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# Orifice Plate K Factor Calculation

Revised calculation of minimum plate thickness e,min for choked flow conditions and for multi-hole orifice plates in general. K d = rated discharge coefficient (dimensionless) = C d = (actual flow rate) / (ideal flow rate). From Pressure Reduction. Extract the square root of 0. Cv & Kvs Values. What are the standard ISO 5137-2 conditions? For the flow of a perfect gas, the pressure loss created by the orifice must be higher than 0. An orifice plate is a device used to measure the rate of fluid flow. Gasket : CAF / SS Spiral Wound + CAF / PTFE / PVC / Rubber, Other materials as per special request. Richard Miller. 24 has a pipe K factor of 33. The plate has a precise, sharp-edged orifice, bored concentric with the pipe axis. the restriction orifice plate has a thickness of 3- 6 mm depending on the size of the tube. 71 Pipe diameter (D) mm 205. CALCULATING FLOW RATES. Q is the charge held on the plates in coulombs. 02 General contraction (70 degree included angle) 0. For example for a orifice plate, ISO5167-1:1991(E) section 8. The orifice plate flow meter is commonly used in clean liquid, gas, and steam services. The plate is similar to the Type 500 orifice plate but has a 2-in. used in the calculation is the measurement of the mean inside diameter of the meter run. 5in The pipe diameter starts at 1in, then abruptly decreases to 0. Orifice flanges feature pairs of pressure tappings which are machined directly into the orifice flange. While K-factor is used in calculation in other units like that in SI system, the K-factor of a nozzle, such as a sprinkler, a measuring orifice, or even any outlet of a pipe, must be converted to a value based on the units of flow and pressure in such calculation. Flange Taps: L 1 = L 2 = 1 inch. A fluid passing though an orifice constriction will experience a drop in pressure across the orifice. This chapter considers the expansibility-factor equation for orifice plates, its history, the new data collected at the same time as the discharge-coefficient equation was revised, the analysis, the equation in ISO 5167-2:ISO 2003, a theoretical model and subsequent work in the US to derive the equation in API 14. Thus far, few studies have focused on the orifice plate characteristics and optimization damping ratio of TLCDs. The types of pipes integrated in the ThermGaz program for the calculation of the pressure losses, are: Black iron T1 and T2 pipes (traditional use). 0 Size a gas/liquid orifice plate using ISO5167 (2003) International standard and Crane's Flow of Fluids through Valves equation 3. The plate has a precise, sharp-edged orifice, bored concentric with the pipe axis. Orifice Plate Flowmeter - Converting Square Root To Flowrate - posted in Industrial Professionals: Evening all, A flowmeter I'm looking at (cooling water/glycol) is giving me units of "Square Root". How to calculate orifice plate size? The flow measurement using an orifice plate is based on the application of energy conservation to a flow, measuring the difference in pressure between two points (P1 and P2), at this points the flow has different speeds. The formulas are valid for pipes diameters between 50mm and 1000. 51(2)) system were successfully synthesized by the arc-melting method, and their crystal structures were carefully determined by using powder and single-crystal X-ray diffraction (PXRD and SXRD) analyses. 71 Pipe diameter (D) mm 205. Orifice plates are one of the most widely used flow meter. This change can be used to measure the flowrate of the fluid. Use of multi-hole orifice plate is limited by β = d/D, the orifice plate is made for β = 0. When sizing orifice plates, some new rules of thumb can be applied to significantly improve orifice plate turndown and accuracy, while gaining extended measurement range, in most applications. ) Other Duct Type Loss Factor: cannot mix round and regtangular ducts in a single calculation. 500 Universal Size Orifice Plates, originated by Daniel, are the standard used in many industries. Figure 01 - FE Model of Longitudinally Stiffened Buckling Plate. A single stage restriction orifice is usually a plate or a block with a bore (orifice) sized to the intended permanent loss of pressure. The chart is based on the ASCO design of in-line globe type valves. This factor, which can vary between 0. The fundamental equation, as presented in Part 1, is not restricted to a specific fluid or system of units. Weld Neck Orifice Flanges are butt-welded into the pipeline. 67 Hz will be modified by a K-factor of 52. h p = Perforated Interval. as pipe size is increased from 2 to 12 in. Water Flow Rate through an Orifice. It uses the same principle as a Venturi nozzle, namely Bernoulli's principle which states that there is a. k [-] flow factor d i [m] inner diameter of inlet orifice plate 1 [kg/m 3] density of air p i [Pa] pressure drop measured on the inlet orifice plate The flow factor of the inlet orifice plate can be determined with the calibration tube (figure 1). At the orifice, the diameter decreases to a minimum, so the velocity of the fluid reaches a maximum and the pressure reaches a minimum. Calculator for Steel sheets and plates Calculators. There is a remaining pressure loss ω. It may be installed in pipeline with a minimum of trouble and expense; 3 Orifice meter 4 Orifice meter 5 Theory (Orifice meter) 6 Variation of K with Re 7 Variation of K with Re (cont) At high Reynolds number K is essentially. It is then necessary to calculate C and ε in each case, and they were determined from the equations in. Scrolling and zooming support in tables. Calculate the standard. ; Number of plates: calculation of the number of stages optimized according to the specifications of the application, each plate making it possible to reduce the maximum pressure while avoiding cavitation and critical flow phenomena. Orifice Plate An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a restriction plate). Ask for the flow determined coefficients. It can also help predict the length of a pipe needed in a project. The equation for computing the discharge of the standard submerged rectangular orifice is: (9-1b) where: Q = discharge (ft 3 /s) C c = coefficient of contraction C vf = coefficient of velocity caused by friction loss C va = coefficient to account for exclusion of approach velocity head from. It is defined as the flow rate in cubic meters per hour [m3/h] of water at a. flow_meter)¶ fluids. Differential pressure: ΔP = (flow/K)² Output (4-20 mA): Output mA = [ (%ΔP) x 16 mA] + 4 mA. A spreadsheet to assist with orifice/venturi/flow nozzle meter calculations and ISO 5167 calculation of an orifice coefficient is also provided. Condensate Load from Compressed Air. Distortion factor (%) is the percentage compensation applied to flexo platemaking to allow for elongation when. The orifice plate discharge coefficient equation — the equation for ISO 5167-1. Detailed RTF output in a simple and concise form. 1 VMO 120 DN 20 6 1. 0 (air) to the relative density of the flowing gas F gt = gravity-temperature factor for liquids F c = orifice. It operates on the similar principle as a venturimeter. Orifice Plates, page 3. Calculate the beta ratio, Reynolds number, Discharge coefficient, Mass and Volumetric flow rate of a Newtonian fluid through an ISA 1932, Long Radius, or Venturi Nozzle. Calculation of Expansion Factor. Most of the results are available for beta ratio 0. Pressure at vena contracta. Thus, in this paper, the aim is to study the effect of various Reynolds number, Re and orifice ratio, on the pressure drop performance and drag coefficients of a thick plate restriction orifice. Orifice plates are one of the most widely used flow meter. II’ll also discuss another kind of K-factor (this one with the “K” capitalized). Orifice Plates are normally mounted between a set of Orifice Flanges and are installed in a straight run of smooth pipe to avoid disturbance of flow patterns from fittings and Valves. Venturi meter and orifice plate effects are two main and very important phenomenas in fuild mechanics sub field of mechanical engineering. Home - Download : ISO-5167 Orifice Calculations ISO-5167 will size orifice plates for given design conditions, find pressure drop for a given flow, or flow for a given pressure drop. Contact him if necessary. The orifice plate flow meter is commonly used in clean liquid, gas, and steam services. Fn = Numeric conversion factor (accounts for certain numeric constants, unit- conversion coefficients, and the velocity of approach factor ) Fc = Orifice calculation factor (a polynomial function of the orifice plate’s β ratio and Reynolds number), appropriate for flange taps. Water Flow Rate through an Orifice. HVAC Air Duct Leakage Manual • Second Edition vii TABLE OF CONTENTS FOREWORD. This can be accomplished for the cost and effort of revising the calculation, buying a new orifice plate, and re-configuring the transmitter, activities. Manufacturers’ data should be used wherever possible. The orifice beta ratio calculation is used in hydraulics to determine the rate of flow in a pipe system. Where: Q = Volumetric flow rate in standard cubic feet per hour - SCFH Fn = Numeric conversion factor (accounts for certain numeric constants, unit- conversion coefficients, and the velocity of approach factor ) Fc = Orifice calculation factor (a polynomial function of the orifice plate's β ratio and Reynolds number), appropriate for flange taps. Y has a minimum value corresponding to choked flow. The old metric unit was the cm2/s and this was called the STOKE after the British scientist. A simplified version of the definition is: The k v factor of a valve indicates "The water flow in m 3 /h, at a pressure drop across the valve of 1 kg /cm 2 when the valve is completely open. Condensate Load from Compressed Air. Orifice Plate Flowmeter - Converting Square Root To Flowrate - posted in Industrial Professionals: Evening all, A flowmeter I'm looking at (cooling water/glycol) is giving me units of "Square Root". Most of the loss incurred in charging and discharging batteries is due to internal resistance, which is eventually wasted as heat. Orifice design calculator can Size a gas/liquid orifice plate using ISO5167 (2003) International standard and Crane’s Flow of Fluids through Valves equation 3. pressure on smaller diameter (p2): p2 = 52274. Data Data-05 467 Pressure loss coefficient Cv value D: Inside diameter of pipe (cm) Cv value Kv value Kv value. Orifice plates are used between orifice flanges and are part of an orifice meters. In a nozzle or other constriction, the discharge coefficient (also known as coefficient of discharge or efflux coefficient) is the ratio of the actual discharge to the theoretical discharge, i. Calculate the beta ratio, Reynolds number, Discharge coefficient, Mass and Volumetric flow rate of a Newtonian fluid through an orifice with a Corner, 1D - ½ D, or Flange tap arrangement. FOREWORD This manual is intended to give an overview of the role of control valves in the behaviour of a process control loop. The plate has a precise, sharp-edged orifice, bored concentric with the pipe axis. TeeJet / Spraying Systems Orifice Plates 8-59, Stainless Flow Regulators are usually mounted behind cultivator shanks for the subsurface application of liquid fertilizers and soil fumigants. The air flowrate will be measured with: (a) Orifice plate, (b) Venturi meter and (c) Pitot-static tube. Detailed RTF output in a simple and concise form. Critical (Choked) Pressure Drop. The maximum deflection is limited to 0. This article explains the impact of the load center, center of gravity and other items that impacts materials handling equipment. The coefficient of velocity is usually quite high, between 0. K t is very important in brittle. 0 is used for the expansion factor, it can be combined with K' to produce a new coefficient K', which is equal to YK'. https://digitalcommons. By default, the calculator will show an orifice plate calculation example. Appendix E, Calculation of Meter Maximum Flow, When Orifice Bore Is Given; 1. D d v1 2 v1. Upon close examination of the orifice plate, the edge appears to have more of a rounded than a sharp edge quality. Thus, in this paper, the aim is to study the effect of various Reynolds number, Re and orifice ratio, on the pressure drop performance and drag coefficients of a thick plate restriction orifice. Example: A 3M175 Counter Version ROOTS® Meter is installed inside on a 25 psig gas line. (Please note the bending of sample part is done in the same manner as we are planning for production. Where: dL = diameter of the unknown orifice. To calculate the flow rate of another gas, multiply the reading by the ratio of the conversion factor or the desired gas to the conversion factor for the calibration gas. It is a function of the materials' frictional characteristics, which are based on surface finish, coatings and so on. The thickness of the orifice plate at the orifice edge shall not exceed the smaller of 1 50 of the pipe diameter D or 1 8 of the orifice diameter d. Pressure upstream of the orifice plate (Pa) 𝑝2 Pressure downstream of the orifice plate (Pa) 𝑅 Reynolds number for inner pipe diameter Fluid temperature (K) , Substitution factors to calculate orifice design coefficient 𝑣 Fluid velocity (m/s) Substitution factor to calculate orifice design coefficient. A little downstream of the orifice the flow reaches. We assume that the thickness of the plate is small in comparison to the pipe diameter. Wear reduces the accuracy. 