1. CHAPTER 6 Post Dryout Heat Transfer Liquid VaporLiquid droplets Liquid deficient region Vapor flow Film boiling (inverted annular flow) Topics to be discussed -Film boiling heat transfer (Inverted annular flow) -Heat transfer in the liquid deficient region (High void fraction film boiling) -Minimum film boiling temperature -Transition boiling heat transfer

2. 6.1 Film Boiling with Inverted Annular Flow Pattern –Film boiling on vertical surface Brmoley’s model Ref.: Chemical Engineering Progress, vol.46, No.5, p.221-227 G.E. for vapor flow Boundary conditions Liquid u g Vapor y x

3. Solution for u(y) -Heat transfer It is assumed that heat travels through vapor film by conduction. Thus x x+dx T=T sat

5. Boundary condition: δ=0 at x=0 Thus, the local heat transfer coefficient is:

6. The average heat transfer coefficient over the vertical plate is: For case a C=2 C 1 =0.943 (Zero stress at film surface) case b C=1 C 1 =0.666 (Zero velocity at film surface) Considering the sensible heat of vapor: vapor properties evaluated at film temperature(T f =(T w +T sat )/2)

7. -Film boiling heat transfer from a horizontal surface Berenson’s model Ref.: J. Heat Transfer, vol.83, pp.351-358,1961 R H VvVv Liquid Solid Vapor Actual shape of liquid-vapor interface Physical model of film boiling from a horizontal surface

8. Where the subscript vf stands for vapor properties evaluated at T f =(T w +T sat )/2 Analysis in Berenson’s model Vapor film H

9. -Momentum equation in the vapor film Boundary condition

10. So the average velocity The force balance requires that

11. Similarly, for u(z=δ)=0, From the energy balance and assuming that heat transfer through the film is by conduction (i.e. the flow is laminar)

12. U(r) Noting that Assume

13. where“1.4”is to account for the ratio of total area /the area between bubble

14. The experimental data suggest that

15. -Film boiling on horizontal cylinders Following similar procedure to that for a vertical surface, Bromley obtained the following equation for film boiling heat transfer coefficient on horizontal cylinders. Breen & Westwater correlation (1962)

16. 布林與韋斯特瓦特的膜沸騰經驗式 (Breen & Westwaater, 1962)

17. -Film boiling on spheres Dhir & Lienhard -Film boiling heat transfer considering thermal radiation

18. Application of boundary layer theory for inverted annular flow film boiling Ref: Sakurai et al., 1990, “A General Correlation for Pool Film Boiling Heat Transfer From a Horizontal cylinder to subcooled liquid Part I & Part II.” J. Heat Transfer, vol. 112 pp.430-450.

19. Governing Equations ， vertical surface ， horizontal cylinder Boundary Conditions At y=0

20. At y=δ At y=∞ Where G i is the interfacial mass transfer rate,

21. Similarity Transform Sakura et al. defined the following similarity variables. where

22. Thus, and interfacial conditions. Numerical solutions is required.

23. 次冷度對池膜沸騰的效應（ Sakurai et al., 1990 ）

24. Approximate solutions -Neglect convective terms in all the conservation equation -Vapor flow is driven by buoyancy force -Liquid flow is driven by vapor flow.

25. Consider energy balance at the interface

26. Thus, Where

27. -Integration of above two equations yields Combined above two equations and solve for Where

28. Thus, The heat flux distribution is given by: Thus, the average heat flux can be obtained as The average Nusselt number for vertical surfaces can then be obtained as:

29. For a horizontal cylinder Where

30. Note that

31. Thus, for a vertical surface and for a horizontal cylinder These expressions are same as those derived by the simplied model of Bromley.

32. 6.2 Heat Transfer in the Liquid Deficient Region From: M.Andreani & G. Yadiaroglu “Prediction Methods for Dispersed Flow Film Boiling” Int. J. Multiphase Flow, vol.20 suppl. Pp.1-51, 1994.

