lecture 22 - Thermal and Fluids Engineering I Lecture 22...

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Thermal and Fluids Engineering I Lecture 22 Page 1 Lecture 22 – Forced Convection in Internal Flow Laminar Convection in Pipes When a fluid in a pipe is hotter or colder than the pipe wall, both a velocity boundary layer and a thermal boundary layer form:
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Thermal and Fluids Engineering I Lecture 22 Page 2 Two common boundary conditions for convection in pipes are - constant heat flux - constant wall temperature We begin with constant heat flux. In a fully developed flow, the velocity profile does not change with downstream location. Furthermore, the shape of the temperature profile does not change with downstream location, although the average value of the temperature does change. To derive a formula for the fully-developed temperature profile in the pipe in laminar flow, we draw an annular control volume within the pipe:
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Thermal and Fluids Engineering I Lecture 22 Page 3 From the first law for an open system: out in ee ii QW m h m h −= ∑∑ & & && The work done by friction (called viscous dissipation) is usually small and will be neglected here. Also, we assume that conduction is only significant in the radial direction, not the axial direction. In the axial direction, conduction is usually overwhelmed by convection. With these assumptions, the first law becomes () "" rs i i rr r qA mh mh +∆ For an incompressible flow ( ) 22 x x x x x r qr x x A h h ππ ρ ∆− = V 2 2 2 2 x A r r r r π = = + ∆ +∆ In the limit as 0, r ∆→ the term 2 r decreases more rapidly than , r so it may be neglected and area becomes 2 x A ∆= 2 2 x xx x r rq rq rh h rx πρ ⎡⎤ ⎣⎦ ∆∆ V Taking the limit as 0 and 0, " 1 dd h rq rdr dx V For an incompressible fluid, enthalpy is given by p dh c dT = From Fourier’s law
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Thermal and Fluids Engineering I Lecture 22 Page 4 " r dT qk dr =− 1 xp TT rk c rr r x ρ ∂∂ ⎛⎞ = ⎜⎟ ⎝⎠ V Assuming a constant thermal conductivity, 1 where x p k r r x c α =≡ V From Chap 9, the fully developed velocity profile is () 2 21 xm
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lecture 22 - Thermal and Fluids Engineering I Lecture 22...

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