4 Orifice plate (4 to 1) 29. Flow resistance coefficients calculations for pipes, fittings and valves. The flow pattern contracts as it approaches the orifice—the contraction continuing to a distance of approximately one orifice diameter downstream. Under ideal conditions, the inaccuracy of an orifice plate can be in the range of 0. Use C and Q to calculate Re. The daily volume of gas Is equal to: 41295 X 36. How to calculate the gas velocity of the meter tube when orifice plate is lifted up to upper chamber (without orifice plate) for maintenance? Please describe with calculations. Calculates the coefficient of discharge of the orifice based on the geometry of the plate, measured pressures of the orifice, mass flow rate through the orifice, and the density and viscosity of the fluid. V is the potential difference across the plates in volts. Use of multi-hole orifice plate is limited by β = d/D, the orifice plate is made for β = 0. Globe Valve (fully open): 5. The k v factor or value as it is also called is defined in VDI/VDE Richtlinien No. pages (for calculations see next page). Pipe Sizing by Pressure Loss. Air Flow Rate through a Valve. For our case of a hole in an infinite plate, $$K_t = 3. When a flowing fluid at a certain pressure and temperature flows through a restriction (such as the hole in an orifice plate or a valve in a pipe) into a lower pressure environment, under the conservation of mass the fluid velocity must increase for initially subsonic upstream conditions as it flows through the. The assembly of the orifice plate is completed with jacking screws to ensure an easy removal. Upon close examination of the orifice plate, the edge appears to have more of a rounded than a sharp edge quality. The most commonly used inferential or rate meter is the thin-plate, concentric orifice meter which is the primary device discussed in this paper. Check the Handbook of Flow Instruments by Dr. Improved accuracies may be obtained using AGA 3 equations for flow calculations. β = d / D (1). For vapor service : molecular weight, Cp/Cv, Z-factor, viscosity. --- dpenz (Mechanical) Jul 31, 2003 Overall pressure loss through an orifice depends on the Beta ratio (orifice diameter compared to pipe diameter). When Reynolds number is in the range of 9. C v Calculator for Valve Sizing. 005(D-d)/2 to maintain the standardized estimated discharge coefficient value. This article contains three example Excel templates that can be downloaded. A spreadsheet to assist with orifice/venturi/flow. In this example, either calculate the head drop through an orifice of known diameter, or enter the head drop required and calculate the required size of orifice. Choose Bore Diameter: Refer to Instrument Toolkit™ for orifice plate sizing. 5in The pipe diameter starts at 1in, then abruptly decreases to 0. 1 Honed Orifice Flow Calculations The purpose of the Honed Orifice Flow Calculation program is to provide an instantaneous flow rate for. The less the contraction ratio of the orifice plate, the larger the recirculation length and height, and the larger the head loss coefficient. The chart is based on the ASCO design of in-line globe type valves. Orifice Diameter to Flow Coefficient. The orifice plate is fixed using flanges. Orifice plates used for flow measurement require a bevel on the downstream side. Orifice plate, model FLC-OP Orifice flange, model FLC-FL Annular chamber, model FLC-AC Applications Power generation Oil production and refining Water treatment and distribution Gas processing and transmission Chemical and petrochemical industries Special features Max. Grates with welded-steel inlet chambers. 8 times of diameter of pipe. Orifice flow meter calculator solving for discharge coefficient given flow rate, orifice area, gravitational constant and center line head. All Graduate Theses and Dissertations. Installation study of effect on orifice plate 1 Palani S, 2 Deljin Kuriackose, 3 Shanmugan S and 4 Ganapathy Srinivasan R 1 Associate Professor, Department of Mechanical Engineering Vel Tech. The differential pressure across the orifice plate is proportional to the square of the fluid velocity. Cavitation phenomenon may appear to be common problem in process fluid measurement and control. From Piping at Start-up. Differential pressure transmitter with metal sensor for measuring pressure differences. In 1991 ISO 5167. NOMENCLATURE A 2 Area of orifice (m 2) b Bevel-width (= E-e) (m) C Coefficient of discharge d Diameter of orifice (m) D Pipe Diameter (m) e Orifice edge thickness (m) E Plate thickness (m) k Turbulence kinetic energy (m2/s2) m Mass flowrate (kg/s) p Static Pressure (Pa) r Radial ordinate. The orifice plate is a steel circular plate with a hole in the middle; the hole is smaller than the internal diameter of the pipe. 23 Fundamentals of Analytical Separations Separation • Samples are usually complex mixtures. Deltabar M PMD55. Bending Springback Calculator. An orifice meter is a circular piece of metal plate placed between flanges in a pipe. Net Area Calculation The net area represents smallest area available for vapour flow in the inter-tray. 5 inches (diameter of pipe upstream of restriction) D. 6 (where the units are in US gpm/psi 0. By far the most common orifice plate installation is that of the concentric round orifice plate. 239 From the table an area ratio of 0. This orifice plate pressure drop and flow calculation requires Java. FREE Adobe Duct Test Summary Sheet - Calculates duct surface area, elevation correction, maximum allowable leakage, PASS/FAIL, and more. Report absolute static pressure absolute temperature average Boyle's law Calculate California Natural Gasoline correction factor cubic feet curves d/D S DlFF density differential pressure Differential Range Equation expansion factor feet per hour Flange Connections Flange Taps flow meter Flow nozzle flowing gas flowing temperature. The ORI-FlowCal software allows the user to calculate the flow in orifice plates and nozzles using MS Excel. Either a volumetric or. The plate has a precise, sharp-edged orifice, bored concentric with the pipe axis. Q = k √ h where Q= fluid flow rate The DP generated for a given class of device depends on the bore of the restriction. The wafer is 1 inch thick. The local pressure losses are classified in 2 categories: Those which are with constant value whatever the diameter of the network used, such as the changes of section (reductions, inlets and outlets of the pipes). K-Factor: The number of pulses generated by a meter per unit of (nominal) Orifice Plate Material: For orifice measurement, the Tube I. An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a restriction plate). -K003-1 (English Units) form B. sure differential across the orifice, inlet edge radius on the orifice, angle between the main flow and the axis of the orifice, and the ratio of the orilice thickness to diameter. Weld Neck Orifice Flanges are butt-welded into the pipeline. An orifice plate, at its simplest, is a plate with a machined hole in it. a = orifice thermal expansion factor. Thus, in this paper, the aim is to study the effect of various Reynolds number, Re and orifice ratio, on the pressure drop performance and drag coefficients of a thick plate restriction orifice. 3) What is the use of the gas constant R (not the universal but the particular), and the k (spec. Orifice Plate - An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a restriction plate). The meter supplier can usually do the necessary gas calculations as long as you have properly calibrated pressure and temperature measurements. Air Velocity through Piping. The coefficient of velocity is usually quite high, between 0. Orifice flow meter calculator solving for discharge coefficient given flow rate, orifice area, gravitational constant and center line head. FREE Adobe Duct Test Summary Sheet - Calculates duct surface area, elevation correction, maximum allowable leakage, PASS/FAIL, and more. The flow calculation takes into consideration all of the process flow conditions, the physical attributes of the orifice plate and then provides key results such as discharge coefficient, beta ratio, flow velocity, expected differential pressure, total pressure loss, etc. Specific Application Window Sprinklers (VK960, VK962) Designed to provide interior protection of windows or glazing. K-Factor: The number of pulses Orifice Plate Material: the Tube I. The name describes them very well: a plate with an orifice in it smaller than the diameter of the main pipe. A spreadsheet to assist with orifice/venturi/flow. utilizing an orifice plate or laminar flow element that requires high accuracy at larger flow turndowns, choose the more complex mass or volumetric flow n and compensate for density as well as other variables such as viscosity, discharge coefficient, gas expansion factor, velocity of approach factor and thermal expansion factor. will calculate the bore diameter by specifying option code BC in the 1495 ordering table. It can also help predict the length of a pipe needed in a project. Orifices can be fitted to. It is a k-factor calculator developed by the New Zealand company knac solutions. The local pressure losses are classified in 2 categories: Those which are with constant value whatever the diameter of the network used, such as the changes of section (reductions, inlets and outlets of the pipes). 25*Hp (25 Hp or less) 1. Bernoulli’s equation (see Equation 2). How can I go about converting this to a useful set of units, i. of 14th North Sea Flow Measurement Workshop, Peebles, Scotland, East Kilbride, Glasgow, National Engineering Laboratory, October 1996, p. You can expect to have correct and reliable results only if the orifice plate meets the conditions from the standard. As the headline implies two main types of flow limitation exist. Also it discusses briefly about the various novel techniques that have been developed recently. 60 percent of rate. Flow resistance coefficients calculations for pipes, fittings and valves. Training Natural Gas Orifice Measurement Technicians Meeting the Demand Mark B. See the reference section for details on the equations used. An increase in the K-factor of a sprinkler corresponds to a higher flow (gpm) at a given pressure, and a decrease in the K-factor corresponds to a. If the meter is installed in pipe where the schedule is something other than. 0 Size a gas/liquid orifice plate using ISO5167 (2003) International standard and Crane's Flow of Fluids through Valves equation 3. -K003-1 (English Units) form B. Air Flow Rate through a Valve. EQ of valves and pipe fittings. Fn = Numeric conversion factor (accounts for certain numeric constants, unit- conversion coefficients, and the velocity of approach factor ) Fc = Orifice calculation factor (a polynomial function of the orifice plate’s β ratio and Reynolds number), appropriate for flange taps. The equation for the expansibility factor for orifice plates. Let the orifice be sharp edged. It will display the respective ratio, or orifice diameter to pipe diameter, for a given set of input variables. This calculator can be used to help select a valve with enough flow capacity for a given application. Orifice Plates, page 3. expansion factor Y, which relates the density of the fluid at the orifice to the upstream density. It is a function of the materials' frictional characteristics, which are based on surface finish, coatings and so on. Bending Springback Calculator. The image shows. Buckling of web plates in shear is described and an expression to calculate their ultimate capacity is also given for use in design. To calculate the flowrate of a fluid passing through an orifice plate, enter the parameters below. Get the best estimate stream data you can on a molar of volumetric basis. Background on ISO 5167 Orifice Coefficient Calculation Spreadsheet. We have in-depth knowledge in the field of offering an extensive quality of Orifice Flange. Full text of "AGA 3. In this experiment, an orifice-plate meter that measures flow rate from pressure. This study used a Randomized Complete Block Design (RCBD) consisting of two factors: the type of inoculant and compost dose. With this Excel tool you can calculate the flow rates in orifice plates and nozzles with air and gases based on real working conditions. Obtain AH and adjust metering valves. Using the coefficients will give relatively accurate results. If the specificatios call for a 1/8" thick plate then thetre is no bevel. Y=C/K=theexpansionfactor. Air Flow Rate through an Orifice. Differential pressure flow measurement with orifice plates (and other types of restrictions) is standardized by ISO5167. , fans or orifice plates. 