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A force is required to accelerate a fluid through a duct system and overcome opposing forces from the system. On entering a system, fluid accelerates from zero velocity in the surrounding atmosphere to the system inlet velocity. When the fluid is expelled from the system, it decelerates from the system discharge velocity to zero velocity in the surrounding atmosphere. Between the system inlet and discharge, the fluid will accelerate and decelerate at each change in duct cross sectional area. Finally, as the fluid flows through the system it rubs against the system walls and fittings which impede the flow -- the system supplies a force in opposite direction of flow.

In a duct system, a fan is used to supply the necessary forces to bring the fluid from rest to the system velocity and overcome friction forces. The force exerted by the fan is the fan total force; or in terms of pressure, the fan total pressure. The total pressure is divided into two vector components. The first component, velocity pressure, is in the direction of flow and whose magnitude is positive and proportional to the velocity. The second component, normal to the direction of flow, is the static pressure. Static pressure may be positive, exerting outward from the frame of reference, or negative, exerting inward. Velocity pressure is always positive, and the sum of static and velocity pressure is the total pressure.

System Resistance
When a fluid flows through a system it encounters resistance due to friction and dynamic losses. Friction losses, as the name suggests, are due to the fluid rubbing against the walls of the system. The loss is dependent on the size and shape of the duct, the duct material, the fluid viscosity, and the fluid velocity. Dynamic losses include: shock, heat, and any other losses due to the conversion of energy. Dynamic losses are dependent on the size and shape of the duct, the fluid viscosity, and the fluid velocity. The sum of both friction and dynamic losses combine to form the total loss of mechanical energy due to resistance.

The total of all pressure losses in a system must be overcome by expending energy from an external force. This is usually done with a fan which delivers air at a pressure large enough to overcome all system resistances. A fan should be selected with a rating high enough to pull the air into the system (exhaust fans) or push the air out of the system (supply fans) and overcome all system resistance. The energy required to pull the air into and push it out of the system is discussed below under Entry and Exit Conditions.

The sum of pressure losses in each component in the system is the total system loss. This includes straight duct sections, elbows, transitions, filters, static regain, entry and exit losses, etc. However, in multiple branch systems, only the path with the largest pressure loss should be considered. This is because the pressure at any junction is equal at the leg of both branches, and therefore the pressure drop in all branches will ultimately be equal. Finally, when selecting a branch, the velocity pressure lost at the system discharge must also be considered.

Conservation of Energy
The laws of conservation of energy state that energy is always conserved in one form or another. In the case of a fluid system, the total pressure at any two points will be equal with exception to the energy that is dissipated through heat, sound, and shock. The dissipated energy is a loss in useful mechanical energy to the system – it is converted to a form which cannot be regained by the system. Therefore, the equation relating two points in a system is:

Total Pressure at A = Total Pressure at B + System Losses

Since total pressure is the sum of velocity pressure and static pressure:

Static Pressure at A + Velocity Pressure at A = Static Pressure at B + Velocity Pressure at B + System Losses

Velocity Pressure
Velocity pressure is the component of total pressure in the direction of flow and whose magnitude is positive and proportional to the velocity. The velocity pressure is equal to the kinetic energy of the fluid in terms of head. The units of kinetic energy, like potential energy, are feet of head of that fluid. In HVAC applications it is customary to convert this to inches of water. Following is the derivation of velocity pressure for a fluid in terms of inches of water.

 Pv = (rgas / rh2o) * v^2 / 2g = rgas * v^2 / (rh2o * 2g) Rearrange = rgas * v^2 / (rh2o * 2 * 32.174 * 60^2) Substitute g and convert to ft/min^2 = rgas * v^2 / (5.193 *  64.348 * 60^2) Substitute density of H2o = rgas * v^2 / (334.159 * 60^2) Combine terms = rgas * v^2 / (18.28^2 * 60^2) Rewrite term as squared = rgas * v^2 / (1097^2) Combine terms = rgas * (v / 1097)^2 Combine again

Derivation of velocity pressure equation

Example:

Given a 30 inch diameter duct with 10,000 CFM of standard air, find the velocity pressure.

Pv = 0.075 (2037 / 1097)^2 = 0.258 inches of water

Static Pressure
Static pressure is the component of total pressure normal to the direction of flow and is equal to total pressure minus velocity pressure. Static pressure may be positive, exerting outward from the frame of reference, or negative, exerting inward.

Entry Conditions
On entering a duct system, the air accelerates from zero velocity in the surrounding atmosphere to the duct velocity. This process involves the transformation of pressure energy into kinetic energy, which means that, as the velocity pressure goes up, the static pressure will go down. In addition to the transformation of energy, there will be a pressure loss whose magnitude depends on the configuration of the entrance fitting; this loss will decrease the total pressure. If the surrounding atmospheric pressure is taken to be zero, the velocity pressure in the duct will be positive, the static pressure will be negative, and the total pressure will fall between them. In this case the static pressure is referred to as the static suction and is measured near the entrance of the hood.

