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engineering
chemical engineering
Questions and Answers of
Chemical Engineering
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
Nuclear reactors are accompanied by conversion of mass into energy according to the famous equation \(E=m c^{2}\). Hence the mass balance does not hold in such systems. Calculate the mass loss in a
Indicate some situations where the continuum models are unlikely to apply.
How many molecules are there in a cube of size \(1 \mathrm{~nm}\) in each direction for(a) air under standard conditions and(b) water under standard conditions? If you are modeling transport in a
The viscosity data for water as a function of temperature are given in Table 1.3. Fit an equation of the type\[\mu=A \exp (B / T)\]to represent the data. This equation is called the Andrade equation
Viscosity data for air as a function of temperature are shown in Table 1.1. Fit a power-law model of the type\[\mu=A T^{n}\]to represent the data. Note that the viscosity of gases increases with
A tank is being filled by having a flow at the rate of \(\dot{m}\) into the system. Unfortunately there is a tiny hole at the bottom and the water is leaking out of the system. Develop a basic model
Verify the following relations by (i) graphical construction of vectors and the parallelogram law and (ii) by transforming the vectors to Cartesian coordinates and then taking the
How are the unit vectors in cylindrical coordinates related to unit vectors in Cartesian coordinates? Write this in a vector-matrix form as\[E=L E^{*}\]where \(E\) are the components of the unit
State or derive the relation between partial derivatives such as \(\partial / \partial x\) and those in cylindrical coordinates. Use this or some other method to derive a formula for \(abla T\) in
The Laplacian of a scalar is important in mass and heat transfer applications, and is obtained by applying the operator \(abla \cdot\) the gradient of a scalar, e.g., to \(abla T\). By applying this
Derive the following corresponding expression for the Laplacian in spherical coordinates:\[\begin{equation*}abla^{2} T=\frac{1}{r^{2}} \frac{\partial}{\partial r}\left(r^{2} \frac{\partial
Flow experiments are done with water in a pipe of diameter \(40 \mathrm{~cm}\). Find the flow rate at which the flow ceases to be laminar.
Note that every equation has to be dimensionally consistent. The units on the RHS of any equation have to be the same as the units on the LHS. Verify that the Hagen-Poiseuille equation Eq. (1.20) is
Glycerine is flowing in a small capillary of diameter \(0.25 \mathrm{~cm}\) and length \(30 \mathrm{~cm}\). For a pressure difference of 3 bars applied across the tube a flow rate of \(0.001
The linear relation given by the Hagen-Poiseuille equation, Eq. (1.20), is reminiscent of Ohm's law for current as a function of voltage in a current-carrying wire,\[I=\frac{V}{\Omega}\]where \(I\)
A \(32 \mathrm{~km}\)-long pipeline delivers petroleum at the rate of \(2000 \mathrm{~m}^{3}\) per day. The pressure drop of the pipe across the system is \(34.5 \mathrm{MPa}\). Find the resistance
Repeat the analysis for the above problem if the flow is turbulent. Note that the resistance concept is not useful here. Why?Above problem:A \(32 \mathrm{~km}\)-long pipeline delivers petroleum at
An Alaska pipeline is 48 in in diameter and 800 miles long. It carries crude oil from Prudhoe bay to Valdez and is designed to deliver about 2.1 million barrels per day. Use viscosity data from the
A heat exchanger is operated at a flow rate of \(30 \mathrm{~kg} / \mathrm{s}\) and water enters at \(20{ }^{\circ} \mathrm{C}\) and leaves at \(80^{\circ} \mathrm{C}\). Find the length of exchanger
Derive the integrated equations for the concentrations in a plug-flow reactor if the reaction is of (i) second order and (ii) zeroth order.
The speed of molecules according to kinetic theory is given by the Boltzmann distribution function \(f\). Thus \(f(c) d c\) represents the probability that \(c\) lies between \(c\) and \(c+d c\) and
What is the relation between the square of the mean velocity and the mean of the velocity squared in the context of the Boltzmann distribution?
