SCALE UP OF COMBUSTION POT BEHAVIOR BY DIMENSIONAL ANALYSIS

SCALE UP OF COMBUSTION POT BEHAVIOR BY DIMENSIONAL ANALYSIS T. R. SATYANARAYANA RAO G. GELERNTER R. H. ESSENHIGH The Pennsylvania State University University Park, Pennsylvania ABSTRACT The purpose this paper is to establish the nature dimensionless groups characterizing combustion behavior in an incinerator. In particular these groups should determine the influence scale in transposing results from the combustion pot to the incinerator. The analysis indica!:s that if.f A is the burning rate per unit area grate and k is an effective reactivity coefficient, then the ratio (F A i'k) can be expressed as a function a minimum four n groups involving nine parameters. From inspection these groups it would seem that their values may well be the same in both systems for the same range dimensional parameters in the two systems. It follows from this that the scale factor involved should be unity, i.e., combustion and burning rate data obtained in a combustion pot should transpose to an incinerator bed without change. For other things then being equal, the area burning rate (F A ) can only be increased if k is increased. INTRODUCTION The combustion pot is a microcosm a representative section solid fuel bed. Its value is that, being much smaller, it is more controllable and more experiments can be performed in a shorter time. Its nominal disadvantage is that it is only representative a full size combustor or incinerator and any data obtained have in principle to be scaled up in some way. The purpose this paper is to apply the methods dimensional analysis to such a system to show that the scale factor should normally be about unity. The establishment such a significant point will mean that combustion pot studies are more directly related to full scale combustors than might appear at first sight, and such units can therefore be used to advantage to obtain relevant data on behavior incinerators in a shorter time and at lower cost than would otherwise be possible. ANALYSIS The Physical System The physical system is a bed burning refuse. The bed is porous, with tortuous channels through which air for the reaction is flowing. The burning rate is F A lb/hr sq ft. The reaction has an upper rate limited by the intrinsic reactivity the material which is assumed to have the same temperature dependence as the reaction rate constant. To analyze the system we assume the following equivalent model: that the bed can be represented by a sequence parallel pores in a solid block, that there are N pores per unit area the bed cross section, that the heat losses from the wall and the convection loss from the hot gases can be combined into a single heat transfer coefficient (h), that the statistical mean properties such as the density the material the bed a and the density 232

packing (Le., porosity ) are constant, that the reactivity behavior the material can be averaged by a constant overall rate reaction (k), and that the bed temperature is steady with steadystate assumed for the analysis. This is represen ted in Fig. 1. The temperature prile through the bed may be approximated by a stepfunction (a fairly common assumtion) implying that the temperature through the reacting part the bed is constant. We can therefore identify the following parameters: (1) where 01 h lvopcb, 02 = = cb6.tpovoiq, 03 = vplk, 04 = vivo, 05 = Plcb 6.T, 06 = Blcb 6.T, 07 = a, 08 = Po 2' 09 = CplCb' 010 = plpo' 011 = pia, 012 =, 013 = LId, 014 = NLd. Some the groups are connected by the following: 011 = pia = 1 = 1 012 SYMBOL DIMENSION PHYSICAL VARIABLE B L N P02 P Q T To 6.T v a v a p Btu/lb Btu/lb Btu/lb ft lbs/hr ft 2 Btu/hr ft 2 lbs/unit pore surface hr heat reaction specific heat bed material specific heat gases mean pore diameter area burning rate heat transfer coefficient intrinsic reactivity ft bed depth 1/ft 2 number pores per ft 2 non dimensional oxygen partial pressure ratio Btu/lb Btu/hr ft 2 ft 3 /hr ft 2 bed surface ft/sec lb/lb non dimensional heat pyrolysis heat loss mean bed temperature ambient temperature temperature difference (TTo) air supply rate mean pore flow velocity stoichiometric air for 1 lb fuel porosity bed density cold air density solid density Analysis The above parameters may be related by application either the Rayleigh Method dimensional analysis or the Buckingham Pi Theorem [1, 2, 3]. I t results in the following general, but rather complicated statement the relationship physical characteristics the system. So equation (1) reduces to the following: This functional equation merely means that the relation the area burning rate and intrinsic reactivity in the system is a function at least eleven ratios various physical characteristics which affect the system. To make use this requires some simplification. All modeling techniques must meet one fundamental requirement before they can be used as an aid to the prototype behavior. The requirement is that model results must be directly comparable or translatable to the prototype. The condition for this is which is the complete similarity criterion. For this to be satisfied all the groups should be the same for the model and prototype, but to determine the functional relationship with eleven groups would be virtually impossible. However, partial modeling may be satisfactory here because appropriate selection our experimental and operating conditions can either reduce some the groups to constants, or to secondary importance, in the process we are considering. In the first place we may use the same material in both the model and the prototype. Such factors as the stoichiometric ratio and the solid density are therefore constants. By correct packing the bed, the bed density and porosity can also be considered constants. Continuing in this way, equation (1) may be reduced to (2) (3) 233

