A mixture of 60 % helium and 40 % nitrogen by mass enters a...

A mixture of $60 \%$ helium and $40 \%$ nitrogen by mass enters a turbine at $1 \mathrm{MPa}$ and $800 \mathrm{K}$ at a rate of $2 \mathrm{kg} / \mathrm{s}$. The adiabatic turbine has an exit pressure of $100 \mathrm{kPa}$ and an isentropic efficiency of $85 \% .$ Find the turbine work.

Key Concepts

Properties of Ideal Gas Mixtures

When dealing with mixtures of gases, it is important to determine the overall thermodynamic properties of the mixture, such as specific heats and gas constants, based on the contribution of each component. This concept allows for the effective application of ideal gas assumptions in performance calculations and energy balances, ensuring accurate predictions of the behavior of the mixed fluid through devices like turbines.

Adiabatic Process

An adiabatic process is one in which there is no heat transfer between the system and its surroundings. In the context of turbines, this means that any change in the energy of the working fluid is due solely to work interactions, simplifying the energy analysis since only enthalpy changes and work need to be accounted for.

Steady-Flow Energy Equation

The steady-flow energy equation is an application of the first law of thermodynamics to systems where mass continuously enters and exits, such as turbines. This principle is used to relate the energy differences between inlet and outlet streams (including enthalpy, kinetic, and potential energies) and the work done by or on the system, forming the foundation for performance calculations in flow devices.

Isentropic Efficiency

Isentropic efficiency is a performance metric that compares the actual work output of a device to the work output that would be achieved if the process were ideal (isentropic and reversible). It quantifies the losses due to irreversibilities within the device, providing a measure of how efficiently the turbine converts the available energy into work.

Transcript

00:01 Hi, everybody.

00:01 So we are asked, the antibiotic turbine has the exit pressure at 100k pascal is an isotropic efficiency.

00:09 Find the turbine work, okay? so for this, we can find the r mix, okay, which again, you find all this information, all the number information in your book, 0 .6, and that's where 60%, 2 .071, plus 0 .4 to 40%.

00:30 0 .2968 equals 1 .3648, 1 .36498, kilojoules per kilogram times kelvin.

00:44 And now we have your cp mix, which equals 0 .6 .5 .193 plus 0 .41 .042 equals 3 .536 kilojoules per kilogram times kelvin.

01:09 And now we have your cv mix, which is just going to mean your cp minus your r mix.

01:16 So we have 3 .536 minus 1 .36498, which equals 2 .16, which equals 2 .16498, which equals 2 .16 .128, which equals 2 .16 .6.

01:28 7 6 to kilojoules per kilogram times kelvin.

01:35 And now we can do the ratio specific heat.

01:38 So ratio specific keys is k of the cp over cb.

01:43 And it's going to be 3 .5, 326 divided by 2 .16762, which equals 1 .6297.

01:55 Okay.

01:57 And now we can do the equation for it.

02:01 So temperature 2 divided by temperature 1 equals pressure 2 minus pressure 1.

02:08 And we got k minus 1.

02:10 For the specific heat.

02:14 And now we have 2t divided by 800 because we're solved for the temperature 2.

02:21 And we got 100 over 1 ,000.

02:25 Divided by 1 .6297 minus 1 divided by 1 .6297.

02:33 And your t2 is going to equal to 3 to 8 .624 -7...

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