Mixing And Agitation PDF

# Mixing and Agitation ## Content - Introduction - Mechanism of Solid mixing - Mixing index and mixing - Mixers for dry powders - Mixers for cohesive solids - Liquid mixing: flow patterns, Types of agitator - Power requirement for liquid mixing ## Introduction - **Agitation**: establishment of a p...

# Mixing and Agitation ## Content - Introduction - Mechanism of Solid mixing - Mixing index and mixing - Mixers for dry powders - Mixers for cohesive solids - Liquid mixing: flow patterns, Types of agitator - Power requirement for liquid mixing ## Introduction - **Agitation**: establishment of a particular flow pattern within the liquid, usually a circulatory motion within a container. - **Mixing**: random distribution, throughout a system, of two or more initially separate ingredients. - Single homogeneous material can be agitated but can't be mixed until some other material is added to it. - Image describing Agitation: A tank of liquid is being stirred by a motor. - Image describing Mixing: A powder is being poured into a liquid from a beaker. ## Objectives - To increase the homogeneity of material in bulk. - To bring about intimate contact between different species in order for a chemical reaction to occur. - To enhance heat and mass transfer. - To change the texture. - Dispenses a liquid which is immiscible with the other liquid by forming an emulsion or suspension of few drops. - Suspends relatively lighter solid particles. ## Applications - Liquid Blending - Solids Suspension - Gas Dispersion - Dissolving Solids - Preparation of: - Emulsions - Pastes - Creams - Image describing Applications: There are two pictures, one showing a handheld mixer blending a pink liquid, and the other one depicting a bowl being hand-whipped for making meringue. ## Mixing Mechanisms - Three basic mechanisms: - **Convection:** Movement of groups of particles because of the direct action of an impeller or a moving device. Ex: trough mixer with spiral ribbon - **Diffusion:** Diffusion refers to random dispersion of individual particles in the inter particle void spaces throughout the mixer. Ex: simple barrel mixer - **Shear mixing:** Groups of particles are mixed through the formation of slipping planes developed by the action of blade. Newly formed slipping planes in turn allow particles to diffuse through new void spaces. - Other Classifications for Mixing Mechanisms - According to the type of motion applied to a bulk - Mixing within bulk material - Centrifugal mixing - Mixing in a fluidized bed - Mixing solids in a suspended condition - Free fall mixing due to gravity ## Degree of mixing - **Mixing index (M):** A dimensionless fractional measure of variance or standard deviation that can be correlated with time. - $M=\frac{s_0^2-s^2}{s_0^2-s_\infty^2}$ - Where: - M - mixing index in fraction - $S^2$ - variance at any given time, - $S = \sqrt{\frac{\sum_{i=1}^n(x-x)^2}{n-1}}$ - 'n' is the number of samples taken, - $X_1, X_2, ......., X_n$ are the fractional compositions of component X in the 1, 2...... n samples - Image describing Degree of mixing: There are two circles, one labeled as Sample X that is filled with 'p' and the other is labeled as Sample Y filled with 'q'. - For an unmixed system of two separate components: - $S_0^2=p(1-p)$ - Variance after complete mixing, - $S^2= p(1-p)/N$ - Where, N-number of particles in a mixed sample. If sample is large quantity then N is also large (infinite), then - $S^2 = 0$ - A biscuit dough is prepared by mixing flour and other ingredients along with tracer material (2% mass). After 10 minutes of mixing 6 random samples are collected & their composition (% of tracer material) is given below: - After 10 min 2.021 1.925 1.826 2.125 2.210 2.015 - Calculate the mixing index after 10 min of mixing. - Solution: p=0.02, q=0.98, n =6 - Avg composition of tracer material=0.0202 - $S = \sqrt{\frac{\sum_{i=1}^n(x-x)^2}{n-1}}$ - S = 1.8762×10-4 - $S_0^2 = p(1- p)= 0.0196$ - For large sample: N= infinite, $S^2 = p(1-p)/N = 0$ - $M=\frac{s_0^2-s^2}{s_0^2-s_\infty^2}$ - Mixing index (M) after 10 min = 0.99 ## Rate of mixing - Rate of mixing at any time under constant working conditions ought to be proportional to the extent of