34 mathematical simulation of corn drying

7/30/2019 34 Mathematical Simulation of Corn Drying

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Matllematical Simulation of Corn DryingA New Model T. L. Thompson, R. M. Peart and G. H. FosterAssoc. MEMBER ASAE MEMBER ASAE MEMBER ASAEDRYH,C is a continuous process withchanges In mOIsture content, air

and grain temperature, an d the humidity of the air all occurring simultaneously. I n fact, thesl: changes vary fordiflerent drying methods and for d ifferent locations in the drying bed. Th eobjective of the study reported in thispaper was to develop a mathematicalprocedure whereby grain-drying prediCtIons could be made with infinitelymany sets of drying condi tions andwi th nonconventional as we ll as conventional convection grain-drying methods. Th e mathematical dryinrr modeldeve loped incorporates manyb of thefactors that affect grain drying, and iscapable of determining the e ffec t ofmany dryin(1 pa rameters on the dryingresults, esp .cially with high-temperature drying.

VOl' thi s study, the ba sic approachused to describe the continuous dryingprocess was to div ide the process intomany small processes and simu late themby conseeuhve ly calculating the changesthat occur during short increments oftime. Th e basic simulation approachused was to calculate the drying performed on a thin layer of grain andtll('n combine many thin layers to formthe grain bed. This approach permitscomplete simulation of less conventionaldrying methods such as concurrent Row(pa rallel) and counterflow, as well ascrossflow drying methods.

The simulation model described inthis paper (11) can be used to predict the results of any of these dryingmethods and also predict transient re sults (such as humidity of tbe air,equilibrium moisture content, and manyothers) throughout the process. Ofcourse, computer capability is necessary, bu t present-day computer tech nology makes thi s type of solution fea sible and practical. An advantage of thi stype of computer simulation is that,

Pa per No. 67 -313 prQse nted at the Annualr C l i n g of t he Am ericell Society of Af(ricwlUIalE ngin t!crs llled...in g jointly with trw Canadian

o c j t of " ' i c u t r a l Eng-inecrin ,:; at Saskatoon, Sask.. Canada, J t ~ C 19-67, tilt [L prograrn, ) i \ ~ I I by the El{'ctn c Power and Processing

Th e autho rs T. L. l ' H O M P S O ~ R MPE ART anrl C. H. FOSTER- arc as; o'ciate I ' r o ~f 'fSso ( of l t u r a l e.agi nC'r rir,1g , Ulliversity ofL c bra ska. Lmcolo; professor at a"ricultu..ra! eng in ,erj ng, Purdue' t i niv ersity, an d agricultural

n g l o e transportat.lOn an d facilities reseru-chrl ivision . . 11.S, USDA . Apl'roved for I'ublicati on. 5 pilp "r :-.10 . 3213 of the journal series of thel'urd ue t:niversity Agricultural Experiment Stat iu n.a Numb e rs jn parentheses refer to the ap refm'el lces.

~ ~TeeF" .. T- bT. F

Humidity ... U + .'Iii , . w t ! . dr y a iri i ec r u Aft .( Dr)' if1f,Coct'! .!!.fore Pr yi ng ' i i . of At:a l s t ... H, % d . b . Q i. - ~ n r . .. : - ' : ' d .h .G, - f. L-_ _ _ ___ . Ie!!' ;:. - C -Ii

Dry!n s lli. .. '[, - r

?u-t t!t-rv .., H, lb . "'/ltl!l .'i b . d l'y .,1 rFIG. 1 Schematic diagram of basic simulation approach .when new truths are learned about thedrying process, it wou ld be a simplematter to incorporate them into thesimulation mode l.Th e following sections develop thismathematical drying model. A thoroughunderstanding of the changes that occur during the drying of a thin layerof grain is necessary before predictionscan be made about deep bed drying.If calculations can be made describingtbese changes, then groups of thin layers can be combined to form the deepbed.T hin Layer Dry ing

Briefly, the d ry ing of a thin layer ofgrain was simulated by conSidering thechanges that occur in the corn and thedrying air as shown in Fig. l.D rying air (T deg F , H Ib water perIb dry air) is passed through a thinlayer of corn (M percent moisture, Cdeg F temperature) for a drying timeinterval, M. During this interva l b.Mpercent moisture is evaporated fromthe corn into the air increasing its absolute hum id ity to H + b.H lb water perlb dry air. During drying the temperature of the drying air is decreased(b.T deg F) in proportion to the tem perature increase of the corn (b.G degF) and the evaporative cooling accompanying the moisture evaporation. Th eamount of dry ing performed was calculated by a tbin -Iayer drying equationwith cons tants dependent on the drying air temperature. Complete heat balances were used to calculate the finalair an d grain temperature consistentwith the evaporat ive cooling accom panying the moisture evaporation andwith the initial temperature of the drying air and the grain. A detailed analysis of th 'ese calculations follows. Th ereader who is not interested in the de

tails of the model, may refer directly tothe section on "deep bed drying" without any loss of continuity.The' following assumptions or relationships were used in the development

of the mathematica l model; justificationand the de velopm ent of e.ach relationship follow the statement:Assump tions

