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Research Papers

Turbomachine Design for Supercritical Carbon Dioxide Within the sCO2-HeRo.eu ProjectOPEN ACCESS

[+] Author and Article Information
Alexander Hacks

Chair of Turbomachinery,
University of Duisburg-Essen,
Lotharstraße 1,
Duisburg 47057, Germany
e-mail: alexander.hacks@uni-due.de

Sebastian Schuster

Chair of Turbomachinery,
University of Duisburg-Essen,
Lotharstraße 1,
Duisburg 47057, Germany
e-mail: s.schuster@uni-due.de

Hans Josef Dohmen

Chair of Turbomachinery,
University of Duisburg-Essen,
Lotharstraße 1,
Duisburg 47057, Germany
e-mail: hans-josef.dohmen@uni-due.de

Friedrich-Karl Benra

Chair of Turbomachinery,
University of Duisburg-Essen,
Lotharstraße 1,
Duisburg 47057, Germany
e-mail: friedrich.benra@uni-due.de

Dieter Brillert

Chair of Turbomachinery,
University of Duisburg-Essen,
Lotharstraße 1,
Duisburg 47057, Germany
e-mail: dieter.brillert@uni-due.de

1Corresponding author.

Manuscript received June 22, 2018; final manuscript received July 6, 2018; published online November 29, 2018. Editor: Jerzy T. Sawicki.

J. Eng. Gas Turbines Power 140(12), 121017 (Nov 29, 2018) (8 pages) Paper No: GTP-18-1295; doi: 10.1115/1.4040861 History: Received June 22, 2018; Revised July 06, 2018

Abstract

The paper aims to give an overview over the keystones of design of the turbomachine for a supercritical CO2 (sCO2) Brayton cycle. The described turbomachine is developed as part of a demonstration cycle on a laboratory scale with a low through flow. Therefore, the turbomachine is small and operates at high rotational speed. To give an overview on the development, the paper is divided into two parts regarding the aerodynamic and mechanical design. The aerodynamic design includes a detailed description on the steps from choosing an appropriate rotational speed to the design of the compressor impeller. For setting the rotational speed, the expected high windage losses are evaluated considering the reachable efficiencies of the compressor. The final impeller design includes a description of the blading development together with the final geometry parameters and calculated performance. The mechanical analysis shows the important considerations for building a turbomachine with integrated design of the three major components: turbine, alternator, and compressor (TAC). It includes different manufacturing techniques of the impellers, the bearing strategy, the sealing components, and the cooling of the generator utilizing the compressor leakage. Concluding the final design of the TAC is shown and future work on the machine is introduced.

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Introduction

Against the background of the changed energy market, the supercritical carbon dioxide cycle has attracted considerable attention due to its compactness and proposed thermodynamic efficiency. Therefore, several companies and institutes are working in the field of development of sCO2 turbomachinery. In general, the main focus is to develop a more economically efficient turbomachine for the production of electricity in a power plant or for shipboard propulsion. Contrary to the production of electricity, the main objective of the turbomachine (TAC), consisting of turbine, alternator and compressor, described within this paper is to propel a closed, unrecuperated Brayton cycle for transportation of residual decay heat from a nuclear reactor core to an ultimate heat sink (Benra et al. [1]). However, the challenges in design are similar, especially because the compactness of the turbomachine becomes an issue in a size of a small demonstrator.

The turbomachine is developed as part of the supercritical CO2 heat removal system (sCO2-HeRo) project, which aims to develop and proof the concept of a new self-launching, self-propelling, and self-sustaining safety system for nuclear power plants (Benra et al. [1]). After a station blackout, the system transports the residual decay heat of the nuclear reactor core to an ultimate heat sink. The system can also be retrofitted to existing power plants. Unlike other safety systems, it does not need external energy supply, but it even produces electrical energy. To have a compact turbomachine and compact heat exchangers and therefore be able to retrofit the system to existing power plants, the closed unrecuperated Brayton cycle is realized using sCO2 as working fluid. In the sCO2-HeRo project, a demonstrator unit is developed and experimentally tested at the Gesellschaft für Simulatorschulung (GfS) in order to prove the concept and assess technology readiness level 3. Furthermore, the cycle shall be used to gain experience on the design, performance, and operation of sCO2 loops and the consisting components. Table 1 contains the thermodynamic design parameters for the TAC, which relate to the use of the sCO2-HeRo system as a demonstrator for a safety system to remove residual heat in nuclear power plants. The application in the safety cycle raises requirements regarding the stable operation, also at off-design conditions, with a minimum number of auxiliary units.

Design Overview

The sCO2-HeRo TAC (Fig. 1) has an integrated design with a single-stage compressor, a single-stage turbine, and a generator in one casing. This represents the most compact machine and theoretically allows a hermetic system without any external leakage. The contour lines in the Cordier diagram in Fig. 22 indicate the expected efficiency for turbocompressors with conventional blading design at given design parameters, while the dashed line indicates the curve of optimum efficiency according to Cordier. The sketched impellers in Fig. 2 illustrate the preferable impeller type. The dots represent the compressor parameters of several small-scale TAC systems either tested in the past years or still under investigation. All compressors are centrifugal compressors:

• Sandia National Lab (SNL) [3]

• Bechtel Marine Propulsion Corporation (BMPC) [47]

• Korea Institute of Energy Research (KIER) [810]

• Tokyo Institute of Technology (TIT) [11]

• Korea Atomic Energy Research Institute (KAERI) (Information regarding the diameter could not be retrieved. Therefore it is not included in Fig. 2) [12]

• University of Duisburg-Essen, HeRo-project (HeRo) [1,13] and Table 2.

The compressors from SNL, BMPC, and TIT apply an unshrouded compressor design, while the impellers of KIER and KAERI have a shrouded design in order to overcome thrust issues, which showed up during testing of the SNL compressor (Cho et al. [8]). The compressors developed in these projects tend to be on the right of the region of best efficiency (Fig. 2). This is caused by a low flow coefficient due to the high density of sCO2 combined with the small power output (small mass flow rate) of the demonstration cycles. The TACs in Fig. 2 operate at similar compressor inlet pressures and temperatures, except for the one at BMPC, which operates further away from the critical point. The major difference of the sCO2-HeRo compressor (star) is the significantly lower mass flow in the sCO2-HeRo cycle resulting in an even smaller flow coefficient and a smaller machine. Thus, challenges related to the small size of the components become more severe, limiting the achievable efficiency.

Aerodynamic Design

The aerodynamic design must account for the properties of sCO2 close to the critical point. The high density of sCO2 means that windage losses become significant and the overall dimensions of this machine are rather small.

The optimum rotational speed is a compromise between stage efficiency and windage losses. The windage losses plus compressor power must not exceed the expected turbine power output. Because windage losses are approximately proportional to the parameters in Eq. (1), the rotational speed has to be limited to a value of around 50,000 rpm. In contrast, the optimum rotational speed for maximum stage efficiency for this small machine is according to Fig. 2 around 200,000 rpm Display Formula

(1)$Pf∼D4,L,ρ,n3$

Besides limiting the rotational speed, windage losses can be reduced by lowering the pressure and thus the density in the central housing. SNL, BMPC, KIER, and KAERI use this approach and reduce the pressure below the compressor inlet conditions. The resulting leakage flow is returned to the cycle with a CO2 pump. This is not a valid option for the sCO2-HeRo cycle, because it limits the self-sustainability. The design therefore aims for a hermetic design similar to the TIT-design or a direct leakage feedback. The latter allows reducing the pressure in the cavity to compressor inlet conditions.

