All Matches
Solution Library
Expert Answer
Textbooks
Search Textbook questions, tutors and Books
Oops, something went wrong!
Change your search query and then try again
Toggle navigation
FREE Trial
S
Books
FREE
Tutors
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Ask a Question
Search
Search
Sign In
Register
study help
engineering
chemical engineering
Questions and Answers of
Chemical Engineering
Extend the analysis of gas absorption with reaction to the case of absorption of two gases with a common liquid-phase reactant. A detailed study of this topic has been published in an award-winning
Condensation rates for a binary vapor mixture (adapted from Taylor and Krishna (1993)). Here we apply Eq. (10.22) to the condensation of a binary mixture of ethylene dichloride (A) and toluene (B)
Code for pore diffusion with CHEBFUN is shown below. Note how compact the code is compared with BVP4C. The backslash operator is used as an overloaded operator here.Test the code which is applicable
A membrane separator is \(3 \mathrm{~mm}\) in diameter, and the membrane permeability was estimated as \(2 \times 10^{-6} \mathrm{~m} / \mathrm{s}\). The solute being transported has a diffusivity of
Develop a simple backmixed model to evaluate the performance of a gas-separation system. Here a feed gas enters a high-pressure chamber with a mole fraction of A of \(x_{\mathrm{Af}}\) and leaves
Pervaporation: a case-study problem. Pervaporation refers to removal of the permeate as vapor and represents an intermediate case between purely gas transport and purely liquid transport in a
Find the acceleration of a fluid particle for the following velocity profiles:(a) \(v_{x}=A\left(1-y^{2}\right) ; v_{y}=0 ; v_{z}=0\)(b) \(v_{x}=A x ; v_{y}=-A y ; v_{z}=0\)(c) \(v_{r}=0 ;
Derive an expression for \(D T / D t\) in Cartesian coordinates.Also show that the same relation holds for a vector; i.e.,\[\frac{D \boldsymbol{v}}{D t}=e_{x} \frac{D v_{x}}{D t}+e_{y} \frac{D
In cylindrical coordinates, the temperature field (a scalar field) can be expressed as\[T=T(r, \theta, z, t)\]Based on this, derive an expression for \(D T / D t\) in cylindrical coordinates.
In spherical coordinates, the temperature field can be expressed as\[T=T(r, \theta, \phi, t)\]Derive from this an expression for \(D T / D t\) in spherical coordinates.
Derive an expression for \(D \boldsymbol{v} / D t\) in cylindrical coordinates.Define the nabla operator in cylindrical coordinates and let it be dotted with a vector. Expand and use the chain rule,
Derive an expression for \(D \boldsymbol{v} / D t\) in spherical coordinates. Follow the same procedure as for cylindrical coordinates. Some unit vectors have derivatives. The components in Table 3.2
Derive the expressions for the divergence of a vector in cylindrical and spherical coordinates. Thereby verify the results in the text.
Prove that \(abla \cdot(abla \times A)=0\); i.e., the divergence of the curl of a vector is zero. Use this result to prove that the vector potential of velocity satisfies the continuity equation.
Show that the necessary and sufficient condition for the flow to be irrotational is the existence of a scalar velocity potential.
Write the units for the (i) vorticity, (ii) circulation, (iii) velocity potential, and (iv) streamfunction.
By direct differentiation using Cartesian coordinates verify the following relations.(a) The divergence of the curl of any vector is zero.\[abla \cdot(abla \times \boldsymbol{A})=0\](b) The curl of
Show by direct differentiation that the divergence of the vorticity vector is zero.\[abla \cdot \boldsymbol{\omega}=0\]
Derive expressions for the components of the curl operator in cylindrical polar coordinates. Using these relations, find the vorticity field for the flow described by the followingvelocity
Show that for a 2D flow or plane confined to the \((x, y)\) plane only the \(z\)-component of the vorticity is non-zero. This component, \(\omega_{z}\), is simply abbreviated as \(\omega\) and
Consider a 2D plane flow that is now represented in terms of the polar coordinates. The flow has then only \(v_{r}\) and \(v_{\theta}\) components and no \(v_{z}\) component. How is the
Write out the components of the vorticity tensor, \(\tilde{W}\) in Cartesian coordinates. Show that the tensor is antisymmetric, i.e., \(W_{i j}=-W_{j i}\), and has only three distinct
Consider the simple shear flow described as \(v_{x}=\dot{\gamma} y\) and \(v_{y}=0\), where \(\dot{\gamma}\) is the rate of strain.Verify that the rate of strain has only shear components as shown in
Derive expressions for the gradient of velocity in cylindrical and spherical coordinates. Define the nabla operator in cylindrical/spherical coordinates and let it act on a vector. Use the chain
Is \(\boldsymbol{n} \cdot \tilde{\tau}\) equal to \(\tilde{\tau} \cdot \boldsymbol{n}\) in general? When will they be the same?
Indicate the direction and the plane over which the following stress quantities act.Rectangular: \(\tau_{-x, x} ; \tau_{-y,-y} ; \tau_{z,-y}\).Polar: \(\tau_{-r, \theta} ; \tau_{\theta,
A stress tensor in two dimensions has the following components at a given point: \(\tau_{x, x}=3\), \(\tau_{x, y}=2\), and \(\tau_{y, y}=2\). Find the stress vector on a plane that is inclined at an
The operator \(abla\) can be considered to be a vector operator.Here \(\tau\) is considered as a dyadic operator defined as\[\tau=\sum_{i} \sum_{j} e_{i} e_{j} \tau_{i j}\]With these definitions it
The average ocean depth is \(2 \mathrm{~km}\). Compute the pressure at this point. Assume a constant density.The change in density of water with pressure is small, and can be represented using the
Derive Eq. (4.13) for pressure variation in the atmosphere with elevation. Find the pressure at Shangri-La, which is about \(3000 \mathrm{~m}\) above the sea level. (In the Tibetan language
Consider a lighter solid of density \(ho_{\mathrm{s}}\) floating on the surface of a liquid of density \(ho_{1}\). Derive an expression for the volume fraction for the solid submerged inside the
Consider a circular viewing port on an aquarium. This window has a radius of \(R\), and the center of this port is at a depth \(d+R\) from the water surface. Find the force and the center of pressure
A process requires the delivery of drops of volume \(3.2 \times 10^{-8} \mathrm{~m}^{3}\). A liquid has a density of \(900 \mathrm{~kg} / \mathrm{m}^{3}\) and a surface tension of \(0.03 \mathrm{~N}
Calculate the settling velocity of a spherical particle of diameter \(2.2 \mathrm{~cm}\) with a density of \(2620 \mathrm{~kg} / \mathrm{m}^{3}\) in a liquid of density of 1590 and a viscosity of
Consider the motion of the particle in the initial stages, i.e., before it reaches the terminal velocity. Include the acceleration terms in the momentum balance of the particle and derive the
Write the divergence of the dyad \(ho \boldsymbol{v} \boldsymbol{v}\) in index notation. Expand the derivatives using the chain rule.Write the continuity equation in index notation and use this in
Write the equation of motion in cylindrical coordinates using the various vector operations. Also write the equation of continuity. These equations together are the needed equations for the solution
Repeat for spherical coordinates.
Write the stress vs. rate-of-strain relations in cylindrical and spherical coordinates for Newtonian fluids.
Unidirectional flows in 2D Cartesian coordinates (also known as channel flows) are defined as systems with only one velocity component, say \(v_{x}\). How does the continuity simplify for such
Unidirectional flows can also be posed as \(v_{x}\) as a function of \(y\) and \(z\). A flow in a square duct (away from the entrance region) is an example. How does the Navier-Stokes equation
Simplify the Navier-Stokes equation for flows in a circular pipe with only the axial velocity \(v_{z}\) as the non-vanishing component.
Determine the flow rate of water at \(25^{\circ} \mathrm{C}\) in a 3000 -m-long pipe of diameter \(20 \mathrm{~cm}\) under a pressure gradient of \(20 \mathrm{kPa}\). Assume a relative roughness
What form does the Colebrook-White equation take for a smooth pipe? Compare this with the Prandtl formula by plotting the values on the same graph.
Verify the steps leading to the vorticity-streamfunction formulation of the \(\mathrm{N}-\mathrm{S}\) equation for 2D flow.If the flow has no vorticity it is called irrotational flow. What form does
Derive the vorticity-velocity formulation of the \(\mathrm{N}-\mathrm{S}\) equation and discuss the advantages and disadvantages of using this formulation. Take the cross product of the NavierStokes
Show that the vorticity transport equation can also be written as\[\frac{D \omega}{D t}=v abla^{2} \omega+[\omega \cdot abla] \boldsymbol{v}\]How does it simplify for 2D flows? How does it simplify
Show all the steps leading to the pressure Poisson equation.
