\(\newcommand{\footnotename}{footnote}\)
\(\def \LWRfootnote {1}\)
\(\newcommand {\footnote }[2][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand {\footnotemark }[1][\LWRfootnote ]{{}^{\mathrm {#1}}}\)
\(\newcommand \ensuremath [1]{#1}\)
\(\newcommand {\LWRframebox }[2][]{\fbox {#2}} \newcommand {\framebox }[1][]{\LWRframebox } \)
\(\newcommand {\setlength }[2]{}\)
\(\newcommand {\addtolength }[2]{}\)
\(\newcommand {\setcounter }[2]{}\)
\(\newcommand {\addtocounter }[2]{}\)
\(\newcommand {\cline }[1]{}\)
\(\newcommand {\directlua }[1]{\text {(directlua)}}\)
\(\newcommand {\luatexdirectlua }[1]{\text {(directlua)}}\)
\(\newcommand {\protect }{}\)
\(\def \LWRabsorbnumber #1 {}\)
\(\def \LWRabsorbquotenumber "#1 {}\)
\(\def \mathchar {\ifnextchar "\LWRabsorbquotenumber \LWRabsorbnumber }\)
\(\def \mathcode #1={\mathchar }\)
\(\let \delcode \mathcode \)
\(\let \delimiter \mathchar \)
\(\let \LWRref \ref \)
\(\renewcommand {\ref }{\ifstar \LWRref \LWRref }\)
\(\newcommand {\intertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\mathllap }[2][]{{#1#2}}\)
\(\newcommand {\mathrlap }[2][]{{#1#2}}\)
\(\newcommand {\mathclap }[2][]{{#1#2}}\)
\(\newcommand {\mathmbox }[1]{#1}\)
\(\newcommand {\clap }[1]{#1}\)
\(\newcommand {\LWRmathmakebox }[2][]{#2}\)
\(\newcommand {\mathmakebox }[1][]{\LWRmathmakebox }\)
\(\newcommand {\cramped }[2][]{{#1#2}}\)
\(\newcommand {\crampedllap }[2][]{{#1#2}}\)
\(\newcommand {\crampedrlap }[2][]{{#1#2}}\)
\(\newcommand {\crampedclap }[2][]{{#1#2}}\)
\(\newenvironment {crampedsubarray}[1]{}{}\)
\(\newcommand {\crampedsubstack }{}\)
\(\newcommand {\smashoperator }[2][]{#2\limits }\)
\(\newcommand {\adjustlimits }{}\)
\(\newcommand {\SwapAboveDisplaySkip }{}\)
\(\require {extpfeil}\)
\(\Newextarrow \xleftrightarrow {10,10}{0x2194}\)
\(\Newextarrow \xLeftarrow {10,10}{0x21d0}\)
\(\Newextarrow \xhookleftarrow {10,10}{0x21a9}\)
\(\Newextarrow \xmapsto {10,10}{0x21a6}\)
\(\Newextarrow \xRightarrow {10,10}{0x21d2}\)
\(\Newextarrow \xLeftrightarrow {10,10}{0x21d4}\)
\(\Newextarrow \xhookrightarrow {10,10}{0x21aa}\)
\(\Newextarrow \xrightharpoondown {10,10}{0x21c1}\)
\(\Newextarrow \xleftharpoondown {10,10}{0x21bd}\)
\(\Newextarrow \xrightleftharpoons {10,10}{0x21cc}\)
\(\Newextarrow \xrightharpoonup {10,10}{0x21c0}\)
\(\Newextarrow \xleftharpoonup {10,10}{0x21bc}\)
\(\Newextarrow \xleftrightharpoons {10,10}{0x21cb}\)
\(\newcommand {\LWRdounderbracket }[3]{\underset {#3}{\underline {#1}}}\)
\(\newcommand {\LWRunderbracket }[2][]{\LWRdounderbracket {#2}}\)
\(\newcommand {\underbracket }[1][]{\LWRunderbracket }\)
\(\newcommand {\LWRdooverbracket }[3]{\overset {#3}{\overline {#1}}}\)
\(\newcommand {\LWRoverbracket }[2][]{\LWRdooverbracket {#2}}\)
\(\newcommand {\overbracket }[1][]{\LWRoverbracket }\)
\(\newcommand {\LATEXunderbrace }[1]{\underbrace {#1}}\)
\(\newcommand {\LATEXoverbrace }[1]{\overbrace {#1}}\)
\(\newenvironment {matrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {pmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {smallmatrix*}[1][]{\begin {matrix}}{\end {matrix}}\)
