Understanding Heat Transfer in Fluids without Phase Change

1. Chapter12 Heat transfer to fluids without phase change

2. Regimes of heat transfer in fluids • A fluid being heated or cooled may be flowing in different flow patterns. Also, the fluid may be flowing in forced or natural convection.

3. At ordinary velocities the heat generated from fluid friction is negligible in comparison with the heat transferred between the fluids.

4. Because the situations of flow at the entrance to a tube differs from those well downstream from the entrance, the velocity field and associated temperature field may depend on the distance from the tube entrance

5. The properties of the fluid-viscosity, thermal conductivity, specific heat, and density are important parameters in heat transfer. Each of these, especially viscosity, is temperature-dependent.

6. Heat transfer by forced convection in turbulent flow • Perhaps the most important situation in heat transfer is the heat flow in a stream of fluid in turbulent flow.

7. Since the rate of heat transfer is greater in turbulent flow than in laminar flow, most equipment is operated in the turbulent range.

8. A dimensional analysis of the heat flow to a fluid in turbulent flow through a straight pipe yields dimensionless relations. (12-27)

9. The three groups in Eq(12-27) are recognized as the Nusselt(Nu), Reynolds(Re), and Prandtl (Pr) numbers respectively.

10. The Nusselt number for heat transfer from a fluid to a pipe or from a pipe to a fluid equals the film coefficient multiplied by d/k The film coefficient h is the average value over the length of the pipe

11. Prandtl number Pr is the ratio of the diffusivity of momentum μ/ρto the thermal diffusivity k/ρcp

12. The Prandtl number of a gas is usually close to 1（0.69 for air, 1.06 for steam). The Prandtl number of gases is almost independent of temperature because the viscosity and thermal conductivity both increase with temperature at about the same rate.

13. Empirical equation For heat transfer to and from fluids that follow the power-law relation, the dimensionless relation becomes To use the dimensionless relation, the constant c and index m, n must be known.

14. A recognized empirical correlation, for long tubes with sharp-edged entrances, is the Dittus-Boelter equation Where n is 0.4 when the fluid is being heated and 0.3 when it is being cooled.

15. A better relationship for turbulent flow is known as the Sieder-Tate equation (12-32)

16. Equation(12-32) should not be used for Reynolds numbers below 6000 or for molten metals, which have abnormally low Prandtl number.

17. Effect of tube length • Near the tube entrance, where the temperature gradients are still forming, the local coefficient hx is greater than h for fully developed flow.

18. In entrance, hx is quite large, but hx value drops rapidly toward h in a comparatively short length of tube. Average value of hi in turbulent flow.Since the temperature of the fluid changes from one end of the tube to the other and fluid properties µ, cp and k are all function of temperature, the local value of hi also varies from point to point along the tube.

19. The relation of local heat transfer coefficient hi and long tube h  is as follows • When L approaches infinite, hi is close to the h  of long tube.

20. For laminar flow, the relation of Nu and Pr and Re is • (12.25)

21. For gases the effect of temperature on hi is small. The increase in conductivity and heat capacity with temperature offset the rise in viscosity, giving a slight increase in hi.

22. For liquids the effect of temperature is much greater than for gases because of the rapid decrease in viscosity with rising temperature.

23. The effects of k, cp, and µ in Eq(12-36) all act in the same direction, but the increase in hi with temperature is due mainly to the effect of temperature on viscosity.

24. In practice, an average value of hi is calculated and used as a constant in calculating the overall coefficient U.

25. the average value of hi is computed by evaluating the fluid properties k, cp, and µ at average fluid temperature, defined as the arithmetic mean between the inlet and outlet temperatures.

26. Estimation of wall temperaturetw The estimation of tw requires an iterative calculation based on the resistance equation

27. To determine tw the wall resistance can usually be neglected

28. Substituting Uo, gives (12-38)

29. Cross sections other than circular To use Eq(12-30) for cross section other than circular it is only necessary to replace the diameter in both Reynolds and Nusselt number by the equivalent diameter de.

30. deis defined as 4 times the hydraulic radius rH. The method is the same as that used in calculating friction loss.

