Introduction to Chemical Engineering Thermodynamics

J. M. Smith, Hendrick C Van Ness, Michael Abbott, Hendrick Van Ness

Chapter 7 APPLICATIONS OF THERMODYNAMICS TO FLOW PROCESSES - all with Video Answers

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Chapter Questions

Problem 1

Air expands adiabatically through a nozzle from a negligible initial velocity to a final velocity of $325 \mathrm{~m} \mathrm{~s}^{-1}$. What is the temperature drop of the air, if air is assumed to be an ideal gas for which $C_P=(7 / 2) R$ ?

Prashant Bana

Numerade Educator

01:57

Problem 2

In Ex. 7.5 an expression is found for the Joule/Thomson coeficient, $\mu=(\partial T / \partial P)_H$, that relates it to a heat capacity and equation-of-state information. Develop similar expressions for the derivatives:
(a) $(\partial T / \partial P)_S ;(b)(\partial T / \partial V)_U$.
What can you say about the signs of these derivatives? For what types of processes might these derivatives be important characterizing quantities?

Dan Ni

Numerade Educator

01:22

Problem 3

The thermodynamic sound speed $c$ is defined in Sec. 7.1. Prove that:
$$
c=\sqrt{\frac{V C_P}{\mathcal{M} C_{V K}}}
$$
where $V$ is molar volume and $\mathcal{M}$ is molar mass. To what does this general result reduce for: (a) An ideal gas? (b) An incompressible liquid? What do these results suggest qualitatively about the speed of sound in liquids relative to gases?

Michael Mackenzie

Numerade Educator

02:04

Problem 4

Steam enters a nozzle at $800 \mathrm{kPa}$ and $553.15 \mathrm{~K}\left(280^{\circ} \mathrm{C}\right)$ at negligible velocity and discharges at a pressure of $525 \mathrm{kPa}$. Assuming isentropic expansion of the steam in the nozzle, what is the exit velocity and what is the cross-sectional area at the nozzle exit for a flow rate of $0.75 \mathrm{~kg} \mathrm{~s}^{-1}$ ?

Naman Kumar

Numerade Educator

02:04

Problem 5

Steam enters a converging nozzle at $800 \mathrm{kPa}$ and $553.15 \mathrm{~K}\left(280^{\circ} \mathrm{C}\right)$ with negligible velocity. If expansion is isentropic, what is the minimum pressure that can be reached in such a nozzle and what is the cross-sectional area at the nozzle throat at this pressure for a flow rate of $0.75 \mathrm{~kg} \mathrm{~s}^{-1}$ ?

Naman Kumar

Numerade Educator

01:19

Problem 6

A gas enters a converging nozzle at pressure $P_1$ with negligible velocity, expands isentropically in the nozzle, and discharges into a chamber at pressure $P_2$. Sketch graphs showing the velocity at the throat and the mass flowrate as functions of the pressure ratio $P_2 / P_1$.

Manik Pulyani

Numerade Educator

02:33

Problem 7

For a convergingldiverging nozzle with negligibleentrance velocity in which expansion is isentropic, sketch graphs of mass flowratem, velocity $u$, and area ratio $A / A_1$ vs. the pressure ratio $P / P_1$. Here, A is the cross-sectional area of the nozzle at the point in the nozzle where the pressure is $P$, and subscript $l$ denotes the nozzle entrance.

Chai Santi

Numerade Educator

01:19

Problem 8

An ideal gas with constant heat capacities enters a convergingldiverging nozzle with negligible velocity. If it expands isentropically within the nozzle, show that the throat velocity is given by:
$$
u_{\text {throat }}^2=\frac{\gamma R T_1}{M}\left(\frac{2}{\gamma+1}\right)
$$
where $T_1$ is the temperature of the gas entering the nozzle, $\mathrm{M}$ is the molar mass, and $\mathrm{R}$ is the molar gas constant.

Manik Pulyani

Numerade Educator

03:20

Problem 9

Steam expands isentropically in a convergingldiverging nozzle from inlet conditions of $1400 \mathrm{kPa}, 598.15 \mathrm{~K}\left(325^{\circ} \mathrm{C}\right)$, and negligible velocity to a discharge pressure of $140 \mathrm{kPa}$. At the throat the cross-sectional area is $6 \mathrm{~cm}^2$. Determine the mass flowrate of the steam and the state of the steam at the exit of the nozzle.