\(\newenvironment {psmallmatrix*}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix*}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix*}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix*}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix*}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newenvironment {psmallmatrix}[1][]{\begin {pmatrix}}{\end {pmatrix}}\)
\(\newenvironment {bsmallmatrix}[1][]{\begin {bmatrix}}{\end {bmatrix}}\)
\(\newenvironment {Bsmallmatrix}[1][]{\begin {Bmatrix}}{\end {Bmatrix}}\)
\(\newenvironment {vsmallmatrix}[1][]{\begin {vmatrix}}{\end {vmatrix}}\)
\(\newenvironment {Vsmallmatrix}[1][]{\begin {Vmatrix}}{\end {Vmatrix}}\)
\(\newcommand {\LWRmultlined }[1][]{\begin {multline*}}\)
\(\newenvironment {multlined}[1][]{\LWRmultlined }{\end {multline*}}\)
\(\let \LWRorigshoveleft \shoveleft \)
\(\renewcommand {\shoveleft }[1][]{\LWRorigshoveleft }\)
\(\let \LWRorigshoveright \shoveright \)
\(\renewcommand {\shoveright }[1][]{\LWRorigshoveright }\)
\(\newenvironment {dcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {dcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {rcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {drcases*}{\begin {cases}}{\end {cases}}\)
\(\newenvironment {cases*}{\begin {cases}}{\end {cases}}\)
\(\newcommand {\MoveEqLeft }[1][]{}\)
\(\def \LWRAboxed #1!|!{\fbox {\(#1\)}&\fbox {\(#2\)}} \newcommand {\Aboxed }[1]{\LWRAboxed #1&&!|!} \)
\( \newcommand {\LWRABLines }[1][\Updownarrow ]{#1 \notag \\}\newcommand {\ArrowBetweenLines }{\ifstar \LWRABLines \LWRABLines } \)
\(\newcommand {\shortintertext }[1]{\text {#1}\notag \\}\)
\(\newcommand {\vdotswithin }[1]{\hspace {.5em}\vdots }\)
\(\newcommand {\LWRshortvdotswithinstar }[1]{\vdots \hspace {.5em} & \\}\)
\(\newcommand {\LWRshortvdotswithinnostar }[1]{& \hspace {.5em}\vdots \\}\)
\(\newcommand {\shortvdotswithin }{\ifstar \LWRshortvdotswithinstar \LWRshortvdotswithinnostar }\)
\(\newcommand {\MTFlushSpaceAbove }{}\)
\(\newcommand {\MTFlushSpaceBelow }{\\}\)
\(\newcommand \lparen {(}\)
\(\newcommand \rparen {)}\)
\(\newcommand {\ordinarycolon }{:}\)
\(\newcommand {\vcentcolon }{\mathrel {\mathop \ordinarycolon }}\)
\(\newcommand \dblcolon {\vcentcolon \vcentcolon }\)
\(\newcommand \coloneqq {\vcentcolon =}\)
\(\newcommand \Coloneqq {\dblcolon =}\)
\(\newcommand \coloneq {\vcentcolon {-}}\)
\(\newcommand \Coloneq {\dblcolon {-}}\)
\(\newcommand \eqqcolon {=\vcentcolon }\)
\(\newcommand \Eqqcolon {=\dblcolon }\)
\(\newcommand \eqcolon {\mathrel {-}\vcentcolon }\)
\(\newcommand \Eqcolon {\mathrel {-}\dblcolon }\)
\(\newcommand \colonapprox {\vcentcolon \approx }\)
\(\newcommand \Colonapprox {\dblcolon \approx }\)
\(\newcommand \colonsim {\vcentcolon \sim }\)
\(\newcommand \Colonsim {\dblcolon \sim }\)
\(\newcommand {\nuparrow }{\mathrel {\cancel {\uparrow }}}\)
\(\newcommand {\ndownarrow }{\mathrel {\cancel {\downarrow }}}\)
\(\newcommand {\bigtimes }{\mathop {\Large \times }\limits }\)
\(\newcommand {\prescript }[3]{{}^{#1}_{#2}#3}\)
