Error propagation for triangular weirs/notches: For a V-notch (triangular notch), discharge varies with head as Q ∝ H^(5/2). What is the ratio of the percentage error in discharge to the percentage error in measured head over the notch?
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A2/3
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B3/2
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C2/5
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D5/2
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E4/3
Answer
Correct Answer: 5/2
Explanation
Introduction / Context:Flow measurement over notches relies on empirical–theoretical relations between head (H) and discharge (Q). Understanding how measurement errors in H propagate into errors in Q is crucial for instrumentation accuracy and calibration protocols.
Given Data / Assumptions:
- Triangular (V-notch) weir with Q = K H^(5/2), where K includes notch angle and discharge coefficient.
- Small measurement errors relative to the nominal head (linearized error propagation).
- Steady flow; proper approach conditions.
Concept / Approach:
For Q = K H^n, fractional errors relate by dQ/Q = n dH/H. Thus, the percentage error in Q is n times the percentage error in H. For a V-notch, n = 5/2, significantly amplifying head measurement errors.
Step-by-Step Solution:
Start with Q = K H^(5/2).Take logarithms: ln Q = ln K + (5/2) ln H.Differentiate: dQ/Q = (5/2) dH/H.Therefore, %error(Q) / %error(H) = 5/2.Verification / Alternative check:
Compare with rectangular notch where Q ∝ H^(3/2): the ratio would be 3/2, confirming the exponent-rule consistency.
Why Other Options Are Wrong:
(a), (b), (c), and (e) correspond to other exponents or are arbitrary; only 5/2 matches the V-notch exponent.
Common Pitfalls:
Using absolute errors instead of fractional; forgetting that the exponent multiplies the fractional error.
Final Answer:
5/2