## Rheology in the design of process plants for the Food Industry

The purpose of **rheology** is to determine the “resistance that opposes a fluid to flow”. Its importance in the design of any processing plant is crucial since the sizing of many of the elements composing it is very dependent on this “resistance”, also taking special relevance with **food products**: **heat exchangers**, pipes, valves, pumps, mixers, etc.

With the aim to set an objective approach of how to evaluate this “resistance to flow” it is necessary to previously resort two concepts:

**Shear stress****Speed of deformation**

## Shear Stress

In a fluid that is in circulation and confined in a certain geometry, the velocity of each particle will be in general different from that of the rest of the particles and will depend on the distance from the particle with respect to the walls of the pipe. For example, in a **fluid** flowing with **laminar flow **inside a circular pipe, the velocity profile has a parabolic shape:

This difference between the velocity of a particle and the adjacent one is due to the internal force * F* (called shear force) that acts in the plane in which the particle is moving. The shear stress is then obtained by dividing this force between the surface

*on which it acts.*

**A**## Shear Rate

The * deformation ý* is understood as the change in size or shape experienced by a body subjected to a stress. Thus, the

*is defined as the variation of this deformation with respect to time.*

**shear rate ý**In the case of a **fluid**, the **shear rate** is calculated by dividing the velocity by a characteristic length of the geometry in which the fluid circulates. For example, in a pipe of circular section it is:

being * U* the average speed and

*the inside diameter.*

**D**In this way, the rheological behaviour of a fluid is characterized by the relationship that exists between the * shear rate, ý* and the

*necessary to produce it, disclosing the internal structural changes that the product suffers during processing.*

**shear stress, τ**With this in mind, a fluid can be classified as follows:

**Newtonian fluid.****Non-Newtonian fluid.****Viscoelastic fluid.**

### Newtonian Fluid

In a **Newtonian fluid**, the ration **τ=τ(ý)** is a ratio of proportionality, that is, in order to double the **shear rate**, it is necessary to double the **shear stress **as well.

The proportionality constant obtained from the previous graph corresponds to the **dynamic viscosity µ**.

### Non-Newtonian Fluid

**Non-Newtonian fluids** are those that do not satisfy the above-mentioned ratio of proportionality (known as Newton’s law). In general, food products are characterized by non-Newtonian behaviour, due to the complexity of their internal structure. Some typical examples are:

In view of the multiple **behaviours** that can be found in different **food products**, it is necessary to turn to the concept of **apparent viscosity**, thus developing useful engineering tools for design.

The apparent viscosity is then defined as the slope of the curve **τ=τ(ý)**, that is:

We can find in the specialized literature many mathematical models that describe the relations * τ=τ(ý)* and

*. Some examples are the Casson, Ellis, Herschel-Bulkley, Carreau, Cross or Powell-Eyring models. In general, these models fit properly only to a certain type of fluid and in a certain range of*

**η=η(ý)****shear rate**.

In practice, it is common to use the Ostwald-de Waele potential model, also called Power-Law, from which calculation tools are developed to determine the main design parameters of a food plant: pressure drop in pipelines, valves and other equipment, **heat transfer** in **heat exchangers**, power consumption in pumps and mixers, etc.

**Power-Law Model:**

being * K* the

**consistency index**and

*the*

**n****flow behaviour index**.

### Viscoelastic Fluid

When a **viscoelastic fluid** is deformed, it presents both viscous characteristics (inherent to a fluid) and elastic characteristics (inherent to a solid). This behaviour usually appears in fluids having a **macromolecular nature**, that is, with a high molecular weight. That is, it has an intermediate behaviour between an ideal newtonian fluid and an ideal solid.

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