Analogies among Mass, Heat, and Momentum Transfer

1. Analogies among Mass, Heat, and Momentum Transfer

2. Analogies Heat  Mass  (sometimes) Momentum Analogies are useful tools An aid to understand transfer phenomena A sound means to predict behavior of systems for which limited quantitative data are available

3. Molecular Transport Equations RECALL: HEAT MOMENTUM MASS Fourier’s law Newton’s law Fick’s law

4. Analogous quantities in transport phenomena

5. Reynolds Analogy The general transport equation can be written in the form where ψ = flux of a property at any value of x δ = molecular diffusivity E = eddy diffusivity Г = volume concentration of transferent property

6. Turbulent diffusion equations

7. Transfer coefficient for momentum In cylindrical geometry, Integrating the above equation and multiplying by A to get a rate equation, where A = cross-sectional area perpendicular to flow = mean eddy diffusivity = ratio of the difference in concentration of transferent property between the wall and the mean value and the mean value of the fluid to the maximum difference between the wall and the center D = diameter

8. Transfer coefficient for momentum The transfer coefficient is then defined as Substituting and rearranging,

9. Transfer coefficient for momentum The transfer coefficient is then defined as Substituting and rearranging,

10. Transfer coefficient for momentum For momentum transfer,

11. Transfer coefficient for momentum If we divide by , At the wall, v1 = 0 so that,

12. The Reynolds analogy For turbulent transport, For heat transfer, For momentum transfer, We assume that α and μ/ρ are negligible, and that

13. The Reynolds analogy Dividing the momentum equation by the heat equation then gives

14. The Reynolds analogy Substituting and

15. The Reynolds analogy Stanton number

16. Dimensionless Groups

17. The Reynolds analogy Experimental results show that the above equation Correlate data approximately for gases in turbulent flow DOES NOT correlate experimental data for liquids in turbulent flow DOES NOT correlate experimental data for any fluids in laminar flow * 0.6 < NPr for gases < 2.5 It was concluded that the Reynolds analogy is valid ONLY at NPr = 1

18. The Reynolds analogy In a similar manner, we can relate mass transfer with momentum transfer For turbulent transport And the complete Reynolds analogy is

19. The Reynolds analogy Experimental results show that the above equation Correlate data approximately for gases in turbulent flow DOES NOT correlate experimental data for liquids in turbulent flow DOES NOT correlate experimental data for any fluids in laminar flow * NSc for gases ~ 1.0 It was concluded that the Reynolds analogy is valid ONLY at NSc = 1

20. The Reynolds analogy CONCLUSIONS At NPr = NSc = 1, the mechanisms for mass, heat, and momentum are identical For other fluids, transfer processes differ in some manner functionally related to the Pr and Sc numbers.

21. The Reynolds analogy Note that the Reynolds analogy assumes that the turbulent diffusivities are equal and the molecular diffusivities are negligible. When are these assumptions not valid? For other fluids, where  usually the case for liquids We CANNOT neglect molecular diffusivities  in the boundary layer where diffusion, conduction, and viscosity are important