A Retrospective on Early Analysis and Simulation of Freeze and Thaw Dynamics. Henry M. Paynter, Life M.ASCE

Transcription

1 A Retrospective on Early Analysis and Simulation of Freeze and Thaw Dynamics 1 Henry M. Paynter, Life M.ASCE Abstract This paper recapitulates certain studies of cycles of freezing and thawing in the ground sponsored by the US Corps of Engineers and begun over 45 years ago by the writer and H.P. Aldrich, Jr. Because the latent heat of the soil moisture renders such problems nonlinear, these two MIT professors, together with R.F. Scott and other students, pioneered the use of digital and analog computers for such research. This involved the construction of unique hydraulic and electronic analog devices which have long since become CRREL museum pieces. However, a more lasting contribution was the provision of a new formula for computing the frost depth in temperate climates as well as annual thaw in permafrost zones. This Modified Berggren Formula remains in wide use today. Introduction In the Fall of 1952 the Arctic Construction and Frost Effects Laboratory [ACFEL] of the New England Division, Corps of Engineers, contracted with Harl P. Aldrich, Jr., and the writer, to investigate a variety of problems concerning freezing and thawing under pavements, both in temperate and permafrost regions. These studies continued for several years, extending into the time of the merger of ACFEL and SIPRE into the contemporary CRREL. Typical of the tasks we faced was the prediction of ground isotherms and thaw penetration under Arctic airfield pavements, as indicated in Figure 1. 1 Emeritus Professor, MIT, POB 568, Pittsford, VT 05763, HankusP@aol.com 1 Paynter

2 Figure 1 Measured Isotherms under Permafrost conditions 2 Paynter

3 The first studies of this situation were made by Josef Stefan in 1889, in connection with ice formation and melting in the Polar oceans. He furnished the first and simplest prediction formula, expressed in contemporary form as 0.5 X STEFAN = (48 k nf / L). (1) Here nf is the effective freezing index in degree-days, k is the thermal conductivity and L is the latent heat. Because this formula neglects volumetric heat, it tends to overestimate frost depth in temperate zones. The essential feature of this Stefan problem is the existence of a moving boundary surface separating unfrozen and frozen soil. At this interface, the latent heat of the soil moisture is either being liberated or absorbed proportional to the directed velocity of the moving boundary. In all practical cases the soil properties each side of the boundary can be substantially different due to the thermal properties of ice and liquid water, resulting in the essential nonlinear thermal energy characteristic of Figure 2. Figure 2. Thermal Energy Diagram It is difficult for younger people to conceive of the situation before the advent of PCs and FEM, and before the widespread interest of mathematicians in this problem upon the translation of Rubenstein s The Stefan Problem in Thus in the 1950s this task was left to engineers like ourselves to make what progress was then possible on this difficult but important problem. An Improved Formula for Predicting Frost & Thaw Penetration Until our studies, the most generally successful formula for frost prediction appeared to be one of those developed by ACFEL, namely X = [48 k f /( L + C ( v + F/t ))]. (2) 0 However, this expression neglects volumetric heat in the unfrozen ground, which substantially reduces frost depths in the milder climates. Accordingly, we undertook to derive an improved rational formula. This would make maximum use of the Neumann solution and would take special advantage of the simplifications obtained assuming identical properties for the frozen and unfrozen ground. 3 Paynter

4 So in order to go beyond the classic Stefan formula, we first obtained a solution to the particular thermal problem of a semi-infinite soil mass of uniform properties and at a uniform initial temperature which was then subjected to surface temperature changes, assumed to be uniform over the surface extent so as to yield a one-dimensional problem. Figure 3 illustrates the nomenclature and variables employed. Essential to progress was the employment of the following dimensionless parameters: α = v C / v C ; δ = ( a / a ) ; µ = v C / L ; Z = X / 2 ( a t ) (3) 0 u s f f u s f f Figure 3. Thermal Conditions during Frost Penetration Thus the Neuman solution provided a simple way to obtain a precise estimate of the effect of volumetric heat on reducing depth of freeze as computed by the original Stefan formula. For the above simplest case, α = v 0 / v s and δ = 1. As a result one could plot λ vs µ, using α as parameter. Figures 4A and 4B directly compare this simplified Neumann solution result with the plot originally presented in Aldrich and Paynter (1953a, 1953b), using for the first plot the analytic expressions: 2 2 λ = Z (2/µ) ; µ = (π) Z / [exp(-z ) / erf( Z ) - α exp ( - Z ) / erfc ( Z ) ] (4) Then by using this final dimensionless coefficient as λ ( α, δ, µ ), one obtains a final value for X ( t ) with time measured in days: X = λ [ 48 k v t / L ] (5) f s Recognizing that v s t are the freezing degree days, n F, and that k f = k, we obtained the final formula: X = l [ 48 k n F / L ] (6) 4 Paynter

