2.3.3: ISA equations
- Page ID
- 78096
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Considering Equation (2.3.1.1), Equations (2.3.1.2)-(2.3.1.3), and Equation (2.3.2.2), the variations of \(\rho\) and \(p\) within altitude can be obtained for the different layers of the atmosphere that affect atmospheric flight:
Troposphere (\(0 \le h < 11000 [m]\)): Introducing Equation (2.3.1.1) and Equation (2.3.1.2) in Equation (2.3.2.2), it yields:
\[\dfrac{d p}{dh} = -\dfrac{p}{R (T_0 - \alpha h)} g. \nonumber \]
Integrating between a generic value of altitude \(h\) and the altitude at sea level (\(h = 0\)), the variation of pressure with altitude yields:
\[\dfrac{p}{p_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha}}.\label{eq2.3.3.2} \]
With the value of pressure given by Equation \ref{eq2.3.3.2}, and entering in the equation of perfect gas (2.3.1.1), the variation of density with altitude yields:
\[\dfrac{\rho}{\rho_0} = (1 - \dfrac{\alpha}{T_0} h)^{\tfrac{g}{R\alpha} - 1}. \nonumber \]
Introducing now the numerical values, it yields:
\[T[k] = 288.15 - 0.0065 h [m]; \nonumber \]
\[\rho [kg/m^3] = 1.225 (1 - 22.558 \times 10^{-6} \times h [m])^{4.2559}; \nonumber \]
\[p [P a] = 101325 (1 - 22.558 \times 10^{-6} \times h[m])^{5.2559} \nonumber \]
Tropopause and Inferior part of the stratosphere (\(11000 [m] \le h < 20000 [m]\)): Introducing Equation (2.3.1.1) and Equation (2.3.1.3) in Equation (2.3.2.2), and integrating between a generic altitude (\(h > 11000 [m]\)) and the altitude at the tropopause (\(h_{11} = 11000 [m]\)):
\[\dfrac{p}{p_{11}} = \dfrac{\rho}{\rho_{11}} = e^{-\tfrac{g}{RT_{11}} (h - h_{11})} \nonumber \]
Figure 2.17: ISA atmosphere. © Cmglee / Wikimedia Commons / CC-BY-SA-3.0.
Introducing now the numerical values, it yields:
\[T[k] = 216.65; \nonumber \]
\[\rho [kg/m^3] = 0.3639e^{-157.69 \cdot 10^{-6} (h[m] - 11000)}; \nonumber \]
\[p [P a] = 22632 e^{-157.69 \cdot 10^{-6} (h[m] - 11000)}. \nonumber \]