How to calculate freezing time for a moist food by-product (75% water)

[Duales]

TL;DR Summary: I’m working on a university project to estimate the freezing time of brewer’s spent grain with 75% moisture. The material is filled into blocks (1 m² base, 30–100 cm height) and frozen from 35 °C in a –18 °C cold room. I want to calculate the time until the core reaches 0 °C and full freezing is complete. Is using Biot and Fourier numbers suitable for this, and how should the phase change be modeled? I also struggle to find reliable thermal properties,any tips for data sources and methods?

For a university project, I need to estimate how long it takes to freeze a food by-product with 75% moisture (brewer’s spent grain – fibrous, paste-like consistency). The material is filled into blocks with a 1 m² base area and different heights: 30 cm, 80 cm, and 100 cm. Initial temperature is 35 °C, and the blocks are placed in a cold storage room at –18 °C. The goal is to determine when the center of the block first reaches 0 °C, and then how long full freezing takes throughout the volume.

My questions are:

1. Is it appropriate to use Biot and Fourier number approaches for estimating the cooling time until 0 °C, or is there a better method for such high-moisture, heterogeneous food materials?
2. What is a suitable calculation method or formula for the actual freezing phase (the phase change from water to ice), especially in thick blocks?
3. How would you calculate this in general (with calculation way)?
4. Since the material is unusual, I’m not sure where to get accurate thermal properties like thermal conductivity, specific heat capacity, and density. Are there reliable sources or databases for materials like this, or for comparable slurries/pastes?

Any suggestions for formulas, literature sources, or practical approaches would be really helpful!

 

[erobz]

TL;DR Summary: I’m working on a university project to estimate the freezing time of brewer’s spent grain with 75% moisture. The material is filled into blocks (1 m² base, 30–100 cm height) and frozen from 35 °C in a –18 °C cold room. I want to calculate the time until the core reaches 0 °C and full freezing is complete. Is using Biot and Fourier numbers suitable for this, and how should the phase change be modeled? I also struggle to find reliable thermal properties,any tips for data sources and methods?

For a university project, I need to estimate how long it takes to freeze a food by-product with 75% moisture (brewer’s spent grain – fibrous, paste-like consistency). The material is filled into blocks with a 1 m² base area and different heights: 30 cm, 80 cm, and 100 cm. Initial temperature is 35 °C, and the blocks are placed in a cold storage room at –18 °C. The goal is to determine when the center of the block first reaches 0 °C, and then how long full freezing takes throughout the volume.

My questions are:

1. Is it appropriate to use Biot and Fourier number approaches for estimating the cooling time until 0 °C, or is there a better method for such high-moisture, heterogeneous food materials?
2. What is a suitable calculation method or formula for the actual freezing phase (the phase change from water to ice), especially in thick blocks?
3. How would you calculate this in general (with calculation way)?
4. Since the material is unusual, I’m not sure where to get accurate thermal properties like thermal conductivity, specific heat capacity, and density. Are there reliable sources or databases for materials like this, or for comparable slurries/pastes?

Any suggestions for formulas, literature sources, or practical approaches would be really helpful!

Before you work on your refined model, it would probably be a good idea to have a ballpark model to compare it against. Do you already have that?

My ideas for a ballpark time:
1) It's mostly water, so assume it is.
2) While cooling from its initial temp to freezing, assume uniform temperature distribution in the body - lumped capacitance method.
3) Assume it freezes in differential layers (via convection) across an ice wall of instantaneous thickness $x$ from the exterior to the center, I would just focus on the $1 \text{m}^2$ faces as the heat transfer area for this - ignore sidewalls.

Probably not great - but just thought I'd spitball some analytical ideas for you to chew on.

 

[haruspex]

Because of the shape, you can probably ignore heat transfer through the sides.
As @erobz notes, you can treat the material as just water, but without convection.
The trickiest part is transfer to the surrounding air. Can the air reach both large surfaces freely? It might freeze faster if you can stand the slabs on their edges so that the air convects freely up the larger faces.

I found https://www.researchgate.net/public...e_Foodstuffs_by_an_Improved_Analytical_Method, but it’s surely paywalled.

 

[Chestermiller]

You are dealing with a composite, composed of wheat grain and water. Hit the Composites literature to approximate the thermal conductivity of this composite.

 

[kuruman]

Is you project theoretical, experimental or both? Are you required to come up with an estimate or a calculation? You mention both, but they have different meanings.

I think that you should make some measurements first, see what Nature wants and then model it. In addition to @Chestermiller's recommendation, I suggest that you familiarize yourself with the attempts to pinpoint and investigate the parameters affecting the Mpemba effect. The experimental plot in the Wikipedia article is quite convincing that something is going on. A good amount of research has gone into it and it seems that the answer to the question "does warm water freeze faster than cold water?" is "It depends on a multidimensional array of parameters."

In other words, the freezing time of water seems to depend on numerous factors that, so far, are poorly understood. Unless one understands which factors are important and how they affect each other, one cannot make an accurate calculation of how fast a given mass of water will freeze. Thus, if an a priori calculation in the case of freezing just water is not easy, how easy can it be in the case of spent grain with 75% moisture?

That is why I suggest that you measure first and attempt to explain with a calculation later. Who knows? Maybe you will be able to demonstrate the Mpemba effect in spent grain.

 

[Duales]

Is you project theoretical, experimental or both? Are you required to come up with an estimate or a calculation? You mention both, but they have different meanings.

I think that you should make some measurements first, see what Nature wants and then model it. In addition to @Chestermiller's recommendation, I suggest that you familiarize yourself with the attempts to pinpoint and investigate the parameters affecting the Mpemba effect. The experimental plot in the Wikipedia article is quite convincing that something is going on. A good amount of research has gone into it and it seems that the answer to the question "does warm water freeze faster than cold water?" is "It depends on a multidimensional array of parameters."

In other words, the freezing time of water seems to depend on numerous factors that, so far, are poorly understood. Unless one understands which factors are important and how they affect each other, one cannot make an accurate calculation of how fast a given mass of water will freeze. Thus, if an a priori calculation in the case of freezing just water is not easy, how easy can it be in the case of spent grain with 75% moisture?

That is why I suggest that you measure first and attempt to explain with a calculation later. Who knows? Maybe you will be able to demonstrate the Mpemba effect in spent grain.

This is a theoretical model aimed at showing that freezing is likely not a feasible preservation method for this material. The calculations should be transparent and reproducible, but they can be based on several theoretical assumptions. If I can plausibly demonstrate that complete freezing would take several days, that will be sufficient for the purpose of this project. The topic is intended to be addressed in a short chapter, rather than as a core focus.

 

[Chestermiller]

Another way of estimating what you are looking for might be to bound the answer, aiming at the maximum time it would take for the center to get to 0 C. You would choose physical properties and external heat transfer coefficients that lead to the longest time. This would involve use to minimum estimated thermal conductivity of the composite, maximum estimated heat capacity and maximum estimated density.