5 4 A 4B Figure 4. Lambda Coefficient Values Using this generic form, the above ACFEL formula could be expressed as: λ = 1 / [ 1 + µ ( α )] (7) ACFEL which allows a direct comparison to the Neumann-derived result. After this formula had been developed it was discovered to be identical in nature to a formula introduced by Berggren in a somewhat different form, which was given an unfavorable later report by Shannon, mostly due to using erroneous values for the thermal properties. Accordingly, we entitled this new equation the Modified Berggren Formula. Computing Machine Solutions of Freeze and Thaw Due to the inherent latent heat nonlinearity (as in Figure 2 ), together with the normally heterogeneous nature of the soil and the variability of surface temperature distributions, it is not possible to get practically significant solutions of the freeze/thaw Stefan problem in closed analytical form. In order to overcome this handicap to analysis and prediction, several useful analog and digital computer approaches and devices were developed for this research. In addition, besides verifying values estimated using the Modified Berggren Formula, these computer solutions were directed toward determining the effect of the shape of the surface temperature-time curves on both the depth of freeze/thaw as well as the corresponding shapes of X (t) curves, all as indicated in Figure 5. 5 Paynter

6 Figure 5. Shape of Depth-Time Curves As indicated previously, the computational methods currently used to solve Stefan problems have been significantly influenced by developments all of which occurred some two decades subsequent to the our earlier work: (1) the advent of modern PCs and supercomputers; (2) the evolving use of FEM methods and software; (3) the interest of and role played by mathematicians. So it was that the computing techniques and apparatus used for our studies was quite different from that now commonly used for moving boundary problems. Instead, for all computer investigations, we employed a lumped element approach best understood for the simplest case of multi-layered strata. Thus for each layer beginning, say, from an unfrozen state, as heat is removed the temperature throughout the layer will drop until the freezing point is reached. Then the layer temperature will remain at this value until all soil moisture in the layer is completely frozen. Only then will the temperature continue to drop below freezing. Because this technique may be less familiar to the reader, Figure 6 depicts a contemporary reconstruction of a 7-layer system subjected to a sinusoidal temperature swing about a mean temperature slightly above the freezing point. As we found then, an excellent estimate of the moving boundary can be obtained by plotting the time when each layer just reaches and just leaves the freezing point; this method is indicated by Figure 7. The linear interpolation assumes that the boundary moves through the layer in direct proportion to the fractional change in latent heat. 6 Paynter

7 Figure 6. Computed Temperature-Time Curves The Hydraulic Analog Computer Figure 7. Computed Frost Depth-Time Curves Hydraulic analog computers to study transient thermal conduction originated simultaneously in both Russia and the US in the 1930s. At the University of Michigan, A.D. Moore s Hydrocal was built primarily to study heat flow through composite building walls, while, at the same time in the Russian Institute of Roads and Construction, V.S. Lukyanov built a Hydraulic Apparatus for studying a variety of problems, including soil freeze and thaw. From these precedents, a new hydraulic machine was designed and constructed by R.F. Scott, primarily to solve complex 1-dimensional problems as encountered in multi-layered pavements and soils subjected to freezing and thawing, both during repeated annual temperature cycles as well as during construction programs. The final computer is portrayed in Figure 8. 7 Paynter

8 Figure 8. The Hydraulic Analog Computer In this computer, each soil layer was represented by a vertical glass tube or standpipe containing a mid-height lateral expansion well. The changing water level in each standpipe represented the varying temperature in the corresponding soil layer, the combined capacity of the standpipe plus expansion being analogous to the thermal energy versus temperature characteristic of the soil ( as in Figure 2). The effective cross-section above and below the expansion well set the volumetric heat value while a trombone-slide in the well set the latent heat value. A typical characteristic is presented in Figure 9. 8 Paynter

9 Figure 9. Latent Heat Well Characteristic The transient flow of heat between any two layers was represented by the flow of water between the corresponding standpipes, where the conductance was controlled by a fluid rheostat, formed from capillary channels of slide-adjustable length. The surface temperature variations were provided by the programming device shown at the left side in the diagram and picture of Figure 8. This device also could readily represent variations in volumetric heat and thermal conductivity with temperature. Local variations in freezing point were handled simply by raising or lowering the latent heat wells. The final machine was marked by its simplicity of programming and for providing direct visualization of the entire progress of the solution. These features proved to be particularly useful for exploratory and tutorial purposes and explained the continued use after it was eventually moved to CRREL. 9 Paynter

10 The Electronic Analog Frost Computer [ EAFCOM ] The hydraulic device just described becomes unduly complex for twodimensional problems, due primarily to the difficulties of interconnecting the chambers while still permitting access and direct observation. For 3-dimensional situations, such an approach seems out of the question. These difficulties are not present in an electronic embodiment operating in directly analogous fashion. Thus the twodimensional array we constructed was able to be installed in two relay-racks ( Figure 12). Large-scale passive electrical RC networks had been used for unsteady heat conduction problems ever since the pioneering work of Beuken (1933) in Holland. Victor Paschkis, who had collaborated with Beuken, then brought this technology to the US in the 1930's, where he built a very large general purpose machine at Columbia University. But to successfully incorporate inhomogeneous latent heat effects demanded the form of an active or electronic implementation. The basic computing elements were configured primarily to solve the 2-d lumped equations. The active circuit for a single lump realizing the latent heat effects as in Figures 2 and 9 is indicated in Figure 10. Figure 10. EAFCOM Component Circuit For each lump the locally variable part of this circuitry was adjoined to a plugin package as shown in Figure 11. The 4 black cylinders on the top of the box contained resistors representing the local directed conductances, while the 2 lightcolored cylinders on the bottom of each box contained the local thermal capacity and the latent heat value. For any given case the element boxes had the appropriate elements plugged into each box which is then attached to an operational amplifier by the jacks indicated. The several boxes were then appropriately thermally interconnected by spider cables inserted on the exposed faces. These then electrically enforced the same relations as for the hydraulic analog. The final array, interconnected to solve a particular problem, is shown in Figure 12.The entire EAFCOM was capable of representing 100 lumps of variegated soil in 1- and 2-dimensional arrangements. 10 Paynter

11 Figure 11. EAFCOM Soil Lumps Figure 12. EAFCOM at ACFEL in Paynter

12 Digital Computer Solutions As a further means of obtaining reliable data on the freeze/thaw process prior to the availability of the hydraulic and electronic computers, a battery of digital computer solutions were obtained using the IBM Card Programmed Calculator at MIT. For historical interest only, we show as Figure 13 one sheet of the typical tabular output corresponding to the case of permafrost thaw with α = 0 and µ = 1 and where the temperature varied sinusoidally about zero degrees. Clearly, re-freeze has just begun at the bottom of the sheet. Once the two analog devices were available there was no further need for such digital solutions. Figure 13. Typical IBM Solution Tabulation 12 Paynter

13 Conclusion Recounting the above archaic methodology surely draws attention to the extraordinary analytical and computing progress made in the following five decades. At the time of our work, ubiquitous electronic analog computers were the counterpart of today s PCs. Also another two decades elapsed before FEM methods addressed this Stefan problem. So until and unless this small bit of history becomes recorded, preferably by a participant, it might otherwise elude discovery. With apologies to all the many neglected investigators, this task has been attempted here. References Aldrich, HP (1955), Frost penetration below highway and airfield pavements, Highway Research Board. Aldrich, HP and Nordal, RS (1957), Frost penetration in multilayer soil profiles, ACFEL Tech Report 67. Aldrich, HP and Paynter, HM (1953a), Analytical studies of freezing and thawing in soils, ACFEL Tech Report 42. Aldrich, HP and Paynter, HM (1953b), Calculation of depths of freezing and thawing under pavements (discussion of Carlson and Kersten paper ) Highway Research Board, Bulletin 71, Aldrich, HP and Paynter, HM (1966), Depth of frost penetration in non-uniform soil, CRREL Special Report 104. Aldrich, HP and Scott, RF (1956), Hydraulic analog computer aids in solution of frost problems, Civil Engineering, , August. Berggren, WP (1943), Prediction of temperature distribution in frozen soils, Trans. AGU, Part 3, 71-77, November. Hawk, R and Lamb, W (1963), Hydraulic analog study of periodic heat flow in typical building walls, CRREL Tech Report 135. Lukyanov, VS (1939), Hydraulic apparatus for engineering computations, Akad Nauk USSR (ACFEL translation Jun 1955). MIT CE Dept (1956), Design and operation of an hydraulic analog computer for studies of freezing and thawing of soils, ACFEL Report, May Moore, AD (1936), The hydrocal, Ind & Eng Chem, v28, 704. Nordal, R (1955), An investigation of frost penetration in soil by hydraulic analog computer, MIT SM Thesis. Paynter, HM (1955), Analog solutions of thermal problems, (in A Palimpsest on the Electronic Analog Art ), Philbrick Researches Rubenstein, LI (1971), The Stefan problem, Translations of Math Monographs, v27, American Math Soc ( 1967 Russian original translated by A. Solomon ). Scott, RF (1955), Investigation of factors affecting heat transfer at the air-ground interface with special reference to freezing and thawing problems under airfield pavements, MIT ScD Thesis. Scott, RF (1969), Predicted depth of freeze or thaw by climatological analysis of cumulative heat flow, CRREL Tech Report 195. Stefan, J (1889), On the theory of ice formation, particularly ice formation in the Arctic ocean (in German ), S-B Wien Akad, v98, Paynter

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15 The Investigators in 1952 R F Scott H M Paynter H P Aldrich

16 The Investigators Today RFS HMP HPA