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Unformatted text preview: C HAPTER 8: I NTERNAL C ONVECTION In internal convection, boundary layer development may be restricted by the diameter of the pipe or conduit within which the fluid flows b Possible that u ∞ is never reached External convection: Internal convection: δ u u ∞ u ∞ u max < u ∞ u ∞ Midline Laminar or turbulent? Laminar or turbulent? Fully developed or not? Thus, to calculate h for internal flow cases, we need to evaluate the effect of constraining the fluid flow on the development of the thermal and velocity boundary layers. ≠ NOTE: A flow is considered internal if the fluid is completely contained within a conduit (for any boundary layer development) OR if fluid flows externally between two bodies (i.e. two parallel plates) and incomplete boundary layer development occurs between those two bodies. For the latter case, you should check the boundary layer thicknesses: Laminar flow between plates: Turbulent flow between plates: If 2 δ < plate spacing b external, plate If 2 δ > spacing b internal,square duct Evaluate Re x at the end of the plate ( maximum δ development b point of highest likelihood of boundary layer overlap) 2 . Re 37 . x x = δ 5 . Re 5 x x = δ Velocity Boundary Layer Development in a Circular Tube Two regions of flow exist in this case: Entrance Region (0 < x < x fd,h ): • Frictional drag induces boundary layer development • Velocity profile, boundary layer thickness vary with x Fully Developed Region ( x > x fd,h ) • “Steadystate” flow occurs • Velocity profile, Re , boundary layer thickness do not change with x o Compare to external flow– Re=f(x) at all x values Hydrodynamic Entrance Region Fully Developed Region x fd,h x u Inviscid flow region Boundary layer region u(r,x) δ δ r o r The length of the entrance region x fd,h depends on the nature of the flow Generally, for internal flow, μ ρ D u m D = Re u m = mean velocity of fluid within pipe For incompressible fluids, c m A Q u = Q = volumetric flow rate of fluid through conduit A c = crosssectional area of conduit Flow is LAMINAR if Re D ≤ 2300 In this case, D h fd D x Re 05 . , ≈ and 2 D = δ Flow is TURBULENT if Re D > 2300 (although full turbulent flow does not occur until Re D > 10000) In this case, 60 10 , < < D x h fd and 2 D < δ b assume fully developed turbulent flow at x fd,h /D >10 The velocity profile within the pipe can be derived based on the definition of the mass flow rate through the pipe. c m A u m ρ = & On this basis, for a cylindrical pipe, The mass flow rate m can also be expressed as the integral of the mass flux ( ρ u ) over the crosssection: c A dA x r u m c ) , ( ∫ = ρ & Thus, the mean velocity u m can also be written as: Or, for an incompressible fluid in a circular tube: m = mass flow rate through tube A c = crosssectional area of tube ρ = fluid density ( T dependent) c c A m A dA x r u u c ρ ρ ) , ( ∫ = ∫ = o r o m rdr x r u r u 2 ) , ( 2 ....
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This note was uploaded on 01/24/2010 for the course CHEM ENG 2A04 taught by Professor Toddhoare during the Winter '10 term at McMaster University.
 Winter '10
 TODDHOARE

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