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S. Suzanne Nielsen (ed.)Food AnalysisFood Science Text Serieshttps://doi.org/10.1007/978-3-319-45776-5_29

29. Rheological Principles for Food Analysis

Helen S. Joyner (Melito)¹ and Christopher R. Daubert²

(1)

School of Food Science, University of Idaho, Moscow, ID 83843-2312, USA

(2)

Department of Food, Bioprocessing & Nutrition Sciences, North Carolina State University, Raleigh, NC 27695-7624, USA

Helen S. Joyner (Melito) (Corresponding author)

Email: hjoyner@uidaho.edu

Christopher R. Daubert

Email: cdaubert@ncsu.edu

Keywords

RheologyViscosityTexture

29.1 Introduction

29.1.1 Rheology and Texture

Food scientists are routinely confronted with the need to measure physical properties related to sensory texture and behavior during processing. These properties are determined by rheological methods; rheology is a science devoted to the deformation and flow of all materials. Rheological properties should be considered a subset of the textural properties of foods, because sensory evaluation of texture encompasses factors beyond rheological properties. Specifically, rheological methods accurately measure force, deformation, and flow, and food scientists and engineers must determine how best to apply this information. For example, the flow of salad dressing from a bottle, the snapping of a candy bar, or the pumping of cream through a homogenizer are each related to the rheological properties of these materials. In this chapter, we describe fundamental concepts pertinent to the understanding of food rheology and discuss typical examples of rheological tests for common foods. A glossary is included to define and summarize the rheological terms used throughout the chapter.

29.1.2 Fundamental and Empirical Methods

Rheological properties are determined by applying and measuring forces and deformations as a function of time. Both fundamental and empirical measurement methods may be used. Fundamental methods account for the magnitude and direction of forces and deformations while placing restrictions on acceptable sample shapes and composition. Fundamental tests have the advantage of being based on known concepts and equations of physics. Therefore, fundamental tests performed on different testing equipment with different sample geometries yield comparable results. When sample composition or geometry is too complex to account for all forces and deformations, empirical methods are often used. These empirical methods are typically descriptive in nature and ideal for rapid analysis. However, the results of empirical tests are dependent on the equipment and sample geometry, so it may be difficult to accurately compare data among different samples. Empirical tests are of value especially when they correlate with a property of interest, whereas fundamental tests determine true physical properties.

29.1.3 Basic Assumptions for Fundamental Rheological Methods

Two important assumptions for fundamental methodologies are that the material is homogeneous and isotropic. Homogeneity implies a well-mixed and compositionally similar material, an assumption generally valid for fluid foods provided they are not a suspension of large particles, such as vegetable soup. For example, milk, infant formula, and apple juice are each considered homogeneous and isotropic. Homogeneity is more problematic in solid foods. For example, frankfurters without skins can be considered homogeneous. However, when particle size is significant, such as fat particles in some processed meats like salami, one must determine if homogeneity is a valid assumption. Isotropic materials display a consistent response to a load regardless of the applied direction. In foods such as a steak, muscle fibers make the material anisotropic so the response varies with the direction of the force or deformation.

29.2 Fundamentals of Rheology

Rheology is concerned with how all materials respond to applied forces and deformations, and food rheology is the material science devoted to foods. Basic concepts of stress (force per area) and strain (relative deformation) are key to all rheological evaluations. Special constants of proportionality, called moduli, link stress with strain. Materials that are considered ideal solids (e.g., gelatin gels) obey Hooke’s Law, where stress is related directly with strain via a modulus. Materials that are considered ideal fluids (e.g., water, honey) obey Newtonian principles, and the proportionality constant is commonly referred to as viscosity, defined as an internal resistance to flow. These principles for solid and fluid behavior form the foundation for the entire chapter and are described in this section, with additional detail available in Steffe and Daubert [1].

29.2.1 Concepts of Stress

Stress (σ) is always a measurement of force. Defined as the force (F, Newtons) divided by the area (A, meters ²) over which the force is applied, stress is generally expressed with units of Pascals (Pa). To illustrate the notion of stress, imagine placing a water balloon on a table as opposed to placing it on the tip of a pin (Fig. 29.1). Obviously the tip of the pin has a considerably smaller surface area, causing the stress, or weight (force due to gravity) per unit area of contact, to be larger when compared with the stress of a balloon resting on a tabletop. Although the force magnitude – the weight of the water balloon – is a constant value for each case, the final outcomes will be very different.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig1_HTML.jpg — figure 29.1
Concepts of stress

The direction of the force, with respect to the surface area impacted, determines the type of stress. For example, if the force is directly perpendicular to a surface, a normal stress results and can be achieved under tension or compression. Should the force act in parallel to the sample surface, a shear stress is experienced. Examples of normal stresses include the everyday practices of pressing the edges of a piecrust together and biting through a solid food. Examples of shear forces, on the other hand, occur when spreading butter over a slice of toast, brushing barbecue sauce on chicken, or stirring milk into a cup of coffee.

