THEORETICAL ASPECTS OF THE EJRYING OF FOR/5.65 WAF’ERS ”Hum: ‘0!- H10 Dogma of pk. D. MICH‘EGAN STATE UNWERSITY Frederik Wilte Bakker-Arkema 1964 /”3/ 9960/ lllllllHl MIMI”lullllllllll‘llHUllHWI | MIL 31293 00180 3281 HlGAN STATE UNNERSHY M‘C UBRARY ~ "' MICHIGAN STATE UNIVERSII‘.’ LIBRARY I.| ...II.IIIII -II ABSTRACT THEORETICAL ASPECTS OF THE DRYING OF FORAGE WAFERS by Frederik Wilte Bakker-Arkema The high labor requirement of existing forage harvesting systems is a major expense in the production of forages. Hay wafer- ing is a recently developed forage harvesting process in which the complete mechanization of the hay crop may be more nearly realized. The drying behavior of forage wafers was investigated under different drying conditions. Rectangular shaped wafers of different densities and dimensions were utilized. An apparatus was developed for weighing the test samples con- tinuously in the drying oven. A recording potentiometer was used to obtain the wafer temperatures. The air velocities were measured with a hot-wire anemometer. ' Diffusion is the physical mechanism controlling the drying behavior of forage wafers. The diffusion coefficients of wafers (92 percent alfalfa) in the density range from 0. 30 to l. 0 g/cm3 were determined for temperatures from 1200 to ZOOOF. Forage wafers, unlike many other hygroscopic agricultural products, display a constant diffusion coefficient in the moisture con- tent range from 40 to 5 percent, d.b. The effect of temperature on the diffusion coefficients of forage wafers can be expressed by an Arrhenius plot. The empirical equation for the diffusion coefficient of non-cracked wafers with a density of 0.45 g/cm3 is given by Frederik Wilte Bakker-Arkema D = 0.114 exp (- 10,810/RT) where D is the diffusion coefficient in cmZ/sec, R the universal gas constant and T the absolute temperature in C)R of the drying air. The wafer temperature rather than the ambient air temperature should be used to analyze theoretically the drying behavior of forage wafers. Wafers dried in the temperature range from lZO-ZOOOF do not reach the ambient air temperature until they are dry enough for safe storage. A semi-theoretical method is utilized to account for the increasing temperatures and diffusion coefficients of drying forage wafers. The second order differential equation for diffusion with constant initial and boundary conditions does not properly describe the drying behavior of forage wafers. The wafer drying surfaces do not reach the static moisture equilibrium instantaneously as usually assumed, but rather change with time. An exponential type function describes the changing boundary condition. A forage wafer can be treated as a semi-infinite body during the major part of drying. A drying solid with one directional moisture gradient behaves as the sum of two semi-infinite bodies until 65 per- cent of the original moisture in the product has been removed. Approved MM M Majo r Professor THEORETICAL ASPECTS OF THE DRYING OF FORACE WAFERS BY Frederik Wilte Bakker-Arkema A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Agricultural Engineering 1964 To Paula, Peter, Irma and Erik Mr. Mr. and Mrs. P. W. Bakker-Arkema and Mrs. P. de Mol ii ACKNOWLEDGMENTS The author wishes to express his sincere appreciation to Dr. Carl W. Hall (Agricultural Engineering), a truly inspiring teacher and counselor, for his guidance and friendship, making this investigation rewarding and enjoyable. The personal interest and unfailing assistance of Dr. Erwin J. Benne (Biochemistry), who helped the author obtain some insight into the fascinating field of biochemistry, is gratefully appreciated. Thankful acknowledgment is extended to Dr. Fred H. Buelow (Agricultural Engineering) and Dr. Thomas H. Edwards (Physics) for serving on the author’s guidance committee. The author is indebted to Dr. A. W. Farrall, chairman of the Department of Agricultural Engineering and Dr. A. H. Mark, Chief Engineer Advance Design, Massey-Ferguson Inc. , for making the undertaking of the investigation possible. Thanks to Barry A. Kline and Fred A. Lankton for assisting with the calculations and laboratory experiments, and to Donna L. McRea for typing parts of the manuscript. >:< >1: a}: >:< )‘5 >§< 3:: >:< >}< a}: >:< >:< iii TABLE OF CONTENTS I.INTRODUCTION....................... 1.10bjective........ ....... . ....... 1.2 Topics Covered in the Dissertation . . . . . . . . . II.THEORYOFDRYING.................... III.EXPERIMENTAL...................... 3.1Forage Wafers. . . . . ...... . . ....... 3.2DryingSetup................ ..... IV. RESULTS AND DISCUSSION. . . . . . . . . . . ...... 4. l Diffusion in Thin Wafers with Constant Boundary Conditions . .......... . ....... 4. 2 Effect of Wafer Density and Drying Air Tempera- ture on the Diffusion Coefficient ..... . . . . . 4. 3 Diffusion in Brick- Shaped Wafers with Constant Boundary Conditions ....... . . . . . . 4. 4 Diffusion in Thin Wafers with Changing Boundary Conditions ........... . ..... . . . . 4. 5 Diffusion in Semi-Infinite Solids ........... V. SUMMARY AND CONCLUSIONS. . . . . . . . . ...... 5.15ummary..... ........... 5.2 Conclusions. . . . . ........... . ..... SUGGESTIONS FOR FURTHER STUDY . . . . ........ REFERENCES................. ........ APPENDIX....... ............ ..... iv 12 12 15 18 18 28 35 46 55 60 60 62 63 64 69 LIST OF TA BLES TABLE Page 1. Feed analysis of forage wafers used in the drying tests, percentoven-drybasis. . . . . . . . . . . . . . . . . .14 2. Moisture contents of surface layers of forage wafers after different drying periods. . . . . . . . . . . . . . . 22 3. Diffusion coefficients of a wafer at different moisture contents and drying times, assuming different moisture equilibrium contents . . . . . . . . . . . . . . . . . . . 26 4. Densities of wafer slices cut from the same forage Wafer. O O O ........ O O O O O O ...... O O 0 O 29 5. Minimum and maximum diffu'Si‘on coefficients at differ- ent temperatures of forage wafers of different density . 31 6. Diffusion coefficients of several hygroscopic products. . 34 7. Diffusion coefficients of wafers as calculated from eql‘la'tiOI.1 (4. 1. 3) O O O O O O O O O O I O O O O O O O O O O O 36 8. Drying parameters for the calculation of the theoretical moistureratiosofawafer. . . . . . . . . . . . . . . . 44 9. Diffusion coefficients of a wafer, treating the wafer as a semi-infinite solid (eqn. 4. 5. 2) and as a slab (eqn. 4.10 3). o o o o o o o o o o o o o o o o o o o o o o o o o o 57 FIGURE 1. 10. ll. 12. 13. LIST OF FIGURES Experimental Massey~Ferguson Wafering machine (MSU Photo No. 621923-5). . . . . . . . . . . . . . . . . Close-up view of wafering dies (MSU Photo No. 621923-3)......................... . Different size forage wafers used in drying tests (MSU PhotoNo.64573-12)................... . Wafers covered on four sides with moisture-proof material (MSU Photo No. 64573-13). . . . . . . . . . . . Moisture-proof shell plus forage wafer (MSU Photo No. 64501-6)......................... . Experimental drying setup (MSU Photo No. 64573-14) . . Close-up view of drying oven (MSU Photo No. 64573-6) . Moisture loss after one hour drying time at different drying temperatures as a function of initial wafer mois- turecontent............... ....... . . Moisture ratio versus time for a wafer, assuming dif- ferent moisture equilibrium values ....... . . . . . o Drying curve for a wafer at 140 F . . . . ....... Relationship between the diffusion coefficient and the reciprocal of the absolute drying air temperature for different wafer density ranges . . . . . . ...... . Average temperature versus time for a brick- shaped wafer . . . O Q . C C C . U C O O O U C C O O O C O . C O . Diffusion coefficient versus time for a brick- shaped Wafer O O O O 0 O O O O C O O O OOOOOOOOOOOOOO vi Page 13 13 16 16 l6 17 17 21 25 27 33 38 41 LIST OF FIGURES - Continued FIGURE Page 14. Drying curves for a brick-shaped wafer and a thin wafer, assuming a D = 1.65 x 10"5 cmz/sec ....... 43 15. Drying curve for a brick-shaped wafer, assuming a changing diffusion coefficient .............. 43 16. Calculated drying curves, assuming the surface concen- tration C + (Cin - C ) e t for different values of rq fizz/D333...”.................. 49 17. Drying curves of thin forage wafers ...... . . . . . 51 18. [3 as a function of the wafer half-thickness; D = 0. 75 x 10'5cmz/sec... .......... 53 19. B as a function of the diffusion coefficient; wafer thick- ness2cm... ........ .. ........... 54 BTU cm ABBREVIATIONS AND SYMBOLS British Thermal Units concentration, g per cm3 centimeter diffusion coefficient, cmz per sec a constant, cm2 per sec, defined by equation (4. 2. l) as a subscript, dry basis as a subscript, equilibrium error-function error-function complement ex energy of activation, BTU per mole degrees Fahrenheit feet mass rate of flow, g per sec gram hour as a subscript, initial dry weight density, g per cm3 dimensionless coefficient defined by equation (4.4. 1) distance, cm infinite series defined by equation (4. I. 3) as a subscript, liquid moisture content, percent dry basis moisture equilibrium content, percent dry basis vapor pressure, g per cmz viii vap w. b. X,y,Z $5 (t) universal gas constant, BTU per mole per degree Rankine degrees Rankine mass rate of flow, g per sec second time or, as a subscript at time t absolute temperature, degrees Rankine as a subscript, vapor as a subscript, wet basis rectangular coordinates, cm 1, defined by equation (4. 4. 2) coefficient, sec' boundary condition at time t Reynolds number ix I. INTRODUCTION Hay has always been and continues to be one of the most important agricultural crops grown in the United States of America. Zimmerman (1964) reported that hay ranks second in both acres (65 millions) and value (2. 