1. Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras
Advanced Transport Phenomena
Module 4 Lecture 17
Momentum Transport: Illustrative Problems
Dr. R. Nagarajan
Professor
Dept of Chemical Engineering
IIT Madras

2. Momentum Transport: Illustrative Problems

3. PROBLEM 3
Consider the steady axisymmetric flow of hot air in a straight circular tube of radius aw and cross sectional area A Conditions (at exit): p=1 atm (uniform) T= 1500 K (uniform) aw= 5 mm

4. PROBLEM 3
Suppose it has been observed that the axial-velocity profile is, in this case, well described by the simple equation:
where U=103 cm/s. Using this observation, the conditions above, answer the following questions:
a. If the molecular mean-free-path in air is approximately given by the equation:

5. PROBLEM 3
Estimate the prevailing mean free path l and the ratio of l to the duct diameter- i.e., relevant Knudsen number for the gas flow: What conclusions can you now draw concerning the validity of the continuum approach in this case? b. Calculate the convective mass flow rate (expressed in g/s) through the entire exit section. For this purpose assume the approximate validity of the “perfect “ gas law, viz,:

6. PROBLEM 3
Here p is the pressure (expressed in atm), M is the molecular weight (g/g-mole)(28.97 for air), R=82.06 (univ gas const), and T is the absolute temperature (expressed in kelvins). Also note that for this axisymmetric flow a convenient area element is the annular ring sketched below ( where is the unit vector in the z- direction).

7. PROBLEM 3
c. Also calculate the average gas velocity at the exit section and the corresponding Reynolds’ number;

8. PROBLEM 3
d. Calculate the convective axial momentum flow rate (expressed in g. cm/s2) through the exit section. Is your result equivalent to Why or why not? e. Calculate the convective kinetic-energy flow rate (expressed in g. cm2/s3). Is your result equivalent to Why or why not? f. If, in a addition to the axial component of the velocity vz, the air in the duct also has a swirl component how would this influence your previous estimates ( of mass flow rate, momentum flow rate, kinetic energy flow rate)? Briefly discuss.

9. PROBLEM 3
g. If the local shear stress is given by the following degenerative form of Newton’s law: at what radial location does maximize? Calculate the maximum value of and express your result in dyne cm-2 and Newton m-2. Calculate the skin-friction coefficient, cf (dimensionless), at the duct exit. At what radius does take on its minimum value? Can- be regarded as the radial diffusion flux of axial momentum? Why or why not? Does the rate at which work is done by

10. PROBLEM 3
the stress maximize at either of the two locations found above? Why or why not? h. Characterize this flow in terms of flow descriptors and defend your choices.

11. SOLUTION 3

12. SOLUTION 3 b.

13. SOLUTION 3
c. d.

14. SOLUTION 3
But since e.

15. SOLUTION 3
Note that:
that is,
f would not influence
Discuss.

16. SOLUTION 3
g. Thus

17. SOLUTION 3
therefore h. Descriptors: Continuum Laminar “Incompressible” Quasi-one-dimensional Newtonian (viscous) Internal Steady Single-Phase

18. PROBLEM 4
Estimate the drag force (Newtons) per meter of length for each of the following long objects of transverse dimension 5 cm if placed in a heated air stream with the following properties.
a. A circular cylinder.
b. A thin “plate” perpendicular to the stream (i.e., at 90o incidence).
c. A thin plate aligned with the stream (i.e., at 0o incidence).

19. PROBLEM 4
In each case qualitatively discuss how the drag is apportioned between “form” (pressure difference) drag and “ friction” drag
For Part ( c), is the application of laminar boundary-layer theory likely to be valid? (Briefly discuss your reasoning) If so, what would be the estimated BL thickness, , at the trailing edge of the plate, i.e., at x=L? Suppose two such adjacent plates were separated by a distance much greater than would they strongly “interact” with respect to momentum transfer?

20. PROBLEM 4
d. Justify the use of an incompressible Newtonian fluid CD(Re, shape) curve to solve Part (a) ( involving the gas air) by showing that is small enough under these conditions to neglect

21. SOLUTION 4
Momentum Transfer to (Drag on) Immersed Objects Drag/meter of axial length=? for objects of transverse dimension 5 cm. in U =10 m/s, 1 atm., 1200 K. a. Cylinder in Cross flow

22. SOLUTION 4

23. SOLUTION 4

24. SOLUTION 4
Frontal area/meter=5 cm x 100 cm = 5 x 102 cm2 /m Therefore Drag/Length

25. SOLUTION 4
Most of this drag is due to the p(q) distribution- that is, “ form” drag. b. Plate Normal to flow: check literature c. Plate Aligned with flow:

26. SOLUTION 4
In this case and Since ReL<106 (approx.) we expect flow in the momentum defect Boundary Layer to be laminar. Then where

27. SOLUTION 4
and But total wetted area/meter=(2)(5x102)=103 cm2/m. Therefore or

28. SOLUTION 4
This drag is entirely due to i.e., it is “friction drag”

29. PROBLEM 5
Reconsider the sonic jet test facility specified in problem1 from the viewpoint of turbulent jet momentum exchange (mixing) with the surrounding atmosphere, and the entrainment of that atmosphere
a. Calculate the Reynolds’ number at the nozzle exit and compare it to the value above which such jets almost certainly lead to turbulent mixing with the surrounding atmosphere.

30. PROBLEM 5
b. Estimate the appropriate value of by assuming that the relevant density is about the arithmetic mean between . How much larger is the effective turbulent momentum diffusivity, , than the intrinsic momentum diffusivity of the jet fluid ? c. Estimate the downstream distance at which the time-averaged velocity (axial momentum per unit mass) along the jet centerline will be reduced to 10% of the initial jet velocity (axial momentum per unit mass) as a result of momentum diffusion. Compare this to the result that would have been obtained had the jet

31. PROBLEM 5
remained laminar (with kinematic viscosity ). d. At this location what is the approximate ratio between the entrained (laboratory air) mass flow and the “primary” (combustion-heated air) jet?

32. SOLUTION 5
Momentum Transfer: Turbulent Round Jet

33. SOLUTION 5
Momentum Transfer: Turbulent Round Jet is determined by . Using we obtain:

34. SOLUTION 5
Therefore that is, Where does drop to Uj/10. For a turbulent jet:

35. SOLUTION 5
We find: z=88.6 cm =0.89 m (This would have been 0.46 km if there had been no turbulent enhancement in momentum diffusion.) where

36. SOLUTION 5
therefore so that at cf. Therefore

37. SOLUTION 5
Exercises: 1. Calculate the time-averaged profile at this location. 2. Can you estimate the centerline, time averaged temperature and CO2(g) concentration at this point? (Itemize and discuss the underlying assumptions.)