convection2

# convection2 - MATTER TRANSPORT (CONTINUED) There seem to be...

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MATTER TRANSPORT (CONTINUED) There seem to be two ways to identify the effort variable for mass flow gradient of the energy function with respect to mass is “matter potential”, µ — (molar) specific Gibbs free energy power dual of mass flow appears to be (molar) specific enthalpy, h The coupling between mass flow and entropy flow apparently reconciles these h = µ + Ts C AUTION : Enthalpy is NOT analogous to voltage or force it is not always an appropriate power dual for mass flow Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 1

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E XAMPLE : vertically oriented piston & cylinder with exiting mass flow How do you model the exit flow orifice? consider kinetic energy transported Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 2
Net power flow: consider three terms flow work rate internal energy transport rate kinetic energy transport rate P v 2 P net = ρ dN/dt + u dN/dt + 2 dN/dt v 2 P net = h + dN/dt 2 Power balance: subscript c: chamber, t: throat of orifice v c 2 v t 2 h c + h t + dN c /dt = dN t /dt 2 2 Mass balance: dN c /dt = dN t /dt Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 3

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Assume that velocity at the throat, v t , is much greater that velocity in the chamber, v c . v c << v t v c 2 2 0 h c – h t = v 2 t 2 v t = 2 h c – h t Mass flow rate dN t /dt = ρ t A t v t Thus dN t /dt = ρ t A t 2 h c – h t Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 4
D OES THIS MAKE PHYSICAL SENSE ? Does enthalpy difference drive mass flow? S NAG : Orifice flow is a typical “throttling” process Throttling is commonly assumed to occur at constant enthalpy Assume an ideal gas Pv = RT u = c v T h = u + Pv = (c v + R)T = c p T Thus enthalpy is proportional to temperature The model above implies that mass flow is initiated by temperature difference i.e., it predicts that mass flow must be zero at thermal equilibrium — NOT TRUE! Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 5

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WHERE DID WE GO WRONG? Enthalpy, h, is not an effort in the sense of a gradient that initiates a flow P net = h dN/dt is a composite of distinct power flows S OLUTION : model the coupling between components of power flow Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 6
ENERGY-BASED APPROACH IDENTIFY ( EQUILIBRIUM ) ENERGY STORAGE FUNCTION variable arguments identify ports gradients identify efforts hence P,T,µ for -V,S,N respectively no dynamics yet IDENTIFY COUPLING BETWEEN ELEMENTS (junction structure) some coupling may be “embedded” in “dissipation” phenomena IDENTIFY ( STEADY STATE ) DISSIPATION FUNCTION Mod. Sim. Dyn. Sys. Matter transport and enthalpy page 7

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EXAMPLE: two chambers connected by a throttling valve
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## convection2 - MATTER TRANSPORT (CONTINUED) There seem to be...

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