J. Chem. T ech. Biotechnol. 1997, 68, 377È380

Economic Design of Biofilter Systems*

Andrew Mark Gerrard

Division of Chemical Engineering, University of Teesside, Middlesbrough, Cleveland TS1 3BA, UK

(Received 12 April 1996 ; accepted 2 September 1996)

Abstract : This paper o†ers an easy-to-use design model for the minimum netpresent cost of a bioÐlter system, allowing for compaction of the bed. It com-pares the two-variable graphical optimisation approach with short-cut methodshaving analytical solutions.

Key words : bioÐlter, cost, optimisation, design

NOTATION

a Constant in Blasius equationa1, a2 , a3 Constants in cost correlationsA Bed areab Constant in Blasius equationb1, b2 , c1 Constants in cost equationc1Èc8 Constants deÐned in equationsCe Cost of electricityd Duct diametere Voidagef Friction factorF Pressure drop increase factor, eqn 8h BioÐlter heightk Constant in pressure drop, eqn 8L Length of ductNPW Net present worthp, q Constants in cost correlation, eqn 10Q Volumetric Ñow rateRe Reynolds numberr Continuous discount rateS SpeciÐc surfacet TimeT Operating timeu Duct velocityv Bed velocityy Present value of total costs

*p Pressure dropj Factor for installation costsk Gas viscosity

* Presented in part at Clean Tech Ï96, London, 19È21 June1996.

o Densityq Residence time

1. INTRODUCTION

The removal of volatile organic compounds (VOC)from gas streams is often a legal requirement. Thisrequirement can be met by using clean-up processessuch as scrubbing, adsorption, incineration and bio-logical reaction. Included in the last category are bio-Ðlter systems. These are particularly appropriate fortreating low inlet concentrations of substances whichbreak down relatively easily.1 Information about thebiotechnology of these units and their place in anoverall odour reduction policy can be found in papersby Bohn2 and Siegell.3

BioÐltration involves passing the VOC-laden airstream through a packed bed, whose resident micro-organisms digest the pollutants. Such a bed must bekept moist in order to ensure the well-being of thebiomass. Typically, then, the Ðlter is sprayed with wateror the inlet gases are humidiÐed. Over time, a moist bedconstructed from a natural material like heather cancollapse under its own weight. This can lead toincreased pressure drop, etc. To accommodate compac-tion in the design, a simple Ðrst order model4 was Ðttedearlier to data from Dragt and Ottergraf.5

The purpose of this work was to give the processengineer a simple equation which will adequately rep-resent the system and allow an economic optimisationto be attempted. To this end, the complexities of thedetailed rector design are bypassed. Therefore, it wasnecessary to determine at the outset a knowledge of thedesired residence time, q, for the needed VOC removal

377J. Chem. T ech. Biotechnol. 0268-2575/97/$17.50 1997 SCI. Printed in Great Britain(

378 A. M. Gerrard

rate. The residence time can be found from pilot plantor fundamental studies. An example of such a study isthe work reported by Ottengraf and Diks6 where theMonod rate equation was found to Ðt the data, and,indeed, the equation often reduced to zero order.

2 THE MODEL

Figure 1 illustrates the bioÐlter with its associatedducting and fan. The diagram shows the nine variableswhose values need to be calculated : bed height, bedarea, pressure drop in the bed, bed velocity, duct diam-eter, duct velocity, friction factor, Reynolds number,and pressure drop in the duct. The factors which it isassumed are known are the gas Ñow rate, bed residencetime, physical properties and cost factors. Among thesenine variables, there are seven relationships : the deÐn-itions of average velocity, Reynolds number, bed veloc-ity, friction factor and area ; and the force balances overthe duct and bed. Accordingly, there are 9[ 7 \ 2design variables to select. By inspection, bed height andduct velocity were determined to be the independentvariables. These were selected in order to minimise thepresent value of the systemÏs total costs.

Thus the equations named earlier can be written interms of design variables, h and u.

d \ (4Q/nu)0Õ5\ c1u~0Õ5 (1)

Re\ (ouc1u~0Õ5)/k \ c2 u0Õ5 (2)

v\ h/q (3)

f \ a(Re)b \ a(c2 u0Õ5)b \ c3 u(0Õ5b) (4)

A\ Qq/h \ c4h~1 (5)

In the above, the residence time was used to calculatebed velocity, v, and area, A. The Blasius equation7 isutilised to calculate the friction factor, f, from the Rey-nolds number.

Fig. 1. BioÐlter, fan and ducting.

The pressure drop terms are deÐned thus :

*pduct\ 2f (L /d)ou2\ 2c3u0Õ5bL ou2/(c1u~0Õ5)

\ c5 u(2Õ5`0Õ5b) (6)

*pbed, original\ 5S2(1 [ e)2khv/(e3) (7)

Equation (7) uses the Kozeny equation, which must bemodiÐed to allow for the collapse of the bed materialswith time. As noted above, the following equation wasfound useful to represent such behaviour :4

*pbed(t) \ *pbed, original(F] (1 [ F)exp([kt)) (8)

Therefore

*pbed(t) \ (5S2(1[ e)2kh2/(qe3))(F] (1[ F)exp([kt))

\ c6(F[ (1 [ F)exp([kt))h2 (9)

The objective function must now be minimised.Capital costs include the Ðlter, ducting and fan, and theoperating costs are principally the fanÏs power cost. Byadding all the operating costs in time T and discountingthem back to todayÏs values, these can be added to thecapital costs and thus, the total can be minimised.

