Air System Fans[LINK]

Overview[LINK]

Three input objects (Fan:ConstantVolume, Fan:VariableVolume, and Fan:OnOff) provide models for fans – the prime movers in most of the air loop and zonal air conditioning systems in EnergyPlus. For these input objects, EnergyPlus uses a simple polynomial-based curve-fit model to describe the relation between the volumetric flow rate and the fan electric power, with no explicit modeling of fan pressure rise. Consequently, duct-static-pressure reset strategies cannot be modeled using this approach. For CAV and VAV central air-handling systems, a fourth input object (Fan:ComponentModel) provides a simple physics-based model for flow-dependent fan pressure rise and detailed models for fan, belt, motor, and variable-frequency-drive efficiencies and energy use. This latter input object provides the capability to analyze the effects of duct-static-pressure reset strategies, as well as changes in fan system element performance. All of these fan models calculate the fan energy usage, which is often a large fraction of HVAC energy use and a significant portion of the building energy consumption. They also calculate the temperature rise in the air stream caused by the static pressure increase as the air goes through the fan. If the motor and belt are in the air stream, waste heat from the motor and belt also acts to raise the air stream temperature.

Model[LINK]

Each of the four fan models is a forward type: the model inputs describe the fan characteristics and the conditions of the air at the fan inlet; the outputs are the fan electrical power consumption and the conditions of the air at the fan outlet. The model algorithms and data are contained in the Fans module in EnergyPlus.

Inputs and Data[LINK]

For the Fan:ConstantVolume, Fan:VariableVolume, and Fan:OnOff models, the user describes the fan by entering values for the design pressure rise across the fan, the design volumetric flow rate, the fan total efficiency, and the fan motor efficiency. The user also needs to specify the fraction of the fan waste heat that will enter the air stream (usually 0 or 1). For the Fan:VariableVolume model, the user must also enter the coefficients of a 4^th order polynomial that relates the fan mass flow rate to the fan power consumption. The independent variable is the volumetric flow fraction; the dependent variable is the fan power part load ratio. For multi-speed fans, the user must enter a fan power ratio as a function of speed ratio performance curve name. Multi-speed fans can only be simulated in a parent object which allows multiple fan speeds (e.g., AirLoopHVAC:Unitary:Furnace:HeatCool, ZoneHVAC:PackagedTerminalAirConditioner, etc.). An optional efficiency curve may also be used when simulating multi-speed fans to adjust the fan total efficiency as the fan speed changes.

For the Fan:ComponentModel object, the user describes the fan in more detail, and also describes the duct system characteristics as seen by the fan. In particular, the user specifies a pressure rise curve with four coefficients that relates the fan total pressure rise to the volumetric flow through the fan, the duct-static-pressure set-point, and the static pressure of the spaces surrounding the ducts. If duct-static-pressure reset is used, the user enters a linear curve with two coefficients that relates the pressure set-point to the volumetric flow through the fan. For the fan itself, the user specifies the fan geometry (wheel diameter and outlet area), maximum efficiency, the Euler number corresponding to the maximum efficiency, the maximum dimensionless flow, the names of four curves with several coefficients that describe the variation of fan efficiency and dimensionless flow with the Euler number in the normal and stall operation regions of the fan, and a sizing factor (applied to the maximum flow through the fan). For the belt, the user specifies the maximum efficiency (or a curve with five coefficients that defines the maximum efficiency as a function of maximum fan shaft input power), three curves with three coefficients each that relate the belt part-load efficiency to belt fractional output torque, the motor/fan pulley diameter ratio, the belt output torque capacity, and a sizing factor (applied to the maximum output torque of the belt). For the motor, the user specifies the maximum efficiency (or a curve with three coefficients that define the maximum efficiency as a function of maximum belt input power), a curve with three coefficients that relate the motor part-load efficiency to motor fractional output power, and a sizing factor (applied to the maximum output power of the motor). For the variable-frequency-drive (VFD), the user specifies a curve with three coefficients that relate the VFD part-load efficiency to motor fractional input power or to motor fractional speed, and a sizing factor (applied to the maximum output power of the VFD).

Control[LINK]

The models must decide whether the fan is on or off. The primary on/off trigger is the fan schedule. This is an on/off schedule associated with each fan: a value of 1 indicates the fan is on; a value of 0 indicates the fan is off. The fan schedule can be overruled by flags set by system availability managers. If the flag TurnFansOn is true, a zero fan schedule value will be overridden and the fan will be turned on. If the flag TurnFansOff is true the fan will be forced off. The inlet air mass flow rate must be greater than zero for the fan to be on.

Generally the fan is a passive component: it accepts the mass flow on its inlet node, uses it in its calculations of energy consumption and temperature rise, and passes it to the outlet node. However the fan maximum and minimum airflow rates act as absolute limits on the airflow rate.

