Solving Adiabatic Process in Diesel Engine: Final Temp of Air-Fuel Mixture

[menco]

Homework Statement

In a diesel engine, the piston compresses the air-fuel mixture from an initial volume of 630 cm^3 to a final volume of 30cm^3. If the initial temperature of the air-fuel mixture is 45 degrees C and the process is occurring adiabatically, determine the final temperature. Comment on the significance of the result.

Homework Equations

V(initial) = 6.3x10^-4 m^3
V(final = 3.0x10^-5 m^3
T(initial) = 318.15 K
T(final) = ?
y = 1.4

T(final) = T(initial)*(V(initial)/V(final))^y-1

The Attempt at a Solution

T(final) = 318.15(6.3x10^-4/3.0x10^-5)^0.4
T(final) = 1075.28 K or 802.13 degrees C


Is that on the right track? I found the equation online but I am a little confused how the equation is actually formed from T(f) = P(f)*V(f) / nR which I have in my textbook.

 

[ehild]

During an adiabatic process, no heat exchange occurs, so the change of internal energy is do to the work alone: dU=-PdV. PV=nRT is valid but the temperature and pressure changes during the process: $P=nRT/V$. For an ideal gas, the internal energy is $U=nC_vT$, $dU=nC_vdT$, so $nC_vdT=-(nRT/V) dV$. n cancels. Collect the like terms and integrate

$$\int{\frac{dT}{T}}=\int{-\frac{R}{C_v}\frac{dV}{V}}$$

$$\ln(T)=-\frac{R}{C_v} \ln(V) +const$$.

R/C_v can be written in terms of γ=C_p/C_v: $\frac{R}{C_v}=\gamma -1$.

You can rewrite the equation as $$\ln(TV^{\gamma-1})=const$$,

that is $$TV^{\gamma-1}=const$$ .

ehild