Since the generator cavity contributes for the greater part to the windage losses, it is used to determine the permissible rotational speed of the shaft. As stated by Wright et al. [3], standard windage models are applicable also for supercritical fluids. Thus, the windage losses in the generator cavity are calculated according to the model of Bilgen and Boulos [14] for the generator dimensions depicted in Fig. 3. The figure shows the rapid increase of the windage losses Pf with rotational speed n for three different pressure levels of CO2 within the cavity. For the sCO2-HeRo, the pressure range between 78.3 bar and 117.5 bar is desired as it allows feedback of the leakage without an additional pump. The design speed is set to 50,000 rpm leading to windage losses of 1.6 kW at the generator. For the resulting overall dimensions of the rotor including the impellers and shaft, the windage losses are expected to be twice as high leaving a surplus power of around 4 kW. If the speed is increased further, e.g., to 60,000 rpm, the surplus power is reduced to 1 kW, which is considered to be an insufficient margin. By reducing the rotational speed, the point of compressor operation is shifted to lower specific speed and higher specific diameter causing lower efficiency (further to the right in Fig. 2). To reach efficiencies similar to SNL and BMPC, the sCO2-HeRo compressor would, compared to the actual design, need to rotate at a 3.5 times higher speed having a 2.5 times smaller outer diameter. Investigations showed that the windage losses at this high speed exceed the surplus power of the turbine.

Compressor Design

This paragraph contains a short description of the compressor development steps and gives detailed information on the final design including geometry and calculated performance. The boundary conditions are summarized in Table 1. The rotational speed is 50,000 rpm. With help of the Cordier diagram in Fig. 2, the general impeller type is determined to be radial only. Therefore, a two-dimensional radial blading is applied, which is also easy to manufacture. Because the volume flow is small, an unshrouded design requires very small tip clearances to reduce tip leakage losses. Very small clearances raise the risk of rubbing between impeller and casing because of axial displacement of the rotor. Therefore, a cover disk is applied in combination with labyrinth sealing (similar to KIER and KAERI) allowing a significant reduction of clearance losses and simultaneously reducing the risk of contact between impeller and casing. Using a cover disk also reduces thrust forces, because the pressure difference between hub and shroud is reduced. This is especially useful during start-up, when the pressure differences between turbine and compressor can be large. The compressor consists of one stage with a vaneless parallel wall diffuser. The blade angle at the compressor trailing edge is set to β2 = 90 deg instead of backward curved blades to increase the circumferential velocity component and be able to reach the required pressure ratio.

Simulations are carried out with ansyscfx to study the blading performance. The impeller is modeled together with the vaneless parallel wall diffuser and the volute. This simulation setup neglects leakage effects and friction in the impeller cavities in favor of reduced computational time. For selected working points, the front and back cavities and labyrinth seals on hub and shroud are considered. This allows a comprehensive simulation of the compressor incorporating leakage losses and friction. The computational domain, including the labyrinth seals and cavity, is shown in Appendix A. All simulations have a swirl-free inflow towards the impeller. According to Behafarid and Podowski [15] and Gong et al. [16], assuming sCO2 as incompressible offers a good approximation for the compressor design. Schuster et al. [13] showed that this is also sufficient for the design procedure of the sCO2-HeRo compressor. Therefore, preliminary calculations are carried out by applying constant properties of CO2 for averaged conditions between inflow and outflow. The results are then applied as initial values for simulations considering compressibility and real gas effects using tabulated sCO2 data. More information on the adopted real gas properties are given by Schuster et al. [13]. Due to rapid property changes of sCO2 near the critical point, convergence is hard to achieve. Pressure clipping improves convergence without a considerable influence on the overall performance, since it occurs only in a small region at the blade leading edge. So far, no detailed investigations on rotating stall are carried out.

Performance Analysis

Figure 6 shows a contour plot of the velocity within the rotor and diffuser at mid span. The blading is a compromise between good guidance of the flow in between adjacent blades and the reduction of the unavoidable flow separation caused by the thick blades. The flow separations are close to the trailing edge, as indicated. Losses are mainly induced by the flow separations and the small inflow angle into the diffuser α2 causing high losses in the diffuser. The resulting design pressure ratio and isentropic efficiency are π = 1.44 and η = 73%, respectively.

Figure 7 gives a glimpse of the compressor performance map. For the design inlet conditions, the pressure ratio π is plotted versus the volume flow rate at the inlet Q. As a result of the small flow coefficient φopt, the pressure ratio does not vary much with the volume flow rate Q. Further numerical and experimental studies will also focus on changing inlet conditions to present a broad spectrum of operation points for the design evaluation.

Due to time restrictions, the previously described blading geometry is built and will be tested, although it does not fully meet the requirement of π = 1.5. Further investigations were carried out in order to reach a higher pressure ratio and efficiency. For once the inflow angle into the parallel plate diffuser α2 < 20 ° suggests the use of diffuser blading. Different types were investigated, showing that the pressure ratio and isentropic efficiency in the design point could be further increased by using profiled diffuser blading. Investigations are also carried out for the use of splitter blades for the compressor with the goal of reducing the flow separations shown in Fig. 6 and increasing α2. The design improvements can be retrofitted to the machine due to its modular design.

Mechanical Design

Various challenges occur in the design of a very compact turbomachine for sCO2, which range from manufacturing strategies to selecting components. Material compatibility, especially for lubricants and polymers, is also of great importance for sCO2.

Manufacturing of the Impellers

In the design of the sCO2-HeRo, the manufacturing of the impellers is a special challenge because of the small dimensions. Even small tolerances in the manufactured geometry can cause significant deviations compared to the calculated performance. This holds especially for the joining of the milled blades on the hub with the cover plate, which causes changes in channel height. Thus, two joining techniques are tested.

Diffusion Welding.

In this process, hub and shroud are pressed against each other at high temperature. To achieve good results, plastic deformation of the parts is required. Tolerances in the used equipment caused unacceptable differences in the channel height of up to 0.1 mm due to the plastic deformation in the welding process. More accurate equipment may reduce the error.

High Temperature Vacuum Brazing.

In this process, brazing material is applied to the top of the blade or the shroud plate and both parts are joined at high temperature. After joining, no difference in the channel height could be detected. However, due to capillary forces, the applied brazing material tends to flow into the flow channel. This was observed in one of the two manufactured impellers of the compressor while no blocking of the flow path could be found for the other. This is an acceptable result for a first attempt.

Bearings

Several bearing types were investigated, from roller bearings over foil bearings to magnetic bearings. Investigation of the rotor dynamic behavior showed the eigenfrequencies of the rotor. Due to the integrated design with two overhung impellers, it has eigenfrequencies distinguished by the oscillation of each impeller. These eigenfrequencies lie close together and exclude a wide range of speeds if the machine would operate above the first critical speed. As the machine should allow a wide range of speeds, it runs below the first critical speed. Therefore, limitations due to eigenfrequencies are avoided but high stiffness of the rotor and the bearings is required. For the rotor, this especially means to have the impellers as close as possible to the bearings and have bearings of high stiffness. Furthermore, the project requires to have a self-launching and self-sustainable system. Among the investigated bearings, magnetic bearings were dismissed because of the necessary auxiliary systems. Foil bearings cannot be used because of their limited thrust capacity and their start-up behavior requiring a minimum lift-off speed. This strongly limits the testing of the self-launching capability. Additionally, these bearing types could not provide the intended bearing stiffness for the required sizes. Finally, grease-lubricated (for lifetime) and heat-stabilized hybrid angular ball bearings with an extra-large grease reservoir, supplied by Cerobear, were chosen. It might be that sCO2 dissolves the grease and therefore reduces the service life of the bearings. This will be tested during the measurement campaign.

Motor/Generator.

According to the findings of Wright et al. [3], it is not expected that there will be any significant effect of the sCO2 on the insulation of the generator. A four pole asynchronous motor manufactured by E + A is used. The used frequency invertor, supplied by Control Techniques, allows switching from motor to generator operation on the fly.

Sealing.

The integrated machine design allows building a hermetic turbomachine that has no outgoing shaft. Therefore, O-rings are used for sealing of the casing. Clementoni and Cox [5] say that they found no issues with the sCO2 interaction for O-rings made of ethylene propylene diene monomer rubber (EPDM). Therefore, ethylene propylene diene monomer rubber is used for temperature regimes below 100 °C. For higher temperatures, O-rings made from perfluoroelastomer (FFKM) are used.

