Documentation

# Fan (G)

Mechanical apparatus for powering flow in gas systems

• Library:
• Simscape / Fluids / Gas / Turbomachinery

## Description

The Fan (G) block models a mechanical apparatus for powering flow in gas systems: a rotor mounted on a drive shaft (often) enclosed in a case. The fan is parameterized in terms of performance metrics commonly reported in technical data sheets (and specified here in tabulated form). The metrics used depend on the parameterization chosen but can include measures of pressure rise, flow rate, rotor speed, and fan efficiency. The parameterizations are valid solely in the normal operating region (for which data is reported).

Normally, an electrical motor, though sometimes another power source, spins the rotor (port R) against its case (C). The rotor transfers the power drawn to the gas stream, causing it to move (and its pressure to rise) from inlet (A) to outlet (B). The direction of the flow depends on the blades of the rotor and it can, in a real fan, be radial, axial, crossed, or mixed. Such effects are assumed to reflect entirely in the performance data specified for the fan.

The mechanical ports belong to the Simscape Rotational domain. To turn the rotor, they must connect to ports of the same domain. Rotational source blocks from the Simscape Foundation library are a simple way to apply the necessary torque. These are idealized models, without friction or other real-world effects. Simscape Driveline blocks are another option, richer in detail for more accurate simulations, among it the effects ignored in their idealized counterparts (when relevant to the model).

### Mechanical Orientation

Two directions play into the block computations: those of the turning of the rotor and of the movement of the gas stream.

The shaft of the rotor can technically turn forward or backward. For the rotor to generate flow, however, only the direction indicated in the Mechanical orientation parameter will do. That direction can be `Positive` or `Negative`. (In other words, the fan is unidirectional in operation.) If the rotor shaft should turn in the counter direction, the fan goes into idle and power is no longer supplied to the stream. You can think of the rotor as having disengaged from the shaft (for example, by use of a one-way clutch).

The stream, on the other hand, when powered by the fan, must flow from inlet (port A) to outlet (B). This direction is designated as positive in the block calculations. It is possible for the flow to reverse, but not by the action of the rotor. Instead, an event must occur to flip the pressure gradient across the fan and so force the flow back toward the inlet. Such occurrences are considered abnormal and, when they take place, are generally transient and short-lived.

Note that the direction of the flow is independent of the Mechanical orientation setting of the block. This setting serves merely as a means to reverse the sweep of the rotor blades. Clockwise and counterclockwise rotors can both generate positive flow—and in fact must if the remainder of the fan requires it—but the way they must turn will differ. (One will generate positive flow when spun in the positive direction; the other, when spun in the negative direction.)

#### Numerical Smoothing

The saturation has the effect of splitting the rotor speed domain into two regions: below the saturation threshold, the rotor speed is fixed at the threshold value; above the saturation threshold, it is a variable determined by calculation. The transition between the regions has one drawback: without modification, it is sharp and its slope discontinuous.

Slope discontinuities pose a challenge to variable-step solvers (the sort commonly used in Simscape models). To precisely capture a sharp transition, the solver must reduce its time step, pausing briefly at the time of the transition in order to recompute the Jacobian matrix for the model (a representation of the dependencies between state variables and their time derivatives).

This solver strategy is efficient and robust when discontinuities are present—it makes the solver less prone to convergence errors—but it can considerably extend the time needed to finish a simulation. An alternative approach, used here, is to remove the slope discontinuities altogether, by smoothing them over a small range of motor speeds.

The smoothing, which adds a slight distortion to the transition, ensures that the fan eases into its saturated state rather than snap abruptly to it. The shape and scale of the smoothing derive from the cubic polynomial expression:

`$\lambda =3{\left(\frac{\omega }{{\omega }_{\text{Th}}}\right)}^{2}-2{\left(\frac{\omega }{{\omega }_{\text{Th}}}\right)}^{3},$`

where ƛ is the smoothing polynomial. The subscript `Th` denotes the threshold value at which to saturate the rotor speed (specified in the Shaft speed threshold for flow reversal block parameter). The smoothing adds a third, transitional, region between the saturated and variable regions of the rotor speed domain:

`${\omega }^{*}=\left\{\begin{array}{ll}{w}_{\text{Th}},\hfill & \omega <0\hfill \\ \left(1-\lambda \right){\omega }_{\text{Th}}+\lambda \omega ,\hfill & \omega <{\omega }_{\text{Th}}\hfill \\ \omega ,\hfill & \omega \ge {\omega }_{\text{Th}}\hfill \end{array},$`

where the asterisk (*) denotes the smoothed value.

The figure shows the effect of smoothing on the speed of the rotor. The plot to the left corresponds to a fan with positive mechanical orientation; the plot to the right, to one with negative mechanical orientation. Region I is the saturated state, region II the transition state, and region III the original state (with neither smoothing nor saturation).

### Mass Balance

The fan, which creates flow but neither stores nor carries its contents, is a relatively compact device. The gas column that surrounds its rotor is thin and, on the scale of the larger components—the pipes, chambers, and reservoirs that often abut the fan—of negligible volume. This volume and its mass are ignored in this block. (While the stream can pass through the fan, its contents cannot accumulate there, as they might in a pipe, say, if it is dilated, heated, or compressed.)

Conservation of mass then requires only that the mass flow rates through the inlet and outlet be of the same magnitude:

`${\stackrel{˙}{m}}_{\text{A}}+{\stackrel{˙}{m}}_{\text{B}}=0,$`

where $\stackrel{˙}{m}$ is the mass flow rate into the valve through the port indicated by the subscript. Port A is always the inlet and port B always the outlet.

### Energy Balance

The smaller the gas volume, the faster its transient response to pressure and temperature disturbances, and the shorter its time spent between steady states. If that volume is trivially small, as it is here, at least in proportion to the volumes of neighboring components, the transient response can be considered instantaneous. The fan then behaves as if it were always at a steady state (one given by the instantaneous conditions at the ports). Such components are known as quasi-steady.

A quasi-steady fan, having no gas volume, has also no energy contents to vary over time. Energy can flow through its ports, and, as mechanical work, from the rotor, but it cannot accumulate inside. In other words, what comes in must simultaneously go out. Conservation of energy then reduces to a sum of energy flow rates:

`${\varphi }_{\text{A}}+{\varphi }_{\text{B}}+{W}_{\text{F}}=0,$`

where ϕ is the total energy flow rate into the fan through the port indicated by the subscript, and WF is the rate of work done on—and the power supplied to—the stream by the rotor.

#### Total Efficiency

This is not to say that the fan is lossless. Energy can dissipate between the ports by interactions involving mechanical components, the gas stream, or both. The rotor shaft, for example, is subject to friction; the stream must pass through restrictions, elbows, and fittings. In this block, the losses due to all such factors are assumed to reflect in the total efficiency of the fan—the ratio of the (pneumatic) power output to the (mechanical) power input:

`${\eta }_{\text{T}}=\frac{{W}_{\text{F}}}{{W}_{\text{M}}},$`

where ηT is the total efficiency and W is the power delivered, in the term denoted by subscript `M`, by the drive shaft to the fan rotor (before any portion of that power can dissipate in the fan).

(The total efficiency is obtained during simulation from the tabulated data of the block. See "Fan Parameterizations" for the block calculations.)

The mechanical power input is defined in terms of the domain variables at the mechanical ports:

`${W}_{\text{M}}=\tau {\omega }^{*},$`

where τ is the torque delivered to the drive shaft (port R) relative to the case (C) and ɷ* is the smoothed shaft velocity, also relative to the case, as defined in "Numerical Smoothing".

The power supplied to the stream, always a fraction ηT of that delivered by the drive shaft, acts to raise the total enthalpy of the gas from the inlet to the outlet. This power is defined here for ideal and nonideal gases alike as:

`${W}_{\text{F}}=\stackrel{˙}{m}\left({h}_{\text{T,B}}-{h}_{\text{T,A}}\right),$`

where hT is the specific total enthalpy at the outlet (subscript B) and at the inlet (subscript A). The specific total enthalpy is calculated from the specific enthalpy (the sum of specific internal energy with the product of pressure and specific volume):

`${h}_{\text{T}}=h+\frac{{v}^{2}}{2},$`

where the fraction on the right is the specific kinetic energy of the stream and v is the velocity of the same, both at the gas port being considered (A or B).

(The power WF is frequently calculated in incompressible fluids as the product of the volumetric flow rate through and the pressure rise across the fan; the use of a compressibility factor allows a variant of the same expression to be used in ideal gases but, for nonideal gases, especially those subjected to large pressure rises, neither expression will give accurate results).

### Inlet Density Definition

The fan parameterizations rely partly on the so-called fan affinity laws. These are expressions of proportionality used to relate fans of the same type but of different size and at different operating conditions—reflected partly in the density of the gas.

As the fan is modeled without gas volume, that density is obtained from the nearest upstream computational node. That node is port A during normal operation and port B if the stream should be forced in the reverse direction:

`$\rho =\left\{\begin{array}{ll}{\rho }_{\text{A}},\hfill & \stackrel{˙}{m}>0\hfill \\ {\rho }_{\text{B}},\hfill & \stackrel{˙}{m}>0\hfill \end{array},$`

where ρ is the gas density at the port indicated by the subscript.

#### Numerical Smoothing

For numerical robustness, when the mass flow rate drops below a small threshold, the gas density is artificially smoothed between the values at the ports. The smoothing is based on the transcendental function:

`$\alpha =\text{tanh}\left(\frac{4\stackrel{˙}{m}}{{\stackrel{˙}{m}}_{\text{Th}}}\right),$`

where α is the smoothing factor and `tanh` is the hyperbolic tangent function. The subscript `Th` denotes a threshold value (a very small number hardcoded in the block). In terms of the smoothing factor, in the transitional region below the mass flow rate threshold, the gas density becomes:

`$\rho ={\rho }_{\text{A}}\left(\frac{1+\alpha }{2}\right)+{\rho }_{\text{B}}\left(\frac{1-\alpha }{2}\right).$`

This expression adds a third row to the piecewise definition of gas density, which is finally defined as:

`$\rho *=\left\{\begin{array}{ll}{\rho }_{\text{A}},\hfill & \stackrel{˙}{m}>{\stackrel{˙}{m}}_{\text{Th}}\hfill \\ {\rho }_{\text{A}}\left(\frac{1+\alpha }{2}\right)+{\rho }_{\text{B}}\left(\frac{1-\alpha }{2}\right),\hfill & |\stackrel{˙}{m}|\le {\stackrel{˙}{m}}_{\text{Th}}\hfill \\ {\rho }_{\text{B}}\hfill & \stackrel{˙}{m}<-{\stackrel{˙}{m}}_{\text{Th}}\hfill \end{array}.$`

The asterisk denotes a smoothed value. The mass flow rate is the same at port A and port B. The value used in the conditions of the gas density expression can refer to either port.

### Fan Parameterizations

The relationships between pressure rise, flow rate, fan efficiency, and rotor speed derive from the tabulated data specified in the block. The form of this data and the calculations that it supports depend on the fan parameterization chosen. Of parameterization options there are four. One is simple and based on one-dimensional tables; three are more complete and based on two-dimensional tables. All are restricted to the normal operating region (given in the technical data sheets of the fan).

#### `1D tabulated data - static pressure and total efficiency table vs. flow rate`

The pressure rise is calculated from the second fan affinity law, which, modified for fans of the same size, states:

`${p}_{\text{SF}}={\left(\frac{\omega *}{{\omega }_{\text{R}}}\right)}^{2}\left(\frac{\rho *}{{\rho }_{\text{R}}}\right){p}_{\text{SF,R}}.$`

The subscript `R` denotes a reference value (and the asterisk a smoothed quantity, as described previously). The reference density and the reference shaft speed are each obtained from a block parameter (of the same name). The static pressure rise at reference conditions is obtained from the tabulated data of the block:

`${p}_{\text{SF,R}}={p}_{\text{SF}}\left({q}_{\text{R}}\right),$`

where qR is the volumetric flow rate at reference conditions (and the independent variable in the tabulated data). Its value is calculated from the first fan affinity law:

`${q}_{\text{R}}=\left(\frac{{\omega }_{\text{R}}}{{\omega }^{*}}\right)q,$`

where q is the volumetric flow rate through the fan, calculated from the mass flow rate at the ports as:

`$q=\frac{\stackrel{˙}{m}}{{\rho }^{*}}.$`

(The first fan law is used here to capture the effect of shaft speed on volumetric flow rate (and therefore also static pressure rise). This calculation is omitted in the ```2D tabulated data - static pressure and total efficiency vs. shaft speed and flow rate``` parameterization, where the effect of shaft speed is taken into account directly in the tabulated data of the block.)

The total fan efficiency, described in "Total Efficiency", is likewise obtained from the tabulated data:

`${\eta }_{\text{T}}={\eta }_{\text{T}}\left({q}_{\text{R}}\right).$`

#### `2D tabulated data - static pressure and total efficiency vs. shaft speed and flow rate`

With the rotor shaft speed now specified in the tabulated data, its reference value can be set to its instantaneous (and smoothed) value. The ratio of actual to reference shaft speeds reduces to `1` and the second affinity law gives:

`${p}_{\text{S,F}}=\left(\frac{{\rho }^{*}}{{\rho }_{\text{R}}}\right){p}_{\text{SF,R}},$`

where pSF,R is obtained from the tabulated data as a bivariate function of the (smoothed) rotor shaft speed and of the volumetric flow rate through the fan:

`${p}_{\text{SF,R}}={p}_{\text{SF}}\left(\omega ,q\right).$`

The total fan efficiency is obtained in a similar manner from the tabulated data:

`${\eta }_{\text{T}}={\eta }_{\text{T}}\left(\omega ,q\right).$`

#### `2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure`

The mass flow rate through the fan is obtained from the tabulated data of the block as:

`$\stackrel{˙}{m}={\rho }_{\text{R}}{q}_{\text{R}},$`

where qR is the volumetric flow rate through the fan at reference density:

`${q}_{\text{R}}=q\left(\omega ,{p}_{\text{SF,R}}\right),$`

The static pressure rise at the same reference condition is calculated as:

`${p}_{\text{SF,R}}=\left(\frac{{\rho }_{\text{R}}}{{\rho }^{*}}\right){p}_{\text{SF}}.$`

The total fan efficiency is defined as before:

`${\eta }_{\text{T}}={\eta }_{\text{T}}\left(\omega ,{p}_{\text{SF,R}}\right).$`

#### `2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure ratio`

The mass flow rate is obtained from the tabulated data of the block as:

`$\stackrel{˙}{m}={\rho }_{\text{R}}{q}_{\text{R}},$`

where the volumetric flow rate at reference density is computed no longer from the static pressure rise but from its ratio to its maximum rated value:

`${q}_{\text{R}}=q\left(\omega ,{r}_{\text{p,R}}\right).$`

The static pressure rise ratio, rp, is specified in the tabulated data of the block. It is defined at reference density as:

`${r}_{\text{p,R}}=\frac{{p}_{\text{SF,R}}}{{p}_{\text{SF,Max}}},$`

which, using the first fan affinity law, becomes:

`${r}_{\text{p,R}}=\left(\frac{{\rho }_{\text{R}}}{{\rho }^{*}}\right)\left(\frac{{p}_{\text{SF}}}{{p}_{\text{SF,Max}}}\right),$`

The maximum static pressure rise, pSF,Max, can vary with the rotor shaft speed. The relationship between the two is specified in a separate, one-dimensional lookup table. A plot of flow rate against static pressure rise need not be rectangular in shape, giving this parameterization a versatility not found in its alternatives.

The total fan efficiency is defined in terms of the same variables as:

`${\eta }_{\text{T}}={\eta }_{\text{T}}\left(\omega ,{r}_{\text{p}}\right).$`

## Ports

### Conserving

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Opening for the gas stream to enter the fan.

Opening for the gas stream to exit the fan.

Drive shaft on which the fan rotor is to be mounted.

Case in which the fan rotor is to be housed.

## Parameters

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Set of performance metrics on which to base the fan model. These metrics are commonly reported in technical data sheets as plots or tables. The 1-D parameterization uses the first fan affinity law to obtain the rotor shaft speed data. (See the block description for the affinity laws.)

Direction in which the fan rotor must turn in order to generate flow. The fan is unidirectional and the gas stream flows only in the positive direction in each case. Use this parameter to capture different rotor designs—for example, those with forward-swept and backward-swept blades, which must turn in different directions relative to the flow.

Volumetric flow rates, as measured from inlet to outlet, at which to specify the active fan characteristics (those used by the selected parameterization). These include static pressure and total efficiency for the fan, specified either as 1-D or 2-D tables. The default vector depends on the parameterization of the block.

The number of elements in the vector must equal the number of elements in the 1-D table or the number of rows in the 2-D table. (The tables are obtained from the Static pressure rise vector and Static pressure rise vector parameters or from the Static pressure rise table and Total efficiency table parameters. The vector elements must increase monotonically from left to right.

The fan generates flow in the forward, or positive, direction only. Negative values, which are counter to this direction, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```1D tabulated data - static pressure and total efficiency vs. flow rate```.

Static pressure differentials from inlet to outlet at the specified volumetric flow rates. The number of elements in this vector must equal that in the Volumetric flow rate vector parameter. The fan generates flow in the forward direction only (from inlet to outlet). Negative static pressure differentials, which generally map to reverse flows, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```1D tabulated data - static pressure and total efficiency vs. flow rate```.

Total fan efficiencies at the specified volumetric flow rates. The total efficiency is the ratio of the output power (transferred to the gas stream) to the inlet power (supplied by the rotor shaft).

As is typical of efficiency parameters, this parameter is defined as a fraction between `0` and `1`. The number of elements in this vector must equal that in the Volumetric flow rate vector parameter. Negative values, which imply a fan capable of extracting power from (instead of supplying it to) the gas stream are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```1D tabulated data - static pressure and total efficiency vs. flow rate```.

Rotor shaft speeds, measured against the fan case, at which to tabulate the active fan characteristics. Those vary with the fan parameterization but include total efficiency and either static pressure of flow rate. The default vector depends on the parameterization selected.

The number of elements in the vector must equal the numbers of rows in the tables of fan characteristics. (The tables are obtained from the Static pressure rise table, Total efficiency table, and Flow rate table block parameters, whichever happen to be active.)

The vector elements must increase monotonically in value from left to right. Negative values, which imply a fan capable of powering flow in reverse, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to a 2-D parameterization.

Static pressure differentials, as measured from inlet to outlet, at which to specify the active fan characteristics (those used by the selected parameterization). Here, those characteristics include flow rate and total efficiency.

Static pressure is that measured when the dynamic pressure, generally due to flow, is zero or subtracted from the (total) pressure reading. Fan characteristics are commonly reported in terms of it, though some manufacturers use the total pressure instead.

The number of elements in the vector must equal the numbers of rows in the tables of fan characteristics. (The tables are obtained from the Flow rate table and Total efficiency table parameters.) The vector elements must increase monotonically in value from left to right. Negative values, which imply a fan capable of powering flow in reverse, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure```.

Normalized static pressure differentials, as measured from inlet to outlet, at which to specify the active fan characteristics (those used by the selected parameterization). Here, those characteristics include flow rate and total efficiency.

The normalization is with respect to the maximum pressure rise allowed at a given volumetric flow rate (and computed during simulation from the same). In other words, this parameter is the fraction of the static pressure rise over its calculated maximum. See the block description for more detail on this parameter.

The number of elements in the vector must equal the numbers of rows in the tables of fan characteristics. (The tables are obtained from the Flow rate table and Total efficiency table parameters.) The vector elements must increase monotonically in value from left to right. Negative values, which imply a fan capable of powering flow in reverse, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure ratio```.

Static pressure differentials, as measured from inlet to outlet, at the breakpoints specified in the block. (The breakpoints are given in this parameterization by the Shaft speed vector and Flow rate vector block parameters.)

Static pressure is that measured when the dynamic pressure, generally due to flow, is zero or subtracted from the (total) pressure reading. Fan characteristics are commonly reported in terms of it, though some manufacturers use the total pressure instead.

The number of rows in the table must equal the number of elements in the first vector specified (Shaft speed vector); the number of columns, the number of elements in the second vector specified (Flow rate vector). The fan generates flow in the forward, or positive, direction only. Negative pressure differentials, which could force the flow in reverse, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```2D tabulated data - static pressure and total efficiency vs. shaft speed and flow rate```.

Total fan efficiencies at the breakpoints specified in the block. (The variables used to define the breakpoints depend on the fan parameterization; they can include shaft speed, flow rate, static pressure, and/or static pressure ratio.)

The total efficiency is the ratio of the output power (transferred to the gas stream) to the inlet power (supplied by the rotor shaft). As is typical of efficiency parameters, this parameter is defined as a fraction, normally between `0` and `1`.

The number of rows in the table must equal the number of elements in the first vector specified (Shaft speed vector); the number of columns, the number of elements in the second vector specified (Flow rate vector, Static pressure rise vector, or Static pressure rise ratio vector).

Negative values, which imply a fan capable of extracting power from (instead of supplying it to) the gas stream are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to a 2-D parameterization.

Volumetric flow rates, measured from inlet to outlet, at the breakpoints specified in the block. (The variables used to define the breakpoints depend on the fan parameterization; they can include shaft speed, static pressure rise, and static pressure rise ratio.)

The number of rows in the table must equal the number of elements in the first vector specified (Shaft speed vector); the number of columns, the number of elements in the second vector specified (Static pressure rise vector, or Static pressure rise ratio vector).

The fan generates flow in the forward, or positive, direction only. Negative values, which are counter to this direction, are disallowed.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure``` or ```2D tabulated data - flow rate and total efficiency vs. shaft speed and static pressure ratio```.

Gas density established in the measurement of the tabulated reference data specified—the volumetric flow rate, static pressure rise, and total efficiency for the fan. This parameter is used in the calculation of the true static pressure rise across the fan.

Rotor shaft speed used in the measurement of the tabulated reference data specified—the volumetric flow rate, static pressure rise, and total efficiency for the fan. This parameter is used in the calculation of the true static pressure rise across the fan.

#### Dependencies

This parameter is active and exposed in the block dialog box when the Fan specification parameter is set to ```1D tabulated data - static pressure and total efficiency vs. flow rate```.

Lower bound of the rotor shaft speed range at which the fan will generate flow. This parameter is always positive, no matter the Mechanical orientation setting.

For use in the flow rate calculations, the rotor speed obtained from ports R and C is saturated at this value. The saturation is smooth, with the degree of smoothing dependent partly on this parameter. The smoothing is based on a cubic polynomial (given in the block description).

Area normal to the flow at the inlet of the fan. The inlet and outlet need not be the same in size. For best results, the value specified here should match the opening area of the component connected to the inlet.

Area normal to the flow at the outlet of the fan. The inlet and outlet need not be the same in size. For best results, the value specified here should match the opening area of the component connected to the outlet.