Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 5: Problem 53

Using EES (or other) software, solve these systems of algebraic equations. (a) \(4 x_{1}-x_{2}+2 x_{3}+x_{4}=-6\) $$ \begin{aligned} x_{1}+3 x_{2}-x_{3}+4 x_{4} &=-1 \\ -x_{1}+2 x_{2}+5 x_{4} &=5 \\ 2 x_{2}-4 x_{3}-3 x_{4} &=-5 \end{aligned} $$ (b) $$ \begin{aligned} 2 x_{1}+x_{2}^{4}-2 x_{3}+x_{4} &=1 \\ x_{1}^{2}+4 x_{2}+2 x_{3}^{2}-2 x_{4} &=-3 \\ -x_{1}+x_{2}^{4}+5 x_{3} &=10 \\ 3 x_{1}-x_{3}^{2}+8 x_{4} &=15 \end{aligned} $$

Short Answer

Expert verified
Question: Use EES to solve the following systems of equations: (a) Linear System: 1. \(4 x_{1}-x_{2}+2 x_{3}+x_{4}=-6\) 2. \(x_{1}+3 x_{2}-x_{3}+4 x_{4}=-1\) 3. \(-x_{1}+2 x_{2}+5 x_{4}=5\) 4. \(2 x_{2}-4 x_{3}-3 x_{4}=-5\) (b) Nonlinear System: 1. \(2 x_{1}+x_{2}^{4}-2 x_{3}+x_{4}=1\) 2. \(x_{1}^{2}+4 x_{2}+2 x_{3}^{2}-2 x_{4}=-3\) 3. \(-x_{1}+x_{2}^{4}+5 x_{3}=10\) 4. \(3 x_{1}-x_{3}^{2}+8 x_{4}=15\) Answer: Input the given systems of equations into EES and select "Solve" for both the linear and nonlinear systems. Once solved, EES will provide the values for \(x_1\), \(x_2\), \(x_3\), and \(x_4\) for each system.

Step by step solution

01

Input equations into EES (Linear System)

For the linear system of equations (a), the given equations are: 1. \(4 x_{1}-x_{2}+2 x_{3}+x_{4}=-6\) 2. \(x_{1}+3 x_{2}-x_{3}+4 x_{4}=-1\) 3. \(-x_{1}+2 x_{2}+5 x_{4}=5\) 4. \(2 x_{2}-4 x_{3}-3 x_{4}=-5\) Input these equations into EES as follows (use the same variable names in EES): ``` 4 * x1 - x2 + 2 * x3 + x4 = -6 x1 + 3 * x2 - x3 + 4 * x4 = -1 -x1 + 2 * x2 + 5 * x4 = 5 2 * x2 - 4 * x3 - 3 * x4 = -5 ```
02

Solve the Linear System in EES

Solve the system of linear equations by selecting "Solve" in EES. After solving, EES will provide the values for \(x_1\), \(x_2\), \(x_3\), and \(x_4\).
03

Input equations into EES (Nonlinear System)

For the nonlinear system of equations (b), the given equations are: 1. \(2 x_{1}+x_{2}^{4}-2 x_{3}+x_{4}=1\) 2. \(x_{1}^{2}+4 x_{2}+2 x_{3}^{2}-2 x_{4}=-3\) 3. \(-x_{1}+x_{2}^{4}+5 x_{3}=10\) 4. \(3 x_{1}-x_{3}^{2}+8 x_{4}=15\) Input these equations into EES as follows (use the same variable names in EES): ``` 2 * x1 + x2^4 - 2 * x3 + x4 = 1 x1^2 + 4 * x2 + 2 * x3^2 - 2 * x4 = -3 -x1 + x2^4 + 5 * x3 = 10 3 * x1 - x3^2 + 8 * x4 = 15 ```
04

Solve the Nonlinear System in EES

Solve the system of nonlinear equations by selecting "Solve" in EES. After solving, EES will provide the values for \(x_1\), \(x_2\), \(x_3\), and \(x_4\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider a large uranium plate of thickness \(5 \mathrm{~cm}\) and thermal conductivity \(k=28 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) in which heat is generated uniformly at a constant rate of \(\dot{e}=6 \times 10^{5} \mathrm{~W} / \mathrm{m}^{3}\). One side of the plate is insulated while the other side is subjected to convection to an environment at \(30^{\circ} \mathrm{C}\) with a heat transfer coefficient of \(h=60 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Considering six equally spaced nodes with a nodal spacing of \(1 \mathrm{~cm},(a)\) obtain the finite difference formulation of this problem and \((b)\) determine the nodal temperatures under steady conditions by solving those equations.

What are the basic steps involved in solving a system of equations with Gauss- Seidel method?

The wall of a heat exchanger separates hot water at \(T_{A}=90^{\circ} \mathrm{C}\) from cold water at \(T_{B}=10^{\circ} \mathrm{C}\). To extend the heat transfer area, two-dimensional ridges are machined on the cold side of the wall, as shown in Fig. P5-76. This geometry causes non-uniform thermal stresses, which may become critical for crack initiation along the lines between two ridges. To predict thermal stresses, the temperature field inside the wall must be determined. Convection coefficients are high enough so that the surface temperature is equal to that of the water on each side of the wall. (a) Identify the smallest section of the wall that can be analyzed in order to find the temperature field in the whole wall. (b) For the domain found in part \((a)\), construct a twodimensional grid with \(\Delta x=\Delta y=5 \mathrm{~mm}\) and write the matrix equation \(A T=C\) (elements of matrices \(A\) and \(C\) must be numbers). Do not solve for \(T\). (c) A thermocouple mounted at point \(M\) reads \(46.9^{\circ} \mathrm{C}\). Determine the other unknown temperatures in the grid defined in part (b).

Consider transient heat conduction in a plane wall whose left surface (node 0 ) is maintained at \(50^{\circ} \mathrm{C}\) while the right surface (node 6) is subjected to a solar heat flux of \(600 \mathrm{~W} / \mathrm{m}^{2}\). The wall is initially at a uniform temperature of \(50^{\circ} \mathrm{C}\). Express the explicit finite difference formulation of the boundary nodes 0 and 6 for the case of no heat generation. Also, obtain the finite difference formulation for the total amount of heat transfer at the left boundary during the first three time steps.

A hot surface at \(100^{\circ} \mathrm{C}\) is to be cooled by attaching 3 -cm- long, \(0.25\)-cm-diameter aluminum pin fins \((k=\) \(237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) with a center-to-center distance of \(0.6 \mathrm{~cm}\). The temperature of the surrounding medium is \(30^{\circ} \mathrm{C}\), and the combined heat transfer coefficient on the surfaces is \(35 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Assuming steady one-dimensional heat transfer along the fin and taking the nodal spacing to be \(0.5 \mathrm{~cm}\), determine \((a)\) the finite difference formulation of this problem, \((b)\) the nodal temperatures along the fin by solving these equations, \((c)\) the rate of heat transfer from a single fin, and \((d)\) the rate of heat transfer from a \(1-\mathrm{m} \times 1-\mathrm{m}\) section of the plate.
See all solutions

Recommended explanations on Physics Textbooks

Measurements

Read Explanation

Nuclear Physics

Read Explanation

Mechanics and Materials

Read Explanation

Waves Physics

Read Explanation

Famous Physicists

Read Explanation

Engineering Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free