Extended Bernoulli application: when work is done on the fluid between two sections (e.g., a pump adds head), this term should be...

Introduction / Context:Real systems often include devices that add or remove energy (pumps, turbines). The extended Bernoulli equation accounts for these by adding a pump head h_p (work done on the fluid) or subtracting a turbine head h_t (work extracted from the fluid) between two sections.

Given Data / Assumptions:

• Steady, incompressible flow.
• Sections 1 (upstream) and 2 (downstream) connected through a system with possible head losses h_f.
• Sign convention: head added to the fluid is positive h_p; head extracted is h_t.

Concept / Approach:

A common, consistent form is: p1/γ + v1^2/(2g) + z1 + h_p − h_t − h_f = p2/γ + v2^2/(2g) + z2. Thus, “work done on the flow system” (pump head) is added to the left-hand side (together with the upstream energy terms) to balance the downstream head. Equivalent rearrangements are possible, but the standard teaching convention places h_p on the upstream (left) side as an addition.

Step-by-Step Solution:

Verification / Alternative check (if short method exists):

Moving h_p to the right-hand side yields identical results; however, the pedagogical standard is the LHS addition convention used above.

Why Other Options Are Wrong:

(b) Uses a different convention than the one requested; (c) ignores essential energy transfer; (d) is nonsensical for balancing; (e) is unnecessary.

Common Pitfalls (misconceptions, mistakes):

Forgetting sign conventions; mixing pump head with turbine head; neglecting head losses in long pipelines.

Final Answer:

added on the left side of the energy equation with the upstream section