Solving Equations and Optimizing Functions

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1 Solving Equations and Optimizing Functions At the end of this lecture, you will be able to: solve for the roots of a polynomial using polyroots. obtain approximate solutions to single equations from tracing a graph. obtain a solution to a single non-linear equation using the root function. solve systems of non-linear equations using Solve Blocks 1. Solving for the roots of a polynomial. Description The polyroots() function is simple and powerful. It finds ALL the roots of a polynomial, both real and imaginary. Demonstration Find all the roots, both real and imaginary, of the following equation. x 6 x 5 3x 4 3x 3 x x = Step 1: Define the input matrix containing the coefficients of each term. coeff Key Points: 1. The coefficient matrix is a single column with n+1 rows where n is the order of the polynomial.. The coefficient matrix is ordered from x to x n. 3. Polynomial must be place in form where RHS =. Step : Use the polyroots function. Practice polyroots( coeff ) i i 1.71 Solve for all the roots, both real and imaginary, of the following equation. x 4 x = coeff polyroots coeff i.7.933i.856

2 . Solving Single Equations Using the root() Function. Description The root() function offers a method of finding the solution to single equations of any type, linear or non-linear. Demonstration Find the roots of the following equation: 1 sin( x) = x Step 1: Put the equation in the form f(x) =. 1 sin( x) x = Step : The root function requires a guess value so first plot the function. 1 1 sin( x) x x Step 3: Use root function with different guesses to find all the roots. Guesses Solutions a 6 a 3 a a 3 a 6 root( 1sin( a) aa ) root( 1sin( a) aa ) root( 1sin( a) aa ) root( 1sin( a) aa ) root( 1sin( a) aa ) Alternate Method: Define a function. f( x) 1sin( x) x Guess Solution b 6 root( f ( b) b) Key Points: 1. The root function needs a guess value. You can get the guess value from a graph of the function.. The guess value is the second argument of the function. The first argument is the function written in terms of the guess value. 3. Solving Systems of Nonlinear Equations Using Solve Blocks (Given/Find Blocks) Description Solve blocks (given/find blocks) can be used to solve systems of non-linear equations. If you have a system of linear equations, matrix math should be used to obtain the solutions. If the equations are not linear, the only other way to

3 solve them in Mathcad is using a solve block. The Procedure 1. Define the guess value for each unknown.. Initiate a Solve Block with the key word Given 3. Enter the equations to be solved. a. You must use a bold equals to define the equations. b. You must also use the variable names used for the guess values. 4. Complete the solve block with a find statement. a. find(x,y,...) b. The arguments to the find statement are the unknowns. Demonstration for Single Equation Use a Solve Block to find the largest solution of the equation: x 6 x 5 x 4 = Step : Plot the function to get guesses. f ( x) x 6 x 5 x 4 1 Step 1: Guess x g Step : Given Given f( x) Step 3: Equation x g x g x g = 1 Step 4: Find ans Find x g ans x Key Points: 1. Use := Outside of the given block and bold = inside the given block.. The equations inside the given block must be written in terms of the guess variable. Demonstration for Multiple Equations Find a solution that satisfies the following equations: x y =.9 y = cos π x Step : Plot the functions to get guesses.

4 1 Step 1: Guesses x g.5 y g.9 Step : Given Given Step 3: Equations x g y g =.9 y g = cos πx g.9x cos πx x Step 4: Find xx yy Find x g y g xx.76 yy.98 Key Points: 1. You need a guess value for each unknown.. Any variable not placed as an argument inside the find() is considered a constant. Other Solve Block Tips 1. The fewer number of equations inside the given block, the easier it is for the solution to converge.. You can include Boolean operators (<, >, etc.) inside the given block to create constraints. 3. If you want to obtain other solutions to the same system of equations, you can copy/paste the entire solve block to a new location on the sheet and just change the guesses. 4. A solve block can also be terminated with the minerr(), maximize() and minimize() statements. Minerr Demonstration Background: Sometimes, the solve block cannot converge to a solution, but you want the best solution to equation. Minerr can be used to find the values of the unknowns that minimizes the error. Find the x and y that best satisfy the following expressions. y = x y =.5 x Step : Plot the functions to get guesses. 4 Step 1: Guesses x g 1 y g 1 x.5x Step : Given Given Step 3: Equations y g x g = 4 4 x y g =.5 x g