mixing remaining to be done at that time. - $\frac{dM}{dt}=K(1 – M)$ - Where (M) is the mixing index and K is a constant, and on integrating from t = 0 to t = t during which (M) goes from 0 to (M), - $\int_0^M\frac{1}{1-M}dM=\int_0^t Kdt$ - $-ln(1 – M) = Kt$ - $1-M = e^{-Kt}$ - In a batch mixer, blending starch and dried, powdered vegetables for a soup mixture, the initial proportions of dried vegetable to starch were 40:60. The variance of the sample compositions measured in terms of fractional compositions of starch was found to be 0.0823 after 300s of mixing. For how much longer should the mixing continue to reach the specified maximum sample composition variance of 0.02? Assume that sample contains 24 particles. - Given: p=0.6, q=0.4, N=24 - $S_0^2 = p(1-p)= 0.24$ - $S^2 = p(1-p)/N = 0.24/24=0.01$ - $M = \frac{s_0^2-s^2}{s_0^2-s_\infty^2}$...... (1) - After 300s: $S^2=0.0823$ then M=0.685 (from eqn.1) - Substituting in eqn $1-M = e^{-Kt}$ - K=3.85×10^-3 - If $S^2$=0.02, then M=0.957 (from eqn.1) - t=820s from eqn $1-M = e^{-Kt}$ ## Powder Mixers - **Tumbling mixer:** - Types: horizontal drum, double-cone, V-cone, Y-cone, and cube. - Operated in batch mode being partially filled with solids - Tumbling mixers are run at a fraction of the critical speed required for centrifugation with a practical maximum speed of about 100 rpm. - Such mixers may have baffles fitted to the inner walls which help to lift solids or alternatively may be fitted with ploughs to assist convection. - Image describing Tumbling mixer: A picture of a horizontal cylinder is given. - a. Horizontal cylinder : - b. **Double cone blender:** The double cone blender consists of two cone-shaped sections, typically with 45° slopes. - Image describing Double cone blender: A simple diagram of a double cone blender is shown. - c. **V-cone blender:** V-cone blender consists of two large diameter pipe sections cut at a 45° and welded together to form a V. - d. **Y-cone blender:** In the same way, the Y-cone blender has a third section that extends the volume of the blender in a bisectional direction with respect to the other pipe sections. - Image describing Tumbling mixer: Three pictures are shown, the first one depicting a double cone blender, the second one displaying a V-cone blender, and the last one showing a Y-cone blender. - **Ribbon blender:** - It consists of a trough & shaft with two helical screws - Inner helical ribbon moves the solids slowly in one direction, while the outer one moves it quickly in the opposite direction. - There is a resultant movement of solids in one direction due to difference in peripheral speed of ribbons. - Image describing Ribbon blender: Pictorial representation of a ribbon blender is depicted. - Radial mixing also achieved due to rotational motion of ribbons - Mixing is strongly convective and segregation is far less pronounced than in tumbling mixers - **Dry applications:** cake and muffin mixes, flour, bread improvers, cereals, trail mixes, snack bars, spices & herbs, tea & coffee - When dry blending food products, relatively small amounts of liquid may be added to the solids in order to coat or absorb coloring, flavoring, oils or other additive solutions. - **Vertical screw mixer:** - Cylindrical or cone shaped vessel. - The screw may be mounted centrally or orbit around the central axis of the vessel near the wall. - Materials are lifted from the bottom are then exchanged with materials on the way up. - Useful for mixing small quantities of additives into large masses of material. - Image describing Vertical screw mixer: A diagram showcasing the functioning of a vertical screw mixer is presented. - **Fluidized bed mixer:** - The resulting turbulence of passing air through a bed of particulate material causes material to blend. - Materials to be mixed have to be relatively fine and fairly narrow in their size distribution, as well as not too cohesive - Mixing times required in fluidized beds are significantly lower than those required in conventional powder mixers. ## Mixers for Cohesive Solids - Mixing elements cannot generate flow currents - High viscosity - High power consumption - **Types:** - Change-Can mixer - Kneaders, Dispersers & Masticators - Mixer extruders & Mixing rolls - Mixing is by combination of low speed - Shear, Smearing, Folding, Stretching & Compressing. ## Change Can Mixer - **Principle:** Change can mixers work by the relative motion of the blades and the can - **Types:** pony mixer & beater mixer - In the pony mixer, the rotating agitator carries several vertical blades positioned near the vessel wall. - The can is driven by a turntable in a direction opposite to that of the agitator. - Image describing Change Can mixer: A picture showing the working principle of a change can mixer. - **Beater mixer:** - Can or vessel is stationary - Agitator has a planetary motion - Beaters are shaped to pass with close clearance over the side & bottom of mixing vessel - Image describing Beater mixer: A diagram showing the working principle of a beater mixer is depicted. ## Kneader Mixer - **Working Principle:** Some kneaders achieve their mixing action by squashing the mass flat, folding it, and squashing it again. Others tear the mass apart and shear it between a moving blade and a stationary surface. - It consist of two contra-rotating arms which fold and shear the material. - The arms rotate at differential speeds (nearly 3:2). - Cooling is provided commonly - Large energy requirements - **Sigma blade:** - Used for general purpose kneading - Edges are serrated to give a shredding action - **S-type Double-naben (fish-tail blade):** - Effective with heavy plastic materials - Develop high shear force - **Z-type Disperser blade:** - Heavier and develop high shearing forces - Disperse powders or liquids into rubbery masses. - Image describing Kneader mixer: A picture showing the working principle of a kneader mixer is depicted. - Image describing Different Types of Blades: Three pictures are shown depicting different types of blades of a kneader mixer. ## Mixer Extruder - The discharge of continuous kneader is restricted by covering it with an extrusion die - Pressure is built by reducing the pitch of helix or by reducing the diameter of chamber or both. - Material is cut and folded and subjected to additional shear - Heating jacket is provided. - Image describing Mixer Extruder: A diagram of a mixer extruder is given. ## Liquid Mixing - Liquids are mixed usually by impellers, which produce shear forces for inducing the necessary flow pattern in the mixing container. - Mixing occurs due to the resultant effect of 3 components acting on liquid: - Radial component - Tangential / Circular component - Axial / Longitudinal component - The type of flow depends on: - Type of impeller - Characteristic of fluid - Size proportion of tank, baffle & impellers ## Flow Patterns - **Radial component:** - Direction: acts in the direction perpendicular to the impeller shaft. - Effect: excessive radial flow takes the material to the container wall, then the material falls to the bottom and rotates as a mass beneath the impeller. - Image describing Radial component: A top view of a mixing tank showing the flow pattern in a radial direction. - **Tangential component:** - Direction: acts in the direction tangent to the circle of rotation around the impeller shaft. - Effect: if shaft is placed vertically & centrally, tangential flow follows a circular path around the shaft & creates a vortex in the liquid. - Image describing Tangential component: A top view of a mixing tank showing the flow pattern in a tangential direction. - **Axial component:** - Direction: acts in the direction parallel to the impeller shaft. - Effect: inadequate longitudinal component causes the liquid and solid to rotate in layers without mixing. - Image describing Axial component: A diagram showing the flow pattern in the axial direction is depicted. ## Types of Agitator/Impeller - Type of agitator / impellor - Paddle, propeller, turbine. - **Paddle agitator:** - Consisting of a pair of flat blades mounted on a shaft. - Paddles rotate at a low speed of 100rpm. - They push the liquid radially and tangentially with almost no axial action unless blades are pitched - Image describing Paddle agitator: A simple diagram of a paddle agitator is shown. - In deep tanks several paddles are attached one above the other on the same shaft. - Advantages: Vortex formation is not