Chemistry Theory of Machines - ebook
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Chemistry Theory of Machines - ebook
This book encompasses the essential range of information on technical aspects of mechanical design. It was written primarily for the students and staff of chemistry faculties of technical universities, yet it may also be utilized by everyone, who ether would like to try or already enjoys designing, but cannot take advantage of typical stress & machine construction handbooks. These handbooks often require familiarity with the concepts of mechanics and higher mathematics, mastery of which is gained after graduating from technical universities.
A short study of material strength contained within is limited only to the basic, statically determinable cases and singular loads. Emphasis has been put on explaining various types of stresses and differences in their influences. The amount of formulas has been limited only to the essential minimum, which still allows assessing component dimensions in such a way that these elements are not damaged or excessively deformed, and are durable & stable. Each chapter is preceded by a short theoretical introduction, followed by examples of solved tasks, including: basic equations, symbols, and a description of the solving process.
In the final chapter of the book, two examples of technical design problems have been solved. The first one describes stable interaction forces and stresses through a handle, mounting and suspension of any fixed element. The second one presents dynamic effects in the drive shaft of a hypothetical mechanical device (agitator, conveyor, pump, fan, etc.).
| Kategoria: | Chemia |
| Język: | Angielski |
| Zabezpieczenie: |
Watermark
|
| ISBN: | 978-83-01-19206-8 |
| Rozmiar pliku: | 10 MB |
FRAGMENT KSIĄŻKI
3 Tensile stresses in the vessels
Cylindrical vessels consist of a cylindrical (boiler drum) part and two bottoms (ends), welded onto it. The ends may be circular flat, dished, spherical or conical. They may also be welded to the flange and then – in a form of screw cap – to the tank. Different types of containers have been presented in figure 3.1.
Fig. 3.1. Types of vessels: a) vertical, cylindrical, closed, b) cylindrical, with flat lid and conical bottom, c) cylindrical, horizontal, with two spherical lids, d) spherical with hatch.
The tensile stresses occurring in the tanks can be divided, depending on where they occur, into stress: in the walls, in flange bolts, in tank’s fixture (suction and discharge piping, measuring connectors, valves, liquid level indicators, glass sighting tubes, lifting eye bolts etc.).
The theoretical basis concerning both tensile σ_(r) and admissible k_(r) stress together with described by the Hooke’s law elongations ∆l, presented in the previous chapter, are still valid, but only for linear loads and longitudinal elements, such as rods, bolts, or pipes.
Walls of the vessels, which are classified as flat and spatial elements, require individual theoretical introduction concerning stress and strength of thin-walled coatings.
A vessel is considered thin-walled if the actual thickness of the walls g_(rz) is less than the thickness limit g_(gr), which is linked to the diameter of the vessel D through the following relation:
(3.1)
3.1. Stresses in cylindrical walls – barrels
Cylindrical vessel consist of a cylindrical part called a barrel and two bottoms. If there is pressure p within the vessel, then it acts both on the surface of the barrel, as well as bottom surfaces. In case of barrel coating, shown in Fig. 3.2, there are two types of stress, circumferential σ_(ro) and longitudinal σ_(rw). They stretch the coating as shown in an example of its rectangular element (Fig. 3.2).
Fig. 3.2. The tensile stresses in the cylindrical axisymmetric shell
In order to analyze the value of these stresses, let us imagine a vessel being cut by two planes perpendicular to each other: one perpendicular to the base – along the symmetry axis (Fig. 3.3) and second parallel (Fig. 3.4).
Fig. 3.3. Circumferential stress.
Fig. 3.4. Longitudinal breaking tensions
3.1.1. Tensile circumferential stresses
According to the definition, stress σ_(ro) is the ratio of force F to the cross-sectional area A of the vessel, where the force is a product of pressure p and area of projected vessel surface parallel to the cross-section of that surface A_(p).
(3.2)
thus, for D ≈ D_(w) and admissible tension stress k_(r): stresses σ_(ro) and wall coating thickness g:
(3.3)
3.1.2. Longitudinal tensile stresses
Similar reasoning carried out for an imaginary longitudinal cross-section (Fig. 3.4) returns the dependence of longitudinal stress in cylindrical vessel walls.
After the imaginary cutting of the vessel with a plane, parallel to its base, through the action of a force F, the longitudinal tensile stress σ_(rw), arises at a plane A.
