Double-wrap electric cord heating device
A 3D resistive element in the form of a flat winding with a holding structure and equipotential bonding addresses thermal expansion and loss issues, ensuring efficient and uniform radiant heating with rapid temperature rise.
Patent Information
- Authority / Receiving Office
- FR · FR
- Patent Type
- Utility models
- Current Assignee / Owner
- APLINOV
- Filing Date
- 2024-06-11
- Publication Date
- 2026-06-12
AI Technical Summary
Existing heating systems using resistive elements for radiant heating face issues such as thermal expansion, deformation, mechanical stress, inefficiency due to thermal losses, and inhomogeneous power emission, particularly when rapid temperature rise is required.
A 3D resistive element in the form of a flat winding with a holding structure and equipotential bonding means, minimizing thermal losses and ensuring uniform power distribution without the need for a reflector, using a flat ribbon with a common winding and holding structure that accommodates turns and allows for tensioning to compensate for thermal expansion.
The solution provides efficient, uniform radiant heating without thermal losses and mechanical stress, enabling rapid temperature rise and maintaining consistent power emission across the heating surface.
Smart Images

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Abstract
Description
Title of the invention: Electric heating device with large emissive surface at moderate temperature.
[0001] FIELD OF THE INVENTION The present invention relates to a device for the electric heating of premises where the heat source is a resistive electric cord and where the heating of the premises occurs preferentially by radiation from the resistive cord heated to a low temperature
[0002] BACKGROUND OF THE INVENTION Heating premises using electrical energy relies primarily on the Joule effect in an electrical resistive element powered by electrical connectors from an electrical energy source. Heat transfer to the room, the air, and the occupants occurs through thermal conduction, convection, and / or infrared radiation emitted by the electrical resistive element due to its temperature.
[0003] When it is desired to promote radiant heating from a resistive element heated to a low temperature (for example, between 37°C and 150°C), it is necessary that the radiating resistive element have a radiating surface area that is all the larger as the temperature of the resistive element is lower. Indeed, the thermal radiation flux varies with the fourth temperature of the radiating source.
[0004] To achieve this objective, the prior art known or imaginable is to produce T electrical resistive element in a generally 2D form, for example in the form of a thin layer of material, for example of dimensions 60cm*60cm, engraved so as to present several spiral or zigzag turns of conductive material, located in a plane or on a slightly convex or concave surface, of which, for example, the front face, i.e. the face oriented towards the room to be heated, is treated so as to present an emissivity close to 1 and the rear face undergoes a preparation so that its emissivity is as low as possible, preferably with a space between turns significantly smaller than the width of the turns to maximize the emissive surface. Among the drawbacks of this type of design are the problems of thermal expansion, particularly if a rapid response time is required (temperature rise to reach the chosen radiation temperature); it is indeed difficult to prevent deformation of the thin layer, or even tearing due to the sudden onset of mechanical stresses. Among the disadvantages is also the need to rest the thin layer on a support structure towards which some of the thermal power is lost, thus reducing the efficiency of the heating system. Among the disadvantages there is also the fact that the spiral or zigzag conductive track has turns, a variable radius of curvature and a width that is difficult to conceive as constant, which can induce an inhomogeneity of the power emitted on the surface.
[0005] To overcome these drawbacks, a 3D electrical resistive element may be chosen, in particular in the form of a winding (coil) of a resistive cord on a core, for example made of ceramic. The core may, for example, be cylindrical or flattened in shape. The resistive cord may have a substantially circular cross-section or be in the form of a thin ribbon, for example 30mm wide and 0.1mm thick, made of stainless steel or aluminum alloy or polymeric material such as polypropylene or PET or Teflon, 50 micrometers thick, covered with a 5 micrometer thick layer of aluminum. The advantage of choosing a resistive element in the form of a 3D winding is that it allows for a conductive track of constant width without bends or variable radius of curvature and potentially with a very small contact area and therefore thermal losses with a support, due to the possibility of tensioning the resistive element between protruding support parts. In the case where the heating cable is in the form of a large flat coil to create a large emissive surface area, for example, a flat coil of conductive tape with overall dimensions of 50cm x 50cm x 3cm, the rear sections of the coils are heated by the current flowing through the coil. The rear section of the coils or winding is defined as the part of the coils or winding that faces the supporting wall and / or ceiling rather than the room. Conversely, the front section is defined as the part of the coils or winding that faces the room. The dividing lines are the two common boundaries between the front and rear sections. It is desirable to avoid wasting the corresponding thermal power emitted by the rear section, which naturally does not emit its power into the room.One solution is to place a reflector that intercepts the power emitted by the rear sections and redirects it forward, i.e., into the room. However, when the emitting surface is large, the reflector must be positioned at a significant distance from the rear of the winding and its dimensions must be significantly larger than those of the winding. This is unfavorable in terms of size, manufacturing complexity, and cost.