5 billion dollars) on the list of agricultural crops. Hay can be harvested in long-loose, chopped, baled or wafered forms. The United States Department of Agriculture (Agri- cultural Statistics, 1963) estimated that about 85 percent of the hay crop is baled, 4 percent chopped and 10 percent handled as long-loose hay. No data are available on the amount of wafered hay produced in 1963. However, it is known that the interest in wafering is growing rapidly (Zimmerman, 1964). Hay wafering is a process whereby the hay is tightly compressed into small high density packages. The shape, size and density of the individual wafers varies according to the make of the wafering machine. The major benefit of wafering is the relatively low labor requirement of the system compared to other available hay harvesting methods. Wafers make a completely mechanized hay harvesting and feeding system possible, because, if properly made, they will flow through augers, conveyors and other types of transporting equipment. Additional advantages of wafers are the high bulk density and nutritional value of the product. Matthies (1963) reported that the bulk density of wafers is two and one-half times as large as that of bales and four times as large as that of loose hay. The animal consumption of wafers is significantly greater than of baled hay while the feeding losses are reduced and the selective eating by the animals is eliminated (Ross e_t ail. , I963). Before wafers can be stored they must be dried. The maximum moisture content of long hay for safe storage is about 17 percent wet basis (Hall, 1957). However, the moisture content of forage wafers has to be reduced to 12 or 13 percent wet basis to prevent molding after three to four weeks (Lamp e_t a_1_1. , 1963). It is not clear why long hay and forage wafers develop mold at different moisture contents. It seems most likely that the mechanical treatment of the hay during wafering changes the sorption characteristics of the forage, resulting in a shift of the moisture equilibrium curve. Notwithstanding the rapid molding of forage wafers, no precise data are available for predicting the drying behavior of forage wafers under different conditions. Holdren e_t a_l. (1962) performed drying tests with alfalfa wafers stored in a round steel grain bin fourteen feet in diameter. The results of this investigation were inconclusive as far as the effect of the wafer density and ambient air temperature were concerned. It can be expected that the necessity for drying wafers will become more pronounced in the future, because of the expected development of new wafering machines which will be able to process forages with an initial moisture content of over fifty percent wet basis (Farrall, 196 3). 1.1 Objectives The objectives of this study were to investigate the physical laws describing the drying behaviour of forage wafers and to relate wafer size, wafer density and ambient air temperature to the drying parameters of the individual forage wafers. 1. 2 Topics Covered in the Dissertation parts: The work reported in this study may be divided into five 10 The treatment of a forage wafer as a two dimensional body with constant boundary conditions. The effect of drying air temperature and wafer density on the diffusion coefficients of forage wafers. The treatment of a forage wafer as a three dimensional body with a changing diffusion coefficient. . The treatment of a forage wafer as a two dimensional body with changing boundary conditions. .- The treatment of a forage wafer as a one dimensional body. II. THEORY OF DRYING The history of theoretical drying goes back to 1904 when the Russian scientist Kossowitsch published his thesis on the molecular mechanisms of moisture movement in capillary products (Toponitzki, 1949). The first serious attempt in the English speaking world to explain the mechanisms of drying did not come until almost twenty years later when Lewis (1921) published his now famous treatise on the rate of drying in solids. One person who deserves more credit for the early development of the theory of drying than any other researcher is T. K. Sherwood. In the period from 1929 to 1936 Sherwood published a series of articles which have become the basis of the diffusion theory in drying (1929, 1930, 1931, 1932, 1933, 1936). Although the economical importance of drying is well recognized, no textbooks have been published in the English language on the theo- retical aspects of drying. One exception is the book by Van Ardsdel e_t a_1_1. (1963) in which some theoretical drying principles are briefly reviewed. In order to study the subject extensively a knowledge of German is essential. Several books have been published rather recently in Germany dealing exclusively with the principles of drying (Lykov, 1955; Lebedev, 1960; Maltry e_t a_._1. , 1962; Krischer, 1963). Since these works are not available to the average English reader, they will be referred to rather frequently in this study. The physical and chemical make-up of a product determines to a large extent the drying behavior of that product. Most materials which are dried in the chemical engineering industry are non-hygro- scopic. - On the other hand, most agricultural products are classified as hygroscopic. The difference between hygroscopic and non-hygro- scopic materials is based on the final moisture content to which a material can be dried in a particular mass of air. Hygroscopic materials, in contrast with non-hygroscopic materials, do not dry to zero percent moisture content in air of‘which the vapor pressure is greater than zero, but approach an equilibrium condition at which the moisture content is larger than zero. At the moisture equilibrium content the vapor pressure of the product is equal to the vapor pressure of the surrounding air. Several physical mechanisms, namely, osmosis, capillary forces and diffusion, can cause the flow of moisture in a capillary- hygrosc0pic product during drying. The importance of each mechanism is limited to a particular moisture content range of the drying product. Lykov (1955) showed that osmosis plays an important role in drying only when the moisture content of the product is above the hygroscopic point. The hygroscopic point is defined as the product moisture con— tent corresponding to a relative humidity of 100 percent. Since the initial moisture content range of forage wafers is well below the hygro- sc0pic point (Bakker-Arkema e_t a_l. , 1962), the mechanism of osmosis does not have to be considered in this study. The drying of sand and other similar non-hygroscopic products is controlled mainly by capillary forces due to the molecular attraction between the liquid and the solid. Ceaglske and Hougen (1937) main- tained that all drying processes are based on capillary behavior. Krischer (1963), among others, pointed out that only during the constant rate drying period, when the flow of moisture away from the drying surface limits the rate of drying, capillary forces determine the rate of moisture transfer through a capillary-hygroscoPic product. Since no constant rate drying period has been observed in forage wafers, it can be assumed that capillary forces can not be the mechan- ism responsible for the drying behavior of forage wafers. If neither osmosis nor capillary forces describes the drying behavior of forage wafers, the third of the three physical mechan- isms, diffusion, must be the determining factor controlling the process. The exact nature of the potential causing drying has been the cause of much debate among researchers. Carling (1956) distinguishes between: (a) liquid diffusion due to a concentration gradient; and (b) vapor dif- fusion due to a vapor pressure gradient. Which mode of diffusion applies to the drying of a particular material is difficult to determine. In general, it can be stated that liquid diffusion only takes place in the moisture content range corresponding to relative humidity values between 85 and 100 percent, while vapor diffusion becomes important at lower moisture contents (Krischer, 1963). Although it is of considerable academic interest which type of diffusion is the controlling factor in the drying of forage wafers, this knowledge is not required for writing the differential equation of the general process. The diffusion equation which describes the drying behavior has the form: bc = - -——-— . 1.1 o D bx (2 ) For liquid diffusion equation (2. l. 1) becomes: bcliq = - D -——-—— 2. 1. 2 C.Iliquid liq b x ( ) Since the liquid concentration is equal to the dry weight density times the moisture content dry basis, equation (2. 1. 2) can also be written as: . b (M.C.) = - o 1. Gliquid Dliq J bx (2 3) where j is the dry weight density of the material. In the case of vapor diffusion equation (2.1. 1) becomes: ac o = -D ————"—a—9 (2.1.4) vapor vap Z x Treating water vapor as an ideal gas and applying the ideal gas law gives for equation (2.1.4): 3P 1 vap = - D 010 vapor vap R T a x (2 5) vap b Pvap/bx can be written as [a Pvap/a (M. C.)] [a (M. C.)/b x] and so equation (2. 1.5) becomes: 1 b 1Dvap a (M.C.) (2 l 6) ax o O G - D * vapor vap Rvap T 3(M.C.) As Lykov (1955) pointed out, the importance of equations (2. 1. 3) and (2. l. 6) lies in the fact that both types of mass transfer, liquid diffusion and vapor diffusion, are due to the same driving potential, namely, the moisture content gradient of the drying material. This means that the use of the general diffusion equation for the drying of forage wafers is justified: o=-D 994°C" (2.1.7) X where D a b p . v A va D‘ Dlqu+R T a(M.c.) (2'1'8) vap It is possible to obtain values for D, the diffusion coefficient, by the use of equation (2. 1. 7). The disadvantage of this equation is that a static condition must be reached before the values of G and b (M. C. )/5 x can be measured and D be calculated. A static condition was never reached when the method was tried with forage wafers because the wafers started crumbling and molding long before a constant value for G was obtained. If a mass balance is written for a cube using equations (2.1. l) and (2. 1. 7), respectively, the following second order differential equations result: %%=DI—'z—+ aiyzC—i'g'r: (2.1.9) and aux/1c.) = D[ blame.) + blame.) + 52(M.C.)](2110) a t E x7 & y2 5 Kat - . In both equations (2. 1.9) and (2. 1. 10) the assumptions are made that the diffusion process is isothermal and that the diffusion coefficient is constant. The equations are simplified if drying takes place from just two opposite surfaces. Only the concentration gradient in one direction has to be considered in that case. The solutions to equations (2.1. 