For multi-speed fans, the parent object determines the fan speed ratio (i.e., the selected speed of the fan motor) and uses this value in conjunction with the fan power ratio performance curve to calculate the full load fan power. This full load fan power is then used to determine the part-load performance of the fan and motor assembly.

Simulation[LINK]

Simple (Single Speed) Fan Model

The following equations define the model for this fan:

$${\dot{Q}}_{tot}=\dot{m}\Delta P/\left(e_{tot}{\rho }_{air}\right)$$

$${\dot{Q}}_{shaft}=e_{motor}{\dot{Q}}_{tot}$$

$${\dot{Q}}_{toair}={\dot{Q}}_{shaft}+({\dot{Q}}_{tot}-{\dot{Q}}_{shaft})f_{motortoair}$$

$$h_{out}=h_{in}+{\dot{Q}}_{toair}/\dot{m}$$

$$w_{out}=w_{in}$$

$$T_{out}=PsyTdbFnHW(h_{out},w_{out})$$

On/Off Fan Model

The on/off fan model is similar to the simple fan model with the exception that the on/off fan may cycle on and off during a simulation time step. The cycling rate of the fan is known as the run time fraction. The calculation of run time fraction accounts for the part-load losses of other equipment used in the HVAC system. A part-load factor (a.k.a. part-load ratio) is first calculated for the fan as the ratio of the actual operating mass flow rate to the maximum fan mass flow rate. The run time fraction is then calculated as the part-load factor divided by the part-load fraction. The part-load fraction is determined by other HVAC equipment in the simulation (Ref. DX coil) for use by this specific fan model.

$$f_{flow}=\frac{\dot{m}}{{\dot{m}}_{max}}$$

$$RTF=\frac{f_{flow}}{PLF}$$

The total fan power is then calculated as the maximum fan power multipled by the run time fraction.

$${\dot{Q}}_{tot}=RTF[\dot{m}\cdot \Delta P/\left(e_{tot}{\rho }_{air}\right)]$$

The remaining calculations are the same as those described in the simple single-speed fan mode.

Multi-Speed Fan Model

The model used to simulate a multi-speed fan relies on the general fan laws to correct for speed changes in the fan’s motor and the corresponding change in fan power and fan total efficiency. Two performance curves are used to define the change in power and efficiency as shown below. The power ratio curve must be used to simulate a multi-speed fan. The power ratio curve modifies the fan power based on a change in fan speed according to the cubic fan law. Exponents other than 3 are allowed. The efficiency ratio curve is used to correct for changes in nominal efficiency at alternate fan speeds. If either or both of these curves are not provided, the ratio is assumed to be 1.

The power ratio term is evaluated using an exponent performance curve. The form of the exponent curve equation is shown below.

$$ExponentCurveOutput=C1+C2\ast x^{C3}$$

The exponent performance curve is used to evaluate the power ratio as a function of speed ratio (x in the equation above). For typical fan performance, coefficient C1 = 0 and coefficient C2 = 1. The exponent (C3) is typically equal to 3 (cubed fan law) but other values are allowed.

$$PowerRatio=f{\left(N_{ratio}\right)}^n=ExponentCurveOutput$$

The efficiency ratio curve is a quadratic or cubic curve used to evaluate the efficiency ratio as a function of speed ratio (x in the following equation). The cubic equation form is shown below.

$$EfficiencyCurveOutput=C1+C2{\left(x\right)}^2+C3{\left(x\right)}^3$$

The curve coefficients are determine by correlating the normalized fan total efficiency (i.e., the actual fan total efficiency divided by the nominal fan total efficiency at a speed ratio of 1) to the speed ratio (i.e., the ratio of actual air flow rate through the fan to the maximum fan air flow rate).

$$EfficiencyRatio=f\left(N_{ratio}\right)=EfficiencyCurveOutput$$

To account for the change in fan speed and to ensure that the run time fraction is accurately calculated, an adjustment to the previous calculation for run time fraction is made. This adjusts the calculated run time fraction to account for times when the fan speed has been reduced and the new speed is now considered the fan’s maximum speed. The speed adjusted run time fraction is then used in the fan power calculation.

$$RTF=\frac{RTF}{N_{ratio}}$$

$${\dot{Q}}_{tot}=RTF\left[\frac{\stackrel{\bullet }{m}\Delta P}{e_{tot}{\rho }_{air}}\right]\left(\frac{PowerRatio}{EfficiencyRatio}\right)$$

Each of the performance curves described above may be used to model the performance of a multi-speed fan motor, however, the power ratio curve must be used to envoke the multi-speed simulation. These curves are used when the fan is used in an HVAC system having multiple flow rates (i.e., different flow rates in cooling and heating mode). If an HVAC system operates at the same speed in either cooling or heating mode, these curves are not required. When these curves are not used, the associated ratio term in the equation above is assumed to be 1. The remaining calculations are identical to the simple single-speed fan model described above.

