Automotive Regenerative Gas-Turbine — Schematic, T-s Diagram, Efficiency, Air Flow: 17-Phys-B6 Applied Thermodynamics and Heat Transfer, December 2018, Q3

Question 3: Automotive Regenerative Gas-Turbine — Schematic, T-s Diagram, Efficiency, Air Flow

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A split-shaft regenerative Brayton-cycle automotive gas turbine: air enters the compressor at 30°C, exits at 425°C, is preheated in the regenerator to 1025°C, fuel is burned raising it to 1700°C entering the compressor-drive turbine (which is on the SAME shaft as the compressor, so its work exactly supplies the compressor's work), then expands further through a separate free power turbine to the rear axle, exiting at 1200°C into the regenerator's hot side, which is exhausted at 500°C. No pressures are printed anywhere on the page — every temperature is read directly off the diagram.

Given data (stations per the figure)
Station	Description	$T$
1	Compressor inlet (ambient)	$30\,{}^{\circ}\text{C}$
2	Compressor exit	$425\,{}^{\circ}\text{C}$
3	Regenerator (cold side) exit / combustor inlet	$1025\,{}^{\circ}\text{C}$
4	Combustor exit / compressor-turbine inlet	$1700\,{}^{\circ}\text{C}$
6	Power-turbine exit / regenerator (hot side) inlet	$1200\,{}^{\circ}\text{C}$
7	Regenerator (hot side) exit / exhaust	$500\,{}^{\circ}\text{C}$

Find. (a) A schematic of the plant; (b) the cycle on a $T$–$s$ diagram; (c) the thermal efficiency; (d) the air flow rate for 50 kW at the rear axle.

(a) Schematic: ambient air → compressor → regenerator (cold side) → combustor → compressor turbine (drives the compressor via a shaft) → power turbine (drives the rear wheels) → regenerator (hot side) → exhaust.

(b) The cycle on $T$–$s$ coordinates (schematic, not to scale): 1–2 compression, 2–3 regenerative preheat, 3–4 combustion (external heat input), 4–5 compressor-turbine expansion, 5–6 power-turbine expansion (net work output), 6–7 regenerator heat recovery to the incoming air.

Approach. Station 5 (between the two turbines) is not printed — find it from an energy balance on the shaft: the compressor-turbine's work output must exactly equal the compressor's work input (both on the same shaft, cold-air-standard, constant $c_p$). The regenerator recovers heat internally and is NOT counted as external heat input; the only external heat added is in the combustor (3→4), and the only useful net work is the power turbine's (5→6), since the compressor turbine's work is fully consumed driving the compressor.

Station 5 — shaft energy balance (compressor turbine = compressor work). $$c_p(T_2-T_1)=c_p(T_4-T_5)\quad\Rightarrow\quad T_5=T_4-(T_2-T_1)$$ $$T_5=1973.15-(698.15-303.15)$$ $$\boxed{T_5=1578.2\text{ K}=1305.0\,{}^{\circ}\text{C}}$$
Thermal efficiency. Net work = power-turbine work only; heat input = combustor only: $$\eta_{th}=\frac{\dot W_{net}}{\dot Q_{in}}=\frac{c_p(T_5-T_6)}{c_p(T_4-T_3)} =\frac{T_5-T_6}{T_4-T_3}=\frac{1578.2-1473.2}{1973.2-1298.2}$$ $$\boxed{\eta_{th}=\frac{105.0}{675.0}=0.1556=15.6\%}$$
Air flow rate for 50 kW at the rear axle. The rear-axle power is exactly the power-turbine's net output: $$\dot W_{pt}=\dot m\,c_p(T_5-T_6)\quad\Rightarrow\quad\dot m=\frac{\dot W_{pt}}{c_p(T_5-T_6)}$$ $$\dot m=\frac{50}{1.005\times105.0}$$ $$\boxed{\dot m=0.474\text{ kg/s}}$$

Question 3 — results
Quantity	Value
Station 5 temperature (compressor-turbine exit)	1305.0°C (1578.2 K)
(c) Thermal efficiency $\eta_{th}$	15.6%
(d) Air flow rate $\dot m$	0.474 kg/s

Topic. Regenerative, split-shaft (free-turbine) Brayton cycle — the standard automotive/turboshaft gas-turbine architecture.

Key relations. Shaft power balance $\dot W_{turbine}=\dot W_{compressor}$ for a gas-generator turbine; cold-air-standard energy balance $\dot Q=\dot mc_p\Delta T$; $\eta_{th}=\dot W_{net}/\dot Q_{in}$.

Why this works. Splitting the expansion across two turbines on independent shafts (one purely to drive the compressor, one purely to drive the load) is what lets a gas turbine deliver USEFUL shaft power at all — a single-shaft turbine's work is entirely consumed running the compressor at the design point. Because every station temperature is already printed, the whole problem reduces to bookkeeping energy balances; no isentropic/pressure-ratio relation is needed (consistent with no pressure data being given anywhere on the page).

Common pitfall. Counting the regenerator's heat recovery (2→3, 6→7) as part of $\dot Q_{in}$ — it is recycled INTERNAL heat, not heat supplied from outside the cycle, so only the combustor's 3→4 heat addition belongs in the efficiency denominator; also computing net work as (compressor-turbine work) + (power-turbine work) instead of recognising the compressor turbine's work never leaves the shaft loop.

Source. Y. A. Çengel and M. A. Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 9 (Brayton cycle with regeneration; multistage/split-shaft gas turbines).

Question 3 of 8: Automotive Regenerative Gas-Turbine — Schematic, T-s Diagram, Efficiency, Air Flow

Question 3: Automotive Regenerative Gas-Turbine — Schematic, T-s Diagram, Efficiency, Air Flow