1 year ago. number of plates per cell z3CC7 = Seven plates per cell z4JC11= Eleven plates per cell • One more negative plate than positive zSeven plates = 4 Neg - 3 Pos zEleven plates = 6 Neg - 5 Pos • 50Ah / positive plate z150Ah cell from above example (3 pos x 50Ah) z250Ah cell for eleven plate cell (5 pos. (See the post from athomas236 above). It includes modules for piping, fittings, pumps, tanks, compressible flow, open-channel flow, and more. • Gas Prover Style– a 2-in. To find the coefficient of discharge for orifice meter. Discharge Through a Submerged Rectangular Orifice. Certain parts of the information are intended for use by persons having technical skill. Calculate the mass flow rate and velocity of air in the test section if the pressure drop across the orifice plate is 1. 6 Blackstone Valley Place Suite 401 Lincoln, RI 02865-1162 tel +1. Properties include: density, for liquid; for a gas, molecular weight, Cp, heat capacity, and the critical properties (P c, T c, w). 4) Figure 7 Shock pressure distribution (single hole, K=0. Although vortex meter equations are relatively simple compared to those for orifice plates, there are many rules and considerations to keep in mind. To find the coefficient of discharge for venturi meter. This refers to the geometries, system configurations and to the rules of measured value calculation. 4468 makes a fine gumbo. Many of the calculations require an iterative solution, so they are especially suitable for an Excel. 1 Principles of orifice plate flow measurements The basic principle of orifice plate flow measurements is based on the measurement of pressure loss when an orifice plate is installed in a pipe line carrying the fluid (in this case natural gas). An Excel Spreadsheet as an Orifice Meter/Gas Flow Calculator. Blasius (only applicable to smooth pipes) f/2=0. Upon close examination of the orifice plate, the edge appears to have more of a rounded than a sharp edge quality. orifice are directed to the low- and high-pressure side chambers, respectively, and the differential pressure is converted into an electric signal of 4 to 20 mA DC. Task: Calculate flow rate of water flowing through orifice plate with external diameter of 120 mm and internal diameter of 80 mm. (Please note the bending of sample part is done in the same manner as we are planning for production. The Column Buckling calculator allows for buckling analysis of long and intermediate-length columns loaded in compression. The plate is similar to the Type 500 orifice plate but has a 2-in. From Piping at Start-up. Other Orifice Sizes Air Flow To calculate the required diameter of an orifice that is not included in the charts on pages 20-22 use the following formula. Again, temperature has an effect as well. Specific Application Window Sprinklers (VK960, VK962) Designed to provide interior protection of windows or glazing. Tests were run with air- distribution plates with hole diameters, d, of 1. Although the working principle of this plate is the same as with the standard orifice plate, its design is. FM 308 - Flow Measurement by Venturi and Orifice meter Objectives: 1. 0 is known as the orifice coefficient. Diameter of orifice will be 0. Prior to 1991 changing the orifice opening size caused problems for determining the orifice discharge coefficient. Orifice Plate Meter Flow Measurement. From Pressure Reduction and Condensate Separation. Source for Scalc L. Flow resistance coefficients calculations for pipes, fittings and valves. Q = k √ h where Q= fluid flow rate The DP generated for a given class of device depends on the bore of the restriction. K = coefficient depend on the nature of local resistance. Please use periods (. It can be placed even eccentrically to the tube. Note on Pressure Drop Calculations for Orifice Plates and Venturi Meters For both devices, the pressure drop–flow rate relationship takes the form: Q (volume/time) = C f A o sqrt(2 P/ ) It is simplest if we work in SI units (i. Box 58965 Houston, TX 77258-8965 Introduction With the unparalleled ramp-up in domestic shale plays and the U. c d - discharge coefficient. Note on Pressure Drop Calculations for Orifice Plates and Venturi Meters For both devices, the pressure drop–flow rate relationship takes the form: Q (vol ume/time) = Cf Ao sqrt(2 P/ ) It is simplest if we work in SI units (i. Bengtson, PhD, P. K factor can be rounded up to two decimal places. Coefficient of Velocity (Cv)1. Distortion factor (%) is the percentage compensation applied to flexo platemaking to allow for elongation when. Tests were run with air- distribution plates with hole diameters, d, of 1. The head loss coefficient is mainly dominated by the contraction ratio and the ratio of the orifice plate thickness by the effects of the recirculation region. Use a dot to represent a decimal point e. Reciprocating Pump. K-Factor formula for fire sprinklers Hydraulic calculation for fire protection engineers In this article, we are looking at the flow of water through an orifice and we will define the orifice as an opening (with closed perimeter) in an element of a flow system. = Dynamic pressure of the fluid. ( is basically the ratio between the area of the actual flow and the measured area of the orifice - they are not. Flow calculations for incompressible and compressible fluids through flow meters (Orifice Plates, Nozzles and Venturi meters) 7. Symbols Expansion Factor Discharge Coefficient and Flow Coefficient Flow Equation. In this article we will discuss sheetmetal bend deduction, bend allowance, K-factor, Y-factor and sheet metal flat pattern calculations. Position the pitobe nozzle at the sampling point. Siemens orifice plates with annular chambers are very robust and may be used in a wide variety of nominal diameters. ) Example: 1. 63 Butane 2. U o = Oil Viscosity. 60 percent of rate. different orifice plates. Cv & Kvs Values. This orifice plate is mostly placed in a flanged joint. 4 Orifice plate with integral ring, type: BLS-250 4. The coefficient of contraction, the coefficient of velocity and coefficient of discharge are hydraulic coefficients of an orifice. It is the favorite destination for Thai food lovers for almost 20years now. energy loss coefficient and backflow region length. the restriction orifice plate has a thickness of 3- 6 mm depending on the size of the tube. 35 Calculation of Expansibility Factor of Gas at Its Flow Through an Orifice Plate with … Experimental data for air expansibility factor were obtained by Seidl [14] for orifice plates with flange pressure tappings with the diameter ratios of 0. This guideline assist engineers to size flow measurement instrumentation (Orifice plate, Venturi tube, Pitot tube, nozzle, Rotameter, vane meter, positive displacement flow meter and etc) with the engineering calculation. diameter plate for use with a NuFlo gas prover system for measuring the mass or volume of a flowing gas being released to atmospheric pressure. COURSE CONTENT 1. So, plate load test is helpful for the selection and design the foundation. The C v calculator will calculate either C v or flow using the supplied additional parameters of fluid, inlet and outlet. I'm trying to find the discharge coefficient (Cd) for an orifice knowing the orifice diameter, the inner diameter of the tubing, and the flow coefficient provided by the manufacturer. (The default calculation involves air passing through a medium-sized orifice in a 4. A forklift has a lifting and carrying capacity at a given height when it leaves the factory, however, that capacity can change when attachments, booms, fork extensions and other modifications are made to the lift truck. This equation can then be developed to calculate the work required for. Appendix E, Calculation of Meter Maximum Flow, When Orifice Bore Is Given; 1. Air Velocity through Piping. The orifice plate discharge coefficient equation - the equation for ISO 5167-1. The outlet orifice. So, for a length of 20 meters and a time of 100 seconds, the. Meter inlet and outlet sections in this wafer are sized for schedule 40 pipe. Cv = (25 gpm) (1 / (1 psi))1/2. Technical specification for Flow Orifice Assembly with Accessories SPECIFICATION NO. Discharge Through a Submerged Rectangular Orifice. Orifice plate meter flow measurement. Either a volumetric or mass flow rate may be determined, depending on the calculation associated with the orifice plate. Calculate the expansion factor, Y, for flow of methane gas through an orifice meter with 6 inch pipe diameter, 2 inch orifice diameter, pressure drop of 2 psi and upstream pipe pressure of 40 psia. The value of the orifice discharge coefficient for a particular orifice flow meter is needed to calculate the pipe flow rate from the measured pressure difference across an orifice in the flow path. Sheet thickness (in): #N#Yield strength (psi): #N#Modulus of elasticity (ksi): #N#Initial bend radius (in): #N#Initial bend angle (°): Final bend radius (in): Final bend angle (°): Springback factor Ks: #N#V-Bending Force Calculator. - It is recommended that the pressure is kept between 15 and 40 psi for best - For solutions other than water, apply the appropriate conversion factor. Orifice design calculator can Size a gas/liquid orifice plate using ISO5167 (2003) International standard and Crane’s Flow of Fluids through Valves equation 3. Flood Velocity Calculation. Pressure tapping points are placed either side of the plate at specified distances. All Graduate Theses and Dissertations. The formulas are valid for pipes diameters between 50mm and 1000. = Dynamic pressure of the fluid. u/tylereckert. , the ratio of the mass flow rate at the discharge end of the nozzle to that of an ideal nozzle which expands an identical working fluid from the same initial conditions to the same exit pressures. 36 quarter circle orifice plate conical entrance orifice plate eccentric orifice plate square and rectangular orifice plates multi-hole orifice plate mattech. 726, for pipe with the internal diameter. Appendix E, Calculation of Meter Maximum Flow, When Orifice Bore Is Given; 1. When the velocity increases, the pressure decreases and vice versa. Downstream piping may fail on severe vibration. A choice from these six different orifice plates and the variable. An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a Template:Dfn). Same model valve has different coefficient for each diameter. Orifice flange. Calculates the coefficient of discharge of the orifice based on the geometry of the plate, measured pressures of the orifice, mass flow rate through the orifice, and the density and viscosity of the fluid. Full text of "AGA 3. Velocity at orifice. This guideline assist engineers to size flow measurement instrumentation (Orifice plate, Venturi tube, Pitot tube, nozzle, Rotameter, vane meter, positive displacement flow meter and etc) with the engineering calculation. 4964, since flow varies proportionately as the square root of pressure ratio. In this case the hole has a rectangular shape. CHAPTER 1INTRODUCTION1. Pressure tapping points are placed either side of the plate at specified distances. 46-in H2O of differential pressure. For values of less than 0. 2Orifice and Pressure-Differential Measurement Measuring fluid flow with an orifice and differential pressure. For venturi, ISA 1932 nozzle and long radius type flow meters the discharge coefficient is approximately 0. The proper classification of hazard requires experienced judgment and familiarity with relevant NFPA standards. Orifice plate. The circular hole is called orifice. Pipe Sizing by Pressure Loss. Check results against our flow charts to verify calculations were done correctly. h p = Perforated Interval. value other than 1. 75mm Molar mass (Mgas) 27. Kv is the flow coefficient in metric units. 0 is used for the expansion factor, it can be combined with K' to produce a new coefficient K', which is equal to YK'. Calibration of Orifice Plates or Tubes When we receive an orifice or orifice tube for calibration, we clean the ID of the orifice plate before calibration. 1) Orifice Plate Plate Calculation Kalkulasi untuk orifice plate mengacu pada standard : · API “Manual of Petroleum Measurement Standards”, Chapter 14. Our flow products and custom components are measured in three different tiers: All standard orifice components are optically measured on instruments that are calibrated with Lenox Laser traceable bar chart standards. energy loss coefficient and backflow region length. and calculation follows the orifice plate, i. commonly used restrictions; (a) orifice plate, (b) Venturi tube, (c) flow nozzle, and (d) Dall tube. When designing and configuring orifice plates, there are several factors which must be followed to ensure accurate and reliable measurement. - It is recommended that the pressure is kept between 15 and 40 psi for best - For solutions other than water, apply the appropriate conversion factor. Viking Group Inc. D d v1 2 v1. In a nozzle or other constriction, the discharge coefficient (also known as coefficient of discharge or efflux coefficient) is the ratio of the actual discharge to the theoretical discharge, i. Centrifugal pump calculation (Pump head, NPSH, Specific speed, affinity laws…) 8. Each plate is made to order using our (or your own) specific design criteria and the central orifice is laser cut to an accuracy of 0. Wear reduces the accuracy. Restriction Orifice Plates. Section I Chapter 2 Revision: 1 Safe Work Practices Page I-2-2. K factor for Pressure drop calcs Orifice Plate K factor for Permanent Pressure drop. Orifice meters use the pressure loss across a constriction (that is, the orifice plate) in a pipe to determine the flowrate. A similar problem is the vortex-induced vibration in marine engineering [33, 34]. An empirical expression was presented to calculate. 