Variable Speed Fan Model

The model for the variable speed fan is similar to the simple single-speed fan model except for a part load factor that multiplies the fan power consumption.

$$f_{flow}=\dot{m}/{\dot{m}}_{design}$$

$$f_{pl}=c_1+c_2\cdot f_{flow}+c_3\cdot f_{flow}^2+c_4\cdot f_{flow}^3+c_5\cdot f_{flow}^4$$

$${\dot{Q}}_{tot}=f_{pl}{\dot{m}}_{design}\Delta P/\left(e_{tot}{\rho }_{air}\right)$$

The rest of the calculation is the same as for the simple fan.

Nomenclature for Simple Models[LINK]

${\dot{Q}}_{tot}$ is the fan power in watts;

$\dot{m}$ is the air mass flow in kg/s;

${\dot{m}}_{design}$ is the design (maximum) air flow in kg/s;

$\Delta P$ is the fan design pressure increase in Pascals;

$e_{tot}$ is the fan total efficiency;

${\rho }_{air}$ is the air density at standard conditions in kg/m³;

$e_{motor}$ is the motor efficiency;

${\dot{Q}}_{shaft}$ is the fan shaft power in watts;

${\dot{Q}}_{toair}$ is the power entering the air in watts;

$N_{ratio}$ is the ratio of actual fan flow rate (or speed) to maximum fan flow rate (or speed)

$h_{in},h_{out}$ are the inlet and outlet air stream specific enthalpies in J/kg;

$w_{in},w_{out}$ are the inlet and outlet air stream humidity ratios;

$T_{out}$ is the outlet air temperature in degrees C;

$PsyTdbFnHW$ is the EnergyPlus psychrometric routine relating enthalpy and humidity ratio to temperature;

$f_{flow}$ is the flow fraction or part-load ratio;

$f_{pl}$ is the part load factor.

Component Fan Model

The Fan:ComponentModel object is based upon combining: modified forms of fan, belt, motor, and variable-frequency-drive (VFD) element models (Stein and Hydeman 2004); a simplified fan pressure rise model with distribution system leakage, duct static pressure, and conditioned space pressure control effects included (Sherman and Wray 2010); and a diagnostic-based simplified duct static-pressure reset (SPR) scheme for air-handling systems (Federspiel 2004, 2005).

Fan electric power depends on fan air power (product of the airflow through and pressure rise across the fan), mechanical efficiencies (fan and belt), and electrical efficiencies (motor and drive). For systems with variable flows, none of these parameters is constant and all are interrelated. For example, Figure 208 shows that fan efficiency strongly depends on fan flow as well as pressure rise, although this is not obvious from the manufacturer’s power and speed performance map (top left). Fan efficiency maps can be derived from the manufacturer’s performance map using a commercially-available software-based data extraction tool such as DigitizeIt and by applying the following equation to the extracted data:

${\eta }_{fan}=\frac{\Delta P_{fan}\cdot Q_{fan}}{H_{fan}}$ ()

where ΔP_fan is the fan pressure rise (Pa); Q_fan is the fan flow at standard conditions (m³/s); and H_fan is the fan shaft power (W). Fan speed values also can be derived from the manufacturer’s performance map using a similar software-based data extraction tool.

Fan pressure rise must be sufficient to overcome the air-handling system pressure drop, which depends on duct static pressure, on duct and equipment leakage, and on pressure drops across duct and duct-like elements (e.g., dampers, fittings), coils, and filters that are connected to the fan. Duct and duct-like pressure drops increase approximately as the square of the flow through them. However, pressure drops across coils and filters behave differently: they are proportional to the flow raised to a power n, which can approach one for high-efficiency filters and wet coils (Liu et al. 2003, Trane 1999).

The relation between system pressure drop and flow defines what is commonly called a “system curve”. When system characteristics change, such as when the duct static pressure set point is varied, a family of system curves results. The intersections of these curves with fan curves (e.g., power as a function of pressure rise and flow) on a pressure versus flow plot define one or more loci of unique fan operating points. Each of these points has an associated fan efficiency, power, and speed.

Fan Pressure Rise Model: To calculate fan pressure rise based on flow through the fan, Sherman and Wray (2010) have developed a simple physics-based data-driven four parameter duct system model for the purpose of simulating its system curve. The embodiment of the model for a fixed outdoor air fraction that can be applied to constant- or variable-volume central air-handling systems is:

$\Delta P_{fan,tot}=A_{fpr}{Q}_{fan}^2+B_{fpr}{Q}_{fan}+C_{fpr}{Q}_{fan}\sqrt{P_{sm}-P_o}+D_{fpr}\left(P_{sm}-P_o\right)$ ()

where ΔP_fan,tot is the fan total pressure rise (Pa); Q_fan is the fan flow at standard conditions (m³/s); P_sm is the duct static pressure set point (Pa); P_o is the static pressure of the spaces surrounding the ducts (Pa); and A_fpr, B_fpr, C_fpr, and D_fpr are constant coefficients that represent different aspects of the fan pressure rise model as described below.

