Uniform Flow - Fundamental equations
Fundamental equations
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
Although first developed for solid bodies they are equally applicable to fluids. Brief descriptions of the concepts are given below.
Conservation of matter
This says that matter can not be created nor destroyed, but it may be converted (e.g. by a chemical process.) In fluid mechanics we do not consider chemical activity so the law reduces to one of conservation of mass.
Conservation of energy
This says that energy can not be created nor destroyed, 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 ?t 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 ?t : Mass flow entering = mass flow leaving
Considering the control volume above which is a short length of open channel of arbitrary cross- Section then, if ? is the fluid density and Q is the volume flow rate then section then, if mass flow rate is ? Q and the continuity equation for steady incompressible flow can be written
pQentering = pQleaving
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:
Qentering = u1A1
Similarly at the downstream face, face 2, where mean velocity is u2 and the cross-sectional area is A2 then:
Qleaving= u2A2
Therefore the continuity equation can be written as
u1A1 = u2A2
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
Work done = p1A1L
The mass entering through face 1 is
Mass entering = p1A1L
Therefore the kinetic energy of the system is:
KE = mu2 = p1A1Lu12
If z1 is the height of the centroid of face 1, then the potential energy of the fluid entering the control volume is :
PE = mgz = p1A1Lgz1
The total energy entering the control volume is the sum of the work done, the potential and the kinetic energy:
Total energy = p1A1L + p1A1Lu12 + p1A1Lgz1
We can write this in terms of energy per unit weight. As the weight of water entering the control volume is ?1 A1 L g then just divide by this to get the total energy per unit weight:
Total energy per unit weight = p1/p1g + u12 + z1
At the exit to the control volume, face 2, similar considerations deduce
Total energy per unit weight = p2/p2g + u22 + z2
If no energy is supplied to the control volume from between the inlet and the outlet then energy leaving = energy entering and if the in compressible
p1/p1g + u12 + z1 + p2/p2g + u22 + z2 = H = constant
This is the Bernoulli equation.
Note:
1. In the derivation of the Bernoulli equation it was assumed that no energy is lost in the control volume - i.e. the fluid is frictionless. To apply to non frictionless situations some energy loss term must be included.
The dimensions of each term in equation 1.2 has the dimensions of length ( units of meters). For this reason each term is often regarded as a "head" and given the names
P/pg=pressure head
U2/2g = velocity head
Z=velocity or potential head
3. Although above we derived the Bernoulli equation between two sections it should strictly speaking be applied along a stream line as the velocity will differ from the top to the bottom of the section. However in engineering practise it is possible to apply the Bernoulli equation with out reference to the particular streamline
The momentum equation (momentum principle)
Again consider the control volume above during the time ?t
Momentum entering = p ?Q1?tu1
Momentum entering = p ?Q2?tu2
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
Fx = pQ(V2x - V1x)
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 the velocity must be constant (equal to V) over the whole cross-section or constant along a stream-line.
Clearly this will 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 respectively.
where V is the mean velocity.
And the Bernoulli equation can be rewritten in terms of this mean velocity:
P/pg + aV2/2g + z = constant
And the momentum equation becomes:
Fx = pQB(V2x - V1x)
The values of ? and 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 will normally be below 1.05. We will see an example below where their inclusion is necessary to obtain accurate results.