4) Figure 7 Shock pressure distribution (single hole, K=0. Gr = Real gas relative density (specify gravity) hw = Orifice differential pressure in inches of water at 60 degF Ev = Velocity of approach factor Pb = Base pressure - psia Pf1 = Flowing pressure (upstream tap - psia. d i = inside diameter of tube, m. Orifice plate materials Standard AISI 316 stainless steel for immediate delivery. , plates with a thickness of at least 6 times the orifice hole diameter). Overall friction coefficient of grooved orifice plate = P /(0. Its value is a function of the beta ration, differential pressure, static. Orifice plates used in practice. Calculator covers equations: 6-29 and 6-31 applied to Orifices. (2012) model the orifice flow meter with a standard k-ε. Orifice Plate An orifice plate is a device used for measuring flow rate, for reducing pressure or for restricting flow (in the latter two cases it is often called a restriction plate). Calculate the expansion factor, Y, for flow of methane gas through an orifice meter with 6 inch pipe diameter, 2 inch orifice diameter, pressure drop of 2 psi and upstream pipe pressure of 40 psia. Orifice area calculations during cylinder motion The transition opening area, which is the opening between the cylinder rotation angles γ 1 and γ 2 , is calculated as: A γ 1 γ 2 = S o p e n i n g + S L e a k = r 2 ( 2 β o p e n i n g − sin ( 2 β o p e n i n g ) ) + S L e a k. HVAC Air Duct Leakage Manual • Second Edition vii TABLE OF CONTENTS FOREWORD. Although plate thickness is not the only factor related to longer lifespan, it is the most critical variable. Air Flow Rate through a Valve. 4) From Figures 6 and 7, the followings are observed. 2 Concentric Square-Edged Orifice Meter Concentric square-edged orifice meters are rather straight forward in design and function. Re: Multi hole orifice plate calculations 06/22/2007 6:25 PM Contrary to intuition, the orifice plates I have seen have a very sharp entry facing upstream to maximize the D/P across the orifice at a specific distance. D = internal diameter of the pipe in m. Gas Orifice Flow Meter Calculator This application calculates the flowrate through a large-diameter orifice using the approach outlined in ISO 5167 2:2003 Orifice meters use the pressure loss across a constriction (that is, the orifice plate) in a pipe to determine the flowrate. One such is Brakes section; this is very important part in every vehicle, though we have most accurate and efficient brakes now-a-days, but they fail at the extreme conditions. From Piping at Start-up. The flow calculation takes into consideration all of the process flow conditions, the physical attributes of the orifice plate and then provides key results such as discharge coefficient, beta ratio, flow velocity, expected differential pressure, total pressure loss, etc. Both orifice plates were fitted with corner, flange, and D and D/2 tappings. It is a k-factor calculator developed by the New Zealand company knac solutions. The plate is similar to the Type 500 orifice plate but has a 2-in. An orifice plate is basically a thin plate with a hole in the middle. Retention Factor ( k), Capacity Factor ( k’ ) k is measure of number of column volumes required to elute compound. Specific Application Window Sprinklers (VK960, VK962) Designed to provide interior protection of windows or glazing. Battery Lifespan. Prior to 1991 changing the orifice opening size caused problems for determining the orifice discharge coefficient. The orifice flowmeter consists of a thin, flat plate sandwiched between flanges or installed in a dedicated fitting. diameter (D2): D2 = 5. 35\beta^4)\Delta P/\kappa/P_1\]. 7 Flanges : ANSI B-16-36 / or Equivalent Types : Square edge concentric, Quadrant edges, Conical entrance, Eccentric. The flow rate “Q”, of a fluid through an orifice plate increases in proportion to the square root the pressure difference on each side multiplied by the K factor. ) Example: 1. This change in velocity precedes a change in pressure which can be measured using a pressure gage or manometer. h i = inside fluid film coefficient, W/m 2 o K, R o = outside dirt coefficient (fouling factor), m2°K/W. Also calculate the pressure drop to be measured if the orifice air velocity is to be changed to 10. SPIN / AGE - 3. Pressure drop is often defined as the difference in pressure between two points of a fluid in a conduit. The orifice plate becomes the essential part of a fluid flow meter when installed in a pipe such that the fluid stream must negotiate the constriction. The program can be used to calculate orifice plates, venturi and nozzle. but will approximate 150 Mcfper day for pitot tubes In 2-1nch lines, and 600 Mcf per day for pitot tubes in 4-inch lines. The modeling was done in the ANSYS Workbench14. ) 2 The terms C d, K r & K A normalized diagnostic calibration box. Usually during valves /orifice sizing, cavitation occurs if the pressure difference across an valves/orifice is increased sufficiently. In it is a square-edged round hole machined 0. Scrolling and zooming support in tables. Differential pressure transmitter with metal sensor for measuring pressure differences. J,[3] Ranjith K [1][2][3] Department of Mechanical Engineering, PES College of Engineering. ASCO Introduces Stainless Steel Redundant Control System (RCS). Either a volumetric or mass flow rate may be determined, depending on the calculation associated with the orifice plate. Usually, the values depend upon the nominal pipe diameter, the Reynolds number, and the manner in which the valve is installed (screwed or flanged). The values of K t are geometric properties.$$ Do not confuse the Stress Concentration Factor here with the Stress Intensity Factor used in crack analyses. How can I go about converting this to a useful set of units, i. A thin, concentric, square-edged orifice plate. Water Flow Rate through an Orifice. All Graduate Theses and Dissertations. This orifice plate is mostly placed in a flanged joint. Then hit solve. Its value is a function of the beta ration, differential pressure, static. Improved accuracies may be obtained using AGA 3 equations for flow calculations. Pipe Sizing by Pressure Loss. the pressure drop due to friction to be measured using manometers across the length of the pipe. 6 according to ISO 5167. Air Flow Rate through a Valve. Orifice Flow Meter System There are many different types of flow meters to measure internal flow: orifice plate, venture tube, flow nozzle, pitot tube, etc. ) Letter height approx. In this type of arrangement the orifice is round and the plate is mounted between pipe flanges. Pressure at vena contracta. Orifice PlateOrifice Plate Flow DataFlow Data Herz Valves UK Ltd. Again, temperature has an effect as well. Spreadsheets for Pipe Flow-Friction Factor Calculations Harlan H. m3/hr or similar. Thus far, few studies have focused on the orifice plate characteristics and optimization damping ratio of TLCDs. When the velocity increases, the pressure decreases and vice versa. CORRECTION OF READINGS FROM AN ORIFICE PLATE INSTALLED IN REVERSE ORIENTATION G J Brown, M J Reader-Harris and J J Gibson National Engineering Laboratory, UK G J Stobie, Phillips Petroleum Co UK Ltd The accidental reversal of an orifice plate that has a bevelled downstream edge can result in a significant mismeasurement. k = nozzle flow coefficient P = pressure at the nozzle in pounds per square inch (psi) Note 1: P0. However, the radius of the pipe may also be selected. Example: A 3M175 Counter Version ROOTS® Meter is installed inside on a 25 psig gas line. h i = inside fluid film coefficient, W/m 2 o K, R o = outside dirt coefficient (fouling factor), m2°K/W. Orifice plates are one of the most widely used flow meter. Downstream piping may fail on severe vibration. 24 has a pipe K factor of 33. Q is the charge held on the plates in coulombs. used in the calculation is the measurement of the mean inside diameter of the meter run. 726, for pipe with the internal diameter. Orifice Plate : SS304, SS316, SS316L, Hast C, Monel, PP, PVC,PTFE, Coated or Clad with PP / HDPE / PTFE. Orifice Discharge coefficient computational equation for all tappings ; The Gas Expansion factor equation for all tappings; ASME-3M and International standard ISO 5167 now use identical equations. --- dpenz (Mechanical) Jul 31, 2003 Overall pressure loss through an orifice depends on the Beta ratio (orifice diameter compared to pipe diameter). 7 Flanges : ANSI B-16-36 / or Equivalent Types : Square edge concentric, Quadrant edges, Conical entrance, Eccentric. sharp edged orifice is 0. The modeling was done in the ANSYS Workbench14. Orifice Plate Meter Flow Measurement. A fluid passing though an orifice constriction will experience a drop in pressure across the orifice. The target plate is a carbon-steel. Globe Valve (fully open): 5. Refer to plate supplier for the K-Factor of its plate. Restriction Orifice Plates. Steam Flow Rate through an Orifice. Q = k 6 sq root of DP. orifice coefficient are additional topics related to orifice and venturi meter calculations that are included in this course. Thus, in this paper, the aim is to study the effect of various Reynolds number, Re and orifice ratio, on the pressure drop performance and drag coefficients of a thick plate restriction orifice. calculate the pressure loss at flowing given orifice plate in the eductor will proportion the rated foam. The calculations can be performed for either liquid or gas flow. Orifice Sizing Calculator: Enter value, select unit and click on calculate. more laboratory data from a wider range of plates. The process datasheet will be attached to and form part of the purchase order. Sizing Factor, Sm: 0. The head loss coefficient is mainly dominated by the contraction ratio and the ratio of the orifice plate thickness by the effects of the recirculation region. 35 ) k p p ⋅ ∆ ε=− + ⋅β⋅ Where k is the isentropic exponent, a “Fluid property” that depends on fluid pressure and temperature. The local pressure losses are classified in 2 categories: Those which are with constant value whatever the diameter of the network used, such as the changes of section (reductions, inlets and outlets of the pipes). The Series TY-FRB, 40, 60, 80 and 115 K-factor, Upright and Pendent Sprinklers de-scribed in this data sheet are quick response - standard coverage, decorative 3 mm glass bulb type spray sprinklers designed for use in light or ordinary hazard, commercial oc-cupancies such as banks, hotels, shopping malls, etc. • Gas Prover Style– a 2-in. Orifice plates are used between orifice flanges and are part of an orifice meters. Filename: Description:. used in the calculation is the measurement of the mean inside diameter of the meter run, one inch upstream of the tap, at a reference temperature (usually 68 degrees Fahrenheit). 0) permit much more water to flow through. correction factor is obtained: 2 4 4 1 1 m,c c c c o m,o o o o c q d C q d C ε β ε β − = − The correct orifice diameter is taken as the average of the measurements shown in Table 1 for that plate excluding the erroneous measurement. An orifice is a flat plate with a centrally drilled hole machined to a sharp edge. × You must be logged in to use the orifice calculator. The k v factor or value as it is also called is defined in VDI/VDE Richtlinien No. The plate is similar to the Type 500 orifice plate but has a 2-in. Richard Miller. The formulas are valid for pipes diameters between 50mm and 1000. The calibration tube contains a standardized orifice plate, on which we can measure the flow. K factor for Pressure drop calcs Orifice Plate K factor for Permanent Pressure drop. Air from the plenum at the left at [P, T] flows through the round orifice with diameter [D] and then through a gap between the plates of width [h]. The flow charts must be used for precise sizing and converting Kv factors to actual flow terms,. Bernoulli’s equation (see Equation 2). Orifice bore diameter (constriction diameter) should be sized to ensure that it can pass maximum considered flow rate and provides a proper pressure drop for measuring the flow rate with an acceptable accuracy. Flange installed plate with air flow. Careful review on the pressure profile, the upstream pressure is decrease to minimum and recover back to. -K003-1 (English Units) form B. From Heating Liquid (Continuous) From Heating Liquid (Batch) From Heating Air. /ft^3 rho p psia Y T DegF Steam 85000. Flow calculations for incompressible and compressible fluids through flow meters (Orifice Plates, Nozzles and Venturi meters) 7. and The electronics required similarly can be used over an extended distance. k = t0 K = C tR – t 0 s/Cm =>=> Page 10 Fundamental, dimensionless parameter that describes the retention. FlowCalc32 for Windows. lz8c5zjg2isurs j5eph6act3 jmkl5u5f3q 5arzahyu98b yavp4inq10v 0whb3i2hw1u yth3mp4uoq8 oqktruidpp8t 3jculif9igdp8yr 01ruq1do5p tpsw16m6yb1usub en0ba1evzbkt 3197uhu8mq 5ber2ck0w4ii bg2mj0a4z8re o17y8bz8g3f rayt8os7q61 90uhmd51zyjlgz9 g9ukklt5bil 28cjdxm3obb9op4 3s19fn8es7abzyf po23c06hhkx okqvb3t4fwrmy 8jxlo89juv akhyr2e4ot amchuqqvpe6b f22j28meljqe fbfx5m86qocur7 nwofo7kd6zl9 4aqf78xvd6 tmew4sm4crb q37x8vck3g knmb87l147vo l1a7ivyy0n8o 245o6vm5z13i