since lis... lin are now considered constants or nearly so. Experimental results obtained on a combustion pot can be assumed to obey a power functional relationship the following form where K is a constant which contains all the scale effects. The power indices, nl' n2' n J ' n4 may be either integers or fractions and may be positive or negative. Such experiments therefore determine a particular value II corresponding to specified numerical values II I ' II2, II J, II4. Now by the theory modeling we can say that this will be (approximately) the same value II for the prototype if II I ' II2, II3, II4 have the same values. Inspection the II groups shows this condition can generally be satisfied in practice by the same real values the parameters in both prototype and the model. This means that the behavior in the combustion pot should map into the full scale bed directly without need for any scale factors or scale changes. Even if there are some scale changes due to the secondary effects and so on, these could be small and the scale effect factor (K) would be very near to unity. HEAT AND MASS BALANCE The result the above analysis can now be compared with that obtained from a heat and mass balance across the bounding surface enclosing the bed. The analysis assumes steadystate with a reacting region wherein active combustion is taking place. When the products leave the reaction zone the combustion is assumed incomplete because oxygen is completely used up, so the products carry away some gasified but only partially burnt fuel. It is assumed that the fuel pyrolyzes before igniting. We can establish an equation for the burning rate by considering what is essentially an oxygen balance in the burning zone. We have: Therefore Total mass input oxygen = po.vo P 02 Total weight fuel burnt F A P o v o P 02 laf where f = amount fuel completely burnt per pound fuel disappearing. To determine f, we now consider a heat balance. We have: Heat pyrolysis = F A.P Heat liberated = ff A.B Heat lost from the top the bed (4 ) Cross Sectional Area A I Set Equivalent Parallel Pores N Per t Area Randomly Packed Real Fuel Bed Incinerator Bed Depth L Incinerator Wall Real Temperature Prile y. Step Function Equivalent Temperature Prile Grate Bars Cold Air Supply t t FIG. 1 REPRESENTA TlON OF COMBUSTION POT OR SOLID FUEL BED BETWEEN PARALLEL WALLS ON A GRATE ILLUSTRA TlNG BOTH A RANDOML Y PACKED REAL BED AND A SET OF EOUIVALENT PARALLEL PORES AS BASIS FOR THE ANAL YSIS. 234

Equating the heat generated to the sum the heat lost and the heat pyrolysis (which is the heat balance) and rearranging (5) From equations (4) and (5), elimination f and rearrangement gives [(B/cp.llT) (P02/a) 1) (Q/Cp.llT.povo) [(Plcp.llT) + 1) (6) Rewriting the above relation in terms the dimensionless groups obtained in equation (1), we get [ (B/CbllT )(Cb/Cp) (Po 2/a) 1) (Q/ cb lltpo Vo )(cb/cp) [(P/cb llt ) (cb/cp) + 1) (7) which is an alternate functional expression for (F A /k) and can be written [(116.118/117119) 1) 1/112119 [115/119 + 1) (8) DISCUSSION The results the analyses may now be compared. We note first that nine the 11 groups in equation (1) are connected by the relationship given by (8). To this extent a similar result has been obtained by the two different approaches. Then by the theory functions we can say that function the other five 11 groups is equal to unity, i.e., g(111, 1111, 111 2,111 3,111 4 ) = 1. Now for a given bed in both model and prototype we can select the following parameters, pia, LId, NLd as constants so or This relationship simply means that the average heat transfer coefficient is a function velocity and so has a limit when the heat carried away will exceed the heat produced by the reaction. Then the reaction will be quenched because the dependence the reaction on temperature as given by the chemical rate theory (k = koee/rt), where E is the activation energy. From equation (6) we see that the area burning rate decreases as the heat loss, Q, increases. The conclusions following from this are self evident (and also common sense). For higher area burning rate the wall losses should be minimized; ideally, this requires an adiabatic chamber. In practice this means heavy insulation the walls which naturally has the effect increasing the bed temperature and thus the reactivity, which means faster burnf. The effect adequate and uniform porosity on the burning rate is also important, since channeling the flow through the bed may otherwise occur. Also, the less porous the bed, the less oxygen can diffuse into the reacting mass which is detrimental to a higher burning rate as desired. However, there is an optimum. If the bed is too porous the less material there is to use the oxygen so combustion space will be wasted. Ideally we need macro pores distributed throughout the volume the bed to carry the bulk the flow, with perhaps a distributive tributary system micropores. For deriving a relationship between burning rate and porosity we could use Then to obtain the empirical functional relationship between 11 and 1111 by the standard procedure, we would plot the area burning rate ratio 11 against the porosity term (pia) (i.e., 1111), with the parameter (LID) (i.e., 111 3 ) maintained constant through a given run, but with 11 and 1111 measured for a series values 111 3, 235

CONCLUSIONS From this analysis we conclude that, with all parameters or n factors taken into account, then 1) The scale factor for mapping data obtained on combustion pots onto a full scale bed should be unity. 2) That the consequence neglecting certain n factors as small will be to introduce a scale effect, but this should also be close to unity. 3) We have, then, that the dimensionless firing rate group (n) should be a prime function only four others (n 1, n 2, n 3, fl4) 4) The conclusions from the dimensional analysis are generally supported by a separate analysis based on a heat and mass balance. REFERENCES [1] Langhaar, Henry L., Dimensional Analysis and Theory Models, John Wiley, New York, 1965. [2] Bridgeman, P. W., Dimensional Analysis, Yale University Press, 1922. [3] Van Driest, E. R., "On Dimensional Analysis and the Presentation Data in Fluid Flow Problems," J. Applied Mechanics, 13, 1946, p. 34. n 1 = hlvopcb n 2 = cbd..tpovolq n 3 = vplk n4 n5 n 6 BlcbD..T n7 = a vivo n 8 = P0 2 PlcbD..T n 9 = cplcb nlo= plpo n 1 1= pia n12= n13= LId n14= NLd DEFINITION OF PI GROUPS (For definitions symbols see identification parameters under Analyses) ACKNOWLEDGMENT This paper has been prepared as part the program research on Incinerator Emissions, sponsored by the Department Health, Education, and Welfare (Public Health Service) under Grant Number 5ROIAP0039703. 236