Fast-neutron reactor fuel rod

Abstract

A fast-neutron nuclear reactor fuel assembly having fuel rods. Each fuel rod has nuclear fuel disposed in a sealed housing in the form of a tubular steel shell and end parts. A steel spacer element is wound in a coil with a specific pitch on the outside surface of the shell and is fastened on the end parts. The spacer element is in the form of a metallic band twisted around its longitudinal axis. The width of said band is approximately equal to the minimum distance between adjacent fuel rods in the fuel assembly. A transverse cross-sectional area of the band is within a range from 0.10 to 0.50 times the area of a circle described around the width of the band.

Claims

1. A fast-neutron lead-cooled nuclear reactor fuel assembly, comprising: a plurality of fuel rods, each fuel rod comprising nuclear fuel arranged in a sealed housing in the form of a tubular steel shell and end parts, and a spacer element wound in a coil with a specific pitch on an outside surface of the tubular steel shell and fastened to ends of the fuel rod on the end parts, wherein the spacer element is in the form of a band twisted around a longitudinal axis of the band, the width of the band transverse to the longitudinal axis being equal to the minimum distance between adjacent tubular steel shells in the fuel assembly of the nuclear reactor, a transverse cross-sectional area of the band being within a range from 0.1 to 0.5 times the area of a circle described around the width of the band, wherein a band twist pitch is calculated by the formula $S_2=\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{N},$ where S.sub.2 is the band twist pitch of the band twisted around the longitudinal axis; S.sub.1 is the winding pitch of the spacer element on the tubular steel shell; d is the outer diameter of the tubular steel shell; δ is the transverse width of the twisted band; N is the number of full turns of the twisted band per pitch S.sub.1.

2. The fuel rod assembly according to claim 1, wherein the direction of the spacer element winding on the tubular steel shell and the direction of band twist around the longitudinal axis are the same.

3. The fuel rod assembly according to claim 1, wherein the direction of the spacer element winding on the tubular steel shell and the direction of band twist around the longitudinal axis are opposite to each other.

4. The fuel rod assembly according to claim 1, wherein the spacer element is made of steel.

5. The fuel rod assembly according to claim 4, wherein a winding direction around which the spacer element is wound on each tubular steel shell is the same as a twist direction that each band is twisted around the longitudinal axis.

6. The fuel rod assembly according to claim 4, wherein a winding direction around which the spacer element is wound on each tubular steel shell is opposite a twist direction that each band is twisted around the longitudinal axis.

7. A fast-neutron lead-cooled nuclear reactor fuel assembly comprising: a plurality of fuel rods, each fuel rod comprising nuclear fuel arranged in a sealed housing formed of a tubular steel shell and end parts, and a plurality of spacer elements, wherein each spacer is formed of a band twisted about the longitudinal axis of the band, wherein each spacer element is wound in a coil having a pitch on the outside surface of the tubular steel shell and fastened to the end parts, wherein the width of the band transverse to the longitudinal axis being generally equal to the minimum distance between adjacent fuel rods, the transverse cross-sectional area of the band being within a range from 0.1 to 0.5 times the area of a circle described around the width of the band, wherein a band twist pitch is calculated by the formula $S_2=\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{N},$ where S.sub.2 is the band twist pitch of the band twisted around the longitudinal axis; S.sub.1 is the winding pitch of the spacer element on the tubular steel shell; d is the outer diameter of the tubular steel shell; δ is the transverse width of the twisted band; N is the number of full turns of the twisted band per pitch S.sub.1.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) The present invention is explained by drawings shown in FIGS. 1-11.

(2) FIG. 1 shows a twisted band prior to been wound on a fuel rod. Section A-A shows the transverse cross-section of the band having a width 8.

(3) FIG. 2 shows a fuel rod where a spacer element is wound and a band is twisted in the same directions.

(4) FIG. 3 shows seven fuel rods with spacer elements in the form of twisted bands of rectangular cross-section arranged therebetween.

(5) FIG. 4 shows a schematic view of spacing a fuel rod with respect to an adjacent fuel rod where spacer element winding and band twist directions match.

(6) FIG. 5 shows a schematic view of spacing a fuel rod with respect to an adjacent fuel rod where spacer element winding and band twist directions are opposite.

(7) FIG. 6 shows unidirectional winding.

(8) FIG. 7 shows oppositely directed winding.

(9) FIG. 8 shows determination of the “hollow” width.

