Chapter 9.12, Problem 64P

An air-standard cycle, called the dual cycle, with constant specific heats is executed in a closed piston– cylinder system and is composed of the following five processes:

1-2 Isentropic compression with a compression ratio, r = V₁/V₂

2-3 Constant-volume heat addition with a pressure ratio, rp = P₃/P₂

3-4 Constant-pressure heat addition with a volume ratio, rc V₄/V₃

4-5 Isentropic expansion while work is done until V₅ = V₁

5-1 Constant-volume heat rejection to the initial state

1. (a)   Sketch the P-ν and T-s diagrams for this cycle.
2. (b)   Obtain an expression for the cycle thermal efficiency as a function of k, r, rc, and rp.
3. (c)   Evaluate the limit of the efficiency as rp approaches unity, and compare your answer with the expression for the Diesel cycle efficiency.
4. (d)   Evaluate the limit of the efficiency as rc approaches unity, and compare your answer with the expression for the Otto cycle efficiency.

To determine

Draw the $P-v$ and $T-s$ diagrams for the given cycle.

Answer to Problem 64P

The $P-v$ and $T-s$ diagrams for the given cycle are shown as in Figure (1).

Explanation of Solution

Draw the $P-v$ and $T-s$ diagram for the given cycle.

Thus, the $P-v$ and $T-s$ diagrams for the given cycle are shown as in Figure (1)

To determine

The expression for the back work ratio as a function of k and r.

Answer to Problem 64P

The expression for the back work ratio as a function of $k,r,r_c,\text{and}{r}_p$ is $\underset{\_}{1-\frac{r_c^k{r}_p-1}{r^{k-1}\left(r_p-1\right)+k{r}_p{r}^{k-1}\left(r_c-1\right)}}$.

Explanation of Solution

Apply first law to the closed system for processes 2-3, 3-4, and 5-1 to get the expression of $q_{\text{in}}\text{and}{q}_{\text{out}}$.

$q_{\text{in}}=c_v\left(T_3-T_2\right)+c_p\left(T_4-T_3\right)$

$q_{\text{out}}=c_v\left(T_5-T_1\right)$

Here, heat added to the system and heat rejected from the system is $q_{\text{in}}\text{and}{q}_{\text{out}}$, constant volume specific heat is $c_v$, temperature at state 1, 2, 3, 4, and 5 are $T_1,T_2,T_3,T_4,\text{and}{T}_5$, and constant pressure specific heat is $c_p$.

Express the cycle thermal efficiency.

${\eta }_{\text{th}}=1-\frac{q_{\text{out}}}{q_{\text{in}}}$ (I)

Conclusion:

Process 1-2: Isentropic

Calculate the ratio of $T_2/T_1$.

$\begin{array}{c}\frac{T_2}{T_1}={\left(\frac{v_1}{v_2}\right)}^{k-1}\\ =r^{k-1}\end{array}$

Here, volume at states 1 and 2 is $v_1,v_2$, compression ratio is r, and specific heat ratio is k.

Process 2-3: Constant volume

Calculate the expression for $T_3/T_2$.

$\begin{array}{l}\frac{T_3}{T_2}=\frac{P_3{v}_3}{P_2{v}_2}\\ \frac{T_3}{T_2}=\frac{P_3}{P_2}=r_p\end{array}$

Here, pressure at state 1 and 2 is $P_1\text{}\text{and}{P}_2$, and pressure ratio is $r_p$

Process 3-4: Constant pressure

Calculate the expression for $T_4/T_3$.

$\begin{array}{l}\frac{P_4{v}_4}{T_4}=\frac{P_3{v}_3}{T_3}\\ \frac{T_4}{T_3}=\frac{v_4}{v_3}\\ \frac{T_4}{T_3}=r_c\end{array}$

Here, compression ratio is $r_c$.

Process 4-5: Isentropic

Calculate the expression for $T_5/T_4$.

$\begin{array}{l}\frac{T_5}{T_4}={\left(\frac{v_4}{v_5}\right)}^{k-1}\\ \frac{T_5}{T_4}={\left(\frac{v_4}{v_1}\right)}^{k-1}\\ \frac{T_5}{T_4}={\left(\frac{r_c{v}_3}{v_1}\right)}^{k-1}\\ \frac{T_5}{T_4}={\left(\frac{r_c{v}_2}{v_1}\right)}^{k-1}\end{array}$

$\frac{T_5}{T_4}={\left(\frac{r_c}{r}\right)}^{k-1}$

Process 5-1: Constant volume

Calculate the expression for $T_5/T_1$.

$\frac{T_5}{T_1}=\frac{T_5}{T_4}\frac{T_4}{T_3}\frac{T_3}{T_2}\frac{T_2}{T_1}$

Substitute ${\left(\frac{r_c}{r}\right)}^{k-1}$ for $\frac{T_5}{T_4}$, $r_c$ for $\frac{T_4}{T_3}$, $r_p$ for $\frac{T_3}{T_2}$, and $r^{k-1}$ for $\frac{T_2}{T_1}$.

