THE VALUE OF VENTILATION FROM THE WEBER-FECHNER LAW

Size: px

Start display at page:

Download "THE VALUE OF VENTILATION FROM THE WEBER-FECHNER LAW"

Ethelbert George
6 years ago
Views:

1 Topic B12: Productivity and economics THE VALUE OF VENTILATION FROM THE WEBER-FECHNER LAW Arne JÖNSSON * Secondary Vocational Training, Härnösand, Sweden * Corresponding arne.jonsson@harnosand.se Keywords: Weber-Fechner, value, ventilation, economics SUMMARY If the value of ventilation or the relation between outdoor air rates and loss of production is known then the air rate can be optimized. If the loss of production can be related directly to different odors like body, building and cigarette odor then the removal of odor sources can be optimized. There are two methods to find the value of ventilation. One method is studies where the work rate is measured in an experiment at determined outdoor air rates and source strength of pollution. The other method is to calculate the marginal cost of outdoor air and assume that the marginal cost equals the marginal loss. The assumption of optimality is based on that the long time use of ventilation has taught us what the optimal outdoor air rate is. The marginal cost method, with costs from a Swedish office with constant 100 % outdoor air ventilation, gives the loss of production at 60 % dissatisfied. This was measured in a work rate study from Sweden. The value of ventilation in offices determined with indirect or marginal cost methods and direct measurements in work rate studies are showing the same result. INTRODUCTION The function for percentage dissatisfied in body odor gives the percentage at an outdoor air rate. With the odor threshold c*1 = 50 ppm CO 2 for the most sensible person (1.3 %) and a generation rate of 16 l/h pers CO 2 then a relation between percentage dissatisfied PD and the number of dilutions x1 to the odor threshold for body odor can be calculated. The relation is assumed to be valid for building odor and cigarette smoke odor too. So if body odor, building odor and cigarette odor separately give the same loss of production then they have the same x1. According to Fechner (1860) the strength of the feeling is proportional to the logarithm of the relation between the concentration and the odor threshold. The relation between the concentration and the odor threshold is the number of dilutions to the odor threshold. The constant of proportionality a can be determined in both dimensions, against the measured loss and against the marginal cost. 1 = 9 SEK.

2 METHODOLOGIES The loss of production for a collective The individuals in a collective have a distribution of odor thresholds. Here the distribution of dissatisfied PD(i) is used instead. For an odor that has normally distributed odor thresholds, lognormal N(ln(x1), 4, 1.8) the function in equation 1 from Fanger (2004) approximates the normal distribution. Frequency distribution of dissatisfied pd(i) equation 2 is the differential of PD(i) with regard to i. The individuals in interval 1 between 50*(1 ± 0.5) ppm are 0.02 of the population according to equation 2 and their thresholds for body odor are approximated to 50 ppm. In interval 2 between 50*(2 ± 0.5) ppm there are also 0.02 of the population and their threshold is approximated to 100 ppm CO 2 above outdoors. The demand and the loss over the intervals are added in equation 3 and equation 4. 1 < i < x 1. (1) (2) The Dem(x1) equation 3 is the function for demand of improvement for a collective. The Loss(x1) equation 4 is the function for loss of production for the collective in an air with the numbers of dilutions for the most sensible x1. The functions are calculated numerically and the result is given in Table 1. (3) (4) Addition of odors The x1 for a mixture of odors is equation 5. q* b outdoor air rate necessary to dilute body odor, from one person to the threshold for the most sensible person and q* bu outdoor air rate necessary to dilute the odor from the building surrounding one person to the threshold for the most sensible person, l/s pers. q is the outdoor air rate l/s pers. This corresponds to the calculation of pol according to Fanger (2004) where the olf is the q*. (5)

3 Table 1. Distribution of dissatesfied, diss at x1 = c/c*1, c*1 = 50 ppm CO 2 above outdoor air. Demand of improvement, Dem and Loss of production, Loss at x1. Outdoor air rate q l/s pers to make diss % dissatiesfied with body odor from one person Jönssson (2013). Optimal outdoor air rate x1 diss Dem Loss q x1 diss Dem Loss q % l/s pers % l/s pers The outdoor air rate that minimizes the sum of cost for outdoor air and loss equation 6 is the optimal outdoor air rate. It has an optimal dilution x1 from equation 5. l is the fixed cost and m is the marginal cost of outdoor air at the design of the building, SEK/ l/s yr. n is the number of persons present in office. p wh is the price of a work hour, SEK/h pers and τ is the time of operation of the office, h/yr. Equation 5 is inserted in equation 6. Equation 7 is differentiated and put equal to zero in equation 8. The optimal x1 can be solved from equation 8. (6) (7) (8) Cost of ventilation in an office The cost of ventilation SEK/yr is the sum of cost for the investment and the cost of operation in a Swedish office with constant, 100 % outdoor air ventilation during the time of operation 2000 h/yr. A linear relation between concentration and loss gave the loss at 10 l/s person Jönsson (1995). Investments An air heater for 1.25 m 3 /s including heat capacity from district heating costs phc = 2000 SEK/kW and if the design outdoor temperature is -20 C then the investment is SEK.

4 The cooling capacity for air conditioning depends on the outdoor air rate. The cooling capacity costs pcc = SEK/kW including the cooling coil. With an indoor design temperature of 27 C and a latent/ sensible relation of 1.5 the investment for 1.25 m 3 /s is SEK. The investments are added in Table 2. Half of the investment from Table 2 depends on the air rate at the design of the system. The annuity is with 4 % interest and 20 years service life. Marginal cost of outdoor air Table 2. Investments in the ventilation system, ksek. Office Investment Heat recovery No heat rec ksek ksek Heater Cooling Heat recovery Rest Ducts, inletts Sum The ventilation system with heat recovery has an investment that depends on the air rate of SEK / 1250 l/s = 460 SEK/ l/s. The annuity gives a marginal cost of 34 SEK/ l/s yr. To include the cost of the space in the building 50 % is added. The no heat recovery system has an investment per l/s of SEK/ 1250 l/s = 350 SEK/ l/s yr. The annuity gives the marginal cost 26 SEK/ l/s yr plus 50 % for space. The cost for the investment and the cost of operation are added in Table 3. Cost of operation for heat in a Swedish climate with degree hours/yr, heat price ph = 0,5 SEK/kWh and time of operation during work hours τ = 2000 h/yr gives the marginal cost of operation 16 SEK / l/s yr. Heat recovery with 75 % efficiency η = 75 % reduces the marginal cost of operation to 16 * (1-0.75) = 4 SEK/ l/s yr. Cost of electricity for two fans at the design pressure drop of 700 Pa. The price of electricity is pe = 1 SEK/kWh. The fan and electric motor efficiency is η = 0.6 which gives the marginal cost of operation to 5 SEK/ l/s yr. Table 3. Marginal cost of outdoor air rate SEK/ l/s yr. Marg. cost Heat recover No heat rec SEK/ l/s yr SEK/ l/s yr Operation Heat 4 16 Electricity 5 4 Investment Sum 60 59

5 The marginal cost of outdoor air or the cost to increase the outdoor air rate 1 l/s at the design of the building will be 60 SEK/ l/s yr with heat recovery. Determination of a If the used air rate is optimal, then the marginal cost equals the marginal loss and a can be solved from equation 8. If the optimal outdoor air rate is 12 l/s pers with q*b = 90, q*bu = 90 l/s pers, then x1 = 180/12 = 15, τ = 2000 h/yr gives Dem(15) = 0.1 from Table 1. m/pwh = 60 SEK/ l/s yr / 250 SEK/h = 0.24 h/ l/s yr in equation 8 gives a = If the optimum is 15 l/s pers then a = The air rates l/s pers for use during 2000 h/yr are high compared to the ASHRAE recommendations 8 l/s pers but the recommended air rate is used during a shorter period. Calculated loss The loss in decimals Lossd(x1) is calculated as function of x1 with equation 9. The curve in Figure 1 shows the Lossd(x1) as function of the PD. The relation between x1 and PD from Table 1. The PD is what can be determined about mixtures of odors. Loss from work rate studies (9) The calculated loss is compared with the measured loss from two work rate studies that used the same carpet as pollutant. Wargocki et al (2000), W and Lagercrantz et al (2000), L used an ordinary office with furniture s as test room. A carpet with a history of sick building syndromes was used as pollutant in both tests. Results from W in Table 4. The percentage dissatisfied, PD at first entrance with carpet is for building, bu and carpet, ca odor. The PD for body, b building, bu and carpet, ca is at reentrance. The study made in Copenhagen had 30 test persons in 5 groups Table 4. Percentage dissatisfied PD, production and loss at different outdoor air rates, W. Outdoor Dissatisfied Speed, norm. b+bu+ca air rate bu+ca, first b+bu+ca, ree Typing Add Pro read l/s pers % % : loss : loss : loss : : : : : : The PD was the same at both 10 and 30 l/s pers this gives the same number of dilutions x1 at 10 and at 30 l/s witch is unlikely.

