by G. D. Gatletly and E. G. Chihon Shell Development Co.
I
EQUIPMENT AND DESIGN A
Steel Shell Impervious Membrane
Non-Mctallk Liner
W O R K B O O K
Design of Prestressed Cylinders with Nonmetallic Linings Vessels for highly corrosive reactions at elevated tempera tures and pressures have been built with steel shells and nonmetallic linings. A design method will determine re quired w a l l thickness and prestress, so that the liner does not crack or separate from the shell during operation The Problem
Figure 1 , A. A cross section o f α typical vessel under discussion shows position of steel shell, impervious membrane, and nonmetallic liner
I N T H I S A G E of a t o m i c reactors a n d h i g h t e m p e r a t u r e - h i g h pressure proc esses the " h o w t o " i n construction o f these vessels assumes increasing i m p o r t a n c e i n the d a i l y life o f the design engineer. Corrosion-resist ant, n o n s h r i n k i n g cements a n d n o n c o r r o d i n g plastic membranes have m a d e p r a c t i c a l the experimenter's recommended operating conditions. Processes t h a t have l a i n d o r m a n t for lack of the " h o w " are b e i n g used to advance technology of this era. A f a b r i c a t i o n t e c h n i q u e , devel o p e d to a h i g h degree o f perfection by a G e r m a n f i r m , is to b u i l d u p the vessel w a l l of a steel shell to p r o v i d e strength, a n i m p e r v i o u s m e m b r a n e to p r e v e n t chemicals f r o m r e a c h i n g the steel, a n d a l i n i n g to insulate the m e m b r a n e f r o m reaction temperatures ( F i g u r e 1). T h e l i n e r is m a d e of bricks cemented together w i t h a patented cement t h a t expands on setting. E x p a n s i o n produces a n i n i t i a l stress i n the l i n e r m a t e r i a l , e l i m i n a t i n g cracks a n d p r e v e n t i n g the liner a n d steel shell f r o m sepa r a t i n g . M a t z has analyzed this type of b r i c k - l i n e d pressure vessel.
Ε C
F E A T U R E
T o find the m i n i m u m w a l l thickness of the steel shell a n d nonmetallic l i n i n g a n d the m a g n i t u d e of the initial prestress, such that: 1. A l l o w a b l e w o r k i n g t e m p e r a t u r e , of the insulating m e m b r a n e , Tit is not exceeded 2. M a x i m u m allowable w o r k ing stress of the steel a n d l i n e r m a t e r i a l is not exceeded u n d e r a n y o p e r a t i n g conditions 3. Steel shell a n d l i n e r do not separate d u r i n g operation
temperature-dependent within the r a n g e of t e m p e r a t u r e s to be con sidered 3. L i n e r is homogeneous elastic material 4. N o axial shearing stresses exist at the liner-steel interface For the purpose of d e f i n i n g the physical m a k e - u p of the vessel, assume also : 1. Vessel has open ends 2. L i n e r a n d steel shell a r e j o i n e d at their ends, so that they u n d e r g o the same axial déforma-
Prestress can be o b t a i n e d b y h e a t i n g the shell a n d a l l o w i n g i t t o cool after i t has been placed over the liner, b y using the e x p a n d i n g cement m e n t i o n e d a n d i n other ways. T h e calculations assume t h a t the t e m p e r a t u r e at w h i c h the vessel was constructed is the same as the a m bient t e m p e r a t u r e , T0. A d d i t i o n a l stresses p r o d u c e d b y a change i n the a m b i e n t t e m p e r a t u r e can be cal culated.
KNOWN O p e r a t i n g conditions Internal diameter of vessel Operating temperature
Assumptions M a d e in A n a l y s i s
Operating pressure
As i n a n y c a l c u l a t i o n of this type, certain s i m p l i f y i n g assumptions must be made, i n o r d e r to m a k e the a n a l ysis t r a c t a b l e . These assumptions seem reasonable.
