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engineering
chemical engineering
Questions and Answers of
Chemical Engineering
Derive the solutions for transient concentration profiles in the two-bulb apparatus (Example 21.5 in the text) for the binary case, and show that the multicomponent case can be derived as an
When two gases A and B are forced to diffuse through a third gas C, there is a tendency of A and \(\mathrm{B}\) to separate because of the difference in their diffusivities in gas \(\mathrm{C}\).
Show that another way of writing the flux due to pressure diffusion is in terms of partial molar volume and the total mixture density is\[J_{\mathrm{A}}^{*}(\text { pressure diffusion })=-C
Estimate the steady-state concentration profile when a typical albumin solution is subjected to a centrifugal field of 30000 times the force of gravity under the following conditions: the cell length
A mixture of \(\mathrm{H}_{2}\) and \(\mathrm{D}_{2}\) is contained in two bulbs connected by a porous plug. The bulbs are maintained at different temperatures of 200 and \(600 \mathrm{~K}\). The
Verify that both the pressure and the vorticity field satisfy the Laplace equation for Stokes flow.Verify that the velocity field satisfies the biharmonic equation\[\begin{equation*}abla^{4}
Show that the force acting on a control surface of any arbitrary control volume is equal to the force on a larger regularly shaped control volume enclosing the given body. In order to do this you
Confirm that the definition of the streamfunction given in the text satisfies the continuity equation.Use this in the N-S equation and derive the form for the \(E^{4}\) operator in spherical
Consider the flow in cylindrical coordinates with no dependence on \(\theta\). How is the streamfunction defined? What is the non-vanishing component of vorticity. Show that the \(E^{2}\) operator
Derive the general solution for \(\psi\) given in the text (Eq. (15.9)). 0 (r,n) = A++ Bnr "+1 + Cnr -" + Dnr"1Qn(n) 2-n (15.9) n=1
What are units for \(C_{1}\) in Eq. (15.10)? Confirm that the units agree on both sides. F = 4 (15.10)
A sphere of radius \(R\) is rotating in an infinite fluid at an angular velocity of \(\Omega\). Derive the following expression for the velocity field:\[v_{\phi}=r \Omega \sin \theta
Find the solution to Stokes flow past a sphere where the far-field velocity satisfies the elongational flow defined as \(v_{x}=\dot{\gamma} x, v_{y}=\) \(\dot{\gamma} y\), and \(v_{z}=-2 \dot{\gamma}
The general solution to Stokes flow in 2D Cartesian coordinates. For the 2D case the governing equation is \(abla^{4} \psi=0\). The operator \(abla\) may be applied either in Cartesian \((x, y)\) or
The case of flow past a cylinder of infinite length normal to the axis was also studied by Stokes. In view of the 2D nature of the problem, it is more convenient to work in \(r, \theta\) coordinates.
Verify the vector identity\[\begin{equation*}abla^{2} \boldsymbol{v}=abla(abla \cdot \boldsymbol{v})-abla \times(abla \times \boldsymbol{v}) \tag{15.83}\end{equation*}\]
Show that, for irrotational axisymmetric flows in cylindrical coordinates, the streamfunction satisfies the following equation:\[\frac{\partial^{2} \psi}{\partial r^{2}}-\frac{1}{r}
Indicate the type of flow given by \(\phi=\sqrt{r} \cos (\theta / 2)\). Calculate and plot typical streamlines.
Consider a source of flow in \(3 \mathrm{D}\) of strength \(S\) (dimensions \(L^{3} / T\) ), Then the flow is axisymmetric and independent of the \(\theta\)-direction in the spherical coordinate
Verify the principle of superposition, which states that if \(\phi_{1}\) and \(\phi_{2}\) are solutions to potential flow then \(\phi=c_{1} \phi_{1}+c_{2} \phi_{2}\) is also a solution.Also show that
Verify by direct substitution that the functions \(r^{n} \cos (n \theta)\) etc. shown in the text satisfy the Laplace equation in polar coordinates.Calculate the streamfunction for the
State the form of the Laplace equation in axisymmetric spherical coordinates.Verify that the following functions satisfy this equation:\[r \cos \theta ; \quad \cos \theta / r^{2}\]A linear
A cylinder of diameter \(1.2 \mathrm{~m}\) and length \(7.5 \mathrm{~m}\) rotates at \(90 \mathrm{r}\).p.m. with its axis perpendicular to an air stream with an approach velocity of \(3.6 \mathrm{~m}
Model equations similar to those for potential flow arise in flow in porous media, which has a wide variety of applications, e.g., in groundwater treatment, water-purity remediation, filtration, flow
One way of cleaning up environmentally polluted underground water is to pump it out of the ground, treat it with some equipment (by catalytic or photochemical oxidation, for example), and then pump
Consider the flow past a flat plate. Here we examine the effect of a superimposed velocity \(v_{y 0}\) at the plate surface. This can be done by the integral method with only minor changes in the
Find the value of \(m\) for which the wall shear stress is independent of the principal flow direction.Find the value of \(m\) for which the boundary-layer thickness is a constant.
