Euler’s network theorem

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Presentation transcript:

Euler’s network theorem Node/Vertice (V) V + R – L = 1 Arc / Lines (L) Loop / Region (R) 3 + 1 – 3 = 1 1 + 0 – 0 = 1 1 + 1 – 1 = 1(add a line) 1 + 1 – 1 = 1(add a line) 2 + 0 – 1 = 1(add a node)

Euler’s network theorem applied to heat exchanger networks A more general formulation of Euler's theorem : V + R – L = S S = subgraphs/networks Formulation adapted for heat exchanger networks: U = N + L - S U = number of units, i.e. heat exchangers and heaters and coolers L = heat load loops N = Number of streams and utilities Heat exchangers connected so that the heat loads of the units can be adjusted without affecting the target temperatures comprise a heat load loop. -x +x +x -x

Umin, mer ≤ (N-1)above pinch + (N-1)below pinch Upper bound for the number of units needed for a maximum heat recovery (MER) network U = N + L - S U = number of units, L = heat load loops N = Number of streams and utilities S = Sub networks Umin, mer ≤ (N-1)above pinch + (N-1)below pinch 150 kW 275 kW N = 4 Umin ≤ 4 - 1 = 3 150 kW 150 kW 275 kW 275 kW

Minimum hot utility: 130 kW Minimum cold utility: 130 kW m·cp = 1.5 kW/K 300°C 100°C H1 m·cp = 5.0 kW/K 200°C 100°C H2 250°C m·cp = 4.0 kW/K 50°C C1 ΔTmin = 20°C Pinch: 190°C Minimum hot utility: 130 kW Minimum cold utility: 130 kW Available 1 hot and 1 cold utility

U = 3 – 1 = 2 U = 4 – 1 = 3 H1 H2 C1 Pinch 130 kW 130 kW 150 kW 500 kW m·cp = 1.5 200°C 100°C H2 m·cp = 5.0 250°C 217.5°C 180°C 55°C 50°C 130 kW C1 m·cp = 4.0 150 kW 500 kW 20 kW U = 3 – 1 = 2 U = 4 – 1 = 3

U = 5 – 1 = 4 H1 H2 C1 -15 kW +15 kW 130 kW 130 kW 165 kW 500 kW 5 kW m·cp = 1.5 130 kW 200°C 100°C H2 m·cp = 5.0 250°C 217.5°C 176.3°C 51.3°C 50°C C1 m·cp = 4.0 130 kW 165 kW 500 kW 5 kW -15 kW +15 kW U = 5 – 1 = 4

U = 5 – 1 = 4 H1 H2 C1 -20 kW +20 kW 130 kW 130 kW 170 kW 500 kW 0 kW m·cp = 1.5 130 kW 200°C 100°C H2 m·cp = 5.0 250°C 217.5°C 175°C 50°C 50°C C1 m·cp = 4.0 130 kW 170 kW 500 kW 0 kW -20 kW +20 kW U = 5 – 1 = 4

300°C 195°C 100°C H1 m·cp = 1.5 142.5 kW 200°C 100°C H2 m·cp = 5.0 250°C 214.4°C 175°C 50°C C1 m·cp = 4.0 142.5 kW 157.5 kW 500 kW 20 kW