WWNW ‘\ MN W L \lWl‘WM 0—; com _(D->O TH Thesis gor The Degree of DH. D. MICEEGAN STATE UEEVERSITY Jaw-kai Wang 1958 TTTTTT ITTTI TTTTTTTITTTTTTTTT . 31293 00692 1013 This is to certify that the thesis entitled THE THEORY OF DRYING presented by Jaw-Kai Wang has been accepted towards fulfillment of the requirements for Ph. D. degree in Agrl. Engineering WMW Major professor 0-169 L I B R A R Y Michigan State Umvcrsity . . .. 1.4.1... y. kn: .J-qfllaa E» .s ,4 .Iunfl r9“! N, .4: Hail!!! A s nah.-. usual“... ... t. Infill.“ NR ...9“ . tlkpnflnhb .....Iut. ....llllr fishnevarflhtd E». If”... ..Jth....hH...« .r..... .. . - . .... H7 . .Wrtt... ..w.nl«m..wc:rml. ...ln ...c .. , .l ...... - .u ... s! 44 THEORY OF DRYING BY Jaw-kai wang AN ABSTRACT Submitted to Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering Year 1958 flMM W, h». ten/952? Approved Jaw-kai Wang ABSTRACT Based on the information gathered by previous workers, a theory, concerning the moisture movement inside a kernel of grain during drying process, has been proposed. Mathe- matical equations have been used to express the proposed dry- ing theory and were solved under various drying conditions. Calculated and experimental drying data for corn were compared. Over a time span of seventy hours and for six different drying conditions, the maximum deviation was found to be less than ten percent of the difference between the initial and final moisture of the corn. It was concluded, contrary to previous beliefs, that the vapor diffusivity of corn could be regarded as a constant for the temperature range of 60 to 100 deg F and moisture content from 30 to 15 percent db. THEORY OF DRYING By Jaw-kai Vang A THESIS Submitted to Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering Year 1958 _,ACKNOULEDGMENTS It is with great pleasure that the author expresses his gratitude to Dr. Carl H. Hall for his inspiring guidance, unfailing interest and continuing encouragement. The author is indebted to Dr. Arthur W. Farrall, Head of the Department of Agricultural Engineering, for the graduate assistantship that enabled him to undertake the investigation. The author also wishes to express_his sincere acknowledg- ments to: Dr. Charles P. Hells, Department of Mathematics, for his reviewing of the mathematical development and serving on 'the guidance committee. Dr. Merle L. Esmay, Department of Agricultural Engineer- ing, and Dr. Ralph M. Dotty, Department of’Mechanical Engin- eering, for serving on the guidance committee. Dr. Douglas H; Hall, Department of Mathematics, for his consultations. T The author would like to record his grateful appreciation to his wife whose sympathetic patience and understanding helped to make the preparation of this thesis more enjoyable. This thesis is dedicated to Professor and Mrs. Shu—ling Hang, Parents of the author. VITA Jaw-kai wang Candidate for the degree of Doctor of PhilosOphy Final Examination: November 13, 1958, 8:00 A.M., Room 218 Agricultural Engineering Building Dissertation: Theory of Drying Outline of Studies: Major Subject: Agricultural Engineering Minor Subjects: Mathematics Mechanical Engineering Biographical Items: Born: March h. 1932, Nanking, Republic of China. Undergraduate Studies: National Taiwan University l9h9-53, B.Sc. Graduate Studies: Michigan State University 1955-58, M.S. 1956 Experience: Military Service, 1953-5h. Navy, Republic of China. Discharged with the rank of Ensign. Instructor in farm machinery, Provincial Taoyuan Institute of Agriculture, l95k-55. Graduate Assistant, Michigan State Univ- ersity, 1955-58. Honorary Societies: Pi.Mu Epsilon (Member) Sigma Xi (Associate Member) Professional T Affiliations: Chinese Society of Agricultural Engineering (Associate Member) American Society of Agricultural Engineering (Associate Member) TABIE 0! CONTENTS Page LIST 01' FIGURE eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee V m 0? LITERATURE ..................................ee 1 INTRODWTION eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee6 THE 6mm SOLUTION 0! THE DmERENTIAL EQUATIONS COMING THE WISTURI WT THROW HYGRO- Scone MATERIAIS .e ”tn-mom "Maurine the “##9QFA991aW1vHee 1° “1°. "9°” Diffusion Egan”:'99999223929?239299-2- 1° The Heat Diffusion Equation ....................... 