[0006] DETAILED DESCRIPTION OF THE INVENTION The invention described below makes it possible to heat a room with at least one heating cable in the form of a flat 3D coil, without the aforementioned drawback, in particular not requiring a reflector at the rear of the coil. Indeed, the rear part of the coils of the heating device that is the subject of the invention emits only a very low residual power. The electric heating device of the invention comprises at least one flat winding of heating cord where the heating cord is in the form of a flat ribbon, where by flat winding it is meant that it fits into a parallelepiped volume of small thickness along the Y coordinate compared to the lateral dimensions along the X and Z coordinates, where Z is the axis of the winding, installed on a common winding and holding structure suitable for accommodating all the turns of heating cord of the device, i.e. suitable for accepting the installation on itself of all the turns of heating cord of the device. For any flat winding, the number of turns of heating cord is defined as N, where N is at least 2. The electrical beginning and end of the turns, the front sections of the turns, and the rear sections of the turns are conventionally defined as being located on one of the lines, called demarcation lines, between the front and rear sections of the winding. One of the winding's power supply poles is connected to the beginning of the front section of turn numbered 1 by convention, and the other power supply pole is connected to the end of the front section of turn numbered N. The beginning and end of the front section of the turn refer to the direction of current flow. Thus, on the electrical conduction path between the two poles are located N front sections of turns, called concatenary front sections, and Nl rear sections of turns, called concatenary rear sections.The Nth rear part is not concatenated and can possibly be shortened. The device of the invention comprises, for each of the Nl concatenary rear parts of a turn of any winding, a pair of electrical connection means between the ends of each of the concatenary rear parts and an electrical means dedicated to the said concatenary rear part, called the electrical equipotential bonding means, capable of imposing a substantially equipotential bond between the connection means of each pair.
[0007] An example of a device according to the invention may, for example, comprise a common winding and holding structure of approximately 60cm*60cm*5cm accommodating 3 identical flat windings of 50-turn cupro-nickel ribbon, 19cm*60cm *5cm positioned side by side with a 1cm separation between them and suitable for being powered by a three-phase alternating electrical system. Another example of a device according to the invention may include, for example, a common winding and holding structure with overall dimensions of approximately 100cm*100cm*5cm accommodating one flat winding of aluminium tape of 98cm*98cm*5cm suitable for being powered by a 24V continuous electrical system.
[0008] In a particular embodiment of said common winding and holding structure, in order to minimize the thermal losses of the ribbon from any winding to said common winding and holding structure, the latter has on at least a part of its external surface a set of projections suitable for receiving said ribbon in such a way that it has contact with said projections but does not have contact with said common structure between the projections. In this embodiment, the contact surface of the protrusions is treated so that the ribbon can still slide in the direction perpendicular to the winding axis, thus allowing the ribbon to elongate or contract according to temperature variations without resulting in significant stresses or deformations. In this embodiment, the common winding and holding structure includes, for any given winding, two means for fixing the winding terminations.
[0009] One embodiment of an electrical equipotential bonding means according to the invention is a shunt resistor with a resistance value much lower than that of said concatenated rear part and connected in parallel with said concatenated rear part A specific example of how to achieve this shunt resistance, for instance with a 20-micrometer-thick stainless steel ribbon, is by depositing an electrolytic copper layer on the back, adjacent portion of the coil, which is 100 micrometers thick. The resulting overall resistance is thus 200 times lower than that of the stainless steel ribbon alone.