33. Thermal nonequilibrium vapor is superheated Major heat transfer mechanisms Heat transfer from the surface to liquid droplet by either “wet” collisions or “dry” collision Convective heat transfer from the surface to the bulk vapor Convective heat transfer from the bulk vapor to suspended droplets in the vapor core Liquid deficient region Vapor flow

34. Bounding surface temperatures -Upper bound: complete departure from equilibrium all the heat added to fluid goes intosuperheating the vapor (toward this situation at low pressures and low velocity) -Lower bound: compete thermodynamic equilibrium all the heat added to fluid goes to evaporate the liquid droplets and T g =T sat. Surface temperature decreases as a result of increasing the vapor flow due to evaporation. Toward this situation at high pressure approaching the critical condition and high flow rates (>3000 kg/m 2 s)

35. From: Collier, 1981

36. Thermodynamic quality vs. actual quality (for uniform heat flux) -Thermodynamic quality (equilibrium quality) x e -Actual quality (for vapor superheat of T v -T sat present) -relation between x d0 and z d0 dryout z d0 x d0

37. Semi-theoretical model of Bennett et al (Ref. :Collier, 1981) -Assume -Assume the droplet number flow rate (N) is constant along the tube that is no drop breakup or coalescence occurs. But r d may be changed due to evaporation. -Mass balance gives -Energy balance equation r d : radius of the droplet

38. -Momentum eq. For liquid droplets -Evaporation of droplets (Ryley’s model) u v : vapor velocity ;u d : droplet velocity M: molecular weight R: universal gas constant

39. -Using the Runge-Kutta method to solve this problem P=69bar, vertical tube, L=5.8m, i.d=12.6mm 2r d =0.3mm(r d is an assumed value) (From: Collier, 1981)

40. Empirical correlations for liquid deficient film boiling heat transfer -Dougall & Rohsenow Correlation see more correlations in Groeneveld & Snoek, 1986. 6.3 The minimum film boiling temperature Ref. A. Sakurai, “Film Boiling Heat Transfer”, Proc. of 9 th Int. Heat Transfer Conference, vol.1, pp.157-186. T I : contact interface temperature ;P cr in kPa

41. It is expected that the minimum film boiling temperature on a solid surface with any thermal physical properties in ay liquid can be evaluated by above eq.(Sakurrai, 1990) Minimum temperature vs. system pressure for cylinders of 1.2 and 3.0 mm diameters in Freon-11 Minimum temperature vs. system pressure for platinum and gold cylinders of 2.0 and 3.0 mm diameters in water.

42. 6.4 Transition boiling Transition boiling Nucleate boiling Film boiling

43. Typical surface temperature history Typical local surface temperature history for a liquid-solid contact. From: Lee, (J.C.) Chen and Nelson.

44. Model of Pan et al (1989) Ref.: Pan et al, “The Mechanism of Heat Transfer in Transition Boiling”, Int. J. Heat Mass Transfer, vol.32 No.7 pp.1337-1349, 1989. (a) bubble departing(b) transient conduction (c) boiling incipience and heat transfer (d) macrolayer evaporation (e) vapour covering

45. Vapor bubble Liquid film (macrolayer) Vapor stem Heated surface Vapor bubble Liquid film vapor structure near heated surface at high heat fluxes Pool boiling at high fluxes on a horizontal flat plate τ: vapor bubble hovering time. t me : time interval of macrolayer evaporation. τ<t me : nucleate boiling. τ=t me : critical heat flux.(Haramura & Katto’s model, 1983) τ>t me : transition boiling / film boiling.

46. From M.Shoji, 1992

47. From: Katto, 1992

48. -Wall temperature during transient contact conduction --Effective liquid thermal conductivity

49. --Incipience criterion -Macrolayer evaporation heat flux -Macrolayer evaporation time δ me : from Haramura & Katta’s theory

50. -Vapor hovering period The average volumetric growth rate of the bubble in the period of macrolayer evaporation and vapour covering is given by where λ D is the most dangerous Taylor wavelength In above equation, it is assumed that the unit heater area participating in the growth of one vapour bubble is

51. The average heat flux within these two periods is given as -Liquid contact time fraction -Transition boiling heat flux From: Shoji, 1992

53. From: Pan er al., 1989From: Shoji, 1992

54. Empirical Correlation for Transition Boiling (for low quality) -Auracher’s correlation Ref.: Int. Froid 1988., vol.11, pp.329-335 Nucleate boiling during liquid contact is the major mode of heat transfer. h 0.3 : reference heat transfer coefficient at reduced pressure of 0.3 From: Auracher, 1988