1. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras Advanced Transport Phenomena Module 7 Lecture 33 1 Similitude Analysis: Illustrative Problems

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3. For ease of manufacture, compactness, and maintenance, the convective heat exchanger tubes in a proposed fossil- fuel combustion power station will contain clusters of four two-inch (O.D.) tubes with sld = 1.5, inclined at the angle = 15 o from the horizontal, exposed to the vertical flow of 1600 K combustion products at = 10 m/s, p = I atm. PROBLEM 1 3

4. However, information in the heat-transfer literature on the performance of such clusters is deemed inadequate, so that the project engineer authorizes you to perform such calculations or experiments as necessary to accurately predict the forced- convection heat-transfer performance of such inclined clusters of long, straight isothermal tubes. PROBLEM 1 4

5. The quantity of maximum interest is the total heat- transfer rate (gas-to-wall) per unit length of the tube cluster. Because of the geometric and fluid-dynamic complexity, theoretical calculations (e.g., finite differences, or finite elements) are rejected in favor of scale-model experiments, if the latter are feasible using available laboratory facilities. PROBLEM 1 5

6. a. Based on the similarity principles in this module, can you select a working fluid, and practical scale-model experimental conditions such that heat-transfer measurements made on the model (m) configuration can be used to make reliable quantitative predictions of the performance of the above mentioned prototype (p)? (Or, as one skeptical official of the corporation has pessimistically concluded, will it be necessary to do expensive and time-consuming full-scale experiments in a prototype "laboratory" furnace?) PROBLEM 1 6

7. b. Demonstrate that it is reasonable to expect that additional (new) phenomena will not enter your scale-model experiments, complicating the relationship between model and prototype performance. c. Under the model experimental conditions you choose, what would the relation be between observed model (m) and predicted prototype ( p) heat- transfer rates? PROBLEM 1 7

8. d. On what "coordinates" should your scale-model heat transport data be plotted so that the results are apt to apply to similar configurations at "any" scale? e. The corrosion engineers are concerned about the condensation of certain salts on the surface of each such cooled tube bundle. What information and assumptions would allow you to predict the maximum possible rate of vapor transport and condensation onto the tube bundle, based on your small-scale heat-transfer measurements? PROBLEM 1 8

9. SOLUTION Model Tests on Heat- Exchanger Tube Bundle “Prototype” Conditions: 9

10. SOLUTION 10

11. Questions: 1. Can useful “model” experiments be done? 2. What is the relation between model and full-scale heat- transfer behavior? Note: If SOLUTION 11

12. Possible “ Model” Conditions (m) Consider small-scale model tested in water flow: e.g., take: SOLUTION 12

13. This ensures geometric similarity. To ensure dynamic similarity, we test under conditions such that: i.e., SOLUTION 13

14. But for water near room temperature: SOLUTION 14

15. For strict thermal similarity, we should have Pr m =0.706; However: Pr m =Pr(H 2 O(l)) 7. We can approximately account for this since we know that for an isolated cylinder in cross-flow the Pr-dependence of (Churchill and Bernstein, ASME J. Heat Transfer 99, 300 (1977)) is: SOLUTION 15 ~

16. (slightly more elaborate than ). We now assume that this Pr-dependence will also hold approximately for a tube bank. Thus, if we test at the same Re, we expect: or SOLUTION 16

17. and the scale factor is: In the prototype: Suppose we test with: SOLUTION 17

18. Now, Therefore SOLUTION 18

19. and since: and therefore SOLUTION 19

20. i.e., under these conditions: Consider Heat- Transfer Rates per Unit Axial Length: SOLUTION 20

21. Discussion questions 1. Are there other “model” possibilities? 2. Do the following phenomena interfere with the geometric, dynamical and /or thermal similarity: a. natural convection ( examine ) b. radiative transfer; c. gas compressibility; (Ma) p =? 3. Mass transfer implications? 4. Other necessary similarity conditions (e.g., mainstream turbulence level, turbulence scale, etc.)? SOLUTION 21