Method of optimizing filter life cycle between replacements and system for monitoring a ventilation system

Abstract

The present disclosure provides a method of determining an optimal filter life cycle between replacements of a filter in a ventilation system. The method comprising performing the following steps in a processing device: receiving at least one filter hardware value, receiving at least one filter use value, receiving a plurality of measured data points, and determining the optimal filter life cycle by minimizing a total filter resource consumption composed of a first factor, a second factor, providing a plurality of predicted data points, grouping successive measured data points into windows, and assigning the identified maximum pressure drop as a maximum pressure drop for all measured data points of that window.

Claims

1. A method of determining an optimal filter life cycle between replacements of a filter in a ventilation system, wherein the method comprises: using at least one pressure sensor to measure pressure change in the filter of the ventilation system, and wherein the method further comprises performing the following steps in a processing device: receiving at least one filter hardware value (C.sub.filter), representing an amount of a resource associated with at least production of the filter, receiving at least one filter use value (C.sub.use), representing an amount or rate of said resource associated with use of the filter, receiving a plurality of measured data points (M[i]) associated with the measured pressure change from the at least one pressure sensor, each representing a measured pressure drop (M[i].sub.p) over the filter at a respective time (M[i].sub.t), and determining the optimal filter life cycle by minimizing a total filter resource consumption comprising: a first factor (Filter.sub.annual_cost[i]), according to which the resource consumption is inversely proportional to the filter life cycle, a second factor (Energy.sub.annual_cost[i]), according to which the resource consumption is directly proportional to the filter life cycle, providing a plurality of predicted data points, each representing an predicted pressure drop over the filter at a respective future point in time, grouping successive measured data points (M[i]) into windows (W), each window (W) comprising at least one of the measured data points (M[i]), for each window (W) identifying a maximum pressure drop, and assigning the identified maximum pressure drop as a maximum pressure drop (max_values[i]) for all measured data points (M[i]) of that window; for each measured data point (M[i]) estimating an air flow (Q[i]) based on the maximum pressure drop (max_values[i]) and the measured pressure drop (M[i].sub.p) of the respective measured data point (M[i]), and deriving the second factor (Energy.sub.annual_cost[i]) based on the filter use value (C.sub.use), the measured data points (M[i]), the predicted data points and the estimated air flows (Q[i]).

2. The method as claimed in claim 1, wherein providing a plurality of predicted data points comprises calculating a respective predicted pressure drop through the formula P(t)=start.sub.Pa*e.sup.b*t, wherein start.sub.Pa is a starting pressure drop of the prediction, selected from the measured pressure drops (M[i].sub.p) of the measured data points (M[i]), b is an environmental coefficient and t is the time from the time when the starting pressure drop (start.sub.Pa) is measured until the point in time for which the pressure drop is predicted.

3. The method as claimed in claim 2, wherein the starting pressure drop (start.sub.Pa) is a value derived based on the measured pressure drop of the latest available measured data point.

4. The method as claimed in claim 2, wherein the starting pressure drop (start.sub.Pa) is a value derived based on the maximum pressure drop of the latest available measured data point or the latest available window.

5. The method as claimed in claim 1, wherein each window (W) comprises at least two measured data points (M[i]).

6. The method as claimed in claim 1, wherein the windows comprises a predetermined or dynamic number of measured data points (M[i]).

7. The method as claimed in claim 1, wherein the maximum pressure drop for each window (W) is determined with regard to an average value of the measure pressure drops (M[i].sub.p) of at least some of the measured data points (M[i]) of the window.

8. The method as claimed in claim 1, wherein the maximum pressure drop for each window (W) is determined with regard to a measured pressure drop (M[i].sub.p) and/or a maximum pressure drop of at least one preceding window.

9. The method as claimed in claim 1, wherein estimating the air flow for each measured data point (M[i]) comprises deriving the estimated air flow (Q[i]) of each measured data point (M[i]) based on a measured pressure drop (M[i].sub.p) of that measured data point and a characteristic function of the filter, wherein the characteristic function describes the air flow (Q) as a function of the pressure drop (P) for the filter.

10. The method as claimed in claim 9, wherein estimating the air flow for each measured data point (M[i]) comprises providing a scaling factor based on the maximum pressure drop (max_values[i]) and the measured pressure drop (M[i].sub.p) of the measured data points (M[i]), and using the scaling factor to adjust a characteristic function of the filter when it is clean.