Efficient sealing of the shaft toward the generator needs to be considered in regard of the efficiency of the compressor and turbine. Because the leakage from the compressor is utilized for generator cooling, a minimum leakage flow is required. The shaft sealing is realized by stepped labyrinth seals with 3 fins on the shroud side and 4 fins on the side of the hub (see Appendix A). The stepped geometry allows easy assembly while reducing the leakage in comparison to a see-through labyrinth seal. Optimized seals with changed chamber geometry resulting in 6 and 7 fins, respectively, are developed as well, but are not yet implemented in the machine. Further optimization of the sealing efficiency on the side of the hub is possible by applying other sealing techniques e.g., dry gas seals. The sealing on the side of the shroud is difficult to optimize, because only labyrinth seals seem applicable. An accumulated leakage mass flow of 0.1 kg/s–0.15 kg/s is expected at the outlet in Fig. 8 depending on the pressure within the central housing.

Leakage Design Toward the Central Housing

The leakage control is very important for two reasons. First reason is that even small leakage losses lead to substantial reduction in performance. Second, the leakage is utilized for cooling purposes. For the leakage, basically three design options exist, which are distinguished by the pressure in the central housing and therefore strongly interact with the windage losses (see Fig. 3) and the cooling. First option is to keep the pressure in the central housing subcritical, reducing the friction losses while increasing the cooling of the generator. The compressor and turbine losses are increased and an additional pump for leakage feedback is required. This possibility was applied for the cycles of SNL, BMPC, KIER, and KAERI. The second option is to have a pressure in the central housing, which is equal or above the low pressure side of the loop (compressor inlet pressure) allowing direct feedback without any additional pump. The third option is to have an internal leakage within the turbomachine, as applied in the turbomachine of TIT. The leakage flows from the compressor toward the turbine caused by the pressure losses in the heater, the pipe between compressor and turbine and the pressure reduction in the turbine stator. Therefore, this option does not even require the additional pipe and pressure regulation valve from the central housing toward the compressor inlet. The last two options have sCO2 in the central housing, which might reduce the bearing service life. Also, the windage losses increase, while the leakage is reduced. To investigate the effect of sCO2 in the central housing, the testing starts with the option 1 and the leakage flow will be reduced step-by-step.

Cooling of the Generator

The compactness of the turbomachine sets some requirements to the cooling of the machine, in particular of the generator. The required cooling of the generator is realized by a leakage flow through the generator. This avoids external cooling devices and supports the self-sustainable operation. The heat transfer analysis shows that the cooling by the leakage is sufficient when combined with cooling fins in the casing. In fact, the heat-up of the CO2 due to the generator losses is less than 20 °C as visible in Fig. 8, which shows the results of a simplified conjugated heat transfer calculation. The leakage flow efficiently cools the generator and also takes out sufficient heat to avoid a large heat transfer from the turbine to the compressor side.

Final Design of the Supercritical CO2 Heat Removal System Turbine, Alternator and Compressor

Figure 9 shows a picture of the final turbomachine during cold-run air tests. The overall width of the casing is 340 mm and the length of the shaft including impellers is 270 mm. The casing has a maximum diameter of 250 mm, which is required to allow screwing the different parts together. Thermocouples and a vibration sensor monitor the behavior of the machine and especially the bearings. The picture shows an additional cooling jacket that can be used to provide additional cooling and also lead the extracted heat to an outside cooler as not to heat up the room. Figure 10 shows the compressor impeller with a maximum diameter of 40 mm. The right part of the figure presents the stepped geometry of the labyrinth seal.

First tests of the machine included an overspeed test of the impellers at 120% of design speed and a pressure test of the casing followed by cold-run tests with air at pressures up to 40 bar. The air tests verified that the turbomachine can reliably reach its design speed and that all control signals are correct. Vibrations were monitored to check the vibration level. All tests were successfully conducted. Performance testing with sCO2 is done at Centrum výzkumu Řež (CVR) in Řež, Czech Republic.

Summary and Future Work

This paper describes the development of the turbomachinery for the sCO2-HeRo project—a promising approach to bring nuclear power plant safety to an even higher level. The challenges in designing small turbomachinery for carbon dioxide are explained and are also applicable for other carbon dioxide cycles.

The proposed design allows a self-sustained operation without the need of supplementary aggregates. The announced challenges regarding lubricant and O-ring material will be verified by performance test.

A suitable compressor blading design is presented and the geometry is fully described. This gives the possibility to validated calculation tools once the performance tests are completed, and hence allows further detailed analysis and optimization of this geometry and future geometries with validated design tools.

Funding Data

• H2020 European Research Council and Training Program 2014–2018 Under Grant Agreement No 662116.

Nomenclature

• cm =

meridional speed (m/s)

• D =

diameter (m)

• h =

channel height (m)

• k =

blocking factor

• L =

length (m)

• $m˙$ =

mass flow (kg/s)

• n =

rotational speed (1/s)

• p =

pressure (Pa)

• P =

power (kW)

• Q =

volume flow (m3/s)

• s =

• t =

gap width (m)

• T =

temperature (° C)

• u =

circumferential speed (m/s)

• y =

specific flow work (J/kg)β

• =

• δopt =

$(ψopt1/4/φopt)=D×2×y/Q024×(π/2)$, specific speed

• η =

efficiency (%)

• π =

pressure ratio

• ρ =

density (kg/m3)

• σopt =

$=(φopt/ψopt3/4)=n×(Q0/2×y3/4)×2×π$ , specific speed

• φopt =

$=(cm/u)=(4×Q0π2×D3×n)$, flow coefficient

• Ψopt =

$=(2×y/u2)=(2×y/π2×D2×n2)$, head coefficient

Subscript
• C =

compressor

• T =

turbine

• 0 =

Inlet

• 1 =

impeller inlet

• 2 =

impeller outlet

• 3 =

diffuser outlet

Abbreviations
• sCO2 =

supercritical CO2

• TAC =

turbine, alternator, compressor system

Appendices

Appendix A: Cavity and Labyrinth Geometry

Figure 11 contains a sketch of the compressor with dimensions for the impeller cavities and labyrinth seals in the detailed views A and B, respectively. The inflow is a nozzle reducing the diameter from 15.8 mm at the compressor inlet to 12 mm at the impeller. The cavity has an outer diameter of 41 mm and width of up to 4 mm. At a diameter of 84 mm, the flow exits the diffuser into the volute with a constant with of 7.5 mm. The height of the volute increases linearly from 0 mm to 8.5 mm with one revolution. Then the flow enters into a circular pipe with diffuser, which increases the diameter from 7.5 mm to 15.8 mm. Detail B shows a chamber of the stepped labyrinth seal. As all chambers are equal, dimensions are only given for one. Each chamber is 3.5 mm high and 5 mm wide. The clearance is 0.1 mm with a fin being 0.25 mm wide at the tip.

Figure 12 and Table 4 contain the coordinates to the blading curves on pressure and suction side on the side of the cover disk. As a two-dimensional blading is applied, replacing the z-coordinate with 0 mm gives the curve on the side of the shroud. The axis of rotation is the z-axis.

