reference is first made to fig1 , which shows diagrammatically a typical cylindrical heat exchanger 2 . the heat exchanger 2 has a cylindrical shell 4 having an inlet conduit 6 and an outlet conduit 8 for fluid which is to be heated or cooled . located within the shell 4 are a number of annular or donut shaped baffles 10 which extend to and are adjacent to the wall of the shell 4 and which have central apertures 12 . located between each pair of donut baffles 10 is a disc - shaped baffle 14 , of smaller diameter than that of the shell 4 and therefore leaving an annular gap 16 extending there around . both sets of baffles 10 , 14 , are intersected by all the tubes 18 of the heat exchanger . the tubes 18 extend parallel to the shell 4 and at right angles to the baffles 10 , 14 . heating or cooling fluid ( liquid or gas ) from a source not shown , is directed into the tubes 18 of the heat exchanger from outside one tube sheet 20 and leaves the tubes 18 at the outside of the other tube sheet 22 . fluid ( liquid or gas ) from the conduit 6 passes through the heat exchanger in the path indicated by arrows 24 and is warmed or cooled by the fluid in the tubes 18 . in some cases the central aperture 12 and the annular gap 16 are made sufficiently large that the baffles 10 , 14 intersect only some of the tubes 18 . reference is now made to fig2 , which shows a set of tubes 18 according to the prior art shown as being located in rings identified by their radii , namely rings r 1 , r 2 , r 3 , r 4 and r 5 . the design parameters used to lay out the tubes 18 include the following . firstly , the diagonal distance between each tube in any ring and its adjacent tubes in the neighboring ring is a constant distance h ( referred to as the ligament size or ligament width ). secondly , the shortest distance between two adjacent tubes in the same ring ( such distances are identified by reference characters d 1 , d 2 , etc .) is a constant in each ring but varies from ring to ring and is always greater than or equal to 2h . thirdly the number of tubes in each ring is always the same . however , the radial distance between rings is varied so that the ligament size h between a tube in one ring and its adjacent tubes in each neighboring ring is as mentioned always the same . a mathematical design procedure for calculating the various radii will be set forth shortly . it will be seen from fig2 that so long as the ligament gaps h adjacent to a tube 18 are no more than half as large as the gaps d 1 , d 2 , etc ., the ligament gaps and not the gaps d 1 , d 2 , etc . will determine the maximum fluid velocity near that tube . this is the opposite of the conventional concentric ring arrangement in which the tube count is varied for constant distance between the tubes of a ring . it will also be seen from fig2 that the total minimum flow area through which fluid must pass as it travels radially inwardly through rings r 1 , r 2 , r 3 , etc . is the distance 2h multiplied by the number of tubes per ring ( the product is termed the area factor constant or afc ) multiplied by the distance between baffles . as indicated , since the afc is never greater than the sum of the distances between the tubes of any one ring , the maximum velocity through the rings is determined by the afc , which is constant between each pair of adjacent rings in the set . a more complete tube sheet drawing is shown in fig3 . fig3 illustrates portions of two sets of circular rings , indicated at 26 and 28 . in set 26 the ligament size h 1 between each tube 18 and its adjacent tubes 18 in each neighboring ring is always the same constant distance , and the number of tubes 18 in each ring r 1 to r 7 is the same . in tube set 28 , the ligament size h 2 between each tube 18 and its adjacent tubes in each neighboring ring is also a constant , but ligament size distance h 2 is greater than ligament size h 1 . the number of tubes in each ring r 8 to r 11 is constant , but this number is less than the number of tubes in each ring r 1 to r 7 . however the controlling flow distance or afc between the tubes of any two adjacent rings of set 26 is the same as the controlling flow distance or afc between the tubes of any two adjacent rings of set 28 . in other words distance h 1 multiplied by the number of tubes in any ring of set 26 is equal to distance h 2 multiplied by the number of tubes in any ring of set 28 . therefore fluid flowing through tube sets 26 , 28 will always be subject to the same controlling afc and the flow velocities through both sets of rings 26 , 28 will be nearly constant . the afc between the adjacent rings of sets 26 , 28 will of course normally be greater than the afc of each of the two sets . a mathematical procedure for laying out the tubes according to the prior art as given hereinbelow , with reference to fig4 . as shown in fig4 , the following quantities have the following meanings : h is the diagonal distance between each tube and the adjacent tubes in each neighboring ring , or in other words is the ligament width , r 1 , r 2 , r 3 - r n are the ring radii , α is the angle between radii directed through the centers of adjacent tubes in a ring , a n is a chord of the circle having radius