Theoretical Analysis of Flow Characteristics and Bearing Load for Mass-produced External Gear Pump

Size: px
Start display at page:

Download "Theoretical Analysis of Flow Characteristics and Bearing Load for Mass-produced External Gear Pump"

Transcription

1 TECHNICAL PAPE Teoretical Analysis of Flow Caracteristics and Bearing Load for Mass-produced External Gear Pump N. YOSHIDA Tis paper presents teoretical equations for calculating pump flow rate and bearing applied load wic are important caracteristics witin external gear pump design. In te external gear pump mass-produced at JTEKT, te eccentricity of te gear ousing greatly affects its caracteristics. Taking tis fact into consideration, I conducted teoretical analysis of leakage troug gear tip clearances. Tis paper describes te equations for te pysical model and te analytical solutions of te equations as well as te numerical calculation metod for calculating te eccentricity position. It also contains a comparison of te values calculated using my proposed teoretical equations wit previous teoretical values and te experimental values. Furtermore, tis paper sows examples of calculated results for te teoretical optimum conditions for te eccentricity position. Key Words: fl uid power system, ydraulic pump, external gear pump, leakage fl ow rate, bearing load, teoretical analysis 1. Introduction Te external gear pump composed of circumscribing twin gears as been used for various macines, suc as industrial equipment and automobiles, since ancient times. Its flow rate performance is very ig in comparison wit oter gear type ydraulic pumps, as it can be operated at ig pressures over 10 MPa. Terefore it as an advantage in regards to energy saving and device downsizing. JTEKT as been mass-producing te external gear pump driven by an electric motor for te automobile ydraulic power steering system known as H-EPS (Hydraulic-Electric Power Steering), and opes tat it will be used in te future for automotive driveline systems requiring a ig pressure ydraulic power source. In te external gear pump suc as tat mass-produced by our company wic is described in tis paper, it is known from experience tat te eccentricity of te gear ousing greatly influences pump performance and tat te delivery flow rate performance increases as te eccentricity increases to a certain extent. Te eccentricity state varies due to te following two reasons. One is dispersion of te parts dimensions caused by macining. Te oter is te use of journal bearings, wic are low in cost. Due to te insufficient supporting rigidity of te journal bearing, te rotating center position of te pump gear varies according to te operating conditions (especially delivery pressure). Previous studies 1)-3) took no account of te influence of te eccentricity of te gears witin te detailed teoretical analysis of te flow rate caracteristics of te pump, wit te exception of Icikawa s study 4), wic analyzed it teoretically wit simple equations. Icikawa s analysis is straigtforward and its equations can be calculated easily. However, it is insufficient in accuracy because te equations used were simple, and matematical approximations were applied under several assumptions. Additionally, altoug te eccentricity direction is considered to be te same as te load direction in is analysis, it is not so in our pump wit journal bearings. Terefore, previous teoretical analyses are not appropriate for pumps suc as our external gear pump. For te purpose of obtaining teoretical equations to accurately represent te caracteristics of te delivery flow rate and bearing load for an actual pump, I ave developed a new teoretical analysis of te leakage troug te clearances between gear tips wit eccentricity and te ousing, and te pressure state wic is distributed along te gear circumference. Tis new teoretical analysis is useful in te prediction of te optimization of flow rate performance and its dispersion according to te tolerance of te dimensions, as well as te prediction of bearing load wic is very important witin te design stage of rotary macines. It also as te ability to elucidate pysical mecanisms wic significantly influence pump caracteristics and contribute to te development of superior new products. JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016) 45

2 2. Nomenclature D D D D a e e g l e D : face widt of gear (axial lengt) : central direction force acting on gear circumference (gear tips and toot spaces) : reaction force of journal bearing wic supports saft : force of engagement between two gears : amount of clearance between gear tip and ousing (gear tip clearance) : fundamental amount of gear tip clearance (clearance between gear tips and ousing witout eccentricity) : circumference lengt of gear toot addendum (gear tip lengt) : rotational speed of gear : pressure at engagement closing : pressure of fluid in toot space : pressure of fluid in gear tip clearance : pressure differential across pump : cange amount of pressure across gear tip clearance : actual pump flow rate : total leakage flow rate in pump : leakage flow rate for single gear : radius of gear addendum : radius of ousing internal circumference : radial distance of position of gear engagement contact point : teoretical torque of pump : relative velocity of one wall of gear tip clearance : fluid velocity in gear tip clearance : teoretical pump displacement per revolution : number of teet in gear : number of sealed teet in gear : pressure angle of contact point : eccentricity ratio of gear center to ousing center : eccentricity ratio of saft center to journal bearing center : volumetric efficiency of pump : fluid viscosity : angle of eccentricity direction of gear center to ousing center : angle of eccentricity direction of saft in journal bearing : angle of inlet side edge of gear tip clearance : angle of point were distance between te teet surfaces in te engagement closing space is smallest : angle of contact point : amount of angle range of gear tip clearance in circumference direction d : distance between gear center and ousing center (eccentricity amount) Subscripts : place number of te gear tip clearance or toot space witin te seal area from te inlet space side : direction component of force : direction component of force o : central direction component of force 3. Structure of pump Journal Bearing Saft Side Plate Drive Gear Driven Gear Housing Inlet Outlet : Te transport route of te fluid Fig. 1 Structure of external gear pump manufactured by JTEKT Te structure of te external gear pump studied in tis paper is sown in Fig. 1. Te gears are assembled in te ousing ole, wic as an inner diameter sligtly larger tan te gear addendum diameter. During operation, te drive gear is driven by an electric motor and rotates, and te driven gear rotates by engagement wit te drive gear. Te gear saft is supported by a journal bearing. Working fluid (oil) is transported from te inlet space to te outlet space troug te toot spaces of te rotating gear via te route of te side opposite te gear engagement, as sown in Fig. 1. It is ten pressurized and discarged by te decreasing volumetric cange caused by te engagement of te gears. 46 JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016)