Saurabh Kumar Gupta

Numerade Educator

04:08

Problem 10

Steam expands adiabatically in a nozzle from inlet conditions of 9 bar, $488.15 \mathrm{~K}\left(215^{\circ} \mathrm{C}\right)$, and a velocity of $70 \mathrm{~m} \mathrm{~s}^{-1}$ to a discharge pressure of 2.4 bar where its velocity is $609.6 \mathrm{~m} \mathrm{~s}^{-1}$. What is the state of the steam at the nozzle exit, and what is $\dot{S}_{G, \text { total }}$ for the process?

Arun Bana

Numerade Educator

05:43

Problem 11

Air discharges from an adiabatic nozzle at $288.15 \mathrm{~K}\left(15^{\circ} \mathrm{C}\right)$ with a velocity of $580 \mathrm{~m} \mathrm{~s}^{-1}$. What is the temperatureat the entrance of the nozzle if the entrance velocity is negligible? Assume air to be an ideal gas for which $C_P=(7 / 2) R$.

Prashant Bana

Numerade Educator

01:19

Problem 12

Cool water at $288.15 \mathrm{~K}\left(15^{\circ} \mathrm{C}\right)$ is throttled from $5 \mathrm{~atm}$ to $1 \mathrm{~atm}$, as in a kitchen faucet. What is the temperature change of the water? What is the lost work per kilogram of water for this everyday household happening? At $288.15 \mathrm{~K}\left(15^{\circ} \mathrm{C}\right)$ and $1 \mathrm{~atm}$, the volume expansivity $\beta$ for liquid water is about $1.5 \times 10^{-4} \mathrm{~K}^{-1}$. The surroundings temperature $T_\sigma$ is $293.15 \mathrm{~K}\left(20^{\circ} \mathrm{C}\right)$. State carefully any assumptions you make. The steam tables are a source of data.

Narayan Hari

Numerade Educator

01:50

Problem 13

A gas at upstream conditions $\left(T_1, P_1\right)$ is throttled to a downstream pressure of 1.2 bar. Use the Redlich/Kwong equation to estimate the downstream temperature and $A S$ of the gas for one of the following:
(a) Carbon dioxide, with $T_1=350 \mathrm{~K}$ and $P_1=80 \mathrm{bar}$.
(b) Ethylene, with $T_1=350 \mathrm{~K}$ and $P_1=60 \mathrm{bar}$.
(c) Nitrogen, with $T_1=250 \mathrm{~K}$ and $P_1=60 \mathrm{bar}$.
(d) Propane, with $T_1=400 \mathrm{~K}$ and $P_1=20$ bar.

Anand Jangid

Numerade Educator

01:55

Problem 14

A gas at upstream conditions given by one of the parts of $\mathrm{Pb} .7 .13$ is throttled to a pressure of $1.2 \mathrm{bar}$. Use the Soave/Redlich/Kwong equation to estimate the downstream temperature and $A S$ of the gas.

Manik Pulyani

Numerade Educator

01:50

Problem 15

A gas at upstreamconditions given by one of the parts of $\mathrm{Pb}, 7.13$ is throttled to a pressure of 1.2 bar. Use the Peng/Robinson equation to estimate the downstream temperatureand $A S$ of the gas.

Anand Jangid

Numerade Educator

04:30

Problem 16

For a pressure-explicitequation of state, prove that the Joule/Thompson inversion curve is the locus of states for which:
$$
T\left(\frac{\partial Z}{\partial T}\right)_\rho=\rho\left(\frac{\partial Z}{\partial \rho}\right)_T
$$
Apply this equation to (a) the van der Waals equation; (b) the Redlich/Kwong equation. Discuss the results.

