Chapter 7, Problem 7.27P

Interpretation Introduction

(a)

Interpretation:

Power output of the expander and the temperature of the exhaust stream for the given set of operating conditions.

Concept Introduction:

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

Answer to Problem 7.27P

Power output of the expander: 2218.23 kW.

Temperature of exhaust stream: 186⁰C.

Explanation of Solution

Given Information:

An expander operates adiabatically with nitrogen. Following conditions are given:

$T_1$ =480⁰C, $P_1$ =6 bar, $\dot{n}$ = 200 mol/s, $P_2$ =1 bar, $\eta$ =0.8

Here, $P_1,T_1$are inlet pressure and temperature respectively and $\dot{n}$ is the molar flow rate. $P_2$ is the exhaust pressure and $\eta$ is the efficiency of the expander.

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

To calculate power output, specific enthalpy of nitrogen will be calculated at the given inlet and exhaust pressure and temperature condition respectively. Theoretical power will be calculated equal to the difference in specific enthalpy at inlet and exhaust condition respectively. Actual power output will be calculated by dividing theoretical power by the given expander efficiency. It is to be noted that molecular weight of nitrogen is 28.

Calculation:

At $T_1$ =480 ⁰C and $P_1$ = 6 bar inlet conditions for nitrogen,

Specific entropy, $S_1$ : 7.29 kJ/kg K

Specific enthalpy, $H_1$ : 484.75 kJ/kg

Exhaust pressure is 1 bar.

To have the same specific entropy, exhaust temperature, $T_2$ :186 ⁰C

At this exhaust temperature, enthalpy, $H_2$ : 167.86 kJ/kgTheoretical work for adiabatic expansion process, $\dot{W}$ = ( $H_2$ - $H_1$ ) kJ/kg

  $\begin{array}{l}\begin{array}{l}=\left(167.86-484.75\right)kJ/kg\\ =-316.89kJ/kg\\ =-316.89kJ/kg\times 28kg/kmol\end{array}\hfill \\ =-8872.92kJ/kmol\hfill \\ =-8872.92J/mol\hfill \end{array}$

Negative sign implies work done by expander.

Total theoretical output power requirement = molar flow rate x theoretical power

  $\begin{array}{l}=200mol/s\times 8872.92J/mol\hfill \\ =1774584J/s=1774584W=1774.58kW\hfill \end{array}$

Actual output power requirement = theoretical power / efficiency

  $\begin{array}{l}=\frac{1774.58}{0.8kW}\hfill \\ =2218.23kW\hfill \end{array}$

Interpretation Introduction

(b)

Interpretation:

Power output of the expander and the temperature of the exhaust stream for the given set of operating conditions.

Concept Introduction:

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

Answer to Problem 7.27P

Power output of the expander: 1451.75 kW.

Temperature of exhaust stream: 157 ⁰C.

Explanation of Solution

Given Information:

An expander operates adiabatically with nitrogen. Following conditions are given:

$T_1$ =400⁰C, $P_1$ =5 bar, $\dot{n}$ = 150mol/s, $P_2$ =1 bar, $\eta$ =0.75

Here, $P_1,T_1$are inlet pressure and temperature respectively and $\dot{n}$ is the molar flow rate. $P_2$ is the exhaust pressure and $\eta$ is the efficiency of the expander.

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

To calculate power output, specific enthalpy of nitrogen will be calculated at the given inlet and exhaust pressure and temperature condition respectively. Theoretical power will be calculated equal to the difference in specific enthalpy at inlet and exhaust condition respectively. Actual power output will be calculated by dividing theoretical power by the given expander efficiency. It is to be noted that molecular weight of nitrogen is 28.

Calculation:

At $T_1$ = 400⁰C and $P_1$ =5 bar inlet conditions for nitrogen,

Specific entropy, $S_1$ : 7.22 kJ/kg K

Specific enthalpy, $H_1$ :396.68 kJ/kg

Exhaust pressure is 1 bar.

