1{2 "document_id": "lb_hotpotqa_2bf8384417cec2f1a2d43a2602326d90a97622d87898b1cc",3 "family": "public_hotpotqa",4 "split": "dev",5 "text": "Passage 1:\nPower factor\nIn electrical engineering, the power factor of an AC power system is defined as the ratio of the real power absorbed by the load to the apparent power flowing in the circuit. Real power is the average of the instantaneous product of voltage and current and represents the capacity of the electricity for performing work. Apparent power is the product of RMS current and voltage. Due to energy stored in the load and returned to the source, or due to a non-linear load that distorts the wave shape of the current drawn from the source, the apparent power may be greater than the real power, so more current flows in the circuit than would be required to transfer real power alone. A power factor magnitude of less than one indicates the voltage and current are not in phase, reducing the average product of the two. A negative power factor occurs when the device (which is normally the load) generates real power, which then flows back towards the source.\nIn an electric power system, a load with a low power factor draws more current than a load with a high power factor for the same amount of useful power transferred. The higher currents increase the energy lost in the distribution system and require larger wires and other equipment. Because of the costs of larger equipment and wasted energy, electrical utilities will usually charge a higher cost to industrial or commercial customers where there is a low power factor.\nPower-factor correction increases the power factor of a load, improving efficiency for the distribution system to which it is attached. Linear loads with a low power factor (such as induction motors) can be corrected with a passive network of capacitors or inductors. Non-linear loads, such as rectifiers, distort the current drawn from the system. In such cases, active or passive power factor correction may be used to counteract the distortion and raise the power factor. The devices for correction of the power factor may be at a central substation, spread out over a distribution system, or built into power-consuming equipment.\n\nGeneral case\nThe general expression for power factor is given by\n\n \n \n \n \n \n power factor\n \n \n =\n P\n \n /\n \n \n P\n \n a\n \n \n \n \n {\\displaystyle {\\mbox{power factor}}=P/P_{a}}\n \n\n \n \n \n \n P\n \n a\n \n \n =\n \n I\n \n r\n m\n s\n \n \n \n V\n \n r\n m\n s\n \n \n \n \n {\\displaystyle P_{a}=I_{rms}V_{rms}}\n where \n \n \n \n P\n \n \n {\\displaystyle P}\n is the real power measured by an ideal wattmeter, \n \n \n \n \n I\n \n r\n m\n s\n \n \n \n \n {\\displaystyle I_{rms}}\n is the rms current measured by an ideal ammeter, and \n \n \n \n \n V\n \n r\n m\n s\n \n \n \n \n {\\displaystyle V_{rms}}\n is the rms voltage measured by an ideal voltmeter. Apparent power, \n \n \n \n \n P\n \n a\n \n \n \n \n {\\displaystyle P_{a}}\n , is the product of the rms current and the rms voltage.\nIf the load is sourcing power back toward the generator, then \n \n \n \n P\n \n \n {\\displaystyle P}\n and \n \n \n \n \n \n power factor\n \n \n \n \n {\\displaystyle {\\mbox{power factor}}}\n will be negative.\n\nPeriodic waveforms\nIf the waveforms are periodic with the same period which is much shorter than the averaging time of the physical meters, then the power factor can be computed by the following\n\n \n \n \n \n \n power factor\n \n \n =\n P\n \n /\n \n \n P\n \n a\n \n \n \n \n {\\displaystyle {\\mbox{power factor}}=P/P_{a}}\n \n\n \n \n \n \n P\n \n a\n \n \n =\n \n I\n \n r\n m\n s\n \n \n \n V\n \n r\n m\n s\n \n \n \n \n {\\displaystyle P_{a}=I_{rms}V_{rms}}\n \n\n \n \n \n P\n =\n \n \n 1\n T\n \n \n \n ∫\n \n \n t\n ′\n \n \n \n \n t\n ′\n \n +\n T\n \n \n i\n (\n t\n )\n v\n (\n t\n )\n d\n t\n \n \n {\\displaystyle P={\\frac {1}{T}}\\int _{t'}^{t'+T}i(t)v(t)dt}\n \n\n \n \n \n \n I\n \n r\n m\n s\n \n \n 2\n \n \n =\n \n \n 1\n T\n \n \n \n ∫\n \n \n t\n ′\n \n \n \n \n t\n ′\n \n +\n T\n \n \n \n \n i\n (\n t\n )\n \n \n 2\n \n \n d\n t\n \n \n {\\displaystyle I_{rms}^{2}={\\frac {1}{T}}\\int _{t'}^{t'+T}{i(t)}^{2}dt}\n \n\n \n \n \n \n V\n \n r\n m\n s\n \n \n 2\n \n \n =\n \n \n 1\n T\n \n \n \n ∫\n \n \n t\n ′\n \n \n \n \n t\n ′\n \n +\n T\n \n \n \n \n v\n (\n t\n )\n \n \n 2\n \n \n d\n t\n \n \n {\\displaystyle V_{rms}^{2}={\\frac {1}{T}}\\int _{t'}^{t'+T}{v(t)}^{2}dt}\n where \n \n \n \n i\n (\n t\n )\n \n \n {\\displaystyle i(t)}\n is the instantaneous current, \n \n \n \n v\n (\n t\n )\n \n \n {\\displaystyle v(t)}\n is the instantaneous voltage, \n \n \n \n \n t\n ′\n \n \n \n {\\displaystyle t'}\n is an arbitrary starting time, and \n \n \n \n T\n \n \n {\\displaystyle T}\n is the period of the waveforms.\n\nNonperiodic waveforms\nIf the waveforms are not periodic and the physical meters have the same averaging time, then the equations for the periodic case can be used with the exception that \n \n \n \n T\n \n \n {\\displaystyle T}\n is the averaging time of the meters instead of the waveform period.\n\nLinear time-invariant circuits\nLinear time-invariant circuits (referred to simply as linear circuits for the rest of this article), for example, circuits consisting of combinations of resistors, inductors and capacitors have a sinusoidal response to the sinusoidal line voltage. A linear load does not change the shape of the input waveform but may change the relative timing (phase) between voltage and current, due to its inductance or capacitance.\nIn a purely resistive AC circuit, voltage and current waveforms are in step (or in phase), changing polarity at the same instant in each cycle. All the power entering the load is consumed (or dissipated).\nWhere reactive loads are present, such as with capacitors or inductors, energy storage in the loads results in a phase difference between the current and voltage waveforms. During each cycle of the AC voltage, extra energy, in addition to any energy consumed in the load, is temporarily stored in the load in electric or magnetic fields then returned to the power grid a fraction of the period later.\nElectrical circuits containing predominantly resistive loads (incandescent lamps, heating elements) have a power factor of almost 1, but circuits containing inductive or capacitive loads (electric motors, solenoid valves, transformers, fluorescent lamp ballasts, and others) can have a power factor well below 1.\nIn the electric power grid, reactive loads cause a continuous ebb and flow of nonproductive power. A circuit with a low power factor will use a greater amount of current to transfer a given quantity of real power than a circuit with a high power factor thus causing increased losses due to resistive heating in power lines, and requiring the use of higher-rated conductors and transformers.\n\nDefinition and calculation\nAC power has two components:\n\nReal power or active power (\n \n \n \n P\n \n \n {\\displaystyle P}\n ) (sometimes called average power), expressed in watts (W)\nReactive power (\n \n \n \n Q\n \n \n {\\displaystyle Q}\n ), usually expressed in reactive volt-amperes (var)Together, they form the complex power (\n \n \n \n S\n \n \n {\\displaystyle S}\n ) expressed as volt-amperes (VA). The magnitude of the complex power is the apparent power (\n \n \n \n \n |\n \n S\n \n |\n \n \n \n {\\displaystyle |S|}\n ), also expressed in volt-amperes (VA).