\(\newenvironment {lgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newenvironment {rgathered}{\begin {gathered}}{\end {gathered}}\)
\(\newcommand {\splitfrac }[2]{{}^{#1}_{#2}}\)
\(\let \splitdfrac \splitfrac \)
\(\newcommand {\LWRoverlaysymbols }[2]{\mathord {\smash {\mathop {#2\strut }\limits ^{\smash {\lower 3ex{#1}}}}\strut }}\)
\(\newcommand{\alphaup}{\unicode{x03B1}}\)
\(\newcommand{\betaup}{\unicode{x03B2}}\)
\(\newcommand{\gammaup}{\unicode{x03B3}}\)
\(\newcommand{\digammaup}{\unicode{x03DD}}\)
\(\newcommand{\deltaup}{\unicode{x03B4}}\)
\(\newcommand{\epsilonup}{\unicode{x03F5}}\)
\(\newcommand{\varepsilonup}{\unicode{x03B5}}\)
\(\newcommand{\zetaup}{\unicode{x03B6}}\)
\(\newcommand{\etaup}{\unicode{x03B7}}\)
\(\newcommand{\thetaup}{\unicode{x03B8}}\)
\(\newcommand{\varthetaup}{\unicode{x03D1}}\)
\(\newcommand{\iotaup}{\unicode{x03B9}}\)
\(\newcommand{\kappaup}{\unicode{x03BA}}\)
\(\newcommand{\varkappaup}{\unicode{x03F0}}\)
\(\newcommand{\lambdaup}{\unicode{x03BB}}\)
\(\newcommand{\muup}{\unicode{x03BC}}\)
\(\newcommand{\nuup}{\unicode{x03BD}}\)
\(\newcommand{\xiup}{\unicode{x03BE}}\)
\(\newcommand{\omicronup}{\unicode{x03BF}}\)
\(\newcommand{\piup}{\unicode{x03C0}}\)
\(\newcommand{\varpiup}{\unicode{x03D6}}\)
\(\newcommand{\rhoup}{\unicode{x03C1}}\)
\(\newcommand{\varrhoup}{\unicode{x03F1}}\)
\(\newcommand{\sigmaup}{\unicode{x03C3}}\)
\(\newcommand{\varsigmaup}{\unicode{x03C2}}\)
\(\newcommand{\tauup}{\unicode{x03C4}}\)
\(\newcommand{\upsilonup}{\unicode{x03C5}}\)
\(\newcommand{\phiup}{\unicode{x03D5}}\)
\(\newcommand{\varphiup}{\unicode{x03C6}}\)
\(\newcommand{\chiup}{\unicode{x03C7}}\)
\(\newcommand{\psiup}{\unicode{x03C8}}\)
\(\newcommand{\omegaup}{\unicode{x03C9}}\)
\(\newcommand{\Alphaup}{\unicode{x0391}}\)
\(\newcommand{\Betaup}{\unicode{x0392}}\)
\(\newcommand{\Gammaup}{\unicode{x0393}}\)
\(\newcommand{\Digammaup}{\unicode{x03DC}}\)
\(\newcommand{\Deltaup}{\unicode{x0394}}\)
\(\newcommand{\Epsilonup}{\unicode{x0395}}\)
\(\newcommand{\Zetaup}{\unicode{x0396}}\)
\(\newcommand{\Etaup}{\unicode{x0397}}\)
\(\newcommand{\Thetaup}{\unicode{x0398}}\)
\(\newcommand{\Varthetaup}{\unicode{x03F4}}\)
\(\newcommand{\Iotaup}{\unicode{x0399}}\)
\(\newcommand{\Kappaup}{\unicode{x039A}}\)
\(\newcommand{\Lambdaup}{\unicode{x039B}}\)
\(\newcommand{\Muup}{\unicode{x039C}}\)
\(\newcommand{\Nuup}{\unicode{x039D}}\)
\(\newcommand{\Xiup}{\unicode{x039E}}\)
\(\newcommand{\Omicronup}{\unicode{x039F}}\)
\(\newcommand{\Piup}{\unicode{x03A0}}\)
\(\newcommand{\Varpiup}{\unicode{x03D6}}\)
\(\newcommand{\Rhoup}{\unicode{x03A1}}\)
\(\newcommand{\Sigmaup}{\unicode{x03A3}}\)
\(\newcommand{\Tauup}{\unicode{x03A4}}\)
\(\newcommand{\Upsilonup}{\unicode{x03A5}}\)
\(\newcommand{\Phiup}{\unicode{x03A6}}\)
\(\newcommand{\Chiup}{\unicode{x03A7}}\)
\(\newcommand{\Psiup}{\unicode{x03A8}}\)
\(\newcommand{\Omegaup}{\unicode{x03A9}}\)
\(\newcommand{\alphait}{\unicode{x1D6FC}}\)
\(\newcommand{\betait}{\unicode{x1D6FD}}\)
\(\newcommand{\gammait}{\unicode{x1D6FE}}\)
\(\newcommand{\digammait}{\mathit{\unicode{x03DD}}}\)