References

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References

Benra, F.-K. , Brillert, D. , Frybort, O. , Hajek, P. , Rohde, M. , Schuster, S. , Seewald, M. , and Starflinger, J. , 2016, “ A Supercritical CO2 Low Temperature Brayton-Cycle for Residual Heat Removal,” The Fifth International Symposium—Supercritical CO2 Power Cycles, San Antonio, TX, Mar. 28–31, pp. 1–12.
Bohl, W. , and Elmendorf, W. , 2013, Strömungsmaschinen 1, 11th ed., Vogel Verlag, Würzburg, Germany.
Wright, S. A. , Radel, R. F. , Vernon, M. E. , Rochau, G. E. , and Pickard, P. S. , 2010, “ Operation and Analysis of a Supercritical CO2 Brayton Cycle,” Sandia National Laboratories, Albuquerque, New Mexico, Report No. SAND2010-0171.
Clementoni, E. M. , Cox, T. L. , and Sprague, C. P. , 2014, “ Startup and Operation of a Supercritical Carbon Dioxide Brayton Cycle,” ASME J. Eng. Gas Turbines Power, 136(7), p. 071701.
Clementoni, E. M. , and Cox, T. L. , 2014, “ Practical Aspects of Supercritical Carbon Dioxide Brayton System Testing,” The Fourth International Symposium—Supercritical CO2 Power Cycles, Pittsburgh, PA, Sept. 9–10.
Kimball, K. J. , and Clementoni, E. M. , 2012, “ Supercritical Carbon Dioxide Brayton Power Cycle Development Overview,” ASME Paper No. GT2012-68204.
Kimball, K. J. , Rahner, K. D. , Nehrbauer, J. P. , and Clementoni, E. M. , 2013, “ Supercritical Carbon Dioxide Brayton Cycle Development Overview,” ASME Paper No. GT2013-94268.
Cho, J. , Choi, M. , Baik, Y.-J. , Lee, G. , Ra, H.-S. , Kim, B. , and Kim, M. , 2015, “ Development of the Turbomachinery for the Supercritical Carbon Dioxide Power Cycle,” Int. J. Energy Res., 40(5), pp. 587–599.
Cho, J. , Shin, H. , Ra, H.-S. , Lee, G. , Roh, C. , Lee, B. , and Baik, Y.-J. , 2016, “ Research on the Development of a Small-Scale Supercritical Carbon Dioxide Power Cycle Experimental Test Loop,” The Fifth International Symposium—Supercritical CO2 Power Cycles, San Antonio, TX, Mar. 28–31.
Cho, J. , Shin, H. , Cho, J. , Ra, H.-S. , Roh, C. , Lee, B. , Lee, G. , and Baik, Y.-J. , 2017, “ Development of the Supercritical Carbon Dioxide Power Cycle Experimental Loop With a Turbo-Generator,” ASME Paper No. GT2017-64287.
Utamara, M. , Hasuike, H. , Ogawa, K. , Yamamoto, T. , Fukushima, T. , Watanabe, T. , and Himeno, T. , 2012, “ Demonstration of Supercritical CO2 Closed Regenerative Brayton Cycle in a Bench Experiment,” ASME Paper No. GT2012-68697.
Cha, J. E. , Bae, sW. , Lee, J. , Cho, S. K. L. J. I. , and Park, J. H. , 2016, “ Operation Results of a Closed Supercritical CO2 Simple Brayton Cycle,” The Fifth International Symposium—Supercritical CO2 Power Cycles, San Antonio, TX, Mar. 28–31.
Schuster, S. , Benra, F.-K. , and Brillert, D. , 2016, “ Small Scale sCO2 Compressor Impeller Design Considering Real Fluid Conditions,” The Fifth International Symposium—Supercritical CO2 Power Cycles, San Antonio. TX, Mar. 28–31.
Bilgen, E. , and Boulos, R. , 1973, “ Functional Dependence of Torque Coefficient of Coaxial Cylinders on Gap Width and Reynolds-Numbers,” ASME J. Fluids Eng., 95(1), pp. 122–126.
Behafarid, F. , and Podowski, M. Z. , 2016, “ Modeling and Computer Simulation of Centrifugal CO2 Compressors at Supercritical Pressures,” ASME J. Fluids Eng., 138(6), p. 061106.
Gong, Y. , Carstens, N. A. , Driscoll, M. J. , and Matthews, I. A. , 2006, “ Analysis of Radial Compressor Options for Supercritical CO2 Power Conversion Cycles,” Center for Advanced Nuclear Energy Systems, MIT Department of Nuclear Science and Engineering, Cambridge, MA, Report No. MIT-GFR-034.

Figures

Fig. 1

Cross section of the sCO2-HeRo TAC

Fig. 2

Cordier diagram for turbocompressors [2]

Fig. 3

Windage losses in the generator cavity

Fig. 4

Axis normal section of the compressor impeller (velocity triangles not to scale)

Fig. 5

Meridian sketch and velocity triangles (not to scale)

Fig. 6

Velocity profile

Fig. 7

Compressor performance map

Fig. 8

Temperature distribution in the central housing

Fig. 9

sCO2-HeRo TAC during the cold air tests

Fig. 10

Supercritical CO2 heat removal system compressor impeller (units in cm)

Fig. 11

Cavity and labyrinth geometry

Fig. 12

Tables

Table 1 Thermodynamic parameters for the SCO2-HERO TAC [1]
Table 2 Compressor design parameters
Table 4 Blade curves in coordinates

Errata

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# The Stern-Brocot Tree

 Title: The Stern-Brocot Tree Authors: Peter Gammie and Andreas Lochbihler Submission date: 2015-12-22 Abstract: The Stern-Brocot tree contains all rational numbers exactly once and in their lowest terms. We formalise the Stern-Brocot tree as a coinductive tree using recursive and iterative specifications, which we have proven equivalent, and show that it indeed contains all the numbers as stated. Following Hinze, we prove that the Stern-Brocot tree can be linearised looplessly into Stern's diatonic sequence (also known as Dijkstra's fusc function) and that it is a permutation of the Bird tree. The reasoning stays at an abstract level by appealing to the uniqueness of solutions of guarded recursive equations and lifting algebraic laws point-wise to trees and streams using applicative functors. BibTeX: @article{Stern_Brocot-AFP, author = {Peter Gammie and Andreas Lochbihler}, title = {The Stern-Brocot Tree}, journal = {Archive of Formal Proofs}, month = dec, year = 2015, note = {\url{http://isa-afp.org/entries/Stern_Brocot.html}, Formal proof development}, ISSN = {2150-914x}, } License: BSD License Depends on: Applicative_Lifting

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# Are the Maxwell's equations enough to derive the law of Coulomb?

Are the 8 Maxwell's equations enough to derive the formula for the electromagnetic field created by a stationary point charge, which is the same as the law of Coulomb?

If I am not mistaken, due to the fact that Maxwell's equations are differential equations, their general solution must contain arbitrary constants. Aren't some boundary conditions and initial conditions needed to have a unique solution. How is it possible to say without these conditions, that a stationary point charge does not generate magnetic field, and the electric scalar potential is equal to

$$\Phi(\mathbf{r})=\frac{e}{r}.$$

If the conditions are needed, what kind of conditions are they for the situation described above (the field of stationary point charge)?

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For essentially the opposite question(v3), see this Phys.SE post. –  Qmechanic Nov 17 '12 at 11:27
Yes, I have already read this post, but my question is quite different. –  achatrch Nov 17 '12 at 11:54
Wouldn't it be trivial to apply the divergence theorem to Gauss' law to get it in its integral form. From here it seems easy enough to use the usual tricks to find the electric field of a point charge, and then multiply by some charge to get your force. Surely this is Coulomb's force law? –  Daniel Blay Nov 17 '12 at 12:31
Why 8 Maxwell's equations and not 4? I am missing something? –  DaniH Nov 17 '12 at 12:35
@DanilH: I meant 8 scalar equations. From the 4 Maxwell's equations two are vector equations. –  achatrch Nov 17 '12 at 12:45

The short answer is yes, and in fact you only need one single Maxwell equation, Gauss's law, together with the Lorentz force, to get Coulomb's law.

More specifically, you need Gauss's law in its integral form, which is equivalent to the differential form for well-behaved fields because of Gauss's theorem. Thus, you use the law $$\nabla\cdot\mathbf{E}=\rho/\epsilon_0\quad\Leftrightarrow\quad \oint_S\mathbf{E}\cdot\mathrm{d}\mathbf{a}=Q/\epsilon_0,$$ where $Q$ is the total charge enclosed by the (arbitrary) surface $S$.

