Classical Theory (3): Thermostatics of Continuous Systems with External Forces

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1 Insttute of Flu- & Thermoynamcs Unersty of Segen Classcal Theory (3): Thermostatcs of Contnuous Systems wth External Forces 3/ Σ: Equlbrum State? Isolaton, Inhomogenety External Forces F ϕ Components:... S S x V Max. ( x) U Wp U ( x) ( x) ( x) V const ρ ϕ m ρ x V const,... x 3 Labor System x 0 x F Σ x Equlbrum Conons: T x const px ρ xf µ x ϕ x const,... Bounary Conons Generalzatons: E-M-Fels Chem. Reactons Ref.: J.U. Keller, TIP, Part Thermostatcs, W. e Gruyter, Berln - ew York, 977.

2 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/ Thermoynamcs of Processes () Contnuous System Flu, Components External Forces x 3 σ z x m V φ z F Extense Quantty ( Z) m Zt z x,tvx V m z ( x,t) m( x) ρ z x,t x,t z x,t V m V Balance Equaton Z Φ P x 0 Local Formulaton (Lab) z φ σ t V z z Materal Formulaton ( z ) x ρ z φ ρ σ t m z m z t, t, t x

3 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/3 ( x,t) ( x,t) ( x,t) Balance Equatons of Mass Denstes ρ x,t,... Partal ensty of component ρ ρ w ρ ρ w Total mass ensty Mass concentraton () Partal elocty Velocty of mass element mx,t Z: z ρ, z w V m φ ρ, σ Γ z z Local balance ( ) ρ ρ Γ t Materal balance Γ,... Chemcal proucton rate Γ 0 Mass conseraton J ρ Dffuson flow () () ρ w J Γ

4 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/4 P β ( x,t) F x,t Z: z ρ, z V Balance Equaton of Lnear Momentum Stress Tensor β-component of stress at -surface element External force actng on component () m ewton's equaton of moton of mass ( m) ρ V P O ρf V Internal stress V(t) O(t) β β V(t) Materal balance β P β F ρ ρ 3,, Local balance ρ P ρ ρ F t β β β 3,, Conecte flow of momentum

5 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/5 Balance Equaton of Angular Momentum (AM) zm ε βγ xβγ σ Specfc angular,, 3 momentum at x,t ε σ Γ βγ β (, 0, ) Momentum balance Antsymmetrc unt-tensor Internal angular momentum (IAM) (Spn of atoms, molecules etc.) Tensor of IAM-flux Proucton ensty of IAM ue to external forces ( γ) x, IAM ρ ( ε βγ x β γ σ ) δ( ε βγ xp β γδ δ ) ε x ρ F Γ F ε P P * βγ βγ β γ βγ βγ γβ ε β Balance for IAM (*:x ) x a,a 0 ρ σ δ δ Γ ε σ const : P P ( F ) ( P P ) βγ βγ γβ Balance for external AM: ρ ε βγ x β γ δε βγ xp β γδ βγ ε x ρf βγ β γ γβ

6 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/6 Balance Equaton of Energy Materal balance(s) e u Total energy per unt of mass u φ eρ P J J e β β u u σ ρ F e Internal energy per unt of mass Heat flux Energy Flux Energy supply by mechancal work of external forces t t e ( P β β J u ) F ρ ρ u β J u P β β F ρ ρ Local balance(s) ( u ) e P J eρ β β ρ F u β β F J ρ u J ρ u P F J

7 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/7 Materal balance Entropy Balance ρ s ( φs ρ s ) σ s m: Smple thermoynamc system Exchange of mass, work, heat Prncple of local equlbrum Gbbs Funamental Equaton s u w T T ρ T p ρt ( E D H B ) w ρ ρ,... u...,..., w... ρ µ Entropy flow: φ J µ J ρs s u T Entropy proucton: σ s Ju ( pδβ Pβ ) β T T F J T T µ Γ T µ 0

8 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/8 Process Equatons (Flux-Force-Relatons) (Eckart-Onsager-Mexner-Prgogne) Entropy Proucton (*) TX k k TY L L σ XY 0 Y X Y k k k L k... Forces... Fluxes L X Onsager (Casmer)-Symmetry n 0... Law Lnear Inepenency of Fluxes r k Γk, Ak γ klµ l k T l Y : σ Ju ( ρδβ Pβ ) β T T Heat Transfer Internal Frcton - J ( F F ) T Dffuson A Chemcal Reactons µ µ T

9 Unersty of Segen Insttute of Flu- & Thermoynamcs 3/9 Example: Dffuson an Heat Transfer n Component Flu System (o nternal frcton or chemcal reactons) Thermostatc EOS ( ) ( ) T T u, ρ, ρ p p u, ρ, ρ µ µ u, ρ, ρ,, Balance Equatons ρ ρ 0 w ρ J 0,, u ρ P ρf Process Equatons F µ J L L T T T k k k u k, F µ J L L T T T k k u uk u k UQ :u, ρ, ρ, ρ,, J, J T, µ, µ, p u Bounary- an Intal-Conons.

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