Thermodynamics: An Engineering Approach
8th Edition
ISBN: 9780073398174
Author: Yunus A. Cengel Dr., Michael A. Boles
Publisher: McGraw-Hill Education
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Chapter 1.11, Problem 27P
To determine
Explain the state of the water in the bathtub and explain the process that the water undergoes when it cools.
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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 1 Solutions
Thermodynamics: An Engineering Approach
Ch. 1.11 - What is the difference between the classical and...Ch. 1.11 - The value of the gravitational acceleration g...Ch. 1.11 - One of the most amusing things a person can...Ch. 1.11 - An office worker claims that a cup of cold coffee...Ch. 1.11 - 1–5C What is the difference between kg-mass and...Ch. 1.11 - Explain why the light-year has the dimension of...Ch. 1.11 - What is the net force acting on a car cruising at...Ch. 1.11 - 1–8 At 45° latitude, the gravitational...Ch. 1.11 - What is the weight, in N, of an object with a mass...Ch. 1.11 - A 3-kg plastic tank that has a volume of 0.2 m3 is...
Ch. 1.11 - Prob. 11PCh. 1.11 - Prob. 12PCh. 1.11 - Solve Prob. 113 using appropriate software. Print...Ch. 1.11 - A 4-kW resistance heater in a water heater runs...Ch. 1.11 - A 150-lbm astronaut took his bathroom scale (a...Ch. 1.11 - The gas tank of a car is filled with a nozzle that...Ch. 1.11 - Prob. 17PCh. 1.11 - A large fraction of the thermal energy generated...Ch. 1.11 - Prob. 19PCh. 1.11 - 1–20C A can or soft drink at room temperature is...Ch. 1.11 - What is the difference between intensive and...Ch. 1.11 - Is the number of moles of a substance contained in...Ch. 1.11 - Is the state of the air in an isolated room...Ch. 1.11 - The specific weight of a system is defined as the...Ch. 1.11 - What is a quasi-equilibrium process? What is its...Ch. 1.11 - Define the isothermal, isobaric, and isochoric...Ch. 1.11 - Prob. 27PCh. 1.11 - Prob. 28PCh. 1.11 - 1–29C What is specific gravity? How is it related...Ch. 1.11 - 1–31C What are the ordinary and absolute...Ch. 1.11 - Prob. 32PCh. 1.11 - Prob. 33PCh. 1.11 - Prob. 34PCh. 1.11 - Prob. 35PCh. 1.11 - Prob. 36PCh. 1.11 - Prob. 37PCh. 1.11 - Prob. 38PCh. 1.11 - The temperature of a system drops by 45F during a...Ch. 1.11 - Explain why some people experience nose bleeding...Ch. 1.11 - A health magazine reported that physicians...Ch. 1.11 - Someone claims that the absolute pressure in a...Ch. 1.11 - 1–43C Express Pascal’s law, and give a real-world...Ch. 1.11 - Consider two identical fans, one at sea level and...Ch. 1.11 - A vacuum gage connected to a chambee reads 35 kPa...Ch. 1.11 - Prob. 46PCh. 1.11 - 1–47E The pressure in a water line is 1500 kPa....Ch. 1.11 - 1–48E If the pressure inside a rubber balloon is...Ch. 1.11 - A manometer is used to measure the air pressure in...Ch. 1.11 - 1–50 The water in a tank is pressurized by air,...Ch. 1.11 - 1–51 Determine the atmospheric pressure at a...Ch. 1.11 - A 200-pound man has a total foot imprint area of...Ch. 1.11 - The gage pressure in a liquid at a depth of 3 m is...Ch. 1.11 - The absolute pressure in water at a depth of 9 m...Ch. 1.11 - 1–55E Determine the pressure exerted on the...Ch. 1.11 - 1–56 Consider a 70-kg woman who has a total foot...Ch. 1.11 - Prob. 57PCh. 1.11 - The barometer of a mountain hiker reads 750 mbars...Ch. 1.11 - The basic barometer can be used to measure