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UBC Theses and Dissertations

Study of the rheological behavior of CaO-SiO2-A12O3 and CaO-SiO2-A12O3-CaF2 slags Michel, Jean Raymond 1973

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c A STUDY OF THE RHEOLOGICAL BEHAVIOR of CaO-SiO 2-Al 20 3 and CaO-:Si0 2-Al 20 3-CaF 2 SLAGS by JEAN-RAYMOND MICHEL B.A.Sc. Ecole Polytechnique, Montreal, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE, REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Metallurgy We accept t h i s thesis as conforming to the required standard THE UNIVERSITY.OF BRITISH COLUMBIA May, 1973 In present ing t h i s thes is in p a r t i a l fu l f i lment o f the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f r ee ly ava i l ab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by h is representat ives . It is understood that copying or pub l i ca t ion of th is thes is fo r f i n a n c i a l gain sha l l not be allowed without my wri t ten permission. Department of Metallurgy  The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada Date May 31, 1973 ABSTRACT A viscometer was designed and built to study the rheological behavior of slags over a range of rates of shear of three orders A 2 of magnitude, up to a shear stress of 10 dynes/cm . The follow-ing sla,g compositions were studied (wt.%) 1) 42 Si0 2 - 20 A^O ~ 38 CaO, 2) 40.3 Si0 2 - 19.2 Al Q3 - 36.5 CaO - 4.0 CaF 2, and 3) 37.8 Si0 2 - 18.0 A l ^ - 34.2 CaO - 10.0 CaF 2, at temperatures of 1) 1390°C - 1350°C - 1332°C, 2) 1325°C - 1300°C and, 3) 1300°C respectively. Newtonian behavior was found for a l l compositions at the temperatures investigated. General conclusions are drawn as to the likelihood of Newtonian behavior fqr different compositions in these systems. - i i i -ACKNOWLEDGEMENT The author sincerely appreciates the advice and assistance given by Professor A. Mitchell who directed this work. The author i s greatly indebted to Mr. A. Thomas who built and helped design the experimental apparatus. He also wishes to thank fellow graduate students for helpful discussions. The author would like to acknowledge the assistance of the National Research Council for providing funds through operating Grant no. A-4528. - i v -TABLE OF CONTENTS Page TITLE PAGE i ACKNOWLEDGEMENT i i i ABSTRACT i i i TABLE OF CONTENTS i v LIST OF FIGURES v i i LIST OF TABLES i x LIST, OF SYMBOLS x 1. INTRODUCTION. 1 1.1 V i s c o s i t y and Viscous Behavior.... 1 1.2 Viscous Behavior and Transport Properties 3 1.3 V i s c o s i t y and Structure of S i l i c a t e Slags 5 1.4 Previous Work 7 1.5 Objectives of the Present Work 11 2. BASIC PRINCIPLE 12 2.1 Selection of a Viscometer 12 2.2 Selection of a Suitable M a t e r i a l 13 2.3 Selection of Slag Compositions and Operating Temperatures 14 2.4 Derivation of Basic Equations 15 2.5 C a l i b r a t i o n of the Wires 18 2.6 C a l i b r a t i o n of the Viscometer 19 2.7 I m p l i c i t Assumptions made i n Deriving the Basic Equations 20 - V -Page 2.7.1 Laminar Flow 20 2.7.2 Slippage 21 3. EXPERIMENTAL TECHNIQUE 22 3.1 Description of Apparatus 22 3.2 Gas System 25 3.3 Technique of Measurement 26 3.4 Preparation of Slags 27 4. EXPERIMENTAL RESULTS 28 4.1 Rheological Behavior of the Calcium Alumino-s i l i c a t e Slag 28 4.2 Rheological Behavior of CaF - containing Slags 7 30 4.3 A c t i v a t i o n Enthalpies f o r Viscous Flow 32 5. DISCUSSION 34 5.1 Rheological Behavior of Slags 34 5.1.1 Calcium A l u m i n o s i l i c a t e Slags 34 5.1.2 CaF^ - containing Slags 35 5.2 Comparison with Published Data 38 5.2.1 Rheological Behavior 38 5.2.2 Values of V i s c o s i t y 39 6. CONCLUSIONS 41 7. SUGGESTIONS FOR FUTURE WORK 42 APPENDIX I. C a l i b r a t i o n of the Suspension Wires 44 APPENDIX I I . C a l i b r a t i o n of the Viscometer 53 APPENDIX I I I . Corrections f o r Temperature and Depth of Immersion 56 - v i -Page APPENDIX IV. Accuracy of V i s c o s i t y Measurements 58 REFERENCES 64 - v i i -LIST OF FIGURES Figure Page 1 Laminar flow of a f l u i d between p a r a l l e l plates 67 2 Flow curves for various i d e a l r h e o l o g i c a l bodies 67 3 Ratio of non-Newtonian to Newtonian heat transfer rates as a function of n, the flow behavior index.... 68 4 Ratio of non-Newtonian to Newtonian mass transf e r rates to bubbles as a function of n, the flow behavior index 68 5 A c t i v a t i o n energies i n the Si02-Al20.j-Ca0 ternary.... 69 6 V i s c o s i t y of a b l a s t furnace slag and two open hearth slags as a function of r o t a t i o n a l speed (x by 20 to obtain rates of shear) 69 7 Schematic diagram of a concentric c y l i n d e r viscometer 70 8 Laminar flow of an incompressible f l u i d i n the space between two concentric c y l i n d e r s , the outer one of which i s r o t a t i n g with an angular v e l o c i t y Q 70 9 C r i t i c a l Reynolds number for tangential flow i n annulus; outer c y l i n d e r r o t a t i n g and inner c y l i n d e r stationary 70 10 A schematic diagram of the apparatus f o r v i s c o s i t y measurements 71 11 A close-up view of the drive mechanism 72 12 A close-up view of the upper part of the viscometer as used at a) low speeds 73 b) high speeds 74 13 I n i t i a l dimensions of the inner and outer c y l i n d e r s . . 75 14 Temperature p r o f i l e of the empty furnace 76 15 A close-up view of the bottom part of the viscometer showing the graphite and z i r c o n i a f e l t discs 77 16 A close-up view of the upper part of the viscometer showing the graphite and graphite f e l t d i scs 78 - v i i i -Figure Page 17 The viscosity measurement apparatus 79 18 Plot of the viscosity as a function of log, n rate of shear - slag #1, 1390°C .7 80 19 Plot of the shear stress as a function of the rate of shear - slag #1, 1390°C 81 20 Plot of the viscosity as a function of l o g i n rate of shear - slag #1, 1350°C .7 82 21 Plot of the shear stress as a function of the rate of shear - slag #1, 1350°C 83 22 Plot of the viscosity as a function of log, n rate of shear - slag #1, 1332°C 7. 84 23 Plot of the shear stress as a function of the rate of shear - slag #1, 1332°C 85 24 Plot of the viscosity as a function of log, n rate of shear - slag #2, 1325°C .7 86 25 Plot of the shear stress as a function of the rate of shear - slag #2, 1325°C 87 26 Plot of the viscosity as a function of log, n rate of shear - slag #2, 1300°C 77 88 27 Plot of the shear stress as a function of the rate of shear - slag #2, 1300°C , 89 28 Plot of the viscosity as a function of l o g i n rate of shear - slag #3, 1300°C ! V 90 29 Plot of the shear stress as a function of the rate of shear - slag #3, 1300°C 91 30 Plot of l o g 1 0 n vs. ^  92 31 A close-up view of the five suspension wires 93 32 A close-up view of the torsional pendulum 93 33 Schematic diagram of a mass as used for wire calibration 94 - i x -LIST OF TABLES Table Page I Summary of published work on the r h e o l o g i c a l beha-v i o r of slags 10 II Composition of the slags studied 15 II I T o r s i o n a l constants of the suspension wires 19 IV Maximum Reynolds number attained i n each slag system studied 20 V Summary of r e s u l t s - Calcium a l u m i n o s i l i c a t e s l a g . . . 28 VI Summary of r e s u l t s - CaF^ - containing slags 31 VII Slag compositions investigated by Langhammer and Geek 38 VIII Comparison of present work with that of other workers 39 IX Weight and dimensions of the masses used to c a l i b r a t e the suspension wires 47 X C a l i b r a t i o n r e s u l t s Wire #1 48 XI C a l i b r a t i o n r e s u l t s Wire #2 49 XII C a l i b r a t i o n r e s u l t s Wire #3 50 XIII C a l i b r a t i o n r e s u l t s Wire #4 51 XIV C a l i b r a t i o n r e s u l t s Wire #5 52 XV C a l i b r a t i o n of the viscometer 54 LIST OF SYMBOLS bottom clearance in the viscometer, cm 2 area, cm a dimensionless constant arising in the Arrhenius equation specific heat, cal g °C ^  any characteristic linear dimension, cm diameter of the suspension wires, cm distance from the center of the pendulum to the inside of the annular discs, cm distance from the center of the annular disc to the center of the torsional pendulum, cm activation energy for viscous flow, cal mole ^ force acting on the plate, dyne gravitational acceleration i n cylindrical coordinates in the r,G and z directions respectively, cm sec -^ -2 modulus of torsion, dyne cm activation enthalpy for viscous flow, cal mole ^  -1 -2 -1 heat transfer coefficient, cal sec cm °C 2 moment of inertia, g cm moment o| inertia with respect to the centroidal axis x, g cm thermal conductivity, cal sec ^  °C ^  cm ^  fl u i d consistency index in the power law dyne sec cm torsional constant of the suspension wires, dyne cm rad ^ torsional constant of.the stainless steel shaft, dyne cm rad~l corrected torsional constant of the suspension wires dyne cm rad~l - x i -length, cm effective length of the inner cylinder, cm actual length of the inner cylinder, cm bottom effect, cm stem effect, cm mass of annular disc, g flow behavior index or an integer number of measurements Nusselt number given by h^ • D/k, dimensionless Graetz number given by wc /k£, dimensionless -2 f l u i d pressure, dynes cm period of the torsional pendulum, sec gas constant 1.987 cal mole °K ^  outer radius of the annular discs used to calibrate the suspension wires, cm inner radius of the annular discs used to calibrate the suspension wires, cm radial distance, cm radius of stem, cm radius of inner cylinder, cm radius of crucible, cm temperature, °K or °C thickness of the annular disc used to calibrate the suspension wires, cm time, sec 3 volume of slag in the crucible, cm velocities i n cylindrical coordinates i n the r, 6, and z directions respectively, cm sec~l - x i i -Symbol Av dv r , -1 -— or — rate of shear, sec Ax dx ' W weight of the slag in the crucible, g w mass flow rate, g sec ^  x immersed length of the stem or deviation of laser beam, cm T torque acting on the inner cylinder and the suspension wire, dyne cm n dynamic viscosity, centipoise (.01 poise) -2 u dynamic viscosity, poise = dyne sec cm v kinematic viscosity, stoke = cm^  sec p density, g cm ^  a standard deviation T shear stress in y direction on surface normal to x, x ^ dyne cm"2 T shear stress in 6 direction on surface normal to r, dyne cm-2 x _ shear stress, dyne cm ^  _2 T°Q yield shear stress, dyne cm Tr0yield yield strength in torsion of suspension wires (p angle of twist, rad rotational velocity of the crucible, sec ^  o 1. INTRODUCTION Heat and mass transfer rates in liquid systems can be strongly affected by the rheological behavior of the liquid. So far, i t has been implicitly assumed, in a l l heat and mass transfer calculations on slag-containing systems, that slags were Newtonian liquids. A study of the structure of s i l i c a t e slags correlated with the viscous flow mechanism shows, however, that this assumption may not be true. From this, and considering the limited amount of work published on the rheological behavior of slags, this work was undertaken. 