\nThe VA and var are non-SI units mathematically identical to the watt, but are used in engineering practice instead of the watt to state what quantity is being expressed. The SI explicitly disallows using units for this purpose or as the only source of information about a physical quantity as used.The power factor is defined as the ratio of real power to apparent power. As power is transferred along a transmission line, it does not consist purely of real power that can do work once transferred to the load, but rather consists of a combination of real and reactive power, called apparent power. The power factor describes the amount of real power transmitted along a transmission line relative to the total apparent power flowing in the line.The power factor can also be computed as the cosine of the angle θ by which the current waveform lags or leads the voltage waveform,.\n\nPower triangle\nOne can relate the various components of AC power by using the power triangle in vector space. Real power extends horizontally in the real axis and reactive power extends in the direction of the imaginary axis. Complex power (and its magnitude, apparent power) represents a combination of both real and reactive power, and therefore can be calculated by using the vector sum of these two components. We can conclude that the mathematical relationship between these components is:\n\n \n \n \n \n \n \n \n S\n \n \n \n =\n P\n +\n j\n Q\n \n \n \n \n \n |\n \n S\n \n |\n \n \n \n \n =\n \n \n \n P\n \n 2\n \n \n +\n \n Q\n \n 2\n \n \n \n \n \n \n \n \n \n pf\n \n \n \n \n =\n cos\n \n \n θ\n \n =\n \n \n P\n \n \n |\n \n S\n \n |\n \n \n \n \n =\n cos\n \n \n \n (\n \n arctan\n \n \n \n (\n \n \n Q\n P\n \n \n )\n \n \n \n )\n \n \n \n \n \n \n Q\n \n \n \n =\n P\n \n tan\n \n (\n arccos\n \n (\n \n pf\n \n )\n )\n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}S&=P+jQ\\\\|S|&={\\sqrt {P^{2}+Q^{2}}}\\\\{\\text{pf}}&=\\cos {\\theta }={\\frac {P}{|S|}}=\\cos {\\left(\\arctan {\\left({\\frac {Q}{P}}\\right)}\\right)}\\\\Q&=P\\,\\tan(\\arccos({\\text{pf}}))\\end{aligned}}}\n As the angle θ increases with fixed total apparent power, current and voltage are further out of phase with each other. Real power decreases, and reactive power increases.\n\nLagging, leading and unity power factors\nPower factor is described as leading if the current waveform is advanced in phase with respect to voltage, or lagging when the current waveform is behind the voltage waveform. A lagging power factor signifies that the load is inductive, as the load will consume reactive power. The reactive component \n \n \n \n Q\n \n \n {\\displaystyle Q}\n is positive as reactive power travels through the circuit and is consumed by the inductive load. A leading power factor signifies that the load is capacitive, as the load supplies reactive power, and therefore the reactive component \n \n \n \n Q\n \n \n {\\displaystyle Q}\n is negative as reactive power is being supplied to the circuit.\n\nIf θ is the phase angle between the current and voltage, then the power factor is equal to the cosine of the angle, \n \n \n \n cos\n \n θ\n \n \n {\\displaystyle \\cos \\theta }\n :\n\n \n \n \n \n |\n \n P\n \n |\n \n =\n \n |\n \n S\n \n |\n \n cos\n \n θ\n \n \n {\\displaystyle |P|=|S|\\cos \\theta }\n Since the units are consistent, the power factor is by definition a dimensionless number between -1 and 1. When power factor is equal to 0, the energy flow is entirely reactive and stored energy in the load returns to the source on each cycle. When the power factor is 1, referred to as unity power factor, all the energy supplied by the source is consumed by the load. Power factors are usually stated as leading or lagging to show the sign of the phase angle. Capacitive loads are leading (current leads voltage), and inductive loads are lagging (current lags voltage).\nIf a purely resistive load is connected to a power supply, current and voltage will change polarity in step, the power factor will be 1, and the electrical energy flows in a single direction across the network in each cycle. Inductive loads such as induction motors (any type of wound coil) consume reactive power with the current waveform lagging the voltage. Capacitive loads such as capacitor banks or buried cables generate reactive power with the current phase leading the voltage. Both types of loads will absorb energy during part of the AC cycle, which is stored in the device's magnetic or electric field, only to return this energy back to the source during the rest of the cycle.\nFor example, to get 1 kW of real power, if the power factor is unity, 1 kVA of apparent power needs to be transferred (1 kW ÷ 1 = 1 kVA). At low values of power factor, more apparent power needs to be transferred to get the same real power. To get 1 kW of real power at 0.2 power factor, 5 kVA of apparent power needs to be transferred (1 kW ÷ 0.2 = 5 kVA). This apparent power must be produced and transmitted to the load and is subject to losses in the production and transmission processes.\nElectrical loads consuming alternating current power consume both real power and reactive power. The vector sum of real and reactive power is the complex power, and its magnitude is the apparent power. The presence of reactive power causes the real power to be less than the apparent power, and so, the electric load has a power factor of less than 1.\nA negative power factor (0 to −1) can result from returning active power to the source, such as in the case of a building fitted with solar panels when surplus power is fed back into the supply.\n\nPower factor correction of linear loads\nA high power factor is generally desirable in a power delivery system to reduce losses and improve voltage regulation at the load. Compensating elements near an electrical load will reduce the apparent power demand on the supply system. Power factor correction may be applied by an electric power transmission utility to improve the stability and efficiency of the network. Individual electrical customers who are charged by their utility for low power factor may install correction equipment to increase their power factor so as to reduce costs.\nPower factor correction brings the power factor of an AC power circuit closer to 1 by supplying or absorbing reactive power, adding capacitors or inductors that act to cancel the inductive or capacitive effects of the load, respectively. In the case of offsetting the inductive effect of motor loads, capacitors can be locally connected. These capacitors help to generate reactive power to meet the demand of the inductive loads. This will keep that reactive power from having to flow all the way from the utility generator to the load. In the electricity industry, inductors are said to consume reactive power and capacitors are said to supply it, even though reactive power is just energy moving back and forth on each AC cycle.\nThe reactive elements in power factor correction devices can create voltage fluctuations and harmonic noise when switched on or off. They will supply or sink reactive power regardless of whether there is a corresponding load operating nearby, increasing the system's no-load losses. In the worst case, reactive elements can interact with the system and with each other to create resonant conditions, resulting in system instability and severe overvoltage fluctuations. As such, reactive elements cannot simply be applied without engineering analysis.