\(\newcommand{\deltait}{\unicode{x1D6FF}}\)
\(\newcommand{\epsilonit}{\unicode{x1D716}}\)
\(\newcommand{\varepsilonit}{\unicode{x1D700}}\)
\(\newcommand{\zetait}{\unicode{x1D701}}\)
\(\newcommand{\etait}{\unicode{x1D702}}\)
\(\newcommand{\thetait}{\unicode{x1D703}}\)
\(\newcommand{\varthetait}{\unicode{x1D717}}\)
\(\newcommand{\iotait}{\unicode{x1D704}}\)
\(\newcommand{\kappait}{\unicode{x1D705}}\)
\(\newcommand{\varkappait}{\unicode{x1D718}}\)
\(\newcommand{\lambdait}{\unicode{x1D706}}\)
\(\newcommand{\muit}{\unicode{x1D707}}\)
\(\newcommand{\nuit}{\unicode{x1D708}}\)
\(\newcommand{\xiit}{\unicode{x1D709}}\)
\(\newcommand{\omicronit}{\unicode{x1D70A}}\)
\(\newcommand{\piit}{\unicode{x1D70B}}\)
\(\newcommand{\varpiit}{\unicode{x1D71B}}\)
\(\newcommand{\rhoit}{\unicode{x1D70C}}\)
\(\newcommand{\varrhoit}{\unicode{x1D71A}}\)
\(\newcommand{\sigmait}{\unicode{x1D70E}}\)
\(\newcommand{\varsigmait}{\unicode{x1D70D}}\)
\(\newcommand{\tauit}{\unicode{x1D70F}}\)
\(\newcommand{\upsilonit}{\unicode{x1D710}}\)
\(\newcommand{\phiit}{\unicode{x1D719}}\)
\(\newcommand{\varphiit}{\unicode{x1D711}}\)
\(\newcommand{\chiit}{\unicode{x1D712}}\)
\(\newcommand{\psiit}{\unicode{x1D713}}\)
\(\newcommand{\omegait}{\unicode{x1D714}}\)
\(\newcommand{\Alphait}{\unicode{x1D6E2}}\)
\(\newcommand{\Betait}{\unicode{x1D6E3}}\)
\(\newcommand{\Gammait}{\unicode{x1D6E4}}\)
\(\newcommand{\Digammait}{\mathit{\unicode{x03DC}}}\)
\(\newcommand{\Deltait}{\unicode{x1D6E5}}\)
\(\newcommand{\Epsilonit}{\unicode{x1D6E6}}\)
\(\newcommand{\Zetait}{\unicode{x1D6E7}}\)
\(\newcommand{\Etait}{\unicode{x1D6E8}}\)
\(\newcommand{\Thetait}{\unicode{x1D6E9}}\)
\(\newcommand{\Varthetait}{\unicode{x1D6F3}}\)
\(\newcommand{\Iotait}{\unicode{x1D6EA}}\)
\(\newcommand{\Kappait}{\unicode{x1D6EB}}\)
\(\newcommand{\Lambdait}{\unicode{x1D6EC}}\)
\(\newcommand{\Muit}{\unicode{x1D6ED}}\)
\(\newcommand{\Nuit}{\unicode{x1D6EE}}\)
\(\newcommand{\Xiit}{\unicode{x1D6EF}}\)
\(\newcommand{\Omicronit}{\unicode{x1D6F0}}\)
\(\newcommand{\Piit}{\unicode{x1D6F1}}\)
\(\newcommand{\Rhoit}{\unicode{x1D6F2}}\)
\(\newcommand{\Sigmait}{\unicode{x1D6F4}}\)
\(\newcommand{\Tauit}{\unicode{x1D6F5}}\)
\(\newcommand{\Upsilonit}{\unicode{x1D6F6}}\)
\(\newcommand{\Phiit}{\unicode{x1D6F7}}\)
\(\newcommand{\Chiit}{\unicode{x1D6F8}}\)
\(\newcommand{\Psiit}{\unicode{x1D6F9}}\)
\(\newcommand{\Omegait}{\unicode{x1D6FA}}\)
\(\let \digammaup \Digammaup \)
\(\renewcommand {\digammait }{\mathit {\digammaup }}\)
\(\newcommand {\smallin }{\unicode {x220A}}\)
\(\newcommand {\smallowns }{\unicode {x220D}}\)
\(\newcommand {\notsmallin }{\LWRoverlaysymbols {/}{\unicode {x220A}}}\)
\(\newcommand {\notsmallowns }{\LWRoverlaysymbols {/}{\unicode {x220D}}}\)
\(\newcommand {\rightangle }{\unicode {x221F}}\)
\(\newcommand {\intclockwise }{\unicode {x2231}}\)
\(\newcommand {\ointclockwise }{\unicode {x2232}}\)
\(\newcommand {\ointctrclockwise }{\unicode {x2233}}\)
\(\newcommand {\oiint }{\unicode {x222F}}\)
\(\newcommand {\oiiint }{\unicode {x2230}}\)
\(\newcommand {\ddag }{\unicode {x2021}}\)