possible with paddle impellers because of low speed mixing. - Disadvantages: Mixing of the suspension is poor therefore baffled tanks are required - **Anchor agitator:** Used for high viscous fluids - **Turbine mixer:** - In turbine mixers, the impeller consists of a larger number (four or more) of flat or curved blades, mounted on a (usually vertical) shaft. - They exert considerable shear on the fluid and are therefore suitable in applications involving mass transfer (e.g. oxygen transfer in fermentors) or phase dispersion (e.g. emulsification and homogenization). - The diameter of the impeller is, typically, one-third to one-half of the diameter of the vessel. - Image describing Turbine mixer: Three pictures are shown, one depicting a turbine mixer with flat blades, the second showing a turbine mixer with curved blades, and the last depicting a turbine mixer with a disk. - **Propeller:** - Primarily used to blend low viscosity liquids. - Impeller diameter is much smaller than that of turbine mixers. - The mixer shaft is usually positioned on an angle and off-center. - Two are more propellers are used for deep tank. - Advantages: high mixing capacity - Disadvantages: not effective with liquids of viscosity >5Pa.s. - Image describing Propeller: Two pictures are shown depicting different propellers. ## Vortex Formation - When an impeller rotates in a liquid the liquid is likely to swirl in a mass and a vortex will form. - wastage of energy as the impeller rotates partly in air - Unwanted dissolution of air - **Controlling the vortex formation:** - By use of baffles - Positioning the mixer shaft off center - By use of draft tube - Image describing Vortex formation: Shows the formation of a vortex in a mixing tank because of the impeller rotation. - Image describing Vortex formation: Shows a diagram of a mixing tank with an offset angle mounted impeller used to control vortex formation. - Image describing Vortex formation: Shows a diagram of a mixing tank with an offset vertical mounted impeller used to control vortex formation. - Image describing Vortex formation: Shows a diagram of a mixing tank with baffles used to control vortex formation. - Image describing Vortex formation: Shows a diagram of a mixing tank with a draft tube used to control vortex formation. - Image describing Vortex formation: shows a side view and top view of a mixing tank with baffles. ## Power Requirement - **Shape factors:** - $S_1= Da/D_1 = 1/3$ - $S_2= H/D_1 = 1$ - $S_3= J/D_1 = 1/12$ to $1/18$ - $S_4= B/D_1 = 1/3$ - $S_5= W/Da = 1/5$ - $S_6= L/Da = ¼$ - Where, B= no of baffles - Image describing Power requirement: Shows a schematic diagram of a standard agitation system. - Power requirement depends on shape factors (dimensionless numbers), density, viscosity & velocity of liquid - $P= f (μ,p,D_1,N,g)$ - **From dimensionless analysis:** - $\frac{P}{\rho_f N^3 D_a^5}=\phi(\frac{\rho_f ND_a^2}{\mu},(\frac{N^2D_a}{g})^n$ - $N_p = \phi(N_{Re})^m(N_{Fr})^n$ - Where, $N_p$=Power no, $N_{Re}$=Reynolds no & $N_{Fr}$= Froude no - By including shape factors: $N_p = \phi(N_{Re}, N_{Fr},S_1,S_2...S_n)$ - **Power no:** - $N_p=\frac{external\ force\ per\ unit\ volume}{inertia\ force\ per\ unit\ volume} = \frac{P}{\rho_f N^3 D_a^5}$ - **Reynolds no:** - $N_{Re} = \frac{inertia\ force}{viscous\ force} = \frac{\rho_f ND_a^2}{\mu}$ - **Froude no:** - $N_{Fr} = \frac{inertia\ force}{gravitational\ force}=\frac{N^2D_a}{g}$ - Laminar flow in tank: $N_{Re}<10$ - Transition flow: $10<N_{Re}<10^4$ - Turbulent flow :$N_{Re}>10^4$ - a correlation for impeller used with Newtonian fluid in baffled tank. For same impellor this fig is also used for un baffled tank when $N_{Re}<300$. - For definite type of impeller power consumption remains unaffected between baffled & unbaffled tank for $N_{Re}<300$. - If higher $N_{Re}>300$ (higher impeller speed) power consumption in unbaffled tank attributes to vortex formation. In this region Froude no becomes more prominent. - Image describing Power requirement: Shows a chart with power correlation for Newtonian fluid baffled & un baffled tank. - Image describing Power requirement: Shows a chart with power requirement for different types of impellers. - A flat blade turbine agitator