Diagonal-flow fan wheel with blades of developable surface shape

A blade of the fan wheel of a diagonal-flow fan, which blade should ideally have a shape of a twisted double-curvature or undevelopable surface, is formed from a portion of a combination of a cylindrical plate and a planar plate tangent to the cylindrical plate or of a combination of a pair of mutually circumscribing cylindrical surfaces, which portion constitutes a developable surface. To realize the formation of a blade from the developable surface, lines of intersection between combined cylinder and planar plates or combined cylinders and a number of coaxial imaginary conical surfaces representing streamlines in the fan wheel are used as a basis for design.

BACKGROUND OF THE INVENTION 
This invention relates generally to fans and blowers for gases and more 
particularly to diagonal-flow fans. More specifically, the invention 
relates to the construction of a novel impeller or fan wheel of a 
diagonal-flow fan of the so-called radial-plate type or limit-load type. 
In the fan wheel of an ordinary centrifugal fan of the radial-plate type or 
the limit-load type, the entrance edges and exit edges of the blades are 
respectively parallel to the fan wheel rotational axis. When the fan wheel 
of the radial plate type fan is viewed in its axial direction, each blade 
is arcuately curved near its entrance edge in order to minimize impact 
loss at the entrance edge and then extends radially toward the exit edge. 
When the fan wheel of the limit-load type fan is viewed in its axial 
direction, each blade has a slight S-shaped or reflex curve as it extends 
toward the outer periphery of the fan wheel. However, each blade in either 
type of fan has no twist with respect to the axial direction, and cross 
section of the blades taken in parallel and spaced-apart planes 
perpendicular to the axis appear to be superposed on each other. Thus, 
each blade has a single-curvature or developable curved surface. 
Furthermore, most of the cross sections of these blades with 
single-curvature surface in an ordinary radial-plate or limit-load type 
centrifugal fan have the shape of a single arc, or the shape of two arcs 
joined together. Accordingly, the fabrication of these blades is 
relatively simple. However, even in the case of a blade of this kind, a 
blade cross section shape in which the radius of the arc varies 
progressively along the chord length is close to the ideal shape from the 
viewpoint of fluid dynamics, but the fabrication of blades of such a shape 
is extremely difficult. For this reason, such blades have not as yet been 
reduced to practice except for centrifugal fans having blades of wind 
profiles (airfoil profiles) being manufactured in spite of this difficulty 
in order to utilize the advantages in efficiency and low noise level. 
In contrast to a centrifugal fan as described above, a diagonal-flow fan 
has blades whose entrance edges and exit edges are not parallel to the 
rotational shaft axis, the radial distance from the shaft axis to each 
entrance edge varying progressively from one end of the entrance edge to 
the other, and furthermore, the radial distance from the shaft axis to 
each exit edge also varying progressively from one end of the exit edge to 
the other. In addition, each blade must be provided with a complicated 
double curvature which causes it to have a twist as viewed in the shaft 
axial direction. These and other features of diagonal-flow fans will be 
described in detail hereinafter, particularly in comparison with a 
centrifugal fan. 
Theoretically, a diagonal-flow fan should have excellent performance but 
has not be reduced to practical use because of certain difficulties as 
will be described hereinafter. 
SUMMARY OF THE INVENTION 
It is an object of this invention to provide a fan wheel of a diagonal-flow 
fan of radial-plate type in which, by utilizing a part of a cylinder 
(which is a single-curvature surface or developable surface) and a plane 
for each blade of the fan wheel, an effect equivalent to that of blades of 
double-curvature surfaces which are close to the ideal from the viewpoint 
of fluid dynamics is attained to produce excellent fan performance, and, 
moreover, the difficulties accompanying the fabrication of diagonal-flow 
fan blades are overcome thereby to facilitate the production of the fan 
wheel. 
It is another object of this invention to provide a fan wheel of a 
diagonal-flow fan of limit-load type in which parts of two cylindrical 
surface are used for each blade of the fan wheel thereby to obtain the 
highly desirable results recited above. 
According to this invention, briefly summarized, there is provided a fan 
wheel of a diagonal-flow fan for propelling a flow of a gas, said fan 
wheel comprising a frustoconical main plate coaxially fixed to a 
rotational shaft, a frustoconical side plate spaced apart from the main 
plate and forming therebetween a diagonal flow path for the gas, and a 
plurality of fan blades each fixed at opposite side edges respectively to 
the inner surfaces of the main and side plates and having an inner 
entrance part and an outer exit part, each of said blades being made of a 
plate of a surface shape conforming to a portion of a combinations of 
imaginary developable surfaces joined to each other in an algebraically 
continuous manner, said surfaces having been caused to intersect 
imaginary, spaced apart and coaxial conical surfaces respectively 
corresponding to representative streamlines of the gas in the flow path 
thereby to form mutual intersection lines which substantially coincide 
respectively with smooth curves lying in corresponding conical surfaces of 
the representative streamlines and having respective shapes conforming to 
gas inflow angles of the entrance part and gas outflow angles of the exit 
part of the blade, at least said inflow angles varying progressively in 
accordance with the positions of the representative streamlines within the 
flow path, said smooth curves having radii of curvature which vary 
progressively between the entrance and exit parts, said portion of the 
combined developable surfaces being peripherally defined by the 
intersection lines at the streamlines at the main and side plates and by 
smooth continuous curves respectively passing through the ends of said 
smooth curves respectively at the entrance and exit parts of the blade. 
The nature, utility, and further features of this invention will be more 
clearly apparent from the following detailed description with respect to 
preferred embodiments of the invention when read in conjunction with the 
accompanying drawings, which are briefly described below, and throughout 
which like parts are designated by like reference numerals and characters.

DETAILED DESCRIPTION 
As conducive to a full understanding of this invention, the differences 
between a centrifugal fan and a diagonal-flow fan and certain problems 
accompanying diagonal-flow fans, which were briefly mentioned 
hereinbefore, will first be described more fully. 
Referring first to FIG. 1, the fan wheel shown therein of an ordinary 
centrifugal fan has a number of blades 1, each having an entrance edge 2 
and an exit edge 3 both of which are parallel to the rotational shaft axis 
4. As viewed in the axial direction (arrow direction P), each blade 1 of a 
centrifugal fan of radial-plate type is arcuately curved in the vicinity 
of its entrance edge 2 in order to minimize impact or collision losses at 
the blade inlet and then continuously extends radially toward the exit 
edge 3 as shown in FIG. 2. On the other hand, with respect to a 
centrifugal fan of limit-load type, each blade 1 is, as viewed in the same 
direction P, curved in the shape of an elongated letter S from its 
entrance edge 2 to its exit edge 3 as shown in FIG. 12. However, in either 
type of centrifugal fan, each blade 1 has no twist in the direction of the 
shaft axis 4, and the sections of the blades respectively in spaced apart 
and parallel planes a.sub.1, a.sub.2, . . . a.sub.n intersecting the shaft 
axis 4 at right angles appear to be superposed on each other. That is, 
each blade 1 may be considered to be a single-curvature surface or 
developable surface. 
