the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Brief communication: Betz’s Law: the Zorich Derivation
Richard Zorich
Abstract. In this article, Betz’s law is derived in a new way. A power equation is constructed by accounting for the forces that a machine applies to the air mass that flows through it. By comparing that power equation to the available power in the wind, Betz’s law is validated.
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Richard Zorich
Interactive discussion
Status: closed

RC1: 'Comment on wes202355', Alois Schaffarczyk, 13 Jun 2023

AC1: 'Reply on RC1', Richard Zorich, 15 Jun 2023
To Professor Alois Schaffarczyk,
Thank you for taking time to comment. You are correct in pointing out that my beliefs are not relevant and I should not express them in scientific work. I regret using the word "better". The final reader should make the judgement of good or bad.
My intent is to provide an alternative derivation that gives a new perspective. You are correct: the classic derivation does not include a discussion of the influence of viscosity. If it did, then it could not use Bernoulli’s principle. The standard textbook derivation uses a useful model. But it doesn't give a complete model of the interaction of a machine with the atmosphere. Nor does mine.
Please consider the work of Käsler et al. See their research article titled:
"Wake Measurements of a MultiMW Wind Turbine with Coherent LongRange Pulsed Doppler Wind Lidar"
at
https://journals.ametsoc.org/view/journals/atot/27/9/2010jtecha1483_1.xml
Their figure 3 shows the velocity deficit downstream from the wind turbine derived from a elevation scan. The maximum velocity deficit is in the near wake(this is the velocity predicted by the classic derivation). If we can presume that the deficit goes to zero at a great distance downwind, then my derivation can be used. Perhaps the complete interaction could be described using both derivations.
I do claim that my derivation of Betz's Law is new. I will continue to make that claim until someone cites a previous work that proves me wrong.Citation: https://doi.org/10.5194/wes202355AC1 
AC2: 'Reply on RC1', Richard Zorich, 15 Jun 2023
To Professor Alois Schaffarczyk,
In addition to my first reply, I want to address your use of capitalization on the word continuum. You wrote
"... CONTINUUM mechanical nature of the problem which
is different from Newton’s point mass mechanics."
I agree. My equations are a function of mass flow rate. My line number 66 is
"Thrust equals the mass flow rate through the machine times the velocity changes that that mass flow undergoes."
Why did you emphasize CONTINUUM ?Citation: https://doi.org/10.5194/wes202355AC2 
AC4: 'Reply on RC1', Richard Zorich, 04 Jul 2023
The classic derivation uses a model that considers inertial and pressure forces. My derivation uses a model that considers inertial and frictional forces. We all agree that there is a region of increased pressure in front of the machine and a region of negative pressure behind the machine. Isn't it reasonable to assume that atmospheric pressure exists inside the machine? That assumption allows us to neglect the pressure forces. The model is simple mechanics. The mass flow through the machine is decelerated by the machine and reaccelerate in the wake.
Citation: https://doi.org/10.5194/wes202355AC4

AC1: 'Reply on RC1', Richard Zorich, 15 Jun 2023

RC2: 'Comment on wes202355', Anonymous Referee #2, 01 Jul 2023
The derivation in the paper is based a sign error and an incorrect assumption, Hence, I cannot accept the paper be published in WES. More detailed comments are attached.