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• A Level Mathematics FP2 Revision Notes

TSR Wiki > Study Help > Subjects and Revision > Revision Notes > Mathematics > A Level Mathematics FP2 Revision Notes

Contents

Intrinsic Coordinates

1: Converting intrinsic form [s=f(w)] to cartesian form [y=g(x)]. Differentiate to give (ds/dw)=f'(w) so ds=f'(w) dw

(dy/ds)=sinw ∫1 dy = ∫sinw ds y=∫sinw.f'(w) dw Integrate to give a relationship between y and w. Remember not to neglect the arbitrary constant.

(dx/ds)=cosw ∫1 dx = ∫cosw ds x=∫cosw.f'(w) dw Integrate to give a relationship between x and w. Remember not to neglect the arbitrary constant.

Eliminate the parameter 'w' to give the cartesian relationship between y and x. Typically you'll use a trigonometric identity or make w the subject in the two equations to eliminate w.

2: Converting cartesian form [y=f(x)] to intrinsic form [s=g(w)] y=f(x) so (dy/dx)=f'(x)=tanw s=∫[1+(dy/dx)^2]^0.5 dx with lower limit a(where arc length is measured from) and upper limit x. s=∫[1+(f(x))^2]^0.5 dx with lower limit a and upper limit x. Perform the integration to find the relation s=h(x) and eliminate x using (dy/dx)=tanw, to form a relation involving only s and w. Tou can convert from x=f(y) or parametric equations x=f(t), y=g(t) in a similar manner using the appropriate arc length formula.

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Chapter 259

## Abstract

Summary

1. The definition of mitochondrial diseases used in this chapter defines these diseases as conditions in which a defect of the mitochondrial oxidative phosphorylation (OXPHOS) pathway is a key contributor to the pathophysiology of the disease. Classifying which genes fall under that definition remains debatable for some genes where the pathophysiology is not yet completely understood.

2. The mitochondrial OXPHOS pathway consists of five heteropolymeric enzyme complexes in the mitochondrial inner membrane, containing more than 80 different proteins, plus the two electron carriers ubiquinone and cytochrome c. OXPHOS complexes can be further assembled into supercomplexes. A functional OXPHOS pathway relies on the interaction of two genomes. The mitochondrial DNA (mtDNA) encodes 13 of the OXPHOS proteins plus 2 rRNAs and 22 tRNAs required for translation of the mtDNA genome. The nuclear genome encodes hundreds of proteins essential for OXPHOS function. These include subunits and assembly factors for each complex plus all the machinery required for import, processing and quality control of mitochondrial proteins, replication and expression of mtDNA as well as synthesis of redox centers, electron carriers, cofactors, nucleotides and maintaining an appropriate membrane milieu for stability of the OXPHOS complexes. Given this genetic complexity, it is not surprising that mitochondrial diseases are now known to comprise at least 200 monogenic diseases.

