Resistance To Flow in Open and Closed Conduits

Resistance To Flow in Open and Closed Conduits
Types of flow
1. Open conduits ( open channel flow)
Definition: Open-channel flow is the flow of a liquid in a conduit
with a free surface, which has atmospheric pressure
over it.
? There are many practical examples, both artificial (flumes, spillways, canals, weirs, drainage ditches, culverts) and natural (streams, rivers, estuaries, and floodplains).
? Such flows are dominated by the effects of gravity.
Where the balance of forces is due to gravity and friction.
Since the liquid density is nearly constant, the steady-flow continuity equation reduces to constant-volume flow (Q) along the channel:

Q = V(x)A(x) = constant
Where:
Q= volume flow rate
A=cross sectional area
P= hydraulic parameter
Dh= hydraulic diameter
Dh= 4A/P
Rh= hydraulic radius
Rh= 1/4 Dh
Rh= A/P
Note that pressure on free surface is equal at all sections:
P1= P2= P3= Pa.
The steady state Energy Eq. is:

P_1/?+(?v_1?^2/2g)+z_1= P_2/?+(?v_2?^2/2g)+z_2+ h_f (1)
Where: hf =head losses due to friction

h_f=f (x_2-x_1)/D_h (v_av^2)/2g (2)

Where : f= coefficient of surface friction of the channel walls

For uniform steady flow: V1= V2= V3= V0

hf = z1 – z2= SoL (3)

where: L=the horizontal distance between section 1 and 2.

h_f=f L/D_h (v_O^2)/2g (4)

Combining eq. 3 and 4, the flow velocity will be:
V_0=(8g/f)^(1/2) R_h^(1/2) S_0^(1/2) (5)
For every channel the term:
(8g/f)^(1/2)=C= constant, the velocity V0 will be:

V_0=CR_h^(1/2) S_0^(1/2) (6)

The volume flow rate (discharge) will be:

Q=ACR_h^(1/2) S_0^(1/2) (7)

These are called the Chezy formula.

The Manning formula for uniform-flow velocity

V_0=(?/n)^ R_h^(3/2) S_0^(1/2) (8)
Where :
n= manning factor, dimensionless.

? = conversion factor which changes with the system of units:

? = 1.0 SI units, ? = 1.486 BG units.

EXAMPLE 1:
A finished-concrete 8-ft-wide rectangular channel has a bed slope of 0.5° and a water depth of 4 ft. Predict the uniform flow rate in ft3/s.