Consider the hydrodynamical flow conditions. The cooling of...

Consider the hydrodynamical flow conditions. The cooling of the gas during expansion can be expressed as follows, $\frac{T_0}{T}=1+\frac{M^2}{3}$, where $T_0$ is the temperature before expansion, $T$ is the temperature after expansion, and $M$ is the ratio of the flow velocity $v$ to the velocity of sound $c$ at temperature $T$. (a) Derive the above expression. (b) Derive a corresponding expression for $\frac{p_0}{p}$, and calculate the value of $M$ for a condition where $\frac{p_0}{p}=10^4$. (c) Calculate the value of $T$ for $\frac{p_0}{p}=10^4$ and $T_0=300 \mathrm{~K}$. (d) Find the maximum value of $v$ in the limit $T \rightarrow 0$.

Consider the hydrodynamical flow conditions. The cooling of the gas during expansion can be expressed as follows, $\frac{T_0}{T}=1+\frac{M^2}{3}$, where $T_0$ is the temperature before expansion, $T$ is the temperature after expansion, and $M$ is the ratio of the flow velocity $v$ to the velocity of sound $c$ at temperature $T$.
(a) Derive the above expression.
(b) Derive a corresponding expression for $\frac{p_0}{p}$, and calculate the value of $M$ for a condition where $\frac{p_0}{p}=10^4$.
(c) Calculate the value of $T$ for $\frac{p_0}{p}=10^4$ and $T_0=300 \mathrm{~K}$.
(d) Find the maximum value of $v$ in the limit $T \rightarrow 0$.

Key Concepts

Mach Number

The Mach number, defined as the ratio of the flow velocity to the speed of sound in the medium, is a key dimensionless parameter in compressible fluid dynamics. It is used to characterize the behavior of gas flows, particularly when dealing with phenomena such as shock waves and expansions. In the context of gas cooling during expansion, the Mach number relates how the kinetic energy of the flow is converted to thermal energy changes, affecting the observed temperature and pressure ratios.

Adiabatic Expansion

An adiabatic expansion process in a gas implies that no heat is exchanged with the surroundings. During such a process, the gas undergoes a change in temperature and pressure as it expands, with the relevant changes governed by the conservation of energy. The derived relationships between the stagnation and static temperatures and pressures (such as those involving the Mach number) rely on the assumption of adiabatic expansion, which is a critical simplification in many fluid dynamic analyses.

Stagnation Properties

Stagnation properties, including stagnation temperature and pressure, refer to the conditions that a fluid would attain if it were decelerated isentropically (i.e., without entropy change) to zero velocity. These quantities are essential in fluid dynamics since they provide a reference state that connects the dynamic properties of the flow (represented by the Mach number) to the thermodynamic state of the gas. They are used to derive expressions for cooling and pressure changes during expansion.

Conservation of Energy in Fluid Flows

In fluid dynamics, conservation of energy plays a crucial role when analyzing flow processes, especially under adiabatic conditions. The total energy of a flow, which includes internal energy and kinetic energy, remains constant along a streamline for an ideal gas in an adiabatic process. This principle is applied to derive relationships between temperature, pressure, and velocity (or Mach number) during expansion, thereby explaining gas cooling as kinetic energy increases at the expense of internal energy.

Isentropic Relations for Ideal Gases

Isentropic relations describe processes that are both adiabatic and reversible, meaning that the entropy remains constant. For an ideal gas undergoing an isentropic process, there exist specific relationships among pressure, temperature, and density. These relations form the basis for deriving the mathematical expressions that connect the stagnation and static conditions of a gas flow through functions of the Mach number, effectively linking the thermodynamic and mechanical aspects of fluid dynamics.

Transcript

00:02 Hello, so the speed of sound and i do gas of molar mass m is given by this expression right here.

00:17 So that's a square root of the bulk modules divided by the density of the other gas.

00:26 So the bulk modulus has this formula right there.

00:34 B equals negative volume, the differential of the the pressure with respect to volume okay so that's the formula for the bulk modules so in this case we are dealing with an adiabetic process okay so which means that pv is a constant so we got pv exponent gamma is a constant gamma is the specific specific heat molar specific heat okay so what we're going to do is to differentiate the bulk modulus okay no i'm sorry we're going to differentiate the pv we're going to differentiate this expression right here okay so if we take a full derivative so we're going to take derivative for p and then we do the same thing for v all right so if we do that you're going to end up with this equation right here and if you isolate dp over dv you're going to get negative gamma p over v that's from this so it come this way and then what we're going to do is to substitute so remember that the bulk modulus is this one right here okay we're going to substitute for d p over dv and we're going to end up with that so now this is b then from i do gas so a gas with density row the number of moles is equal to the density that is volume over the molar mass so you remember pv because nrt so you're going to substitute for m sorry for n right here and then the volumes are going to cancel out because there is a volume here and there is volume there is volume there so we're going to also the density if you isolate that you're going to have that so this density because the pressure and the molar mass over r t is a constant the molar gas constant t is the temperature so you remember the velocity of sound in an ideal gas is square root of b over row right so now we've got b that's the bulb modulus so we're just going to do substitution for b and then for density and we're going to end up with this so that's for the first part of your question the second part of the question we know that the speed of sound cannot speak cannot exceed the speed of light right so it's going to be this expression right here so either that is less than or equal to okay so for the maximum you're going to say that vs max is equal to c which is the speed of light okay so that's equal to so that's basically this expression again okay coming here but this time at maximum temperature so with this if you isolate t -max they just have to square both sides right to get rid of the real sign and just isolate t -max you're gonna get that okay so this t -max, then c is for a model atomic gas, the ratio of the molar -specific heat, that's the gamma is 5 over 3.

04:51 Okay, that's a constant.

04:53 So you can just plug it in...

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