Cooling Water Requirement for Kneading

Reference ID: MET-2B5E | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The calculation determines the cooling-water requirement for a dough-kneading jacket and evaluates the heat-transfer capability of a parallel-flow two-phase (air/silicone) tube. In process engineering these calculations are essential for:

Sizing cooling-water pumps and ensuring adequate temperature control of the dough.
Verifying that the jacket operates in the turbulent regime where the Dittus-Boelter correlation is valid.
Assessing the combined convective performance of a gas–liquid mixture inside heat-exchange tubes.

Typical applications include bakery process lines, continuous mixers, and any thermal-controlled kneading operation where the dough temperature must be raised or held within a narrow band while removing excess heat.

Methodology & Formulas

1. Dough heat load

The rate of heat that must be supplied to the dough is obtained from the sensible-heat balance:

\[ Q = \dot m_{d}\,C_{p,d}\,\Delta T_{d} \]

where \(\Delta T_{d}=T_{d,\text{out}}-T_{d,\text{in}}\).

2. Required cooling-water flow (single-phase)

The water mass flow that can absorb the dough heat load is:

\[ \dot m_{w}= \frac{Q}{C_{p,w}\,\Delta T_{w}} \]

with \(\Delta T_{w}=T_{w,\text{out}}-T_{w,\text{in}}\). The corresponding volumetric flow is:

\[ \dot V_{w}= \frac{\dot m_{w}}{\rho_{w}} \]

3. Jacket hydraulic and convective check (single-phase water)

Reynolds number for water flowing in the jacket:

\[ \text{Re}_{w}= \frac{\rho_{w}\,V_{w}\,D_{\text{jacket}}}{\mu_{w}} \]

Prandtl number for water:

\[ \text{Pr}_{w}= \frac{C_{p,w}\,\mu_{w}}{k_{w}} \]

Using the Dittus-Boelter correlation (valid for turbulent internal flow), the Nusselt number is:

\[ \text{Nu}_{w}=0.023\,\text{Re}_{w}^{0.8}\,\text{Pr}_{w}^{0.4} \]

and the convective heat-transfer coefficient becomes:

\[ h_{w}= \frac{\text{Nu}_{w}\,k_{w}}{D_{\text{jacket}}} \]

4. Two-phase (air/silicone) heat-transfer coefficients

Separate single-phase correlations are applied to the liquid (silicone) and gas (air) streams.

Liquid side

\[ \text{Re}_{l}= \frac{\rho_{l}\,V_{l}\,D_{\text{tube}}}{\mu_{l}},\qquad \text{Nu}_{l}=0.023\,\text{Re}_{l}^{0.8}\,\text{Pr}_{l}^{0.4},\qquad h_{l}= \frac{\text{Nu}_{l}\,k_{l}}{D_{\text{tube}}} \]

Gas side

\[ \text{Re}_{g}= \frac{\rho_{g}\,V_{g}\,D_{\text{tube}}}{\mu_{g}},\qquad \text{Nu}_{g}=0.023\,\text{Re}_{g}^{0.8}\,\text{Pr}_{g}^{0.4},\qquad h_{g}= \frac{\text{Nu}_{g}\,k_{g}}{D_{\text{tube}}} \]

The overall two-phase convective coefficient for parallel flow is obtained by harmonic averaging weighted by the void fraction \(\alpha\) (gas volume fraction):

\[ h_{\text{tp}}=\left[\,\frac{\alpha}{h_{g}}+\frac{1-\alpha}{h_{l}}\,\right]^{-1} \]

5. Validity checks (Dittus-Boelter applicability)

Parameter	Valid Range	Check Condition
\(\text{Re}_{w}\)	\(1\times10^{4}\) ≤ \(\text{Re}_{w}\) ≤ \(1\times10^{6}\)	\(\text{Re}_{w}\) within turbulent regime
\(\text{Pr}_{w}\)	\(0.7\) ≤ \(\text{Pr}_{w}\) ≤ \(160\)	Prandtl number suitable for correlation
\(\text{Re}_{l}\)	\(1\times10^{4}\) ≤ \(\text{Re}_{l}\) ≤ \(1\times10^{6}\)	Liquid side turbulent
\(\text{Pr}_{l}\)	\(0.7\) ≤ \(\text{Pr}_{l}\) ≤ \(160\)	Liquid Prandtl range
\(\text{Re}_{g}\)	\(1\times10^{4}\) ≤ \(\text{Re}_{g}\) ≤ \(1\times10^{6}\)	Gas side turbulent
\(\text{Pr}_{g}\)	\(0.7\) ≤ \(\text{Pr}_{g}\) ≤ \(160\)	Gas Prandtl range

6. Use of the overall coefficient

If an external convection coefficient \(h_{\text{ext}}\) and wall resistance \(R_{\text{wall}}\) are known, the overall tube heat-transfer coefficient \(U\) can be assembled as:

\[ \frac{1}{U}= \frac{1}{h_{\text{tp}}}+R_{\text{wall}}+\frac{1}{h_{\text{ext}}} \]

This reference sheet provides the algebraic framework required to size cooling-water systems for dough kneading and to evaluate the combined convective performance of a gas-liquid heat-exchange tube.

