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Thermodynamics: An Engineering Approach
8th Edition
ISBN: 9780073398174
Author: Yunus A. Cengel Dr., Michael A. Boles
Publisher: McGraw-Hill Education
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Question
Chapter 4.5, Problem 88P
To determine
The total rate of heat transfer
Expert Solution & Answer
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Problem 2 (55 pts). We now consider the FEM solution of Problem 1.(a) [5pts] Briefly describe the 4 steps necessary to obtain the approximate solution of thatBVP using the Galerkin FEM. Use the minimum amount of math necessary to supportyour explanations.(b) [20pts] Derive the weak form of the BVP.(c) [10pts] Assuming a mesh of two equal elements and linear shape functions, sketch byhand how you expect the FEM solution to look like. Also sketch the analytical solutionfor comparison. In your sketch, identify the nodal degrees of freedom that the FEMsolution seeks to find.(d) [10pts] By analogy with the elastic rod problem and heat conduction problem considered in class, write down the stiffness matrix and force vector for each of the twoelements considered in (c).(e) [10pts] Assemble the global system of equations, and verbally explain how to solve it.
An aluminum rod of length L = 1m has mass density ρ = 2700 kgm3 andYoung’s modulus E = 70GPa. The rod is fixed at both ends. The exactnatural eigenfrequencies of the rod are ωexactn =πnLqEρfor n=1,2,3,. . . .1. What is the minimum number of linear elements necessary todetermine the fundamental frequency ω1 of the system? Discretizethe rod in that many elements of equal length, assemble the globalsystem of equations KU = ω2MU, and find the fundamentalfrequency ω1. Compute the relative error e1 = (ω1 − ωexact1)/ωexact1.Sketch the fundamental mode of vibration.
Chapter 4 Solutions
Thermodynamics: An Engineering Approach
Ch. 4.5 - An ideal gas at a given state expands to a fixed...Ch. 4.5 - Nitrogen at an initial state of 300 K, 150 kPa,...Ch. 4.5 - 4–3 The volume of 1 kg of helium in a...Ch. 4.5 - 4–4E Calculate the total work, in Btu, for process...Ch. 4.5 - 4–5 A piston–cylinder device initially contains...Ch. 4.5 - A pistoncylinder device with a set of stops...Ch. 4.5 - 4–7 A piston–cylinder device initially contains...Ch. 4.5 - 4–8 A mass of 5 kg of saturated water vapor at 300...Ch. 4.5 - 1 m3 of saturated liquid water at 200C is expanded...Ch. 4.5 - A gas is compressed from an initial volume of 0.42...
Ch. 4.5 - A mass of 1.5 kg of air at 120 kPa and 24C is...Ch. 4.5 - During some actual expansion and compression...Ch. 4.5 - 4–14 A frictionless piston–cylinder device...Ch. 4.5 - Prob. 15PCh. 4.5 - During an expansion process, the pressure of a gas...Ch. 4.5 - A pistoncylinder device initially contains 0.4 kg...Ch. 4.5 - 4–19E Hydrogen is contained in a piston–cylinder...Ch. 4.5 - A pistoncylinder device contains 0.15 kg of air...Ch. 4.5 - 1 kg of water that is initially at 90C with a...Ch. 4.5 - Prob. 22PCh. 4.5 - An ideal gas undergoes two processes in a...Ch. 4.5 - A pistoncylinder device contains 50 kg of water at...Ch. 4.5 - Prob. 26PCh. 4.5 - 4–27E A closed system undergoes a process in which...Ch. 4.5 - A rigid container equipped with a stirring device...Ch. 4.5 - A 0.5-m3rigid tank contains refrigerant-134a...Ch. 4.5 - A 20-ft3 rigid tank initially contains saturated...Ch. 4.5 - Prob. 31PCh. 4.5 - Prob. 32PCh. 4.5 - Prob. 33PCh. 4.5 - An insulated pistoncylinder device contains 5 L of...Ch. 4.5 -
