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# Class Note for BSysE 595 at WSU 07

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Date Created: 02/06/15

BSYSE 595 Groundwater Flow and Contaminant Transport PREREQUISITE KNOWLEDGE 1 Mathematics You should know the following concepts and be able to perform the associated computation 1 Linear interpolation and extrapolation Linear interpolation is to estimate the value of a function at a point on a line between two given points at which the function values are known Linear extrapolation is to estimate the value of a function at a point on a line beyond the two given points at which the function values are known Ifx at point x y is given the x x formula fory 1s y yl fx m 71 2 r 2 Solution of a system of linear equations 3xy11 has a solution ofx l andy 2 The system of linear equatrons 2x y 3 Derivative for a function of a single variable Notations for the derivative of y fx can be x y g fx Dxfx ny and d E flx fx h fx H h 4 Partial derivative for a function of multiple variables If f is a function of several variables to calculate the partial derivative with respect to a certain variable treat the remaining variables as constants and differentiate as usual by using the rules of onevariable calculus Let 2 f x y be a function of two variables the two partial derivatives are denoted g f and g f 3 y or E and 6 6y 5 Inde nite and de nite integration The notation f xdx F x C where F x x and C is an arbitrary constant denotes the family of all antiderivatives of x on an interval The symbol f is an integral sign and f xdx is the inde nite integral of x The expression fx is called integrand and C is the constant of integration The process of nding F x C when given f xdx is referred to as inde nite integration evaluating the integral or integrating x b Let f be de ned on a closed interval a b The definite integral of f from a to b denoted by f x ix b is f x ix lim E w QAxk where P is the largest partition of 11 b and wk is a point within Axk and a Pl 0 I provided the limit exists The process of nding the limit is called definite integration or evaluating the integral and a and b are the limits of integration There are numerous applications of the de nite integral such as estimating arc length area bounded by curvilinear functions and various volumes Two simple examples of inde nite and de nite integration are 2 1 fx c 2 x2 2 22 02 22 2 2 2l 2 fxdx 0 6 Exponentials and logarithms powers and roots Exponential functions involve raising a constantbase to avariable exponent Two examples are f x 10quot and gx x The exponential function with base a is de ned by f x a quot where a gt 0 a s l andx is any real number Laws of exponents if u and v are any two real numbers then auav auv ii new a v iii a quotv a Logarithmic functions are closely related to exponential functions If a is a positive real number other than 1 then the logarithm of x with base a is de ned by y logax if and only if x 11 Properties of logarithms and equivalent exponential forms are i loga10 a l ii logaa 1 1 iii log aquot x a iv logax logax alo A logarithm with base 10 is called common logarithm A logarithm with base e 271 828 1828 is called natural logarithm denoted In x To numerically calculate logarithms with bases other than 10 and e we need x wherexgt 0 and a and b are pos1t1ve real numbers to use the following changeofbase formula 10gb x 1 b oga other than 1 You need to know how to obtain powers 42 5 32 and roots 2273 304 with a calculator 11 Physics The following basic laws of classical physics are important 1 Conservation of mass Mass is neither created nor destroyed 2 Newton 3 laws of motion 1 The momentum of a body remains constant unless the body is acted upon by a net force conservation of momentum 2 The rate of change of momentum of a body is proportional to the net force acting on the body and is in the same direction as the net force Force equals mass times acceleration 3 For every net force acting on a body there is a corresponding force of the same magnitude exerted by the body in the opposite direction 3 Laws of thermodynamics 1 Energy is neither created nor destroyed conservation of energy 2 No process is possible in which the sole result is the adsorption of heat and its complete conversion into work 4 Fick sfirst law ofdi usion A diffusion substance X moves from where its concentration is larger to where its concentration is dCX dz 5 where F X is rate of transfer ofX in direction Z per unit area per unit time also