Mahnoor Amin

Numerade Educator

04:10

Problem 17

Two nonconducting tanks of negligible heat capacity and of equal volume initially contain equal quantities of the same ideal gas at the same T and P. Tank A discharges to the atmosphere through a small turbine in which the gas expands isentropically; $\operatorname{tank} B$ dischargesto the atmosphere through a porous plug. Both devices operate until discharge ceases.
(a) When discharge ceases, is the temperature in tank A less than, equal to, or greater than the temperature in tank B?
(b) When the pressures in both tanks have fallen to half the initial pressure, is the temperature of the gas discharging from the turbine less than, equal to, or greater than the temperature of the gas discharging from the porous plug?
(c) During the discharge process, is the temperature of the gas leaving the turbine less than, equal to, or greater than the temperature of the gas leaving tank A at the same instant?
(d) During the discharge process, is the temperature of the gas leaving the porous plug less than, equal to, or greater than the temperature of the gas leaving tank B at the same instant?
(e) When discharge ceases, is the mass of gas remaining in tank A less than, equal to, or greater than the mass of gas remaining in tank B?

Eileen Sullivan

Numerade Educator

02:15

Problem 18

A steam turbine operates adiabatically at a power level of $3500 \mathrm{~kW}$. Steam enters the turbine at $2400 \mathrm{kPa}$ and $773.15 \mathrm{~K}\left(500^{\circ} \mathrm{C}\right)$ and exhausts from the turbine as saturated vapor at $20 \mathrm{kPa}$. What is the steam rate through the turbine, and what is the turbine efficiency?

Eric Mockensturm

Numerade Educator

01:19

Problem 19

A turbine operates adiabatically with superheated steam entering at $T_1$ and $P_1$ with a mass flow rate $\dot{m}$. The exhaust pressure is $P_2$ and the turbine efficiency is $\eta$. For one of the following sets of operating conditions, determine the power output of the turbine and the enthalpy and entropy of the exhaust steam.
(a) $T_1=723.15 \mathrm{~K}\left(450^{\circ} \mathrm{C}\right), P_1=8000 \mathrm{kPa}, \dot{m}=80 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=30 \mathrm{kPa}, \eta=0.80$.
(b) $T_1=823.15 \mathrm{~K}\left(550^{\circ} \mathrm{C}\right), P_{\mathrm{I}}=9000 \mathrm{kPa}, \dot{m}=90 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=20 \mathrm{kPa}, \eta=0.77$.
(c) $T_1=873.15 \mathrm{~K}\left(600^{\circ} \mathrm{C}\right), P_1=8600 \mathrm{kPa}, m=70 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=10 \mathrm{kPa}, \eta=0.82$.
(d) $T_1=673.15 \mathrm{~K}\left(400^{\circ} \mathrm{C}\right), P_1=7000 \mathrm{kPa}, \dot{m}=65 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=50 \mathrm{kPa}, \eta=0.75$.
(e) $T_1=473.15 \mathrm{~K}\left(200^{\circ} \mathrm{C}\right), P_1=1400 \mathrm{kPa}, \dot{m}=50 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=200 \mathrm{kPa}, \eta=0.75$.
(f) $T_1=755.15 \mathrm{~K}\left(482^{\circ} \mathrm{C}\right), P_1=75.8 \mathrm{bar}, \dot{m}=68 \mathrm{~kg} \mathrm{~s}^{-1}, \quad P_2=0.14 \mathrm{bar}, \eta=0.80$.
(g) $T_1=700.15 \mathrm{~K}\left(427^{\circ} \mathrm{C}\right), P_1=69$ bar, $\quad \dot{m}=45.4 \mathrm{~kg} \mathrm{~s}^{-1}, P_2=0.28 \mathrm{bar}, \eta=0.75$.

Manik Pulyani

Numerade Educator

03:18

Problem 20

Nitrogen gas initially at 8.5 bar expands isentropically to 1 bar and $423.15 \mathrm{~K}\left(150^{\circ} \mathrm{C}\right)$. Assuming nitrogen to be an ideal gas, calculate the initial temperature and the work produced per mole of nitrogen.

Keshav Singh

Numerade Educator

06:30

Problem 21

Combustion products from a burner enter a gas turbine at 10 bar and $1223.15 \mathrm{~K}\left(950^{\circ} \mathrm{C}\right)$ and discharge at 1.5 bar. The turbine operates adiabatically with an efficiency of $77 \%$. Assuming the combustion products to be an ideal-gas mixture with a heat capacity of $32 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}$, what is the work output of the turbine per mole of gas, and what is the temperature of the gases discharging from the turbine?