Fan static pressure rise is determined from the total pressure rise by subtracting the outlet velocity pressure:

$$\Delta P_{fan}=\Delta P_{fan,tot}-\frac{\rho }{2}{\left(\frac{Q_{fan}}{A_{fan,out}}\right)}^2$$

where A_fan,out is the fan outlet area (m²) and ρ is the air density at the fan inlet (kg/m³).

The first term in Equation looks like the common system curve in which the fan pressure rise is proportional to the square of the fan flow, but here it also depends implicitly on supply and return pressure losses, and in part on the fraction of the fan flow that is outdoor air (essentially “leaks” into and out of the return side of the system). Very often it is the only term considered, but that would only be correct with fixed-position dampers, no distribution system leakage, no linear resistance components, and no duct static pressure control.

The second term accounts for significant flow resistances in the system where the pressure difference is linearly proportional to the flow. Some filters and coils in the return may need this term to be adequately described. This term could be ignored if there are no linear components or if their pressure drops are very small compared to the other terms.

The third term, which depends on the fan flow and square root of the supply duct pressure P_sm, accounts in part for air leakage from the supply system when damper positions are fixed or are changed independently of static pressure or fan flow. In this case, reducing or eliminating supply leakage results in a different system curve. This, however, might be only a minor “correction” to the simple system curves generally used. The third term is zero when VAV box dampers are modulated to control flow. Consequently, with variable-position supply dampers, reducing or eliminating supply leakage does not change the system curve.

The last term also accounts in part for leakage from the supply system when damper positions are fixed or are changed independently of static pressure or fan flow. This term indicates that the same fan pressure rise can be achieved by raising the duct pressure and closing dampers. The only change in the system in such a case is that the leakage flow may increase. The coefficient for this term is equal to one when the VAV box dampers are modulated to control flow. In both cases, this term may be the most important “correction” to the simple system curves generally used, especially at low flows.

In principle, especially for research applications of Energy Plus, the four individual fan pressure rise coefficients can be determined using duct design calculation tools for the entire system (e.g., Right-CommDuct, UNI-DUCT, Ductsize, Varitrane Duct Designer, T-Duct). In practice, however, especially for existing buildings, the necessary details may not be known sufficiently. In that case, one can make active measurements at different combinations of fan flows, outside air fractions, damper positions (if they are variable), and duct static pressures and then use non-linear system identification techniques to fit the data and determine the coefficients. If passive measurements can be made over a sufficiently long time so that a wide range of outside air settings, damper positions, and fan flows is obtained, one could instead regress the data to find all of the parameters in the equation. Unfortunately, there are no standardized test procedures available to draw upon, even though some standards appear to be related (e.g., ASHRAE 1999, 2008). Consequently, field test protocols need to be developed to determine the parameters for the new duct system model. These protocols then need to be integrated into standardized data collection and analysis tools such as Pacific Gas and Electric’s “Universal Translator” tool. The California Energy Commission is funding such a project; data, procedures, and tools from this project will support the EnergyPlus implementation of the fan and duct system models described here.

SPR Model: The model for duct-static-pressure reset (SPR) is based on a simple diagnostic procedure and a linear correlation between duct static pressure and supply fan airflow (Federspiel 2004, 2005). The diagnostic method involves measuring the static pressure at the duct static pressure sensor and the velocity pressure at the fan inlet (represents the fan flow) at multiple points over the fan’s operating range, while the VAV box dampers attempt to control flow in response to a constant thermostat setpoint. The goal of the test is to define the lowest duct static pressure where all VAV boxes are still in control (dampers modulating). In the model, the correlation between P_sm and Q_fan is as follows for Q_fan,min ≤ Q_fan ≤ Q_fan,max:

$P_{sm}=P_{sm,min}+\left(P_{sm,max}-P_{sm,min}\right)\ast \frac{\left(Q_{fan}-Q_{fan,min}\right)}{\left(Q_{fan,max}-Q_{fan,min}\right)}=C_1+C_2\ast Q_{fan}$ ()

Where

$C_1=P_{sm,min}-C_2\ast Q_{fan,min}$ and $C_2=\frac{\left(P_{sm,max}-P_{sm,min}\right)}{\left(Q_{fan,max}-Q_{fan,min}\right)}$ ()

For Q_fan < Q_fan,min, P_sm = P_sm,min; for Q_fan > Q_fan,max, P_sm = P_sm,max

Fan Efficiency and Shaft Input Power Model: A dimensionless parameter in the form of an Euler number can be used to simplify the description of fan static efficiency variations:

$Eu=\frac{\left(\Delta P_{fan}\ast D_{fan}^4\right)}{\left(\rho \ast Q_{fan}^2\right)}$ ()

where ΔP_fan is the fan static pressure rise (Pa), D_fan is the fan wheel outer diameter (m), ρ is the air density at the fan inlet (kg/m³), and Q_fan is the fan flow at standard conditions (m³/s). Eu is nominally the ratio of pressure forces across the fan to inertial forces at the fan wheel exit.