To have the same specific entropy, exhaust temperature, $T_2$ :157⁰C

At this exhaust temperature, enthalpy, $H_2$ : 137.44 kJ/kg

Theoretical work for adiabatic expansion process, $\dot{W}$ = ( $H_2$ - $H_1$ ) kJ/kg

  $\begin{array}{l}\begin{array}{l}=\left(137.44-396.68\right)kJ/kg\\ =-259.24kJ/kg\\ =-259.24kJ/kg\times 28kg/kmol\end{array}\hfill \\ =-7258.72kJ/kmol\hfill \\ =-7258.72J/mol\hfill \end{array}$

Negative sign implies work done by expander.

Total theoretical output power requirement = molar flow rate x theoretical power

  $\begin{array}{l}=150mol/s\times 7258.72J/mol\hfill \\ =1088808J/s=1088808W=1088.81kW\hfill \end{array}$

Actual output power requirement = theoretical power / efficiency

  $\begin{array}{l}=\frac{1088.81}{0.75kW}\hfill \\ =1451.75kW\hfill \end{array}$

Interpretation Introduction

(c)

Interpretation:

Power output of the expander and the temperature of the exhaust stream for the given set of operating conditions.

Concept Introduction:

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

Answer to Problem 7.27P

Power output of the expander: 2176.73 kW.

Temperature of exhaust stream: 179 ⁰C.

Explanation of Solution

Given Information:

An expander operates adiabatically with nitrogen. Following conditions are given:

$T_1$ =500⁰C, $P_1$ =7 bar, $\dot{n}$ = 175mol/s, $P_2$ =1 bar, $\eta$ =0.78

Here, $P_1,T_1$are inlet pressure and temperature respectively and $\dot{n}$ is the molar flow rate. $P_2$ is the exhaust pressure and $\eta$ is the efficiency of the expander.

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

To calculate power output, specific enthalpy of nitrogen will be calculated at the given inlet and exhaust pressure and temperature condition respectively. Theoretical power will be calculated equal to the difference in specific enthalpy at inlet and exhaust condition respectively. Actual power output will be calculated by dividing theoretical power by the given expander efficiency. It is to be noted that molecular weight of nitrogen is 28.

Calculation:

At $T_1$ =500⁰C and $P_1$ =7 bar inlet conditions for nitrogen,

Specific entropy, $S_1$ : 7.273 kJ/kg K

Specific enthalpy, $H_1$ :507.01 kJ/kg

Exhaust pressure is 1 bar.

To have the same specific entropy, exhaust temperature, $T_2$ :179⁰C

At this exhaust temperature, enthalpy, $H_2$ : 160.51 kJ/kg

Theoretical work for adiabatic expansion process, $\dot{W}$ = ( $H_2$ - $H_1$ ) kJ/kg

  $\begin{array}{l}\begin{array}{l}=\left(160.51-507.01\right)kJ/kg\\ =-346.5kJ/kg\\ =-346.5kJ/kg\times 28kg/kmol\end{array}\hfill \\ =-9702kJ/kmol\hfill \\ =-9702J/mol\hfill \end{array}$

Negative sign implies work done by expander.

Total theoretical output power requirement = molar flow rate x theoretical power

  $\begin{array}{l}=175mol/s\times 9702J/mol\hfill \\ =1697850J/s=1697850W=1697.85kW\hfill \end{array}$

Actual output power requirement = theoretical power / efficiency

  $\begin{array}{l}=\frac{1697.85}{0.78}kW\hfill \\ =2176.73kW\hfill \end{array}$

Interpretation Introduction

(d)

Power output of the expander and the temperature of the exhaust stream for the given set of operating conditions.

Answer to Problem 7.27P

Power output of the expander: 816.51 kW.

Temperature of exhaust stream: 220 ⁰C.

Explanation of Solution

Given Information:

An expander operates adiabatically with nitrogen. Following conditions are given:

$T_1$ = 450⁰C, $P_1$ =8 bar, $\dot{n}$ = 100mol/s, $P_2$ =2 bar, $\eta$ =0.85

Here, $P_1,T_1$are inlet pressure and temperature respectively and $\dot{n}$ is the molar flow rate. $P_2$ is the exhaust pressure and $\eta$ is the efficiency of the expander.

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

To calculate power output, specific enthalpy of nitrogen will be calculated at the given inlet and exhaust pressure and temperature condition respectively. Theoretical power will be calculated equal to the difference in specific enthalpy at inlet and exhaust condition respectively. Actual power output will be calculated by dividing theoretical power by the given expander efficiency. It is to be noted that molecular weight of nitrogen is 28.

