This image represents a network of flow values on a grid, with several branches denoted by $f_1, f_2, \dots, f_7$. There are also external flows (shown as arrows with values) entering and leaving at various points. The problem is asking us to determine the range of flows in each branch, given that the flow on branch $f_4 = 0$.

To determine the range of flow on the remaining branches, we can apply the principle of conservation of mass (flow) at each junction. Specifically, at each junction (e.g., $A, B, C, D, E, F$), the total flow entering the node must equal the total flow exiting it.

The system of equations can be set up as follows based on the conservation of flow at each junction:

1. At Node $A$: $200 = f_3 + f_1$

2. At Node $B$: $f_1 + f_2 = 150 + 200$ $f_1 + f_2 = 350$

3. At Node $C$: $f_2 = 100$

4. At Node $D$: $f_3 + 200 = f_6 + 100$

5. At Node $E$: $f_6 + 100 = f_4 + f_7$ Since $f_4 = 0$: $f_6 + 100 = f_7$

6. At Node $F$: $f_7 = 150 + 100$ $f_7 = 250$

Now, solving these equations:

Solving for $f_2$ and $f_7$:

• From Node $C$, $f_2 = 100$.
• From Node $F$, $f_7 = 250$.

Solving for $f_1$:

• From Node $B$, $f_1 + 100 = 350$, so $f_1 = 250$.

Solving for $f_3$:

• From Node $A$, $200 = f_3 + 250$, so $f_3 = -50$.

Solving for $f_6$:

• From Node $D$, $f_3 + 200 = f_6 + 100$, substituting $f_3 = -50$: $-50 + 200 = f_6 + 100$ $150 = f_6 + 100$ So, $f_6 = 50$.

Thus, the flows on each branch are:

• $f_1 = 250$
• $f_2 = 100$
• $f_3 = -50$ (negative indicates reverse direction)
• $f_4 = 0$
• $f_5 = 250$ (since $f_7 = f_5$)
• $f_6 = 50$
• $f_7 = 250$

Would you like further details or have any questions on this? Here are some related questions to expand the topic:

1. How does the conservation of mass (flow) principle work in network flow problems?
2. What is the significance of a negative flow value like $f_3 = -50$?
3. How would the system change if $f_4$ was non-zero?
4. What are the possible physical interpretations of this type of flow network?
5. Can flow problems like this be extended to larger, more complex networks?

Tip: Negative flow in network problems often indicates that the assumed flow direction is opposite to the actual direction.