Chapter 8 INTERNAL FORCED CONVECTION

Transcription

1 Heat Transfer Chapter 8 INTERNAL FORCED CONVECTION Universitry of Technology Materials Engineering Department MaE216: Heat Transfer and Fluid

2 bjectives Obtain average velocity from a knowledge of velocity profile, and average temperature from a knowledge of temperature profile in internal flow. Have a visual understanding of different flow regions in internal flow, and calculate hydrodynamic and thermal entry lengths. Analyze heating and cooling of a fluid flowing in a tube under constant surface temperature and constant surface heat flux conditions, and work with the logarithmic mean temperature difference. Obtain analytic relations for the velocity profile, pressure drop, friction factor, and Nusselt number in fully developed laminar flow. Determine the friction factor and Nusselt number in fully

3 RODUCTION uid or gas flow through pipes or ducts is commonly used in heating and ling applications and fluid distribution networks. fluid in such applications is usually forced to flow by a fan or pump through ow section. ough the theory of fluid flow is reasonably well understood, theoretical tions are obtained only for a few simple cases such as fully developed inar flow in a circular pipe. refore, we must rely on experimental results and empirical relations for most flow problems rather than closed-form analytical solutions. For a fixed surface area, the circular tube gives the most heat transfer for the least pressure drop.

4 uid velocity in a pipe changes ero at the wall because of the p condition to a maximum at the enter. d flow, it is convenient to work n average velocity V avg, which ns constant in incompressible hen the cross-sectional area of pe is constant. verage velocity in heating and g applications may change what because of changes in ty with temperature. practice, we evaluate the fluid rties at some average rature and treat them as ants.

5 RAGE VELOCITY AND TEMPERATURE alue of the average (mean) velocity t some streamwise cross-section is: verage velocity for incompressible a circular pipe of radius R is: flow, it is convenient to work with an ge or mean temperature T m, which ns constant at a cross section. The mean rature T m changes in the flow direction ver the fluid is heated or cooled.

6 inar and Turbulent Flow in Tubes in a tube can be laminar or turbulent, depending on the flow tions. flow is streamlined and thus laminar at low velocities, but turns lent as the velocity is increased beyond a critical value. ition from laminar to turbulent flow does not occur suddenly; r, it occurs over some range of velocity where the flow fluctuates en laminar and turbulent flows before it becomes fully turbulent. pipe flows encountered in practice are turbulent. ar flow is encountered when highly viscous fluids such as oils n small diameter tubes or narrow passages. ition from laminar to turbulent flow depends on the Reynolds er as well as the degree of disturbance of the flow by surface ness, pipe vibrations, and the fluctuations in the flow. low in a pipe is laminar for Re < 2300, fully turbulent for Re >

7 s number for flow in a circular tube through noncircular tubes, the s number as well as the Nusselt, and the friction factor are n the hydraulic diameter D h Under most practical conditions, the flow in a pipe is laminar for Re < 2300, fully turbulent for Re > 10,000, and transitional in between.

8 ENTRANCE REGION y boundary layer (boundary layer): The region of the flow in which the effects of the shearing forces caused by fluid viscosity are felt. othetical boundary surface divides the flow in a pipe into two regions: ary layer region: The viscous effects and the velocity changes are significant. onal (core) flow region: The frictional effects are negligible and the velocity remains ally constant in the radial direction. ynamic entrance region: The region from the pipe inlet to the point at which the profile is fully developed. ynamic entry length L h : The length of the hydrodynamic entrance region. ynamically fully developed region: The region beyond the entrance region in which city profile is fully developed and remains unchanged. the entrance region is ydrodynamically ing flow since this is on where the velocity evelops.

9 The ment of uid properties in internal flow are usually evaluated at the bulk mean fluid rature, which is the arithmetic average of the mean temperatures at the nd the exit: T b = (T m, i +T m, e )/2. al entrance region: The region of flow over which the thermal boundary layer s and reaches the tube center. al entry length: The length of the thermal entrance region. ally developing flow: Flow in the thermal entrance region. This is the region the temperature profile develops. ally fully developed region: The region beyond the thermal entrance region in he dimensionless temperature profile remains unchanged. eveloped flow: The region in which the flow is both hydrodynamically and lly developed.

