First, the particle size is not uniform Feed hydrometallurgical leaching operations are typically ground using a ball mill or rod mill, then through the classifier is larger than a certain size particles return regrinding, thus leached solids having a particle size range. This particle size range depends on a number of factors, including the mechanical properties of the material and the minerals it contains. The rate at which the target metal is leached from the feedstock is determined by the set chemical conditions and temperature, depending on the surface area of ​​the solid being immersed. It is therefore necessary to describe the continuous range of particle sizes and the particle size distribution in some way. This can be done by sieving. The weighed dry mill material is passed through a standard set of sieves and the weight of material remaining on each screen is weighed. The smallest screen used is typically 200 or 325 mesh with a mesh diameter of 75 μm or 45 μm. The weight of solids on each screen is converted to a cumulative mass fraction that is less than each given sieve diameter. Thus, the mass of the solid is expressed as a fraction (or percentage) of the sieve component having a particle size smaller than the mesh size. The mass fraction of a set of sieves is generally expressed in terms of the Schuhman equation: y=(d∕K) m Wherein the mass fraction of the y-particle size is less than d; M-distribution constant; K - another constant, called the amount of sieve modulus. m and K are constant for a particular solid. The m and K values ​​of the ore can be determined by plotting lgy against lgd. Second, the heterogeneous mass transfer equation A mathematical model is established assuming that the particles are uniform, non-porous and almost spherical (radius r). The introduction of the shape factor φ corrects the increase in surface area due to deviation from spherical symmetry. The particles are believed to shrink with dissolution without shape change, and the surface is free of insoluble product layers. Mass transfer occurs through three mechanisms: diffusion, convection, and electromigration. The mass transfer flux of each mechanism can be considered separately, and then the three flux equations are combined to obtain a net flux. (1) Diffusion The driving force for diffusion is the chemical level gradient. The flux J d caused by diffusion alone can be expressed as B-absolute molar mobility in the formula: C-concentration; Γ-activity coefficient; Δ-δ operator of spherical coordinates; Îœ-chemical position. If the diffusion coefficient D is defined as Substituting it into the above formula yields the first law of diffusion J d =-Dâ–³c (3) Since the data of each substance considered in this system is not sufficient to calculate the D value, it is assumed here that D takes the value of the infinitely dilute solution regardless of the concentration. (2) Convection The convection term in the mass transfer equation is derived from the overall motion of the fluid relative to the solid particles. The flux caused by convection is J c =-vc (4) Where v is the flow rate of the fluid to the solid particles. The theoretical treatment of convective mass transfer of suspended particles in a stirred reactor has not been fully established, and the fluid along the particles may be in turbulent motion, and the relative velocity v is difficult to estimate. However, for particles in the Stokes sedimentation mode (the Reynolds number should be less than 1, and for ZnO particles this corresponds to a maximum particle size of 74 μm), v can be expressed as Where g-gravity acceleration; r t - the particle radius of time t; Ï l , Ï s - the density of the solution and the solid particles; η l - viscosity of the fluid. Drawn from the above two equations (III) Electromigration The presence of an electric field in the boundary layer causes electromigration of all charged substances. According to Ohm's law, the flux of electromigration is J e = ucE (7) Where u-ion mobility, cm 2 /(V·s); E- electric field strength. Ion mobility is closely related to concentration, but as with diffusion coefficients, the ion mobility in this model is also considered constant due to the lack of data for each ion component. The electric field is related to the concentration of the ion component by the Poisson equation: The dielectric constant of the ε-solution in the formula; The number of different ionic components present in the n-solution. 