(10) FIG. 9 shows trajectories of extreme points of the twisted band cross-section per full turn of the large coil.

(11) FIG. 10 shows a minimal gap calculation model.

(12) FIG. 11 shows dependence between length differences from the origin to the point of intersection with the envelope point and the adjacent circle.

EMBODIMENTS OF THE INVENTION

(13) A fuel rod according to one of the embodiments of the present invention (see FIG. 2, FIG. 4 and FIG. 5) comprises a shell (1) which has ends sealed with end components (3). A spacing element comprising a twisted band (2) and end sections (4) is wound around an outer surface of the shell (1) as a wide-pitch coil. The end sections (4) are welded to the end components (3). Nuclear fuel (5) and, if necessary, other components and materials, such as fuel holders, elements made of non-fissible materials, metal melts in a gap between a fuel and the shell, etc. (not shown), are arranged inside the shell (1).

(14) To ensure that the fuel rod will be effectively spaced from each of the adjacent fuel rods per every pitch of the spacer element winding on the shell, a twist pitch of a twisted band is defined by the formula:

(15) $\begin{array}{cc}{S}_2=\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{N},& \left(1\right)\end{array}$

(16) where S.sub.2 is the band twist pitch of the band twisted around the longitudinal axis, S.sub.1 is the winding pitch of the spacer element on the tubular shell, which is determined based on the condition of vibration strength of a bundle of fuel rods in a fuel assembly; d is the outer diameter of the tubular shell; δ is the transverse width of the twisted band; N is the number of full turns of the twisted band twisted with the pitch S.sub.2 per a pitch S.sub.1, where this number is determined by the following formulas: for the unidirectional winding
N=N.sub.0=1+6.Math.n;  (2) for the oppositely directed winding
N=N.sub.p=1+6.Math.n,  (3)

(17) where n is the number selected from 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. In addition, the larger n provides a smaller possible deviation between the spacer element and adjacent fuel rods, which can be caused by dimensional errors in the manufacture of the fuel rods and fuel assembly.

(18) Appendix 1 shows how formulas (1), (2), and (3) have been derived.

(19) However, the value of n to be selected is constrained by the acceptable relative elongation of the twisted band material during plastic deformation of the material in the region of longitudinal edges when twisting. That is why the value of n must satisfy the condition: for the unidirectional band twist and spacer element winding

(20) $\begin{array}{cc}n\le \frac{\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{\pi .Math.\delta }.Math.\sqrt{{\left(\frac{{\delta }_5\left(t_{\mathrm{manuf}}\right)}{K_{\mathrm{safety}}}+1\right)}^2-1}-1}{6}& \left(4\right)\end{array}$ for the oppositely directed band twist and spacer element winding

(21) $\begin{array}{cc}n\le \frac{\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{\pi .Math.\delta }.Math.\sqrt{{\left(\frac{{\delta }_5\left(t_{manuf}\right)}{K_{\mathrm{safety}}}+1\right)}^2-1}-2}{6}& \left(5\right)\end{array}$

(22) where δ.sub.5(t.sub.manuf) is the breaking elongation for the band material at manufacturing temperature (twisting);

(23) K.sub.safety is the safety coefficient in terms of the permissible elongation during band twisting.

(24) How formulas (4) and (5) have been derived is shown in Appendix 2.

(25) The calculation of the dependence of the possible deviation between the spacer element and adjacent fuel rods using the twisted band is given in Appendix 3.

(26) Exemplary embodiments of the fuel rod according to the present invention are described below.

(27) Example 1. A fuel rod with a shell having an outer diameter 10.5 mm and a thickness 0.5 mm, and a spacer element (2) in the form of a band with a rectangular cross-section 2.6×0.5 mm having rectangle corners rounded with the radius 0.25 mm. The shell (1) of the fuel rod and the spacer element (2) are made of steel, which is highly corrosion-resistant in a lead coolant environment. The band is cooled-twisted around a longitudinal axis at a pitch of 8.2 mm (the pitch is calculated by formulas (1), (2) and (4)), for instance, by drawing it through a rotating nozzle. Meanwhile, the relative deformation of the band material in the region of longitudinal edges is 42.7% (by analogy with a sheet of steel 10X18H9, an adopted maximum allowable relative deformation at 20° C. is 50% and the safety coefficient is 1.1). With such pitch, one coil turn can receive 19 small turns of the twisted band. After that, the band (2) is wound on the tubular shell (1) at a winding pitch of 250 mm and the winding direction matches the direction of the band twist around the central longitudinal axis and is fixed at the end elements (4) to the end components (3) of the fuel rod.

