A compound pipeline consists of two pieces of identical pipes. The equivalent length of the same diameter and the same friction factor for this compound pipeline is L₁ when the pipes are connected in series and is L₂ when the pipes are connected in parallel. Determine the ratio of equivalent lengths, that is, L₁ to L2.

Concept:

The equivalent length of pipes connected in series and parallel can be calculated using the following principles:

- For pipes connected in series, the equivalent length is the sum of the lengths of the individual pipes.
- For pipes connected in parallel, the equivalent length is determined using the relationship of hydraulic resistance, which results in a different calculation compared to series connection.

Calculation:

Given:

- Two identical pipes with the same diameter (d) and length (l).
- The Darcy friction factor is the same for both pipes.

Series Connection:

For two identical pipes connected in series, the equivalent length (L₁) is simply the sum of the lengths of the two pipes.

L₁ = l + l = 2l 

Parallel Connection:

For two identical pipes connected in parallel, the equivalent length (L₂) is calculated using the formula for the equivalent resistance of pipes in parallel. For identical pipes, this relationship can be expressed as:

\(\left( \frac{d_{eq}^5}{L_2} \right)^{\frac{1}{2}} = 2 \left( \frac{d^5}{l} \right)^{\frac{1}{2}} \)

Since the pipes are identical and (d_eq = d), we can simplify the equation:

\(\left( \frac{d^5}{L_2} \right)^{\frac{1}{2}} = 2 \left( \frac{d^5}{l} \right)^{\frac{1}{2}} \)

Squaring both sides, we get:

\(\frac{d^5}{L_2} = 4 \left( \frac{d^5}{l} \right) \)

\(L_2 = \frac{l}{4}\)

Ratio of Equivalent Lengths:

The ratio of the equivalent lengths for series and parallel connections is:

\(\frac{L_1}{L_2} = \frac{2l}{\frac{l}{4}} = \frac{2l \times 4}{l} = 8 \)

Therefore, the ratio of equivalent lengths is: 8:1