CO2 Snow Freezing Calculation

Reference ID: MET-8E42 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

The CO₂ snow freezing calculation is a fundamental energy balance model used in process engineering to determine the refrigerant requirements for continuous belt or tunnel freezers. In these systems, solid carbon dioxide (CO₂ snow) is applied directly to the product surface. As the snow sublimates, it absorbs thermal energy from the product, facilitating rapid cooling and freezing. This method is critical in the food processing and pharmaceutical industries where high-throughput, rapid-freeze cycles are required to maintain product quality and structural integrity. This calculation ensures that the mass flow of CO₂ is sufficient to overcome the product's sensible and latent heat loads while accounting for system-specific thermal losses.

Methodology & Formulas

The calculation follows a steady-state energy balance approach. The total heat load of the product is determined by summing the sensible heat required to reach the freezing point, the latent heat of phase change, and the sensible heat required to reach the final target temperature. The refrigeration capacity is then derived from the enthalpy change of the CO₂ as it transitions from solid snow to exhaust gas.

The total heat load of the product is calculated as:

\[ \dot{Q}_{prod} = \dot{m}_{prod} \cdot \left[ c_{p,unf} \cdot (T_{in} - T_{freeze}) + x_{water} \cdot h_{fus} + c_{p,frz} \cdot (T_{freeze} - T_{out}) \right] \]

The effective refrigeration capacity of the CO₂ is defined by the sum of the latent heat of sublimation and the sensible heat gain of the resulting gas:

\[ \Delta h_{eff} = h_{sub} + c_{p,CO_2} \cdot (T_{exhaust} - T_{sub}) \]

Finally, the required mass flow rate of CO₂ is determined by dividing the total product heat load by the effective refrigeration capacity, adjusted by an empirical loss factor to account for system inefficiencies:

\[ \dot{m}_{CO_2} = \frac{\dot{Q}_{prod}}{\Delta h_{eff}} \cdot (1 + F_{loss}) \]

Parameter	Constraint/Range	Description
T_exhaust	-60.0°C to -40.0°C	Valid range for exhaust gas temperature.
F_loss	0.1 to 0.3	Empirical safety factor for system losses.
Driving Force	(T_out - T_exhaust) ≥ 10.0°C	Minimum temperature gradient for effective heat transfer.
c_p,CO₂	0.8 to 0.9 kJ/kg·K	Valid range for low-temperature CO₂ gas specific heat.

The calculation follows a steady-state energy balance methodology. Follow these steps:

Calculate the total product heat load by summing three components: (1) sensible heat to cool the unfrozen product from its inlet temperature to the freezing point, (2) latent heat of fusion for the water fraction, and (3) sensible heat to cool the frozen product from the freezing point to the target outlet temperature.
Determine the effective refrigeration capacity of the CO₂, which equals the latent heat of sublimation plus the sensible heat gained by the CO₂ gas as it warms from the sublimation temperature (−78.5°C) to the exhaust gas temperature.
Divide the product heat load by the effective refrigeration capacity to obtain the theoretical CO₂ mass flow rate.
Multiply by (1 + F_loss) to apply the empirical loss factor and obtain the required CO₂ mass flow rate for practical operation.

The loss factor F_loss (typically in the range 0.1 to 0.3) is an empirical safety factor that accounts for real-world system inefficiencies not captured by the ideal energy balance. These losses include:

Heat infiltration from the ambient environment into the freezing chamber.
Incomplete CO₂ sublimation before the gas exits the freezer, leaving unused cooling capacity.
Snow particle carryover in the exhaust gas stream.
Non-uniform distribution of CO₂ snow on the product surface, leading to localized under-freezing.

Applying this factor ensures the CO₂ supply is adequate to meet the required freezing duty under practical operating conditions with a suitable safety margin.

A minimum temperature driving force of 10°C (T_out − T_exhaust ≥ 10°C) is essential to maintain effective heat transfer between the product and the CO₂ gas throughout the freezer. Key considerations include:

The CO₂ snow sublimes at −78.5°C, and the resulting cold gas warms as it flows counter-current or cross-current to the product. The exhaust temperature represents the warmest point of the CO₂ stream.
If the exhaust temperature approaches the product outlet temperature, the thermal gradient near the freezer exit diminishes, slowing the final stage of cooling and potentially causing incomplete freezing.
Maintaining this minimum gradient ensures a positive driving force across the entire heat transfer zone, maximizing the utilization of the CO₂'s refrigeration capacity.

