Chapter 7: External Forced Convection Yoav Peles Department of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Objectives When you finish studying this chapter, you should be able to: Distinguish between internal and external flow, Develop an intuitive understanding of friction drag and pressure drag, and evaluate the average drag and convection coefficients in external flow, Evaluate the drag and heat transfer associated with flow over a flat plate for both laminar and turbulent flow, Calculate the drag force exerted on cylinders during cross flow, and the average heat transfer coefficient, and Determine the pressure drop and the average heat transfer coefficient associated with flow across a tube bank for both inline and staggered configurations.
Drag and Heat Transfer in External flow Fluid flow over solid bodies is responsible for numerous physical phenomena such as drag force automobiles power lines lift force airplane wings cooling of metal or plastic sheets. Free-stream velocity the velocity of the fluid relative to an immersed solid body sufficiently far from the body. The fluid velocity ranges from zero at the surface (the noslip condition) to the free-stream value away from the surface.
Friction and Pressure Drag The force a flowing fluid exerts on a body in the flow direction is called drag. Dragis compose of: pressure drag, friction drag (skin friction drag). The drag force F D depends on the density ρ of the fluid, the upstream velocity V, and the size, shape, and orientation of the body.
The dimensionless drag coefficient C D is defined as C D F = = 力 (7-1) 12 D 2 ρv A At low Reynolds numbers, most drag is due to friction drag. The friction drag is also proportional to the surface area. The pressure drag is proportional to the frontal area and to the difference between the pressures acting on the front and back of the immersed body.
The pressure drag is usually dominant for blunt bodies and negligible for streamlined bodies. When a fluid separates from a body, it forms a separated region between the body and the fluid stream. The larger the separated region, the larger the pressure drag.
Heat Transfer The phenomena that affect drag force also affect heat transfer. The local drag and convection coefficients vary along the surface as a result of the changes in the velocity boundary layers in the flow direction. The average friction and convection coefficients for the entire surface can be determined by 1 L CD = CD, xdx L 0 (7-7) h 1 L x = h dx (7-8) L 0
Parallel Flow Over Flat Plates Consider the parallel flow of a fluid over a flat plate of length L in the flow direction. The Reynolds number at a distance x from the leading edge of a flat plate is expressed as Re ρvx Vx x = = (7-10) µ ν
In engineering analysis, a generally accepted value for the critical Reynolds number is ρvx Re cr 5 10 µ cr 5 = = (7-11) The actual value of the engineering critical Reynolds number may vary somewhat from 10 5 to 3 x10 6.
Local Friction Coefficient The boundary layer thickness and the local friction coefficient at location x over a flat plate Laminar: δ C 4.91x = 0.664 = vx, 1/2 Rex f, x 1/2 Rex Re < 5 10 x 5 (7-12a,b) Turbulent: δ C 0.38x = 0.059 = vx, 1/5 Rex f, x 1/5 Rex 5 10 Re 10 5 7 x (7-13a,b)
Average Friction Coefficient The average friction coefficient Laminar: Turbulent: 1.33 5 C f = Re 5 10 1/2 L < (7-14) ReL 0.074 5 7 C f = 5 10 ReL 10 (7-15) Re 1/5 L When laminar and turbulent flows are significant x 1 cr L Cf = Cf, x laminar dx+ Cf, x turbulentdx L 0 xcr 5 Re = cr 5 10 1/5 L (7-16) 0.074 1742 5 7 C f = 5 10 ReL 10 (7-17) Re Re L
Heat Transfer Coefficient The local Nusselt number at location x over a flat plate Laminar: 1/2 1/3 Nu x = 0.332Re Pr Pr > 0.6 (7-19) Turbulent: 0.6 Pr 60 0.8 1/ 3 Nu x = 0.0296Rex Pr 5 7 5 10 Re 10 x (7-20) h x is infinite at the leading edge (x=0) and decreases by a factor of x 0.5 in the flow direction.
