Question

An air standard cycle with constant specific heats is executed in a closed piston-cylinder system and is composed of the following three processes: \(1-2 \quad\) Constant volume heat addition \(2-3 \quad\) Isentropic expansion with an expansion ratio \(r=V_{3} / V_{2}\) \(3-1 \quad\) Constant pressure heat rejection (a) Sketch the \(P\) -v and \(T\) -s diagrams for this cycle (b) Obtain an expression for the back work ratio as a function of \(k\) and \(r\) (c) Obtain an expression for the cycle thermal efficiency as a function of \(k\) and \(r\) (d) Determine the value of the back work ratio and efficiency as \(r\) goes to unity What do your results imply about the net work done by the cycle?

Short answer

(a) The P-v diagram for the cycle consists of a vertical line representing the constant volume heat addition (process 1-2), a curve for the isentropic expansion (process 2-3), and a horizontal line for the constant pressure heat rejection (process 3-1). The T-s diagram shows a horizontal line for the constant volume heat addition (process 1-2), a vertical line for the isentropic expansion (process 2-3), and an inclined line with a negative slope for the constant pressure heat rejection (process 3-1). (b) The back work ratio (BWR) expression for the cycle as a function of specific heat ratio (k) and expansion ratio (r) is: BWR \(= \frac{1 - k \times (1 - r)}{k \times (r - 1)}\) (c) The cycle's thermal efficiency expression as a function of \(k\) and \(r\) is: \(\eta_\text{cycle} = 1 - \text{BWR}\) (d) As \(r \to 1\), the back work ratio is 1 and the cycle thermal efficiency is 0. This means that the work done to compress the gas equals the work done by the cycle, resulting in no net work output.

Step-by-step solution

P-v Diagram

On the \(P-v\) diagram, the constant volume heat addition process (1-2) is shown as a vertical line, while the constant pressure heat rejection process (3-1) is a horizontal line. The isentropic expansion process (2-3) can be represented by a curve between points 2 and 3.

T-s Diagram

On the \(T-s\) diagram, the constant volume heat addition process (1-2) is a horizontal line because there is no change in entropy during the process. The constant pressure heat rejection process (3-1) is shown as an inclined line with a negative slope. The isentropic expansion process (2-3) is a vertical line because it is an isentropic process with no change in entropy. (b) Obtain an expression for the back work ratio as a function of \(k\) (specific heat ratio) and \(r\) (expansion ratio)

Back Work Ratio Expression

The back work ratio (BWR) is the ratio between the net work output of the cycle and the work done during the isentropic expansion process (2-3). Using the equations for the isentropic process and constant pressure heat rejection, we get: BWR \(= \frac{W_{cycle}}{W_{2-3}} = \frac{1 - k \times (1 - r)}{k \times (r - 1)}\) (c) Obtain an expression for the cycle thermal efficiency as a function of \(k\) and \(r\)

Thermal Efficiency Expression

The cycle's thermal efficiency (\(\eta\)) can be calculated as the ratio between the net work output (\(W_{cycle}\)) and the heat addition to the cycle during the constant volume heat addition process (1-2): \(\eta_\text{cycle} = \frac{W_{cycle}}{Q_{1-2}}\) Since the back work ratio is a function of \(k\) and \(r\), we can use it to find the cycle's thermal efficiency: \(\eta_\text{cycle} = 1 - \text{BWR}\) (d) Determine the value of the back work ratio and efficiency as \(r \to 1\)

Back Work Ratio As r Goes To Unity

Taking the limit of the BWR expression as \(r \to 1\), we get: \(\lim_{r \to 1} \left(\frac{1 - k \times (1 - r)}{k \times (r - 1)} \right)= \lim_{r \to 1} \frac{1 - k + kr}{kr - k}\) which simplifies to: BWR \(= 1\)

Efficiency As r Goes To Unity

As BWR \(= 1\), this implies: \(\eta_\text{cycle} = 1 - 1 = 0\) The results indicate that, as \(r \to 1\), the back work ratio is 1 and the cycle thermal efficiency is 0. This means that the work done to compress the gas equals the work done by the cycle, resulting in no net work output.