Consider the problem of a semi-infinite fluid subject to a constant shear at the interface. This can be caused, for instance, by a surface-tension gradient. Show that the following differential
Von Kármán assumed a cubic profile for the integral momentum analysis over a flat plate. Since a cubic has four constants, four conditions were used.(i) \(V_{x}=0\) at \(y=0\).(ii)
Consider a boundary layer with no external pressure gradient (a flat plate). Then, from the Prandtl boundary-layer equation, deduce that\[\mu \frac{d^{2} v_{x}}{d y^{2}}=0 \text { at } y=0\]Now
A cubic approximation is commonly used in conjunction with the von Kármán momentum integral. An alternative form is the sine function:\[v_{x}=\alpha \sin (b y)\]What should the constants \(\alpha\)
Follow up the derivations leading to the Blasius equation leading to\[f^{\prime \prime \prime}+f f^{\prime \prime}=0\]A useful routine to solve this is BVP4C in MATLAB. Solve the Blasius equation
The following system of differential equations arises in the modeling of a stirred tank reactor with an autocatalyic reaction:\[\begin{aligned}& \frac{d x}{d t}=x-x y \\& \frac{d y}{d t}=-y+x
The Lorenz equation is widely used in the theory of non-linear equations and was encountered in modeling of natural convection. While working with these equations Lorenz discovered the phenomena of
Analyze the Frank-Kamenetskii problem for the three standard geometries of slab, cylinder, and sphere. You will need to discretize the operators suitably for the cylinder and sphere. Plot the
Verify the integral obtained by Rayleigh and hence show that the velocity profile needs to have an inflexion point for instability. Show that a simple shear flow is stable. Hence viscosity is needed
Derive the kinematic and dynamic conditions needed in the analysis. Set up the equations to find the constants. Now require that the determinant of the coefficient matrix should be zero to obtain a
A shear layer between two fluids. Assume the following velocity distribution between two shear layers:\[v_{x}(y)=v_{0} \tanh (y / \delta)\]This is known as the Betchov and Criminale form. Calculate
Stability of flow in torsional flow. Taylor determined the critical speed of rotation for flow between concentric cylinders with the inner cylinder rotating. The transition is characterized by a
Would the critical Rayleigh number for flow transition for the Bénard problem increase or decrease with the Prandtl number? Explain why in terms of the physics of the problem.
Stability analysis with heat transfer. Set up the equations for steady state for the Bénard problem.Now perturb the temperature and use an energy equation to derive an equation for the temperature
Chandrasekhar (1961) has shown that the above (Bénard) problem can be solved in terms of the vorticity, which reduces the problem to a sixth-order eigenvalue problem for the perturbation in
Search the literature or web and discuss briefly the principles behind the following flowmeasurement devices: a Pitot tube; a hot-wire anemometer; a laser-Doppler velocity meter; and a
Explain the significance of the following cross-correlation terms:\[\begin{aligned}& \overline{v_{x}^{\prime} v_{y}^{\prime}} \\& \overline{v_{x}^{\prime} T^{\prime}} \\& \overline{v_{x}^{\prime}
The following data were obtained for the velocity profiles in turbulent flow at a specified point in a channel flow as a function of time. The measurements were taken 0.01 seconds apart. The velocity
Verify that the fluctuating component of velocity (2D assumption) satisfies the following equation:\[\begin{equation*}\frac{\partial v_{x}^{\prime}}{\partial x}+\frac{\partial
The velocity profile for turbulent flow in circular pipes is often approximated by the \(1 / 7\) thpower law:\[\bar{v}_{z}=v_{\max }(1-r / R)^{1 / 7}\]Find an expression for the cross-sectionally
Use the equation (17.19) suggested by Pai for the turbulent stress and integrate for the velocity profile. How do the results compare with that of Prandtl? = 0.9835 H+ (1-#)['-(1-#)'] (17.19) H+
Water is flowing through a long pipe of diameter \(15 \mathrm{~cm}\) at \(300 \mathrm{~K}\). The pressure gradient is \(500 \mathrm{~Pa} / \mathrm{m}\).Using the Blasius equation for the friction
Water is flowing in a pipe of diameter \(20 \mathrm{~cm}\) with a pressure gradient of \(3000 \mathrm{~Pa} / \mathrm{m} . \mu=\) \(0.001 \mathrm{~Pa} \cdot \mathrm{s}\).Find the wall shear
Consider a fully developed flow of water in a smooth pipe of diameter \(15 \mathrm{~cm}\) at a flow rate of \(0.006 \mathrm{~m}^{3} / \mathrm{s}\).The pressure drop can be calculated by the Blasius
If the RANS equation (17.5) is subtracted from the unaveraged \(\mathrm{N}-\mathrm{S}\) equation (17.1), we obtain the following equation for the perturbation
Take the product of the perturbation velocity equation given by Eq. (17.53) by any component of the perturbation velocity. This results in an equation for \(v_{x}^{\prime} v_{y}^{\prime}\), which is
Show that, for the isotropic case, (i) the velocity correlation function is now symmetric and (ii) the pressure-velocity correlation is zero.