11. The method as claimed in claim 1, further comprising predicting the air flow for at least one of the predicted data points.

12. The method as claimed in claim 11, wherein the predicted air flow of the predicted data points is a fixed value derived based on the estimated air flows (Q[i]) for at least some of the measured data points (M[i]).

13. The method as claimed in claim 11, wherein the predicted air flow of the predicted data points is a respective value derived based on at least one of the predicted data points.

14. The method as claimed in claim 1, wherein the second factor (Energy.sub.annual_cost[i]) is derived based on the estimated air flow (Q[i]) of each measured data point (M[i]) and/or the predicted air flow for the predicted data points.

15. The method as claimed in claim 1, wherein the second factor (Energy.sub.annual_cost[i]) is calculated as a sum of resource consumptions of each interval (M[i+1].sub.t−M[i].sub.t) between the measured data points (M[i]).

16. The method as claimed in claim 1, wherein the first factor (Filter.sub.annual_cost[i]) is determined as a product of the filter hardware value (C.sub.filter), an inverse of the time (M[i].sub.t) and optionally one or more constants.

17. The method as claimed in claim 1, wherein the second factor (Energy.sub.annual_cost[i]) is determined as a sum of products of the filter use value (C.sub.use), an air flow (Q[i]), a pressure drop (M[i].sub.p), a time interval (M[i+1].sub.t−M[i].sub.t), an inverse of a fan efficiency (η), an inverse of a time (M[i].sub.t), and/or one or more constants.

18. The method as claimed in claim 1, wherein the second factor (Energy.sub.annual_cost[i]) is determined according to the formula ${\mathrm{Energy}}_{\mathrm{annual}\_\mathrm{cost}}\left[i\right]=\underset{k=0}{\overset{i}{.Math.}}E\left[k\right]*\frac{\left(8760*3600\right)}{{M\left[i\right]}_t}*{\mathrm{Price}}_{\mathrm{kWh}}$ wherein Price.sub.kWh is the energy cost per kilowatt hour, M[i].sub.t is the respective point in time and $E\left[i\right]=\frac{\mathrm{Current}Q*{M\left[i\right]}_p*\left({M\left[i+1\right]}_t-{M\left[i\right]}_t\right)}{1000*\eta }$ $i\le {\mathrm{pred}}_{{\mathrm{start}}_{\mathrm{index}}}.fwdarw.\mathrm{Current}Q=Q\left[i\right]$ $i>{\mathrm{pred}}_{{\mathrm{start}}_{\mathrm{index}}}.fwdarw.\mathrm{Current}Q=\stackrel{\_}{Q}$ wherein M[i]p is the pressure drop, M[i+1].sub.t−M[i].sub.t is the time interval between measure data points M[i+1] and M[i], η is the fan efficiency and CurrentQ is the estimated air flow (Q[i]) of the measured data point (M[i]) or the average of at least some of the estimated air flows (Q[i]).

19. The method as claimed in claim 1, wherein the optimal life cycle is determined by minimizing the formula
Tot.sub.annual_cost[i]=Energy.sub.annual_cost[i]+Filter.sub.annual_cost[i] with respect to the time (M[i].sub.t).

20. A system for monitoring a filter in a ventilation system, comprising: at least one pressure sensor configured to measure pressure change in the filter of the ventilation system; and a processing device configured to: receive a filter hardware value, representing an amount of a resource associated with at least production of the filter, receive a filter use value, representing an amount of said resource associated with use of the filter, determine a pressure drop over the filter during use of the ventilation system based on the measured pressure change from the at least one pressure sensor, receive at least one filter hardware value (C.sub.filter), representing an amount of a resource associated with at least production of the filter, receive at least one filter use value (C.sub.use), representing an amount or rate of said resource associated with use of the filter, receive a plurality of measured data points (M[i]), each representing a measured pressure drop (M[i].sub.p) over the filter at a respective time (M[i].sub.t), and determine the optimal filter life cycle by minimizing a total filter resource consumption comprising: a first factor (Filter.sub.annual_cost[i]), according to which the resource consumption is inversely proportional to the filter life cycle, a second factor (Energy.sub.annual_cost[I]), according to which the resource consumption is directly proportional to the filter life cycle, provide a plurality of predicted data points, each representing an predicted pressure drop over the filter at a respective future point in time, group successive measured data points (M[i]) into windows (W), each window (W) comprising at least one of the measured data points (M[i]), for each window (W) identify a maximum pressure drop, and assign the identified maximum pressure drop as a maximum pressure drop (max_values[i]) for all measured data points (M[i]) of that window; for each measured data point (M[i]) estimate an air flow (Q[i]) based on the maximum pressure drop (max_values[i]) and the measured pressure drop (M[i].sub.p) of the respective measured data point (M[i]), and derive the second factor (Energy.sub.annual_cost[i]) based on the filter use value (C.sub.use), the measured data points (M[i]), the predicted data points and the estimated air flows (Q[i]).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 schematically illustrates a ventilation system 1, to which the present disclosure is applicable.

