Mathematical Modeling In Chemical Engineering 1st Edition Anders Rasmuson, Bengt Andersson, Louise Olsson, Ronnie Andersson - Solutions

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Consider the problem of predicting the concentration of a reactant in an isothermal batch reactor, where the reaction kinetics is second order (non-linear problem). The reaction rate, r, is given by r = ky2, where k is the reaction rate constant (k = 1) and y is the concentration of the reactant.
Reaction rates inside a spherical isothermal porous catalyst particle depend on the rate of diffusion and the kinetics. When the reaction is fast compared to the diffusion, e.g.at higher temperatures, the reactants will be consumed near the surface and not all the catalyst will be effectively used.
Consider the problem of prediciting the concentration of a reactant in an isothermal batch reactor, assuming a first-order reaction, where the reaction rate, r, is given by the product of the reaction rate constant, k, and the concentration of the reactant, y:In this equation, we have introduced
A mathematical model of a pneumatic conveying dryer, Figure 1.1, has been developed (Fyhr, C. and Rasmuson; A., AIChE J. 42, 2491–2502, 1996; 43, 2889–2902, 1997)and validated against experimental results in a pilot dryer. Heating steam Auxiliary Moist superheater
In the second example, a mathematical model of a Wurster bed coater, Figure 1.4, has been developed (Karlsson, S., Rasmuson, A., van Wachen, B., and Niklasson Bjorn, I., AIChE J. 55, 2578–2590, 2009; Karlsson, S., Rasmusson, A., Niklasson Bjorn, I., and Schantz, S., Powder Tech. 207, 245–256,
Give some reasons for doing mathematical modeling in chemical engineering.
Explain why the model development often becomes an iterative procedure.
One of the oldest scientific investigations was the attempt to understand gravity. This problem provides a nice illustration of the steps in modeling.
What is the difference between a lumped- and a distributed-parameter model?
Explain the difference between deterministic and stochastic models.
Why is a linear mathematical model tractable for analytical solution?
Describe the continuum concept.
The variables and parameters used in this example are as follows: p = density, cp = heat capacity, q = heat flux, k = thermal conductivity, A = cross-sectional area, S = source strength (J m−3 s−1).Insertion of these terms into the balance equation yields accumulation: in (conduction): out $accumulation: in (conduction): out (conduction): production: PC, (x); (q 4)\, t; (q4)\x+. ; SAx t.$
At what scale is Darcy’s law formulated? Are there alternatives that describe flow in porous materials?
What are the key steps in deriving a transport phenomena model?
The variables and parameters used in this example are as follows: CA = concentration, JA = diffusion flux, v = velocity, RA = source term (mol m−3 s−1). accumulation: in (diffusion): ACAAX AYAZ; x-direction JA,xlx Ay Az At, y-direction JA, yly Ax Az At, z-direction JA,zlz AX AyAt; x-direction
What does a generic balance equation describe?
The energy balance equation for rectangular coordinates is given byIn Example 3.1, there is no convective flow, consequently there is no heat convection, and the following terms are canceled:Example 3.1The variables and parameters used in this example are as follows: p = density, cp = heat $accumulation: in (conduction): out (conduction): production: PC, (x); (q 4)\, t; (q4)\x+. ; SAx t.$
Explain what is meant by a boundary condition, and how they can be classifed.
Let us consider a population of cells flowing through a plug flow reactor. The cells are characterized at time t by their position, x, and their mass, m. They are supposed to grow, to die, and to divide into two daughter cells (with mass conservation).
What is the difference between a balance and a conservation principle?
Let us consider a single incompressible fluid that is poured into the reactor at a constant flow rate Qin. Since there is no outlet stream, E = 1/τ1out = 0. The inlet hydrodynamic time is given by Ti = V/Qin=tQin/Qin = t, and Equation (3.12) reduces to al at The solution of this equation is I
What is a population balance?
What is the balanceable property in residence time distributions?
This scenario has been discussed for more than 50 years. The conditionis under dispute because it introduces a feedback of information from the end of the reactor that does not occur in real reactors. A more realistic formulation for an irreversible reaction is perhaps dC dz = 0 at z = L
A simple problem illustrates how the error is related to step size. Consider the following problem:The numerical solution using the explicit first-order Euler method, and the step size h = 0.2, is shown in Figure 6.2. dy dt y(0) = 1, te [0, 1]. = f(t, y)= y +t,
What is the difference between the local truncation error and the global error, and how are they related?
How can error tolerance be ensured when a true solution does not exist when solving differential equations?
Explain what is meant by adaptive step size methods, and why they are used.