1 Fully exposed , thin-layer drying isrepresented by tbe equation :t A In (MR) + B [ In(MR) J2where

A -1 .862 + 0.00488 '1'B 427.4 e-0033Tt time to dry to MR with dry

ing tempel:ature '1', hI'. . M -M eMR mOIsture ratIO, -:-:--_;-:-_Me -M eM moisture content at time t,

percent dry basisMe initial mois ture conten t . percent dry basisMe equilibrium moisture content,percent dry basis

Henderson and Perry (3 ) reportedthat during the falling -rate drying period, the moisture removal rate is inversely proportional to the moisture to

dMbe removed or - . = - k(M - M )cit t' .Solution of this equation yields the simp le exponen tial drying equation MR= e-k t

Most investigators state that the drying constant k is dependent on thE' dry ing air temperature, bu t they do notexplicitly specjfy tbe relationship. Th efollOW ing experimen tal in v es t igat ion"vas performed to detelmine dryingconstants and this relationsbip for yellow-dent sbelled corn.A series of thin-layer drying testswas performed in the a ~ ~ c l t u r a l en-

Th i s ilJ'ticle is reprinted from the TRANSACTIONS of the ASAE (vol. 11, no. 4, pp. 582, 583, 584, 585 and 586, 1968),the fm nsClctlOns of the AmencCln Society of Agric\1ltmal Engineers , SClint Joseph, Michigan


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FIG. 2 Predicted (cw-ves) versus experimental (points) thin-layer drying results.gineering processing laboratory at Purd ue Ullive:rsity during the fall of 1963.The t8sts were performed in an experime n tal thin layer dryer. Th e dryer consisted of a fan blowing heated airthrough a screened bottom tray holdinga sample of corn.

Th e main variable investigated inthe test series was drying air temperature. A few other exploratory testswere performed to evaluate the effectof airflow rate and corn variety on drying rate.T hroughout the drying tests, th emoistme in the corn was measured byperiodically weighin g the corn sample.This enabled the Illoi'ilure removal rateto be evalu ;,; ted. Each drying test waste rminated wh f'1l the moisture was red uced to a safe storagc level (12 perCf'nt w.b.). Tests were performed with19, 23, and 33 percent w. b. initia lmoistme corn, 20 and 60 cfm/ft2 ofdrying air, :l.IJd with tem peratures ranging from 120 to 340 F.

T he results from each drying testwere analyzed by fitting the experime ntal d JLa to two types of assumeddrying cyuations. On e o[ these was th esim ple exp onential drying equa t ion :i\1 H = c kt.. P lots of the experimentalda ta versus the best fit of the assumedequation showed that the simple exponental equation did not adequatelyrepresent thc experimental results.T he st'cond drying equation used wasa second-order exponential curve (parabola on 'icmilog paper). This equationhad the form:

t = A In(i\1R) + B (In M R ) ) ~Th e results from the test series showcdthat dry -air temperaturc was the mainvariablE' that affected the drying rate.Th e other variables, corn variety andairflow rate, did not significantly affecttltc drying constants for the values investigated. F rom this it was assumedthat the air flow rates, in the rangeconsid


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L' = ( 1094 . - 0.S7T) ( 1.0:. 4..l5e-2K25M) ... .. . [3JTh e data of Thompson an d Shedd

(9 ) \" 'IS fitted to th e eq ua ti on : LL'= 1. + ae-1, \1 bv the me thod of leas tsql lares . 'L he t i o n , L = 10 94 - 0.571' , for the la trllt heat of wa ter (3) an dthe resul ts from the leas t squ ares analysis res ulted in the above eq u ation. Itwas base d on experimental da ta in the10 to 15 perce nt w.b . reg ion . T he authors feel th at extr ap ola tion of thiseq uation was justified, beca use forhi gher moisture conten ts the hea t ofvapo rizatio n approach ed that of freewa ter and for lowe r moistu res it became increas in gly difficult to eva poratewa ter from the corn kerne l.