1. Comprehensive mass transfer equation If all three mass transfer mechanisms are considered, the net flux is If the chemical reaction occurs only at the solid-liquid interface, it can be obtained from conservation of mass. If it is assumed that D, v and u are not related to concentration, then 2, mathematical model Calculating the flux of each chemical substance using equation (11) for the established mathematical model can be simplified as follows. (1) Quasi-steady approximation Although the solid-liquid interface moves at a finite rate as the solid particles dissolve and shrink, there is no steady state, but since the rate of interfacial movement is small relative to the rate at which the substance passes through the boundary layer, it is assumed that the steady state is reasonable. The sample is removed from equation (11) over time, so that for any substance, equation (11) can be rewritten as Equations (10) and (12) provide partial differential equations for the system. In principle, boundary conditions can be used to solve the change pattern and electric field of the concentration, and then the flux and reaction rate of each substance are calculated. However, these equations cannot be solved analytically, and it is time-consuming to find numerical solutions and difficulties. Assuming that it is electrically neutral in the boundary layer and using the semi-theoretical transformation of Sherwood, the partial differential equation can be transformed into a set of algebraic equations. (2) Electrically neutral determination of surface concentration The electric field gradient ΔE generally does not exceed 10 5 V∕cm. Substituting this value into equation (8) yields a value of ∑Z i C i of approximately 10 -12 equ·cm -3 , which is 7 to 9 orders of magnitude smaller than the ionic strength of the solution. Thus, the effect of charge on the interface between the actual concentration of the substance containing hydrogen and zinc can be ignored. Therefore, the total charge neutrality is assumed when calculating the concentration, ie It should be emphasized that although the effect of charge on the surface concentration of the material is negligible, the approximate electric field gradient causes significant migration and thus cannot be ignored in the mass transfer equation. (3) uniform electric field Although the electric field E in the boundary layer is not a constant value in practice, it can be assumed to be a constant value for the mathematical convenience, and the uniform value of E is used in the equation (7) to obtain the electromigration flux of the substance i. (4) Sherwood transform The flux caused by diffusion and convection can be calculated by Sherwood semi-theoretical transformation The mass transfer coefficient K i of the substance i is defined as Where Re is the Reynolds number Where, Sc is the Schmidt number Combine these simplifications into equation (9) to get the total flux of each material as Third, the simulation equation of leaching dynamics A curve representing the metal leaching rate as a function of time, related to the mass transfer rate or chemical reaction rate of the reactants to the solid surface. But for the purpose of determining process control under conditions, the most convenient way is to use a mathematical equation to fit the curve without mentioning these factors. This advancement and retreat of the surface affects the leaching kinetics as the surface of the solid reactant in a leaching system advances or retreats as the reaction progresses. This is also the case when solid reaction products are continuously formed around the reactive ions. Based on this behavior, mathematical equations for several models were obtained and used to describe the shape of the rate curve obtained by leaching minerals under different conditions. The chemical behavior of the leaching system is extremely important when determining the chemical conditions used, especially when developing new processes. (a) shrinkage particle model Exporting a contracted particle model assumes that the particle is a sphere, but the resulting final equation can be applied to any other shape of equal volume particle. Let the number of moles n of unreacted spheres be n=4Ï€r 3 ∕3V (20) Where V-molar volume is equal to m∕Ï, where m is the mass and Ï is the solid density. The reaction rate of the surface of the sphere with radius r can be written as -dn∕dt=4Ï€r 2 Ch' (21) Where k' is the first order reaction rate constant of the reactant at a concentration of C in the solution. The equation (20) is differentiated from time and substituted into equation (21) to obtain a linear rate -dr∕dt=VCk' (23) Where VCk' is the linear rate constant k 1 . If C is the concentration of the reaction solute expressed in mol ∕ cm 3 , the linear rate at which the radius r decreases is expressed in cm ∕ s. If C is left unchanged, equation (23) represents a constant rate of movement of the reaction interface, which is the definition of linear dynamics. If the initial radius of the reactive particles is r 0 and α is the reaction score, then The above formula is different for time Combine equations (23), (24), and (25) For the initial condition t=0, α=0, it can be assumed that C is