Pressure vessel problems



In the following, reference to Australian Standards implies use of the extracts appearing above.

  1. The equation of a parabola in the meridional plane is   2cz = r2 where   c is a constant. Show that the membrane stresses in a paraboloid of revolution, of wall thickness   t and subjected to a pressure differential   p, are :-
    σφ   =   k ( r2 + c2 )1/2       ;       σθ   =   k ( 2r2 + c2 ) / ( r2 + c2 )1/2       where   k   =   p/2t
    An observation dome for an undersea vehicle is made of glass with an allowable compressive strength of   35 MPa. If the dome geometry is defined by   c = 60 mm and   rmax = 150 mm, how thick should it be if the vessel is to operate at depths up to 200 m? The density of sea water may be taken as 10 kN/m3.           [ 8.6 mm]
     
  2. conical tank A conical tank is supported circumferentially around its periphery and filled with a liquid of density ρ to a height h.
    Derive the membrane stress equations :-
    σφ   =   k ( 3h - 2z ) z     ;     σθ   =   6 k ( h - z ) z     where   k   =   ρ sinα / 6t cos2α
                            noting that ( iii) must be modified for liquid weight in the apex.
    A steel tank of this form and semi-angle α = 45o contains water to a depth of 6.5 m. If the design stress is 65 MPa based on the distortion energy failure theory, calculate the wall thickness necessary and the diametral strain at the water level.         [ 2 mm, -71 μstrain]
     
  3. An unfired pressure vessl of 920 mm bore and 2.5 MPa operating pressure has to be manufactured to AS 1210. A design pressure is chosen 10% in excess of the operating pressure and the internal corrosion allowance is 2 mm. The design stress of the steel selected is 108 MPa and the material is available in plate thicknesses (mm) :-
    3     4     5     6     8     10     12     16     20     25     28     32     36     40     45     50     55     60     70     80
    1. Determine a suitable plate thickness, given that the longitudinal and circumferential joint efficiencies are 85 and 75% respectively.         [ 16 mm]
    2. What should be the thickness of a seamless ellipsoidal head (a/b = 2) to suit ?         [ 16 mm]
    3. Calculate the increase in bore at the operating pressure, measured separately on the cylinder and on the head, assuming each is not constrained by the other.         [ 0.28, -0.37 mm]
    4. Repeat (b) with a torispherical head of common proportions.         [ 25 mm]
    5. Design a 250 mm bore nozzle set into the cylinder wall, and reinforce it if necessary.
    6. Repeat (e), but the branch is set onto the head of (d).
    7. Which flange Table from AS 2129 should be used for pad design if the design temperature of the vessel is ( i) 350oC ( ii) 450oC ?         [ J; R]
    8. A 330 x 230 mm elliptical opening in the shell is closed by a flat door whose gasket is effectively 7 mm wide. The minimum seating pressure (the 'y-factor') is 10 MPa. If the design stress for the studding material is 100 MPa, determine the size and number of studs required.         [ 2off M24]
      What plate thickness should be used for the door if the constants in ( xiv) are C1 = 0.4 & C2 = 0.8 ?         [ 32 mm]
    9. Design bridges suitable for the above opening.
       
  4. The table below presents alternative solutions which are proposed by a pressure vessel design program, for a 6.8 m3 pressure vessel with design pressure and stress of 1 and 95 MPa respectively; incorporating 1 mm corrosion allowance and 350 mm nominal size branches which do not require additional reinforcing rings. The ends of class 3 vessels are attached by double fillet welds. Verify a few of the proposals at random, checking in particular the adequacy of intrinsic reinforcement.
    solution   class     RHEEM head   nominal thickness
          diameter     form     head    shell     branch  
    1 3 1118 i.d. s.e. 12 12 5
    2 " 1137 i.d. tor. " " "
    3 " 1143 o.d. tor. 10 " "
    4 " " " " 12 " "
    5 1 1219 o.d. tor. " 8 12
    6 2A " " " " 10 10
    7 2B " " " " " "
    8 3 " " " " 12 8
    9 1 1219 i.d. s.e. 10 8 12
    10 2A " " " " 10 10
    11 2B " " " " " "
    12 3 " " " " 12 8
    13 " " " " 12 " "

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