Feed Throat Cooling Requirement

Reference ID: MET-745B | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The feed throat cooling calculation is a critical process engineering task in extrusion technology. The feed throat, located immediately below the hopper, serves as the transition zone where solid material enters the barrel. Maintaining the inner barrel wall temperature below the material's softening or sticking point is essential to prevent premature melting, which leads to bridging, inconsistent mass flow, and potential mechanical blockages.

This calculation is typically used during the design phase of extruder barrels or when validating cooling system capacity for new material formulations. By balancing the axial heat conduction from downstream processing zones against the convective cooling capacity of the jacket, engineers ensure stable operation and prevent thermal degradation of heat-sensitive materials.

Methodology & Formulas

The calculation follows a systematic approach to determine the required heat transfer area and verify the cooling system's effectiveness.

1. Mass Flow Rate of Coolant

\[ \dot{m} = \frac{\dot{Q}}{c_{p} \cdot (T_{\text{out}} - T_{\text{in}})} \]

2. Fluid Dynamics and Convection

The fluid velocity v and the Reynolds number Re are calculated to determine the flow regime:

\[ v = \frac{\dot{m}}{\rho \cdot A_{c}} \] \[ Re = \frac{\rho \cdot v \cdot D_{h}}{\mu} \]

The Nusselt number Nu is derived using the Dittus-Boelter correlation, which is then used to find the convective heat transfer coefficient h_{c}:

\[ Nu = 0.023 \cdot Re^{0.8} \cdot Pr^{0.4} \] \[ h_{c} = \frac{Nu \cdot k_{\text{water}}}{D_{h}} \]

3. Overall Heat Transfer Coefficient (U)

The overall coefficient accounts for convective resistance, conductive resistance through the barrel wall, and fouling resistance:

\[ \frac{1}{U} = \frac{1}{h_{c}} + \frac{t_{\text{wall}}}{k_{\text{steel}}} + R_{\text{fouling}} \]

4. Log Mean Temperature Difference (LMTD)

\[ \Delta T_{\text{lm}} = \frac{(T_{\text{wall}} - T_{\text{in}}) - (T_{\text{wall}} - T_{\text{out}})}{\ln \left( \frac{T_{\text{wall}} - T_{\text{in}}}{T_{\text{wall}} - T_{\text{out}}} \right)} \]

5. Area Requirement and Safety Factor

\[ A_{\text{req}} = \frac{\dot{Q}}{U \cdot \Delta T_{\text{lm}}} \] \[ SF = \frac{A_{\text{available}}}{A_{\text{req}}} \]

Parameter	Condition/Threshold	Engineering Significance
Reynolds Number	Re > 4000	Ensures turbulent flow for effective heat transfer.
Coolant Delta T	(T_out - T_in) ≤ 10°C	Prevents thermal shock and barrel distortion.
Fouling Factor	R_fouling ≈ 0.0001 - 0.0002	Accounts for long-term performance degradation.
Condensation	T_out > T_dewpoint	Prevents external moisture accumulation on the barrel.

Maintaining a controlled temperature at the feed throat is essential to prevent premature melting of the polymer pellets. If the feed throat becomes too hot, the following issues typically occur:

Bridging of material in the hopper due to surface softening.
Inconsistent mass flow rates caused by premature pellet deformation.
Increased torque loads on the screw drive motor.

Process engineers should monitor the coolant loop to ensure the heat transfer efficiency remains within design specifications. Key considerations include:

Maintaining a steady flow rate to prevent localized hot spots.
Ensuring the coolant temperature is set below the glass transition temperature of the specific resin being processed.
Verifying that the pressure differential across the cooling jacket is within the manufacturer's recommended range.

If you suspect a cooling system malfunction, perform the following diagnostic steps:

Check the coolant inlet and outlet temperatures to verify heat exchange is occurring.
Inspect the cooling jacket for mineral buildup or scale that may be restricting flow.
Verify that the solenoid valve or flow control valve is responding to the controller signal.
Check for air locks in the cooling lines that might be preventing full circulation.

Worked Example: Feed Throat Cooling Requirement

Scenario: A pilot-scale extruder with a 30 mm screw diameter and 200 mm feed throat length processes a heat-sensitive starch. The inner barrel wall must be kept at or below 60 °C to prevent premature melting and bridging. A water-cooled jacket is designed to remove a conduction-dominated heat load of 500 W.