Differing from a centrifugal fan, a diagonal-flow fan has a fan wheel with 
blades 11, whose entrance edges 12 and exit edges 13 are not parallel to 
the rotational shaft axis 14 as shown in FIG. 3, and the radial distance 
from the shaft axis 14 to the entrance edge 12 of each blade progressively 
varies as r.sub.in1, r.sub.in2, . . . r.sub.in.sbsb.n respectively at 
positions corresponding to representative streamlines 15.sub.1, 15.sub.2, 
. . . 15.sub.n in the gas flow path within the fan wheel. Furthermore, the 
radial distance from the shaft axis 14 to the exit edge 13 of each blade 
progressively varies as r.sub.out.sbsb.1, r.sub.out.sbsb.2, . . . 
r.sub.out.sbsb.n. If these radii vary in this manner, the inflow angles at 
the entrance edge 12 for minimizing the collision loss for respective 
streamlines 15.sub.1, 15.sub.2, . . . 15.sub.n and the corresponding 
outflow angles for evening out the pressure head must be progressively 
varied as .beta..sub.11, .beta..sub.12, . . . .beta..sub.1n and 
.beta..sub.21, .beta..sub.22, . . . .beta..sub.2n, respectively, as 
indicated in FIG. 4, which shows a blade of the fan wheel of a 
diagonal-flow fan of radial-plate type, and in FIG. 13, which shows a 
blade of the fan wheel of a diagonal-flow fan of limit-load type (in the 
radial-plate type diagonal-flow fan, the outflow angles .beta..sub.2 are 
often selected to be a constant value such as 90.degree. as shown in FIG. 
4 because it is possible to even out the pressure head by suitably 
selecting the ratios of r.sub.out to r.sub.in on respective streamlines). 
It will therefore be understood that in order to obtain an ideal fan 
performance, the shape of each blade must be made to assume a complicated 
twisted double-curvature surface as viewed in the direction of the axis 
14. 
That is, if the blades 11 of the fan wheel of the diagonal-flow fan were to 
be merely of the shape of a single-curvature surface which has a single 
arcuate curve or a curve comprising two arcuate curves similar to the 
blades 1 in a centrifugal fan as shown in FIG. 1 and FIG. 2 or 12 and were 
to be mounted with inclinations in accordance with the inclination of the 
representative streamlines 15.sub.1, 15.sub.2, . . . 15.sub.n, the fan 
performance would drop except in the case of extremely small fans. If, in 
order to improve the performance, an attempt were to be made to fabricate 
blades 11 of the shape of a twisted, double-curvature surface, the 
fabrication would be very difficult. 
Basically considered, the fan wheels of fans of this character are 
fabricated, not by casting, but by assembling parts principally of rolled 
steel plates. Moreover, fans of a wide variety of dimensions, even up to 
large impellers of diameters of 3 to 4 meters, are produced in a great 
variety of kinds, each in small quantities. For this reason, it is very 
difficult to fabricate fan wheels of blades of the shape of a 
double-curvature surface at respective costs which are not prohibitive. 
Because of the foregoing reasons, centrifugal fans as described have been 
and are being widely produced, whereas diagonal-flow fans requiring 
double-curvature blades 11 as shown in FIGS. 4 and 13 have not been 
reduced to practice in spite of the great expections for their high 
performance. 
Before describing the invention, a geometrical analysis of the theoretical 
shape of the blades of diagonal-flow fans will be made. 
As partly described hereinbefore in conjunction with FIG. 3, a plurality of 
blades 11 are fixed by welding between shroud-like main and side plates 16 
and 17, and the main plate 16 at its radially inner part is secured to a 
hub 18. The representative streamlines 15.sub.1, 15.sub.2, . . . 15.sub.n 
(which are actually "streamsurfaces" but will be herein referred to as 
"streamlines") respectively are in the shapes of conical surfaces of half 
vertex angles .theta..sub.1, .theta..sub.2, . . . .theta..sub.n. Each 
blade 11 begins from entrance points (inlets) M.sub.1, M.sub.2, . . . 
M.sub.n on these conical surfaces and ends at exit points (outlets) 
N.sub.1, N.sub.2, . . . N.sub.n. When the conical surface constituted by 
one (15.sub.1) of the representative streamlines is developed in a planar 
surface, it appears as in FIG. 5, in which a section of only one blade 11 
of the fan wheel of a diagonal-flow fan of radial-plate type is shown. 
This section of the blade 11 in FIG. 5 has a specific inflow angle 
.beta..sub.11 at the entrance point M.sub.1 and a specific outflow angle 
.beta..sub.21 (90.degree. in this case) at the exit point N.sub.1 and, in 
between, has a shape resembling a part of an ellipse with a gradually 
varying radius .rho. of curvature in the vicinity of the entrance point 
M.sub.1 and a straight-line shape extending radially toward the exit point 
N.sub.1. The specific inflow angle .beta..sub.11 and the radius .rho. of 
curvature of this blade 11 continually vary as .beta..sub.12, 
.beta..sub.13, . . . .beta..sub.1n as shown in FIG. 4 in correspondence 
with the transition of the representative streamlines 15.sub.2, 15.sub.3, 
. . . 15.sub.n as shown in FIG. 3. Accordingly, a complicated 
double-curvature surface is required for each blade 11, as was pointed out 
hereinbefore. 
According to this invention, a shape of the blade closely approximating the 
above stated ideal shape of the blade is realized by the use of a 
single-curvature surface without using a complicated double-curvature 
surface. In order to constitute a single-curvature blade which satisfies 
the above stated geometrical requirements, this invention makes use of 
intersections between the above stated conical surfaces constituted by the 
representative streamlines and an imaginary cylindrical surface and an 
imaginary plane tangent to the cylindrical surface in the case of a blade 
of a diagonal-flow fan of radial-plate type and two imaginary cylindrical 
surfaces in the case of a blade of a diagonal-flow fan of limit-load type. 
FIG. 6 is a graphical perspective view indicating intersections between 
conical surfaces 15.sub.11, 15.sub.21, 15.sub.31, . . . 15.sub.n1 
constituted by the representative streamlines 15.sub.1, 15.sub.2, 
15.sub.3, . . . 15.sub.n shown in FIG. 3 and an imaginary cylindrical 
surface 19 of a radius C and an imaginary plane 20 tangent to the 
cylindrical surface, which are newly introduced. In FIGS. 7A, 7B, and 7C, 
the intersections between a conical surface 15.sub.11 constituted by a 
representative streamline 15.sub.1 and the cylindrical surface 19 and 
plane 20 are projectionally shown, only the single conical surface 
15.sub.11 being shown for the sake of simplicity. 
For the following analysis, three-dimensional, rectangular coordinate axes 
U, V, and W as shown in FIGS. 6, 7A, 7B, and 7C are used, the origin of 
this coordinate system being positioned at the vertex E of the conical 
surface 15.sub.11. The W axis is taken to be parallel to the centerline O 
of the cylindrical surface 19 and to form an angle K with the centerline 
axis H of the conical surface 15.sub.11, and the V axis is taken to be 
included in the plane 20 and to be superimposed on the point m.sub.s1 of 
tangency between the cylindrical surface 19 and the plane 20, which point 
is on the curve M.sub.1 N.sub.1 when viewed in the W-axis direction (arrow 
direction Q in FIG. 6) as shown in FIG. 7A. 
