Chapter 7: External Forced ConvectionYoav PelesDepartment of Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute

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ObjectivesWhen 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. Drag is compose of: pressure drag, friction drag (skin friction drag).

The drag force FD depends on the

density of the fluid, the upstream velocity V, and the size, shape, and orientation of the body.

The dimensionless drag coefficient CD is defined asCD FD = = 2 1 2 V A

(7-1)

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 by1 CD = CD , x dx L0 1 h = hx dx L0L L

(7-7)

(7-8)

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 asRe x = Vx

= Vx

(7-10)

In engineering analysis, a generally accepted value for the critical Reynolds number is

Vxcr Recr = = 5 105

(7-11)

The actual value of the engineering critical Reynolds number may vary somewhat from 105 to 3 x106.

Local Friction Coefficient The boundary layer thickness and the local friction coefficient at location x over a flat plate Laminar:4.91x v , x = 1/ 2 Re x Re x < 5 105 0.664 C f , x = 1/ 2 Re x 0.38 x v , x = 1/ 5 Re x 5 105 Re x 107 0.059 C f ,x = Re1/ 5 x

(7-12a,b)

Turbulent:

(7-13a,b)

Average Friction Coefficient The average friction coefficient Laminar: Turbulent:1.33 C f = 1/ 2 Re L 0.074 Cf = Re1/ 5 L Re L < 5 105 5 105 Re L 107

(7-14) (7-15)

When laminar and turbulent flows are significantL xcr 1 C f = C f , x laminar dx + C f , x turbulent dx (7-16) L 0 xcr Recr = 5 105

0.074 1742 Cf = 1/ 5 Re L Re L

5 105 Re L 107 (7-17)

Heat Transfer Coefficient The local Nusselt number at location x over a flat plate Laminar:Nu x = 0.332 Re1/ 2 Pr1/ 30.8 x 1/ 3

Pr > 0.6 0.6 Pr 60 5 10 Re x 105 7

(7-19) (7-20)

Turbulent: Nu x = 0.0296 Re Pr

hx is infinite at the leading edge (x=0) and decreases by a factor of x0.5 in the flow direction.

Average Nusset Number The average Nusselt number Laminar: Turbulent:Nu = 0.664 Re0.5 Pr1/ 3 L Nu = 0.037 Re Pr0.8 L 1/ 3

Re < 5 105 0.6 Pr 60 5 10 Re x 105 7

(7-21) (7-22)

When laminar and turbulent flows are significantL xcr 1 h = hx , laminar dx + hx , turbulent dx L 0 xcr Recr = 5 105

(7-23)

Nu = 0.037 Re 871 Pr0.8 L

(

)

13

(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 = 0.453Re0.5 Pr1/ 3 L 0.6 Pr 60 5 105 Re x 107 (7-32)

(7-31)

Turbulent: Nu x = 0.0308 Re Pr0.8 x

1/ 3

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 Recr=2 x 105. Cross-flow over a cylinder exhibits complex flow patterns depending on the Reynolds number.

At very low upstream velocities (Re1), 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 CD. At low Reynolds numbers (Re5000) pressure drag dominate. At intermediate Reynolds numbers both pressure and friction drag are significant.

Average CD for circular cylinder and sphere Re 1 creeping flow Re 10 separation starts Re 90 vortex shedding starts. 103 < Re < 105 in the boundary layer flow is laminar in the separated region flow is highly turbulent

105 < Re < 106 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 around the periphery of a cylinder subjected to cross flow varies considerably. Small Nu decreases with increasing as a result of the thickening of the laminar boundary layer. 80