Design of agitated vessels and jackets

Why it matters

Batch reactors, crystallisers, neutralisation tanks and blending vessels all depend on an agitator to mix, suspend solids, disperse gas and move heat to a jacket or coil. Sizing the impeller and motor, specifying the shaft and seal, and designing the jacket as a pressure part are everyday tasks in pharmaceutical, fine-chemical and polymer plants. Under-powered agitators give poor yields and hot spots; badly designed jackets buckle the inner shell.

Key ideas

Standard geometry. A widely used reference configuration has: liquid depth H equal to tank diameter D_T; impeller diameter D_a = D_T/3; impeller clearance above the bottom C = D_T/3; four wall baffles of width about D_T/10 to D_T/12. Baffles convert swirl into top-to-bottom circulation and stop vortexing; without them the liquid simply rotates with the impeller.

Impeller types.

• Radial-flow: flat-blade disc (Rushton) turbine — high shear, gas dispersion; power number about 5 in a baffled tank at high Re.
• Axial-flow: pitched-blade turbine (about 1.3), marine propeller and hydrofoils (about 0.3–0.4) — efficient bulk circulation and solids suspension per unit power.
• Close-clearance: anchors and helical ribbons for very viscous liquids (laminar regime), sweeping the wall to improve heat transfer. Power numbers depend on the exact geometry; read them from the N_P–Re curve for your impeller in a data book.

Power. Dimensional analysis gives the power number N_P = P/(ρ·N³·D_a⁵) as a function of the impeller Reynolds number Re = ρ·N·D_a²/μ (and the Froude number in unbaffled vessels with a vortex). In the laminar regime (Re below about 10) N_P = K_L/Re, so P = K_L·μ·N²·D_a³ and power does not depend on density. In the turbulent regime of a baffled tank (Re above about 10⁴) N_P is constant, so P = N_P·ρ·N³·D_a⁵: power rises as speed cubed and diameter to the fifth. The motor is sized above the shaft power to cover seal, gearbox and motor losses.

Scale-up. Geometric similarity is kept and one criterion is held constant: equal power per unit volume (common for dispersion and heat transfer, giving N ∝ D^(−2/3)), equal tip speed π·N·D_a (shear-sensitive systems, N ∝ D^(−1)), or equal blend time. No single rule keeps everything constant.

Mechanical design of the agitator. The shaft transmits torque T = P/(2π·N) and also carries bending from hydrodynamic side loads on the impeller; its diameter is sized for combined shear and bending with a service factor, and its first critical speed must be well away from the operating speed. Long overhung shafts may need a bottom steady bearing. The shaft passes through the vessel head via a stuffing box (gland packing) for low pressure or a mechanical seal for pressure, toxic or flammable service.

Jackets and coils.

• Conventional (plain) jacket: an outer shell around the vessel with an annular gap; simple, but the annulus velocity is low unless agitating nozzles or spiral baffles are fitted. The inner shell must be designed for jacket pressure as external pressure (a buckling check), and the jacket wall for internal pressure.
• Half-pipe coil jacket: half-pipes welded spirally on the shell; small diameter means thin walls can take high pressures, and the fluid velocity is high. Also stiffens the shell.
• Dimple jacket: a thin sheet dimpled and welded to the shell at intervals; light and cheap for moderate pressures.
• Internal coils: larger area and higher coefficients, but obstruct cleaning and add to wetted alloy.

Heat transfer. The vessel-side coefficient correlates as Nu = C·Re^(2/3)·Pr^(1/3)·(μ/μ_w)^0.14, with the constant C depending on impeller type, baffles and whether the surface is a jacket or a coil (take C from your data book). For a batch heated by an isothermal medium (condensing steam) the liquid temperature approaches the steam temperature exponentially.

Formulas

Re = ρ·N·D_a² / μ (impeller Reynolds number) N_P = P / (ρ·N³·D_a⁵) (power number) P = N_P·ρ·N³·D_a⁵ (turbulent, baffled, Re > about 10⁴) P = K_L·μ·N²·D_a³ (laminar, Re < about 10) N2 = N1·(D1/D2)^(2/3) (scale-up at constant P/V, geometric similarity) T = P / (2π·N) (shaft torque); d = (16·T / (π·τ))^(1/3) (shaft diameter, torsion only) t = (M·cp / (U·A))·ln((T_s − T_1) / (T_s − T_2)) (batch heating time, isothermal medium) t_j = P_j·D_j / (2·f·J − P_j) (jacket wall thickness, internal jacket pressure)

• ρ (kg/m³); N = speed (rev/s); D_a = impeller diameter (m); μ (Pa·s); P = power (W); K_L = laminar constant (–, from chart); τ = allowable shear stress of shaft (Pa); M = batch mass (kg); cp (J/kg·K); U (W/m²·K); A = wetted jacket area (m²); T_s = medium temperature; T_1, T_2 = initial and final batch temperatures; P_j = jacket design pressure (MPa); D_j = jacket inside diameter (mm).

Worked examples

Example 1 (standard): power and shaft for a Rushton turbine Given: D_T = 1.8 m (standard geometry, H = D_T), D_a = 0.6 m, N = 120 rpm = 2 rev/s, water-like liquid ρ = 1000 kg/m³, μ = 1 × 10⁻³ Pa·s; N_P = 5.0 (data-book curve, baffled); shaft allowable shear stress 40 MPa (given).

1. Re = 1000 × 2 × 0.6² / 10⁻³ = 7.2 × 10⁵ — fully turbulent, so N_P is constant.
2. P = 5.0 × 1000 × 2³ × 0.6⁵ = 5.0 × 1000 × 8 × 0.07776 = 3110 W (3.11 kW).
3. Volume = π/4 × 1.8² × 1.8 = 4.58 m³; P/V = 0.68 kW/m³.
4. Torque T = 3110 / (2π × 2) = 247.5 N·m.
5. Shaft (torsion only): d = (16 × 247.5 / (π × 40 × 10⁶))^(1/3) = 0.0316 m → about 32 mm; the final size is larger after adding bending, service factor and critical-speed checks. Motor rating above 3.1 kW to cover losses.

Example 2 (GATE level): batch heating time with a steam jacket Given: same vessel holding M = 4000 kg of aqueous solution (cp = 4.0 kJ/kg·K) to a depth of 1.6 m; heated from 25 °C to 85 °C by steam condensing at 120 °C in the jacket; U = 600 W/m²·K; jacket covers the wetted wall and the bottom (treat the bottom as flat).

1. Area: A = π × 1.8 × 1.6 + π/4 × 1.8² = 9.05 + 2.54 = 11.59 m².
2. M·cp/(U·A) = 4000 × 4000 / (600 × 11.59) = 2301 s.
3. ln((120 − 25)/(120 − 85)) = ln(95/35) = 0.998.
4. t = 2301 × 0.998 = 2297 s = 38.3 min. Note how the last few degrees take longest: heating to 110 °C instead would need ln(95/10) = 2.25, more than twice as long.

Common mistakes

• Using rpm instead of rev/s in the power and Reynolds equations.
• Applying the turbulent constant N_P in the laminar regime (or for an unbaffled tank).
• Forgetting the fifth power of D_a: a 10 % larger impeller needs about 61 % more power at the same speed.
• Designing the inner shell only for internal pressure and forgetting jacket pressure as external pressure.
• Using the arithmetic mean temperature difference for unsteady batch heating.
• Scaling up at constant speed, which grossly over-powers the large vessel.

For GATE CH

Expect numericals on Reynolds number, power from a given power number, scale-up at constant power per volume or tip speed, and batch heating or cooling time in a jacketed vessel; conceptual questions on baffles, impeller types and regimes. Practise unit conversions of speed and the logarithmic batch-heating equation.

Quick check

1. At constant N_P, how does power change if speed is doubled?
2. Why are baffles fitted in agitated tanks?
3. A 0.2 m impeller runs at 3 rev/s in water (ρ = 1000, μ = 0.001 Pa·s). What is Re?
4. Scale-up by 3 times in diameter at constant P/V from 120 rpm: new speed?

Answers: 1. It rises eightfold. 2. To prevent swirl and vortexing and create top-to-bottom circulation. 3. 1000 × 3 × 0.04 / 0.001 = 1.2 × 10⁵. 4. 120 × (1/3)^(2/3) ≈ 58 rpm.