Problem Statement:An aluminum pitot probe is placed in a supersonic air stream of temperature 30 ∘C. If the melting temperature of aluminum is about 660 ∘C, determine the flow velocity at which the probe
will begin to melt. Assume air to be an ideal gas.
Equation 1:speed of sound
Equation 2:isentropic realtion
Isentropic relations are important for analyzing the flow of compressible fluids, like gases, especially when there is no heat transfer or friction. They help predict how pressure, temperature, and density change in a flowing fluid without external heat sources or viscous effects.
The term “isentropic” comes from the Greek words isos, meaning equal, and entropia, meaning entropy. Entropy is a measure of disorder or randomness in a closed system.
In real-world applications, the isentropic process is an approximation because it’s not possible to completely isolate a system. However, the concept is useful for simplifying the analysis and calculation of various thermodynamic processes.
Equation 3:energy Equation
The equation developed by Albert Einstein, which is usually given as E = mc2, showing that, when the energy of a body changes by an amount E (no matter what form the energy takes), the mass (m) of the body will change by an amount equal to E/c2.
Equation 4:velocity, temperature
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Thermal velocityThe formula for thermal velocity isvT=3kTmv sub cap T equals the square root of the fraction with numerator 3 k cap T and denominator m end-fraction end-root
𝑣𝑇=3𝑘𝑇𝑚
. Thermal velocity is the square root of the quantity
v2¯v squared bar𝑣2
, which is part of the definition of temperature.
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Speed of soundThe formula for the speed of sound in air isv=γRTMv equals the square root of gamma cap R cap T cap M end-root
𝑣=𝛾𝑅𝑇𝑀√
. In this formula,
γgamma𝛾
is 1.4,
Mcap M𝑀
is 0.02897 kg/mol, and
Rcap R𝑅
is 8.31 J/mol • K. The velocity of sound increases as the temperature increases because the molecules vibrate faster when they gain thermal energy.
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Velocity of an ideal gasThe formula for the velocity of an ideal gas isvrms=3RTMv sub r m s end-sub equals the square root of the fraction with numerator 3 cap R cap T and denominator cap M end-fraction end-root
𝑣𝑟𝑚𝑠=3𝑅𝑇𝑀
. In this formula,
Rcap R𝑅
is the ideal gas constant,
Tcap T𝑇
is the temperature of the gas in Kelvin, and
Mcap M𝑀
is the molar mass of the gas.
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Demonstrating the Relationship between Pressure, Number of …5 Dec 2021 — Step 1: Write down the equation for the velocity of an ideal gas: v r m s = 3 R T M , where R is the ideal gas constant,Study.com
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[Tamil] What is the relation between the velocity and temperature?Doubtnut
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17.3: Speed of Sound – Physics LibreTexts12 Sept 2022 — v=√γRTM. Note that the velocity is faster at higher temperatures and slower for heavier gases. For air, γ = 1.4, M = 0…Physics LibreTexts
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Thermal Velocity – an overview | ScienceDirect TopicsTo describe the velocity of the thermal motion of particles, we can use the square root of the quantity υ 2 ¯ which appears in the…ScienceDirect.com
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Demonstrating the Relationship between Pressure, Number of …5 Dec 2021 — Step 1: Write down the equation for the velocity of an ideal gas: v r m s = 3 R T M , where R is the ideal gas constant,Study.com
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[Tamil] What is the relation between the velocity and temperature?Doubtnut
-
17.3: Speed of Sound – Physics LibreTexts12 Sept 2022 — v=√γRTM. Note that the velocity is faster at higher temperatures and slower for heavier gases. For air, γ = 1.4, M = 0…Physics LibreTexts
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Velocity of the gas : The velocity of an ideal gas, also known as the root-mean squared velocity of the ideal gas, is given by the formula v r m s = 3 R T M , where R is the ideal gas constant, T is the temperature of the gas, M is the Molar mass of the gas.5 Dec 2021