Question 2 of 5: Heat Integration – Pinch Analysis & Heat Exchanger Network

Question 2: Heat Integration – Pinch Analysis & Heat Exchanger Network (24 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.

ΔT_min = 10°C. Utilities: LPS (≈160°C condensing, 2000 kJ/kg), HPS (≈250°C condensing, 1700 kJ/kg), CW (30→45°C, 63 kJ/kg sensible).

Find. (a) Utility type and flow rate per stream with no integration. (b) Pinch temperature and minimum hot/cold utility duties via the problem-table (temperature-interval) method. (c) A feasible heat exchanger network achieving those minimum utility targets.

Approach. Compute each stream's duty CP·ΔT; without integration each stream is served by whichever utility is both hot/cold enough and cheapest, sized by Q = ẓ·ΔH. With integration, shift hot streams down and cold streams up by ΔT_min/2, tabulate the net CP imbalance in each temperature interval between consecutive shifted stream temperatures, cascade the heat downward from zero at the top, and add the minimum heat needed at the top to make every cascade value ≥0 — the point where the cascade first reaches exactly zero is the pinch. A network is then built stream-by-stream honouring the pinch-matching feasibility rule (CP_hot ≤ CP_cold for any match touching the pinch from above).

1. (a) Utility duties without heat integration. Each stream's total duty: $$Q_{H101}=33.0\times(190-50)=4620\ \text{kW},\quad Q_{C102}=48.0\times(175-85)=4320\ \text{kW},\quad Q_{C103}=20.0\times(115-50)=1300\ \text{kW}$$ C102's target (175°C) requires a utility temperature of at least 175+10=185°C to hold a 10°C approach, which LPS ($\approx$160°C condensing) cannot supply, so C102 must be served by HPS; C103's target (115°C) needs only $\ge$125°C, comfortably below LPS's ceiling, so the cheaper LPS serves it; H101 needs only sensible cooling well within the CW range, so CW serves it. Sizing each from ẓ = Q/|ΔH|: $$\dot m_{HPS}=\frac{4320}{1700}=\boxed{2.541\ \text{kg/s}},\qquad \dot m_{LPS}=\frac{1300}{2000}=\boxed{0.650\ \text{kg/s}},\qquad \dot m_{CW}=\frac{4620}{63}=\boxed{73.33\ \text{kg/s}}$$
2. (b) Shifted temperatures and the problem-table cascade. Shift the hot stream down 5°C and the cold streams up 5°C (half of ΔT_min) onto a common scale: H101*: 185→45°C; C102*: 90→180°C; C103*: 55→120°C. The six distinct shifted temperatures (185, 180, 120, 90, 55, 45) bound five internal intervals; in each interval, net = (ΣCP_hot − ΣCP_cold)×ΔT of the streams spanning it: $$[185,180]:\ (33-0)(5)=+165 \quad [180,120]:\ (33-48)(60)=-900 \quad [120,90]:\ (33-48-20)(30)=-1050$$ $$[90,55]:\ (33-20)(35)=+455 \quad [55,45]:\ (33-0)(10)=+330$$ Cascading from Q=0 at the top gives running totals 0, 165, −735, −1785, −1330, −1000 — the most negative value is −1785 kW, so adding Q_H,min = 1785 kW at the top makes every value ≥0. The re-cascaded values are 1785, 1950, 1050, 0, 455, 785 — the cascade hits exactly zero at shifted T*=90°C, which is the pinch (actual hot-stream pinch temperature 95°C, actual cold-stream pinch temperature 85°C), and the final value, Q_C,min = 785 kW, is the minimum cold utility. $$Q_{H,min}=\boxed{1785\ \text{kW (above pinch)}}\qquad Q_{C,min}=\boxed{785\ \text{kW (below pinch)}}$$
3. Energy-balance check. Total process heating demand minus total cooling demand (4320+1300−4620 = 1000 kW) must equal Q_H,min−Q_C,min = 1785−785 = 1000 kW — confirmed. Heat integration recovers 3735 kW of process-to-process exchange (versus 5620+4620 = 10,240 kW of total utility duty with no integration at all), a 74.5% reduction in total utility load.

4. (c) Heat exchanger network design. Above the pinch there is one hot stream (H101, CP=33) and two cold streams (C102, CP=48; C103, CP=20); since N_hot (1) ≤ N_cold (2) above the pinch, no stream split is required. Checking the pinch-matching feasibility rule (CP_hot≤CP_cold for any match touching the pinch), H101 (CP=33) can only be matched against C102 (CP=48≥33) at the pinch — matching directly against C103 (CP=20<33) would violate the rule and cause the approach temperature to shrink below 10°C moving away from the pinch. C102's cold inlet (85°C) sits exactly at the cold-pinch temperature, so E1 is the pinch match: running H101 the full length of its above-pinch range (95→190°C) into C102 transfers $$Q_{E1}=33\times(190-95)=\boxed{3135\ \text{kW}}\ \Rightarrow\ C102: 85^\circ\text{C}\rightarrow\left(85+\frac{3135}{48}\right)=150.3^\circ\text{C}$$ (feasible: 10°C approach at the pinch end, 39.7°C at the hot end). C102's remaining duty up to its 175°C target, $$Q_{E2}=4320-3135=\boxed{1185\ \text{kW}}=Q_{H,min}-Q_{E3},$$ must be supplied by HPS (1185/1700 = 0.697 kg/s), since 175°C is beyond LPS's reach. C103 (85→115°C, CP=20) is not touched by H101 at all (H101 is fully committed to C102 above the pinch), so it is heated entirely by LPS: $$Q_{E3}=20\times(115-85)=\boxed{600\ \text{kW}}\qquad(0.300\ \text{kg/s LPS})$$ confirming $Q_{E2}+Q_{E3}=1185+600=1785\ \text{kW}=Q_{H,min}$. Below the pinch, only H101 (CP=33) and C103's below-pinch segment (50→85°C, CP=20) remain (N_cold=1≤N_hot=1, CP_hot≥CP_cold as required below the pinch), forming the second process match, E4: $$Q_{E4}=20\times(85-50)=\boxed{700\ \text{kW}}\ \Rightarrow\ H101: 95^\circ\text{C}\rightarrow\left(95-\frac{700}{33}\right)=73.8^\circ\text{C}$$ (feasible: 10°C approach at the pinch end, 23.8°C at the cold end). H101's remaining duty down to its 50°C target, $$Q_{E5}=33\times(73.8-50)=\boxed{785\ \text{kW}}=Q_{C,min},$$ is removed by CW (785/63 = 12.46 kg/s), confirming the below-pinch target. This 5-exchanger network (E1, E2-heater, E3-heater, E4, E5-cooler) achieves both minimum utility targets exactly, with no stream split anywhere in the design.