INDUSTRIAL AND ENGINEERING CHEMISTRY
660
mately 10 seconds upn-ard, depending on the nature and state of the product. Commercially, the aseptic process has several advantages over the conventional method. The most important are as follows: 1. A better flavor, texture, and color are produced Kith the flash sterilization method. 2. The flash sterilization method, with aseptic filling and closing, provides a very, wide range between scorching of the product and sufficiently safe sterilization, thus permitting a large factor of safety without sacrificing quality. 3. Regardless of container size the quality of the finished pioduct is the same. ' From this very general comparison of conventional canning and the aseptic canning process it must be realized that the heat exchanger system required to supply a sterile product must have several requisites. The heat exchanger under discussion provides these various requisites and affords many unique advantages. The high ratio of heat transfer surface to the volume of material treated fulfills the requirement for a very high heating rate. The constant and rapid cleaning of the heat transfer surface1000 to 1400 times per minute-prevents scorching and localized overcooking. Turbulence is created by the revolving shaft and blades rather than by forcing the product a t a high velocity through small diameter tube exchangers, thus eliminating the necessity of building the heat exchanger to withstand high pressures. As a matter of fact, the pressure drop across this heat exchanger has to be imposed by a back pressure valve thus making it possible to operate a t a process pressure high enough only to prevent flashing a t the sterilization temperature. Because of the spinning shaft and blades and the flexibility
Einr;g;r
ng
p m e S S
development
Vol. 44, No. 3
of pressure adjustment across the heat exchanger system, the equipment lends itself to the processing of a wide variety of food products irrespective of product viscosity and consistency. Some of the products sterilized with this equipment range from whole fluid milk to pumpkin and cream style corn. The ability to control the pressure drop across the system makes possible the elimination of high pressure pumps, By the proper design of the shaft and control of the annular space, products containing discrete particles have been processed, for example, cream style corn. In respect to sanitation the heads and shaft are readily accessible. All surfaces coming in contact with the product can be visually inspected. In conclusion, this heat exchanger has been proved to be ideally suited for applications where the material is heat sensitive, very viscous, and undergoes a change of state during heating or cooling. Examples may be found in starch cooking and cooling, gelatin cooling, freezing of concentrates, chilling and plasticizing of fats and oils. In short, it has been experienced that, in most cases, if the product to be processed is pumpable it can be processed in this closed, continuous sanitary heat exchanger, LITERATURE CITED
(1) Slaughter, J. E., Jr., and McMichael, C. E.,, J . Am. Oil Chemists' SOC.,26, No. 10, 623-8 (October 1949). (2) Sutton, Mack, and Bond, A., Jr., Am. Paint J.,33, S o . 31, 56, 58, 61-2, 64, 66, 68, 70 (1949). RECEIVED for review May 4, 1951. ACCEPTED September 28, 1951. Presented as part of the Symposium on Chemical Engineering Aspects of Food Technology before the Divisions of Industrial and Engineering Chemistiy, and Agricultural and Food Chemistry, 119th Meeting, A\IERICAN CHEMICAL SOCIETY,Boston, Mass., dpril 1981.
High Purity Spongelike Copper from Waste Pickling Sludge ROBERT L. RUSHER'
AND
G E O R G E W. B L U M
CASE INSTITUTE OF TECHNOLOGY, CLEVELAND, OHIO
A
PORTION of an investigation of possible methods for the utilization of pickling sludge from the brass and copper industries resulted in the development of a method for the preparation of a high purity copper of particularly fine spongelike texture and large surface area. The waste material utilized is that produced in the acid pickling baths of the brass and copper products manufacturers. It is a highly acidic metallic salts sludge, which a t present constitutes a severe water pollution problem, especially in the Connecticut area where one third of the United States copper and brass products are mariufactured. The analysis of the pickle liquor sludge used in this investigation is shown in Table I. With the assumption that all of the nitrogen present exists as nitric acid and that all of the metals are present as sulfates, the analysis of Table I1 was obtained. The effects of such a waste on waters used for public water supplies for industrial, agricultural, and recreational purposes 1 Present address, Grasselli Chemicals Departmenr, E. I. du Pout de Nemours & Co., Inc., Wilmington, Del.
