Dual cycle-Diesel Engine question

[santeria13]

Homework Statement

A diesel engine works on the dual combustion cycle and has a compression ratio of 20/1. At the beginning of the compression process the pressure and temperature are 1.0 bar and 22°C respectively. In the cycle, heat is added at constant volume until the pressure has increased by 50% and then at constant pressure for 7% of the swept volume. For air assume k= 1.4; cv = 0.718 kJ/kg K; cp = 1.005 kJ/kg K.

Calculate for the cycle: a) The temperature at the remaining state points in the cycle; (5 marks)

Homework Equations

V3/V4 = cutoff ratio (rc) V2/V1 = compression ratio (r) = 20/1 P3/P2 = pressure ratio(rp)
T2 = T1 * (r)^k-1
T3 = T1 * (r)^k-1 * rp
T4 = T1 * (r)^k-1 * rp *rc ----------------- confused on how to calculate the cutoff ratio(rc)

The Attempt at a Solution

Calculated all figures up until T4 at which the cutoff ratio is required.

 

[SteamKing]

I am working though a question which states that the compression ratio of a diesel engine is 20/1 . This allows me to find the first 3 temperatures until the constant pressure stage. I do this using the formula :
T2 = T1 * (Compression ratio)^1-k where k is 1.4 .

However, after T3, the cutoff ratio must be used instead of the compression ratio in a similar formula which writes as follows:

T4 = T3 * (cutoff ratio)^1-k where k is again 1.4 .

The cutoff ratio is the ratio of the volume of air when fuel is added over the volume of air once the fuel and air mixture has stopped burning.

The question states that the piston moves 7% of the swept volume during that time (between fuel being added to end of burning).

To get a better idea, the compression ratio is defined as
clearence volume + swept volume / clearance volume

so I assumed the cuttoff ratio would be 0.07 x 19 = 1.33 but I am not sure this is the right way to go.
The reason I did this is because compression ratios are generally given as a number and not a fraction.

There's a definite relationship between a ratio and a fraction. For example, a ratio of 2:1 implies that something is 1/2 as big as something else.

A CR of 20:1 means that the volume inside the cylinder just before the injection of the fuel is 1/20 = 0.05 = 5% of the total volume when the piston is at bottom dead center.

In other words, total volume = swept volume + clearance volume

and $CR = \frac{CV}{SV + CV} = \frac{V_1}{V_2}$ as shown in the Wiki article on the Diesel cycle below

where SV = swept volume and CV = clearance volume, which means V₁ = CV and V₂ = SV + CV

https://en.wikipedia.org/wiki/Compression_ratio

The cutoff ratio is defined:

$α = \frac{V_3}{V_2}$, where V₂ is defined as above and V₃ = CV + 0.07 * SV, in this case

https://en.wikipedia.org/wiki/Diesel_cycle

You can calculate the cutoff ratio knowing the relationship between the CV and the SV given by the CR.