By plotting the normalized fan static efficiency (static efficiency / maximum static efficiency) versus the logarithm base 10 of the normalized Euler number (Eu / Eu at maximum static efficiency), the dimensionless performance of various fan sizes is very similar (as one might expect from the “fan laws”), but so also is the dimensionless performance of different types of fans (e.g., single-inlet plenum fans, double-inlet housed centrifugal fans, mixed flow fans, vane axial fans, fans with backward or forward curved blades). An example of this correlation for the “normal operation” (non-stall) and stall regions of eight fans is shown in Figure 209.

This model uses a continuous function to represent the normalized fan efficiency (η_fan) variation. The normalized exponential-conditioned skew-normal functional relationship is:

$\frac{{\eta }_{fan}\left(x_{fan}\right)}{{\eta }_{fan,max}}=\frac{e^{(-0.5\ast Z_1^2)}[1+\frac{Z_2}{\left|Z_2\right|}\bullet erf\left(\frac{\left|Z_2\right|}{\sqrt{2}}\right)}{e^{(-0.5\ast Z_3^2)}[1+\frac{Z_3}{\left|Z_3\right|}\bullet erf\left(\frac{\left|Z_3\right|}{\sqrt{2}}\right)}$ ()

where

x_fan = log₁₀(Eu / Eu_max)

Z₁ = (x_fan - a_fan) / b_fan; Z₂ = ((e^(cfan^*xfan⁾*d_fan*x_fan) - a_fan) / b_fan; Z₃ = -a_fan / b_fan

For conceptual design when only rough estimates are needed, a generic curve (also shown in Figure 209) can be used and then one only needs to know the maximum efficiency (η_fan,max) and Eu at that maximum (Eu_max) to entirely model the fan efficiency and hence fan power. The dimensionless coefficients for the generic normalized fan efficiency curve are as follows:

a_fan = -2.732094, b_fan = 2.273014, c_fan = 0.196344, d_fan = 5.267518

In this case, the coefficient of determination R² is 0.994. Also, the average and RMS differences between the efficiency values based on extracted data and the fitted generic curve are, respectively, about 0.5% and 1.4%; maximum differences are about ±9%.

If more accuracy is needed for a specific fan, a similarly shaped curve can be developed for that fan (using extracted data, Equation , and least-squares regression techniques), segmented into normal operation and stall regions with specific coefficients fitted for each region. Figure 210 shows an example, using the data from Figure 208.

For this specific fan, the coefficients are:

Normalized Fan Efficiency Coefficients for Specific Backward-Curved Plenum Fan

Operating Region	afan	bfan	cfan	dfan
Normal (Non-Stall)	0.072613	0.833213	0	0.013911
Stall	-2.354091	2.117493	0	2.753264

In this case, the average and RMS differences between the efficiency values based on extracted data and fitted specific curve (R² is 0.999) are, respectively, about -0.1% and 0.3%; maximum differences are about ±0.7%. Field tests are especially needed to determine the coefficients for installed fans because fans are susceptible to “system effects” (e.g., inlet obstructions) that are not included in the manufacturer test data (AMCA 1990a).

To determine fan efficiency and fan shaft power at a particular time step, first calculate the fan pressure rise (ΔP_fan) using the time step fan flow (Q_fan) and Equation (and also using Equations and if there is SPR). Next, calculate Eu using Equation and then x_fan (log₁₀ normalized Eu), which is based on Eu and the specified Eu_max (Eu at maximum efficiency) for the fan. The corresponding normalized efficiency (η_fan(x_fan) / η_fan,max) is obtained using Equation . The fan efficiency (η_fan(x_fan)) therefore is:

${\eta }_{fan}\left(x_{fan}\right)={\eta }_{fan,max}\left[\frac{{\eta }_{fan}\left(x_{fan}\right)}{{\eta }_{fan,max}}\right]$ ()

The fan shaft input power (mechanical, W) is:

$H_{fan}=\frac{\Delta P_{fan}\cdot Q_{fan}}{{\eta }_{fan}\left(x_{fan}\right)}$ ()

Fan Shaft Speed and Torque Modeling: For rotating elements, power (H) is the product of torque (τ) and rotational speed (ω), or conversely, torque is power divided by rotational speed (τ = H / ω).

The Stein and Hydeman variable-frequency-drive (VFD) component model correlates VFD efficiency as a linear function of VFD fractional output power (i.e., motor input power). Available data for about 50 drives from Saftronics were apparently used to develop their model, but those data represent VFD efficiency as a function of motor fractional speed (ω_motor / ω_motor,max, or nominally, drive output frequency divided by maximum output frequency, if motor slip is ignored).