9) and (2.1.10) of the concen- tration and the moisture content as a function of time and position, will depend on the initial and boundary conditions of the drying material under study. Newman (1931) derived solutions of the one dimensional diffusion equation for the drying of four cases: a) initial moisture content uniform and surface moisture content constant b) initial moisture content uniform and the rate of drying at the surface constant c) initial moisture content uniform and the rate of evaporation at the surface a function of the surface moisture content d) initial moisture content parabolic and the surface moisture content constant Case b is only of interest in the constant rate drying period, which has not been observed for forage products. Case c is of im- portance for products of which the mass transfer coefficient is finite due to a thick boundary layer. This condition will only occur during natural convection air drying; forced convection air drying will soon make the mass transfer coefficient infinite. The parabolic initial moisture content distribution of case (1 is of importance in products with an initial moisture content above the hygroscopic point. During the drying of such materials the moisture content distribution becomes parabolic when the hygroscopic moisture content is reached (Lykov, 1955). Due to the low initial moisture contents of forage wafers, this case does not have to be considered in this study. Forced air drying of forage wafers is best represented by case a. The mass transfer coefficient in forced air drying is very large which accounts for the assumption that the surface of a product attains the moisture equilibrium content as soon as the drying has started (Lebedev, 1961). ' Many solutions to equations (2. 1.9) and (2. 1.10), representing different initial and boundary conditions, can be found in the heat transfer literature. The book on conduction heat transfer by Carslaw and Jaeger (1959) is especially useful in this respect. When the solutions for the concentration or moisture content distribution, obtained from equations (2. 1.9) and (2.1. 10), are inte- grated with respect to time, values for the amount of moisture dif- fused in or out of the product, are obtained. Crank (1956) gives such solutions for many different initial and boundary conditions. Several investigators (Babbit, 1949; Becker e_t a_1_l. , 1955; Hustrulid e_t 31. , 1959; Pabis e_t a_1. , 1961), have used the solutions of the diffusion equation given by Lewis (1921), Sherwood (1929) and 10 Newman (1931) in the drying research of agricultural products. The predicted drying curves agreed reasonably well with the experi- mental data. However, when the experimental curves were examined closely, divergence from the theoretical curves can be observed in most cases. The divergence is usually most noticeable during the first and last stages of drying. The theoretical drying rate at small times is higher than is observed from the experimental data. Pabis e_t_ a_._1. (1961) explained this phenomenon by taking the temperature change of the drying product into account. The divergence of the theoretical drying curve from the experi- mental curve during the last stages of drying of many organic products, is probably a result of a change in the value of the diffusion coefficient (Van Arsdel, 1947). Becker e_t a_1. (1955) found that in the case of wheat the calculated and experimental values agreed well for moisture contents from 20. 5 to 12 percent dry basis. In the moisture content range below 12 percent smaller diffusion coefficients had to be used to make the theoretical curve fit the experimental values. No extensive study has been made on agricultural products as to the exact behavior of the diffusion coefficient in the moisture content range from 5 to 10 percent wet basis as has been done for other products. King (1945) studying the drying behavior of keratin proteins found more than a hundred-fold change in the diffusion coefficient during the drying from 12 to 2 percent wet basis. Fish (1958) reported a decrease in the diffusion coefficient from a value of 1. 23 x 10-7 cmZ/sec to a value of 0. 05 x 10'7 cmz/sec for starch gel in the moisture content range from 30 to 0 percent dry basis. It should be stressed that not all drying products show a con- tinuous decrease in the diffusion coefficients at low moisture contents. Jason (1958) found that the analytical solution to the diffusion equation for the drying of fish muscle described the drying process if two 11 constant diffusion coefficients were used; his data did not show a change in the diffusion coefficient in the range from 10 to 2 percent moisture content dry basis. Krischer (1963) reported that soap and paper are two products which appear to have constant diffusion co- efficients over the full moisture content span in which diffusion is the rate controlling drying mechanism. The drying of forage wafers reported in this study is limited to single wafer drying. This can be compared with single layer drying of grain. No attempt has been made to investigate multi-layer drying of forage wafers. The literature of deep bed drying, therefore, will not be reviewed. III. EXPERIMENTAL 3.1 Forage Wafers The forage wafers employed in this study were made with an experimental wafering machine manufactured by the Massey-Ferguson company. The machine is shown in Figure 1. The mechanics of the machine have been described by Lundell ft 31. (1961). The only dif- ference between the Lundell and the experimental Massey-Ferguson machine is the position of the wafering dies. Figure 2 is a close-up of the horizontal wafering dies of the Massey-Ferguson machine. The cross-sectional area of each die, with zero p. 3.1. die pressure applied, is 6. 0 x 6. 5 cm = 39 cmz. The length of the wafers varied from 1 to 8 cm. Schoedder e_t a_1. (1964) reported that the average size of the Massey-Ferguson wafers is 6. 2 x 6. 3 x 6.7 cm. The density of the wafers varied from 0. 25 to 1. 00 g/cm3. Wafers with a density below 0. 4 g/cm3 were not tested due to their non-uniform character. The average density of alfalfa wafers made with the Massey-Ferguson machine is 0.435 g/cm3 (Schoedder eta}. , 1964). The wafer densities are expressed on a dry matter basis. The wafers used in this study were made from a chopped alfalfa- grass mixture (92 percent alfalfa, 8 percent grass) with an initial moisture content of 18 percent dry basis. The average length of the chopped forage was two inches. Moisture was added to the forage before it was wafered. The moisture content of the wafers was between 20 and 40 percent dry basis. Some of the wafers employed in the drying tests were chemically analyzed. Table 1 gives the results. 12 l3 Figure 1. Experimental Massey-Ferguson wafering machine. Figure 2. Close-up View of wafering dies. 14 Table 1. Feed analysis of forage wafers used in the drying tests, percent oven-dry basis. Crude Ether N-free Sample Protein fiber extract Ash extract 1 16.56 22.99 1.40 7.41 51.64 2 16.63 28.05 1.45 7.06 46.81 3 16.50 25.52 1.45 6.75 49.78 Immediately after the wafers were made they were placed in plastic bags and stored at 33°F. The wafers were left in storage for at least three months to assure an uniform moisture distribution in each wafer before it was used as a drying sample. A total of 216 wafers were dried and their drying curves analyzed. The average wafer produced by the Massey-Ferguson wafering machine was evenly shaped except at the two ends. The non-flat wafer ends can be seen sticking out of the wafering dies in Figure 2. Since it is very difficult to treat an irregularly- shaped body mathe- matically, the wafers were cut in brick- shaped samples on a bandsaw. Figure 3 shows some wafers of different thicknesses which were obtained from irregularly- shaped parent wafers. When it was desired to limit the moisture gradient in a drying wafer to one dimension, four sides of the wafer were coated with a moisture-proof material. This material had to have the following properties: a) moisture-proof; b) heat resistent; c) expandable; d) adhesive, and e) inert. After testing many materials a metal :1: . mender, consisting of 50 percent polyester-resm and 50 percent >1: White Velvet manufactured by ABC Autoparts, Inc. , Lansing, Michigan. 15 inert filler, was found to have the required properties. The only disadvantage of the material was that the weight decreased somewhat when placed in the oven at high temperatures. The weight loss of the material was accounted for in the wafer drying curves. Figure 4 shows some wafers coated with the metal mender. A full-size wafer along with its moisture-proof shell can be seen in Figure 5. 3. 2 Drying Setup An overall view of the instrumental setup used in obtaining the drying curves for forage wafers is shown in Figure 6. The equip- ment consists of a drying oven, analytical balance, automatic record- ing potentiometer and a hot-wire anemometer. The balance, located on top of the oven, was fitted with a brace on which a mesh-wire basket was suspended. A sample was placed in the basket during a drying test. This setup made it possible to weigh the sample without removing it from the oven. . Figure 7 is a close-up view of the interior of the oven. The illustration shows, in addition to the drying basket and sample, the thermocouples and the hot-wire anemometer probe for measuring the wafer temperatures and oven air flows, respectively. A Precision Scientific (cat. no. 31058) oven (range 100° to 500°F) served as the drying apparatus. The velocity of the oven air was adjustable between 75 and 250 feet per minute. The oven temperature was thermostatically controlled. A sensitive microswitch was added to limit the temperature span to 40F. A Mettler balance (type k7, accuracy .1: 0. 03 g) was employed to obtain the drying wafer weights. The wafer temperatures were sensed by means of copper-constantan (1938 calibration, 24 B and S gage) thermocouples and recorded on a Brown (Type 153 Electronik) 12 point potentiometer. The oven air velocities were measured with a hot-wire anemometer (Flow Corporation, Model HWB3). The moisture contents of the forage wafers were determined after drying the samples for 48 hours at 2000F. 16 . ___§‘. Figure 3. Different size forage wafer used in drying tests. Figure 4. Wafers covered on four sides with moisture-proof material. "Q | 2‘ ' . Figure 5. Moisture-proof shell plus forage wafer. 17 Figure 6. Experimental drying set up .— .