[0010] Another embodiment of an electrical equipotential bonding means, the beginning and end of each concatenary rear part of each turn of a first winding are connected in a cross manner by conductors, called crossed conductors, which are quasi-equipotential, i.e. of very low resistance, to the end and beginning of the concatenary rear part of the conjugate turn of a second winding and vice versa, where the first and second windings are part of the device of the invention, are supplied with the same voltage and are identical and where by conjugate turns we mean turns which are equidistant from a pole of a given sign on the conduction path of both windings. In this example, the preferred configuration is the antiparallel architecture where the two windings are interleaved. This means that the turns of one winding are sandwiched between the turns of the other, are parallel to each other, and the direction of current flow from one pole to the other of one winding is the reverse of that of the other winding. The advantage of this architecture is that the terminations (beginning or end) of the concatenated rear turns, which must be connected by an equipotentially bonded crossed conductor, are located on the same dividing line. This avoids the need for all the crossed conductors to cross the winding from one dividing line to the other.
[0011] Preferably the common winding and holding structure comprises two uprights oriented substantially along Z, of length along Z compatible with accommodating all the turns of ribbon of the device of the invention, spaced along X, and held apart by a spacing distance along X by a holding and spacing means, where said spacing distance is significantly greater than the spacing along Y between the front longitudinal part and the rear longitudinal part of each turn, where the holding and spacing means is capable of being compressed slightly elastically along the direction X,in order to generate a tension force on the longitudinal parts of the ribbon turns so as to compensate for variations in the length of the longitudinal parts due to the phenomenon of expansion. This property can be particularly advantageous when installing a winding on the common winding and holding structure; indeed, it can be beneficial to compress the holding and spacing means during this phase and then release the compression so that the ribbon is under tension. By slight compression, we mean compression capable of allowing a relative elastic alteration of the length of the longitudinal part on the order of 0 to 10*E-3. The turns of any winding comprise two rounded parts and two longitudinal parts, not necessarily flat, of a flat conductive ribbon extending longitudinally in the X direction where each turn has no direct electrical contact with any other turn other than the start and end contact. The ribbon of the winding makes substantially half a turn around the upright and therefore the ribbon is in contact with the uprights in the part of the circumference of the uprights facing outwards from the winding.
[0012] In a particular, but not limiting, embodiment where the upright must accommodate a winding of 20 turns of 18mm wide and 50 micrometers thick ribbon, the length of the upright along Z is approximately 400mm and the radius of the rounded part in contact with the ribbon is 10mm. The uprights can be in a monolithic cylindrical or partially cylindrical form. For example, it can be semi-cylindrical, monolithic, or the result of an assembly, and its overall geometry can be a sawtooth profile of revolution, resulting in a multi-truncated conical upright where each truncated conical section accommodates a turn of ribbon. For a truncated conical shape with a semi-vertical angle alpha, the front longitudinal sections of the winding turns can be oriented perpendicular to the axis of the upright, creating a rectangular heating surface. The rear longitudinal sections then form an angle Theta with the perpendicular to the upright, where Theta = Pi sin(alpha). As an example, with alpha = 1.67 degrees, the angle Theta is 5.25 degrees. The structure of the uprights is utilized in a particular embodiment called an integrated upright to not only fulfill its primary function of mechanically supporting the windings, but also to provide individual electrical contact for each turn of each winding on and near the demarcation line. In this particular embodiment, each upright is structured into unit sections, the number of which corresponds to the number of turns. These sections have a cylindrical or frustoconical peripheral surface, are made of a conductive material, or have at least one conductive peripheral portion designed to establish electrical contact with the conductive tape. Each unit section has no direct electrical contact with the following or preceding unit section. Each integrated upright has a cylindrical core made of insulating material along its length. Along its periphery, these cores have longitudinal grooves designed to accommodate an electrically conductive strip, called a conduction strip, made of, for example, copper, brass, or aluminum, with a cross-section of 3 x 1 mm. Each strip has at least one embossed shape along a segment of its length, corresponding to the position of the coil with which an electrical connection is to be established, and its length is less than or equal to the width of the tape. Note that the term "embossed" refers to only one possible method for creating this shape. Optionally, some conduction strips can be designed to extend beyond the end of the insulating core with an electrical connection tip. This type of strip is called a conduction and connection strip. The overall thickness of the lamellae is such that, except at the level of the stamped shapes, the conductive lamella remains set back from the geometric envelope of the insulating core. An insulating shell, whose inner diameter matches the diameter of the insulating cylindrical core, surrounds the insulating cylindrical core fitted with all the fins. Open sections in the insulating shell, where the width of the opening matches the width of the conductive fin, are positioned at the level of level of each stamped section. Each stamped section fits into the corresponding open portion and has a slight overhang (for example 0.15mm) relative to the precise flushness with the outer geometric envelope of the insulating shell. The upright is equipped with tubular and conductive sections of length corresponding to the width of the tape, positioned longitudinally so as to coincide with the position of the turns at the level of the upright and insulated from each other, preferably by means of insulating washers positioned between said tubular sections.The inner diameter of each conductive tubular section corresponds to the outer diameter of the insulating shell; each stamped portion of a conductive strip is thus in intimate electrical contact with a conductive tubular section; this is due in particular to the elasticity of the stamped shape which allows it to fit into the corresponding open portion, and which naturally has a slight protrusion (for example 0.15mm) relative to the precise flushness with the outer envelope of the insulating shell, and due to its compression by the placement of the tubular section exerts a bearing pressure on the inner surface of the latter.