\n\nAn automatic power factor correction unit consists of a number of capacitors that are switched by means of contactors. These contactors are controlled by a regulator that measures power factor in an electrical network. Depending on the load and power factor of the network, the power factor controller will switch the necessary blocks of capacitors in steps to make sure the power factor stays above a selected value.\nIn place of a set of switched capacitors, an unloaded synchronous motor can supply reactive power. The reactive power drawn by the synchronous motor is a function of its field excitation. It is referred to as a synchronous condenser. It is started and connected to the electrical network. It operates at a leading power factor and puts vars onto the network as required to support a system's voltage or to maintain the system power factor at a specified level.\nThe synchronous condenser's installation and operation are identical to those of large electric motors. Its principal advantage is the ease with which the amount of correction can be adjusted; it behaves like a variable capacitor. Unlike with capacitors, the amount of reactive power supplied is proportional to voltage, not the square of voltage; this improves voltage stability on large networks. Synchronous condensers are often used in connection with high-voltage direct-current transmission projects or in large industrial plants such as steel mills.\nFor power factor correction of high-voltage power systems or large, fluctuating industrial loads, power electronic devices such as the static VAR compensator or STATCOM are increasingly used. These systems are able to compensate sudden changes of power factor much more rapidly than contactor-switched capacitor banks and, being solid-state, require less maintenance than synchronous condensers.\n\nNon-linear loads\nExamples of non-linear loads on a power system are rectifiers (such as used in a power supply), and arc discharge devices such as fluorescent lamps, electric welding machines, or arc furnaces. Because current in these systems is interrupted by a switching action, the current contains frequency components that are multiples of the power system frequency. Distortion power factor is a measure of how much the harmonic distortion of a load current decreases the average power transferred to the load.\n\nNon-sinusoidal components\nIn linear circuits having only sinusoidal currents and voltages of one frequency, the power factor arises only from the difference in phase between the current and voltage. This is displacement power factor.Non-linear loads change the shape of the current waveform from a sine wave to some other form. Non-linear loads create harmonic currents in addition to the original (fundamental frequency) AC current. This is of importance in practical power systems that contain non-linear loads such as rectifiers, some forms of electric lighting, electric arc furnaces, welding equipment, switched-mode power supplies, variable speed drives and other devices. Filters consisting of linear capacitors and inductors can prevent harmonic currents from entering the supplying system.\nTo measure the real power or reactive power, a wattmeter designed to work properly with non-sinusoidal currents must be used.\n\nDistortion power factor\nThe distortion power factor is the distortion component associated with the harmonic voltages and currents present in the system.\n\n \n \n \n \n \n \n \n \n \n distortion power factor\n \n \n \n \n \n =\n \n \n \n I\n \n 1\n \n \n \n I\n \n r\n m\n s\n \n \n \n \n \n \n \n \n \n \n =\n \n \n \n I\n \n 1\n \n \n \n \n I\n \n 1\n \n \n 2\n \n \n +\n \n I\n \n 2\n \n \n 2\n \n \n +\n \n I\n \n 3\n \n \n 2\n \n \n +\n \n I\n \n 4\n \n \n 2\n \n \n +\n ⋯\n \n \n \n \n \n \n \n \n \n =\n \n \n 1\n \n 1\n +\n \n \n \n \n I\n \n 2\n \n \n 2\n \n \n +\n \n I\n \n 3\n \n \n 2\n \n \n +\n \n I\n \n 4\n \n \n 2\n \n \n +\n ⋯\n \n \n I\n \n 1\n \n \n 2\n \n \n \n \n \n \n \n \n \n \n \n \n \n =\n \n \n 1\n \n 1\n +\n T\n H\n \n D\n \n i\n \n \n 2\n \n \n \n \n \n \n \n \n \n \n \n {\\displaystyle {\\begin{aligned}{\\mbox{distortion power factor}}&={\\frac {I_{1}}{I_{rms}}}\\\\&={\\frac {I_{1}}{\\sqrt {I_{1}^{2}+I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\\cdots }}}\\\\&={\\frac {1}{\\sqrt {1+{\\frac {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\\cdots }{I_{1}^{2}}}}}}\\\\&={\\frac {1}{\\sqrt {1+THD_{i}^{2}}}}\\\\\\end{aligned}}}\n \n \n \n \n \n \n \n THD\n \n \n \n i\n \n \n \n \n {\\displaystyle {\\mbox{THD}}_{i}}\n is the total harmonic distortion of the load current. \n\n \n \n \n T\n H\n \n D\n \n i\n \n \n =\n \n \n \n \n \n ∑\n \n h\n =\n 2\n \n \n ∞\n \n \n \n I\n \n h\n \n \n 2\n \n \n \n \n \n I\n \n 1\n \n \n \n \n =\n \n \n \n \n I\n \n 2\n \n \n 2\n \n \n +\n \n I\n \n 3\n \n \n 2\n \n \n +\n \n I\n \n 4\n \n \n 2\n \n \n +\n ⋯\n \n \n I\n \n 1\n \n \n \n \n \n \n {\\displaystyle THD_{i}={\\frac {\\sqrt {\\displaystyle \\sum _{h=2}^{\\infty }I_{h}^{2}}}{I_{1}}}={\\frac {\\sqrt {I_{2}^{2}+I_{3}^{2}+I_{4}^{2}+\\cdots }}{I_{1}}}}\n \n \n \n \n \n I\n \n 1\n \n \n \n \n {\\displaystyle I_{1}}\n is the fundamental component of the current and \n \n \n \n \n I\n \n \n rms\n \n \n \n \n \n {\\displaystyle I_{\\mbox{rms}}}\n is the total current – both are root mean square-values (distortion power factor can also be used to describe individual order harmonics, using the corresponding current in place of total current). This definition with respect to total harmonic distortion assumes that the voltage stays undistorted (sinusoidal, without harmonics). This simplification is often a good approximation for stiff voltage sources (not being affected by changes in load downstream in the distribution network). Total harmonic distortion of typical generators from current distortion in the network is on the order of 1–2%, which can have larger scale implications but can be ignored in common practice.The result when multiplied with the displacement power factor (DPF) is the overall, true power factor or just power factor (PF):\n\n \n \n \n \n \n PF\n \n \n =\n \n \n \n cos\n \n \n φ\n \n \n \n 1\n +\n T\n H\n \n D\n \n i\n \n \n 2\n \n \n \n \n \n \n \n {\\displaystyle {\\mbox{PF}}={\\frac {\\cos {\\varphi }}{\\sqrt {1+THD_{i}^{2}}}}}\n\nDistortion in three-phase networks\nIn practice, the local effects of distortion current on devices in a three-phase distribution network rely on the magnitude of certain order harmonics rather than the total harmonic distortion.\nFor example, the triplen, or zero-sequence, harmonics (3rd, 9th, 15th, etc.) have the property of being in-phase when compared line-to-line. In a delta-wye transformer, these harmonics can result in circulating currents in the delta windings and result in greater resistive heating. In a wye-configuration of a transformer, triplen harmonics will not create these currents, but they will result in a non-zero current in the neutral wire. This could overload the neutral wire in some cases\nand create error in kilowatt-hour metering systems and billing revenue. The presence of current harmonics in a transformer also result in larger eddy currents in the magnetic core of the transformer. Eddy current losses generally increase as the square of the frequency, lowering the transformer's efficiency, dissipating additional heat, and reducing its service life.Negative-sequence harmonics (5th, 11th, 17th, etc.) combine 120 degrees out of phase, similarly to the fundamental harmonic but in a reversed sequence. In generators and motors, these currents produce magnetic fields which oppose the rotation of the shaft and sometimes result in damaging mechanical vibrations.