mech7220-lam-int-flow - 5 Laminar Internal Flow 5.1...

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5 Laminar Internal Flow 5.1 Governing Equations These notes will cover the general topic of heat transfer to a laminar flow of fluid that is moving through an enclosed channel. This channel can take the form of a simple circular pipe, or more complicated geometries such as a rectangular duct or an annulus. It will be assumed that the flow has constant thermophysical properties (including density). We will first examine the case where the flow is fully developed . This condition implies that the flow and temperature fields retain no history of the inlet of the pipe. In regard to momentum conservation, FDF corresponds to a velocity profile that is independent of axial position x in the pipe. In the case of a circular pipe, u = u ( r ) where u is the axial component of velocity and r is the radial position. The momentum and continuity equations will show that there can only be an axial component to velocity (i.e., zero radial component), and that u ( r ) = 2 u m 1 - r R · 2 (1) with the mean velocity given by u m = 1 A Z A u dA = 2 R Z R 0 u r dr (2) The mean velocity provides a characteristic velocity based on the mass flow rate; ˙ m = ρ u m A (3) where A is the cross sectional area of the pipe. The thermally FDF condition implies that the dimensionless temperature profile is independent of axial position. The dimensionless temperature T is defined by T = T - T s T m - T s = func( r ) , TFDF (4) in which T s is the surface temperature (which can be a function of position x ) and T m is the mean temper- ature, defined by T m = 1 u m A Z A u T dA (5) This definition is consistent with a global 1st law statement, in that ˙ m C P ( T m, 2 - T m, 1 ) = ˙ Q 1 - 2 (6) with C P being the specific heat of the fluid. A differential form of this law is ˙ mC P dT m dx = q 00 s dA s dx = q 00 s P (7) in which q 00 s is the surface heat flux, which may be a function of x , and A s and P are the pipe surface area and perimeter. Note that integration of Eq. ( 7 ) over the length of the pipe results in Eq. ( 6 ). The thermally FDF condition, in Eq. ( 4 ), also states that the heat transfer coefficient h is constant. The convection law for internal flow is defined by q 00 s = h ( T s - T m ) = k ∂T ∂r fl fl fl fl R = - k ( T s - T m ) R d T d r fl fl fl fl 1 (8) with r = r/R being the dimensionless r coordinate. There is no negative sign in Fourier’s law, because the radial coordinate points toward the surface. Rearranging, Nu D = hD k = - 2 d T d r fl fl fl fl 1 = constant (9) The governing DE for the temperature distribution in the fluid is ρ C P u ∂T ∂x = k 2 T = k 1 r ∂r r ∂T ∂r + 2 T ∂x 2 (10) 1
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with boundary conditions (for the circular pipe case) ∂T ∂r fl fl fl fl 0 = 0 , T ( r = R ) = T s or - k ∂T ∂r fl fl fl fl R = q 00 s (11) The specific form of the wall BC would obviously depend on the given conditions, i.e., specified temperature, heat flux, or something more complicated. A convection–type BC is not an option here, as we are trying to determine the heat transfer coefficient h from a detailed formulation of convective–diffusive energy transport.
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