6 The test room in L was built in an office space from 1984 with ordinary furniture s. The floor was linoleum and the walls of gypsum board. The test room was in Östersund, Sweden. The Result from the tests is in Table 5. The study had 30 test persons. Table 5. Percentage dissatisfied, production and loss at 10 l/s pers outdoor air rate and mixtures of different pollutants, L. Air rate Dissatisfied Productivity 10 l/s pers first ree Typing Add Pollutant Speed correct bu 36 : loss : loss b+bu bu+ca 60 b+bu+ca : :0.012 L got a higher percentage dissatisfied but a lower loss of production than W. b+bu+ca with 10 l/s pers in L and b+bu+ca with 3/s pers in W both gives 60 % dissatisfied. 60 % dissatisfied gave loss in W but only loss in L. "As all surface materials are 15 years old it was assumed that the primary emission of pollutants in the office was at a very low level." The office chamber may explain the high percentage dissatisfied without carpet in Table 5. Comparison between the calculated loss and loss from work rate studies The measured loss, dots from Wargocki et al. (2000) W and Lagercrantz et al. (2000) L as function of percentage dissatisfied PD % and the calculated Lossd curve if optimum is 12 l/s pers a = If the optimum is 15 l/s pers then a = Figure 1. Calculated Lossd (loss in decimals) as function of the percentage dissatisfied, PD %, curve. The measured loss from work rate studies, dots.

7 The calculated Lossd is at 60 % dissatisfied. The measured loss at 62 % dissatisfied is in L and the loss at 58 % dissatisfied is in W. Jönsson (1995) used a linear relation between concentration and loss. He calculated the loss at the optimal outdoor air rate 10 l/s pers to The calculated Lossd in Figure 1 shows that it is not profitable to reduce the PD to zero, because the Lossd goes to zero at PD = 0. The calculated Lossd at 24 % and 33 % dissatisfied is resp The Lossd = has the value of 4 work hours per pers and year. (1000 SEK/yr) If pollutants can be removed so that 20 % is dissatisfied then the Lossd is reduced to Then 3 more work hours per pers and year will be available and = or 750 SEK/pers yr will be gained. The loss curve is convex since both the loss per person and the percentage dissatisfied are reduced at a low concentration. The convex curve for the loss per person shows that the marginal loss increases at higher x1. This is realistic since at x1 = 600 or 3 % CO2 a person will faint. There are contradictions in the measured percentage dissatisfied. L measured a higher percentage dissatisfied but measured a lower loss of production than W. RESULTS AND DISCUSSION If the used outdoor air rates is optimal then the marginal cost of the outdoor air rate is put equal to the marginal loss at normally used outdoor air rates l/s pers then a constant can be solved. The constant makes it possible to calculate the loss of production with the integrated Weber-Fechner Law. The marginal cost from a Swedish office with constant, 100 % outdoor air ventilation, during an operation of 2000 h/yr gives the calculated loss of production at 60 % dissatisfied. The loss was measured in a work rate study from Sweden, Lagercrantz et al. (2000). The assumption of optimality is based on that the long time use of ventilation and the long time research about ventilation has taught us what the optimal outdoor air rate is. The measured losses have a higher variation at 60 % dissatisfied than the variation in the calculated loss at 60 % dissatisfied from different assumptions about the optimal outdoor air rate 12 or 15 l/s pers. There are contradictions in the measurements of the percentage dissatisfied. CONCLUSIONS The value of ventilation in offices determined with indirect or marginal cost methods and direct measurements in work rate studies are showing the same result. The calculated loss of production at 60 % dissatisfied and the measured loss of production at 60 % dissatisfied are the same. There is however other studies with the same pollutant showing up to loss at 60 % dissatisfied.

8 The value of ventilation in offices determined with indirect or marginal cost methods and direct measurements in work rate studies are showing the same result. REFERENCES Fanger PO (2004) Perceived air quality and ventilation requirements, Ch. 22 Indoor air quality handbook, McGraw-Hill, (2004) ( Fechner GT (1860) Elemente der Psychophysik, zweiter teil, Breitkopf und Härtel, Leipzig 1860, p Google Books (German) Jönsson A (1995) Economic analysis of ventilation in an Office Building, Indoor air quality in practice, Oslo, Norway, p Jönsson A (2012) Odour nuisance and the loss of work hours according to the Weber-Fechners Law, Ventilation 2012, Paris, France Jönsson A (2013) Improvement of the Integrated Weber-Fechner Law for Odors, Clima 2013, Prague, Czech Republic Lagercrantz L, Wistrand M, Willen U, Wargocki P, Witterseh T, Sundell J (2000) Negative impact of air pollution on productivity: Previous Danish findings repeated in new Swedish test room, Healthy Buildings 2000, Vol. 1 p , Esbo, Finland Wargocki P, Wyon DP, Fanger PO (2000) Productivity is affected by the air quality in offices, Healthy Buildings 2000, Vol. 1 p , Esbo, Finland

Improvement of the Integrated Weber-Fechner Law for Odors

Improvement of the Integrated Weber-Fechner Law for Odors Arne Jönsson Secondary Vocational Training Härnösand, Sweden arne.jonsson@harnosand.se Abstract The Weber-Fechner law gives the odor nuisance as