CHAPTER 6CIRCULATION, VELOCITIES AND POWER CONSUMPTION IN AGITATED VESSELSCHE503 FLUID FLOW

OUTLINESIntroduction

Geometric Similarity & Scale up of stirred vessel

Power Consumption in Stirred Vessel - Low viscosity

INTRODUCTIONImportant factor in design of agitated vessel power required to drive the impellerEmpirical correlation have been developed to predict the power requiredDue to different flow pattern & mixing mechanisms involved, power consumption is considered separately low & high viscosity.

SCALE-UP OF STIRRED VESSELS.

Problem to satisfy the arrangement from experiments with small units to a large unit .

To attain the same kind of flow pattern in two units geometrical, kinematic, and dynamic similarity and identical boundary conditions must be maintained.

GEOMETRIC SIMILARITY OF A TANKGeometric similarity prevails between two systems of different sizes if all counterpart length dimensions have a constant ratio. Thus the following ratios must be the same in two systems:

Thus, for typical mixing tank

EXAMPLEA solution of sodium hydroxide of density 1650 kg/m3 and viscosity 50 mN s/m2 is agitated by a propeller mixer of 0.5 m diameter in a tank of 2.28 m diameter, and the liquid depth is 2.28 m. The propeller is situated 0.5 m above the bottom of the tank. Determine the geometric similarity of the particular tank.

SCALE-UP OF STIRRED VESSELSFor similarity in two mixing systems, it is important to achieve geometric kinematic and dynamic similarity.

Geometric similarity prevails between two systems of different sizes if all counterpart length dimensions have a constant ratio. Thus the following ratios must be the same in two systems: and so on.

Kinematic similarity exists in two geometrically similar units when the velocities at corresponding points have a constant ratio. Also, the paths of fluid motion (flow patterns) must be alike.

Dynamic similarity occurs in two geometrically similar units of different sizes if all corresponding forces at counterpart locations have a constant ratio.

POWER CONSUMPTION IN STIRRED VESSELSPower consumption is perhaps the most important parameter in the design of stirred vessels.Due to different flow patterns and mixing mechanisms involved, it is convenient to consider the power consumption in low and high viscosity systems separately.

LOW VISCOSITY SYSTEMSVertical cylindrical tankH/D = 1.5 2.0Fitted with an agitatorDiameter of propeller = 1/3 of tank diameterRotational speed at 10 25 Hz

LOW VISCOSITY SYSTEM - POWER CONSUMPTION

Where;P = power (J/s or W)Np = power number = density (kg/m3)N = rotational speed (rev/s)Da = diameter of agitator ( m)

LOW VISCOSITY SYSTEM - POWER NUMBER

Where:Reynold number (Re) = NDa2/Froude Number (Fr) = N2Da/g

Note: Fr is neglected if Re < 300

Depend upon type of impeller/vessel design, baffle arrangement, flow regime of the fluid, fluid properties

Simplest form is power law giving; Np = K Reb Frc K, b and c determined from experimental measurements

if Re < 300, region Fr has no significant effect on Np thus; Np = K Reb

if Re< 10, b = -1 ( inverse of Re value) Np = K Re-1 thus; P = KN2D3 Highly Re Number, P= KN3D5 K depends on the type impeller/vessel arrangement and fitted baffles.LOW VISCOSITY SYSTEM - POWER NUMBER

LOW VISCOSITY SYSTEM - POWER CORRELATION

LOW VISCOSITY SYSTEM - POWER CORRELATION (2)

POWER NO. AS A FUNCTION OF REYNOLD NO.if Re < 300, region Fr has no significant effect on Np thus; Np = K Reb

if Re< 10, b = -1 ( inverse of Re value) Np = K Re-1 thus; P = KN2D3

EXAMPLE 7.1On the assumption that the power required for mixing in a stirred tank is a function of the variables given in equation 7.12, obtain the dimensionless groups which are important in calculating power requirements for geometrically similar arrangements.

EXAMPLE 7.2A solution of sodium hydroxide of density 1650 kg/m3 and viscosity 50 mN s/m2 is agitated by a propeller mixer of 0.5 m diameter in a tank of 2.28 m diameter, and the liquid depth is 2.28 m. The propeller is situated 0.5 m above the bottom of the tank. What is the power which the propeller must impart to the liquid for a rotational speed of 2 Hz?

HIGH VISCOSITY SYSTEMNeed special designed impellers close clearances with vessel wallsInterest in processing industries exhibit non-Newtonian behaviour & Newtonian fluids (glycerol & lubricating oils)

HIGH VISCOSITY SYSTEM (2)For pseudoplastic liquids, average angular shear rate;

Limitation for a given geometry:

HIGH VISCOSITY SYSTEM- POWER CONSUMPTIONPrediction of Power Consumption for non-Newtonian fluids:Estimate the average shear rateEvaluate corresponding apparent viscosity flow curve / modelEstimate Re and then Np and hence P (curve in Figure 7.8)

TYPICAL POWER CONSUMPTION

SCALE UP TECHNIQUE-GENERAL GUIDE1. Constant tip speed- Where suspended solids are involved- Where heat is transferred to a coil or jacket- For miscible liquids

2. Constant power per unit volume- Immiscible liquids- Emulsions- Pastes- Gas liquid systems

EXAMPLE 7.3A reaction is to be carried out in an agitated vessel. Pilot scale tests have been carried out under fully turbulent conditions in a tank 0.6 m in diameter, fitted with baffles and provided with a flat-bladed turbine, and it has been found that satisfactory mixing is obtained at a rotor speed of 4 Hz when the power consumption is 0.15 kW and the Reynolds number 160,000. What should be the rotor speed in order to achieve the same degree of mixing if the linear scale of the equipment if increased by a factor of 6 and what will be the Reynolds number and the power consumption?