10 Boundary Conditions for Shallow Bed Drying ........ 18 eeeeeeeeeeeeeeeeeeeeeeeeee Solution .......................................... 13 CALCULATIQI ............................'................ 14 DISCUSSION so stunner ...............,................................ 24 momma FOR rmn'rma STUDY as The Temparture Distribution In A Composite Sphere With Thin Skin Vhen Moisture EvaporationAnd Heat Exchange TakePlaoe At The Surface ................ 2? The Application Of Drying Theory To Deep Layer Drying ............................................ 29 Wells ....DOOOOCOOOOO.......OOCOOOOOOOOOOOOOO...... 31 LIST OF FIGURES Figure Page 1. Comparison of Experimental and Calculated Data on Drying Shelled Corn at 100 deg F............... 17 2. Comparison of Experimental and Calculated Data on Drying Shelled Corn at 86 deg F................ 18 3. Comparison of Experimental and Calculated Data on Drying Shelled Corn at 60 deg F................ 19 REVIEW OF LITERATURE Drying in agricultural.work refers to the removal of moisture until the moisture content of the product is in equilibrium'with the surrounding air, usually twelve to forteen percent, wet basis (wb). (Hall, 1957) Since the introduction of mechanized harvesting, consi- derable emphasis has been placed on the drying of agricultural products which has led to the study of the movement of water through hygroscopic materials. However, much of the research work in drying of agricultural products reported from.l9h0 to l9h5 has not delved into the theory of drying but has been. concerned mainly with field results. (Hall, 1957) Among the few researchers Sherwood(l936), Newman(l931), Ceaglske'(l937), Hougen(l9h0), Henry(l939), Babbitt(l9b0), Simmond(l953), Van Arsdel(l9k7), Fenton(l9hl), Hukill(19h7), Hall and Rodriguez- Arias(l958), who have concerned themselves with the theoret- ical aspect of the study of water movement in hygroscopic materials, various hypothesis have been advanced to explain the mechanism.by which the moisture moves. It is now generally agreed by researchers, that the drying of a hygroscopic mate- rial can be divided into two or more distinct phases. The earlier of these, characterised by phenomena similiar to evaporation from.a free liquid surface, is known as the constant-rate period. One or more later periods, character- ised by a steadily-decreasing drying rate, is known as the falling-rate period. (Van Arsdel, 19h?) (Hall, 1957) Drying of agricultural products usually falls in the falling-rate period. Babbitt(19h0) has proved beyond doubt that the vapor pressure is the main driving force for the moisture movement during a drying process. In his experiment, the sample had the vapor pressure gradient in one direction and the moisture concentration gradient in the opposite direction. He found the moisture moved against the moisture concentration grad- ient and that the total moisture migration is proportional to the vapor pressure difference. In its differential form, the relationship expressed by him is, 3Tr= D'%;—::. equation 1 X where Mfl= moisture content p'-_-. vapor pressure D =diffusion constant 1'. =time x = distance Edward and Hray(l936) showed in their experiment, in ‘which vapor pressure difference accross the sample was kept constant while the temperature was varied over a wide range, that the rate of moisture migration was virtually constant irrespective of temperature. It is important to bear in mind that equation 1 is true only when the vapor pressure is the dominant factor control- ling the internal mOvement of moisture. At higher or lower moisture content other than that indicated by Babbitt(l9hO), other factors, like, for instance, capillary force might become the governing mechanism.and equation 1 will no longer hold. Vapor pressure of agricultural products, almost without exception, is a function of