29.2.2 Concepts of Strain and (Shear) Strain Rate

When a stress is applied to a food, the food deforms or flows. Strain is a dimensionless quantity representing the relative deformation of a material, and the direction of the applied stress with respect to the material surface will determine the type of strain. If the stress is normal (perpendicular) to a sample surface, the material will experience normal strain (ε). Foods show normal strains when they are compressed (compressive stress) or stretched (tensile stress).

Normal strain (ε) can be calculated as a true strain from an integration over the deformed length of the material (Fig. 29.2):

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig2_HTML.png — figure 29.2
Normal strain in a cylinder in (a) compression and (b) tension

$\varepsilon =\underset{L_i}{\overset{L_i+\varDelta L}{{\displaystyle \int }}}\frac{dL}{L}= \ln \left(1+\frac{\varDelta L}{L}\right)$

(29.1)

According to Steffe [2], a true strain is more applicable to larger deformations such as may occur during texture profile analysis testing (see Sect. 29.4.1.1.2). Strain calculations result in negative valves for compression and positive values for extension (tensile strains). Rather than expressing a negative strain, many typically record the absolute value of the strain and denote the compressive test mode:

$\varepsilon =-0.05={0.05}_{\mathrm{compresion}}$

(29.2)

On the other hand, when a sample encounters a shear stress, such as the pumping of tomato paste through a pipe, a shear strain (γ) is observed. Figure 29.3 shows how a sample deforms when a shear stress is applied. Shear strain is determined from applications of geometry as:

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig3_HTML.png — figure 29.3
Shear strain in a cube

$\tan \left(\gamma \right)=\frac{\varDelta L}{h}$

(29.3)

or:

$\gamma ={ \tan}^{-1}\left(\frac{\varDelta L}{h}\right)$

(29.4)

where h is the specimen height. For simplification, during exposure to small strains, the angle of shear may be considered equal to the shear strain:

$\tan \left(\gamma \right)\approx \gamma$

(29.5)

When the material is a liquid, this approach for strain quantification is a bit more challenging. As coffee is stirred, water is pumped, or milk is pasteurized, these fluids all are exposed to shear and display irrecoverable deformation. Therefore, a (shear) strain rate $\left(\dot{\gamma}\right)$ , often called shear rate, is typically used to quantify strain during fluid flow. Shear rate is the amount of deformation (strain) per unit time with units of s⁻¹:

$\gamma =\frac{\varDelta L}{h}$

(29.6)

$\frac{d\gamma}{ d t}=\frac{d\left(\frac{\varDelta L}{h}\right)}{ d t}=\dot{\gamma}$

(29.7)

$\dot{\gamma}=\frac{1}{h}\frac{d}{ d t}\left(\varDelta L\right)$

(29.8)

$\dot{\gamma}=\frac{U}{h}$

(29.9)

To picture the shear rate concept, consider a fluid filling the gap between two moveable, parallel plates separated by a known distance, h, as illustrated in Fig. 29.4. Now, set one plate in motion with respect to the other at a constant horizontal velocity, U.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig4_HTML.png — figure 29.4
Shear flow between parallel plates

The shear rate for this system can be approximated by dividing the plate velocity by the fluid gap height, producing a value with units of s⁻¹. This shear rate may be more easily understood through the deck of cards analogy. Imagine a stack of playing cards, with each card representing an infinitely thin layer of fluid. When the top card is pushed with some force, the entire deck deforms to some degree proportional with the magnitude of the force. This type of movement is commonly called simple shear and may be defined as a laminar deformation along a plane parallel to the applied force.

29.2.3 Solids: Elastic and Shear Moduli

Hooke’s Law states that when a solid material is exposed to a stress, it experiences an amount of deformation or strain proportional to the magnitude of the stress. The constants of proportionality, used to equate stress with strain, are called moduli:

$\mathrm{Stress}\left(\sigma \right)\propto \mathrm{Strain}\left(\varepsilon \kern0.5em \mathrm{or}\kern0.5em \gamma \right)$

(29.10)

$\mathrm{Stress}=\mathrm{Modulus}\times \kern0.5em \mathrm{Strain}$

(29.11)

If a normal stress is applied to a sample, the proportionality constant is known as elastic modulus (E), often called Young’s modulus:

$\sigma =\frac{F}{A}= E\varepsilon$

(29.12)

Likewise, if the applied stress is shear stress, the constant is the shear modulus (G):

$\sigma = G\gamma$

(29.13)

These moduli are inherent properties of the material and have been used as indicators of quality. Moduli of selected foods and materials are provided in Table 29.1.