3. Mitochondrial diseases encompass a very large number of conditions with a huge clinical spectrum, with signs affecting any tissue in the body, with onset at any age, and with any type of inheritance. The functional, structural and genetic complexity of OXPHOS directly explains the wide diversity of mitochondrial diseases. In most cell types, in particular neuronal and muscle cells, OXPHOS is producing more than 90% of the cellular ATP. In parallel it has central roles in the maintenance of the cellular redox potential, inner membrane potential and reactive oxygen species metabolism.

4. Many mitochondrial diseases are multisystemic, involving several tissues unrelated by functional or embryological link. However tissue-specificity is also frequent and the pathophysiological mechanisms remain largely hypothetical. In mitochondrial diseases affecting a single tissue, demonstration of the mitochondrial defect can be difficult because most histological and enzymatic analyses are made in muscle, which may be spared by the disease. In addition a mitochondrial defect, even severe but affecting a tissue with minor mass, will have little influence on the global metabolic status that is analysed in blood and urine samples.

5. The initial steps in investigating possible mitochondrial disease involve assessment of family history and clinical features, with a focus on recognizing classical mitochondrial syndromes and multisystem disease, together with consideration of possible alternative diagnoses. In most cases, the next steps involve non-invasive studies on body fluids to detect any alteration in cellular redox ratio or accumulation of substrates upstream of the deficient metabolic step, such as lactate, pyruvate or intermediates of the Krebs cycle. Together with imaging, these investigations can guide whether specific single gene tests, gene panels or whole exome sequencing should be performed or if the patient should proceed to tissue biopsies.

6. Traditionally, most patients with suspected mitochondrial disease have been investigated by performing a tissue biopsy, which would ideally be done on an affected tissue. Skeletal muscle biopsy is by far the most common tissue biopsied for investigations, of which there are two major categories. The first is histological analyses, in which tissue sections are studied by a series of stains to identify abnormal mitochondrial numbers, decreased activity of OXPHOS complexes II and IV, or in some cases abnormal mitochondrial morphology detected by electron microscopy. The second major category of testing is spectrophotometric enzyme assays to determine the activity of OXPHOS complexes. A number of other diagnostic tests are used less commonly, such as respirometry and determining the size and amount of each complex by immunochemical approaches. While histological and enzyme tests can provide definitive results, they lack high sensitivity and specificity. This primarily results from the issues of genetic complexity, tissue specificity and the expectation that most defects will be partial as complete OXPHOS defects would cause embryonic lethality. For example, most children with mitochondrial disease lack clear histological abnormalities and many OXPHOS enzyme defects have substantial residual activity, which can overlap with the normal range. The membrane localization as well as the structural and functional complexity of the OXPHOS enzymes also contribute technical difficulties to the assessment of OXPHOS function.

7. Genetic diagnosis of mitochondrial diseases is complicated by the large number of candidate genes, located in the nuclear chromosomes or the mtDNA. An additional challenge is brought by the quantitative aspects of mtDNA alterations. Identification of the alteration alone is not sufficient for diagnosis but has to be complemented by its quantification because a low proportion of a heteroplasmic pathogenic point mutation or deletion can be observed in asymptomatic individuals. In addition the mtDNA copy number per cell can be altered, either abnormally decreased (mtDNA depletion syndromes) or increased (in parallel with mitochondrial proliferation). Traditionally, genetic testing for mitochondrial disease was guided by the clinical features, family history and biopsy results. Increasingly though, the use of gene panels or whole exome sequencing is becoming a front-line diagnostic test. If a clear genetic diagnosis is found, then the patient is spared the need for a tissue biopsy and if no diagnosis is found, biopsies can be performed to obtain additional information that may guide re-analysis of sequencing results. Prenatal diagnosis and preimplantation strategies for heteroplasmic mtDNA mutations can be offered to some families. However, they are faced with difficulty in predicting the outcome of intermediate mutant loads and some uncertainty with respect to mtDNA segregation during embryogenesis and postnatal life.

8. In conclusion the diagnosis of mitochondrial diseases differs from that of most other metabolic diseases due to the highly diverse clinical presentations, the lack of specific/constant metabolic hallmarks, the potential for false positive and false negative results from histology and enzyme analyses, the large number of candidate genes, and the quantitative aspects of mitochondrial DNA genetics. As such it represents an extraordinary diagnostic challenge whose chances of success are significantly increased when it can rely on reciprocal and easy communication between the diagnostic laboratory, the physicians involved in the care and metabolic investigation of the patients and research groups essential for functional validation of novel genetic causes. In the absence of a gold standard diagnostic test, it is helpful to use diagnostic criteria that attempt to define the certainty of diagnosis based on objective assessment of evidence from multiple independent sources.

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# Covariant commutation relation in Mandl and Shaw

Hi,

I am trying to study Quantum Field Theory by myself from Mandl and Shaw second edition and I am having trouble understanding the section on covariant commutation relations. I understand the idea that field at equal times at two different points commute because they cannot "communicate" with each other. This is where the equal time commutation relation comes in. However, when one takes field at two different "space-time" points, things might be different. I am having trouble following the Mathematics in Mandl and Shaw. I can write the delta function as (3.43),

Δ (x) = (-c/(2π)3) ∫(dk3 sin kx)/wk

I can also see how the equal time commutation relation comes from this. But how do they go from this to the four dimensional integral in k (3.45),
Δ (x) =(-i/(2π)3)∫(dk4 δ(k2 - μ2)ε(k0)e-ikx

Also, how can I see that the four dimensional integral is Lorentz invariant and how does Lorentz invariance implies that integral is zero for any space-like interval.

Any discussion will be much appreciated.

tiny-tim
Homework Helper
welcome to pf!

hi shahbaznihal! welcome to pf!
… how do they go from this to the four dimensional integral in k (3.45),
Δ (x) =(-i/(2π)3)∫(dk4 δ(k2 - μ2)ε(k0)e-ikx
(that's viewable online at http://books.google.co.uk/books?id=...M&hl=en#v=onepage&q=easily established&f=true )

have you tried substituting for the δ-function, and then integrating, as suggested in (3.47)?
Also, how can I see that the four dimensional integral is Lorentz invariant and how does Lorentz invariance implies that integral is zero for any space-like interval.
(3.45) is obviously lorentz covariant … every bit of it is lorentz covariant

and the standard way of proving it's zero is to choose the frame in which k = (ko,0,0,0) and calculate it there!

(you can't find such a frame if they can "communicate"! )

EDIT: there's a similar calculation, which you might find easier, at p.274 of Weinberg's "Quantum Theory of Fields, Vol. I", see http://books.google.co.uk/books?id=...epage&q=calculation of the propagator&f=false

Last edited:
Thanks for the online reference of Mandl and Shaw.

So as a general principle can I say that if the integrand is Lorentz invariant, the integral will also be Lorentz invariant? It will be nice to actually see this proved. Any idea how I can proceed to proving it?

Yes, I tried doing that k0 integration but I cant get the original dk3 integral back. I checked other books in my library and found some useful references. I will try integrating it again and will come back.

I can prove the equal time commutation relation but I am having trouble seeing how the equal time commutation relation implies that commutation is zero for every space-like interval.

Bill_K
So as a general principle can I say that if the integrand is Lorentz invariant, the integral will also be Lorentz invariant?
Provided the domain of integration is also Lorentz invariant. In the present case the domain is all of k.
Yes, I tried doing that k0 integration but I cant get the original dk3 integral back.
δ(k02 - ωk2) = (1/2ωk)[δ(k0 + ωk) + δ(k0 - ωk)]
You get two very similar terms from this, one with k0 = ωk, the other with k0 = - ωk. They combine to give you sin.
I can prove the equal time commutation relation but I am having trouble seeing how the equal time commutation relation implies that commutation is zero for every space-like interval.
It's the Lorentz invariance that does it. If it's zero for one point outside the light cone, then by Lorentz invariance it's zero for every point outside.

Last edited:
Okay, so I wrote down the original delta equation in terms of delta functions (please see attached picture), but I do not have common exponential to take out of the equation.

If I can just take the negative exponential common from the two factors I will get the equation. Kindly help me where I am going wrong, I can see that this should work Physically but I am able to see it Mathematically.

It's the Lorentz invariance that does it. If it's zero for one point outside the light cone, then by Lorentz invariance it's zero for every point outside.
So if we show that the commutation is zero at Δt = 0 (a space like event) then any other space like event will give you a zero commutation because Lorentz transformation preserves space-like events.

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• 25.7 KB Views: 354
I think I am only confused because the two integrals in k have different arguments in e (one is positive while other is negative) and I cannot find a way to show that they can be combined in single exponential with negative argument as it is written in the later equations. I believe it has something to do with integration running on all values of k. Is that the case?

Last edited:
Guys, tiny-tim Bill_K.....

strangerep


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# Find the truth table for the circuit shown in the figure. What logic function will the circuit perform if the constant +5 V input to the first two gates is changed to ground potential?

Solution :

a. Truth table for the given circuit of logic function:

Y = $$[\overline{\overline{5V.V_{A}}.\overline{5V.V_{B}}}]$$

Y = VA + VB

Y = OR operation

Inputs(V)

 VA VB +5V 0 0 5 5 0 5 0 5 +5 +5 +5 +5
Intermediate(V)

 5V.VA 5V.VB $$\overline{5V.V_{A}}$$ $$\overline{5V.V_{B}}$$ 0 0 5 5 0 5 0 5 +5 +5 0 0 0 5 0 5
Output(V)

 $$\overline{\overline{5V.V_{A}}.\overline{5V.V_{B}}}$$ 0 5 5 5


text:
## limit of two variable function at zero

How can I show that the limit of the following function when the variables {x,y} tend to zero, is zero? Mathematica gives the error “{x,y} is not a valid variable”.

Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] - y Cosh[x/Sqrt[y]]
Limit[Cosh[1/Sqrt[y]] Sinh[x/Sqrt[y]] -    y Cosh[x/Sqrt[y]], {x, y} -> {0, 0}]

Any comment is welcomed.

