Basic Probability (Chapter 2, W.J.Decoursey, 2003) Objectives: -Define probability and its relationship to relative frequency of an event. -Learn the basic rules of combining probabilities. -Understand the concepts of mutually exclusive / not mutually exclusive and independent / not independent events. Summary

Basic Probability Summary of Combining Rule When doing combined probability problems, ask yourself: 1.Does the problem ask the logical OR or the logical AND? 2.If OR, ask your self are the events mutually exclusive or not? If yes, Pr [ A U B ] = Pr [A] +Pr [B], other wise Pr [ A U B ] = Pr [A] +Pr [B] – Pr [A ∩ B] 3.If AND, use the multiplication rule and remember conditional probability. A probability tree may be helpful. 4.Fault Tree Analysis

Descriptive Statistics Objectives: (Chapter 3, Decoursey) - To understand the definition of mean, median, variance, standard deviation, mean absolute deviation and coefficient variation and calculate these quantities. - To calculate some of these quantities using the statistical functions of Excel.

Descriptive Statistics

Probability Distributions, Discrete Random Variables Objectives: (Chapter 5, DeCoursey) - To define a probability function, cumulative probability, probability distribution function and cumulative distribution functions. - To define expectation and variance of a random variable. - To determine probabilities by using Binomial Distribution.

Probability Distributions, Discrete Random Variables

Binomial Distribution: Let p = probability of “success” q=probability of “failure” = 1-p n = number of trails r = number of “success” in “n” trails Then the probability of r successes for n trials is given by the following general formula: Probability Distributions, Discrete Random Variables

Probability Distributions, Continuous Variables Objectives: (Chapter 6, DeCoursey) - To establish the difference between probability distribution for discrete and continuous variables. - To learn how to calculate the probability that a random variable, X, will fall between the limits of “a” and “b”.

Continuous Variable Discrete Variable

Normal Distribution f(z) z (X variable)

Use Normal Distribution Statistical Inference for the Mean S. D. of population σ is known and unchanged Test at a specified Level of significance S. D is from the sample Use t Distribution Table A1 Table A2 Confidence Interval: Compare to μ

Use t Distribution Statistical Inference for the Mean Unpaired samples: independent test Test at a specified Level of significance Paired samples: dependent test d=x 1 -x 2 Use t Distribution Compare

Test of Significance: - State the null hypothesis in terms of the mean difference - State the alternative hypothesis in terms of the same population parameters. - Determine the mean and variance. - Calculate the test statistic t or z of the observation. - Determine the degrees of freedom for t-test. - State the level of significance – rejection limit. - If probability falls outside of the rejection limit, we reject the Null Hypothesis, which means the difference of the two samples are significant. Assume the samples are normally distributed. Statistical Inference for the Mean

Procedures for regression: 1. Assume a regression equation. 2. If the equation is simple linear form, use least squares method to determine the coefficients. If not, convert it to the form linear in coefficients and then use least squares method. Or use Excel functions such as Solver, Trendline or Linest. 3. Evaluate the regression by statistical analysis Regression and Correlation of Data Summary

Simple Linear Regression: Regression and Correlation of Data Summary Method of Least Squares: a and b are determined by minimizing the sum of the squares of errors (SSE), deviation, residuals or difference between the data set and the straight line that approximate it.

Centroidal point: Regression and Correlation of Data Summary

Convert the equation to the form containing the original variables. Use least square method to determine the coefficients. Regression and Correlation of Data Summary Other forms linear in coefficients: e.g. Forms transformable to linear in coefficients: e.g.