both the moisture content and temperature. (Dexter, 1955) (Hall, 1958) Consequently, in general, equation 1 is an equation with two unknowns and to solve it an additional equation would be needed. Henry(l939) has suggested a general approach for solving certain type of simultaneous partial differential equations. Up to now, most of the researchers have avoided the task of solving simultaneous partial differential equations by making certain assumptions, some of them have since been known to be wrong and others may be shown to be incorrect. Van Arsdel(19h7) has cited the works by various researchers who took the difference in moisture concentration as the driv- ing force and concluded that the diffusivity Dc in the follow- ing equation is highly de epsndent upon the moisture content. on _D a M at cdx‘ where Dc is the diffusivity when moisture concentration equation 2 is used for the driving force. Van Arsdel realized that the vapor pressure difference should be taken as the main governing mechanism for moisture diffusion through a hygroscopic material, but in his work he assumed that temperature throughout the medium was uniform during the drying process, or in other words, that the diffu- sion process is an isothermal one. The temperature variation inside a spherical meduim 'where surface evaporation is taking place has been solved by wang1Appendix). In general, except for extremely low dry- ing rate, it can be shown that the temperature inside a ker- nel of grain is not uniform. For materials that have smaller individual dimensions than a grain, King and Cassie ( l9hO, Part I )have shown that during the uptake of moisture by wool fibres the temp- erature of wool fibres were anything but in equilibrium with its surroundings. They have stated in their paper "The result of this paper showed that slow rate(of the uptake of moisture by wool fibres) is entirely due to external factors and have no relation to the surface structure of the colloids". Equation 3 has been used by many research workers to correlate drying data (Simmond, 1953) ( Hall and Rodriguez- Arias, 1958). M "' Me _‘ e_k9 .. e nation 3 M0 M6 <1 When applied to the heating or cooling of a substance, equation 3 has been known as the Newton's equation and it applies only when the temperature can be regarded as uniform throughout the substance and the heat exchange between the substance and its surroundings is preportional to their temp- erature difference. In using this equation to drying, the effect of internal resistance to moisture movement has been entirely.neglected. Simmond(l953) has indicated that in forced air shallow bed drying, the air velocity usually has only neglible effect on the drying rate. Since air velocity is of great importance in determining the film coefficient between the grains and its surroundings, it is therefore apparent that the internal resistance must be the main factor in determining drying rate. - A modified form.of equation 3 has been suggested by Hall (1958b) and it can be used to approximate the experimental drying curves to great accuracy. M—Me N -kna M°_MO=Z|IA.,€ equation A N 1, 2, 3, eeee' N ’1, Ne Since both equation 3 and A are not derived from basic considerations, it is therefore little surprise that the values of drying constant, k, have to be determined exper- imentally and no effort has been successful in predicting, under various drying conditions, the values of drying constant. It is worthwhile to note, however, that the theoretically derived drying equation, under certain conditions, has a form very close to that of equation A. (See, for instance, equation 21.) INTRODUCTION A rigorous treatment of the diffusion of water vapor through hygroscopic materials would require the usage of statistical mechanics or kinetic theory. However, as in many engineering problems, the primary interest of this thesis is not in the motions of individual water molecules, but rather in the phenomenological result. Thus, to avoid the forbidable task of cumbersome calculation which.would result if the statistical