The average Nusselt number Average Nusset Number Laminar: Nu = 0.664Re Pr Re < 5 10 (7-21) 0.5 1/3 5 L Turbulent: 0.6 Pr 60 0.8 1/3 Nu = 0.037 ReL Pr 5 7 5 10 Re 10 x (7-22) When laminar and turbulent flows are significant x 1 cr L h= hx, laminar dx+ hx, turbulentdx L 0 xcr 5 Re = cr 5 10 (7-23) ( ) L 0.8 1 3 Nu = 0.037 Re 871 Pr (7-24)
Uniform Heat Flux When a flat plate is subjected to uniform heat flux instead of uniform temperature, the local Nusselt number is given by Laminar: Nu = x 0.453Re Pr 0.5 1/ 3 L (7-31) Turbulent: 0.6 Pr 60 0.8 1/ 3 Nu x = 0.0308Rex Pr 5 7 5 10 Re 10 x (7-32) These relations give values that are 36 percent higher for laminar flow and 4 percent higher for turbulent flow relative to the isothermal plate case.
Flow Across Cylinders and Spheres Flow across cylinders and spheres is frequently encountered in many heat transfer systems shell-and-tube heat exchanger, Pin fin heat sinks for electronic cooling. The characteristic length for a circular cylinder or sphere is taken to be the external diameter D. The critical Reynolds number for flow across a circular cylinder or sphere is about Re cr =2 x 10 5. Cross-flow over a cylinder exhibits complex flow patterns depending on the Reynolds number.
At very low upstream velocities (Re 1), the fluid completely wraps around the cylinder. At higher velocities the boundary layer detaches from the surface, forming a separation region behind the cylinder. Flow in the wake region is characterized by periodic vortex formation and low pressures. The nature of the flow across a cylinder or sphere strongly affects the total drag coefficient C D. At low Reynolds numbers (Re<10) friction drag dominate. At high Reynolds numbers (Re>5000) pressure drag dominate. At intermediate Reynolds numbers both pressure and friction drag are significant.
Average C D for circular cylinder and sphere Re 1 creeping flow Re 10 separation starts Re 90 vortex shedding starts. 10 3 < Re < 10 5 in the boundary layer flow is laminar in the separated region flow is highly turbulent 10 5 < Re < 10 6 turbulent flow
Effect of Surface Roughness Surface roughness, in general, increases the drag coefficient in turbulent flow. This is especially the case for streamlined bodies. For blunt bodies such as a circular cylinder or sphere, however, an increase in the surface roughness may actually decrease the drag coefficient. This is done by tripping the boundary layer into turbulence at a lower Reynolds number, causing the fluid to close in behind the body, narrowing the wake and reducing pressure drag considerably.
Heat Transfer Coefficient Flows across cylinders and spheres, in general, involve flow separation, which is difficult to handle analytically. The local Nusselt number Nu θ around the periphery of a cylinder subjected to cross flow varies considerably. Small θ Nu θ decreases with increasing θ as a result of the thickening of the laminar boundary layer. 80º<θ <90º Nu θ reaches a minimum low Reynolds numbers due to separation in laminar flow high Reynolds numbers transition to turbulent flow.
θ >90º laminar flow Nu θ increases with increasing θ due to intense mixing in the separation zone. 90º<θ <140º turbulent flow Nu θ decreases due to the thickening of the boundary layer. θ 140º turbulent flow Nu θ reaches a second minimum due to flow separation point in turbulent flow.
Average Heat Transfer Coefficient For flow over a cylinder (Churchill and Bernstein): Nu cyl 12 1/3 58 hd 0.62Re Pr Re = = 0.3+ 1 2/3 14 + k 1 282,000 + ( 0.4 Pr) Re Pr>0.2 The fluid properties are evaluated at the film temperature [T f =0.5(T +T s )]. 45 (7-35) Flow over a sphere (Whitaker): 14 hd 12 23 0.4 µ Nusph = = 2 + 0.4 Re 0.06 Re Pr k + µ s (7-36) The two correlations are accurate within 30%.