1.1 Viscosity and Viscous Behavior The viscosity of a liquid can be defined as a measure of i t s resis-tance to flow, due to internal f r i c t i o n forces. This internal resistance results in a transfer of translational energy from one layer of liquid to the next layer when the fl u i d i s set in motion by an applied force. The net result is a transfer of momentum between adjacent layers in the liquid. Consider a liquid between two parallel plates where a shearing force F is applied to the top plate, the bottom one remaining stationary. As shown in Figure 1, the liquid layer adjacent to the top plate moves at a velocity equal to the velocity of that plate; this layer, in turn, drags the next lower layer and transfers some of i t s momentum to that layer causing i t to move with a slightly smaller velocity than the upper - 2 -layer. In this manner, each layer drags the adjacent lower layer causing i t to move with a smaller velocity u n t i l one reaches the liquid layer adjacent to the bottom plate which is motionless. It i s found for most liquids that the force per unit area, F/A, required to push one liquid layer relative to the next one is propor-tional to the ratio of the change i n velocity to the change in the distance perpendicular to flow, Av/Ax. Mathematically, F _ Av c\ 1 \ A ~ ^ Ax I1-1* or, xy Ax M dx It i s important to differentiate u, the coefficient of dynamic viscosity, from v , the coefficient of kinematic viscosity. The latter originates from the timing of the flow of a given volume of liquid, the flow being caused only by the hydrostatic head of liquid. As one would expect v is a function of both u and p . In fact, v is equal to u / p . The unit _2 of dynamic viscosity i s the poise (dyne sec cm ) whereas the unit of 2 -1 kinematic viscosity i s the stoke (cm sec ). Fluids for which u does not vary with the rate of shear, i.e. which obey equation (1.2), are called Newtonian fluids. Conversely, fluids for which u varies with the rate of shear are called non-Newtonian fluids. Typical flow curves, i.e. shear stress vs. rate of shear curves, for various ideal rheological bodies are presented in Figure 2. A is a Newtonian liquid; B, a pseudoplastic f l u i d , C, a dilatant f l u i d ; D a Bingham plastic; E, a pseudoplastic material with a yield value; and F, - 3 -a d i l a t a n t material with a y i e l d value. T° i n Figure 2 i s the y i e l d value. To a f i r s t approximation, the flow curves of a pseudoplastic and a d i l a t a n t f l u i d can be represented by a power law of the form (if Txy K \dx/ (1.3) where n i s the "flow behavior index." n > 1 f o r d i l a t a n t f l u i d s , n < 1 for pseudoplastic f l u i d s and = 1 for Newtonian f l u i d s . The v i s c o s i t y i s given by n_-^ K(fr) " «•*> K=u when n=l. Most non-Newtonian f l u i d s follow a power law over a wide range of rates of shear. Hence, to f i n d the r h e o l o g i c a l behavior of a f l u i d , one has to measure the shear stress as a function of the rate of shear and f i n d the value of n which best f i t s the data. 1.2 Viscous Behavior and Transport Properties Several studies pertaining to transport properties i n non-Newtonian systems have shown that a n a l y t i c a l and empirical trans f e r equations applicable to Newtonian systems are not v a l i d f o r non-Newtonian f l u i d s . Pigford"^ was one of the f i r s t to introduce a factor i n a heat tr a n -s f e r equation to account f o r non-Newtonian behavior. His isothermal s o l u t i o n f o r heat trans f e r to both Newtonian and non-Newtonian f l u i d s i n laminar flow through c y l i n d r i c a l tubes i s given by «.„ - 1.75 (^LpW 1 ' 3 ( 1.5) where N„ = h • D/k, N 0 = wc /k£ and n i s the "flow behavior index" Nu m Gz p as defined i n equation (1.3). - 4 -Pigford's equation i s only v a l i d f o r n > 0.1 . Metzner et a l extended Pigford's s o l u t i o n to values of n smaller than .1. Their r e s u l t s are shown i n Figure 3. As can be seen, heat trans f e r rates can be strongly dependent upon the r h e o l o g i c a l behavior of a f l u i d , e s p e c i a l l y at high Graetz number. For more information on heat tran-s f e r i n non-Newtonian systems, the reader i s ref e r r e d to comprehensive 3 ^ 5 6 works by Metzner ' , Christiansen and Craig and Skelland . Mass transf e r rates have not received as much study as heat tr a n -s f e r rates. One of the very few research works on mass tr a n s f e r i n non-Newtonian systems has been c a r r i e d out by Hirose and Moo-Young^. These authors have studied mass transf e r rates to bubbles moving i n two f l u i d s for which n was equal to .85 and .7, and one Newtonian f l u i d (n=l). Their r e s u l t s shown i n Figure 4 indi c a t e that mass tran-s f e r rates can also be affected by the r h e o l o g i c a l behavior of the f l u i d . Other factors which are affected by r h e o l o g i c a l behavior include momentum transfer rates, mixing, and the v e l o c i t y of droplets f a l l i n g through a f l u i d . These factors are affected because they depend on v i s c o s i t y which f or non-Newtonian l i q u i d s varies with the rate of defor-mation. Hence, apart from objective s c i e n t i f i c i n t e r e s t , the study of the viscous behavior of s i l i c a t e slags i s also of p r a c t i c a l importance since i t a f f e c t s the transfer properties of the s l a g . U n t i l now, i n a l l heat and mass transfer c a l c u l a t i o n s i n slag containing systems, i t has been i m p l i c i t l y assumed that slags were Newtonian l i q u i d s . - 5 -1.3 V i s c o s i t y and Structure of S i l i c a t e Slags E a r l y t h e o r e t i c a l and experimental work on room temperature l i q u i d s , 8 9 p a r t i c u l a r l y by Andrade and Eyring , has shown that the temperature dependancy of v i s c o s i t y can be expressed by an Arrhenius type equation n = A 1 exp (E /RT) (1.6) This equation has since proved adequate f o r most l i q u i d s i n c l u d i n g s i l i c a t e slags. The a c t i v a t i o n energy for viscous flow, E ^ , i s a very important parameter since i t can be re l a t e d to the structure of the l i q u i d through the mechanism of viscous flow. The flow of a viscous body i s usually described microscopically i n terms of two elementary steps; holes are formed i n the l i q u i d and sub-sequently , p a r t i c l e s commonly c a l l e d flow units jump in t o these holes. In order to jump into a hole, a flow unit has to pass over an energy b a r r i e r of height equal to the a c t i v a t i o n energy f o r viscous flow. The magnitude of t h i s b a r r i e r i s made up of two parts; the energy necessary to form a hole compatible with the shape and s i z e of the flow unit and, the energy required to detach the flow unit from i t s surroundings and move i t i n t o the hole. This l a t t e r f actor may include the energy necessary to break one or more bonds to produce the flow u n i t as w e l l as the energy to overcome i o n i c i n t e r a c t i o n s . I t i s now widely accepted that l i q u i d s i l i c a t e s are p o l y i o n i c melts 2+ 2-containing metal cations (e.g. Ca ), oxygen anions (0 ) and an array of complex s i l i c a t e anions i n a state of dynamic chemical equilibrium. 3+ 3+ If other "network formers" are present (e.g. A l , Cr ), these w i l l also enter into the s i l i c a t e network forming for example,aluminosilicate - 6 -anions. Conductivity and v i s c o s i t y measurements on simple i o n i c m e l t s ^ ' ^ have shown that viscous flow i s almost e n t i r e l y an anionic e f f e c t , the cations being too small to a f f e c t the flow p r o p e r t i e s . From t h i s and the high value of for s i l i c a t e slags, i t can be concluded that, i n these melts, the flow units are complex anions or parts of them. From the above discussion, i t follows that i f a change i n the sys-tem has the e f f e c t of modifying the siz e and/or shape of the complex anions or the i n t e r a c t i o n s between ions present i n the s l a g , i t w i l l have an important e f f e c t on the viscous behavior of the system. 12-19 Several authors have investigated the viscous behavior of calcium a l u m i n o s i l i c a t e melts (CaO-A^O^-SiC^) and t h e i r r e s u l t s i n d i -14 cate that E^ varies strongly with composition as shown i n Figure 5 The reason for t h i s dependance i s e a s i l y understood since i t has been 20-21 13 shown by Bockris and others that the s i z e and shape of the complex anions i s strongly dependent upon the 0/(E network formers) r a t i o i n the melt. S i m i l a r l y , change i n i o n i c i n t e r a c t i o n s have also been proposed to 22 a f f e c t the viscous behavior of slags U n t i l now, only composition has been shown to a f f e c t the viscous behavior of slags. We suggest i n t h i s work that v a r i a t i o n s i n rates of shear could also a f f e c t the viscous behavior of slags. The possible e f f e c t s of change i n rates of shear are as follows: 1. High rates of shear may favor the depolycondensation of large s i l i c a t e anions. 2. Changes i n rate of shear may cause the cations to relocate - 7 -themselves so as to weaken c e r t a i n Si-0 or Al-0 bonds and favor the bond rupture process. 3. Variations i n rate of shear may also change the p o s i t i o n of the non-network anions and free cations present i n the melt thus changing the forces acting on the flow u n i t s . Furthermore, the structure of s i l i c a t e slags i s somewhat s i m i l a r to the structure of solutions and melts of polymers which are known to 23 be non-Newtonian. M e r r i l l has suggested that t h i s r h e o l o g i c a l beha-v i o r i s due to progressive disentanglement of large molecules under the influence of shearing forces. In t h i s s i t u a t i o n , the p a r t i c l e s tend to orient themselves i n the d i r e c t i o n of shear thus lowering the resistance to flow. A s i m i l a r mechanism could apply when s i l i c a t e slags are sub-jected to shear forces. Hence, i t i s not u n l i k e l y that the viscous behavior of s i l i c a t e slags could be shear dependent, i . e . that slags could be non-Newtonian. 1.4 Previous Work Because of the t h e o r e t i c a l i n t e r e s t with regard to s i l i c a t e s t r u c -tures as. w e l l as the p r a c t i c a l importance of s i l i c a t e s with regard to smelting reactions and glass technology, many in v e s t i g a t i o n s have been c a r r i e d out on the v i s c o s i t y of molten and glassy s i l i c a t e s . R e l a t i v e l y few, however, have dealt with possible Newtonian or non-Newtonian beha-v i o r of these melts. 24 Quincke was the f i r s t to report v i s c o s i t y measurements on s i l i -cate melts made at d i f f e r e n t rates of shear. He investigated the sys-tem (wt.%) 70 Si0_ - 10 CaO - 20 Na o0 as w e l l as the system 74 SiO - 7 - 8 -(A1 20 3 + Fe-jO-j) - 7 CaO - 12 Na 20 over a range of rates of shear of h a l f an order of magnitude using a concentric c y l i n d e r viscometer (C.C.V.) and found Newtonian behavior. 25 Bockris , using the same type of apparatus, measured v i s c o s i t y i n the system CaO-Si0 2 (42-70 mole % SiO,,) using four d i f f e r e n t rates of shear between 1.3 and 13 sec ''"at each composition and reported that non-Newtonian behavior was absent. 26 Euler and Winkler l a t e r measured the v i s c o s i t y of rock and s i l i -cate melts (>45% S i 0 2 ) at three d i f f e r e n t rates of shear namely 5, 15 and 46 sec ^ and stated that they confirmed the r e s u l t s of the two previous authors. This statement was, however, not substantiated by data i n the p u b l i c a t i o n . 