Eric Mockensturm

Numerade Educator

04:34

Problem 22

Isobutane expands adiabatically in a turbine from $5000 \mathrm{kPa}$ and $523.15 \mathrm{~K}\left(250^{\circ} \mathrm{C}\right)$ to $500 \mathrm{kPa}$ at the rate of $0.7 \mathrm{kmol} \mathrm{s}^{-1}$. If the turbine efficiency is 0.80 , what is the power output of the turbine and what is the temperature of the isobutane leaving the turbine?

Rashmi Sinha

Numerade Educator

04:08

Problem 23

The steam rate to a turbine for variable output is controlled by a throttle valve in the inlet line. Steam is supplied to the throttle valve at $1700 \mathrm{kPa}$ and $498.15 \mathrm{~K}\left(225^{\circ} \mathrm{C}\right)$. During a test run, the pressure at the turbine inlet is $1000 \mathrm{kPa}$, the exhaust steam at $10 \mathrm{kPa}$ has a quality of 0.95 , the steam flow rate is $0.5 \mathrm{~kg} \mathrm{~s}^{-1}$, and the power output of the turbine is $180 \mathrm{~kW}$.
(a) What are the heat losses from the turbine?
(b) What would be the power output if the steam supplied to the throttle valve were expanded isentropically to the final pressure?

Hariprasad Annamalai

Numerade Educator

01:51

Problem 24

Carbon dioxide gas enters an adiabatic expander at 8 bar and $673.15 \mathrm{~K}\left(400^{\circ} \mathrm{C}\right)$ and discharges at $1 \mathrm{bar}$. If the turbine efficiency is 0.75 , what is the discharge temperature and what is the work output per mole of $\mathrm{CO}_2$ ? Assume $\mathrm{CO}_2$ to be an ideal gas at these conditions.

Hast Aggarwal

Numerade Educator

05:11

Problem 25

Tests on an adiabatic gas turbine (expander) yield values for inlet conditions $\left(T_1, P_1\right)$ and outlet conditions $\left(T_2, P_2\right)$. Assuming ideal gases with constant heat capacities, determine the turbine efficiency for one of the following:
(a) $T_1=500 \mathrm{~K}, P_1=6 \mathrm{bar}, T_2=371 \mathrm{~K}, P_2=1.2 \mathrm{bar}, C_P / R=7 / 2$.
(b) $T_1=450 \mathrm{~K}, P_1=5 \mathrm{bar}, T_2=376 \mathrm{~K}, P_2=2 \mathrm{bar}, C_P / R=4$.
(c) $T_1=525 \mathrm{~K}, P_1=10 \mathrm{bar}, T_2=458 \mathrm{~K}, P_2=3 \mathrm{bar}, C_P / R=11 / 2$.
(d) $T_1=475 \mathrm{~K}, P_1=7 \mathrm{bar}, T_2=372 \mathrm{~K}, P_2=1.5 \mathrm{bar}, C_P / R=9 / 2$.
(e) $T_1=550 \mathrm{~K}, P_1=4 \mathrm{bar}, T_2=403 \mathrm{~K}, P_2=1.2 \mathrm{bar}, C_P / R=5 / 2$.

James Kiss

Numerade Educator

01:39

Problem 26

The efficiency of a particular series of adiabatic gas turbines (expanders) correlates with power output according to the empirical expression:
$$
\eta=0.065+0.080 \ln |\dot{W}|
$$

Here, $|\boldsymbol{W}|$ is the absolute value of the actual power output in $\mathrm{kW}$. Nitrogen gas is to be expanded from inlet conditions of $550 \mathrm{~K}$ and 6 bar to an outlet pressure of $1.2 \mathrm{bar}$. For a molar flowrate of $175 \mathrm{~mol} \mathrm{~s}^{-1}$, what is the delivered power in $\mathrm{kW}$ ? What is the efficiency of the turbine? What is the rate of entropy generation $\dot{S}_G$ ? Assume nitrogen to be an ideal gas with $C_P=(7 / 2) R$.