(2) FIG. 2 is a plot of characteristic filter curves for a clean filter and the scaled characteristic curves for filters used to different extents.

(3) FIG. 3 is a plot of a scaled filter curve illustrating air flow Q(p) as a function of pressure drop over the filter.

(4) FIG. 4 is a plot of a scaled characteristic filter curve illustrating air flow Q(p) as a function of pressure drop over the filter.

(5) FIG. 5 illustrates measured pressure drops over time of a filter.

DETAILED DESCRIPTION

(6) FIG. 1 schematically illustrates a ventilation system 1, which can be used to provide air to/from the rooms of e.g. a building. The system 1 comprises ventilation ducts 20, 22, a fan 21 for driving air through the ventilation ducts and a filter module 10, adapted for receiving a replaceable filter cartridge 11. In the filter module 10, there is provided a measurement device 12a, 12b for measuring a pressure drop over the filter. For example the measurement device may comprise first and second pressure sensors 12a, 12b. The measurement device may be connected to a controller 30, which may be adapted for receiving measurement data from the pressure sensors 12a, 12b.

(7) The controller 30 may be arranged to receive a respective pressure value from the sensors 12a, 12b and to calculate the pressure drop. As an alternative, the sensors may be arranged to directly measure the pressure drop, and thus to provide a single value to the controller 30.

(8) The controller 30 may be arranged to read values from the sensors 12a, 12b continuously or at predetermined intervals and to store the received data in a memory. As an alternative, the controller 30 may be arranged to read values from the sensors only when being polled.

(9) The controller 30 may be arranged to communicate with a remote unit 31, which may be a computer, a mobile terminal, etc.

(10) In one embodiment, the controller 30 may be provided in the form of a dedicated unit having a sensor interface and a communications device, which may be arranged to communicate via e.g. a text messaging service (“SMS”—Short Message Service) or an e-mail through communication protocols, such as 2G, 3G, 4G, 5G, Bluetooth®, Wi-Fi, Zigbee, WLAN, etc. The unit may be arranged to send sensor data at predetermined intervals or only when polled. For example, the unit may be arranged to automatically reply to an incoming text message by sending the current sensor data. The unit thus need not have any memory at all, but merely the necessary interfaces and an a/d converter to convert the received sensor values into data for further processing and/or communication. In this embodiment, all data storage and processing may take place in the remote unit 31, possibly with a backup function being provided.

(11) In another embodiment, a software may be provided for performing the methods disclosed herein, which is either in the form of a program stored on the controller, or in a computer which is communicatively connected with the controller, and which is accessible by the remote unit, e.g. via a web browser.

(12) As another option, the software may be provided in the form of a downloadable application software (known as an “app”), which is downloaded to the remote unit. In one embodiment, the software is an application software which is configured to run on a mobile terminal, such as an iPhone®, or tablet-type PC, such as an iPad®, which is communicatively connected with the controller and which has a backup function provided either through a docking function with a host computer or through a cloud-based service (such as iCloud®).

(13) In the following description, the following references will be used.

(14) M—a list of measured pressure drop over time (t, p), sorted by time value.

(15) P.sub.max—max pressure drop

(16) Price.sub.kWh—Cost/kWh

(17) Pricefilter—Total cost for a setup of filter including labour cost for the switch

(18) Q.sub.max—Max flow that the HVAC unit runs at

(19) η—Fan efficiency

(20) M is grouped into timeframe windows of size MaxWindowTime. This is done to determine a running cycle. MaxWindowTime is set to 24 h but could be changed to better suit current filter installation. Here is one example of measured data points M[i] and windows W.

(21) $\begin{array}{cc}M=\left[\left[1,49.1\right],\left[2,50.3\right],\left[3,48.7\right],\mathrm{.Math.}\left[998,99.1\right],\left[999,98.2\right],\left[1000,101.5\right]\right]& \left(\mathrm{equation}1\right)\\ W=\left[\left[\left[1,49.1\right],\left[2,50.3\right],\left[3,48.7\right]\right],\mathrm{.Math.}\left[\left[998,99.1\right],\left[999,98.2\right],\left[1000,101.5\right]\right]\right]& \left(\mathrm{equation}2\right)\\ W\left[a\right]\left[b\right]=\begin{array}{c}M\left[\left(\underset{k=1}{\overset{a}{.Math.}}.Math.W\left[k-1\right].Math.\right)+b\right],a>1\\ W\left[1\right]\left[b\right]=M\left[b\right],a=1\end{array}\}1<a<.Math.W.Math.& \left(\mathrm{equation}3\right)\end{array}$

(22) For each M[i], a max value max_values[i] is determined, which is the value considered as max for that time. As described above, the measured values are grouped into windows. For each window, an analysis is done whether the window has a value which should be considered as a new max pressure drop. This is done by iterating the values of the window and comparing the values to the previous max. Another criteria that has to be met is that the max is not much higher than the mean pressure of current window. The mean value is only calculated for values in the window above the median value of the window. These criteria exist to avoid spikes and noise.

(23) $\begin{array}{cc}{W\left[a\right]}_{\max }=\max \left(\left\{w\left[a\right]\left[b\right].Math.w\left[a\right]\left[b\right]\ge \stackrel{\_}{\left\{w\left[a\right]\left[b\right].Math.w\left[a\right]\left[b\right]\ge \right\}}\right\}*\mathrm{AverageScale}\right),0<b<.Math.w\left[a\right].Math.& \left(\mathrm{equation}4\right)\end{array}$

(24) AverageScale is set to 130%, but could be set to different values to either allow more or less difference in the max value.

(25) $\begin{array}{cc}\begin{array}{c}\mathrm{max\_values}\left[i\right]={W\left[a\right]}_{\max },i=1,{W\left[a\right]}_{\max }>\mathrm{max\_values}\left[i-1\right]*\mathrm{PreMaxScale}\\ \mathrm{max\_values}\left[i\right]=\mathrm{max\_values}\left[i-1\right],{W\left[a\right]}_{\max }\le \mathrm{max\_values}\left[i-1\right]\end{array}\}\begin{array}{c}\left(\underset{k=1}{\overset{a}{.Math.}}.Math.W\left[k-1\right].Math.\right)<i<\left(\underset{k=1}{\overset{a}{.Math.}}.Math.W\left[k-1\right].Math.\right)+.Math.w\left[a\right].Math.,a>1\\ 1<i<.Math.w\left[1\right].Math.,a=1\end{array}\}1<a<.Math.W.Math.& \left(\mathrm{equation}5\right)\end{array}$

(26) PreMaxScale is set to 60%, but could be set to different values to either allow more or less difference in the max value. The reason why the old max is scaled at all is to allow the max to decrease, since this is a common behavior for certain states of filter installation.

(27) The identified max_values[i] are used for an exponential regression to predict future pressure drops. Only the most recent max_values are used instead of all the identified max_values. This is done to give more weight to most recent data, since it tells the most of how the pressure drops will develop. A dynamic number of max values can be used, but a good amount is about 14, which if the pressure drop keep rising in every window will correspond to 14 days.

(28) The formula p(t)=a*e.sup.b*t obtained by the exponential regression is used to calculate future pressure drop, added to the parameter M, with steps of t=24 h. The provided parameter P.sub.max is used as a higher limit of calculated pressure drop. If the calculated value exceeds P.sub.max, the value is set to P.sub.max. This is done to avoid exorbitant pressure values.

(29) It is common for filter manufacturers to measure the pressure drop at certain flow levels for a clean filter. This could be considered as the filters characteristic for pressure drop development.

(30) The characteristic function is given by multiple linear regressions on a set of values for a clean filter. It could also be done using a polynomial, but to ensure 100% fit to the data, this method is chosen.

(31) The flow values are given as a percentage of Q.sub.nominal (nominal flow for the filter, e.g. 3400 m3/h).