the...Ch. 1.11 - Prob. 61PCh. 1.11 - A gas is contained in a vertical, frictionless...Ch. 1.11 - Reconsider Prob. 158. Using appropriate software,...Ch. 1.11 - Both a gage and a manometer are attached to a gas...Ch. 1.11 - Reconsider Prob. 161. Using appropriate software,...Ch. 1.11 - A manometer containing oil ( = 850 kg/m3) is...Ch. 1.11 - A mercury manometer ( = 13.600 kg/m3) is connected...Ch. 1.11 - Repeat Prob. 165 for a differential mercury height...Ch. 1.11 - The pressure in a natural gas pipeline is measured...Ch. 1.11 - Repeat Prob. 167E by replacing air with oil with a...Ch. 1.11 - Blood pressure is usually measure by wrapping a...Ch. 1.11 - The maximum blood pressure in the upper arm of a...Ch. 1.11 - Prob. 73PCh. 1.11 - Consider a U-tube whose arms are open to the...Ch. 1.11 - Consider a double-fluid manometer attached to an...Ch. 1.11 - Prob. 76PCh. 1.11 - Prob. 77PCh. 1.11 - Calculate the absolute pressure. P1, of the...Ch. 1.11 - Consider the manometer in Fig. 173. If the...Ch. 1.11 - Consider the manometer in Fig. 173. If the...Ch. 1.11 - Consider the system shown in Fig. 177. If a change...Ch. 1.11 - What is the value of the engineering software...Ch. 1.11 - Determine a positive real root of this equation...Ch. 1.11 - Solve this system of three equations with three...Ch. 1.11 - Solve this system of three equations with three...Ch. 1.11 - The reactive force developed by a jet engine to...Ch. 1.11 - A man goes to a traditional market to buy a steak...Ch. 1.11 - What is the weight of a 1-kg substance in N, kN,...Ch. 1.11 - A hydraulic lift is to be used to lift a 1900-kg...Ch. 1.11 - Prob. 92RPCh. 1.11 - Prob. 93RPCh. 1.11 - Prob. 94RPCh. 1.11 - Prob. 95RPCh. 1.11 - Prob. 96RPCh. 1.11 - It is well known that cold air feels much colder...Ch. 1.11 - Reconsider Prob. 1116E. Using appropriate...Ch. 1.11 - A vertical pistoncylinder device contains a gas at...Ch. 1.11 - An air-conditioning system requires a 35-m-long...Ch. 1.11 - The average body temperature of a person rises by...Ch. 1.11 - Balloons are often filled with helium gas because...Ch. 1.11 - Reconsider Prob. 1101. Using appropriate software,...Ch. 1.11 - Determine the maximum amount of load, in kg, the...Ch. 1.11 - The lower half of a 6-m-high cylindrical container...Ch. 1.11 - A vertical, frictionless pistoncylinder device...Ch. 1.11 - A pressure cooker cooks a lot faster than an...Ch. 1.11 - Prob. 108RPCh. 1.11 - Consider a U-tube whose arms are open to the...Ch. 1.11 - Prob. 110RPCh. 1.11 - A water pipe is connected to a double-U manometer...Ch. 1.11 - A gasoline line is connected to a pressure gage...Ch. 1.11 - Repeat Prob. 1110 for a pressure gage reading of...Ch. 1.11 - The average atmosphere pressure on earth is...Ch. 1.11 - Prob. 115RPCh. 1.11 - Prob. 116RPCh. 1.11 - Consider the flow of air through a wind turbine...Ch. 1.11 - The drag force exerted on a car by air depends on...Ch. 1.11 - An apple loses 3.6 kJ of heat as it cools per C...Ch. 1.11 - Consider a fish swimming 5 m below the free...Ch. 1.11 - The atmospheric pressures at the top and the...Ch. 1.11 - Consider a 2.5-m-deep swimming pool. The pressure...Ch. 1.11 - During a heating process, the temperature of an...Ch. 1.11 - At sea level, the weight of 1 kg mass in SI units...
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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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