On Determining the Dimensions of Carriers for Pneumatic Mail
Abstract Generated abstract
This paper addresses the problem of selecting maximum carrier dimensions for pneumatic mail systems whose pipeline routes include bends of small radius, a situation in which graphical sizing methods are laborious and existing semiempirical formulas may be overly conservative. The authors derive analytical relations connecting bend radius, pipe diameter, carrier length, sleeve diameter, sealing belt dimensions, and clearances for a commonly used carrier shape, using geometric tangency conditions between the carrier and the toroidal inner surface of a pipe bend. The resulting formulas define limiting carrier dimensions for unobstructed passage through ideal bends, include conditions for choosing the appropriate relation, and recommend a compensating belt curvature for manufacturing deviations. The paper also notes that, for the same bend parameters, an alternative carrier form can provide greater useful volume.
Full Text
ON DETERMINING THE DIMENSIONS OF CARRIERS FOR PNEUMATIC MAIL
G. P. Borisenko, A. E. Belyaev
(Presented by the scientific seminar of the Department of Applied Mechanics)
When constructing pneumatic-mail systems for already operating installations and equipment, it is often necessary to include bends of small radii in the pipeline route. In some cases the bend radius has to be made close to, or even equal to, the minimum permissible radius under the conditions for obtaining a high-quality bend [5, 6, 7, 8].
For such conditions it is very important to determine the maximum possible dimensions of the carriers, since this is necessary in order to answer the question of whether pneumatic mail can be used under the conditions of the route.
Fig. 1. Forms of carriers used
The maximum dimensions of a carrier whose form is shown in Fig. 1, a, depending on the parameters of the bent section of the pipeline, are determined relatively simply by a graphical method. However, this method becomes labor-consuming and gives inaccurate results when determining the dimensions of carriers whose forms are shown in Fig. 1, b, c.
In [1] a semiempirical formula is given for determining the bend radius as a function of the carrier dimensions. We have established that the results obtained from this formula are 2–3 times greater than the minimum possible ones and, consequently, it cannot be used to determine the limiting values of the above-mentioned parameters. We know of no other methods for determining the limiting parameters of the bent section of a pneumatic-mail pipeline.
In the present work an attempt is made to obtain analytical expressions for the maximum possible dimensions of carriers whose shapes are shown in Fig. 1, б, as a function of the smallest parameters of the bend included in the pneumatic-mail pipeline.
Figure 2 schematically shows a bent section of pipe with bend radius \(R\) and passage cross-section diameter \(d_0\), and a carrier located in it. According to Fig. 2, one of the limiting conditions for unobstructed passage of a carrier whose shape is shown in Fig. 1, б, through this section of pipe is written as follows:
\[ \frac{d}{2}=y_0-\left(R-\frac{d_0}{2}\right), \tag{1} \]
where \(y_0\) is the distance from the carrier axis along the \(OY\) axis in the selected coordinate system, which is determined from the system of equations describing the conditions of tangency of the belt with the inner surface of the bend in the plane
Fig. 2. Toward determining the dimensions of a carrier
\[ \left(\sqrt{\frac{l^2}{4}+y^2}-R\right)^2+Z^2=\frac{d_0^2}{4}; \tag{2} \]
\[ (y-y_0)^2+Z^2=\frac{D^2}{4}; \tag{3} \]
\[ \frac{ y\left(\sqrt{\frac{l^2}{4}+y^2}-R\right) }{ \sqrt{\frac{l^2}{4}+y^2}\sqrt{\frac{d_0^2}{4}-\left(\sqrt{\frac{l^2}{4}+y^2}-R\right)^2} } = \frac{ y-y_0 }{ \sqrt{\frac{D^2}{4}-(y-y_0)^2} }, \tag{4} \]
where equation (2) is the equation of the curve obtained by sectioning the toroidal surface (the inner surface of the bent section of the pipeline) by the plane \(X=\frac{l}{2}\) in the adopted coordinate system (\(l\) is one of the dimensions of the carrier, see Fig. 1, б); equation (3) is the equation of the circle obtained by sectioning the cylindrical part of the carrier (diameter \(D\)) by the plane \(X=\frac{l}{2}\) in the adopted coordinate system; equation (4) is the condition of tangency of curves (2) and (3); the tangents to these curves must coincide at the point of tangency.
