\) problem?Example 14.4:Consider flow in a rectangular
The problem of potential flow past a circular object is a well-studied problem in fluid mechanics governed by the Laplace equation. We wish to study the flow around a circle which is slightly
Write out in detail all the terms for \(\tilde{\tau}: abla v\) in rectangular Cartesian coordinates.Now assume a Newtonian fluid; use the generalized version of Newton's law of viscosity for the
Verify that the viscous dissipation term is always positive, indicating that this is an irreversible conversion into internal energy.
Comment and elaborate on the following statement from the BSL book: for viscoelastic fluids the term \(\tilde{\tau}: abla \boldsymbol{v}\) does not have to be positive since some energy may be stored
For fully developed flow in a pipe the contributions to viscous dissipation are from \(\tau_{r z}\) and \(d v_{z} / d r\). What is the form of the viscous generation term for a fully developed
Verify the following thermodynamic relation:\[\begin{equation*}\left(\frac{\partial \hat{H}}{\partial P}\right)_{T}=\hat{V}-T\left(\frac{\partial \hat{V}}{\partial T}\right)_{P}
How does the temperature equation simplify if there is no flow? Write this out in detail in all of the three coordinate systems.
Write in detail the expression for \((\boldsymbol{v} \cdot abla) T\) in cylindrical coordinates. Also write in detail the expression for the Laplacian, and thereby complete the temperature equation
Write in detail the expression for \((\boldsymbol{v} \cdot abla) T\) in spherical coordinates. Also write in detail the expression for the Laplacian, and thereby complete the temperature equation for
A turkey is being roasted in a microwave oven. How would you calculate the internal heat generation term, \(\dot{Q}_{\mathrm{V}}\) ?
Derive equations for the temperature in a slab if the thermal conductivity (a) is constant, (b) varies linearly as \(k(T)=k_{0}+a\left(T-T_{0}\right)\), and (c) varies as a quadratic function
Find the rate of heat flow in the radial direction through a spherical shell of inner radius \(r_{\mathrm{i}}\) and outer radius \(r_{\mathrm{o}}\) for the case where the thermal conductivity varies
Show that a variable transformation known as the Kirchhoff transformation,\[F(T)=\int_{0}^{T} k(s) d s\]where \(s\) is a dummy variable, reduces the heat equation to \(abla^{2} F=0\) for the variable
Consider the case of linear heat generation with a linear variable-thermal conductivity.Express the governing equation in terms of dimensionless form. What are the number of dimensionless parameters
Consider heat conduction with generation in a slab and a sphere geometries. Derive the solution similar to Eq. (8.18) for slab and sphere cases.For all three cases (slab, cylinder, and sphere) find
For the linear generation, use the Robin condition at the surface rather than the Dirichlet condition used in the text. Derive an expression for the temperature profile, the maximum temperature, and
Verify the solutions in the text for the temperature distribution in a square slab with constant generation of heat. Generate illustrative contour plots for the temperature profiles.
Second-order elliptic equations of the Laplace and Poisson type can be solved by centraldifference-based finite-difference schemes. Thus, if we employ a square mesh, show that the temperature at a
The square problem can also be fitted using boundary collocation. For this we need functions that satisfy the Laplace equation in an exact manner. The solution can then be expanded in terms of these
Show that, if purely Neumann conditions are imposed along the boundary, then the integral of the temperature gradient over the perimeter, \(\Gamma\),\[\int_{\Gamma} \frac{\partial T}{\partial n} d
State the governing equation and the dimensionless parameters needed to characterize the problem.Verify that the temperature profile is given by\[\theta=\theta_{\max } \frac{J_{0}(\zeta
Repeat the above problem for a spherical geometry. Find the temperature profile: What is the limit for explosion for this case? The answer is \(\sqrt{\beta}
Repeat the analysis for the above two problems if a Robin condition is applied at the surface.