Derive the following form of the pressure Poisson equation shown in the book by Saffman (1993) on vortex dynamics:\[abla^{2} p=ho(\tilde{W}: \tilde{W}-\tilde{E}: \tilde{E})\]where \(\tilde{W}\) is
Consider the case of channel flow where the top plate at \(y=H\) is moving at a velocity of \(U_{1}\) and the bottom plate at \(y=0\) is moving with a velocity of \(U_{0}\). We also impose a pressure
Consider the flow in a conduit whose cross-section has the shape of an equilateral triangle. State the differential equations and the associated boundary conditions.Show that the following expression
Consider the flow in a conduit whose cross-section has the shape of an ellipse. State the differential equations and the associated boundary conditions. Show that the following expression satisfies
The following solution is presented for fully developed flow in a square duct that has a half width of \(B\) in the \(y\)-and \(z\)-directions.\[v_{x}=\frac{G B^{2}}{4
A simple model for a tornado is a central core of radius \(R\) rotating at an angular velocity of \(\Omega\) and an outer region. The flow is assumed to be tangential in both regions. Derive an
Verify the expression (6.28) for torque for flow between two cylinders with the outer cylinder rotating. What would the corresponding result be if the inner cylinder were rotating? T = 4R ( (6.28) K
For flow in a square channel the shear stress is not a constant along the perimeter, unlike in a cylindrical channel. Obtain the stress distribution in a rectangular channel. At what point is the
Show that the superposition of a radial velocity field and a torsional field results in a spiral flow.Sketch typical streamlines. Spiral flows are good prototypes for tornadoes.Explain why the
Find the point where the pressure is a maximum and find the value of this pressure.Calculate the tangential and normal stresses on the top plate.Calculate the tangential stress on the bottom plate.
Consider the flow of water in a pipe of length \(2 \mathrm{~m}\) with an imposed pressure difference of \(1000 \mathrm{~Pa}\). Find the flow rate if (a) the pipe has a uniform cross-section of
Two parallel disks of radius \(R\) are separated by a distance \(H\). The space between them is filled with an incompressible fluid. The top plate is moved towards the bottom at a constant velocity,
Consider the flow over a cone as shown in Fig. 6.21. Find the film thickness as a function of distance along the surface of the cone. Use the lubrication approximation. Repeat for the solid sphere in
A solution of \(13.5 \%\) by weight of polyisoprene has the following power-law parameters: \(\Lambda=5000 \mathrm{~Pa} \cdot \mathrm{s}^{n}\) and \(n=0.2\).Consider the flow of such a solution in a
A power-law fluid has the following rheological constants: \(n=0.5\) and \(\Lambda=0.8 \mathrm{~Pa} \cdot \mathrm{s}^{n}\).It is pumped in a pipe of diameter \(2 \mathrm{~cm}\) and length \(5
A \(1-\mathrm{m}\) long pipe delivers a fluid with a power-law index of 0.5 at the rate of \(0.02 \mathrm{~m}^{3} / \mathrm{s}\). The pressure drop of the pipe across the system is \(3.5 \times
The model equation takes the following form for pipe flow:\[\begin{equation*}\sqrt{-\tau}=\sqrt{\tau_{0}}+s \sqrt{-\left(d v_{z} / d r\right)} \tag{6.93}\end{equation*}\]since both the shear stress,
The three-constant Ellis model (Eq. (5.43)) can describe a wide range of experimental data for many fluids. The paper by Matsuhisa and Bird (1965) provides a detailed analysis of this case. the
Sketch the velocity profiles for the magnetohydrodynamic for the two cases of insulating and conducting walls, and compare the profiles.
Show that the steady-state exit concentration for a first-order reaction is \(1 /(1+D a)\). What is the corresponding expression if the reactor is modeled as a plug-flow reactor? Produce a comparison
Bolus injection is a very useful and important tool in pharmacokinetic analysis. Repeat the analysis for a bolus injection of a tracer that undergoes a first-order reaction. Sketch typical
Consider a reactor modeled as \(N\) interconnected tanks in series. Derive an expression for the exit response for a bolus injection of a tracer.Show that the maximum value of the tracer
Consider the system of IVP in the matrix form which is repeated here for convenience:\[\frac{d \boldsymbol{y}}{d t}=\tilde{A} \boldsymbol{y}+\boldsymbol{R}\]Eigenvalue representation is a useful and
Consider again the IVPs shown above.If \(R\) is time-varying, i.e., \(R(t)\), then show that the solution can be formally written in terms of the exponential matrix as\[\boldsymbol{y}=\exp (\tilde{A}
Consider the series reaction scheme in a constant batch reactor:\[\mathrm{A} \xrightarrow{k_{1}} \mathrm{~B} \xrightarrow{k_{2}} \mathrm{C} \xrightarrow{k_{3}} \mathrm{D}\]Assuming all reactions are
The following data were found in response to a drug injected as a pulse:How good is the fit to a one-compartment model? What is the time constant?How good is the fit to a two-compartment model? What
CHEBFUN helps to make the MATLAB codes easier, and allows you to work with solutions as though they were analytic functions. You will find it very useful once you start using these. I will walk you
This code shows how CHEBFUN can be used with MATLAB ODE45 to give a "symbolic" look to your answer. The presence of the calling argument domain in the calling statement creates a Chebyshev polynomial
The velocity profile in laminar flow in a pipe is a parabolic function of \(r\) and can be represented as\[v_{z}=v_{\mathrm{c}}\left[1-(r / R)^{2}\right]\]The parameter \(v_{\mathrm{c}}\) appearing
The velocity profile in turbulent flow in a pipe is often approximated by a one-seventh power law:\[v_{z}=v_{\mathrm{c}}[1-(r / R)]^{1 / 7}\](a) Calculate the following: \(\langle vangle,\left\langle
A turbine discharges \(40 \mathrm{~m}^{3} / \mathrm{s}\) of water and generates \(42 \mathrm{MW}\) of power. The rotational speed is 24 r.p.s. Fluid enters at a radius of \(1.6 \mathrm{~m}\) with a
A compressor delivers natural gas through a straight pipe of length \(L\). Find the pressure at the exit. Include frictional losses in the pipe.The energy balance should now include the frictional
Water flows from an elevated reservoir through a conduit to a turbine at a lower level through a conduit of the same diameter. The elevation difference is \(90 \mathrm{~m}\). The inlet pressure is
Momentum analysis for a rocket. A rocket has a mass of \(m_{\mathrm{R}}\) and carries fuel with mass \(m_{\mathrm{f}}\) at a given instant of time. Thus the total mass of the system at the current
Integrate the equation in the above problem analytically to find the velocity of the rocket as a function of time.Also write a MATLAB code to integrate the system numerically using the ODE45 solver.
