Question 11.SP.1: Air at 20°C is flowing through the pipe shown in Fig. S11.1,......

Air at 20°C is flowing through the pipe shown in Fig. S11.1, resulting in pressure gage readings of 70.2kPa at A and 71.1kPa at B. Atmospheric pressure is 684mmHg. (a) Find the air velocity u. (b) What is the largest pressure difference (kPa) between the two gages for which compressibility effects can be safely neglected?

S11.1
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Table A.5 for air: R=287 m2/(s2K)R=287 \mathrm{~m}^{2} /\left(\mathrm{s}^{2} \cdot \mathrm{K}\right)

(a)   pabs=pat+pgage=101.32(684/760)+70.2=161.4kPap_{\mathrm{abs}}=p_{\mathrm{at}}+p_{\mathrm{gage}}=101.32(684 / 760)+70.2=161.4 \mathrm{kPa}

Eq. (2.5):                  γ=gpRT=9.81(161.4)287(273+20)=0.01883kN/m3\quad \gamma=\frac{g p}{R T}=\frac{9.81(161.4)}{287(273+20)}=0.01883 \mathrm{kN} / \mathrm{m}^{3}

Eq. (11.6):                 u=2(9.81)(71.170.20.01883)=30.6 m/s\quad u=\sqrt{2(9.81)\left(\frac{71.1-70.2}{0.01883}\right)}=30.6 \mathrm{~m} / \mathrm{s} \quad

(b) Sec. 11.3, footnote 3: We need M<0.1\mathbf{M}<0.1

Sec. 5.4: M=V/c\mathbf{M}=V / c; at 20C,c=345 m/s20^{\circ} \mathrm{C}, c=345 \mathrm{~m} / \mathrm{s} [or use Eq. (13.15)].

So we need V<0.1c=umax=34.5 m/sV<0.1 c=u_{\max }=34.5 \mathrm{~m} / \mathrm{s}

From Eq. (11.5):     Δp=p0p=γu22g \Delta p=p_{0}-p=\gamma \frac{u^{2}}{2 g}

So                             Δpmax=γ(umax)22g=0.01883(34.5)22(9.81)=1.142kPa\Delta p_{\max }=\frac{\gamma\left(u_{\max }\right)^{2}}{2 g}=\frac{0.01883(34.5)^{2}}{2(9.81)}=1.142 \mathrm{kPa}

Note: This Δpmax\Delta p_{\max } compares with the observed Δp\Delta p of 71.170.2=0.9kPa71.1-70.2=0.9 \mathrm{kPa}. Therefore it was safe to neglect compressibility effects in part (a).

γ=gpRT \gamma=\frac{g p}{R T}       (2.5)

u=Cui=C2g(p0γpγ) u=C u_i=C \sqrt{2 g\left(\frac{p_0}{\gamma}-\frac{p}{\gamma}\right)}        (11.6)

c=kRT=kpρ c=\sqrt{k R T}=\sqrt{\frac{k p}{\rho}}        (13.15)

hL12=(1Cv21)V222g h_{L_{1-2}}=\left(\frac{1}{C_v^2}-1\right) \frac{V_2^2}{2 g}             (11.15)

TABLE A.5   Physical properties of common gases at standard sea-level atmospheric pressure a  { }^a   .
Gas Chemical formula Molar mass ,  M  \mathbf{ M} Density  ρ  \rho   , Absolute viscosity, b{}^ b Gas constant,   R  \boldsymbol{R} Specific heat,  cpcv  c_p \quad c_v Specific heat ratio,k=cp/cv k=c_p / c_v
  at 68F  \textbf { at } 68^{\circ} \mathbf{F}  slug/ slug-mol     slug/ft 3 \textbf { slug/ft }{ }^3 106lbsec2ft2 10^{-6} \mathbf{lb} \cdot \mathbf{sec}^2 \mathbf{ft}^2  ftlb/( slug R)=ft2/(sec2R)  \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} ftlb/( slug R)=ft2/(sec2R) \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 CO2 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 H2 H_2 2.016 0.000162 0.189 24,680 86,390 61,710 1.40
Methane  CH4  CH_4 16.04 0.00129 0.280 3,100 13,400 10,300 1.30
Nitrogen  N2  N_2 28.02 0.00226 0.368 1,773 6,210 4,437 1.40
Oxygen O2 O_2 32.00 0.00258 0.418 1,554 5,437 3,883 1.40
Water vapor  H2O  H_2O 18.02 0.00145 0.212 2,760 11,110 8,350 1.33
  at 20C  \textbf { at } 20^{\circ} \mathbf{C} kg / kg-mol  kg/m3  \mathbf{kg} / \mathbf{m}^3  106 Ns/m2 10^{-6} \mathbf{~N} \cdot \mathbf{s} / \mathbf{m}^2   Nm/(kgK)=m2/(s2K)  \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}  Nm/(kgK)=m2/(s2K) \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 CO2 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 H2 H_2 2.016 0.0839 9.0 4120 14450 10330 1.40
Methane  CH4  CH_4 16.04 0.668 13.4 520 2250 1730 1.30
Nitrogen  N2  N_2 28.02 1.16 17.6 297 1040 743 1.40
Oxygen O2 O_2 32.00 1.33 20.0 260 909 649 1.40
Water vapor  H2O  H_2O 18.02 0.747 10.1 462 1862 1400 1.33

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