r n extending between the centers of two adjacent tubes on the circle of radius r n , d o is the outer diameter of each tube , assumed to be the same for all tubes , n tr is the number of tubes per ring , assumed to be the same for all rings in each set of rings , p is the pitch , i . e . the distance between the centers of adjacent tubes in adjacent rings , and is to be constant . h = p - d o ( 1 ) α 2 = 180 ⁢ ° n tr ( 2 ) a n 2 = p · cos · θ n = r n · sin ⁢ ⁢ α 2 ( 3 ) r n + 1 + b n = r n · cos ⁢ ⁢ α 2 ( 4 ) in practice , the design may be started by selecting the required area for flow , i . e . the afc , which is 2h . n tr . if a ligament width h is chosen , this determines the number of tubes for the first ring of radius r 1 , which is laid out adjacent the shell 4 of the heat exchanger . once n tr is chosen , this yields a value for α / 2 and for chord a 1 , which with the value of h sets a value for θ 1 . since b 1 = p sin θ 1 , this yields a value for b 1 , so that r 2 can be calculated . there are certain limits applicable to the values that may be chosen . firstly , as discussed , the minimum flow area between adjacent rings is to be limited by the ligaments h and not by the gaps d 1 , d 2 , etc . therefore r min ≥ do + 2 ⁢ h 2 · sin ⁢ ⁢ 180 ⁢ ° n tr ( 6 ) equation ( 6 ) gives the minimum ring radius which may be used in order to satisfy equation ( 5 ). the derivation of equation ( 6 ) is as follows with reference to fig5 . therefore ⁢ ⁢ ⁢ a n 2 = r n · sin ⁢ ⁢ α 2 hence ⁢ ⁢ 2 ⁢ ⁢ r n · sin ⁢ ⁢ α 2 ≥ d o + 2 ⁢ h ⁢ ⁢ or ⁢ ⁢ r min ≥ do + 2 ⁢ h 2 · sin ⁢ ⁢ 180 ⁢ ° n tr ( 6 ) if the minimum ring radius is less than r min , the chord distance between two adjacent tubes in the same ring will be less than twice the ligament width , so that the minimum flow area will no longer be governed by the ligaments , which is undesirable . it will however be appreciated that when a number of rings of tubes are to be packed into a heat exchanger , and if space considerations so demand , one or more of the inner rings can be more tightly packed , so that the chord distance between two adjacent tubes in ring is in fact less than 2h . this of course has the disadvantage that the flow through these rings will not behave as ideally as the flow through the rings laid out as described . such rings , where the chord distance is less than 2h , would not be considered as being members of the set of rings laid out according to the invention . similarly an outer ring or rings can be provided near the shell with tube spacing other than those described , to provide higher or lower heat transfer near the shell wall . the second limit for tubes laid out as described is as follows . it is normally necessary to ensure that the radial distance between any two rings which are separated by one ring is greater than the pitch , i . e . that r n − r n − 2 ≧. p . this results in the limit ( 7 ) θ n ≥ 30 ⁢ ° - 180 n tr degrees for the largest radius ring , i . e . ring r 1 . the derivation of equation ( 7 ) is as follows . since it has been postulated , with reference to fig5 , that ⁢ ϕ = 90 ⁢ ° - ( θ 1 + α 2 ) ⁢ ⁢ c 1 = p ⁢ ⁢ cos ⁡ [ 90 ⁢ ° - ( θ 1 + α 2 ) ] = p ⁢ ⁢ sin ⁡ ( θ 1 + α 2 ) ⁢ ⁢ a 2 2 = p · sin ⁢ ⁢ φ = p · cos ⁡ ( θ 1 + α 2 ) ⁢ ⁢ p ⁢ ⁢ sin ⁡ ( θ 1 + α 2 ) + p ⁡ [ 1 - cos 2 ⁡ ( θ 1 + α 2 ) ] 1 / 2 ≥ p ⁢ ⁢ θ 1 + α 2 ≤ 90 ⁢ ° , ⁢ 2 ⁢ sin ⁡ ( θ 1 + α 2 ) ≥ 1 ⁢ ⁢ sin ⁡ ( θ 1 + α 2 ) ≥ 1 / 2 ⁢ ⁢ θ 1 ≥ 30 ⁢ ° - 180 n tr ( 7 ) equation ( 7 ) represents a normal limit on how closely the rings can be spaced without unduly weakening the tube sheets 20 , 22 and the baffles 10 , 14 . in some special cases it may be possible to achieve slightly closer spacing . the minimum flow area in the space between adjacent baffles 10 and 14 is where two sets of rings are used , each with its own ligament size , as shown in fig3 , then the afc of each set is as discussed normally held the same as that of the other set . if ring n is the last ring in one set and ring n − 1 is the first ring in the second set , this is accomplished by maintaining n tr ⁢ ⁢ n - 1 = n trn ⁡ ( p n - d o p n - 1 - d o ) ( 8 ) where p n is the pitch for ring n and p n − 1 is the pitch for ring n − 1 . this ensures that the mass flow velocity is nearly constant throughout the tube bundle . if in special cases it is desired to have a different afc in each set of tube rings , for example more rapid flow through the outer set than through the inner set , then the afc can be made larger in the outer set than the inner set . it will be seen from fig4 that the tubes 18 are laid so that each tube is located circumferentially midway between the two adjacent tubes in each neighboring arc , so that the centers of such three tubes form an isosceles triangle . as shown in fig3 , this results in the tubes of each set of rings 26 , 28 being laid out in a spiral configuration . this facilitates cleaning , which may be accomplished by inserting a corresponding shaped tool through the tubes between the spirals . when two sets of rings are used , as shown in fig3 , then since each set of tubes has a different spiral configuration , it is necessary to clean the outer set of rings by a tool inserted from the outside , and the inner set of rings by a tool inserted from the inside . in the typical embodiment shown in fig3 , each ring r 1 to r 7 contains 68 tubes ( total 476 ), and the radii are the tube outer diameter is 1 . 