To make use of the available data, the Stein and Hydeman linear correlation must intrinsically make an assumption that motor speed and torque have some fixed relationship. Although not documented, their assumption might be the common belief that fractional torque (τ / τ_max) for a motor is simply the square of its fractional speed. For fans serving duct systems with components such as filters and coils, with relatively low pressure drops elsewhere in the system, and for systems that have a non-zero controlled duct static pressure, this assumption may be inappropriate. Consequently, to make use of the available data and to avoid such assumptions, one needs to know the fraction of full speed at which fan components operate.

For the fan, dimensionless flow (φ) can be defined as (ASHRAE 1993):

$$\phi =\frac{Q_{fan}}{{\omega }_{fan}\cdot D_{fan}^3}$$

where ω_fan is the fan speed (rad/s). This parameter can be calculated for each operating point on the fan manufacturer’s performance map that represents fan speed as a function of flow and fan pressure rise. To simplify the data representation to a single curve (as we have done for fan efficiency), one can plot normalized dimensionless flow (dimensionless flow divided by maximum dimensionless flow, (φ(x_fan) / φ_max) as a function of the log base 10 of the normalized Euler (Eu) parameter (x_fan). Figure 211 shows an example plot derived from the manufacturer’s data in Figure 208.

Equation describes the sigmoidal functional form for normalized φ:

$$\frac{\phi \left(x_{fan}\right)}{{\phi }_{max}}=(A_{spd}+B_{spd}/((1+e^{\left[(C_{spd}-x_{spd})/D_{spd}\right]}{)}^{E_{spd}})$$

where A_spd, B_spd, C_spd, D_spd, and E_spd are coefficients for the fan. Separate sets of coefficients for the normal operation and stall regions can be specified. For the curve shown in Figure 211, these coefficients are:

Dimensionless Flow Coefficients - Specific Backward-Curved Fan

Operating Region	Aspd	Bspd	Cspd	Dspd	Espd
Normal (Non-Stall)	0	1.001423	0.123935	-0.476026	1
Stall	0	5.924993	-1.916316	-0.851779	1

For a generic centrifugal fan with backward-curved blades, the coefficients are:

Dimensionless Flow Coefficients - Generic Backward-Curved Fan

Operating Region	Aspd	Bspd	Cspd	Dspd	Espd
Normal (Non-Stall)	-0.551396	1.551467	-0.442200	-0.414006	0.234867
Stall	0.000608	0.586366	0.021775	-0.063218	0.072827

For any operating point of flow (Q_fan) and fan pressure rise (ΔP_fan), one can calculate the corresponding Euler number (Eu) for the given fan, and then determine the corresponding normalized dimensionless flow (φ(x_fan) / φ_max) from the sigmoidal dimensionless flow function (Equation ). With φ(x_fan) / φ_max determined and knowing φ_max for the fan, the dimensionless flow is:

$$\phi \left(x_{fan}\right)={\phi }_{max}\left[\frac{\phi \left(x_{fan}\right)}{{\phi }_{max}}\right]$$

With φ(x_fan) determined, the fan rotational speed (rad/s) is:

$${\omega }_{fan}=\frac{Q_{fan}}{\left(\phi \left(x_{fan}\right)\ast D_{fan}^3\right)}$$

Fan shaft torque (N∙m), which the belt drive must supply to the fan shaft, is then:

$${\tau }_{fan}=\frac{H_{fan}}{{\omega }_{fan}}$$

The fraction of full-load driven torque for the belt (τ_fan / τ_belt,max), which is typically called “belt load”, is thus τ_fan divided by the belt torque capacity (τ_belt,max). For a particular belt type and cross-section, belt torque capacity can be determined from manufacturer’s information such as a shaft speed versus power chart.

Ignoring belt slip, motor shaft speed (ω_motor, rad/s) can then be determined using the fan speed (ω_fan) and the motor/fan pulley diameter ratio (D_pulley,motor / D_pulley,fan), which is typically called the “drive” ratio:

$${\omega }_{motor}=\frac{{\omega }_{fan}}{\left(\frac{D_{pulley,motor}}{D_{pulley,fan}}\right)}$$

Belt, Motor, and Variable-Frequency-Drive Efficiency and Input Power Models - Overview: The models for belt drives, motors, and VFDs that Stein and Hydeman included in their air-handling system model represent maximum efficiency as a function of power input to the adjacent downstream component (e.g., the power input to a fan shaft by a belt drive), but they do not include part-load models. The part-load models are needed, because part-load efficiency for these components can fall off rapidly at low load (e.g., to zero at zero load) and many systems operate occasionally (and some much of the time) at low loads, in part because of current practices that result in substantial oversizing of components.

Belt Efficiency and Input Power Model: Figure 212 shows three maximum efficiency (η_belt,max) curves for belts (low, medium, and high) as a function of maximum fan shaft torque. This set of efficiency curves is based on belt drive loss data from AMCA Publication 203 (1990b), which reportedly is an aggregation of data from over 400 tests.