It is well-known that the cost of many items of plantcan be linked to their size by a power-law equation. Forexample, Garrett8 gives data linking fan costs tothrough-put and pressure drop, and ducting cost todiameter. It can also be assumed that the bioÐlter costsdepend on bed area and height raised to powers. There-fore :

y \ capital cost of fan, duct and bed

] NPW of total operating costs

\ j(a1Qb1*ptotC1] a2 L Db2 ] a3Aphq)

]P0

T*ptot QCe/exp(rt) dt (10)

where

*ptot\ *pbed(t) ] *pduct (11)

and r is the continuous discounting rate with j as theinstallation cost factor. Using eqns (6), (9) and (11) inthe integral above, we Ðnd

Integral \ (c5QCe(1 [ exp([rT )/r))u(2Õ5`0Õ5b)

]c6QCeM(F[ F exp([rT ))/r

] ((1[ F)[ (1 [ F)

] exp([(k ] r)T ))/(k ] r)Nh2 (12)

\ c7 u(2Õ5`0Õ5b)] c8 h2 (13)

BioÐlter design 379

Fig. 2. Cost optimisation : cost versus height and duct velocity.

The objective function can now be assembled in termsof h and u.

y \ ja1Qb1Mc5 u(2Õ5`0Õ5b)] c6 h2NC1] ja2 L c1b2u(~0Õ5b2)] ja3(c4 h~1)phq] c7 u(2Õ5`0Õ5b)] c8 h2 (14)

3 DISCUSSION

Equation 14 is now an explicit two-variable functionwhich is to be minimised. The Ðrst partial derivatives

can be easily formed but, because u and h both occur inthe Ðrst term, the two resultant equations are highlynon-linear, so a numerical solution is needed. A simpleralternative is to do a direct search on eqn (14). This ismost e†ectively done by graphical means where y isplotted against h with u as a parameter. Using thenominal data from Table 1, Fig. 2 was obtained. As canbe seen, the minimum cost arises with a duct velocity of14 m s~1 and a bed height of 1É9 m. Fortunately, eachcurve is fairly Ñat around the local minimum so the

380 A. M. Gerrard

TABLE 1Data for Example

Parameter V alue Unit

a 0É0792a1 340É0 £ size~1a2 216É1 £ size~1a3 108É4 £ m~2(p~q)b [0É25b1 0É71b2 1É08C1 0É3Ce 1É3889 ] 10~8 £ J~1e 0É8F 5k 1É876 ] 10~8 s~1L 50É0 mp 0É9q 0É3Q 6É11 m3 s~1r 0É1 year~1 (or 3É022 ] 10~9 s~1)S 16000 m~1T 10 year (or 3É1536 ] 108 s)j 2É0k 0É000018 kg m~1 s~1o 1É0 kg m~3q 60 s

solution is not too sensitive to changes in h. In addition,inspection of various curves for di†erent duct velocitiesshows that there is not much penalty in operating at18 m s~1 either.

For the example, the optimal bed height is notstrongly inÑuenced by the other variable. This wouldsuggest an alternative short-cut strategy. If duct and fancosts are ignored, and the net present worth of the Ðlterand power costs only minimised, then the objectivefunction becomes

y \ ja3(c4 h~1)phq] c8 h2 (15)

This can be easily di†erentiated to give

dy/dh \ ja3 c4p(q [ p)h(q~p~1)] 2c8 h (16)

Setting this equal to zero gives

hopt \ M[2c8/(ja3 c4p(q [ p))N(1@(q~p~2)) (17)

By substituting the appropriate numbers we get h \2É0 m, which is pleasingly close to the exact answer.Equation (17) is equivalent to that derived in Ref. 4.

Similarly, by focusing attention on the ducting alone,the following is obtained :

y \ ja2 L c1b2u(~0Õ5b2)] c7 u(2Õ5`0Õ5b) (18)

leading to

uopt\ M(ja2 L b2 c1b2)/(2c7(2É5 ] 0É5b))N(1@(2Õ5`0Õ5b`0Õ5b2)

(19)

This expression gives u \ 19É8 m s~1, which is a goodestimate of the value needed to start a more detailedstudy using the graphical approach explained above.

In summary, this paper describes a simple model of abioÐlter system which enables the designer to size thenecessary equipment so as to minimise total costs. Agraphical approach leads to the most economicarrangement. For a short cut to the approximate valuesof the design variables, two equations are presented thatget close to the global optimum.

REFERENCES

1. Martin, A. M., Nolen, S. L., Martin, A. M., Nolen, S. L.,Gess, P. S. & Baesen, T. A., Control odors from CPI facili-ties. Chemical Engineering Process, 88 (1992) 53È61.

2. Bohn, H., Consider bioÐltration for decontaminating gases.Chemical Engineering Progress, 88 (1992) 34È40

3. Siegell, J. H., Solve plant odor problems. Chemical Engin-eering Progress 92 (1996) 35È41

4. Gerrard, A. M., Economically optimal design of bioÐlters.Paper given to AlChemE Summer National meeting,session 40, Boston, USA, 1 August 1995.

5. Dragt, A. J. & Ottengraf, S. P. P., Fundamental aspects ofbioÐltration and some practical experiences. Paper given toconference on Advanced Technology and Biotechnologyfor Environmental Protection, Syracuse, Italy, 22 April1991.

6. Ottengraf, S. P. P. & Diks, R., Biological puriÐcation ofwaste gases. Chimicaoggi, May (1990) 41È5.

7. Coulson, J. M. & Richardson, J. F., Chemical Engineering,volumes 1 and 2, Pergamon, Oxford, 1990.

8. Garrett, D. E., Chemical Engineering Economics. Van Nos-trand Reinhold, New York, 1989.