table 29.1

Elastic and shear moduli for common materials

Material	E, elastic moduli (Pa)	G, shear moduli (Pa)
Apple	1.0 × 10⁷	0.38 × 10⁷
Potato	1.0 × 10⁷	0.33 × 10⁷
Spaghetti, dry	0.27 × 10¹⁰	0.11 × 10¹⁰
Glass	7.0 × 10¹⁰	2.0 × 10¹⁰
Steel	25.0 × 10¹⁰	8.0 × 10¹⁰

29.2.4 Fluid Viscosity

For the case of the simplest kind of fluid, the viscosity is constant and independent of shear rate and time. In other words, Newton’s postulate is obeyed. This postulate states that if the shear stress is doubled, the velocity gradient (shear rate) within the fluid is also doubled. Typically for fluids, the shear stress is expressed as some function of shear rate and a viscosity term that provides an indication of the internal resistance of a fluid to flow. For Newtonian fluids, the viscosity function is constant and called the coefficient of viscosity or Newtonian viscosity (μ):

$\sigma =\mu \dot{\gamma}$

(29.14)

However, for most liquids the viscosity term is not constant, but rather changes as a function of shear rate, and the material is considered non-Newtonian. A function called the apparent viscosity (η) is defined as the shear-dependent viscosity. Mathematically, the apparent viscosity function is the result of the shear stress divided by the shear rate:

$\eta = f\left(\dot{\gamma}\right)=\frac{\sigma}{\dot{\gamma}}$

(29.15)

Table 29.2 provides Newtonian viscosities for some common items at 20 °C [2, 3]. Temperature is a very important parameter when describing rheological properties. Viscosity generally decreases as temperature increases.

table 29.2

Newtonian viscosities for common materials at 20 °C

Material	Viscosity, μ (Pa s)
Honey	11.0
Rapeseed oil	0.163
Olive oil	8.4 × 10⁻²
Cottonseed oil	7.0 × 10⁻²
Raw milk	2.0 × 10⁻³
Water	1.0 × 10⁻³
Air	1.81 × 10⁻⁵

Adapted from [2, 3]

29.2.5 Fluid Rheograms

The flow behavior of a food can be described with fluid rheograms or appropriate rheological models (see Sect. 29.3). A rheogram is a graphical representation of flow behavior, showing the relationship between stress and strain or shear rate. Much can be learned from inspection of rheograms. For example, if a plot of shear stress versus shear rate results in a straight line passing through the origin, the material is a Newtonian fluid, with the slope of the line equaling the Newtonian viscosity (μ), shown in Fig. 29.5. Many common foods exhibit this ideal response, including water, milk, vegetable oils, and honey.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig5_HTML.png — figure 29.5
Rheogram for three different Newtonian fluids

As established earlier, the majority of fluid foods do not show Newtonian flow behavior. The flow changes with shear rate (i.e., mixing speed) or with time at a constant shear rate. Time-independent deviation from ideal Newtonian behavior will cause the relationship between shear stress and shear rate to be nonlinear. If the viscosity decreases as shear rate increases, the material is referred to as shear thinning or pseudoplastic. Examples of pseudoplastic food items are applesauce and pie fillings. On the other hand, if the viscosity of the material increases with increased shear rate, the sample is called shear thickening or dilatent. Cornstarch slurries are well known for dilatent behavior.

Pseudoplasticity and dilatency are time-independent properties. Materials that thin and thicken with time are known as thixotropic and rheopectic liquids, respectively. These fluids are easily detected by monitoring the viscosity at a constant shear rate with respect to time. If pumpkin pie filling is mixed at a constant speed, the material thins (thixotropy) with time due to the destruction of weak bonds linking the molecules. Table 29.3 summarizes the terminology for time-dependent and independent responses.

table 29.3

Summary of shear-dependent terminology

	Time independent	Time dependent
Thinning	Pseudoplastic	Thixotropic
Thickening	Dilatent	Rheopectic (anti-thixotropic)

Many fluids do not flow at low magnitudes of stresses. In fact, a certain catsup brand once staked numerous marketing claims on the “anticipation” of flow from the bottle. Often, additional force was applied to the catsup container to expedite the pouring. The minimum force, or stress, required to initiate flow is known as a yield stress (σ _o). Because Newtonian fluids require the stress-shear rate relationship to be a continuous straight line passing through the origin, any material with a yield stress is automatically non-Newtonian. A few common foods possessing yield stresses are catsup, yogurt, mayonnaise, and salad dressing.

Many foods are explicitly designed to include a certain yield stress. For example, if melted cheese did not have a yield stress, the cheese would flow off a cheeseburger or pizza. If salad dressings flowed at the lowest of applied stresses, the force of gravity would cause the dressing to run off the salad leaves. There are many fascinating rheological features of foods that consumers may never consider!