are obvious. It causes corrosion and deterioration of the sewage treatment plants to take place and seriously affects the biological treatment of sewage. The acids and metal salts are toxic to aquatic life, even in concentrations as small as 2 p.p.m. for copper and zinc. Although it was realized that the underlying emphasis of this problem is normally directed toward the study of pickling wastes for the abatement of pollution, this work was a study of the utilization of the sludge, because the possibility arises for the recovery of copper and zinc which are again critical metals in the present world situation. During the course of preliminary laboratory investigation of the problem, four methods of utilization were considered. 1. The method of thermal decomposition, whereby one third of the copper was recovered from the sludge due to a solubility difference after roasting a t 670" C. for 1 hour, was considered economically not feasible because of the high cost of construction, operation, and maintenance of a rotary kiln operating at 670' C. 2. The use of the acid-free sludge plus additives, such as so-
,
March 1952
INDUSTRIAL AND ENGINEERING CHEMISTRY
TABLE I.
661
SLUDGECOMPOSITION AS RECEIVED
Total acid (as HBOd % Total nitrogen aa E@), Total copper, Total zinc % Total sdf& as 803 % Inerts ( i n o d e s PbSbd, %
d,
20.85 4.69 10.60 11. so 48.00 2.5
TABLE11. ANALYSISOF PICKLE LIQUORSLUDGE Bulfuric acid % Nitric acid, b/, Copper sulfate % Zinc sulfate ./, Inerts (inclddes PbSOl) % Water (by difference), b/,
18.51 4.69 26.6 27.9 2.6 19.8
dium cyanide and sodium carbonate, as a brass plating bath was considered highly impractical because of the variation in composition of the sludge from different sources and the meticulous concern of the plating industry regarding the composition of its plating baths. 3. A study of the work of Guyer and Perren (6) and other literature on froth flotation (3,Q) was made, but it was believed that this method would require lengthy fundamental research with little likelihood of a successful separation of the soluble metal salts, owing t o the complex and varying composition of the sludge. 4. The method of separation, best described by the name fractional crystallization, was considered, whereby it was hoped the copper and zinc sulfate present in the sludge could be separated because of their difference in solubilities a t different temperatures. A study of the copper sulfate-zinc sulfate-water phase diagram (11) showed that neither salt could be separated by crystallization from the other in a high degree of purity. Laboratory tests showed that 3 to 5 recrystallizations and much recycle of mother liquors are necessary to obtain separation of these salts in a degree of purity approaching technical grade compounds. Because of the complexity of the sludge system copper sulfate-zinc sulfatelead sulfate-water-sulfuric acid-nitric acid, its great variation in composition (14) depending upon its source, and the complex recycle system needed for efficient recovery of the salts, this method was also abandoned a~not feasible, although of the four methods investigated it is considered the best. COPPER-ZINC DISPLACEMENT
The copper-zinc displacement reaction is referred t o in the literature variously aa a couple, a precipitation, and a cementation ( I , 6, 7, It?, IS). Hereafter the following reaction for the formation of sponge copper will be referred to as displacement CuSO, (aqueous)
+ Zn HzSO, ZnSOc (aqueous) + Cu -+
Figure 1. Sponge Copper Photographed by Reflected Light
finely divided sponge structure has been found to be dependent upon the presence of from 4 to 77, of sulfuric acid in the copper sulfate solution. If the acid is not present the copper will plate out on the zinc in solid sheet form; if excess acid is present more than the minimum amount of zinc required for displacement and gas formation will be used. Even under optimum conditions about 40% of the zinc used in the displacement reaction goes into the zinc-sulfuric acid reaction. I n order t o reduce the zinc-acid reaction, the suggestion was made that an increase of hydrogen pressure on the system might be effective. This assumption is in accord, in part, with Ipatev and Ipatev who report that pressures of 50 to 600 atmospheres a t temperatures of 60" to 300" C. will cause the reversal of the reaction of sulfuric acid on copper (6, 7 ) . To determine the effect of pressure on the undesirable (from an economic viewpoint) reaction of sulfuric acid on zinc during the displacement reaction, an actual laboratory high pressure displacement reaction was run. The reaction vessel used was a borosilicate glass test tube 5 inches long and 11/2 inches in diameter, which fit snugly inside a stainless steel pressure bomb. In the glass tube was placed a known weight of 33% sludge solution from which the insoluble matter had been removed by filtration. A known mass of 99.99% zinc in the form of a small cylinder of known surface area was suspended immersed in the solution for 2 hours a t room tem erature and under 1000 pounds per square inch gage (67 atmospferes) of hydrogen pressure. Another reaction, the same in all respects rn
(1)
By this method of pickling sludge utilization, about 70% of the acids contained in the raw sludge was removed by vacuum filtration, the remaining acid being an essential factor for the formation of the spongelike structure of the copper. The role of the acid in this reaction may be postulated as follows: Simultaneously with the copper-zinc displacement reaction the acid present reacts with the zinc
Zn
+ H&O4 -+ Hz + ZnSO,
(2)
It is believed that the hydrogen, which under the proper conditions forms in minute bubbles giving the solution a turbid appearance, is principally responsible for t h e fine division of the cop.per, its formation acting as a physical force t o separate the forming of copper. The acid also appears t o undercut the formed copper in its attack on the zinc, causing further division of the copper. The displacement of the copper by the zinc in the form of a
Figure 2. Sponge Copper Photographed by Transmitted Llght
INDUSTRIAL AND ENGINEERING CHEMISTRY
662
the one described above, was run a t atmospheric pressure as a control. Keither solution was agitated. From the loss in weight of zinc in each case it was concluded that under the test conditions 67 atmospheres of hydrogen pressure does not retard the action of sulfuric acid on zinc.