The distribution of the energy of the molecules is also of importance in the kinetics of chemical reactions. The fraction of molecules with energy in the range between \(E\) and \(E+d E\) is given
Find the following properties for oxygen at \(298 \mathrm{~K}\) and \(1 \mathrm{~atm}\) based on the kinetic theory of gases.(a) The average speed of the molecules.(b) The r.m.s. value of the
The simple formula \(C_{p}=(5 / 2) R\) (molar units) is valid only for monatomic gases. A simple extension that has been suggested is\[C_{p}=\left(5+N_{\mathrm{r}}\right) \frac{1}{2} R\]where
Show the details leading to the equation (1.52) in the text,\[D_{\mathrm{AA}}=\frac{1}{3} \bar{c} \lambda\]where \(D_{\mathrm{AA}}\) is the self-diffusion coefficient.
Estimate the thermal conductivity of water if the speed of sound in water is \(1498 \mathrm{~m} / \mathrm{s}\). Compare your answer with the experimental values.
For most liquids other than water and substances with polyhydroxy groups, the thermal conductivity decreases with an increase in temperature. Water shows an anomalous effect. The conductivity, \(k\),
We are interested to produce $P$ in the reaction $A \rightarrow P$ using a continuous reactor at $v=240$ liters/ hr with $C_{A_{0}}=3$ moles/liter. However, it is noticed that there is a second
For installation and operation of a pipeline for an incompressible fluid, the total cost (in dollars per year) can be represented as follows:\[C=C_{1} D^{1.5} L+C_{2} m \Delta p / ho\]where$C_{1}=$
A fertilizer producing company purchases nitrates, phosphates, potash, and an inert chalk base and produces four different fertilizers A, B,C, and D. The cost of these nitrates, phosphates, potash,
Heavy fuel oil, initially semisolid at $15^{\circ} \mathrm{C}$ is to be heated and pumped through a $15 \mathrm{~cm}$ diameter (inside) pipe at the rate of $20000 \mathrm{~kg} / \mathrm{h}$. The pipe
The topological optimization is discussed in chapter 1. Here, we will consider a topological optimization problem for a chemical process plant. The layout of the chemical process plant has been shown
Two grade of coal $(A, B)$ are mixed to get a coal $(C)$ for blast furnace. The composition of coals and cost of coals are given in the table belowTable 2.3Composition and cost of coalCoalComposition
An oil refinery has three process plants, and four grades of motor oil have been produced from these plants. The refinery is liable to meet the demand of customers. The refinery incurs a penalty for
An adiabatic two-stage compressor is used to compress a gas, which is cooled to the inlet gas temperature between the stages, the theoretical work can be expressed by the following
A refinery produce three major products: gasoline, jet fuel and lubricants by distilling crude petroleum from two sources, Venezuela and Saudi Arabia. These two crudes have different chemical
A chemical company has acquired a site for their new plant. They required to enclose that field with a fence. They have 700 meter of fencing material with a building on one side of the field where
Why single-variable unconstrained optimization is important for chemical engineers? Give some examples of the single-variable unconstrained optimization problem.
What is the difference between interpolation and elimination methods?
Describe the Fibonacci numbers.
What is a unimodal function? What are the different methods for optimizing the unimodal function?
Prove that a convex function is unimodal.
Determine the value of $x$ within the interval $(0,1)$ that minimizes the function $f=(x-1)(x-2.5)$ to within $\pm 0.025$ by (a) the Fibonacci method and (b) the golden section method.
Find the minimum value of the function $f=x^{4}-x+1$ using secant method.
Write an algorithm for finding the minimum resistance of heat transfer using Eq. (2.50). Use Newton method for the same. R k In- + 1 2Lk hr 2Lk 2Lk k In- +1 br (2.50)
Find the quadratic approximation of the function around the point (1,1) R(C, t) 77.92 + 9.41C + 3.86t-3.53C-7.33
What are the advantages of trust region methods over line search methods?
Discuss the effect of initial trust region radius (ITRR) on the performance of TR algorithm.
Define trust region fidelity.
