THE UNIV ER S I TY OF MI CHI GAN COLLEGE OF ENGINEERING Department of Mechanical Engineering Progress Report DIFFUSION OF WATER VAPOR INTO ARTIFICIAL SOILS Richard Wilcox ORA Project 03026 under contract with: DEPARTMENT OF THE ARMY ORDNANCE CORPS DETROIT ORDNANCE DISTRICT CONTRACT NO. DA-20-018-ORD-14620 DETROIT, MICHIGAN administered through: OFFICE OF RESEARCH ADMINISTRATION November 1961 ANN ARBOR

LIST OF FIGURES Figure Page 1. Plot of the diffusion of water vapor into pure clay, 5 2. Plot of the diffusion of water vapor into 10o Glycol-clay mixture, 6 3. Plot of Fig. 1 from analytical work. 7 4. Plot of Fig. 2 from analytical work. 8 5. Plot of the determination of Kw for Fig. 1. 9 6. Plot of the determination of Kw for Fig. 2, 10 iii * *

LIST OF SYMBOLS - depth ratio L L depth of artificial soil, inches x P- - Po0 — L dimensionless vapor-pressure ratio Po - P P vapor pressure of medium surrounding soil P0 intrinsic vapor pressure of soil at the bottom of the soil bed Q %o gain of water/time at time 9 QO % gain of water/time at the equilibrium pt. of soil Bi Biot modulus Fo Fourier modulus W fractional water in clay-glycol-water mixture fractional water in mixture vapor pressure of water 9 time in days v

INTRODUCTION This report is a first step in attempting to reach a solution to the problem of whether a certain soil will support a given vehicle under specific weather conditions. A set of tables and analyzing equipment to determine if the ground soil has the necessary properties to support a given vehicle, and what action, if any is needed to correct it, would be the ideal solution. Hence this problem compels us to start with the soil itself; i.e., to study the diffusion of water vapor into the soil. PRELIMINARY DIFFUSION-RATE STUDIES IN ARTIFICIAL SOILS The purpose of these tests was to determine, both analytically and experimentally, the rates at which artificial soils reach an equilibrium point. Consideration must be given to such factors as temperature, vapor pressure, relative humidity, soil composition, experimental conditions, boundary and surface conditions of soil, etc. Some of the factors concerning the experiment must be considered to be constant to allow an analytic correlation between idealized and experimental data. It is hoped that later these constant factors may be considered as variables under both experimental and analytical treatment. Vapor pressure and temperature will be considered as constants in the desiccators where the experiments took place. This is a feasible assumption since the laboratory temperature was 77~F ~ 3~ and the relative humidities in the desiccators, controlled by chemical solution, are believed to be reasonably accurate. Soil composition was assumed to be homogeneous throughout, since clay (Bentonite) was the only substance used in these initial diffusion-rate studies. Another assumption made was that the petrie dishes in which the soil was placed had perfectly insulated boundaries except, of course, at the open surface. EXPERIMENTAL PROCEDURES The initial weight of the soil samples was determined by weighing the petrie dishes first, then weighing the combined weight of the dish and the 1

soil sample, and subtracting the difference between the dish weight and combined weight. The following relative humidities were used in the experiment: 100, 85, 75, 62, and 32%. The average laboratory temperature was 77~F. The samples were weighed daily, when possible, and their changes in weight were noted on a data sheet, a sample of which is attached. Care was taken in the weighing process to keep the daily weight error to a minimum. The surface of the petrie dish is 9.28 sq in., approx. The depth of the petrie dish is approx. 1/2 in. 2

R. H. 100 85 75 62 1 32 Date Dish Weight Temp., ~F 33 35 28.91 32.10 28.87 28.49 Total Weight, gm 100lo Bentonite -Large Diffusion-Rate Studies 10-12-60 10-13-60 10-14-60 10-18-60 10-19-60 10-20-60 10-21-60 10-25-60 10-27-60 10-28-60 11-1 -6o 11-3 -60 11-15-60 11-17-60 11-22-60 11-29-60 113 65 117.56 119.57 123.22 123.77 124.20 124.63 126.04 126.52 127.56 127.61 127.89 129.15 129.28 129.67 130. 03 119.10 120.92 121.91 123.42 123.55 123.61 123.69 124. 00 124.07 124. 07 124.13 124.27 124.45 124.38 124.42 124.42 Samples For 116.80 118.15 118.82 119.74 119.78 119.81 119.81 119.93 119.93 119.93 119.95 120.04 120.10 120. 07 120.10 120. 09 110.46 111.14 111.49 111.91 111.89 111.88 111.85 111 93 111.89 111.89 111.85 111.81 111.91 111.86 111.92 111.85 100.70 100.30 100.05 99.69 99.67 99.63 99.61 99.59 99.585 99.56 99.54 99.56 99.60 99.58 99.58 99.58 82 82 8o 78.5 78 78 78 77.5 79 80 78.5 76 78 77 77 77 3