TABLE
111. 33%
SLUDGE SOLUTION -4NALYSIS
Total acid (as H&O4), % Copper. % Zinc, %
6.75 3.43 3.66
A series of displacement reaction tcsts were conducted under fairly standardized conditions in order to determine the effect of temperature and agitation on reaction rate and type of copper formed, and the relation of the area of exposed zinc to rate of reaction.
All tests were run in a 100-ml. beaker which was fitted with a small air-powered stirrer to provide agitation. The zinc used was 99.99% pure and in the shape of small cylinders 0.75 inch in diameter and 1.2 inches long. The zinc cylinders were therefore of known initial surface area and weight and were reweighed after the test to determine zinc loss. The cylinders were suspended in the sludge solution by means of a small diameter glass rod cemented in a hole in the end of the cylinder. All tests \!-ere run at room temperature with the only heat coming from the exothermic reactions involved. The test solution used consisted of a 33% solution of vacuumfiltered sludge dissolved in water and filtered to remove inerts and lead sulfate. This concentration of sludge in water is near saturation a t room temperature and could easily be prepared on it commercial scale. This solution mas found to have the composition shown in Table III.
Vol. 44, No. 3
solution. Since both the displacement and acid-zinc reactions are highly exothermic, and since the reactions are run a t room temperature with no provision for cooling, the greater the zinc area the greater the reaction rate. Increased reaction tempcrature then causes a further increase in reaction rate, because of lack of heat dissipation provisions. The result is essentially the same as too vigorous agitation-that is, too rapid gas evolution, resulting in the formation of flake copper of reduced purity. Optimum zinc reaction area for 100 grams of 3370 sludge solution was found to be 10 square inches for the formation of a '/r-inch copper sponge layer in about 40 minutes. The copper sample depicted in Figures 1 to 4 was made under conditions considered most favorable for the production of a pure, fine spongelike copper. Three strips of standardizing zinc, approximately 3 X 1.5 inches, were imbedded in three parallel slots cut in the larger end of a large cork. The free ends of the zinc sheets were then immersed in the 33y0 sludge solution contained in a 100-ml. beaker; a shaft \vas inserted in a hole centered in the other end of the cork to serve as a support for the zinc and as a center of rotation, so the sheets could be revolved slowly in the solution to provide agitation. Copper was deposited on the zinc sheets in a 1/4-inch layer of fine meshed, dusty rose-colored sponge over a period of 40 minutes. The reaction was run a t room temperature (27' C.) and the solution temperature attained a maximum of 58' C. about three fourths of the way to completion of reaction; the reaction was considered complete when the sludge solution had changed in color from blue-green to clear white, although it was found that a white solution could contain 1to 2% of copper. The deposited sponge was cut along the edges of the zinc and peeled off, washed in distilled water several times followed by two spray rinses with C.P. methanol, and dried for 5 minutes in an oven which had open doors to provide air circulation, a t 85' C. Drying of the copper sponge should be accomplishcd as rapidly as possible t o prevent excessive oxidation \ihicli is readily evidenced by darkening of the color of the sponge. CHARACTERISTICS OF COPPER FORMED
Figure 3. 27,000 X Electron Micrograph of Sponge Copper, Showing Existence of Sponge Structurc Down to Ultimate Particle Size
The degree of agitation was found to be second in importance only to the presence of sulfuric acid as a factor on which the sponge copper formation is dependent. Agitation rate is proportional to reaction temperature and rate of hydrogen evolution; an increase in the agitation rate of the sludge solution during the displacement reaction decreases reaction time but increases reaction temperature and rate of hydrogen evolution, causing the forming copper to be blown apart as finely divided flake copper of low purity. The optimum agitation condition necessary to form a fine spongelike copper of 99+y0 purity is a very slow rate of solution flow past the zinc surface, sufficient to bring fresh solution in contact with the zinc and give a fine cloud of evolving hydrogen, but not so fast as t o cause the forming copper to be disintegrated by the rapidly forming gas bubbles or the force of a rapidly moving solution. The desired degree of slow stirring was finally accomplished in the laboratory by revolving the zinc slowly in the solution instead of circulating the solution. The reaction time and the quality of the copper displaced is also controlled by the area of reacting zinc per weight of sludge
