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Lecture_Notes_11_21

# Lecture_Notes_11_21 - One-dimensional isentropic ﬂow The...

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Unformatted text preview: One-dimensional isentropic ﬂow The previous relations apply, but conservation of matter has not yet been used. For duct ﬂow ' (,ouA)1 = (pa/02 = m = constant m=puA=LuA= “ we MP/Po 719A A RT yRT 4/7RT _ 4/er JT/TO . _ y+l - m: gpoAM(1+—7“1M2)2(7‘D m 2 The results from this relation are easier to put into a table by using a reference state which is the star-state Where M=l. y+1 . [9014* (7/ +1)_2(7—1) a? Dividing this equation into the previous one ' 7+1 — 2—1 €21,2(1+y‘1M2)<7> A My+l 2 Where, A = A* for M=l m: 1 R 6W4 wwwwwwwwww wwé It is seen that -A is large for small M. W“ f,» —A decreases to a minimum A : 14* at M21. —A increases as M increases Mach ’ greater than M=l. For a given area there are two solutions, a subsonic (M<l) ﬂow and a supersonic ﬂow Where M>l. Example For the conditions given atsection 1, ﬁnd the mach number, velocity and pressure at section 2. A1 = 37.5cm2 W A2 = 25cm2 , M1: 0.3 f 4 T1 :100C W [91 = SOkPa ' With = 0.3 the isentropic ﬂow table gives ‘4; = 2.035, I1— : 0.9823 and ﬂ = 0.9395 A . 76 Po ' Then 24* =2 A1 /2.035 = 37.5/2.035 = 18.42 A2 25 * =—=1.357 A 18.42 And With‘AZ/A* 21.36 5 = 0.954 and ﬂ = 0.848 ’ g . To P0 To = 11/9823 = 283/.9823 = 288.1 ' the table gives M2 = 0.49, T2 2 %T0 ‘= .954(288.1) = 274.8 0 . a2 = JyRTZ = ./1.4(287)(274.8) = 332m/S V2 : M2012 = =163m/S p0 = p1/0.9395 = 50/09395 = 53.22 p2 = @190 = .848(53.22) = 45.1kPa Po 9 ~ Flow changes with area change dp = —deV = azdp a = W/a’p/a’,0 for s = constant m = pVA —> AVdp+deA+pAdV= O .pVa’V AV(—- az j+deA+pAdV = 0 1/2 dV dA dV dV 1 dA —2 + + =0 —= 2 — a V A V V M—lA M<1 lM>1 dA>0 dV<0, dp>0 I dV>0, dp<0 dA<0 I dV>0, dp<0 l dV<0, dp>0 { For M = 1, dA=O for ﬁnite dV M = at a throat dA<0 dA>0 M<1 . M>1 dV>O dV>O dA<O dA>0 M>1 ' M<1 dV<O dV<0 Converging Nozzle ﬂow - * Pb p0 :2 2a;\ To = 200C e ' A8 = 25cm2 a) Find the ﬂow rate for pb 21.5mm. Take pe 2 pl, . 