Question

In a diesel engine, the fuel is ignited without a spark plug. Instead, air in a cylinder is compressed adiabatically to a temperature above the ignition temperature of the fuel; at the point of maximum compression, the fuel is injected into the cylinder. Suppose that air at (20°C) is taken into the cylinder at a volume (V_1) and then compressed adiabatically and quasi-statically to a temperature of (600°C) and a volume (V_2). If (gamma=1.4), what is the ratio (V_1/V_2)? (Note: In an operating diesel engine, the compression is not quasi-static.)

a) Calculate the ratio (V_1/V_2) for the given conditions.
b) The compression being quasi-static has no impact on the solution.
c) The solution requires knowledge of the specific heat at constant volume.
d) The value of (gamma) is irrelevant to the solution.

Answer 1

To calculate the volume ratio V1/V2 in a diesel engine's adiabatic compression, convert temperatures to Kelvin and use Poisson's law. Given γ=1.4, T1=293.15 K, and T2=873.15 K, the ratio is approximately V1/V2 ≈ 2.98.

Step-by-step explanation:

In diesel engines, rather than using a spark plug, the process of air compression raises the temperature enough to ignite the fuel. This compression of air is an adiabatic process, which is a transformation that occurs without exchange of heat with the surroundings. To calculate the ratio of the original volume V1 to the compressed volume V2 under adiabatic conditions, you can use the equation derived from Poisson's law:

V1 / V2 = (T2 / T1)1/(γ-1)

where γ is the ratio of specific heats (Cp/Cv), T1 is the initial temperature in Kelvins, and T2 is the final temperature in Kelvins. Since we are given temperatures in degrees Celsius, we need to convert them to Kelvins by adding 273.15 to each. Thus:

T1 = 20°C + 273.15 = 293.15 K

T2 = 600°C + 273.15 = 873.15 K

And given that γ = 1.4, we can plug the values into the formula:

(V1 / V2) = (873.15 K / 293.15 K)1/(1.4-1) ≈ 2.98

The ratio of V1 to V2 is approximately 2.98, which implies that the original volume is almost three times larger than the compressed volume under these ideal conditions.