A porous solid is in the shape of a hollow sphere and has temperatures of \(T_{\mathrm{i}}\) and \(T_{\mathrm{o}}\) at the radii of \(r_{\mathrm{i}}\) and \(r_{\mathrm{o}}\), respectively. Find an
Apply the lumped model for cooling of a solid in the form of (i) a long slab and (ii) a long cylinder. Also apply the modified lumped model shown in the text, and show what correction is needed for
Show that mole fractions can be converted to mass fractions by the use of the following equation:\[\begin{equation*}y_{i}=\left[\omega_{i} / M_{i}\right] \bar{M} \tag{9.57}\end{equation*}\]Derive an
Show that mass fractions can be converted to mole fractions by the use of the following equation:\[\begin{equation*}\omega_{i}=\left[y_{i} M_{i}\right] / \bar{M} \tag{9.58}\end{equation*}\]Derive an
At a point in a methane reforming furnace we have a gas of the composition \(10 \% \mathrm{CH}_{4}, 15 \%\) \(\mathrm{H}_{2}, 15 \% \mathrm{CO}\), and \(10 \% \mathrm{H}_{2} \mathrm{O}\) by
The Henry's-law constants for \(\mathrm{O}_{2}\) and \(\mathrm{CO}_{2}\) are reported as \(760.2 \mathrm{l} \cdot \mathrm{atm} / \mathrm{mol}\) and \(29.41 \mathrm{l}\). \(\mathrm{atm} /
Solubility data for \(\mathrm{CO}_{2}\) are shown below as a function of temperature:Fit an equation of the type\[\ln H=A+B / T\]What is the physical significance of the parameter \(B\) ? 280
The Antoine constants for water are \(A=8.07131, B=1730.63\), and \(C=233.426\) in the units of \(\mathrm{mm} \mathrm{Hg}\) for pressure and \({ }^{\circ} \mathrm{C}\) for temperature.Convert this to
Given the Antoine constants for a species, can you calculate the heat of vaporization of that species?
Is \(\boldsymbol{n}_{\mathrm{A}}\) equal to \(M_{\mathrm{A}} \boldsymbol{N}_{\mathrm{A}}\) ? Why?Is \(j_{\mathrm{A}}\) equal to \(M_{\mathrm{A}} \boldsymbol{J}_{\mathrm{A}}\) ? Why?
Show the validity of the following relations between the species velocities \(v_{\mathrm{A}}\) and \(v_{\mathrm{B}}\) referred to a stationary frame of
Show for a binary mixture the validity of the relation\[\boldsymbol{j}_{\mathrm{A}}=\frac{M_{\mathrm{A}} M_{\mathrm{B}}}{M} \boldsymbol{J}_{\mathrm{A}}\]
Show the validity of the following form of Fick's law:\[j_{\mathrm{A}}=-\frac{C^{2}}{ho} M_{\mathrm{A}} M_{\mathrm{B}} D_{\mathrm{AB}} abla x_{\mathrm{A}}\]where \(x_{\mathrm{A}}\) is the mole
Show the validity of the following equation based on Fick's law:\[\begin{equation*}abla x_{\mathrm{A}}=\frac{x_{\mathrm{A}} N_{\mathrm{B}}-x_{\mathrm{B}} N_{\mathrm{A}}}{C D_{\mathrm{AB}}}
Velocity based on volume-fraction weighting. If the volume fraction \(\phi_{\mathrm{a}}\) is used as the weighting factor, one can define an average velocity \(\boldsymbol{v}^{\mathrm{V}}\)
Knudsen diffusion is a phenomenon whereby transport takes place by gas-pore wall collisions rather than by gas-gas collisions. Knudsen diffusion is modeled using concepts similar to the kinetic
Hindered diffusion is a phenomenon of diffusion of a solute in narrow liquid-filled pores when the size of the pores is comparable to the size of the diffusing molecule. The hindered diffusion is
Verify that the concentration profile in a slab is linear, whereas that in a hollow cylinder is logarithmic, and for a spherical shell it is an inverse function of \(r\) for the three geometries in
Derive an expression for the fall in the liquid level during evaporation using a quasi-steadystate approach. Show that\[\frac{d H}{d t}=\frac{M_{\mathrm{A}}}{ho_{\mathrm{L}}} N_{\mathrm{A}}\]where
Benzene is contained in an open beaker of height \(6 \mathrm{~cm}\) and filled to within \(0.5 \mathrm{~cm}\) of the top. The temperature is \(298 \mathrm{~K}\) and the total pressure is \(1
For the above problem find the time for the benzene level to fall by \(2 \mathrm{~cm}\). The specific gravity of benzene is 0.874 .For this condition find the mole-fraction profile of benzene in the
A liquid is contained in a tapered conical flask with a taper angle of \(30^{\circ}\). The radius in the flask for the liquid level at the bottom is \(7 \mathrm{~cm}\) and the vapor height above this
Two bulbs are connected by a straight tube of diameter \(0.001 \mathrm{~m}\) and length \(0.15 \mathrm{~m}\). Initially one bulb contains nitrogen and the bulb at the other end contains hydrogen. The
Derive an expression for the case of mass transfer with second-order surface reaction based on a low-flux model. The resistance concept does not hold, unlike for a first-order reaction.
