Heating Ventilating and Air Conditioning: Analysis and Design
6th Edition
ISBN: 9780471470151
Author: Faye C. McQuiston, Jeffrey D. Spitler, Jerald D. Parker
Publisher: Wiley, John & Sons, Incorporated
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Chapter 1, Problem 1.10P
A room with dimensions of
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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
Heating Ventilating and Air Conditioning: Analysis and Design
Ch. 1 - Convert the following quantities from English to...Ch. 1 - Convert the following quantities from SI to...Ch. 1 - A pump develops a total head of 50 ft of water...Ch. 1 - A fan is observed to operate with a pressure...Ch. 1 - The electric utility rate for a facility during...Ch. 1 - For the business whose monthly electrical energy...Ch. 1 - Detmine the interest rate at which the project in...Ch. 1 - How much could a company afford to spend on an...Ch. 1 - Make the following volume and mass flow rate...Ch. 1 - A room with dimensions of 31020m is estimated to...
Ch. 1 - Compute the heat transferred from water as it...Ch. 1 - Air enters a heat exchanger at a rate of 5000...Ch. 1 - Water flowing at a rate of 1.5 kg/s through a heat...Ch. 1 - Air at a mean temperature of 50 F flows over a...Ch. 1 - Repeat Problem 1-10 for air at 10 C, a tube with...Ch. 1 - Air at 1 atm and 76 F is flowing at the rate of...Ch. 1 - Air flowing at the rate of 1000 cfm and with a...Ch. 1 - A chiller is providing 5 tons of cooling to an air...Ch. 1 - Air is delivered to a room at 58 F and the same...Ch. 1 - A chiller is to pro;ide 12 tons of cooling to a...Ch. 1 - Air is being rnished to a 30-ft by 40-ft by 12-ft...Ch. 1 - If cold outside air at 20 F is leaking into a...Ch. 1 - A Btu-meter is a device that measures water flow...Ch. 1 - A heat pump uses a 100,000-gallon swimming pool as...Ch. 1 - A heat pump uses a 100,000-gallon swimming pooi as...
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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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