called ux ofX DX is the diffusivity of X in the medium and CX is the concentration of X smaller at a rate that is proportional to the spatial gradient of concentration that is Fl X DX 111 General Properties of Water Water is an unusual substance with anomalous properties It is necessary for you to know the following features of water 1 Structure of water Chemical formula H20 strong covalent bond molecular structure asymmetric resulting in polarity which produces a hydrogen bond Having high melting 0 and boiling 100 temperature compared to other hydrides of the Group VIa elements existing in three physical states at earthsurface temperature having a lower density in the solid state than in the liquid Liquid water molecules dissociated into hydrogen ions H and hydroxide ions OH39 acidity measured with pH pH E log10H pH7 for pure water 9973 of all water consists of normal 1H2160 2 Density of water mass density p llj3 l gcm3 weight density yw pg 980 dyncm3 9800 Nm3 3 Thermal capacity of water Thermal capacity CF is de ned as the amount of heat energy AH absorbedreleased by a mass M of a substance when its temperature is raised or lowered by an amount AT CF E AHM AT For water at 0 CF 4216 Jkg K 1007 calg C CF for water is very high compared to the CF values for other substances due to water s strong hydrogen bonds 4 Latent heat of water Latent heat is the energy released or absorbed when a given mass of substance undergoes a change of phase no temperature change Water s latent heat is very high as compared with other substances Latent heat of fusion heat energy absorbed or released when a unit mass melts or freezes For water latent heat of fusion hf 334X105 Jkg 797 calg Latent heat of vaporization heat energy absorbed or released when a unit mass vaporizes or condenses For water latent heat of vaporization A 2495X1016 Jkg 5959calg 5 Solvent power of water Due to the polar structure and hydrogen bonds almost every substance is soluble in water to some degree Importance of such power to biogeochemical processes virtually all life processes take place in water and depend on the delivery of nutrients and removal of wastes in solution in water erosion processes the rst steps are dissolution and aqueous alteration of minerals furthermore a signi cant portion of all the materials transported by rivers from land to oceans is carried in solution IV Probability and Statistics You should be familiar with the concepts of mean variance standard deviation probability probability density function pdf and cumulative distribution function cdf for continuous variables and normal and t distributions 1 Normal distribution x 60quotquot1 1 for 00 lt x lt 00 is the pdf of normal distribution where p and 0 are the a 21 population mean and standard deviation respectively 2 t distribution If sample mean and standard deviation are 1 and s respectively then tquot has a tdistribution with s n n e 1 degree of freedom where n is the sample size Let 06 be the signi cance level the decision rules for testing alternative hypotheses of H0 1 g 10 and H p gt 10 are if t tloc nil conclude H0 if t gt tloc nil conclude H V Others 1 Dimensions of hydrologic quantities The dimension of a hydrologic quantity can always be written as M Lb T 8d or Fquot Lf T3 8quot where M F L T and 9 refer to the dimensions of mass force length time and temperature respectively and a b h are rational numbers 2 Units and unit conversion Units are the arbitrary standards in which measurable quantities are expressed Three systems are commonly used Syst rne International SI centimetergramsecond cgs and English system Unit conversion is a common practice in science and engineering Rules for unit conversion 1 Start with given or known entity 2 Work toward desired entity 3 Always multiply 4 Write all conversion factors with units as fractions for appropriate canceling 5 Cancel appropriately 6 Carefully practice unit conversion for 100 times anybody can do it fast and correctly Examples 1 Convert 352 in to mm as follows 352m 352in X 254cm X10mm 894mm lin lcm 2 Convert 100 mih to ms as follows 100 g 100miX 1609m X 1h 447mS 1h lmi 36005 3 Numerical precision of hydrologic quantities Rule 1 Assume no more than threesignificantfigure precision in hydrologic quantities unless greater precision is warranted Rule 2 Always round off to the appropriate number of signi cant gures at the end Rule 3 Computers and calculators do not know anything about significant figures

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