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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 2.8, Problem 111RP
To determine
The heating system with the lowest energy cost.
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 2 Solutions
Thermodynamics: An Engineering Approach
Ch. 2.8 - What is total energy? Identify the different forms...Ch. 2.8 - List the forms of energy that contribute to the...Ch. 2.8 - How are heat, internal energy, and thermal energy...Ch. 2.8 - What is mechanical energy? How does it differ from...Ch. 2.8 - Natural gas, which is mostly methane CH4, is a...Ch. 2.8 - Portable electric heaters are commonly used to...Ch. 2.8 - Prob. 7PCh. 2.8 - Prob. 8PCh. 2.8 - 2–9E Calculate the total potential energy, in Btu,...Ch. 2.8 - Prob. 10P
Ch. 2.8 - Prob. 11PCh. 2.8 - At a certain location, wind is blowing steadily at...Ch. 2.8 - A water jet that leaves a nozzle at 60 m/s at a...Ch. 2.8 - Prob. 14PCh. 2.8 - Prob. 15PCh. 2.8 - Consider a river flowing toward a lake at an...Ch. 2.8 - When is the energy crossing the boundaries of a...Ch. 2.8 - Consider an automobile traveling at a constant...Ch. 2.8 - A gas in a pistoncylinder device is compressed,...Ch. 2.8 - A room is heated by an iron that is left plugged...Ch. 2.8 - A room is heated as a result of solar radiation...Ch. 2.8 - Prob. 23PCh. 2.8 - A small electrical motor produces 5 W of...Ch. 2.8 - Prob. 25PCh. 2.8 - 2–26C Lifting a weight to a height of 20 m takes...Ch. 2.8 - Prob. 27PCh. 2.8 - Prob. 28PCh. 2.8 - Prob. 29PCh. 2.8 - Prob. 30PCh. 2.8 - Prob. 31PCh. 2.8 - Prob. 32PCh. 2.8 - Prob. 33PCh. 2.8 - A ski lift has a one-way length of 1 km and a...Ch. 2.8 - The engine of a 1500-kg automobile has a power...Ch. 2.8 - Prob. 36PCh. 2.8 - What are the different mechanisms for transferring...Ch. 2.8 - On a hot summer day, a student turns his fan on...Ch. 2.8 - Prob. 39PCh. 2.8 - A vertical pistoncylinder device contains water...Ch. 2.8 - At winter design conditions, a house is projected...Ch. 2.8 - A water pump increases the water pressure from 15...Ch. 2.8 - Prob. 43PCh. 2.8 - Prob. 44PCh. 2.8 - A university campus has 200 classrooms and 400...Ch. 2.8 - Prob. 46PCh. 2.8 - Consider a room that is initially at the outdoor...Ch. 2.8 - Prob. 48PCh. 2.8 - 2-49 The 60-W fan of a central heating system is...Ch. 2.8 - Prob. 50PCh. 2.8 - An escalator in a shopping center is designed to...Ch. 2.8 - Prob. 52PCh. 2.8 - How is the combined pumpmotor efficiency of a pump...Ch. 2.8 - Prob. 54PCh. 2.8 - Can the combined turbinegenerator efficiency be...Ch. 2.8 - Consider a 2.4-kW hooded electric open burner in...Ch. 2.8 - Prob. 57PCh. 2.8 - Prob. 58PCh. 2.8 - Prob. 59PCh. 2.8 - A geothermal pump is used to pump brine whose...Ch. 2.8 - Prob. 62PCh. 2.8 - Prob. 63PCh. 2.8 - The water in a large lake is to be used to...Ch. 2.8 - A 7-hp (shaft) pump is used to raise water to an...Ch. 2.8 - At a certain location, wind is blowing steadily at...Ch. 2.8 - Reconsider Prob. 265. Using appropriate software,...Ch. 2.8 - Water is pumped from a lake to a storage tank 15 m...Ch. 2.8 - Prob. 69PCh. 2.8 - A hydraulic turbine has 85 m of elevation...Ch. 