--- title: "Loss due to sudden enlargement and contraction" subject: "Hydraulics" chapter: 1 key: "semester_4/hydraulics:sudden-enlargement-contraction-loss" source: https://www.bctengineer.com/q/semester_4/hydraulics/sudden-enlargement-contraction-loss site: BCT Engineering --- # Loss due to sudden enlargement and contraction Hydraulics · Chapter 1: Pipe flow - Asked in: 2079 Chaitra, 8 marks - Video: [Head Loss Due to Sudden Enlargement](https://www.youtube.com/watch?v=RCKfQgnp5sU) — TutorialsPoint - Answer checked by Claude Sonnet 5.5 on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/hydraulics/sudden-enlargement-contraction-loss - Whole chapter: https://www.bctengineer.com/md/semester_4/hydraulics/chapter-1 ## Question Derive an expression for the loss due to sudden enlargement in pipe flow and therefrom deduce the loss due to sudden contraction. ## Short answer Loss due to sudden enlargement is h_e = (V1 - V2)²/(2g) = (1 - A1/A2)² V1²/(2g). Loss due to sudden contraction is deduced by applying the enlargement formula at the vena contracta: h_c = (1/Cc - 1)² V2²/(2g), with Cc ≈ 0.62 giving k_c ≈ 0.375 (commonly ~0.5). ## Answer #### Loss due to sudden enlargement ```text 1 2 _______ _________ Pipe area A1 -> A2 (A2 > A1) | ~ ~ ~ | Eddies form in the corner; V1,p1 | eddy | V2,p2 the pressure on the annular face _______| zone |_________ equals p1. ``` Consider section 1 (just before the enlargement) and section 2 (downstream, where the flow has spread over the full area $A_2$). Assume the pipe is horizontal and neglect wall friction between 1 and 2. The pressure on the annular shoulder is taken as $p_1$ (eddy zone). Momentum equation: $$ p_1A_1 + p_1(A_2 - A_1) - p_2A_2 = \rho Q (V_2 - V_1), \qquad Q = A_2V_2 $$ $$ \frac{p_1 - p_2}{\gamma} = \frac{V_2(V_2 - V_1)}{g} \quad \cdots (1) $$ Energy equation (head loss $h_e$): $$ \frac{p_1}{\gamma} + \frac{V_1^2}{2g} = \frac{p_2}{\gamma} + \frac{V_2^2}{2g} + h_e \quad\Rightarrow\quad h_e = \frac{p_1-p_2}{\gamma} + \frac{V_1^2 - V_2^2}{2g} $$ Substituting (1): $$ h_e = \frac{V_2^2 - V_1V_2}{g} + \frac{V_1^2 - V_2^2}{2g} = \frac{V_1^2 - 2V_1V_2 + V_2^2}{2g} $$ $$ \boxed{h_e = \frac{(V_1 - V_2)^2}{2g} = \left(1 - \frac{A_1}{A_2}\right)^2\frac{V_1^2}{2g}} $$ #### Loss due to sudden contraction (deduced) ```text 1 c 2 Jet contracts to the vena contracta c ________ (area Cc A2), then expands to fill \ ~~~ _ the smaller pipe A2. V1 \___|___ V2 Loss is only in the expansion c -> 2. ________/ Vc ``` In a sudden contraction from $A_1$ to $A_2$ the stream contracts to a vena contracta of area $A_c = C_c A_2$, then expands suddenly to fill the pipe $A_2$. The only significant loss is this expansion from $c$ to 2. Applying the enlargement result between $c$ and 2: $$ h_c = \frac{(V_c - V_2)^2}{2g}, \qquad V_c = \frac{A_2V_2}{A_c} = \frac{V_2}{C_c} $$ $$ \boxed{h_c = \left(\frac{1}{C_c} - 1\right)^2\frac{V_2^2}{2g} = k_c\frac{V_2^2}{2g}} $$ With $C_c \approx 0.62$, $k_c = (1/0.62 - 1)^2 \approx 0.375$; a common value used is $k_c \approx 0.5$ for a large area ratio. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What pressure is assumed on the annular shoulder in the sudden enlargement derivation? A. p2 B. p1, due to eddies in the corner zone C. Atmospheric D. Zero Answer: B. The eddy zone keeps the shoulder pressure equal to p1, giving the momentum equation. (Key point: Pressure on shoulder) 2. What is the final loss for a sudden enlargement? A. (V1 + V2)^2/2g B. (V1^2 - V2^2)/2g C. 0.5 V1^2/2g D. (V1 - V2)^2/2g Answer: D. Combining momentum and energy gives h_e = (V1 - V2)^2/2g = (1 - A1/A2)^2 V1^2/2g. (Key point: Enlargement loss) 3. How is the sudden contraction loss deduced? A. Loss occurs in the contraction to the vena contracta B. As the enlargement loss between the vena contracta c and section 2 C. By reversing the sign of the enlargement loss D. By assuming zero loss Answer: B. The jet contracts to a vena contracta with little loss, then expands suddenly into pipe A2, so the enlargement formula applies from c to 2. (Key point: Contraction loss idea) 4. What is the loss formula for the contraction in terms of Cc? A. (1 - Cc)^2 V2^2/2g B. (1/Cc - 1)^2 V2^2/2g C. Cc V2^2/2g D. (1/Cc + 1)^2 V2^2/2g Answer: B. Vc = V2/Cc, so (Vc - V2)^2/2g = (1/Cc - 1)^2 V2^2/2g. 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