Non-Equilibrium Thermodynamics for Engineers

Transcription

1 Non-Equilibrium Thermodynamics for Engineers How do we find the otimal rocess unit? Signe Kjelstru, Chair of Engineering Thermodynamics Deartment of Process and Energy TU Delft ecture no. 7

2 Why is the entroy roduction imortant? The work outut from the maximum available w = w max wlost Guy-Stodola s theorem (889, 9): w lost = T ( dsirr / dt) > dsirr / dt = σdv ecture no. 7

3 Mathematical methods for constrained otimisation Euler agrange otimisation: Constraint examles: P= P T = T a ds irr = + λi dt i Conservation equations are included in the objective function Cf. Course on Engeering Fundamentals ectured by rof. J. Gross P i Control theory H =σ (,) z t + λ (,) z t f i dt Energy balance ft = =... d Momentum balance f = =... dξ Mass balance fξ = =... Extra conditions, i.e. = const. ocal control of conservation equations Defined control variables give a ractical handle Mathematically robust An autonomous Hamiltonian is constant along the ath i ext i ecture no. 7 3

4 Otimal isothermal exansion () Find the external ressure in a one ste rocess that gives N moles of ideal gas in a iston mininmum lost work, when the ressure of the system changes from V to V wmax = NRT ln V w= extdv = NRT V ( ) wlost = w wmax = NRT ext ( ) + ln The iston moves with time: dv () t dt f = [ ] ( () () ) ext t t t () ecture no. 7 4

5 Otimal isothermal exansion () Which ext gives minimum entroy roduction, given the values of and? Min N moles of ideal gas in a iston w = NRT max ln ds irr = NR ext ( ) + ln dt Given, Solution: ext ( / ) f ex( ) NRT V w= extdv NRT ( ) ecture no. 7 = 5 V

6 Otimal isothermal exansion (3) The ressure variation giving minimum lost work (stiled line) can be obtained from control theory General objective function: 5 Max. work θ dsirr dv ( t) = ( ext ( t) ( t)) dt T dt Constraint: dv () t dt ext t () f = Solution: [ ] ( () () ) ext t t t () NRT = + ln fθ = t / θ t / θ Presssure / bar Volume / 3 m 3 The driving force is constant along the otimal ath, and so is σ! ecture no. 7 6

7 Otimal exansion (4): Continuous Exansion of Gases in a Turbine Multistage gas turbine a realization of the results derived for the K-ste exansion case? ecture no. 7 7

8 Otimal heat exchange () Find the temerature rofile T(z) that gives minimum entroy roduction, when a given amount of heat is transferred from the hot fluid Constraints By fixing T and T we fix the heat transferred h, in h, out Fixed heat transferred means fixed: T and T h, in h, out The energy balance must be obeyed: FC dt z = J z Δy h( ) q( ) ecture no. 7 8

9 Otimal heat exchange (3): Is work is obtainable by heat exchange? dq dq T dq = dq = Jq ( z) Δy T ( z) c Carnot machine dw η dw = Cdq =Δy Jq Tc ( z) T T (z) c T (z) h dq Cold fluid Hot fluid T J q w=δy Jq = FoutHout FinHin ΔyT T ( ) ( ) c z T c z Entroy roduction from transfer to cold side ecture no. 7 9

10 Otimal heat exchange (): The entroy roduction (,) q(,) q d xz J xz dx T (,) x z J x z J z (,) q() z y x z dx yj z () (,) q() T () z T () z h c By fixing T and T we fix the heat transferred h, in h, out J q ds dt = lqqδ T irr =Δ σ ( ) =Δ ( qq ) q y z y l J ecture no. 7

11 Otimal heat exchange (5): Solution Carnot machine T (z) c T (z) h dq dq dw Cold fluid Hot fluid Exact solution: Constant entroy roduction (EoEP) Aroximate solution: Constant thermal force (EoF) ds dt irr The entroy balance for the hot fluid Jq = FoutSout FinSin Δy T ( ) c z The entroy roduction for heat exchange ds dt irr δ =Δ y σ( xzdx, ) =Δy σ( z ) =Δy lqq ( Th ( z)) Δ T Temerature / K T h, EoEP T c, EoEP T h, EoF T c, EoF Position / m ecture no. 7

12 Reasons to minimize the entroy roduction We obtain a realistic target for the efficiency: The most energy efficient oeration for the real system We find a zero on a yardstick that measures lost work We can find rules for rocess design: Rules of thumb, energy efficient design A turbine with equiartition of forces Heat exchange with equiartition of forces ecture no. 7

13 Energy efficient design means that:. The ath of minimum entroy roduction has been used, given the boundary conditions. This oerating ath has constant entroy roduction if the system has sufficient degrees of freedom 3. Constant driving forces seems to be a good aroximation to a state with constant entroy roduction ecture no. 7 3