36— : is = 0.75 —> Me = 0.654, Ze— = 0.926 p0 2.0 T0 1;:9261‘0 = 271% aé = m ’= 330m/S Ve = Meae = .654(330) = 216m/s __ p _1.5(101000) _ R]; _ , 287(271) m : peAeVe :1.9S(25x10_4)(216)21.05kg/S pg 21.95kg/m3 b) Pressure pb to just Cause the maximum ﬂow rate; , Maximum ﬂow rate occurs when Me = l. pe 2 pb Be. 2 0,528, pe = 528(2) =1.056az‘m For Me :11. i0 . Fe = 0.833, I; = .833(293) = 244% 0 a6: yRTe =313m/S VezMeaezl.O(3l3)=3l3m/S I p 21056001000) R2; 287(244) . m : pé'AeVe = 21.19am pe =1.’52kg/m3 For pb <1.056az‘m, pe > pb and there are expansion waves in the jet and M. is still unity at the nozzle eXit. Nozzle is choked. ConVerging—diverging nozzle ﬂow V 190 = 2az‘m e To = 200C p At =12.5cm2 A6 = 25cm2 P0 6 21) Find V6 and Mt, p. if 1952192100. . pe = pb lie. 2 Q = 0.95 —> Me 20.27, 5- = 0.985, A: = 2.24 p0 2.0 To A Te = .9852‘0 = 2880K ae = 77122; _= 340m/S Ve = Meae = 27(340) 2. 91.8m/S 24*: A6 =11.16 —>A’=-1—2—'——5—~:1.12 2.24 A* 11.16 ' gives Mr E 0.66, [917120 2 0.747 pl. 2 .747(2) :1.49az‘m b)’Back pressure pb to just cause sonic ﬂow at the throat. For Ml. =1 —>At =A* 212.5 14:: i: 2 —’>Me '= 0.305, &= 0.937 A 12.5 . ' P0 pb z pe 2.937190 2.937(2) 21.87am c) Back pressure to cause smooth supersonic ﬂow in the nozzle ,andjet. Mt21 ——>At:A*=12.5 ’ A: '= 2—5: 2 —> Me = 2.20, & = 0.124 pl, 2 p6 2.1124190 : .124(2) 2 0.248az‘m Continued 2.50 2.64 0.059 0.132 0.444 1.74 1.38 0.191 0.306 0.623 1.76 1.40 0.185 0.300 0.617 2.52 2.69 0.057 0.129 0.441 1.78 1.42 0.179 0.293 0.612 2.54 2.74 0.055 0.126 0.437 1.80 1.44 0.174 0.287 0.607 2.56 2.79 0.053 0.123 0.433 1.82 1.46 0.169 0.281 0.602 2.58 2.84 0.052 0.121 0.429 1.84 1.48 0.164 0.275 0.596 2.60 2.90 0.050 0.118 0.425 1.86 1.51 0.159 0.269 0.591 2.62 2.95 0.049 0.115 0.421 1.88 1.53 0.154 0.263 0.586 2.64 3.01 0.047 0.113 0.418 :8 1.90 1.56 0.149 0.257 0.581 2.66 3.06 0.046 0.110 0.414 1.92 1.58 0.145 0.251 0.576 2.68 3.12 0.044 0.108 0.410 1.94 1.61 0.140 0.246 0.571 2.70 3.18 0.043 0.106 0.407 1.96 1.63 0.136 0.240 0.566 2.72 3.24 0.042 0.103 0.403 1.98 1.66 0.132 0.235 0.561 2.74 3.31 0.040 0.101 0.400 2.00 1.69 0.128 0.230 0.556 2.76 3.37 0.039 0.099 0.396 2.02 1.72 0.124 0.225 0.551 2.78 3.43 0.038 0.097 0.393 2.04 1.75 0.120 0.220 0.546 2.80 3.50 0.037 0.095 0.389 _ 2.06 1.78 0.116 0.215 0.541 2.82 3.57 0.036 0.093 0.386 2.08 1.81 0.113 0.210 0.536 2.84 3.64 0.035 0.091 0.383 2.10 1.84 0.109 0.206 0.531 2.86 3.71 0.034 0.089 0.379 2.12 1.87 0.106 0.201 0.526 2.88 3.78 0.033 0.087 0.376 2.14 1.90 0.103 0.197 0.522 2.90 3.85 0.032 0.085 0.373 2.16 1.94 0.100 0.192 0.517 2.92 3.92 0.031 0.083 0.370 2.18 1.97 0.097 0.188 0.513 2.94 4.00 0.030 0.081 0.366 2.20 2.01 0.094 0.184 0.508 2.96 4.08 0.029 0.080 0.363 2.22 2.04 0.091 0.180 0.504 2.98 4.15 0.028 0.078 0.360 2.24 