\n\nSwitched-mode power supplies\nA particularly important class of non-linear loads is the millions of personal computers that typically incorporate switched-mode power supplies (SMPS) with rated output power ranging from a few watts to more than 1 kW. Historically, these very-low-cost power supplies incorporated a simple full-wave rectifier that conducted only when the mains instantaneous voltage exceeded the voltage on the input capacitors. This leads to very high ratios of peak-to-average input current, which also lead to a low distortion power factor and potentially serious phase and neutral loading concerns.\nA typical switched-mode power supply first converts the AC mains to a DC bus by means of a bridge rectifier. The output voltage is then derived from this DC bus. The problem with this is that the rectifier is a non-linear device, so the input current is highly non-linear. That means that the input current has energy at harmonics of the frequency of the voltage. This presents a problem for power companies, because they cannot compensate for the harmonic current by adding simple capacitors or inductors, as they could for the reactive power drawn by a linear load. Many jurisdictions are beginning to require power factor correction for all power supplies above a certain power level.\nRegulatory agencies such as the EU have set harmonic limits as a method of improving power factor. Declining component cost has hastened implementation of two different methods. To comply with current EU standard EN61000-3-2, all switched-mode power supplies with output power more than 75 W must at least include passive power factor correction. 80 Plus power supply certification requires a power factor of 0.9 or more.\n\nPower factor correction (PFC) in non-linear loads\nPassive PFC\nThe simplest way to control the harmonic current is to use a filter that passes current only at line frequency (50 or 60 Hz). The filter consists of capacitors or inductors and makes a non-linear device look more like a linear load. An example of passive PFC is a valley-fill circuit.\nA disadvantage of passive PFC is that it requires larger inductors or capacitors than an equivalent power active PFC circuit. Also, in practice, passive PFC is often less effective at improving the power factor.\n\nActive PFC\nActive PFC is the use of power electronics to change the waveform of current drawn by a load to improve the power factor. Some types of the active PFC are buck, boost, buck-boost and synchronous condenser. Active power factor correction can be single-stage or multi-stage.\nIn the case of a switched-mode power supply, a boost converter is inserted between the bridge rectifier and the main input capacitors. The boost converter attempts to maintain a constant voltage at its output while drawing a current that is always in phase with and at the same frequency as the line voltage. Another switched-mode converter inside the power supply produces the desired output voltage from the DC bus. This approach requires additional semiconductor switches and control electronics but permits cheaper and smaller passive components. It is frequently used in practice.\nFor a three-phase SMPS, the Vienna rectifier configuration may be used to substantially improve the power factor.\nSMPSs with passive PFC can achieve power factor of about 0.7–0.75, SMPSs with active PFC, up to 0.99 power factor, while a SMPS without any power factor correction have a power factor of only about 0.55–0.65.Due to their very wide input voltage range, many power supplies with active PFC can automatically adjust to operate on AC power from about 100 V (Japan) to 240 V (Europe). That feature is particularly welcome in power supplies for laptops.\n\nDynamic PFC\nDynamic power factor correction (DPFC), sometimes referred to as real-time power factor correction, is used for electrical stabilization in cases of rapid load changes (e.g. at large manufacturing sites). DPFC is useful when standard power factor correction would cause over or under correction. DPFC uses semiconductor switches, typically thyristors, to quickly connect and disconnect capacitors or inductors to improve power factor.\n\nImportance in distribution systems\nPower factors below 1.0 require a utility to generate more than the minimum volt-amperes necessary to supply the real power (watts). This increases generation and transmission costs. For example, if the load power factor were as low as 0.7, the apparent power would be 1.4 times the real power used by the load. Line current in the circuit would also be 1.4 times the current required at 1.0 power factor, so the losses in the circuit would be doubled (since they are proportional to the square of the current). Alternatively, all components of the system such as generators, conductors, transformers, and switchgear would be increased in size (and cost) to carry the extra current. When the power factor is close to unity, for the same kVA rating of the transformer more load current can be supplied.Utilities typically charge additional costs to commercial customers who have a power factor below some limit, which is typically 0.9 to 0.95. Engineers are often interested in the power factor of a load as one of the factors that affect the efficiency of power transmission.\nWith the rising cost of energy and concerns over the efficient delivery of power, active PFC has become more common in consumer electronics. Current Energy Star guidelines for computers call for a power factor of ≥ 0.9 at 100% of rated output in the PC's power supply. According to a white paper authored by Intel and the U.S. Environmental Protection Agency, PCs with internal power supplies will require the use of active power factor correction to meet the ENERGY STAR 5.0 Program Requirements for Computers.In Europe, EN 61000-3-2 requires power factor correction be incorporated into consumer products.\nSmall customers, such as households, are not usually charged for reactive power and so power factor metering equipment for such customers will not be installed.\n\nMeasurement techniques\nThe power factor in a single-phase circuit (or balanced three-phase circuit) can be measured with the wattmeter-ammeter-voltmeter method, where the power in watts is divided by the product of measured voltage and current. The power factor of a balanced polyphase circuit is the same as that of any phase. The power factor of an unbalanced polyphase circuit is not uniquely defined.\nA direct reading power factor meter can be made with a moving coil meter of the electrodynamic type, carrying two perpendicular coils on the moving part of the instrument. The field of the instrument is energized by the circuit current flow. The two moving coils, A and B, are connected in parallel with the circuit load. One coil, A, will be connected through a resistor and the second coil, B, through an inductor, so that the current in coil B is delayed with respect to current in A. At unity power factor, the current in A is in phase with the circuit current, and coil A provides maximum torque, driving the instrument pointer toward the 1.0 mark on the scale. At zero power factor, the current in coil B is in phase with circuit current, and coil B provides torque to drive the pointer towards 0. At intermediate values of power factor, the torques provided by the two coils add and the pointer takes up intermediate positions.Another electromechanical instrument is the polarized-vane type. In this instrument a stationary field coil produces a rotating magnetic field, just like a polyphase motor. The field coils are connected either directly to polyphase voltage sources or to a phase-shifting reactor if a single-phase application. A second stationary field coil, perpendicular to the voltage coils, carries a current proportional to current in one phase of the circuit. The moving system of the instrument consists of two vanes that are magnetized by the current coil. In operation, the moving vanes take up a physical angle equivalent to the electrical angle between the voltage source and the current source. This type of instrument can be made to register for currents in both directions, giving a four-quadrant display of power factor or phase angle.