The total pressure loss on entering the duct system is the sum of two components: the transformation of energy from pressure to kinetic, sometimes referred to as acceleration loss, and the pressure loss due to the entry fitting. Since the acceleration loss involves the transformation of zero velocity pressure in the surrounding atmosphere to the velocity pressure in the system, it is equal to the velocity pressure in the system. The other component is the hood loss and is equal to a loss coefficient times the velocity pressure just past the hood.

The pressure loss Pl and static suction Psuction in the duct can be determined by:

Pl = Kl PV1

Psuction  = Pv1 / Ke^2

Pl is the pressure loss in the duct element

Psuction is the static pressure at the duct entrance

Kl is the loss coefficient (see tables below)

Ke is the coefficient of entry (see tables below)

Pv1 is the velocity pressure at point 1 (depicted in the figure below) Kl = 0.90 Kl = 0.50 Kl = 0.05 Kl = 0.05 See Table Below Ke = 0.73 Ke = 0.82 Ke = 0.98 Ke = 0.98 See Table Below

Coefficients for Entry Conditions

The table below depicts coefficients for the converging taper where the total included angle between the sides of the hood is shown in the Angle column – in degrees. The coefficients are approximate and include both round and rectangular ductwork.

 q  (degrees) KL KE 20 0.10 0.95 40 0.05 0.98 60 0.10 0.95 80 0.15 0.94 100 0.20 0.92 120 0.25 0.90 140 0.30 0.86 160 0.35 0.84 180 0.40 0.80

Converging Duct Coefficients

When a fan is located at the entrance of a duct system, the entrance losses become part of the fan work and should not be added to the system pressure losses. However, when this is not the case, entrance losses must be added.

For example:

·        entrance losses on a supply fan without inlet ductwork should not be calculated

·        entrance losses on an exhaust fan where the system is connected to the fan inlet must be calculated

Example:

Given a Bell-Mouthed entry to a 30 inch diameter duct with 10,000 CFM of standard air, find:

1.     The entry loss

2.     The static suction

From the velocity pressure example and the tables above:
Pv = 0.258 w.g., Kl = 0.05 and Ke = 0.98

1.     Pl = Kl Pv = 0.05 (0.258) =  0.0129 w.g.

2.     Psuction  = Pv / Ke^2 = 0.258 / 0.98^2 = 0.269 w.g.

Therefore, if the hood were part of an exhaust system, the total pressure loss would be equal to the entry loss 0.0129 w.g. plus the velocity pressure (acceleration loss) 0.258 w.g. or 0.271 inches of water.

Exit Conditions
On exiting from a duct system, the air decelerates from the duct velocity to zero velocity in the surrounding atmosphere. This process involves dissipation of the kinetic energy of the air stream. The static pressure in the stream will be equal to the surrounding atmospheric pressure. If the atmospheric pressure is taken to be zero, the total pressure at the exit will equal the velocity pressure at the exit, and the exit loss will also equal the velocity pressure.

Pl = Pv1

Pstatic = Pv1

Straight Ducts
The resistance to flow through a straight duct can be determined using the velocity pressure Pv, the duct length L, the Darcy friction factor f, and the equivalent diameter D.

Pl = f Pv L / D

Elbows
The resistance to flow through elbows and bends can be determined with the velocity pressure Pv, and a loss coefficient for the elbow.

Pl = Kl Pv

The loss coefficient is dependent upon the whether the elbow is round, square, or rectangular; the angle of the elbows bend; the curve ratio; and to some degree the aspect ratio if the elbow is rectangular. The curve ratio is the inside radius divided by the outside radius. The aspect ratio is the depth along the axis of the bend, divided by the width in the plane of the bend.

AR = D / W

CR = R1 / R2

For an elbow with throat radius R1 equal to the duct width W:

Rcl = 3/2 W, R1 = W, and R2 = 2W

CR = 1/2

For an elbow with throat radius R1 equal to 1/2 the duct width W:

Rcl = W, R1 = 1/2 W, and R2 3/2 W

CR = 1/3

Note that both the radius ratio, and curve ratio are directly proportional to the throat radius.

