Page 281 - Boiler_Operators_Handbook,_Second_Edition

266 Boiler Operator’s Handbook

CYLINDERS UNDER INTERNAL PRESSURE force on the two thicknesses of metal (2T) and the stress
in the metal (S) so the mathematical formula for a cylin-
The basic calculations for determining the required der under pressure is P × D = 2T × S. Substitute known
thickness of a cylinder under internal pressure (like a values for any three of the letters and you can calculate
boiler tube or drum or shell or piping) is best explained the fourth using simple algebra. If you don’t know alge-
by looking at a cross section of the cylinder like that in bra then here’s what you do for the four options:
Figure 9-4. The Figure shows half the cylinder with ar-
• To determine the stress on the metal you multiply
rows beside where we imagine that we cut through the
pressure times the diameter and divide that result
cylinder. When we’re evaluating that view we make the
by twice the metal thickness. S = (P × D)/(2 × T)
section over a unit length of the cylinder, normally one
inch. So imagine the dark line is a piece of metal that’s
• To determine the minimum thickness of the metal
one inch deep into the page. Any inch along the length
you multiply pressure times the diameter, divide
of a cylinder would be the same so we can work with
that result by the allowable stress and divide that
one inch and it applies to the whole length. The gray ar-
result by two.
rows show the direction of the forces that are applied.
T = ((P × D)/S)/2
The pressure is inside the cylinder trying to get
out and pushing against the area that is equal to the • To determine the maximum diameter for a cylinder
inside diameter (I.D.) of the cylinder. The area equals of a given thickness at a selected operating pres-
the diameter because the width is unity (one inch). The sure you multiply the thickness and the allowable
pressure times the diameter equals the force produced stress, that result is multiplied by two and you fin-
by the internal pressure (p×d). We’re applying a pres- ish by dividing by the pressure. D = (T × S × 2)/P
sure, pounds per square inch, against an area measured
in square inches so the overall force can be measured in • To determine the maximum pressure for a cylinder
pounds. That force has to be balanced and the balance of a given thickness, diameter, and material, you
is the force produced by the metal cylinder. If the force multiply the thickness and the allowable stress,
were not balanced the cylinder would rupture. The area that result is multiplied by two and you finish by
of the metal in the cylinder is equal to twice the metal dividing by the diameter. P = (T × S × 2)/P
thickness so we can determine the stress in a known
thickness of metal. Alternatively, we can calculate the The ASME Code isn’t quite as simple and it’s
minimum thickness of the metal for a given stress be- because the overall length of the material around the
cause the forces have to be equal. cylinder gets larger as the thickness increases. The code
The force from pressure equals the pressure (P) formula is:
times the diameter (D) and it must be equaled by the
T = (P × D)/(2 × S × E+2 × Y × P) + C 8

to determine the thickness and

P = (2 × S × E) × (T - C)/(D - 2 × Y) × (T - C)

to determine the maximum allowable pressure for a
given thickness. There are values in addition to those in
the more simple explanation above represented by C for
corrosion allowance, E for a factor that depends on the
method of welding (sometimes called weld efficiency)
and Y which is a coefficient that depends on maximum
operating temperature and the type of steel. These for-
mulas are for power boilers. The ones for heating boilers
and pressure vessels are a little different.
For your purposes the simple formulas should
be fine. As long as you know there’s a little difference
Figure 9-4. Cylinder analyzed for pressure stress between them and the actual code formulas it’s okay.