5 T he speCific he at of corn was assum ed to be represe nted by th e equ ali on :c - 0.350 + 0.008 51 M ", . . . [4J

where ,H ," ~ moi sture co nten t percentwet basis.Kaza rian and Hall (5 ) foun d th a tthe speCific hea t of soft whi te wh eatan d yellow d f'lI t corn was linearly depe nc.l ent on the moi sture co ntent. Fromthpir expe rim ental evalua tions on yel lo w dent corn , they determined th eabove equation.Thin Layer Simulation

Th e l' ma in der of tlli s sec ti on on thinlaye r dr y ing is a de ve lopme nt of a procedu re to c a l c u b l t the average ch angesd uring c.lry in g when a thill layer ofcorn is d ri ed for a time increment !',.t .Tlip. dr ying process was co nsid ered tohe divided in to separa te pr ocesses ( including lcm pcrature e quil ibr ium betwc 'I l tlte ~ r : , l i n and air, moistu re remo val , ami evaporati ve cooling of th eair and the gra in ) for this deve lopm entand th e res ulting ca lculations. Th esepr ocesses actu a lJy occu r Simultaneously,b u t the process was div ided up to simp lify th e simu la tion . The hea t balanceswe re w ritLcn in te rms of Btu pc r lb ofdr y ai r Howing th rough the layer. Agrain-to-a ir ra tio was used to con vertlhe specific hcat of corn to these units.

Pr edict ion of the am ount of dryingthat occurs in a thin layer of corn canbe mad e 11 )' conSidering the initial airand gra in conditions, using a thin-laye rllrying eq uation and complete hea t balances to predict th e final air an d graincondition s.D rying-Air Tc mp e la tu re

Th e equilibrium temp era ture of th edr ying ai r an d the corn was calculatedby performing a sensible hea t balancean d \va:, used as the drying-air temperature , Drying-air temperature as used

fused with the te m p er a tu re of th eheated air before it enters the dryingc:olumn. Th is heat balance is on ly anin term ediate calculation to detennineth e drying air temperature and dol" notinclude moisture evaporation. NonnalJyth e corn does no t a ttain this temperature since evapora tive cooling accompanies the sensible hea t transfer.

Fo r the heat ba lance, the speCifiche a t of corn was conve rted to Btu pe rlb air deg F .

Th us C = (0.350 -I- 0.00851 M,, ) Rwhere R = air -to-gra in ratio , Ib ai r pe rlb corn .T he eq uili br iLUll tempera tur e of thecorn an d the air beforc dr ying was determined w ith the follOW ing heat balance :0.24 T o + Ho (1060.8 + 0.45 T o)+ C Co = 0.24 T o + Ho (1060.8+ 0.45 T e) + C T .,

where th e subsc ri p t 0 refe rs to originaland e to equilibrium va lu es of air temper a ture, T , g rain temp erature, C, an dabsolute humidity, H .The first two tenus on each side ofthe equation represen t the initial an dequi librium hea t con tent of the air , an dthe third terms are the initial an d equilibrium heat content of the corn. Solving this equation for the unkIlow,equilibrium tempe ra tu re :

T = (0.24 /-- 0.45 Ho) To + C 0.24 + 0.45 Ho -I- C. . . . . . . . [5JMoisture Removed

Th e equi librium moi sture con ten t,Me , of the co rn was ca lculated by determining the relative humidity of theair an d using th e equilibrium temperatu re from the above he a t balance in theequilibrium moisture content equation.Th us - - In (1 - RH )

Me=1(3.82 X 10 5 ) ( T e + 50 d ['6JIn a deep-bed dr ying p rocess the drying air tempera ture, T e, a t one location30

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in th e bed usually changes as dr yin gprogresses. A new d rying curve is specifi ed when th e dr ying temp e r a turech ang es, and the amount of drying onthe old c:urve ha s [ 0 be transformed tothe new curve. Th is transform ationwa s m ade by calculating an "equi valenLdr ying time. " This wa s calculated withthe drying equation:

t = A In (MR) + 13 ( In (M R ) )2using th e new va lues of A, 13, a lld M R( calcula ted with the new i\ l . an d th epr esen t IVI ). Th is is the eqUivalent timeth at wo uld be needc rl on th e new cur veto d ry to the p rese nt m ois ture cont (' ut.Th e mois tu re ra lio at th e en d of theprese nt drying pe riod was calcula Ledby solv in g the thin laye r eq uation for M R. and using a ti me, t , of tlweq u iva lent dr ying timc p lus the dry ingtime interva l !',.t . Th e final moi stu re con tent or th e laye r was then calculatedfrom th e mois ture ra tio.Fi nal Air an d Gra in Te mpera tu re

