Stagnation temperature

In thermodynamics and fluid mechanics, stagnation temperature is the temperature at a stagnation point in a fluid flow. At a stagnation point the speed of the fluid is zero and all of the kinetic energy has been converted to internal energy and is added to the local static enthalpy. In incompressible fluid flow, and in compressible flow, the stagnation temperature is equal to the total temperature at all points on the streamline leading to the stagnation point.[1] See gas dynamics.

Derivation

Stagnation temperature can be derived from the First Law of Thermodynamics. Applying the Steady Flow Energy Equation [2] and ignoring the work, heat and gravitational potential energy terms, we have:

${\displaystyle h_{0}=h+{\frac {V^{2}}{2}}\,}$

where:

${\displaystyle h_{0}=\,}$ stagnation (or total) enthalpy at a stagnation point
${\displaystyle h=\,}$ static enthalpy at the point of interest along the stagnation streamline
${\displaystyle V=\,}$ velocity at the point of interest along the stagnation streamline

Substituting for enthalpy by assuming a constant specific heat capacity at constant pressure (${\displaystyle h=C_{p}T}$) we have:

${\displaystyle T_{0}=T+{\frac {V^{2}}{2C_{p}}}\,}$

or

${\displaystyle {\frac {T_{0}}{T}}=1+{\frac {\gamma -1}{2}}M^{2}\,}$

where:

${\displaystyle C_{p}=\,}$ specific heat capacity at constant pressure
${\displaystyle T_{0}=\,}$ stagnation (or total) temperature at a stagnation point
${\displaystyle T=\,}$ temperature (or static temperature) at the point of interest along the stagnation streamline
${\displaystyle V=\,}$ velocity at the point of interest along the stagnation streamline
${\displaystyle M=\,}$ Mach number at the point of interest along the stagnation streamline
${\displaystyle \gamma =\,}$ Ratio of Specific Heats (${\displaystyle C_{p}/C_{v}}$), ~1.4 for air at ~300 K

${\displaystyle h_{02}=h_{01}+q}$
${\displaystyle T_{02}=T_{01}+{\frac {q}{C_{p}}}}$
q = Heat per unit mass added into the system

Strictly speaking, enthalpy is a function of both temperature and density. However, invoking the common assumption of a calorically perfect gas, enthalpy can be converted directly into temperature as given above, which enables one to define a stagnation temperature in terms of the more fundamental property, stagnation enthalpy.

Stagnation properties (e.g. stagnation temperature, stagnation pressure) are useful in jet engine performance calculations. In engine operations, stagnation temperature is often called total air temperature. A bimetallic thermocouple is often used to measure stagnation temperature, but allowances for thermal radiation must be made.

Solar thermal collectors

Performance testing of solar thermal collectors utilizes the term stagnation temperature to indicate the maximum achievable collector temperature with a stagnant fluid (no motion), an ambient temperature of 30C, and incident solar radiation of 1000W/m2. The aforementioned figures are just arbitrary values and do not possess any true meaning without a proper context.[3]