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03_Transport_web

# 03_Transport_web - ENU 4134(Deriving/Driving Towards 1-D...

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ENU 4134 – (Deriving/Driving Towards) 1-D Transport D. Schubring Fall 2011

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Learning Objectives I 1-b-i Use the general balance equation to develop integral mass, momentum, and energy balances for two-phase flow I 1-b-ii Use the concept of a differential volume to reduce integral balances to one-dimensional transport equations for mass and momentum I 1-b-iii Identify and explain the significance of all terms in the one-dimensional differential mass, momentum, and energy equations
Approaches to the Transport Equations T&K attack the problem this way: I Develop mixture equations in 1-D (Section 5-5), for mass, momentum, & energy I Develop integral transport equations in 3-D (Section 5-6), for mass, momentum, & energy I Use of integral 3-D transport equations in volume Δ zA z to produce 1-D equations (Section 5-7), for mass, momentum & energy The final equations in 1-D are shown in Table 5-3 in T&K. (THE MIXTURE MOMENTUM EQUATION IS INCORRECT. Use Equation 5-140 instead.)

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Approaches to the Transport Equations (2) We’ll proceed this way in the next two days: I Develop integral transport equations in 3-D, including definition of jump conditions I Use of integral 3-D transport equations in volume Δ zA z to produce 1-D equations I Use 1-D equations to produce mixture equations (for momentum and mass – take energy equation from book without proof) I Mass, momentum, energy in turn – all the way through the calculations
General Balance Equation For any quantity X in a volume: dX dt = ˙ X in - ˙ X out + ˙ X generation - ˙ X dissipation (1) For mass/momentum/energy, ˙ X in - ˙ X out is referred to as the convection (or advection) term. It is usual to put this on the LHS of the equation: dX dt + ˙ X out - ˙ X in = ˙ X generation - ˙ X dissipation (2) In which case, ˙ X out - ˙ X in is said to be the convection term. For mass, ˙ X generation and ˙ X dissipation are zero. For momentum, these are forces, typically either body forces or pairs of forces grouped together as a derivative ( e.g. , in the Navier-Stokes equation). For energy, the term includes heat generation, viscous dissipation, etc.

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Integral Transport – Mass The total mass of phase k in a volume, V , is computed by: ZZZ V ρ k α k dV = ZZZ V k ρ k dV (3) The convection term ˙ X out - ˙ X in is: ZZ S k ρ k ( ~ v k - ~ v s ) · ~ ndS (4) Where S k is the surface of the control volume for phase k , including both solid boundaries and inter-phase interfaces. ~ v k is the fluid velocity at these interfaces. ~ v s is the velocity of the interface itself, and ~ n is an outward-pointing normal vector from the control volume.
Integral Transport – Mass – Convection Term ZZ S k ρ k ( ~ v k - ~ v s ) · ~ ndS (5) There are two types of interfaces in two-phase flow: those with fixed control volume boundaries and those between phases.

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03_Transport_web - ENU 4134(Deriving/Driving Towards 1-D...

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