energy concepts applied to streamflow

This powerpoint is not designed for presentation to a group, but rather for individual viewers to learn about streamflow energy.

outline (with hyperlinks) 1) energy defined (the scientific meaning)energy defined 2) the origin of energy conceptsthe origin of energy concepts 3) derivation from Newton’s 1 st Law (calculus)derivation from Newton’s 1 st Law 4) spreadsheet model of mechanical energyspreadsheet model of mechanical energy 5) discussion of mechanical energy as a relative conceptdiscussion of mechanical energy as a relative concept 6) other types of energyother types of energy 7) streamflow energy– a small volume of flowstreamflow energy– a small volume of flow 8) including pressure energyincluding pressure energy 9) effect of channel slopeeffect of channel slope 10) including the effect of velocity distributionincluding the effect of velocity distribution 11) streamflow energy– interchangeable termsstreamflow energy– interchangeable terms 12) the Bernoulli equationthe Bernoulli equation 13) head losshead loss 14) calculating head loss with Mannings equation and conveyancecalculating head loss with Mannings equation and conveyance 15) what Mannings equation cannot dowhat Mannings equation cannot do 16) similarity between energy and momentum conceptssimilarity between energy and momentum concepts

energy defined (in physics) energy is... 1) a property of a body or physical system which enables it to move against a force. 2) an amount of work required to move a mass through a distance, where work, in physics, is the applied force multiplied by the distance moved. back to outline

energy (in physics) a bit of history Where did the idea of energy come from? Is energy an abstract idea, or do things really “contain” energy?

a bit of history energy as a scientific conception It was “discovered” when fiddling around with calculus and Newton’s 2 nd law. 1666: Newton proposes law of gravitation 1686: Newton proposes 3 laws of motion same timeframe: Newton & Liebniz invent calculus back to outline

The origin of the idea of energy... Newton’s 2 nd law: F = ma Force = mass X acceleration take a simple case of a falling rock: it has a mass and is being accelerated by gravity

The origin of the idea of energy... so F = ma = mg where g is earth’s gravitational pull near the surface g = m/sec 2 back to outline

The origin of the idea of energy... ma = mg let’s use calculus (down is negative) integrate with limits of zero to any time or velocity

The origin of the idea of energy...

now include the height of the rock h

The origin of the idea of energy... integrate again, with limits of zero to any time and height h

The origin of the idea of energy... h from our earlier integration: so... substitute in:

The origin of the idea of energy... h simplify:

The origin of the idea of energy... h The early physicists looked at this result and saw two big facts

The origin of the idea of energy... h 1) time is not a variable 2) the sum is a constant

The origin of the idea of energy... h they said, “let’s call this energy !”

The origin of the idea of energy... h “this term involves velocity, so let’s call it kinetic energy !”

The origin of the idea of energy... h “this term involves position, so let’s call it potential energy !”

The origin of the idea of energy... h the sum of kinetic energy and potential energy is a constant!

The origin of the idea of energy... h namely, the starting potential energy back to outline

Let’s run a test on this equation h In a spreadsheet, let’s model a falling rock and calculate this equation with timesteps as the rock falls.

h Let m = 1 kg h start = 500 m timestep = 0.5 sec h start

h m = 1 kg h start = 500 m timestep = 0.5 sec E k = kinetic energy E p = potential energy energy units: Joules h start

h constant back to outline

energy consider the earlier question: Is energy an abstract idea, or do things really “contain” energy? answer:abstract idea! (at least for mechanical energy) The energy of the rock is in reference to the earth’s surface. If we were dropping the rock into a deep hole it would have a different total energy!

energy with references to other datums: The earth rotates, so our rock, even at rest on the ground, has a velocity depending on latitude. photo by NASA

energy with references to other datums: rock velocity at equator: 11,130 m/sec photo by NASA

energy with references to other datums: rock velocity with earth’s revolution: 30,000 m/sec diagram by NASA back to outline

energy what about non-mechanical energy? Physicists found the energy concept to be very useful. They found that work could be done by other forces than mechanical. quick list of other energy sources: thermal, chemical, electrical, electrochemical, electromagnetic, sound, nuclear

energy what about non-mechanical energy? But they kept in mind the very convenient finding that when dealing with mechanical forces (caused by gravity and weight) sum of the energy terms should be a constant, given a defined datum. back to outline

energy of streamflow

starting with a simple hypothetical case, a profile with flat slopes, no flow

energy of streamflow consider the water pressure on a small volume at some depth, h, above a datum, at a distance, y.

