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Unformatted text preview: ENU 4134 (Deriving/Driving Towards) 1D Transport D. Schubring September 3, 2010 Learning Objectives I 1bi Use the general balance equation to develop integral mass, momentum, and energy balances for twophase flow I 1bii Use the concept of a differential volume to reduce integral balances to onedimensional transport equations for mass and momentum I 1biii Identify and explain the significance of all terms in the onedimensional differential mass, momentum, and energy equations Approaches to the Transport Equations T&K attack the problem this way: I Develop mixture equations in 1D (Section 55), for mass, momentum, & energy I Develop integral transport equations in 3D (Section 56), for mass, momentum, & energy I Use of integral 3D transport equations in volume zA z to produce 1D equations (Section 57), for mass, momentum & energy The final equations in 1D are shown in Table 53 in T&K. (THE MIXTURE MOMENTUM EQUATION IS INCORRECT. Use Equation 5140 instead.) Approaches to the Transport Equations (2) Well proceed this way in the next two days: I Develop integral transport equations in 3D, including definition of jump conditions I Use of integral 3D transport equations in volume zA z to produce 1D equations I Use 1D 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 NavierStokes equation). For energy, the term includes heat generation, viscous dissipation, etc. Integral Transport Mass The total mass of phase k in a volume, V , is computed by: ZZZ V k k dV = ZZZ V k k dV (3) The convection term X out X in is: ZZ S k 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 interphase interfaces. ~ v k is the fluid velocity at these interfaces. ~ v s is the velocity of the interface itself, and ~ n is an outwardpointing normal vector from the control volume. Integral Transport Mass Convection Term ZZ S k k ( ~ v k ~ v s ) ~ ndS (5) There are two types of interfaces in twophase flow: those with fixed control volume boundaries and those between phases....
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 Spring '11
 Schubring

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