#### Basic Duct Hydraulics

Duct air moves according to three fundamental laws of physics: conservation of mass, conservation of energy, and conservation of momentum.

Conservation of mass simply states that an air mass is neither created nor destroyed. From this principle it follows that the amount of air mass coming into a junction in a ductwork system is equal to the amount of air mass leaving the junction, or the sum of air masses at each junction is equal to zero. In most cases the air in a duct is assumed to be incompressible, an assumption that overlooks the change of air density that occurs as a result of pressure loss and flow in the ductwork. In round ductwork, the law of conservation of mass means a duct diameter can be recalculated for a new air velocity using the simple equation:

Diameter2 = Diameter1 (Velocity1 / Velocity2)0.5

or the air velocity for a different diameter can be calculated using:

Velocity2 = Velocity1 (Diameter1 / Diameter2)2

where,

Velocity = (Flow) / (Free area)

Conservation of mass is often referred to as "conservation of flow," which is confusing because of the change of flow that can result from change in pressure and temperature (air density). Only the best commercially available computer programs such as T-Duct (1994) can perform mass rate balancing instead of flow balancing. These programs will show that volume air-flow rates at the fan suction and the fan discharge are not equal because of the pressure and temperature differences, but the mass flow rate is constant.

It is convenient to calculate pressures in ducts using as a base an atmospheric pressure of zero. Mostly positive pressures occur in supply ducts and negative pressures occur in exhaust/return ducts; however, there are cases when negative pressures occur in a supply duct as a result of fitting effects.

Airflow through a duct system creates three types of pressures: static, dynamic (velocity), and total. Each of these pressures can be measured. Air conveyed by a duct system imposes both static and dynamic (velocity) pressures on the duct's structure. The static pressure is responsible for much of the force on the duct walls. However, dynamic (velocity) pressure introduces a rapidly pulsating load.

Static pressure is a measure of the potential energy of a unit of air in the particular cross section of a duct. Air pressure on the duct wall is considered static. Imagine a fan blowing into a completely closed duct; it will create only static pressure because there is no air flow through the duct.

Dynamic (velocity) pressure is the kinetic energy of a unit of air flow in an air stream. Dynamic pressure is a function of both air velocity and density:

Dynamic pressure = (Density)x(Velocity)2 / 2

The static and dynamic pressures are mutually convertible; the magnitude of each is dependent on the local duct cross section, which determines the flow velocity.

Total pressure, the sum of static and dynamic pressures in a particular duct cross section, is:

Total pressure = (Static pressure) + (Dynamic pressure)

Pressure loss is the loss of total pressure in a duct or fitting. There are three important observations that describe the benefits of using total pressure for duct calculation and testing rather than using only static pressure (Williams, 1995):

• Only total pressure in ductwork always drops in the direction of flow. Static or dynamic pressures alone do not follow this rule.
• The measurement of the energy level in an air stream is uniquely represented by total pressure only. The pressure losses in a duct are represented by the combined potential and kinetic energy transformation, i.e., the loss of total pressure.
• The fan energy increases both static and dynamic pressure. Fan ratings based only on static pressure are incorrect.

The law of energy conservation states that energy cannot disappear; it is only converted from one form to another. This is the basis of one of the main expression of aerodynamics, the Bernoulli equation. Bernoulli's equation in its simple form shows that, for an elemental flow stream, the difference in total pressures between any two points in a duct is equal to the pressure loss between these points, or:

(Pressure loss)1-2 = (Total pressure)1 - (Total pressure)2

Conservation of momentum is based on Newton's law that a body will maintain its state of rest or uniform motion unless compelled by another force to change that state. This law is useful to explain flow behavior in a duct system's fitting.

As mentioned above, pressure loss in ductwork has two components, frictional losses along duct walls and dynamic losses in fittings and duct-mounted equipment.

Frictional losses in duct sections are result from air viscosity and momentum exchange among particles moving with different velocities. Frictional losses are calculated using the Darcy equation:

Frictional loss =

where Friction Factor depends on: duct surface roughness; duct diameter; joint type and spacing; and the Reynolds number, a hydraulic criterion, which specifies the laminar or turbulent characteristics regime of the air flow.