— - - - - Figure 7. Close up view of drying oven. IV. RESULTS AND DISCUSSION 4. l Diffusion in Thin Wafers with Constant Boundary Conditions The forage wafers used for the drying tests in the first two sections of this study were coated with the moisture-proof compound on all but two sides. This caused the moisture content gradients in the y and z directions, 5 (M. C. 1/3 y and a (M. C. )/é z, to be zero. When the thickness of a wafer is 2 g, with the surfaces located at + Z and - Z and a constant diffusion constant D is assumed, the general diffusion equation describing the isothermal drying process becomes: 2 RC = D C (4. 1.1) 5 t x If the boundary and initial conditions are Cx ifl = Ceq at t 2 o and Ctzo = Cin for - Z \ N ..\"’ 5 9°: 1.0 \ \ 9x 0-9 \V \ O 0.8 \ \ OJ \\ \ A \ O (16 ‘\\\\‘\\!‘N ‘~\“‘\\L 0.5 \\ \\ °'“ e—e' Density 040-050 g/cm° ' \ o—n Density0.65-0.75.g/cms \ 0.3 0—0 Density=0.90-I.OO g/cm’ \ 0.2 L50 L55 L60 L65 I/T, °R )1 IO‘3 L70 L75 ' L80 34 brick- shaped wafer is not constant, but will increase slowly during the course of drying. This means that the diffusion coefficient will also change. If the diffusion coefficient of a particular wafer is known at one temperature (this value can be found by analyzing a thin slice of the wafer), the point representing the wafer can be located on Figure 11. A straight line drawn through this point parallel to the other lines in Figure 11 will then make it possible to find the values of the diffusion coefficient at any time assuming that the wafer temperature history is known. It is interesting to compare the values of the diffusion coefficients of forage wafers obtained in this study with the diffusion coefficients reported by other researchers for different hygroscopic products. In Table 6 some of the values available in the literature are tabulated. The diffusion coefficients for wood and potatoes are in the same range as those of forage wafers. Table 6. Diffusion coefficients times 10“5 for several hygroscopic products. D, Temp. Product cmzjsec oF. Investigator Comments Wheat 0.69 68 Becker e_t a_1. (1955) M. E.=dynamic M. E. Wheat 27.70 176 Becker e1: a_1. (1955) M. E.=dynamic M. E. Corn 2.78 115 Pabis e_t: a1. (1961) M. E.=dynamic M. E. Corn 5.56 152 Pabis at; al. (1961) M.E.=dynamic M. E. Wood(pine) 0.47 122 Stamm _e_t a_Ll. (1961) tangential diffusion Wood(pine) 1. 22 190 Stamm gt a_1. (1961) tangential diffusion Wood(pine) 0. 53 122 Stamm _e_t a_1. (1961) radial diffusion Wood(pine) 2. 56 190 Stamm e_t a_1. (1961) radial diffusion Cottonbale 3.60 59 Henry (1939) density 0. 2 g/cm3 Cottonbale 0.79 59 Henry (1939) density 0.4 g/cm3 Cottonbale o. 22 59 Henry (1939) density 0.6 g/om3 Catfish 0. 36 86 Jason (1958) Cod 0. 34 86 Jason (1958) Potato 1.0 130 Saravacos e_t a_1. (1962) potatoes were Potato 2.4 156 Saravacos e_t l. (1962) scalded 35 In this section it was shown that the diffusion coefficients of forage wafers are dependent upon the drying air temperature and the wafer density. Arrhenius plots relating these three variables were ” established for the slowest drying samples. An increase in the drying air temperature and a decrease in wafer density resulted in an in- crease of the diffusion coefficient. 4. 3 Diffusion in Brick Shaped Wafers with Constant Boundary Conditions In section 4. 1 the drying of thin wafers with an initial tempera— ture equal to the ambient air temperature was analyzed. It is questionable if the results of that section are of direct practical value since the average thickness of a Massey-Ferguson wafer is 6. 2 cm while the initial temperature is well below that of the ambient drying air. In this part of the study the influence of these two factors on the drying behavior of forage wafers will be investigated. Brick- shaped, non-insulated wafers of uniform density were selected and dried. An one cm slice was cut from each wafer, insu- lated on all but two sides and dried at the same temperature as its parent, the brick-shaped wafer. The initial temperature of the thin wafer was equal to that of the ambient drying air; the brick- shaped wafer temperature was equal to the room temperature. The drying curves of the brick- shaped wafer 92A and the thin wafer 92B will be analyzed. The dimensions of wafer 92A and 92B were 6.2 x 5.7 x 6.1 cm and 6.2 x 5.7 x1.0 cm, respectively; the densities, 0. 518 g/cm3 and 0.510 g/cm3. The initial temperature of 92A was 79°F, the initial temperature of 92B, 1380F. The drying oven temperature was 1400F. The diffusion coefficients were calcu- lated using the method of section 4. 1. The results are tabulated in Table 7. 36 Table 7. Diffusion coefficients times 10"5 for wafers 92A and 92B¥ as calculated from equation (4. 1. 3). M.C., % d.b. 92A 923 42.22 .30 .89 39.13 .58 1.59 36.37 .65 1.65 31.60 .93 1.65 23.60 1 45 1.66 19.75 1.50 1.64 17.20 1.61 1.66 15.00 1 71 1.65 9.80 1 82 1.67 8.85 1 84 1.65 92A 92B * . o o Dry1ng temperature 140 F 140 F Dimensions 6.2x5.7x 6.1cm 6.2x5.7x1.0 cm Density .518 g/cm3 .510 g/cm3 Insulated surfaces none four Initial temperature 79°F 1380F It is obvious from Table 7 that the initial temperature and the wafer thickness have a pronounced effect on the diffusion coefficients of forage wafers. The thin wafer displays an almost constant diffusion coefficient soon after the start of drying, while the diffusion coefficient for the brick- shaped wafer increases steadily in value. It will be shown that the difference in the drying behavior between wafers 92A and 92B, which were cut from the same parent wafer, is largely due to the different temperatures of the wafers during drying. Pabis 3t a_1. (1961) considered the change in temperature of corn kernels during drying. They assumed that the ratio of the external to the internal heat transfer was so large that the temperature gradients in the drying kernels could be neglected. The measurement of the temperature as the center of a corn kernel was therefore sufficient to 37 obtain the temperature history of the drying product. The problem is more complicated for drying forage wafers. The temperature gradient in brick-shaped wafers is substantial and has to be taken into consideration in the analysis of the drying curves. Before the moisture can evaporate from the surface of a drying solid, it must travel a certain distance through the internal parts of the solid. The distance covered by each moisture molecule is differ- ent, since their adsorption sites on the internal surfaces of the solid are different. Due to the temperature gradient in the drying brick- shaped wafer, each water molecule in the wafer will traverse through a different path-length and a different temperature differential. The overall effect of this condition on the drying behavior of forage wafers appeared to be accounted for best by considering the change in the average temperature during drying. This in turn leads to the con- sideration of the average diffusion coefficient of forage wafers. To obtain the average temperatures of wafers 92A and 92B at different times during drying, nine thermocouples were connected in parallel and distributed through each wafer. The location of the thermocouples is calculated in Appendix A. l. The temperature read- ings showed that the average temperature of the thin wafer 92B re- mained almost constant at 1380F while the thicker wafer 92A did not reach this temperature until it had become practically dry. In Figure 12 the average temperature of wafer 92A is plotted versus time. The temperature increased very slowly after a fast initial rise. After twelve hours of drying the temperature difference between the wafer and the ambient drying air was still 40F . It was shown previously that the diffusion coefficient is a function of temperature, with an increase in temperature resulting in an in- crease of the diffusion coefficient. Therefore the increase in D of wafer 92A as reported in Table 7 is very likely caused by the slow in- crease of the wafer temperature. .2; 65:. 38 m. x N. o. m m o N o mm cm 09 .o__ m... w d a m M m. > 8..» 0 on. a 1m 82“."; EM... .303. , C 222362 cmSC O C < < a t _ a .< .NH oksmfm 39 Although it is clear from Table 7 that the diffusion coefficient of wafer 92A is not constant due to the change in wafer temperature, the 'H theoretical moisture ratios were calculated using one value for D as was done in section 4. 1. Since the constant diffusion coefficient of wafer 92B was 1.65 x 10‘5 cmZ/sec, this value was chosen to obtain the moisture ratio values of wafer 92A. The theoretical curves as well as the experimental values of both wafers are plotted in Figure 14. As could be expected, the experimental points of 92A fall well above the theoretical drying curve; wafer 92B follows the theoretical curve closely. The solution of the basic diffusion equation for cases with a variable diffusion coefficient is generally rather complicated. However, in the case of drying wafers the diffusion coefficient is a function of time due to the change in the average wafer temperature in the course of drying. This makes the solution comparatively easy. The diffusion equation for a brick- shaped wafer is: b(M.c.) z‘blxm‘t’ o\o(1:.c.)]+$%[mt) 3%4;c.)]+55_ bit 2 [D(t) baa/:0) ] (4.3.1) or since D is not a function of x, y and z: a (M1,:C.) = Du” 9???) + Eggs.) + 55:11:43.) 1 (4.3.2) Making the transformation: dT = D(t) dt (4.3.3) equation (4. 3. 2) becomes: 2 2 2 biNfrfi-J ._. %%"°') + 9393,39) + 9:11:59) (4.3.4) 40 The solution to equation (4. 3.4) for the average moisture content, assuming a moisture gradient in one direction only, then becomes: (M.C.)T - (M.C.) 9‘1 (M.C.). - (M.C.) 1n eq 8 oo = Z (2n+1)qr exp [-(2n+1)2n2T/4ZZ] n=o (4. 3. 5) Comparing equations (4. 1.4) and (4. 3. 5), it can be seen that equation (4. 3. 5) can be obtained by replacing Dt in equation (4. 1. 4) by T. The moisture ratio values for the brick- shaped wafer can be calcu- lated by multiplying three series of the type on the right hand side of equation (4. 3. 5), each series representing a value for Ag (Newman, 1931). The table in Appendix A. 1 can be used for the evaluation of each series. Before equation (4. 3. 5) can be employed for calculating the theoretical moisture ratios, the relationship between T, D and time as expressed by equation (4. 3. 3) must be solved. Unfortunately the integral in equation (4. 3. 