\nDigital instruments exist that directly measure the time lag between voltage and current waveforms. Low-cost instruments of this type measure the peak of the waveforms. More sophisticated versions measure the peak of the fundamental harmonic only, thus giving a more accurate reading for phase angle on distorted waveforms. Calculating power factor from voltage and current phases is only accurate if both waveforms are sinusoidal.Power Quality Analyzers, often referred to as Power Analyzers, make a digital recording of the voltage and current waveform (typically either one phase or three phase) and accurately calculate true power (watts), apparent power (VA) power factor, AC voltage, AC current, DC voltage, DC current, frequency, IEC61000-3-2/3-12 Harmonic measurement, IEC61000-3-3/3-11 flicker measurement, individual phase voltages in delta applications where there is no neutral line, total harmonic distortion, phase and amplitude of individual voltage or current harmonics, etc.\n\nMnemonics\nEnglish-language power engineering students are advised to remember:\nELI the ICE man or ELI on ICE – the voltage E, leads the current I, in an inductor L. The current I leads the voltage E in a capacitor C.\nAnother common mnemonic is CIVIL – in a capacitor (C) the current (I) leads voltage (V), voltage (V) leads current (I) in an inductor (L).\nPassage 2:\nWomersley number\nThe Womersley number (\n \n \n \n α\n \n \n {\\displaystyle \\alpha }\n or \n \n \n \n \n Wo\n \n \n \n {\\displaystyle {\\text{Wo}}}\n ) is a dimensionless number in biofluid mechanics and biofluid dynamics. It is a dimensionless expression of the pulsatile flow frequency in relation to viscous effects. It is named after John R. Womersley (1907–1958) for his work with blood flow in arteries. The Womersley number is important in keeping dynamic similarity when scaling an experiment. An example of this is scaling up the vascular system for experimental study. The Womersley number is also important in determining the thickness of the boundary layer to see if entrance effects can be ignored.\nThe square root of this number is also referred to as Stokes number, \n \n \n \n \n Stk\n \n =\n \n \n Wo\n \n \n \n \n {\\displaystyle {\\text{Stk}}={\\sqrt {\\text{Wo}}}}\n , due to the pioneering work done by Sir George Stokes on the Stokes second problem.\n\nDerivation\nThe Womersley number, usually denoted \n \n \n \n α\n \n \n {\\displaystyle \\alpha }\n , is defined by the relation\n\nwhere \n \n \n \n L\n \n \n {\\displaystyle L}\n is an appropriate length scale (for example the radius of a pipe), \n \n \n \n ω\n \n \n {\\displaystyle \\omega }\n is the angular frequency of the oscillations, and \n \n \n \n ν\n \n \n {\\displaystyle \\nu }\n , \n \n \n \n ρ\n \n \n {\\displaystyle \\rho }\n , \n \n \n \n μ\n \n \n {\\displaystyle \\mu }\n are the kinematic viscosity, density, and dynamic viscosity of the fluid, respectively. The Womersley number is normally written in the powerless form\n\nIn the cardiovascular system, the pulsation frequency, density, and dynamic viscosity are constant, however the Characteristic length, which in the case of blood flow is the vessel diameter, changes by three orders of magnitudes (OoM) between the aorta and fine capillaries. The Womersley number thus changes due to the variations in vessel size across the vasculature system. The Womersley number of human blood flow can be estimated as follows:\n\n \n \n \n α\n =\n L\n \n \n (\n \n \n \n ω\n ρ\n \n μ\n \n \n )\n \n \n \n 1\n 2\n \n \n \n \n .\n \n \n {\\displaystyle \\alpha =L\\left({\\frac {\\omega \\rho }{\\mu }}\\right)^{\\frac {1}{2}}\\,.}\n \nBelow is a list of estimated Womersley numbers in different human blood vessels:\n\nIt can also be written in terms of the dimensionless Reynolds number (Re) and Strouhal number (St):\n\nThe Womersley number arises in the solution of the linearized Navier–Stokes equations for oscillatory flow (presumed to be laminar and incompressible) in a tube. It expresses the ratio of the transient or oscillatory inertia force to the shear force. When \n \n \n \n α\n \n \n {\\displaystyle \\alpha }\n is small (1 or less), it means the frequency of pulsations is sufficiently low that a parabolic velocity profile has time to develop during each cycle, and the flow will be very nearly in phase with the pressure gradient, and will be given to a good approximation by Poiseuille's law, using the instantaneous pressure gradient. When \n \n \n \n α\n \n \n {\\displaystyle \\alpha }\n is large (10 or more), it means the frequency of pulsations is sufficiently large that the velocity profile is relatively flat or plug-like, and the mean flow lags the pressure gradient by about 90 degrees. Along with the Reynolds number, the Womersley number governs dynamic similarity.The boundary layer thickness \n \n \n \n δ\n \n \n {\\displaystyle \\delta }\n that is associated with the transient acceleration is inversely related to the Womersley number. This can be seen by recognizing the Womersley number as the square root of the Stokes number.\nwhere \n \n \n \n L\n \n \n {\\displaystyle L}\n is a characteristic length.\n\nBiofluid mechanics\nIn a flow distribution network that progresses from a large tube to many small tubes (e.g. a blood vessel network), the frequency, density, and dynamic viscosity are (usually) the same throughout the network, but the tube radii change. Therefore, the Womersley number is large in large vessels and small in small vessels. As the vessel diameter decreases with each division the Womersley number soon becomes quite small. The Womersley numbers tend to 1 at the level of the terminal arteries. In the arterioles, capillaries, and venules the Womersley numbers are less than one. In these regions the inertia force becomes less important and the flow is determined by the balance of viscous stresses and the pressure gradient. This is called microcirculation.Some typical values for the Womersley number in the cardiovascular system for a canine at a heart rate of 2 Hz are:\nAscending aorta – 13.2\nDescending aorta – 11.5\nAbdominal aorta – 8\nFemoral artery – 3.5\nCarotid artery – 4.4\nArterioles – 0.04\nCapillaries – 0.005\nVenules – 0.035\nInferior vena cava – 8.8\nMain pulmonary artery – 15It has been argued that universal biological scaling laws (power-law relationships that describe variation of quantities such as metabolic rate, lifespan, length, etc., with body mass) are a consequence of the need for energy minimization, the fractal nature of vascular networks, and the crossover from high to low Womersley number flow as one progresses from large to small vessels.