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Homework answers / question archive / 1)A laminar reactor with a power-law non-Newtonian fluid

1)A laminar reactor with a power-law non-Newtonian fluid. Extend the analysis of the laminar-flow reactor in Section 12.3 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.

2)Find the exit concentration and the conversion for a laminar-flow reactor under the following conditions using the segregated model: radius 1 cm, length 500 cm, mass flow rate 0.1 kg/s, density 1000 kg/m3, and first-order rate constant k = 0.1 s−1. Also model the reactor (a) as a plug-flow reactor and (b) as a completely backmixed reactor.

3)Derive Eq. (12.34) leading to the exponential integral solution of the segregated flow model.Natural convection: air. Show that β = 1/T for an ideal gas. Repeat the example above for air instead of water. Which fluid generates more circulation? Suggest a reason for this.

4)The effect of viscous dissipation: the Brinkman problem. 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 readily obtained using MATLAB functions or other numerical methods. Set up and solve the problem, for example using PDEPE. Also set this up using CHEBFUN as an eigenvalue problem, and solve by the method of series solution.

5)Consider the oil flowing in a pipe in Problem 10. The inlet temperature is 300 K and the pipe is now heated electrically at a rate of 76 W/m2. Plot the wall temperature, the center temperature, and the bulk temperature as a function of position using the asymptotic solution for large length.Verify that the cup mixing temperature in dimensionless units is equal to 4ζ .

6)A pipe of diameter 2 cm with an oil flow rate of 0.02 kg/s heated by a wall temperature of 400 K. The inlet temperature is 300 K. Find and plot the radial temperature profile at a distance of 5 cm from the entrance. Find and plot the radial temperature profile at a distance of 100 cm from the entrance. If an outlet temperature of 380 K is needed, what is the length of the heat exchanger needed? Use an average Nusselt-number value of 3.66. The following values for the physical properties are applicable: ν = 4 × 10−5, cp = 2120, k = 0.14, and ρ = 1000. All are in SI units.

7)Repeat the analysis if instead a gas such as oxygen is being absorbed into the liquid film.

8)Consider a liquid flowing down a vertical wall at a rate of 1 × 10−5 m2/s per meter unit width. Find the concentration at a height 25 cm below the entrance for a dissolving wall such as a wall coated with benzoic acid as a function of perpendicular distance from the wall. Also find the local mass transfer coefficient. Find the average mass transfer coefficient for a wall of height 50 cm. Assume D = 2 × 10−9 m2/s, and for other properties take those of water. Use CA,s = 20 mol/m3 and the physical properties of water.

9)Show all the steps leading to Eq. (12.15). Integrate once, using the substitution p = dθ/dη. Now integrate a second time to get the temperature distribution given by Eq. (12.16) in terms of the incomplete gamma function. Verify the expression for the Nusselt number given by (12.18).

10)Constant heat-flux analysis for pipe flow. Show all the steps leading to (12.12) and verify that the Nusselt number has a value of 48/11 for this case.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?

11)Stretching transformation. Consider the differential equation given by Eq. (12.2). Use a coordinate transformation ζ = z∗Pea, where a is some index to be chosen suitably. Show that in the transformed equation there are no free parameters in the convection term and the radial conduction term if the index a is chosen as one. Also verify that the axial conduction term has a leading coefficient of 1/Pe. Hence justify the neglect of the axial conduction term if Pe is, say, greater than 10.

12)A case-study problem: oscillatory flow of a Casson fluid. The rheology of blood is nonNewtonian and is often represented by the Casson fluid model which was discussed in Section 5.7. Your project is to examine the blood flow with this model and examine the conditions under which the deviation from Newtonian fluid behavior may be expected. The mathematics is horrendous except for my students, and you may wish to refer to the paper by Rohlf and Tenti (2001), who studied the problem in detail.

13)Pipe flow with periodic pressure variation. Verify the result for the velocity profile in the complex domain shown in Section 11.11.6. Write MATLAB code to find the real and imaginary parts. Use this code to plot the velocity profile for various values of Wo as a function of position and time. Also derive an expression for the volumetric flow rate. Plot this as a function of time and also superimpose on the plot the pressure variation with time. Show that there is a phase lag between pressure and flow for high values of Wo.

14)Transient Couette flow. The problem is similar to Section 11.11.1 and is obtained by 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 separation of variables after subtraction of the steady-state solution.

15)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 obviously satisfies the differential equation. Fit the boundary conditions for each side. Now use the flux balance at the reaction plane and derive an expression for λ as a function of time. How does this model compare with the film model?

16)A gas stream with CO2 at partial pressure 1 atm is exposed to liquid in which it undergoes a first-order reaction for 0.01 s. The total amount of gas absorbed during this time was measured as 1.5 × 10−4 mol/m2. Estimate the rate constant for this reaction. Use CA,s = 30 mol/m3 and DA = 1.5 × 10−9 m2/s.