Manik Pulyani

Numerade Educator

05:34

Problem 27

A turbine operates adiabatically with superheated steam entering at 45 bar and 673.15 $\mathrm{K}\left(400^{\circ} \mathrm{C}\right)$. If the exhaust steam must be "dry," what is the minimum allowable exhaust pressure for a turbine efficiency, $\eta=0.75$ ? Suppose the efficiency were 0.80 . Would the minimum exhaust pressure be lower or higher? Why?

Vipender Yadav

Numerade Educator

06:05

Problem 28

Turbines can be used to recover energy from high-pressure liquid streams. However, they are not used when the high-pressure stream is a saturated liquid. Why? Illustrate by determining the downstream state for isentropic expansion of saturated liquid water at 5 bar to a final pressure of 1 bar.

Sanu Kumar

Numerade Educator

02:01

Problem 29

Liquid water enters an adiabatic hydroturbine at $5 \mathrm{~atm}$ and $288.15 \mathrm{~K}\left(15^{\circ} \mathrm{C}\right)$, and exhausts at $1 \mathrm{~atm}$. Estimate the power output of the turbine in $\mathrm{J} \mathrm{kg}^{-1}$ of water if its efficiency is $\eta=0.55$. What is the outlet temperature of the water? Assume water to be an incompressible liquid.

Eric Mockensturm

Numerade Educator

05:11

Problem 30

An expander operates adiabatically with nitrogen entering at $T_1$ and $P_1$ with a molar flow rate A. The exhaust pressure is $P_2$, and the expander efficiency is $\eta$. Estimate the power output of the expander and the temperature of the exhaust stream for one of the following sets of operating conditions.
(a) $T_1=753.15 \mathrm{~K}\left(480^{\circ} \mathrm{C}\right), P_1=6 \mathrm{bar}$, $\dot{n}=0.2 \mathrm{kmol} \mathrm{s}^{-1}, \quad P_2=1 \mathrm{bar}, \eta=0.80$.
(b) $T_1=673.15 \mathrm{~K}\left(400^{\circ} \mathrm{C}\right), P_1=5 \mathrm{bar}$, $\mathrm{A}=0.15 \mathrm{kmol} \mathrm{s}^{-1}, P_2=1 \mathrm{bar}, \eta=0.75$.
(c) $T_1=773.15 \mathrm{~K}\left(500^{\circ} \mathrm{C}\right), P_1=7$ bar, $\quad \mathrm{A}=0.175 \mathrm{kmol} \mathrm{s}^{-1}, P_2=\mathrm{I}$ bar, $\eta=0.78$.
(d) $T_1=723.15 \mathrm{~K}\left(450^{\circ} \mathrm{C}\right), P_1=8 \mathrm{bar}, \quad \dot{n}=0.1 \mathrm{kmol} \mathrm{s}^{-1}, \quad P_2=2 \mathrm{bar}, \eta=0.85$.
(e) $T_1=755.15 \mathrm{~K}\left(482^{\circ} \mathrm{C}\right), P_1=6.55 \mathrm{bar}, \dot{n}=0.23 \mathrm{kmol} \mathrm{s}^{-1}, P_2=1.03 \mathrm{bar}, \eta=0.80$.

James Kiss

Numerade Educator

00:58

Problem 31

What is the ideal-work rate for the expansion process of Ex. 7.6? What is the thermodynamic efficiency of the process? What is the rate of entropy generation $\hat{S}_G$ ? What is $\dot{W}_{\text {lost }}$ ? Take $T_\sigma=300 \mathrm{~K}$.

