Steam at 7 MPa and 400^∘ C enters a two-stage adiabatic turbine at a rate of 15 kg / s. Ten per

Steam at 7 MPa and 400^∘ C enters a two-stage adiabatic...

Key Concepts

Isentropic Efficiency

Isentropic efficiency is a measure of the real performance of a turbine compared to an ideal, reversible (isentropic) process. It quantifies the losses due to irreversibilities such as friction, turbulence, and other non-idealities within the turbine. By comparing actual work output to the maximum possible work, this efficiency parameter is critical for assessing the performance and deriving the potential for energy recovery or wasted power.

Multi-Stage Turbine Operation with Extraction

Multi-stage turbine operation involves dividing the expansion process into separate stages, often with intermediate extraction of a portion of the fluid for various uses. This concept is important because it affects the thermodynamic properties and subsequent work potential in each stage. Extraction at an intermediate pressure alters the mass flow and energy balance, requiring careful analysis to accurately assess overall turbine performance and lost work potential.

Exergy Analysis

Exergy analysis evaluates the maximum useful work that can be extracted from a system when it is brought into equilibrium with its environment. This method accounts for energy quality in addition to quantity, and is particularly useful for quantifying losses due to irreversibilities. In turbine systems, exergy analysis helps identify where energy is degraded, allowing for an estimation of wasted power potential relative to the environment's baseline state.

Adiabatic Process

An adiabatic process is one in which no heat is exchanged with the surroundings. In turbine analysis, this means that any changes in the energy of the working fluid are solely due to work interactions and internal energy changes, not heat transfer. This assumption simplifies calculations by ensuring that energy losses or gains are attributed only to the work done by or on the fluid.

Irreversibilities and Wasted Power Potential

Irreversibilities refer to the losses in a thermodynamic system that prevent it from reaching its maximum work potential. These losses, arising from factors like friction, non-equilibrium heat transfer, and internal dissipation, result in wasted power potential. The concept ties directly to exergy analysis, where the difference between the ideal and actual work output represents the exergy destruction or lost work due to these irreversibilities.

Transcript

00:01 And this problem, we have a two -stage turbine, where we have steam coming in for the first stage.

00:09 And then coming out of the first stage, we actually split the line.

00:13 And they say for other uses, we take 10 % of the mass of steam coming out of the first stage and we use it for other things.

00:24 And then 90 % of it, we feed back into the second stage.

00:29 And then that comes out there.

00:31 And so obviously these are connected.

00:34 And it's very much similar to the tap we had in the first thing, where we just, you could think of that problem, not in the first thing.

00:40 In another problem we had a, we said we had a tap in here or a bleed line.

00:46 This is kind of like a bleed line, basically, but if you just look at this whole thing as one turbine, it's kind of like a bleed line coming out here.

00:55 So anyway, we have, then the rest of this, 90 % of it going into the same.

01:00 Second stage and then coming out.

01:02 Coming in, we have 7 megapascals and 400 degrees c.

01:08 This is a mass ratio.

01:09 So the mass flow out at 2 here is 10 % of what's coming in here.

01:18 And then the mass flow that goes through here winds up at 3 and into here is 0 .9 times the mass flow rate of here.

01:28 We're told that both stages have an is centropic efficiency of 88%.

01:33 We're also told that the pressure coming out of the first stage is 1 .8 megapascals.

01:40 So if we make a control volume around both these stages here, and again, it's running a steady state so we can use the entropy and figure out the entropy balance, entropy generated.

01:53 So the rate of entropy generation is the rate of entropy flowing out at three, the rate of entropy flowing out at two, and minus the rate of entropy flowing out at two, and minus the rate of entropy flowing out.

02:04 Of entropy flowing in at 1.

02:06 And again, substituting these relationships for the mass flow rates, we get that this is, you know, m .t times 0 .9 times the entropy at this specific entropy here, plus 0 .1 times the specific entropy here, and minus the specific entropy at 1.

02:25 And again, once we know this, we can find the exigy destroyed.

02:29 And i guess they gave us the surrounding temperature, was, what did they have at 25? if i had to do you see, there we come.

02:41 So we need to find these guys.

02:46 Well, at one, that one's easy because we have two thermodynamic properties.

02:50 So we can get the entropy and the entropy there.

02:55 Now, we know the efficiencies.

02:59 So in the isentropic case, the entropy at two would be the entropy at 1, which would give us the iscentropics.

03:08 Anthropic enthalpy at 2, because we'll take pressure at 2 and then the isotropic enthalpy.

03:15 And so we get the isotropic entropy at 2.

03:21 Now because we know the efficiency and we know how the efficiency relates to the enthopies, the real and then the isotropic case, we can then solve for the actual enthalpy at 2.

03:34 And that turns out to be 2 ,870, about 2 ,871 kilojoules per kilogram.

03:43 And now that we know the actual enthalpy at 2, and we know the pressure at 2, we can get the entropy at 2...

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