The rapid color change of the copper on exposure to air indicates that the freshly prepared metal is highly surface active, and since the spongelike structure provides a large surface area, it is thought that this material might serve as a gas or liquid phase reaction catalyst. The copper analysis of this representative sample showed it to be 99% pure, the 170 impurity probably being zinc. The physical structure of this specimen is roughly shown in the photographs of Figures 1 and 2, actual examination from several angles being required to reveal its fine intricate structure. It is strong cnough to support about 25 times its own weight, abovc which i t crumbles into a fine p o d e r . The electron micrographs of Figures 3 and 4 mere made a t 27,000 X magnification. Figure 3 reveals that the sponge structure exists even under this magnification, the agglomerate shown being 0.00375 mm. in its greatest dimension. Figure 4 reveals the ultimate particle size of the copper as being 3.12 X 10-jmm. in diameter (312 A , ) , the ultimate particles being the smaller diameter group of specks, the larger plainly being agglomerates. The ultimate particle size of this copper, examined by Powell of Case Institute of Technology, is shown to be about 100 times smaller than any copper normally employed for catalytic purposes or powder metallurgy. ZINC RECOVERY
The solution resulting from the displacement reaction contains a maximum of 4% copper; this is removed by zinc dust treatment and filtration prior to electrolysis t o obtain the recovery of the zinc. Conditions for the low current density process of zinc electrode position were maintained in the laboratory work; the
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1952
type of reaction, such as hydrogenation, deliydrogenation, dehydration, reduction, oxidation, etc. (2, 4 ) ; it appeared to be the most common catalytic material in use today. A typical dehydration catalyst was found to have the composit i o n 0 . 3 ~ i r o n , 0 . 2 ~ s i l i c o n1.3%zinc, , and 1.40J,aluminum. The zinc content is of particular interest as the product of this design project contains about 1%zinc.
TABLE IV. COMPOSITION OF SOLUTION RESULTINGFROM DISPLACEMENT REACTION Total acid (as HzSOd, % Zinc % W'atLr, %
0.48
10.3 68.8
.
analysis of this solution, given in Table IV, shows it to be of the approximate composition required for this process. The low current density process of zinc electrode position requires the use of a 0.5% sulfuric acid solution containing 10% zinc to be electrolyzed in a cell using aluminum cathodes and lead anodes. The current density is maintained a t 35 to 40 amperes per square foot and, under these conditions, 1 ton of zinc will require about 3550 kw.-hr. (8,IO). The spent electrolyte solution contained 11%sulfuric acid and 2.5% zinc. Obviously this solution cannot be discarded because of its high zinc and acid content. On a cominercial scale the electrolysis step could be carried out by an established electrolytic zinc producer, the spent electrolyte being used to leach zinc ores, thereby utilizing the acid and recovering all of the zinc. METHODS OF ANALYSIS
The following analytical scheme was employed to determine quantitatively total acid, nitrogen, sulfur, lead, copper, and zinc:
PREPARATION 08 THE SAMPLE.A representative portion of the sludge was weighed in a tared sample bottle on an analytical balance. The known weight of sludge was transferred without loss to a 1-liter volumetric flask and diluted to volume. DETERMINATION O F TOTAL -4CIDITY. T O determine total acidity, 50 ml. of the stock solution were titrated with a standard sodium hydroxide solution to a methyl red end peint. Acidity was calculated as per cent of sulfuric acid. DETERMINATION OF TOTAL NITROGEN. To analyze nitrogen content the standard Kjeldahl flask method was employed, using approximately 25 grams of De Varda's alloy for 50 ml. of the stock solution for the indication of the end point. Total nitrogen as nitric acid is equal to (ml. HCl X N
.
