# Question 2.SP.6: (a) Calculate the density, specific weight, and specific vol......

(a) Calculate the density, specific weight, and specific volume of oxygen at 100°F and 15 psia (pounds per square inch absolute; see Sec. 2.7). (b) What would be the temperature and pressure of this gas if we compressed it isentropically to 40% of its original volume? (c) If the process described in (b) had been isothermal, what would the temperature and pressure have been?

Step-by-Step
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Table A.5 for oxygen $\left(\mathrm{O}_{2}\right)$ : Molar mass M=32.0, k=1.40

(a) Sec. 2.7: $\quad R \approx \frac{R_{0}}{M}=\frac{49,709}{32.0}=1553 \mathrm{ft} \cdot \mathrm{lb} /\left(\mathrm{slug} \cdot{ }^{\circ} \mathrm{R}\right) \quad$ (as in Table A.5)

From Eq. (2.4): $\quad \rho=\frac{p}{R T}=\frac{15 \times 144 \mathrm{lb} / \mathrm{ft}^{2}}{\left[1553 \mathrm{ft} \cdot \mathrm{lb} /\left(\mathrm{slug} \cdot{ }^{\circ} \mathrm{R}\right)\right]\left[(460+100)^{\circ} \mathrm{R}\right]}$

$=0.00248 \mathrm{slug} / \mathrm{ft}^{3} \quad$

With $g=32.2 \mathrm{ft} / \mathrm{sec}^{2}, \gamma=\rho g=0.00248(32.2)=0.0800 \mathrm{lb} / \mathrm{ft}^{3} \quad$

Eq. (2.2): $\quad v=\frac{1}{\rho}=\frac{1}{0.00248}=403 \mathrm{ft}^{3} / \mathrm{slug}$

(b) Isentropic compression: $v_{2}=40 \% v_{1}=0.4(403)=161.1 \mathrm{ft}^{3} / \mathrm{slug}$

$\rho_{2}=1 / v_{2}=0.00621 \mathrm{slug} / \mathrm{ft}^{3}$

Eq. (2.6) with $n=k$ : $p v^{k}=(15 \times 144)(403)^{1.4}=\left(p_{2} \times 144\right)(161.1)^{1.4}$

\begin{aligned}& p_{2}=54.1 \text { psia } \end{aligned}

From Eq. (2.4): $\quad p_{2}=54.1 \times 144$ psia $=\rho R T=0.00621(1553)\left(460+T_{2}\right)$

$T_{2}=348^{\circ} \mathrm{F} \quad$

(c) Isothermal compression: $T_{2}=T_{1}=100^{\circ} \mathrm{F} \quad$

$p v=$ constant: $(15 \times 144)(403)=\left(p_{2} \times 144\right)(0.4 \times 403)$

$p_{2}=37.5 \mathrm{psia} \quad$

$v=\frac{1}{\rho}$          (2.2)

$\frac{p}{\rho}=p v=R T$       (2.4)

$p v^n=p_1 v_1^n=\text { constant }$      (2.6a)

$\frac{p}{p_1}=\left(\frac{\rho}{\rho_1}\right)^n=\text { constant }$      (2.6b)

 TABLE A.5   Physical properties of common gases at standard sea-level atmospheric pressure${ }^a$  . Gas Chemical formula Molar mass , $\mathbf{ M}$ Density $\rho$  , Absolute viscosity, ${}^ b$ Gas constant,  $\boldsymbol{R}$ Specific heat, $c_p \quad c_v$ Specific heat ratio,$k=c_p / c_v$ $\textbf { at } 68^{\circ} \mathbf{F}$ – slug/ slug-mol $\textbf { slug/ft }{ }^3$ $10^{-6} \mathbf{lb} \cdot \mathbf{sec}^2 \mathbf{ft}^2$ \begin{aligned}& \mathbf{f t} \cdot \mathbf{l b} /\left(\text { slug } \cdot{ }^{\circ} \mathbf{R}\right) \\& =\mathbf{f t}^2 /\left(\sec ^2 \cdot{ }^{\circ} \mathbf{R}\right)\end{aligned} \begin{aligned}& \mathbf{f t} \cdot \mathbf{l b} /\left(\text { slug } \cdot{ }^{\circ} \mathbf{R}\right) \\& =\mathbf{f t}^2 /\left(\sec ^2 \cdot{ }^{\circ} \mathbf{R}\right)\end{aligned} – Air carbon 28.96 0.00231 0.376 1,715 6,000 4,285 1.40 dioxide carbon $CO_2$ 44.01 0.00354 0.310 1,123 5,132 4,009 1.28 quad monoxide CO 28.01 0.00226 0.380 1,778 6,218 4,440 1.40 Helium He 4.003 0.000323 0.411 12,420 31,230 18,810 1.66 Hydrogen $H_2$ 2.016 0.000162 0.189 24,680 86,390 61,710 1.40 Methane $CH_4$ 16.04 0.00129 0.280 3,100 13,400 10,300 1.30 Nitrogen $N_2$ 28.02 0.00226 0.368 1,773 6,210 4,437 1.40 Oxygen $O_2$ 32.00 0.00258 0.418 1,554 5,437 3,883 1.40 Water vapor $H_2O$ 18.02 0.00145 0.212 2,760 11,110 8,350 1.33 $\textbf { at } 20^{\circ} \mathbf{C}$ – kg / kg-mol $\mathbf{kg} / \mathbf{m}^3$ $10^{-6} \mathbf{~N} \cdot \mathbf{s} / \mathbf{m}^2$ \begin{aligned}& \mathbf{N} \cdot \mathbf{m} /(\mathbf{kg} \cdot \mathbf{K}) \\& =\mathbf{m}^2 /\left(\mathbf{s}^2 \cdot \mathbf{K}\right)\end{aligned} \begin{aligned} & \mathbf{N} \cdot \mathbf{m} /(\mathbf{k g} \cdot \mathbf{K}) \\ & =\mathbf{m}^2 /\left(\mathbf{s}^2 \cdot \mathbf{K}\right) \end{aligned} – Air carbon 28.96 1.205 18.0 287 1003 716 1.40 dioxide carbon $CO_2$ 44.01 1.84 14.8 188 858 670 1.28 quad monoxide CO 28.01 1.16 18.2 297 1040 743 1.40 Helium He 4.003 0.166 19.7 2077 5220 3143 1.66 Hydrogen $H_2$ 2.016 0.0839 9.0 4120 14450 10330 1.40 Methane $CH_4$ 16.04 0.668 13.4 520 2250 1730 1.30 Nitrogen $N_2$ 28.02 1.16 17.6 297 1040 743 1.40 Oxygen $O_2$ 32.00 1.33 20.0 260 909 649 1.40 Water vapor $H_2O$ 18.02 0.747 10.1 462 1862 1400 1.33

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