From the manner in which the W is taken, the angle K of inclination of the 
cylindrical surface 19 (i.e., of the centerline O thereof) with respect to 
the conical surface 15.sub.11 can be represented by the angle between the 
W axis and the centerline axis H of the conical surface 15.sub.11. This 
conical surface 15.sub.11 is taken to be the same as the conical surface 
constituted by the representative streamline 15.sub.1 in FIG. 3. The 
intersection line between this conical surface 15.sub.11 and the 
cylindrical surface 19 and the plane 20, that is, that portion of the line 
of intersection which extends from the entrance point M.sub.1, through the 
tangent point m.sub.s1, to the exit point N.sub.1, is indicated by a thick 
line. The view shown in FIG. 7C, which is a development of the conical 
surface 15.sub.11 is equivalent to the representation in FIG. 5. 
More specifically, in FIG. 5, the blade 11 has a specific inflow angle 
.beta..sub.11 and a specific outflow angle .beta..sub.21 (of 90.degree. in 
this case) on the conical surface 15.sub.11 of one representative 
streamline 15.sub.1 and therebetween has a sectional profile in the shape 
of a smooth curve having a radius of curvature .rho. varying progressively 
in the vicinity of the entrance point M.sub.1 and thereafter of a straight 
line extending radially. This sectional profile can be obtained 
geometrically by determining the coordinates u.sub.o and v.sub.o of the 
centerline O of the cylindrical surface 19 along the axes U and V, the 
inclination angle K, and the radius C shown in FIGS. 7A and 7B by a method 
described hereinafter. Here, it is to be noted that since the plane 20 
includes the element 22 (FIG. 6) of the conical surface 15.sub.11, the 
outflow angle .beta..sub.21 at the exit point N.sub.1 is 90.degree.. 
These relationships will now be geometrically studied. An arbitrary point m 
on the curve M.sub.1 N.sub.1 constituting one part of the intersection 
between the conical surface 15.sub.11 of the representative streamline 
15.sub.1 and the cylinder 19 will be considered. This point m has 
coordinates (u,v) in FIG. 7A, coordinates (v,w) in FIG. 7B, and 
coordinates (x,y) in FIG. 7C, the coordinates (x,y) being based on 
orthogonal coordinate axes X and Y having their origin on the centerline 
axis H as shown in FIG. 7C. The axis Y is at the angle .theta..sub.1 
relative to the axis H and passes through the tangent point m.sub.s1 and 
the exit point N.sub.1. 
In this case, the following relationships were found to exist as a result 
of our mathematical and geometrical analysis 
EQU x=f (.theta..sub.1, u, r) (1) 
EQU y=f (.theta..sub.1, u, r) (2) 
EQU u=f (U.sub.o, V.sub.o, K, .theta..sub.1, C, r) (3) 
EQU .phi.=f (.theta..sub.1, u, r) (4) 
Here, r is the distance of the point m from the centerline axis H as shown 
in FIG. 7B, and .phi. is the angle between the axis Y and a straight line 
passing through the point m(x,y) and the origin E of the axis Y as shown 
in FIG. 7C. Therefore, by substituting the equations (1) through (4) 
respectively into the relationships 
##EQU1## 
EQU .beta.=tan.sup.-I (dy/dx)+.phi. (6) 
which are derived through differential analysis known in the art, the 
radius of curvature .rho. and the flow angle .beta. at the point m in FIG. 
7C are obtained. 
When the point m is at the entrance point M.sub.1, the corresponding angle 
.beta. coincides with the inflow angle .beta..sub.11. When this point m is 
at the tangent point m.sub.s1 of the cylindrical surface 19 and the plane 
20, the corresponding angle .beta. coincides with the outflow angle 
.beta..sub.21 (of 90.degree. in this case). Similarly, in the case where 
the arbitrary point m is on the straight line m.sub.s1 N.sub.1, which is 
one part of the mutual intersection between the plane 20 and the conical 
surface 15.sub.11 constituted by the representative streamline 15.sub.1, 
the coordinate u expressed by the above Eq. (3) becomes as indicated in 
Eq. (3)' given below, irrespective of the position of the point m. 
EQU u=0 (3)' 
Furthermore, Eqs. (5) and (6) respectively become as follows. 
EQU .rho.=.infin. (infinity) (5)' 
EQU .beta.=.beta..sub.s1 =.beta..sub.21 (90.degree. in this case) (6)' 
The reason why the value of the flow angle .beta..sub.s1 at the tangent 
point m.sub.s1 comes out the same (90.degree. in this case) whether it is 
derived by calculation with respect to the cylindrical surface 19 (i.e., 
the curve M.sub.1 m.sub.s1) or whether it is derived by calculation with 
respect to the plane 20 (i.e., the straight line m.sub.s1 N.sub.1) is that 
the cylindrical surface 19 and the plane 20 are tangent at the cylindrical 
element S.sub.1 -S.sub.2 (FIG. 6) including the tangent point m.sub.s1. As 
a result, the intersection line from the entrance point m.sub.s1, and to 
the exit point N.sub.1 is algebraically continuous. 
The radius .rho. of curvature varies gradually from the entrance point 
M.sub.1 toward the tangent point m.sub.s1. Therefore, the curve from the 
entrance point M.sub.1 to the tangent point m.sub.s1 becomes an ideal 
smooth curve in contrast to the blades of the fan wheel of a known 
centrifugal fan of the radial-plate type in which each blade has a curve 
comprising a single arc or, at the most, two arcs of different radii in 
the vicinity of the entrance point M.sub.1. 
Thus, the representative streamline 15.sub.1 shown in FIG. 3 is obtained as 
indicated in outline form in FIG. 6. In the same manner, the 
representative streamlines 15.sub.2, 15.sub.3, . . . 15.sub.n are obtained 
respectively from the intersections of the cylinder 19 and plane 20 and 
the conical surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1. 
FIG. 8A shows a projection of this state as viewed in the arrow direction Q 
(FIG. 6). This projection corresponds to FIG. 7A. Furthermore, FIG. 8B is 
a projection corresponding to FIG. 7B. These intersection lines can be 
readily computed by carrying out with respect to the conical surfaces 
15.sub.21, 15.sub.31, . . . 15.sub.n1 operations similar to that with 
respect to the conical surface 15.sub.11. 
That is, FIGS. 8A and 8B are similar to FIGS. 7A and 7B but further have 
conical surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1 having a common 
centerline axis H with the conical surface 15.sub.11 and respectively 
having half vertex angles .theta..sub.2, .theta..sub.3, . . . 
.theta..sub.n. These n conical surfaces 15.sub.11, 15.sub.21, . . . 
15.sub.n1 are arranged in the same manner as the n conical surfaces 
constituted by the representative streamlines 15.sub.1, 15.sub.2, . . . 
15.sub.n in FIG. 3, and, moreover, the balde 11 shown in FIG. 3 is 
obtained as a part of the cylinder 19 of radius C and the plane 20 shown 
in FIG. 8. 
In any fans, including diagonal-flow fans, if the gas flow rate, gas 
pressure and rotational speed are given, the radial distances from the 
shaft axis to the entrance and exit edges of each blade, inflow and 
outflow angles at the entrance and exit edges, and blade width in the 
direction transverse to the gas flow direction can be determined, as 
values on representative streamlines in the fan wheel, as a result of 
fluid-dynamic analyses. How the above values are determined is explained 
in available textbooks relating to fans. 