(32) $\begin{array}{cc}Q=\left[\begin{array}{c}{q}_1\\ \mathrm{.Math.}\\ q_n\end{array}\right]& \left(\mathrm{equation}6\right)\\ P=\left[\begin{array}{c}{p}_1\\ \mathrm{.Math.}\\ p_n\end{array}\right]& \left(\mathrm{equation}7\right)\\ P\left(q\right)=\{\begin{array}{c}{k}_j*q+m_j,Q_j<q<Q_{j+1}\\ k_n*q+m_n,q>Q_n\end{array}& \left(\mathrm{equation}8\right)\\ n=\mathrm{length}\mathrm{of}Q=\mathrm{length}\mathrm{of}P& \\ k_j=\frac{P_{j+1}-P_j}{Q_{j+1}-Q_j}\}1<j<n-1& \left(\mathrm{equation}9\right)\\ k_j=\frac{P_n-P_{n-1}}{Q_n-Q_{n-1}}\}j=n& \left(\mathrm{equation}10\right)\\ m_j=k*Q_j-P_j& \left(\mathrm{equation}11\right)\end{array}$

(33) FIG. 2 is a plot of characteristic filter curves for a clean filter and the scaled characteristic curves for filters used to different extents.

(34) The values for the characteristic function is given for a clean filter, but the pressure values is then scaled so that the 100% flow value gives the value of the current value considered as max pressure drop.

(35) $\begin{array}{cc}\mathrm{scale}=\frac{P\left(100\right)}{\mathrm{max\_values}\left[i\right]}& \left(\mathrm{equation}12\right)\\ P_{\mathrm{scaled}}=\mathrm{scale}*P& \left(\mathrm{equation}13\right)\end{array}$

(36) To determine the flow Q[i] of M[i], the measured pressure drop, M[i]p and max_values[i] is used together with the characteristic values.

(37) To determine the flow Q[i], instead of looking at the pressure as a function of flow P(Q), the relationship of P and Q is inverted to look at flow as a function of pressure Q(p). It can be written as Q(M[i]p).

(38) $\begin{array}{cc}Q\left[i\right]=\{\begin{array}{c}{k}_j*{M\left[i\right]}_p+m_j,P_{{\mathrm{scaled}}_j}<{M\left[i\right]}_p<P_{{\mathrm{scaled}}_{j+1}}\\ k_n*{M\left[i\right]}_p+m_n,{M\left[i\right]}_p>P_{{\mathrm{scaled}}_n}\end{array}& \left(\mathrm{equation}14\right)\\ n=\mathrm{length}\mathrm{of}Q=\mathrm{length}\mathrm{of}{P}_{\mathrm{scaled}}& \\ k_j=\frac{Q_{j+1}-Q_j}{P_{{\mathrm{scaled}}_{j+1}}-P_{{\mathrm{scaled}}_j}}\}1<j<n-1& \left(\mathrm{equation}15\right)\\ k_j=\frac{Q_n-Q_{n-1}}{P_{{\mathrm{scaled}}_n}-P_{{\mathrm{scaled}}_{n-1}}}\}j=n& \left(\mathrm{equation}16\right)\\ m_j=k*P_{{\mathrm{scaled}}_i}-Q_i& \left(\mathrm{equation}17\right)\end{array}$
Whole Procedure as One Expression

(39) $\begin{array}{cc}Q\left[i\right]=\{\begin{array}{c}\frac{Q_{j+1}-Q_j}{P_{{\mathrm{scaled}}_{j+1}}-P_{{\mathrm{scaled}}_j}}*{M\left[i\right]}_p+Q_j-\frac{Q_{j+1}-Q_j}{P_{{\mathrm{scaled}}_{j+1}}-P_{{\mathrm{scaled}}_j}}*P_{{\mathrm{scaled}}_j},P_{{\mathrm{scaled}}_j}<{M\left[i\right]}_p<P_{{\mathrm{scaled}}_{j+1}}\\ \frac{Q_n-Q_{n-1}}{P_{{\mathrm{scaled}}_n}-P_{{\mathrm{scaled}}_{n-1}}}*{M\left[i\right]}_p+Q_n-\frac{Q_n-Q_{n-1}}{P_{{\mathrm{scaled}}_n}-P_{{\mathrm{scaled}}_{n-1}}}*P_{{\mathrm{scaled}}_n},{M\left[i\right]}_p>P_{{\mathrm{scaled}}_n}\end{array}& \left(\mathrm{equation}18\right)\end{array}$

(40) Table 1 below provides an example of pressure drops and airflows, wherein the airflows are expressed in % of nominal airflow

(41) TABLE-US-00001 TABLE 1 Example data for characteristic function Pressure drop (Pa) Q (% of nominal) 1 1 20 25 40 50 65 75 90 100 125 125

(42) FIG. 3 is a plot of a scaled filter curve illustrating air flow Q(p) as a function of pressure drop over the filter.