- ml. KaOH X N ) X 0.06302 X sample weight
663
20 X 100 (3)
DETERMINATION OF TOTAL SULFUR. A convenient-sized sample of the stock solution was analyzed for total sulfur by precipitation of the sulfate as barium sulfate and weighing in the usual manner. ANALYSISOF TOTAL LEAD. The lead was precipitated in a convenient-sized sample as lead sulfate and weighed in the usual manner. DETERMINATION OF COPPER. The copper was determined electrolytically on platinum electrodes. After analysis the catalyst was washed, dried, and weighed. The difference in weight may be recorded as copper. ANALYSISOF ZINC. The zinc was determined volumetrically by titrating a standard solution of potassium ferrocyanide past the pea-green end point. Back titration was then made to the purple end point using a standard zinc sulfate solution. EVALUATION OF PRODUCT
The copper product of the displacement and electrolysis process herein described has sufficient properties of high purity, large surface area, small ultimate particle size, and an unusual spongelike structure for it to be considered for use as a catalyst. In order to evaluate it for such a use the more recent literature was surveyed in order to determine the characteristics of the copper catalysts now in use. It was found that copper is used as a catalyst in such forms as finely divided powder, pellets of various diameter, fine ribbon and wire, and supported on an inert material. No mention was found of a sponge-structured copper having been prepared or used as a catalyst, although it was emphasized that a large surface was desirable. Copper was found to be used as a catalyst for every conceivable
__
* &
c
V
e . I
1
Figure 4. 27,000 X Electron Micrograph of Sponge Copper, Showing Ultimate Particle Size as Being 312 A. Units in Diameter Large irregular masses are agglomerates
Small amounts of nickel and cobalt were found to increase the activity of copper in such reactions as the hydrogenation of benzene. For example, 1yonickel was found to increase the hydrogenation efficiency from 0 to 79% for 7 to 1 mixtures of hydrogen and benzene a t 225" C. and 1 atmosphere. The copper was found to be poisoned by small amounts of such materials as bismuth, cadmium, lead, mercury, tin, sodium chloride, and sodium sulfate. The catalyst was completely deactivated by 0.2% of bismuth, cadmium, or lead, but lead in amounts of less than 0.1% acts as a weak activity promoter (8). These facts indicate that sponge copper, such as is produced by the displacement method described herein, might be suitable for use as a catalyst for gas and liquid phase reactions. The question of economic feasibility of the suggested methods are naturally subject to some question. It is important to note that the work was undertaken primarily as a laboratory investigation with a full realization that the economics of the processes involved would vary considerably in terms of a normal copper market and a normal copper supply. A special interest results today, because of the possibility of recovering strategic metals which appear to be in increasing short supply. In view of the uncertain industrial availability of these materials, a study of possible recovery was thought to be desirable. Zinc will replace other elements besides copper; therefore, the original composition of the brass-treating sludge material must be previously determined. The possibility of other metallic constituents would require additional study for their prior elimination. The electrodeposition of copper is a more normal operation industrially, but it was found in preliminary study that the desirable characteristics of the fine spongelike copper of large surface area were not readily obtainable by these electrodeposition methods. Scrap iron has been used in the past to remove copper from the pickling solution but difficulties of purity, which arise with the copper deposited, limit its potential application as a pure copper catalyst. In addition, the use of scrap iron, which is also present as a major constituent of the pickled material, would completely nullify the success of zinc recovery. The paper is predicated upon the supposition that facilities for the recovery of the zinc by the current density process would be available. This would be desirable because of the large percentage of valuable zinc present. Obviously, if such facilities were not available, the zinc and acid would by necessity be wasted to the sewer.