Knowns

Heat load, \( \dot{Q} = 500 \, \text{W} \)
Coolant inlet temperature, \( T_{\text{in}} = 15.0 \, ^\circ\text{C} \)
Coolant outlet temperature, \( T_{\text{out}} = 20.0 \, ^\circ\text{C} \)
Target inner wall temperature, \( T_{\text{wall,target}} = 60.0 \, ^\circ\text{C} \)
Water specific heat capacity, \( c_{p} = 4180 \, \text{J/(kg} \cdot \text{K)} \)
Water density, \( \rho = 1000 \, \text{kg/m}^3 \)
Water dynamic viscosity, \( \mu = 0.001 \, \text{kg/(m} \cdot \text{s)} \)
Water Prandtl number, \( Pr = 7.0 \)
Water thermal conductivity, \( k_{w} = 0.6 \, \text{W/(m} \cdot \text{K)} \) (assumed constant)
Fouling resistance, \( R_{f} = 0.0001 \, \text{m}^2\!\cdot\!\text{K/W} \)
Hydraulic diameter of jacket channel, \( D_{h} = 0.005 \, \text{m} \)
Flow cross-sectional area, \( A_{c} = 2.5 \times 10^{-5} \, \text{m}^2 \)
Inner barrel diameter, \( D_{i} = 0.03 \, \text{m} \)
Jacket length, \( L_{j} = 0.2 \, \text{m} \)
Barrel wall thickness, \( t_{\text{wall}} = 0.01 \, \text{m} \)
Barrel thermal conductivity, \( k_{s} = 50 \, \text{W/(m} \cdot \text{K)} \)

Step-by-Step Calculation

Mass flow rate: \[ \dot{m} = \frac{\dot{Q}}{c_{p}\,(T_{\text{out}}-T_{\text{in}})} = \frac{500}{4180 \times 5} = 0.02392 \; \text{kg/s} \]
Coolant velocity: \[ v = \frac{\dot{m}}{\rho A_{c}} = \frac{0.02392}{1000 \times 2.5\times10^{-5}} = 0.9568 \; \text{m/s} \approx 0.957 \; \text{m/s} \]
Reynolds number: \[ Re = \frac{\rho v D_{h}}{\mu} = \frac{1000 \times 0.957 \times 0.005}{0.001} = 4785 \]
Nusselt number (Dittus-Boelter): \[ Nu = 0.023 \, Re^{0.8} \, Pr^{0.4} = 0.023 \,(4785)^{0.8} \,(7.0)^{0.4} = 44.0 \]
Convective heat transfer coefficient: \[ h_{c} = \frac{Nu\, k_{w}}{D_{h}} = \frac{44.0 \times 0.6}{0.005} = 5280 \; \text{W/(m}^2\!\cdot\!\text{K)} \]
Overall heat transfer coefficient: \[ \frac{1}{U} = \frac{1}{h_{c}} + \frac{t_{\text{wall}}}{k_{s}} + R_{f} = \frac{1}{5280} + \frac{0.01}{50} + 0.0001 \] \[ \frac{1}{U} = 0.0001894 + 0.0002 + 0.0001 = 0.0004894 \; \text{m}^2\!\cdot\!\text{K/W} \quad \Rightarrow \quad U = 2043 \; \text{W/(m}^2\!\cdot\!\text{K)} \]
Log-mean temperature difference (LMTD): \[ \Delta T_{\text{lm}} = \frac{(T_{\text{wall,target}}-T_{\text{in}})-(T_{\text{wall,target}}-T_{\text{out}})} {\ln\!\left(\frac{T_{\text{wall,target}}-T_{\text{in}}}{T_{\text{wall,target}}-T_{\text{out}}}\right)} \] \[ = \frac{(60-15)-(60-20)}{\ln\left(\frac{45}{40}\right)} = \frac{5}{0.117783} = 42.45 \; ^\circ\text{C} \]
Required heat transfer area: \[ A_{\text{req}} = \frac{\dot{Q}}{U\,\Delta T_{\text{lm}}} = \frac{500}{2043 \times 42.45} = 0.00577 \; \text{m}^2 \] (rounded to \(0.0058\;\text{m}^2\))
Available heat transfer area: \[ A_{\text{avail}} = \pi D_{i} L_{j} = \pi \times 0.03 \times 0.2 = 0.01885 \; \text{m}^2 \]
Safety factor: \[ SF = \frac{A_{\text{avail}}}{A_{\text{req}}} = \frac{0.01885}{0.00577} = 3.27 \]

Validation Checks

Reynolds number: \( Re = 4785 > 4000 \) → Dittus-Boelter correlation is valid.
Coolant temperature rise: \( \Delta T = 5 \,^\circ\text{C} < 10 \,^\circ\text{C} \) → no risk of thermal shock.
Target wall temperature: \( T_{\text{wall,target}} = 60 \,^\circ\text{C} < 70 \,^\circ\text{C} \) → safe margin for starch.

Final Answer: The available jacket area of \( 0.01885 \, \text{m}^2 \) exceeds the required area of \( 0.0058 \, \text{m}^2 \) by a factor of 3.27. This ensures the feed throat inner wall remains below the 60 °C limit under the specified heat load and coolant conditions.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle