Backwater curve length from specific-energy and slope data If E1 is the specific energy at the start of a backwater curve and E2 at the maximum water-surface rise, and i_b and i_e are bed slope and energy-grade-line slope respectively, the backwater length L is

Difficulty: Medium

Correct Answer: L = (E2 − E1) / (i_b − i_e)

Explanation:


Introduction / Context:
Gradually varied flow (GVF) profiles like backwater curves are evaluated using the specific-energy form of the GVF equation. Over a reach, changes in specific energy relate directly to the difference between bed slope and energy grade line slope, allowing estimation of the water-surface profile length.


Given Data / Assumptions:

  • Steady, gradually varied open-channel flow.
  • Specific energies E1 and E2 at the start and at the section of maximum rise.
  • Bed slope i_b and energy-grade-line slope i_e approximately uniform over the reach.


Concept / Approach:

The differential form is dE/dx = i_b − i_e. Integrating from section 1 to section 2 gives E2 − E1 = ∫(i_b − i_e) dx ≈ (i_b − i_e) L if slopes are roughly constant. Hence L = (E2 − E1)/(i_b − i_e).


Step-by-Step Solution:

Start from GVF: dE/dx = i_b − i_e.Assume i_b and i_e constant over the small reach.Integrate: E2 − E1 = (i_b − i_e) L ⇒ L = (E2 − E1)/(i_b − i_e).


Verification / Alternative check:

Backwater curves have i_b > i_e; thus denominator is positive, giving a positive L for E2 > E1, consistent with a rise in the water surface along the flow.


Why Other Options Are Wrong:

(b) Changes sign incorrectly; (c) ignores bed slope; (e) is dimensionally inconsistent. (d) is unnecessary since (a) is correct.


Common Pitfalls:

Confusing specific energy with total head; overlooking that i_e includes both friction and minor losses where relevant.


Final Answer:

L = (E2 − E1) / (i_b − i_e)

Discussion & Comments

No comments yet. Be the first to comment!
Join Discussion