(28) Example 2. A fuel rod is produced with a spacer element (2) having dimensions of Example 1. For the production purposes, a band having dimensions of Example 1 is used. The band is twisted around the longitudinal axis at a pitch of 7.9 mm and is wound on the tubular shell (1) at a winding pitch of 250 mm and the winding direction is opposite to the direction of the band twist around the central longitudinal axis, and is fixed at the end elements (4) to the end components (3) of the fuel rod.

(29) Example 3. A fuel rod is produced with a spacer element in the form of a twisted band with provision for possible extreme deviations of geometrical dimensions of the fuel rod and spacer element which are included in a fuel rod and fuel assembly design for the reactor system BN-1200. The rectangular cross-section of the band is 2.56×0.5 mm. The outer diameter of the fuel rod is 10.53 mm. The winding pitch of the spacer element on the tubular shell is 258.3 mm. The band is twisted around the longitudinal axis at the pitch of 20.10 mm and is wound on the tubular shell. In this case, according to the procedure in Appendix 2, it can be seen that in the most unfavorable combination of the fuel rod and fuel assembly manufacturing errors, a maximum possible displacement of an adjacent fuel rod from its nominal position will be 0.16 mm.

(30) The inventive configuration allows the considerable reduction of metal consumed per fuel rod by means of a spacer element made in the form of a twisted band. For instance, in a reactor core comprising fuel rods with a shell outer diameter of 10.5 mm, where said fuel rods are spaced within triangular array sites at the pitch of 13.1 mm, for spacing a twisted band with a cross-section of 2.6×0.5 mm, the relative metal consumption (the ratio of spacer element volumes) of the twisted band compared to a spacer wire having a diameter of 2.6 mm is 24.5%. The relative metal consumption of the twisted band compared to a tube of 0 2.6×0.5 mm will be 39.4%, and compared to a tube of 0 2.6×0.3 mm will be 60.0%.

(31) Moreover, an important additional technical effect of the present invention is an additional turbulization of coolant flow in a fuel assembly consisted of fuel rods which use spacer elements in the form of twisted bands. The additional turbulization of the coolant flow is generated due to the twist of the bands around axes thereof, and it allows reducing the risk of formation and a surface area of coolant stagnant regions and, accordingly, “hot spots” on the tubular shell.

(32) Said technical effects help to improve neutron and physical characteristics and the reliability of the fast-neutron reactor core. Said features of the technical solution allow assuming the possibility of its practical application in the manufacture of a fuel rod and fuel assemblies for lead-cooled fast-neutron reactors.

Appendix 1

(33) Let's derive a formula for winding pitches depending on the geometrical parameters of a fuel rod bundle and a spacer element. Generally, there are several possible situations: Spacing only in one direction, and a contact in the same direction is reproduced over each pitch of the large coil; Spacing is ensured in three directions, i.e., at one pitch of the large coil a wire contacts alternately a fuel rod in the direction of 0°, a fuel rod in the direction of 120°, and a fuel rod in the direction of 240°; Spacing is ensured in six directions, i.e., at one pitch of the large coil a wire contacts alternately all adjacent fuel rods (the last case is of particular interest).

(34) For a start, let's derive some equations of geometrical parameters. Let's associate a fuel rod to a coordinate system. Analyze one turn of a large coil. A parametric equation will be as following:

(35) $\begin{array}{cc}\{\begin{array}{c}r=\frac{s}{2}\\ \phi \left(t\right)=2.Math.\pi .Math.t^{\prime },\mathrm{where}0\le t\le 1\\ z\left(t\right)=s_1.Math.t\end{array},& \left(1\right)\end{array}$

(36) where s=d|δ is the pitch of the fuel rod array; d is the outer diameter of the tubular shell; δ is the larger dimension of the twisted band cross-section; S.sub.1 is the pitch of the full turn of the large coil (the winding pitch on the tubular shell); t is the equation parameter.

