Introduction to Heat and Mass Transfer. Week 14

Size: px

Start display at page:

Download "Introduction to Heat and Mass Transfer. Week 14"

Oscar Clement Chandler
5 years ago
Views:

1 Introduction to Heat and Mass Transfer Week 14

2 Next Topic Internal Flow» Velocity Boundary Layer Development» Thermal Boundary Layer Development» Energy Balance

3 Velocity Boundary Layer Development Velocity Boundary Layer Development r x Hydrodynamic Entrance Region Fully-developed Region

4 Velocity Boundary Layer Development (contd.) Laminar vs. Turbulent Boundary layer growth due to viscous effects near walls; velocity profile changes when fluid enters a channel or pipe Hydrodynamic Entrance Region x fd, hydrodynamic D laminar turbulent Re 2300 Re laminar 0.05Re Boundary layers on each surface merge at a certain point and the velocity profile does not change beyond that point Fully Developed Region D D D x fd, hydrodynamic D turbulent 10

5 Mean Velocity Mean velocity can be written as: Mean velocity for an incompressible flow inside a circular pipe: u m 2 r o u r, x rdr r For a fully-developed, laminar, incompressible flow inside a u r circular pipe, the velocity profile is given as: u o dp r r 1 4 dx r o m 2 o A c u r, x da 2 A c c u m 2 r dp o 8 dx

6 Friction Factor Moody (or Darcy) Friction Factor:» Laminar Flow:» Turbulent Flow: f laminar 64 Re Re D Re D D f f f dp D dx 1 2 um 2 turbulent turbulent / 4 Re D /5 Re D

7 Moody Digram

8 Thermal Boundary Layer Development Thermal Boundary Layer Development r x Thermal Entrance Region Fully-developed Region

9 Thermal Boundary Layer Development (contd.) Boundary layer growth due to difference in temperatures; temperature profile changes when fluid enters a channel or pipe Thermal Entrance Region x fd, thermal D laminar 0.05Re Pr The shape of temperature profile does not change beyond a certain point Fully Developed Region D x fd, thermal D turbulent 10 T x T r, x s x T x T x, s m fd thermal 0

10 Mean Temperature Mean temperature can be written as: Mean temperature for an incompressible flow inside a circular pipe: T m T m A c u 2 r o ut r, x rdr ur 2 0. mc ct da c m o T m increases with distance if T s > T m ; T m decreases with distance if T s < T m

11 Temperature Profile In thermally fully-developed flow, local convection coefficient is constant For constant heat flux, axial temperature gradient does not depend on radial position d d T r, x T x T x m s x dx dx fd, thermal fd, thermal fd, thermal For constant wall temperature, axial temperature gradient depends on radial position T T r, x d s T r, x T x m x fd, thermal T T dx s m fd, thermal

12 Energy Balance Using conservation of energy, we can write: For constant heat flux: " dt x qp P m s h T T s m dx mc mc p p For constant wall temperature:.. " qp s T x T. x m m, inlet mc T T x Px exp h s m. T T s m, inlet mcp p

13 Energy Balance (contd.)

14 Energy Balance (contd.) For constant wall temperature, the average temperature difference gives the convective heat transfer rate conv conv s LMTD Arithmetic Mean Temperature Difference Logarithmic Mean Temperature Difference q T T h AT AMTD LMTD T inlet T ln outlet T 2 outlet T T outlet T inlet inlet

15 Energy Balance (contd.) Complementary Materials

16 Mean Temperature Determination of Determination of the Mean Temperature Tm x is an essential feature of an internal flow analysis. Determination begins with an energy balance for a differential control volume. dq mc T dt T mc dt conv p m m m p m Integrating from the tube inlet to outlet, q mc T T conv p m o m i,, (1)

17 Mean Temperature (cont) A differential equation from which T x may be determined is obtained by substituting for dq q Pdx h T T Pdx m. conv s s m dtm qsp P h T s T m dx mc mc Special Case: Uniform Surface Heat Flux dt dx m s p qp mc p, p f x qp s Tm x Tm i x mc p 2 Why does the surface temperature vary with x as shown in the figure? Total heat rate: q conv q PL s

location, Ts Tm x Px exp h x Ts Tm, i mcp h x 1 x x 0 h dx Overall Conditions: T x

18 Mean Temperature (cont) Special Case: Uniform Surface Temperature From Eq. (2), with T Ts Tm d T dtm P h T dx dx mc Integrating from x=0 to any downstream location, Ts Tm x Px exp h x Ts Tm, i mcp h x 1 x x 0 h dx Overall Conditions: T x T p To s m, o PL h As exp h exp Ti Ts T m, i mcp mcp q h A T conv s m T m To Ti 1n T / T o i 3