29.3 Rheological Fluid Models

Once data for shear stress and shear rate are collected, rheological models can be used to gain a greater understanding of the flow response. Rheological models are mathematical expressions relating shear stress to shear rate, providing a “flow fingerprint” for a particular food. In addition, the models permit prediction of rheological behavior across a wide range of processing conditions. Figures 29.6 and 29.7 show typical rheograms of stress versus shear rate and viscosity versus shear rate for several rheological models.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig6_HTML.png — figure 29.6
Typical stress versus shear rate behavior for various flow models in (a) normal and (b) log scale

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig7_HTML.png — figure 29.7
Typical viscosity versus shear rate behavior for various flow models in (a) normal and (b) log scale

29.3.1 Herschel-Bulkley Model

For most practical purposes, the Herschel-Bulkley model can account for the steady-state rheological performance of many fluid foods:

$\sigma ={\sigma}_o+ K{\dot{\gamma}}^n$

(29.16)

K and n represent material constants called the consistency coefficient and flow behavior index, respectively. The flow behavior index provides an indication of Newtonian or non-Newtonian flow, provided the material has no yield stress. Table 29.4 illustrates how the Herschel-Bulkley model is used to identify specific flow characteristics.

table 29.4

Manipulation of Herschel-Bulkley model to describe flow behavior

Fluid type	σ _o	n
Newtonian	0	1.0
Non-Newtonian
Pseudoplastic	0	<1.0
Dilatent	0	>1.0
Yield stress	>0	Any

σ _o yield stress; n flow behavior index

Table 29.5 displays Herschel-Bulkley model data obtained for a variety of food products. Take, for example, the data for peanut oil. Table 29.5 reports no yield stress and a flow behavior index of 1.00 – the scenario for a Newtonian fluid. Accordingly, the Herschel-Bulkley model does in fact collapse to the special Newtonian case. The following models are considered as simple modifications to the Herschel-Bulkley model [2].

table 29.5

Herschel-Bulkley model data for common foods

Product	Temp. (°C)	Shear rate (s⁻¹)	K (Pa sⁿ)	n (−)	σ _o (Pa s)
Orange juice (13 °Brix)	30	100–600	3.2	0.79	–
Orange juice (13 °Brix)	30	100–600	0.06	0.86	–
Orange juice (22 °Brix)	30	100–600	3.2	0.79	–
Orange juice (22 °Brix)	30	100–600	0.12	0.79	–
Orange juice (33 °Brix)	30	100–600	0.14	0.78	–
Apple juice (35 °Brix)	25	3–2,000	0.001	1.00	–
Ketchup	25	50–2,000	6.1	0.40	–
Apple sauce	32	–	200	0.42	240
Mustard	25	–	3.4	0.56	20
Melted chocolate	46	–	0.57	1.16	1.16
Peanut oil	21.1	0.32–64	0.065	1.00	–

Adapted from [2, 4]

29.3.2 Newtonian Model [n = 1; K = μ; σ _o = 0]

For Newtonian fluids, Eq. 29.17 is manipulated with the flow behavior index (n) equaling 1.0 and the consistency coefficient (K) equaling the Newtonian viscosity (μ):

$\sigma =0+\mu {\dot{\gamma}}^1$

(29.17)

$\sigma =\mu \dot{\gamma}$

(29.18)

29.3.3 Power Law Model [σ _o = 0]

Power law fluids show no yield stress (σ _o) and a nonlinear relationship between shear stress and shear rate. Pseudoplastic and dilatent fluids may be considered power law fluids, each with different ranges for flow behavior index values; refer to Table 29.4:

$\sigma =0+ K{\dot{\gamma}}^n$

(29.19)

$\sigma = K{\dot{\gamma}}^n$

(29.20)

29.3.4 Bingham Plastic Model [n = 1; K = μ _pl]

Bingham Plastic materials have a distinguishing feature: a yield stress is present. Once flow is established, the relationship between shear stress and shear rate is linear, explaining why n = 1.0 and K is a constant value known as the plastic viscosity, μ _pl. Caution: The plastic value is not the same as the apparent viscosity (η) or the Newtonian viscosity (μ)!:

$\sigma ={\sigma}_o+{\mu}_{\mathrm{pl}}{\dot{\gamma}}^1$

(29.21)

$\sigma ={\sigma}_o+{\mu}_{\mathrm{pl}}\dot{\gamma}$

(29.22)

29.4 Rheometry

Rheometers are devices used to determine viscosity and other rheological properties of materials. Relationships between shear stress and shear rate are derived from physical values of system configurations, pressures, flow rates, and other applied conditions.

29.4.1 Compression, Extension, and Torsion Analysis

The rheological properties of solid foods are measured by compressing, extending, or twisting the material and can be accomplished by two general approaches called small- or large-strain testing. In small-strain tests, the goal is to apply the minimal amount of strain or stress required to measure the rheological behavior while at the same time preventing (or at least minimizing) damage to the sample. The goal in large-strain and fracture tests is the opposite. Samples are deformed to an extent at which the food matrix is significantly strained, damaged, or possibly fractured. Small-strain tests are used to understand properties of a food network, whereas large-strain tests give an indication of sensory texture or product durability. In general, compressive and extensional tests are generally performed using large strains. Torsional (shear) tests may be performed in using small or large strains.