(37) The equation in the Cartesian coordinates (1) will be as follows

(38) $\begin{array}{cc}\{\begin{array}{c}x\left(t\right)=\frac{s}{2}.Math.\cos \left(2.Math.\pi .Math.t\right)\\ y\left(t\right)=\frac{s}{2}.Math.\sin \left(2.Math.\pi .Math.2\right),\mathrm{where}0\le t\le 1\\ z\left(t\right)=S_1.Math.t\end{array}.& \left(2\right)\end{array}$

(39) By developing the coil turn, an equation for the length along the large coil depending on the t parameter can be obtained:
L(t)=√{square root over (S.sub.1.sup.2+(π.Math.s).sup.2.Math.t)}.  (3)

(40) A tilt of a tangent of the large coil to a plane of the fuel rod cross-section (in other words, an angle of coil axis tilt to the horizon) will be:

(41) $\begin{array}{cc}\alpha =\arcsin \left(\frac{1}{\sqrt{1+{\left(\frac{\pi .Math.s}{S_1}\right)}^2}}\right)& \left(4\right)\end{array}$

(42) The equation of a turning angle of the twisted band cross-section around the coil when moving along the turn is as follows:

(43) $\begin{array}{cc}\theta \left(t\right)=2.Math.\pi .Math.\frac{L}{S_2}.Math.t,& \left(5\right)\end{array}$

(44) where S.sub.2 is the pitch of one turn of the twisted band (the band winding pitch). To achieve the perfect contact, at each pitch of large coil the twisted band must contact alternately all 6 adjacent fuel rods every 1/6 of the pitch. Meanwhile, the angle of turning of the cross-section must be as follows:

(45) $-0\mathrm{for}a\mathrm{first}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{\pi }{3}+2.Math.\pi .Math.n_1,\mathrm{where}{n}_1\mathrm{is}a\mathrm{random}\mathrm{integer}\text{-}\mathrm{for}a\mathrm{second}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{2.Math.\pi }{3}+2.Math.\pi .Math.n_2\text{-}\mathrm{for}a\mathrm{third}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\pi +2.Math.\pi .Math.n_3\text{-}\mathrm{for}a\mathrm{fourth}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{4.Math.\pi }{3}+2.Math.\pi .Math.n_4\text{-}\mathrm{for}a\mathrm{fifth}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{5.Math.\pi }{3}+2.Math.\pi .Math.n_5\text{-}\mathrm{for}a\mathrm{six}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod}.$

(46) Consider the first turn of the large coil. For each contact point, the t is 0, 1/6, 2/6, 3/6, 4/6 5/6. By substituting these values of the parameter for the equation (5), the following equations defining contact conditions will be obtained:

(47) $\begin{array}{cc}\frac{1}{3}.Math.\frac{L}{S_2}=\frac{1}{3}+2.Math.n_1;\frac{2}{3}.Math.\frac{L}{S_2}=\frac{2}{3}+2.Math.n_2;\frac{3}{3}.Math.\frac{L}{S_2}=1+2.Math.n_3;\frac{4}{3}.Math.\frac{L}{S_2}=\frac{4}{3}+2.Math.n_4;\frac{5}{3}.Math.\frac{L}{S_2}=\frac{5}{3}+2.Math.n_5.& \left(6\right)\end{array}$

(48) In addition, the parameter of s can be expressed via the fuel rod diameter and the wire section length δ. Keep in mind here, too, that the wire section length projected onto a horizontal plane δ.sub.w is somewhat less than an actual wire section length (because the coil is somewhat inclined with respect to the horizontal plane):

(49) 0${\delta }_w=\delta .Math.\sin \left(\alpha \right)=\delta .Math.\frac{1}{\sqrt{1+{\left(\frac{\pi .Math.s}{S_1}\right)}^2}}.$
In this case, the equality s=d.sub.fr+δ.sub.w, is satisfied, where d.sub.fr is the outer diameter of the tubular shell.

(50) Based on the equations (6) a pitch of the band turning can be derived:

(51) $\begin{array}{cc}{S}_2=L.Math.\frac{1}{1+6.Math.n_1};S_2=L.Math.\frac{1}{1+3.Math.n_2};S_2=L.Math.\frac{1}{1+2.Math.n_3};S_2=L.Math.\frac{2}{2+3.Math.n_4};S_2=L.Math.\frac{5}{5+6.Math.n_5}.& \left(7\right)\end{array}$

(52) In the result, the following series for band twist pitch can be obtained:

(53) $\frac{S_2}{L}=\frac{1}{1+6.Math.n}.$

(54) Consider a variant when the winding direction is opposite to the large coil turning direction. In this case cross-section turning angles must be as follows:

