An ideal-gas mixture consists of 60 percent helium and 40...

An ideal-gas mixture consists of 60 percent helium and 40 percent argon gases by mass. The mixture is now expanded isentropically in a turbine from $400^{\circ} \mathrm{C}$ and $1.2 \mathrm{MPa}$ to a pressure of $200 \mathrm{kPa}$. The mixture temperature at the turbine exit is (a) $56^{\circ} \mathrm{C}$ (b) $195^{\circ} \mathrm{C}$ (c) $130^{\circ} \mathrm{C}$ (d) $112^{\circ} \mathrm{C}$ (e) $400^{\circ} \mathrm{C}$

An ideal-gas mixture consists of 60 percent helium and 40 percent argon gases by mass. The mixture is now expanded isentropically in a turbine from $400^{\circ} \mathrm{C}$ and $1.2 \mathrm{MPa}$ to a pressure of $200 \mathrm{kPa}$. The mixture temperature at the turbine exit is
(a) $56^{\circ} \mathrm{C}$
(b) $195^{\circ} \mathrm{C}$
(c) $130^{\circ} \mathrm{C}$
(d) $112^{\circ} \mathrm{C}$
(e) $400^{\circ} \mathrm{C}$

Key Concepts

Ideal Gas Behavior

Ideal gases obey the equation of state PV = nRT, where the internal energy is independent of volume and depends only on temperature. In the context of mixtures, the individual gas species are assumed to satisfy the ideal gas law, and properties like pressure, temperature, and volume are applied to the mixture as a whole.

Gas Mixtures

When dealing with mixtures of gases, properties are often determined from the weighted contributions of each species. In many cases, such as for ideal gas mixtures, mass fractions or mole fractions are used to compute average properties like specific heat capacities or the specific gas constant, which are crucial for analyzing thermodynamic processes.

Isentropic Processes

An isentropic process is a thermodynamic process that is both adiabatic and reversible, leading to constant entropy throughout the process. In ideal gas systems, isentropic relations provide a way to connect pressure and temperature changes, which is especially useful in analyzing turbines and compressors.

Isentropic Expansion in Turbines

In turbines, the working fluid often undergoes an isentropic expansion, where its pressure drops and temperature decreases as it performs work. The relationship T2 = T1 (P2/P1)^((??1)/?) helps predict the exit temperature when the process is assumed to be isentropic, with ? representing the ratio of specific heats.

Specific Heat Ratio (?)

The specific heat ratio, ?, is the ratio of the specific heat at constant pressure to that at constant volume (Cp/Cv). For a mixture, an effective ? is often calculated from the contributions of the individual gases, taking into account their individual specific heats. This value is essential for accurately using isentropic relations and analyzing energy transformations in turbine processes.

Transcript

00:01 Question 107 is a multiple choice question, where it states that a mixed gas mixture or ideal gas mixture contains 60 % helium and 40 % argon by mass.

00:13 This mixture is expanded isentropically in a turbine from 400 degrees celsius and 1 .2 megapascals to a pressure of 200 kilopascals.

00:23 We want to find the ending temperature, the exit temperature, in this turbine.

00:29 So if remember, or we can look up at least, the relationship between.

00:33 Change in temperature for an expanding isentropic scenario.

00:39 You can know that this is given by the initial temperature times the pressure ratio to the power of k minus 1 over k, which is k is our specific heat ratio.

00:55 So because we know k is that specific heat ratio, i'm just going to go ahead and rewrite it out where my first two terms don't change p2 over p1.

01:07 So k, the specific heat ratio, it's cp over cv, but since we're, if we look how k minus 1 over k would expand out, it'd be 1 minus 1 over k.

01:24 So we can just rewrite this numerator to be, so the exponent term to be 1 minus cv over cp, specific heat concept volume, or we're specifically heat over time constant pressure.

01:37 And so because we're doing different mass fractions, we can just write out what this exponent term would look like in terms of mass fractions.

01:45 For each of our gases.

01:47 So first time again doesn't change.

01:51 We're now our exponent term 1 minus our mass fraction times specific heat volume of helium plus that's mass fraction, specific heat volume of argon.

02:09 And similarly for the pressure term, mfcp of helium, plus the mass fraction cp of argon.

02:23 So these specific heat, heat values for pressure and temperature are ones just to look up, but everything else we have...

Please give Ace some feedback