To derive Coulomb's law, consider the electric field of a single point particle, with nothing else in the universe. Because of isotropy (which must be added as an additional postulate), the electric field at a sphere of radius $r$ centred on the charge must be radial and with the same magnitude throughout. That means the integral is trivial and the electric field must be $$\mathbf{E}=\frac{Q}{4\pi\epsilon_0 r^2}\hat{\mathbf{r}}.$$

Coupled with Lorentz's force law at zero velocity for the test particle (since Coulomb's law only holds in electrostatics) this yields Coulomb's law.

It is not obvious that this highly symmetric situation can give the general electrostatic force for multiple particles. This follows from the superposition principle, which is very much at the heart of classical electrodynamics, and which can be obtained from the linearity of Maxwell's equations. This gives you the field for a single source; add the fields for all the individual sources and you'll get the field for the collection of sources.

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The exact derivation goes as follows. You start from Gauss' Law, integrate on both sides over some volume V:

$$\stackrel{\tiny div}{\vec{\nabla}}\cdot\vec{\mathbf{E}}=\frac{1}{\epsilon_0}\rho \,\,\,\,\,\,\,\,\,\,\,\Big/\iiint\limits_V\,d^3\vec{r}$$ Then switch to integration over a closed surface, and also note that total charge inside this volume is Q: $$\iiint\limits_V\stackrel{\tiny div}{\vec{\nabla}}\cdot\vec{\mathbf{E}}\,\,d^3\vec{r}=\oint\vec{\mathbf{E}}\cdot d\vec\sigma=\iiint\limits_V\frac{1}{\epsilon_0}\rho\,\,d^3\vec{r}=\frac{Q}{\epsilon_0}$$ Now you need to note that the volume of integration is quite arbitrary and so is the surface, so we will use a sphere. You can describe the integral over a sphere using: $$\frac{Q}{\epsilon_0}=\oint\vec{\mathbf{E}}\cdot d\vec\sigma=\int\limits_{\phi=0}^{\phi=2\pi}\int\limits_{\theta=0}^{\theta=\pi}\mathbf{E}\hat{\vec{n}}\cdot\hat{\vec{n}}\,R\,d\phi\,R\,d\theta=4\pi R^2\mathbf{E}\,\,\,\,\,\,\Big/\frac{1}{4\pi R^2}$$ And so you obtain: $$\mathbf{E}=\frac{Q}{4\pi \epsilon_0 R^2}$$ It should be: $$\vec{\mathbf{E}}=\frac{Q}{4\pi\epsilon_0 R^2}\hat{\mathbf{r}}$$ But I lost the normal vector along the way (I hope that someone can correct this and edit this post).

Now you use the Lorentz Force law (where $\vec{\mathbf{B}}=\vec 0$): $$\vec{\mathbf{F}}_{lor}=q \vec{\mathbf{E}}+q \vec{\mathbf{V}}\times\vec{\mathbf{B}}=\frac{q\,Q}{4\pi\epsilon_0 R^2}\hat{\mathbf{r}}$$ And so you obtain the Coulomb force law.

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No you didn't lost the unit vector. You calculated the charge, which is not a vector. It's a geometric argument(already given in the previous answer physics.stackexchange.com/a/44423/16689 : the electric field at a sphere of radius r centred on the charge must be radial and with the same magnitude throughout) that allow you to recover the vector form of the electric field. –  FraSchelle Jun 25 '13 at 15:42
thanks, I was hoping for a more strict (mathematically speaking) method of recovering the unit vector. It is true that charge is not a vector, so it is impossible to obtain a unit vector in this calculation. But simply "adding" (writing) it after derivation is complete is not what I prefer. Maybe we can do it strictly mathematically if we use the $\hat{\vec{z}}$ axis which currently is used only to define the $\theta$ angle inside the integral. –  cosurgi Jun 25 '13 at 16:04

Given the Gauss law AND the Lorentz force, yes it's possible to derive Coulomb's law as it already been answered. So I think the question is if it's possible to derive it given ONLY the four Maxwell equations (and not the Lorentz force). The answer is still yes since the Lorentz force is equivalent to the Faraday law and it can be derive from it. The relation between Faraday law and Lorentz force is not trivial in 3D since the Faraday paradox arises (check wiki). On the other hand when electromagnetism is expressed in the covariant formulation no such paradox exists.

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Lorentz force formula does not follow from Maxwell's equations, nor from the Faraday law $\nabla \times \mathbf E = -\frac{\partial \mathbf B}{\partial t}$, which is part of Maxwell's equations. –  Ján Lalinský Jan 16 '14 at 1:22

If you are asking about the Coloumb's law for the electric fields, yes, which you can see others' answers.

If you are asking about the Coloumb's law for the electric force,

$$\text{NO!}$$

Maxwell equations do NOT tell you about how the force acting on the charges $q$ or currents $\vec{J}$.

Simply speaking, to FULLY understand classical E&M (i.e. one can determine the physics from a initial value problem to determine all its consequences - physics is all about to determine/predict the future), you need BOTH:

(1) Maxwell's equations

(2) Lorentz force law (Newtonian mechanics, E&M equivalence of Newtonian gravitation force.)

Punch line I: (1) and (2) are absolute different things.

Lagrangian and variational principle E.O.M. viewpoint

However, if you start from a Lagrangian viewpoint, writing down the action: $$S=\int (-\frac{1}{2} |f|^2 + A \wedge *J)=\int d^3xdt (-\frac{1}{4} f_{\mu\nu} f^{\mu\nu} + A_\mu J^\mu)$$ with the 2-form field strength $f_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu$ gives you E and M fields, you can determine Maxwell's equation from the equations of motions(E.O.M.) by doing the variation principle on the 1=form gauge field $A$. The source is a 1-form current $J=(\rho,\vec{J})$.

Maxwell's equations: E.O.M. respect to varying 1-form gauge field $A$

You obtain Maxwell equations by varying $A$: $$d*F=J \;\;\;\text{-Gauss law for electricity, Maxwell-Ampere's law}$$ and $$dF=d^2A=0 \;\;\;\text{-Gauss's law for magnetism, Maxwell–Faraday equation}$$

How about the Lorentz force law? You can do variation respect to the spatial coordinate $x^\mu=(t,x)$, and you need to specify which massive particle with mass $m$ experiencing the force $F$, which is $F=m \ddot{\vec{x}}$ by Newtonian mechanics. To specify massive particle in the Lagrangian/action, you just need to add its kinetic energy $\frac{1}{2}m \dot{\vec{x}}^2$.

Lorentz force law: E.O.M. respect to varying spacetime coordinates $x^\mu=(t,x)$

$$S=\int d^3xdt (-\frac{1}{4} f_{\mu\nu} f^{\mu\nu} + A_\mu \wedge J^\mu+\frac{1}{2}m \dot{\vec{x}}^2) \to \int d^3xdt (+ q\Phi - q \dot{\vec{x}} \cdot \vec{A}+\frac{1}{2}m \dot{\vec{x}}^2)$$

you will derive Lorentz force law $$m \ddot{\vec{x}}=q\vec{E}+q \dot{\vec{x}} \cdot \vec{B}$$

Punch line II: The action and Variational principles are very powerful to unite (1) Maxwell's equations and (2) Lorentz force law, in the same framework.

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The above sketch of derivation of force on point particles appears in good books, but has a fundamental flaw for point particles. For such particles, the interaction field term has no value, since the field is singular at the particle, and the pure field term is infinite, which invalidates any formal differentiation. The result, although it seems correct, is wrong: as vector potential, the total electric and magnetic fields are not defined at the place of the particle $\vec{x}$. –  Ján Lalinský Jan 22 '14 at 5:08
@ Ján Lalinský, thanks but you miss my point. The field is NOT from the singular-size point particle, but from the external source surrounding it. Such as external current, electric planes, etc. The derivation is carried out the external EM effects on the point particle. –  Idear Jan 22 '14 at 5:29
So your comment is actually NOT to my point. –  Idear Jan 22 '14 at 5:32
In the first part of your derivation, you talk about deriving Maxwell equations, so there $f_{\mu\nu}, A_\mu$ are total fields produced by prescribed sources. In the second part after "How about the Lorentz force law? ", if by $f_{\mu\nu}, A_\mu$ you refer to external fields, the derivation is correct, but please consider making this clear in your text: due to the first part and even without it, when most people see these symbols, they assume they refer to the total field. –  Ján Lalinský Jan 22 '14 at 8:37

## protected by Qmechanic♦Jan 15 '14 at 22:42

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How could "groundhogging" work, and what explanation would require the least handwaving?

