thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...

and

fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...

, stagnation temperature is the

temperature Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer. Thermometers are calibrated in various temperature scales that historically have relied o ...

at a

stagnation point In fluid dynamics, a stagnation point is a point in a flow field where the local velocity of the fluid is zero.Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London. A plentiful, albeit surprising, example of such points seem ...

in a fluid flow. At a stagnation point the speed of the fluid is zero and all of the

kinetic energy In physics, the kinetic energy of an object is the energy that it possesses due to its motion. It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its acce ...

has been converted to

internal energy The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

and is added to the local static enthalpy. In both compressible and

incompressible fluid flow In fluid mechanics or more generally continuum mechanics, incompressible flow ( isochoric flow) refers to a flow in which the material density is constant within a fluid parcel—an infinitesimal volume that moves with the flow velocity. An eq ...

, the stagnation temperature is equal to the ''total temperature'' at all points on the streamline leading to the stagnation point. See

gas dynamics Compressible flow (or gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density. While all flows are compressible, flows are usually treated as being incompressible when the Mach number (the ...

Derivation

Adiabatic

Stagnation temperature can be derived from the

First Law of Thermodynamics The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant am ...

. Applying the Steady Flow Energy Equation and ignoring the work, heat and gravitational potential energy terms, we have: :

h_0 = h + \frac\,

where: :

h_0 =\,

mass-specific stagnation (or total) enthalpy at a stagnation point :

h =\,

mass-specific static enthalpy at the point of interest along the stagnation streamline :

V =\,

velocity at the point of interest along the stagnation streamline Substituting for enthalpy by assuming a constant specific heat capacity at constant pressure (

h = C_p T

) we have: :

T_0 = T + \frac\,

or :

\frac =  1+\fracM^2\,

where: :

C_p =\,

specific heat capacity In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...

at constant pressure :

T_0 =\,

stagnation (or total) temperature at a stagnation point :

T =\,

temperature (or static temperature) at the point of interest along the stagnation streamline :

V = \,

velocity at the point of interest along the stagnation streamline :

M =\,

Mach number at the point of interest along the stagnation streamline :

\gamma =\,

Ratio of Specific Heats In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...

(

C_p/C_v

), ~1.4 for air at ~300 K

Flow with heat addition

h_ = h_ + q

T_ = T_ + \frac

: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 A jet engine is a type of reaction engine discharging a fast-moving jet (fluid), jet of heated gas (usually air) that generates thrust by jet propulsion. While this broad definition can include Rocket engine, rocket, Pump-jet, water jet, and ...

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 collector A solar thermal collector collects heat by absorbing sunlight. The term "solar collector" commonly refers to a device for solar hot water heating, but may refer to large power generating installations such as solar parabolic troughs and s ...

s 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 'worst case scenario values' that allow collector designers to plan for potential overheat scenarios in the event of collector system malfunctions.

References

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Bibliography