Worked Example: CO₂ Snow Freezing for a Belt Freezer

Scenario: A continuous belt freezer treats 1000 kg/hr of pure water, cooling it from 0°C water to -18°C ice. Solid CO₂ snow is the refrigerant. The exhaust gas temperature is estimated at -50°C, and a 15% loss factor accounts for inefficiencies. All parameters are within the empirical ranges defined by the calculation blueprint.

Knowns:

Product mass flow rate: \(\dot{m}_{prod} = 1000.0 \, \text{kg/hr}\)
Inlet temperature: \(T_{in} = 0.0 \, ^\circ\text{C}\)
Freezing temperature: \(T_{freeze} = 0.0 \, ^\circ\text{C}\)
Outlet temperature: \(T_{out} = -18.0 \, ^\circ\text{C}\)
Water mass fraction: \(x_{water} = 1.0\)
Specific heat of unfrozen product: \(c_{p,unf} = 4.18 \, \text{kJ/kg·K}\)
Specific heat of frozen product: \(c_{p,frz} = 2.04 \, \text{kJ/kg·K}\)
Latent heat of fusion of water: \(h_{fus} = 334.0 \, \text{kJ/kg}\)
Exhaust gas temperature: \(T_{exhaust} = -50.0 \, ^\circ\text{C}\)
Loss factor: \(F_{loss} = 0.15\)
CO₂ sublimation latent heat: \(h_{sub} = 571.0 \, \text{kJ/kg}\)
CO₂ sublimation temperature: \(T_{sub} = -78.5 \, ^\circ\text{C}\)
CO₂ gas specific heat: \(c_{p,CO_2} = 0.85 \, \text{kJ/kg·K}\)

Step-by-Step Calculation:

Product Heat Load (\(\dot{Q}_{prod}\)) \[ \begin{aligned} \dot{Q}_{sens,unf} &= \dot{m}_{prod} \cdot c_{p,unf} \cdot (T_{in} - T_{freeze}) = 1000.0 \cdot 4.18 \cdot (0.0 - 0.0) = 0.0 \, \text{kJ/hr} \\ \dot{Q}_{lat} &= \dot{m}_{prod} \cdot x_{water} \cdot h_{fus} = 1000.0 \cdot 1.0 \cdot 334.0 = 334000.0 \, \text{kJ/hr} \\ \dot{Q}_{sens,frz} &= \dot{m}_{prod} \cdot c_{p,frz} \cdot (T_{freeze} - T_{out}) = 1000.0 \cdot 2.04 \cdot (0.0 - (-18.0)) = 36720.0 \, \text{kJ/hr} \\ \dot{Q}_{prod} &= \dot{Q}_{sens,unf} + \dot{Q}_{lat} + \dot{Q}_{sens,frz} = 0.0 + 334000.0 + 36720.0 = 370720.0 \, \text{kJ/hr} \end{aligned} \]
Effective CO₂ Refrigeration (\(\Delta h_{eff}\)) \[ \Delta h_{eff} = h_{sub} + c_{p,CO_2} \cdot (T_{exhaust} - T_{sub}) = 571.0 + 0.85 \cdot (-50.0 - (-78.5)) = 571.0 + 0.85 \cdot 28.5 = 571.0 + 24.225 = 595.225 \, \text{kJ/kg} \]
Theoretical CO₂ Mass Flow Rate \[ \dot{m}_{CO_2,theo} = \frac{\dot{Q}_{prod}}{\Delta h_{eff}} = \frac{370720.0}{595.225} = 622.823 \, \text{kg/hr} \]
Required CO₂ Mass Flow Rate (with Loss Factor) \[ \dot{m}_{CO_2,req} = \dot{m}_{CO_2,theo} \cdot (1 + F_{loss}) = 622.823 \cdot (1 + 0.15) = 622.823 \cdot 1.15 = 716.247 \, \text{kg/hr} \]
Validity Check
- Exhaust temperature: \(T_{exhaust} = -50.0 \, ^\circ\text{C}\) is within the empirical range \([-60.0, -40.0] \, ^\circ\text{C}\).
- Driving force: \(T_{out} - T_{exhaust} = -18.0 - (-50.0) = 32.0 \, ^\circ\text{C} > 10.0 \, ^\circ\text{C}\), ensuring effective heat transfer.
- All input parameters satisfy the blueprint's constraints.

Final Answer: The required CO₂ snow mass flow rate is \(\dot{m}_{CO_2} = \boldsymbol{716.247 \, \text{kg/hr}}\). The corresponding CO₂-to-product mass ratio is approximately 0.716, consistent with efficient direct-contact snow freezers.

"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

CO2 Snow Freezing Calculation

Introduction & Context

🚀 Skip the Manual Math!

Methodology & Formulas

Worked Example: CO₂ Snow Freezing for a Belt Freezer