(3.4)
For thin-walled vessels (3.1), where D >> g, D_(z) = D_(w) + 2g ≈ D_(w) = D, circular ring area A equals approximately:
(3.5)
and then, after substituting (3.5) to (3.4) and shortening, we obtain:
(3.6)
3.2. Stresses and wall thickness of cylindrical vessels
A comparison of equations defining circumferential (3.3) and longitudinal (3.6) stresses shows that the former is, for identical conditions (p, D, k_(r)), twice as high σ_(ro) = 2σ_(rw), therefore the wall thickness of the cylindrical vessel with prescribed pressure p and diameter D is equal to the thickness calculated from the circumferential stress g_(cyl) = g_(ro) and hence – should be established from equation (3.3).
Exceptions include cases where the vessel contains a layer of liquid of height H and density γ, with hydrostatic pressure p_(h) = H ∙ γ, which is comparable to or greater than the gas pressure p inside the vessel. In such cases, longitudinal stresses σ_(rw) might have equal, or even greater value than the circumferential σ_(ro). In this scenario, the thickness must be calculated from the higher one of these values, through one of the following relationships:
(3.7)
3.3. Stress and wall thickness of spherical vessels
The stress occurring in a tank wall or in the bottoms of spherical vessel, results from the symmetry condition for stresses in spherical coatings, where there is only one type of stress – the longitudinal tension stress σ_(rw).
The longitudinal stresses occurring in the cross-section of the vessel and their directions have been illustrated in Fig. 3.5 and relations (3.4) – (3.6), which show that wall thickness of a spherical vessel or bottom g_(d) can be written as:
(3.8)
Fig. 3.5. Tensile stresses in the spherical shell
3.4. Calculation of bolts for flange connections in vessels
Example 3.1.
Calculate how many studs n_(sr) with a diameter d = 36 mm (Figure 3.6) must be used to connect the cover with the vessels drum of diameter D = 2.2 m, if the pressure inside the vessel is p = 16 at, and k_(r) = 130 MPa.
Fig. 3.6. Flange joint with a stud
SOLUTION:
Data: D = 2.2 m
p = 16 at ≈ 1.6 MPa
d = 36 mm = 3.6 ∙10⁻² m
k_(r) = 130 MPa
Searches: n_(sr)
An upward force F acts on the cover of the vessel – it is derived from pressure, and equal to the product of the cross-section of the vessel and pressure. This force in studs cross-section, with a total area A_(sr) generates tensile stress:
For technical reasons, the nearest value which is a multiple of 3 or 4 is chosen: n_(sr) = 48.
Example. 3.2.
Fig. 3.7. The joint flange with hexagonal bolt screwed into the flange
Calculate how many steel bolts n_(sr), with k_(r) = 120 MPa, and diameter d = 16 mm (M16) must twist the tank cap, of a diameter D = 1.4 m, wherein the pressure inside is p = 20 at.
SOLUTION:
Data: D = 1.4 m
p = 20 at = 2 MPa
k_(r) = 120 MPa
d = 16 mm = 0.016 m
Searches: n_(sr)
The number of bolts in the cover n_(sr) can be calculated from tensile stress σ_(r), which on their surface A_(sr) causes a force F, generated by a pressure p_(maks), to act on the vessel lid A_(p).
Cylindrical vessels consist of a cylindrical (boiler drum) part and two bottoms (ends), welded onto it. The ends may be circular flat, dished, spherical or conical. They may also be welded to the flange and then – in a form of screw cap – to the tank. Different types of containers have been presented in figure 3.1.
Fig. 3.1. Types of vessels: a) vertical, cylindrical, closed, b) cylindrical, with flat lid and conical bottom, c) cylindrical, horizontal, with two spherical lids, d) spherical with hatch.
The tensile stresses occurring in the tanks can be divided, depending on where they occur, into stress: in the walls, in flange bolts, in tank’s fixture (suction and discharge piping, measuring connectors, valves, liquid level indicators, glass sighting tubes, lifting eye bolts etc.).
The theoretical basis concerning both tensile σ_(r) and admissible k_(r) stress together with described by the Hooke’s law elongations ∆l, presented in the previous chapter, are still valid, but only for linear loads and longitudinal elements, such as rods, bolts, or pipes.
Walls of the vessels, which are classified as flat and spatial elements, require individual theoretical introduction concerning stress and strength of thin-walled coatings.
A vessel is considered thin-walled if the actual thickness of the walls g_(rz) is less than the thickness limit g_(gr), which is linked to the diameter of the vessel D through the following relation:
(3.1)
3.1. Stresses in cylindrical walls – barrels
Cylindrical vessel consist of a cylindrical part called a barrel and two bottoms. If there is pressure p within the vessel, then it acts both on the surface of the barrel, as well as bottom surfaces. In case of barrel coating, shown in Fig. 3.2, there are two types of stress, circumferential σ_(ro) and longitudinal σ_(rw). They stretch the coating as shown in an example of its rectangular element (Fig. 3.2).
