Chapter8_B - Chapter 8 - Section B - Non-Numerical...

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Unformatted text preview: Chapter 8 - Section B - Non-Numerical Solutions 8.12 ( a ) Because Eq. (8.7) for the efficiency η Diesel includes the expansion ratio , r e ≡ V B / V A , we relate this quantity to the compression ratio , r ≡ V C / V D , and the Diesel cutoff ratio , r c ≡ V A / V D . Since V C = V B , r e = V C / V A . Whence, r r e = V C / V D V C / V A = V A V D = r c or 1 r e = r c r Equation (8.7) can therefore be written: η Diesel = 1 − 1 γ · ( r c / r ) γ − ( 1 / r ) γ r c / r − 1 / r ¸ = 1 − 1 γ ( 1 / r ) γ 1 / r µ r γ c − 1 r c − 1 ¶ or η Diesel = 1 − µ 1 r ¶ γ − 1 r γ c − 1 γ( r c − 1 ) ( b ) We wish to show that: r γ c − 1 γ( r c − 1 ) > 1 or more simply x a − 1 a ( x − 1 ) > 1 Taylor’s theorem with remainder, taken to the 1st derivative, is written: g = g ( 1 ) + g ( 1 ) · ( x − 1 ) + R where, R ≡ g 00 [1 + θ( x − 1 ) ] 2! · ( x − 1 ) 2 ( < θ < 1 ) Then, x a = 1 + a · ( x − 1 ) + 1 2 a · ( a − 1 ) · [1 + θ( x − 1 )...
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Chapter8_B - Chapter 8 - Section B - Non-Numerical...

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