3 To understand te actual flow rate of te displacement type pump, it is necessary to take into account not only te teoretical displacement volume obtained from te geometric form and te rotational speed, but also te leakage flow rate. Te leakage flow rate refers to te part of te delivered flow rate wic flows backward into te inlet space. It is caused by leakage flow from te ig-pressure outlet space to te low-pressure inlet space troug gaps between te parts of te pump. Taking into account te leakage flow rate D, te volumetric efficiency representing te flow rate performance is defined by te following equation 5). D g 4. Teoretical analysis 4. 1 Equations for calculating leakage flow rate troug gear tip clearances wit eccentricity Figure 2 sows te designated dimensions and definitions of te coordinates (circumference direction position and te Cartesian coordinates and ). Here, te directions of te angle of circumference position and gear rotation are te same, and te positive direction of te leakage flow rate is te opposite direction. Eac toot and toot space in te sealed area between te gear tips and te ousing ave been given consecutive numbers starting from te inlet space side sown in Fig. 2. In tis analysis, te inlet pressure is defined as 0. Wen eccentricity of te gear ousing arises (eccentricity ratio e and eccentricity direction e ), te gear tip clearance between te gear tip and te ousing inner diameter canges according to position angle. Wen te pump operates (i.e. wen te gear rotates), during te state in wic oil as filled te pump space, te outlet pressure increases relatively by only D in respect to te inlet pressure. Te gear tip clearance functions to seal oil leakage from te outlet side to te inlet side. In tis analysis, te pressure pulsation in te outlet space and te flow unsteadiness are considered to be negligible, and terefore static analysis is conducted wit consideration to a steady state. It is assumed tat pressure inside eac toot space is uniform, and tat tere is no inlet/outlet pressure loss at te edge of eac gear tip clearance and no cange in te viscosity and density of te fluid. 1) Equations for calculating leakage flow rate troug single gear tip clearance Toot space of te low-pressure side Toot space Gear tip Moving velocity of wall surface Leakage flow rate D Housing Toot space of te ig-pressure side Toot space D Fig. 3 Spatial configuration and fluid condition in te toot tip clearance D Direction of pump flow rate Gear addendum diameter d e Inlet space 0 Outlet space D Drive gear Driven gear Enlarged view of engagement Outer diameter of saft Inner diameter of journal bearing otational direction of te gear Toot space Gear tip clearance Housing inner diameter (sealing area) It is assumed tat, for te flow in te space of a single gear tip clearance as sown in Fig. 3, te wall surface of te ousing can be displayed as a linear plane as te curvatures of te gear and ousing are extremely large in respect to te gap amount. Wen a sealed toot exists in te position at te arbitrary angle, te gap amount at te gear tip can be approximated by te following equation, e e were te variables are as follows. d e Fig. 2 Scematic diagram of gear and ousing of pump (Cross section of pump saft) Closing space Te position in te flow direction at eac gap is expressed by te following equation. Te following equations are also introduced. D JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016) 47

4 p D D To consider te influence of te flow velocity in te gap caused by te gear rotation, it is assumed tat te gear wall surface moves in te direction sown in Fig. 3 at te constant velocity expressed by te following equation. p In tis space, te position in te flow direction is determined from te edge of te upstream side (inlet space side) in te flow passage sown in Fig. 3, and te gap amount canges according to. As te pressure is different witin eac toot space, te static pressure differential D exists between te two edges of te flow passage. In tis flow passage space, it could be considered tat te flow is laminar as te eynolds number is very low, due to te fact tat te gap amount is muc smaller tan te flow passage lengt. Furtermore, it is assumed tat te velocity component of te teet widt direction is negligible, since te flow passage widt is muc larger tan te gap amount. Furtermore, it is assumed tat te velocity component of te direction can be overlooked since te cange in te gap amount in respect to is extremely small. Te equation of motion, te boundary conditions and te equation of continuity for te state of pressure and velocity of a Newtonian fluid in suc a space are expressed by te following simultaneous equations. l D D By solving equation by te variable separation metod and integrating te solution into equation I obtained te following equation of te leakage flow rate, wic depends on te pressure differential D across bot edges of te gear tip clearance. D l D 2) Equation for calculating leakage flow rate in entire pump Te pressure differential D across te inlet space and te outlet space equals te total loss of pressure from te several sealing gear tip clearances. Terefore, te following equation was obtained. By solving equation for pressure differential D, te following equation was obtained. D l l D Substituting tis in equation and I solved for D, tus obtaining te following equation of te leakage flow per gear. D l D l l D D l D Te total leakage flow rate of te pump is expressed as D D, since tere are two gears Calculation of te bearing load force Tere are four kinds of pressure acting on te gear circumference, wic are divided by teir position ranges on te circumference. Toot space closed by te sealing internal diameter of te ousing p p D Gear tip sealed by te internal diameter of te ousing p p D Outlet space in te pump D ange of closing space formed by te engagement of two gears Next, I described te derivations of te above mentioned equations to 1) Equations for calculating pressure distribution in sealed areas I obtained te following equation on te pressure in eac toot space tat is closed by te sealed inner 48 JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016)

5 circumference of te ousing. D From tis, te following equation is obtained by substituting equation. D l l D D D Pressure distribution in te gear tip clearance is affected by a penomenon called te edge effect. As a result, pressure in te flow passage becomes to iger tan tat of bot edges due to te inflow of oil caused by te movement of te wall in te flow passage direction. Consequently, te pressure in te gap does not simply drop linearly in te flow direction. Te following equation obtained from te solution of equations and expresses te pressure distribution caused by te penomenon. l D l D Analytical solution of te integral calculus sown in te following equations sould be applied to te two kinds of definite integrals in equations, and obtained above 6),7), e e e e e e e e e e e e e e e were te domain of function is p p, as it is not continuous at p. Furtermore, bot a and b of equations and are constants. 2) Oter forces acting on gear circumference Te forces acting on te gear circumference excluding te distributed pressure in te area sealed by te ousing are as follows: te force due to te pressure D in te outlet space, te closing pressure, and te engagement force. Two closed spaces are formed by te gear circumference around te two contact points of te gears. Te volume of tese closed spaces canges due to gear rotation, owever te pressure inside can become to be iger tan tat of te pump outlet. Tis is due to te difficult compressibility of te fluid, and because a flow passage wit insufficient area forms between te two toot surfaces or in a notc of te side plate. Tis paper omits description of tis teoretical analysis. Wen two gears engage eac oter wit contact at one or two points as sown in Fig. 2, te engagement force acts in te circumference direction () and its magnitude is expressed in te following equation, were te torque is te torque received at eac engagement point. Te actual pump torque is expressed by te following equation, in wic several losses are not taken into account because te dominant factor is te act of ig-pressure pumping. D p Te central direction component of te engagement force is expressed by te following equation, were a is te engagement pressure angle of te gear. a Te following equations, wic present te component of te direction and tat of te direction in te Cartesian coordinates of te central direction component of te engagement forces, are derived from equations to. p D a p D a 3) Equations for calculating force of bearing load Te force applied to te bearing is te central direction component of te force acting on te gear circumference and can be calculated using te pressure distribution () expressed in equations to and te engagement forces expressed in and. Te direction component, te direction component and te magnitude of tese forces are expressed in te following equations. JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016) 49

