(i). Consider equations 3 and 4. Show that these equations yield a result that h

Question

(i). Consider equations 3 and 4. Show that these equations yield a result that has the same dimensions as energy. To do this you should plug in the base dimensions of pressure and volume. Remember that k is unitless.

(ii). What are the units (in both SI and AES) and base dimensions of energy?

(iii). What are the units (in both SI and AES) and base dimensions of power?

(iv). Relate each labeled state of the idealized process in figure b, to the corresponding step in the actual four stroke engine process (figure a). To do this, first locate each labeled (1 through 4) black dot on the plot in figure b, then determine what physical process is happening in the actual 4 stroke cycle by comparing figure b to figure a.

(v). Use the following properties for this idealized process: State Property Values

State 1: 1 P1= 1.00 atm T1 = 17.0°C V1 = 152 mL

State 3: 3 T3 = 1977 °C V3 = 19.1 mL

In the calculations below, be sure to leave your work in variables until the very last step.

a. Use the information given above for state 1 and the ideal gas law to calculate the amount (n, in units of moles) of air-fuel mixture in the idealized cycle.

b. Use your result from (a) to calculate the maximum pressure in the cycle. To determine the maximum pressure in the cycle reference figure b and find the labeled state (black dot) that corresponds to the highest pressure.

c. Assuming k = 1.4, use equations 1 and 2 above together with the ideal gas equation and observations from the PV diagram to calculate the properties Temperature (T), Pressure (P) and Volume (V) at states 2 and 4.

6. Use equations 3 and 4 to calculate the energy input and output required for the compression and power strokes respectively. You will need to convert your answer to units of Joules. Track your units carefully and ask questions if you get stuck. Calculate the net energy produced (per cycle) by adding these two values for work together (Note: the compression stroke should have negative work).

7. In question 5a above you calculated the moles of air-fuel mixture used in the cycle. In order to determine the moles of gasoline used in the cycle, you need to determine the stoichiometric ratio of gasoline to air-fuel mixture. You can use the following assumptions:

Model Gasoline as Octane (C8H18)

Model air as 3.76 moles of Nitrogen (N2) for every 1 mole of Oxygen (O2)

The resulting ideavaed thermodynami cycle (oto Cyclel is shawn below: The expansion and compression processes can be moceled hy what is called a polytropic process with the following equations from Thermadynamics: Equation 1 Equation 2: The esponent is dimensionless and can be determined knowing the operating conditionsof the enineor assumed besed un idealng the process Think fil s ueril delemined by the lype uf prucess wou are analyaing. For our purpases wo wll ue k-1.4 which ibased n an idealized process. The mechanical energy called the Wark, canbe caldated usine equation 3 bekw Work will be negative for the ompression strake since energy has to be put in to campress the sandIt wil be posltive tor the power stroke since work is heing done by the engire. Equation 3 (Compressian Stroc): The sibscn and2 reter to the initial and tinal stares at the gas in the process. In ather words, you coadd alsa write this same equation to determine th work rrom stage 3 to slee 4 as quation 4 rStroke):

Explanation / Answer

i) given equation 3 :
   W = (P2V2 - P1V1)/(1 - k)
   given equation 4 :
   W = (P4V4 - P3V3)/(1 - k)

   now base dimension of pressure, P = Force/Area = Kg m/s^2 * m^2 = Kg / ms^2
   Base dimension of volume, V = m^3
   so in both the equaitons, k is dimension less
   so the base equation of energy comes out to be
   PV = kg*m^3 /ms^2 = kg*m^2/s^2

   now energy = FOrce*distance = kg * m^2/s^2
   so the answer from the equation has the same base dimension as energy
ii) SI unit of energy = Joules, symbol J
   AES unit of energy = foot pound force, symbol ft . lb
   base unit of energy in SI unit = kg m^2/s^2
   base unit in AES system = ft . lb
   base dimebnsion = [M L^2 T^(-2)]
iii) SI unit of power = Watt, symbol W
   AES unit of power = foot pound force per second, symbol ft . lb /s
   base unit of power in SI unit = kg m^2/s^3
   base unit in AES system = ft . lb / s
   base dimebnsion = [M L^2 T^(-3)]
iv) consider step 1 to 2
    the process is isentropic expansion
    this can be related to step 3->4 of 4 stroke engine
    consider step 2 to 3
    the process is isochoric heat rejection
    this can be related to step 4->1 of 4 stroke engine
    consider step 3 to 4
    the process is polytropic compression
    this can be related to step 1->2 of 4 stroke engine
    consider step 4 to 1
    the process is isochoric heat intake
    this can be related to step 2->3 of 4 stroke engine
  
   

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