PREREQUISITE KNOWLEDGE. You should know the following concepts and be able to perform the associated computation.

Transcription

1 BSYSE 595 Ground-water Flow and Contaminant Transport I. Mathematics PREREQUISITE KNOWLEDGE 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. If x at point (x, y) is given, the formula for y is. 2. Solution of a system of linear equations The system of linear equations has a solution of x = 1 and y = Derivative for a function of a single variable Notations for the derivative of can be, and. 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 one-variable calculus. Let be a function of two variables, the two partial derivatives are denoted and, or and. 5. Indefinite and definite integration The notation where FN(x) = f(x) and C is an arbitrary constant denotes the family of all antiderivatives of f(x) on an interval I. The symbol I is an integral sign, and is the indefinite integral of f(x). The expression f(x) is called integrand, and C is the constant of integration. The process of finding, when given, is referred to as indefinite integration, evaluating the integral, or integrating f(x). Let f be defined on a closed interval [a, b]. The definite integral of f from a to b, denoted by, 1

2 is, where P is the largest partition of [a, b] and w k is a point within )x k, and provided the limit exists. The process of finding 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 definite integral, such as estimating arc length, area bounded by curvilinear functions, and various volumes. Two simple examples of indefinite and definite integration are (1) (2) 6. Exponentials and logarithms, powers and roots Exponential functions involve raising a constant base to a variable exponent. Two examples are and. The exponential function with base a is defined by where a > 0, a 1, and x is any real number. Laws of exponents: if u and v are any two real numbers, then (i) (ii) (iii) 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 defined by if and only if. Properties of logarithms and equivalent exponential forms are (i) (ii) (iii) (iv) A logarithm with base 10 is called common logarithm. A logarithm with base e= is called natural logarithm, denoted ln x. To numerically calculate logarithms with bases other than 10 and e, we need to use the following change-of-base formula, where x > 0, and a and b are positive real numbers other than 1. You need to know how to obtain powers (4 2.5 = 32) and roots ( ) with a calculator. 2

3 II. Physics The following basic laws of classical physics are important. 1. Conservation of mass Mass is neither created nor destroyed. 2. Newton s 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 s first law of diffusion A diffusion substance X moves from where its concentration is larger to where its concentration is smaller at a rate that is proportional to the spatial gradient of concentration, that is,, where F z (X) is rate of transfer of X in direction z per unit area per unit time (also called flux of X), D X is the diffusivity of X in the medium, and C(X) is the concentration of X. III. 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 S Chemical formula H 2 O, strong covalent bond, molecular structure asymmetric resulting in polarity which produces a hydrogen bond S Having high melting (0 ) and boiling (100 ) temperature compared to other hydrides of the Group VIa elements, existing in three physical states at earth-surface temperature, having a lower density in the solid state than in the liquid S Liquid water molecules dissociated into hydrogen ions H + and hydroxide ions OH -, acidity measured with ph (ph / -log 10 ([H + ]), ph=7 for pure water) S 99.73% of all water consists of normal 1 16 H 2 O. 3

4 2. Density of water S mass density D [ML!3 ] = 1 g/cm 3 S weight density ( w = Dg = 980 dyn/cm 3 = 9,800 N/m 3 3. Thermal capacity of water Thermal capacity c p is defined as the amount of heat energy )H absorbed/released by a mass M of a substance when its temperature is raised or lowered by an amount )T, c p / )H/(M )T). For water at 0, c p = 4,216 J/(kg K) = cal/(g C ). c p for water is very high compared to the c p 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. S S Latent heat of fusion: heat energy absorbed or released when a unit mass melts or freezes. For water, latent heat of fusion 8 f = 3.34x10 5 J/kg = 79.7 cal/g. Latent heat of vaporization: heat energy absorbed or released when a unit mass vaporizes or condenses. For water, latent heat of vaporization 8 v = 2.495x10 16 J/kg = 595.9cal/g. 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 first steps are dissolution and aqueous alteration of minerals, furthermore, a significant 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 population mean and standard deviation, respectively. 2. t-distribution for - 4 < x < + 4 is the pdf of normal distribution, where : and F are the If sample mean and standard deviation are and s respectively, then has a t-distribution with n!1 degree of freedom, where n is the sample size. Let " be the significance level, the decision rules for testing alternative hypotheses of H 0 : : # : 0 and H a : : > : 0 are 4

5 if t * # t(1-"; n 1), conclude H 0 if t * > t(1-"; n 1), conclude H a V. Others 1. Dimensions of hydrologic quantities The dimension of a hydrologic quantity can always be written as [M a L b T c 1 d ] or [F e L f T g 1 h ], where M, F, L, T, and 1 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: Système International (SI), centimeter-gram-second (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 35.2 in to mm as follows: 35.2 in = = 894 mm (2) Convert 100 mi/h to m/s as follows 100 = = 44.7m/s 3. Numerical precision of hydrologic quantities Rule 1 Assume no more than three-significant-figure precision in hydrologic quantities unless greater precision is warranted. Rule 2 Always round off to the appropriate number of significant figures at the end. Rule 3 Computers and calculators do not know anything about significant figures. 5