Consider an exothermic reaction occurring in a batch reactor, as shown in Figure 6.10. The reaction mechanism is a first-order reaction, and the Arrhenius reactionrate law applies, i.e.The system is described by the material balance,and the energy balance,with the initial conditions CA(0) = CA0,
What characterizes multi-step methods, and why are they often used?
In this example we will consider the transformation of a second-order ODE to a system of two first-order ODEs. The second-order differential equation is given byThis system of first-order differential equations is integrated to solve the second-order ODE Equation (6.34), using any of the methods
How do predictor–corrector pairs work, and why are they used?
Consider the potentially stiff systemFor large α values, the system becomes stiff as it contains components that vary with different speeds, i.e. y2(t) approaches zero much faster than y1(t) does. The solutions to this problem for α = 2 and α = 200 000 are shown in Figure 6.11. By analyzing
How is stability related to explicit and implicit methods, and what is a conditionally stable method?
Cooling fins, Figure 6.15, are frequently used to improve heat transfer from walls to gases, because gases have low thermal conductivity. The fin efficiency factor, η, is defined as the ratio between the actual heat transfer and the heat transfer if the entire fin has the wall temperature Tw. The
Explain what is meant by stiff differential equations, and what category of ODE solvers is suitable for stiff systems.
Consider solving the following problem:The solution of the heat equation problem (D = 1.0), using the explicit forward method, is shown in Figure 6.22. Here, σ 2/D, the temperature profile is expected. ди ²u = D. at Əx²¹ x e [0, 1], t > 0, and the initial condition with the simple Dirichlet
Why are different solution methods needed for IVPs and BVPs, and how can BVPs be solved?
Explain the principle behind the finite difference method.
Find a suitable dimensionless model for heat transfer between a fluid and a spherical body, as seen in Figure 4.1. Heat transfer to a particle in a turbulent flow is a complex phenomenon involving convection and conduction. The objective is to develop an empirical correlation that can describe the
Assume that the chemical reaction A + B → products is scaled up from a 10 l pilot plant to 10 m3 full-scale production unit. Temperature control is usually a minor problem in stirred tank reactors; instead the critical property is the rate of mixing. The characteristic reaction time can be
The time constants in an unsteady catalytic tubular reactor may be very different for different phenomena. The concentration may change in seconds, the temperature in minutes, and the catalyst activity in weeks.
What properties are essential in an equation that may be solved independently of the remaining equations in a system of equations?
The conversion of a second-order reaction in a tubular reactor with axial dispersion,can be overestimated by replacing the second-order reaction by a first-order reaction using the maximal concentration in the reactor,where CIN is the inlet concentration. A better approximation is obtained if CIN
Why are the net mass and heat transfer through a symmetry boundary condition zero?
The equation for a pendulum is given by d²w dt² || bol- g L sin @.
What are the maximum number of symmetry planes in a cube, cylinder, and sphere?
An object is falling in air according to dv m- dt - mg + CDA: = 0. pv² 2
In Equation (5.9), what is the difference if the left-hand side or the right-hand side is set to zero? ac a2C at ах2 accumulation dispersion. = Deal - ac v- - k(T,t). C". ax convection reaction
Find a criterion for neglecting the axial dispersion of a first-order reaction in a tubular reactor. A first-order reaction in a tubular reactor without dispersion is described by Dea - U² d²C d C dz² dispersion dz convection - KC = 0. reaction
What do the collocation and finite element methods have in common, and how do they differ?
Set up the normal equations and derive expressions for b0 and b1 in a linear model.
How can the parameters in vector b be determined using the matrix form?
Why are weighted residuals used?
How do you determine confidence intervals and regions, respectively?
What are the differences between confidence intervals and confidence bands?
Describe the ANOVA table and how to calculate the different values in the table.
What is the correlation between parameters and how do you determine it?
Why is the correlation matrix symmetrical?
What makes a model non-linear?
Give an example of an intrinsically linear model.
How do you determine confidence intervals for a non-linear model?
Give examples of residual plots when the residuals are evenly distributed and when there are clear trends. Also describe how the models should be changed when clear trends are discernible.
How is R2 determined in order to get information about how much of a variation can be described by a model?

Mathematical Modeling In Chemical Engineering 1st Edition Anders Rasmuson, Bengt Andersson, Louise Olsson, Ronnie Andersson - Solutions

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