3) could not be evaluated formally for the case under study and so the relationship between T, D and time had to be obtained graphically. A similar procedure was followed by Pabis e_t a_1. (1961) in analyzing the drying curve of shelled corn. The relationship between the diffusion coefficient and the wafer temperature was obtained previously in section 4. 2. Since the history of the average temperature of the brick- shaped wafer 92A is known (see Figure 12), a plot can be drawn of the average diffusion co- efficient of the wafer versus time. Figure 13 shows this relationship. It is interesting to note that the diffusion coefficient has not reached the theoretical value of 1. 65 x 10"5 cmZ/sec after 16 hours of drying. The actual value at that time is only 1. 58 x 10'5 cmZ/sec due to the o , . . 4 difference between the amb1ent air and the average wafer temperature. 41 o— S .2; .25 N. o. . m L. m w wd ad 0.. S O 0 C) m._ . udofioamwooo :onDHEQ .2 23E (oaS/Zw0)9,0( x g 42 After the average diffusion coefficients at different drying times are known, equation (4. 3. 3) can be evaluated graphically. The values for T are given in Table 8. The series on the right hand side of equation (4. 3. 5) can now be calculated for any values of Z with the aid of Appendix Table A. 1; in the case of the brick- shaped wafer 92A /1 = 3.05, Z; = 2.85, and £3 = 3.1 cm. The series are called L1, L2, and L3 corresponding to the spatial dimensions £1, 4, and 4, respectively and are tabulated along with the theoretical and experimental moisture ratio values in Table 8. The theoretical moisture ratios are obtained by multiplying the three series L1, L2, and L3. The theoretical drying curve for wafer 92A is drawn in Figure 15 along with the experimentally found points. The experimental data fit the theoretical curve much better in Figure 15 than in Figure 14. This could have been predicted since the temperature change during drying of the wafer is only accounted for in the construction of the theoretical drying curve of Figure 15. The experimental points in Figure 15 fall above the theoretical drying curve during the initial drying period. This trend can also be observed in Figure 10, the theoretical drying curve for a thin wafer. An explanation of this phenomenon will be given in section 4. 3. Recognition of the fact that the temperature of large pieces of drying material changes only gradually, has important implications for the drying of at least one other agricultural commodity, namely, high density hay bales. It has been reported by the Aerovent Com- pany (1964) that difficulties have been encountered in obtaining dry bales before molding starts. High density hay bales are made from the same basic material as forage wafers, but the density of the bales is lower than that of wafers. It is likely that the slope of the line on the Arrhenius plot for high'density bales will be equal to the lines in 43 1.0 °'9§I\ 0.8 O .1\\ 0 \. \\ o .9 06 \ o“? \ \ o 0.5 8 \ O {—1 \0 \ '5 0.4 \R Q 2 \>\ O 0.3 923 — Theoretical 0 Experimental .. P l I ' 0 r 2 3 4 5 6 7 a 9 I0 (I 12 Time, hrs. Figure 14. Drying curves for brick- shaped wafer 92A and thin wafer 92B assuming a D = 1.65 x10'5 cmz/sec. (.0 0.0 \Q or \ .9 0.6 \O‘\\ a? \e.\ 3 0.5 :3 O 5 0.4 - \ . \k — Theoretical \ 0 Experimental 0.3 0'2 I I2 3 4 5 6 7 a 9 )0 II )2 Time, hrs. Figure 15. Drying curve for brick- shaped wafer 92A, assuming a changing diffusion coefficient. 44 Table 8. Drying parameters for the calculation of the theoretical moisture ratios of wafer 92A. Time, Eemp” Dx10+5., T, min. F. cmz/sec cm2 L1 L2 L3 M.Rth M'R'exp 55 89 0.900 0.706 .951 .943 . .953 . .856 .879 92 105 0.106 1.307 .921 .916 .922 .780 .810 135 115 1.200 2.104 .896 .890 .897 .714 .739 225 124 1.350 4.034 .859 .850 .862 .632 .648 350 130 1.446 6.975 .815 .803 .815 .533 .540 440 133 1.495 9.214 .786 .773 .791 .481 .482 585 134 1.530 12.850 .748 .733 .753 .412 .410 690 135 1.550 15.574 .722 .704 .726 .369 .. 371 805 136 1.560 18.563 .697 .678 .698 .330 .321 Figure 11. When the location of this line in Figure 11 and the tempera- ture history of the bales are known, it is possible to predict the time required for drying the bales to a particular moisture content. Also, if the maximum time to prevent molding is given, the required drying air temperature can be calculated. It is somewhat surprising that only in two drying investigations concerning agricultural products (Becker and Sallans, 1955, and Pabis st :11. , 1961), the product temperature rather than the drying air temperature, is used as the drying variable. Simmonds e_t a_1. (1953), in studying the drying of wheat, eliminated the heating up period by making the initial grain temperature equal to the drying air tempera- ture. No one, however, has seemingly recognized that large products with low thermal conductivities do not reach the drying air tempera— ture before they are considered dry enough for storage. A serious drawback of the theoretical treatment presented in this section is that it was not possible to calculate the average wafer temperatures during drying. The equations describing heat and mass 45 transfer are basically the same and so the same solutions can be used for obtaining the average wafer temperatures and the average wafer moisture contents. Equation (4. 3. 4) was not solved for the average wafer temperatures at different times, because the relation- ship between the thermal conductivity and the moisture content for forage material is not known. If a plot of the thermal conductivity versus time, similar to Figure 13, could have been drawn, equation (4. 3. 5) would have furnished the temperature history of the drying wafer, which in this study had tobe found experimentally (see Figure 12). Although the relationship between the thermal conductivity and the moisture content was not known for forage wafers, a theoretical curve was plotted for the average temperature of a wafer versus time, assuming a constant value for the thermal conductivity of 0.30 BTU/hr it°F. The value of 0. 30 BTU/hr ftoF was reported by Krischer (1963) for paper which appears to have approximately the physical characteristics as forage wafers. The resulting theoretical temperature curve agreed rather well with the experimentally found points for the first two drying hours, but deviated substantially from the experimental points during the rest of the drying period. Two factors caused this deviation. In equation (4. 3.5) the heat required for evaporation of the moisture is not taken into consideration. Furthermore the assumption of a constant thermal conductivity is in- correct. Krischer (1963) has reported the thermal conductivities of some materials in the dry and wet state and concluded that the thermal conductivity of a material during drying may change by as much as 150 percent. The wet material always showed a higher thermal conductivity value than the dry material. It has been shown that the temperature of a drying product is an important factor in determining its drying behaviour. It cannot be 46 assumed that forage wafers reach the ambient air temperature shortly after the drying starts. A method has been described to account for the changing wafer temperature in the analysis of a brick- shaped wafer. The theoretical curve agreed rather well with the experi- mental points. The temperature history of a drying wafer could not be determined theoretically because the relationships between the thermal conductivity and the wafer moisture content are not known. 4. 4 Diffusion in Thin Wafers with Changing Boundary Conditions In the previous section it was concluded that the diffusion equa- tion with constant initial and boundary conditions describes the drying process correctly over all but the initial drying period. The explana- tion for the deviation from the theoretical drying curve for small drying times may be twofold. A slight temperature change of the wafer during drying will be caused by the fact that drying is not a true isothermal diffusion process. This is true even where the initial wafer and drying air temperature are the same. The result is a changing of the diffusion coefficient , although the effect appears to be insignificant. A more important fact which may cause a deviation of the experimental drying curve from the theoretical drying curve, is an improper choice of the boundary conditions used in solving the diffusion equation. It is this possibility which will be investigated in the following section. The assumption has been made in section 4. 1 that the surfaces of a forage wafer immediately attain the moisture content equilibrium after drying starts, or in mathematical terms, that the boundary con- dition for diffusion in a wafer is described by Cx-ig = Ce for t 2 o. q Carslaw and Jeager (1959) reported that if a steel bar is suddenly 47 dropped in a hot bath, the surfaces of the bar do not reach the temperature of the heating fluid instantaneously. Consequently the boundary condition used in the solution of equation (4. 1. 1), namely, that the moisture equilibrium is reached at t 3 0, may have been an approximation rather than a true mathematical fact. There are several solutions available in the heat transfer literature for the temperature distribution in solids with variable boundary conditions. Carslaw and Jaeger (1959) list solutions for the temperature distribution in a plane with linearly and exponentially changing surface temperatures. Crank (1956) has adapted these solu- tions for diffusion problems and has calculated the total amount of diffusing substance in a plane sheet as a function of time by integrating Carslaw and Jaeger's solutions with respect to time. The experi- mental drying curves of forage wafers will be compared with the solu- tions obtained by Crank for diffusion in a plane sheet with variable surface concentrations. In the case that the moisture concentration at the two surfaces of a plane wafer varies linearly with time, i. e. , 01 (t) = 02(t) = kt, the total amount, Qt’ of diffusing moisture is given by the expression (Crank, 1956): DQt 2Dt 64 00 exp -D(2n+1)°1r°t/4 £7- - -2: H’— " T 3 +73 n30 (2n+1)T ”'4'” In comparing the experimental drying curves of several wafers with the theoretical curves obtained from equation (4.4. 1) for different values of k, the experimental values were found to differ very sub- stantially from the theoretical drying points. Therefore it was con- cluded that a linearly changing boundary condition does not describe the behavior of a drying forage wafer surface correctly. 