27 Z i l ' b e r has proposed the existence of a t r a n s i t i o n temperature above which slags would be Newtonian and below which they would show a marked deviation from i d e a l viscous flow. This t r a n s i t i o n temperature, however, i s f o r a l l compositions, very closed to the l i q u i d u s temperature (- 20°C) below which s o l i d p a r t i c l e s are present i n the melt. The pres-ence of such p a r t i c l e s i s known to induce non-Newtonian behavior i n s i l i -i 28,29 cate slags 30 Tsimermanis et a l have noted that at low temperatures, the viscous flow of slags often followed the Bingham model i . e . the following flow equation ^ E " T ° 0 = v(dv/dx) (1.7) 31 Langhammer and Geek investigated the viscous behavior of two open hearth slags and a b l a s t furnace slag at d i f f e r e n t temperatures and rates - 9 -of shear using both the outflow method and the concentric c y l i n d e r v i s -cometer method. The r e s u l t s of t h e i r i n v e s t i g a t i o n with t h i s l a t t e r method are shown i n Figure 6. As can be seen, these melts showed a marked deviation from i d e a l viscous flow at low rates of shear. The outflow method confirmed t h i s non-ideal viscous behavior. More recently, Saito et a l published a seri e s of papers on the viscous behavior of calcium s i l i c a t e based melts. They measured v i s -c o s i t i e s at d i f f e r e n t speeds using a concentric c y l i n d e r viscometer and 32 33 found the following systems to be Newtonian: CaO-SiC^ , CaO-SiO^-CaS and CaO-Si0 2-MF 2 . The systems C a O - S i O ^ C r ^ and Ca0-Si0 2"10% C r ^ -29 FeO showed a marked deviation from i d e a l viscous behavior. They as-cribed the n o n - i d e a l i t y of the l a s t two systems to the presence of sus-pensions of Cr^O^ p a r t i c l e s and chromite p a r t i c l e s r e s p e c t i v e l y . Their r e s u l t s and those of the previous authors are summarized i n Table I. A l l the experiments reported here have a weakness i n common i n that none of them (with the possible exception of Langhammer's) were designed to study the r h e o l o g i c a l behavior of slags. They cover a very small range of rates of shear, i n most cases less than an order of magnitude. This i s an important shortcoming since even for melts or solutions of polymers, non-Newtonian behavior can often not be detected over an order of magnitude v a r i a t i o n i n rates of shear. Furthermore, a l l the experi-ments which have lead to the conclusion of Newtonian behavior have been 3 2 c a r r i e d out at shearing stresses lower than 10 dynes/cm , the lower 35 l i m i t f o r non-Newtonian behavior of the longest known chain polymers. I t i s quite apparent from the above discussion and the r e s u l t s presented i n Table I that more work i s required before any conclusion TABLE I Summary of Published Work on the Rheological Behavior of Slags Range of rate Shearing V i s c o s i t y of shear stress Viscous Investigator Slag system Method (poise) covered (sec -^) (dynes/cm^) behavior 70% S i 0 2 - 10% CaO - 20% Na 20 C.C.V. 1.35 x 102 (1) (1) Newtonian Quincke 74% Si02 - 7%(A1 20 3 + Fe 20 3) o - 7% CaO - 12% Na 20 C.C.V. 3.78 x 10 2 (1) (1) Newtonian Bockris CaO - S i 0 2 (30-50 mole % CaO) C.C.V. .5 to 20 1.3 - 13 up to 260 Newtonian Euler 45-65% S i 0 2 ; 12-17% A1 20 3 ; and 4-11% CaO ; balance Na 20, k 20 C.C.V. 10 to 4000 5, 15, 46 (2) Newtonian Winkler MgO, F e 2 0 3 , FeO (-20% of tot a l ) Langhammer Two open hearth slags and one B.F. slag C.C.V. 1.2 to 21 5 to 160 200 to 1000 Non-Newtonian Geek CaO - S i 0 2 - M F 2 C.C.V. 1.6 to 13 33, 66 & 100 up to 1100 Newtonian CaO - S i 0 2 - CaS C.C.V. 43, 87 & 131 up to 1100 Newtonian Saito et a l 14, 20, 41, 61 CaO - S i 0 2 - C r 2 0 3 C.C.V. up to 1500 Non-Newtonian CaO - S i 0 2 - 10% C r 2 0 3 - FeO C.C.V. 14, 20, 41, 61 up to 1500 Non-Newtonian CaO - S i 0 2 C.C.V. .6 to 9 up to 50 Newtonian (1) Cannot be calculated because the dimensions of the apparatus were not given. (2) These authors said that t h e i r slags were Newtonian without giving any information on how they arr i v e d to t h i s conclusion. - 11 -can be drawn about the r h e o l o g i c a l behavior of s i l i c a t e melts. 1.5 Objectives of the Present Work Although much research has been done pertaining to the v i s c o s i t y of slags, l i t t l e work has been published concerning t h e i r r h e o l o g i c a l behavior. The aims of the present work are: 1. To design and b u i l d a viscometer which w i l l permit v i s c o s i t y measurements over a wide range of rates of shear and up to shear stresses 2 i n excess of 1,000 dynes/cm . 2. To investigate the r h e o l o g i c a l behavior of s i l i c a t e slags. 3. To study the e f f e c t of CaF^ additions on the viscous behavior of s i l i c a t e slags. - 12 -2. BASIC PRINCIPLE 2.1 Sele c t i o n of a Viscometer Although numerous methods are av a i l a b l e f o r the measurement of v i s -c o s i t y at room temperature, t h e i r successful a p p l i c a t i o n becomes pro-g r e s s i v e l y r e s t r i c t e d as the temperature i s increased. Methods of mea-surement req u i r i n g a long and uniform hot zone, as f o r instance, those methods based on the f a l l i n g body p r i n c i p l e , as w e l l as those based on the f a m i l i a r P o i s e u i l l e p r i n c i p l e of viscous flow i n c a p i l l a r i e s could 36 not be considered. The logarithmic decrement method ( i n i t s various modifications, u t i l i z i n g r e s p e c t i v e l y an o s c i l l a t i n g d i s c , sphere or c r u c i b l e ) i n e v i t a b l y involves a continuously varying rate of shear and i s therefore unsuitable f or non-Newtonian l i q u i d s . This l e f t two methods, the cone and plate method, and the concentric c y l i n d e r method. The l a t -ter was selected on the following b a s i s : 1. Concentric cy l i n d e r viscometers are the most common for v i s c o s -i t y measurements at high temperature because of the r e l a t i v e l y small i s o -thermal zone needed. 2. This type of viscometer i s easy to operate and permits accurate v i s c o s i t y measurements over a wide range of rates of shear. 3. The mathematical formulation f o r the rate of shear of a non-37-39 Newtonian l i q u i d i n a concentric c y l i n d e r viscometer i s av a i l a b l e A schematic diagram of a concentric c y l i n d e r viscometer i s shown i n - 13 -Figure 7. There are two variants of the concentric cy l i n d e r method, one i n which the outer cy l i n d e r i s kept stationary and the inner c y l i n -der rotated by a va r i a b l e torque, and the other i n which the outer cyl i n d e r i s rotated at a va r i a b l e speed while the torque produced on the inner cy l i n d e r i s measured usually by measuring the angle of twist of an e l a s t i c suspension. A l l i n d u s t r i a l models are of the former type. The l a t t e r , however, was selected f o r the following reasons: 1. I t i s much easier to b u i l d . The various i n d u s t r i a l types l i k e , f o r example, the Brookfield model Syncro-Lectric could not be used be-cause of the small range of rates of shear which they cover. 2. Furthermore, the onset of turbulent flow with the ro t a t i n g inner c y l i n d e r type occurs at a much lower r o t a t i o n a l speed. The cen-t r i f u g a l force s t a b i l i z e s the flow when the outer cy l i n d e r i s rotated. 2.2 Sele c t i o n of a Suitable M a t e r i a l As i s often the case with most measurements at elevated temperature, a primary l i m i t a t i o n f o r v i s c o s i t y measurements i s the a v a i l a b i l i t y of a s u i t a b l e r e f r a c t o r y container. This container should s a t i s f y the three following conditions 1. I t must be machinable to exact and reproducible dimensions required f o r v i s c o s i t y measurements. 2. I t must be i n e r t to chemical attack with respect to the sla g and the environment. 3. I t must be wetted by the slag otherwise slippage occurs and the values of v i s c o s i t y thus obtained become meaningless. - 14 -For the present study, these conditions were met by molybdenum. 2.3 Selection of Slag Compositions and Operating Temperatures The s e l e c t i o n of a calcium a l u m i n o s i l i c a t e slag and of operating temperatures was subject to the following c o n s t r a i n t s : 1. The slag had to have as high an 0/(Si+Al) r a t i o as possible i n order to give large a l u m i n o s i l i c a t e s anions. As discussed i n the introduction, the larger the r a t i o , the more l i k e l y i t i s to observe non-Newtonian behavior. 2. The composition had to be such as to permit the measurement of v i s c o s i t i e s at d i f f e r e n t temperature to detect, i f possible,the change from Newtonian to non-Newtonian viscous behavior reported by 31 Langhammer and Geek (see Figure 2) as the temperature decreased. 3. The maximum operating temperature was set at 1400°C because of apparatus l i m i t a t i o n s . The minimum temperature had to be at l e a s t 25°C above the l i q u i d u s to make sure that no s o l i d p a r t i c l e s would be present i n the melt. Hence, a slag with a low l i q u i d u s temperature was necessary i n order to be able to vary the temperature as proposed i n 2. 4. F i n a l l y , the v i s c o s i t y of the slag had to be such as to y i e l d high shear stresses. Since the longest known chain polymers are New-3 2 tonian up to shear stress of 10 dynes/cm , i t appeared that shear stresses i n excess of that l i m i t should be produced i n order to detect possible non-Newtonian behavior of slags. The composition chosen i s given i n Table II along with the compo-s i t i o n s of two CaF - containing slags which were also studied. - 15 -TABLE II Composition of the Slags Studied (weight %) S i 0 2 CaO A 1 2 ° 3 C a F 2 Slag 1 42.0 38.0 20.0 Slag 2 40.3 36.5 19.2 4.0 Slag 3 37.8 34.2 18.0 10.0 These slags have an O/Si+Al r a t i o equal to 2.44. The l i q u i d u s of „ 40 Slag 1 i s 1265 C. Slag compositions 2 and 3 were obtained by adding the necessary quantity of CaF 2 to Slag 1. Although no data are a v a i l -able pertaining to the l i q u i d u s temperatures of Slags 2 and 3, i t i s very l i k e l y that they w i l l not be f a r from 1265°C. 2.4 Derivation of Basic Equations The general equations of motion i n c y l i n d r i c a l coordinates may be 41 written i n the following way r-component f 9 v r 9v v Q 3v y . v r 9 r 0 , + r T — + — T - r - + 9r r 3 0 r v 3v z r 3p 3r '1 3 ( r v r y 1 3 2v 3z 2 3v ) -3r \ r 3r + ~2 ^ 2 2 r 3 6 r 3 6 s 2 3 v 3z 0-component 3 v 0 ^ V r 9 v 0 J . V 0 8 v 0 4 . V r V 0 + v 9 v ( 3t 3r r 30 r 3z (2.1) 1 3p \ 1 3P / = ~ r 3 l + M ' 3 / 1 3 , .\ 1 32v_ 2 3v 3r V r 3r / r 3 6 r 30 3 2v, + 3z + Pg f (2.2) - 16 -z-component / 9v 9v v„ 9v 3v ^  z + B z + z 9r r 90 V z 9z, 1 9 r 9r 9v 9r, 1 9 2v + 2 2 r 9 6 9 2v 9z 9p = " 3z + P g , (2.3) For the tangential flow of an incompressible f l u i d in a concentric cylinder viscosmeter, the velocity components v^ , and are zero and there i s no pressure gradient in the 6-direction. Under those conditions, the equations of motion reduce to 2 r-component - P 9p 9r (2.4) 6-component 0 = d _ / l _d (rv e)) dr I r dr ' (2.5) z-component 0 = - 4r + P S OZ Z (2.6) Equation (2.5) may be integrated with respect to r, which gives I _ d (rv Q) = r dr (2.7) then, V C (2.8) The integration constants and may be determined by using the boundary conditions: at r = r^, v Q = 0; at r = r c > v g = fi0rc ( s e e Figure 8). The result i s v~ = 0 r ~ o c 2 2 r, - r c L c r -r (2.9) - 17 -The component ^ of the stress tensor f or Newtonian f l u i d s i n c y l i n -d r i c a l coordinates i s given by 42 T = - V r 6 r 3 /v r r \ r / r 8 6 (2.10) Rec a l l i n g that v r=0 and s u b s t i t u t i n g equation (2.9) into equation (2.10), one obtains a f t e r d e r i v a t i o n the shear stress d i s t r i b u t i o n i n the l i q u i d as a function of r. r 0 1 r 1 x = 2(ifi r r 6 o c 2 2 r - r b c. (2.11) The torque required to hold the inner c y l i n d e r i s given by 2?rr, L * r, b b (2.12) r=r. Thus T = 4iTuLft r 2 r c r - r. (2.13) The torque T i s obtained by measuring the angle of twist <p of a suspension wire. The r e l a t i o n between T and <p i s as follows r = Gfrd 16 a (2.14) Equations (2.12), (2.13), and (2.14) may then be combined to y i e l d the f i n a l expression for the shear stress at the bob i n a con-c e n t r i c cylinder viscometer. The r e s u l t i s - 18 -GTTd . 