Carnot and Otto Engines

This shows a thermoelectric device which takes heat from one cup and powers the spinning disk, and the other side acts as a cold reservoir. The disc spins in whichever direction the flow goes. If the hot cup is on the left and the cold on the right the disk will go clockwise, and visa versa.

When the sensors are plugged into a power supply one side heats up and one side cools down.

We assumed that the direction of the spinning disk would change when the cups were swapped. We also assumed the heat would flow to one side of the device while cooling the other side.

We used the formulas Q=n*Cv*dT and E=3/2*n*R*T and Cv = 3/2*R to come up with the equation of 3/2*R = Cp - R, meaning that 5/2*R = Cp.

We show that n*dT = (p*dV + V*dp)/(Cp-Cv) by first finding that Cp-Cv = R and subbing that into the ideal gas formula where there is both a change in volume and pressure.

The picture above shows the simplification of the of the previous equation to show that dp/p + (dV/V)*(Cp/Cv) = 0. With that we were able to derive another equation by moving the V's to the other side, taking the integral of both sides, then take the exponential function of both sides and we then find the equation relating the pressures and the volumes.

We derive a formula for temperatures and volumes, by subbing for p values determined from the ideal gas law(nR cancel), on the left of the board and we derive a formula for work in an adiabatic process on the right.

We used the derived formula for work in an adiabatic process to find the work in this situation.

This is a table of a Carnot cycle The top and bottom are isothermal and the left and right sides are adiabatic. We found work on the adiabatic with the formula W=PiVi^5/3[(Vf^-2/3 - Vi^-2/3)/(-2/3)] and we found work in the isothermal stages with W=nRT*ln(Vf/Vi). 5/3 is gamma and -2/3 is 1 - gamma. We were able to fill out the rest of the table since change of energy is zero in isothermal processes and Q is zero in adiabatic processes.

This device is an example of how an Otto engine works. Air is brought into the cylindrical shell when the piston is drawn down. Then the piston goes up quickly to increase pressure and reduce volume to cause the temp to rise. The spark plug then ignites causing and explosion where the piston is forced down. As that happens the exhaust valve opens and releases the burnt air and fuel. The process is then repeated.

The left side of the board shows what we could do to improve the work output of an engine and the right side of the board shows the extra degrees of freedom in a diatonic gas.