[0013] The integrated winding architecture is particularly advantageous if the number N has at least two submultiples p and q, thus N = p*q, for implementing a parallel power supply to the winding. For this power supply architecture, the winding is considered to have p groups of q turns in series, each group beginning or ending on a dividing line where the next group begins or ends, thanks to the fact that the concatenated rear sections of the turns are equipotential. Each group can be supplied with the same voltage, but the direction of the +V-V polarity changes alternately from one group to the next. In one embodiment, one of the uprights has p conduction and connection strips connecting on an upright, unit section number 1 for the +V connection, and regularly distributed unit sections number 1+q for connection to -V, l+2*q for connection to +V, l+3*q, for connection to -V, etc., up to l+(pl)*q, the other upright has a conduction and connection strip connecting unit section number N for the -V connection.
[0014] If (p-1) is divisible by two, another embodiment is possible; and is This is particularly interesting because the two uprights then have a symmetrical architecture. One of the uprights has (p -l) / 2 +1 conduction and connection strips connecting unit section number 1 for the +V connection, and regularly distributed unit sections number 1+q for connection to -V, l+3*q for connection to -V, l+5*q for connection to -V, etc., up to l+(pl)*q. The other upright comprises (p - 1) / 2 + 1 conduction and connection strips connecting to one upright, the unit section numbered N for the -V connection, and regularly distributed unit sections numbered Nq for connection to +V, N-3*q for connection to +V, N-5*q for connection to V, etc., up to N-(pl)*q. An example of this implementation is N = 12, p = 3, q = 4, which gives (pl) = 2, therefore divisible by two, and thus allows for the creation of two symmetrical uprights, each containing only two conduction and connection strips. This creates an electrical partition of the winding into three groups that can be powered electrically in parallel, as opposed to the winding as a whole where the turns are powered in series. This allows the injected power to be multiplied by 3 squared, or 9, which can be particularly useful during the warm-up phases when, for example, very short rise times are desired, on the order of a second.
[0015] The integrated upright architecture is also particularly interesting for the so-called anti-parallel winding architecture. In this case, for windings of N turns each, for each upright, each of Nl conduction or conduction and connection strips has two stamped shapes which are called conjugate stamped shapes, corresponding to two conjugate unit sections, numbered "i" and respectively "2N+3-i", for i ranging from 3 to 2N, that is to say contacting two rear parts of conjugate turns, two other strips have only one stamped shape each corresponding to unit sections 1 and 2, one establishing the connection with one of the supply poles of one of the windings and the other with the supply pole of opposite sign of the other winding. Conductive tubular sections can be monolithic or preferably composed of a first conductive tube section, called the interface tube section, in close contact with a stamped portion of a conductive strip, and a second cylindrical or frustoconical tubular conductive section, called the peripheral section, coaxial with the interface tube section. The peripheral section's internal diameter is adjusted to ensure gentle friction around the interface tube section, guaranteeing good electrical contact and rotational freedom. This rotational freedom can be useful for allowing the longitudinal sections of the strip a slight displacement, enabling them to adjust in length during variations resulting from thermal expansion or contraction.
