87 lines
2.7 KiB
TeX
87 lines
2.7 KiB
TeX
\section{Derivation of \cref{EQINFWELL}}
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\label{AINFWELL}
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\begin{equation*}
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\pi_{w}=\int_{t=0}^{T}\left[ \left(P_{ur}\cdot q_{i}\cdot e^{-Dt}-C_{op}\right)e^{-rt}\right] \,dt-C_{Drill}-C_{Res}\cdot e^{-rT} \
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\end{equation*}
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\begin{equation*}
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\frac{d \pi_{w}}{dT}= \left(P_{ur}\cdot q_{i}-C_{op}e^{-rT}+r\cdot C_{Res}\right)e^{-rT}=0 \
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\end{equation*}
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\begin{equation*}
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P_{ur}\cdot q_{i}\cdot e^{-DT}=C_{op}-r\cdot C_{Res}
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\end{equation*}
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\begin{equation*}
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e^{-DT}=\frac{C_{op}-r\cdot C_{Res}}{ P_{ur}\cdot q_{i}}
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\end{equation*}
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\begin{equation*}
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-DT=ln(\frac{C_{op}-r\cdot C_{Res}}{ P_{ur}\cdot q_{i}})
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\ln( P_{ur}\cdot q_{i})
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\left(\ln( P_{ur})+\ln(\cdot q_{i})\right)
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\end{equation*}
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\begin{equation*}
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-DT=\ln(C_{op}-r\cdot C_{Res})-\ln( P_{ur})-\ln(\cdot q_{i})
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\end{equation*}
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\begin{equation*}
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T^{\star}=\frac{\ln(P_{ur})+\ln(q_{i})-\ln(C_{op}-r C_{Res})}{D}
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\end{equation*}
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\section{Derivation of \cref{EQPROD}}
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\label{APROD}
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\begin{equation}
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\Delta Q =\int_{T_{1}^{\star}}^{T_{1}^{\star}+\Delta T} q_{i}\cdot e^{-rt}\,dt
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\end{equation}
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\begin{equation*}
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\Delta Q = -\frac{q_{i}\cdot e^{-D \left(T_{1}^{\star}+\Delta T\right)}}{D}+\frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}
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\end{equation*}
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\begin{equation*}
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\Delta Q = -\frac{q_{i}\cdot e^{-D T_{1}^{\star}}\cdot e^{-D \Delta T}}{D}+\frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}
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\end{equation*}
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\begin{equation*}
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\Delta Q = q_{i}\cdot e^{-D T_{1}^{\star}}\frac{-e^{-D \Delta T}+1}{D}
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-D \Delta T}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-D \frac{\ln(C_{op}-r C_{Res})-\ln(C_{op})}{D}
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}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-e^{-\ln(C_{op}-r C_{Res})+\ln(C_{op})}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}\cdot e^{-D T_{1}^{\star}}}{D}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}\cdot e^{-D \frac{\ln(P_{ur})+\ln(q_{i})-\ln(C_{op})}{D} }}{D}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{q_{i}}{D}\cdot e^{-\ln(P_{ur})-\ln(q_{i})+\ln(C_{op})}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = q_{i}\frac{C_{op}}{D\cdot P_{ur}\cdot q_{i}}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
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\end{equation*}
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\begin{equation*}
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\Delta Q = \frac{C_{op}}{D\cdot P_{ur}}\left(1-\frac{C_{op}}{C_{op}-r C_{Res}}\right)
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\end{equation*}
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