Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 9 Lecture 39 1 Students Exercises: Numerical Questions (Modules 1-5)

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NUMERICAL PROBLEMS Module 2: Consider the use of the species and element mass-balance equations to design a steady-flow chemical reactor for carrying out the homogeneous thermal decomposition of a feed vapor under conditions such that the kinetics are simple and well established from previous measurements. “Phosphine”,PH 3 (g)(the phosphorous analog of ammonia, with a molecular weight,, of 34 kg/kg-mole) under the conditions of interest here, is known to thermally decompose according to the stoichiometry; 3

NUMERICAL PROBLEMS With the simple first-order irreversible reaction rate law: where the reaction rate “constant” has been experimentally found to be (where T is in Kelvins and is the phosphine number density (kg-mole/m 3 )). 4

NUMERICAL PROBLEMS Complete the following preliminary design of a steady- flow reactor vessel that maintains the gas mixture temperature at 953 K (upper limit set by readily available materials of construction) and at about 1 atm pressure, but with negligible mixing in the streamwise direction. a. How large(total volume) a reactor vessel would be required to continuously decompose 68 percent of a feed-flow rate of 16 kg/hr of pure phosphine (Assume the validity of the ideal gas mixture equation of state under these proposed operating conditions)? 5

NUMERICAL PROBLEMS b. Using the thermochemical data assembled below for the ideal gases predict the rate, at which energy would have to be added (removed?) to maintain the- decomposition reactor isothermal at 953 K (express your result in kW) 6

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 Ans: For any fraction decomposed, f, and operating conditions (T,p), we find: For NUMERICAL PROBLEMS 8

Module 3: Consider the properties of a “binary gas” mixture comprised of 0.2  m diameter SiO 2 (c) particles present in the mass fraction    with combustion products (with properties not very different from N 2 (g)) present in mass fraction   =0.67. Treating the silica “aerosol” as a high- molecular weight “vapor,” and at T=1600 K, p=1 atm: a. Calculate the mean molecular weight, M, of this binary mixture. NUMERICAL PROBLEMS 9

b. Calculate the specific heat, c p, of this binary mixture. c. Estimate the effective dynamic viscosity,  of this mixture, using both the “square-root rule” and the semi- theoretical (Wilke) equation: where NUMERICAL PROBLEMS 10

d. Estimate the effective thermal conductivity, k, of this mixture, using both the “cube-root rule” and the semi- theoretical (Mason-Saxena) equation: identical in structure to that given above (in the viscosity formula) e. Estimate the binary Fick diffusivity, D 12, under these conditions (using an equivalent “hard-sphere” diameter for each N 2 molecule) NUMERICAL PROBLEMS 11

f. Predict the effective Prandtl number characterizing this pseudo-single-phase mixture. How does it compare to that of pure N 2 (g)? g. Predict the Schmidt number characterizing this pseudo-single –phase mixture. h. Calculate the ratio of the N 2 (g) mean free path to the SiO 2 (c ) particle diameter. Is this (“ particle Knudsen number,” Kn p ) ratio large enough to validate your treatment of this mixture as a pseudo-single-phase “gas mixture”? NUMERICAL PROBLEMS 12

If at T=1600 K the pressure level was 20 atm (rather than 1 atm), would this approach remain equally valid? i. Despite the fact that the silica fume is present with an appreciable mass fraction (0.33), it is a “trace species” on a number (mole-fraction) basis. Study your results for the predicted effect of the SiO 2 -fume on each of the mixture properties (M, C p, , k) and discuss the possible significance / implications of your conclusions. NUMERICAL PROBLEMS 13

Module 4: A domestic automobile manufacturer recently announced that it has developed a passenger car with a drag coefficient, C D, (at cruise conditions) of only This is to be compared with the values 0.30 to 0.35 typical of most production passenger cars. a. At a cruise speed of 60 mph, compare the power (in kW and HP) needed to overcome the aerodynamic drag for both a new and conventional car with a frontal area of 3.75 m 2. b. What is the Reynold’s number (per meter) at cruise speed when p=1 atm, T=300 K? NUMERICAL PROBLEMS 14

c. How do the aforementioned C D - values compare to the drag coefficient of a sphere in the same Reynolds’ number range? d. What would be the corresponding power and drag coefficient for a 2 m high by 6 m long thin plate aligned with a 60 mph flow? NUMERICAL PROBLEMS 15

Module 5: Select any problem in turbulent-flow heat- (and/or mass) transfer in the engineering literature for which a quantitative “prediction” has been made, and answer as many of the following questions as possible: a. How have the turbulent transport terms (i.e., Reynolds’ fluxes and associated diffusivities ) been mathematically “modeled”? b. What experimental data have been used in the turbulent transfer law (e.g., what and how many “universal” constants have been used)? In what ways are the predicted configurations similar to the configurations for NUMERICAL PROBLEMS 16

which the empirical constants have been derived? c. What assumption has been made concerning the turbulent diffusivity ratios (i.e., turbulent Prandtl numbers)? With what justification? d. Are there laminar regions embedded somewhere within the upstream or near-wall regions of flow field? If so, at what point has "transition to turbulence" been assumed to occur? What criterion has been used to define transition to turbulence? NUMERICAL PROBLEMS 17

e. Estimate a representative scale of turbulence in your problem. How does it compare to macroscopic geometric dimensions appearing in the problem? How does it compare with the prevailing intermolecular spacing or mean-free-path? f. Can the formulation accommodate (or utilize) known upstream and/or boundary conditions on the free-stream turbulence level, or turbulence intensity? g. Are there realizable conditions when the turbulent- transport model used would predict no turbulent mixing? NUMERICAL PROBLEMS 18

h. Do the eddy-transport coefficients unrealistically vanish (or "blow up") anywhere in the flow field? Is it necessary or possible to avoid this difficulty? i. If a second-order irreversible homogeneous chemical reaction were occurring in the flow field, how might you estimate the local correlation, appearing in the expression for the time-averaged reaction rate, Can the "intensity" of temperature and/or concentration changes be "predicted" or related to the "intensity" of velocity disturbances? NUMERICAL PROBLEMS 19