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# SM_8B - Chapter 8 Section B Non-Numerical Solutions 8.12(a...

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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 r γ c 1 r c 1 or η Diesel = 1 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 [1 + θ( x 1 ) ] 2! · ( x 1 ) 2 ( 0 < θ < 1 ) Then, x a = 1 + a · ( x 1 ) + 1 2 a · ( a 1 ) · [1 + θ( x 1 ) ] a 2 · ( x 1 ) 2 Note that the final term is R . For a > 1 and x > 1, R > 0. Therefore: x a > 1 + a · ( x 1 ) x a 1 > a · ( x 1 ) and r γ c 1 γ( r c 1 ) > 1 ( c ) If γ = 1 . 4 and r = 8, then by Eq. (8.6): η Otto = 1 1 8 0 . 4 and η Otto = 0 . 5647 r c = 2 η Desiel = 1 1 8 0 . 4 2 1 . 4 1 1 . 4 ( 2 1 ) and η Diesel = 0 . 4904 r c = 3 η Desiel = 1 1 8 0 . 4 3 1 . 4 1 1 . 4 ( 3 1 ) and η Diesel = 0 . 4317 579

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8.15 See the figure below. In the regenerative heat exchanger, the air temperature is raised in step B B , while the air temperature decreases in step
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SM_8B - Chapter 8 Section B Non-Numerical Solutions 8.12(a...

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