A more compact correlation for flow across cylinders Nu cyl hd Re m n = = C Pr (7-37) k where n = 1 / 3 and the experimentally determined constants C and m are given in Table 7-1. Eq. 7 35 is more accurate, and thus should be preferred in calculations whenever possible.
Flow Across Tube Bank Cross-flow over tube banks is commonly encountered in practice in heat transfer equipment such as heat exchangers. In such equipment, one fluid moves through the tubes while the other moves over the tubes in a perpendicular direction. Flow through the tubes can be analyzed by considering flow through a single tube, and multiplying the results by the number of tubes. For flow over the tubes the tubes affect the flow pattern and turbulence level downstream, and thus heat transfer to or from them are altered.
Typical arrangement in-line staggered The outer tube diameter D is the characteristic length. The arrangement of the tubes are characterized by the transverse pitch S T, longitudinal pitch S L, and the diagonal pitch S D between tube centers. In-line ( 列 ) Staggered ( 列 )
As the fluid enters the tube bank, the flow area decreases from A 1 =S T L to A T = (S T -D)L between the tubes, and thus flow velocity increases. In tube banks, the flow characteristics are dominated by the maximum velocity V max. The Reynolds number is defined on the basis of maximum velocity as ρvmaxd VmaxD Re D = = µ ν (7-39) For in-line arrangement, the maximum velocity occurs at the minimum flow area between the tubes V max = S T ST V D (7-40)
In staggered arrangement, for S D >(S T +D)/2 : for S D <(S T +D)/2 : V V max max = = 2 S T ST V D S T ( S D) D V (7-40) (7-41) The nature of flow around a tube in the first row resembles flow over a single tube. The nature of flow around a tube in the second and subsequent rows is very different. The level of turbulence, and thus the heat transfer coefficient, increases with row number. there is no significant change in turbulence level after the first few rows, and thus the heat transfer coefficient remains constant.
Zukauskas has proposed correlations whose general form is Nu hd m n = = C Re Pr ( Pr Pr ) 0.25 (7-42) k D D s where the values of the constants C, m, and n depend on Reynolds number. The average Nusselt number relations in Table 7 2 are for tube banks with 16 or more rows. Those relations can also be used for tube banks with N L provided that they are modified as Nu D N The correction factor F values are given in Table 7 3., L = F Nu (7-43) D
Pressure drop the pressure drop over tube banks is expressed as: 2 max P N f ρv = L χ (7-48) 2 f is the friction factor and χ is the correction factor. The correction factor (χ) given in the insert is used to account for the effects of deviation from square arrangement (in-line) and from equilateral arrangement (staggered).
流 度 S T T i T e = D T dx L = L ( ) dq T s T e dt T i x
. Q= ( mc pc ) ( T e T i ) = total heat transfer rate = 量 " dx", (i) Energy Balance (cold fluid) : dq = ( mc ) dt (ii) Heat Transfer (tube surface to fluid) : dq = h( pdx)( T T ). p c s Combined (i) and (ii), T T e i L ' dt hp Te Ts hpl' = [ ] dx;, ln( ) =.. ( T Ts) 0 ( mc ) Ti Ts ( mc ) Te T s hpl' ha s = exp = exp Ti Ts ( mc p) c ( mc p) c p c p c π DNL h( ) L' exp L ' hπ DNL π DhN = = exp = exp ( ρu Afr ) cp ρu ( NT ST L) cp ρu ( N S ) c T T p
p = 度 L' = 度 ( ) pl' = N L N T = = total total tube = number : heat number length = of transfer of tubes area = 數 = ( πdl) N tubes in one. Q = ( m c p ) c ( T e = total heat row (perpendicular to x - direction) T i ) transfer rate Ts Ti hpl' ln( ) =. Ts Te ( mc ) p c 兩 Q = ( Ts Ti ) ( Ts Te ) h( pl') = h( As )( LMTD) Ts T i ln( ) Ts Te : LMTD ( T s Ti ) ( Ts Ts Ti ln( T T s e T ) e ) = Log Mean Temperature Difference T ln
Example 7-7