19 Closure Coverage thus far..» discussed velocity and thermal boundary layer concepts related to internal flow» talked about mean velocity and temperature» treated constant heat flux and constant wall temperature boundary conditions related to internal flow

20 Example Velocity and temperature profiles for laminar flow in a tube of radius r o = 10 mm are given by: u(r) = 0.1[1-(r/r o ) 2 ] and T(r) = (r/r o ) (r/r o ) 4.» What is the mean velocity at this axial position?» Determine the mean temperature at this axial position

21 Example Is hydrodynamic entrance length higher in turbulent flow? Consider laminar and turbulent flow in a circular pipe. Is the friction factor higher at the inlet or at the exit? Why? Consider laminar and turbulent forced convection in a circular pipe. Is the heat flux higher at the inlet or at the exit? Why?

23 Closure Velocity boundary layer development related to internal flow» Hydrodynamic entrance region and fully-developed region and dependence on Re» Mean velocity based on mass flow rate Thermal boundary layer development related to internal flow» Thermal entrance region and fully-developed region and dependence on Re» Mean (bulk) temperature based thermal energy transport Energy balance related to constant heat flux and constant wall temperature Concept of logarithmic mean temperature difference

24 Next Topic Internal Flow» Laminar Correlations» Turbulent Correlations

25 Laminar Flow For constant heat flux: In thermally fully-developed region, velocity and thermal boundary layer approximations satisfied exactly Nu hd k laminar 4.36 D " q constant For constant wall temperature: In thermally fully-developed region, velocity boundary layer approximations valid; the thermal boundary layer approximation reasonably accurate Nu All properties evaluated at mean temperature fluid hd k s laminar D T constant fluid 3.66

26 Laminar Flow (contd.) In thermal entrance region, velocity and temperature profile can be still developing; leads to complicated problem For constant surface temperature:» Hausen: Nu» Sieder-Tate: D Ts constant hd k Nu D Ts fluid s D L Re Pr D L Re Pr 1/ constant hd Re Pr D 1.86 k L D fluid s 0.48 Pr D D 2/3

27 Turbulent Flow For fully-developed region (either constant heat flux or constant wall temperature), we can write: Dittus-Boelter:» n = 0.4 (heating); n = 0.3 (cooling) Sieder-Tate: Nu Nu D D hd k hd k turbulent /5 n Re Pr D» All properties at mean temperature except s fluid turbulent 4/5 1/ Re Pr D fluid s 0.14

28 HW # 7 prob. 1 Engine oil is heated by flowing through a circular tube of diameter 40 mm and length 20 m. The temperature of the tube is maintained at 150C. At the inlet, the flowrate and temperature are 0.4 kg/s and 20C, respectively.» What is the mean temperature of oil at the outlet?» Determine the total heat transfer rate for the tube

29 HW # 7 prob. 2 Hot water at 350 K flows through a steel pipe (thermal conductivity 14 W/m-K) of 100 mm outside diameter and 10 mm wall thickness. During winter, air at -10C flows at 5 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?

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

31 Closure (contd.) Thermally fully developed, laminar internal flow» Constant heat flux: Nu» Constant wall temperature: hd k laminar 4.36 D " q constant 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

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

33 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

34 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

35 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

36 Heated Vertical Flat Plate

37 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

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

39 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 L Gr 3 g T T L L2 1 L s 2 Nu f Gr, Pr Re Gr L2 1 Re L Gr L 2 1 L Nu Nu L L f Re, Pr Re L L f Gr, Re, Pr L L L

40 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: 1/ 4 Gr x Nu f Pr x 4» On average basis: 1/ 4 4 Gr L Nu f Pr L 3 4

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

42 Complementary Materials

43 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

T and : Velocity boundary layer thickness 5 for

44 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

Vertical Plates (cont) Nusselt Numbers Nu Nu x x and Nu L : 1/ 4 1/ 4 Grx hx Grx dt k 4 d 4 0 1/ 2 0.75 Pr g Pr 0 Pr 1/ 2 0.609 1.221 Pr 1.

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

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

49 HW # 7 prob. 3 Consider an object of characteristic length 1 cm and temperature difference between the object and ambient fluid is 20ºC.» Determine the Rayleigh number when ambient fluid is air at 1 atm and 300 K» What is the Rayleigh number when ambient fluid is water at 300 K?

50 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 22C. Each plate is 10 cm long and the plates are initially at 52C.» What is the minimum spacing that prevents interference between their free convection boundary layers?

51 HW # 7 will due on 1/3 (Thursday), right before the class! Late HW will not be accepted!!

52 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?

53 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

54 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 L2 1 Re L Nu L f Gr, Pr L

Introduction to Heat and Mass Transfer. Week 14

Introduction to Heat and Mass Transfer Week 14 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,