PCRT.I.E.04

PCRT.I.E

04 Einmal linksherum im Kreis in einer (S,V;N=const)-Ebene

Eine Ebene schreiben wir als (x,y;z=const)-Ebene. Die Koordinaten einer Ebene schreiben wir in umgekehrter Reihenfolge (z=const;y,x). So können wir die Zustandübergänge leichter überschauen. Denn diese Notation ist im Einklang mit der Notation für die Integration ausgehend vom Ausgangangszustand beim Raufklettern im Kreisprozess von der unteren Grenze (z,y1,x1) zur oberen Grenze (z,y1,x2) und von der unteren Grenze (z,y1,x2) zur oberen Grenze (z,y2,x2) und dann beim Runterklettern von der unteren Grenze (z,y2,x2) zur oberen Grenze (z,y2,x1) und von der unteren Grenze (z,y2,x1) zur oberen Grenze (y1,x1) zurück zum Ausgangszustand.

A Wärme hinein, Qa > 0: (N,V1,S1) -> (N,V1,S2)

$\qquad \qquad$	$Q_{a}$	$=$	$Q_{V1,N}(S_{2},S_{1})>0$	$\qquad (a:V,S;V_{1}S_{1}\rightarrow V_{1}S_{2}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\,\left[\exp \left({\frac {2}{3}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{3}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (V,S;V_{1}S_{1}\rightarrow V_{1}S_{2}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[T(N,V_{1},S_{2})-T(N,V_{1},S_{1})\right]$	$\qquad (T;T_{11}\rightarrow T_{12}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{1},S_{2})-U(N,V_{1},S_{1})$	$\qquad (U;U_{11}\rightarrow U_{12}),\,dN=0\qquad$

B Arbeit hinaus, Wb < 0: (N,V1,S2) -> (N,V2,S2)

$\qquad \qquad$	$W_{b}$	$=$	$W_{S2,N}(V_{2},V_{1})<0$	$\qquad (b:V,S;\,V_{1}S_{2}\rightarrow V_{2}S_{2}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}-\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\right]\,\exp \left({\frac {2}{3}}{\frac {S_{2}-S_{0}}{n\,R}}\right)$	$\qquad (S,V;V_{1}S_{2}\rightarrow V_{2}S_{2}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[\,T(N,V_{2},S_{2})-T(N,V_{1},S_{2})\right]$	$\qquad (T;\,T_{12}\rightarrow T_{22}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{2},S_{2})-U(N,V_{1},S_{2})$	$\qquad (U;U_{12}\rightarrow U_{22}),\,dN=0\qquad$

C Wärme hinaus, Qc < 0: (N,V2,S2) -> (N,V2,S1)

$\qquad \qquad$	$Q_{c}$	$=$	$Q_{V2,N}(S_{1},S_{2})<0$	$\qquad (c:V,S;V_{2}S_{2}\rightarrow V_{2}S_{1}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}\,\left[\exp \left({\frac {2}{3}}{\frac {S_{1}}{n\,R}}\right)-\exp \left({\frac {2}{3}}{\frac {S_{2}}{n\,R}}\right)\right]$	$\qquad (V,S;V_{2}S_{2}\rightarrow V_{2}S_{1}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[T(N,V_{2},S_{1})-T(N,V_{2},S_{2})\right]$	$\qquad (T;T_{22}\rightarrow T_{21}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{2},S_{1})-U(N,V_{2},S_{2})$	$\qquad (U;U_{22}\rightarrow U_{21}),\,dN=0\qquad$

D Arbeit hinein, Wd > 0: (N,V2,S1) -> (N,V1,S1)

$\qquad \qquad$	$W_{d}$	$=$	$W_{S1,N}(V1,V2)>0$	$\qquad (d:V,S;V_{2}S_{1}\rightarrow V_{1}S_{1}),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}-\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}\right]\,\exp \left({\frac {2}{3}}{\frac {S_{1}-S_{0}}{n\,R}}\right)$	$\qquad (V,S;V_{2}S_{1}\rightarrow V_{1}S_{1}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[\,T(N,V_{1},S_{1})-T(N,V_{2},S_{1})\right]$	$\qquad (T;\,T_{21}\rightarrow T_{11}),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{1},S_{1})-U(N,V_{2},S_{1})$	$\qquad (U;U_{21}\rightarrow U_{11}),\,dN=0\qquad$

E Wärme gesamt, Qa + Qc > 0: (N,V1,S1) -> (N,V1,S2), (N,V2,S2) -> (N,V2,S1)

$\qquad \qquad$	$Q_{a}+Q_{c}$	$=$	$Q_{V2}(S1,S2)+Q_{V1}(S2,S1)>0$	$\qquad (a+c:V,S;11\to 12,22\to 21),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}-\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\times \left[\exp \left({\frac {2}{3}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{3}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (V,S;11\to 12,22\to 21),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,\left[T(N,V_{2},S_{1})-T(N,V_{2},S_{2})\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,{\frac {3}{2}}\,n\,R\,\left[T(N,V_{1},S_{2})-T(N,V_{1},S_{1})\right]$	$\qquad (T;11\to 12,22\to 21),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{2},S_{1})-U(N,V_{2},S_{2})$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,U(N,V_{1},S_{2})-U(N,V_{1},S_{1})$	$\qquad (U;U_{11}\rightarrow U_{12},\,U_{22}\rightarrow U_{21}),\,dN=0\qquad$
$\qquad \qquad$	$\oint _{V,S}T\,dS$	$=$	$\int _{V_{1},S_{1}}^{V_{1},S_{2}}T\,dS+\int _{V_{1},S_{2}}^{V_{2},S_{2}}T\,dS$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\int _{V_{2},S_{2}}^{V_{2},S_{1}}T\,dS+\int _{V_{2},S_{1}}^{V_{1},S_{1}}T\,dS$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$
$\qquad \qquad$	$0$	$=$	$\int _{V_{1},S_{2}}^{V_{2},S_{2}}T\,dS+\int _{V_{2},S_{1}}^{V_{1},S_{1}}T\,dS$	$\qquad (V,S;12\to 22,21\to 11),\,dN=0\qquad$
$(02)\qquad$	$\oint _{V,S}T\,dS$	$=$	$\int _{S,V_{1},V_{2}}T\,dS$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$

