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| 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]] (adiabatically) and is added to the local [[Enthalpy|static enthalpy]]. In [[incompressible flow|incompressible fluid flow]], and in [[isentropic]] [[compressible flow]], the stagnation temperature is equal to the ''total temperature'' at all points on the streamline leading to the stagnation point.
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| <ref>Van Wylen and Sonntag, ''Fundamentals of Classical Thermodynamics'', section 14.1</ref> See [[gas dynamics]].
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| ==Derivation==
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| ===Adiabatic===
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| Stagnation temperature can be derived from the [[First Law of Thermodynamics]]. Applying the Steady Flow Energy Equation
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| <ref>Van Wylen and Sonntag, ''Fundamentals of Classical Thermodynamics'', equation 5.50</ref> and ignoring the work, heat and gravitational potential energy terms, we have:
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| :<math>h_0 = h + \frac{V^2}{2}\,</math>
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| where:
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| :<math>h_0 =\,</math> stagnation (or total) enthalpy at a stagnation point
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| :<math>h =\,</math> static enthalpy at the point of interest along the stagnation streamline
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| :<math>V =\,</math> velocity at the point of interest along the stagnation streamline
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| Substituting for enthalpy by assuming a constant specific heat capacity at constant pressure (<math>h = C_p T</math>) we have:
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| :<math>T_0 = T + \frac{V^2}{2C_p}\,</math>
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| or
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| :<math>\frac{T_0}{T} = 1+\frac{\gamma-1}{2}M^2\,</math>
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| where:
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| :<math>C_p =\,</math> [[specific heat capacity]] at constant pressure
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| :<math>T_0 =\,</math> stagnation (or total) temperature at a stagnation point
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| :<math>T =\,</math> temperature (or static temperature) at the point of interest along the stagnation streamline
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| :<math>V = \,</math> velocity at the point of interest along the stagnation streamline
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| :<math>M =\,</math> Mach number at the point of interest along the stagnation streamline | |
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| :<math>\gamma =\,</math> [[Heat capacity ratio|Ratio of Specific Heats]] (<math>C_p/C_v</math>), ~1.4 for air at ~300 K
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| ===Flow with Heat Addition===
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| :<math>h_{02} = h_{01} + q </math> | |
| :<math>T_{02} = T_{01} + \frac{q}{C_p} </math>
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| :q = Heat per unit mass added into the system
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| Strictly speaking, enthalpy is a function of both temperature and density. However, invoking the common assumption of a calorically perfect gas, enthalpy can
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| be converted directly into temperature as given above, which enables one to define a stagnation temperature in terms of the more fundamental property,
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| stagnation enthalpy.
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| 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.
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| ==Solar Thermal Collectors==
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| Solar Thermal Collector performance testing 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/m^2. The aforementioned figures are just arbitrary values and do possess any true meaning without a proper context.
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| ==See also== | |
| *[[Stagnation point]]
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| *[[Stagnation pressure]]
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| *[[Total air temperature]]
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| ==References==
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| *Van Wylen, G.J., and Sonntag, R.E. (1965), ''Fundamentals of Classical Thermodynamics'', John Wiley & Sons, Inc., New York
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| {{Reflist}}
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| [[Category:Fluid dynamics]]
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