(55) $-0\mathrm{for}a\mathrm{first}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\left(-\frac{2.Math.\pi }{3}\right)+2.Math.\pi .Math.n_1,\mathrm{where}{n}_1\mathrm{is}a\mathrm{random}\mathrm{integer}\text{-}\mathrm{for}a\mathrm{second}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\left(-\frac{4.Math.\pi }{3}\right)+2.Math.\pi .Math.n_2\text{-}\mathrm{for}a\mathrm{third}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\left(-2.Math.\pi \right)+2.Math.n_3\text{-}\mathrm{for}a\mathrm{fourth}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{8.Math.\pi }{3}+2.Math.\pi .Math.n_4\text{-}\mathrm{for}a\mathrm{fifth}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod};-\frac{10.Math.\pi }{3}+2.Math.\pi .Math.n_5\text{-}\mathrm{for}a\mathrm{six}\mathrm{adjacent}\mathrm{fuel}\mathrm{rod}.$

(56) The formula for the angle of turning will be as follows:

(57) $\begin{array}{cc}\theta \left(t\right)=-2.Math.\pi .Math.\frac{L}{S_2}.Math.t.& \left(8\right)\end{array}$

(58) The following system can be obtained:

(59) $\begin{array}{cc}{S}_2=L.Math.\frac{1}{2-6.Math.n_1};S_2=L.Math.\frac{1}{2-3.Math.n_2};S_2=L.Math.\frac{1}{2-2.Math.n_3};S_2=L.Math.\frac{2}{4-3.Math.n_4};S_2=L.Math.\frac{5}{10-6.Math.n_5}.& \left(9\right)\end{array}$

(60) In the result, the following series for the band twist pitch around the longitudinal axis can be obtained

(61) $\frac{S_2}{L}=\frac{1}{2+6.Math.n}.$

(62) The obtained series of numbers 1+6.Math.n and 2+6.Math.n can be expressed as N. This number shows how many small turns of the twisted band are on one large coil turn.

Appendix 2

(63) FIGS. 6 and 7 show a top view of 7 fuel rods with wires (the following dimensions are used: s=12.98 mm; d=10.53 mm; S.sub.big=258.3 mm; S=2.45 mm; S.sub.small=30.61 mm (the values are selected considering possible geometrical tolerances). FIG. 6 shows the embodiment when the directions of the band twist and the coil winding on the tubular shell are the same, and FIG. 7 shows the embodiment when these directions are opposite to each other. The section is made at a height of 260 mm (i.e., one full turn of the large coil is shown).

(64) Comparison of these pictures demonstrates that, generally, the unidirectional winding is more preferable because it provides a smoother envelope of the spacer band (if viewed from above). Moreover, it can be noticed that the smaller the pitch of the band twist around the longitudinal axis is, the closer the envelope is to the circle. FIG. 6 shows that, despite the tolerances and dimension deviations, the spacing is achieved for all adjacent fuel rods. The maximum possible gap is equal to the transverse width of a “hollow” between the perfect circle and the band envelope. We shall evaluate this dimension.

(65) Let's take a look at FIG. 8. Derive polar equations for points 1 and 2 (extreme points of the spacer band):

(66) $\begin{array}{cc}\phi \left(t\right)=2.Math.\pi .Math.t;& \left(1\right)\\ \theta \left(t\right)=2.Math.\pi .Math.\frac{L}{S_2}.Math.t;& \left(2\right)\\ R_1=\frac{d}{2}+\frac{\delta }{2}.Math.\left(1-\cos \left(\theta \right)\right);& \left(3\right)\\ R_2=\frac{d}{2}+\frac{\delta }{2}.Math.\left(1-\cos \left(\theta \right)\right);& \left(4\right)\end{array}$

(67) $\begin{array}{cc}{\phi }_1=\phi -\arcsin \left(\frac{\delta .Math.\sin \left(\theta \right)}{d+\delta .Math.\left(1-\cos \left(\theta \right)\right)}\right);& \left(5\right)\end{array}$$\begin{array}{cc}{\phi }_2=\phi +\arcsin \left(\frac{\delta .Math.\sin \left(\theta \right)}{d+\delta .Math.\left(1+\cos \left(\theta \right)\right)}\right).& \left(6\right)\end{array}$

(68) Let's build lines based on the polar equations (3)-(6). The following dimensions will be used: d=10.53 mm; S.sub.big=258.3 mm; S=2.45 mm; S.sub.small=30.61 mm. The lines are represented in FIG. 9. FIG. 9 shows how to define the “hollow” dimension. To do this, the radius of the intersection point is required. By making radii from the equations (3) and (4) equal, the l values corresponding to the intersection points can be found:

(69) $\begin{array}{cc}{t}_{\mathrm{inters}}=\left(\frac{1}{2}+k\right).Math.\frac{S_2}{2.Math.L_{\mathrm{turn}}}& \left(7\right)\end{array}$

(70) where k is the natural number.