Provoked by a comment by Steve Mangiameli.

In my short story series, Penniless Joe is a down on his luck archaeologist/(al?)chemist that is struggling to get employment.

One day he goes to a pawn shop sell some stuff. The pawn shop owner convinces him to buy some weird liquor bottle he found. Joe is intrigued by the ceramic vial, with some uncommon characters he barely recognizes. He buys it for \$1.99 and takes it home to decipher it.

He finds out that it is a "Groundhog day potion" that supposedly allows one to re-live one single day.

Joe decides to use the potion to get out of his wealth predicament. His choice is to win the state lottery.

Unknown to him, he made a mistake translating the inscriptions on the potion, and become a permanent "groundhogger".

The rules so far are:

• A groundhogger re-lives each day from 3 to 5 times before moving on to the next day. The amount of repetitions is random and cannot be predicted.
• A "repeat day" is day that will be erased and repeated. Days 1 and 2 of a cycle are always a "repeat day", day 3 is a repeat day ~33% of the time, and so on.
• The "last day" or "real day" is the day where "tomorrow" is "calendar tomorrow".
• If a groundhogger stays awake until midnight of a repeat day, everything goes black exactly at midnight, and the day begins anew at his first conscious moment of that day.
• If a groundhogger stays awake until midnight of a last day, he does not black out, and tomorrow's repeat days begin exactly at midnight.
• Anything that happens on a repeat day has no permanent effect. He just wakes up again next(this) morning. Just like Bill Murray's, even death is not permanent. He just awakes again this morning (to his subjective perception, immediately after losing consciousness and dying).
• That includes internal body changes, like catching a disease or impregnation (for a female groundhogger).
• Physical fatigue/wounds are completely erased. Mental fatigue is relieved completely if he wakes up from sleep. but only partially if he stays awake overnight. For that reason, they prefer to sleep.
• Since days 1 and 2 are always repeat, they are more daring on these days.
• He probably becomes permanently dead if he dies on a "last day". But he did not try that so it is unknown.
• He is not bound to a specific location. So he can travel to Vegas on a "last day" and wake up in Vegas tomorrow.
• He retains full knowledge/memories of what happened in the repeat days, to the limit of his own cognitive capabilities. No physical objects (paper notes, files on a USB drive, etc) are retained or changed on a repeat day.

Later on he finds other groundhoggers. It is dead easy to spot one when you meet him, because it is the only thing that changes on repeat days.

So on the interaction between groundhoggers:

• Both groundhoggers freely and independently retain memories of repeat days.
• Both of them can act on their own free will and do whatever they want on repeat days. Even avoid/escape meeting the other guy again.
• If a groundhogger is killed by another groundhogger on a repeat daynote1, the dead one awakes next morning (to his subjective perception, immediately).
• If a groundhogger kills another one in a last day, the effect is unknown. Probably permanent, but nobody tried.
• There is no compulsion over groundhoggers (no, there can be more than one) to act or interact with each other. There may be curiosity or kinship, but most avoid each other out of fear.
• Even on repeat days, the actions of one groundhogger can cause changes on the environment of another (thus causing the repeat days not to be equal). This (things not happening exactly the same way) is usually the tip to the presence of another groundhogger.
• It is unknown if all of them experience the same amount of repeat days for each day.

Given the rules above that already fleshed out on the written stories (and considered canon), what is the best explanation to how it works? I thought of some:

• Physical Time travel: groundhoggers do travel back in time. This does not explain how their bodies are repaired/recovered.
• Subjective space/precognition: they aren't actually repeating the days. They gain a very precise precognition ability, and are only mentally re-living that in their minds. This does little to explain how groundhoggers can interact with each other independently.
• Mental Time travel: groundhoggers send their consciousness back in time. This has the problem of explaining how their consciounsness are replaced.
• Alternate dimensions: each repeat day happens in an alternate dimension. This burns even more handwavium than the first one, above.

The list above is not meant to be exhaustive. Other explanations might exist.

So, what explanation to the effect described above requires the least handwaving, what its pros and cons, and where does it fall short?

1. They know days 1 and 2 are always repeat. So they can be rather daring and open to experiments on these days.
• Are you feeling very strongly regarding the potion aspect? What if it's a more meta-physical explanation and the groundhogger has a little more control of the day(s) to be relived? You could do this with a talisman or device (alien?) that has the ability to anchor consciousness to a particular space/time plane, lasting 24 hours before reset. Whether you do this with the potion or device, I'm having trouble reconciling the rule regarding location - how does everything else reset except for the location? May 25, 2016 at 18:54
• The talisman option could add to the story and why there is an organization hunting down groundhoggers, collecting the finite number of available (not able to be duplicated/synthesized) devices in order to shape history as they see fit or some other such end. May 25, 2016 at 18:57
• @Steve: I believe that the OP is saying that location isn't an exception — on a "repeat day", everything resets (including location); on a "last day", nothing resets (not even location). May 26, 2016 at 1:21
• Why not just have the main character be insane. Jun 22, 2016 at 19:42

Since travelling back in time, swapping consciousness, etc. are all pretty "magical" explanations and you're looking for some realism, I'd like to propose an unpleasant possibility:

These groundhoggers are not human beings with amazing capabilities. Instead, they are aspects of a Virtual Reality which are bugging out.

Terminally Ill

Joe is a terminally ill patient hooked up to a Virtual Reality machine - his body is broken beyond repair, but his family is paying for him to live out his days in a fantastical VM universe where he can enjoy his last few days.

The scenario for which they paid is one in which Joe gets to use his business knowledge to manipulate the world in a setting where the day repeats, but he's the only one who realizes it - they know it would be a fun challenge for him!

Economic Sim

This was actually the subject of a sci-fi book from the 80's which I read about a decade ago:

An accident at a nearby chemical plant releases a toxin into a small town which puts all of the inhabitants in a coma - they will soon die. The government seizes the opportunity! They go in and copy each person's consciousness, then makes a miniature reproduction of their city, and tiny robots that look like the people, etc. (ridiculous, I know), then uploads their memories, and use that setup to run social experiments on them. They reset the robots at the end of each experiment, but one day they miss one. He begins to relive the same day over and over again, with no explanation of what the heck is going on. He also witnesses the others being reset, etc.

You could explain your story in a similar sense, but in a virtual world, not using tiny robots. Joe is an uploaded consciousness, or even an AI which is bugging out, and thinks the VM is real. He is not being properly reset when the VM reloads.

• The VR option reminds me too much of Vanilla Sky... I really like (and it surprised me) the lateral thinking you showed us. Thanks for your feedback. May 25, 2016 at 19:19

You're not going back in time, you're going forward

Scientifically, it's hard to argue you can go back in time and everything resets (except your memory). It's a lot easier to suggest your mind explores the day before it actually happens (and that it can do so a couple of times before the actual day).

So let's say some scientific phenomenon says that there's a kind of pseudo-quark potential energy (PQSE) that radiates off all matter in anticipation of how it will act and interact with the rest of the universe.

When groundhoggers sleep their consciousness begins to read this PQSE and experience living in the day that hasn't happened yet. Time, the length of their "day", the mental state of the groundhogger etc. could determine how many times they can repeat before the actual day dawns and they wake up, this time to actually live the day.

Since all groundhoggers tap into the same PQSE, their actions effect what each other see, so they can interact with each other.

To the groundhogger, the real days wouldn't seem much different to the PQSE day, the only difference is that the changes of this day carry on into the future days.

