Loss due to sudden enlargement and contraction

Loss due to sudden enlargement

  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)

  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.