29.4.1.1 Large-Strain Testing

Compression and tension (i.e., extension) tests are used to determine large-strain and fracture food properties. Compression tests are generally selected for solid or viscoelastic solid foods when a tight attachment between sample and the testing fixture is not required, thereby simplifying sample preparation. Tension and torsion tests are well suited for highly deformable foods when a high level of strain is needed to fracture the sample. The main disadvantage to tension and torsion tests is that the sample must be attached to the test fixture [5]. Hamann et al. [6] provides a detailed comparison and analysis of large deformation rheological testing.

29.4.1.1.1 Determining Stress, Strain, and Elastic Modulus (E) in Compression

There are several assumptions to consider when doing compression testing. Along with the previously mentioned considerations of a homogeneous and isotropic material, the assumption that the food is an incompressible material greatly simplifies matters. An incompressible material is one that changes in shape but not volume when compressed. Foods such as frankfurters, cheese, cooked egg white, and other high-moisture, gel-like foods generally are considered incompressible. The calculations for strain are as discussed previously (Eq. 29.1).

During compression, the initial cross-sectional area (A _i) increases as the length decreases. To account for this change, a correction term incorporating a ratio of the cylinder lengths (L/L _i) is applied to the stress calculation:

$\sigma =\frac{F}{A_i}\left(\frac{L}{L_i}\right)$

(29.23)

In compression testing one should use a cylindrical shaped sample with a length (L) to diameter ratio of >1.0. The sample should be compressed between two flat plates with diameters exceeding the lateral expansion of the compressed sample (i.e., the entire sample cross-section should be in contact with the plates during testing). The equations are based on the sample maintaining a cylindrical shape when compressed. If this is not the case, the contact surface between the plate and the sample may need lubrication. Water or oil can be used, and one should pick the fluid that provides the desired lubrication without causing any deleterious effects to the sample.

A cylinder of Cheddar cheese 3 cm in length (L _i), with an initial radius (R _i) of 1 cm, was compressed at a constant rate to 1.8 cm (L) and recorded a force of 15 N. Then:

$\varepsilon = \ln \kern0.5em \left[1+\left(-0.4\right)\right]=-0.5={0.5}_{\mathrm{compresion}}$

(29.24)

${A}_i=\pi {R}_i^2=0.000314\ {\mathrm{m}}^2$

(29.25)

$\begin{array}{l}\sigma =\frac{F}{A_i}\left(\frac{L}{L_i}\right)=\frac{15\ N}{0.000314\ {\mathrm{m}}^2}\times \left(\frac{1.8}{3.0}\right)=28,700\ {\mathrm{Pa}}_{\mathrm{compresion}}\\ {}=28.7\kern0.5em {\mathrm{kPa}}_{\mathrm{compresion}}\end{array}$

(29.26)

$E=\frac{\sigma}{\varepsilon}=\frac{28,700\ \mathrm{kPa}}{0.5}=57.4\ \mathrm{kPa}$

(29.27)

If the material compressed is a pure elastic solid, the compression rate does not matter. However, if the material is viscoelastic (as is the case with most foods), then the values for stress, strain, and elastic modulus may change with the speed of compression. A complete characterization of a viscoelastic material requires determining these values at a variety of compression rates. Another factor to consider is the level of compression. The sample can be compressed to fracture, or some level below fracture. The goal should be compression to fracture if correlating rheological with sensory properties.

29.4.1.1.2 Texture Profile Analysis

Texture profile analysis (TPA) is an empirical technique using a two-cycle compression test. It is typically performed using a Universal Testing Machine (Fig. 29.8a) or a Texture Analyzer (Fig. 29.8b). This test was developed by a group of food scientists from the General Foods Corporation and is compiled as force during compression and time. Data analyses correlated numerous sensory parameters, including hardness, cohesiveness, and springiness, with texture terms determined from the TPA test curve. For example, the peak force required to fracture a specimen has been strongly related to sample hardness. Bourne [7] provides a more detailed description of TPA.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig8_HTML.png — figure 29.8
Photos of (a) Universal Testing Machine (Courtesy of Instron®, Norwood, MA) and (b) Texture Analyzer (Courtesy of Texture Technologies Corp., Hamilton, MA)

29.4.1.2 Fracture Testing

Fracture tests are large-strain tests carried out to the point of sample failure. Generally, stress and strain at the failure point are determined by the sudden decrease in stress as the sample fractures. Fracture tests may be performed in compressive, tensile, or shear (torsion) modes. Compressive or shear modes are usually used for food materials, since difficulties in gripping the sample without damaging it make tensile fracture testing difficult.

