Critical state of flow in an open channel
Critical state of flow in an open channel has the following properties:
1. The specific energy is a minimum for a given discharge.
2. The discharge is a maximum for a given specific energy.
3. The specific force is minimum for a given discharge.
4. The velocity head is equal to half the hydraulic depth.
5. The found number is unity.
6. The slope of the channel that sustains a discharge of a uniform depth equal to
critical depth is called critical slope.
7. A control section is one where a definitive stage (depth) discharge.
8. A critical flow section is excellent control section because, at that section.
For analysis various geometric properties of the channel cross-sections are required.
The commonly needed geometric properties are:
Depth(y)–the vertical distance from the lowest point of the channel section to the free surface.
Stage (z) – the vertical distance from the free surface to an arbitrary datum.
Area (A) – the cross-sectional area of flow is normal to the direction of flow.
Wetted perimeter (P) – the length of the wetted surface measured normal to the direction of flow.
Surface width (B) – width of the channel section at the free surface.
Hydraulic radius (R) – the ratio of area to wetted perimeter ( A/P ).
Hydraulic mean depth (Dm) – the ratio of area to surface width ( A/B ).
1.4 Fundamental equation
The equations which describe the flow of fluid are derived from three fundamental laws of physics:
1. Conservation of matter (or mass) 2. Conservation of energy 3. Conservation of momentum
Conservation of energy:
This says that energy can not be created nor de stroyed, but may be converted form one type to another (e.g. potential may be converted to kinetic energy). When engineers talk about energy "losses" they are referring to energy converted from mechanical (potential or kinetic) to some other form such as heat. A friction loss, for example, is a conversion of mechanical energy to heat. The basic equations can be obtained from the First Law of Thermodynamics but a simplified derivation will be given below.
Conservation of momentum:
The law of conservation of momentum says that a moving body cannot gain or lose momentum unless acted upon by an external force. This is a statement of Newton's Second Law of Motion: Force = rate of change of momentum In solid mechanics these laws may be applied to an object which is has a fixed shape and is clearly defined. In fluid mechanics the object is not clearly defined and as it may change shape constantly. To get over this we use the idea of control volumes. These are imaginary volumes of fluid within the body of the fluid. To derive the basic equation the above conservation laws are applied by considering the forces applied to the edges of a control volume within the fluid.
The Continuity Equation (conservation of mass):
For any control volume during the small time interval δt the principle of conservation of mass implies that the mass of flow entering the control volume minus the mass of flow leaving the control volume equals the change of mass within the control volume.If the flow is steady and the fluid incompressible the mass entering is equal to the mass leaving, so there is no change of mass within the control volume.
So for the time interval δt : Mass flow entering = mass flow leaving
A small length of channel as a control volume
Considering the control volume above which is a short length of open channel of arbitrary cross- Section then, if ρ is the fluid density and Q is the volume flow rate then section then, if mass flow rate is ρ Q and the continuity equation for steady incompressible flow can be written
As, Q, the volume flow rate is the product of the area and the mean velocity then at the
upstream face (face 1) where the mean velocity is u and the cross-sectional area is A1
then:
Similarly at the downstream face, face 2, where mean velocity is u2and the cross-sectional area is A2 then:
Therefore the continuity equation can be written as
The Energy equation (conservation of energy):
Consider the forms of energy available for the above control volume. If the fluid moves from the upstream face 1, to the downstream face 2 in time d t over the length L.
The work done in moving the fluid through face 1 during this time is
Where p1 is pressure at face 1
The mass entering through face 1 is
Therefore the kinetic energy of the system is:
If z1 is the height of the centroid of face 1, then the potential energy of the fluid entering the control volume is :
The total energy entering the control volume is the sum of the work done, the potential and the kinetic energy:
We can write this in terms of energy per unit weight. As the weight of water entering the control volume is ρ1 A1 L g then just divide by this to get the total energy per unit weight:
At the exit to the control volume, face 2, similar considerations deduce
If no energy is supplied to the control volume from between the inlet and the outlet
then energy leaving = energy entering and if the fluid is incompressible
This is the Bernoulli equation.
The momentum equation (momentum principle):
Again consider the control volume above during the time δt
By the continuity principle : = d Q1 = dQ 2 = dQ And by Newton's second law Force = rate of change of momentum
It is more convenient to write the force on a control volume in each of the three,x, y and z direction e.g. in the x-direction
Integration over a volume gives the total force in the x-direction as
As long as velocity V is uniform over the whole cross-section.
This is the momentum equation for steady flow for a region of uniform velocity.
Energy and Momentum coefficients
In deriving the above momentum and energy (Bernoulli) equations it was noted that thevelocit y must be constant (equal to V) over the whole cross-section or constant along a stream-line.Clearly this wil l not occur in practice. Fortunately both these equation may still be used even for situations of quite non-uniform velocity distribution over a section. This is possible by the introduction of coefficients of energy and momentum, a and ß respectively.
These are defined:
where V is the mean velocity.
And the Bernoulli equation can be rewritten in terms of this mean velocity:
And the momentum equation becomes:
The values of α and ß must be derived from the velocity distributions across a cross-section. They will always be greater than 1, but only by a small amount consequently they can
often be confidently omitted – but not always and their existence should always be remembered.
For turbulent flow in regular channel a does not usually go above 1.15 and ß will
normally be below 1.05. We will see an example below where their inclusion is necessary to
obtain accurate results.