Chapter 16, Expansion and flow processes Video Solutions, Thermodynamics: A complete undergraduate course

Problem 1

Use equation (16.3) to derive the relation between pressure and volume for an isentropic expansion of an ideal gas with $\gamma$ independent of temperature.

Sheh Lit Chang

University of Washington

02:37

Problem 2

Derive equations (16.1) and (16.2). Verify that for an ideal gas $\left.\frac{\partial T}{\partial V}\right|_U=0$ and $\left.\frac{\partial T}{\partial p}\right|_H=0$.

Mukesh Devi

Numerade Educator

02:37

A gas obeys the equation $p(V-b)=R T$ and has $C_V$ independent of temperature. Show that (a) the internal energy is a function of temperature only, (b) the ratio $\gamma=C_p / C_V$ is independent of temperature and pressure, (c) the equation of an adiabatic change has the form $p(V-b)^\gamma=$ constant.

Pritesh Ranjan

Numerade Educator

03:35

Problem 4

Show that the inversion curve for the Dieterici gas is given by $V=$ $4 /(8-T)$ in critical units [Hint: differentiate the equation of state and then replace $(\partial V / \partial T)_p$ by $\left.V / T\right]$. Hence obtain the inversion curve
$$
p=(8-T) \exp \left(\frac{5}{2}-\frac{4}{T}\right) \text {. }
$$

Linh Vu

Numerade Educator

02:31

Problem 5

Show that as $p \rightarrow 0$, the van der Waals inversion curve intercepts the $T$ axis at $T=2 a / R b$ and $T=2 a / 9 R b$ (i.e. 6.75 and 0.75 in critical units).

Harshita Goel

Numerade Educator

04:07

Problem 6

Show that, for a gas with second virial coefficient $B=a-b e^{c / T}$, where $a, b, c$ are constants, the maximum inversion temperature is a solution of the equation $a T=b(c+T) e^{c / T}$. Hence find the maximum inversion temperatures of helium and argon, for which $[a, b, c]=$ $[114.1,98.7,3.245 \mathrm{~K}]$ and $[154.2,119.3,105.1 \mathrm{~K}]$, respectively.

Mukesh Devi

Numerade Educator

13:22

Problem 7

Use equation (16.13) to roughly estimate the temperature change in a Joule-Kelvin expansion, as follows. First, assume we are comfortably in the region where cooling occurs, so the $b$ term can be neglected. Then the equation is easily integrated. Show that the ratio of final to initial temperature, after a pressure change $\Delta p$, is
$$
\frac{T_f}{T_i}=\sqrt{1+\frac{4 a \Delta p}{R C_p T_i^2}}
$$
where $\Delta p<0$ so $T_f<T_i$. Putting in values for a mole of nitrogen ( $a=0.137 \mathrm{Jm}^3, C_p=3.5 R$ ), find $T_f / T_i$ for a starting temperature of $300 \mathrm{~K}$ and a pressure drop of 100 atmospheres. (For a more accurate estimate, consult Figure 16.2.)

Sheh Lit Chang

University of Washington

02:06

Problem 8

Consider the Linde process for nitrogen, with parameters as in Figure 16.4. Assuming the gas leaves the cold end of the heat exchanger at $160 \mathrm{~K}$, trace the path of a small fixed mass of gas on a TS diagram such as Figure 16.3 as it passes once around the process. Estimate the heat leaving a unit mass of gas in (i) the compressor and (ii) the heat exchanger, and estimate the fraction that liquifies in the expansion.

Manik Pulyani

Numerade Educator

02:08

Problem 9

Roughly trace the path on a TS diagram of a small fixed mass of nitrogen passing several times around the Linde process, with the whole apparatus starting at $300 \mathrm{~K}$.