1 16 X, 2frr 2 L (2.15) The expression for the rate of shear at the inner c y l i n d e r i s obtained by so l v i n g equation (2.11) f o r r=r^ and by d i v i d i n g by -u since T = u dv Q/dr. The r e s u l t i s r r) u dv dr 2tt r 2  o c 2 2 b c b (2.16) The v i s c o s i t y i s calculated from the following equation: y = T R Q / dv 0/dr (2.17) or, G-rrd 16A / 2 27 c b (2.18) 2.5 C a l i b r a t i o n of the Wires Because of the wide range of torque which was covered, several sus-pension wires were used. The value of the wire constant, given by K = G^d 4 (2.19) w 16 Jl could not be evaluated with enough p r e c i s i o n due mainly to the uncertainty i n G. In order to get very accurate values for K w, the suspension wires were c a l i b r a t e d using a t o r s i o n a l pendulum. The technique used i s pre-sented i n Appendix I. The constants of the f i v e wires which were used are l i s t e d i n Table I I I . Their accuracy i s believed to be better than - 19 -TABLE III Torsional Constants of the Suspension Wires wire # K (dyne cm rad ) w 1 54,960 2 123,200 5,079,000 2,062,000 530,700 (1) Corrected values ju s t i f i e d in Appendix I. 2.6 Calibration of the Viscometer It should be noted that L is the effective or equivalent length of the inner cylinder. The value of L is always greater than the actual 35 length of the inner cylinder because of end and stem effects . These effects are constant provided that the inner cylinder i s well centered and always immersed to the same depth, and that the bottom clearance is kept constant. The quantity L i s obtained by calibrating the visco-meter with a Newtonian f l u i d of known viscosity u , then This calibration i s reported in Appendix II. The value of L at room temperature was found equal to 5.92 cm. This value of L is valid at room temperature for a constant bottom L = 2 2rrr b y (dv Q/dr) r (2.20) b - 20 -clearance and depth of Immersion. During the experiments, the bottom clearance was kept constant at 1.27 cm it .005 cm but the depth of immersion changed due to the addition and removal of s l a g . Furthermore, the actual length of the inner c y l i n d e r increased because of thermal expansion. Corrections to the value of L f o r these two factors are given i n Appendix I I I . An err o r analysis on L i s presented i n Appendix IV. 2.7 I m p l i c i t Assumptions Made i n Deriving the Basic Equations Two i m p l i c i t assumptions were made i n the foregoing d e r i v a t i o n of the equations: 1. Laminar flow 2. No slippage at the slag-molybdenum i n t e r f a c e . 2.7.1 Laminar Flow The t r a n s i t i o n Reynolds number for flow i n a concentric c y l i n d e r viscometer i s strongly dependent upon the r a t i o of the annulus t h i c k -ness to the radius of the outer c y l i n d e r ( ( r c ~ r ^ ) / r ). The t r a n s i t i o n 2 Reynolds number, defined as (^ Q R C P / P ) i s shown i n Figure 9 as a function 43 of (r -r, )/r . The maximum Reynolds number attained i n each system c b c investigated i n the present study i s given i n Table IV. TABLE IV Maximum Reynolds Number attained i n each slag system studied. Slag System Slag 1 Slag 1 Slag 1 Slag 2 Slag 2 Slag 3 (1390°C) (1350°C) (1332°C) (1325°C) (1300°C) (1300°C) Re 30.4 20.8 10.5 30.7 20.2 60.5 < V c p / y ) - 21 -The r a t i o ( r c _ r ^ ) / r c i n o u r viscometer was equal to .175. By comparing values i n Table IV to Figure 9, i t i s seen that the assump-t i o n of laminar flow i s v a l i d . 2.7.2 Slippage Molybdenum has been used i n numerous high temperature viscometers. Nobody has yet reported any slippage i n slag-molybdenum systems. Recen-44 t l y , Amosov et a l have measured the angle of contact of a binary oxide I^O-SiO^ on molybdenum. In the v i c i n i t y of 40 weight % Si02» they r e -ported contact angles of approximately 30° which indicates good wetting of the molybdenum by t h i s s l a g . The author i s not aware of any other paper published on wetting of molybdenum by slags. Slippage can be found and allowances f o r i t can be made i n con-45 c e n t r i c c y l i n d e r viscometers by Mooney's technique . This method how-ever involves the measurement of v i s c o s i t y with two d i f f e r e n t inner c y l -inders and c r u c i b l e s . This was p r a c t i c a l l y impossible to do i n the pre-sent study. Although we do not have concrete evidences that molybdenum i s wetted by the slags studied i n the present work, we believe that the assumption of no slippage i s reasonable. - 22 -3. EXPERIMENTAL TECHNIQUE 3.1 Description of Apparatus The v i s c o s i t y apparatus and furnace are shown i n Figure 10. The viscometer assembly can be divided i n two parts: 1. The bottom part or r o t a t i n g part included the molybdenum cr u c i b l e A, mounted on a molybdenum shaft B, connected to a v a r i a b l e speed motor C (Boston Gear R a t i o t r o l 1/8 H.P. motor) through a speed reducing assembly D. Figure 11 gives a close-up view of the drive mechanism. By changing the s i z e of the smaller p u l l e y or by coupling the motor d i r e c t l y to the 7/1 speed reducer, a reduction r a t i o of any-where from 7/1 to approximately 170/1 could be obtained. This arrange-ment permitted r o t a t i o n a l speeds from =.3 to 450 r.p.m. The c r u c i b l e and shaft were supported by two bearings 4 inches apart located i n a s t a i n l e s s s t e e l water cooled bearing housing E. The r o t a t i o n a l speed was measured by means of a s l o t t e d d i s c F (Figure 11) attached to the molybdenum shaft d i r e c t l y below the bearing housing. A p h o t o - c e l l G was positioned so as to produce a pulse f o r every re v o l u t i o n . The out-put from the phot o - c e l l was connected to an o s c i l l o s c o p e and, by using the c a l i b r a t e d time base, the r o t a t i o n a l speed could be obtained. 2. The upper part or torque measuring part consisted of an inner molybdenum cyl i n d e r H attached to the end of an alumina tube I by a molybdenum sleeve J . The alumina tube was f i x e d to the end of a s t a i n -- 23 -less s t e e l tube K. This arrangement was used i n order to obtain a s t r a i g h t shaft to hold the inner c y l i n d e r while minimizing heat losses by conduction. A thermocouple L was inserted i n t o the inner c y l i n d e r through the alumina and s t a i n l e s s s t e e l tubes. The e n t i r e assembly was attached to the suspension wire M by a s t a i n l e s s s t e e l frame N carrying the mirror 0. The leads of the thermocouple L were brought out through holes i n N and connected to copper and a l l o y 11 wires P supported i n s t a i n l e s s s t e e l rings Q. When the inner molybdenum c y l -inder H was immersed i n the molten s l a g , the copper and a l l o y 11 wires were immersed i n separate pools of mercury R. Copper and a l l o y 11 leads coming out of that pool were connected to a potentiometer (Met 17), through a 0°C junction, f or temperature measurements. The upper part of the s t a i n l e s s s t e e l frame N was machined to permit the p o s i t i o n i n g of a bearing assembly S. This bearing assembly was used at speeds above 15 to 50 r.p.m., depending on the v i s c o s i t y of the l i q u i d , to s t a b i l i z e the inner c y l i n d e r . Figures 12 a-b give a close-up view of the upper part of the viscometer. The viscometer assembly within the furnace was enclosed i n an alumina tube T. The s t a i n l e s s s t e e l frame N and the wire M were en-closed i n a brass tube U with a water cooled base V. The side openings i n the upper part of the brass tube were used to change the suspension wires when operating with the bearing assembly S. While measurements were taken, these openings were sealed o f f with an aluminum f o i l . The whole upper part of the viscometer rested on a support W which could be l e v e l l e d and moved h o r i z o n t a l l y to permit centering of the inner c y l i n -der into the c r u c i b l e . The inner cy l i n d e r could be positioned v e r t i c a l l y - 24 -by r a i s i n g or lowering the shaft X. The s t a i n l e s s s t e e l l i d Y had an argon i n l e t . Another argon i n l e t was located i n the brass tube at the mirror l e v e l . The i n i t i a l dimensions of the inner c y l i n d e r and the c r u c i b l e are shown i n Figure 13. These optimum dimensions were deter-mined a f t e r several experiments i n glycerine with several shapes and sizes of inner c y l i n d e r . The furnace was a resistance type employing two super kanthai heating elements (Z) and a power supply. The temperature p r o f i l e of the "empty" furnace i s shown i n Figure 14. The p o s i t i o n of the c r u c i b l e and inner cy l i n d e r are also indicated i n that Figure. The furnace had a very f l a t temperature p r o f i l e (+ 2.5°C) over the three inch c r i t i c a l length. In order to improve t h i s p r o f i l e , graphite and z i r c o n i a f e l t discs were stacked around the shaft from the bottom plate to the c r u c i b l e . Simi-l a r l y , a molybdenum r a d i a t i o n s h i e l d was placed above the inner c y l i n -der and graphite and graphite f e l t d i scs were stacked on top of t h i s r a d i a t i o n s h i e l d . These arrangements are shown i n Figure 15 and Figure 16 r e s p e c t i v e l y . We believe that, with these additions and considering the mixing i n the s l a g , the temperature gradient within the slag i n the c r i t i c a l zone was l e s s than 2°C. A f t e r t h i s furnace had reached thermal equilibrium, the temperature could be c o n t r o l l e d to within i 1.5°C. The detection thermocouple was made from P T - P T 10% Rh wires. I t was enclosed i n a twin bored alumina sheath and placed i n the center of the inner c y l i n d e r . Before v i s c o s i t y measurements were s t a r t e d , the inner c y l i n d e r was lowered into the molten slag and the furnace was held at constant temperature f o r a minimum of one hour. Under these - 25 -conditions, the reported temperatures are believed to be accurate to t 4%. A helium-neon l a s e r (Spectra Physics model 155, 0.5mW) was used as a l i g h t source to obtain a small r e f l e c t e d beam from the mirror 0. Figure 17 shows the experimental equipment. The d e f l e c t i o n of the beam of l i g h t caused by the torque exerted on the suspension wire was measured on a scale shown i n Figure 17. During the actual experiments, t h i s 2 meter-long scale was placed at approximately 2 meters from the apparatus. As seen i n Figure 17, a mirror was fi x e d to the scale at i t s zero point. This mirror was used to r e f l e c t the beam back to the mirror 0 of the upper part of the viscometer i n order to ensure that the angle between the scale and the beam was 90° at zero d e f l e c t i o n . The angle was obtained by c a l c u l a t i n g the tangent knowing the d e f l e c t i o n and the distance from the scale to the mirror. 3.2 Gas System High p u r i t y argon was passed through the furnace at a flow rate of approximately 100 scc/min. The gas was introduced at the top or at the middle of the brass tube. Since the bottom end of the furnace was open, a i r was c e r t a i n l y penetrating into the furnace. The oxygen entering the furnace e i t h e r through the opening at the bottom or with the argon reacted with the graphite and graphite f e l t and no appreciable oxidation of the molybdenum parts occurred. - 26 -3.3 Technique of Measurement A f t e r the Mo c r u c i b l e containing the slag was positioned i n the furnace and the upper part was properly centered, the system was flushed with argon f o r at l e a s t 2 hours before the furnace was heated. The furnace temperature was raised to 400-500°C over 3-4 hours and the gas flow rate was reduced to approximately 100 scc/min. The furnace temp-erature was then brought up to the experimental temperature over a per-iod of approximately 8 hours. The inner c y l i n d e r was lowered and p o s i -tioned i n s i d e the molybdenum c r u c i b l e . The distance between the bottom of the inner cy l i n d e r and the bottom of the c r u c i b l e was kept constant at 1.27 cm + .005 cm. The inner c y l i n d e r was then recentered (JT .05 cm) and the c r u c i b l e was rotated. Once the furnace had reached.thermal equilibrium at constant power input and had been maintained there f o r at l e a s t one hour, the measure-ments were started. The d e f l e c t i o n of the l a s e r beam was measured as a function of the r o t a t i o n a l speed and temperature measurements were taken every 2 to 5 minutes. Once the maximum measurable torque had been reached for a given wire, the inner c y l i n d e r was r a i s e d , and the wire was changed. The inner c y l i n d e r was then lowered and kept at a constant temperature for at l e a s t one hour while the c r u c i b l e was rotated before further measure-ments were taken. Since the bearing assembly could not be positioned while the furn-ace was at high temperature, the furnace was shut o f f , the bearing assembly was positioned, and the above procedure was repeated f o r high - 27 -speed measurements. In order to have a re lat ively stable rotational ve loc i ty , the crucible and shaft assembly had to be re lat ively free to rotate. The bearings supporting this assembly were oxidized because of the high temperature involved ( 2 0 0 - 5 0 0 ° C ) . The bearing l i f e was of the order of 48 hours at high temperatures. After that they had to be replaced because of their detrimental effect on the s t ab i l i ty of the rotational velocity. 