Lecture 4 Ideal cycles III Polytropic efficiencies Exercise Reheat cycle Intercool cycle The WR21 engine Polytropic efficiencies Exercise Problem 2.1, 2.3 and 2.9

Reheat cycle/Reheat with heat exchanger Split expansion into a high pressure and a low pressure step and reheat in between

Selection of pressure ratio – reheat cycle

Theory 4.1 - Selection of optimal pressure ratio – reheat cycle Introduce auxiliary variable β according to:

Theory 4.1 - Selection of optimal pressure ratio – reheat cycle Insert result into power formula:

Efficiency for reheat cycle (at pressure division for max Efficiency for reheat cycle (at pressure division for max. power output)

Cycle changes due to reheat Figure 2.5 or better from Cengel You introduce an “additional cycle” operating at lower pressure ratio. We have already derived what we want to know!!! Decreasing pressure ratio in simple cycle => efficiency decreases.

Reheat/reheat with heat exchange compared to single cycle Simple reheat Power output increases Decrease in efficiency (added cycle is worse than underlying cycle, since simple cycle efficiency decreases with pressure ratio) Reheat with heat exchange Increase in efficiency. Heat is added at a higher average temperature and removed at a lower temperature than in simple cycle. See figure to the right. Simple cycle Reheat with heat exchange

Intercooling Bulky and requires large amounts of cooling water Compactness and self-containedness of gas turbine is lost What about efficiency and power output of cycle ?.... Try to draw a T-S diagram and make some arguments. Check with CRS.

The WR 21 ICR cycle - Intercooled Recuperated Cycle Improved part load performance => 30% reduction in fuel burn for a typical operating profile 25 MW output Fits in footprint of current naval engines of similar power. LM2500 ηth=37. ICR ηth=43. Starts in two minutes instead of 4 hours for comparable steam engine. Greater power for given space when compared with steam/diesel. The WR 21

The WR 21

Polytropic efficiencies - motivation If we study multistage designs the isentropic efficiency for high pressure compressors tend to be lower than for low pressure compressors. Why? Memorize these! You are going to be using them all the time. Assume ηs (stage efficiency) constant, the overall temperature rise ΔT is obtained by:

Polytropic efficiency - motivation Thus, the total efficiency is always less than the stage efficiency. But: Remember that ΔT is proportional to the inlet temperature for compressors and turbines. Preheat effect: as you go through the stages you move to the right in the T-s diagram. Isobars diverge in that direction!

Polytropic efficiency – “preheat independence” Define the polytropic efficiency (differential stage efficiency) as: We have (second revision question – lecture 1 – before integrating): Divide with T in (1) and use differential isentropic formula (1) + (2) produces:

Polytropic efficiency – “preheat independence” Similarly for a turbine: First guess for preliminary design work Polytropic efficiencies are useful for preliminary design, when many compressor concepts with different pressure ratios may be evaluated for a given application.

Recommendation to get started with the course Work through Example 2.1 and 2.2 at home (page 74-78). Derive the optimal pressure ratio (for maximum power) for the simple cycle gas turbine (Theory 2.1) Derive the efficiency for the heat-exchange cycle (Theory 3.1) Read ”very important” sections as stated in course PM (so far section 2.1, 2.2 and 2.3) Start with Design Task 1 !!! If you have time: Read all ”important” sections as well. Work through example 2.3 and attempt to solve problem 2.5.

Learning goals Know how to show (by arguments or T-S diagrams) how the efficiency of the reheat cycle with and without heat exchange changes in comparison with the simple cycle (ideal case) Be able to derive the optimal pressure division in ideal reheat cycles Be familiar with the polytropic efficiency concept and state reasonable loss levels for turbine and compressors