10 ynamically fully developed: ally fully developed: Variation of the friction factor and the convection heat transfer coefficient in the flow direction for flow in a tube (Pr > 1). heat flux: hermally fully developed region of a e local convection coefficient is nt (does not vary with x). ore, both the friction (which is related shear stress) and convection ients remain constant in the fully ped region of a tube. essure drop and heat flux are higher in rance regions of a tube, and the effect

11 ths Nusselt numbers and thus h values are much higher in the entrance region. Nusselt number reaches a constant value at a distance of less than 10 eters, and thus the flow can be assumed to be fully developed for x > 10D. Nusselt numbers for uniform surface perature and uniform ace heat flux itions are identical e fully developed ons, and nearly tical in the entrance ons. iation of local Nusselt umber along a tube in turbulent flow for both

12 ERAL THERMAL ANALYSIS f heat transfer e heat flux e local heat transfer coefficient. The thermal conditions at the surface can be approximated to be constant surface temperature (T s = const) or constant surface heat flux (q s = const). The constant surface temperature condition is realized when a phase change process such as boiling or condensation occurs at the outer surface of a tube. The constant surface heat flux condition is realized when the tube is subjected to radiation or electric resistance heating uniformly from all directions. We may have either T s = constant or q = constant at the surface of a tube,

13 tant Surface Heat Flux (q s = constant) f heat transfer: fluid temperature tube exit: e temperature: Variation of the tube surface and the mean fluid

14 r tube: Energy interactions for a differential control volume in a tube.

15 stant Surface Temperature (T s = constant) f heat transfer to or from a fluid flowing in a tube uitable ways of expressing T avg : rithmetic mean temperature difference garithmic mean temperature difference metic mean temperature difference: mean fluid temperature: T b = (T i +T e )/2 sing arithmetic mean temperature difference, we assume that the mean temperature varies linearly along the tube, which is hardly ever the case n T s = constant. simple approximation often gives acceptable results, but not always.

16 ating from x = 0 (tube inlet, T i ) to x = L (tube exit, T m =T e ) The variation of the mean fluid temperature along the tube for the case of constant temperature.

17 Log mean temperature difference umber of transfer units. A re of the effectiveness of the ansfer systems. U = 5, T e =T s, and the limit for ansfer is reached. l value of NTU indicates more unities for heat transfer. an exact representation of the e temperature difference n the fluid and the surface. T e differs from T i by no more percent, the error in using the tic mean temperature ce is less than 1 percent. n NTU greater than 5 indicates that

18 INAR FLOW IN TUBES

19 The maximum velocity occurs at the centerline, r = 0:

20 tity of interest in the analysis of pipe flow is the pressure drop P since ectly related to the power requirements of the fan or pump to maintain flow. Pressure Drop In laminar flow, the friction factor is a function of the Reynolds number only and is independent of the roughness of the pipe surface. Head loss losses are

21 d loss h L represents the additional height that the fluid be raised by a pump in order to overcome the frictional n the pipe. The head loss is caused by viscosity, and it is related to the wall shear stress. uired pumping power to e the pressure loss: Poiseuille s law The average velocity for laminar flow pecified flow rate, the pressure drop and e required pumping power is proportional ength of the pipe and the viscosity of the ut it is inversely proportional to the fourth of the radius (or diameter) of the pipe.

22 perature Profile and the Nusselt Number te of net energy transfer to the The differential volume element

23 stant Surface Heat Flux ing the boundary conditions = 0 at r = 0 (because of etry) and T = T s at r = R: Therefore, for fully developed laminar flow in a circular tube subjected to constant surface heat flux, the Nusselt number is a constant. There is no dependence on the Reynolds or the Prandtl numbers.