R. H. 100 8 75 62 32 Date Dish Weight, gm Temp., ~F 28.88 37.76 33.30 1 28.86 j 29.32 Total Weight, gm 10%o Glycol-Bentonite, Large Samples May 10 107.43 109.56 106.09 99.22 93.34 May 11 113.77 112.10 109.63 100.24 94.40 Average May 12 116.70 115.94 112.36 102.84 953.7 Temp. May 13 Sample 117.77 113.92 104.65 95.77 75 May 15 was 118.16 114.22 104.81 95.81 + 4 May 17 Accidentally 119.48 115.02 106.01 96.48 May 19 Saturated, 120.06 115.23 106.17 96.62 May 24 Could 120.41 115.40 106.41 96.72 May 26 Not 120.77 115.38 106.37 96.72 June 8 Be 120.77 June 9 Restored June 10 New 100%o R. H. Sample (Glycol was added to sample after the sample was dried) June 19 128.72 July 20 145323 June 21 129.92 July 21 143.57 June 23 131.01 July 25 145.10 June 27 133.01 July 30 145.87 June 30 134.51 Aug. 3 148.41 July 3 135.67 Aug. 9 150.78 July 7 137.80 Aug. 10 151.10 July 11 139.57 Aug. 11 151.50 July 13 140.43 Aug. 14 152.45 July 14 140.83 July 18 142.49

0i ~cr>O~~~~~~~~~~~ 1100%R. L.H. z 14 10 - 8 Y6 ~ y85%R.L.H. 4 75 5%R. L.H. 2 0 u 32 %R.L.H. 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 TIME (days) Fig. 1. Plot of the diffusion of water vapor into pure clay.

10% GLYCOL - CLAY MIXTURE SAMPLES 12 11 10 9 75 8 2O ~~~~~~~~~~~62:I 7, 0 2 4 6 8 10 12 14 16 18 20 22 24 TIME (days) v6i 3 0 2 4 6 8 10 12 14 16 18 20 22 24 TIME (days) ---- --- 5% R.H. 5 %R. H. 2 % R. H. 2%R. H. 26 30 32 34 Fig. 2. Plot of the diffusion of water vapor into 109 Glycol-clay mixture.

16 14 -^1 m 12 (5 10 8 6 CD e- A TIME ( DAYS) Fig. 3. Plot of Fig. 1 from analytical work.

22 20 18 16 01 o 14 C) Ocz 12 10 85% R.H. C., s 1 75 % R. H. 8 62 % R. H. 6 4 L A 32% R.H. 2 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 TIME (days) 32 34 36 38 40 Fig. 4. Plot of Fig. 2 from analytical work.

/ K.0106 w 10 E -,/ Method of computatior V, 6L/ base mixture =. 88 cl u 16 -6/. 12wa X3/ a1 14- clay water CD/.88.12.835.165 <12- / It l I~l ^o -~~~~10 |,f r ^^~~~~1 lb. mixture 8 6. 4 2 0.05.10.15.20 This is with respect to base mixture of.88 clay -. 12 water ( Fraction of water in water - clay mixture ) 100 % CLAY SAMPLES Fig. 5. Plot of the determination of Kw for Fig. 1.

10% GLYCOL - CLAY SAMPLES I E Ir LL C):) li LUi 0 ra_ 1. a_ 26 24 22 20 18 16 14 12 10 8 6 4 2 0 85 % 75% 62% 32%.05 K - - 005 w 10 Base Mixture 1~:0 Clay-Glycol H20 Mixture at equilibrium Clay-Glycol H,0 I.907 1093 I I I I I I I I I I.02.04.06.08.10.12.14.16.18.20.22.24 FRACTI ON OF WATER IN WATER-CLAY MIXTURE Fig. 6. Plot of the determination of Kw for Fig. 2.

