ATMOSPHERE_1976

Overview

The ATMOSPHERE_1976 function calculates standard atmospheric properties at a given altitude using the U.S. Standard Atmosphere 1976 model. This empirical model, developed jointly by NOAA, NASA, and the U.S. Air Force, defines idealized atmospheric conditions from -610 meters below sea level up to 86,000 meters altitude, making it essential for aerospace engineering, aviation, and atmospheric science applications.

This implementation uses the fluids library, a comprehensive Python package for fluid dynamics and chemical engineering calculations. For detailed documentation, see the fluids.atmosphere module.

The model returns seven key atmospheric properties at the specified elevation:

  • Temperature (T): Calculated from standardized lapse rates across atmospheric layers
  • Pressure (P): Derived from barometric equations
  • Density (ρ): Computed using the ideal gas law relationship:
\rho = \frac{P \cdot MW}{T \cdot R \cdot 1000}

where MW is the molecular weight of air (28.9644 g/mol) and R is the gas constant (8314.32 J/kmol/K).

  • Gravity (g): Adjusted for altitude using:
g = g_0 \left(\frac{r_0}{r_0 + Z}\right)^2

where g_0 is sea-level gravity and r_0 is Earth’s radius.

  • Dynamic viscosity (μ): Temperature-dependent correlation
  • Thermal conductivity (k): Temperature-dependent correlation
  • Speed of sound (v_sonic): Calculated from the thermodynamic relationship:
c = \sqrt{\frac{\gamma R T}{MW}}

The optional temperature offset parameter (dt) allows modeling non-standard atmospheric conditions, useful for hot-day or cold-day performance analyses. Up to 32 km, this model is identical to the International Standard Atmosphere (ISA) and the World Meteorological Organization (WMO) standard atmosphere.

This example function is provided as-is without any representation of accuracy.

Excel Usage

=ATMOSPHERE_1976(z, dt)
  • z (float, required): Elevation above sea level in meters (valid range -610 to 86000 m).
  • dt (float, optional, default: 0): Temperature offset from standard conditions in Kelvin.

Returns (list[list]): [[T, P, rho, g, mu, k, v_sonic]] where T is temperature (K), P is pressure (Pa), rho is density (kg/m^3), g is gravity (m/s^2), mu is dynamic viscosity (Pa·s), k is thermal conductivity (W/m/K), and v_sonic is speed of sound (m/s). str: Error message if input is invalid or out-of-range.

Examples

Example 1: Standard atmosphere at 5,000 m elevation

Inputs:

z dt
5000 0

Excel formula:

=ATMOSPHERE_1976(5000, 0)

Expected output:

Result
255.68 54048.29 0.7364 9.7912 0.00001628 0.02273 320.55

Example 2: Sea level standard atmosphere

Inputs:

z dt
0 0

Excel formula:

=ATMOSPHERE_1976(0, 0)

Expected output:

Result
288.15 101325 1.225 9.8067 0.00001789 0.02533 340.29

Example 3: Below sea level atmosphere (-500 m)

Inputs:

z dt
-500 0

Excel formula:

=ATMOSPHERE_1976(-500, 0)

Expected output:

Result
291.4 107478 1.2849 9.8082 0.00001805 0.02558 342.21

Example 4: Custom temperature offset at 10,000 m

Inputs:

z dt
10000 -5

Excel formula:

=ATMOSPHERE_1976(10000, -5)

Expected output:

Result
218.25 26499.9 0.423 9.7759 0.0000143 0.01964 296.16

Python Code

import micropip
await micropip.install(["fluids"])
from fluids.atmosphere import ATMOSPHERE_1976 as fluids_atmosphere_1976

def atmosphere_1976(z, dt=0):
    """
    Calculate standard atmospheric properties at a given altitude using the US Standard Atmosphere 1976 model.

    See: https://fluids.readthedocs.io/fluids.atmosphere.html

    This example function is provided as-is without any representation of accuracy.

    Args:
        z (float): Elevation above sea level in meters (valid range -610 to 86000 m).
        dt (float, optional): Temperature offset from standard conditions in Kelvin. Default is 0.

    Returns:
        list[list]: [[T, P, rho, g, mu, k, v_sonic]] where T is temperature (K), P is pressure (Pa), rho is density (kg/m^3), g is gravity (m/s^2), mu is dynamic viscosity (Pa·s), k is thermal conductivity (W/m/K), and v_sonic is speed of sound (m/s). str: Error message if input is invalid or out-of-range.
    """
    try:
        z_val = float(z)
        dt_val = float(dt)
    except (TypeError, ValueError):
        return "Error: z and dt must be numeric values."

    if not (-610 <= z_val <= 86000):
        return "Error: z must be between -610 and 86000 meters."

    try:
        atm = fluids_atmosphere_1976(z_val, dT=dt_val)
        result = [
            atm.T,
            atm.P,
            atm.rho,
            atm.g,
            atm.mu,
            atm.k,
            atm.v_sonic
        ]
        return [result]
    except Exception as e:
        return f"Error: {str(e)}"

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