6 p p It is obvious tat te bearing caracteristics of te external gear pump obey te previous establised teory. In tis paper, te bearing performance numerical values used in te calculation are referenced from a separate document 7) Metod for calculating pump caracteristics wit consideration to gear support by journal bearings From te aforementioned equations, te pump caracteristics can be calculated wen te rotational center position of te gear is known. However, wen using journal bearings, te position of te gear rotational center is not known, and must terefore be calculated. Tis is because bot te bearing load force vector (wic depends on pressure distribution) and te rotational center position of te gear (wic depends on te bearing load force vector) are dependent on eac oter. It is terefore necessary to conduct numerical iteration calculation to solve te equations, under te condition tat te bearing load force vector and journal bearing supporting force vector are equal. Figure 4 sows te flow cart of te calculation program., amount of eccetricity ) e D eac gear tip gap Pressure in eac toot space Determination of calculation conditions gap amount of eac gear tip leakage flow rate of pump closing pressure force of engagement D e saft in te journal bearing angle of eccentricity direction e centra direction force vector of gear ( ) Judgment: (, e Counter force vector from journal bearing ( ) ( ) Yes pump flow rate force of journal bearing load Fig. 4 Flow cart for calculating caracteristics of journal bearing support structure pump No 5. esults of calculations Table1 Geometry of actual gear pump (Typical values) l D 5. 1 Comparison wit values from previous teory A comparison between te calculated values of te volumetric efficiency according to te proposed teory and tose of te previous teory 4) is sown in Fig. 5. Te calculation conditions are te dimensions and operation conditions described in Table 1. As sown in Fig. 5, te values between te previous teory and te proposed teory ave very little difference only wen te eccentricity ratio is small. Also, te value of volumetric efficiency of te proposed teory differs greatly from tat of te previous teory wen te nonlinearity of its cange in te proposed teory, in regard to te eccentricity ratio, is large. An eccentricity ratio equal to 1 pysically contradicts a volumetric efficiency tat is not equal to 1, since tis refers to te occurrence of leakage. In te derivation of te equations in te previous teory, te fundamental equations are based on a teoretical equation based on constant gap amount, and iger order components in te equations are omitted in order to easily obtain solutions. Terefore, as sown in Fig. 5, approximate accuracy is poor and tere are large errors wen te conditions are not limited to a lower eccentricity ratio, or to a small nonlinearity of te cange in volumetric efficiency in regard to te eccentricity ratio. As a result, te proposed teoretical equations ave muc iger accuracy tan te previous equations in te case of a large eccentricity ratio. Te bearing load caracteristic is te same as well, since its calculation equations depend on te leakage flow rate. 50 JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016)

7 Volumetric efficiency of pump g Dotted line :Proposed teoretical vallue :Previous teoretical value e e, Fig. 5 Comparison of proposed teoretical values and conventional teoretical values 5. 2 Comparison between teoretical values and experimental values Figure 6 sows te results of a comparison between experimental values from an actual pump and te teoretical values. Typical pump conditions are described in Table 1. Te experimental values sown in Fig. 6 (a) are measured under te condition were te pump speed is constant and te pump outlet pressure as been canged by altering te outlet side pipe resistance. Te two tested pumps differ only in te dimensions wic influence te amount of gear tip clearances. Te values sow tat te calculation results of te pump flow rate using te proposed teoretical equations are larger tan te experimental one. Tis difference is derived from te fact tat te leakage occurs troug gaps oter tan tat of te gear tip on te actual pump. Inferring tat leakage troug oter gaps wit constant clearance besides te gear tip clearance is directly proportional to te pressure, it would be pysically justifiable tat tis difference increases wit an increase in pressure. To validate te accuracy of te teoretical value of only te leakage troug gear tip clearances, Fig. 6 (b) sows te investigated results obtained by canging te y direction position of te center of te journal bearing sown in Fig. 2 so tat only te gear tip clearances cange. On te axis of ordinate, te cange in leakage flow rate is plotted against te leakage flow rate wen te abscissa is at zero. Te leakage flow rate is calculated troug equation after solving for D and substituting te pump flow rate measured in te actual macine. According to tis grap, it is evident tat te teoretical value nearly coincides wit te experimental value. Tis proves tat te proposed teoretical equations can predict loss due to te leakage of te actual pump flow rate from te gear tip clearances. Pump delivery flow rate, L/min Cage amount of leakage flow rate, L/min Solid line Dotted line :Experimental measured value :Teoretical calculated value Tested pump No.1 No Pressure differential across a pump D, MPa Experimental Teoretical Discussion of teoretical optimum conditions Figure 7 sows te teoretical calculation results of te pump flow rate and te bearing load obtained by canging te eccentricity direction and te eccentricity ratio. Te pump flow rate caracteristic is expressed as te volumetric efficiency, and bearing load is expressed as te normalized amount wen te value of te eccentricity ratio is zero. Te reason tat te volumetric efficiency may be over 1 is tat fluid oil is dragged and transported by te gear tip outer surface due to te viscosity of te oil. Tese results indicate te following facts: te caracteristics of te flow rate are influenced greatly by te eccentricity ratio, but not so muc by te eccentricity direction, and te eccentricity direction is optimum at 180 degrees wen te eccentricity ratio is small. It is obvious tat te bearing load decreases wen eccentricity arises in te direction of te inlet space side. On te oter and, te bearing load increases significantly wen eccentricity arises in te direction around 180 degrees. Tis is because, wen sealing is conducted on te inlet space side, te direction components of te direction position of journal bearing, µm (b) Amount of cange due to component dimension affecting toot tip clearance Fig. 6 Comparison of experimental values and teoretical values of pump flow rate JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016) 51

