CHAPTER 1 INTRODUCTION TO NUMERICAL METHOD

Transcription

1 CHAPTER 1 INTRODUCTION TO NUMERICAL METHOD Presenter: Dr. Zalilah Sharer 2018 School of Chemical and Energy Engineering Universiti Teknologi Malaysia 16 September 2018

2 Chemical Engineering, Computer & Numerical Methods Role of Chemical Engineers Chemical engineering covers basic skill in mathematics, chemistry, physic and biology, also engineering practical aspect. Its definition was purposely general because chemical engineers can work in many types of industry. Chemical engineers involve in chemical process that transform raw material into product. It covers all aspect of design, testing, scale-up, operation, control and optimizations. These processes involve solution to huge system of algebraic equation, nonlinear and complex equation, which are difficult to be solved analytically.

3 Chapter 2: Approximation and Errors Chapter 3: Roots of equations - a variable or parameter that satisfies a single nonlinear equation Chapter 4: Linear algebraic equations - a set of values that satisfies a set of linear algebraic equations Chapter 5: Curve Fitting - to fit curves to data points Chapter 6: Numerical differentiation and integration - - area under a curve Chapter 7: Ordinary differential equations - many engineering applications used rate of change Approximations and round-off errors Bracketing methods Linear algebraic equations Least-Squares Regression Newton-cotes integration of equations Runge-Kutta methods Taylor series Open methods Gauss Elimination Interpolation Numerical differentiation Engineering Applications Engineering applications LU decomposition & matrix inversion Engineering Applications Engineering Applications Gauss Seidel and Engineering Applications 3

4 Differentiation & Integration Linear algebraic Numerical Methods Roots of equation Curve fitting Ordinary differential equations

5 Numerical methods Techniques by which mathematical problems are formulated so that they can be solved with arithmetic operations involve large numbers of calculation

6 Numerical methods Example: Integration Analytical solution: Computer as usual as in calculus Numerical method: Use trapezoidal rule or Simpson s rules

7 Advantages 1. Powerful problem-solving tools capable of handling large systems of equations, nonlinearities and complicated geometries that are often impossible to solve analytically 2. Able to design and develop own programs without having to buy or commission expensive software 3. Able to reduce higher mathematics to basic arithmetic operations

8 Computers and Software MATLAB, Mathematica, Dynafit etc are some software packages to implement numerical methods. Help to solve engineering problem in numerical methods. Else. MS-EXCEL also can be used to solve Numerical problems.

9 Problem Solving Process

10 Problem Solving Process

11 Mathematical model Equations that expresses the essential features of a physical systems Represented as a functional relationship in the form of Dependent Variables = f (independent, parameters, forcing function, variables ) Dependent Variables - Reflects the behavior or state of the system Independent Variables - Dimensions, such as time and space Parameters - Reflective of the system s properties or composition Forcing Function - External influence acting upon it

12 Newton s 2 nd law of Motion States that the time rate change of momentum of a body is equal to the resulting force acting on it. The model is formulated as F = ma (eqn 1.2) F=net force acting on the body (N) m=mass of the object (kg) a=its acceleration (m/s 2 )

13 Newton s 2 nd law of Motion Equation 1.2 can be written as: a = F / m eqn 1.3 simple algebraic equation that can be solved analytically

14 To determine the terminal velocity of a free-falling body near the earth s surface using Newton 2 nd law. Express acceleration as the time rate of change of the velocity (dv/dt) and substituting into eq. (1.3) to yield or d /dt = F/m (eqn. 1.4) F = m (d /dt) F +ve : accelerate F -ve : decelerate F = 0 (constant velocity)

15 Express the net force in term of measurable variables and parameters, in which the net force is composed of 2 opposing forces: The downward pull of gravity F D and the upward force of air resistance F u : F u F = F D + Fu (eqn. 1.5) F D If downward force is +ve, 2 nd law can be used to formulate the force due to gravity, as F D = mg (eqn. 1.6) g = 9.8 m/s 2 The air resistance that acts in an upward direction; F u = -c (eqn. 1.7) c = drag coefficient (kg/s)