5 Step 4: Find ans Minerr x g y g ans.5.15 Using Functions in Solve Blocks The heat, q, needed to change the temperature of a gas at low pressures from T 1 to T is given by q = n T c p ( T) dt T 1 where c p is the heat capacity of the gas. If 1, BTU is removed from 5 moles of ethane initially at 8 F, calculate the final temperature of the gas. The heat capacity for ethane is given below. Please report your answe Rankine. Equation c p ( t) t K t K t K t K erg molk Known Quantities: T 1 ( ) R q 1BTU n 5mol Guess T g ( ) R T g Given q = n c p () t dt T FindT g T R T 1 Key Points: 1. Notice that the variable given to c p is the variable of integration, not the guess.. By labeling the guess at T g, you can use T to label the final answer. 3. By using a function for the heat capacity, you can use c p for later calculations. Reducing the Number of Equations in a Solve Block. Background Numerical solvers, such as the solver in Excel or Given..Find Blocks in Mathcad, make use of sophisticated n algorithms to find answers to systems of equations. When using solvers, it is best to reduce the number of eq and unknowns. It is generally easier to solve for two unknowns than three unknowns.

6 Techniques to Reduce Equations 1.. Substitute simple relationships when possible. For example, mole fractions must sum to one, so instead of and x, use x 1 and (1-x 1 ) in your equations. Often, physical properties are correlated as a function of another variable such as temperature. In this case function to reduce the number of equations. Demonstration/Practice Imagine mixing liquid benzene(species 1) and toluene(species ) together in an initially empty contain temperatures and pressures at equilibrium, some of the liquid from both species will evaporate into the and some will be left liquid phase. Raoult's law may be used to describe the distribution of species in e for ideal cases. For the specific example mentioned above, Raoult's law gives the following two expre describing the equilibrium state y 1 P = x 1 Psat 1 y P = x Psat where y i is the mole fraction of species i (either 1 or ) in the vapor phase, x i is the mole fraction of spe liquid phase, Psat i is the vapor pressure of species i at the system temperature T, and P is the system pr vapor pressure of benzene and toluene can be found using the Antoine Equation ln( Psat) = A B T C where Psat is in kpa, T is in C, and A, B, and C are found in the table below. Antione Constants for Acetone and Methanol Compound A B C Benzene Toluene If a mixture of benzene and toluene has y 1 =.33 and P = 1 kpa, find x 1. Solution Method 1 (Poor) Step 1: Define the knowns: y 1.33 P 1 Step : List the unknowns: y x 1 x Psat 1 Psat Step 3: Define guesses for each unknown: y g.5 x1 g.5 x g.5 Psat1 g 1 Psat g 1 Step 4: Use Given..Find to solve for six unknowns. Given x1 g x g = 1 y 1 y g = 1 y 1 P = x1 g Psat1 g y g P = x g Psat g

7 x 1 x y Psat 1 Psat T ln Psat1 g = ln Psat T g g = T g Find x1 g x g y g Psat1 g Psat g T g x x.87 y.67 Questions: 1. What don't you like about the above setup?. How can functions be used to improve the situation? 3. What simple relationships can be used to improve the situation? Psat Psat T Solution Method (Better) ORIGIN 1 A B C B i ( ) expa i t K Psat it C i kpa y 1.33 x1 g.5 T g 4K P 1kPa Given x1 g Psat 1T g = y 1 P 1 x1 g PsatT g = 1 y 1 P x 1 T Find x1 g T g x T K

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12 using x 1, define a er. At vapor phase each phase ession ecies i in the ressure. The T T g 5

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Mathcad Lecture #6 In-class Worksheet Solving Equations and Optimizing Functions

Mathcad Lecture #6 In-class Worksheet Solving Equations and Optimizing Functions At the end of this lecture, you will be able to: solve for the roots of a polynomial using polyroots. obtain approimate