Find the maximum of the function using Trust region method \(f=15 x_{1}+8 x_{1} x_{2}+5 x_{2}\) subject to \(x_{1}+x_{2} \leq 10\)
Calculate the reduction ratio \(r_{k}=\frac{f\left(X_{k}\right)-f\left(X_{k}+s_{k}\right)}{m_{k}\left(X_{k}\right)-m_{k}\left(X_{k}+s_{k}\right)}\) for the function given in problem 1 .
Is the result of any Trust Region method depends on the initial guess value \(X_{0} \in \mathbb{R}^{n}\) ? Justify your answer.
Discuss the convergence criteria of Trust Region method, show how the other parameters \(\left(\tau_{1}, \tau_{2}, \tau_{3}\right.\) and \(\left.\tau_{4}\right)\) affect the convergence.
When combination of other methods with Trust region method is advantageous?
Show that one iteration is sufficient to find the minimum of a quadratic function by using the Newton's method.
Give some examples of unconstrained optimization in the field of chemical engineering.
Find the of the function\[f(X)=2 x_{1}^{2}+x_{2}^{2}-2 x_{1} x_{2}\]using Random Jumping method.
The dye removal by $\mathrm{TiO}_{2}$ adsorption (at $\mathrm{pH}$ 5.5) is given by (Anupam K. et al. 2011)\[\text { dye removal }=13.08\left(\mathrm{TiO}_{2}\right)+15.77(\text { Time
What are basic differences between Exploratory move and Pattern move?
Consider the following minimization problem\[f(X)=x_{1}^{2}+2 x_{1} x_{2}+x_{2}^{2}+3 x_{1}\]Find the minimum using Steepest Descent method.
Solve the problem 6 using Newton's method.
What are advantages of Marquardt method over Steepest Descent and Newton's method.
The BFGS method can be considered as a Quasi-Newton, conjugate gradient, and variable metric method- Justify this statement with proper example.
In Quasi-Newton method, the matrix $\left[B_{i}\right]$ is updated using the formula\[\left[B_{i+1}\right]=\left[B_{i}\right]+\frac{\lambda_{i}^{*} s_{i} s_{i}^{T}}{s_{i}^{T}
Is it possible to convert a constrained optimization problem to an unconstrained optimization problem? Explain your answer with proper example.
Maximize $f=5 x_{1}+3 x_{2}$subject to the constraints$2 x_{1}+x_{2} \leq 1000$$x_{1} \leq 400$$x_{2} \leq 700$$x_{1}, x_{2} \geq 0$
Convert the following problem to its dual formMaximize $f=400 x_{1}+200 x_{2}$subject to the constraints\[\begin{aligned}& 18 x_{1}+3 x_{2} \leq 800 \\& 9 x_{1}+4 x_{2} \leq 600 \\& x_{1}, x_{2} \geq
A refinery distills two crude petroleum, A and B, into three main products: jet fuel, gasoline and lubricants. The two crudes differ in chemical composition and thus, yield different product mixes
What are the advantages of Khachiyan's ellipsoid method over simplex method?
Minimize $f=-3 x_{1}+2 x_{2}$Subject to $x_{1}+x_{2} \leq 9$$x_{2} \leq 6$$x_{1}, x_{2} \geq 0$make the "off-center" point to an "equidistant" from the coordinate axes in a transformed feasible
Solve the following problem by branch and bound method.Maximize $f=x_{1}+x_{2}$subject to the constraints$2 x_{1}+5 x_{2} \leq 16$$6 x_{1}+5 x_{2} \leq 30$$x_{1}, x_{2} \geq 0$ and Integer
Solve the problem using direct substitution methodMaximize $f=3 x_{1}^{2}+2 x_{2}^{2}$subject to the constraints\[x_{1}+3 x_{2}=5\]
Solve the problem using Lagrange multiplier methodMinimize $f=3 x_{1}^{2}+4 x_{2}^{2}+x_{1} x_{3}+x_{2} x_{3}$subject to\[\begin{aligned}& x_{1}+3 x_{2}=3 \\& 2 x_{1}+x_{3}=7\end{aligned}\]
Find the minimum value of the function$f=x_{1}^{2}+x_{2}^{2}-x_{1} x_{2}$subject to the inequality constraint\[x_{1}+3 x_{2} \geq 3\]
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