# Energy changes in a duct system

When air flows through a system of duct and plant, the prime source of energy to make good the losses incurred by friction and turbulence is the total pressure of the air stream, defined by a simplified form of Bernoulli’s theorem:

Pt = Ps+Pv (15.18)

Where

Pt = total pressure in Pa

Ps = static pressure in Pa

Pv = velocity (or dynamic) pressure in Pa

The velocity pressure corresponds to the kinetic energy of the airstream and the static pressure corresponds to its potential energy.

For the airstream to flow through the system of plant and ductwork, in spite of the losses from friction and turbulence, the total energy on its upstream side must exceed the total energy on its downstream side. If a pressure, p, propels a small quantity of air, 8q, through a duct in a short time, 81, against a frictional resistance equal and opposite to p, then the rate at which energy must be fed into the system to continue the flow is p(8q/8t). This is termed the air power, wa, and is delivered to the airstream by the fan impeller.

Air power = force x distance per unit time

= pressure in N m”2 x area in m2 x velocity in m s_1 = fan total pressure in N rrf2 x volumetric airflow rate in m3 s’1 w3L = ptFQ (15.19)

Where

P{F = fan total pressure in Pa or kPa (see equation (15.21))

Q = volumetric airflow rate in m3 s_1

In its passage through a fan the airstream suffers various losses. These are similar to those occurring in a centrifugal compressor and are illustrated in Figure 12.13. In addition, there

Are bearing losses. It follows that the power input to the fan shaft must exceed the output from the impeller to the airstream. The ratio of impeller output to shaft input is termed the total fan efficiency, r|t and this provides a definition of fan power:

W{ = wa/r|t (15.20)

The air power delivered to the airstream provides for the sum of the following:

The acceleration of outside air from rest to the velocity in the air intake, the frictional resistance of the air inlet louvres, the energy losses incurred by turbulence formed in the vena-contracta at entry, the frictional resistance by each item of plant, frictional resistance in the ducts and duct fittings, the frictional resistance of the index grille, losses incurred by the presence of turbulence anywhere in the system and the kinetic energy loss from the system (represented by the mass of moving air delivered from the index grille).

The energy loss by friction and turbulence would cause a temperature rise in the airstream if it were not exactly offset by the fall in temperature resulting from the adiabatic expansion accompanying the pressure drop. The only temperature rise occurs at the fan, where adiabatic compression takes place.

The power absorbed by the electric driving motor must exceed the fan power, during steady-state operation, because of the loss in the drive between the fan and the motor and because the efficiency of the motor is less than 100 per cent. There is also the matter of margins, discussed later in section 15.18.

The size of the fan depends on the airflow rate and the type of fan depends on the application, but the speed at which the fan must run and the size of the motor needed to drive the fan depend on the total pressure loss in the system of duct and plant. Hence it is necessary to calculate energy losses in the system. The following principles and definitions relate to such calculations:

This is a simplification of Bernoulli’s theorem, stating that, in an airstream, the total energy of the moving air mass is the sum of the potential and kinetic energies. Energy is the product of an applied force and the distance over which it is acting. Hence, since pressure is the intensity of force per unit area, total pressure may be regarded as energy per unit volume of air flowing. (This is seen if the unit for pressure, N/m2, has its numerator and denominator multiplied by metres, yielding Nm/m3, which equals J/m3.) Similarly, static pressure can be considered as potential energy per unit volume and velocity pressure as kinetic energy per unit volume.

A conclusion drawn from the above is that energy loss through a system corresponds to fan total pressure.

(b) Energy loss corresponds to a fall of total pressure

It is a corollary of Bernoulli’s theorem that a fall in energy should correspond to a fall in total pressure and so, when assessing the energy loss through a system, it is the change in total pressure that must be calculated. On the suction side of a fan the total pressure upstream exceeds that at fan inlet and on the discharge side of the fan the total pressure at fan outlet exceeds that downstream. The fan impeller replaces the energy dissipated, by elevating the total pressure between the fan inlet and the fan outlet.

(c) Velocity pressure is always positive in the direction of airflow

For a given volumetric airflow rate in a duct of constant cross section, the mean velocity of airflow and the corresponding kinetic energy must be constant. If a loss of energy occurs, because of friction or turbulence, the corresponding fall in total pressure can only appear as an equal fall in static pressure. Thus it is the static pressure of an airstream that is the source of energy for making good losses. If the kinetic energy of the airstream is to be drawn upon then it is first necessary to reduce the velocity by expanding the duct section, in order to convert kinetic energy to potential energy (in the form of static pressure), as described by Bernoulli’s theorem.

This is defined by

Piv = Рю — Pu (15.21)

Where pt0 is the total pressure at fan outlet and pti the total pressure at fan inlet.

(e) Fan static pressure, psF This is defined by

PsF ~ Pso ~ Pti (15.22)

Where pso is the static pressure at fan outlet.

(f) Fan static pressure by virtue of equation (15.18)

Fan static pressure is also defined by

PsF — PtF — Pvo (15.23)

Where pvo is a velocity pressure at fan outlet based on a notional mean velocity vfo defined by

Vfo =Q! Af0 (15.24)

Where Q is the volumetric airflow rate and Afo is the area across the flanges at fan outlet.

The velocity distribution over the outlet area of a fan is very turbulent and not easy to measure with accuracy. Hence a notional mean velocity at fan outlet, defined by equation (15.24), is determined and the corresponding velocity pressure is added to fan static pressure to define fan total pressure indirectly, by means of equation (15.23). Static pressure may be above or below atmospheric pressure, acting as a bursting or collapsing influence on the system. Hence the zero chosen for the expression of pressure is atmospheric pressure, and static and total pressures are given positive or negative values.

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