8 forces acting due to te pressure distributed along te circumference of te gear cancel eac oter out in te second and tird quadrants of te coordinate space in Fig. 2, and te direction components counteract te closing pressure and engagement force and decrease. In addition, te bearing load increases wit an increase in te eccentricity ratio because te pressure is distributed unevenly. Taking into account tat te life of te rolling element type bearing is generally inversely proportional to approximately te cube of te load 8), attention must be given during design because even a cange in te load as small as tat in Fig. 7 (b) greatly influences te reliability of te product. Volumentric efficiency of pump g Bearing load (Dimensionless amount) Eccentricity direction e Eccentricity ratio e, Eccentricity ratio e 1 (1/2) 1 1 (1/2) 2 1 (1/2) 3 1 (1/2) Eccentricity direction, e Tis teory takes into account te eccentricity of te gear more correctly tan te previous teory, and its calculation results are muc different from tose of te previous teory, except for te local conditions. Wen te experimental values are compared wit te calculated values according to te proposed teory, te calculated values are confirmed as valid. Te leakage flow rate troug te gear tip clearances greatly canges according to te eccentricity ratio. Te bearing load canges greatly due to te eccentricity direction. eferences 1) Atsusi Miyadzu: Optimum adial Tip Clearance of Gear Pump, Transactions of te JSME Vol. 17, No. 56 (1951) 36 (in Japanese). 2) Syo ju Itaya, Tsuneo Icikawa: Leakage in Gear Pump (1st eport), Transactions of te JSME Vol. 17, No. 60 (1951) 162 (in Japanese). 3) Tsuneo Icikawa: Leakage in Gear Pump (4t eport), Transactions of te JSME Vol. 18, No. 66 (1952) 160 (in Japanese). 4) Tsuneo Icikawa: On Unusual Caracteristics in Gear Pump, Transactions of te JSME Vol. 24, No. 137 (1958) 28 (in Japanese). 5) Te Japan Fluid Power System Society: Hydro- Pneumatic Hand Book, Omsa. (1989) 205 6) Yukio Hori: Hydrodynamic Lubrication, Yo kendo. (2002) 31 (in Japanese). 7) Yukio Hori: Hydrodynamic Lubrication, Yo kendo. (2002) 79 (in Japanese). 8) Te Japan Society of Mecanical Engineers: JSME Mecanical Engineers Hand Book, Maruzen. (1987) B1-36 (in Japanese). (b) Bearing applied load caracteristics Fig. 7 Eccentricity ratio of teoretical value and canges in eccentric direction 6. Conclusion I ave proposed new teoretical equations for te external gear pump to calculate te bearing load and te leakage flow rate caused by gear tip clearances. In addition, I ave created a program tat calculates te numerical values of te unknown eccentricity position of te gear wen te bearing type of te pump is a journal bearing. By investigating te proposed teoretical equations, te following conclusions were drawn. N.YOSHIDA * * Hydraulic System Engineering Dept., Driveline Systems Operations Headquarters 52 JTEKT ENGINEEING JOUNAL Englis Edition No. 1013E (2016)

A = h w (1) Error Analysis Physics 141

A = h w (1) Error Analysis Physics 141 Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.

More information

Chapters 19 & 20 Heat and the First Law of Thermodynamics

Chapters 19 & 20 Heat and the First Law of Thermodynamics Capters 19 & 20 Heat and te First Law of Termodynamics Te Zerot Law of Termodynamics Te First Law of Termodynamics Termal Processes Te Second Law of Termodynamics Heat Engines and te Carnot Cycle Refrigerators,

More information

Model development for the beveling of quartz crystal blanks

Model development for the beveling of quartz crystal blanks 9t International Congress on Modelling and Simulation, Pert, Australia, 6 December 0 ttp://mssanz.org.au/modsim0 Model development for te beveling of quartz crystal blanks C. Dong a a Department of Mecanical

More information

Hydraulic validation of the LHC cold mass heat exchanger tube.

Hydraulic validation of the LHC cold mass heat exchanger tube. Hydraulic validation o te LHC cold mass eat excanger tube. LHC Project Note 155 1998-07-22 (pilippe.provenaz@cern.c) Pilippe PROVENAZ / LHC-ACR Division Summary Te knowledge o te elium mass low vs. te

More information

Analysis of Static and Dynamic Load on Hydrostatic Bearing with Variable Viscosity and Pressure

Analysis of Static and Dynamic Load on Hydrostatic Bearing with Variable Viscosity and Pressure Indian Journal of Science and Tecnology Supplementary Article Analysis of Static and Dynamic Load on Hydrostatic Bearing wit Variable Viscosity and Pressure V. Srinivasan* Professor, Scool of Mecanical

More information

6. Non-uniform bending

6. Non-uniform bending . Non-uniform bending Introduction Definition A non-uniform bending is te case were te cross-section is not only bent but also seared. It is known from te statics tat in suc a case, te bending moment in

More information

Comment on Experimental observations of saltwater up-coning

Comment on Experimental observations of saltwater up-coning 1 Comment on Experimental observations of saltwater up-coning H. Zang 1,, D.A. Barry 2 and G.C. Hocking 3 1 Griffit Scool of Engineering, Griffit University, Gold Coast Campus, QLD 4222, Australia. Tel.:

More information

How to Find the Derivative of a Function: Calculus 1

How to Find the Derivative of a Function: Calculus 1 Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te

More information

The derivative function

The derivative function Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative

More information

Part 2: Introduction to Open-Channel Flow SPRING 2005

Part 2: Introduction to Open-Channel Flow SPRING 2005 Part : Introduction to Open-Cannel Flow SPRING 005. Te Froude number. Total ead and specific energy 3. Hydraulic jump. Te Froude Number Te main caracteristics of flows in open cannels are tat: tere is

More information

Simulation and verification of a plate heat exchanger with a built-in tap water accumulator

Simulation and verification of a plate heat exchanger with a built-in tap water accumulator Simulation and verification of a plate eat excanger wit a built-in tap water accumulator Anders Eriksson Abstract In order to test and verify a compact brazed eat excanger (CBE wit a built-in accumulation