The best explanation, I think, would be to give no explanation at all. You are making up a set of rules about time travel that does not work in reality and as you said, any explanation would be a hand wave. Just having the potion work with no explanation to its origins is a valid strategy. This is what the groundhog day movie did - they used to have scenes showing that the looping was caused by a curse by the main characters' ex-girlfriend, but the movie worked better with those scenes removed.

That said, if you do want to have an explanation here is my ideas. Two thing you wrote stick out to me. One are some of the rules: the reset is always at midnight, and the number of repeats of each day number from 2-4 with an even distribution. This sounds like rules that are purposefully designed by a human to be useful and intuitive but not too exploitable, and not what you'd do if you wanted to use the potion yourself if you could help it. The other is that you mentioned that Joe is an alchemist, meaning there is already magic in the setting (or at least alchemy). The most believable explanation is the one that requires the fewest new assumptions, so the explanation would preferably be magical as well.

Looping explanation: the potion loosens the grip your soul has on your body, making it detach and attach again at certain times. Your soul holds your memories, but you body is what grounds you in reality - including time. When you've experienced a repeat day, your soul is pulled back and merges with the previous version of you, giving him the memories. This functions even if you died, since the soul still hangs around after death.

Potion explanation (less necessary to include, but the reason for its existence): A whole bunch of these was made by a powerful wizard/alchemist/genie/god/other that wanted entertainment. Such a potion is complicated and expensive to make, but he/she/it made and distributed several all over the world to be sold cheap to observe what fun things those who drank it would get up to and interact with each other - while being able to observe everything from some cosy pocket dimension or other space outside of time.

The problem with a potion is it is ingested and wears off in time as the body metabolizes the potion. The strength of the potion and the speed of an individual's metabolism would effect the duration differently, dose to dose and person to person. But eventually it would stop working. You see this in many other story lines and makes a lot of sense. As such, your protagonist would need to constantly be ingesting the potion with no way of knowing how long it will last, but somewhere between 3 - 5 days.

The potion doesn't turn back time, per se, but causes the user's consciousness to "anchor" to a point in time. When the consciousness gets 24 hours out from the anchor, the user is yanked back to the anchor point. It is only when the potion wears off that the anchor is released and the consciousness is allowed to continue down the linear path of time and space as we know it.

This mechanic allows for a little more reality in that it isn't a "forever" effect and can be controlled, to a point, by the user. Additionally, the user is bound by the 24 hour rule for 3 - 5 days, but not by an arbitrary, already done, midnight. This would give the user a great deal more flexibility in the usage of the potion.

Taking the potion allows the user, known as a Groundhogger, to learn "quicker", be extremely "lucky", and the "ability" to predict future events. These are the tell tale signs other Groundhoggers use to spot each other, of which, there are few others.

Joe, a brilliant, young chemist, has trouble holding a job due to his intellect and intuition when it comes to chemical bonding. His superiors are threatened and his peers think he's just weird. Joe happened on the potion at a science expo, perusing the latest advancements. He runs into a frightened, desperate man who identifies, begging for help from unknown pursuers. Before Joe can respond, the man bolts off again, leaving behind several vials of potion in Joe's coat pocket. Joe finds these later and reverse engineers them - finding all of the ingredients odd, but safe for consumption.

The potion isn't really a potion, but rather a by-product of time travel research. The researchers aren't part of any government agency and their exact existence and purpose are a mystery. Suffice it to say, there is a lot of power and wealth that comes with the ability to know the future, even if it is only 24 hours out. This agency will stop at nothing to protect their research and get back what was taken.

Some unknowns at this point that may need to be fleshed out and could come in handy as plot twists or guides:

• How exotic are the ingredients? While known and consumable, maybe one or two are difficult to come by
• Does the Groundhogger continue to age while looping? A 30 year old that looks 60 could be another "indicator" of a Groundhogger. Of course a billionaire 20 something with the wisdom of a century could be as well. Just depends on whether or not they age normally
• Are there any after effects of the taking the potion? Beside possible aging effects, does taking the potion constrict blood vessels or put undue stress on the heart, or over time cause consciousness hold in time/space to be lost

Lots of questions! But this sounds like a really fun project. Best of luck!

• Welcome to the site Steve. If you have questions on the site check out the help center and feel free to visit us in Worldbuilding Chat May 26, 2016 at 14:06


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2.2k views
In an $11-bit$ computer instruction format, the size of address field is $4-bits.$ The computer uses expanding OP code technique and has $5$ two-address instructions and $32$ one-address instructions. The number of zero-address instructions it can support is ________
retagged | 2.2k views

No. of possible instruction encoding $=2^{11} = 2048$

No. of encoding taken by two-address instructions $=5 \times 2^4 \times 2^4 = 1280$

No. of encoding taken by one-address instructions $=32 \times 2^4 = 512$

So, no. of possible zero-address instructions $=2048 - (1280 + 512) = 256$
selected by
0
what is the logic behind the calculation of no. of 2 address and 3 address instructions ?

@arjun sir @habibkhan
+10
0

Could you generalize this concept for all  the problems because by using this method I am unable to find out the answer to all the questions .

Plz see this.

0

Start at 07:32

Out of 11 bits, 3 bits are for opcode in 2-Address. So 2^3=8 instructions(2-Address) are possible.

But 5 are being used. So 3 are free.

We use these 3 for 1-address instruction by using 4 bits of 1 of the addresses i.e. 3*16=48

Now out of 48 1-address instructions, 32 are being used. So we are left with 16.

Again use these 16 for 0-address instruction by using 4 bits of address i.e. 16*16=256
0
why have you done 16*16, can you explain this in more detail

1


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# How to Multiply a Fraction?

## Introduction on How to Multiply a Fraction

Last updated date: 19th Mar 2023
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A fraction is a number that is a component of a whole. By breaking a whole into a number of parts, it is evaluated. If a fraction is expressed as a/b, then a and b are its components, with a serving as the fraction's numerator and b serving as its denominator. For example, if 1/2 is a fractional value then 1 is the numerator and 2 is the denominator and this value represents half of a whole number.

Introduction on how to Multiply Fraction

The product of a fraction by another fraction or an integer or a set of variables is referred to as the multiplication of a fraction.

Multiplication is actually a process of repetition of addition. Look at the image below to understand this. After reading this article one will be able to answer and solve questions like finding the product of fractions, how to multiply unlike fractions, etc.

## How to Multiply Fraction

The steps for multiplication of fractions are as follows:

• Multiply the numerator with the numerator

• Multiply the denominator with the denominator

• Simplify the fractions(reduce the resultant fraction to its lowest term), if required.

## Fractional Multiplication

Now, we'll talk about fractional multiplication by whole numbers, fractional numbers, and mixed fractional numbers.

## Multiplication of Fraction by a Whole Number

To multiply a fraction with a whole number, first multiply the numerator of the fraction by the whole number and then reduce the fraction to the lowest terms, if required.

Let us multiply 2/3 by 4 to understand the multiplication of fractions by whole numbers. We use the rule of repeated addition to solve it.

Multiplication of Fraction by a Whole Number

That means, we can say that 2/3 of 4 is 8/3.

## Multiplication of Fraction by another Fractional Number

When multiplying two or more fractions, the denominators are multiplied to obtain the product's denominator and the numerators are multiplied to obtain the new numerator of the product.

Let us multiply 1/4 by 1/2 to understand the multiplication of fractions by other fractional numbers.

To do so, first multiply the numerators, 1 by 1, and then multiply the denominators, 4 by 2.

Multiplication of Fraction by another Fractional Number

That means, we can say that 1/4 of 1/2 is 1/8.

## Multiplication of Fraction by a Mixed Number

To multiply a fractional number by a mixed number first convert the mixed number into an improper fractional number and then simply apply the rule we discussed above to multiply a fraction with another fractional number. Look at the example given below to understand this concept.

Let’s multiply 6 by 3$\frac{1}{4}$,

To find the product of the fraction and the mixed number, we will first simplify the mixed number 3 ¼ as;

(4 × 3 + 1)/4 = 13/4

Then,

6 × 13/4 = 78/4.

