Introduction to Heat and Mass Transfer. Week 14

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1 Introduction to Heat and Mass Transfer Week 14

2 HW # 7 prob. 2 Hot water at 50C flows through a steel pipe (thermal conductivity 14 W/m-K) of 100 mm outside diameter and 8 mm wall thickness. During winter, air at -5C flows at 3 m/s across the pipe. The water flow at a mean velocity of 0.5 m/s inside the pipe. The cost of producing hot water is $0.05 per kw.hr.» What is the daily cost of heat loss from 1 km length of the above pipe?

3 Closure Coverage thus far..» presented correlations for calculation of heat and mass transfer parameters for laminar and turbulent internal flow

4 Closure (contd.) Thermally fully developed, laminar internal flow» Constant heat flux: Nu» Constant wall temperature: hd k D q constant laminar 4.36 " Entrance region of laminar internal flow the related complications and correlations Thermally fully developed, turbulent internal flow Nu D hd k fluid turbulent fluid Nu hd k s laminar D T constant fluid 4/5 n Re Pr D 3.66

5 Next Topic Free Convection» Physical Mechanism» Governing Equations» Similarity Solution: Vertical Plate

6 Physical Mechanism Forced convection is owing to imposed motion of fluid On the other hand, free convection is due to density gradient and body force Combination of density gradient and body force gives rise to buoyancy force resulting in motion of fluid We will consider» Density gradient caused by temperature gradients within the fluid» Body force assumed to be gravitational

7 Physical Mechanism (contd.) T 1 Heat transfer via free convection Unstable; Buoyancy forces lead to circulation T 2 T 2 > T 1 T 1 Gravitational Force Stable; No circulation Heat transfer via conduction only T 2 T 2 < T 1

8 Volumetric Thermal Expansion Coeffecient Change in density of fluid due to temperature change at constant pressure; certainly important for free convection In general: 1 T 1 T T p=constant For ideal gas: ideal gas 1 T

9 Heated Vertical Flat Plate

10 Continuity: Momentum: Energy: Governing Equations u 0 x y 2 u u g u u x y y u g T T x y y 2 T T T u x y y 2 2 u u u 2 Unlike in forced convection, flow field is strongly coupled to temperature field 2

11 Similarity Solution Using non-dimensional variables» Velocity u u u» Temperature T T Ts» Spatial Coordinate ref T T x x u ref y y» u ref is an arbitrary reference velocity

12 Similarity Solution (contd.) Grashof Number represents the ratio of buoyancy forces to viscous forces Grashof Number related to free convection like Reynolds Number related to forced convection Free Convection Forced Convection Mixed Convection Gr Gr 3 g T T 2 1 s 2 Nu f Gr, Pr Re Gr 2 1 Re Gr 2 1 Nu Nu f Re, Pr Re f Gr, Re, Pr

13 Similarity Solution (contd.) Using an appropriate similarity variable, we can solve the governing equations for laminar free convection on a vertical plate and obtain:» On local basis: Nu x Grx 4 1/ 4 f Pr» On average basis: Nu 4 Gr 3 4 1/ 4 f Pr

14 Similarity Solution (contd.) Rayleigh number, Ra Ra Gr Pr g Ts T 3 Gr 3 g T T s 2 Pr

15 Complementary Materials

16 Vertical Plates (cont) Similarity Solution Based on existence of a similarity variable,, through which the x-momentum equation may be transformed from a partial differential equation with twoindependent variables ( x and y) to an ordinary differential equation expressed exclusively in terms of. y Gr x x 4 1/ 4 Transformed momentum and energy equations: 2 f 3 ff 2 f T 0 T 3Pr ft 0 *" *' df x 1/ 2 T T f Grx u T d 2 T T s

17 Vertical Plates (cont) Numerical integration of the equations yields the following results for f T and : Velocity boundary layer thickness 5 for Pr 0.6 1/ 4 Grx Pr 0.6 : 5x 7.07 x 1/ 4 4 Gr x x 1/ 4

