An air-standard dual cycle has a compression ratio of 10. The pressure and temperature at the beginning of compressing are 1 bar and 27 $^{\circ}$ C. The temperature after constant volume heat addition is 987 $^{\circ}$ C and ned heat supplied to the cycle is 878 kJ/kg. What will be the cut-off ratio of the cycle?

$[T a k e : c_{p} = 1.004 k J / k g K a n d c_{v} = 0.717 k j / k g K a n d γ = 1.4]$

2.40

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2.8

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3.40

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1.40

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Solution

The correct option is D 1.40

$Heat supplied/kg = c_{v} (T_{3} - T_{2}) + c_{p} (T_{4} - T_{3}) . . . (i)$

$\frac{T_{2}}{T_{1}} = {(\frac{V_{2}}{V_{1}})}^{γ - 1} = (10)^{1.4 - 1}$

$T_{2} = (10)^{0.4} \times 300$

$753.57 K = {480.57}^{\circ} C$

Now, from equation (i)

$878 = 0.717 (987 - 480.57) + 1.004 (T_{4} - 987)$

$T_{4} = {1499.84}^{\circ} C = 1772.84 K$

$Cut-off ratio, γ_{c} = \frac{V_{4}}{V_{3}}$

$For constant pressure process(3-4)$

$\frac{V_{4}}{V_{3}} = \frac{T_{4}}{T_{3}}$

$s o, γ_{c} = \frac{T_{4}}{T_{3}} = \frac{1772.84}{1260} = 1.407$

$[∴ T_{3} = 987^{\circ} C = 1260 K]$

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