More information

Numerical Differentiation

Numerical Differentiation Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function

More information

INTRODUCTION DEFINITION OF FLUID. U p F FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION

INTRODUCTION DEFINITION OF FLUID. U p F FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION INTRODUCTION DEFINITION OF FLUID plate solid F at t = 0 t > 0 = F/A plate U p F fluid t 0 t 1 t 2 t 3 FLUID IS A SUBSTANCE THAT CAN NOT SUPPORT SHEAR FORCES OF ANY MAGNITUDE WITHOUT CONTINUOUS DEFORMATION

More information

Taylor Series and the Mean Value Theorem of Derivatives

Taylor Series and the Mean Value Theorem of Derivatives 1 - Taylor Series and te Mean Value Teorem o Derivatives Te numerical solution o engineering and scientiic problems described by matematical models oten requires solving dierential equations. Dierential

More information

232 Calculus and Structures

232 Calculus and Structures 3 Calculus and Structures CHAPTER 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS FOR EVALUATING BEAMS Calculus and Structures 33 Copyrigt Capter 17 JUSTIFICATION OF THE AREA AND SLOPE METHODS 17.1 THE

More information

3. Using your answers to the two previous questions, evaluate the Mratio

3. Using your answers to the two previous questions, evaluate the Mratio MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MECHANICAL ENGINEERING CAMBRIDGE, MASSACHUSETTS 0219 2.002 MECHANICS AND MATERIALS II HOMEWORK NO. 4 Distributed: Friday, April 2, 2004 Due: Friday,

More information

Study of Convective Heat Transfer through Micro Channels with Different Configurations

Study of Convective Heat Transfer through Micro Channels with Different Configurations International Journal of Current Engineering and Tecnology E-ISSN 2277 4106, P-ISSN 2347 5161 2016 INPRESSCO, All Rigts Reserved Available at ttp://inpressco.com/category/ijcet Researc Article Study of

More information

Material for Difference Quotient

Material for Difference Quotient Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient

More information

A general articulation angle stability model for non-slewing articulated mobile cranes on slopes *

A general articulation angle stability model for non-slewing articulated mobile cranes on slopes * tecnical note 3 general articulation angle stability model for non-slewing articulated mobile cranes on slopes * J Wu, L uzzomi and M Hodkiewicz Scool of Mecanical and Cemical Engineering, University of

More information

Optimal Shape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow

Optimal Shape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow THE 5 TH ASIAN COMPUTAITIONAL FLUID DYNAMICS BUSAN, KOREA, OCTOBER 7 ~ OCTOBER 30, 003 Optimal Sape Design of a Two-dimensional Asymmetric Diffsuer in Turbulent Flow Seokyun Lim and Haeceon Coi. Center

More information

Consider the element shown in Figure 2.1. The statement of energy conservation applied to this element in a time period t is that:

Consider the element shown in Figure 2.1. The statement of energy conservation applied to this element in a time period t is that: . Conduction. e General Conduction Equation Conduction occurs in a stationary medium wic is most liely to be a solid, but conduction can also occur in s. Heat is transferred by conduction due to motion

More information

The Basics of Vacuum Technology

The Basics of Vacuum Technology Te Basics of Vacuum Tecnology Grolik Benno, Kopp Joacim January 2, 2003 Basics Many scientific and industrial processes are so sensitive tat is is necessary to omit te disturbing influence of air. For

More information

Velocity distribution in non-uniform/unsteady flows and the validity of log law

Velocity distribution in non-uniform/unsteady flows and the validity of log law University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 3 Velocity distribution in non-uniform/unsteady

More information

Click here to see an animation of the derivative

Click here to see an animation of the derivative Differentiation Massoud Malek Derivative Te concept of derivative is at te core of Calculus; It is a very powerful tool for understanding te beavior of matematical functions. It allows us to optimize functions,

More information

Problem Solving. Problem Solving Process

Problem Solving. Problem Solving Process Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and

More information

Derivation Of The Schwarzschild Radius Without General Relativity

Derivation Of The Schwarzschild Radius Without General Relativity Derivation Of Te Scwarzscild Radius Witout General Relativity In tis paper I present an alternative metod of deriving te Scwarzscild radius of a black ole. Te metod uses tree of te Planck units formulas:

More information

Combining functions: algebraic methods

Combining functions: algebraic methods Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc

More information

Finding and Using Derivative The shortcuts

Finding and Using Derivative The shortcuts Calculus 1 Lia Vas Finding and Using Derivative Te sortcuts We ave seen tat te formula f f(x+) f(x) (x) = lim 0 is manageable for relatively simple functions like a linear or quadratic. For more complex

More information

Distribution of reynolds shear stress in steady and unsteady flows

Distribution of reynolds shear stress in steady and unsteady flows University of Wollongong Researc Online Faculty of Engineering and Information Sciences - Papers: Part A Faculty of Engineering and Information Sciences 13 Distribution of reynolds sear stress in steady

More information

APPENDIXES. Let the following constants be established for those using the active Mathcad

APPENDIXES. Let the following constants be established for those using the active Mathcad 3 APPENDIXES Let te following constants be establised for tose using te active Matcad form of tis book: m.. e 9.09389700 0 3 kg Electron rest mass. q.. o.6077330 0 9 coul Electron quantum carge. µ... o.5663706

More information

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT

LIMITS AND DERIVATIVES CONDITIONS FOR THE EXISTENCE OF A LIMIT LIMITS AND DERIVATIVES Te limit of a function is defined as te value of y tat te curve approaces, as x approaces a particular value. Te limit of f (x) as x approaces a is written as f (x) approaces, as

More information

MVT and Rolle s Theorem

MVT and Rolle s Theorem AP Calculus CHAPTER 4 WORKSHEET APPLICATIONS OF DIFFERENTIATION MVT and Rolle s Teorem Name Seat # Date UNLESS INDICATED, DO NOT USE YOUR CALCULATOR FOR ANY OF THESE QUESTIONS In problems 1 and, state

More information

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225

THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225 THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:

More information

5.1 We will begin this section with the definition of a rational expression. We

5.1 We will begin this section with the definition of a rational expression. We Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go

More information

CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY

CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY CALCULATION OF COLLAPSE PRESSURE IN SHALE GAS FORMATION AND THE INFLUENCE OF FORMATION ANISOTROPY L.Hu, J.Deng, F.Deng, H.Lin, C.Yan, Y.Li, H.Liu, W.Cao (Cina University of Petroleum) Sale gas formations

More information

CHAPTER 2 MODELING OF THREE-TANK SYSTEM

CHAPTER 2 MODELING OF THREE-TANK SYSTEM 7 CHAPTER MODELING OF THREE-TANK SYSTEM. INTRODUCTION Te interacting tree-tank system is a typical example of a nonlinear MIMO system. Heiming and Lunze (999) ave regarded treetank system as a bencmark

More information

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.

