Is Power Conserved in Energy Systems?

[Ali Durrani]

Ok i have this very basic question, Is power in a system conserved or not just like energy is conserved? because for example there is a spring compression setup where i am supplying 2 joules/sec of power for around 10 mins to compress the spring, now after 10 mins i leave it abruptly and it jumps and looses all of its energy in a fraction of a second, now i know the total energy is conserved and is 1200 joules but what about the power? it will be 1200joules/ fraction of a second which means a huge amount of power? can someone guide me please

 

[DrClaude]

Power is not a conserved quantity.

 

[Ali Durrani]

ok then i was right hahaha thanks, i was in a talk with a senior person i don't know why was he arguing with me on that but he said energy is not conserved and he said he can prove it, then he said he give me 2 kW of power and i can give you 10 kW of power :P and i was like yes its possible but that 10 kW would not be for a long while. well thanks for clearing my ambiguity

 

[Mike Snider]

The law of conservation of energy states: "In a closed system, the total energy in all forms is equal to a constant." Power is the time rate of energy change, ie. Force times distance divided by the change in time that the force acted over the distance. Dividing both sides of the energy equation (potential, kinetic, electromagnetic, chemical, nuclear, mc**2, equals a constant) by the change per unit time yields the power law for a closed system: "The time rate of change from one energy form to another must sum to zero" Constant energy means there is not change in total energy over time. Kirchoff's Voltage Law (The sum of the voltages around a closed loop is zero) is the most common manifestation.

If you could release this stored energy kx instantaneously without any external forces applied to the spring this energy becomes kinetic. One half mass of the spring times velocity squared. A compressed automobile spring is very dangerous!

 

[anorlunda]

As @DrClaude said, power is not conserved, only energy. But it is easy to bet confused by the bookkeeping. The way to write this is

Energystoreddt=Powerin−Powerout

If the change in energy stored is zero, then Power In = Power Out.

If someone is sloppy and forgets to mention stored energy, then it appears that power must be conserved.

 

[Mike Snider]

So the total energy in your closed system increased magically? or was a potential form of energy increased such as heat, pressure, height, energy to mass conversion by a photon stopping as examples? The time rate of the conversion from kinetic to a potential energy must be in the bookkeeping. Power is conserved!

 

[russ_watters]

So the total energy in your closed system increased magically?

The example the OP gave is an open system where power and energy both vary over time.

 

[Mike Snider]

Conservation of power applies only when energy is in the process of changing form. In the case of the spring, the power supplied by the spring compressor equals the power put into the spring as potential energy. Force of the spring compressor acting over the distance of compression in the time period the motion occurs equals the spring constant times the distance of compression times the distance of compression in the time interval of compression. (kxx/t) Power into the spring equals power out of the compressor. When the stored energy in the compressed spring is instantaneously released it becomes kinetic energy very rapidly. The total power of the release is zero. Potential energy in the spring equals its kinetic energy after the release. The instantaneous power is much higher than the compression of the spring but it occurs in a much shorter time so the total energy is equal to compressor supplied energy. Energy and Power are conserved!

 

[Nugatory]

Conservation of power applies...

Never - there's no such thing.

Compressing a spring adds energy $k\Delta{x}^2/2$ to the spring. I can do this with low power by compressing the spring slowly, or with high power by compressing it quickly - I'm varying the $\Delta{t}$ in $P=k\Delta{x}^2/2\Delta{t}$ to vary the power while keeping the energy constant. Likewise, releasing the spring recovers this energy, and again I can release the energy quickly or slowly to get a short period of high power or a longer period of lower power.

 

[Mike Snider]

Precisely Nugatory! The power added to the spring equals the power absorbed by the spring--conservation of power. Note that k delta x is the force. The energy is the force times the displacement or k delta x squared. When the spring is perfectly released, no outside forces acting on it, all the energy will be use to accelerate it rapidly. Potential kxx becomes kinetic 1/2 mass of spring v^2 in a short period of time. Compression and releasing the spring are two distinct events in time. Each conserving power during that events occurrence. Should an outside force act on the spring during its release power would be absorbed by that force effecting the both the power balance and the time of the event. Energy and Power are both conserved!

Consider riding a bicycle. The rider must overcome inertia (mass of rider and bicycle), rolling resistance, aerodynamic drag and any change in grade. At any point in time during the riders trip leg power will equal all the other forces or the bike will slow or speed up. At every point in time both power and energy are conserved!

 

[russ_watters]

Precisely Nugatory! The power added to the spring equals the power absorbed by the spring--conservation of power.

That's clearly not what he said.

Compression and releasing the spring are two distinct events in time. Each conserving power during that events occurrence. Should an outside force act on the spring during its release power would be absorbed by that force effecting the both the power balance and the time of the event.