\(\newcommand {\P }{\unicode {x00B6}}\)
\(\newcommand {\copyright }{\unicode {x00A9}}\)
\(\newcommand {\dag }{\unicode {x2020}}\)
\(\newcommand {\pounds }{\unicode {x00A3}}\)
\(\newcommand {\iddots }{\unicode {x22F0}}\)
\(\newcommand {\utimes }{\overline {\times }}\)
\(\newcommand {\dtimes }{\underline {\times }}\)
\(\newcommand {\udtimes }{\overline {\underline {\times }}}\)
\(\newcommand {\leftwave }{\left \{}\)
\(\newcommand {\rightwave }{\right \}}\)
\(\newcommand {\toprule }[1][]{\hline }\)
\(\let \midrule \toprule \)
\(\let \bottomrule \toprule \)
\(\newcommand {\cmidrule }[2][]{}\)
\(\newcommand {\morecmidrules }{}\)
\(\newcommand {\specialrule }[3]{\hline }\)
\(\newcommand {\addlinespace }[1][]{}\)
\(\newcommand {\LWRsubmultirow }[2][]{#2}\)
\(\newcommand {\LWRmultirow }[2][]{\LWRsubmultirow }\)
\(\newcommand {\multirow }[2][]{\LWRmultirow }\)
\(\newcommand {\mrowcell }{}\)
\(\newcommand {\mcolrowcell }{}\)
\(\newcommand {\STneed }[1]{}\)
\( \newcommand {\multicolumn }[3]{#3}\)
\(\newcommand {\tothe }[1]{^{#1}}\)
\(\newcommand {\raiseto }[2]{{#2}^{#1}}\)
\(\newcommand {\ang }[2][]{(\mathrm {#2})\degree }\)
\(\newcommand {\num }[2][]{\mathrm {#2}}\)
\(\newcommand {\si }[2][]{\mathrm {#2}}\)
\(\newcommand {\LWRSI }[2][]{\mathrm {#1\LWRSInumber \,#2}}\)
\(\newcommand {\SI }[2][]{\def \LWRSInumber {#2}\LWRSI }\)
\(\newcommand {\numlist }[2][]{\mathrm {#2}}\)
\(\newcommand {\numrange }[3][]{\mathrm {#2\,\unicode {x2013}\,#3}}\)
\(\newcommand {\SIlist }[3][]{\mathrm {#2\,#3}}\)
\(\newcommand {\SIrange }[4][]{\mathrm {#2\,#4\,\unicode {x2013}\,#3\,#4}}\)
\(\newcommand {\tablenum }[2][]{\mathrm {#2}}\)
\(\newcommand {\ampere }{\mathrm {A}}\)
\(\newcommand {\candela }{\mathrm {cd}}\)
\(\newcommand {\kelvin }{\mathrm {K}}\)
\(\newcommand {\kilogram }{\mathrm {kg}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\mole }{\mathrm {mol}}\)
\(\newcommand {\second }{\mathrm {s}}\)
\(\newcommand {\becquerel }{\mathrm {Bq}}\)
\(\newcommand {\degreeCelsius }{\unicode {x2103}}\)
\(\newcommand {\coulomb }{\mathrm {C}}\)
\(\newcommand {\farad }{\mathrm {F}}\)
\(\newcommand {\gray }{\mathrm {Gy}}\)
\(\newcommand {\hertz }{\mathrm {Hz}}\)
\(\newcommand {\henry }{\mathrm {H}}\)
\(\newcommand {\joule }{\mathrm {J}}\)
\(\newcommand {\katal }{\mathrm {kat}}\)
\(\newcommand {\lumen }{\mathrm {lm}}\)
\(\newcommand {\lux }{\mathrm {lx}}\)
\(\newcommand {\newton }{\mathrm {N}}\)
\(\newcommand {\ohm }{\mathrm {\Omega }}\)
\(\newcommand {\pascal }{\mathrm {Pa}}\)
\(\newcommand {\radian }{\mathrm {rad}}\)
\(\newcommand {\siemens }{\mathrm {S}}\)
\(\newcommand {\sievert }{\mathrm {Sv}}\)
\(\newcommand {\steradian }{\mathrm {sr}}\)
\(\newcommand {\tesla }{\mathrm {T}}\)
\(\newcommand {\volt }{\mathrm {V}}\)
\(\newcommand {\watt }{\mathrm {W}}\)
\(\newcommand {\weber }{\mathrm {Wb}}\)
\(\newcommand {\day }{\mathrm {d}}\)
\(\newcommand {\degree }{\mathrm {^\circ }}\)
\(\newcommand {\hectare }{\mathrm {ha}}\)
\(\newcommand {\hour }{\mathrm {h}}\)
\(\newcommand {\litre }{\mathrm {l}}\)