Air in HVAC ducts generally becomes turbulent where the Reynolds number is larger than 4,000. There are many formulas for calculating friction factor for turbulent conditions. Colebrook (1938 ref168) developed the best-known formula, which requires iterations, in 1938. A simplified theoretical formula for calculating friction factor in turbulent conditions was developed by Altshul and modified by Tsal (ASHRAE, 1997). This formula does not require iterations.

The easiest way of defining frictional loss per unit length is by using the Friction Chart (ASHRAE, 1997); however, this chart should be used for elevations no higher of 500 m (1,600 ft), air temperature between 5°C and 40°C (40°F and 100°F), and ducts with smooth surfaces. For different ranges and/or greater accuracy, use the Altshul-Tsal equation.

Dynamic losses are the result of changes in direction and velocity of air flow. Dynamic losses occur whenever an air stream makes turns, diverges, converges, narrows, widens, enters, exits, or passes dampers, gates, orifices, coils, filters, or sound attenuators. Velocity profiles are reorganized at these places by the development of vortexes that cause the transformation of mechanical energy into heat. The disturbance of the velocity profile starts at some distance before the air reaches a fitting. The straightening of a flow stream ends some distance after the air passes the fitting. This distance is usually assumed to be no shorter then six duct diameters for a straight duct. Dynamic losses are proportional to dynamic pressure and can be calculated using the equation:

Dynamic loss = (Local loss coefficient) (Dynamic pressure)

where the Local loss coefficient, known as a C-coefficient, represents flow disturbances for particular fittings or for duct-mounted equipment as a function of their type and ratio of dimensions.

A local loss coefficient can be related to different velocities; it is important to know which part of the velocity profile is relevant. The relevant part of the velocity profile is usually the highest velocity in a narrow part of a fitting cross section or a straight/branch section in a junction.

A large number of tables, charts, and formulas are used to calculate local loss coefficients [Brooks, 1993]. The most advanced work in developing and collecting C-coefficients was done by Idelchik and Steinberg (1994) ref184. A Computer Duct Fitting Database (DFDB) was developed by ASHRAE in 1993 (ASHRAE, 1994). The DFDB includes 228 fittings organized by fitting function (supply, exhaust, common), duct shape (round, rectangular, flat oval), and category (entries, exits, elbows, transitions, junctions, obstructions, fan and system interactions, dampers, hoods, and duct-mounted equipment).

Pressure loss in a duct section is calculated as the sum of frictional and dynamic losses using the Darcy-Weisbach equation:

Pressure loss = ( Frictional loss + ΣC ) (Dynamic pressure)

where, ΣC is the sum of all C-coefficients for fittings and duct-mounted equipment in a single duct. A single duct that has the same shape, cross section, and mass flow is called a duct section or just a section.

For ducts with shapes that are not round, an equivalent diameter is used to calculate pressure loss. The three types of equivalent diameters are "by friction," "by velocity," and "by flow." Dynamic pressure must be calculated using air velocity based on the equivalent by-velocity diameter, and friction loss must be calculated using the equivalent by-friction diameter. For non-round ducts, the Darcy-Weisbach equation should be used instead of the friction chart, and equivalent by-friction diameter should be calculated using what is called the hydraulic diameter, which applies to any duct shape. It is calculated as:

Hydraulic diameter = 4 (Free area) / (Perimeter)

In general, the function between pressure loss and flow can be represented by the following parabola:

Pressure loss = (Hydraulic constant) (Flow)2

where "Hydraulic constant" depends on duct shape, size, length, material, air density, and the sum of C-coefficients. All of these variables are presented in the previous equations above.

A similar function between pressure loss and diameter holds true for round ducts where the pressure loss can be approximated by the following equation:

Pressure loss = (Constant) (Diameter)n

where,

n = between 4 and 5.

Following is the recommended procedure for calculating total pressure loss in a single duct section:

• Gather input data: air flow, duct shape, duct size, roughness, altitude, air temperature, and fittings;
• Calculate air density as a function of altitude and temperature;
• Calculate air velocity as a function of air flow and cross section;
• Calculate the Reynolds number;
• Calculate friction factor using the Altshul-Tsal equation as a function of duct surface roughness and Reynolds number;
• Calculate local C-coefficients for each fitting used; and
• Calculate pressure loss using the Darcy-Weisbach equation.

An excellent way to analyze the hydraulics of a ductwork system is to utilize pressure gradient diagrams to show the pressure-loss characteristics of the system in the design phase [Graham, 1966]. The diagram will also pinpoint critical factors when a system is not operating properly.