For the case under consideration:
\[ y_{0_1}=\sqrt{\left(R+\frac{d_0}{2}\right)^2-\frac{l^2}{4}}-\frac{D}{2}, \tag{5} \]
\[ y_{0_2}=\sqrt{R^2-\left(\frac{l}{2}-\sqrt{\frac{d_0^2}{4}-\frac{D^2}{4}}\right)^2}. \tag{6} \]
Substituting (5) and (6) into (1), we obtain:
\[ d_1=\sqrt{(2R+d_0)^2-l^2}-D-(2R-d_0), \tag{7} \]
\[ d_2=\sqrt{4R^2-\left(l-\sqrt{d_0^2-D^2}\right)^2}-(2R-d_0). \tag{8} \]
The maximum possible diameter of the cartridge sleeve \(d\), as a function of the selected cartridge length \(l\), is determined by formula (7) if
\[ l\leq(2R+d_0)\sqrt{1-\left(\frac{D}{d_0}\right)^2} \tag{9} \]
and by formula (8) if
\[ l\geq(2R+d_0)\sqrt{2-\left(\frac{D}{d_0}\right)^2}. \tag{10} \]
Conditions (9) and (10) were obtained from the fact that \(y_{0_1}\leq y_{0_2}\). If it is required to determine the maximum possible cartridge length “\(l\)” for a selected sleeve diameter “\(d\),” then, solving equations (7) and (8) with respect to “\(l\),” we obtain:
\[ l_1=\sqrt{(2R+d_0)^2-(d+2R-d_0+D)^2}, \tag{11} \]
\[ l_2=\sqrt{4R^2-(2R+d-d_0)^2}+\sqrt{d_0^2-D^2}. \tag{12} \]
In this case it is necessary to use formula (11) if
\[ d\geq d_0-\frac{2R}{d_0}(d_0-D), \tag{13} \]
and formula (12) if
\[ d\leq d_0-\frac{2R}{d_0}(d_0-D). \tag{14} \]
The difference “\(d_0-D\)” in equations (7), (8), (11), (12), (13), (14) is equal to the clearance “\(\delta\)” between the sealing band of the cartridge and the inner surface of the tube. According to recommendations [1, 2, 3, 4], \(\delta=1\div3\) mm.
Taking into account that
\[ d_0-D=\delta, \tag{15} \]
\[ d_0-d=\delta_1, \tag{16} \]
we transform equations (7), (8), (11), (12), (13), (14):
\[ d_1=\sqrt{(2R+d_0)^2-l^2}-(2R-\delta), \tag{7¹} \]
\[ d_2=\sqrt{4R^2-\left(l-\sqrt{(d_0+D)\delta}\right)^2}-2R+d_0, \tag{8¹} \]
\[ l_1=\sqrt{(\delta+\delta_1)(4R+D+d)}, \tag{11¹} \]
\[ l_2=\sqrt{\delta_1(4R-\delta_1)}+\sqrt{(d_0+D)\delta}. \tag{12¹} \]
\[ d \ge d_0 - \frac{2R\delta}{d_0}, \tag{13'} \]
\[ d \le d_0 - \frac{2R\delta}{d_0}. \tag{14'} \]
For the unobstructed passage of a carrier whose shape is shown in Fig. 1, b, one more condition must be satisfied:
\[ b = \frac{l}{2} - X_0, \tag{17} \]
where \(b\) is the width of the sealing belt of the carrier;
\(X_0\) is determined from the system of equations (18), (19), (20), which describes the conditions of tangency of the seal belt in the plane \(X=X_0\), in the adopted coordinate system (Fig. 2), with the inner surface of the bend:
\[ \left\{ \begin{aligned} \left(\sqrt{X_0^2+y^2}-R\right)^2+Z^2 &= \frac{d_0^2}{4}; \\[4pt] (y-Y_0)^2+Z^2 &= \frac{D^2}{4}; \\[4pt] \frac{y\left(\sqrt{X_0^2+y^2}-R\right)} {\sqrt{X_0^2+y^2}\sqrt{\dfrac{d_0^2}{4}-\left(\sqrt{X_0^2+y^2}-R\right)^2}} &= \frac{y-Y_0} {\sqrt{\dfrac{D^2}{4}-(y-Y_0)^2}}. \end{aligned} \right. \tag{18--20} \]
where equation (18) is the equation of the curve obtained by cutting the toroidal surface (the inner surface of the bend) with the plane \(X=X_0\) in the adopted coordinate system; equation (19) is the equation of the circle obtained by cutting the cylindrical portion of the carrier (diameter \(D\)) with the plane \(X=X_0\) in the adopted coordinate system; equation (20) is the condition of tangency of curves (18) and (19); the tangents to these curves must coincide at the point of tangency.