4–35 A piston–cylinder device initially...Ch. 4.5 - Prob. 37PCh. 4.5 - A 40-L electrical radiator containing heating oil...Ch. 4.5 - Steam at 75 kPa and 8 percent quality is contained...Ch. 4.5 - Prob. 40PCh. 4.5 - An insulated tank is divided into two parts by a...Ch. 4.5 - Is the relation u = mcv,avgT restricted to...Ch. 4.5 - Is the relation h = mcp,avgT restricted to...Ch. 4.5 - Is the energy required to heat air from 295 to 305...Ch. 4.5 - A fixed mass of an ideal gas is heated from 50 to...Ch. 4.5 - A fixed mass of an ideal gas is heated from 50 to...Ch. 4.5 - A fixed mass of an ideal gas is heated from 50 to...Ch. 4.5 - Prob. 49PCh. 4.5 - What is the change in the enthalpy, in kJ/kg, of...Ch. 4.5 - Prob. 51PCh. 4.5 - Prob. 52PCh. 4.5 - Prob. 53PCh. 4.5 - Determine the internal energy change u of...Ch. 4.5 - Prob. 55PCh. 4.5 - Prob. 56PCh. 4.5 - Is it possible to compress an ideal gas...Ch. 4.5 - A 3-m3 rigid tank contains hydrogen at 250 kPa and...Ch. 4.5 - A 10-ft3 tank contains oxygen initially at 14.7...Ch. 4.5 - 4–60E A rigid tank contains 10 Ibm of air at 30...Ch. 4.5 - 4–61E Nitrogen gas to 20 psia and 100°F initially...Ch. 4.5 - An insulated rigid tank is divided into two equal...Ch. 4.5 - 4–63 A 4-m × 5-m × 6-m room is to be heated by a...Ch. 4.5 - 4-64 A student living in a 3-m × 4-m × 4-m...Ch. 4.5 - A 4-m 5-m 7-m room is heated by the radiator of...Ch. 4.5 - 4–66 Argon is compressed in a polytropic process...Ch. 4.5 - An insulated pistoncylinder device contains 100 L...Ch. 4.5 - 4–68 A spring-loaded piston-cylinder device...Ch. 4.5 - An ideal gas contained in a pistoncylinder device...Ch. 4.5 - Air is contained in a variable-load pistoncylinder...Ch. 4.5 - Prob. 71PCh. 4.5 - Prob. 72PCh. 4.5 - Prob. 74PCh. 4.5 - Prob. 75PCh. 4.5 - Prob. 76PCh. 4.5 - 4–77 Air is contained in a piston-cylinder device...Ch. 4.5 - A pistoncylinder device contains 4 kg of argon at...Ch. 4.5 - The state of liquid water is changed from 50 psia...Ch. 4.5 - During a picnic on a hot summer day, all the cold...Ch. 4.5 - Consider a 1000-W iron whose base plate is made of...Ch. 4.5 - Stainless steel ball bearings ( = 8085 kg/m3 and...Ch. 4.5 - In a production facility, 1.6-in-thick 2-ft 2-ft...Ch. 4.5 - Prob. 84PCh. 4.5 - An electronic device dissipating 25 W has a mass...Ch. 4.5 - Prob. 87PCh. 4.5 - 4–88 In a manufacturing facility, 5-cm-diameter...Ch. 4.5 - Prob. 89PCh. 4.5 - Is the metabolizable energy content of a food the...Ch. 4.5 - Is the number of prospective occupants an...Ch. 4.5 - Prob. 92PCh. 4.5 - Prob. 93PCh. 4.5 - Consider two identical 80-kg men who are eating...Ch. 4.5 - A 68-kg woman is planning to bicycle for an hour....Ch. 4.5 - A 90-kg man gives in to temptation and eats an...Ch. 4.5 - A 60-kg man used to have an apple every day after...Ch. 4.5 - Consider a man who has 20 kg of body fat when he...Ch. 4.5 - Consider two identical 50-kg women, Candy and...Ch. 4.5 - Prob. 100PCh. 4.5 - Prob. 101PCh. 4.5 - Prob. 102PCh. 4.5 - Prob. 103PCh. 4.5 - Prob. 104PCh. 4.5 - Prob. 105PCh. 4.5 - Prob. 106PCh. 4.5 - Prob. 107RPCh. 4.5 - Consider a pistoncylinder device that contains 0.5...Ch. 4.5 - Air in the amount of 2 lbm is contained in a...Ch. 4.5 - Air is expanded in a polytropic process with n =...Ch. 4.5 - Nitrogen at 100 kPa and 25C in a rigid vessel is...Ch. 4.5 - Prob. 112RPCh. 4.5 - Prob. 113RPCh. 4.5 - Prob. 114RPCh. 4.5 - 4–115 A mass of 12 kg of saturated...Ch. 4.5 - Prob. 116RPCh. 4.5 - Prob. 117RPCh. 4.5 - Prob. 118RPCh. 4.5 - Prob. 119RPCh. 4.5 - Prob. 120RPCh. 4.5 - Prob. 121RPCh. 4.5 - Prob. 122RPCh. 4.5 - Prob. 123RPCh. 4.5 - Prob. 124RPCh. 4.5 - Prob. 125RPCh. 4.5 - Prob. 126RPCh. 4.5 - Prob. 127RPCh. 4.5 - Prob. 128RPCh. 4.5 - A well-insulated 3-m 4m 6-m room initially at 7C...Ch. 4.5 - Prob. 131RPCh. 4.5 - Prob. 133RPCh. 4.5 - Prob. 134RPCh. 4.5 - An insulated pistoncylinder device initially...Ch. 4.5 - Prob. 137RPCh. 4.5 - Prob. 138RPCh. 4.5 - A pistoncylinder device initially contains 0.35 kg...Ch. 4.5 - Prob. 140RPCh. 4.5 - 4–141 One kilogram of carbon dioxide is compressed...Ch. 4.5 - Prob. 142RPCh. 4.5 - Prob. 143RPCh. 4.5 - Prob. 144FEPCh. 4.5 - A 3-m3 rigid tank contains nitrogen gas at 500 kPa...Ch. 4.5 - Prob. 146FEPCh. 