the data given in the table, obtain a relation for the variation of
density with elevation, and calculate the density at an elevation
of 7000 m. (b) Calculate the mass of the atmosphere using the
correlation you obtained. Assume the earth to be a perfect
sphere with a radius of 6377 km, and take the thickness of the
atmosphere to be 25 km.
z, km r, kg/m
3
6377 1.225
6378 1.112
6379 1.007
6380 0.9093
6381 0.8194
6382 0.7364
6383 0.6601
6385 0.5258
6387 0.4135
6392 0.1948
6397 0.08891
6402 0.04008
ANS
1-25 EES The variation of densityof atmospheric air withelevation isgiven intabular form. A relation for
the variation of densitywithelevation istobe obtained, the densityat7 kmelevation istobe calculated,
and the mass of the atmosphere using the correlation istobe estimated.
Assumptions1Atmospheric air behaves as an ideal gas. 2 The earthisperfectlysphere with a radius of
6377 km, and the thickness of the atmosphere is25 km.
PropertiesThe densitydataare given intabular formas
r, km z, km ρ, kg/m3
6377 0 1.225
6378 1 1.112
6379 2 1.007
6380 3 0.9093
6381 4 0.8194
6382 5 0.7364
6383 6 0.6601
6385 8 0.5258
6387 10 0.4135
6392 15 0.1948
6397 20 0.08891
6402 25 0.04008
![Analysis Using EES, (1) Define a trivialfunction rho= a+z inequation window, (2) selectnew parametric table fromTables, and type the dataina two-columntable, (3) selectPlotand plot the data, and (4) select plotand click on “curve fit”togetcurve fitwindow. Then specify2 nd order polynomialand enter/edit equation. The results are: ρ(z) = a + bz + cz 2 = 1.20252 – 0.101674z + 0.0022375z 2 for the unitof kg/m 3 , (or, ρ(z) = (1.20252 – 0.101674z + 0.0022375z 2 )×10 9 for the unitof kg/km 3 ) where zis the vertical distance fromthe earth surface at sea level. At z = 7 km, the equation would give ρ= 0.60 kg/m 3 . (b) The mass of atmosphere can be evaluated byintegration tobe []5 / 4 / ) 2 ( 3 / ) 2 ( 2 / ) 2 ( 4 ) 2 )( ( 4 ) ( 4 ) ( 5 4 0 3 2 0 0 2 0 0 2 0 2 0 2 0 2 0 2 0 2 0 ch h cr b h cr br a h br a r h ar dz z z r r cz bz a dz z r cz bz a dV m h z h z V + + + + + + + + = + + + + = + + + = = ∫ ∫ ∫ = = π π π ρ where r0 = 6377 kmisthe radius of the earth, h= 25 kmisthe thickness of the atmosphere, and a = 1.20252, b = -0.101674, and c= 0.0022375 are the constants in the density function. Substituting and multiplyingbythe factor10 9 for the densityunitykg/km 3 , the mass ofthe atmosphere is determinedtobe m = 5.092×10 18 kg DiscussionPerforming the analysis with excel would yield exactly the sameresults.](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhMuAuTGQ7HJsY8Utu1YOHqUDlv0ymplcznzPDKApHPmPM4UO68lPPNrFTwb-f-_1Mr8Rze-jTA1HQlFQn5liCxcFukkPapxXiQCVIar-m8FpP2ci1_blzPEgfsMY-ruzSI_vMZ2NWoKKE/s1600/1-25-1.PNG "Analysis Using EES, (1) Define a trivialfunction rho= a+z inequation window, (2) selectnew parametric table fromTables, and type the dataina two-columntable, (3) selectPlotand plot the data, and (4) select plotand click on “curve fit”togetcurve fitwindow. Then specify2 nd order polynomialand enter/edit equation. The results are: ρ(z) = a + bz + cz 2 = 1.20252 – 0.101674z + 0.0022375z 2 for the unitof kg/m 3 , (or, ρ(z) = (1.20252 – 0.101674z + 0.0022375z 2 )×10 9 for the unitof kg/km 3 ) where zis the vertical distance fromthe earth surface at sea level. At z = 7 km, the equation would give ρ= 0.60 kg/m 3 . (b) The mass of atmosphere can be evaluated byintegration tobe []5 / 4 / ) 2 ( 3 / ) 2 ( 2 / ) 2 ( 4 ) 2 )( ( 4 ) ( 4 ) ( 5 4 0 3 2 0 0 2 0 0 2 0 2 0 2 0 2 0 2 0 2 0 ch h cr b h cr br a h br a r h ar dz z z r r cz bz a dz z r cz bz a dV m h z h z V + + + + + + + + = + + + + = + + + = = ∫ ∫ ∫ = = π π π ρ where r0 = 6377 kmisthe radius of the earth, h= 25 kmisthe thickness of the atmosphere, and a = 1.20252, b = -0.101674, and c= 0.0022375 are the constants in the density function. Substituting and multiplyingbythe factor10 9 for the densityunitykg/km 3 , the mass ofthe atmosphere is determinedtobe m = 5.092×10 18 kg DiscussionPerforming the analysis with excel would yield exactly the sameresults.")
a=1.2025166
b=-0.10167
c=0.0022375
r=6377
h=25
m=4*pi*(a*r^2*h+r*(2*a+b*r)*h^2/2+(a+2*b*r+c*r^2)*h^3/3+(b+2*c*r)*h^4/4+c*h^5/5)*1E+9
No comments:
Post a Comment