Studies on Magnifying and Improvement of the Perforated Dust Head for the Pipe Duster (II)

Theoretical Investigation for Predicting Pressure Variation in the Dust Head

Abstract

The movement of fluid in p. d. heads (an abbreviation of perforated dust heads) can be thought as a single-dimensional flow. Based on the fact that the change of static pressure in p. d. heads is derived from the frictional head loss by the pipe wall and from branching flow rate, the theory of predicting the static pressure was described in a formula. Here, Darcy & Weisbach's formula was introduced to calculate the frictional head loss.
(1) Theory of pressure gradients for incompressible flow.
The pressure variation (P_I-P₀) in an arbitrary division which has perforations and a length of L can be calculated from the following formula.
P_I-P₀γ_a=fL/24gM(V_I²+V_IV₀+V₀²)-K/g(V_I²-V₀²)…(1)
Where, V_I and V₀ are mean air velocities at the upper reaches and at the lower reaches, respectively. f=friction coefficient, g=acceleration of gravity, M=hydraulic diameter. The formula may be transformed as the next.
P_I-P₀/γ_a=1/3{fLV_I²/D⋅2g+fLV_IV₀/D⋅2g+fLV₀²/D⋅2g}-2K{V_I²/2g-V₀²/2g}
Consequently, the change of static pressure between the both reaches equal to the sum of the arithmetic mean of frictional head losses which calculated with V_I², V₀² and V_IV₀ and the change of the kinetic energy multiplied by a constant (-2K).
If V_I and V₀ equal to V, the formulae returns to Darcy & Weisbach's formula by substituting V=V_I=V₀. Therefore the formula can be applied to arbitrary division of pipes and ducts with branch or without branch.
(2) Theory of pressure gradients for compressible flow.
Considering the aerial compressibility between the upper and the lower reaches, the change of static pressure can be calculated from the following formula.
P_I-P₀=fL(P+P₀)/24gMRT(V_I²+V_IV₀+V₀²)-K(P+P₀/gRT)(V_I²-V₀²)/1-fL/24gMRT(3V_I²+2V_IV₀+V₀²)+KV_I²/gRT
This is a quotient of the precding formula (1) devided by the denominator. And if the denominator approximate to 1, the formula (2) is the same as the former. Where, P=atmospheric pressure, R=gas constant, T=absolute temperature.