Helpful information: Interior Length is L – w Interior Volume = pi *(R-t/2)^2 *

Question

Helpful information:

Interior Length is L – w
Interior Volume = pi *(R-t/2)^2 * (L-w)
For endcap the Radius is (R +t/2)
Height of each end cap is w
Volume of 2 endcap is 2* pi* (R+t/2)^2 * w
Inner Radius of Cyl wall is (R-t/2)
Outer Radius if Cyl wall is (R+t/2)
Volume of the Cylindrical shell is [pi*(R+t/2)^2 –pi*(R-t/2)^2] * (L-w)
This can be simplified to 2*pi*R*t*(L-w)
Mass of the solid volume = [2* pi* (R+t/2)^2 * w +2*pi*R*t*(L-w)]*Rho
Or [2 end cap volume + cyl shell volume ]*Rho

In the design of a closed-end, thin-walled cylindrical pressure vessel shown in the figure below, the design objective is to select the mean radius R and wall thickness t to minimize the total mass. The vessel should contain at least 30.0 m3 of gas at an internal pressure of P = 5.0 MPa. It is required that the circumferential stress in the pressure vessel not exceed 300 MPa and the circumferential strain not exceed 2.0x103.The circumferential stress and strain are calculated from the equations: PR(2-v) e2Et PR where ,is the circumferential stress (in Pa) and is the circumferential strain. Parameters describing the vessel are = density of vessel = 7850 kg/m3 E Young's modulus 210 GPa v= Poisson's ratio L= Length of vessel 10.0 m w Thickness of endcap 5.0 cm = 0.30 5.0 cm Gas 10.0 m (a) Carefully set up the problem and concisely describe the optimization problenm (b) Solve the problem graphically (c) Solve the problem using Excel (d) Solve the problem using Matlab

Explanation / Answer

OBJECT IS CLOSED ENDED THIN WALLED CYLINDER DECISION VARIABLES MEAN RADIUS = R M WALL THICK NESS T M DEPENDENT FUNCTIONS …. TOTAL VOLUME =V= PI* L * [(R-0.5T)^2 ] …………………………………………………………….1 CIRCUMFERENTIAL STRESS = Sc = P*R / T ………………………………………………….2 CIRCUMFERENTIAL STRAIN = Ec = P*R*(2-v) /(2*E*T) ………………………………………..3 HOPE THE FORMULAS GIVEN BY YOU ARE DIMENSIONALLY CONSTANT …. ANY EMPIRICAL CONSTANTS ARE IN CONSISTENCY WITH UNITS GIVEN FOR THE DATA DIMENSIONAL CONSISTENCY IS ESSENTIAL FOR EXAMPLE WALL THICK NESS OF END CAP IS IN CM … I CHANGED IT TO METERS …BUT OTHERS I HAVE NOT MADE ANY CHANGES E - YOUNGS MODULUS IS IN GPA …. I CHAGED IT TO MPA OPTIMIZING FUNCTION ….MASS = M OUTER WALL RADIUS = R + 0.5 * T INNER WALL RADIUS = R - 0.5*T RADIUS OF TOP & BOTTOM ENDS = R+0.5 * T LENGTH OF VESSEL = L = TOTAL AREA OF MATERIAL = 2*PI*[(L*R)+(R+0.5)^2] TOTAL MASS OF MATERIAL    =    M   = D * [ 2*PI*{L*R*T)+ W*(R+0.5)^2}] WHERE D=DENSITY OF MATERIAL = GAMMA AS GIVEN BY YOU W = THICKNESS OF END CAP GIVEN DATA .. D= GAMMA = 7850 KG/CUM E = YOUNGS MODULUS = 210 GPA 210000 MPA v = POISSONS RATIO = 0.3 L = LENGTH OF VESSEL = 10 M W = THICKNESS OF END CAP 5 CM 0.05 M P= 5 MPA FORMULAS WITH THE GIVEN DATA … V= 31.41592654 *[(R-0.5T)^2 ] …………………….1 Sc = P*R / T 5 * [R / T ] ……………...………………………….2 Ec = P*R*(2-v) /(2*E*T) 2.02381E-05 * [R / T ] …….……………………………..3 M = 7850*[ 2*3.1416*{(10*R*T)+ 0.05*(R+0.5)^2}] ……………………….4 OPTIMIZING FUNCTION MINIMIZE ……M = 7850*[ 2*3.1416*{(10*R*T)+ 0.05*(R+0.5)^2}] CONSTARAINTS …. V > = 30 CUM Sc < = 300 MPA Ec < = 0.002

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