To size the cooling water system for a kneader, follow these steps:

Determine the total heat load (Q) that must be removed. This includes the heat generated by mechanical work and any exothermic reactions.
Choose the allowable temperature rise (ΔT) for the cooling water, typically 5–10 °C for most processes.
Use the formula Q = ṁ · Cp · ΔT, where ṁ is the mass flow rate (kg /h) and Cp is the specific heat capacity of water (≈4.18 kJ /kg·°C).
Rearrange to find ṁ: ṁ = Q / (Cp · ΔT).
Convert the mass flow rate to volumetric flow rate (L /h) by dividing by water density (≈1 kg/L).
Apply a safety factor of 1.1–1.2 to accommodate fouling and variations in operating conditions.

The acceptable temperature rise depends on the process sensitivity and water-side equipment limits. Generally:

For standard food-grade kneaders, a ΔT of 5–8 °C is typical.
If the product is temperature-sensitive, keep ΔT ≤ 5 °C to avoid hot spots.
When using a closed-loop chiller, a ΔT up to 10 °C may be permissible if the chiller can maintain the setpoint.

Always verify the chosen ΔT against the manufacturer’s recommendations for the specific kneader model.

Higher dough viscosity increases the mechanical work needed to achieve the same degree of mixing, which raises the internal heat generation. Consequently:

Viscosity ↑ → Mechanical power consumption ↑ → Heat load ↑ → Cooling water flow rate ↑.
For very high-viscosity formulations, consider a 10–20 % increase in water flow over the baseline calculation.
Monitoring torque or power draw can provide a real-time indicator to adjust cooling water flow dynamically.

Process engineers typically employ one or more of the following monitoring techniques:

Flow meters (magnetic or ultrasonic) installed on the water supply line to record real-time volumetric flow.
Temperature sensors at inlet and outlet of the heat exchanger to calculate actual heat removal.
Energy balance software that integrates power draw, water flow, and temperature data to flag deviations.
Periodic water balance audits to compare recorded usage against calculated requirements.

Integrating these measurements into the control system enables automatic adjustments and early detection of fouling or pump issues.

Worked Example: Cooling Water Requirement for Dough Kneader

A bakery is scaling up production of a high-moisture dough in a 100 L sigma-blade kneader. During the 8-minute mix cycle the mechanical energy raises the dough temperature from 30 °C to 70 °C. To protect yeast activity the operator must remove this heat with a jacket fed by 15 °C municipal water, allowing the water to leave the jacket no hotter than 25 °C. Determine the required cooling-water volumetric flow rate.

Knowns

Dough mass, \(m_d\) = 0.140 kg
Dough specific-heat capacity, \(C_{p,d}\) = 3.0 kJ kg⁻¹ °C⁻¹
Dough temperature rise, \(\Delta T_d\) = 40 °C
Water inlet temperature, \(T_{w,in}\) = 15 °C
Water outlet temperature, \(T_{w,out}\) = 25 °C
Water specific-heat capacity, \(C_{p,w}\) = 4.18 kJ kg⁻¹ °C⁻¹
Water density, \(\rho_w\) = 998 kg m⁻³

Step-by-step calculation

Calculate the heat that must be removed from the dough. \[Q = m_d\,C_{p,d}\,\Delta T_d = 0.140 \times 3.0 \times 40 = 16.8\ \text{kJ}\]
Determine the cooling-water mass flow rate required to absorb this heat. \[\dot m_w = \frac{Q}{C_{p,w}\,(T_{w,out}-T_{w,in})} = \frac{16.8}{4.18 \times 10} = 0.402\ \text{kg}\]
Convert the mass flow rate to volumetric flow rate. \[\dot V_w = \frac{\dot m_w}{\rho_w} = \frac{0.402}{998} = 4.03 \times 10^{-4}\ \text{m}^3\text{ s}^{-1}\]
Express the result in litres per minute, the unit commonly read from a rotameter. \[\dot V_w = 4.03 \times 10^{-4} \times 1000 \times 60 = 24.2\ \text{L min}^{-1}\]

Final Answer

The kneader jacket requires a continuous cooling-water flow of 24 L min⁻¹ at 15 °C to keep the dough below 70 °C during the mix cycle.

"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