To determine η_belt,max if data for a specific belt are not available, first use the maximum fan shaft input power (H_fan,max) for the load spectrum to calculate the natural logarithm of belt power capacity:

x_belt,max = ln(H_fan,max) with H_fan,max expressed in terms of hp()

Then, use Equations and , along with a choice of low, medium, or high efficiency coefficients from Table 68, to calculate η_belt,max.

The quartic polynomial curves in Figure 212 and their coefficients are as follows:

$${\eta }_{belt,max,\ln }=c_1+c_2\cdot x_{belt,max}+c_3\cdot x_{{}_{belt,max}}^2+c_4\cdot x_{{}_{belt,max}}^3+c_5\cdot x_{{}_{belt,max}}^4$$

Belt Maximum Efficiency Curve Coefficients

Efficiency Class (Loss)	c1	c2	c3	c4	c5
High (Low Loss)	-6.502E-2	2.475E-2	-6.841E-3	9.383E-4	-5.168E-5
Medium (Medium Loss)	-9.504E-2	3.415E-2	-8.897E-3	1.159E-3	-6.132E-5
Low (High Loss)	-1.422E-1	5.112E-2	-1.353E-2	1.814E-3	-9.909E-5

The belt maximum efficiency is therefore:

$${\eta }_{belt,max}=e^{{\eta }_{belt,max,\ln }}$$

The normalized part-load belt efficiency model is provided to modify the maximum efficiency of the belt. This model is based on part-load data for belts in ACEEE’s handbook on motor systems (Nadel et al. 2002). To determine the normalized belt efficiency at part-load (η_belt(x_belt) **/ η_belt,max), use the fan shaft (belt output) fractional torque (x_belt = τ_belt / τ_belt,max;τ_belt = τ_fan) as the belt fractional load in Equation or (select the equation for the region that contains x_belt) and use coefficients from Table 69 that correspond to that region and the belt type: V-Belt or synchronous (toothed). Figure 213 shows a graphical representation of the curves defined by these equations and coefficients.

The belt normalized (part-load) efficiency curves in Figure 213 for Regions 1 and 3 (single rectangular hyperbola type 2) and Region 2 (exponential decay), respectively, and their coefficients are as follows:

Region 1 (0 <= x_belt < x_belt,trans) and Region 3 (x_belt > 1):

$\frac{{\eta }_{belt}\left(x_{belt}\right)}{{\eta }_{belt,max}}=\left[\frac{a_{belt}\cdot x_{belt}}{b_{belt}+x_{belt}}\right]+c_{belt}\cdot x_{belt}$ ()

Region 2 (x_belt,trans <= x_belt <= 1):

$\frac{{\eta }_{belt}\left(x_{belt}\right)}{{\eta }_{belt,max}}=a_{belt}+b_{belt}\cdot e^{\left(c_{belt}\cdot x_{belt}\right)}$ ()

where x_belt = $\tau$ _belt / $\tau$ _belt,max

Belt efficiency therefore is:

${\eta }_{belt}\left(x_{belt}\right)={\eta }_{belt,max}\left[\frac{{\eta }_{belt}\left(x_{belt}\right)}{{\eta }_{belt,max}}\right]$ ()

Belt input power (mechanical, W) at the motor shaft is:

$H_{belt}=\frac{H_{fan}}{{\eta }_{belt}\left(x_{belt}\right)}$ ()

Belt Normalized Efficiency Curve Coefficients

Belt Type	xbelt,trans	Region	abelt	bbelt	cbelt
V-Belt	0.167	1	0.920797	0.026269	0.151594
-	-	2	1.011965	-0.339038	-3.436260
-	-	3	1.037778	0.010307	-0.026815
Synchronous	0.137	1	0.982167	0.049135	0.158164
-	-	2	1.002134	-0.531885	-5.295707
-	-	3	1	0	0

Motor Efficiency and Input Power Model: Figure 214 shows three maximum efficiency (η_motor,max) curves for motors (low, medium, and high) as a function of rated motor output power (belt input power). This set of efficiency curves is based on maximum efficiency data in DOE’s MotorMaster+ database (2003) from about 800 tests. It is provided for use in the absence of manufacturer’s data.

To determine η_motor,max if data for a specific motor are not available (e.g., as listed in Table 71), first use the maximum belt input power (H_belt,max) for the load spectrum (multiplied by whatever oversizing factor may be desired) to calculate the natural logarithm of belt power capacity:

x_motor,max = ln(H_belt,max) with H_belt,max expressed in terms of hp()

Then, use Equation , along with a choice of low, medium, or high efficiency coefficients from Table 70, to calculate η_motor,max.

The maximum motor efficiency curves in Figure 214 (single rectangular hyperbola type 1) and their coefficients are as follows:

$${\eta }_{motor,max}=\left[\frac{a_{motor,max}\cdot x_{motor,max}}{b_{motor,max}+x_{motor,max}}\right]+c_{motor,max}$$

Motor Maximum Efficiency Curve Coefficients

Case	amotor,max	bmotor,max	cmotor,max
High-Efficiency	0.196205	3.653654	0.839926
Mid-Efficiency	0.292280	3.368739	0.762471
Low-Efficiency	0.395895	3.065240	0.674321

The normalized part-load motor efficiency model is provided to modify the maximum efficiency of the motor. This model is based on part-load data for motors from DOE’s MotorMaster+ database (2003). To determine the normalized motor efficiency at part-load (η_motor(x_motor) **/ η_motor,max), use the motor fractional power output (x_motor = H_belt / H_belt,max) as the fraction of motor output power in Equation and use coefficients from Table 71, or for a specific motor, determined from DOE MotorMaster+ data or from manufacturer’s data. Figure 215 shows a graphical representation of eight example curves defined by these equations and coefficients.

The example motor normalized efficiency curves (single rectangular hyperbola type 2) and their coefficients as a function of motor load fraction in Figure 215 are determined from DOE MotorMaster+ data and are as follows:

$\frac{{\eta }_{motor}\left(x_{motor}\right)}{{\eta }_{motor,max}}=\left[\frac{a_{PLmotor}\cdot x_{motor}}{b_{PLmotor}+x_{motor}}\right]+c_{PLmotor}\cdot x_{motor}$ ()

Example Motor Normalized Efficiency Curve Coefficients

Poles	Motor RatedOutput (hp)	MaximumEfficiency	aPLmotor	bPLmotor	cPLmotor
8	1	0.6675	1.096694	0.097126	0.002011
4	1	0.7787	1.092165	0.082060	-0.007200
-	5	0.8400	1.223684	0.084670	-0.135186
-	10	0.8745	1.146258	0.045766	-0.110367
-	25	0.8991	1.137209	0.050236	-0.089150
-	50	0.9129	1.088803	0.029753	-0.064058
-	75	0.9259	1.077140	0.029005	-0.049350
-	100	0.9499	1.035294	0.012948	-0.024708
-	125	0.9527	1.030968	0.010696	-0.023514

Motor efficiency therefore is:

$${\eta }_{motor}\left(x_{motor}\right)={\eta }_{motor,max}\left[\frac{{\eta }_{motor}\left(x_{motor}\right)}{{\eta }_{motor,max}}\right]$$

Motor input power (electrical, W) is:

$$H_{motor}=\frac{H_{belt}}{{\eta }_{motor}\left(x_{motor}\right)}$$

VFD Efficiency and Input Power Model: For VFDs, published performance data are limited. Data from DOE (2008) suggest using a functional relation similar to that used for motors to represent VFD efficiency (η_VFD) as a function of the fraction of full-load motor input power (x_VFD = H_motor / H_motor,max), as shown in Figure 216, or as a function of the fraction of full-load speed (x_VFD = ω_motor / ω_motor,max), as shown in Figure 217.

The performance curve (single rectangular hyperbola type 2) used here for VFD efficiency is:

$${\eta }_{VFD}=\left[\frac{a_{VFD}\cdot x_{VFD}}{b_{VFD}+x_{VFD}}\right]+c_{VFD}\cdot x_{VFD}$$

Example coefficients derived from the DOE data as a function of the fraction of full-load motor input power are listed in Table 72.

Example VFD Efficiency Curve Coefficients

VFD RatedOutput Power (hp)	avfd	bvfd	cvfd
3	0.978856	0.034247	-0.007862
5	0.977485	0.028413	-0.002733
10	0.978715	0.022227	0.001941
20	0.984973	0.017545	-0.000475
30	0.987405	0.015536	-0.005937
50	0.987910	0.018376	-0.001692
60	0.971904	0.014537	0.011849
75	0.991874	0.017897	-0.001301
100	0.982384	0.012598	0.001405
>= 200	0.984476	0.009828	-0.004560

VFD input power (electrical, W) is:

$$H_{VFD}=\frac{H_{motor}}{{\eta }_{VFD}}$$

System Total Efficiency: The combined efficiency of the fan system components (i.e., fan, belt, motor, and VFD) is:

$${\eta }_{TOTAL}={\eta }_{fan}\cdot {\eta }_{belt}\cdot {\eta }_{motor}\cdot {\eta }_{VFD}\left(=\frac{Q_{fan}\cdot \Delta P_{fan}}{H_{VFD}}\right)$$

Heat Loss to Air: To calculate the temperature rise from waste heat entering the air stream from the fan, belt, and motor, it is assumed that the user-specified “motor in air fraction” applies to the belt and motor but not to the VFD. The power “lost” to the air (W) is:

$${\dot{Q}}_{toair}=H_{fan}+\left(H_{motor}-H_{fan}\right)\cdot f_{motortoair}$$

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