Pure Clay Samples - Analysis Data Fx aO PL P Water = W Relative PL F L P Misc. L F P - jPo H20O/day Humidity Lj ~ j J- o. o.00.1 92.01 100% temp. 75~F.20.1.88.024 " P = 11.1, Po 1.40.1.80.032 " 0 = 2 days.60..62.051 Zw/6.050.8o..535. o80 1.00.1.20.096 " (/Q0) = 355.00.35.54.06 100o PO = 11.1, Po = 1.20.35.50.064 " o = 6 days.40.35.45.07 Z" w/6..083.60.35.32.083 i.8o 355.17.099 i" (Q/QO) =.56 1.00.55.00.12 i.oo.9.134.10 100oo P = 11.1, Po = 1.20.9.13.103 " = 22 days.40.9.12.105 w/6.111.6o.9 o.118.80.9 0.118 1.00.9 0.118 (Q/Qo) =.894.00.32.6o.033 85% PO = 7.8.20.32.58.034 i P0 = 0 40o 32.50. 041 i Zw/6 -.051.60.32.35.054 t" = 2 days.80.32.18.067 1.00.52 0.o8o it (Q/Qo) =.52.00.70.225.064 85% Po = 7.8.20.70.22.064 i" PG = o.40.70.185.067 " o = 6 days.60.70.15.071 i" w/6 o 072.8o.7o 0o.o8o 1.00.700.o8o (Q/Q) 1.oo. 7o o.o80 1, (Q/Qo) =.84 11

Pure Clay Samples (Continued) x x aG p - Water = W Relative M F _ p- PO Misc. L o L20L - % H20/day Humidity Po - P00 jk.00 1.15 0.80 85% Poo = 7.8.20 1.15 0.080 " Pq = 0.40 1.15 0.080 " = 22 days.60 1.15 0. o80 " Zw/6 =.080.80 1.15 0.080 " 1.00 1.15 0.80 "/ (Q/Qo) = 95.00.57.55.04 75* PO = 5.1.20.37.51.027 i P =.40.37.46.029" = 2 days.60.37.33.036 Zw/6 -.035.8o0.7.18.043 " temp. 75~F 1.00.37 0.054 (Q/Q) =.60.00.80.1 x 7.044 75 P, = 5.1.20.80.1 x 6.045 P =.40.80.14.047" 9 = 6 days.60.80.11.048 " Zw/6.049.80.80 0.054 " (Q/Qo)=.89 1.00.8o 0.054.00 1.2 0.054 75 Poo = 5.1.20 1.2 0.054 P =O.40 1.2.0 54" 0 = 22 days.60 1.2 0.054 Zw/6 =.054.8o 1.2 0.054 1.00 1.2 0 54 " (Q/Q) =.96.00.31.60.012 62% P, = 2.7.20.31.58.012 i PG=o = 0.40.31.50.014 i 0 = 2 days.60.31 35.018 it Zw/6 0.018.80.31.18.023 " temp. 75~F 1.00.31 0.028 (Q/Q) =.50 12

Pure Clay Samples (Concluded) x x aF = aG P - P Water = W Relative M L - -- L co HuMisc. L ~ L2 % ---- p H20/day Humidity Po - P~.00.20.40.60.80 1.00 2.0 2.0 2.0 2.0 2.0 2.0 0 0 0 0 0 0.028.c28.028.028.028.028 62% tt It it it I1 PO = 2.7 P = 0 0 O = 6 days Zw/6 -.028 (Q/Qo) = 1 13

10% Glycol-Clay Samples x x F _a0G - P0 Water = W Relative Misc. L ~- L2 -o 2 L Humidity PO=o = Poo.00.35.54.o43 85Poo = 18.9.20.35.50.047 I P = 0.40.35.45.052 " = 2 days.60o 35.32.064 temp. 75~F.80.35.17.078," E/6 -.063 1.00.5 0.094 (Q/Qo) = *56.00.68.21.074 85% P = 18.9.20.68.19.078 P = 0.40.68.16.079 1 = 6 days.60.68.11.084 " Zw/6..084.80.68 0.094 1.00.68 0.094 " (Q/Qo) =.81.00 2.0 0.094 85 P = 18.9.20 2.0 0.094 " P =.4 20 2.0.094 " = 22 days.60 2.0 0.094- i Ew/6.094.8o 2.0 0.094 1.00 2.0 0.094 " (Q/Qo) =.99.00.42.48.043 75 Po = 16.6.20.42.43.047 P = ~.40.42.38.051 temp. 75~F.60.42.28.060o w/6..059.80.42.15.071 " = 2 days 1.00.42 0.083 (Q/Qo) =.68.00 1.15 0.08 75 Poo = 16.6.20 1.15 0.o83 Po = 0.40 1.15.0 83 " temp. 75~F.60 1.15.0853 " Zw/6 o.083.80 1.15.083 "8 = 6 days 1.00 1.15 0.83 " (Q/Qo) =.94 14