mechanics or kinetic theory approach were used, the concept of continuum is introduced. Using this concept, vapor pressure and moisture concentration or moisture content can be defined at a point. This makes the description of the diffusion process in mathematical terms a much easier job. It would be well to bear in mind, however, that the use of a continuum concept to construct a working theory unifying a large body of observational knowledge and permitting the deduction of useful conclusions is only justi- fied by its outcomes. But this is nothing new in modern physics. Indeed, even the most advanced atomic and nuclear theories are regarded as useful conceptual schemes rather than as reality. The moisture content or moisture concentration in dry basis (db) is expressed by the ratiom , where 3111 is the weight of water ins v , 3v is the “531:. and p is the specific weight of the dry material. It is obvious when 3" becomes so small as to contain relatively few molecules, the moisture content fluctuates substantially with time as molecules pass into and out of the volume, so it is impossible to speak about a definite value of 2 8'?! . If the smallest volume which can be regarded as continuous is 8v’, then moisture content (db) at a point is defined as M’= lim 8m 8V—08V. P 8V It is known that the vapor pressure exerted by most agr- icultural products is dependent upon its moisture content and temperature. (Hall, 1957) Considering any 8V9 all the water contained inside the volume 8v may either be absorbed by the hygroscopic material or exist in vapor form in the pore space of the medium. If 8 V is small enough, the vapor pres- sure exerted by the medium should be always in equilibrium with the vapor pressure existing in the surrounding pore space. Furthermore, if 8v is small enough, the moisture content and temperature can be regarded as uniform throughout 8 v at all time and therefore, vapor pressure at a point can be defined in terms of M'and T' in the following way, P‘ = avian P. (£82;- . T‘) As cited in the Review of Literature, previous work shows 'that for the moisture range involved in the drying of agricul- tural products, that the moisture migration between two points in a hygroscopic material is proportional to the vapor pressure difference between the two points. (Babbitt, l9h0) (Edward and Wray, 1936) Thus, the drying process can be regarded as the diffusion of water vapor through a hygroscopic material which may set free and release or absorb and immobilize some of the diff- using vapor, depend on whether the vapor pressure in its sur- rounding is greater or smaller than that exerted by itself. When absorption takes place, heat will be evolved; when eva— poration takes place, heat will be needed as latent heat. Heat produced or taken up in this fashion will diffuse through the same medium and thus affect the extent to which the vapor will diffuse. The vapor is thought of reaching from one point of the medium to another by diffuse through the pore space. The pores are envisaged as a continuous net work of space included in the solid, the dimension of the pore is small as compared with 8" and its distribution is uniform, so that the whole medium can be regarded as a continuum. THE GENERAL SOLUTION OF THE DIFFERENTIAL EQUATIONS GOVERNING THE MOISTURE MOVEMENT THROUGH HYGROSCOPIC MATERIAL NOMENCLATURE: P' Vapor pressure, psia M' Moisture content,lb of water/ lb of dry matter Average moisture content of the kernel Mo Initial moisture content Me Equivalent moisture content of drying air T' Temperature, degree Fahrenheit (deg F) r Radius, feet (ft) x Heat conductivity, BTU/ deg F-ftz-hour/-ft D' Vapor permeability, lb/ psia-ftz-hour/ ft k Heat diffusivity, x/pc, ftz/ hour D Vapor diffusivity, D'/p, ft2/ hour-psia t Time, hour q Latent of moisture, BTU/ lb of moisture p Specific weight of medium, lb/ rt3 c Specific heat of medium, BTU/ deg F-lb °n Constants, n l, 2, 3, ... h Film coefficient on Eigen values Eigen values 10 ASSUMPTIONS UNDERLYING THE MATHEMATICAL ANALYSIS In order to derive