(43) With a clean filter, 90 Pa is supposed to be the pressure drop at 100% flow. When the filter is running it is stuffed with catched particles, which makes the pressure drop rise. The pressure drop considered as 100% is changing and the other values for the characteristic function is scaled to fit it.

(44) Nominal flow: 3400 m3/h

(45) New max pressure drop (max_values[i]): 143 Pa

(46) Pressure drop to calculate flow for (M[i]p): 66 Pa

(47) Scale factor: 143/90=1,588889

(48) The scaled values of Table 1 are illustrated in Table 2 below.

(49) TABLE-US-00002 TABLE 2 Scaled values for the characteristic function Pressure drop (Pa) Q (% of max flow) j 1,588889 0 1 31,77778 25 2 63,55556 50 3 103,2778 75 4 143 100 5 198,6111 125 6

(50) FIG. 4 is a plot of a scaled characteristic filter curve illustrating air flow Q(p) as a function of pressure drop over the filter.

(51) 0$\begin{array}{cc}63.55556<66<103.2778.fwdarw.j=3& \left(\mathrm{equation}19\right)\\ k=\frac{75-50}{103.2778-63.55556}\approx 0.6294& \left(\mathrm{equation}20\right)\\ m=50-0.6294*63.55556\approx 10& \left(\mathrm{equation}21\right)\\ Q\left(66\right)=0.6294*66+10\approx 51.5& \left(\mathrm{equation}22\right)\end{array}$

(52) The scaled Q(p) gives the flow 51.5% of nominal flow, which if the nominal flow is 3400 m3/h is calculated as 1734 m3/h.

(53) Each measured pressure drop value and 6 months forward of predicted pressure drop values are used to calculate a one year normalized LCC for the switch interval until that point. For predicted pressure values, a mean value of Q is used.

(54) $\begin{array}{cc}E\left[i\right]=\frac{\mathrm{Current}Q*{M\left[i\right]}_p*\left({M\left[i+1\right]}_t-{M\left[i\right]}_t\right)}{1000*\eta }i\le {\mathrm{pred}}_{{\mathrm{start}}_{\mathrm{index}}}.fwdarw.\mathrm{Current}Q=Q\left[i\right]i>{\mathrm{pred}}_{{\mathrm{start}}_{\mathrm{index}}}.fwdarw.\mathrm{Current}Q=\stackrel{\_}{Q}& \left(\mathrm{equation}23\right)\end{array}$

(55) Equation 23 provides the energy cost for the time period from M[i].sub.t to M[i+1].sub.t

(56) $\begin{array}{cc}{\mathrm{Energy}}_{\mathrm{annual}\_\mathrm{cost}}\left[i\right]=\underset{k=0}{\overset{i}{.Math.}}E\left[k\right]*\frac{\left(8760*3600\right)}{{M\left[i\right]}_t}*{\mathrm{Price}}_{\mathrm{kWh}}& \left(\mathrm{equation}24\right)\end{array}$

(57) Equation 24 provides the total annual energy consumption for switch interval M[i].sub.t

(58) $\begin{array}{cc}{\mathrm{Filter}}_{\mathrm{annual}\_\mathrm{cost}}\left[i\right]={\mathrm{Price}}_{\mathrm{filter}}*\frac{\left(8760*3600\right)}{{M\left[i\right]}_t}& \left(\mathrm{equation}25\right)\end{array}$

(59) Equation 25 provides the total annual cost for filters for switch interval M[i].sub.t
Tot.sub.annual_cost[i]=Energy.sub.annual_cost[i]+Filter.sub.annual_cost[i]   (equation 26)

(60) Equation 26 provides the total annual cost for the installation for switch interval M[i].sub.t

(61) The Tot.sub.annual_cost is calculated with equation 26 and then the minimum cost Tot.sub.annual_cost[index_of_min] is identified, and from that the optimal switch interval M[index_of_min].sub.t is determined.

(62) It is recognized that in the calculations above, the fan efficiency is still set as a fixed value, but in the real world it varies due to different flows. To improve the accuracy of the algorithm this could either be calculated for each measure point by adding a specification sheet of the equipment in use or by adding a wattmeter to the fan.