\nPassage 3:\nWeissenberg number\nThe Weissenberg number (Wi) is a dimensionless number used in the study of viscoelastic flows. It is named after Karl Weissenberg. The dimensionless number compares the elastic forces to the viscous forces. It can be variously defined, but it is usually given by the relation of stress relaxation time of the fluid and a specific process time. For instance, in simple steady shear, the Weissenberg number, often abbreviated as Wi or We, is defined as the shear rate \n \n \n \n \n \n \n γ\n ˙\n \n \n \n \n \n {\\displaystyle {\\dot {\\gamma }}}\n times the relaxation time \n \n \n \n λ\n \n \n {\\displaystyle \\lambda }\n . Using the Maxwell model and the Oldroyd-B model, the elastic forces can be written as the first Normal force (N1).\n\n \n \n \n \n Wi\n \n =\n \n \n \n \n elastic forces\n \n \n viscous forces\n \n \n \n \n =\n \n \n \n \n τ\n \n x\n x\n \n \n −\n \n τ\n \n y\n y\n \n \n \n \n τ\n \n x\n y\n \n \n \n \n =\n \n \n \n 2\n λ\n μ\n \n \n \n \n γ\n ˙\n \n \n \n \n 2\n \n \n \n \n μ\n \n \n \n γ\n ˙\n \n \n \n \n \n \n =\n 2\n \n \n \n γ\n ˙\n \n \n \n λ\n .\n \n \n \n {\\displaystyle {\\text{Wi}}={\\dfrac {\\mbox{elastic forces}}{\\mbox{viscous forces}}}={\\frac {\\tau _{xx}-\\tau _{yy}}{\\tau _{xy}}}={\\frac {2\\lambda \\mu {\\dot {\\gamma }}^{2}}{\\mu {\\dot {\\gamma }}}}=2{\\dot {\\gamma }}\\lambda .\\,}\n Since this number is obtained from scaling the evolution of the stress, it contains choices for the shear or elongation rate, and the length-scale. Therefore the exact definition of all non dimensional numbers should be given as well as the number itself.\nWhile Wi is similar to the Deborah number and is often confused with it in technical literature, they have different physical interpretations. The Weissenberg number indicates the degree of anisotropy or orientation generated by the deformation, and is appropriate to describe flows with a constant stretch history, such as simple shear. In contrast, the Deborah number should be used to describe flows with a non-constant stretch history, and physically represents the rate at which elastic energy is stored or released.\nPassage 4:\nEricksen number\nIn the study of liquid crystals, the Ericksen number (Er) is a dimensionless number used to describe the deformation of the director field under flow. It is defined as the ratio of the viscous to elastic forces. In the limit of low Ericksen number the elastic forces will exceed the viscous forces and so the director field will not be strongly affected by the flow field. The Ericksen number is named after American mathematics professor Jerald Ericksen of the University of Minnesota. The number is defined:\n\n \n \n \n \n \n E\n r\n \n \n =\n \n \n \n μ\n v\n L\n \n K\n \n \n \n \n {\\displaystyle {\\mathit {Er}}={\\frac {\\mu vL}{K}}}\n where\n\n \n \n \n μ\n \n \n {\\displaystyle \\mu }\n is the fluid's dynamic viscosity,\n\n \n \n \n v\n \n \n {\\displaystyle v}\n is a characteristic scale for the fluid's velocity,\n\n \n \n \n L\n \n \n {\\displaystyle L}\n is a characteristic scale length for the fluid flow,\n\n \n \n \n K\n \n \n {\\displaystyle K}\n is an elasticity force, for example the elasticity modulus times an area.\nPassage 5:\nQualified Person for Pharmacovigilance\nIn the European Union, the Qualified Person Responsible For Pharmacovigilance (QPPV) is an individual, usually an employee of a pharmaceutical company, who is personally responsible for the safety of the human pharmaceutical products marketed by that company in the EU. This function was established in 2004 by article 23 of regulation (EC) No 726/2004. The article establishes that the holder of a marketing authorization for a drug for human use must have a QPPV. When a company submits an application for permission to bring a medicinal product onto the market, the company submits a description of its system for monitoring the safety of the product in actual use (a pharmacovigilance system) and proof that the services of a QPPV are in place.\"The holder of an authorisation for a medicinal product for human use granted in accordance with the provisions of this Regulation shall have permanently and continuously at his disposal an appropriately qualified person responsible for pharmacovigilance.\"\n\nResponsibilities\nPer Article 23, the QPPV is responsible for:\n\nEstablishing and maintaining a pharmacovigilance system\nPreparing pharmacovigilance reports as defined by regulations\nAnswering requests from Health Authorities\nProviding Health Authorities with any other information relevant to product safetyThe Good Pharmacovigilance Practices (GVP) provide further guidance regarding these responsibilities of the QPPV (Module I, Section I.C.1.3: Role of the qualified person responsible for pharmacovigilance in the EU. Per GVP, the QPPV is responsible for the establishment and maintenance of the marketing authorisation holder's pharmacovigilance system and therefore shall have sufficient authority to influence the performance of the quality system and the pharmacovigilance activities and to promote, maintain and improve compliance with the legal requirements. The QPPV is specifically responsible for:\n\nEstablishing and maintaining/managing the pharmaceutical company's pharmacovigilance system\nHaving an overview of the safety profiles and any emerging safety concerns for the company's drugs\nActing as a single contact point for the Health Authorities on a 24-hour basis\n\nCriteria for QPPV role\nThe QPPV must reside in the EU, and should be permanently and continuously at the disposal of the MAH. Each company (i.e. Applicant/Marketing Authorisation Holder or group of Marketing Authorisation Holders using a common pharmacovigilance system) should appoint one QPPV responsible for overall pharmacovigilance for all medicinal products for which the company holds marketing authorisations within the EU.\nDetailed information on the role and responsibilities of the QPPV, and guidance for a Marketing Authorisation Holder on how to adequately support the QPPV are specified in Guideline on good pharmacovigilance practices (GVP; Module I – Pharmacovigilance systems and their quality systems).\" At a minimum the QPPV should be appropriately qualified, with documented experience in all aspects of pharmacovigilance in order to fulfil the responsibilities and tasks of the post. If the QPPV is not medically qualified, access to a medically qualified person should be available.\nIn addition, there are national regulations in some EU member states that also require a nominated individual in that country who has specific legal obligations as a QPPV at a national level.\n\nSee also\nPharmacovigilance\nEuropean Medicines Agency\nPassage 6:\nJoseph Fourier\nJean-Baptiste Joseph Fourier (; French: [fuʁje]; 21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series, which eventually developed into Fourier analysis and harmonic analysis, and their applications to problems of heat transfer and vibrations. The Fourier transform and Fourier's law of conduction are also named in his honour. Fourier is also generally credited with the discovery of the greenhouse effect.\n\nBiography\nFourier was born in Auxerre (now in the Yonne département of France), the son of a tailor. He was orphaned at the age of nine. Fourier was recommended to the Bishop of Auxerre and, through this introduction, he was educated by the Benedictine Order of the Convent of St. Mark. The commissions in the scientific corps of the army were reserved for those of good birth, and being thus ineligible, he accepted a military lectureship on mathematics. He took a prominent part in his own district in promoting the French Revolution, serving on the local Revolutionary Committee. He was imprisoned briefly during the Terror but, in 1795, was appointed to the École Normale and subsequently succeeded Joseph-Louis Lagrange at the École Polytechnique.