Consider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx. Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions

More information

Continuity and Differentiability of the Trigonometric Functions

Continuity and Differentiability of the Trigonometric Functions [Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te

More information

Nonlinear correction to the bending stiffness of a damaged composite beam

Nonlinear correction to the bending stiffness of a damaged composite beam Van Paepegem, W., Decaene, R. and Degrieck, J. (5). Nonlinear correction to te bending stiffness of a damaged composite beam. Nonlinear correction to te bending stiffness of a damaged composite beam W.

More information

University Mathematics 2

University Mathematics 2 University Matematics 2 1 Differentiability In tis section, we discuss te differentiability of functions. Definition 1.1 Differentiable function). Let f) be a function. We say tat f is differentiable at

More information

2.8 The Derivative as a Function

2.8 The Derivative as a Function .8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open

More information

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12.

Notes: Most of the material in this chapter is taken from Young and Freedman, Chap. 12. Capter 6. Fluid Mecanics Notes: Most of te material in tis capter is taken from Young and Freedman, Cap. 12. 6.1 Fluid Statics Fluids, i.e., substances tat can flow, are te subjects of tis capter. But

More information

Experimental Analysis of Heat Transfer Augmentation in Double Pipe Heat Exchanger using Tangential Entry of Fluid

Experimental Analysis of Heat Transfer Augmentation in Double Pipe Heat Exchanger using Tangential Entry of Fluid ISR Journal of Mecanical & Civil Engineering (ISRJMCE) e-issn: 2278-1684,p-ISSN: 2320-334X PP 29-34 www.iosrjournals.org Experimental Analysis of Heat Transfer Augmentation in Double Pipe Heat Excanger

More information

Continuity and Differentiability Worksheet

Continuity and Differentiability Worksheet Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;

More information

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point

1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note

More information

De-Coupler Design for an Interacting Tanks System

De-Coupler Design for an Interacting Tanks System IOSR Journal of Electrical and Electronics Engineering (IOSR-JEEE) e-issn: 2278-1676,p-ISSN: 2320-3331, Volume 7, Issue 3 (Sep. - Oct. 2013), PP 77-81 De-Coupler Design for an Interacting Tanks System

More information

The total error in numerical differentiation

The total error in numerical differentiation AMS 147 Computational Metods and Applications Lecture 08 Copyrigt by Hongyun Wang, UCSC Recap: Loss of accuracy due to numerical cancellation A B 3, 3 ~10 16 In calculating te difference between A and

More information

LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION

LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y

More information

ADCP MEASUREMENTS OF VERTICAL FLOW STRUCTURE AND COEFFICIENTS OF FLOAT IN FLOOD FLOWS

ADCP MEASUREMENTS OF VERTICAL FLOW STRUCTURE AND COEFFICIENTS OF FLOAT IN FLOOD FLOWS ADCP MEASUREMENTS OF VERTICAL FLOW STRUCTURE AND COEFFICIENTS OF FLOAT IN FLOOD FLOWS Yasuo NIHEI (1) and Takeiro SAKAI (2) (1) Department of Civil Engineering, Tokyo University of Science, 2641 Yamazaki,

More information

Introduction to Derivatives

Introduction to Derivatives Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))

More information

Vane pump theory for mechanical efficiency

Vane pump theory for mechanical efficiency 1269 Vane pump theory for mechanical efficiency Y Inaguma 1 and A Hibi 2 1 Department of Steering Engineering, Toyoda Machine Works Limited, Okazaki, Japan 2 Department of Mechanical Engineering, Toyohashi

More information

Copyright c 2008 Kevin Long

Copyright c 2008 Kevin Long Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula

More information

Section 15.6 Directional Derivatives and the Gradient Vector

Section 15.6 Directional Derivatives and the Gradient Vector Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,

More information

Microstrip Antennas- Rectangular Patch

Microstrip Antennas- Rectangular Patch April 4, 7 rect_patc_tl.doc Page of 6 Microstrip Antennas- Rectangular Patc (Capter 4 in Antenna Teory, Analysis and Design (nd Edition) by Balanis) Sown in Figures 4. - 4.3 Easy to analyze using transmission

More information

Derivatives of Exponentials

Derivatives of Exponentials mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function

More information

Development and experimental verification of a numerical model for optimizing the cleaning performance of the doctor blade press roll tribosystem

Development and experimental verification of a numerical model for optimizing the cleaning performance of the doctor blade press roll tribosystem Development and experimental verification of a numerical model for optimizing te cleaning performance of te doctor blade press roll tribosystem M. Rodríguez Ripoll, B. Sceicl,, D. Bianci, B. Jakab, F.

More information

HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS

HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS HOW TO DEAL WITH FFT SAMPLING INFLUENCES ON ADEV CALCULATIONS Po-Ceng Cang National Standard Time & Frequency Lab., TL, Taiwan 1, Lane 551, Min-Tsu Road, Sec. 5, Yang-Mei, Taoyuan, Taiwan 36 Tel: 886 3

More information

AN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES

AN IMPROVED WEIGHTED TOTAL HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES AN IMPROVED WEIGHTED TOTA HARMONIC DISTORTION INDEX FOR INDUCTION MOTOR DRIVES Tomas A. IPO University of Wisconsin, 45 Engineering Drive, Madison WI, USA P: -(608)-6-087, Fax: -(608)-6-5559, lipo@engr.wisc.edu

More information

Work and Energy. Introduction. Work. PHY energy - J. Hedberg

Work and Energy. Introduction. Work. PHY energy - J. Hedberg Work and Energy PHY 207 - energy - J. Hedberg - 2017 1. Introduction 2. Work 3. Kinetic Energy 4. Potential Energy 5. Conservation of Mecanical Energy 6. Ex: Te Loop te Loop 7. Conservative and Non-conservative

More information

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.