Extend the analysis of the laminar-flow reactor for a power-law fluid. Perform some computations using the PDEPE solver, and show how the power-law index affects the conversion in the reactor.
Perform the order-of \(-\epsilon^{2}\) approximation for the problem of diffusion with reaction in a catalyst of variable activity. Compare the flux with that obtained from the BVP4C solver in
Express the Henry's-law constants reported in Table 8.1 as \(H_{i, p c}\) and \(H_{i, c p}\). Dimensionless temperature, 0.9 0.8 0.7 0.6 K=1+ (0-1/2) 0.5 0.4 0.3 0.2 0.1 K = 1 0 0.2 0.4 0.6 0.8
It is common to rearrange Eq. (6.57) to a friction-factor and Reynolds-number form similar to that for a Newtonian fluid. The friction factor is defined as usual by Eq. (1.22):\[f=\frac{1}{4}
Find the binary pair diffusivity for the system methane (A)-ethane (B) at \(293 \mathrm{~K}\) and \(1 \mathrm{~atm}\) by using the Lennard-Jones method given by Eq. (1.53) by the following
Consider expansion of a function in terms of a series \(F_{n}(x)\) in the following form:\[f(x)=\sum_{n} A_{n} F_{n}(x)\]If the functions \(F_{n}\) are orthogonal then this property helps to unfold
The fact that the eigenfunctions are orthogonal can be verified easily using the symbolic calculations in MATLAB or MAPLE. But the underlying theory is based on the SturmLiouville problem. You may
There is nothing magical about the eigenfunctions being orthogonal. This can be shown by integration by parts twice.Consider two eigenfunctions \(F_{n}\) and \(F_{m}\), both of which satisfy the
Analyze the transient problem with the Dirichlet condition for a long cylinder and for a sphere. Derive expressions for the eigenfunctions, eigenconditions, and eigenvalues. Find the series
A concrete wall \(20 \mathrm{~cm}\) thick is initially at a temperature of \(20^{\circ} \mathrm{C}\), and is exposed to steam at pressure \(1 \mathrm{~atm}\) on both sides. Find the time for the
A finite cylinder is \(2 \mathrm{~cm}\) in diameter and \(3 \mathrm{~cm}\) long and at a temperature of \(200^{\circ} \mathrm{C}\), and is cooled in air at \(30^{\circ} \mathrm{C}\). The convective
For the Biot problem in a slab by expanding the sin and cos term and keeping only terms up to \(\lambda^{2}\) the following approximate relation can be obtained for the eigenvalues for small Biot
Consider the problem of transient heat transfer with a constant heat source in a slab.Show that the governing equation in dimensionless form is\[\begin{equation*}\frac{\partial \theta}{\partial
Eigenvalues without pain: CHEBFUN code. Eigenfunctions can be derived using the CHEBFUN with MATLAB since it has an overloaded eig function. The following code solves for the eigenfunctions of the
A hot dog at \(5^{\circ} \mathrm{C}\) is to be cooked by dipping it in boiling water at \(100^{\circ} \mathrm{C}\). Model the hot dog as a long cylinder with a diameter of \(20 \mathrm{~mm}\). Find
Assume that the asympotic solution has a time-dependent part, which is linear in time, and a positiondependent solution, which is an unknown function. Thus the solution should be of the following
A slab has a thermal diffusivity of \(5 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\) and is fairly thick. The initial temperature is \(300 \mathrm{~K}\) and the surface temperature is raised to
Consider the problem of transient diffusion in a composite slab with two different thermal conductivities. Thus region 1 extending from 0 to \(\kappa\) has a thermal conductivity \(k_{1}\), while the
A large tank filled with oxygen has an initial concentration of oxygen of \(2 \mathrm{~mol} / \mathrm{m}^{3}\). The surface concentration on the liquid side of the interface is changed to \(9
Consider the drug-release problem in the text, but now the drug is in the form of a cube with sides \(0.652 \mathrm{~cm}\). Find the center concentration after \(48 \mathrm{~h}\) and the fraction of
A porous cylinder, \(2.5 \mathrm{~cm}\) in diameter and \(80 \mathrm{~cm}\) long, is saturated with alcohol and maintained in a stirred tank. The alcohol concentration at the surface of the cylinder
A gas bubble of diameter \(3 \mathrm{~mm}\) is rising in a pool of a liquid. What is the mass transfer coefficient if the diffusion coefficient is \(2 \times 10^{-5} \mathrm{~cm}^{2} / \mathrm{s}\)
Consider again the periodic variation of temperature in a semi-infinite domain analyzed. Sketch or plot the temperature for three values of time. Show that a wave type of propagation of the
