speed of sound,isentropic Relation, mach number,mass flow rate

Problem Statement:Helium gas at 250 kPa and 100 ∘C flows through a rectangular duct of cross-section 40 cm × 50 cm, with a velocity of 500 m s−1.Determine the Mach number, stagnation
temperature, stagnation pressure, and the mass flow rate. Assume helium to be an ideal gas.

Equation 1: Speed Of Sound.

More simply, the speed of sound is how fast vibrations travel. At 20 °C (68 °F), the speed of sound in air is about 343 m/s (1,125 ft/s; 1,235 km/h; 767 mph; 667 kn), or 1 km in 2.91 s or one mile in 4.69 s. It depends strongly on temperature as well as the medium through which a sound wave is propagating.

Equation 2:Isentropic Relations

Isentropic relations are thermodynamic relationships that describe how a fluid behaves during an isentropic process. An isentropic process is a reversible adiabatic process where entropy remains constant. 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. — Isentropic relations are thermodynamic relationships that describe how a fluid behaves during an isentropic process. An isentropic process is a reversible adiabatic process where entropy remains constant.
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: Mach Number

Mach number is calculated by finding the ratio of speed of an object to the speed of sound in the surrounding medium. Mach number is unitless (dimensionless quantity). The formula for calculating Mach number is M = v/c , where M is Mach number, v is the velocity of object (in meters per second, feet per second, etc.)

Equation 4: Mass Flow Rate:

In physics and engineering, mass flow rate is the rate at which mass of a substance changes over time. Its unit is kilogram per second (kg/s) in SI units, and slug per second or pound per second in US customary units. The common symbol is m ˙ {\displaystyle {\dot {m}}} (ṁ, pronounced "m-dot"), although sometimes μ (Greek lowercase mu) is used. Sometimes, mass flow rate as defined here is termed "mass flux" or "mass current".[a] Confusingly, "mass flow" is also a term for mass flux, the rate of mass flow per unit of area.[2] Formulation Mass flow rate is defined by the limit[3][4] m ˙ = lim Δ t → 0 Δ m Δ t = d m d t , — In physics and engineering, mass flow rate is the rate at which mass of a substance changes over time. Its unit is kilogram per second (kg/s) in SI units, and slug per second or pound per second in US customary units. The common symbol is
m
˙
{displaystyle {dot {m}}} (ṁ, pronounced “m-dot”), although sometimes μ (Greek lowercase mu) is used.
Sometimes, mass flow rate as defined here is termed “mass flux” or “mass current”.[a] Confusingly, “mass flow” is also a term for mass flux, the rate of mass flow per unit of area.[2]
Formulation
Mass flow rate is defined by the limit[3][4]
m
˙
=
lim
Δ
t
→
0
Δ
m
Δ
t
=
d
m
d
t
,

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speed of sound,isentropic Relation, mach number,mass flow rate

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