48 Much better success was obtained by assuming that the surface concentration varies exponentially with time. Crank (1956) gives a solution of the total amount of moisture absorbed in time t for a plane sheet of thickness 2 Z, of which the concentration is initially zero and of which both the surfaces are approaching the equilibrium concen- tration Ceq exponentially. This boundary condition can be written as: , - t a, (t) = (mt) = ceq (1-e °) (4.4.2) It was found that Crank's solution contains an error. The correct solu- tion appeared to be: Q 1 t _ = 1- exp [-(Dt/ZZ) (oil/D) tan (D/I3 £2121 ' Zceq 8 00 exp [—(2n+1)?‘rrz DtL4 £21 _172' Z (2n+1)°“[1-(2n+1)Z Dw‘74i552] (4'4'3) n=o It is obvious that the boundary condition given in equation (4. 4. 2) applies to adsorption rather than to desorption. The proper boundary condition for a drying wafer is: -Bt : : + - . . (mt) 912m Ceq (Cin ceq) e (4 4 4) The equation for the total amount of moisture diffused out of the wafer will, however, be similar to equation (4.4. 3) except that the term Ce has to be replaced by (Cin - Ceq). To facilitate the use of equation (4. 4. 3) a graph was plotted of Qt/Z (Cin - Ceq) versus (Dt/gzfi- for different values of aZZ/D. The curves applicable to the drying of biological materials are plotted in Figure 16. The curves are of the same nature as those obtained by C rank for adsorption. Figure 16. Calculated drying curby c + (c. - c ) 6 eq 1n eq 49 km. es assuming the surface concentration for different values of (3 0.85 i / 0.8 0.7 0.6 0.5 0: 1 (cm- 0..) 0.4 0.3 0.2 0.| 0.0 75 0.2 0.4 0.6 0.8 50 The use of equation (4.4. 3) is not as direct as that of the solu- tion to the diffusion equation with constant boundary conditions (equation 4. 1. 3), since equation (4.4. 3) has two unknowns, 8 and D, while equation (4.1. 3) has only one unknown, D. To obtain the magni- tude of 8, the value of D was assumed to be equal to that obtained from the diffusion equation with constant boundary conditions at large times. From Figure 16 it can be noted that this assumption is a proper one. The curve 8 Zz/D = 00 corresponds to the boundary x=+ 4.1. The MVD: oo curve practically coincides with the 8 z/D = 75 condition C = C eq at t _>_ 0, which was employed in the section curve for large values of (Dt/Z°)°—. Since the diffusion coefficients calculated in the preceding section are obtained at relative large values of t, the diffusion coefficient calculated from equations (4. 1. 3) and (4.4. 3) do not differ much. For wafer 34A this meant that a value for D of 0.75 x 10'5 cmZ/sec could be assumed in equation (4.4.3). In Figure 17 the experimental values Q ti/Z(Cn - Ce q) versus (Dt/zez) 2"are plotted for several wafers of thicknesses between 0. 8 to l. 2 cm dried at 120 and 1400 F. Also drawn are the theoretical curves of equation (4.4. 3) for values of 8ZZ/D == 00 and 75. The important conclusion can be drawn from this figure that the drying curves of forage wafers do not coincide with the 8ZZ/D = 00 curve but with the 8K2/D = 75 curve. This means that the surface concentrations of drying wafers do not drop instantaneously to the moisture equilibrium concentration Ceq’ but that the value of Ceq is approached exponentially with time. Since the experimental values for the drying of several thick- nesses of thin forage wafers all fall on or closely to the 8Zz/D = 75 curve (see Figure 17), it can be concluded that the value of (Biz/D is constant for the particular conditions used in this study for the drying of forage wafers. This means that the value of 8 depends on the diffusion Qt /I (Cin- C") Figure 17. 51 Drying curves for thin forage wafers. 0.8 // 0.7 0.6 0.5 Theorefical 8t CIO.8c ,I20°EI 34 Ail.Ocm l40°E1 102 EILZCm, IZO”F.) 0.4 (or/r2)"15 0.6 0.8 52 coefficient and the thickness of a particular wafer. In Figure 18 8 has been plotted on log-log paper for a diffusion coefficient of 0. 75 x 10'5 cmz/sec as a function of the wafer half-thickness. The value of 8 decreases from a value of 8. 20 x 10'3 see"1 for a 0. 5 cm thick wafer to a value of 1.41 x 10" sec"1 for a 4.0 cm thick wafer. In other words the surfaces of a thin wafer will approach their moisture equilibrium faster than the surfaces of thick wafers. 8 has been plotted in Figure 19 as a function of the diffusion coefficient for a 2 cm thick wafer. An increase in the diffusion co- efficient increases the value of 8. The physical significance of this fact is that a wafer surface will approach its equilibrium moisture content sooner for a large than for a small D value. 8 changes from a value of 0. 38 x 10'3sec'1 at D a 0. 5 x10'5cm3/sec to a value of 3.0 x 10‘33ec'l at D: 4. 0 x10'5cmZ/sec for a 2 cm thick wafer. At this point the question should be raised if the values of 8 ob- tained for forage wafers in this study are representative for all single wafer drying conditions. Since 8 is a value pertaining to a wafer surface, it could be possible that the velocity of the drying air, or the Reynolds number of the drying fluid, will affect the values of 8. To investigate this possibility three wafers of the same thickness and density were dried at the same temperature but in air flows with dif- ferent Reynolds numbers. The average free stream drying air velocities used in this part of the study were 1.15 ft/sec, and 3.96 ft/sec, corres- ponding to the Reynolds numbers 1252 and 4320, respectively. The experimental points of the three wafers fell on or close to the same theoretical drying curve for 8 Zz/D = 75. This means that the values of 8 are not dependent on the drying air velocity for Reynolds numbers above 1252. This result is not surprising. Lykov (1955), Krisher (1963), Simmonds e_t a_1., (1953), and Jason (1958) all reported that air velocity has no influence on drying during the falling-rate period if the air Figure 18. 8 as a function of wafer half thickness; D = 0.75 x 10"5 cmZ/sec. B x l0‘lsec") P.- . 0.1 53 C’NOQO "7° 08 0.6 0.5 0.4 0.2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 LO Item) 20 3.0 54 Aumm\uEu 7.9 x Q s. on on o._ . . m o o.— U 8 m. m. o.m .50 N mmofiodfl pummsvficowoflwooo £033.36 mafi— mo £03093 o no a .m: unamrm 55 velocity is kept above a certain minimum value of about 12 ft/min. The minimum air velocity used in this study is well above that figure. It is interesting to speculate about the importance of the values of 8 obtained for wafers in this study. Do they hold for all hygroscopic products? Unfortunately this seems highly unlikely. The chemical and physical make-up of a product determines the adsorption and desorption rates of that product. Different individual fibers, for instance, do not have the same desorption rates (Nordon e_t a_._l. , 1960). One of the reasons that Becker and Sallans (1955) and other researchers (Hustrulid gt a_1. , 1959; Pabis e_t .11. , 1961) found constant diffusion coefficients during the initial drying period of grain may be due to the larger values of 8 and consequently to the higher desorption rates of the outer surface of the grain than of forage wafers. However, if some drying points had been determined sooner after the start of dry- ing by these investigators (after minutes instead of hours as is usually done), the physical phenomenon of an exponential moisture content drop would probably have been observed. It has been shown that the surfaces of a hygroscopic product do not obtain the moisture equilibrium content directly after the start of drying as has been assumed in prior drying investigations. The sur- face moisture equilibrium is reached exponentially. 4. 5 Diffusion in Semi-Infinite Solids When it became obvious that the diffusion equation with constant initial and boundary conditions did not describe the drying behavior of wafers during the initial drying period correctly, it was decided to investigate how long a plane wafer could be considered a semi-infinite solid. Since the solutions to most second order differential equations with changing boundary conditions or varying intrinsic properties are 56 easier obtained for a semi-infinite solid than for a slab, it would be a real advantage if the in the literature available semi-infinite solid solutions to the diffusion equation could be used for a plane wafer. When the same initial and boundary conditions are assumed for the drying of a wafer as in section 4. 1, the solution to the diffusion equation for the concentration distribution in a semi-infinite solid, bounded by the plane x = o and extended to infinity in the direction of xpositive, becomes (Crank, 1956): C. -C t,x x 1n 1n eq The total amount of moisture which has diffused out of the semi- infinite solid per unit area will then be: Q = 2(c. -C ) Pat—)3— (4°51) t 1n eq Equation (4. 5. 2) makes it possible to calculate the diffusion co- efficient at different moisture contents or drying times. Wafer 34A was used to test how long equations (4. 1. 2) and (4. 5. 2) would give the same values for D. In Table 9 the D values calculated at different times with equations (4. 1. 3) and (4. 5. 2) are tabulated. The data show the surprising result that the solutions do not diverge until the moisture content of the wafer has decreased from 22. 9 to at least 6. 98 percent dry basis. It appears that for almost five hours a 1 cm thick wafer can therefore be treated as a semi-infinite solid; more than 65 per- cent of the original moisture had been diffused out of the slab by that time. The experimental result obtained for wafer 34A was confirmed when other thin wafers were treated as a slab and semi-infinite solid and their solutions compared. The length of time that a wafer can be 57 :1: Table 9. Diffusion coefficients times 10’!5 of wafer 34A, treating the wafer as a semi-infinite solid (eqn. 4. 5. 2) and as a slab (eqn. 4.1. 