Several considerations are needed when performing fracture testing. The geometry of the sample must be controlled, especially in fundamental fracture testing. Certain fracture methodologies specify the shape of the sample, for example, a cylinder, capstan, or beam shape. The homogeneity of the sample must also be considered. Anisotropic samples such as meats may have different fracture behaviors depending on their orientation. Testing parameters such as strain rate can affect fracture behavior as well. Hamann et al. [6] provides a more detailed discussion of fracture testing.

29.4.2 Rotational Viscometry

For fluids, the primary mode for rheological measurement in the food industry is rotational viscometry, providing rapid and fundamental information. Rotational viscometry involves a known test fixture (geometrical shape) in contact with a sample, and through some mechanical, rotational means, the fluid is sheared by the fixture. Primary assumptions are made for the development of constitutive equations (relationships between shear stress and rate) and include the following:

Laminar flow. Laminar flow is synonymous with streamline flow. In other words, if we were to track velocity and position of a fluid particle through a horizontal pipe, the path would only be in the horizontal direction without a shift toward the pipe wall.
Steady state. There are no net changes to the system over time.
No-slip boundary condition. When the test fixture is immersed in the fluid sample, the walls of the fixture and the sample container serve as boundaries for the fluid. This condition assumes that, at whatever speed either boundary is moving, an infinitely thin layer of fluid immediately adjacent to the boundary is moving at precisely the same velocity.

Rotational rheometers may operate in two modes: steady shear or oscillatory. The next few sections consider steady shear rotational viscometry. Steady shear is a condition in which the sheared fluid velocity, contained between the boundaries, remains constant at any single position. Furthermore, the velocity gradient across the fluid is a constant. Three test fixtures most often used in steady shear rotational viscometry are the concentric cylinder, cone and plate, and parallel plates.

29.4.2.1 Concentric Cylinders

This rheological attachment consists of a cylindrical fixture shape, commonly called a bob with radius R _b, suspended from a measuring device that is immersed in a sample fluid contained in a slightly larger cylinder, referred to as the cup with radius R _c (see Figs. 29.9 and 29.10). Torque (M) is an action that generates rotation about an axis and is the product of a force and the perpendicular distance (r), called the moment arm, to the axis of rotation. The principles involved can be described relative to changing tires on a car. To loosen the lugnuts, a larger tire iron is often required. Essentially, this longer tool increases the moment arm, resulting in a greater torque about the lugnut. Even though you are still applying the same force on the iron, the longer device provides greater torque!

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig9_HTML.jpg — figure 29.9
Concentric cylinder geometry

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig10_HTML.png — figure 29.10
Photo of cup and bob test fixtures (concentric cylinder geometry) for rheological measurements (Courtesy of TA Instruments, New Castle, DE)

To derive rheological data from experiments, equations for shear stress and shear rate are used. Shear stress at the surface of the bob (σ _b) may be calculated from a force balance as:

${\sigma}_b=\frac{M}{2\pi {R}_b^2}$

(29.28)

Therefore, to determine shear stress, all we need to know is the bob geometry (h and R _b) and the torque response (M) of the fluid on the measuring sensor.

A simple shear approximation commonly calculates a shear rate at the bob surface and assumes a constant shear rate across the fluid gap. This approximation is valid for small gap widths where R _c/R _b ≤ 1.1:

$\dot{\gamma}=\frac{\varOmega {R}_b}{R_c-{R}_b}$

(29.29)

This calculation requires the rotational speed, or angular velocity (Ω), of the bob, typically expressed in radians per second. Converting units of revolutions per minute (rpm) to radians per second is simply achieved by multiplying by 2π/60, and the following example converts 10 rpm to radians per second:

$\begin{array}{l}\left(\frac{10\kern0.5em \mathrm{revolution}\mathrm{s}}{1\kern0.5em\min}\right)\left(\frac{1\kern0.5em\min }{60\kern0.5em \mathrm{s}}\right)\left(\frac{2\pi \kern0.5em \mathrm{radians}}{1\kern0.5em \mathrm{revolution}}\right)\\ {}=\frac{1.047\kern0.5em \mathrm{radians}}{\mathrm{second}}\end{array}$

(29.30)

The food industry uses several rheological devices to measure viscosity. These devices include the Brookfield viscometer (Fig. 29.11), the Bostwick consistometer (Fig. 29.12), and Zahn cup (Fig. 29.13). The Brookfield viscometer is one of the most common rheological devices found in the food industry. This apparatus uses a spring as a torque sensor. The operator selects a rotational speed (rpm) of the bob, attached to the spring. As the bob moves through the sample fluid, the viscosity impedes free rotation, causing the spring to wind. The degree of spring windup is a direct reflection of the torque magnitude (M), used to determine a shear stress at the bob surface. Newer Brookfield models convert torque to viscosity automatically using Eq. 29.28, while older models provide conversion factors for calculating viscosity from torque.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig11_HTML.jpg — figure 29.11
Photo of Brookfield viscometer (Courtesy of Brookfield AMETEK, Middleboro, MA)

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig12_HTML.jpg — figure 29.12
Photo of Bostwick consistometer (Courtesy of Cole-Parmer, Vernon Hills, IL)

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig13_HTML.jpg — figure 29.13
Photo of Zahn cup (Courtesy of Paul N. Gardner Co., Inc., Pompano Beach, FL)

The Bostwick consistometer uses a given amount of sample poured into one end of the instrument behind a gate. A timer is started as the gate is quickly lifted and the time to reach a certain mark on the ramp is measured. The distance the fluid travels is called the consistency. Note that consistency and viscosity are not interchangeable.