16 The net force is the difference between the downward (FD) and upward (FU). By combining eqs. (1.4) through (1.7) to yield: d /dt = (mg c )/m (eqn. 1.8) or simplifying the right side, d /dt = g (c/m)v (eqn. 1.9) Eq. (1.9) is a differential equation. The exact solution of eq. (1.9) cannot obtained by simple algebraic manipulation, which needs calculus to obtain an exact or analytical solution. If = 0 at t=0, calculus can be used to solve eq. (1.9) for (t)= (gm/c)[1-e -(c/m)t ] (eqn. 1.10)

17 This is a differential equation and is written in terms of the differential rate of change dv/dt of the variable that we are interested in predicting. If the parachutist is initially at rest (v=0 at t=0), using calculus Independent variable v( t) Dependent variable gm c 1 e Forcing function ( c/ m) t Parameters 1.10

18 Eq is called analytical/exact solution because it exactly satisfies the original differential equation. (t) t = dependent variable = independent variable c & m = parameters g = the forcing function However, many mathematical models cannot be solved as shown in eqn The only alternative is to develop a numerical solution that approximates the exact solution i.e. numerical method.

19 Force on a falling parachute Mainly from second law of thermodynamics ==> F = ma The model then can be derived with The force acting on the body : F = FU + FD

20 Example 1 Analytical Solution to the Falling Parachutist Problem A parachutist of mass 68.1 kg jumps out of a stationary hot air balloon. Use equation 1.4 to compute velocity for every 2 seconds. The drag coefficient is equal to 12.5 kg/s and g = 9.8 m/s 2

21 Solution Inserting the parameter into eq. (1.10) yields v(t) gm c 1 e v(t) (9.8)(68.1) 12.5 c t m 1 e v(t) e t 12.5 t 68.1 which can be used to compute terminal velocity.

22 Solution

23 Terminal velocity, u t The terminal velocity of a falling body occurs during free fall when a falling body experiences zero acceleration.

24 Using Numerical Method Approach The time rate of change of velocity can be approximated using: dv dt v t v( t ) v( ) i 1 t i 1 t i t i eqn. 1.11

25 Substituted into eq. (1.9) to give: v( ti 1) v( ) i c g m t i 1 t i t v t i Eq is called a finite divided difference approximation of the derivative time t i. This eq. then be rearranged to yield: c v( ) v( ) g - v( ) - m t 1 t t t 1 t i i i i i eqn The term in [brackets] is the differential equation in eq.(1.9). This provides a means to compute the rate of change or slope of. Eq can be used to determine the velocity at t i +1(new value of velocity) using slope and initial value for velocity at sometime t i. New value = old value +(slope x step size)

26 Example 2 Perform the same computation as in Example 1 but use Equation 1.12 to compute the velocity. Employ a step size of 2 s for the calculation c v( ) v( ) g - v( ) - m t 1 t t t 1 t i i i i i Eqn. 1.12

27 Solution At start of the computation (t i =0), the velocity of the parachutist is zero. First interval (from t=0 to 2s) v m/s 68.1 For next interval, use t = 2 to 4s 12.5 v m/s 68.1 The calculation is continued in a similar fashion to obtain additional value

28 v, m/s Solution t,s v,m/s Approximate, numerical solution Terminal velocity Exact, analytical solution t,s

29 Analytical vs numerical solution Equation 1.4 is called analytical or exact solution exactly satisfies the original differential equation (no error) Unfortunately, many mathematical models cannot be solved exactly numerical methods approximate the exact solution

30 Equation 1.12 can be used to determine the velocity at time t i+1 if an initial value for velocity at time t i is given. This new value of velocity at t i+1 can in turn be employed to extend the computation to velocity at t i+2 and so on. In general: New value = old value + (slope x step size) This approach is formally called Euler s method

31 Question?

32 THE END Thank You for the Attention