Statistical analysis of the regression: Sum of the squares of errors (SSE), Estimated standard deviation or standard error of the points from the line: The degrees of freedom=n data points – the number of estimated coefficients Estimated variance of the points from the line: Regression and Correlation of Data Summary

Correlation Coefficient: r =1: the points (xi,yi) are in a perfect straight line and the slope of that line is positive; r =-1: the points (xi,yi) are in a perfect straight line and the slope of that line is negative; r close to +1 or -1: X and Y follow a linear relation affected by random errors. r=0: there is no systematic linear relation between X and Y. Regression and Correlation of Data Summary

Coefficient of determination: The coefficient of determination is the fraction of the sum of squares of deviations in the y-direction from is explained by the relation between y and x given by regression. Regression and Correlation of Data Summary

Statistical analysis of the regression: Residuals (also used for graphical checks): Regression and Correlation of Data Summary Percentage Error: Absolute Percentage Error:

Modeling with Unsteady State Material and Energy Balance Objectives: (Mainly in Chapters 6 & 22, Himmelblau) -To review the material and energy balance concept. -To discriminate between unsteady state and steady state material balance. -To perform unsteady state material and energy balance and solve the resulting first order linear differential equations.

System - You can define the limits of the system by drawing the system boundary, namely a line that encloses the portion of the process that you want to analyze. System : any arbitrary portion of or a whole process that you want to consider for analysis. - You can define a system such as a reactor, a section of a pipe, or an entire refinery by stating in words what the system is. (Himmelblau, D., 2004, p.136) C, P V, t f o =100 L/hr f i =100 L/hr

Closed System Closed system : That material neither enters nor leaves the vessel, that is, no material crosses the system boundary. - Changes can take place inside the system, but for a closed system, no mass exchange occurs with the surroundings. (Himmelblau, D., 2004, p.136) C, P V, t

Open System Open system : also called a flow system because material cross the system boundary. (Himmelblau, D., 2004, p.136) f o =100 L/hr f i =100 L/hr

Basic laws: conservation law Modeling with Unsteady State Material and Energy Balance - mass - energy - momentum - rate expression for generation or consumption Accumulation within system Input through system boundaries output through system boundaries Generation within system Consumption within system =- + -

Basic laws: conservation law Modeling with Unsteady State Material and Energy Balance Steady State: accumulation is zero. The quantity does not change with time. Unsteady State: The quantity changes with time. Accumulation within system Input through system boundaries output through system boundaries Generation within system Consumption within system =- + -

Modeling with Unsteady State Material and Energy Balance Procedures: - Specify the system - Bring together all variables of interest - Set down finite and differential elements - Develop equations by conservation law and/or rate expression - Generate differential equations as the differential elements shrink in the limit. - Specify boundary conditions

Basic laws: Modeling with Unsteady State Material Balance Mass Accumulation within system Mass Input through system boundaries Mass output through system boundaries Mass generation within system Mass consumption within system =- + - Liquid, gas and solid systems Closed or open system

Basic energy balance: Modeling with Unsteady State Energy Balance Energy Accumulation within system Energy Input through system boundaries Energy output through system boundaries Energy generation within system Energy consumption within system =- + - Open or closed systems: T vs t

Modeling with Unsteady State Energy Balance E t+ ∆ t – E t = (U + K.E.+ P.E.) t+ ∆ t - (U + K.E.+ P.E.) t = Q + W + (H+K.E.+P.E.) in - (H+K.E.+P.E.) out + ∆ H r In most chemical engineering processes, the change of K.E. and P.E. can be negligible, the above equation for a open system becomes E t+ ∆ t – E t = (U) t+ ∆ t - (U) t = Q + W + (H) in - (H) out + ∆ H r For a closed system, the above equation becomes E t+ ∆ t – E t = (U) t+ ∆ t - (U) t = Q + W + ∆ H r dU = mC v dT dH = mC p dT M: mass (g), C v : heat capacity at constant volume (J/g o C); C p : heat capacity at constant pressure (J/g o C).

SOLVE ODEs Approach: separation of variables (Adams, Calculus, 1999, p.525) If a 1 st order differential equation has the form = g(x)h(y) Then separate and integrate both sides Apply initial condition, unique solution to y can be determined. (See in-class example)

Summary of Integrating Factor Write the general equation of 1 st order linear ODE Determine integrating factors I (x) Determine dependent variable or unknown function y Arbitrary constant C is determined by the boundary condition.