energy of streamflow The pressure, P, is a force, Fp on the face of the small volume, V.

energy of streamflow This pressure force, moving the unit distance is an expression of work. Thus pressure results in another form of energy!

energy of streamflow Pressure energy is caused by the fact that, under atmospheric pressure, and its own weight, liquid water can exert a force in any direction. atmospheric pressure

energy of streamflow Water pressure depends on the depth: pressure energy is: ( V is volume) water pressure back to outline

energy of streamflow where

energy of streamflow To express these energies as “head” or in terms of vertical lengths, divide out the volume and density.

energy of streamflow pressure head velocity head elevation head

energy of streamflow We said the water was not moving, so the velocity head is zero!

energy of streamflow Let’s get more real. In flowing water, each little volume has different values for energy types ( E wp E k E p ). How do we handle a sloped bed and the entire flow column (not just the little volume at some particular depth)?

Here’s a flowing stream with a bed slope, theta. Now the pressure head is the full depth. The active flow now creates a velocity head, which can be added to the diagram.

Variable names have been changed to the conventional ones. Notice that streambed slope causes the pressure head, y, to vary slightly from depth measured perpendicular to the bed. As long as theta is about six degrees or less (about a 10% profile slope) then the difference can be ignored. back to outline

Remember, too, that flow velocity can vary a lot within a stream cross-section, as shown in the above sketch, showing blue lines of equal velocity, increasing toward the top and center. Each small area of flow within the cross-section has a velocity head related to it. (It has its own kinetic energy!) So how can we represent a total velocity head?

We can add an adjustment coefficient, alpha, that accounts for velocity variation in the cross-section. Alpha is computed by proportions of flow velocity in different parts of the cross-section. back to outline

interchangeable terms elevation head depth pressure head static head due to potential energy due to pressure energy

interchangeable terms velocity head energy head due to kinetic energy

interchangeable terms water surface hydraulic grade line

interchangeable terms energy grade line energy slope friction slope back to outline

backwater computations Given two cross-sections, we can use a relation called the Bernoulli equation.

backwater computations But what’s this new term h e ? This is “energy head loss” which accounts for energy used between the two cross-sections to overcome friction, obstructions, and changes in cross-section shape.

backwater computations How to calculate head loss, h e... back to outline

computing head loss Since head loss is caused by boundary friction between cross-sections, along with change in channel size and shape, obstructions, etc... L = discharge weighted reach length S f = friction slope between cross-sections C = expansion or contraction loss coefficient ∆ h v = change in energy head btwn cross-sections

computing head loss To calculate friction slope between two cross-sections......we can make use of an empirically-derived 19 th century formula that computes flow in open channels by considering cross-sectional area and boundary friction, namely, Mannings equation

Mannings equation (English units) Q = flow discharge A = cross-sectional flow area v = average velocity R = wetted perimeter / flow area S = streambed slope n = empirical “roughness” coefficient given:then:

“conveyance” This is a very convenient idea that keeps all the Mannings variables except profile slope. In a single cross-section it can help with proportioning flow, velocity, or friction loss between different parts of a cross-section. Mannings equation assumes parallel slopes for streambed, water surface, and friction slope.

“conveyance” HecRAS Hydraulic Reference Manual

Mannings equation gives us a way to evaluate the effect of boundary friction on conveyance in different parts of a cross-section. Total K and Q at a given cross-section is the sum of individual K s and Q s for channel and overbanks with different n values.

Between two cross-sections the HecRAS default method to compute friction slope is: back to outline

what Mannings equation cannot do... We cannot compute water surface profiles that account for backwater with Mannings equation because it assumes “normal depth”, or “uniform flow”, and parallel slopes between bed, water surface, and energy grade line.

Why not Mannings equation? n o r m a l d e p t h backwater That’s why we need the Bernoulli equation! back to outline

Remember: the energy approach was derived from Newton’s force equation, so it should not be radically different from the momentum approach.

To understand the momentum approach view that slideshow. back to outline

end of info on energy