3.4 Preparation of Slags The i n i t i a l slag (slag 1) was prepared by mixing weighed amounts of coarse high purity CaO and kl^O^ powders with f inely crushed SiO^ obtained from fused quartz. This mixture was then melted in a platinum crucible, kept molten for approximately one hour and poured onto a metal plate. This procedure was adopted because not enough powder could be packed into the crucible for one experiment. Furthermore, a ir 21 entrapment in viscous slags has been reported in a case where a pow-dered mixture was used. The prepared slag was broken into very coarse particles and placed into the viscometer. Subsequent changes in composition were achieved by adding the necessary powdered material on top of the so l id i f ied slag. Since the additions were less than 7% i t i s believed that no bubbles were trapped in the slag. - 28 -4. EXPERIMENTAL RESULTS 4.1 Rheological Behavior of the Calcium Aluminosllicate Slag The v i s c o s i t y of t h i s slag was measured over a range of rates of shear of 2.5, 3.1 and 2.7 orders of magnitude at temperatures of 1390°C, 1350°C and 1332°C r e s p e c t i v e l y . Graphs of the v i s c o s i t y as a function of l°g^ n rate of shear are presented i n Figures 18, 19 and 20 f o r these three systems. Corresponding pl o t s of the shear stress as a function of the rate of shear are presented i n Figures 21, 22 and 23. The log scale was used f o r the former seri e s i n order to show the experimental points obtained at low rates of shear. These points are not very c l e a r on an a r i t h m e t i c a l scale as seen on the shear stress vs. rate of shear p l o t s . A summary of the r e s u l t s along with the equations of l i n e A,B,C and D i s given i n Table V. TABLE V Summary of Results - Calcium Aluminosilicate Slag T - 1390°C Range of rates of shear: 2.5 orders of magnitude Average v i s c o s i t y : 17.7 poises Number of points 207 2a i n t e r v a l .55 poise 2a % 3.1% - 29 -TABLE V (continued) T = 1350°C T = 1332°C Line A " T r e = 17.7 ( d v e / d r ) Line B - T r e = 3.5 + 17.7 ( d v j d r ) Line C - T r e = 17.7 ( d v e / d r ) ° -9 9 Line D - T r e = 17.7 ( d V g / d r )1 ' 0 1 Range of rates of shear: 3.1 orders of magnitude Average v i s c o s i t y : 28.5 poises Number of points 200 2o i n t e r v a l : .93 poise 2a % : 3.3% Line A - Tre = 2 8 ' 5 (dv / d r ) Line B - Tre= 1 ' 0 + 28. 5 (dv Q/dr) Line c - x r e = 28.5 (dv 3/dr) Line D - x r e - 28.5 (dv ,/dr) Range of rates of shear: 2.7 orders of magnitude Average v i s c o s i t y : 44.1 poises Number of points 2o i n t e r v a l 1 7 3 2.0 poises 2a % 4.6% Line A " T r 9 = 44 .1 ( d v Q / d r ) Line B " T r e = 17 + 43.0 ( d v Q / d r ) Line c - T r e = 44 .1 ,, ,, .0.99 ( d V g / d r ) Line D - T r 9 = 44 .1 .1.01 ( d v Q / d r ) - 30 -Line A i s f o r a Newtonian f l u i d with a v i s c o s i t y equal to the average measured v i s c o s i t y . Line B i s a regression l i n e through the experimental points. Lines C and D represent a + 1% change i n the value of n which i s equal to 1 f o r a Newtonian f l u i d . 4.2 Rheological Behavior of CaF^-Containing Slags Two CaF2~containing slags were investigated i n order to determine whether or not the change i n the structure of the slag r e s u l t i n g from CaF^ additions would have any e f f e c t on the viscous behavior. A f i r s t a d dition of only 4% CaF^ was made i n order to be able to study the system at two d i f f e r e n t temperatures, 1325°C and 1300°C, and s t i l l maintain a high enough v i s c o s i t y to obtain a shear stress i n excess of 2 1,000 dynes/cm . Subsequently, the CaF^ content was increased to 10%. This was the maximum l i m i t because CaF^ losses s t a r t to become s i g n i -f i c a n t above t h i s l e v e l and the v i s c o s i t y of the slag becomes too low, even at 1300°C. Graphs of the v i s c o s i t y as a function of log rate of shear are presented i n Figures 24, 25 and 26 for the systems 4% CaF2-1325°C, 4% CaF 2-1300°C, and 10% CaF 2 1300°C r e s p e c t i v e l y . Corresponding pl o t s of the shear stress as a function of the rate of shear are presented i n Figures 27, 28 and 29. A summary of these r e s u l t s along with the equa-tions of l i n e s A,B,C and D i s given i n Table VI. - 31 -TABLE VI Summary of Results - CaF 2 containing Slags T = 1325°C, 4% CaF 2 Range of rates of shear: 3.2 orders of magnitude Average v i s c o s i t y : 18.1 poises Number of points 2a i n t e r v a l 191 .7 poise 2a % 3.8% Line A — T . , r £ = 18. 1 (dv Q/dr) Line B " T r e = -6. + 18.6 ( d v j d r ) Line C - T r t = 18. 1 ( d v e / d r )0 ' " Line D " T r e - 18. 1 ( d v e / d r )1 , 0 1 T = 1300°C, 4% CaF 2 Range of rates of shear: 3 orders of magnitude Average v i s c o s i t y : 23.9 poises Number of points: 196 2a i n t e r v a l : .9 poise 2a J { : 3.8% Line A - T r e =23.9 (dv @/dr) Line B -T r t = -5. + 24.4 (dv Q/dr) Line C - T r e , = 23.9 (dv e/dr)°-9 9 Line D - T r e = 23.9 ( d v e / d r )1 , 0 1 TABLE VI (continued) - 32 -T = 1300°C, 10% CaF 2 Range of rates of shear: 2.7 orders of magnitude Average v i s c o s i t y : 9.5 poises Number of points: 144 2 i n t e r v a l : .37 2 % • 3.9% Line A — T -r e = 9.5 (dv /dr) Line B " T r e = 0.4 + 9. ,5 (dv /dr) Line c " T r e = 9.5 i (dv IA \0.99 /dr) Line D " T r e = 9.5 1 (dv ,1.01 /dr) The accuracy of the r e s u l t s i s of the order of 3 to 4%. A d e t a i l e d discussion of the errors involved i n these experiments as w e l l as a math-ematical evaluation i s presented i n Appendix IV. 4.3 A c t i v a t i o n Enthalpies f o r Viscous Flow The r e l a t i o n s h i p viscosity-temperature i s given by n = A exp (E^/RT) (4.1) l o g 1 0 n = l o g 1 0 A +2.3 E^/RT (4.2) Hence, a p l o t of l o g ^ n as a function of 1/T has a slope of 2.3 H/R. These pl o t s are shown i n Figure 30 for the calcium a l u m i n o s i l i c a t e and the 4%-CaF^ slags. The enthalpies of a c t i v a t i o n were calculated to be approximately 50 Kcal/mole and 40 Kcal/mole r e s p e c t i v e l y . - 33 -While i t i s evident that accurate values of enthalpies of a c t i -vation cannot be obtained from Figure 30 because of i n s u f f i c i e n t data, the calculated values are w i t h i n an acceptable l i m i t from published values such as 40 Kcal/mole obtained from Figure 5 1 4 , 50 Kcal/mole for a 43% CaO - 43% S i 0 2 - 14% A l ^ s l a g 1 2 and, 45 Kcal/mole 1 2 f o r a slag of composition s i m i l a r to slag 2. (4% CaF ) - 34 -5. DISCUSSION 5.1 Rheological Behavior of Slags 5.1.1 Calcium A l u m i n o s i l i c a t e Slags The calcium a l u m i n o s i l i c a t e slag c l e a r l y shows Newtonian behavior at 1390°C and 1350°C. The r e s u l t s obtained at 1332°C seem to i n d i c a t e the presence of a y i e l d point at zero shear (Table V) and a s l i g h t negative deviation from the Newtonian l i n e c l e a r l y v i s i b l e i n Figure 23 at high rates of shear. However, these two observations are i n t e r r e l a t e d . I t i s because the v i s c o s i t y apparently decreases at high rates of shear that these two deviations are observed. The 2a l i m i t reported i n Table V for slag 1 at 1332°C i s also higher than the values reported f o r slag 1 at 1390°C and 1350°C. Part of t h i s increase (e.g. from 3.2% to 3.9%) i s due to a greater v a r i a t i o n of v i s c o s i t y with temperature at 1332°C than at 1350°C and 1390°C as discussed i n Appendix IV. The other part (e.g. 3.9 to 4.6) must be ascribed to the apparent decrease i n v i s c o s i t y at high rates of shear. No s a t i s f a c t o r y explana-t i o n has been found to account for t h i s deviation. Nevertheless, the r e s u l t s f a l l between l i n e s C and D which represent a s l i g h t non-Newton-ian behavior. We conclude from Figures 18-23 that the calcium a l u m i n o s i l i c a t e slag investigated i s a Newtonian f l u i d w ithin the range of shear stress and temperature covered. No conclusion can be drawn however, as to i t s - 35 -behavior at lower temperatures and/or at shear stresses i n excess of 4 2 10 dynes/cm . On the other hand, i t i s very l i k e l y that t h i s s l a g w i l l behave i n a Newtonian manner at higher temperatures wi t h i n the range of shear stress investigated. Other slag compositions i n the Ca0-Al20^-Si02 ternary for which the 0/(Si+Al) r a t i o i s greater than 2.44 w i l l have an array of smaller 2-complex anions than the slag studied due to the presence of more 0 46 ions , These complex anions w i l l be le s s susceptible to depolyconden-sation under a given shear stress than the anions i n the system i n v e s t i -gated because they are smaller. Hence, we can reasonably conclude that any Ca0-Si02 _Al20^ slag composition with an 0/(Si+Al) r a t i o greater than 2.44 w i l l be Newtonian above i t s l i q u i d u s temperature when subject 4 2 to shear stresses - 10 dynes/cm . 5.1.2 CaF2-Containing Slags The viscous behavior of two CaF2~containing slags was investigated because any conclusion drawn as to the viscous behavior of calcium a l u m i n o s i l i c a t e slags would not have been applicable to CaF2~containing slags. One reason f o r t h i s i s the f a c t that the complex anions i n a CaF2~containing slag could have d i f f e r e n t structures, s i z e d i s t r i b u t i o n and, shapes than the a l u m i n o s i l i c a t e anions i n slag 1. In a d d i t i o n , the nature and the strength of the i n t e r a c t i o n forces between the complex anions are most probably affected by additions of CaF^. Therefore, CaF^' containing slags may not have the same viscous behavior as calcium alumino-s i l i c a t e slags. - 36 -The r e s u l t s presented i n Figures 27 and 28 show a s l i g h t p o s i -t i v e deviation from the Newtonian l i n e at high rates of shear. The small negative shear stress at zero shear reported i n Table VI as we l l as the higher 2a l i m i t obtained as compared to the 2a l i m i t ob-tained at 1390°C and 1350°C with Slag 1, can also be ascribed to t h i s p o s i t i v e deviation. This deviation cannot be at t r i b u t e d to the pres-ence of s o l i d p a r t i c l e s i n the melt because,although the liquidus temp-eratures of the two slag compositions are not known, they are very u n l i k e l y to be higher than 1265°C, since CaF^ i n quantities up to 10 wt.% w i l l reduce the liquidus temperature. One phenomenon which could account f o r t h i s s l i g h t deviation from the Newtonian l i n e i s loss of f l u o r i n e from the sla g . Such losses have been reported i n a very s i m i l a r experimental system at s l i g h t l y higher 12 temperatures . Fluorine losses from the present system would probably occur according to the following reaction S i 0 2 + 2CaF 2 = SiF ^ t + 2CaO (5.1) which i s known to occur i n calcium f l u o r i d e slags containing lime and .... 