F Arbeit gesamt, Wb + Wd < 0: (N,V1,S2) -> (N,V2,S2), (N,V2,S1) -> (N,V1,S1)

$\qquad \qquad$	$W_{b}+W_{d}$	$=$	$W_{S1,N}(V1,V2)+W_{S2,N}(V2,V1)<0$	$\qquad (b+d:V,S;12\to 22,21\to 11),\,dN=0\qquad$
$(01)\qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\,T_{0}\,\left[\left({\frac {V_{0}}{V_{2}}}\right)^{2/3}-\left({\frac {V_{0}}{V_{1}}}\right)^{2/3}\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$\times \left[\exp \left({\frac {2}{3}}{\frac {S_{2}}{n\,R}}\right)-\exp \left({\frac {2}{3}}{\frac {S_{1}}{n\,R}}\right)\right]$	$\qquad (V,S;12\to 22,21\to 11),dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	${\frac {3}{2}}\,n\,R\left[\,T(N,V_{1},S_{1})-T(N,V_{2},S_{1})\right]$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,{\frac {3}{2}}\,n\,R\left[\,T(N,V_{2},S_{2})-T(N,V_{1},S_{2})\right]$	$\qquad (T;12\to 22,\,21\to 11),\,dN=0\qquad$
$\qquad \qquad$	${\displaystyle }$	$=$	$U(N,V_{1},S_{1})-U(N,V_{2},S_{1})$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$+\,U(N,V_{2},S_{2})-U(N,V_{1},S_{2})$	$\qquad (U;U_{12}\rightarrow U_{22},\,U_{21}\rightarrow U_{11}),\,dN=0\qquad$
$\qquad \qquad$	$-\oint _{V,S}p\,dV$	$=$	$-\int _{V_{1},S_{1}}^{V_{1},S_{2}}p\,dV-\int _{V_{1},S_{2}}^{V_{2},S_{2}}p\,dV$	$\qquad \qquad$
$\qquad \qquad$	${\displaystyle }$	${\displaystyle }$	$-\int _{V_{2},S_{2}}^{V_{2},S_{1}}p\,dV-\int _{V_{2},S_{1}}^{V_{1},S_{1}}p\,dV$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$
$\qquad \qquad$	$0$	$=$	$-\int _{V_{1},S_{1}}^{V_{1},S_{2}}p\,dV-\int _{V_{2},S_{2}}^{V_{2},S_{1}}p\,dV$	$\qquad (V,S;11\to 12,22\to 21),\,dN=0\qquad$
$(02)\qquad$	$-\oint _{V,S}p\,dV$	$=$	$-\int _{V,S_{2},S_{1}}p\,dV$	$\qquad (V,S;12\to 22,\,21\to 11),\,dN=0\qquad$

G Verhältnis und Summe der Integrale von dW = -pdV und dQ = +TdS

$(01)\qquad$	$-\int _{V,S_{2},S_{1}}p\,dV$	${\Big /}$	$\int _{S,V_{1},V_{2}}T\,dS$	$=$	${\frac {W_{b}+W_{d}}{Q_{a}+Q_{c}}}$	$=$	$-1$	$=$	$-100\%$	${\displaystyle }$	$\qquad (V,S;11\to 12\to 22\to 21\to 11)\,dN=0\qquad$
$(02)\qquad$	$-\oint _{S,V}p\,dV$	${\Big /}$	$\oint _{S,V}T\,dS$	$=$	${\frac {W_{b}+W_{d}}{Q_{a}+Q_{c}}}$	$=$	$-1$	$=$	$-100\%$	${\displaystyle }$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$
$(03)\qquad$	$-\int _{V,S_{2},S_{1}}p\,dV$	$+$	$\int _{S,V_{1},V_{2}}T\,dS$	$=$	$Q_{a}+W_{b}+Q_{c}+W_{d}$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$
$(04)\qquad$	$-\oint _{S,V}p\,dV$	$+$	$\oint _{S,V}T\,dS$	$=$	$Q_{a}+W_{b}+Q_{c}+W_{d}$	$=$	$0$	${\displaystyle }$	${\displaystyle }$	${\displaystyle }$	$\qquad (V,S;11\to 12\to 22\to 21\to 11),\,dN=0\qquad$

Übung a

Wir berechnen das für die Technologie interessantere Verhältnis $(W_{b}+W_{d})/Q_{a}$ für einen Linksumlauf in der (S,V;N=const)-Ebene. Von welchen Parametern hängt es ab? Wie müssen die Parameter verändert werden, damit das Verhältnis $(W_{b}+W_{d})/Q_{a}$ grösser wird? Welche Vorstellung von $Q_{b}$ könnte uns bei weniger theoretischen und mehr technologischen Betrachtungen ausgehend vom Verhältnis $(W_{b}+W_{d})/(Q_{a}+Q_{b})$ zum Verhältnis $(W_{b}+W_{d})/Q_{a}$ führen?

Übung b

Wenn wir Gl. (E.02) und Gl. (F.02) addieren, welcher Vorstellung entspricht die Summe der beiden Gleichungen im (S,V,U)-Koordinatensystem mit der Entropie S als x-Achse, dem Volumen V als y-Achse, der inneren Energie U als z-Achse und N=const?