\(\newcommand {\liter }{\mathrm {L}}\)
\(\newcommand {\arcminute }{^\prime }\)
\(\newcommand {\minute }{\mathrm {min}}\)
\(\newcommand {\arcsecond }{^{\prime \prime }}\)
\(\newcommand {\tonne }{\mathrm {t}}\)
\(\newcommand {\astronomicalunit }{au}\)
\(\newcommand {\atomicmassunit }{u}\)
\(\newcommand {\bohr }{\mathit {a}_0}\)
\(\newcommand {\clight }{\mathit {c}_0}\)
\(\newcommand {\dalton }{\mathrm {D}_\mathrm {a}}\)
\(\newcommand {\electronmass }{\mathit {m}_{\mathrm {e}}}\)
\(\newcommand {\electronvolt }{\mathrm {eV}}\)
\(\newcommand {\elementarycharge }{\mathit {e}}\)
\(\newcommand {\hartree }{\mathit {E}_{\mathrm {h}}}\)
\(\newcommand {\planckbar }{\mathit {\unicode {x210F}}}\)
\(\newcommand {\angstrom }{\mathrm {\unicode {x212B}}}\)
\(\let \LWRorigbar \bar \)
\(\newcommand {\bar }{\mathrm {bar}}\)
\(\newcommand {\barn }{\mathrm {b}}\)
\(\newcommand {\bel }{\mathrm {B}}\)
\(\newcommand {\decibel }{\mathrm {dB}}\)
\(\newcommand {\knot }{\mathrm {kn}}\)
\(\newcommand {\mmHg }{\mathrm {mmHg}}\)
\(\newcommand {\nauticalmile }{\mathrm {M}}\)
\(\newcommand {\neper }{\mathrm {Np}}\)
\(\newcommand {\yocto }{\mathrm {y}}\)
\(\newcommand {\zepto }{\mathrm {z}}\)
\(\newcommand {\atto }{\mathrm {a}}\)
\(\newcommand {\femto }{\mathrm {f}}\)
\(\newcommand {\pico }{\mathrm {p}}\)
\(\newcommand {\nano }{\mathrm {n}}\)
\(\newcommand {\micro }{\mathrm {\unicode {x00B5}}}\)
\(\newcommand {\milli }{\mathrm {m}}\)
\(\newcommand {\centi }{\mathrm {c}}\)
\(\newcommand {\deci }{\mathrm {d}}\)
\(\newcommand {\deca }{\mathrm {da}}\)
\(\newcommand {\hecto }{\mathrm {h}}\)
\(\newcommand {\kilo }{\mathrm {k}}\)
\(\newcommand {\mega }{\mathrm {M}}\)
\(\newcommand {\giga }{\mathrm {G}}\)
\(\newcommand {\tera }{\mathrm {T}}\)
\(\newcommand {\peta }{\mathrm {P}}\)
\(\newcommand {\exa }{\mathrm {E}}\)
\(\newcommand {\zetta }{\mathrm {Z}}\)
\(\newcommand {\yotta }{\mathrm {Y}}\)
\(\newcommand {\percent }{\mathrm {\%}}\)
\(\newcommand {\meter }{\mathrm {m}}\)
\(\newcommand {\metre }{\mathrm {m}}\)
\(\newcommand {\gram }{\mathrm {g}}\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\of }[1]{_{\mathrm {#1}}}\)
\(\newcommand {\squared }{^2}\)
\(\newcommand {\square }[1]{\mathrm {#1}^2}\)
\(\newcommand {\cubed }{^3}\)
\(\newcommand {\cubic }[1]{\mathrm {#1}^3}\)
\(\newcommand {\per }{/}\)
\(\newcommand {\celsius }{\unicode {x2103}}\)
\(\newcommand {\fg }{\femto \gram }\)
\(\newcommand {\pg }{\pico \gram }\)
\(\newcommand {\ng }{\nano \gram }\)
\(\newcommand {\ug }{\micro \gram }\)
\(\newcommand {\mg }{\milli \gram }\)
\(\newcommand {\g }{\gram }\)
\(\newcommand {\kg }{\kilo \gram }\)
\(\newcommand {\amu }{\mathrm {u}}\)
\(\newcommand {\nm }{\nano \metre }\)
\(\newcommand {\um }{\micro \metre }\)
\(\newcommand {\mm }{\milli \metre }\)
\(\newcommand {\cm }{\centi \metre }\)
\(\newcommand {\dm }{\deci \metre }\)
\(\newcommand {\m }{\metre }\)
\(\newcommand {\km }{\kilo \metre }\)
\(\newcommand {\as }{\atto \second }\)
\(\newcommand {\fs }{\femto \second }\)
\(\newcommand {\ps }{\pico \second }\)
\(\newcommand {\ns }{\nano \second }\)