24 stant Surface Temperature thermal conductivity k for use in the Nu relations should be evaluated e bulk mean fluid temperature. aminar flow, the effect of surface roughness on the friction factor and eat transfer coefficient is negligible. minar flow in a tube with constant ce temperature, both the friction Laminar Flow in Noncircular Tubes Nusselt number relations are given in Table 8-1 for fully developed laminar flow in tubes of various cross sections. The Reynolds and Nusselt numbers for flow in these tubes are based on the hydraulic diameter D h = 4A c /p. Once the Nusselt number is available,

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26 loping Laminar Flow in the Entrance Region circular tube of length L subjected to constant surface temperature, the ge Nusselt number for the thermal entrance region can be determined from: verage Nusselt number is larger at the entrance region, and it aches asymptotically to the fully developed value of 3.66 as L. the difference between the surface and the fluid temperatures is large, y be necessary to account for the variation of viscosity with temperature: All properties are evaluated at the bulk mean fluid temperature, except for s, which is evaluated at the surface temperature. verage Nusselt number for the thermal entrance region of etween isothermal parallel plates of length L is

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28 BULENT FLOW IN TUBES Chilton Colburn analogy First Petukhov equation Colburn equation Dittus Boelter equation the variation in properties is large due to a large temperature difference:

29 Second Petukhov equation Gnielinski relation relations above are not very sensitive to the thermal conditions at the surfaces and can be used for both T s = constant and q s = constant.

30 gh Surfaces iction factor in fully developed turbulent pipe flow depends on the lds number and the relative roughness /D, which is the ratio of the height of roughness of the pipe to the pipe diameter. y chart is given in the appendix as Fig. A 20. Colebrook equation ents the Darcy friction factor for pipe flow as a function of the Reynolds er and /D over a wide range. An approximate explicit relation for f was given by S. E. Haaland. bulent flow, wall roughness increases the heat transfer coefficient h factor of 2 or more. The convection heat transfer coefficient for rough can be calculated approximately from the Gnielinski relation or

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32 loping Turbulent Flow in the Entrance Region ntry lengths for turbulent flow are typically short, often just 10 tube ters long, and thus the Nusselt number determined for fully developed ent flow can be used approximately for the entire tube. imple approach gives reasonable results for pressure drop and heat er for long tubes and conservative results for short ones. lations for the friction and heat transfer coefficients for the entrance regions ailable in the literature for better accuracy. ulent Flow in Noncircular Tubes ure drop and heat transfer cteristics of turbulent flow in tubes are ated by the very thin viscous sublayer the wall surface, and the shape of the egion is not of much significance. rbulent flow relations given above for r tubes can also be used for cular tubes with reasonable accuracy In turbulent flow, the velocity profile is nearly a straight line in

33 through Tube Annulus Hydraulic diameter of annulus minar flow, the convection coefficients for the and the outer surfaces are determined from A double-tube heat exchanger that consists of two concentric tubes. lly developed turbulent flow, h i and h o proximately equal to each other, and the nnulus can be treated as a noncircular ith a hydraulic diameter of D h =D o D i. usselt number can be determined from a le turbulent flow relation such as the ski equation. To improve the accuracy, lt numbers can be multiplied by the ng correction factors when one of the alls is adiabatic and heat transfer is h the other wall:

34 Transfer Enhancement s with rough surfaces have much r heat transfer coefficients than with smooth surfaces. transfer in turbulent flow in a tube e increased by as much as 400 nt by roughening the surface. hening the surface, of course, ncreases the friction factor and he power requirement for the or the fan. onvection heat transfer cient can also be increased by ing pulsating flow by pulse ators, by inducing swirl by ing a twisted tape into the tube,

35 ummary Introduction Average Velocity and Temperature Laminar and Turbulent Flow in Tubes The Entrance Region Entry Lengths General Thermal Analysis Constant Surface Heat Flux Constant Surface Temperature Laminar Flow in Tubes Pressure Drop Temperature Profile and the Nusselt Number Constant Surface Heat Flux Constant Surface Temperature Developing Laminar Flow in the Entrance Region Turbulent Flow in Tubes Rough Surfaces Developing Turbulent Flow in the Entrance Region Turbulent Flow in Noncircular Tubes