4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these. Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra

More information

A Reconsideration of Matter Waves

A Reconsideration of Matter Waves A Reconsideration of Matter Waves by Roger Ellman Abstract Matter waves were discovered in te early 20t century from teir wavelengt, predicted by DeBroglie, Planck's constant divided by te particle's momentum,

More information

Squeeze film effect at elastohydrodynamic lubrication of plain journal bearings

Squeeze film effect at elastohydrodynamic lubrication of plain journal bearings Science, Juliana Engineering Javorova, & Education, Anelia Mazdrakova 1, (1), 16, 11- Squeeze film effect at elastoydrodynamic lubrication of plain journal bearings Juliana Javorova *, Anelia Mazdrakova

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS INTODUCTION ND MTHEMTICL CONCEPTS PEVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips of sine,

More information

Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature

Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates with Ramped Wall Temperature Volume 39 No. February 01 Effects of Radiation on Unsteady Couette Flow between Two Vertical Parallel Plates wit Ramped Wall Temperature S. Das Department of Matematics University of Gour Banga Malda 73

More information

Grade: 11 International Physics Olympiad Qualifier Set: 2

Grade: 11 International Physics Olympiad Qualifier Set: 2 Grade: 11 International Pysics Olympiad Qualifier Set: 2 --------------------------------------------------------------------------------------------------------------- Max Marks: 60 Test ID: 12111 Time

More information

REVIEW LAB ANSWER KEY

REVIEW LAB ANSWER KEY REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g

More information

Numerical analysis of a free piston problem

Numerical analysis of a free piston problem MATHEMATICAL COMMUNICATIONS 573 Mat. Commun., Vol. 15, No. 2, pp. 573-585 (2010) Numerical analysis of a free piston problem Boris Mua 1 and Zvonimir Tutek 1, 1 Department of Matematics, University of

More information

Differential Calculus (The basics) Prepared by Mr. C. Hull

Differential Calculus (The basics) Prepared by Mr. C. Hull Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit

More information

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative

Mathematics 5 Worksheet 11 Geometry, Tangency, and the Derivative Matematics 5 Workseet 11 Geometry, Tangency, and te Derivative Problem 1. Find te equation of a line wit slope m tat intersects te point (3, 9). Solution. Te equation for a line passing troug a point (x

More information

CFD Analysis and Optimization of Heat Transfer in Double Pipe Heat Exchanger with Helical-Tap Inserts at Annulus of Inner Pipe

CFD Analysis and Optimization of Heat Transfer in Double Pipe Heat Exchanger with Helical-Tap Inserts at Annulus of Inner Pipe IOR Journal Mecanical and Civil Engineering (IOR-JMCE) e-in: 2278-1684,p-IN: 2320-334X, Volume 13, Issue 3 Ver. VII (May- Jun. 2016), PP 17-22 www.iosrjournals.org CFD Analysis and Optimization Heat Transfer

More information

Lecture 10: Carnot theorem

Lecture 10: Carnot theorem ecture 0: Carnot teorem Feb 7, 005 Equivalence of Kelvin and Clausius formulations ast time we learned tat te Second aw can be formulated in two ways. e Kelvin formulation: No process is possible wose

More information

1watt=1W=1kg m 2 /s 3

1watt=1W=1kg m 2 /s 3 Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory

More information

Polynomial Interpolation

Polynomial Interpolation Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x

More information

158 Calculus and Structures

158 Calculus and Structures 58 Calculus and Structures CHAPTER PROPERTIES OF DERIVATIVES AND DIFFERENTIATION BY THE EASY WAY. Calculus and Structures 59 Copyrigt Capter PROPERTIES OF DERIVATIVES. INTRODUCTION In te last capter you

More information

Chapter 2 Limits and Continuity

Chapter 2 Limits and Continuity 4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(

More information

WYSE Academic Challenge 2004 Sectional Mathematics Solution Set

WYSE Academic Challenge 2004 Sectional Mathematics Solution Set WYSE Academic Callenge 00 Sectional Matematics Solution Set. Answer: B. Since te equation can be written in te form x + y, we ave a major 5 semi-axis of lengt 5 and minor semi-axis of lengt. Tis means

More information

Chapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3.

Chapter 1 Functions and Graphs. Section 1.5 = = = 4. Check Point Exercises The slope of the line y = 3x+ 1 is 3. Capter Functions and Graps Section. Ceck Point Exercises. Te slope of te line y x+ is. y y m( x x y ( x ( y ( x+ point-slope y x+ 6 y x+ slope-intercept. a. Write te equation in slope-intercept form: x+

More information

INTRODUCTION AND MATHEMATICAL CONCEPTS

INTRODUCTION AND MATHEMATICAL CONCEPTS Capter 1 INTRODUCTION ND MTHEMTICL CONCEPTS PREVIEW Tis capter introduces you to te basic matematical tools for doing pysics. You will study units and converting between units, te trigonometric relationsips

More information

1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible.

1. Which one of the following expressions is not equal to all the others? 1 C. 1 D. 25x. 2. Simplify this expression as much as possible. 004 Algebra Pretest answers and scoring Part A. Multiple coice questions. Directions: Circle te letter ( A, B, C, D, or E ) net to te correct answer. points eac, no partial credit. Wic one of te following

More information

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine?

1 Power is transferred through a machine as shown. power input P I machine. power output P O. power loss P L. What is the efficiency of the machine? 1 1 Power is transferred troug a macine as sown. power input P I macine power output P O power loss P L Wat is te efficiency of te macine? P I P L P P P O + P L I O P L P O P I 2 ir in a bicycle pump is

More information

Some Review Problems for First Midterm Mathematics 1300, Calculus 1

Some Review Problems for First Midterm Mathematics 1300, Calculus 1 Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,

More information

Why gravity is not an entropic force

Why gravity is not an entropic force Wy gravity is not an entropic force San Gao Unit for History and Pilosopy of Science & Centre for Time, SOPHI, University of Sydney Email: sgao7319@uni.sydney.edu.au Te remarkable connections between gravity

More information

Quantum Mechanics Chapter 1.5: An illustration using measurements of particle spin.