AC3: 'Reply on RC2', Richard Zorich, 02 Jul 2023
To Anonymous Referee #2,
thank you for taking time to comment.
I guess that my explaination was not clear.
In my equation number (15), I have defined the velocity at the end of the wake as Vf.
By "wake" I am referring to what is described at
https://en.wikipedia.org/wiki/Wake_(physics)
The model for the classic derivation does NOT include a wake. The classic derivation uses a stream tube as you have shown in your comment.
In that stream tube, downwind of the machine, there is region that has a lower power flow than that of the upwind region and the rest of the atmosphere.
That difference in the flow of power is what the machine is capturing. That region of lower power flow continues downwind for infinity.
That is OK because the model allows equations to be constructed using Bernoulli's principle and for Betz's Law to be derived.
In your "Downstream CV analysis" equation you have included my Vf. But, Vf does not exist in the classic derivation. I think you have mistaken my Vf to mean the final velocity in your stream tube and in the classic model.
Please consider the work of Käsler et al. See their research article titled:
"Wake Measurements of a MultiMW Wind Turbine with Coherent LongRange Pulsed Doppler Wind Lidar"
at
https://journals.ametsoc.org/view/journals/atot/27/9/2010jtecha1483_1.xml
Their figure 3 shows the velocity deficit downstream from the wind turbine derived from a elevation scan. The maximum velocity deficit is in the near wake(this is the final velocity predicted by the classic derivation using Bernoulli's principle). Since their figure shows data from a real machine, the velocity deficit is reduced as the flow moves downwind. If we can presume that the deficit goes to zero at a great distance downwind, then my derivation can be used.
You wrote
"...by assuming that Vf = Vo , which is the second erroneous assumption in the paper."
I don't use that equation. What I said was Vf approaches Vo as A approaches 0(see equation number 20).
Please consider the work of Aitken et al. See their research article titled:
"Quantifying Wind Turbine Wake Characteristics from Scanning Remote Sensor Data"
at
https://journals.ametsoc.org/view/journals/atot/31/4/jtechd1300104_1.xml
Their figure 1 presents an equation to model a wake as
VD = 56x^(0.57)
In their equation, the velocity deficit never quite goes to zero. But far downwind, it gets very close to zero.
You wrote
"in the paper the second downstream equation has a wrong sign"
In light of the misunderstanding concerning the final wake velocity, I am not sure about how to respond to that.
In your equation, you subtract the negative pressure that is on the downwind side of the machine from the positive pressure on the upwind side.
That is how you account for the total thrust on a machine. For argument's sake, let's say that they add to equal 2p. But my equation adds the push from upwind and the pull from downwind. I don't why I would have the signs any other way.
You wrote
"It is too vague just to phrase some general statements regarding the use of Newton’s law."
Do you object to my equation number 10?
Thrust equals the mass flow rate through the machine times the velocity changes that that mass flow undergoes
Do you say that is vague? Would it be clearer to say that
thrust equals the change of momentum of the air that flows through the machine over the change in time?
I thought it was simpler to leave time out of the equation.Citation: https://doi.org/10.5194/wes202355AC3 
AC5: 'Reply on RC2', Richard Zorich, 04 Jul 2023
The classic derivation uses a model that considers inertial and pressure forces. My derivation uses a model that considers inertial and frictional forces. We all agree that there is a region of increased pressure in front of the machine and a region of negative pressure behind the machine. Isn't it reasonable to assume that atmospheric pressure exists inside the machine? That assumption allows us to neglect the pressure forces. The model is simple mechanics. The mass flow through the machine is decelerated by the machine and reaccelerate in the wake.
Citation: https://doi.org/10.5194/wes202355AC5

AC3: 'Reply on RC2', Richard Zorich, 02 Jul 2023
Interactive discussion
Status: closed

RC1: 'Comment on wes202355', Alois Schaffarczyk, 13 Jun 2023

AC1: 'Reply on RC1', Richard Zorich, 15 Jun 2023
To Professor Alois Schaffarczyk,
Thank you for taking time to comment. You are correct in pointing out that my beliefs are not relevant and I should not express them in scientific work. I regret using the word "better". The final reader should make the judgement of good or bad.
My intent is to provide an alternative derivation that gives a new perspective. You are correct: the classic derivation does not include a discussion of the influence of viscosity. If it did, then it could not use Bernoulli’s principle. The standard textbook derivation uses a useful model. But it doesn't give a complete model of the interaction of a machine with the atmosphere. Nor does mine.
Please consider the work of Käsler et al. See their research article titled:
"Wake Measurements of a MultiMW Wind Turbine with Coherent LongRange Pulsed Doppler Wind Lidar"
at
https://journals.ametsoc.org/view/journals/atot/27/9/2010jtecha1483_1.xml
Their figure 3 shows the velocity deficit downstream from the wind turbine derived from a elevation scan. The maximum velocity deficit is in the near wake(this is the velocity predicted by the classic derivation). If we can presume that the deficit goes to zero at a great distance downwind, then my derivation can be used. Perhaps the complete interaction could be described using both derivations.
I do claim that my derivation of Betz's Law is new. I will continue to make that claim until someone cites a previous work that proves me wrong.Citation: https://doi.org/10.5194/wes202355AC1 
AC2: 'Reply on RC1', Richard Zorich, 15 Jun 2023
To Professor Alois Schaffarczyk,
In addition to my first reply, I want to address your use of capitalization on the word continuum. You wrote
"... CONTINUUM mechanical nature of the problem which
is different from Newton’s point mass mechanics."
I agree. My equations are a function of mass flow rate. My line number 66 is
"Thrust equals the mass flow rate through the machine times the velocity changes that that mass flow undergoes."
Why did you emphasize CONTINUUM ?Citation: https://doi.org/10.5194/wes202355AC2 
AC4: 'Reply on RC1', Richard Zorich, 04 Jul 2023
The classic derivation uses a model that considers inertial and pressure forces. My derivation uses a model that considers inertial and frictional forces. We all agree that there is a region of increased pressure in front of the machine and a region of negative pressure behind the machine. Isn't it reasonable to assume that atmospheric pressure exists inside the machine? That assumption allows us to neglect the pressure forces. The model is simple mechanics. The mass flow through the machine is decelerated by the machine and reaccelerate in the wake.
Citation: https://doi.org/10.5194/wes202355AC4