\(\newcommand {\us }{\micro \second }\)
\(\newcommand {\ms }{\milli \second }\)
\(\newcommand {\s }{\second }\)
\(\newcommand {\fmol }{\femto \mol }\)
\(\newcommand {\pmol }{\pico \mol }\)
\(\newcommand {\nmol }{\nano \mol }\)
\(\newcommand {\umol }{\micro \mol }\)
\(\newcommand {\mmol }{\milli \mol }\)
\(\newcommand {\mol }{\mol }\)
\(\newcommand {\kmol }{\kilo \mol }\)
\(\newcommand {\pA }{\pico \ampere }\)
\(\newcommand {\nA }{\nano \ampere }\)
\(\newcommand {\uA }{\micro \ampere }\)
\(\newcommand {\mA }{\milli \ampere }\)
\(\newcommand {\A }{\ampere }\)
\(\newcommand {\kA }{\kilo \ampere }\)
\(\newcommand {\ul }{\micro \litre }\)
\(\newcommand {\ml }{\milli \litre }\)
\(\newcommand {\l }{\litre }\)
\(\newcommand {\hl }{\hecto \litre }\)
\(\newcommand {\uL }{\micro \liter }\)
\(\newcommand {\mL }{\milli \liter }\)
\(\newcommand {\L }{\liter }\)
\(\newcommand {\hL }{\hecto \liter }\)
\(\newcommand {\mHz }{\milli \hertz }\)
\(\newcommand {\Hz }{\hertz }\)
\(\newcommand {\kHz }{\kilo \hertz }\)
\(\newcommand {\MHz }{\mega \hertz }\)
\(\newcommand {\GHz }{\giga \hertz }\)
\(\newcommand {\THz }{\tera \hertz }\)
\(\newcommand {\mN }{\milli \newton }\)
\(\newcommand {\N }{\newton }\)
\(\newcommand {\kN }{\kilo \newton }\)
\(\newcommand {\MN }{\mega \newton }\)
\(\newcommand {\Pa }{\pascal }\)
\(\newcommand {\kPa }{\kilo \pascal }\)
\(\newcommand {\MPa }{\mega \pascal }\)
\(\newcommand {\GPa }{\giga \pascal }\)
\(\newcommand {\mohm }{\milli \ohm }\)
\(\newcommand {\kohm }{\kilo \ohm }\)
\(\newcommand {\Mohm }{\mega \ohm }\)
\(\newcommand {\pV }{\pico \volt }\)
\(\newcommand {\nV }{\nano \volt }\)
\(\newcommand {\uV }{\micro \volt }\)
\(\newcommand {\mV }{\milli \volt }\)
\(\newcommand {\V }{\volt }\)
\(\newcommand {\kV }{\kilo \volt }\)
\(\newcommand {\W }{\watt }\)
\(\newcommand {\uW }{\micro \watt }\)
\(\newcommand {\mW }{\milli \watt }\)
\(\newcommand {\kW }{\kilo \watt }\)
\(\newcommand {\MW }{\mega \watt }\)
\(\newcommand {\GW }{\giga \watt }\)
\(\newcommand {\J }{\joule }\)
\(\newcommand {\uJ }{\micro \joule }\)
\(\newcommand {\mJ }{\milli \joule }\)
\(\newcommand {\kJ }{\kilo \joule }\)
\(\newcommand {\eV }{\electronvolt }\)
\(\newcommand {\meV }{\milli \electronvolt }\)
\(\newcommand {\keV }{\kilo \electronvolt }\)
\(\newcommand {\MeV }{\mega \electronvolt }\)
\(\newcommand {\GeV }{\giga \electronvolt }\)
\(\newcommand {\TeV }{\tera \electronvolt }\)
\(\newcommand {\kWh }{\kilo \watt \hour }\)
\(\newcommand {\F }{\farad }\)
\(\newcommand {\fF }{\femto \farad }\)
\(\newcommand {\pF }{\pico \farad }\)
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frühe Windnutzung: Mensch nutzt den Wind schon seit mehr als 4000 Jahren, die Ägypter trieben mit ihm Segelboote an
chinesische Segel-Karussell-Windmühle: sog. Vertikalläufer, diese haben gegenüber horizontalen Achse den Vorteil, dass sie von der Windrichtung unabhängig sind, andererseits laufen die Blätter auf einer Hälfte des Umfangs stets gegen den Wind, es gibt zwar die Möglichkeit, den Wind auf einer Seite abzusperren, andererseits strömt die Luft dann eher um die Anlage herum (z. B. persische Windmühle bei Neh ab 1271), außerdem waren die chinesischen Windmühöem Langsamläufer mit großen Widerstandsflächen, daher schlechte Leistung