[0016] BRIEF DESCRIPTION OF THE DRAWINGS Other objects, features and advantages of the invention will be better understood in light of the detailed description that follows, with reference to the accompanying drawings in which:
[0017] Figure 1 shows in the upper left quadrant a top view of one of the two windings (13) of the invention. It shows in the upper right quadrant the two uprights (11) with the spacer core (12) between them. It illustrates in the lower left quadrant how the two windings (13) and (14) are designed to interlock both geometrically and electrically. It shows in the lower right quadrant the two windings (13) and (14) installed on the uprights and spacer core assembly. Figure 2 presents the electrical connection diagram of the turns according to the invention, to facilitate understanding, in the form of a network of connected resistors. The resistors (211) represent the rear parts of winding 1 while the resistors (221) represent the rear parts of winding 2. The resistors (212) represent the front parts of winding 1 while the resistors (222) represent the front parts of winding 2. The drivers (23) represent the drivers who crossed.
[0018] Fig. 3 shows a double anti-parallel winding (31) and (32) of ribbon according to the invention with the equipotential bonds (33) between the turns representing said crossed conductors shown in symbolic form of lines. Figure 4 shows a double winding (41) and (42) of ribbon according to the invention extending from one upright (43) to the other upright (44), where the uprights are of the integrated upright type. To facilitate understanding, each upright is also shown at a distance from the double winding. On each of these windings, the set of sections (45) of conductive tube is visible, positioned longitudinally so as to coincide with the position of the turns. Figures 5 and 5-2 present the upright architecture of the integrated upright type for a double anti-parallel winding where N=20. In these representations, to increase the visibility of the details, the longitudinal dimensions have been reduced by a factor of four. Shown are the insulating core (51) with its eleven grooves (511), a set of two strips, one conductive (521) and the other conductive and connecting (522), each with their two conjugate stamped forms (523).
[0019] The set (53) of the eleven strips (9 conduction strips, including one conduction and connection strip, and two connection strips) are presented in the positions they are intended to occupy on the insulating core The insulating core is shown (512) equipped with the eleven strips installed in the corresponding grooves. The two insulating half-shells are shown (54) positioned at a distance. The insulating core fitted with the lamellae and fitted with the two insulating half-shells is shown (55). The insulating core fitted with the lamellae, fitted with the two insulating half-shells and fitted with the 20 interface tube sections is presented (56) The insulating core equipped with the lamellae, equipped with the two insulating half-shells, equipped with the 20 interface tube sections and equipped with part of the 20 peripheral truncated conical tube sections is shown (57). Fig. 6 illustrates the use of uprights (631), (632) having frustoconical unit sections (here with a half-angle at the apex of 1.67 degrees) to make two windings (61) and (62) whose front parts (612) of the turns are horizontal, i.e. perpendicular to the axis of the uprights and whose rear parts (611) of the turns make an angle (here 5.25 degrees) with respect to the horizontal.
Claims
Demands
1. Electric heating device characterized in that: - said device comprises at least one flat winding of N turns of heating cord, composed of N front concatenary parts and Nl rear concatenary parts, where by concatenary part of a turn is understood to be the part of a turn located on the conduction path between the connection to the positive terminal and the connection to the negative terminal of the electric generator; - said device comprises a common winding and support structure capable of accommodating all the turns of heating cord of the device; - flat winding means that the winding is contained within a parallelepiped volume of small thickness compared to the lateral dimensions; - said heating cord is a flat ribbon of which at least a part of the thickness is electrically conductive.-the said device comprises, for each of the Nl concatenary rear parts of the coil, a pair of electrical connection means between the ends of each of the concatenary rear parts and an electrical means dedicated to the said concatenary rear part, called the equipotential bonding electrical means, capable of imposing a substantially equipotential bond between the two connection means of each pair.
2. Electric heating device according to claim 1 wherein said common winding and holding structure has on at least a part of its external surface a set of projections suitable for receiving said ribbon such that the ribbon has contact with said projections but does not have contact with said common structure between the projections and wherein the contact surface of the projections is treated so that sliding of the ribbon in the direction perpendicular to the winding axis remains possible and wherein said common winding and holding structure comprises for any winding two means for fixing the terminations of said winding.