The Zahn cup also used a set amount of sample. The sample is put into the cup and allowed to drain from a hole in the bottom of the cup. The time from start of flow to the first break in the stream of liquid is recorded and can be converted to viscosity using conversion factors and the specific gravity of the fluid.

The Brookfield viscometer, the Bostwick consistometer, and the Zahn cup are suitable for quick quality control measurements, as they provide rapid measurements and are easy to use and clean. However, these are empirical instruments and are not as precise as a rheometer. Furthermore, they are generally used to measure viscosity at a single shear rate, which can lead to incorrect assumptions about flow behavior. For example, what might happen if the Newtonian and the n < 1 Herschel-Bulkley fluid in Fig. 29.7 were tested only at the shear rate at which their viscosities are equivalent? It is recommended that these rheological devices be used with fluids that are Newtonian or close to Newtonian to remove shear rate effects from the measurements.

Following progression through a series of rotational speeds, a rheogram can be created showing shear stress (σ) versus shear rate $\left(\dot{\gamma}\right)$ . The importance of rheograms has been discussed, with a primary significance being apparent viscosity determination (Eq. 29.15). The following tomato catsup data in Table 29.6 were collected with a standard cup and bob system (R _c = 21 mm, R _b = 20 mm, and h = 60 mm). Using Eqs. 29.15, 29.28, and 29.29 you should verify the results.

table 29.6

Rheological data of tomato catsup collected using a concentric cylinder test fixture

rpm	Torque (N m)	Shear rate (s⁻¹)	Shear stress (Pa)	Apparent viscosity (Pa s)
1.0	0.00346	2.09	22.94	10.98
2.0	0.00398	4.19	26.39	6.30
4.0	0.00484	8.38	32.10	3.83
8.0	0.00606	16.76	40.18	2.40
16.0	0.00709	33.51	47.02	1.40
32.0	0.00848	67.02	56.23	0.84
64.0	0.01060	134.04	70.29	0.52
128.0	0.01460	268.08	96.82	0.36
256.0	0.01970	536.16	130.63	0.24

29.4.2.2 Cone and Plate and Parallel Plate

Another popular system for rotational measurement is the cone and plate configuration (Figs. 29.14 and 29.15). Its special design permits the shear stress and shear rate to remain constant for any location of sample in the fluid gap. Test quality is best when the cone angle (θ) is small, and large errors may be encountered when the gap is improperly set or not well maintained.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig14_HTML.png — figure 29.14
Cone and plate geometry

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig15_HTML.jpg — figure 29.15
Photos of cone and plate and parallel plate test fixtures for rheological measurements (Courtesy of TA Instruments, New Castle, DE)

The shear stress may be determined for a cone and plate configuration as:

$\sigma =\frac{3 M}{2\pi {R}^3}$

(29.31)

while the shear rate is calculated as:

$\dot{\gamma}=\frac{ r\varOmega}{r \tan \theta}=\frac{\varOmega}{ \tan \theta}$

(29.32)

A primary advantage of the cone and plate test fixture is that shear stress and rate are independent of position – constant throughout the sample. However, if there are large particles in the sample, they may become trapped under the point of the cone, invalidating the viscosity measurement. In this case, a parallel plate configuration may be used (Fig. 29.16), as the standard gap in this configuration is 1.0 mm.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig16_HTML.png — figure 29.16
Parallel plate geometry

Unlike the cone and plate, shear stress and strain are not constant between the parallel plates. For consistency, the outer edge of the plate is set as the measurement location. Therefore, care must be taken to ensure the outer edge of the sample does not change during testing.

