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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?

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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
2.6

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