A water jet is deflected through \(180^{\circ}\) by a vane that is attached to a cart, which is free to move. Assume that the flow along the vane is frictionless and neglect gravity. The jet enters
Consider the flow in a U-bend in a pipe. Water is flowing at a rate of \(80 \mathrm{~cm}^{3} / \mathrm{s}\) in a pipe of diameter \(10 \mathrm{~cm}\). The inlet pressure is \(1.2 \mathrm{~atm}\). The
Consider the draining of a tank with three different flow arrangements at the exit as shown in Fig. 2.25. Develop an equation to calculate the level of liquid in the tank as a function of time for
A compressor is to be designed to provide a compression ratio of 4 . The inlet temperature is \(27^{\circ} \mathrm{C}\). The \(C_{p}\) may be assumed to be a constant, \(38.9 \mathrm{~J} /
Show that for the windmill problem (Example 2.11) the maximum value of power for a constant wind incoming velocity is achieved when \(v_{2}=v_{1} / 3\). Find the corresponding power and the
Kundu and Cohen (2008) make the following statement which is counter-intuitive: Friction can make the fluid go faster in an adiabatic flow in a channel of constant cross-section. Develop a model for
Consider a natural-gas pipe-line of inner diameter \(60 \mathrm{~cm}\) and length \(16 \mathrm{~km}\). The pressure at the inlet is \(7 \mathrm{~atm}\) and the temperature is \(20^{\circ}
Derive the following equation for the conditions across a shock shown in Example 2.13.\(M a_{1}\) and \(M a_{2}\) are the Mach numbers before and after the shock.Example 2.13:A normal shock wave
An aluminum plate is at a temperature of \(25^{\circ} \mathrm{C}\). It is then suddenly subjected to a uniform and continuing heating at the rate of \(6 \mathrm{~W} / \mathrm{m}^{2}\). Develop a
A compressible gas of nitrogen is contained in a tank and is discharged thorough a small convergent nozzle. The pressure at the exit is maintained at \(p_{2}\).Find the instantaneous discharge rate
Nuclear reactor analysis. A nuclear fuel rod is generating heat and is in the shape of a cylinder of radius \(R\). The surface is losing heat with a heat transfer coefficient of \(h\) to the
Sketch the qualitative time-temperature profiles in the sphere for the following limiting cases.(a) Cooling is such that the Biot number is much less than one.(b) Cooling is such that the Biot number
A fluid is flowing in an annular region formed by two concentric cylinders. The inner cylinder has radius \(R_{\mathrm{i}}\) and the outer cylinder has radius \(R_{0}\). Develop an expression for the
For the problem of heat transfer in a pin fin, solve the equations for the temperature profile for a fin at temperature \(T_{0}\) at the base and with no heat loss at the edge of the fin.
Develop mesocopic models for heat transfer from extended surfaces of various shapes shown in Fig 2.26. Circular Tapered Annular disk Figure 2.26 Various arrangement of extended surfaces to be modeled
A dynamic model for a tank with jacket cooling or heating. The mathematical representation for the fluid in the tank is the same with \(T_{\mathrm{S}}\) replaced by \(T_{\mathrm{C}}\) in Example 2.15
Explain briefly the following terminologies.- Continuum models- Control volume and control surface- Conservation laws- Constitutive models or laws- Differential or microscopic models- 1D or
Showing 300 - 400
of 895
1
2
3
4
5
6
7
8
9