Windwagen, Ventomobil: gegen den Wind fahren ist möglich
Windanlagen im Mittelalter: Savonius-Vorläufer des Mittelalters, 1567 – 1617
Holläder-Windmühlen: Wasserpumpen mit Windkraft, Horizontalachse treibt Wasserpumpe an (z. B. archimedische Spirale), Bockwindmühlen sind auf Drehscheiben, damit man sie in den Wind drehen kann, Holländer verwendeten Holzzahnräder, die besser austauschbar war als Gusseisen (wenn ein Zahn ausbricht, einfach neuen Zahn einkeilen, bei Gusseisen musste das ganze Rad ausgetauscht werden)
Seitenradantrieb: automatische Nachführung des Hauptrotos
Segelwindmühlen auf Mykonos: Sturmsicherung durch einrollbare Segel, Hochebene von Lassithi, Kreta: Wasserpumpen für Wasser in trockenen Zeiten
archimedische Spirale: Wassertransport um bis zu 2 m, mehr schafft die Anlage wegen großer Reibung nicht, Langsamläufer, Vielblattturbine, Blätter aus Blech (Western Mill)
Unterschied USA – D: in Deutschland gibt es zuerst viele Papierstudien, bevor gebaut wird, in den USA ist es umgekehrt: „einfach mal bauen“
Gittermasten: billigste und schnellste Bauweise, wegen vielen sich wiederholenden Elementen für die Massenfertigung geeignet
Hermann Honnef: Vorschlag von drei Rotoren pro Mast, ineffizient, am besten ist ein Rotor pro Mast, außerdem Vorschlag von Riesen-Doppelrotoren, die sich mit Magneten gegeneinander drehen und so einen Generator betreiben, Durchmesser von 180 m, Leistung \(3 \cdot \SI {20}{\mega \watt }\)
Offshore-Windenergienutzung: Honnef hat 1932 schwimmende Plattformen mit je zwei Rotoren vorschlagen, automatische Windnachführung, da mit Seilen ein einem fest im mehr verankerten Pfahl befestigt, Offshore-Windenergienutzung ist wesentlich effektiver in der Nordsee als in der Ostsee, da in der Ostsee der Wind durch das Land abgeschwächt wurde
Windanlage auf dem Feldberg: 1955, später abgebaut, „Inselanlage“
Allgaier WE-10: auch in Südafrika aufgebaut
Windcharger: Jacobs Windcharger zur Stromversorgung von Farmen
Yalta, Krim: Zweiflügler mit Windnachführung durch einen Wagen am Boden auf Schienen, Experiment: Klappen zur Vergrößerung des Auftriebs (wie bei Flugzeugen), hat nichts gebracht, da die Klappen hier im Gegensatz zum Flugzeug rotieren, außerdem Problem der dauerhaften, langfristigen Befestigung
Putnam, USA: 1937 – 1945 stand dort eine Windkraftanlage mit 1250 kW Leistung und 52 m Durchmesser (Smith, Zweiflügler), im Sturm hat es eines Nachts einen Flügel von 8 t weggerissen, 1 km weit geflogen
Enfield-Andreau-Anlage: „aerodynamisches“ Getriebe, da Schlitze in der Flügeln, wo Luft aus dem Inneren der Flügel bei der Rotation austritt, in Schlitzen unten am Boden kann die Luft in den Turm nachströmen, dort befindet sich im Turm ein zweiter Generator, war in Marokko in Betrieb, aber nicht sehr effizient (zu hoher Widerstand im Turm)