4.3.4.1: The Basic Analysis

There are situations when the main change of the density results from other effects. For example, when the temperature field is not uniform, the density is affected and thus the pressure is a location function (for example, the temperature in the atmosphere is assumed to be a linear with the height under certain conditions.). A bit more complicate case is when the gas is a function of the pressure and another parameter. Air can be a function of the temperature field and the pressure. For the atmosphere, it is commonly assumed that the temperature is a linear function of the height. Here, a simple case is examined for which the temperature is a linear function of the height as
$\dfrac{dT}{dh} = - C_x \label{static:eq:Tdz} \tag{91}$
where h here referred to height or distance. Hence, the temperature–distance function can be written as
$T = Constant - C_x\, h \label{static:eq:TxxIassss} \tag{92}$
where the Constant is the integration constant which can be obtained by utilizing the initial condition. For $$h=0$$, the temperature is $$T_0$$ and using it leads to

Temp variations

$\label{static:eq:TxxI} T = T_0 - C_x\, h \tag{93}$

Combining equation (93) with (11) results in
$\dfrac {\partial P} {\partial h} = - \dfrac{g \,P }{R \left( T_0 -C_x\, h\right) } \label{static:eq:TxxIaa} \tag{94}$
Separating the variables in equation (94) and changing the formal $$\partial$$ to the informal $$d$$ to obtain
$\dfrac {d\,P} {P} = - \dfrac{g \,dh }{R \left( T_0 -C_x\, h\right) } \label{static:eq:TxxG} \tag{95}$
Defining a new variable as $$\xi = (T_0 -C_x\, h)$$ for which $$\xi_0 = T_0 -C_x\, h_0$$ and $$d/d\xi = - C_x\,d/dh$$. Using these definitions results in
$\dfrac {d\,P} {P} = \dfrac{g }{R C_x} \dfrac{d\xi}{ \xi } \label{static:eq:ThhG} \tag{96}$
After the integration of equation (95) and reusing (the reverse definitions) the variables transformed the result into
$\ln \dfrac {P} {P_0} = \dfrac{g }{R \,C_x} \ln \dfrac{ T_0 -C_x\, h}{T_0} \label{static:eq:TxxIbb} \tag{97}$
Or in a more convenient form as

Pressure in Atmosphere

$\label{static:eq:TxxSi} \dfrac {P} {P_0} = \left(\dfrac{ T_0 -C_x\, h}{T_0} \right) ^ {\left( \dfrac{g }{R \,C_x} \right) } \tag{98}$

It can be noticed that equation (98) is a monotonous function which decreases with height because the term in the brackets is less than one. This situation is roughly representing the pressure in the atmosphere and results in a temperature decrease. It can be observed that $$C_x$$ has a "double role'' which can change the pressure ratio. Equation (98) can be approximated by two approaches/ideas. The first approximation for a small distance, $$h$$, and the second approximation for a small temperature gradient. It can be recalled that the following expansions are
$\dfrac{P}{P_0} = \lim_{h -> 0} {\left(1 - \dfrac{C_x}{T_0}\,h\right) }^{\dfrac{g}{R\,C_x}} = \\ { 1-\overbrace{\dfrac{g\,h}{T_0\,R}}^{\dfrac{g\,h\,\rho_0}{P_0}}- \overbrace{\dfrac{\left( R\,g\,C_x-{g}^{2}\right) \,{h}^{2}}{2\,{T_0}^{2}\,{R}^{2}}}^{\text{correction factor}} } - \cdots \quad \label{static:eq:aproxTonPh} \tag{99}$
Equation (99) shows that the first two terms are the standard terms (negative sign is as expected i.e. negative direction). The correction factor occurs only at the third term which is important for larger heights. It is worth to point out that the above statement has a qualitative meaning when additional parameter is added. However, this kind of analysis will be presented in the dimensional analysis chapter. The second approximation for small $$C_x$$ is
$\dfrac{P}{P_0} = \lim_{C_x -> 0} {\left(1 - \dfrac{C_x}{T_0}\,h\right) }^{\dfrac{g}{R\,C_x}} = \\ \text{ e}^{-\dfrac{g\,h}{R\,T_0}} - \dfrac{g\,{h}^{2}\,C_x}{2 \,{T_0}^{2}\,R} \text{e}^{-\dfrac{g\,h}{R\,T_0}} - \cdots \quad \label{static:eq:aproxTonPCx} \tag{100}$
Equation (100) shows that the correction factor (lapse coefficient), $$C_x$$, influences at only large values of height. It has to be noted that these equations (99) and (100) are not properly represented without the characteristic height. It has to be inserted to make the physical significance clearer. Equation (98) represents only the pressure ratio. For engineering purposes, it is sometimes important to obtain the This relationship can be obtained from combining equations (98) and (93). The simplest assumption to combine these equations is by assuming the ideal gas model, equation (Ideal Gas), to yield
$\dfrac{\rho}{\rho_0} = \dfrac{P\,T_0}{P_0\,T} = \overbrace{\left( 1 - \dfrac{C_x\, h}{T_0} \right) ^ {\left( \dfrac{g }{R \,C_x} \right) } } ^{\dfrac{P}{P_0}} \overbrace{\left( 1 + \dfrac{C_x\, h}{T} \right)} ^{\dfrac{T_0}{T}} \label{static:eq:TxxRho} \tag{101}$