Fig. 3.2. The tensile stresses in the cylindrical axisymmetric shell
In order to analyze the value of these stresses, let us imagine a vessel being cut by two planes perpendicular to each other: one perpendicular to the base – along the symmetry axis (Fig. 3.3) and second parallel (Fig. 3.4).
Fig. 3.3. Circumferential stress.
Fig. 3.4. Longitudinal breaking tensions
3.1.1. Tensile circumferential stresses
According to the definition, stress σ_(ro) is the ratio of force F to the cross-sectional area A of the vessel, where the force is a product of pressure p and area of projected vessel surface parallel to the cross-section of that surface A_(p).
(3.2)
thus, for D ≈ D_(w) and admissible tension stress k_(r): stresses σ_(ro) and wall coating thickness g:
(3.3)
3.1.2. Longitudinal tensile stresses
Similar reasoning carried out for an imaginary longitudinal cross-section (Fig. 3.4) returns the dependence of longitudinal stress in cylindrical vessel walls.
After the imaginary cutting of the vessel with a plane, parallel to its base, through the action of a force F, the longitudinal tensile stress σ_(rw), arises at a plane A.
(3.4)
For thin-walled vessels (3.1), where D >> g, D_(z) = D_(w) + 2g ≈ D_(w) = D, circular ring area A equals approximately:
(3.5)
and then, after substituting (3.5) to (3.4) and shortening, we obtain:
(3.6)
3.2. Stresses and wall thickness of cylindrical vessels
A comparison of equations defining circumferential (3.3) and longitudinal (3.6) stresses shows that the former is, for identical conditions (p, D, k_(r)), twice as high σ_(ro) = 2σ_(rw), therefore the wall thickness of the cylindrical vessel with prescribed pressure p and diameter D is equal to the thickness calculated from the circumferential stress g_(cyl) = g_(ro) and hence – should be established from equation (3.3).
Exceptions include cases where the vessel contains a layer of liquid of height H and density γ, with hydrostatic pressure p_(h) = H ∙ γ, which is comparable to or greater than the gas pressure p inside the vessel. In such cases, longitudinal stresses σ_(rw) might have equal, or even greater value than the circumferential σ_(ro). In this scenario, the thickness must be calculated from the higher one of these values, through one of the following relationships:
(3.7)
3.3. Stress and wall thickness of spherical vessels
The stress occurring in a tank wall or in the bottoms of spherical vessel, results from the symmetry condition for stresses in spherical coatings, where there is only one type of stress – the longitudinal tension stress σ_(rw).
The longitudinal stresses occurring in the cross-section of the vessel and their directions have been illustrated in Fig. 3.5 and relations (3.4) – (3.6), which show that wall thickness of a spherical vessel or bottom g_(d) can be written as:
(3.8)
Fig. 3.5. Tensile stresses in the spherical shell
3.4. Calculation of bolts for flange connections in vessels
Example 3.1.
Calculate how many studs n_(sr) with a diameter d = 36 mm (Figure 3.6) must be used to connect the cover with the vessels drum of diameter D = 2.2 m, if the pressure inside the vessel is p = 16 at, and k_(r) = 130 MPa.
Fig. 3.6. Flange joint with a stud
SOLUTION:
Data: D = 2.2 m
p = 16 at ≈ 1.6 MPa
d = 36 mm = 3.6 ∙10⁻² m
k_(r) = 130 MPa
Searches: n_(sr)
An upward force F acts on the cover of the vessel – it is derived from pressure, and equal to the product of the cross-section of the vessel and pressure. This force in studs cross-section, with a total area A_(sr) generates tensile stress:
For technical reasons, the nearest value which is a multiple of 3 or 4 is chosen: n_(sr) = 48.
Example. 3.2.
Fig. 3.7. The joint flange with hexagonal bolt screwed into the flange
Calculate how many steel bolts n_(sr), with k_(r) = 120 MPa, and diameter d = 16 mm (M16) must twist the tank cap, of a diameter D = 1.4 m, wherein the pressure inside is p = 20 at.
SOLUTION:
Data: D = 1.4 m
p = 20 at = 2 MPa
k_(r) = 120 MPa
d = 16 mm = 0.016 m
Searches: n_(sr)
The number of bolts in the cover n_(sr) can be calculated from tensile stress σ_(r), which on their surface A_(sr) causes a force F, generated by a pressure p_(maks), to act on the vessel lid A_(p).
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