3. An electric heating device according to claim 1, wherein the electrical equipotential bonding means is a shunt resistor with a resistance value much lower than that of said part rear concatenary and connected in parallel to said rear concatenary part
4. An electric heating device according to claims 1, 2 and 3, wherein the common winding and holding structure comprises two uprights oriented substantially along Z, the length of which is suitable to accommodate all the turns of the device of the invention, of homogeneous cylindrical peripheral shape or of peripheral geometry consisting of the repetition along Z of peripheral parts with the same geometric shape, called shape patterns, each of said shape patterns being suitable to accommodate a turn, wherein the uprights are spaced along X, and maintained at a spacing distance by a holding and spacing means, wherein said spacing distance is significantly greater than the spacing along Y between the front longitudinal part and the rear longitudinal part of each turn, wherein the spacing maintenance means is positioned either between the uprights and is called the inner core,either on the outside is then called outer frame, where each turn has no direct electrical contact with any other turn other than the start and end contact, where any winding is composed of turns having two rounded parts and two longitudinal parts not necessarily flat, of a flat conductive ribbon extending longitudinally in the X direction.
5. Electric heating device according to claim 4 wherein the shape pattern is a truncated cone
6. Electric heating device according to claim 4 wherein the shape pattern is freely rotating about the Z axis
7. Electric heating device according to claim 2 wherein at least one of said means for fixing any winding is capable of exerting traction on the end of the ribbon of said winding while accepting a slight displacement of said end.
8. An electric heating device according to claim 4, wherein the separation and spacing means is elastically compressible along the X direction
9. An electric heating device according to claim 6, wherein said shape pattern is made of conductive material and is insulated from other shape patterns
10. Electric heating device according to claim 9 wherein said shape pattern comprises an axial cylindrical bore fitted with very low friction on a cylindrical part called interface tube section, integral with the upright, preferably tubular, insulated from other interface tube sections, not movable in rotation, made of conductive material, suitable for allowing rotation of said shape pattern around said interface tube section while maintaining electrical contact with the latter.
11. An electric heating device according to claims 1 to 10, wherein the heating device comprises two identical windings of N-turn heating cord supplied with the same voltage, and wherein said electrical equipotential bonding means consists, for each winding, of the cross connection of each said pair of electrical connection means to the conjugate pair of electrical connection means of the other winding, wherein two conjugate pairs are defined as those relating to the concatenated rear turns of both windings located at the same distance from the same-polarity electrical pole of each winding.
12. Electric heating device according to claim 11 characterized in that the two said identical windings have their axis along the Z direction, each of N turns where N is at least equal to two, where the windings are parallel to each other and intercalated along the Z direction without direct electrical contact between them and where the relative position along Z of the positive and negative poles of the electrical supply of one winding is the inverse of that of the other winding.
13. Electric heating device according to claims 9 and 10 wherein each shape pattern of number i, for i between 3 and 2N, of one of the uprights corresponding to a turn of one of the windings is electrically equipotentially connected to the conjugate shape pattern 2N+3-i of the same upright, corresponding to a turn of the other winding.
14. Electric heating device according to claim 13 wherein on the same amount all the electrical connection pairs between each pattern of shape number i of the amount and each pattern of shape 2N+3-i of the same amount is ensured each by a longitudinal electrically conductive strip called a conduction strip forming together a bundle of conduction strips preferentially radially and regularly distributed around the Z axis, electrically insulated from each other, installed inside the volume of the upright where each conductive strip is electrically insulated from all the shape patterns of the upright other than the shape pattern i and the shape pattern 2N +3-i by the internal surface of said shape pattern or contact with the internal surface of said interface tube section and where optionally at least one of said connecting strips, then called connecting and linking strips, of each upright extends so as to emerge from the end of said upright.
15. Electric heating device according to claim 4 wherein each of the two uprights comprises for a winding of N turns where N=p*q and p-1 is divisible by 2, (pl) / 2 +1 electrically insulated conductive and connecting strips installed inside the volume of the upright and each of the connecting strips for one of the uprights the shape patterns of number 1, 1+q, l+3*q, l+5*q, up to l+(pl)*q, and for the other upright the shape patterns of number N, Nq, N-3*q, N-5*q up to N-(pl)*q, to the external connecting end of said strip, wherein each of said strips is electrically insulated from all the shape patterns except the one it is intended to connect.
16. Electric heating device according to claim 4 wherein the shape patterns are frustoconical with a half-angle at the apex alpha and wherein the front longitudinal parts of the winding turns are installed perpendicular to the upright and the rear longitudinal parts are then installed making the angle Theta with the perpendicular to the upright, where Theta=Pi*sin(alpha)