(71) If K−0, t.sub.inters=0.036 can be obtained. This t corresponds to the radius R=6.49 mm. Then, the transverse width of the “hollow” will be ΔR=1.22 mm.

(72) It can be shown that the “hollow” transverse width is independent of the number of times the band is twisted around the axis of the band on the large coil pitch. If the value of the t.sub.inters from the equation (7) is substituted for the equation (3) or (4), then

(73) 0$R_{\mathrm{hol}}=\frac{d}{2}+\frac{\delta }{2}.$
In this case,

(74) $\Delta R=\frac{\delta }{2}.$

(75) However, although the “hollow” transverse width is constant, when the pitch of the band twist is small, only a part of the transverse width of an adjacent fuel rod can penetrate therein, that is why an actual minimal possible gap will be much smaller than that defined according to the (8). Let's find this gap.

(76) Consider a first intersection point of curves in FIG. 9. The parameter of t for this point is equal to

(77) $\begin{array}{cc}{t}_{\mathrm{inters}1}=\frac{s_2}{4.Math.L_{\mathrm{turn}}},& \left(9\right)\end{array}$

(78) and a corresponding angle is co:
φ.sub.int ers=2.Math.π.Math.t.sub.int ers1.  (10)

(79) Consider a circle with a diameter equal to the diameter of the fuel rod, the center of a beam exiting the origin at an angle φ.sub.inters. Let this circle is spaced from the intersection point of the curve envelopes by the distance Δ.sub.gap. Then, the distance from the origin (the center of the fuel rod of interest) to a center of an adjacent circle (a hypothetic fuel rod in the “hollow”) will be;

(80) $\begin{array}{cc}{R}_{center}=\left(R_{\mathrm{fr}}+\frac{\delta }{2}\right)+{\Delta }_{gap}+R_{fr}=2.Math.R_{fr}+\frac{\delta }{2}+{\Delta }_{gap}.& \left(11\right)\end{array}$

(81) Draw form the origin a tangent to the circle which corresponds to an adjacent fuel rod. An angle between this tangent and a beam connecting centers of the circles will be

(82) $\begin{array}{cc}{\mathrm{\Delta \phi }}_{\mathrm{tang}}=\arcsin \left(\frac{R_{\mathrm{fr}}}{R_{\mathrm{center}}}\right).& \left(12\right)\end{array}$

(83) Draw another beam at an angle Δφ to the beam connecting the centers of the circles and find distances from the origin to the intersection point of the beam and enveloping curves and to the intersection point with the adjacent circle. A calculation model is shown in FIG. 10.

(84) Using FIG. 10, the following geometric relationships can be made:
L.Math.cos(Δφ)+R.sub.fr.Math.cos(ρ′)=R.sub.center;  (13)
R.sub.center.Math.sin(ρ.sub.int ers)+R.sub.fr.Math.sin(φ′−φ.sub.int ers)=L.Math.sin(φ.sub.int ers+Δφ);  (14)
L.Math.sin(Δφ)=R.sub.fr.Math.sin(φ′).  (15)

(85) Based on these equations, a quadratic equation of a relative length L can be obtained, solving which an equation for L will be obtained:

(86) Using Mathcad, set several values of angles Δφ and find differences of lengths from the origin to enveloping curves and to the intersection with the circle. Build a graph of these dependencies and find a value of Δ.sub.gap, at which the circle contacts the envelope. Consider, for instance, the following dimensions: δ=2.56 mm; R.sub.fr=5.265 mm; S.sub.big=258.3 mm; δ−2.45 mm; S.sub.small=1/7.Math.L.sub.turn=37.365 mm, Δ.sub.gap=0.91 mm. With these dimensions, the graph of dependencies between length differences and the beam angle shown in FIG. 11 will be obtained. As can be seen in FIG. 11, with the defined Δ.sub.gap the adjacent circle contacts the band envelope.