## Multiplication of Fractions Examples

Let’s understand the concept better with some more multiplication of fractions examples.

Example 1: Multiply 4 by 2/8.

Ans: Multiply the numerator by 4.

4 × 2 = 8

That means,  4 × 2/8 = 8/8

Now reduce the resultant fraction to its lowest term,

8/8 = 1

Therefore, 4 by 2/8 = 1.

Example 2: Multiply 2/3  by 5/9.

Ans:

Step1. Multiply the numerators,

2 × 5 = 10

Step 2. Multiply the denominators,

3× 9 = 27

Therefore, 2/3 by 5/9 = 10/27.

Example 3: Multiply 2 1/3  by 5 2/5.

Ans:

Step1. Convert the mixed numbers to improper fractions,

(2 × 3+1)/3 × (5 ×5+2)/5 = 7/3 × 27/5

Multiply and simplify;

7/3 × 27/5 = 189/15.

Therefore, 2 1/3  by 5 2/5 = 189/15.

Q1. Multiply 2/7 by 3.

(Ans. 6/7)

Q2. Multiply 1 1/2 by 4 4/7.

(Ans. 96/14)

Q3. Multiply 5 by 8/10.

(Ans. 4)

## Summary

Multiplication of fractions refers to the operation of multiplying one fraction by another fraction, an integer, or a collection of variables. We can easily find the product of fractions by following a simple three-step procedure, first, multiply the numerators then the denominators and finally reduce the product to its lowest term if required.

## FAQs on How to Multiply a Fraction?

1. How to multiply unlike fractions and explain it with the help of an example?

Just as we multiply two fractions i.e a fractional number with another fractional number, we can also multiply unlike fractions (the fractional numbers with different numerators or denominators or both). This can be better understood by the following example.

Consider if we have to multiply 2/5 by 1/2,

To do so, first, we will multiply the numerators i.e. multiply 2 by 1, and then the denominators i.e. multiply 5 by 2.

Hence, the product of the 2/5 by 1/2 is 1/5.

2. What is the difference between proper fractions and improper fractions?

A portion of a whole is described by a fraction. The denominator, which is represented by the number at the bottom of the fraction, indicates the number of equally-sized pieces into which the whole is divided. The numerator, which is the number at the top of the fraction, indicates how many parts we are taking. The fraction 5/4, for instance, means "5 of 4 equal portions." The numerator is 5, and the denominator is 4.

A fraction with a smaller numerator than the denominator is said to be a correct fraction. A fraction whose numerator is the same as or bigger than its denominator is said to be inappropriate. Proper fractions include 1/4, 3/7, and 9/17, while improper fractions include 15/2, 28/15, and 25/11.


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My Math Forum relationship operator

 Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion

 April 14th, 2015, 03:15 AM #1 Newbie     Joined: Feb 2014 Posts: 15 Thanks: 0 relationship operator Hi there, I have come across a notation that I cannot break, I hope you might help me The notation is Pg:Pcr:Pm=1:S:1-s I know that they are that one term is directly proportional to other how would I write in other way, maybe more meaningful?
 April 14th, 2015, 08:25 AM #2 Senior Member   Joined: Apr 2014 From: Glasgow Posts: 2,111 Thanks: 706 Math Focus: Physics, mathematical modelling, numerical and computational solutions It looks to me like a ratio $\displaystyle P_g : P_{cr} : P_m = 1 : S : 1-S$ The total = $\displaystyle 1 + S + (1-S) = 2$ So let's say, just for example, that you have £200 to share among $\displaystyle P_g$, $\displaystyle P_{cr}$ and $\displaystyle P_m$. In this case: $\displaystyle P_g$ gets £100 $\displaystyle P_{cr}$ gets some amount between 0 and £100 depending on what $\displaystyle S$ is. $\displaystyle P_m$ gets the rest. Note that this is assuming $\displaystyle S$ is less than 1 so that $\displaystyle 1-S$ is positive. It depends on the situation you are looking at.

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# Quark Fragmentation and Hadron Formation in Nuclear Matter

Abstract : The hadron formation is, in the framework of the quantum chromodynamics theory (QCD), a non-perturbative process; this characteristic leads to important theoretical challenges. This is why experimental measurements of fragmentation in nuclei are a necessity in order to obtain substantial progress in our understanding of the mechanisms of hadron formation. The thesis begins with the introduction of theoretical background, followed by an overview of theoretical models. The thesis continues with the analysis of Jefferson Lab data obtained with a 5 GeV electron beam incident on various targets ($^2$H, C, Al, Fe, Sn and Pb).The reaction products are measured with the CLAS spectrometer of Hall B. The main results are: (a) multi-dimensional analysis of the measured observables, which permits a better confrontation with theoretical models and the extraction of temporal information on fragmentation, and (b) the observation of a non linear hadronic attenuation as a function of the target's nuclear radius. The PyQM event generator, developed to reproduce the data from the HERMES collaboration, is also presented. The results are ambivalent, the theoretical basis used does not seem to apply to the studied case, however, some characteristics of the data are reproduced allowing to understand their origin, which is sometimes unexpected. Finally, the possibilities for future experiments, at Jefferson Lab and at an Electron-Ion Collider (EIC), are explored.
keyword :
Document type :
Theses
Nuclear Experiment [nucl-ex]. Université Claude Bernard - Lyon I, 2011. English

https://tel.archives-ouvertes.fr/tel-00670293
Contributor : Sylvie Flores <>
Submitted on : Wednesday, February 15, 2012 - 10:30:48 AM
Last modification on : Thursday, March 22, 2012 - 12:43:38 PM

### Identifiers

• HAL Id : tel-00670293, version 1

### Citation

Raphaël Dupré. Quark Fragmentation and Hadron Formation in Nuclear Matter. Nuclear Experiment [nucl-ex]. Université Claude Bernard - Lyon I, 2011. English. <tel-00670293>

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# Alayna and Vowels

National Girls' Programmi...
Limits 1s, 512 MB

Alayna loves playing. Every night before going to sleep, she plays different games with Baba. Now Alayna seems bored with the old games she played many times. So she asked Baba to invent new games.

Baba has now made an exciting game with vowels of the English alphabet (Do not underestimate Alayna, she is a big girl now, and she knows about vowels!).

The game is that Baba gives a string that contains English letters and other characters. Alayna's work is to find the number of vowels from the given string.

As Maa is stringent regarding Baba and Alayna's sleep time, Alayna and Baba don't get much time for playing, and Alayna cannot finish counting when Baba gives her a large string.

So little Alayna needs your help. You have to make a vowel calculator for Alayna, which will give the number of vowels from a given string.

## Input

A single line contains a non-empty word. This word consists of lowercase and uppercase English letters and other characters (excluding space characters). The length of the word will not exceed $10^3$.

## Output

Output the number of vowels from the given string.

## Sample

InputOutput
Bangladesh

3


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# Centriod of T-Section

Centroid of T-section can be found easily with respect to reference x-axis. This is so because T section is symmetrical about Y-axis.

### Centroid of T Section Example

Find the centroid of T-section as shown below. All units are in cm.

Centriod of T-section

### Solution

T-section is comprised of two rectangles.

Area of Large Rectangle = 20×5 = 100 cm.

Area of smaller Rectangle = 12.5×5 = 62.5 cm.

Sum of Area = 100 + 62.5 = 162.5 cm.

Centroid of Large Rectangle with respect to reference x-axis = Y = 12.5/2 = 6.25 cm.

Centroid of small rectangle with respect to reference x-axis = Y = 5/2 + 12.5 = 15 cm.

Moment of Area of Large rectangle = M1 =  62.5×6.25 = 390.25 cm.

Moment of Area of small rectangle = M2 =  100×15 = 1500 cm.

Sum of Moments = 390.25 + 1500 = 1890.625 cm.

$$\bar{Y}=\frac{Sum\space\ of\space\ of\space\ Moments}{Sum\space\ of\space\ of\space\ Area}$$

$$\bar{Y}=\frac{1890.625}{162.5}=11.63cm$$

As shown below;