In the case of diagonal-flow fans, the determination of the half vertex 
angle (diagonal-flow angle) .theta. is 90.degree. in the case of a 
radial-flow fan and 0.degree. in the case of an axial-flow fan, the angle 
.theta. being determined as a value between 90.degree. and 0.degree. in 
the case of a diagonal-flow fan, by dynamical and mathematical analyses 
and/or on the basis of various texts. 
If the various values as mentioned above have been determined temporarily 
with respect to representative streamlines by the above described 
procedures, a streamline shape as shown in FIG. 3 of the present 
application is determined temporarily. On the basis of this temporarily 
determined streamline shape, the above values temporarily determined are 
considered again and somewhat changed. More specifically, a theoretical 
analysis is made on the basis of the temporarily determined streamline 
shape so that gas collision will not occur at the blade entrance edge and 
discharge gas pressure will be distributed as required along the blade 
exit edge, and, as a result of this analysis, the final values of the 
radial distances from the shaft axis to the blade entrance and exit edges, 
inflow and outflow angles and so on are determined, which in turn makes it 
possible to determine the streamline shape finally. 
The procedure will be explained more fully below. For purposes of 
simplicity, the streamline 15.sub.1 is taken as a reference streamline. 
The radial distance r.sub.s1 of the point m.sub.s1, which is a tangent 
point between the planar portion and curved portion of the blade 11, is 
determined as a result of theoretical analysis of gas flow and/or on the 
basis of experimental tests. For example, the value of (r.sub.in1 
+r.sub.out)r/.sub.s1 is ordinarily taken approximately between 1.8 and 
2.5. 
There are the following relations between the variables C and K and the 
inflow and outflow angles .beta..sub.1 and .beta..sub.2. 
EQU .beta..sub.1 (at r.sub.in1)=f(.theta..sub.1, r.sub.s1, C,K) (7) 
EQU .beta..sub.2 (at r.sub.out1)=g(.theta..sub.1, r.sub.s1, C,K) (8) 
Here the values of .theta..sub.1 and r.sub.s1 have already been determined, 
so that the variables C and K can be determined as a combination of C and 
K by solving the simultaneous equations (7) and (8) and by substituting 
.beta..sub.11 for .beta..sub.1 and .beta..sub.21 for .beta..sub.2. 
If the variables C and K are determined, the line of intersection M.sub.1 
-N.sub.1 between the cylindrical surface 19 and the plane 20 can be 
calculated from the coordinates (u,v,w) of the point m. In FIG. 6, the 
portion M.sub.1 -m.sub.s1 of the intersection M.sub.1 -N.sub.1 is in the 
cylindrical surface 19 and the portion m.sub.s1 -N.sub.1 in the plane 20. 
After determining the line of intersection M.sub.1 -N.sub.1 with respect to 
the representative streamline 15.sub.1, the next streamline 15.sub.2 is 
taken for determination. The tangent point m.sub.s2 (FIG. 8B) between the 
cylindrical surface 19 and the plane on the streamline 15.sub.2 has a 
radial distance r.sub.s2. This radial distance r.sub.s2 and the position 
of the point m.sub.s2 can be determined mathematically as a function of C 
and K, the axial distance of the streamline 15.sub.2 from the streamline 
15.sub.1, and the angle .theta..sub.2. Thus, but substituting m.sub.s2, 
r.sub.s2, C and K in the equations (7) and (8), the following equations 
are obtained. 
EQU .beta..sub.1 =f(r.sub.in) (9) 
EQU .beta..sub.2 =g(r.sub.out) (10) 
On the other hand, the inflow angle .beta..sub.1 at which the inflow gas 
collision at the streamline 15.sub.2 is theoretically zero at an entrance 
edge radial distance r.sub.in is expressed by the following equation: 
EQU .beta..sub.1 =tan.sup.-1 (r.sub.in1 .multidot.tan .beta..sub.11/ r.sub.in) 
(11) 
where r.sub.in1 is the radial distance of the entrance edge on the 
streamline 15.sub.1 and .beta..sub.11 is the inflow angle on the 
streamline 15.sub.1. By solving the simultaneous equations (9) and (11), 
the radial distance r.sub.in2 and inflow angle .beta..sub.12 on the 
streamline 15.sub.2 are determined. 
The overflow angle .beta..sub.2 at which the gas discharge pressure on the 
streamline 15.sub.2 is at a given value at an exit edge radial distance 
r.sub.out is expressed by the following equation: 
EQU .beta..sub.2 =g(r.sub.out, r.sub.in2, Z, etc.) (12) 
where r.sub.in2 is the radial distance of the entrance edge on the 
streamline 15.sub.2 and Z is the number of the blade 11. 
Therefore, by substituting given values for r.sub.in2, Z and so on and by 
solving the simultaneous equations (10) and (12), the exit edge radial 
distance r.sub.out2 and outflow angle .beta..sub.22 on the streamline 
15.sub.2 can be determined. 
By carrying out similar procedures with respect to the streamlines 15.sub.3 
-15.sub.n, the lines M.sub.1 -m.sub.s1 -N.sub.1 -N.sub.2 - . . . -N.sub.n 
-m.sub.sn -M.sub.n -M(n-1)- . . . -M.sub.1 as shown in FIG. 6 can be 
determined to enable the cutting of a blade. Of the blade thus, cut, the 
portion defined by M.sub.1 -m.sub.s1 -M.sub.s2 - . . . -N.sub.n -m.sub.sn 
-m.sub.s1 is formed from the plane 20. 
In the above explanation, the streamline 15.sub.1 was taken as a reference 
streamline. However, any one of the streamlines could be made a reference 
streamline. For example, a streamline in the middle of the streamlines 
could be made a reference line. In any case, the equations (9)-(12) can be 
used to obtain similar results with respect to the streamlines other than 
the reference streamline. Moreover, even in the case of using a 
cylindrical surface C.sub.2 in place of a plane, as shown in FIG. 15, a 
similar procedure can be taken. 
As is apparent from FIGS. 6 and 8A, when the group of n conical surfaces 
inclined as shown is viewed in the axial direction of the cylinder 19 (the 
arrow direction Q in FIG. 6), the intersection lines, that is, the blade 
11, coincides with a part of the single-curvature surface comprising the 
cylinder 19 of the radius C and the plane 20 and has no twist, appearing 
as a superimposition with the same sectional profile. By the absence of 
twist in the developable surfaces 19 and 20, progressively varying inflow 
and outflow angles at the entrance and exit parts are obtained, because 
the developable blade is cut obliquely as shown in FIG. 6 and in FIG. 10D, 
and because the thus cut blade is installed in the main and side plates 16 
and 17 with its entrance and exit edges disposed at specific relations to 
the stream surfaces. When the conical surface 15.sub.11 is developed into 
a planar surface, it becomes as shown in FIG. 7C, and the other conical 
surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1 also can be similarly 
developed. The intersections due to these developments are not shown in 
FIG. 8, but, as indicated in outline form in FIG. 6, they respectively 
start at points M.sub.2, M.sub.3, . . . M.sub.n, pass through the tangent 
points m.sub.s2, m.sub.s3, . . . m.sub.sn, and end at point N.sub.2, 
N.sub.3, . . . N.sub.n, having inflow angles .beta..sub.12, .beta..sub.22, 
. . . .beta..sub.1n and an outflow angle .sub.21, the inflow angle 
respectively differing slightly from the inflow angles .beta..sub.11 at 
the streamline 15.sub.1. Between the entrance and tangent points, the 
intersection lines are in the form of smooth curves having gradually 
varying radii .rho. of curvature. 