3). Time, min M. C. , %d.b. D, eqn (18) D, eqn (4) 10 20.80 .33 .32 30 18.02 .58 .58 60 15,28 .69 ‘.70 90 13.23 .75 .74 150 10.38 .75 .75 210 8.44 .75 .75 270 6.98 .73 .74 300 6.47 .68 .75 430 4.33 .60 .75 * Drying temperature 1400F; wafer thickness 1 cm; wafer density 0.92 g/cm3; insulated surfaces 4. treated as a semi-infinite solid depends on its thickness. The time period was found to be equal to the time required for removing about 65 percent of the original moisture in the wafer. In the following part of this section the mathematical reasons will be explored why a plane drying wafer can be treated as a semi- infinite solid during the major part of its drying period. It is clear that initially both sides of a drying plane will act as semi-infinite solids. This condition will continue at least as long as the concen- tration at the midplane has not significantly changed. The time in which the concentration ratio at the midplane in wafer 34A had de- creased by the arbitrary small amount of 5 percent will be calculated with equation (4. 5. 1). The solution to the diffusion equation for the concentration ratio of a plane wafer with one end at x = 0 and the other end at x = 2 Z can be obtained from equation (4. 5. 1), by assuming that both ends behave 58 as independent semi-infinite solids. Since the sum of two solutions of a linear differential equation is itself a solution, the concen- tration ratio of a plane wafer becomes: C. - C 1n t,x C. -C 1n e 'X x 2 — erfc W ‘I" erfc 2 Di; (4.5.3) q At the midplane, at x = Z , the concentration ratio will be: c:in - Ct,x=£ / C, - Ce = Z erfc TF1)? (4.5.4) 1n q The time at which the 5 percent change at the midplane moisture ratio occurs can now be calculated from equation (4. 5.4): 2 0.05 = 2 erfc W (4-5-5) Using Jahnke and Ende's Tables on Error Functions, it is found that Ig/Z N/BTL— = 1. 57. Thus, the time at which a 5 percent change in the midplane concentration ratio of wafer 34A occurs is equal to 0. 95 hours. The fact that the concentration ratio at the midplane of the wafer has changed 5 percent after 0.95 hours might seem to indicate that equation (4. 5. 3) can be used for only a limited period of time. There- fore, the question remains to be answered on what theoretical grounds the treatment of the plane wafer 34A as a semi-infinite solid for 5 hours can be justified. From equation (4. 1. 3) it is clear that the total amount of moisture which diffuses out of a wafer, is a function of the concentration gradient 3 C/bx at the surface. This means that the concentration at the mid- plane does not serve as the criterion for the length of time a plane wafer can be treated as a semi-infinite solid, but the concentration at the surface. In other words, the concentration ratios at x = 0 and 59 x = 2 2 will determine how long the plane wafer will behave as the sum of two semi-infinite solids. The concentration ratio of the surface at x = 0 will not be effected to any great extent by the surface at x = 2x until the term erfc (2 Z - x)/2\I Dt at x = 0 becomes significant (see eqn. 4.5. 3). Thus, as long as Z < < 1 . . erfc T2 t (4 5 6) a plane wafer will behave as a semi-infinite solid. Allowing for a 5 percent change in the concentration ratio at the surface x = 0 due to the surface at x = 2 Z , gives: 2 erfc W - 0.05 (4.5.7) From equation (4. 5. 7) it is found that the time period in which the plane wafer can be treated as a semi-infinite solid is equal to 4. 6 hours or in the moisture content range from 22.9 to at least 6. 98 per- cent dry basis. This result is approximately the same that had been obtained experimentally (see Table 9). In this section it has been shown that a plane wafer while drying can be treated as a semi-infinite solid over a large portion of its drying period. Since the differential equation describing diffusion often can be solved much easier for a semi-infinite solid than for a plane, this result has important applications for future drying investi- gations. V. SUMMARY AND CONCLUSIONS 5. 1 Summary The proper drying of forages is one of the major problems encountered in modern hay harvesting systems. This is especially the case with forage wafers due to the high densities of the individual wafers. An effective solution to the drying problem demanded a more thorough understanding of the variables affecting the drying behaviour of forage wafers. Forage wafers (92 percent alfalfa), made with an experimental Massey-Ferguson wafering machine, were dried at five different temperatures between 1200 and 2000F. The wafer densities varied from 0. 30 to 1.0 g/cm3; the wafer size from 6. 2 x 6.0 x 5.7 cm to 6.2 x6.0 x0.6 cm. An apparatus was developed for weighing the samples without re- moving them from the drying oven. The wafer temperatures were measured with copper-constantan thermocouples and recorded on an automatic recording potentiometer. The air velocities were measured with a hot-wire anemometer. Three major points were studied in this investigation: (i) the effect of wafer density on the diffusion coefficient of forage wafers; (ii) the effect of temperature on the diffusion coefficient of forage wafers; and (iii) the physical laws describing the drying behaviour of forage wafers. . The effect of density and temperature on the diffusion co- efficient of forage wafers was established. The minimum diffusion coefficient values for certain wafer density ranges were plotted versus the reciprocal of the absolute temperature on an Arrhenius plot. The minimum rather than the maximum values were reported because of 60 61 the practical importance of the slowest drying samples in the design of drying apparatus. The equation for the Arrhenius line of the most common wafer density range (0.4 - 0. 5 g/cm3) was D = 0.114 exp (-10, BIO/RT). An increase in temperature and/or a decrease in wafer density resulted in an increase of the diffusion coefficient of forage wafers. The solution to the differential equation for diffusion with con- stant initial and boundary conditions described the drying behaviour of forage wafers well in the latter stages of drying. The diffusion coefficients remained constant during this period until at least a wafer moisture content of five percent dry basis had been reached. The experimental moisture ratios fell above the theoretical drying curve during the initial drying period. In solving the diffusion equation it was imperative to use the experimentally found static moisture equilibrium values for the boundary condition. If the dynamic moisture equilibria were employed the diffusion equation did not describe the drying of forage wafers. It was shown that the deviation between the experimental and theoretical drying curves during the initial drying stages was caused by the change of the wafer-surface moisture concentration. The sur- face concentration varied exponentially with time according to the equation Q-(t) = Ceq + (Cin - Ceq) exp (- 8t). The value of 8 was a function of the wafer thickness and the diffusion coefficient. For a 4 cm thick wafer with a diffusion coefficient of 0. 75 x 10'5cmz/sec the value of 8 was equal to 1.41 x 10“sec'l. The temperature of the wafer rather than the ambient air temperature should be used to analyze theoretically the drying be- haviour of the product. A wafer dried at an oven temperature of 1400F had reached a temperature of only 136OF by the time it had be- come dry enough for safe storage. A semi-theoretical method was 62 developed which accounted for the changing wafer temperature during drying. The theoretical and experimental drying curves agreed well. A forage wafer was treated as a semi-infinite solid during the major part of the drying operation. It was found, and mathematically substantiated, that a two dimensional drying solid behaves as the sum of two semi-infinite bodies until 65 percent of the original moisture in the product has been removed. 5. 2 Conclusions 1. Diffusion is the physical mechanism controlling the drying behaviour of forage wafers. The diffusion coefficient is constant down to a wafer moisture content of at least 5 percent dry basis. 2. The wafer drying surfaces do not reach the static moisture equilibrium instantaneously, but exponentially with time. The boundary condition describing this physical phenomenon is of the form Q(t) = Ceq + (Cin ness and of the wafer diffusion coefficient. - Ceq) exp (-8t_). 8 is a function of the wafer thick- 3. The effect of temperature on the drying behaviour of forage wafers can be expressed by an Arrhenius plot. The equation for a.0.45 g/Cm3 wafer-density, relating the diffusion coefficient and temperature, is D = 0.114 exp (~10, 810/RT). The diffusion coefficients of forage wafers decrease in value with an increase in density and decrease in ambient air temperature. 4. The temperature of forage wafers slowly rises during drying, resulting in an increase of the diffusion coefficient. By making a rather simple transformation the solution to the diffusion equation with constant D can be used for obtaining the theoretical moisture ratio values of forage wafers. 5. A forage wafer can be treated as a semi-infinite solid until 65 percent of the total initial moisture has been removed. SUGGESTIONS FOR FURTHER STUDY Further investigations should be conducted in the following areas: 1. Rewetted versus Non-Rewetted Wafers. The diffusion co- efficients of rewetted and non-rewetted forage wafers should be compared when a wafering machine becomes available which is able to process high moisture forages. 2. Forage Properties. The influence of type of forage, length of cut and chemical analysis on the diffusion coefficients of forage wafers needs to be investigated. 3. Thermal Conductivity. The thermal conductivity of forage wafers as affected by density, moisture content and type of forage should be evaluated. 4. Heat of Vaporization. The heat of vaporization of forage wafers as a function of wafer moisture content is needed in order to study the analytical solution of the differential equation for diffusion with a negative heat sink. 63 REFERENCES Allen, J. R. (1960) Application of grain drying theory to the drying of maize and rice. Journal of Agricultural Engineering Research, 5(4), 363. Babbit, J. D. (1949) Observations on the adsorption of water vapor by wheat. Canadian Journal of Research, 32, Sect. F, 55. Babbitt, J. D. (1951) An unified picture of diffusion. Canadian Journal of Research 28 (A), 427. Bakker-Arkema, F. W., Hall, C. W. and Benne, E. J. (1962) Equilibrium moisture content of alfalfa. Quarterly Bulletin of the Michigan State University Experiment Station, 44, 492. Becker, H. A. and Sallans, H. R. (1955) A study of internal moisture movement in the drying of the wheat kernel. Cereal Chemistry, 32(3), 212. Becker, H. A. (1959) A study of diffusion in solids of arbitrary shapes, with application to the drying of the wheat kernel. Journal of Applied Polymer Science, 1 (2), 212. Carslaw, H. S. and Jaeger, C. J. (1959) Conduction of Heat in Solids. Oxford, England: Clarendon Press. Ceaglske, N. H. and Hougen, O. A. (1937) The drying of granular solids. Transactions of the American Institute of Chemical Engineers, 33, 283. Crank, J. (1956) The Mathematics of Diffusion. Oxford, England: Clarendon Press. Fan, L. T., Chung, D. S. and Schellenberger, J. A. (1961) Diffusion coefficients of water in wheat kernels. Cereal Chemistry, 38(3), 540. Farrall, A). W. (1963) Agricultural Engineering in the 60's. Lecture given at Agricultural Engineering meeting at Braunschweig- VESlkenrode, Germany, April 5, 1963. 