AC1: 'Reply on RC1', Richard Zorich, 15 Jun 2023

RC2: 'Comment on wes202355', Anonymous Referee #2, 01 Jul 2023
The derivation in the paper is based a sign error and an incorrect assumption, Hence, I cannot accept the paper be published in WES. More detailed comments are attached.

AC3: 'Reply on RC2', Richard Zorich, 02 Jul 2023
To Anonymous Referee #2,
thank you for taking time to comment.
I guess that my explaination was not clear.
In my equation number (15), I have defined the velocity at the end of the wake as Vf.
By "wake" I am referring to what is described at
https://en.wikipedia.org/wiki/Wake_(physics)
The model for the classic derivation does NOT include a wake. The classic derivation uses a stream tube as you have shown in your comment.
In that stream tube, downwind of the machine, there is region that has a lower power flow than that of the upwind region and the rest of the atmosphere.
That difference in the flow of power is what the machine is capturing. That region of lower power flow continues downwind for infinity.
That is OK because the model allows equations to be constructed using Bernoulli's principle and for Betz's Law to be derived.
In your "Downstream CV analysis" equation you have included my Vf. But, Vf does not exist in the classic derivation. I think you have mistaken my Vf to mean the final velocity in your stream tube and in the classic model.
Please consider the work of Käsler et al. See their research article titled:
"Wake Measurements of a MultiMW Wind Turbine with Coherent LongRange Pulsed Doppler Wind Lidar"
at
https://journals.ametsoc.org/view/journals/atot/27/9/2010jtecha1483_1.xml
Their figure 3 shows the velocity deficit downstream from the wind turbine derived from a elevation scan. The maximum velocity deficit is in the near wake(this is the final velocity predicted by the classic derivation using Bernoulli's principle). Since their figure shows data from a real machine, the velocity deficit is reduced as the flow moves downwind. If we can presume that the deficit goes to zero at a great distance downwind, then my derivation can be used.
You wrote
"...by assuming that Vf = Vo , which is the second erroneous assumption in the paper."
I don't use that equation. What I said was Vf approaches Vo as A approaches 0(see equation number 20).
Please consider the work of Aitken et al. See their research article titled:
"Quantifying Wind Turbine Wake Characteristics from Scanning Remote Sensor Data"
at
https://journals.ametsoc.org/view/journals/atot/31/4/jtechd1300104_1.xml
Their figure 1 presents an equation to model a wake as
VD = 56x^(0.57)
In their equation, the velocity deficit never quite goes to zero. But far downwind, it gets very close to zero.
You wrote
"in the paper the second downstream equation has a wrong sign"
In light of the misunderstanding concerning the final wake velocity, I am not sure about how to respond to that.
In your equation, you subtract the negative pressure that is on the downwind side of the machine from the positive pressure on the upwind side.
That is how you account for the total thrust on a machine. For argument's sake, let's say that they add to equal 2p. But my equation adds the push from upwind and the pull from downwind. I don't why I would have the signs any other way.
You wrote
"It is too vague just to phrase some general statements regarding the use of Newton’s law."
Do you object to my equation number 10?
Thrust equals the mass flow rate through the machine times the velocity changes that that mass flow undergoes
Do you say that is vague? Would it be clearer to say that
thrust equals the change of momentum of the air that flows through the machine over the change in time?
I thought it was simpler to leave time out of the equation.Citation: https://doi.org/10.5194/wes202355AC3 
AC5: 'Reply on RC2', Richard Zorich, 04 Jul 2023
The classic derivation uses a model that considers inertial and pressure forces. My derivation uses a model that considers inertial and frictional forces. We all agree that there is a region of increased pressure in front of the machine and a region of negative pressure behind the machine. Isn't it reasonable to assume that atmospheric pressure exists inside the machine? That assumption allows us to neglect the pressure forces. The model is simple mechanics. The mass flow through the machine is decelerated by the machine and reaccelerate in the wake.
Citation: https://doi.org/10.5194/wes202355AC5

AC3: 'Reply on RC2', Richard Zorich, 02 Jul 2023
Richard Zorich
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