Properties of materials of construction T h e r m a l conductivity Coefficient of expansion M o d u l u s of elasticity
1. Ratio of inside diameter of vessel to thickness of l i n e r is suffi ciently large that t h i n - w a l l shell t h e o r y can b e used 2. Physical and mechanical properties of the materials are not I/EC
Thickness of insulating membrane Maximum allowable operating temperature of m e m brane Outside air temperature
W
Poisson 's ratio Allowable stresses H e a t transfer coefficient
O R K B O O K FELA.TXJR.E3S
65 A
Equations Used in Calculation H e a t Transfer Relations
where
Q-^-™-JU(T.'
-
ι
d
-
"^-s-ËJr To) (1) Compression in Liner
Α
r . ' - ^ r +^ r . ; * ϋ ί Γ + Γ Τ Τ Γ · . +
W . J = j - [/-* + dloxT
» τ *, Stresses Due to Temperature Alone Inner surface (r = a)
_
(24)
=
_
7,
APT = - Ay/2bH +
σ Γ
'
~ W ,
7, ('p'' ~ p + Pp) + "lT ~ mt
INDUSTRIAL AND ENGINEERING CHEMISTRY
(14)
(15)
APT' = Δ Ύ / 2 / /
Ay =
(Φ^ΓΪ)
[φ(2αϊ _ α , ) + αι/3]
(33) (34)
(35)
EQUIPMENT AND DESIGN
Figure 1 , B. Temperature profile through the three layers shows path heat takes during o p e r a t i o n
turns when subjected to internal temperature and pressure 3. Initial prestress is applied in both radial and axial directions 4. Section to be analyzed is away from the ends, with no attach ments or openings nearby M e t h o d of A n a l y s i s
W i t h the basis established on which to make the calculation, the problem is: A. From heat transfer considerations, equations are obtained relating the tem perature of the liner and the steel shell in terms of the various thicknesses and inside and outside temperatures. B. Using the theory of elasticity, equations are obtained which ensure that displacements of the liner and steel shell are always equal during operation—• i.e., they do not separate. C. Stresses in the steel and at the inner and outer fibers of the liner are then calculated using the results of A and B. Parameters which admit the exist ence of a solution (design of an uncracked vessel) are found by imposing the requirement that the stresses found i n C do not exceed their allowable values (Figure 2). I f a solution exists, optimum design for a specific liner temperature, Ti} is given by the intersection of the lower bound for the liner material with the upper bound for the steel. A criterion is also given for determining whether a solution really exists. I f one does not exist, it is not possible to design a vessel which w i l l not crack subject to the given conditions.
The vessel can still be built, although the liner may crack and the analysis given has to be modified. Stepwise procedure for calculating the solu tion is given i n a box accompanying this article, as are the equations and definition of terms relating to the calculations. However, certain ex planations should be made to define clearly the basis on which a solution will be reached. Heat Transfer Relations. Be cause of the high thermal conduc tivity of the steel and its relatively small thickness in the usual construc tion, Ts = Ts'. Pressure at Shell-Liner Interface. Cylindrical vessels in the "as con structed" position will be subjected to both radial and axial prestress, the magnitudes of which have to be determined. When internal pres sure and temperature gradient are imposed on the vessel, the liner and the steel cylinder will want to de form by different amounts i n both the radial and axial directions. As the cylinders are joined at their ends, their axial displacements must remain the same. Cylinders some times tend to separate radially because of temperature ; this w i l l be prevented by prestress. Temperature Gradient. As the liner and steel cylinder would have different axial and radial displace ments, it is necessary to apply axial and radial forces to bring them back together. Tension in Liner. M a x i m u m tensile stresses occur i n the outer surface of the liner. M a x i m u m axial tensile stresses occur w i t h prestress
.
A Workbook Feature
EE33
plus temperature. I n the circum ferential direction, maximum tensile stresses occur during operation—i.e., prestress plus pressure plus tempera ture. Compression in Liner. Maxi mum compressive stresses occur at the inner surface of the liner. I f PT is positive—i.e., a pressure and not a tension—in the circumferential direction maximum compressive stresses occur with prestress plus temperature. I f PT is negative, maximum stress can occur under initial prestress alone. Whether i t occurs or not depends on the magni tudes of the second and third terms on the right side of Equation 17. This contingency should be checked in designing the vessel. Tension in Steel Cylinder. Max imum tensile steel stresses usually occur during operation. Upper a n d Lower Bounds for Initial Prestress
Denoting allowable stresses in the liner by
