02 May '11 02:311 edit

I felt like figuring this out, but the units of measure seem to be flawed at my final result, so If you could/will, point out my flaw(s)

I'm attemting to find the density as a function of height of a column of air

Im going to assume ideal gas law

P =p*R*T eq(1)

P = pressure

p= density

R=universal gas constant

T= temperature(absolute)

assuming acceleration due to gravity & Area are constant, and the fluid(air) is in static equilibrium

A= Area(constant)

dP = dF/A eq(2)

dF= dm*g eq(3)

m= mass of air above section

m=p*V eq(4)

where

p=density

V=volume

thus

dm = V*dp+p*dV eq(4'ðŸ˜‰

Back substitution until eq(2) yeilds

A*dP = g(V*dp+p*dV) = g(A*h*dp + p*A dh)

where the Area (A) divides out &

h=height

giving

dP = g(h*dp+p*dh) eq(5)

Taking the derivative with respect to density from eq(5) gives

dP/dp = g( h + p* dh/dp ) eq(6)......(This equation could be wrong, not sure if Im taking the derivative correctly)

taking the derivative of eq(1) with respect to density (Assuming temp is constant) and substituting for left side of eq(6) gives after some rearrangement

h + p*dh/dp = RT/g= constant

here I take the laplace transform of both sides (could be where the mistake lies)

after solving for F(s) I come to

F(s) = (RT/g*s- h(0))/(s*(s-1)) which is in basic form

= (C*s - A)/(s*(s-1))

which breaks into after some fuddleing

C/(s-1) - A/(s-1) + A/s

The inverse Laplace of this yeilds

h(p) = (RT/g)*(e^p) - h(0)*(e^p) + h(0)

does anyone else agree with the above equation?

I'm attemting to find the density as a function of height of a column of air

Im going to assume ideal gas law

P =p*R*T eq(1)

P = pressure

p= density

R=universal gas constant

T= temperature(absolute)

assuming acceleration due to gravity & Area are constant, and the fluid(air) is in static equilibrium

A= Area(constant)

dP = dF/A eq(2)

dF= dm*g eq(3)

m= mass of air above section

m=p*V eq(4)

where

p=density

V=volume

thus

dm = V*dp+p*dV eq(4'ðŸ˜‰

Back substitution until eq(2) yeilds

A*dP = g(V*dp+p*dV) = g(A*h*dp + p*A dh)

where the Area (A) divides out &

h=height

giving

dP = g(h*dp+p*dh) eq(5)

Taking the derivative with respect to density from eq(5) gives

dP/dp = g( h + p* dh/dp ) eq(6)......(This equation could be wrong, not sure if Im taking the derivative correctly)

taking the derivative of eq(1) with respect to density (Assuming temp is constant) and substituting for left side of eq(6) gives after some rearrangement

h + p*dh/dp = RT/g= constant

here I take the laplace transform of both sides (could be where the mistake lies)

after solving for F(s) I come to

F(s) = (RT/g*s- h(0))/(s*(s-1)) which is in basic form

= (C*s - A)/(s*(s-1))

which breaks into after some fuddleing

C/(s-1) - A/(s-1) + A/s

The inverse Laplace of this yeilds

h(p) = (RT/g)*(e^p) - h(0)*(e^p) + h(0)

does anyone else agree with the above equation?