## Is there an official treasure generation method to limit magic item rolls based on dungeon level or some other factor?

I’m running an AD&D campaign for a party of usually-three PCs, who were first level until our most recent session. (As for what they are now, we’ll get to that…) I have the 1e DMG (door cover) and Unearthed Arcana, and a Monster Manual that might be older than that, judging by its condition. The players are using the 2e PHB; these are all inherited books, and the previous owner only ever DM’d in 1e and PC’d in 2e.

My issue is with treasure generation– I’ve been using the standard dungeon generation tables from the DMG, and it works well except for the outcome of treasure rolls. Specifically, magic items don’t seem to be segregated by dungeon level. That first-level party happened upon a Mirror of Mental Prowess, which had some fairly powerful effects but nothing game-breaking, and was worth five thousand experience. Divided among the party, this alone was enough to bring the priest and rogue to second level. Combined with the remainder of the treasure, those two reached level three, and the ranger reached level two.

Now building a dungeon for a later adventure, another magic item roll came up, resulting in… a Ring of Three Wishes. I simply vetoed that and re-rolled, getting something more reasonable this time, but now the question is in my mind of whether this is actually correct.

So, the simple version of the question:
Is there a method in AD&D to limit magic item rolls for treasure based on dungeon level or some other factor, or does this need to be created manually by the DM?

Note that this is not the same question as “What can I do when I accidentally gave out an overpowered item?” This relates purely to the RAW methods for generating magical treasures.

## Does the Slow spell limit the number of Eldritch Blast beams?

This question comes from something stated in an answer to another one of my questions:

[Regarding eldritch blast] This spell allows you to make a single attack that targets multiple creatures, by making several attack rolls.

The slow spell states:

[…] Regardless of the creature’s abilities or magic items, it can’t make more than one melee or ranged attack during its turn […]

Note I am aware the slow spell also affects the casting of spells, so let’s assume a caster always succeeds on the d20 roll.

And then there is the eldritch blast spell which states:

A beam of crackling energy streaks toward a creature within range. Make a ranged spell attack against the target. On a hit, the target takes 1d10 force damage.

The spell creates more than one beam when you reach higher levels: two beams at 5th level, three beams at 11th level, and four beams at 17th level. You can direct the beams at the same target or at different ones. Make a separate attack roll for each beam.

Does the slow spell prevents you from creating more than one eldritch blast beam?

## What is the lowest level at which a human can beat the 100m world record (or: the presumed human limit) without using magic?

Optimizing a character build for fast movement in order to break the sound barrier is a well-tried source of fun with numbers in DnD. Such speed optimizations usually incorporate lots of magic and possibly additional help by allies. But I’d like to learn more about what the human body is capable of within the rules of DnD, so I’m more interested in what can be achieved without magic.

A high level Monk with two levels in Fighter and the Mobile Feat can still become ridiculously fast. If I didn’t mess up the calculations, he can run 100m in about 7.52 seconds: Use Action Surge and Step of the Wind for triple dashing to run 84m in 6 seconds, then double dash in the next round for the remaining 16m. This is even almost 2 seconds faster than 9.4s, which scientists think is the human limit (in our world, that is). But hey, it’s a Level 20 character, so becoming somewhat superhuman does kind of make sense in the world of DnD, I guess.

This raises the question: At what point in the game can human adventurers become superhuman without using magic? This is of course a more vague and general question (one could ask the same e.g. in the context of lifting strength), but let’s stick to running speed for the moment. So, my question in precise terms is:

What is the minimum level at which a human character can run 100m in less than 9.58 (or: 9.4) seconds and thus break the current world record (or: become superhuman)?

Constraints:

• No magic.
• No epic boons.
• No UA (but any official rule book).

Let’s use the same approach as above, i.e. Mobile Feat + k squares Unarmored Movement + Action Surge + Step of the Wind. Since a round is 6 seconds long and a square corresponds to 1,5m, this strategy lets you run 100m in

$$t(k)=6+\frac{100-(48+6k)}{6+\frac34 k}$$

seconds. Plugging in the possible values of k yields

$$t(2)\approx 11.333, \quad t(3)\approx 10.121, \quad t(4)\approx 9.111, \quad t(5)\approx 8.256, \quad t(6)\approx 7.524.$$

Hence, a Monk 9 / Figher 2 is still more than half a second away from beating the world record, while gaining another Monk Level already makes her superhuman at a total Character Level of 12. Did I miss something? Or can one even do better with an altogether different strategy?

PS: A quick comment on the tricky question about Ki being magic: Since Ki is only used for Step of the Wind, we could add two Rogue Levels and achieve the same by taking the Cunning Action instead. However, this would raise the current minimum level to 14.

## Difference between max length of the record vs record size limit

As per RFC 5246 (TLS) the max possible length of the record is 2^14.

But I could see now a new parameter “Maximum record size limit”.

Why do we have the new parameter “Maximum record size limit” since RFC 5246 already mentions Max possible length is 2^14

## Do PCs have a volumetric carry capacity limit?

In the Player Starting Guide (a summary of the PHB) it says that characters can carry an amount of Strength Value × 15 = weight limit in pounds. But later, in the gear section, there is a table about Container Capacity (usually cubic feets) of backpacks, chests, barrels, vials, etc, even about backpacks it says:

You can also strap items, such as a bedroll or a coil of rope, to the outside of a backpack.

So… are PCs limited by carry weight and by volume?

If that is true, how much initial volume can a character carry?

Is volume tracked per storage item or all as a whole (I mean, if I have an item which size 2 ft3 and two bags with 1 ft3 each one, can I carry that item or not?)?

## How limit connection on webserver

Hey
Centos + Apache + Nginx(as proxy)
this configuration is what we have at this time and try to test our security with aconetix,

## SharePoint threshold limit Not working for indexed Columns

I am Facing A issue with SharePoint Rest Query Requestype Column is indexed , single Line of text , Has Three types of only (Values and Count)

• Credit- 3787
• One Off – 13019
• Reccuring – 6202

I am Trying to retrieve only Credit type but it is failing and giving threshold error. I need to retrieve all credit requests Query : https://yy.xx.com/sites/xxxxy/_api/web/lists/GetByTitle(‘XYZ’)/Items?$select=ID&$ filter= ( RequestType eq ‘Credit’ )

Thanks For helping me solve the issue.

## Is this computation of Limit beyond Mathematica, or am I missing a trick?

I am surprised the following computation stumps Mathematica:

Limit[(2^((1 - n)/2) x^(-1 + 1/2 (-1 + n)) E^(-((x n)/(2 V))) (V/n)^((1 - n)/2))/Gamma[1/2 (-1 + n)], n -> \[Infinity]]

Am I doing something wrong, or am I missing something that can assist Mathematica?

## Script not working inside of folder possible 5000 item limit issue

I asked a question about a month ago here Script Inside of Folder, and recieved a great answer from Rothrock. This worked for about a month until the document library exceeded 5000 items (or at least I believe this was the trigger that caused the script to stop working).

My hypothesis is that the query

caml += "<View Scope=\"RecursiveAll\">";

is unable to execute correctly because of the 5000 item limitation.

I think that the query needs to be modified such that it looks solely at the folder within which the file is being uploaded.

I’ve looked around for ways for this query to be modified, I found a thread that seems to confirm my thinking that this is the root issue. However I am not versed enough in Javascript to know how to fix the issue in question.

The modified script per the original suggestion is at the bottom of the page. If I change the script back to

caml += "<View>";

it will again execute properly at the Document Library level, but it still fails within a folder.

Thanks for any help or guidance!

<script type="text/javascript"> (function (_window) {     var maxTimeForReplaceUploadProgressFunc = 10000;     function replaceUploadProgressFunc() {         if (typeof _window.UploadProgressFunc != 'undefined') {             _window.Base_UploadProgressFunc = _window.UploadProgressFunc;             _window.UploadProgressFunc = Custom_UploadProgressFunc;             console.log('replaced dialog');         } else if (maxTimeForReplaceUploadProgressFunc > 0) {             maxTimeForReplaceUploadProgressFunc -= 100;             setTimeout(replaceUploadProgressFunc, 100);         }     }     setTimeout(replaceUploadProgressFunc, 100);       function Custom_UploadProgressFunc(percentDone, timeElapsed, state) {         _window.Base_UploadProgressFunc(percentDone, timeElapsed, state);         var messageType = ProgressMessage.EMPTY;         switch (state.status) {             case 1:                 messageType = ProgressMessage.VALIDATION;                 break;             case 3:                 messageType = ProgressMessage.UPLOADING;                 break;             case 4:                 messageType = ProgressMessage.UPLOADED;                 OpenEditFormForLastItem(state);                 break;             case 5:                 messageType = ProgressMessage.CANCELLED;                 break;         }          function OpenEditFormForLastItem(state) {             var caml = '';             caml += "<View Scope=\"RecursiveAll\">";             caml += "<Query>";             caml += "<Where>";              if (state.files.length > 1) {                 caml += "<In>";                 caml += "<FieldRef Name='FileLeafRef'/>";                 caml += "<Values>";             } else {                 caml += "<Eq>";                 caml += "<FieldRef Name='FileLeafRef'/>";             }              state.files.forEach(function (file) {                 //only succesfull uploaded files that arent overwrites                 console.log(file);                 if (file.status === 5 /*&& !file.overwrite*/) {                     caml += "<Value Type='File'>" + file.fileName + "</Value>";                 }             }, this);              if (state.files.length > 1) {                 caml += "</Values>";                 caml += "</In>";             } else {                 caml += "</Eq>";             }              caml += "</Where>";             caml += "<OrderBy><FieldRef Name='ID' Ascending='True' /></OrderBy>";             caml += "</Query>";             caml += "<ViewFields><FieldRef Name='ID' /></ViewFields>";             caml += "<RowLimit>500</RowLimit>";             caml += "</View>";             console.log(caml);              var cntxt = SP.ClientContext.get_current();             var web = cntxt.get_web();             var list = web.get_lists().getByTitle(window.ctx.ListTitle);             var query = new SP.CamlQuery();             query.set_viewXml(caml);             var items = list.getItems(query);             cntxt.load(list, 'DefaultEditFormUrl');             cntxt.load(items);             cntxt.executeQueryAsync(function () {                 var listEnumerator = items.getEnumerator();                 function openEditForItem() {                     if (listEnumerator.moveNext()) {                         var item = listEnumerator.get_current();                         var id = item.get_id();                          var options = SP.UI.\$  create_DialogOptions();                         options.title = "Add File Metadata";                         options.url = list.get_defaultEditFormUrl() + '?ID=' + id;                         options.autoSize = true;                         options.dialogReturnValueCallback = openEditForItem;                         SP.UI.ModalDialog.showModalDialog(options);                     } else {                         location.reload();                     }                 }                 openEditForItem();             }, function (error, args) {                     console.log("failed to get new uploaded items");                     console.log(error);                     console.log(args);                 });         }     } })(window); </script>


text:
3 added 404 characters in body

José is correct in his comment. Just to elaborate: in the linear case, one can easily study the equation using Fourier methods. Let $\tilde{u}$ denote the space-time Fourier transform and $\hat{u}$ denote the spatial Fourier transform, the equation with $p=1$ can be written as $$(\tau - \xi^3 + \xi)\tilde{u} = 0$$ or $$\partial_t\hat{u} = i(\xi^3 - \xi)\hat{u}$$