The outflow angles of the intersections, that is, the representative 
streamlines 15.sub.2, 15.sub.3, . . . 15.sub.n, are 90.degree. (constant 
value) since the intersecting plane 20 passes through elements of the 
conical surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1. The intersection 
lines, of course, are continuous curves in the algebraic sense also at the 
tangent points m.sub.s2, m.sub.s3, . . . m.sub.sn of the cylindrical 
surface 19 and the plane 20. That the inflow angles .beta..sub.11, 
.beta..sub.12, . . . .beta..sub.1n respectively differ slightly from each 
other is a natural result of the variation of the radial distance r.sub.in 
at the entrance point of each of the representative streamlines 15.sub.1, 
15.sub.2, . . . 15.sub.n as described hereinbefore with respect to FIG. 3. 
When all intersection lines, that is, all representative streamlines 
15.sub.1, 15.sub.2, . . . 15.sub.n have been determined by calculation as 
described above, the figure enclosed by the curve M.sub.1 m.sub.s1 at the 
representative streamline 15.sub.1, the curve M.sub.n m.sub.sn at the 
representative streamline 15.sub.n, and the curve M.sub.1 M.sub.n and the 
straight line m.sub.s1 m.sub.sn straddling the remaining representative 
streamlines and the figure enclosed by the straight line m.sub.s1 N.sub.1 
at the representative streamline 15.sub.1, the straight line m.sub.sn 
N.sub.n at the representative streamline 15.sub.n, and the straight line 
m.sub.s1 m.sub.sn and the curve N.sub.1 N.sub.n straddling the remaining 
representative streamlines are respectively cut out from a cylindrical 
blank corresponding to the cylindrical surface 19 of radius C and a planar 
plate blank corresponding to the plane 20. The locus of this cutting out 
can be readily understood from the coordinates of the point m, that is, m 
(u, v, w), in FIG. 7. 
On another hand, the cutting out locus in the case of planar development 
can also be readily understood from the m point coordinates m (x, y). 
Accordingly, the figure enclosed the curves M.sub.1 N.sub.1, N.sub.1 
N.sub.n, M.sub.n N.sub.n, and M.sub.1 M.sub.n may be cut out from a steel 
sheet, and the part from the entrance points M.sub.1, M.sub.2, . . . 
M.sub.n to the tangent points m.sub.s1, m.sub.s2, . . . m.sub.sn may be 
curved to a radius of C. In this case, since the tangent line of the 
cylindrical surface 19 of radius C and the plane 20 coincides with an 
element S.sub.1 -S.sub.2 of the cylindrical surface 19, the fabrication of 
the blade by bending the steel sheet by means of rolls, for example, can 
be easily carried out. 
In the above described manner, the blade 11 is cut out from the cylindrical 
surface 19 and the plane 20. Alternatively, a steel sheet cut out 
beforehand is curved to a radius c at its part corresponding the region 
near the entrance points. Then, as indicated in FIG. 9, blades 11 thus 
formed are assembled with the main plate 16 and the side plate 17 thereby 
to form a fan wheel. Thus, without using blades having double-curvature 
surfaces, which have been considered a requisite for diagonal-flow fans, a 
fan wheel with blades producing a performance equivalent to that of 
double-curvature blades is easily fabricated. 
The two developable surfaces, such as a cylindrical surface 19 and a planar 
surface 20, are chosen depending on the type, performance and dimensions 
of a fan wheel to be produced. Because there are a number of predetermined 
standards of types, performances and dimensions of diagonal-flow fans, the 
choice of the two developable surfaces can be determined on the basis of 
such standards. How the blade is oriented with respect to the conical 
surfaces will be apparent from the discussion in the following paragraphs. 
In designing a fan wheel according to this invention of a diagonal-flow fan 
of radial-plate, the representative streamlines 15.sub.1 through 15.sub.n 
as shown in FIG. 3 are first determined. From these, the half-vertex 
angles .theta..sub.1 through .theta..sub.n of the conical surfaces are 
determined. Standard values based on common practice of the ratio of the 
inner and outer diameters of each blade have been determined in accordance 
with the gas flow rate and delivery pressure, and, therefore, the 
distribution of the inflow angle .beta..sub.1 along the blade entrance 
edge 12 is determined from the rotational speed of the fan wheel. 
The radial distance r.sub.s of the tangent point m.sub.s of the curved part 
and the straight-line part of the blade 11 is also made to equal a 
standard value based on experience. The distances u.sub.o and v.sub.o 
shown in FIGS. 6 and 7 are determined at once from the radius distance 
r.sub.s1 of the tangent point m.sub.s1 (FIG. 7B) when the inclination 
angle K and the cylindrical surface radius C have been determined. 
Accordingly, the remaining variables are K and C. These two variables K 
and C are so adjusted that the inflow angle .beta..sub.1 at the entrance 
edge 12 will become a specific value. It is to be noted that the specific 
value of the inflow angle B.sub.1 is predetermined. After thus finally 
determining the angle K and the radius C as well as the coordinates 
U.sub.o and V.sub.o, it is now possible to plot the entrance and exit 
points M.sub.1 and N.sub.1 and the tangent point m.sub.s1 and to draw the 
curve 15.sub.1 on a blank cylinder 19. This curve 15.sub.1 can be readily 
determined from the coordinates of the point m, that is, m(u,v,w). 
The thus determined positions of the entance and exit points M.sub.1 and 
N.sub.1 on the cylinder become basic reference points from which the 
plotting of the other entrance and exit points M.sub.2, M.sub.3, . . . 
M.sub.n and N.sub.2, N.sub.3, . . . N.sub.n starts. The next procedure is 
to determined the positions of the adjoining entrance and exit points 
M.sub.2 and N.sub.2 on the line of intersection or curve 15.sub.2. The 
determination of the position of the point M.sub.2 is made by so adjusting 
the inner radial distance thereof from the shaft axis with respect to the 
conical surface 15.sub.21, in which the intersection line 15.sub.2 lies, 
on the basis of the determined values of the angle K, the radius C and the 
coordinates U.sub.o and V.sub.o as to obtain the predetermined inflow 
angle .beta..sub.12. If the thus determined position of the point does not 
coincide substantially with an expected position, a different combination 
of the values of K and C is adopted and the same procedure as above stated 
is repeated. The same procedure is repeated for the other conical 
streamline surfaces to determine the positions of the other points. It 
will be understood that the determination of the exit points can be easily 
made since the outflow angle is constant. 
For convenience in design, data may be prepared in advance in the above 
described manner as design information so that, when the inflow angle and 
the ratio of the inner and outer diameters of the fan wheel are given, the 
essential dimensions can be immediately determined. For example, in the 
case of an inner-to-outer diameter ratio .lambda. and a conical half 
vertex angle .theta., a graph with the inclination angle K as the 
abscissa, the inflow angle .beta..sub.1 as the ordinate, and the 
cylindrical surface radius C as a parameter may be prepared beforehand. 