Calculation:

At $T_1$ = 450⁰C and $P_1$ = 8 bar inlet conditions for nitrogen,

Specific entropy, $S_1$ : 7.159 kJ/kg K

Specific enthalpy, $H_1$ :451.54 kJ/kg

Exhaust pressure is 1 bar.

To have the same specific entropy, exhaust temperature, $T_2$ :220⁰C

At this exhaust temperature, enthalpy, $H_2$ : 203.67 kJ/kg

Theoretical work for adiabatic expansion process, $\dot{W}$ = ( $H_2$ - $H_1$ ) kJ/kg

  $\begin{array}{l}\begin{array}{l}=\left(203.67-451.54\right)kJ/kg\\ =-247.87kJ/kg\\ =-247.87kJ/kg\times 28kg/kmol\end{array}\hfill \\ =-6940.36kJ/kmol\hfill \\ =-6940.36J/mol\hfill \end{array}$

Negative sign implies work done by expander.

Total theoretical output power requirement = molar flow rate x theoretical power

  $\begin{array}{l}=100mol/s\times 6940.36J/mol\hfill \\ =694036J/s=694036W=694.036kW\hfill \\ \hfill \end{array}$

Actual output power requirement = theoretical power / efficiency

  $\begin{array}{l}=\frac{694.036}{0.85kW}\hfill \\ =816.51kW\hfill \end{array}$

Interpretation Introduction

(e)

Interpretation:

Power output of the expander and the temperature of the exhaust stream for the given set of operating conditions.

Concept Introduction:

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

Answer to Problem 7.27P

Power output of the expander: 1973.86 Btu/s.

Temperature of exhaust stream: 352.13 ⁰F

Explanation of Solution

Given Information:

An expander operates adiabatically with nitrogen. Following conditions are given:

$T_1$ =900⁰F, $P_1$ =95 psia, $\dot{n}$ = 0.5 (lbmol)/s, $P_2$ =15 psia, $\eta$ =0.80

Here, $P_1,T_1$are inlet pressure and temperature respectively and $\dot{n}$ is the molar flow rate. $P_2$ is the exhaust pressure and $\eta$ is the efficiency of the expander.

Expansion through an expander is an isentropic process which implies entropy at the inlet and exhaust will be the same. For the given inlet pressure and temperature condition, specific entropy of nitrogen will be determined. Then exhaust temperature will be determined for the given outlet pressure condition.

To calculate power output, specific enthalpy of nitrogen will be calculated at the given inlet and exhaust pressure and temperature condition respectively. Theoretical power will be calculated equal to the difference in specific enthalpy at inlet and exhaust condition respectively. Actual power output will be calculated by dividing theoretical power by the given expander efficiency. It is to be noted that molecular weight of nitrogen is 28.

Calculation:

At $T_1$ = 900⁰F and $P_1$ = 95 psia inlet conditions for nitrogen,

Specific entropy, $S_1$ : 1.737 Btu/lbm⁰R

Specific enthalpy, $H_1$ : 209.55 Btu/lbm

Exhaust pressure is 15 psia.

To have the same specific entropy, exhaust temperature, $T_2$ :352.13⁰F= 178⁰C

At this exhaust temperature, enthalpy, $H_2$ : 68.56 Btu/lbm

Theoretical work for adiabatic expansion process, $\dot{W}$ = ( $H_2$ - $H_1$ ) kJ/kg

  $\begin{array}{l}\begin{array}{l}=\left(68.56-209.55\right)Btu/lbm=-140.99Btu/lbm\\ =-140.99Btu/lbm\times 28lbm/lbmol\end{array}\hfill \\ =-3947.72Btu/lbmol\hfill \end{array}$

Negative sign implies work done by expander.

Total theoretical output power requirement = molar flow rate x theoretical power

  $\begin{array}{l}\begin{array}{l}=0.5lbmol/s\times 3947.72Btu/lbmol\hfill \\ =1973.86Btu/s\hfill \end{array}\\ =2082.53kW\end{array}$

Actual output power requirement = theoretical power / efficiency

  $\begin{array}{l}=\frac{2082.53}{0.8kW}\hfill \\ =2603.16kW\hfill \\ \hfill \end{array}$