5 inches and the pitch is 2 . 0 inches . in addition each ring r 8 to r 11 contains 43 tubes , and the radii are the tube outer diameter remains 1 . 5 inches and the pitch is 2 . 29 inches . the values given for fig3 are exemplary only and will of course vary depending on the application . in fig3 it is assumed that each set of rings 26 , 28 extends through a full circle of 360 degrees , i . e . that each ring r 1 to r 11 is a closed circle . however if desired the sets of rings 26 , 28 may be arranged not as closed rings but as sections of annuli . this arrangement is shown in fig6 , where the heat exchanger 2 is shown in section as a section of an annulus and the tubes 18 are arranged along concentric arcs where the arcs do not extend through a full 360 degrees . the fig6 arrangement of tubes is in fact simply a portion of the fig3 set 26 , and the same radii r 1 to r 7 are shown in the drawings . the shell of the heat exchanger is shown at 40 . in the annulus arrangement shown in fig6 , all of the relationships previously described remain applicable , except that the arcs may not all have the same number of tubes 18 . in fig6 the odd numbered arcs have ten tubes each and the even numbered arcs have nine tubes each . this is because the end walls 42 , 44 of the shell are straight and because of the location of such end walls . if end wall 44 were moved to the location shown in dotted lines at 46 , then each arc would have the same number of tubes ( nine tubes in the fig6 embodiment ). thus , when the tube layout has the form of a section of an annulus , the number of tubes in each arc will be either the same as the number in each other arc or may differ from the number of tubes in each other arc by not more than one . in addition the end tubes in the odd numbered arcs do not of course form an isosceles triangle with the two adjacent tubes of each neighboring arc , because of the end walls 42 , 44 , but these walls are sufficiently close to the end tubes of the odd numbered arcs to prevent “ punch - through ”. in the appended claims , reference is made to the distance between tubes . such distance refers to the distance between the outer diameters of the tubes . with reference to fig7 , this shows generally as 50 a diagrammatic isometric view of a segment , in the form of part of an annulus for use in the construction of a heat exchanger to comprise seven such segments . the segment has a radial tube layout of tubes 52 between tube sheet end plates 54 and a pair of longitudinal baffle plates 56 , 58 ( with only 56 being seen ), extending the length between end plates 54 . radial edges 60 , 62 of each of end plates 54 subtend on angle β of about 51 . 4 ° in the embodiment shown . in field assembly , the longitudinal baffle plate 56 a adjacent to end plate 54 is not present . a pair of segments 50 is positioned longitudinally adjacent one to the other with a radial edge 60 abutting radial edge 62 of adjacent segment 50 . the segments are welded together adjacent the edges . thereafter , baffle plate 56 a is welded in place adjacent edge 62 . the method is repeated with the other segments 50 until the cylindrical exchanger having a diameter of over 5 m is constructed . the maximum width of each segment is about 2 . 4 m and is suitable for transportation in standard containers for truck , rail and cargo shipment . fig8 shows generally as 60 , a end view of a heat exchanger having a plurality of segment tube arrays 50 welded together and having a pair of adjacent longitudinal baffle plates 56 between adjacent arrays according to the invention . fig9 shows generally as 70 a plurality of seven segments 60 in the process of construction of the complete tube 52 layout with longitudinal baffle plates 56 , 58 and end plates 54 . fig1 shows the resultant connection between two adjacent segments 50 , welded in situ in the field to provide a “ double plate ” arrangement of a pair of parallel plates intermittently welded , one to the other , at at least one end thereof . the plates are so spaced apart as to provide a gap “ g ” between a plate 56 , 56 a , 58 , 58 a and its adjacent row of tubes , as to allow gas flow between each of the tubes and the plates , while preventing gas bypass between the adjacent row of tubes of adjacent bundles . fig1 shows generally in end view 80 , the angles α for both the segments and the offset arcs . dimension “ f ” is the offset distance between the center of shell 82 and the center of the “ petal ” arcs 84 . dimension “ m ” denotes a maximum deviation between centered arc 86 and petal arc 84 . this figure also shows adjacent arcs 88 and non - adjacent arcs 90 . although this disclosure has described and illustrated certain preferred embodiments of the invention , it is to be understood that the invention is not restricted to those particular embodiments . rather , the invention includes all embodiments which are functional or mechanical equivalence of the specific embodiments and features that have been described and illustrated .