FUNNEL BASED ON BIVARIATE NORMAL DISTRIBUTION

Abstract

A funnel based on bivariate normal distribution is provided, and belongs to the field of daily tools. The funnel is formed by connecting a funnel body in a shape of a bivariate normal distribution with an outlet tube. The funnel body can be formed by rotating a normal distribution curve around the axis of symmetry z with a positive direction pointing to a deep part of the funnel, towards which the fluid flows. An outlet tube is a conical structure with a wide upper part and a narrow lower part. The surface of the funnel body has arbitrary order smoothness at everywhere. The design is simple and the fluid experiences little resistance when it flows through the inner surface. Thus, the funnel can effectively prevented blockage during the transport of fluid, powder and granules and the transport efficiency is remarkably improved.

Claims

1. A bivariate normal distribution funnel, wherein the funnel is formed by connecting a funnel body in a shape of a bivariate normal distribution with an outlet tube; the funnel body formed by rotating a normal distribution curve around an axis of symmetry z with a positive direction pointing to a deep part of the funnel, towards which a fluid in the funnel flows: $z=\frac{k}{2\pi {\sigma }^2}\exp \left(-\frac{x^2+y^2}{2{\sigma }^2}\right)$ wherein, x represents a first variable varying along a first direction in a horizontal plane, y represents a second variable varying along a second direction perpendicular to the first direction in the horizontal plane and the axis of symmetry z is perpendicular to the horizontal plane, sigma subscript x represents a standard deviation σ.sub.x of the first variable x, sigma subscript y represents a standard deviation σ.sub.y of the second variable y and k represents correction coefficient.

2. The bivariate normal distribution funnel according to claim 1, wherein the outlet tube is a conical tubular structure with a narrow lower part and a wide upper part.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0009] FIG. 1 is a front view of a bivariate normal distribution funnel;

[0010] FIG. 2 is a perspective view of the bivariate normal distribution funnel; and

[0011] FIG. 3 is a top view of the funnel body.

DETAILED DESCRIPTION OF THE EMBODIMENTS

[0012] The present disclosure is further described in detail below with reference to the attached figures.

[0013] FIG. 1 is a front view of a bivariate normal distribution funnel in the present disclosure, and FIG. 2 is a perspective view of a bivariate normal distribution funnel in the present disclosure. FIG. 3 shows a funnel body with a diameter of 10 cm at the entrance and a diameter of 2 cm at the connection with the outlet. The whole device is formed by smoothly connecting a funnel body 1 in the shape of a bivariate normal distribution with an outlet tube 2. Furthermore, the outlet tube 2 is a conical tubular structure with a wide upper part and a narrow lower part. The funnel body 1 can be formed by rotating a normal distribution curve around the axis of symmetry z with a positive direction pointing to a deep part of the funnel, towards which the fluid in the funnel flows, and expressed by the following equation:

[00002]$z=f\left(x,y\right)=\frac{k}{2\pi {\sigma }^2}\exp \left(-\frac{x^2+y^2}{2{\sigma }^2}\right)$

[0014] where x represents a first variable varying along a first direction in a horizontal plane, y represents a second variable varying along a second direction perpendicular to the first direction in the horizontal plan, and the axis of symmetry z is perpendicular to the horizontal plane, sigma subscript x represents a standard deviation σ.sub.x of the first variable x, sigma subscript y represents a standard deviation σ.sub.y of the second variable y, and k represents a correction coefficient.

[0015] Based on this, the funnel is smooth at everywhere, for the fluid flowing through the inlet of the funnel body 1. Therefore, the fluid experiences little resistance when flowing through the funnel due to the existence and continuity of arbitrary order derivatives of the surface function that describes the funnel body 1:

[00003]$z=f\left(x,y\right)=\frac{k}{2\pi {\sigma }_x{\sigma }_y}\exp \left[\left(-\frac{x^2}{2{\sigma }_x^2}+\frac{y^2}{2{\sigma }_y^2}\right)\right]$

[0016] According to the general equation, the first-order partial derivatives are:

[00004]$\frac{\partial f}{\partial x}=\frac{-kx}{2\pi {\sigma }_x^3{\sigma }_y}\exp \left[-\left(\frac{x^2}{2{\sigma }_x^2}+\frac{y^2}{2{\sigma }_y^2}\right)\right]$$\mathrm{and}$$\frac{\partial f}{\partial y}=\frac{-ky}{2\pi {\sigma }_y^3{\sigma }_x}\exp \left[-\left(\frac{x^2}{2{\sigma }_x^2}+\frac{y^2}{2{\sigma }_y^2}\right)\right]$

[0017] Now suppose that (m+n)th-order partial derivatives exist for nonnegative integers m and n,

[00005]$\frac{{\partial }^{m+n}f}{\partial x^m\partial y^n}=\underset{i}{.Math.}{P}_i\left(x,y\right)\exp \left[-\left(\frac{x^2}{2{\sigma }_x^2}+\frac{y^2}{2{\sigma }_y^2}\right)\right]$

[0018] in which P.sub.i are some polynomials, then (m+n+1)th-order partial derivatives can be written as:

[00006]$\frac{{\partial }^{m+n+1}f}{\partial x^{m+1}\partial y^n}=\underset{i}{.Math.}{\tilde{P}}_i\left(x,y\right)\exp \left[-\left(\frac{x^2}{2{\sigma }_x^2}+\frac{y^2}{2{\sigma }_y^2}\right)\right],$

[0019] in which the function {tilde over (P)}.sub.i are polynomials as well with an expression:

[00007]${\tilde{P}}_i\left(x,y\right)=\frac{\partial }{\partial x}{P}_i\left(x,y\right)-\frac{x}{{\sigma }_x^2}{P}_i\left(x,y\right),$

and thus are continuous and differentiable. Similarly, it can be proved that

[00008]$\frac{{\partial }^{m+n+1}f}{\partial x^m\partial y^{n+1}}$

exists and is differentiable. Therefore arbitrary order smoothness of the funnel surface is proved recursively.

[0020] The funnel in the present disclosure can be applied to various types of discharge pipes in production and daily life, such as sewers, counter basins and oil pipeline, and is used for guiding various substances such as water bodies, oils and silt and mixtures thereof. When the funnel is used, the lower end of the outlet tube 2 is connected into an inlet of a corresponding container, and when the fluid, powder and granules are filled into the container, in view of the everywhere smoothness characteristic of the funnel surface, the device designed in this disclosure can effectively prevent blockage and improve transport efficiency.