1. 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)

2. 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

3. 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 ≤ = 3
150 kW
150 kW
275 kW
275 kW

4. 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

5. 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

6. 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

7. 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

8. 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