INDUSTRIAL AND ENGINEERING CHEMISTRY
664
LITERATURE CITED
Capriati, E., Ital. Patent 422,665 (June 20, 1947). Carson, B. B., and Ipatieff, V. N., J . PhzJs. Chem., 45, 431-40 (1941). “Denver SubA Flotation,” Bull. F10.B29,Denver Equipment Co., 1400-17th St., Denver, Colo. Faucounau, L., BUZZ.soc. china., [SI, 4 , 55-67 (1937). Cuyer, A., and Perren. R., Helc. Chi7n. Acta, 25, 1179-97 (1942). Ipatev, V., and Ipatev, V., .Jr., He?.., GZB, 856-90 (1929). Ipatev, V.. and T’erhorukii, V.,Ibid., 44, 1755-5 (1911). Perry, J. H., ed., “Chemical Engineers’ Handbook,” 3rd ed., 1). 1508, Kew York. McGraw-Hill Book Co.. 1950. P o i , L. 8 . ,“Froth Flotation, Industrial and Chemical Applica-
Vol. 44, No. 3
tion,” BUZZ. FIO.B46, Denver Equipment Co., Denver 17, Colo. (10) Bamans, C. H., “Engincering Metals and Their Alloys,” p. 152, New York, Maemillan Co., 1949. (11) Seidell, “Solubilities of Inorganic and Metal Organic Chemical Compounds,” I, 3rd ed., New York, D. Van Nostrand Co..
1940. (12) Shengle and Smith, J . A m . C h m . Soc., 21, No. 2 , 932-3 (1899). (13) Sneed, M. C., and Maynard, 3’. L., “General Inorganic Chemistry,” 4th ed., p. 790, New York, D. Von Nostrand Co., 1943. (14) Wise, W. S., Dodge, B. F., and Bliss, H., IND.
[email protected] 39, No. 5 , 632-6 (May 1947). RECEIVED for review April 28, 1951.
ACCEPTED September 28, 1951.
EngFnTring Bocess development I
RALPH E. PECK, RUSSELL T. GRIF~ITH’,AND K. NAGARAJA RA02 ILLINOIS INSTITUTE OF TECHNOLOGY, CHICAGO, ILL.
M
-4NY equations have been developed to calculate the time of drying to reach any given average moisture concentration for various drying conditions, but most of those equations are not applicable to materials of varying diffusivity or capillarity. The aim of this work is to develop equations relating the effective diffusivity t o the average total moisture content of material and the drying conditions. These relations can be applied in design calculations. The term effective diffusivity denotes the value of the proportionality factor not only for diffusion but for capillarity or any other force which may cause migration of moisture through a solid. This factor is a function of moisture concentration and thus varies throughout the slab. As all the data were obtained on a total veight basis, only average values of the effective diffusivity were obtained. Sherwood (10-15) was one of the first to approach drying calculations by utilizing the analogy of unsteady heat conduction in a slab. He considers that drying of fibrous materials takes place by diffusion of the moisture to the surface. Newman ( 7 , 8) applied the diffusion equation to the drying of solids of various shapes and various surface conditions. The latter equations take the surface film resistance into account, but are not explicit in concentration as a function of time and require a trial and error solution. Kamei and Siomi (5)found the diffusivity of moisture through paper, clay, and soap to change somewhat with temperature, humidity, and air velocity. Bateman, Hohf, and Stainm ( 1 ) reported drying curves for wood cylinders. Van -4rsdel ( 1 6 ) studied the effect of varying diffusivity in solids and outlined an approvimate numerical procedure to calculate drying rates. THEORETICAL ANALYSIS
The general differential equation for the unidirectional diffusion of the moisture through the sides of a rectangular slab is given by 1
Present addreaa, General American Transportation Corp., East Chicago,
Ind. 2 Present address, Consultant Djakarta, Indonesia.
(ECA)
Government
of
Indonesia,
Assuming D as constant, the solution of Equation 1 (5,4 ) can be written in the form (21
The constant represented by A, can be evaluated by the initial condition e = 0, C = COfor all values of 2. A,, then, is given ( 4 ) by
1
.
-4 sin 2ct, R
+ -21 a,R
The constants represented by a,, may be evaluated by applying the boundary condition a t the surface of the slab. At x = T
Applying Equation 4 to Equation 2 it follows that (5)
For large values of 0, i t has been shown (IS)that the series represented by Equation 2 is a rapidly converging series and for most practical purposes only the f i r s t term is important. Then AI and a1 need be evaluated. Equation 5 has to be solved by trial and error or by the help of trigonometric charts. However, approximations can be made t o evaluate a, for certain general cases. Where the solid offers the major resistance cot alR will approach H
2
- alR.
?r
This is seen to be true for small values of ,j - a,R by
reference to trigonometric tables. Therefore