10 Glycol-Clay Samples (Continued) x x F = a - Water = W RelativeMisc. L ~ 2 Humidity Po=o = POO o00 2.0 0.083 75 Poo = 16.6.20 2.0 0.083 P= 0.40 2.0 0.083 Zw/6.. 08.60 2.0 0.083 i.80 2.0 0.083 " o = 22 days 1.00 2.0 0.08 (Q/Q) = 1. 0.00.22.77.016 62* P, = 15.8.20.22.71.020 t P= 0.40.22.64.025 " temp. 75~F.60.22.47. 037 " Zw/6 -.037.80.22.24.053 0 = 2 days 1.00.22 0.069 i (Q/Q) =.43.00.80.18.057 62* Poo = 13.8.20.8o.17.057 "P = 0.40.80.14.059g " w/6 -. 64.6o.80.10.062.80.80 0.069 " = 6 days 1.00.80 0.069 " (Q/Qo) =.90.00 2.0 0.069 62 Po = 13.8.20 2.0 0.069 " P = 0.40 2.0 0.069 I w/6 I.069.60 2.0 0.069.80 2.0 0.069g " = 22 days 1.00 2.0 0.069 (Q/Qo) = 1.00.38.56.016 32% P = 7.10.20.38.52.017i P = 0.40.38.47.019 temp. 75~F.60o.38 33.024 I" Zw/6..024.80.38.19.029 it o = 2 days 1.00o.8 0.036 it (Q/Q) =.61 15

10% Glycol-Clay Samples (Concluded) x F=ag pX - Water W Relative Misc. - F- L Water = W Misc. L ~ L2 L Humidity P = Pw G=o 00.70.23.027 32% Po = 7.10.20.70.21.028 P =.40.70.18.029,, temp. 75~F.60.70 13.031 " Zw/6.031.80.70 0.036 ~ = 6 days 1.00.70 0.036 (Q/Q) =.86.00 2.0 0.036 32% P. = 7.10.20 2.0 0.036 i P = 0.40 2.0 0.036 temp. 75~F.60 2.0 0.036 t Zw/6 -.036.80 2.0 0.036 " = 22 days 1.00 2.0 0.o36 (Q/Q) = 1 16

The theory behind the analysis curves to the experimental data lies in an analogy between a heat-conduction problem through a flat plate of certain thickness and diffusion of water vapor into an artificial soil of certain thickness. Of course in such an analogy certain terms must be re-defined and assumptions made to make the theory possible. ASSUMPTIONS AND DEFINITIONS (a) The unit surface conductance between the soil and surrounding medium was considered to be infinite. (b) The soils were considered to be homogeneous throughout. (c) The bottom of the petrie dish was analogous to the plane x = L in the flat plate, since there was no diffusion taking place here. Tx/L - T Px/L - PO (d) - = dimensionless temp. ratio = -/ TG=o T P=o- p P = dimensionless vapor-pressure ratio fractional H2Q here P/L = Kw = Kw =vapor pressure in mixture of H20 (e) Vapor pressures of H20 at 75~F 10Oo R.H, 85% R.H, 75% R.H, 62% R.H. 32% R.H, 22.2 mm Hg 18.9 t 16.6 " " 13.8 7.10 " (f) The solutions of this type of problem involves trigonometric relations. The analysis curves used in this report were resolved from graphical solutions used by Frank Kreith in his book, Principles of Heat Transfer. The graphs used, on pages 138 and 139, represent numerical data parametrized to non-dimensionalize the functional relationships. CONCLUSION In view of the many assumptions involved, the analogy between heat transfer and diffusion appears to be quite close as shown by the comparison of the experimental and calculated diagrams. 17

Additional work along the lines outlined appears to be justified, using clay and expanding to natural soils. REC OMMENDATI ONS While the analysis curves fit the experimental data very closely, particularly in the samples which were dried initially, the results can hardly be generalized to include diffusion of water vapor into all possible combinations of artificial soils. A great deal more experimental data are needed, including many combinations of soils and humidities. Under existing conditions, the total time to run an experiment (approximately 2 months) is a severe handicap in collecting data. Expenditures in this area are necessary to resolve the diffusionrate problem completely. 18

REFERENCES Frank Kreith, Principles of Heat Transfer, 1958. L. M. K. Boelter, V. H. Cherry, H. A. Johnson, and R. C. Martinelli, Heat Transfer Notes, 1946. 19

UNIVERSITY OF MICHIGAN I ll9 flll 0352 Mll 3 9015 03527 5067