Th e final air an d grain state p oint sco nsistent with th e am oun t of u rvingp erformed on a thin hl ye r .of graining tim e int erval were calc ula ted byth e foll ow ing procedu re :(1(, - 11 [) percentage points of mois.ure were remov ed from the co rn ~ l l l Jevap or ated into the ail'; thus Lil e ab solu te hwnidity of th e air \\ :1> incre asedby an am oun t

AI!. Mn --: ,.JJ Ilu. and 100 Hr = Ho + t,./ITh e final tcmp era ture was de termin ed with th e foll owing heat halance:0 .21T e + Ho(l 0 60.8 ~ 0.451'0)+ CG o -i-- !',.H( G, ,32 ) c .. 0.24T, + II , ( 10GO .8 -I- 0. 45Tf) + GTr + !',.L t,./-, where C o - Te from equation [ 5 J. Th e first two tcrm s on cach side ofth e eq uation art' th e initial and finalhea t cOlltent of the a ir ; the third termis the in itia l an d fin al !tea t content ot

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." 20 0w0:;=a::w0.:EZ 100


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the corn. T he fourth term on the left Th e above description of the matheside of the equ at ion is the hcat content matical drying model presents the stepsof the waler lh a t was evaporated , and that were ne cessary to calcul ate th ethe last term in th e equation is the heat final air and grain conditions afterof vapo rization required to evapora te drying for a time interval M on a singlemoistu re from the corn above that re laye r of corn.quired to evapo rate the same amoun t All of the steps presented in this secd free water.

(0 .24 -:.. 0.45Ho)1'e - b..H(1060.S + b..L + 32 - G(.) + CGeSo lving, Tr = 0.24 + 0.45He + C . . . . [7Jis the final alance has two unknowns of corn onto the next. E ach layer wasth e fina l tcmpera ture, T f , and the ex dried [or a sho rt time interval usi llgll a(Jst humidi ty, [-If In terpolation wa s the exh a ust air from one layer as theused on the rela tive humidity versus input drying air to the next. The proctem pe rature relationsh ip to convcrg e to ess was then repe;,tcd w ith a secoml,a rclati \(.: humidity of 100 pe rcen t and

third , . . . , drying tim e interva l untilto dc tennine ll e and T r. Usin g a me thod the average final moisture was as dede\' loped to fi nd th , zero of unknown sired.fu nctions ( ) .3 ) , thi s interpolation re Assuming no app reCiab le mixing, aquired only lhree or four trials to ob con tinuous crossflow dryer operates thet,lin ;, rel;,tive hum idi ty between 99 same as the batch-t ype dryer just deand 100 percent. The wa ter that w;,s scrib ed. Infinitely long layers of cornremoved from the ;'lr w;,s condensed pass through th e dryer, with the dr yinginto the corn, or: a ir flow ing in a direc tion normal to the0 0 ( H - H ~ . . [8 J corn layers . The air passes from oneR layer of corn onto the next. Th e time3 0 30 0

W200z 0:.., 2qcrw"2:100

0:EOUIUBRiUM MO(STURE CON TENT '"

f fU DEPTH FEET2o L-__ ~ " - - - _ ~ _ " - - - _ ~ - - ,o

TIME HOUR S TI , .E HOURS

that each kernel is sub jcctpd to thedr ying air and the depth of thc groupof thin layers are the two variab Ics usedto ca lculate the


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0-!u cn]tlu'al C'1l l!itlccri ng: . Jo hn \Yil f'Y !sSOliS , In c., 1 ew Yor k, ]Q!j 5...J H enderSo ll , S . ..L :1 nd S. T tl\' il i r("t rain. temperatu re : : t t i o n ~ Jo ur . of .\ g-I' .E n g. I\ese'Hch 7, (I ) Hi62.5 Ka"L.:..lri an . A. :1nd Hall. C . \V. Tl wn li:11propc rti ps o f g rrt ill . Transad ion s of th e ASA ES (l)'13 - 7, 4 8 , 19 65 .6 KeeHan , J. H. a.ud F . C . T iHTII1(l dy nami c ~ r t i v . ; ; ' of sLeam . Juh u \ \,liey ex. SOilS .York, 193 6.7 Rorlrig llez - ri as, J. B ., IInll, C. W .. ;lndBakk t!r-!ukcttUl. . .F'. \V. Ih ..l t o f \ ;;)' p or il :lt lO lllor shelled Gor f! . Cd f'al Ch emistry 4 0: lh i 1UG3 .8 St roh.rntm , H. D. H' It! Ym 'r gpr, n . n. A IIC Wcquilibr.iu nl mojslurc-conllln t cq llat ion. TI' :l11 "iac.:lions of th e ASA.E HL (5W/ .5 , 7, 19 67 .9 Thomp.

34 mathematical simulation of corn drying

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