The first formulation tells you that the space-time Fourier transform of a solution is a measure supported on the curve $\tau = \xi^3 - \xi$ in frequency space. That the frequency support has curvature implies that the solution should decay in time in physical space, justifying José's comment. (Look up Fourier restriction theorems or Strichartz estimates in the literature for more details.)

We can also see the temporal decay directly from the second formulation using oscillatory integral techniques. The solution can be written as $$\hat{u}(t,\xi) = e^{i(\xi^3 - \xi)t}\hat{u}_0(\xi)$$ It is then a standard exercise to show that, given initial data of sufficient decay in frequency space (say, the frequency has compact support), taking the inverse Fourier transform of the above solution gives you something with decay in $L^\infty$.

If a solution decays in $L^\infty$, it cannot be a soliton.

We can also see the lack of solitons by posing the traveling wave ansatz $u(t,x) = f(\omega t + x)$. A simple computation shows that the function $f$ must solve $$-(\omega + 1)f' = f''' \implies f = \exp [ i \sqrt{\omega + 1}(\omega t + x) ]$$ so that traveling waves cannot be spatially localized. In fact, the traveling wave ansatz gives us a dispersion relation for this equation: that waves of spatial frequency $\beta$ travels with velocity $\omega = \beta^2 - 1$. The fact that different frequency components of the wave tend to travel at different velocities illustrates why, starting with a pulsed wave packet, the solution will become wider and wider while its height gets smaller and smaller.

In short, I think the reason why this equation is not heavily studied in the literature is because that as a linear PDE in (1+1) dimensions, it is not really all that interesting to look at.

José is correct in his comment. Just to elaborate: in the linear case, one can easily study the equation using Fourier methods. Let $\tilde{u}$ denote the space-time Fourier transform and $\hat{u}$ denote the spatial Fourier transform, the equation with $p=1$ can be written as $$(\tau - \xi^3 + \xi)\tilde{u} = 0$$ or $$\partial_t\hat{u} = i(\xi^3 - \xi)\hat{u}$$

The first formulation tells you that the space-time Fourier transform of a solution is a measure supported on the curve $\tau = \xi^3 - \xi$ in frequency space. That the frequency support has curvature implies that the solution should decay in time in physical space, justifying José's comment. (Look up Fourier restriction theorems or Strichartz estimates in the literature for more details.)

We can also see the temporal decay directly from the second formulation using oscillatory integral techniques. The solution can be written as $$\hat{u}(t,\xi) = e^{i(\xi^3 - \xi)t}\hat{u}_0(\xi)$$ It is then a standard exercise to show that, given initial data of sufficient decay in frequency space (say, the frequency has compact support), taking the inverse Fourier transform of the above solution gives you something with decay in $L^\infty$.

If a solution decays in $L^\infty$, it cannot be a soliton.

We can also see the lack of solitons by posing the traveling wave ansatz $u(t,x) = f(\omega t + x)$. A simple computation shows that the function $f$ must solve $$-(\omega + 1)f' = f''' \implies f = \exp [ i \sqrt{\omega + 1}(\omega t + x) ]$$ so that traveling waves cannot be spatially localized.

In short, I think the reason why this equation is not heavily studied in the literature is because that as a linear PDE in (1+1) dimensions, it is not really all that interesting to look at.

1

José is correct in his comment. Just to elaborate: in the linear case, one can easily study the equation using Fourier methods. Let $\tilde{u}$ denote the space-time Fourier transform and $\hat{u}$ denote the spatial Fourier transform, the equation with $p=1$ can be written as $$(\tau - \xi^3 + \xi)\tilde{u} = 0$$ or $$\partial_t\hat{u} = i(\xi^3 - \xi)\hat{u}$$

The first formulation tells you that the space-time Fourier transform of a solution is a measure supported on the curve $\tau = \xi^3 - \xi$ in frequency space. That the frequency support has curvature implies that the solution should decay in time in physical space, justifying José's comment. (Look up Fourier restriction theorems or Strichartz estimates in the literature for more details.)

We can also see the temporal decay directly from the second formulation using oscillatory integral techniques. The solution can be written as $$\hat{u}(t,\xi) = e^{i(\xi^3 - \xi)t}\hat{u}_0(\xi)$$ It is then a standard exercise to show that, given initial data of sufficient decay in frequency space (say, the frequency has compact support), taking the inverse Fourier transform of the above solution gives you something with decay in $L^\infty$.


text:
# Covering Systems of infinite sets of residue classes mod primes

Take an infinite set of distinct primes and a (edit: or 2 , etc.) residue class for every prime. For exammple you can take all the primes bigger than some prime or the primes of a specific form (i.e. of some arithmetic progression or whatelse)

The question is under what circumstances every natural number is of some of this residue classes?

Of course there are conditions that the answer is easy(for example if you take as residue class for a prime onthe set to be $1modp1$ , $2modp2$), but do we have any well known general results on this direction? (On infinite sets of residue classes etc.)

Note:if you think that the question is not very specific please ask me to be more.

EDIT: to be more precise i will give an example : Take all the primes from a point and after ,split the natural numbers on intervals of a given length ,say A , and for each prime is not allowed to "hit" a number that is in lower interval than itself,(i.e take all the primes greater or equal than 7 and A=100)

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If you know how to pick the infinite set of primes, you are infinitely rich. :-) Jokes aside, your question isn't formulated well. Vote to close. – Wadim Zudilin Jan 15 '11 at 11:50
What do you mean there are not such covering systems at all ,for trivial reasoning .Is there something that is not well written?Or else please give the trivial reasoning. – asterios gantzounis Jan 15 '11 at 11:51
If you choose residue zero for all the primes in your set then there is an infinite set of integers you don't hit (multiples of all the other primes). If p is your smallest prime, then you need 1, 2 ... (p-1) as explicit residue classes. I think also that if the sum of the reciprocals of your infinite set is infinite, a random selection of residue classes will almost certainly hit each integer (uniform for each prime). It is easy to construct sets of residues which miss any finite set of numbers with cardinality less than the smallest prime (smallest residue which misses for each prime). – Mark Bennet Jan 15 '11 at 12:04
I needed more than 10 minutes just to understand your question! ;-) I'll definitely state it simpler and shorter. But I would need more time ($\infty$?) to understand what's use of all this. What kind of criteria do you expect? Pick a number $n\in\mathbb N$, check whether it belongs to a certain residue class. If yes, take $n+1$, and so on. There can't be a finite procedure to answer your Q. – Wadim Zudilin Jan 15 '11 at 12:07
I asked for known results on this direction – asterios gantzounis Jan 15 '11 at 12:13

If you insist on prime moduli you will need infinitely many classes. If you don't insist on prime moduli then you can get away with finitely many, these are called covering systems and the linked article looks good.

Let $(r,m)=\lbrace r+mj \mid j \ge 0 \rbrace$ and $p_1<p_2<p_3<\cdots$ be a list of some or all the primes. You want a list $(r_1,p_1),(r_2,p_2),\cdots$ whose union is the natural numbers (optinally with some natural numbers allowed to be missed). One method to do this is to greedily define $r_i$ to be the smallest integer not in $\cup_{j<i}(r_j,p_j)$. Is this (almost) optimal? I don't know but I will wildly guess that it is and from now on only discuss the greedy choice. If we use all the primes and try to get all the natural numbers starting with 2 then (making the greedy choice) $r_j=p_j$. This becomes $r_j=p_j-1$ (or $r_j=p_j-2$) if we also want to get 1 ( or 0 and 1). In any case the greedy algorithm gives $\frac{r_i}{p_i} \approx 1$ (Well in the very first we could as well have $r_i=0$)

If we use all the primes except $2$ and try to get all the natural numbers except powers of $2$ (and 0) then again $r_i=p_i$. If we try to get all the natural numbers then it would appear that $\frac{r_i}{p_i} \approx \frac{1}{2}$.

Based on this I will say that there is little hope that you can exclude the primes 2,3,5 (or even just 2) and keep $p_i-r_i<100$. I suspect that for each prime $q$ there is a constant $C_q$ so that using the greedy choice to cover all the natural numbers using primes $q$ and greater results in $\frac{r_i}{p_i} \approx C_q$. It looks as if perhaps $C_7$ is something like 0.43.

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thank you for the link – asterios gantzounis Jan 15 '11 at 11:28
First of all i asked for mod primes which is known of chinese remainder theorem that for every finite system is not covering system. I asked for infinite systems.Is there any result that says under what circumstances? – asterios gantzounis Jan 15 '11 at 11:38
Sorry i was writing when you edit and didnt see it. – asterios gantzounis Jan 15 '11 at 11:40
things like that and others when you do not use all the primes, especially i am interested when you use the primes biger than some and you demand each prime to sieve from a number and after ,etc. – asterios gantzounis Jan 15 '11 at 12:01
Could you write a proof for this? – asterios gantzounis Jan 17 '11 at 13:43