2.08 0.088 0.176 0.499 3.00 4.23 0.027 0.076 0.357 2.26 2.12 0.085 0.172 0.495 3.10 4.66 0.023 0.0685 0.342 2.28 2.15 0.083 0.168 0.490 3.20 5.12 0.020 0.062 0.328 2.30 2.19 0.080 0.165 0.486 3.3 5.63 0.0175 0.0555 0.315 2.32 2.23 0.078 0.161 0.482 3.4 6.18 0.015 0.050 0.302 2.34 2.27 0.075 0.157 0.477 3.5 ‘ 6.79 0.013 0.045 0.290 2.36 2.32 0.073 0.154 0.473 3.6 7.45 0.0114 0.041 0.278 2.38 2.36 0.071 0.150 0.469 3.7 8.17 0.0099 0.037 0.2675 2.40 2.40 0.068 0.147 0.465 3.8 8.95 0.0086 0.0335 0.257 2.42 2.45 0.066 0.144 0.461 3.9 9.80 0.0075 0.030 0.247 2.44 2.49 0.064 . 0.141 0.456 4.0 10.72 0.0066 0.028 0.238 2.46 2.54 0.062 0.138 0.452 2.48 2.59 0.060 0.135 0.448 TFor a perfect gas with constant speciﬁc heat, k = 1.4 09999 0.9999 0.999 0.998 0.997 0.995 0.993 0.990 0.987 0.984 0.980 0.976 0.972 0.967 0.962 0.956 0.951 0.944 0.938 0.931 0.924 0.917 0.909 0.902 0.893 0.885 0.877 0.868 0.859 0.850 0.840 0.831 0.821 0.812 0.802 . 0.792 0.781 0.771 0.761 0.750 0.740 0.729 0.719 ‘ 0.999 ' 0.966 0.9999 0.9999 0.9996 0.999 0.998 0.997 0.996 0.995 0.994 0.992 0.990 0.989 0.987 0.985 0.982 0.980 0.977 0.975 0.972 0.969 0.963 0.959 0.956 0.952 0.949 0.945 0.941 0.937 0.933 0.929 0.924 0.920 0.915 0.911 0.906 0.901 0.896 0.891 0.886 0.881 0.876 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30- 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.01 1.01 1.01 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.01 1.01 1.01 1.02 1.02 1.02 1.03 1.04 1.04 1.05 1.06 1.07 1.08 1.08 1.09 1.10 1.11 1.13 1.14 1.15 1.16 1.18 1.19 1.20 1.22 1.23 1.25 1.27 1.28 1.30 1.32 1.34 1.36 0.604 0.591 0.578 ‘ 0.566 0.553 0.541 0.528 0.516 0.504 0.492 0.480 0.468 0.457 0.445 0.434 0.423 0.412 0.402 0.391 0.381 0.371 0.361 0.351 0.342 0.332 0.323 0.314 0.305 0.297 0.289 0.280 0.272 0.265 0.257 0.250 0.242 ' 0.235 0.228 0.222 0.215 0.209 0.203 0.197 0.698 0.687 0.676 0.666 0.655 0.645 0.632 0.623 0.613 0.602 0.592 0.582 0.571 0.561 0.551 0.541 0.531 0.521 0.512 0.502 0.492 0.483 0.474 0.464 0.455 0.446 0.437 0.429 0.420 0.412 0.403 0.395 0.387 0.379 0.371 0.363 0.356 0.348 0.341 0.334 0.327 0.320 0.313 0.865 0.860 0.855 0.850 0.844 0.839 0.833 0.828 0.822 0.817 0.810 0.805 0.799 0.794 0.788 0.782 0.776 0.771 0.765 0.759 0.753 0.747 0.742 0.736 0.730 0.724 . 0.718 0.713 0.707 0.701 0.695 0.690 0.684 0.678 0.672 0.667 0.661 0.656 0.650 0.645 0.639 0.634 0.628 ...
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