In the above description, the line of intersection 15.sub.1 at one end was 
made a reference curve for a purpose of simplicity. However, in practical 
design, the reference curve is selected not from the line of intersection 
at one end but from the line in the middle of the blade. The use of such 
middle line as a reference curve is advantageous because it represents a 
mean streamline. 
In practice, the plotting of the entrance and exits points as well as the 
drawing of the contour line of the blade on a blank can be made manually, 
but this procedure is most advantageously carried out by a computerized 
apparatus. 
In the foregoing disclosure, the case wherein the plane 20 is so set that 
elements of the conical surfaces lie in that plane thereby to set the 
outflow angle .beta..sub.2 at the constant value of 90.degree. has been 
described. If necessary, however, the various dimensions can be determined 
by similar calculation also for the case wherein the outflow angle 
.beta..sub.2 progressively varies. For example, in the case where the 
outflow angle .beta..sub.2 is caused to vary progressively along the exit 
edge 13 for some purpose such as attaining an even more uniform pressure 
head at the exit edge 13 or a improvement in performance, the flow angle 
.beta..sub.s at the tangent point m.sub.s of the cylindrical surface 19 
and the plane 20 is made smaller (or greater) than 90.degree.. The 
intersection drawing corresponding to FIG. 7 in this case is shown in FIG. 
10. Here, the plane 20 is so set that it is parallel to the W axis and, 
moreover, intersects the V axis with a certain angle at a point S.sub.o 
(FIG. 6) on the V axis. 
Thereafter, the intersection lines of the conical surfaces 15.sub.11, 
15.sub.21, . . . 15.sub.n1 and the cylindrical surface 19 and the plane 20 
are obtained by the same method. Then, the outflow angle .beta..sub.2 of 
the balde 11 progressively varies as .beta..sub.21, .beta..sub.22, . . . 
.beta..sub.2n at the intersection points, and, further, as shown in FIG. 
10C, the curve from the tangent point m.sub.s to the exit point N also 
becomes a smooth curve (a rearwardly curved line in this case) wherein the 
radius of curvature varies gradually. Of course, the blade 11 has an 
algebraically continuous curve at the tangent points m.sub.s1 through 
m.sub.sn of the cylindrical surface 19 and the plane 20. 
With respect to FIG. 10, a plate bland such as shown in FIG. 10D is 
prepared. Since this blank is made of developable surfaces (A) and (B), it 
is easy to produce such blank. On the other hand, basic mathematical 
calculations are made from the coordinates of the point m (u,v,w) (FIG. 7) 
on the basis of predetermined values such as those referred to hereinfore, 
and as a result of the calculations the locus of cutting of the blank with 
respect to the origin E of the coordinate system can be determined for 
producing the shape (M.sub.1 -N.sub.1 -N.sub.n -M.sub.n) of the blade 
shown in FIG. 10D. 
Alternatively, the blade shape can be cut from a planar plate and then a 
portion thereof is curved into a cylindrical form to obtain the blade 
shown in the enclosed sketch. In practice the cutting operation is carried 
out by a computerized apparatus. 
FIG. 11 illustrates one example of construction of a fan wheel wherein an 
intermediate plate 21 of conical shape is furthr installed between the 
main plate 16 and the side plate 17 in the fan wheel shown in FIG. 3, and 
all blades 11 are divided by this intermediate plate 21 into sections 
11.sub.1 and 11.sub.2. Depending on the circumstances, a plurality of 
intermediate plates can be similarly installed thereby to divide the 
blades 11 into a greater number of sections. 
The reasons for such a measure is that, in the case where the requirements 
for variations of the inflow angles .beta..sub.11 through .beta..sub.1n 
and the outflow angles .beta..sub.21 through .beta..sub.2n cannot be 
satisfied for all of the representative streamlines 15.sub.1 through 
15.sub.n related to each blade 11 with only a single cylinder 19 and a 
single plane 20, blades produced by intersections with a plurality of 
mutually different cylinders and planes are afforded by this measure. 
Another reason is that, by this construction, the strength of the fan 
wheel itself is increased by the insertion of the intermediate plate 21. 
This invention can be applied also to the fan wheel of a diagonal-flow fan 
of the limit-load type, as will now be described in conjunction with FIGS. 
13 through 18. The general structural features of a fan wheel of a fan of 
this type are similar to those of a fan wheel of a diagonal-flow fan of 
the radial-plate type described in the foregoing disclosure and, 
therefore, will not be described again. 
A planar development of the conical surface 15.sub.11 representing the 
representative streamline 15.sub.1 in FIG. 3 is shown in FIG. 14 and shows 
a chordwise section of a blade 11. This blade section has a specific 
inflow angle .beta..sub.11 at the entrance point M.sub.1 and a specific 
outflow angle .beta..sub.21 at the exit point N.sub.1 and has between 
these two points a curved shape resembling a portion of an ellipse with a 
gradually varying radius .rho. of curvature. The inflow angle 
.beta..sub.11 of this blade 11 varies progressively as .beta..sub.12, 
.beta..sub.13, . . . .beta..sub.1n as indicated in FIG. 13 in 
correspondence to the representative streamlines 15.sub.2, 15.sub.3, . . . 
15.sub.n of FIG. 3 and the radius .rho. of curvature also varies. For this 
reason, the blade 11 is required to have a complicate double-curvature 
surface shape. This double-curvature blade shape is closely approximated 
by the blade 11 according to this invention which is obtained in the 
following manner. 
FIG. 15 is a graphical perspective view showing intersections between 
coaxial conical surfaces corresponding to the representative steamline 
15.sub.1, 15.sub.2, . . . 15.sub.n shown in FIG. 3 and newly introduced 
two imaginary cylindrical surfaces 29 and 30 circumscribing each other. In 
FIGS. 16A, 16B, and 16C, the intersections between a conical surface 
corresponding to the representative streamline 15.sub.1 and the 
cylindrical surfaces 29 and 30 are projectionally indicated. For the 
following analysis, three-dimensional, rectangular coordinate axes U, V, 
and W, similar to those used in the description of the preceding 
embodiment of the invetnion, are used. The origin of this coordinate 
system is positioned at the vertex E of the concial surface 15.sub.11. The 
W axis is made to be parallel to the centerline O.sub.1 of the cylindrical 
surface 29 and to the centerline O.sub.2 of the cylindrical surface 30, 
and the V axis is taken to be superimposed on the point m.sub.s1 of 
tangency between the cylindrical surfaces 29 and 30 on the curve M.sub.1 
N.sub.1 when viewed in the W-axis direction (arrow direction Q in FIG. 15) 
as shown in FIG. 16. 
As indicated in FIG. 15, the coordinates relative to these coordinate axes 
U, V and W of the centerline O.sub.1 of the cylindrical surface 29 of 
radius C.sub.1 in the U-axis and V-axis directions are respectively 
u.sub.o1 and v.sub.o1, while the coordinates of the centerline O.sub.2 of 
the cylindrical surface 30 of radius C.sub.2 in the U-axis and V-axis 
directions are respectively u.sub.o2 and v.sub.o2. Furthermore, the 
centerlines O.sub.1 and O.sub.2 of these two cylindrical surfaces 29 and 
30 are inclined by the same angle K relative to the centerline H of the 
conical surface 15.sub.11 of a half vertex angle of .theta..sub.1. At the 
same time, these two cylindrical surfaces 29 and 30 are mutually tangent 
along a common cylindrical element S.sub.1 S.sub.2 passing through a point 
S on the V axis. 