2.8 - Prob. 71PCh. 2.8 - Water is pumped from a lower reservoir to a higher...Ch. 2.8 - Prob. 73PCh. 2.8 - An oil pump is drawing 44 kW of electric power...Ch. 2.8 - How does energy conversion affect the environment?...Ch. 2.8 - What is acid rain? Why is it called a rain? How do...Ch. 2.8 - Why is carbon monoxide a dangerous air pollutant?...Ch. 2.8 - What is the greenhouse effect? How does the excess...Ch. 2.8 - What is smog? What does it consist of? How does...Ch. 2.8 - Prob. 80PCh. 2.8 - Consider a household that uses 14,000 kWh of...Ch. 2.8 - When a hydrocarbon fuel is burned, almost all of...Ch. 2.8 - Prob. 83PCh. 2.8 - A typical car driven 20,000 km a year emits to the...Ch. 2.8 - What are the mechanisms of heat transfer?Ch. 2.8 - Which is a better heat conductor, diamond or...Ch. 2.8 - How does forced convection differ from natural...Ch. 2.8 - What is a blackbody? How do real bodies differ...Ch. 2.8 - Define emissivity and absorptivity. What is...Ch. 2.8 - Does any of the energy of the sun reach the earth...Ch. 2.8 - The inner and outer surfaces of a 5-m 6-m brick...Ch. 2.8 - The inner and outer surfaces of a 0.5-cm-thick 2-m...Ch. 2.8 - Reconsider Prob. 292. Using appropriate software,...Ch. 2.8 - Prob. 94PCh. 2.8 - Prob. 95PCh. 2.8 - Prob. 96PCh. 2.8 - Prob. 97PCh. 2.8 - For heat transfer purposes, a standing man can be...Ch. 2.8 - Prob. 99PCh. 2.8 - Prob. 100PCh. 2.8 - A 1000-W iron is left on the ironing board with...Ch. 2.8 - A 7-cm-external-diameter, 18-m-long hot-water pipe...Ch. 2.8 - A thin metal plate is insulated on the back and...Ch. 2.8 - Reconsider Prob. 2103. Using appropriate software,...Ch. 2.8 - The outer surface of a spacecraft in space has an...Ch. 2.8 - Prob. 106PCh. 2.8 - A hollow spherical iron container whose outer...Ch. 2.8 - Consider a vertical elevator whose cabin has a...Ch. 2.8 - Consider a homeowner who is replacing his...Ch. 2.8 - Prob. 110RPCh. 2.8 - Prob. 111RPCh. 2.8 - Prob. 112RPCh. 2.8 - 2–113 The U.S. Department of Energy estimates that...Ch. 2.8 - Prob. 114RPCh. 2.8 - Prob. 115RPCh. 2.8 - Prob. 116RPCh. 2.8 - Prob. 117RPCh. 2.8 - Consider a TV set that consumes 120 W of electric...Ch. 2.8 - Water is pumped from a 200-ft-deep well into a...Ch. 2.8 - Prob. 120RPCh. 2.8 - Prob. 121RPCh. 2.8 - In a hydroelectric power plant, 65 m3/s of water...Ch. 2.8 - The demand for electric power is usually much...Ch. 2.8 - The pump of a water distribution system is powered...Ch. 2.8 - On a hot summer day, the air in a well-sealed room...Ch. 2.8 - Prob. 126FEPCh. 2.8 - A 2-kW electric resistance heater in a room is...Ch. 2.8 - A 900-kg car cruising at a constant speed of 60...Ch. 2.8 - Prob. 129FEPCh. 2.8 - Prob. 130FEPCh. 2.8 - Prob. 131FEPCh. 2.8 - A 2-kW pump is used to pump kerosene ( = 0.820...Ch. 2.8 - Prob. 133FEPCh. 2.8 - Prob. 134FEPCh. 2.8 - Prob. 135FEPCh. 2.8 - Prob. 136FEPCh. 2.8 - Prob. 137FEPCh. 2.8 - Heat is transferred steadily through a...Ch. 2.8 - The roof of an electrically heated house is 7 m...
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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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