and solve the partial differential equations governing the diffusion of moisture through hygros- ' cOpic materials, the following assumptions have been made. The validity and applicability of those assumptions in drying process, within its range of temperature and moisture varia- tion, will be presented in the Discussion; (1) The medium is isotrOpic. (2) The rate of moisture migration between two points in the medium is proportional to the vapor pressure difference. Other factors, like capillary force, is assumed to be neglible. (3) The relationship between vapor pressure, temperature and moisture content is linear, or, P'=c1+ “2M"? c3T' (A) The quantities K, D, q,p , c are contants. ' (5) The Structure and volume of the medium remain cons- tant during the drying. A If the above are true, then, for a spherical specimen, THE VAPOR DIFFUSION EQUATION p Qfl_ = _‘__ ( '2 LP. equation 5 a: l' zdr’ a r ‘ 'FHE HEAT DIFFUSION EQUATION 6T, K a” zaT c __= — r qp— tio 6 ("at r 58):) _')" '9‘“ ” Let T=rT', P=rP', u=rm "and using the relationship P = c1r+czll+c3T to eliminate N in equations 5 and 6. 11 k.a_zl = Jud—P.4- +(|_ 23.9.)21 equation? ar‘ cc “6 cc! 5 DazP = _|_ _a_F_’_ + 2 Q1 equation 8 Dar 0361 c; 0? Multiply equation 7 by sx/ k and divide through r-qua‘tion 8 by D, then add, 62 5x_ “X } a 2(P + SXT)=— 97652“ +:-g—c )P+(— we: 33;)1 choose sx such that Dequation 9 ' 5x0 l 0:0 cs _+ _= ._..__ = e nation 10 0C2 kCCZ k kCCg DCsz P'X q eliminate sx in equation 10 and solve for C 1% = (F'x‘ 5'32)“;th _§_2_.L) equation 10a or, .I'LX" -2_-Dk [D(l- %§)+-c-2k _(D(l—%§)+%2)-%fi equation 10b equation lOb gives two roots °fP-x , say,F_, “mp-2° ' Using [.L' ,P-2 and equation 10, corresponding values of Bi and s2 can be found. Equation 9 can now be simplified to 62 . 0r -1(p + SJ) ”1;”, + 5x7) ‘ equation 11 Assume P and T have the following solutions, (I) 2 PX = "EDA" SIN (Inf e-ant/[J—x equation 12 - a = g Bk SIN Xkr e" xkt/F'X equation 13 k=0 when x=l P=P1, T=Tl and x=2, P=P2, T=T2 e 12 Since equation 11 is only a rearrangement of equations 7 and 8, it is clear that any solution of equation 11 must also be a solution to equations 7 and 8. The solution of equation 11 are P+s1T and P+s2T. But when x=l, P=P1 and T=T1 ; when x=2, P=P2 and T=T2 . SO, P+81T= P1+ 811']. equation 1h P+82T= P2+82T2 equation 15 or, P _ §zfi + 5| P2 + 52$|(T|" T2) equation 16 SI:"’ SI 1' = H - P13 + 5'“ - $21.2 equation 17 SI "5;a Equations 16 and 17 are general solution to the simul- taneous partial differential equations, provided that equa- tions 12 and 13 are true. BOUNDARY CONDITIONS FOR SHALLOVI BED DRYING To show that equations 12 and 13 are formal solutions of Px, Tx' it is necessary to show that the eigen valuesan,),k and coefficients An and Bk can be found so that equations 16 and 17 satisfy certain initial and boundary conditions. Suppose a spherical shaped kernel of radius b is in equilibrium with its surroundings and that at time t=0 the - vapor pressure and the temperature of its surroundings are suddenly altered from P6 to Pa and T5 to Ta . Expressing P and T as functions of r and t, or, P==P(r,t) and T=T(r, t), the initial and boundary conditions are, 13 HO, t) =r]_._i.n3 rP'(r, t) = 0 equation T(O, t) ='1_i‘n3 rT'(r, t) = 0 equation P(b, t)= b P'(b, t): o tZO equation ( h+aé',-) T'(b, t) = 0 tZO equation P(r, o) = r Po equation T(r, O) = r To equation where P°=P5 - P‘ , TO=T3 - T‘ . SOLUTION Applying the above conditions to equations 12 and 13 the eigen values and coefficients in those equations were found to be = -. "‘H 2b An ( 1 ) ‘n—' Po an: Lit—L = 1' 2' 3, eeeeee Bk; 4B(SINXkb-b)\'§0$)\kb) K: 1' 2, 3' ...... M2 bkk- sm 2th,) xk are the roots of , b Xk TAN b= x“ l-hb 18 19 20 21 22 23 CALCULATION The average moisture content of a kernel at any time is, i ='clzv_gf(?e-CI-Ca%+Po)dV If the external vapor pressure only were changed, as Rodriguez-Arias has conducted his experiment, then Bk: 0 ad P = SgP.-S.Pz ' '1‘ a _P.2_.-_Pl 32- 8| SE- sI Hence, 3 =dogI(P—:—°fl)dv- -°—' =c‘m‘af‘} (5,9. + c,P. - s. P,- c,P,)dv - %4 =czv