From the manner in which the W axis is taken as described above, the 
inclination angle K of the cylinder 29 can be expressed by the angle 
between the W axis and the centerline H of the conical surface 15.sub.11. 
The conical surface 15.sub.11 is the same as the conical surface 
constituted by the representative streamline 15.sub.1 in FIG. 3. The 
intersection line of this conical surface 15.sub.11 with the two 
cylindrical surfaces 29 and 30, that is, that portion of the tangency line 
from the entrance point M.sub.1, through the tangent point m.sub.s1, to 
the exit point N.sub.1, is indicated by a thick-line curve in the 
development of the conical surface 15.sub.11 in FIG. 16C, and this curve 
is equivalent to the curve of the blade 11 in FIG. 14. 
More specifically, the sectional profile of the blade 11 as shown in FIG. 
14 has specific inflow and outflow angles .beta..sub.11 and .beta..sub.21 
on a conical surface 15.sub.11 of one representative streamline 15.sub.1, 
and the entrance point M.sub.1 and the exit point N.sub.1 are joined by a 
smooth, elongated S-shaped curve having a radius of curvature which varies 
progressively. This sectional profile of the blade 11 can be geometrically 
derived by determining the distances u.sub.o1, v.sub.o1, u.sub.o2, and 
v.sub.o2, the inclination angle K, and the radii C.sub.1 and C.sub.2 by 
the method described hereinafter. 
These relationships can be geometrically considered similarly as described 
hereinbefore in the preceding embodiment of the invention with respect to 
Eqs. (1) through (6) set forth hereinbefore. 
For example, in the case where any point m is disposed on the arcuate curve 
m.sub.s1 N.sub.1, which is a part of the intersection line between the 
conical surface 15.sub.11 constituted by the representative streamline 
15.sub.1 and the cylindrical surface 29, is considered, the same theory 
can be applied directly except that Eq. (3) set forth hereinbefore merely 
changes into the following form. 
EQU u=f (u.sub.o1, v.sub.o1, K, .theta..sub.1, C.sub.1, r) (3a) 
As a result, the radius .rho. of curvature and the flow angle .beta. of the 
point m in FIG. 16C is obtained. When the point m is at the tangent point 
m.sub.s1, the angle .beta. at that time coincides with the flow angle 
.beta..sub.s1 at the point of inflection of the S figure, and when point m 
is at the exit point N.sub.1, the angle .beta. at that time coincides with 
the outflow angle .beta..sub.21. 
Similarly, in the case where any point m is disposed on the arcuate curve 
M.sub.1 m.sub.s1, which is a part of the intersection line between the 
conical surface 15.sub.11 constituted by the representative streamline 
15.sub.1 and the cylindrical surface 30, is considered, the above 
described theory can be applied directly except that Eq. (3) set forth 
hereinbefore merely changes into the following form. 
EQU u=f (u.sub.o2, v.sub.o2, K, .theta..sub.1, C.sub.2, r) (3b) 
Accordingly, when the point m is at the entrance point M.sub.1, the angle 
.beta. at that time coincides with the inflow angle .beta..sub.11, and 
when the point m is at the tangent point m.sub.s1, the angle .beta. at 
that time coincides with the flow angle .beta..sub.s1 at the point of 
inflection of the S figure. Since the two cylindrical surfaces 29 and 30 
are mutually tangent along their elements S.sub.1 -S.sub.2, this flow 
angle .beta..sub.s1 at this tangent point (point of inflection) comes out 
to be the same value whether it is calculated on the basis of its being on 
the cylindrical surface 29 (on the curve m.sub.s1 N.sub.1) or whether it 
is calculated on the basis of its being on the cylindrical surface 30 (on 
the curve M.sub.1 m.sub.s1). As a result, it is evident that the curve 
M.sub.1 N.sub.1 of S shape is an algebraically continuous curve. 
Furthermore, as the point m is considered to move from the entrance point 
M.sub.1 to the exit point N.sub.1, the radius of curvature .rho. varies 
gradually. For this reason, the S-shaped curve from the entrance point 
M.sub.1 to the exit point N.sub.1 is a smooth curve approaching the ideal 
shape, in contrast to the fan wheel of a conventional centrifugal fan of 
limit-load type wherein each of the curved parts of the S-shaped figure 
comprises a single arc or two arcs, at the most, joined together. 
In the above described manner, the representative streamline 15.sub.1 shown 
in FIG. 3 is obtained as indicated in outline form in FIG. 15. In the same 
manner, the other representative streamlines 15.sub.2, 15.sub.3, . . . , 
15.sub.n shown in FIG. 3 are obtained as respective intersection lines 
between the cylindrical surfaces 29 and 30 and the conical surfaces 
15.sub.21, 15.sub.31, . . . , 15.sub.n1. 
FIG. 17A is a projection of this state as viewed in the arrow direction Q 
in FIG. 15. This projection corresponds to FIG. 16A, and, further, FIG. 
17B corresponds to FIG. 16B. These intersection lines can be readily 
obtained through calculation by carrying out, with respect to the conical 
surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1, operations similar to that 
carried out with respect to the conical surface 15.sub.11. 
That is, FIGS. 17A and 17B are equivalent to FIG. 16 with the addition of 
the conical surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1 coaxially 
disposed relative to the conical surface 15.sub.11 with the centerline 
axis H as a common centerline and respectively having half vertex angles 
.theta..sub.2, .theta..sub.3, . . . .theta..sub.n. These n conical 
surfaces 15.sub.11, 15.sub.21, . . . 15.sub.n1 are arranged similarly as 
the n conical surfaces constituted by the representative streamlines 
15.sub.1, 15.sub.2, . . . 15.sub.n of FIG. 3, and, moreover, the blade 11 
of FIG. 3 is substituted into a part of the cylindrical surface 29 of 
radius C.sub.1 and the cylindrical surface 30 of radius C.sub.2 shown in 
FIG. 17. 
As is apparent also from FIGS. 15 and 17A, when the intersection lines on 
the n conical surfaces are viewed in the axial direction of the 
cylindrical surfaces 29 and 30 (arrow direction Q in FIG. 15), the 
intersection lines, that is, the blade 11, is a part of a single-curvature 
(developable) surface constituted by the cylindrical surface of radius 
C.sub.1 and the cylindrical surface of radius C.sub.2, having no twist, 
and appears as a superimposition of the same sectional profiles. When the 
conical surface 15.sub.11 is developed into a plane, it becomes as shown 
in FIG. 16C as mentioned hereinbefore. 
The conical surfaces 15.sub.21, 15.sub.31, . . . 15.sub.n1 can also be 
developed in the same manner. The intersection lines due to these 
developments begin at the entrance points M.sub.2, M.sub.3, . . . M.sub.n, 
pass through the tangent points (inflection points) m.sub.s2, m.sub.s3, . 
. . m.sub.sn, and terminate at the exit points N.sub.2, N.sub.3, . . . 