constant and the integral of equation (26) is obtained. Where k = Ck 1 ∕r 0 (time - 1 ). To plot the left side of equation (27) with t, we should get a straight line with a slope of k, and the dimension of k is 1∕t. The above model is derived from a single particle. If all the ore particles in the slurry have the same initial diameter, the reaction combination rate of many ore particles will also obey this equation. For the slurry with a certain particle size distribution, the mass fraction w i of each particle size distribution must be known. Assuming that the initial average ore radius of the mass fraction w i is r i0 , then the equation (28) becomes Where α i is the reacted fraction with a mass fraction of w i . The total amount of reaction solids is α= If the concentration changes as the reaction progresses, the change in concentration must be considered in the integral basic rate expression. For surface control reactions, or for diffusion through a boundary film of the limit, equation (29) must account for the concentration of the reactants, thus obtaining Wherein C 0 - initial concentration; Σ-stoichiometric coefficient; b = n 0 ∕V 0 C 0 , C 0 , n 0 are the total moles of minerals in the system, respectively. In the specific case of σb=1, the integral is obtained. (2) Shrinking core model In many cases mineral particles contain a variety of metals, while only one metal is eluted when leaching. This situation can result in the formation of a porous solid reaction product around each of the ore particles that are reacting. Only one special case is considered here, that is, the particle radius always remains equal to the original radius r 0 of the particle before the reaction starts. The shrinking core of radius r continues to react at a rate determined by the rate at which the reactants diffuse through the product layer toward the reaction interface. If the reaction particles are spherical, the reaction rate can be written as: The number of moles of unreacted mineral in the n-core; Σ-metering factor; C- the number of moles of diffusing material required to leach 1 mole of metal from the core; The effective diffusion coefficient in the D-product layer is the concentration of the reactants at the interface of the shrinking core. In the steady state condition, from r to r 0 integral, when the concentration C of the reactant on the interface is much smaller than the bulk concentration C 0 , The equations for combining the equations (32) and (33) to obtain the core and reaction product boundary movement rate expressed by the radius r of the unreacted core are as follows Combining equations (34), (24) and (25), obtains a reaction rate equation expressed by the fractional α of the reaction. When the boundary condition t=0, α=0, the integral is obtained. The left side of the above equation is plotted against time to get a straight money, and the slope is proportional to r 0 -2 . Similarly, when it is necessary to consider the change in the concentration of the reactants, the equation (35) becomes a diffusion through the product or the residual layer. In the specific case of σb=1, the integral is obtained. In the general case of σb ≠1, equations (33) and (37) need to be solved by numerical integration. Fourth, mixed dynamics The kinetics observed in the leaching system often may contain more than one rate process. For example, the surface reaction acts simultaneously with the mass transfer process through the diffusion layer on the total kinetics. Then the surface reaction rate can be expressed by the following equation: Where A- total area; C s - surface concentration; K s - the reaction rate constant of the surface reaction. Under steady state conditions, C s can be obtained and brought into the above formula to obtain a mixed kinetic expression: Where Δx is the thickness of the diffusion boundary layer, and the constant k 0 'is represented by the sum of the tandem reaction, ie, the mass transfer of the diffusion layer and the resistance of the surface reaction (the reciprocal of the rate), ie If the area and Δx remain constant during the reaction, equation (40) becomes a simple first-order rate expression whose first-order rate constant k 0 ' contains D and K s with complex temperature relationships. Usually D (solution diffusion coefficient) is less sensitive to temperature changes than K s with greater activation energy. Therefore K s will become much larger than D as the temperature increases. It is apparent from the formula (40) that diffusion at a high temperature is a rate controlling step, and at a low temperature, the surface reaction is a rate controlling step, and at a medium temperature, a mixing rate is controlled. If the reaction causes the surface to form a product layer, diffusion occurs through the product layer, assuming that the surface area is constant, equation (40) becomes Where b(n 0 -n)=σΔx; n 0 is the number of moles of reactants in all ore blocks at the beginning of the reaction; n is the number of moles of residual reactant at any time t; n 0 -n represents The amount of reaction. Equation (42) can be integrated under the conditions of constant concentration and area. This formula represents a linear parabolic rate and the sum rate, wherein the rate constant K p is equal to 2D / b. Equation (45) can also be written as According to this formula, plotting t/Δn on Δn should result in a straight line whose slope contains the reciprocal of the parabolic rate constant K p and the intercept contains the reciprocal of the linear rate constant K s . It is reported that the Fe 3 + equation describes the initial stage of the oxidation leaching of copper ore. In the study of mineral particle leaching process, the kinetic problem can often be treated by steady-state approximation. For example, stirring leaching must include steps of solution diffusion through the liquid boundary film, diffusion through the solid product layer, and surface reaction. For spherical particles, three successive kinetic processes are: Boundary layer diffusion Diffusion through the product layer Surface reaction Under approximate steady-state conditions, each of the above rates is equal, so that a comprehensive expression can be derived (ignoring the inverse reaction): The above formula can also be rewritten to express the rate using the reaction rate: Formula (51) can be applied to a slurry composed of single-grained particles having an average radius of r 0 . For the case of having a particle size distribution characteristic, α = α I , that is, the reaction rate of the first particle having a particle size of r 0i , and then weighting the reaction rate of the individual particle size to obtain a total reaction rate α = It is apparent from equation (52) that the integrated equation includes only the sum of the rate expressions of the boundary layer diffusion, the diffusion through the solid product layer, and the surface reaction. For the case where the concentration changes with time, as previously described, the concentration in equation (52) is C = C 0 (1-σba), at which point the equation must be numerically integrated. V. Examples of leaching kinetics (1) Dissolution of zinc calcine in dilute acid Many acid and alkali leaching of oxides obey the shrinkage core model. A typical example is the dissolution of zinc calcine in dilute acid. It is calculated using equations (53) and (54) based on the ion diffusion coefficient and ion mobility of each chemical involved in the dissolution process. The calculation assumes that the dissolution rate is controlled by mass transfer, so the calculation used can only be used in situations where no chemical reaction is involved. Solving equations (53) and (54) requires several boundary conditions that specify the values ​​of the various parameters in the model and correlate the flux of each species through the metering relationship of the leaching reaction. For the sulfuric acid leaching system, the data used for the calculation include the ion diffusion coefficients and ion mobility of H + , HSO 4 - , SO 4 2 - and Zn 2 + , and the following equilibrium equilibrium constants and activity coefficients. The mass transfer data used in the mathematical model calculation of dilute acid leaching of zinc oxide is listed in the following table. substance Equivalent ion conductance Λ i 0 ∕ (Ω - 1 ·cm 2 ·equ -1 ) Ion diffusion coefficient D∕(cm 2 ·s -1 ) Ion mobility u∕(cm 2 ·V -1 ·s -1 ) H + 348.9 9.3×10 -5 3.6×10 -3 Zn 2 + 53.8 7.2×10 -6 5.6×10 -4 SO 4 2 - 79.0 1.0×10 -5 -8.2×10 -4 HSO 4 - 100.00 2.7×10 -5 -1.6×10 -3 Several boundary conditions are At the solid-liquid interface, ie r=r t , C i =C i s (55) Since the slowest step in the leaching process is mass transfer through the boundary layer, it can be assumed that chemical equilibrium is reached at the interface, resulting in the following boundary conditions. In the formula, In the solution phase, ie r = ∞, E = 0 (59) C i =C i b (60) The bulk concentration is calculated by the mass balance and the chemical equilibrium of the bulk phase. In the formula, [H 2 SO 4 ] and [ZnSO 4 ] are the net concentrations of t-time sulfuric acid and zinc sulfate. Measurement relationship Sulfate flux The mathematical model consists of the written equations (54), equations (53) and the boundary conditions derived above for each substance. Once the flux of each species is known, the dissolution rate of ZnO can be calculated. If the spherical particles of radius r t contain Nmol of ZnO, then In the formula, M w is the molecular weight of ZnO. Since there is no material accumulation in the boundary layer under steady state, all dissolved zinc must be transferred to the bulk phase of the