18 Vertical Plates (cont) Nusselt Numbers Nu Nu x x and Nu : 1/ 4 1/ 4 Grx hx Grx dt k 4 d 4 0 1/ Pr g Pr 0 Pr 1/ Pr Pr 1 h 4 o hdx Nu Nu 3 Transition to Turbulence Amplification of disturbances depends on relative magnitudes of buoyancy and viscous forces. Transition occurs at a critical Rayleigh Number. 3 9 g Ts T x Rax, c Grx, c Pr 10 g 1/ 4 Pr

19 Vertical Plates (cont) Empirical Heat Transfer Correlations 9 aminar Flow Ra 10 : 1/ 4 Nu /16 4 / Ra / Pr All Conditions: Nu / Ra 9/16 8/ / Pr 2

20

21

22 HW # 7 prob. 3 Consider an object of characteristic length 1 cm and temperature difference between the object and ambient fluid is 30ºC.» Determine the Rayleigh number when ambient fluid is air at 1 atm and 400 K» What is the Rayleigh number when ambient fluid is water at 300 K?

23 HW # 7 prob. 4 A number of thin plates are to be cooled by vertically suspending them in a water bath at a temperature of 20C. Each plate is 15 cm long and the plates are initially at 54C.» What is the minimum spacing that prevents interference between their free convection boundary layers?

24 HW # 7 will due on 12/28 (Thursday), right before the class! ate HW will not be accepted!!

25 Questions What is the difference between free convection and forced convection? Consider a boiled egg in a spacecraft. Will the egg cool faster or slower when spacecraft is in space as compared to when it is on the ground?

26 Closure Coverage thus far..» talked about the basic mechanism related to free convection» discussed the governing equations and resulting similarity solution for a vertical flat plate

27 Closure (contd.) Basic physical mechanism related to free convection owing to density gradients caused via temperature gradients and body forces due to gravity Strongly coupled continuity, momentum and energy equations for free convection Grashof Number and its importance in description of free convection process For free convection: Gr 2 1 Re Nu f Gr, Pr

28 Next Topic Empirical correlations for free convection analysis» Vertical Flat Plate» Horizontal Flat Plate» Vertical Cylinder» Horizontal Cylinder» Sphere

29 For isothermal vertical plate: Vertical Plate» C and n listed in following figure (see next page); all properties at film temperature Churchill-Chu: Nu Nu h k Modifications necessary for constant heat flux conditions fluid h k fluid isothermal vertical plate CRa n 0.387Ra / 6 Pr 9/16 8/ 27 2

30

31 Cylinder For vertical cylinders, correlations for vertical flat plate applicable when velocity boundary layer thickness much less than cylinder diameter For isothermal horizontal cylinders» Morgan: C and n listed in Table 9.1; all properties at film temperature» Churchill-Chu: Nu D Nu D hd k fluid hd 0.6 k fluid isothermal cylinder CRa n D 0.387Ra / 6 D Pr 9/16 8/ 27 2

32

33 Sphere/Other Shapes For isothermal spheres:» Churchill: Nu D hd 2 k isothermal sphere fluid 0.589Ra /16 A variety of other shapes have been investigated and correlations available» Horizontal Plates» Inclined Plates» Channels 1/ 4 D Pr 4/9

34

35

36 Complementary Materials

37 Horizontal Plates Horizontal Plates Buoyancy force is normal, instead of parallel, to the plate. Flow and heat transfer depend on whether the plate is heated or cooled and whether it is facing upward or downward. Hot Surface Facing Upward or Cold Surface Facing Downward Ts T Nu 0.54 Ra 10 Ra 10 1/ Nu 0.15 Ra 10 Ra 10 1/ Ts T How does h depend on when Nu Ra 1/ 3?

38 Horizontal Plates (cont) Hot Surface Facing Downward or Cold Surface Facing Upward Ts T Nu 0.27Ra 10 Ra 10 1/ Ts T Why do these flow conditions yield smaller heat transfer rates than those for a heated upper surface or cooled lower surface?

39 Example A household oven door of 0.5 m height and 0.7 m width reaches an average surface temperature of 32ºC during its operation. The ambient air in the room is at 22ºC.» Estimate the heat loss to the room Now consider that the emissivity of door is 1 and isothermal surrounding is at 22ºC» Comment on the heat loss by free convection relative to radiation

40 Closure Coverage thus far..» presented correlations for calculation of heat transfer due to free convection

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