N.sub.n, as indicated in outline form in FIG. 15 although not shown in 
FIG. 17. These intersection lines respectively have inflow angles 
.beta..sub.12, .beta..sub.13, . . . .beta..sub.1n and outflow angles 
.beta..sub.22, .beta..sub.23, . . . .beta..sub.2n which respectively 
differ progressively by small differences from the inflow angle 
.beta..sub.11 and the outflow angle .beta..sub.21 corresponding to the 
representative streamline 15.sub.1, and the entrance points and the 
corresponding exit points are respectively joined by smooth curves of 
radii of curvature .rho. which gradually vary. 
All intersection lines, of course, are algebraically continuous also at the 
tangent points m.sub.s2, m.sub.s3, . . . m.sub.sn of the cylindrical 
surfaces 19 and 20. That the inflow angles .beta..sub.11, .beta..sub.12, . 
. . .beta..sub.1n and the outflow angles .beta..sub.21, .beta..sub.22, . . 
. .beta..sub.2n respectively differ slightly from each other is a natural 
result of the variations of the radial distance r.sub.in at the entrance 
point and the radial distance r.sub.out at the exit point of each of the 
representative streamlines 15.sub.1, 15.sub.2, . . . 15.sub.n as described 
hereinbefore with reference to FIG. 3. 
When all intersection lines, that is, a representative streamlines 
15.sub.1, 15.sub.2, . . . 15.sub.n have been operationally determined, the 
part enclosed by the curve m.sub.s1 N.sub.1 at the representative 
streamline 15.sub.1, the curve m.sub.sn N.sub.n at the representative 
streamline 15.sub.n, and the curve N.sub.1 N.sub.n and the straight line 
m.sub.s1 m.sub.sn straddling all representative streamlines is cut out 
from the cylindrical surface 29 of radius C.sub.1. The part enclosed by 
the curve M.sub.1 m.sub.s1 at the representative streamline 15.sub.1, the 
curve M.sub.n m.sub.sn at the representative streamline 15.sub.n, and the 
curve M.sub.1 M.sub.n and the straight line m.sub.s1 m.sub.sn straddling 
all representative streamlines is cut out from the cylindrical surface 30 
of radius C.sub.2. The path or outline of this cutting out operation can 
be readily determined from the coordinates of the point m, that is, m 
(u,v,w). 
On another hand, the cutting out path in the case of development into a 
planar figure can be readily determined in a similar manner from the 
coordinates of the point m, that is, m (x,y). For this reason, the blade 
11 may be produced by first cutting out from a flat sheet of steel a part 
enclosed by the curves M.sub.1 N.sub.1, N.sub.1 N.sub.n, M.sub.n N.sub.n, 
and M.sub.1 M.sub.n and then curving this cut-out steel sheet with the 
radius C.sub.1 and the radius C.sub.2 thereby to impart the S shape 
thereto. 
In this case, since the line of juncture of the cylindrical surfaces 29 and 
30 of radii C.sub.1 and C.sub.2, respectively, is an element of each of 
these cylindrical surfaces, the blade 11 can be easily fabricated by 
curving the steel sheet by rolling, for example. 
The blade 11 is thus cut out from the cylindrical surfaces 29 and 30 or is 
cut out from a flat steel sheet and then curved into the S shape with the 
radii C.sub.1 and C.sub.2. By assembling a designed number of these blades 
11 together with a main plate 16 and a side plate 17 as indicated in FIG. 
18, there is obtained a diagonal-flow fan of a performance equivalent to 
that of a fan wheel provided with blades of double-curvature surface, 
which were considered to be requisite for the fan wheel of a diagonal-flow 
fan. Thus, this high-performance fan wheel can be easily produced. 
In actually designing a fan wheel according to this embodiment of the 
invention of a limit-load type, diagonal-flow fan, the representative 
streamlines 15.sub.1 through 15.sub.n are first determined. From these, 
the conical surface half vertex angles .theta..sub.1 through .theta..sub.n 
are determined. Standard values of the ratio of the inner and outer 
diameters of each blade have been tentatively determined in accordance 
with the gas flow rate and delivery pressure. Therefore, from the 
rotational speed of the fan wheel, the distribution of the inflow angle 
.beta..sub.1 along the blade entrance edge 12 and the distribution of the 
outflow angle .beta..sub.2 along the blade exit edge 13 are determined. 
Furthermore, for the flow angle .beta..sub.s at the point of inflection 
m.sub.s, a value based on experience has been determined as a standard 
value. When the inclination angle K and the radii C.sub.1 and C.sub.2 of 
the cylindrical surfaces 29 and 30 have been determined, the distances 
u.sub.o1, v.sub.o1, u.sub.o2, and v.sub.o2 are readily determined from the 
radial distance r.sub.s1 (FIG. 16B) of the inflection point m.sub.s1 and 
the flow angle .beta..sub.s1. Accordingly, the remaining variables are K, 
C.sub.1, and C.sub.2. K and C.sub.2 become variables at the entrance point 
M.sub.1, and K and C.sub.1 become variables at the exit point N.sub.1. 
These three variables K, C.sub.1, and C.sub.2 are selected at values such 
that the outflow angle .beta..sub.2 at the exit edge 13 and inflow angle 
.beta..sub.1 at the entrance edge 12 will be of respective specific 
values. 
For convenience in design, similarly as in the example of the diagonal-flow 
fan of radial-plate type described hereinbefore, data may be prepared in 
advance in the above described manner as design information so that, when 
the inflow and outflow angles and the ratio of the inner and outer 
diameters of the fan wheel are given, the essential dimensions can be 
immediately determined. For example in the case of an inner-to-outer 
diameter ratio .lambda., a conical half vertex angle .theta., and a flow 
angle .beta..sub.2 at the inflection point of the S figure, it is 
advantageous to prepare in advance a graph with the cylindrical radius 
C.sub.1 as a parameter, the inclination angle K as the abscissa, and the 
outflow angle .beta..sub.2 as the ordinate and a graph with the 
cylindrical radius C.sub.2 as a parameter, K as the abscissa, and inflow 
angle .beta..sub.1 as the ordinate. In using these two graphs, of course, 
common values of the inclination angle K must be used. 
As in the preceding embodiment of this invention, an intermediate plate 21 
of frustoconical shape can be further installed as illustrated in FIG. 11, 
whereby the various advantages feature described hereinbefore are 
afforded. 
In accordance with this invention, as described above, blades each of a 
single-curvature (developable) surface, which is a portion of a 
cylindrical surface, are used instead of blades each of double-curvature 
(undevelopable) surface, which was heretofore considered to be 
indispensable, in the fan wheel of a diagonal-flow fan, whereby a fan 
performance equivalent to that of a fan provided with ideal 
double-curvature blades can be attained. 
That is, the inflow angles and outflow angles of each blade vary 
progressively in accordance with the positions taken in the gas flow path 
by the representative streamlines within the fan wheel. In addition, each 
curve extending from the corresponding entrance point to the exit point 
also has a shape which is not a simple arc with a single radius of 
curvature or, at the most, a curve formed by joining two arcs as in 
centrifugal fans but is a curve which is close to the ideal according to 
fluid dynamics and has a radius of curvature varying progressively over 
the entire chord length.