Quantum Mechanics Chapter 1.5: An illustration using measurements of particle spin. I Introduction. Quantum Mecanics Capter.5: An illustration using measurements of particle spin. Quantum mecanics is a teory of pysics tat as been very successful in explaining and predicting many pysical

More information

Dedicated to the 70th birthday of Professor Lin Qun

Dedicated to the 70th birthday of Professor Lin Qun Journal of Computational Matematics, Vol.4, No.3, 6, 4 44. ACCELERATION METHODS OF NONLINEAR ITERATION FOR NONLINEAR PARABOLIC EQUATIONS Guang-wei Yuan Xu-deng Hang Laboratory of Computational Pysics,

More information

Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran b

Department of Mechanical Engineering, Azarbaijan Shahid Madani University, Tabriz, Iran b THERMAL SCIENCE, Year 2016, Vol. 20, No. 2, pp. 505-516 505 EXPERIMENTAL INVESTIGATION ON FLOW AND HEAT TRANSFER FOR COOLING FLUSH-MOUNTED RIBBONS IN A CHANNEL Application of an Electroydrodinamics Active

More information

NOTES ON OPEN CHANNEL FLOW

NOTES ON OPEN CHANNEL FLOW NOTES ON OPEN CANNEL FLOW Prof. Marco Pilotti Facoltà di Ingegneria, Università degli Studi di Brescia Profili di moto permanente in un canale e in una serie di due canali - Boudine, 86 OPEN CANNEL FLOW:

More information

Stepped-Impedance Low-Pass Filters

Stepped-Impedance Low-Pass Filters 4/23/27 Stepped Impedance Low Pass Filters 1/14 Stepped-Impedance Low-Pass Filters Say we know te impedance matrix of a symmetric two-port device: 11 21 = 21 11 Regardless of te construction of tis two

More information

2.1 THE DEFINITION OF DERIVATIVE

2.1 THE DEFINITION OF DERIVATIVE 2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative

More information

CFD calculation of convective heat transfer coefficients and validation Part I: Laminar flow Neale, A.; Derome, D.; Blocken, B.; Carmeliet, J.E.

CFD calculation of convective heat transfer coefficients and validation Part I: Laminar flow Neale, A.; Derome, D.; Blocken, B.; Carmeliet, J.E. CFD calculation of convective eat transfer coefficients and validation Part I: Laminar flow Neale, A.; Derome, D.; Blocken, B.; Carmeliet, J.E. Publised in: IEA Annex 41 working meeting, Kyoto, Japan Publised:

More information

The effects of shear stress on the lubrication performances of oil film of large-scale mill bearing

The effects of shear stress on the lubrication performances of oil film of large-scale mill bearing Universit of Wollongong Researc Online Facult of Engineering - Papers (Arcive) Facult of Engineering and Information Sciences 9 Te effects of sear stress on te lubrication performances of oil film of large-scale

More information

FINITE ELEMENT STOCHASTIC ANALYSIS

FINITE ELEMENT STOCHASTIC ANALYSIS FINITE ELEMENT STOCHASTIC ANALYSIS Murray Fredlund, P.D., P.Eng., SoilVision Systems Ltd., Saskatoon, SK ABSTRACT Numerical models can be valuable tools in te prediction of seepage. Te results can often

More information

Exam 1 Review Solutions

Exam 1 Review Solutions Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),

More information

Continuity. Example 1

Continuity. Example 1 Continuity MATH 1003 Calculus and Linear Algebra (Lecture 13.5) Maoseng Xiong Department of Matematics, HKUST A function f : (a, b) R is continuous at a point c (a, b) if 1. x c f (x) exists, 2. f (c)

More information

Journal of Applied Science and Agriculture. The Effects Of Corrugated Geometry On Flow And Heat Transfer In Corrugated Channel Using Nanofluid

Journal of Applied Science and Agriculture. The Effects Of Corrugated Geometry On Flow And Heat Transfer In Corrugated Channel Using Nanofluid Journal o Applied Science and Agriculture, 9() February 04, Pages: 408-47 AENSI Journals Journal o Applied Science and Agriculture ISSN 86-9 Journal ome page: www.aensiweb.com/jasa/index.tml Te Eects O

More information

Quantum Theory of the Atomic Nucleus

Quantum Theory of the Atomic Nucleus G. Gamow, ZP, 51, 204 1928 Quantum Teory of te tomic Nucleus G. Gamow (Received 1928) It as often been suggested tat non Coulomb attractive forces play a very important role inside atomic nuclei. We can

More information

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006

Math 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 Mat 102 TEST CHAPTERS 3 & 4 Solutions & Comments Fall 2006 f(x+) f(x) 10 1. For f(x) = x 2 + 2x 5, find ))))))))) and simplify completely. NOTE: **f(x+) is NOT f(x)+! f(x+) f(x) (x+) 2 + 2(x+) 5 ( x 2

More information

NCCI: Simple methods for second order effects in portal frames

NCCI: Simple methods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames NCC: Simple metods for second order effects in portal frames Tis NCC presents information

More information

DRIVE-TECHNOLOGY INKOMA - GROUP. SDA - Spindle direct drive actuator. Maschinenfabrik ALBERT GmbH Technologiepark 2 A Gampern - Austria

DRIVE-TECHNOLOGY INKOMA - GROUP. SDA - Spindle direct drive actuator. Maschinenfabrik ALBERT GmbH Technologiepark 2 A Gampern - Austria DRIVETECHNOLOGY ALBERTINKOMA GROUP SDA Spindle direct drive actuator Mascinenfabrik ALBERT GmbH Tecnologiepark A Gampern Austria pone: +/(0)0 fax: +/(0)0 email: office@albert.at internet: www.albert.at

More information

GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES

GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES GRID CONVERGENCE ERROR ANALYSIS FOR MIXED-ORDER NUMERICAL SCHEMES Cristoper J. Roy Sandia National Laboratories* P. O. Box 5800, MS 085 Albuquerque, NM 8785-085 AIAA Paper 00-606 Abstract New developments

More information