Alex Bretton

Numerade Educator

11:45

Problem 32

Exhaust gas at $673.15 \mathrm{~K}\left(400^{\circ} \mathrm{C}\right)$ and 1 bar from internal-combustionengines flows at the rate of $125 \mathrm{~mol} \mathrm{~s}^{-1}$ into a waste-heat boiler where saturated steam is generated at a pressure of $1200 \mathrm{kPa}$. Water enters the boiler at $293.15 \mathrm{~K}\left(20^{\circ} \mathrm{C}\right)\left(T_\sigma\right)$, and the exhaust gases are cooled to within $10 \mathrm{~K}\left(10^{\circ} \mathrm{C}\right)$ of the steam temperature. The heat capacity of the exhaust gases is $C_P / \mathrm{R}=3.34+1.12 \times 10^{-3} \mathrm{~T} / \mathrm{K}$. The steam flows into an adiabatic turbine and exhausts at a pressure of $25 \mathrm{kPa}$. If the turbine efficiency $\eta$ is $72 \%$,
(a) What is $\dot{W}_s$, the power output of the turbine?
(b) What is the thermodynamic efficiency of the boilerlturbine combination?
(c) Determine $\hat{S}_G$ for the boiler and for the turbine.
(d) Express $\dot{W}_{\text {lost }}$ (boiler) and $\dot{W}_{\text {lost }}$ (turbine) as fractions of $\mid \dot{W}_{\text {ideal }}$, the ideal work of the process.

Eric Mockensturm

Numerade Educator

04:13

Problem 33

A small adiabatic air compressor is used to pump air into a $20-\mathrm{m}^3$ insulated tank. The tank initially contains air at $298.15 \mathrm{~K}\left(25^{\circ} \mathrm{C}\right)$ and $101.33 \mathrm{kPa}$, exactly the conditions at which air enters the compressor. The pumping process continues until the pressure in the tank reaches $1000 \mathrm{kPa}$. If the process is adiabatic and if compression is isentropic, what is the shaft work of the compressor? Assume air to be an ideal gas for which $C_P=(7 / 2) R$ and $C_V=(5 / 2) R$.

Narayan Hari

Numerade Educator

03:49

Problem 34

Saturated steam at $125 \mathrm{kPa}$ is compressed adiabatically in a centrifugal compressor to $700 \mathrm{kPa}$ at the rate of $2.5 \mathrm{~kg} \mathrm{~s}^{-1}$. The compressor efficiency is $78 \%$. What is the power requirement of the compressor and what are the enthalpy and entropy of the steam in its final state?

Rubeena Zulfiqar

Numerade Educator

Problem 35

A compressor operates adiabatically with air entering at $T_1$ and $P_1$ with a molar flow rate A. The discharge pressure is $P_2$ and the compressor efficiency is $\eta$. Estimate the power requirement of the compressor and the temperature of the discharge stream for one of the following sets of operating conditions.
(a) $T_1=298.15 \mathrm{~K}\left(25^{\circ} \mathrm{C}\right), P_1=101.33 \mathrm{kPa}, \dot{n}=0.1 \mathrm{kmol} \mathrm{s}^{-1}, P_2=375 \mathrm{kPa}, \eta=0.75$.
(b) $T_1=353.15 \mathrm{~K}\left(80^{\circ} \mathrm{C}\right), \quad P_1=375 \mathrm{kPa}, \mathrm{A}=0.1 \mathrm{kmol} \mathrm{s}^{-1}, P_2=1000 \mathrm{kPa}, \eta=0.70$.
(c) $T_1=303.15 \mathrm{~K}\left(30^{\circ} \mathrm{C}\right), P_1=100 \mathrm{kPa}, A=0.15 \mathrm{kmol} \mathrm{s}^{-1}, P_2=500 \mathrm{kPa}, \eta=0.80$.
(d) $T_1=373.15 \mathrm{~K}\left(100^{\circ} \mathrm{C}\right), P_1=500 \mathrm{kPa}, \dot{n}=0.05 \mathrm{kmol} \mathrm{s}^{-1}, P_2=1300 \mathrm{kPa}, \eta=0.75$.
(e) $T_1=300.15 \mathrm{~K}\left(27^{\circ} \mathrm{C}\right), P_1=1.01 \mathrm{bar}, \dot{n}=0.23 \mathrm{kmol} \mathrm{s}^{-1}, P_2=3.8 \mathrm{bar}, \eta=0.75$.
(f) $T_1=339.15 \mathrm{~K}\left(66^{\circ} \mathrm{C}\right), P_1=3.8 \mathrm{bar}, \mathrm{A}=0.23 \mathrm{kmol} \mathrm{s}^{-1}, P_2=9.3 \mathrm{bar}, \eta=0.70$.

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