Chapter 1.11, Problem 115RP

(a)

To determine

The relation for the change of pressure in troposphere using constant $g$.

Answer to Problem 115RP

The relation for the change of pressure in troposphere using constant $g$ is

$P=P_0{\left(1-\frac{\beta z}{T_0}\right)}^{\frac{g}{\beta R}}$.

Explanation of Solution

Write the expression to determine the pressure change $\left(dP\right)$ across a differential fuild layer.

$dP=-\rho gdz$ (I)

Here, density of fluid is $\rho$, acceleration due to gravity is $g$, and the thickness of differential fluid layer is $dz$.

Write the ideal gas relation to determine the density of gas $\left(\rho \right)$.

$\rho =\frac{P}{RT}$

$\rho =\frac{P}{R\left(T_0-\beta z\right)}$ (II)

Here, pressure of the gas is $P$, gas constant is $R$, temperature is $T$, temperature at the sea level is $T_0$, and temperature change per unit thickness is $\beta$.

Conclusion:

Substitute Equation (II) for $\rho$ in Equation (I).

$\begin{array}{l}dP=-\left(\frac{P}{R\left(T_0-\beta z\right)}\right)gdz\\ \frac{dP}{P}=-\frac{gdz}{R\left(T_0-\beta z\right)}\end{array}$

Apply integration for the limits 0 to $z$.

${\displaystyle \underset{P_0}{\overset{P}{\int }}\frac{dP}{P}}=-{\displaystyle \underset{0}{\overset{z}{\int }}\frac{gdz}{R\left(T_0-\beta z\right)}}$

$\ln \frac{P_0}{P}=\frac{g}{R\beta }\ln \left(\frac{T_0-\beta z}{T_0}\right)$ (III)

Rewrite the Equation (III) for the pressure change across fluid layer with constant $g$ as

$P=P_0{\left(1-\frac{\beta z}{T_0}\right)}^{\frac{g}{\beta R}}$

Thus, the relation for the change of pressure in troposphere using constant $g$ is

$P=P_0{\left(1-\frac{\beta z}{T_0}\right)}^{\frac{g}{\beta R}}$.

(b)

To determine

The relation for the change of pressure in troposphere using variable $g$.

Answer to Problem 115RP

The relation for the change of pressure in troposphere using variable $g$ is

$P=P_0\exp \left[-\frac{g_0}{R\left(\beta +k{T}_0\right)}\left(\frac{1}{1+1/kz}+\frac{1}{1+k{T}_0/\beta }\ln \frac{1+kz}{1-\beta z/T_0}\right)\right]$.

Explanation of Solution

Write the expression for the change in pressure when there is variation of $g$.

$dP=\frac{P}{R\left(T_0-\beta z\right)}\frac{g_0}{{\left(1+z/6,370,320\right)}^2}dz$ (IV)

Conclusion:

Apply integration from the limits 0 to $z$ in Equation (IV).

$\begin{array}{l}{\displaystyle \underset{P_0}{\overset{P}{\int }}\frac{dP}{P}}={\displaystyle \underset{0}{\overset{z}{\int }}\frac{g_{0}dz}{R\left(T_0-\beta z\right){\left(1+z/6,370,320\right)}^2}}\\ {\ln P|}_{P_0}^P=\frac{g_0}{R\beta }{\left|\frac{1}{\left(1+k{T}_0/\beta \right)\left(1+kz\right)}-\frac{1}{{\left(1+k{T}_0/\beta \right)}^2}\ln \frac{1+kz}{T_0-\beta z}\right|}_0^z\\ P=P_0\exp \left[-\frac{g_0}{R\left(\beta +k{T}_0\right)}\left(\frac{1}{1+1/kz}+\frac{1}{1+k{T}_0/\beta }\ln \frac{1+kz}{1-\beta z/T_0}\right)\right]\end{array}$

Thus, the relation for the change of pressure in troposphere using variable $g$ is

$P=P_0\exp \left[-\frac{g_0}{R\left(\beta +k{T}_0\right)}\left(\frac{1}{1+1/kz}+\frac{1}{1+k{T}_0/\beta }\ln \frac{1+kz}{1-\beta z/T_0}\right)\right]$.