$\begin{array}{c}\frac{T_5}{T_1}={\left(\frac{r_c}{r}\right)}^{k-1}\left(r_c\right)r_p\left(r^{k-1}\right)\\ =r_c^k{r}_p\end{array}$

Calculate the ratio of $T_3/T_1$.

$\frac{T_3}{T_1}=\frac{T_3}{T_2}\frac{T_2}{T_1}$

Substitute $r_p$ for $\frac{T_3}{T_2}$ and $r^{k-1}$ for $\frac{T_2}{T_1}$.

$\frac{T_3}{T_1}=r_p{r}^{k-1}$

Substitute $c_v\left(T_3-T_2\right)+c_p\left(T_4-T_3\right)$ for $q_{\text{in}}$ and $c_v\left(T_5-T_1\right)$ for $q_{\text{out}}$ in equation (I).

$\begin{array}{c}{\eta }_{\text{th}}=1-\frac{c_v\left(T_5-T_1\right)}{c_v\left(T_3-T_2\right)+c_p\left(T_4-T_3\right)}\\ =1-\frac{T_1\left(T_5/T_1-1\right)}{T_2\left(T_3/T_2-1\right)+k\frac{T_3}{T_1}\left(T_4/T_3-1\right)}\\ =1-\frac{\left(T_5/T_1-1\right)}{\frac{T_2}{T_1}\left(T_3/T_2-1\right)+k\frac{T_3}{T_1}\left(T_4/T_3-1\right)}\end{array}$ (II)

Substitute $r_p{r}^{k-1}$ for $T_3/T_1$, $r_p$ for $\frac{T_3}{T_2}$, $r_c$ for $\frac{T_4}{T_3}$, $r^{k-1}$ for $\frac{T_2}{T_1}$, and $r_c^k{r}_p$ for $\frac{T_5}{T_1}$ in Equation (II).

${\eta }_{\text{th}}=1-\frac{\left(r_c^k{r}_p-1\right)}{r^{k-1}\left(r_p-1\right)+k{r}_p{r}^{k-1}\left(r_c-1\right)}$

Thus, the expression for the back work ratio as a function of $k,r,r_c,\text{and}{r}_p$ is $\underset{\_}{1-\frac{r_c^k{r}_p-1}{r^{k-1}\left(r_p-1\right)+k{r}_p{r}^{k-1}\left(r_c-1\right)}}$.

To determine

The limit of the efficiency as $r_p$ approaches unity.

Answer to Problem 64P

The limit of the efficiency as $r_p$ approaches unity is $1-\frac{1}{r^{k-1}}\left[\frac{r_c^k-1}{k\left(r_c-1\right)}\right]$.

Explanation of Solution

Recall the expression for the back work ratio as a function of $k,r,r_c,\text{and}{r}_p$ and apply the limit as $r_p$ approaches unity.

$\begin{array}{c}\underset{r_p\to 1}{\lim }{\eta }_{\text{th}}=1-\left\{\underset{r_p\to 1}{\lim }\frac{r_c^k{r}_p-1}{r^{k-1}\left(r_p-1\right)+k{r}_p{r}^{k-1}\left(r_c-1\right)}\right\}\\ =1-\frac{1}{r^{k-1}}\left[\frac{r_c^k-1}{k\left(r_c-1\right)}\right]\end{array}$

Thus, the limit of the efficiency as $r_p$ approaches unity is $1-\frac{1}{r^{k-1}}\left[\frac{r_c^k-1}{k\left(r_c-1\right)}\right]$.

The limit of the efficiency as $r_p$ approaches unity is same as the thermal efficiency of diesel cycle.

To determine

The limit of the efficiency as $r_c$ approaches unity.

Answer to Problem 64P

The limit of the efficiency as $r_c$ approaches unity is $1-\frac{1}{r^{k-1}}$.

Explanation of Solution

Recall the expression for the back work ratio as a function of $k,r,r_c,\text{and}{r}_p$ and apply the limit as $r_c$ approaches unity.

$\begin{array}{c}\underset{r_p\to 1}{\lim }{\eta }_{\text{th}}=1-\left\{\underset{r_p\to 1}{\lim }\frac{r_c^k{r}_p-1}{r^{k-1}\left(r_p-1\right)+k{r}_p{r}^{k-1}\left(r_c-1\right)}\right\}\\ =1-\left[\frac{r_p-1}{r^{k-1}\left(r_p-1\right)}\right]\\ =1-\frac{1}{r^{k-1}}\end{array}$

Thus, the limit of the efficiency as $r_c$ approaches unity is $1-\frac{1}{r^{k-1}}$.

The limit of the efficiency as $r_c$ approaches unity is same as the thermal efficiency of Otto cycle.