Pressure Variations in a Gas For a gas of molecular weight Mw (such as the earth’s atmosphere), investigate how the pressure p varies with elevation z if p = p0 at z = 0. Assume that the temperature T is constant. What approximation may be made for small elevation increases? Explain how you would p

Question

Pressure Variations in a Gas

For a gas of molecular weight [latex]M_{\mathrm{w}}[/latex] (such as the earth's atmosphere), investigate how the pressure [latex]p[/latex] varies with elevation [latex]z[/latex] if [latex]p=p_{0}[/latex] at [latex]z=0[/latex]. Assume that the temperature [latex]T[/latex] is constant. What approximation may be made for small elevation increases? Explain how you would proceed for the nonisothermal case, in which [latex]T=T(z)[/latex] is a known function of elevation.

Accepted Answer

Assuming ideal gas behavior, Eqns. (1.30) and (1.7) give:

[latex]\frac{p_{z+d z}-p_{z}}{d z}=-\rho g .[/latex] (1.30)

[latex]\rho=\frac{n M_{\mathrm{w}}}{V}=\frac{M_{\mathrm{w}} p}{R T}[/latex] (1.7)

[latex]\frac{d p}{d z}=-\rho g=-\frac{M_{\mathrm{w}} p}{R T} g.[/latex] (E1.5.1)

Separation of variables and integration between appropriate limits yields:

[latex] \int_{p_{0}}^{p} \frac{d p}{p}=\ln \frac{p}{p_{0}}=-\int_{0}^{z} \frac{M_{\mathrm{w}} g}{R T} d z=-\frac{M_{\mathrm{w}} g}{R T} \int_{0}^{z} d z=-\frac{M_{\mathrm{w}} g z}{R T}, [/latex] (E1.5.2)

since [latex]M_{\mathrm{w}} g / R T[/latex] is constant. Hence, there is an exponential decrease of pressure with elevation, as shown in Fig. E1.5:

[latex] p=p_{0} \exp \left(-\frac{M_{\mathrm{w}} g}{R T} z\right) . [/latex] (E1.5.3)

Since a Taylor's expansion gives [latex]e^{-x}=1-x+x^{2} / 2-\ldots,[/latex] the pressure is approximated by:

[latex]p \doteq p_{0}\left[1-\frac{M_{\mathrm{w}} g}{R T} z+\left(\frac{M_{\mathrm{w}} g}{R T}\right)^{2} \frac{z^{2}}{2}\right].[/latex] (E1.5.4)

For small values of [latex]M_{\mathrm{w}} g z / R T[/latex], the last term is an insignificant second-order effect (compressibility effects are unimportant), and we obtain:

[latex]p \doteq p_{0}-\frac{M_{\mathrm{w}} p_{0}}{R T} g z=p_{0}-\rho_{0} g z,[/latex] (E1.5.5)

in which [latex]\rho_{0}[/latex] is the density at elevation [latex]z=0[/latex]; this approximation---essentially one of constant density---is shown as the dashed line in Fig. E1.5 and is clearly applicable only for a small change of elevation. Problem [latex]1.19[/latex] investigates the upper limit on [latex]z[/latex] for which this linear approximation is realistic. If there are significant elevation changes --- as in Problems [latex]1.16[/latex] and [latex]1.30[/latex]---the approximation of Eqn. (E1.5.5) cannot be used with any accuracy. Observe with caution that the Taylor's expansion is only a vehicle for demonstrating what happens for small values of [latex]M_{\mathrm{w}} g z / R T[/latex]. Actual calculations for larger values of [latex]M_{\mathrm{w}} g z / R T[/latex] should be made using Eqn. (E1.5.3), not Eqn. (E1.5.4).

For the case in which the temperature is not constant, but is a known function [latex]T(z)[/latex] of elevation (as might be deduced from observations made by a meteorological balloon), it must be included inside the integral:

[latex]\int_{p_{1}}^{p_{2}} \frac{d p}{p}=-\frac{M_{\mathrm{w}} g}{R} \int_{0}^{z} \frac{d z}{T(z)}.[/latex] (E1.5.6)

Since [latex]T(z)[/latex] is unlikely to be a simple function of [latex]z[/latex], a numerical method---such as Simpson's rule in Appendix A --- will probably have to be used to approximate the second integral of Eqn. (E1.5.6).

Question 1.5: Pressure Variations in a Gas For a gas of molecular weight M...

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