In a flow distribution network that progresses from a large tube to many small tubes (e.g., a blood-vessel network), the frequency, density, and dynamic viscosity are (usually) the same throughout
Transient heat conduction in citrus fruits is of concern to farmers in Florida and other farmers as well. They must develop strategies to prevent freezing during cold weather.The thermal diffusivity
Use the thermal-diffusivity value estimated from the above problem to find out whether any part of a fruit of diameter \(8 \mathrm{~cm}\) hanging in a tree will freeze if the ambient temperature
A membrane of thickness \(L\) separates two solutions. Both solutions and the membrane have initially a zero concentration of a permeable solute. At time \(t=0\) and thereafter one side is maintained
For gas absorption with a semi-infinite region with reaction show that the following limiting values for the flux into the system can be derived starting from the detailed equation in the
Consider the above problem of gas absorption, but now for the limit for large values of \(k t\), and show that\[N_{\mathrm{A}, \mathrm{s}}=C_{\mathrm{A}, \mathrm{s}} \sqrt{D_{\mathrm{A}} k} \text {
A gas stream with \(\mathrm{CO}_{2}\) at partial pressure \(1 \mathrm{~atm}\) is exposed to liquid in which it undergoes a first-order reaction for \(0.01 \mathrm{~s}\). The total amount of gas
Extend the penetration model for two species reacting instantaneously. Assume that the solution on either side of the reaction front in Fig. 11.8 can be expressed in terms of error function which
Complete the solution to transient channel flow with a pressure gradient using the separation of variables. Also solve the problem numerically using the PDEPE or other solvers and compare the results.
Assuming the pressure gradient to be zero. Set up the problem and state the boundary conditions. Verify that the boundary conditions lead to a non-homogeneity. Obtain the transient solution by
Pipe flow with periodic pressure variation. Verify the result for the velocity profile in the complex domain. Write MATLAB code to find the real and imaginary parts. Use this code to plot the
The rheology of blood is nonNewtonian and is often represented by the Casson fluid model. Your project is to examine the blood flow with this model and examine the conditions under which the
Consider the differential equation given by Eq. (12.2). Use a coordinate transformation \(\zeta=z^{*} P e^{a}\), where \(a\) is some index to be chosen suitably. Show that in the transformed equation
Show that the differential equation for \(F\) given in the text can also be written as\[\begin{equation*}\xi \frac{d^{2} F}{d \xi^{2}}+\frac{d F}{d \xi}+\lambda^{2} \xi\left(1-\xi^{2}\right) F=0
This exercise demonstrates how to find the eigenfunctions using CHEBFUN. The procedure is the same as that for the transient conduction prolbem. The first step is to declare \(\xi\) as a CHEBFUN and
Show all the steps leading to (12.12) and verify that the Nusselt number has a value of 48/11 for this case. 7 8(5,5)=45+52. - (12.12) 4 24
Consider laminar flow between two parallel plates. The plates are electrically heated to give a uniform inward wall flux \(-q_{\mathrm{w}}\). Set a mesoscopic model to find the cup mixing average
Repeat the analysis if only one plate is exposed to the constant flux while the other plate is kept insulated. What is the value of the Nusselt number for this case?
Show all the steps leading to Eq. (12.15).Integrate once, using the substitution \(p=d \theta / d \eta\).Now integrate a second time to get the temperature distribution given by Eq. (12.16) in terms
Consider a liquid flowing down a vertical wall at a rate of \(1 \times 10^{-5} \mathrm{~m}^{2} / \mathrm{s}\) per meter unit width. Find the concentration at a height \(25 \mathrm{~cm}\) below the
A pipe of diameter \(2 \mathrm{~cm}\) with an oil flow rate of \(0.02 \mathrm{~kg} / \mathrm{s}\) heated by a wall temperature of \(400 \mathrm{~K}\). The inlet temperature is \(300
Consider the oil flowing in a pipe in Problem 10.The inlet temperature is \(300 \mathrm{~K}\) and the pipe is now heated electrically at a rate of \(76 \mathrm{~W} / \mathrm{m}^{2}\). Plot the wall
Heat transfer in laminar flow with internal generation of heat is called the Brinkman problem. The additional dimensionless group needed here is the Brinkman number. Numerical solutions can be
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