(87) Consider how the minimum possible gap between the adjacent fuel rod and the wire envelope depends on the pitch of the band twist around the longitudinal axis. Intuitively, it seems obvious that with the reduced twist pitch the envelope approaches a circumscribed circle radius of which is R.sub.fr+δ. Consider several twist pitches: 1/7; 1/10 and 1/13 of the turn length. Calculation results are shown in Table 1.

(88) TABLE-US-00001 TABLE 1 Minimal possible gap calculation     Twist pitch, mm   Minimal possible gap Δ.sub.gap, mm $\mathrm{Ratio}\frac{{\Delta }_{\mathrm{gap}}}{\frac{\delta }{2}}$ $S_{\mathrm{small}}=\frac{1}{7}.Math.L_{turn}$ 0.91 0.711 $S_{\mathrm{small}}=\frac{1}{10}.Math.L_{turn}$ 1.07 0.844 $S_{\mathrm{small}}=\frac{1}{13}.Math.L_{turn}$ 1.14 0.891

(89) As can be seen in this table, the intuitive conclusion has been confirmed. It appears that with a small band twist pitch the spacing will be guaranteed even with the most unfavorable dimension deviations. However, the twist pitch is constrained by the method of twisted band manufacturing.

Appendix 3

(90) When a flat band is twisted into a coil, generatrices thereof passing through the extreme points of the cross-section have the biggest length. Thus, while twisted, the band material undergoes some plastic deformation. The way how a generatrix is elongated with respect to the longitudinal axis of the band can indicate a relative material elongation. By developing the extreme generatrix, the length thereof can be found
L=√{square root over (S.sub.2.sup.2+(π.Math.δ).sup.2)},  (1)

(91) where S.sub.2 is the band twist pitch;

(92) δ is the band transverse width (the greatest distance between the extreme points of the band cross-section, or the diameter of a circle circumscribed around the twisted band).

(93) The initial length of the generatrix (prior to twisting) was
L.sub.o=S.sub.2  (2)

(94) In this case, the maximum relative elongation during band twisting will be

(95) $\begin{array}{cc}{\epsilon }_{\mathrm{ma}x}=\frac{L-L_0}{L_0}=\frac{\sqrt{S_2^2+{\left(\pi .Math.\delta \right)}^2}-S_2}{S_2}=\sqrt{1+{\left(\frac{\pi .Math.\delta }{S_2}\right)}^2}-1.& \left(3\right)\end{array}$

(96) Express the twist pitch in terms of the band winding pitch on a fuel rod and fuel rod dimensions and in terms of the natural number n, according to Appendix 1:

(97) 0$\begin{array}{cc}\frac{S_2}{L}=\frac{S_2}{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}=\frac{1}{1+6.Math.n}& \left(4\right)\end{array}$

(98) for a unidirectional winding.

(99) Rewrite the formula (3) taking into account the (4):

(100) $\begin{array}{cc}{\epsilon }_{\mathrm{ma}x}=\sqrt{1+{\left(\frac{\pi .Math.\delta }{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}.Math.\left(1+6.Math.n\right)\right)}^2}-1.& \left(5\right)\end{array}$

(101) During twisting the following condition must be satisfied:

(102) $\begin{array}{cc}{\epsilon }_{\mathrm{ma}x}\le \frac{{\delta }_5\left(t_{\mathrm{manuf}}\right)}{K_{\mathrm{safety}}}.& \left(6\right)\end{array}$

(103) where δ.sub.5(t.sub.manuf) is the breaking elongation for the band material at manufacturing temperature (twisting);

(104) K.sub.safety is the safety coefficient in terms of the permissible elongation during band twisting.

(105) Based on (5) and (6), the constraint of the allowable value n can be found:

(106) $\begin{array}{cc}n\le \frac{\frac{\sqrt{S_1^2+{\left(\pi .Math.\left(d+\delta \right)\right)}^2}}{\pi .Math.\delta }.Math.\sqrt{{\left(\frac{{\delta }_5\left(t_{manuf}\right)}{K_{\mathrm{safety}}}+1\right)}^2-1}-1}{6}.& \left(7\right)\end{array}$

(107) The similar formula will be obtained for oppositely directed winding, it is just necessary to do

(108) $\begin{array}{cc}\frac{S_2}{L}=\frac{1}{2+6.Math.n}.& \left(8\right)\end{array}$