\nFourier accompanied Napoleon Bonaparte on his Egyptian expedition in 1798, as scientific adviser, and was appointed secretary of the Institut d'Égypte. Cut off from France by the British fleet, he organized the workshops on which the French army had to rely for their munitions of war. He also contributed several mathematical papers to the Egyptian Institute (also called the Cairo Institute) which Napoleon founded at Cairo, with a view of weakening British influence in the East. After the British victories and the capitulation of the French under General Menou in 1801, Fourier returned to France.\n\nIn 1801, Napoleon appointed Fourier Prefect (Governor) of the Department of Isère in Grenoble, where he oversaw road construction and other projects. However, Fourier had previously returned home from the Napoleon expedition to Egypt to resume his academic post as professor at École Polytechnique when Napoleon decided otherwise in his remark\n\n... the Prefect of the Department of Isère having recently died, I would like to express my confidence in citizen Fourier by appointing him to this place.\nHence being faithful to Napoleon, he took the office of Prefect. It was while at Grenoble that he began to experiment on the propagation of heat. He presented his paper On the Propagation of Heat in Solid Bodies to the Paris Institute on 21 December 1807. He also contributed to the monumental Description de l'Égypte.In 1822, Fourier succeeded Jean Baptiste Joseph Delambre as Permanent Secretary of the French Academy of Sciences. In 1830, he was elected a foreign member of the Royal Swedish Academy of Sciences.\nFourier never married.In 1830, his diminished health began to take its toll:\n\nFourier had already experienced, in Egypt and Grenoble, some attacks of aneurysm of the heart. At Paris, it was impossible to be mistaken with respect to the primary cause of the frequent suffocations which he experienced. A fall, however, which he sustained on the 4th of May 1830, while descending a flight of stairs, aggravated the malady to an extent beyond what could have been ever feared.\nShortly after this event, he died in his bed on 16 May 1830.\nFourier was buried in the Père Lachaise Cemetery in Paris, a tomb decorated with an Egyptian motif to reflect his position as secretary of the Cairo Institute, and his collation of Description de l'Égypte. His name is one of the 72 names inscribed on the Eiffel Tower.\nA bronze statue was erected in Auxerre in 1849, but it was melted down for armaments during World War II. Joseph Fourier University in Grenoble was named after him.\n\nThe Analytic Theory of Heat\nIn 1822, Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytical Theory of Heat), in which he based his reasoning on Newton's law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. This book was translated, with editorial 'corrections', into English 56 years later by Freeman (1878). The book was also edited, with many editorial corrections, by Darboux and republished in French in 1888.There were three important contributions in this work, one purely mathematical, two essentially physical. In mathematics, Fourier claimed that any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable. Though this result is not correct without additional conditions, Fourier's observation that some discontinuous functions are the sum of infinite series was a breakthrough. The question of determining when a Fourier series converges has been fundamental for centuries. Joseph-Louis Lagrange had given particular cases of this (false) theorem, and had implied that the method was general, but he had not pursued the subject. Peter Gustav Lejeune Dirichlet was the first to give a satisfactory demonstration of it with some restrictive conditions. This work provides the foundation for what is today known as the Fourier transform.\nOne important physical contribution in the book was the concept of dimensional homogeneity in equations; i.e. an equation can be formally correct only if the dimensions match on either side of the equality; Fourier made important contributions to dimensional analysis. The other physical contribution was Fourier's proposal of his partial differential equation for conductive diffusion of heat. This equation is now taught to every student of mathematical physics.\n\nReal roots of polynomials\nFourier left an unfinished work on determining and locating real roots of polynomials, which was edited by Claude-Louis Navier and published in 1831. This work contains much original matter—in particular, Fourier's theorem on polynomial real roots, published in 1820. Fourier's theorem on real roots of polynomials states that a polynomial with real coefficients has a real root between any two consecutive zeros of its derivative. François Budan, in 1807 and 1811, had published independently his theorem (also known by the name of Fourier), which is very close to Fourier's theorem (each theorem is a corollary of the other). Fourier's proof is the one that was usually given, during 19th century, in textbooks on the theory of equations. A complete solution of the problem was given in 1829 by Jacques Charles François Sturm.\n\nDiscovery of the greenhouse effect\nIn the 1820s, Fourier calculated that an object the size of the Earth, and at its distance from the Sun, should be considerably colder than the planet actually is if warmed by only the effects of incoming solar radiation. He examined various possible sources of the additional observed heat in articles published in 1824 and 1827. However, in the end, because of the large 33-degree difference between his calculations and observations, Fourier mistakenly believed that there is a significant contribution of radiation from interstellar space. Still, Fourier's consideration of the possibility that the Earth's atmosphere might act as an insulator of some kind is widely recognized as the first proposal of what is now known as the greenhouse effect, although Fourier never called it that.In his articles, Fourier referred to an experiment by de Saussure, who lined a vase with blackened cork. Into the cork, he inserted several panes of transparent glass, separated by intervals of air. Midday sunlight was allowed to enter at the top of the vase through the glass panes. The temperature became more elevated in the more interior compartments of this device. Fourier concluded that gases in the atmosphere could form a stable barrier like the glass panes. This conclusion may have contributed to the later use of the metaphor of the \"greenhouse effect\" to refer to the processes that determine atmospheric temperatures. Fourier noted that the actual mechanisms that determine the temperatures of the atmosphere included convection, which was not present in de Saussure's experimental device.\n\nWorks\nSee also\nFourier analysis\nFourier–Deligne transform\nHeat equation\nLeast-squares spectral analysis\nList of things named after Joseph Fourier\nPassage 7:\nMorton number\nIn fluid dynamics, the Morton number (Mo) is a dimensionless number used together with the Eötvös number or Bond number to characterize the shape of bubbles or drops moving in a surrounding fluid or continuous phase, c.\nIt is named after Rose Morton, who described it with W. L. Haberman in 1953.