31. Heat transfer in transition region between laminar and turbulent flow Equation (12-32) applies only for Reynolds numbers greater than 6000. The range of Reynolds numbers between 2100 and 6000 is called the transition region, and no simple equation applies here.

32. A graphical method therefore is used. The method is based on a common plot of the Colburn j factor versus Re, with lines of constant value of L/D

33. The heat transfer coefficient can be calculated by following equation

34. Heating and cooling of fluids in forced convection outside tubes The mechanism of heat flow in forced convection outside tubes differs from that of flow inside tubes. The local value of heat-transfer coefficient varies from point to point around circumference in forced convection outside tube.

35. In Fig12.5, the local value of the Nusselt number is plotted radially for all points around circumference of the tube.

36. Nuθis maximum at the front and back of the tube and a minimum at the sides. In practice, the variations in the local coefficient are often no importance, and average values based on the entire circumference are used.

37. fluids flowing normal to a single tube The variables affecting the coefficient of heat transfer to a fluid in forced convection outside a tube are Do, the outside diameter of the tube; cp, μ, and k, the specific heat, the viscosity, and thermal conductivity, respectively, of the fluid; and G, the mass velocity.

38. Dimensional analysis gives Nusselt number is only a function of the Reynolds number.

39. The experimental data for air are plotted in this way in Fig12.6

40. For heating and cooling liquids flowing normal to single cylinders the following equation is used

41. Natural convection Consider a hot, vertical plate in contact with the air in a room. The density of the heated air immediately adjacent to the plate is less than that of the unheated air at a distance from the plate, and the buoyancy of the hot air causes an unbalance between the vertical layers of air of differing density.

42. Temperature difference between the surface of plate and the air causes a heat transfer. Natural convection in liquid follows the same pattern. The buoyancy of heated liquid layers near a hot surface generates convection currents just as in gases.

43. For single horizontal cylinders, the heat transfer coefficient can be correlated by equation containing three dimensionless groups Nu=f(Pr, Gr) Gr: Grashof number Pr: Prandtl number

44. (12-67) The coefficient of thermal expansion βis a property of fluid

45. Fig 12.8 shows a relationship, which satisfactorily correlates experimental data for heat transfer from a single horizontal cylinder to liquids or gases

47. For magnitudes of log Gr Pr of 4 or more, the line of Fig 12.8 follows closely the empirical equation

48. Natural convection to air from vertical shapes and horizontal plates Equations for heat transfer in natural convectionbetween fluids and solids of definite geometric shape are of the form (12-73) Values of the constants b and n for various conditions are given in Table 12.4

49. A double pipe heat exchanger is used to condense the saturated toluene vapor (2000kg/h) into saturated liquid. The condensation temperature and latent heat of toluene are 110 oC and 363kJ/kg, respectively. The cold water at 20 oC (inlet temperature) and 5000kg/h goes through the pipe (di=50 mm) fully turbulently. If the individual heat transfer coefficient hi of water side is 2100 w/(m2 K), and heat resistances of pipe wall as well as toluene side are much larger than that of water side (this means both resistances can be ignored), find: • Outlet temperature of cold water, in oC. • Pipe length of exchanger. • In order for mass flow rate of toluene to be double, if the mass flow rate of cold water at the same inlet temperature (20 oC) is double, what is the pipe length of new exchanger to be required?

50. Solution: Heat balance q=m1=m2Cp(Tcb-Tca) • 2000363=50004.19(Tcb-20) • (1)Outlet temperature of cold water Tcb=54.65oC • (2)U=h (from the problem) • ∆T1=110-54.65=55.35, ∆T2=110-20=90 • ∆T=(∆T1+∆T2)/2=72.68 (since ∆T2/∆T1<2) • L=q/(Ud∆T)=20003631000/3600/(21000.0572.68)=8.42m • (3) q’=2qm1=2m2Cp(T’cb-Tca) • Outlet temperature of cold water Tcb=54.65oC • ∆T’=(∆T1+∆T2)/2=72.68 • Fully developed turbulent flow, hRe0.8~m0.8~u0.8 • h’/h=20.8 , h’=1.74h • q’=1.74hdL’ ∆T’= 2m1 • q=hdL ∆T= m1 • L’/L=2/1.74 so L’=28.42/1.74=9.68m