For the case under consideration:
\[ X_{0_1}=\sqrt{\left(R-\frac{d_0}{2}\right)^2-\left(Y_{0_1}-\frac{D}{2}\right)^2}, \tag{21} \]
\[ X_{0_2}=\sqrt{R^2-Y_{0_2}^{\,2}}-\sqrt{\frac{d_0^2}{4}-\frac{D^2}{4}}. \tag{22} \]
Substituting equation (5) into (1), and (6) into (22), after some transformations we obtain:
\[ X_{0_1}=\sqrt{(D-d)\left(R-\frac{d_0}{2}-\frac{D}{4}+\frac{d}{4}\right)}, \tag{21'} \]
\[ X_{0_2}=\frac{l}{2}-\sqrt{d_0^2-D^2}. \tag{22'} \]
Substituting (21′) and (22′) into (17), respectively, we obtain:
\[ b_1=\frac{l}{2}-\sqrt{(D-d)\left(R-\frac{d_0}{2}-\frac{D}{4}+\frac{d}{4}\right)}, \tag{23} \]
\[ b_2=\sqrt{d_0^2-D^2}. \tag{24} \]
Equation (23) is used when condition (9) or (13) is satisfied, and equation (24) when condition (10) or (14) is satisfied.
The overall dimensions of the carrier were calculated from the limiting conditions for its unobstructed passage through a bend of ideal shape. Since, in manufacturing a bend, certain deviations from the ideal shape are inevitable, for the unobstructed passage of the carrier through an actual bend it is necessary to introduce a compensating dimension. As such, we take the radius of curvature of the sealing belt of the carrier, \(r\). We recommend taking the value of \(r\) equal to
\[ r = \frac{1}{3} b . \tag{25} \]
To determine the radius of the bend as a function of the selected dimensions of the carrier, whose shape is indicated in Fig. 1, б, we use formula (1):
\[ R = Y_0 + \frac{1}{2}(d_0 - d). \]
Substituting equations (5) and (6), respectively, into (1), we obtain
\[ R_1 = \frac{l^2}{4(\delta+\delta_1)} - \frac{D+d}{4}, \tag{26} \]
\[ R_2 = \frac{(l-\sqrt{(d_0+D)\delta^2})}{4\delta_1} + \frac{\delta_1}{4}. \tag{27} \]
Formula (26) is used under the condition
\[ l < (\delta+\delta_1)\sqrt{\frac{D+d}{\delta}}, \tag{28} \]
and formula (27) under the condition
\[ l > (\delta+\delta_1)\sqrt{\frac{D+d}{\delta}}. \tag{29} \]
Determination of the dimensions of the carrier whose shape is shown in Fig. 1, в, as a function of the bend radius \(R\) of the pipeline is carried out analogously to the method presented above.
Conclusions
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The paper presents analytical relations that make it possible to relate the bend radii of pneumatic-mail pipelines to the dimensions of the carriers most commonly encountered at present (Fig. 1, б).
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For identical parameters of pipeline bends, the carrier shape shown in Fig. 1, в, proves to be more preferable owing to its greater useful volume.
References
-
N. F. Rudenko, F. A. Govorov. Pneumatic transport of documents and small objects in carriers, Mashgiz, Moscow, 1963.
-
A. G. Bogdanchenko. Pneumatic delivery of samples from the steelmaking shop to the laboratory. Metallurgiya, Moscow, 1964.
-
S. I. Gluzman, L. V. Kornyushin, S. S. Orlov. On the system of pneumatic transport of steel samples at metallurgical plants. Factory Laboratory. Metallurgizdat, Moscow, 1957, No. 4.
-
S. A. Nazarov. Pneumatic transport of steel samples at metallurgical plants. Factory Laboratory. Metallurgizdat, Moscow, 1962, No. 11.
-
A. I. Galperin. Bending of Pipes. Gosstroiizdat, Moscow, 1958.
-
A. N. Batenchuk. Fabrication and Installation of Technological Pipelines. Gosstroiizdat, Moscow, 1962.
-
L. P. Ginzburg-Shik. Bending of Pipes under Installation Conditions. Gosenergoizdat, Moscow, 1951.
-
B. S. Dmitriev. Pipe-Bending Works. Mashgiz, Moscow, 1953.