4 7 s x l i c a An increase i n the calcium content of the slag would have the e f f e c t of r a i s i n g the liquidus temperature. I t i s quite possible that, because we are operating very close to the l i q u i d u s , t h i s temperature e f f e c t which tends to rai s e the v i s c o s i t y could more than compensate for the compositional e f f e c t which tends to lower i t . Since the r e s u l t s were obtained going from low to high speed over a period of approximately - 37 -30 hours at or above the experimental temperature, such f l u o r i n e losses would have caused the v i s c o s i t y to increase with the rate of shear as observed i n Figures 27 and 28. One drawback to the above prop o s i t i o n , however, i s the fa c t that no increase i n v i s c o s i t y with time occurred with the 10%-CaF2 slag under s i m i l a r experimental conditions as seen i n Figure 29. Fluorine losses i n t h i s l a t t e r system should have been higher because of the higher a c t i v i t y of CaF^ i n the s l a g . On the other hand, the r e s u l t s were obtained over 15 hours instead of 30. Furthermore, the v i s c o s i t y i s less sensible to f l u o r i n e losses at 10% CaF^ than at 4% CaF^. So i t i s f a r from c e r t a i n that the observed deviations are due to f l u o r i n e losses. No f l u o r i n e analyses were c a r r i e d out because the method of 48 analysis i n s i l i c a t e s contains an uncertainty greater than the change i n composition due to v o l a t i l i z a t i o n . Nevertheless, the r e s u l t s shown i n Figures 27-29 a l l f a l l w ithin l i n e s C and D. From t h i s , we conclude that the two CaF^ containing slags studied were Newtonian within the range of shear stress and temperature covered. Furthermore, we think that t h i s conclusion i s very l i k e l y to apply to other slag compositions, i n the quaternary CaF2-Ca0-Al20^-Si02 with an 0/(Si+Al) r a t i o greater than 2.44, at temperatures above t h e i r l i q u i d u s temperature and at shear 3 2 stresses up to 5 x 10 dynes/cm . The problem of how f l u o r i d e ions are incorporated into the s i l i c a t e l a t t i c e i s s t i l l unsolved. Published work on the topic has produced 49-50 opinions divided between a mechanism producing Si-F bonds i n the anions, and one r e l y i n g on loose coordination structures between Si-0 , CaF + and 2+ 22 Ca . In addition, reaction (5.1) above has complicated the experi mental p i c t u r e i n s t r u c t u r a l studies. However, from the e x i s t i n g evidence we may state that F additions to a s i l i c a t e slag change the anion s t r u c -- 38 -2-ture i n a way which i s not equivalent to the same addi t i o n of hO In s p ite of a d i s t i n c t change i n the flow-unit behavior, as evidenced by changes i n the v i s c o s i t y on F additions, we may conclude from our present findings that the flow-unit involving F i s not changed from the pure s i l i c a t e i n a way which w i l l induce non-Newtonian behavior 3 2 of any kind at shear stresses up to 5 x 10 dyne/cm . 5.2 Comparison with Published Work 5.2.1 Rheological Behavior No increase i,n v i s c o s i t y at low rates of shear occurred i n the present i n v e s t i g a t i o n i n contrast with the r e s u l t s obtained by Lang-31 hammer and Geek . (See Figure 6). The compositions of the slags they studied are given i n Table VII. TABLE VII Slag Compositions Investigated by Langhammer and Geck^l (weight %) FeO F e 2 ° 3 p 2 ° 5 V S n 0 S i 0 2 A 1 2 ° 3 C a 0 M g 0 Open hearth 1 16.3 4.8 3.1 8.2 11.9 4.6 43.4 6.7 Open hearth 2 12.6 3.0 3.5 8.4 11.8 4.3 45.7 9.7 Blast furnace slag 1.12 0.44 1.24 30.72 11.74 45.21 7.49 As can be seen these authors have studied very complex systems for which l i q u i d u s temperatures and phase diagrams are unknown. Hence, they could have measured v i s c o s i t i e s below the l i q u i d u s temperatures - 39 -of t h e i r systems. As mentioned previously, two such systems containing 27-28 s o l i d p a r t i c l e s have been shown by Saito et a l to be non-Newtonian. The presence of more than one l i q u i d phase i n the slag could also account f o r the non-ideal viscous behavior observed. This i s not u n l i k e l y since many s i l i c a t e systems show a m i s c i b i l i t y gap i n the l i q u i d 5.2.2 Values of V i s c o s i t y To the author's knowledge, no r e s u l t s have been published p e r t a i n -ing to the v i s c o s i t y of systems of the same composition as the ones investigated i n the present work. Hence, no d i r e c t comparison i s poss i b l e . The v i s c o s i t y r e s u l t s obtained i n the present work are l i s t e d i n Table VIII along with published data on systems of c l o s e l y s i m i l a r compositions. TABLE VIII Comparison of Present Work with that of Other Workers Composition(weight %) S i 0 2 A1 20 3 CaO CaF 2 V i s c o s i t y i n poise, °C 1400 1350 1325 1300 Reference 42 20 38 16 ( 1 )28.5 present work 43 15 41.5 18.5 38 27 43 14 43 10 24 12 40 20 40 21 - 17 40 20 40 22 13 40 20 40 21 45 52 value corrected for temperature. - 40 -TABLE VIII (continued) Composition(weight %) V i s c o s i t y i n poise, °C S i 0 2 A1 20 3 CaO CaF 2 1400 1350 1325 1300 Reference 40.3 19.2 36.5 4. — — 18 24 present work 39.7 19.7 38.8 2. ~ — 43 57 52 37.8 18.0 34.2 10.0 -- — — 9.5 present work 36.2 17.9 36 9.9 — -- -- 32 52 38.8 12.9 40 8.3 — — — 6 12 I t i s seen from Table VIII that the values reported i n the present work are within the range of published data. - 41 -6. CONCLUSIONS 1. A concentric c y l i n d e r viscometer was found to be su i t a b l e to study the r h e o l o g i c a l behavior of s i l i c a t e slags over a three order of magnitude range i n rates of shear and up to shear stress of 10 dynes/cm 2. 2. The calcium alu m i n o s i l i c a t e slag of composition (weight %) 38 CaO - 20 A^O^ - 42 SiO^ i s Newtonian up to shear stresses of 5 x 10 to 10 dynes/cm at temperatures of 1390°C, 1350°C and 1332°C. 3. Other slag compositions i n the Si02 - CaO - Al^O^ ternary, with an 0/(Si+Al) r a t i o greater than 2.44 should be Newtonian at temperatures A 2 above t h e i r l i q u i d u s and at shear stresses up to 10 dynes/cm . 4. The 4% CaF^ and 10% CaF^ calcium a l u m i n o s i l i c a t e slags studied 3 3 2 are Newtonian up to shear stresses of 5 x 10 and 2.3 x 10 dynes/cm at temperatures of 1325-1300°C and 1300°C r e s p e c t i v e l y . 5. I t i s very l i k e l y that other slag compositions i n the CaF^ - CaO -Si02 - A-l^O^ quaternary for which the r a t i o 0/(Si+Al) i s greater than 2.44 w i l l be Newtonian at temperatures above t h e i r l i q u i d u s and at 3 2 shear stresses up to 5 x 10 dynes/cm . - 42 -7. SUGGESTIONS FOR FUTURE WORK 1. I t has been shown that i t i s very l i k e l y that slag compositions i n the S i 0 2 - A l ^ - CaO ternary and CaF 2 - S i 0 2 - A l ^ - CaO quat-ernary with an 0/(Si+Al) r a t i o 1 2.44 are Newtonian up to shear 3 4 2 stresses of the order of 10 to 10 dynes/cm . I t would be very i n -te r e s t i n g to know the magnitude of the shear stresses encountered i n slag-containing systems. To the author's knowledge, nothing has been published as yet on t h i s matter. 2. I t would also be i n t e r e s t i n g to investigate the viscous behavior of slags at higher shear stresses. To be meaningful such work would have to be c a r r i e d out on slags with a v i s c o s i t y of the order of 50 poise or l e s s . The apparatus described i n t h i s thesis could be used to carry out t h i s i n v e s t i g a t i o n . The following modifications would have to be made, however, i n order to permit higher rates of shear to be attained. 1. Modify the design of the bottom part so that the bearings do not oxidize and that they stay t i g h t on the shaft supporting the c r u c i b l e . This could be accomplished by e i t h e r using molybdenum bear-ings or by making the bottom part of the shaft out of s t a i n l e s s s t e e l and using s t a i n l e s s s t e e l bearings. - 43 -2. Increase the diameter of the inner cylinder. Under those conditions the length (L c) may have to be decreased because of l i m i -tations as to the maximum torque which can be sustained by the apparatus. 3. Modify the design of the upper part so that the bearing assembly could be positioned closer to the inner cylinder, to increase i t s s t a b i l i t y . - 44 -APPENDIX I CALIBRATION OF THE SUSPENSION WIRES The suspension wires used were as follows: Wire #1: 5.4 cm long, .038 cm 0 Tungsten wire. Wire #2: 6.8 cm long, .058 cm 0 Piano wire. This wire as w e l l as wire #3 was annealed, cold drawn (- 50% reduction) and pulled i n an Instrom machine i n order to ensure straightness. Wire #3: 5.4 cm long, .076 cm 0 Piano wire. Wire #4: 4.5 cm long, .104 cm 0 Piano wire as received. Wire #5: 4.2 cm long, .130 cm 0 Piano wire as received. The wires were soldered into brass or s t a i n l e s s s t e e l end pieces as shown i n Figure 31. The end pieces were about .5 cm i n diameter and 1 cm long. The constant, K , of a suspension wire was found by constructing w a t o r s i o n a l pendulum of varying moment of i n e r t i a . A close up view of the apparatus used i s shown i n Figure 32. One or two p a i r of masses were used f o r each wire. D i f f e r e n t moments of i n e r t i a were obtained by moving the masses along the shaft. The period of the system was care-f u l l y measured for each p o s i t i o n of the masses. The period of such a pendulum i s expressed by Now, i f for two moments of i n e r t i a I^ and 1^, the periods are P^ and P , one has (A.I-1) w 2 - 45 -4 T T 2 ( I 1 - I 2 ) 4 T T 2 A I " " * R ^ (4.1-2) By using equation (A.I-2), the part of I which i s constant may be eliminated, and only that v a r i a b l e part due to the masses, need be used i n c a l c u l a t i n g I . Consider the diagram i n Figure 33. The mass moment of i n e r t i a of a composite body i s the sum of the moments of i n e r t i a of the i n d i v i d u a l parts about the same axis. I f or a dis c xx i s given by 1 2 1 2 I = T MR, + Tr- Mtf (A. 1-3) xx 4 1 12 1 = j pt^irR* + - i j p t ^ R 2 t 2 (A. 1-4) I f o r the hole i s xx I = f p^rrR* + ~ p t ^ R 2 t 2 (A.1-5) x x 4 1 2 12 1 2 1 Hence, I for the annular d i s c i s xx xxx = \ PV<4*£> + I I P V f c i ( R r R 2 } ( A - I " 6 ) The mass of the annular d i s c i s given by M = T T R 2 t ± p - irR 2 t p (A. 1-7) Substituting equation (A.1-7) in t o equation (A.1-6), Txx = \ M < R H ) + 12 M t l ( A- I" 8> If the moment of i n e r t i a of a body i s known about a centroSdal axis, i t may be determined about any p a r a l l e l axis from the equation 1 = 1 + Md 2 (A.1-9) xx 2 - 46 -Substituting equation (A.1-8) into equation (A.1-9), one obtains h 2 1 = 1 + M ( d , + 7 T ~ ) (A.I-10) I = M [| (R^+R2) + d 2 + d 1 t 1 + | t 2 ] (A.I-11) where d^, the distance from the center of the pendulum to the inside of the mass, is simply measured with a vernier. The dimensions and weight of the masses used are given in Table IX. P was measured with a timer. An accuracy of better than it.001 sec was obtained by counting 100 cycles. A l l the data for the determination of K for each wire are presented in Tables X-XIV. w v The 2a limit for K defined as 2a measurements//N i s better than w i.2% in a l l cases. A careful examination of the design of the viscosity apparatus (Figure 12a) shows that the suspension wires are attached to a small, long stainless steel part N. This part also acts as a suspension wire in that i t deforms when a torque i s applied. If i s taken as the constant of that shaft, the effective constant of the wire K i s then wc given by 1 " 1 + (A. 1-12) K K Kvr WC W M 11 2 From the dimensions of the shaft using G = 7.3 x 10 dynes/cm , g K^ j was found equal to 2.96 x 10 dynes/cm rad . Applying equation (A.1-12), the following corrected values were obtained for wires #3,4 and 5 - 47 -wire #3 K = 530,700 dynes cm r a d " 1 a .2% change wire #4 K = 2,062,000 dynes cm r a d " 1 a .6% change wire #5 K = 5,079,000 dynes cm r a d " 1 a 1.6% change Corrections f o r wires #1 and 2 were i n s i g n i f i c a n t . The y i e l d strengths i n to r s i o n f o r wire #4 and 5 were obtained 53 from the l i t e r a t u r e . The y i e l d strengths i n torsi o n of wires #1, 2, and 3 were obtained from load vs. elongation curves using the maximum shear stress c r i t e r i o n given by fcr0 y i e l d = y ± e l d s t r e n 8 t h / - 