Manual Headloss Calculation

Manual Headloss Calculation

Assumptions:

The water level in the inlet pipe is assumed to be equal to the water level in the manhole. This implies that the expansion loss at the inlet is automatically assumed. All calculated energy losses are assumed to occur at the outlet pipe, i.e.:

\[ E_{manhole} = E_{pipe} + \Delta H \]

Equation: Energy balance equating manhole energy to inlet pipe energy plus energy loss.

or expressed by using the notation in the sketch defining the terms of the example:

\[ \frac{v_{m}^{2}}{2g}\!+\!y_{m}\!+\!Z_{m}\!=\frac{v_{o u t}^{2}}{2g}\!+\!y_{o u t}\!+\!Z_{o u t}\!+\!\zeta\frac{v_{o u t}^{2}}{2g} \]

Equation: Energy equation relating manhole velocity head and elevation to outlet conditions and energy loss.

z   =   a coefficient expressing the total outlet energy loss (see Section 4.6.3 Headloss calculation for outflowing water (p. 60)).

Data:

Discharge                     Q   =   2.0   m³s^-1 Diameter in outlet pipe                     D_out   =   1.0   m Diameter in manhole                     D_m   =   1.5   m Velocity in outlet pipe (Q is capacity assumed)                     V_out   ~   2.55   ms^-1 Length of outlet pipe                     L   =   50.0   m Manning number                     M   =   70   m^1/3s^-1 Water level in outlet                     H_out   =   15.0   m Bottom level in manhole                     Z_m   =   14.0   m Head “shape” loss coefficient                     K_m   =   0.5

Friction loss in outlet pipe from manhole to outlet:

\[ h_{f} = \frac{Q^{2}L}{M^{2}A^{2}R^{\frac{4}{3}}} = \frac{2.0^{2} \cdot 50.0}{70^{2} \cdot 0.7854^{2} \cdot 0.25^{\frac{4}{3}}} \]

Equation: Friction loss in a pipe outlet using Manning formula with channel geometry and roughness.

The water level in the manhole (H) can be found from:

\[ H + \frac{v_m^2}{2g} = H_{out} + h_f + \frac{v_{out}^2}{2g} + \zeta \frac{v_{out}^2}{2g} \]

Equation: Energy equation across a manhole accounting for pressure, velocity head, and friction losses.

v_m is approximated by assuming that y_m = y_out:

\[ v_{m} = \frac{Q}{A_{m}} = \frac{Q}{D_{m}y_{m}} = \frac{2.0}{1.5 \cdot \left(15.0 + 0.42 -14.0\right)} = 0.94ms^{-1} \]

Equation: Velocity calculated from discharge and cross-sectional area in manhole node.

\[ v_{out} = \frac{Q}{A_{out}} = \frac{2.0}{0.78} = 2.55 ms^{-1}, \quad \frac{v_{out}^2}{2g} = 0.33 m \]

Equation: Example calculation of outlet velocity and kinetic head in a manhole.

\[ \frac{v_{m}^{2}}{2g}~=~0.045~\mathrm{m} \]

Equation: Velocity head at manhole chamber calculated from flow velocity.

\[ \zeta = \zeta_{contr} + \zeta_{dir} = K_m \cdot \left(1 - \frac{A_{out}Q_{in}}{Q_{out}A_m}\right) + \frac{Q_{in}}{Q_{out}} \cdot \frac{\theta^2}{90^2} = 0.5 \cdot \left(1 - \frac{0.78}{1.5 \cdot 1.42}\right) + \frac{45^2}{90^2} = 0.567 \]

Equation: Local headloss coefficient combining contraction and directional change effects at manhole outlet.

Substituting values to the Equation the velocity calculated from discharge and cross-sectional area in manhole node and calculating H yields:

\[ H~=~15.89~\mathrm{m} \]

Equation: Calculated water level result from manhole headloss energy equation.

The deviation between the MOUSE simulation and manual calculation result is due to the fact that MOUSE calculates v_out by using the following area in the pipe:

\(A_{out} = \frac{D_{out}^2 \pi}{4} + \text{Preismann slot area}\) 

The table shows which setups have been used for the calculation, and also which types of headloss are included. z_c0, z_c1, z_c2 are all headloss coefficients due to contraction and correspond to the types (a), (b) and (c).

Table: Test matrix for implementation of Headloss Type.

Test	Setup	Direction zd	Drop zl	Contraction zc (a) zc0	Contraction zc (b) zc1	Contraction zc (c) zc0	Hcalc	HM
1	I)			X			15.81	15.82
2	I)				X		15.87	15.87
3	I)					X	15.87	15.87
4	II)	X		X			15.89	15.90
5	II)	X			X		15.95	15.94
6	II)	X				X	15.87	15.87
7	III)		X	X			15.94	15.94
8	III)		X		X		16.00	15.99
9	IV)	X	X	X			16.02	16.02
10	IV)	X	X		X		16.03	16.02