Sanjeev Kumar

Numerade Educator

06:42

Problem 10

At modest pressures the equation of state of a gas can be written as $p V=R T+B p$, where $B$ is a function of $T$ only. (i) Find an expression for the Joule-Kelvin inversion temperature in terms of $B$ and $\mathrm{d} B / \mathrm{d} T$. (ii) For helium gas between $5 \mathrm{~K}$ and $60 \mathrm{~K}, B$ can be represented as $B=m-(n / T)$, where $m=15.3 \times 10^{-6} \mathrm{~m}^3 \mathrm{~mole}^{-1}$ and $n=352 \times 10^{-6} \mathrm{~m}^3$ $\mathrm{K} \mathrm{mole}{ }^{-1}$. From this information, estimate the inversion temperature for helium. For an expansion starting below the inversion temperature, comment on the efficiency of the process (i.e. cooling per unit pressure change) as the temperature falls. (iii) In a helium liquifier, compressed helium gas at $14 \mathrm{~K}$ is fed to the expansion valve, where a fraction $x$ liquifies and the remaining fraction $(1-x)$ is rejected as gas at $14 \mathrm{~K}$ and atmospheric pressure. If the specific enthalpy of helium gas at $14 \mathrm{~K}$ is given by
$$
h=a+b\left(p-p_0\right)^2
$$
where $a=71.5 \mathrm{~kJ} \mathrm{~kg}^{-1}, b=1.3 \times 10^{-12} \mathrm{~kJ} \mathrm{~kg}^{-1} \mathrm{~Pa}^{-2}$, and $p_0=3.34 \mathrm{MPa}$ $(=33.4 \mathrm{~atm})$, and the specific enthalpy of liquid helium at atmospheric pressure is $10 \mathrm{~kJ} \mathrm{~kg}^{-1}$, determine what input pressure will allow $x$ to achieve a maximum value, and show that this maximum value is approximately 0.18 .

Khoobchandra Agrawal

Numerade Educator

03:17

Problem 11

Derive equation (16.15), first for a flow process with a single entrance and exit pipe, and then, more generally, for a chamber with several entrance and exit pipes containing possibly different fluids at different pressures.

Sheh Lit Chang

University of Washington

06:21

Problem 12

The working of many gas turbines can be modelled by the Brayton cycle, which consists of two adiabatic and two isobaric processes (cf. Figure 16.8). Show that if the working fluid is an ideal gas, then the thermal efficiency is $1-\left(p_1 / p_2\right)^{1-1 / \gamma}$. (Compare this with the Otto cycle, Eq. 11.2.)

Averell Hause

Carnegie Mellon University

01:27

Problem 13

Consider the Mollier or hs chart shown in Figure 16.7.
(i) Why do the isobars get steeper at high $T$ ? Is this a universal property of Mollier charts?
(ii) If the chart only showed one of $T$ or $p$ as a function of $h$ and $s$, then which would it be better to show?
(iii) Use the chart to find the heat capacity at constant pressure of water at $1 \mathrm{bar}, 50^{\circ} \mathrm{C}$ and of steam at $1 \mathrm{bar}, 200^{\circ} \mathrm{C}$.
Figure 16.7 can't copy

Carson Merrill

Numerade Educator

Problem 14

The Rankine cycle (Figure 16.9) is a model suitable for the heat engines widely used in power stations which employ a steam turbine. The cycle involves a working fluid which undergoes a phase change, and consequently allows large amounts of heat transfer without great changes in pressure, and for this reason it is also useful in refrigerator applications. There are four stages: compressor, boiler, turbine, condenser. The compressor raises the pressure of a liquid at approximately constant volume. Because not much volume change takes place, not much work needs to be supplied. This also raises the temperature slightly. Then the liquid is heated in a boiler at constant pressure. Its temperature rises to the boiling point, and then stays constant as large amounts of heat are supplied and there is a large increase in volume. Usually a complete phase change takes place. After this the fluid (now vapour) passes through a turbine which it drives in an approximately adiabatic expansion. Finally, the fluid is cooled at constant pressure by heat exchange so that its volume falls again. In early devices, condensation began in the turbine, but because water droplets can damage the turbine blades, modern power stations avoid this by using a high operating temperature at the input to the turbine, and also a further stage called reheating.
(i) Consider a Rankine cycle with water as the working fluid, as shown in Figure 16.9. Show that the work extracted per unit mass of water is $w=\left(h_c-h_d\right)-\left(h_b-h_a\right)$ and the heat input is $q_{\mathrm{in}}=h_c-h_b$, where $h$ is specific enthalpy.
(ii) Given that the pressure limits are $10 \mathrm{kPa}$ and $5 \mathrm{MPa}$, and the temperature limits are $45^{\circ} \mathrm{C}$ and $400^{\circ} \mathrm{C}$, find the thermal efficiency defined by $\eta=w / q_{\text {in }}$. [Use the $h s$ chart shown in Figure 16.7]
(iii) Calculate the flow rate required for an output power of $1 \mathrm{MW}$.
(iv) Calculate the volume of water vapour entering the turbine per second.
Figure 16.9 can't copy

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02:18

Problem 15

In a refrigerator based on the Rankine cycle, the working fluid goes around the cycle in the opposite direction to the one appropriate for power generation. Briefly describe the main components of such a refrigerator. Where is the work input? How is a net transfer of heat from a colder to a hotter body achieved?

Alexander Lorenzo

Numerade Educator

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