64 65 Fish, B. P. (1958) Diffusion and thermodynamics of water in potato starch gel. In Fundamental Aspects of the Dehydration of Food- stuffs. London, England: Society of Chemical Industry. Carling, P. (1956) Untersuchungen zur Aufkl'arung des Trocknungsver- haltens pflanzlicher Stoffe. In German. VDI Forschungs Heft 458. Gurney, H. P. and Lurie, J. (1923) Charts for estimating tempera- ture distributions in heating or cooling solid shapes. Industrial and Engineering Chemistry, 15 (11), 1170. Hall, C. W. (1957) Drying Farm Crops. Ann Arbor, Michigan: Edwards Brothers. Hall, C. W. and Rodriguez- -Arias, J. H. (1958) Application of Newton's equation to moisture removal from shelled corn at 40- 1400 F. Journal of Agricultural Engineering Research, 3(4), 275. Henderson, S. M. and Pabis, S. (1961) Grain drying theory: 1. Temperature effect on drying coefficient. Journal of Agri- cultural Engineering Research, 6(3), 169. Henderson, S. M. and Pabis, S. (1962) Grain drying theory IV. The effect of airflow rate on the drying index. Journal of Agricultural Engineering Research, 7 (3), 85 Henry, P. S. H. (1939) Diffusion in absorbing media. Proceedings of the Royal Society of London, 171, Ser. A, 215. Holdren, R. D. and Menear, J. R. (1962) Wafering research pro- gress at Beltsvilles. Proceedings of seminar on wafering, sponsored by Massey-Ferguson, Inc. , Chicago, December 10 and 11, 1962. Hudson, R. G. (1959) The Engineers' Manual. New York: John Wiley and Sons. Hukill, W. V. and Schmidt, J. L. (1960) Drying rate of fully exposed grain kernels. Transactions of the American Society of Agri- cultural Engineers, 3(2), 71 Hustrulid, A. and Flikke, A. M. (1959) Theoretical drying curve for shelled corn. Transactions of the American Society of Agricultural Engineers, 2 (l), 112. 66 Jason, A. C. (1958) A study of evaporation and diffusion processes in the drying of fish muscle. In Fundamental Aspects of the Dehydration of Foodstuffs. London, England: Society of Chemi- cal Industry. Jones, C. R. (1951) Evaporation in low vacuum from warm granular material (wheat) during the falling-rate period. Journal of the Science of Food and Agriculture, 2, 565. King, G. (1945) Drying of keratin proteins. Transactions of the Faraday Society, 41 (6), 325. Krischer, O. (1963) Die Wissenschaftlichen Grundlagen der Trocknungs- technik. 2nd ed. In German. Berlin, Germany: Springer—Verlag. Kroll, K. (1959) Trockner und Trocknungsverfahren . In German. Berlin, Germany: Springer-Verlag. Lamp, B. J. and Bartlett, A. B. (1963) Field experiences with wafer- ing. ASAE paper 63-636. Lewis, W. K. (1921) The rate of drying of solid materials. Industrial Engineering Chemistry, 13, 427. Lundell, V. J. and Hull, D. O. (1961) Field production of hay wafers. Agricultural Engineering, 42(8), 412. Lykov, A. W. (1955) Experimentelle und Theoretische Grundlagen der Trocknung. In German. Berlin, Germany: VEB Verlag Technik. Maltry, W. and PBtke, E. (1962) Landwirtschaftliche Trocknungstechnik. In German. Berlin, Germany: VEB Verlag Technik. Matthies, H. J. (1963) Entwicklung und Forschung auf dem Gebiete des Verdichtens von Halmgut. In German. Landtechnische Forschung, l3 (6), 157. McKay, A. T. (1930) Diffusion into an infinite plane sheet subject to a surface condition with a method of application to experimental data. Proceedings of the Physical Society of London, 42, 547. Newman, A. B. (1931) The drying of porous solids: Diffusion and surface emission equations. Transactions of the American Institute of Chemical Engineers, 27, 203. 67 Newman, A. B. (1931) The drying of porous solids: Diffusion calcu- lations. Transactions of the American Institute of Chemical Engineers, 27, 310. Niesper, A. A. (1958) Tontrocknung. In German. Weisbaden, Germany: Bauverlag GMBH. Nordon, P., Mackay, B. H., Downes, J. G., and McMahon, G. B. (1960) Sorption kinetics of water vapor in wool fibers: Evalu- ation of diffusion coefficients and analysis of integral sorption. Textile Research Journal, 30, 761. Pabis, S. and Henderson, 8. M. (1961) Grain drying theory 11. A critical analysis of the drying curve for shelled corn. Journal of Agricultural Engineering Research, 6 (4), 272. Pabis, S. and Henderson, 5. M. (1962) Grain drying theory III. The air/grain temperature relationship. Journal of Agricultural Engineering Research, 7 (2), 21. Ross, R. H., Fourt, D. L., and Gebhart, F. C. (1963) Alfalfa "Hayfers" versus baled alfalfa hay for milk and butterfat production. Mimeograph - Department of Dairy, University of Idaho, Moscow, Idaho. Saravacos, J. B. and Charm, S. E. (1962) A study of the mechan- ism of fruit and vegetable dehydration. Food Technology, 16 (1), 78. Schoedder, F. and Busse, W. (1964) Einsatzversuche mit Aufsammel - Brikettierpresse. In German. Landtechnik, l9 (3), 57. Sherwood, T. K. (1929) The drying of solids I. Industrial Engineer- ing Chemistry, 21, 12. Sherwood, T. K. (1929) The drying of solids II. Industrial Engi- neering Chemistry, 21, 976. Sherwood, T. K. (1930) The drying of solids III. Industrial Engi- neering Chemistry, 22, 132. Sherwood, T. K. (1932) The drying of solids IV. Industrial Engi- neering Chemistry, 24, 307. 68 Sherwood, T. K. and Comings, E. W. (1933) The drying of solids V. Industrial Engineering Chemistry, 25, 311. Sherwood, T. K. (1931) Application of theoretical diffusion equations to the drying of solids. Transactions of the American Institute of Chemical Engineers, 27, 190. Sherwood, T. K. (1936) The air drying of solids. Transactions of the American Institute of Chemical Engineers, 32, 150. Simmonds, W. H. C., Ward, G. T., and McEwen, E. (1953) The drying of wheat grain. Part I: The mechanism of drying. Transactions of the Institution of Chemical Engineers, 31, 265. Sprenger, J. J. I. (1957) Vorschlag fur die Berechnung von Trocknern inbesondere von Darren. In German. De Ingenieur, 7, 29. Stamm, A. J. (1960) Bound-water diffusion into wood in across-the- fiber directions. Forest Products Journal, 10, 644. Toponitzki, P. A. (1949) Priorita't russischer Gelehrter in Trocknungs- fragen. In German. Torfindustrie, 2, 23. Van Arsdel, W. B. (1947) Approximate diffusion calculations for the falling-rate phase of drying. Transactions of the American Institute of Chemical Engineers, 47 (1), 13. Van Arsdel, W. B. and Copley, M. J. (1963) Food Dehydration. Westport, Connecticut: Avi Publishing Company. Williamson, E. D. and Adams, L. H. (1919) Temperature distri- bution in solids during heating or cooling. Physical Review, 14 (2), 99. Zimmerman, M. (1964) Harvesting the hay crops. Implement and Tractor, 79(5), 26. APPENDIX A 69 70 A. 1 Thermocouple Locations for Measuring the Average Temperature in a Cube-Shaped Forage Wafer. Assumptions: (i) the wafer is uniform in density (ii) the physical properties of the forage wafer are the same at every point of the wafer (iii) the wafer can be divided into an arbitrary number of shells each having the same heat content Solution: Assume that the sides of the cube inside the innermost shell are equal to Z cm. The volume of this cube is [3cm. If the thickness of the first shell is called i—x, the volume of the shell becomes: (£+x)3-Z3=x°+3gxz+3zzx (A.l.l) Since the volume of the shell must be the same as the volume of the cube, equation (A. 1.1) has to be equal to Z3 or x3+3[x°+3gzx= Z3 (A.l.2) and x3+3[x°+3zzx- 253:0 (15.1.3) Equation (A. 1. 3) is a cubic equation of the general form x3 + pxz + qx + z = 0, and can be reduced to the form y3 + ay + b = 0 by substituting for x the value (y - p/3) or by setting a equal to 1/3(3q - p2) and b equal to 1/27(2p3 - 9pq + 27r) (Hudson, 1959). Equation (A. 1. 3) then gives: ll 1/3 (9Z3-9Z")=0 (A.l.4) 1/27(54 K3 - 81Z3 - 2733) = -2Z3 (A.l.S) a and b Equation (A. 1. 3) then becomes: Y3-2[3=0 (A0106) 71 The real root of equation (A. 1.6), which is of practical interest, is equal to y = Z {I 2 . The solution to equation (A. 1. 3) is then: xey-/Z=ZF-Z=0.259/ (A.1.7) The locations of the parallel connected thermocouples, which will give the average temperature of a cube-shaped forage wafer, can be calculated from equation (A. 1. 7). 72 Table A. 1 Values of the function L(v) or 8/75 2( e-v + 1/9 8-9V +1/25 8-25" + ----) Adapted from A. T. McKay(1930). v 0.00 0.02 0.04 0.06 0.08 0.00 0.0000 .8984 .8563 .8240 .7068 0.10 .7028 .6918 .7132 .7127 .6952 0.20 .6787 .6633 .6481 .6337 .6199 0.30 .6066 .5936 .5812 .5691 .5573 0.40 .5458 .5346 .5237 .5131 .5028 0.50 .4926 .4827 .4730 .4636 .4543 0.60 .4452 .4363 .4277 .4192 .4108 0.70 .4027 .3947 .3868 .3792 .3716 0.80 .3643 .3570 .3499 .3430 .3362 9.90 .3296 .3230 .3166 .3104 .3042 1.00 .2982 .2923 .2865 .2809 .2751 1.10 .2968 .2645 .2592 .2541 .2491 1.20 .2441 .2393 .2346 .2300 .2253 1.30 .2209 .2165 .2122 .2081 .2039 1.40 .1999 .1959 .1920 .1882 .1845 1.50 .1808 .1773 .1738 .1703 .1670 1.60 .1636 .1604 .1573 .1541 .1511 1.70 .1481 .1452 .1423 .1394 .1367 1.80 .1340 .1313 .1287 .1262 .1237 1.90 .1213 .1188 .1165 .1142 .1119 2.00 .1097 .1076 .1054 .1033 .1012 2.10 .0993 .0973 .0954 .0935 .0916 2.20 .0898 .0880 .0863 .0846 .0829 2.30 .0813 .0797 .0781 .0761 .0751 2.40 .0735 .0721 .0707 .0692 .0678 2.50 .0665 .0653 .0640 .0627 .0614 2.60 .0602 .0590 .0579 .0567 .0556 2.70 .0545 .0534 .0524 .0513 .0503 2.80 .0493 .0483 .0473 .0464 .0455 2.90 .0446 .0437 .0429 .0420 .0412 3.00 .0404 .0396 .0387 .0380 .0373 3.10 .0365 .0357 .0350 .0344 .0336 3.20 .0331 .0324 .0318 .0311 .0306 3.30 .0298 .0293 .0287 .0281 .0276 3.40 .0271 .0265 .0260 .0254 .0250 3.50 .0245 .0240 .0235 .0230 .0226 ENGR. LIB. HICHIGQN STQTE UNIV. LIBRQRIES 31293001803281