Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 9 Lecture 40 Students Exercises: Numerical Questions (Modules 6-8) Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. Students Exercises: Numerical Questions (Modules 6-8)

3. NUMERICAL PROBLEMS Module 6: Convective-diffusion mass transfer plays an important role in the fabrication of small solid-state electronic devices, which often start with the "chemical vapor deposition" (CVD) of micron-thickness films of crystalline Si(s) grown on a sapphire (Al2O3) substrate ("wafer"). a. Use the material studied in this course to make a preliminary quantitative estimate of the average silicon deposition flux (mg/min/cm2) to a flat,

4. NUMERICAL PROBLEMS growing wafer exposed to a parallel flow of SiH4(g)/H2 gas mixture under the following conditions: U =50 cm/s, = 300 K, L = 5 cm Tw= 1273 K, p = 1 atm , = 0.005 (mole fraction). Assume that the Si-deposition flux is determined by SiH4(g) ("silane"-) vapor transport to the wafer surface, at which: That is, assume , but neglect the possible

5. NUMERICAL PROBLEMS complication of SiH4(g)-decomposition in the vapor phase near the hot wafer. List and quantitatively defend each of your important assumptions and property estimates. Identify the "weakest links" in your estimates (for possible future investigation). Ans. 405 mg/min/cm2. b If the estimated equilibrium Si(g) vapor pressure over Si(s) at this surface temperature is about 1.1 x 10-8 atm, can the "physical" sublimation rate of silicon be neglected under these CVD-reactor operating conditions?

6. NUMERICAL PROBLEMS c. Evaluate the natural convection/forced convection ratio parameter and comment on the possible importance of natural convection (caused by heat transfer) in augmenting (or suppressing?) the mass-transfer coefficient if the heated wafer is horizontal, and the gas mixture flows past above it.

7. NUMERICAL PROBLEMS Note: While the Stefan-mass flow ("suction") associated with Si-deposition is negligible in this example (since the "blowing" effect associated with thermal (Soret) diffusion is appreciable (over 20% reduction in ) and should be taken into account (Rosner (1980)).

8. NUMERICAL PROBLEMS Module 7: Application of the Mass/Energy Transfer "Analogy" and Scale-Model Theory. Consider the following solution to the problem of scale-model testing that involves heat-transfer measurements on a cluster of 2 – mm diameter rods in a water tunnel. This is based on the feasibility of making dissolution (convective mass transfer)-rate measurements on a cluster of 5-mm diameter rods exposed to the flow of liquid water at 25°C.

9. NUMERICAL PROBLEMS • a. Given the physical data below, systematically work out how you would use benzoic-acid dissolution-rate data to predict the heat-transfer per­formance of the full-scale tube bundles, also circumventing the need for expensive full-scale furnace tests. Defend the validity of each of your underlying assumptions and itemize the experimental precautions you would have to take. In your opinion, does the present mass-transfer anal­ogy approach have any decisive advantages or drawbacks? Basic data: Pure benzene carboxylic acid (commonly called "benzoic acid") is a solid (mp 122°C,

10. NUMERICAL PROBLEMS density = 1.266 g/cm3) which is sparingly soluble in 25°C water (0.0278 g-moles/liter H2O). Its diffusion coefficient has been measured to be 1.00 x 10-5 cm2/s in 25°C water (Table 2.6 of Sherwood, Pigford, and Wilke (1975)). b. If the simultaneous role of natural convection cannot conveniently be “simulated” in your mass-transfer model experiment, is it possible to dispense with Grm- similarity altogether?

11. NUMERICAL PROBLEMS Examine this possibility on the basis of available experimental data for an isolated cylinder under the simultaneous influence of forced convection and ("opposed") natural convection. (In that case how much is , decreased below its "pure" forced convection counterpart) when is as large as expected in our "prototype"?)

12. NUMERICAL PROBLEMS c. Why in Part (a) above, is it reasonable to test with Rem = Rep, assuming the dependence of on Pr (or Sc) is "known" (from isolated cylinder data)? Would it be reasonable to also dispense with the “set-up rule”: Rem = Rep by assuming that the Re-dependence of (tube bundle) is the same as the Re-dependence of (isolated cylinder)? Are any experimental data available which would allow you to compare the Re-dependence of for both an isolated cylinder and, say, two cylinders "in tandem"?

13. NUMERICAL PROBLEMS d. In a previous solution, no mention was made of the possible importance of simultaneous natural ("free") convection in the full-scale furnace application. Investigate this possibility by first calculating the magnitude of Grh and then forming the important ratio (governing the relative importance of free and forced convection). Is this ratio small enough to consider natural convection negligible, or will it be necessary to also do model testing at the same Grashof number as for the full-scale prototype?

14. NUMERICAL PROBLEMS In the benzoic-acid dissolution model (mass-transfer analog) experiment considered in this exercise, what is the value of (Grm)m given the fact that the density difference between benzoic-acid-saturated water and pure water is only 1.42 x 10-3 g/cm3 at 25°C. To force (Grm)m =(Grh)p what value of dmwould be needed, and what water veloc­ity, Um would now be needed to preserve Rem= Rep? Would the Pr and/or Sc-dependence of be known under these more complicated (combined forced convection and natural convection) conditions?

15. NUMERICAL PROBLEMS Module 8: By stating and where possible, numerically evaluating the relevant dimensionless ratios, quantitatively defend the following approximations made in solving the illustrative problems of heat transfer and ash fouling in a pulverized-coal combustor • Continuum approximation, • Neglect of viscous dissipation, • Neglect of natural convection

16. NUMERICAL PROBLEMS With additional information it would also be possible to defend (or relax) the: d. neglect of mainstream turbulence e. neglect of effects of neighboring heat-exchanger tubes f. neglect of deposition on the cylinder by “eddy impaction” g. neglect of radiative energy gain (from gases and suspended ash)

17. NUMERICAL PROBLEMS 8.2 Suppose, for structural-design reasons, you wanted to estimate the drag force per unit length of the heat-exchanger tube. Could this also be done based on the - relation using the so-called "analogy" between momentum and energy transfer? 8.3 In estimating the submicron particle (ash) mass-transfer rate we exploited the (a) mass-energy transfer "analogy" (i.e., - are the same functions). Can this analogy be derived from "dimensional analysis"? Similitude analysis? (b) proportionality for large Prandtl numbers. Can this proportionality be derived from "dimensional analysis"? Similitude analysis?

18. NUMERICAL PROBLEMS 8.4 Verify the effective "capture efficiency" figures quoted in the course for submicron particle collection by the heat-exchanger tube under the conditions treated above. How does it compare to that estimated above for inertial capture of 20 particles? 8.5 Show that the particle Stokes' number can be re-expressed in terms of the prevailing values of the Mach, Reynolds', and Schmidt numbers in accordance with:

19. NUMERICAL PROBLEMS where is the ratio of individual gas molecule-to-particle mass. Can analogous momentum nonequilibrium phenomena be observed for mixtures containing trace amounts of a high-molecular- weight vapor (e.g., WF6(g)) in a low-molecular-weight carrier gas (e.g., H2(g))under continuum flow conditions (see, e.g., Fernandez de la Mora, J. (1982))?