Question 2 of 8: Heat Recovery Steam Generator

Question 2: Heat Recovery Steam Generator (10 marks)

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.

Find. The temperature profile of both streams, the enthalpy at each key point, the steam mass flow rate, the pinch-point temperature difference, and the physical arrangement that produces that profile.

Approach. Take the total duty from the gas side, since both gas terminal temperatures are given; get the steam enthalpy rise from the steam tables; divide to obtain the steam flow; then split the duty into superheater, evaporator and economiser and walk the gas temperature back from the stack to locate the pinch.

1. Part (b) — enthalpies at the key points. The water side is fixed by the pressure of 1.40 MPa, at which the steam tables give $T_{sat}=195.07\,{}^{\circ}\text{C}$, $h_f=830.30$ kJ/kg and $h_g=2790.0$ kJ/kg (so $h_{fg}=1959.7$ kJ/kg). The feedwater is a compressed liquid at 30 °C, for which $h_d\simeq h_f(30\,{}^{\circ}\text{C})=125.8$ kJ/kg; the pressure correction $v\,\Delta p = 0.001004\times 1395.8 = 1.4$ kJ/kg is negligible against a 3400 kJ/kg rise and is neglected. The superheated outlet at 1.40 MPa and 540 °C is interpolated between the 500 °C and 600 °C table entries, $$h_a=3474.1+\frac{540-500}{100}\,(3694.8-3474.1)=3562.4\ \text{kJ/kg}.$$
2. Collecting the four water-side points. Reading along the water path from inlet to outlet gives the values tabulated below, which are the four numbers part (b) asks for.

   Point	State	Temperature	Enthalpy
   d	compressed liquid feedwater	30 °C	125.8 kJ/kg
   c	saturated liquid, 1.40 MPa	195.07 °C	830.3 kJ/kg
   b	saturated vapour, 1.40 MPa	195.07 °C	2790.0 kJ/kg
   a	superheated steam, 1.40 MPa	540 °C	3562.4 kJ/kg

3. Part (c) — total duty and steam mass flow rate. The gas side is fully determined, so it fixes the duty: $$\dot{Q}=\dot{M}_g\,c_{p,g}\,(T_1-T_4)=125\times 1.148\times(560-130) =143.5\times 430=61\,705\ \text{kW}.$$ The water absorbs all of it (an unfired heat recovery steam generator is adiabatic to atmosphere), and its enthalpy rise is $h_a-h_d=3562.4-125.8=3436.6$ kJ/kg, so $$\boxed{\dot{M}_{steam}=\frac{61\,705}{3436.6}=17.96\ \text{kg/s}} ,$$ about one seventh of the gas flow — a ratio typical of an unfired single-pressure recovery boiler.
4. Part (d) — splitting the duty between the three surfaces. With the steam flow known, each surface's duty follows from the enthalpy interval it covers: $$\dot{Q}_{econ}=\dot{M}_{steam}(h_c-h_d)=17.96\times 704.5=12\,649\ \text{kW},$$ $$\dot{Q}_{evap}=\dot{M}_{steam}\,h_{fg}=17.96\times 1959.7=35\,187\ \text{kW},$$ $$\dot{Q}_{sup}=\dot{M}_{steam}(h_a-h_b)=17.96\times 772.4=13\,869\ \text{kW}.$$ These sum to 61 705 kW, reproducing the gas-side duty exactly, which confirms the steam flow.
5. Locating the pinch. The pinch is at the point where the water first reaches saturation, because that is where the water-side line goes flat while the gas-side line keeps falling. Working the gas back from the stack through the economiser only, $$\Delta T_{gas,econ}=\frac{\dot{Q}_{econ}}{\dot{M}_g c_{p,g}} =\frac{12\,649}{143.5}=88.2\ \text{K},$$ so the gas at the evaporator outlet is at $130+88.2=218.2\,{}^{\circ}\text{C}$. Working forward instead — taking the superheater and evaporator duties off the 560 °C inlet — gives $560-96.6-245.2=218.2\,{}^{\circ}\text{C}$, the same value from the opposite direction, which is the check worth doing.
6. The pinch-point temperature difference. Subtracting the saturation temperature of the boiling water at that same station, $$\boxed{\Delta T_{pinch}=218.2-195.07=23.1\ \text{K}} .$$ Every other terminal difference is larger — 20 K at the superheater end (560 against 540) and 100 K at the stack end (130 against 30) — so 23.1 K is indeed the closest approach anywhere in the unit, and it is comfortably positive, so there is no temperature cross.
7. Part (e) — why the pinch governs the design. The pinch is the tightest approach temperature in the whole exchanger, and since the local heat flux is proportional to the local temperature difference, it is the point where surface area is bought most dearly. Narrowing the pinch recovers more heat: the gas can be cooled further before the economiser, the stack temperature falls, and both the steam flow and the bottoming-cycle output rise. But the surface required to transfer a given duty varies as $1/\Delta T$, so the last few kelvin cost disproportionate tube area, pressure drop and capital. A single-pressure unit is therefore normally designed to a pinch of roughly 10–25 K, and the 23.1 K found here sits at the relaxed end of that band. The pinch also fixes what the plant can do off design: as load falls the gas flow and inlet temperature change, the pinch migrates, and if it were set too tight at design point it can close to zero or invert at part load, at which point the evaporator simply stops generating. Finally, the pinch is the reason modern combined-cycle plants use two or three pressure levels — each additional pressure adds another evaporator with its own pinch lower down the gas path, so the gas can be cooled closer to the feedwater temperature without any single approach becoming impossibly tight.
8. Part (f) — the physical arrangement. The three surfaces must be strung along the gas path in the order that puts the hottest surface in the hottest gas: superheater first, then evaporator, then economiser nearest the stack, with the water flowing in the opposite direction. The evaporator is not a once-through pass but a natural-circulation loop hung off a steam drum, as sketched below.
   Figure 2.2 — Part (f): the actual arrangement. Gas flows upward through the casing past superheater, evaporator and economiser in that order; feedwater enters the economiser at d, leaves it as saturated water at c and passes to the steam drum; the drum feeds the evaporator through downcomers and receives the saturated mixture back through risers at b; dry saturated steam is taken from the top of the drum to the superheater and leaves at a. The letters match Figure 2.1.