# Bernoulli’s equation

Consider an elemental tube in which flow is entirely parallel to the boundaries. For simplicity assume it to have constant cross-section area of 8a (although it can be shown it is not es­sential to do so).

The forces on the element may be equated to the rate of change of momentum. In the direction of flow, the forces are: due to change in pressure:

P5A — (p + 5p)5A = -8p5A

Where AH is the loss of total head between the two sections. This may be rewritten

— + — = — + — + AH + (Vl2 — H. — Pat1 ~Pal21 Equ3.10

2g pg 2g pg I pg J

Now, if Pat represents the atmospheric pressure at a height H above some datum, and pat+ 5pat at a height H + 5H above the same datum, and a column of air of cross-section A is consid­ered,

PstA~(Pat + 5Pat)A = PgA(H + SH) — pgAH from which

-8pat =pgSH Equ 3.11

If pg remains constant, then equation 3.8 may be rewritten

Pat1 + P2 + H2 + H1

Pg

And inserting this in equation 3.10 gives

Vi2.Pi = ^+Pi+aH

Оg pg 2g pg

 Va = Absolute velocity of gas Vf = Relative velocity of gas Vf = Radial velocity of gas vw = Whirl velocity of gas (ie tangential component of Va) u = Peripheral velocity of impeller /3 = Impeller blade angle d = Impeller diameter r = Impeller radius co = Angular velocity m = Mass flow of air gas g = Gravitational constant P = Gas density

 Equ 3.14

 Suffix 1 at inlet of impeller 2 at discharge from impeller

Multiplying throughout by pg gives the equation in terms of pressure:

Ipv^+p! = ^pv2 + p2 + Ap Equ 3.13

Or

Pt1 = Pt2 + AP

In equation 3.13, pi and p2 are known as the static pressures at the two sections and may be positive or negative according to whether the absolute pressure is greater or less than the ambi­ent atmospheric pressure which, as stated above, is the arbi­trary datum or zero to which static pressure is generally re­ferred.

The sum of static pressure and velocity pressure (p + pv2) is

Known as the total pressure pT. Although in many cases the air density remains substantially constant, this may not be so where the height between two parts of a system is consider­able, or if there is a temperature gradient.

Equation 3.13 shows that the resistance of a system of ducting expressed as a pressure loss fora particular flow rate, is equal to the difference between the total pressures at the two ends of the system. In practice the use of this equation to calculate the resistance of a system is complicated by the fact that the veloc­ity nearly always varies considerably between the centre and the duct walls, although the static pressure, except near bends, is often sensibly constant across a section.

In determining the pressure loss it is not correct to calculate the velocity pressure component of the total pressure from the ex­pression:

^pVm

Where:

Vm = the mean velocity and is equal to Q/A

Q = the volume flow

A = cross-sectional area of the airway

Strictly speaking, and neglecting any variations in the static pressure p across the section, the mean velocity pressure must be calculated from the kinetic energy per unit time divided by the volume flow per unit time, that is, in a circular duct:

R R

Pv(mean) =j^pvxv2x27irdr^jvx27trdr

Or

R r

Pv(mean) =4pjVrdr^fvr dr

Ground to Chapter 1 and explain how those characteristic curves match with the fundamental fluid mechanics. A detailed design guide could be written and it would certainly require a similar number of pages to this volume, to do the subject jus­tice.

Posted in Fans Ventilation A Practical Guide