Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
8th Edition
ISBN: 9781305387102
Author: Kreith, Frank; Manglik, Raj M.
Publisher: Cengage Learning
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Chapter 1, Problem 1.77P
Explain each in your own words. (a) What is the mode of heat transfer through a large steel plate that has its surfaces at specified temperatures? (b) What are the modes when the temperature on one surface of the steel plate is not specified, but the surface is exposed to a fluid at a specified temperature?
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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
Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)
Ch. 1 - 1.1 On a cold winter day, the outer surface of a...Ch. 1 - 1.2 The weight of the insulation in a spacecraft...Ch. 1 - 1.3 A furnace wall is to be constructed of brick...Ch. 1 - 1.4 To measure thermal conductivity, two similar...Ch. 1 - To determine the thermal conductivity of a...Ch. 1 - A square silicon chip 7mm7mm in size and 0.5-mm...Ch. 1 - A cooling system is to be designed for a food...Ch. 1 - 1.80 Describe and compare the modes of heat loss...Ch. 1 - Heat is transferred at a rate of 0.1 kW through...Ch. 1 - 1.10 A heat flux meter at the outer (cold) wall of...
Ch. 1 - 1.11 Calculate the heat loss through a glass...Ch. 1 - 1.12 A wall with a thickness is made of a...Ch. 1 - 1.13 If the outer air temperature in Problem is...Ch. 1 - Using Table 1.4 as a guide, prepare a similar...Ch. 1 - 1.15 A thermocouple (0.8-mm-diameter wire) used to...Ch. 1 - Water at a temperature of 77C is to be evaporated...Ch. 1 - The heat transfer rate from hot air by convection...Ch. 1 - The heat transfer coefficient for a gas flowing...Ch. 1 - 1.19 A cryogenic fluid is stored in a...Ch. 1 - A high-speed computer is located in a...Ch. 1 - 1.21 In an experimental set up in a laboratory, a...Ch. 1 - 1.22 In order to prevent frostbite to skiers on...Ch. 1 - Using the information in Problem 1.22, estimate...Ch. 1 - Two large parallel plates with surface conditions...Ch. 1 - 1.25 A spherical vessel, 0.3 m in diameter, is...Ch. 1 - 1.26 Repeat Problem 1.25 but assume that the...Ch. 1 - Determine the rate of radiant heat emission in...Ch. 1 - 1.28 The sun has a radius of and approximates a...Ch. 1 - 1.29 A spherical interplanetary probe with a 30-cm...Ch. 1 - A spherical communications satellite, 2 m in...Ch. 1 - A long wire 0.7 mm in diameter with an emissivity...Ch. 1 - Wearing layers of clothing in cold weather is...Ch. 1 - A section of a composite wall with the dimensions...Ch. 1 - A section of a composite wall with the dimensions...Ch. 1 - Repeat Problem 1.35 but assume that instead of...Ch. 1 - 1.37 Mild steel nails were driven through a solid...Ch. 1 - Prob. 1.38PCh. 1 - 1.39 On a cold winter day, the outside wall of a...Ch. 1 - As a designer working for a major electric...Ch. 1 - 1.41 A heat exchanger wall consists of a copper...Ch. 1 - 1.43 A simple solar heater consists of a flat...Ch. 1 - A composite refrigerator wall is composed of 5 cm...Ch. 1 - An electronic device that internally generates 600...Ch. 1 - 1.47 A flat roof is modeled as a flat plate...Ch. 1 - A horizontal, 3-mm-thick flat-copper plate, 1-m...Ch. 1 - 1.49 A small oven with a surface area of is...Ch. 1 - A steam pipe 200 mm in diameter passes through a...Ch. 1 - 1.51 The inner wall of a rocket motor combustion...Ch. 1 - 1.52 A flat roof of a house absorbs a solar...Ch. 1 - Determine the power requirement of a soldering...Ch. 1 - 1.54 The soldering iron tip in Problem 1.53...Ch. 1 - Prob. 1.55PCh. 1 - A pipe carrying superheated steam in a basement at...Ch. 1 - Draw the thermal circuit for heat transfer through...Ch. 1 - 1.60 Two electric resistance heaters with a 20 cm...Ch. 1 - 1.63 Liquid oxygen (LOX) for the space shuttle is...Ch. 1 - The interior wall of a large, commercial walk-in...Ch. 1 - 1.67 In beauty salons and in homes, a ubiquitous...Ch. 1 - The heat transfer coefficient between a surface...Ch. 1 - The thermal conductivity of fibreglass insulation...Ch. 1 - 1.71 The thermal conductivity of silver at 212°F...Ch. 1 - 1.72 An ice chest (see sketch) is to constructed...Ch. 1 - Estimate the R-values for a 5-cm-thick fiberglass...Ch. 1 - A manufacturer in the United States wants to sell...Ch. 1 - Referring to Problem 1.74, how many kilograms of...Ch. 1 - 1.76 Explain a fundamental characteristic that...Ch. 1 - 1.77 Explain each in your own words. (a) What is...Ch. 1 - What are the important modes of heat transfer for...Ch. 1 - 1.79 Consider the cooling of (a) a personal...Ch. 1 - Describe and compare the modes of heat loss...Ch. 1 - A person wearing a heavy parka is standing in a...Ch. 1 - Discuss the modes of heat transfer that determine...
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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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