5 (A.1-13) The maximum angle of twist which could be measured was calculated from A3 (D - r Q y i e l d (A.i-i4) 4 • K - 6 wc No p l a s t i c deformation of the wires was noticed during the experiments. TABLE IX Weight and Dimensions of the Masses used to Cal i b r a t e the Suspension Wires =200 Weight (g ) 20.127 „ 20.157 39.967 39.955 "•8 6 4=100 99.928 195.869 196.633 O.D. (cm) 1.905 2.606 4.048 5.730 1.907 2.606 4.049 5.738 l.D. (cm) .635 .635 .635 .635 .635 .635 .635 .635 Thickness (cm) 1.013 1.016 1.016 .988 1.013 1.013 1.016 .988 - 48 -No. 1 2 3 4 5 6 7 Combination 1 and 2 1 and 3 1 and 4 1 and 5 2 and 4 2 and 5 6 and 1 6 and 2 6 and 3 6 and 4 6 and 5 6 and 7 1 and 7 3 and 7 TABLE X Calibration Results Wire #1 Mass Period (sec) d^cm) I(g cm 40 4.886 14.110 23,548 40 4.240 10.795 15,340 40 3.400 5.839 6,344 40 2.746 0 781 20 2.693 0 385 100 6.891 13.716 56,324 100 2.913 0 2,074 AI A(P 2) K w 8,208 5.897 54,951 17,204 12.314 55,155 22,767 16.331 55,036 23,163 16.619 55,024 14,559 10.434 55,083 14,954 10.722 55,064 32,775 23.613 54,798 40,984 29.510 54,828 49,980 35.927 54,920 55,542 39.944 54,895 55,938 40.231 54,891 54,250 39.002 54,914 21,474 15.389 54,092 4,270 3.074 54,838 Average K : 54,960 dynes cm rad ^  w 2a : 210 dynes cm rad N : 14 2a % : .1% ave - 49 -TABLE XI Calibration Results Wire #2 No. Mass Period (sec) d^cm) I(g cm2) 1 100 4.698 14.114 & 14.140 59.113 2 100 3.466 8.694 & 8.689 27,727 3 100 2.338 2.916 & 2.924 7,324 4 100 1.945 0 2,074 5 100 3.268 7.620 & 7.584 22,856 6 40 3.231 14.107 & 14.122 23,562 Combination 1 and 2 1 and 3 1 and 4 2 and 3 2 and 4 1 and 5 5 and 4 5 and 3 6 and 1 6 and 3 6 and 4 A l 31,385 51,789 57,039 20,403 25,654 35,551 21,488 16,238 36,256 15,532 20,783 A(P 2) 10,056 16,600 18,283 6,545 8,227 11,386 6,897 5,214 11,629 4,971 6,653 K w 123,218 123,163 123,165 123,078 123,100 123,262 123,006 122,947 123,080 123,365 123,314 Average K^ : 123,200 dynes cm rad 2a : 230 dynes cm rad N : 11 2a % : .05% ave - 50 -TABLE X I I Calibration Results Wire #3 2 No. Mass Period(sec) d^cm) I(g cm ) 1 100 4 .647 30 .173 220,619 2 100 4 .213 26 .032 169 ,097 3 100 3 .639 20 .193 108 ,073 4 100 2 .936 12 .083 45 ,921 5 100 2 .317 0 2 ,074 Combination 1 and 2 1 and 3 1 and 4 1 and 5 2 and 3 2 and 4 2 and 5 3 and 4 3 and 5 4 and 5 A l 51 ,522 112,546 174,697 218,545 61 ,024 123,176 167 ,023 62 ,151 105,999 43 ,847 A(P 2) 3.849 8 .356 12 .981 16 .232 4 .507 9 .132 12 .383 4 .625 7 .876 3 .251 K w 528 ,455 531 ,732 531 ,317 531 ,543 534,532 532 ,524 532 ,503 530 ,567 531 ,343 532 ,446 Average K 2a N w 2a % ave 531 ,700 dynes cm rad 2,970 dynes cm rad 10 .18% -1 -1 - 51 -No. 1 2 3 4 5 TABLE XIII Calibration Results Wire #4 Mass 200 200 200 200 200 Period(sec) 3.113 2.729 2.283 1.709 1.193 d1(cm) 30.381 25.555 19.710 11.455 0 I(g cm ) 438,900 321,480 203,757 83,164 4,442 Combination 1 and 2 1 and 3 1 and 4 1 and 5 2 and 3 2 and 4 2 and 5 3 and 4 3 and 5 4 and 5 Al 117,520 235,144 355,736 434,458 117,724 238,316 317,038 120,592 199,314 78,722 A(P 2) 2.240 4.475 6.766 8.267-2.235 4.527 6.027 2.291 3.792 1.500 K w 2,069,616 2,074,451 2,075,509 2,074,827 2,079,295 2,078,425 2,076,764 2,077,575 2,075,272 2,071,753 Average K w 2a N 2a % L ave 2,075,000 dynes cm rad 5,600 dynes cm rad 10 .09% -1 -1 - 52 -No. 1 2 3 4 5 TABLE XIV Calibration Results Wire #5 Mass 100 & 200 200 200 200 200 Period(sec) 2.368 1.982 1.630 1.170 .755 d1(cm) 30.762 for 200 & 29.746 for 100 30.597 23.525 13.599 0 I(g cm ) 663.938 444,579 277,562 109,338 4,442 Combination 1 and 2 1 and 3 1 and 4 1 and 5 2 and 3 2 and 4 2 and 5 3 and 4 3 and 5 4 and 5 Al 219,360 386,377 554,600 659,496 167,017 335,240 440,136 168,223 273,119 104,896 A(P 2) 1.681 2.953 4.241 5.040 1.272 2.560 3.359 1.288 2.0872 .7992 K w 5.151,481 5,165,079 5,162,502 5,165,488 5,183,048 5,169,739 5,172,497 5,156,594 5,166,067 5,181,332 Average K w 2a N 2a % ave 5,167,000 dynes cm rad 18,700 dynes cm rad 10 .12& -1 - 53 -APPENDIX II CALIBRATION OF THE VISCOMETER The viscometer was c a l i b r a t e d at room temperature with a s i l i -cone f l u i d f o r which the v i s c o s i t y at 25°C was 9.40 poise. The sup p l i e r (Brookfield) states that t h i s value i s accurate to wit h i n t 1 per cent. In order to get very accurate temperature measurements, a copper-con-stantan thermocouple was used instead of the Pt/Pt-10% Rh one. The v i s c o s i t y i n a concentric c y l i n d e r viscometer i s given by (see equation (2.18)). P j 4 2 2 Gud r - r, ^ P = — • -£-5-=^ • ^ (A.II-1) 16£. 4irr r^L ft c b o Rearranging, one obtains G 7 T d A . r c - rb . q_ . i (A.II-2) 16£ 4 i r r 2 r 2 ti y c b o L = The values of <V and fi^were obtained experimentally. The value of u was corrected f o r temperature according to the chart supplied by Brook-f i e l d and and r ^ measured very p r e c i s e l y with a ve r n i e r . The r e s u l t s of the c a l i b r a t i o n are given i n Table XV. Three wires were used: #2,3 and 4. Wires #1 and 5 were not used becaused the speed was not low enough to use the former and the torque not high enough f o r the l a t t e r . As can be seen a r e l a t i v e l y constant value i s obtained f o r L. The average measured value was 5.92 with a 2a l i m i t equal to "t 1.4%. - 54 -TABLE XV C a l i b r a t i o n of the Viscometer D e f l e c t i o n Speed u(corrected f o r T) L (cm) (r.p.m.) (poise) (cm) Wire #1 7.3 2.58 10.00 5.979 23.7 8.59 9.95 5.844 29.8 10.71 9.95 5.881 34.7 12.52 9.95 5.846 38.6 13.80 9.95 5.884 42.0 15.06 9.95 5.856 44.2 15.73 9.95 5.891 52.2 18.45 9.95 5.898 56.5 19.87 9.95 5.907 60.5 21.22 9.95 5.905 64.7 22.58 9.95 5.912 68.3 23.81 9.95 5.898 71.7 24.90 9.90 5.930 75.8 26.21 9.90 5.932 81.0 27.83 9.90 5.939 79.6 27.36 9.90 5.943 83.0 28.29 9.90 5.970 83.5 28.44 9.90 5.972 88.0 29.96 9.90 5.942 91.8 31.04 9.90 5.956 97.5 32.76 9.90 5.951 98.0 32.85 9.90 5.963 Wire #2 59.7 89.85 9.85 5.994 32.3 49.80 9.87 5.946 35.8 55.64 9.87 5.887 40.6 62.60 9.87 5.918 44.8 68.40 9.87 5.961 48.7 73.83 9.87 5.987 51.6 78.63 9.87 5.944 54.7 83.41 9.87 5.925 58.4 88.92 9.87 5.916 - 55 -TABLE XV (continued) Deflection Speed u(corrected for T) L (cm) (r.p.m.) (poise) (cm) (Wire #2) 62.0 94.08 9.87 5.918 64.8 98.51 9.87 5.892 69.3 103.83 9.87 5.953 72.7 108.92 9.87 5.933 77.5 115.02 9.87 5.960 81.3 121.21 9.87 5.908 85.2 126.51 9.85 5.921 89.9 131.85 9.85 5.962 93.0 136.14 9.85 5.951 96.1 139.87 9.85 5.962 98.3 143.52 9.82 5.942 Wire #3 20.1 122.53 9.80 5.916 23.2 142.03 9.80 5.884 26.2 159.05 9.80 5.928 29.2 177.12 9.80 5.925 32.2 194.69 9.80 5.936 35.3 215.31 9.80 5.875 37.6 227.78 9.80 5.908 39.8 240.00 9.80 5.927 41.5 252.64 9.76 5.889 45.0 273.04 9.76 5.895 47.6 288.37 9.72 5.918 50.2 303.87 9.72 5.912 52.6 317.98 9.72 5.909 55.0 332.27 9.72 5.902 57.7 350.66 9.72 5.854 60.3 363.58 9.72 5.887 62.8 379.56 9.72 5.860 65.1 390.47 9.72 5.893 68.1 407.35 9.70 5.904 71.1 424.71 9.65 5.926 75.5 456.52 9.65 5.828 76.0 456.52 9.65 5.863 Average: 5.917 2a : .083 2a% : 1.4% - 56 -APPENDIX I I I CORRECTIONS FOR TEMPERATURE AND DEPTH OF IMMERSION The e f f e c t i v e length of the inner c y l i n d e r , L, can be divided i n three parts 1. The bottom e f f e c t L, b 2. The main part of the inner c y l i n d e r L c 3. The stem e f f e c t L s with L = L + L + L (A.III-1) b c s The value of L c i s equal to the actual length of the inner c y l i n d e r . 54 According to Rait , L g can be approximated by 2 r L = - | • x (A.III-2) r b where r g and r ^ are the r a d i i of the stem and inner c y l i n d e r r e s p e c t i v e l y and x the immersed length of the stem. The value of x f o r the c a l i b r a -t i o n experiment has been determined accurately from volume considerations. Hence, the bottom e f f e c t , L^, can be evaluated. The values of L and i t s three components are given below: L = 5.917 cm; L, = .477 cm; L = 5.080 cm; L = .360 cm. b c s Assuming that i s constant, which i s a f a i r assumption since molybdenum has a low c o e f f i c i e n t of thermal expansion, the value of L experimental becomes 2 r L = L, + L (high T) + —§• • x. (A.II D C Z. r b - 57 -The value of L at high temperature i s obtained by adding to L c the thermal expansion of the molybdenum. The value of x i s obtained as follows: The volume of slag in the crucible i s given by W V " p~ (A.III-4) From the viscometer design we have (see Figure 7) V = ^ r 2 (a+L +x) - r r r 2 • L - ^r 2x (A.III-5) c c b c s which gives W 2 , ,T , , 2 • _ trr (a+L ) + i r r , • L p c c b e x = ^  ^ 2 ( ^ r c - r r r s ) (A.III-6) - 58 -APPENDIX IV ACCURACY OF VISCOSITY MEASUREMENTS In the following discussion we s h a l l d i s t i n g u i s h between two types of errors namely random errors which depend on how accurately somebody can read a measuring device (e.g. read a potentiometer) and systematic errors which are inaccuracies a f f e c t i n g a l l measurements equally, (e.g the accuracy of a thermocouple). We s h a l l d i s t i n g u i s h between the two types because mathematically they are not handled i n the same manner. Consider, f o r instance the equation a = xyz (A.IV-1) If ax, ay and az are the random unc e r t a i n t i e s attached to x, y and z 55 r e s p e c t i v e l y , then the uncertainty aa i s given by z ) 2 (A.IV-2) d i v i d i n g equation (A.IV-2)by a, (A.IV-3) or 6a =* \ / ( 6 x ) 2 + ( 6 y ) 2 + ( 6 z ) 2 (A.IV-4) where 6x, 6y and 6z are r e l a t i v e random er r o r s . On the other hand i f 6x, 6y and 6z are r e l a t i v e systematic e r r o r s , the expression f o r 6a becomes 6a = 6x + 6y + 6z (A.IV-5) - 59 -Thus, the types of errors involved i n the determination of a quantity must be " q u a l i f i e d " before any attempt i s made to evaluate the accuracy of that quantity. IV.1 Accuracy of L, the E f f e c t i v e Length of the Inner Cylinder The value of L i s given by L = K • K • ~ - - - (A.IV-6) wc Q u o where K i s 2 2 r - r, v = _c b . 2 2 4iTr - r, c b The r e l a t i v e errors involved are as follows: (a) Random errors (1) C a l i b r a t i o n wires <SK = +.5% wc (2) cp - p r e c i s i o n of x i s 1% Stp = ± 1% (3) H q - accuracy of reading the scope <5^ o = ±.7% (4) T - accuracy of reading potentiometer and accuracy of 0°C junction T = ± %°C SUj, = 1.5% (b) Systematic errors (1) u - accuracy stated by Brookfield 6u = ± 1% - accuracy of thermocouple ± -j^C Sy = Jt.25% o (2) Time base of the scope; i t s c a l i b r a -t i o n i s accurate to +.7% <5t.b.= i . 7 % (3) K (assuming 6 r c = 6 r b = ±.001 cm) 6K = ±.3% - 60 -The t o t a l r e l a t i v e e r r o r attached to L i s given by = ±V(fiK ) 2 + (Sep)2 + (fin ) 2 + (6y„) 2 + fiy + fiy + fit.b. + wc 6L = + (1.4% + 1% + .25% + .7% + . fiL = ± 3.7% IV.2 Accuracy of y The value of y i s given by y = K • K • ~r^ " * ? (A.IV-7) wc U L o where K i s as defined above. The r e l a t i v e errors involved are as follows (a) Random errors (1) C a l i b r a t i o n wires (SK wc = 5% (2) <P - p r e c i s i o n of x i s 3% ficp = + 3% (3) - accuracy of reading the scope fifi o = + m 7% (4) T - accuracy of reading potentiometer = 5% - accuracy of 0°C junction t h°C = 25% stematic error (1) L 6L = + 3.7% (2) T - accuracy of thermocouple + h% 6y T = + 3.5% (3) Time base - c a l i b r a t i o n fit.b, + 7% (4) K - assuming r c = rb = ±-002 cm 5K = 2% (5) L - error on the actual length of the inner cylinder at high temperature (+.004 cm) <5LT = +.2% - 61 -The random error attached to y i s given by 6y = / ( 6 K w c ) 2 + (Sep)2 + (5fto)2 + (Sy2 + (Si^ )2 6y = V.25 +9'+ .49 + .25 + .06 = / 10.05 6y * 3.2% The total relative error i s then <Sy = 3.2% + 3.7% + .7% + .2% + .2% - + 11.5% Discussion The error due to temperature was introduced because the viscosity is a function of temperature. If the correlation is known, an error can be attached to the viscos i ty . For the calibrating f l u i d , the var-iat ion of y with T was supplied by Brookfield. For the slags, the corre-lat ion was obtained from published results for similar slag compositions. The variation of y with T was approximately 1%/°C. For slag 1 at 1 3 3 2 ° C , no data were available. The variation of y as a function of temperature was obtained from a plot of our y values for T = 1350°C and 1332 °C . We found a variation of approximately 1 p o i s e / ° C i . e . - 2 . 