\n\nDefinition\nThe Morton number is defined as\n\n \n \n \n \n M\n o\n \n =\n \n \n \n g\n \n μ\n \n c\n \n \n 4\n \n \n \n Δ\n ρ\n \n \n \n ρ\n \n c\n \n \n 2\n \n \n \n σ\n \n 3\n \n \n \n \n \n ,\n \n \n {\\displaystyle \\mathrm {Mo} ={\\frac {g\\mu _{c}^{4}\\,\\Delta \\rho }{\\rho _{c}^{2}\\sigma ^{3}}},}\n where g is the acceleration of gravity, \n \n \n \n \n μ\n \n c\n \n \n \n \n {\\displaystyle \\mu _{c}}\n is the viscosity of the surrounding fluid, \n \n \n \n \n ρ\n \n c\n \n \n \n \n {\\displaystyle \\rho _{c}}\n the density of the surrounding fluid, \n \n \n \n Δ\n ρ\n \n \n {\\displaystyle \\Delta \\rho }\n the difference in density of the phases, and \n \n \n \n σ\n \n \n {\\displaystyle \\sigma }\n is the surface tension coefficient. For the case of a bubble with a negligible inner density the Morton number can be simplified to \n\n \n \n \n \n M\n o\n \n =\n \n \n \n g\n \n μ\n \n c\n \n \n 4\n \n \n \n \n \n ρ\n \n c\n \n \n \n σ\n \n 3\n \n \n \n \n \n .\n \n \n {\\displaystyle \\mathrm {Mo} ={\\frac {g\\mu _{c}^{4}}{\\rho _{c}\\sigma ^{3}}}.}\n\nRelation to other parameters\nThe Morton number can also be expressed by using a combination of the Weber number, Froude number and Reynolds number,\n\n \n \n \n \n M\n o\n \n =\n \n \n \n \n W\n e\n \n \n 3\n \n \n \n \n \n F\n r\n \n \n 2\n \n \n \n \n \n R\n e\n \n \n 4\n \n \n \n \n \n .\n \n \n {\\displaystyle \\mathrm {Mo} ={\\frac {\\mathrm {We} ^{3}}{\\mathrm {Fr} ^{2}\\,\\mathrm {Re} ^{4}}}.}\n The Froude number in the above expression is defined as\n\n \n \n \n \n F\n \n r\n \n 2\n \n \n \n =\n \n \n \n V\n \n 2\n \n \n \n g\n d\n \n \n \n \n \n {\\displaystyle \\mathrm {Fr^{2}} ={\\frac {V^{2}}{gd}}}\n where V is a reference velocity and d is the equivalent diameter of the drop or bubble.\nPassage 8:\nKapitza number\nThe Kapitza number (Ka) is a dimensionless number named after the prominent Russian physicist Pyotr Kapitsa (Peter Kapitza). He provided the first extensive study of the ways in which a thin film of liquid flows down inclined surfaces. Expressed as the ratio of surface tension forces to inertial forces, the Kapitza number acts as an indicator of the hydrodynamic wave regime in falling liquid films. Liquid film behavior represents a subset of the more general class of free boundary problems. and is important in a wide range of engineering and technological applications such as evaporators, heat exchangers, absorbers, microreactors, small-scale electronics/microprocessor cooling schemes, air conditioning and gas turbine blade cooling.\nAfter World War II Kapitza was removed from all his positions, including director of his Institute for Physical Problems, for refusing to work on nuclear weapons. He was at his country house and devised experiments to work on there, including his experiments on falling films of liquid.Unlike most dimensionless numbers used in the study of fluid mechanics, the Kapitza number represents a material property, as it is formed by combining powers of the surface tension, density, gravitational acceleration and kinematic viscosity.\n\n \n \n \n K\n a\n =\n \n \n σ\n \n ρ\n (\n g\n sin\n \n β\n \n )\n \n 1\n \n /\n \n 3\n \n \n \n ν\n \n 4\n \n /\n \n 3\n \n \n \n \n \n \n \n {\\displaystyle Ka={\\frac {\\sigma }{\\rho (g\\sin \\beta )^{1/3}\\nu ^{4/3}}}}\n where σ is the surface tension (SI units: N/m), g is gravitational acceleration (m/s2), ρ is density (kg/m3), β is inclination angle (rad), and ν is kinematic viscosity (m2/s).\n\nNotes\nPassage 9:\nFourier number\nIn the study of heat conduction, the Fourier number, is the ratio of time, t, to a characteristic time scale for heat diffusion, td. This dimensionless group is named in honor of J.B.J. Fourier, who formulated the modern understanding of heat conduction. The time scale for diffusion characterizes the time needed for heat to diffuse over a distance, L. For a medium with thermal diffusivity, α, this time scale is td = L2/α, so that the Fourier number is t/td = αt/L2. The Fourier number is often denoted as Fo or FoL.The Fourier number can also be used in the study of mass diffusion, if the thermal diffusivity is replaced by a mass diffusivity.\nThe Fourier number is used in analysis of time dependent transport phenomena, generally in conjunction with the Biot number if convection is present. The Fourier number arises naturally in nondimensionalization of the heat equation.\n\nDefinition\nThe general definition of the Fourier number, Fo, is:\nFor heat diffusion with a characteristic length scale L in a medium of thermal diffusivity, α, the diffusion time scale is td =L2/α, so that\n\nwhere:\n\nα is the thermal diffusivity (m2/s)\nt is the time (s)\nL is the characteristic length through which conduction occurs (m)\n\nInterpretation of the Fourier number\nConsider transient heat conduction in a slab of thickness L that is initially at a uniform temperature, T0. One side of the slab is heated to higher temperature, Th at time t = 0. The other side is adiabatic. The time needed for the other side of the object to show significant temperature change is the diffusion time, td. \nWhen Fo ≪ 1, not enough time has passed for the other side to change temperature. In this case, significant temperature change only occurs close to the heated side, and most of the slab remains at T0.\nWhen Fo ≅ 1, significant temperature change occurs all the way through the thickness L. None of the slab remains at temperature T0.\nWhen Fo ≫ 1, enough time has passed for the slab to approach steady state. The entire slab approaches temperature Th.\n\nDerivation and usage\nThe Fourier number can be derived by nondimenionalizing the time-dependent diffusion equation. As an example, consider a rod of length L that is being heated from an initial temperature, T0, by the imposition at time t = 0 of a higher temperature at x = L, TL (with x along the axis of the rod). The heat equation in one spatial dimension, x, can be applied \n\nwhere T is the temperature for 0 < x < L and t > 0. The differential equation can be scaled into a dimensionless form. A dimensionless temperature may be defined as Θ = (T - TL)/(T0 - TL), and the equation may be divided through by α/L2:\n\nThe resulting dimensionless time variable is the Fourier number, FoL = αt/L2. The characteristic time scale for diffusion, td =L2/α, comes directly from this scaling of the heat equation. \nThe Fourier number is frequently used as the nondimensional time in studying transient heat conduction in solids. A second parameter, the Biot number arises in nondimensionalization when convective boundary conditions are applied to the heat equation. Together, the Fourier number and the Biot number determine the temperature response of a solid subjected to convective heating or cooling.\n\nApplication to mass transfer\nAn analogous Fourier number can be derived by nondimensionalization of Fick's second law of diffusion. The result is a mass transfer Fourier number, Fom defined as:\nwhere:\n\nFom is the mass transfer Fourier number\nD is the mass diffusivity (m2/s)\nt is the time (s)\nL is the length scale of interest (m)The mass transfer Fourier number can be applied to the study of certain time-dependent mass diffusion problems.\n\nSee also\nBiot number\nConvection\nHeat conduction\nHeat equation\nMolecular diffusion",6 "char_len": 74102,7 "document_hash": "2aca78489610e9c932ee91ae7468186892504445e15a50cdb3c3b3a4a70f6500",8 "questions": [9 {10 "question_id": "q00",11 "qtype": "hotpotqa",12 "question": "What was the nationality and profession of the person responsible for the concept of a dimensionless number in physics and engineering?"13 }14 ]15}