An example of a problem where there is severe counter-diffusion of the products is the deposition of \(\mathrm{SiO}_{2}\) from tetraethoxysilane (TEOS) on a solid substrate. The reaction is
9. At a certain point in a mass transfer equipment the bulk mole fractions are \(y_{\mathrm{A}}=0.04\) in the gas phase and \(x_{\mathrm{A}}=0.004\) in the liquid phase. The mass density of the
Consider the diffusion-reaction problem represented in the three geometries.Verify the analytical solutions shown in the text for the three geometries with the Dirichlet condition of
Consider the same problem with now a Robin condition at the surface:\[\left(\frac{d c_{\mathrm{A}}}{d \xi}\right)_{1}=B i\left[1-\left(c_{\mathrm{A}}\right)_{1}\right]\]Derive an expression for the
Consider a second-order reaction in a catalyst in the form of a slab. Show that the differential equation can be expressed as\[\begin{equation*}\frac{d^{2} c_{\mathrm{A}}}{d \xi^{2}}=\phi^{2}
The sulfur compounds present in petroleum fractions such as diesel can be removed by contact with a porous catalyst containing active metals such as Mo in the presence of hydrogen. If the catalyst is
Gas absorption in an agitated tank with a first-order reaction.Oxygen is absorbed in a reducing solution, where it undergoes a first-order reaction with a rate constant of \(40 \mathrm{~s}^{-1}\).
Verify the expression for the critical Thiele modulus for the oxygen concentration to become zero at \(R_{0}\) in the tissue as a function of \(\kappa\), the ratio of the capillary diameter to the
The effective diffusion coefficient of \(\mathrm{H}_{2}\) in a mixture of \(\mathrm{H}_{2}\) and \(\mathrm{CO}\) in a porous catalyst was found to be \(0.036 \mathrm{~cm}^{2} / \mathrm{s}\) at a
Consider a porous catalyst with a series reaction represented as\[\mathrm{A} \rightarrow \mathrm{B} \rightarrow \mathrm{C}\]Write governing equations for A and B. Express them in dimensionless form.
A spherical capsule has an outer membrane thickness with inner and outer radii \(r_{\mathrm{i}}\) and \(r_{\mathrm{o}}\), respectively. A solute is diffusing across this capsule. Consider the case
A pool of liquid is \(10 \mathrm{~cm}\) deep, and a gas \(A\) dissolves and reacts in the liquid. The solubility of the gas is such that the interfacial concentration is equal to \(2 \mathrm{~mol} /
\(\mathrm{CO}_{2}\) is absorbed into a liquid under conditions such that the liquid-side mass transfer coefficient is \(2 \times 10^{-4} \mathrm{~m} / \mathrm{s}\). The diffusion coefficient of
A porous catalyst is used for \(\mathrm{CO}\) oxidation, and the process is modeled as a first-order reaction with a rate constant of \(2 \times 10^{4} \mathrm{~s}^{-1}\). The effective diffusion
For the \(\mathrm{CO}_{2}\) absorption in \(\mathrm{NaOH}\) problem in the text, examine the effects of (i) changing the partial pressure of \(\mathrm{CO}_{2}\) and (ii) the concentration of the
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