1. An air standard cycle is executed within a closed piston-cylinder system and consists of three processes as follows:1-2 = constant heat addition from 100 kPa and 27∘C to 700 kPa 2-3 Isothermal expansion until V3 = 7v23-1 P = constant heat rejection to the initial state2. Assume air has constant properties with cv = 0.718 kJ/kg K, cp = 1.005 kJ/kg K, R = 0.287 kJ/kg K, and k = 1.4.(a) Sketch the P- and T-s diagrams for the cycle.(b) Determine the ratio of the compression work to the expansion work (the back work ratio).(c) Determine the cycle thermal efficiency.

(a) To sketch the P-v and T-s diagrams for the described air standard cycle, we need to understand the process in each step of the cycle.

The cycle described has three processes:

1-2: Constant heat addition which is typically shown as a vertical process on the P-v diagram since pressure is increasing while volume remains same in an idealized model (in reality, the volume would change somewhat during heat addition, but we assume constant volume for simplicity). On the T-s diagram, the constant volume heat addition shows as a curve to the right as the entropy increases with heat addition.

2-3: Isothermal expansion at constant temperature, which is represented as a horizontal line to the right on the T-s diagram, since temperature remains constant and entropy increases during the expansion due to heat addition. On the P-v diagram, this process is shown as a hyperbolic curve (Pv = Constant for an isothermal process in an ideal gas) dropping from the high-pressure point.

3-1: Constant pressure (isobaric) heat rejection, returning to the initial volume and pressure. On the P-v diagram, this is a vertical downward line (when volume is the horizontal axis and pressure is the vertical axis) since pressure is held constant while volume decreases. On the T-s diagram, the constant pressure cooling is shown as a curve to the left as the entropy decreases with heat rejection.

Since we are dealing with an air standard cycle and given that air is treated as an ideal gas, we can use the ideal gas properties provided such as c_v, c_p, R, and k here.

(b) To determine the back work ratio, which is the ratio of the compression work to the expansion work, we can use the first law of thermodynamics for each process:

For process 1-2 (constant volume heat addition), the work done is zero since the volume does not change.

For process 2-3 (isothermal expansion), the work done can be determined using W_2-3 = P_2 * V_2 * ln(V_3/V_2)

For process 3-1 (constant pressure heat rejection), the work done can be calculated as W_3-1 = P_3 * (V_3 - V_1)

The back work ratio is then the absolute value of W_3-1 divided by W_2-3.

(c) The thermal efficiency of the cycle is determined by the net work output divided by the heat input. The net work is the work done during the expansion (2-3) minus the work done during the compression (3-1). The heat input is the heat added during the constant volume process (1-2).

Using q_in = m * c_v * (T_2 - T_1) for the heat added (constant volume, 1-2) and knowing the work done in each process, the thermal efficiency (η) can be calculated as:

η = (W_net) / (q_in)

Here, W_net = W_2-3 - W_3-1

Using the specific heat ratios and ideal gas law, you can solve for temperatures and volumes at each state, which will then allow you to find the work done and the heat added in each process needed to solve these equations.

Extra:

The air standard cycle described in the problem is a theoretical cycle used to simplify the analysis of reciprocating engines, such as internal combustion engines. In these cycles, air is assumed to be the working fluid, and some key assumptions are made: the air behaves as an ideal gas, specific heats (c_v and c_p) are constant, and combustion is replaced by heat addition.

In thermodynamics, the specific heats at constant volume (c_v) and at constant pressure (c_p) represent the amount of energy required to raise the temperature of a unit mass of a substance by one degree at constant volume or pressure, respectively. The specific heat ratio (k) is the ratio of c_p to c_v.

The thermal efficiency of a cycle is an important metric, representing the percentage of heat energy that is converted into work. A higher thermal efficiency means a greater portion of the fuel's thermal energy is being converted into useful mechanical work, which is desirable in any heat engine.

Understanding P-v and T-s diagrams is crucial in thermodynamics as they provide a visual representation of the changes in pressure, volume, temperature, and entropy during thermodynamic processes, helping students grasp how heat engines operate, and how energy is transformed and conserved.