 Rectangular Duct Round Duct Angle Factors AR Factors CR Kl CR Kl Angle Factor AR Factor Miter 1.50 Miter 1.50 30 0.4 0.2 2.1 0.1 0.60 0.1 0.70 60 0.7 0.4 1.6 0.2 0.35 0.2 0.50 90 1.0 0.6 1.2 0.3 0.25 0.3 0.40 120 1.2 0.8 1.1 0.4 0.18 0.4 0.30 150 1.4 1 to 2.5 1.0 0.5 0.15 0.5 0.28 180 1.5 3.0 1.0 0.6 0.12 0.6 0.24 3.5 0.9 0.7 0.11 0.7 0.24 4.0 0.8

Elbow Loss Coefficients and Correction Factors

The tables above give the pressure loss coefficients for various curve ratios; angle of bend and aspect ratio factors. The first two tables list coefficients for various curve ratios, the third lists angle of bend factors, and the last is an aspect ratio factor table.

The angle of bend factor must be multiplied by the loss coefficient for any elbow with an angle that is not 90 degrees. The last table is an aspect ratio table; these factors must be multiplied by the loss coefficient for rectangular duct whose aspect ratio is not with the range 1.0 to 2.5.

Finally, as a rule of thumb in the region of 0.5 curve ratio (1 1/2 centerline radius), the equivalent length of straight duct for a 90 degree round elbow is about 9 to 10 diameters. Similarly, the equivalent length of a 90 degree rectangular elbow is about 6 to 8 equivalent diameters.

Example:

Given a 30 inch diameter 45 degree elbow with 10,000 CFM of standard air, and a 30 inch throat radius, find the pressure loss in the duct.

From the velocity pressure example and the tables above

Pv  = 0.258 w.g.

CR = 0.50

Kl' = 0.15

AngleFactor = 0.55

Kl = Kl AngleFactor = .0825

Pl = Kl Pv = .0825 x 0.258 = .021

Fan Inlet Pressure
When ductwork is connected to the inlet of a fan, the fan must generate enough static pressure to pull the air into the system and reach the velocity pressure of the fan inlet. Unlike the fan discharge, it is the static pressure which is responsible for both pressure losses and velocity. The static pressure at a fan inlet is negative and has a magnitude equal to the fan inlet velocity pressure plus the total resistance to flow through the system. Also, note that the total pressure prior to the system entrance is zero, so the system entrance total pressure minus system losses, is simply the negative of system losses.

There may be transformations of pressure energy to velocity energy between these two points, but the conversions are two-fold; energy is transformed to and from static pressure. Along with these transformations there will be losses, one hundred percent regain is impossible, but these are added to the system loss term.

Fan Inlet Total Pressure = - System Losses

Fan Inlet Static Pressure = - Fan Inlet Velocity Pressure - System Losses

Fan Outlet Pressure
When ductwork is connected to the outlet of a fan there are losses associated to the system and the dissipation of energy at the system discharge. At the end of the system discharge duct, the static pressure is zero and therefore the total pressure at that point is equal to the velocity pressure. Moving from the system discharge to the fan outlet, the total pressure rises in proportion to the system pressure loss. That is, at the fan outlet, the total pressure is equal to the system discharge total pressure plus the system losses between the two points.

Fan Outlet Total Pressure = System Discharge Velocity Pressure + System Losses

Fan Outlet Static Pressure = System Discharge Velocity Pressure + System Losses - Fan Outlet Velocity Pressure

There may be transformations of pressure energy to velocity energy between these two points, but the conversions are two-fold; energy is transformed to and from static pressure. Along with these transformations there will be losses, one hundred percent regain is impossible, but these are added to the system loss term.

Fan Inlet and Outlet Pressure
The total energy delivered to a system by a fan is the fan total pressure. The fan total pressure is equal to the fan outlet total pressure minus the fan inlet total pressure. Fan velocity pressure is defined as the velocity pressure at the fan outlet, not the change in velocity pressure across the fan. Finally, fan static pressure, as expected, is equal to fan total pressure minus fan velocity pressure.

Fan Outlet Total Pressure = System Discharge Velocity Pressure + Discharge System Losses

Fan Inlet Total Pressure = - Inlet System Losses

Fan Total Pressure = Fan Outlet Total Pressure - Fan Inlet Total Pressure

Fan Total Pressure = System Discharge Velocity Pressure + Discharge System Losses - (- Inlet System Losses)

Fan Total Pressure = System Discharge Velocity Pressure + Discharge System Losses + Inlet System Losses

Fan Total Pressure = System Discharge Velocity Pressure + System Losses

Fan Static Pressure = Fan Total Pressure - Fan Velocity Pressure

Fan Static Pressure = Fan Total Pressure - Fan Outlet Velocity Pressure

Fan Static Pressure = System Discharge Velocity Pressure - Fan Outlet Velocity Pressure + System Losses

There are several important points to remember:

1.     The fan velocity pressure is not the change in velocity pressure across the fan, it is equal to the fan discharge velocity pressure

2.     The system loss term includes the entry loss which is a function of the entry fitting and the velocity pressure at that fitting -- do not add a kinetic energy term to this loss. The kinetic energy term is always taken into account at the system exit.