5 % / ° C . Since a value pf 1%/°C was used in the computation of the random error on u, the latter should be corrected to ± 3.9% for slag 1 at 1332 °C . Accord-ingly, the total relative error i s increased to = i 17%. - 62 -The random err o r on L compares quite favorably with the 2a c o n f i -dence i n t e r v a l given i n Table X. The random e r r o r s on y are In general somewhat lower than the 2a l i m i t s reported i n Tables V and VI. There are however, other possible sources of errors which have not been accounted f o r such as d r i f t i n g of the time base c a l i b r a t i o n and changes i n the composition of the sl a g . The calculated t o t a l e r r o r f o r y may seem large. I t should be remembered, however, that the apparatus was designed to study the rheo-l o g i c a l behavior of slags, not to determine absolute v i s c o s i t i e s . As f a r as the former goes, the error i s equal to the random err o r i . e . - 3-4%. Hence very small deviations from i d e a l viscous behavior could have been measured with the present system. In the above an a l y s i s , two other sources of errors were not con-sidered (1) p, the density of the slag (2) Abnormal changes i n the dimensions of the c r u c i b l e . 12 The value of p was obtained from Kato et a l who measured the density of slags of s i m i l a r composition. No er r o r margin was reported i n t h e i r work. Because of the design of the inner cylinder and stem, we consider that any er r o r i n p would have a very small e f f e c t on p. In 3 f a c t , we evaluated t h i s error as .25% per .05 g /cm error i n p. During the f i r s t experiment at high speed at 1350°C, the diameter of the bottom 3 inches of the c r u c i b l e increased by .086 cm(1.7%) on the average. This deformation of the c r u c i b l e can be seen i n Figure 15. No reasons have been found for t h i s p a r t i c u l a r behavior. I t i s possible - 63 -that the deformation was caused by v i b r a t i o n of the c r u c i b l e assembly which was p a r t i c u l a r l y bad during that experiment. The c r u c i b l e may also have deformed because of i n t e r n a l s t r e s s e s . This deformation was accounted f or i n the c a l c u l a t i o n s by increasing the diameter by .086 cm for each measurement, n being the number of measurements taken. We be-l i e v e that t h i s procedure introduced no important errors i n the r e s u l t s Thereafter, the dimensions of the c r u c i b l e remained constant. - 64 -REFERENCES 1. Pigf o r d , R.L., Chem. Engng. Prog. Symp. Ser. (1955) 51 (17), 79. 2. Metzner, A.B., Vaughn, R.D. and Houghton, G.L., A.I.Ch.E. Journal (1957) 3 (1), 92. 3. Metzner, A.B., Advances i n Chem. Eng. (1956) _1, 77. 4. Metzner, A.B., Advances i n Heat Transfer (1965) 2, 357. 5. Christiansen, E.B. and Graig, S.E., A.I.Ch.E. Journal (1962) 8^  (2), 154. 6. Skelland, A.H.P., Non-Newtonian Flow and Heat Transfer, John Wiley & Sons Inc., New York, (1967). 7. Hirose, T. , and Moo-Young, M. , Can. J . Chem. Eng. (1967) 47_, 265. 8. Andrade, E.N. da C , Phylosophical Magazine Series 7, (1934) 17_, 698. 9. Eyring, H., J . Chem. Phys. (1936) 4, 283. 10. Bloom, H. and Hemann E., Proc. Roy. Soc. London (1946-47) A 188, 392. 11. Kato, M. and Minowa, S., Trans. Iron Steel Inst. Japan (1969) £ (1), 39. 12. Kato, M. and Minowa, S., Trans. Iron Steel Inst. Japan (1969) 9_ (1), 31. 13. Kozakevitch, P., Rev. de Metallurgie (1960) 57_, 149. 14. Bossin, R., Bersan,J. and Urbain, G., Rev. Hautes Temper, et Refract. (1964) I (2), 159. 15. Saito, T. and Kawai, Y., Science Reports of the Research I n s t i t u t e s Series A (1957) 3, 491. 16. Sakai, M., Katsuda, T. and Namikawa, R., Shigakenritsu Tanki Daigaku Gakujutsu Zasshi (1967) 8_, 10. 17. Machin, J.S. and Yee, T.B., J . Am. Ceram. Soc. (1948) 31 (7), 200. - 65 -18. S c h l e i e r , J . , Neue Hutte (1958) _3, 282. 19. Johannsen, F. and Bruniar, H ., Z. Erzbergbau u. MetallhUttenw. (1959) 12, 211. 20. Bockris, J . O'M. and Lowe, D.C., Proc. Roy. Soc. (1954) A 226, 423. 21. Bockris, J . O'M., Mackenzie, J.D. and Kitchener, J.A., Trans. Farad. Soc. (1955) 51, 1734. 22. Baak, T., Proc. Conf. Phys. Chem. Iron and Steelmaking, Dedham, Massachussetts (1956) , Wiley New York (1958). 23. M e r r i l l , E.W., Ind. Eng. Chem. (1959) 51, 868. 24. Quincke, J.E., Z. Elektrochem. (1953) 5_7, 715. 25. Bockris, J . O'M., Record of Chem. Prog. (1955) 16 (1), 23. 26. Euler, R. and Winkler, H.G.F., Glastech. Ber. (1957) 30, 325. 27. Z i l ' b e r , M.K., Russ. Met. (1967) 4, 18. 28. Saito, T. and Saeki, K., J . of Jap. Iron. S t e e l . Inst (1965) 51 (10), 1851. 29. Saito, T. and S a i k i , K., J . of Jap. Iron. S t e e l . Inst. (1965) 51 (10), 1855. 30. Z i l ' b e r , M.K., Russ. Met. (1967) 4, 18. (ref.12) 31. Langhammer, H.G. and Geek, H.G., Arch. Eisenhuttenwesen (1967) 38 (9), 691. 32. Tamura, A., S h i r a i s h i , Y. and Saito, T., B u l l . Res. Inst. Min. Dress. Mat., Tohoku U n i v e r s i t y , (1971) 27 (1,2), 169. 33. Tamura, A. and Saito, T., B u l l . Res. Inst. Min. Dress. Met., Tohoku Univ e r s i t y (1966) 22 (1), 7. 34. S h i r a i s h i , Y. and Saito, T., Nippon Kinzoku Gakkaishi (1965) 29 (6), 614. 35. Van Wazer., J.R., Lyons, J.W., Kim, K.Y. and Colwell, R.E., V i s c o s i t y and Flow Measurement, Interscience Publishers (1963), 23. 36. McCaffery, R.S., Amer. Inst. Mining M e t a l l . Engrs. Tech. Publ., (1931) No. 383. 37. Krieger, I.M. and Maron, S.H., J . Appl. Phy. (1952) 23 (1), 147. - 66 -38. Krieger, I.M. and E l r o d , H., J . Appl. Phy. (1953) 24 (2), 134. 39. Krieger, I.M. and Maron, H., J . Appl. Phy. (1954) 25 (1), 72. 40. Levin, E.M. ,Robbins, CR. and McMurdie, H.F., Phase Diagrams f o r Ceramists (1964), 219. 41. B i r d , R.B., Stewart, W.E. and Lighfoot, E.N., Transport Phenomena, John Wiley & Sons, Inc., New York (I960), 85. 42. B i r d , R.B., Stewart, W.E. and Lighfoot, E.N., o p . c i t . , 89. 43. B i r d , R.B., Stewart, W.E. and Lighfoot, E.N., o p . c i t . , 96. 44. Amosov, V.M. et a l , Inorganic Materials (Akad. Nauk. SSSR I s v e s t r i a Aenganicheskie Materialy) (1971) 1_ (2), 292. 45. Mooney, M., J . Rheol. (1931) 2, 210. 46. Masson, C.R., Proc. Roy. Soc. (1965) A 287, 201. 47. M i t c h e l l , A., Trans. Far. Soc. (1967) 6_3, 1408. 48. The United Steel Companies Limited, Standard Methods of Analysis of Iron, Steel and Associated M a t e r i a l s , 5th e d i t i o n (1951), 191. 49. Kozakevitch, P., Rev Metallurgie (1954) 51, 569. 50. Kumar, D., Ward, R.G. and Williams, D., Trans. Faraday Soc. (1966) 61, 1850. 51. Levin, E.M. , Robbins, CR. and McMurdie, H.F., Phase Diagrams for Ceramists (1964). 52. B i l l s , P.M., J . Iron Steel Inst. (1963) 200, 133. 53. Metals Handbook, 8th e d i t i o n , vol.1 (1969), 163. 54. Rait, J.R., Trans. B r i t . Ceram. Soc. (1941) 40_, 157. 55. Baird, D.C, An Introduction to Measurement Theory and Experimental Design, P r e n t i c e - H a l l , Inc., (1962), 48. - 67 -- 68 -j 0.1 0.2 0.3 0.4 0.5 0.6 0.9 1.0 Figure 3. Ratio df non-Newtonian to Newtonian heat trans f e r rates as a function of n, the flow behavior index Figure 4. Ratio of non-Newtonian to Newtonian mass transfer rates to bubbles as a function of n, the flow behavior index. - 69 -(10000 Figure 6. V i s c o s i t y of a b l a s t furnace s l a g and two open hearth slags as a f u n c t i o n of r o t a t i o n a l speed (x by 20 to o b t a i n r a t e s of shear). r 70 -L i q u i d l e v e l Figure 7. Schematic diagram of a concentric cylinder viscometer. A, inner c y l i n d e r ; B, stem; C, outer c y l i n d e r . Figure 8. Laminar flow of an incompressible f l u i d i n the space between two concentric c y l i n d e r s , the outer one of which i s r o t a t i n g with an angular v e l o c i t y ft . I ' i I 1» i i 1 2 5 10 20 50 * io2 Figure 9. C r i t i c a l Reynolds number for tangential flow i n annulus; outer cylinder r o t a t i n g and inner cylinder stationary. - 71 -L J D Figure 10. A schematic diagram of the apparatus for viscosity measurements. (A) molybdenum cylindrical crucible, (B) Mo shaft connected to the motor, (C) variable speed motor, (D) speed reducing assembly, ( E ) bottom plate, (F) slotted disc to measure speed, (G) photo c e l l , (H) Mo Inner cylinder, (I) alumina tube, (J) Mo sleeve, (K) stainless steel tube, (L) thermocouple (Pt/Pt-10% Rh), (M) suspension wire, (N) stainless steel frame, (0) mirror, (P) extension wires, (Q) stainless steel rings to support extension wire, (R) mercury pool, (S) bearing assembly, (T) furnace tube, (U) brass tube, (V) water cooled base, (W) support, (X) shaft providing vertical movement, (Y) stainless steel l i d , (Z) heating elements. Figure 11. A close-up view of the drive mechanism. Figure 12 a. A close-up view of the upper part of the viscometer as used at low speeds. - 74 -Figure 12 b. A close-up view of the upper part of the viscometer as used at high speeds. - 75 -Figure 13. I n i t i a l dimensions of the inner and outer c y l i n d e r s . - 76 -1400 1300 1200 1100 P 1000 900 8,00 700 600 ro-o-cxo 0 6 12 18 24 30 36 42 48 54 60 DISTANCE FROM BOTTOM OF FURNACE CM. Figure 14. Temperature profile of the empty furnace. The position of the inner cylinder (I.C.) and outer cylinder (O.C.) are also indicated. Figure 15. A close-up view of the bottom part of the viscometer showing the graphite and z i r c o n i a f e l t d i s c s . - 78 -Figure 16. A close-up view of the upper part of the viscometer showing the graphite and graphite f e l t d i s c s . Figure 17, The viscosity measurement apparatus. CD ro cn i—i Q_ cn o co t—4 ID CD a -0.07 Figure 18, T E M P 1 3 9 0 ° C S L R G N O 1 i i 1 —1 1 — 1 1 0 . 2 5 0.57 0 . 8 9 1.21 1.53 LOG OF RATE OF SHEAR 1/SEC 1.85 2.17 00 o 2.49 Plot of the viscosity as a function of log-^g r a t e °f shear. f*d TO e VO SHEAR STRESS DYNES/CMXCM (XlO 2 ) 0.0 0 22.0 33.0 44.0 J I L 55.0 - 18 -i n T E M P 1 3 5 0 ° C S L A G N O 1 cn i—i ID CL *A A A 4 / ^ 4 A A 4 A AAAA A M A A * * * ^ A ^A^^^/gt^^ oo — a . 1 CM I — I in o CJ co 1—4 a . -0.7 Figure 20. -0.3 ' 1 1 1— 0.1 0.5 0.9 1.3 LOG OF RATE OF SHEAR 1/SEC 1.7 2.1 2.5 Plot of the viscosity as a function of log-^g rate of shear. RATE OF SHEAR 1/SEC Figure 21. Plot of the shear stress as a function of the rate of shear. r-LO ro CC I 4 Q_ I CD . or: o CJ CO I—^  i n . TEMP 1332 t SLAG NG 1 1 ——i . . - 0 . 3 0 . 0 3 0.36 0.E9 1.02 1.35 LOG 0 C RATE 0 r SHEAR 1/SEC oo 4> 1 . 6 8 2.0] 2.34 Figure 22. Plot of the viscosity as a function of l°g^Q rate of shear 0.0 oo 27.0 5 4-° 108.0 135.0 R A T E OF S H E A R 1/SEC 1 6 2 . 0 189.0 216.0 Figure 23. Plot of the shear stress as a function of the rate of shear. oo ON 0.0 0.4 O.B L O G OF R A T E OF S H E A R 1.2 1/SEC Figure 24. Plot of the viscosity as a function of l o g 1 Q rate of shear - LQ -i n CO I J o Q_ >-CD I"3—I CO o CO ' a OD • TEMP 1300 t SLAG NO 2 A A A A A 4 A AArf* i A A V * A ^ A A * W j » i « f t g * -0.75 -0.3B 0.03 0.42 0.81 1.2 L O G OF R A T E OF S H E A R 1/SEC 1.59 1.98 2.1 Figure 26. Plot of the viscosity as a function of l°g 1 Q rate of shear. oo i 1 r 0.0 28.0 55.0 84.0 112.0 140.0 R A T E O F S H E A R 1/SEC Figure 27. Plot of the shear stress as a function of the rate of shear. 1B8.0 195.0 224.0 ro CD CO I—I 1_ i— CO o CJ CO I—I TEMP SLAG 1300 t NG 3 -0.35 0.0 ~1- 1 1 1— 0.35 0.7 1.05 1.4 L O G OF R A T E OF S H E A R 1/SEC 1.75 V O o 2.1 2.45 Figure 28. Plot of the viscosity as a function of l o g ^ rate of shear. - 16 -6 0 0 0 £ 5 0 0 0 g 4 0 0 0 g 3 0 0 0 t 2 0 0 0 >-CO o o CO > 1000 1400 6.0 TEMPERATURE °C 1350 1300 SLAG I SLAG Z—i-6.1 I 0 4 - I / T 6.2 l/°K 6.3 6.4 Figure 30. Plot of l o g 1 0 n vs. ±. Figure 31. A close-up view of the five suspension wires. Figure 32. A close-up view of the torsional pendulum. - 94 -Figure 33. Schematic diagram of a mass as used for wire calibration. 

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