4.5 - A well-sealed room contains 60 kg of air at 200...Ch. 4.5 - Prob. 148FEPCh. 4.5 - A room contains 75 kg of air at 100 kPa and 15C....Ch. 4.5 - A pistoncylinder device contains 5 kg of air at...Ch. 4.5 - Prob. 151FEPCh. 4.5 - Prob. 152FEPCh. 4.5 - A 2-kW electric resistance heater submerged in 5...Ch. 4.5 - 1.5 kg of liquid water initially at 12C is to be...Ch. 4.5 - An ordinary egg with a mass of 0.1 kg and a...Ch. 4.5 - 4–156 An apple with an average mass of 0.18 kg and...Ch. 4.5 - A 6-pack of canned drinks is to be cooled from 18C...Ch. 4.5 - An ideal gas has a gas constant R = 0.3 kJ/kgK and...Ch. 4.5 - Prob. 159FEPCh. 4.5 - Prob. 161FEP
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- Problem 1 (65 pts, suggested time 50 mins). An elastic string of constant line tension1T is pinned at x = 0 and x = L. A constant distributed vertical force per unit length p(with units N/m) is applied to the string. Under this force, the string deflects by an amountv(x) from its undeformed (horizontal) state, as shown in the figure below.The PDE describing mechanical equilibrium for the string isddx Tdvdx− p = 0 . (1)(a) [5pts] Identify the BCs for the string and identify their type (essential/natural). Writedown the strong-form BVP for the string, including PDE and BCs.(b) [10pts] Find the analytical solution of the BVP in (a). Compute the exact deflectionof the midpoint v(L/2).(c) [15pts] Derive the weak-form BVP.(d) [5pts] What is the minimum number of linear elements necessary to compute the deflection of the midpoint?(e) [15pts] Write down the element stiffness matrix and the element force vector for eachelement.arrow_forwardProblem 1 (35 pts). An elastic string of constant line tension1 T is pinned at x = 0 andx = L. A constant distributed vertical force per unit length p (with units N/m) is appliedto the string. Under this force, the string deflects by an amount v(x) from its undeformed(horizontal) state, as shown in the figure below.Force equilibrium in the string requires thatdfdx − p = 0 , (1)where f(x) is the internal vertical force in the string, which is given byf = Tdvdx . (2)(a) [10pts] Write down the BVP (strong form) that the string deflection v(x) must satisfy.(b) [2pts] What order is the governing PDE in the BVP of (a)?(c) [3pts] Identify the type (essential/natural) of each boundary condition in (a).(d) [20pts] Find the analytical solution of the BVP in (a).arrow_forwardProblem 2 (25 pts, (suggested time 15 mins). An elastic string of line tension T andmass per unit length µ is pinned at x = 0 and x = L. The string is free to vibrate, and itsfirst vibration mode is shown below.In order to find the frequency of the first mode (or fundamental frequency), the string isdiscretized into a certain number of linear elements. The stiffness and mass matrices of thei-th element are, respectivelyESMi =TLi1 −1−1 1 EMMi =Liµ62 11 2 . (2)(a) [5pts] What is the minimum number of linear elements necessary to compute the fundamental frequency of the vibrating string?(b) [20pts] Assemble the global eigenvalue problem and find the fundamental frequency ofvibration of the stringarrow_forward
- I need part all parts please in detail (including f)arrow_forwardProblem 3 (10 pts, suggested time 5 mins). In class we considered the mutiphysics problem of thermal stresses in a rod. When using linear shape functions, we found that the stress in the rod is affected by unphysical oscillations like in the following plot E*(ux-a*T) 35000 30000 25000 20000 15000 10000 5000 -5000 -10000 0 Line Graph: E*(ux-a*T) MULT 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Arc length (a) [10pts] What is the origin of this issue and how can we fix it?arrow_forwardanswer the questions and explain all of it in words. Ignore where it says screencast and in class explanationarrow_forward
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