with disk having six blades operating at 90rpm is used to mix a liquid ($\rho=929kg/m^3$, $\mu=0.01Pa.s$). Calculate the power requirement for mixing. (Given: tank dia =1.83m, agitator dia =0.61m, H=D₁, Width = 0.122m, no of baffles=4, width of each baffle = 0.15m) - Solution: $D_1$=1.83m, $D_a$=0.61m, H= $D_1$, W= 0.122m, B=4, J=0.15m, $\rho$=929kg/m³, $\mu$=0.01Pa.s - $S_3= J/D_1=0.15/1.83= 1/12$ - $S_5=W/D_a=0.122/0.61= 1/5$ - **Reynolds no:** - $N_{Re} = \frac{\rho_f ND_a^2}{\mu}$ - $N_{Re} = \frac{929\times1.5\times0.61^2}{0.01} = 51852.135$ - Using curve 1 (from fig): $N_p$=5 - **Power requirement:** - $P = Np \rho_f N^3 D_a^5$ - A pseudo plastic mixture is prepared by mixing two liquids in the volumetric ratio 60:39.9 along with 0.1% emulsion. A 4 baffled agitation tank (1.2m dia) with 6 blade turbine (0.8m dia) rotating at 200rpm is used for agitation. A draft tube is installed to facilitate high shear rate. Consequently power requirement increase 20% due to installment of draft tube. Calculate the total power requirement. Use empirical relation for non newtanian, 6 blade turbine: $N_p=0.4486 (N_{Re})^{0.4587}$ - | Liquid | Phase | K (Pa.sⁿ) | n | P (kg/m³) | Avg shear rate | |---|---|---|---|---|---| | A | Continuous phase | 8 | 0.65 | 959 | $\gamma = 10N$ | | B | Dispersed phase | 6 | 1.3 | 1050 | $\gamma =38N(D/D_a)^{0.5}$ | - N=200/60 = 3.33rps - For liquid A: - $\gamma = 10N = 33.3\ s^{-1}$ - $\mu_c=Ky_{avg}^{n-1} = 8(33.3)^{0.65-1} = 2.35 \ Pa.s$ - For liquid B: - $\gamma =38N(D_a/D_1)^{0.5} =38\times3.33(0.8/1.2)^{0.5} =103.31\ s^{-1}$ - $\mu_d=Ky_{avg}^{n-1} = 6(103.3)^{1.3-1} = 24.12 \ Pa.s$ - Then for mixture: - $\mu = (\frac{\mu_c}{V_c})(1+\frac{1.5\mu_d V_d}{\mu_c+\mu_d})=\frac{2.35}{0.6}(1+\frac{1.5\times24.12\times0.6}{2.35+24.12}) = 6.05Pa.s$ - $\rho = \rho_c V_c + \rho_d V_d = 0.6 \times 959 + 0.399 \times 1050 = 994.4kg/m^3$ - **Reynolds no:** - $N_{Re} = \frac{\rho_f ND_a^2}{\mu}$ - $N_{Re} = \frac{994.4 \times 3.33 \times 0.8^2}{6.05} = 350.3$ - $N_p=0.4486 (N_{Re})^{0.4587}$ - $N_p=0.4486 (350.3)^{0.4587}=6.59$ - **Power requirement:** - $P = N_p \rho_f N^3 D_a^5$ - $P = 6.59 \times 994.4 \times 3.33^3 \times 0.8^5 = 79.29kW$ - Including draft tube: - $P_{total}=1.2P=95.15kW$ ## Scale Up of Agitator System - **Geometric similarity:** Geometric similarity means that a single ratio between small scale and large scale applies to every length dimension. - **Kinematic similarity:** two geometrically similar unit have constant ratio of velocity at corresponding point along with similar flow pattern. - **Dynamic similarity:** all corresponding forces at counter part location in two geometrical similar units bears a constant ratio. - Objective: a) equal liquid motion b) equal suspension solids c) equal rate of mass transfer - **Scale up ratio:** $S_R=(\frac{V_2}{V_1})^{1/3}$ - $\frac{V_2}{V_1} = \frac{\pi D_2^3 H_2}{\pi D_1^3 H_1} =\frac{D_2^3}{D_1^3}$ - Also H=D, - Where, D= vessel diameter - Using this $S_R$ all new dimensions are calculated. $S_R D_{a1} = D_{a2}$ , $S_R J_1 = J_2$ - To determine agitator speed: $N_2=N_1 (\frac{1}{S_R})^n =N_1 (\frac{D_1}{D_2})^n$ - Where, n=1 for equal liquid motion i.e $\pi D_1^2 N_1 = \pi D_2^2 N_2$ - n=3/4 for equal suspension solids - n=2/3 for equal rates of mass transfer i.e $P_1/V_1 = P_2/V_2$ - Condition of existing agitation system is as follows. $D_1$=1.83m, $D_a$=0.61m, $J_1$=0.15m, $N_1$=1.5rev/s, $\rho$=929kg/m³ & $\mu$=0.01Pa.s.scale up the agitation system for volume 3 times the existing one for following objective. - a) where equal liquid motion is needed - Solution: $D_1$=H, - $V_1= 3.142(1.83)^3/4 = 4.813\ m^3$ - $V_2=3\times 4.813= 14.44\ m^3$ - $S_R=(\frac{V_2}{V_1})^{1/3}=1.442$ - $D_{a2}= S_R D_{a1}=0.88m$ & $J_2 = S_R J_1=0.216m$ - $N_2=N_1 (\frac{1}{S_R})^n$ - For equal liquid motion: n=1 - $N_2=N_1 (\frac{1}{S_R})=1.5/1.442=1.04rev/s$ - **Reynolds no:** - $N_{Re} = \frac{\rho_f ND_a^2}{\mu}$ - $N_{Re}=\frac{929 \times 1.04 \times 0.88^2}{0.01 } = 74819$ - $N_p=5$ - **Power requirement:** - $P = N_p \rho_f N^3 D_a^5$ - $P = 5 \times 929 \times 1.04^3 \times 0.88^5 = 2.757kW$ - Image describing Scale Up of Agitator: Shows a chart with power requirement for different types of impellers.