3.     On a fan with an inlet duct and an open outlet, the system discharge and fan outlet are the same. Therefore the fan static pressure is equal only to the system losses.

4.     These equations are correct for all fans, with and without inlet and or outlet ductwork.

Fan Rating
Fan rating catalogs are in terms of Fan Total Pressure or Fan Static Pressure. Fan total pressure is the increase in pressure across the fan, and fan static pressure is the fan total pressure minus the fan velocity pressure (as described above). When selecting a fan from a catalog, care should be taken to determine the ratings, which are usually in terms of fan static pressure.

Finally, note the chart rating temperature and fluid density. Usually the charts list ratings based on air at 70 degrees F and 0.075 lb. per cubic foot. If this is not the case, the pressures must be corrected for the desired temperature. This can be done by multiplying by the ratio of densities. If the fluid temperature is higher than standard, its density will be smaller, and its pressure loss will also be smaller.

Absolute Roughness
Absolute roughness e, is a measure of how rough a surface is. All surfaces have high and low points, and the average distance between these is the absolute roughness. Relative roughness is the absolute roughness divided by the effective diameter.

The absolute roughness of a pipe or duct section in which a fluid flows is directly proportional to the pressure losses: the larger the absolute roughness, the larger the pressure losses in a system.

For example, if you were to push an object across a polished tile floor, it would slide with very little resistance (small absolute roughness). However, if the same objects were pushed across a concrete floor, the floor would offer greater resistance and the object would be harder to push (large absolute roughness).

 Condition Typical Surface Average Low High Very Smooth Drawn Tubing 0.000005 - - Medium Smooth Aluminum Duct 0.00015 0.0001 0.0002 Average Galvanized Duct 0.0005 0.00045 0.00065 Medium Rough Concrete Pipe 0.003 0.001 0.01 Very Rough Riveted Steel 0.01 0.003 0.03

Absolute Roughness Values in feet

To aid in calculations on round and rectangular ductwork, the concept of hydraulic radius and equivalent diameter is used. The hydraulic radius is the cross-sectional area divided by the perimeter, and the equivalent diameter is 4 times the hydraulic radius.

 Round Duct Rectangular Duct Symbols A = p r2 A = x y A - Area S - Perimeter S = 2p r S = 2 (x + y) D - Diameter R - Radius Re = A/S = r/2 Re = A/S = x y / (2 (x + y)) X - Width Y - Height De = 4 Re = D De = 4 Re = 2 x y / (x + y) Re - Hydraulic rad. De – Equiv. dia.

Derivation of Equivalent Diameter

Kinematic Viscosity
Viscosity is a measure of resistance of a fluid to flow. This resistance acts against the motion of any solid object through the fluid, against the motion of the fluid through any stationary object, and internally within the fluid between slower and faster moving adjacent layers. A fluid is said to be viscous if it has a relatively high resistance to flow, such as molasses. All fluids exhibit viscosity to some degree.

Mathematically, dynamic or absolute viscosity m is the proportionality factor relating sheer stress t to velocity gradient du/dy. Kinematic viscosity n is the absolute viscosity divided by the mass density of the fluid r.

t = m du/dy

n = m / r

The viscosity of a fluid is directly proportional to the pressure losses in a system. The larger the viscosity, the larger the pressure losses will be.

For example, when a fluid flows through a system it resists the flow against the boundaries and against layers of itself. If the fluid were molasses which has a large viscosity, it would resist the flow. However, if it was air, relatively small viscosity, the resistance would not be as large.

 Temperature (F) Kinematic Viscosity ft2/s 40 0.00015 60 0.00016 80 0.00017 200 0.00024 300 0.00031 500 0.00045 1000 0.0009

Viscosity for air at one atmosphere in ft2/s

Darcy Friction Factor
The Darcy friction factor f, is a coefficient that relates pressure loss to pressure velocity, duct length, and an equivalent duct diameter. There are many equations for various flow conditions but the most common for HVAC applications is the Colebrook equation. This equation is dependent on the duct absolute roughness e, the equivalent diameter D, and the Reynolds number Re.

1 / f’ = -2 log (e / (3.7 D) + 2.51 / (Re f’))

f’          - square root of the friction factor

e          - absolute roughness of the ductwork

D          - equivalent diameter of the ductwork

Re         - Reynolds number for the flow

The Reynolds number Re, is the ratio of the inertia force to the shearing force acting on an element of fluid at a given point. For pipe and ductwork, it is the diameter times the velocity of the fluid divided by the kinematic viscosity of the fluid.

Re = D V / n  