When switching between discussing power and energy the way you are, you are changing the system boundary/definition. What you're describing is more of a Newton's 3rd law interaction in the conversion of one form of energy into another, by two different components, inside the system. That's fine, but it isn't the same thing as conservation of energy for a system, and isn't generally described as "conservation of power". In this case, the system is a spring-mass, and energy can go into it and out of it at different rates. That would be different from, say, a control volume, where you typically have a steady-state flow of mass and energy through the system. But people don't look at control volumes and say they follow "conservation of power".

 

[Mike Snider]

The problem defined the spring compression using the input of joules per second and described releasing the stored energy. Two events. Each occurring at different time intervals. No change in boundary/definition. Applying the energy equation requires obeying Newton's third law. Dividing the energy equation by a time interval to get the power equation by definition complies with Newton's third law. I am guilty of choosing a time interval that spans the event! Suppose we are asked to calculate the reaction force the spring applies to whatever it was compressed against as a function of time. The energy equation contains force and position. To get time the equation of motion ( x = x0t + 1/2at^2) would be employed-- a second order equation. The math gets burdensome. When one evaluates the problem in power the equation of motion becomes linear!

Perhaps they should. Turbulence involves spatial and temporal changes in the flow field and simplifying the math would reduce the computational burden significantly. Please see U.S. Patent 10,698,980 it you would like more...

May your thanksgiving be blessed with family and friends.

Mike

 

[russ_watters]

The problem defined the spring compression using the input of joules per second and described releasing the stored energy. Two events. Each occurring at different time intervals. No change in boundary/definition.

But you're looking at the interaction at/across the system boundary, not just the change in internal energy of the system. Those are two different things. In describing the before and after states of compressing a spring, you don't even need to know anything about the compression process other than the total energy absorbed or distance compressed. Time doesn't need to enter into the analysis.

Dividing the energy equation by a time interval to get the power equation by definition complies with Newton's third law. I am guilty of choosing a time interval that spans the event!

Yes. We don't necessarily know it or even need to know it. It's not necessarily relevant to the analysis. A different problem.

It might be worth googling the terms "conservation of power" and "conservation of energy" (with quotes) and seeing what sort of hits you get.

 

[berkeman]

May your thanksgiving be blessed with family and friends.

Hopefully in 2021.

 

[etotheipi]

Thread has confusing premise. Often if you have some intensive conserved quantity $q = \partial Q / \partial V$, then the equation expressing its conservation is$$\partial_t q + \nabla \cdot \boldsymbol{j} = 0$$where $\boldsymbol{j}$ is the flux density [per unit area, per unit time] of that quantity. If you're thinking about energy as a conserved quantity, then the energy density $u = \partial E / \partial V$ satisfies $\partial_t u + \nabla \cdot \boldsymbol{j} = 0$, or in a more recognisable form$$\frac{dE}{dt} = -\oint_{\partial \Omega} \boldsymbol{j} \cdot d\boldsymbol{S}$$which you will recognise as the net power flowing into the region $\Omega$. In other words, the power is just the rate at which energy is being transferred across some boundary. You could quite reasonably split the integral into two terms, $$-\oint_{\partial \Omega} \boldsymbol{j} \cdot d\boldsymbol{S} = \frac{dE_{\text{in}}}{dt} - \frac{dE_{\text{out}}}{dt}$$and if the energy in the region $\Omega$ is constant, then $\dot{E}_{\text{in}} = \dot{E}_{\text{out}}$. But that's not a conservation law, that's just numerical equality of two terms, in a special case.

 

[Dale]

Power is conserved!

Power is not conserved. It can be zero at one point in time and non-zero at some other point.

Conservation of power applies only when energy is in the process of changing form

This doesn’t fix it either. While a motor is changing energy from electrical to mechanical the amount of power can change arbitrarily over time. Power is not conserved.

The power added to the spring equals the power absorbed by the spring--conservation of power.

That is conservation of energy. To show conservation of power you would need to show that power satisfies a continuity equation like $$\frac{\partial \rho}{\partial t}+\nabla \cdot \vec j=0$$ where $\rho$ is the power density and $\vec j$ is the power flux.

Power is not conserved.

 

[sophiecentaur]

So the total energy in your closed system increased magically? or was a potential form of energy increased such as heat, pressure, height, energy to mass conversion by a photon stopping as examples? The time rate of the conversion from kinetic to a potential energy must be in the bookkeeping. Power is conserved!

I can see what's happened here. You have had a 'brainwave' about an alternative way of looking at something that's already very well established. Rather than give up on it, you are continuing to present it in various forms, despite having been corrected several times.

I hope you realize just how confusing this sort of thing is for someone who doesn't know the subject and who is trying to learn something useful. Why not give 'em a break?

 

[berkeman]

Agreed. Thread is closed.