solution. Therefore, the reaction rate can be correlated with the rate of mass transfer of zinc and acid through the boundary layer as follows Wherein J Zn - the net flux of zinc flowing away from the surface; J H - The net flux of acid to the surface. From equation (68) and equation (69) Equation (70) is numerically integrated by the finite interval method to obtain a function of r t versus time. For single-sized particles, the relationship between r t and reaction fraction α is That is, the contracted particle model of the formula (72), r 0 is the initial radius of the solid particles. The case of the particle size distribution can be similarly processed, and the single size fractions of the m initial radii r 0k each constitute a fraction w k of the total mass. Degree of leaching The total leaching rate is determined by the following formula In order to verify the correctness of the model and calculation, it is necessary to study the rate at which the calcined zinc sulfide concentrate is dissolved in four acids such as sulfuric acid, perchloric acid, nitric acid and hydrochloric acid. The selected agitation conditions allowed all solid particles to be suspended and the rate of dissolution independent of the rate of agitation. The experimental curve in perchloric acid and nitric acid solution agrees well with the predicted curve calculated by the model, but it is acceptable before the leaching rate is 80% in the sulfuric acid solution. The reason why the dissolution curve is not satisfactory is due to the solid particles. The dissolution is not as uniform as it is assumed and always remains spherical, and it is actually found that the partially leached calcine particles have large, deep pores. The simplified model does not consider the formation of chloride ions by the formation of zinc chloride and therefore cannot be used to predict the dissolution rate of hydrochloric acid leaching calcine. The model established earlier without considering the contribution of electromigration to mass transfer is seriously deviated even for the kinetics of 0.1 mol 高L perchloric acid leaching, reflecting the role that electromigration cannot be ignored in mass transfer. (2) Alkali leaching dynamics of scheelite Tungsten is a high-yield resource in China. The scheelite (CaWO 4 ) is not leached with caustic as well as the wolframite because it does not decompose in the caustic solution, and is conventionally decomposed with hydrochloric acid. Institute of Process Engineering Kejia Jun et solubility product principle, proposed to add an appropriate amount of soluble phosphate at the caustic leaching, in order to precipitate calcium scheelite is insoluble compounds HAP Ca 5 (PO 4) 3 OH (kap = 1.6 × 10 -58 ), so that the scheelite is continuously dissolved and leached in the form of sodium tungstate. In order to confirm this possibility and the speculative mechanism, the alkali leaching kinetics of scheelite was studied. The disc samples were pressed with scheelite powder of 100% particle size <44 μm (-325 mesh), and the empirical leaching kinetic equation was obtained by the rotating disc method as follows: Where k is the apparent leaching rate constant. From the above formula, it can be seen that the apparent energy E α =42700 J∕mol, indicating that the leaching process is mainly controlled by surface chemical reactions. During the experiment, it was observed that solid suspension appeared in the liquid and the solid precipitate layer appeared on the surface of the disk sample, which is consistent with the conclusion that the leaching process derived from apparent activation energy is mainly controlled by surface chemical reaction. The results of X-ray diffraction and infrared spectroscopy of the solid suspension also confirmed the formation of calcium hydroxyphosphate in the leaching. It can be inferred that the caustic leaching of scheelite is carried out as follows: This is an example of understanding the reaction mechanism through kinetic studies. Plastic Auxiliary Machinery,Plastic Machinery,Plastic Auxiliary Equipment,Plastic Machinery For Sale zhangjiagang sevenstars machinery co.,ltd , https://www.sevenstars-machineries.com
(1) (2) (5) (6) (8) (10) (11) (12) (13) (14) (15) (16) (17) (18) (19) (twenty four) (25) (26) (27) (28) (29) . (30) (31) (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (45) (46) (47) (48) (49) (50) (51) . Under the condition of constant concentration, the integral equation (51) is obtained. (52) (53) (54) (56) (57) (58) , , Representing the equilibrium constants of reactions (a) and (b) and (c), respectively; Q a , Q b , and Q c are the equilibrium constants of reactions (a), (b), and (c), respectively, when concentration is used; γi is a substance. The activity coefficient of i. (61) (62) (63) (64) (65) (66) (67) (68) (69) (70) (71) (72) (73) (74) (75) (76)