The shear stress at the edge of the plate may be determined as:

$\sigma =\frac{M}{2\pi {R}^3}\left(3+\frac{d\left( \ln M\right)}{d\left( \ln {\dot{\gamma}}_R\right)}\right)$

(29.33)

The rheometer performs the calculation for the derivative. For a Newtonian fluid, Eq. 29.33 simplifies to:

$\sigma =\frac{2 M}{\pi {R}^3}$

(29.34)

The shear rate at the edge of the plate is calculated as:

$\dot{\gamma}=\frac{R\varOmega}{h}$

(29.35)

29.4.2.3 Experimental Procedure for Steady Shear Rotational Viscometry

29.4.2.3.1 Test Fixture Selection

Many considerations go into the decision of selecting a fixture for a rheological test. To simplify the process, the information in Table 29.7 should be considered [8].

table 29.7

Advantages and disadvantages of rotational viscometry attachments

Rotational geometry	Advantages	Disadvantages
Concentric cylinder	Good for low-viscosity fluids	Potential end effects
	Good for suspensions	Large sample required
	Large surface area increases sensitivity at low shear rates	Large sample required
Cone and plate	Constant shear stress and shear rate in gap	Large particles interfere with sensitivity
	Good for high shear rates	Potential edge effects
	Good for medium- and high-viscosity samples	Must maintain constant gap height
	Small sample required
	Quick-and-easy cleanup
Parallel plate	Allows measurement of samples with large particles	Shear stress and strain are not constant in gap
	Good for high shear rates	Potential edge effects
	Good for medium- and high-viscosity samples	Must maintain constant gap height
	Small sample required
	Quick-and-easy cleanup

29.4.2.3.2 Speed (Shear Rate) Selection

When performing a rheological test, it is necessary to have an understanding of the process for which the measurement is being performed. From the earlier example of tomato catsup, the apparent viscosity continuously decreased, exhibiting shear thinning behavior, as the shear rate increased. How would one report a viscosity? To answer that question, the process must be considered. For example, if a viscosity for molten milk chocolate is required for pipeline design and pump specification, a shear rate for this process should be known. All fluid processes administer a certain degree of shear on the fluid, and a good food scientist will consider the processing shear rate for proper rheological property determination. Barnes et al. [9] have prepared a list of common shear rates for typical processes, many of which are shown in Table 29.8.

table 29.8

Predicted shear rates for typical food processes

Process	Shear rate (s⁻¹)
Sedimentation of powders in a liquid	10^-6–10⁻⁴
Draining under gravity	10^-1–10¹
Extruding	10⁰–10²
Chewing and swallowing	10¹–10²
Coating	10¹–10²
Mixing	10¹–10³
Pipe flow	10⁰–10³
Spraying and brushing	10³–10⁴

Reprinted from [9], with kind permission from Elsevier Sciences – NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands

29.4.2.3.3 Data Collection

Once the test fixture and shear rate ranges have been selected, the experiment can begin. Record values of torque for each viscometer speed.

29.4.2.3.4 Shear Calculations

Values for shear stress and shear rate are solved based on test fixture, fixture geometry, and angular velocity.

29.4.2.3.5 Model Parameter Determination

Shear stress and shear rate can now be inserted into various rheological models previously described in Sect. 29.3. Rheological model parameters such as viscosity (μ, η, μ _pl), yield stress (σ _o), consistency coefficient (K), and flow behavior index (n) may be analyzed for an even greater understanding of the flow of the material. For example, one may want to know: Does the material have a yield stress? Is the material shear thinning or shear thickening? What is the viscosity at a specific processing rate? Answering these and similar questions gives the food scientist a greater command of the behavior of the material for process design or quality determination.

29.4.3 Oscillatory Rheometry

The goal in oscillatory rheometry is to characterize the viscoelastic properties of a material, generally under small stresses and strains. The term viscoelastic implies the material exhibits both fluid-like (viscous) and solid-like (elastic) behaviors simultaneously. Think of a piece of cheese: it springs back (elastic behavior) if you compress it gently, but not quite to its original height (viscous behavior shown as permanent deformation). Viscoelastic behaviors are determined by applying: (1) a stress or strain in oscillation and measuring the respective strain or stress and phase angle between stress and strain, (2) a constant strain and measuring the decrease (relaxation) in stress, or (3) a constant stress and measuring the rate of deformation (creep). These tests are generally used to evaluate the viscoelastic behaviors of solid and semisolid foods, such as cheese and pudding, although they can also be used to evaluate fluid foods, such as salad dressing. A more detailed description of these techniques may be found in Steffe [2] and Rao [4].

29.5 Tribology

Tribology is a subfield of rheology involving the study of friction, lubrication, and wear behaviors of materials. Originally used to study materials such as engine lubricants, tribology has gained interest from the food industry as a way to investigate friction-related aspects of food texture. Sensory attributes such as mouthcoat, chalkiness, or astringency relate poorly to traditional rheological measurements. However, these attributes all involve a sensation of friction. Tribology may be helpful in determining the mechanisms behind these friction-related textural attributes.

Tribological tests are performed by moving one surface against another, fixed surface. A thin layer of the test material is placed between the two surfaces and acts as a lubricant. Different geometries for tribological testing may be seen in Fig. 29.17. The moving surface is slid along the stationary surface at different speeds. Stribeck curves, or plots of friction coefficient versus sliding speed, are generated from the results (Fig. 29.18). Stribeck curves consist of three different behavioral regimes: the boundary regime (a), in which the sliding surfaces are in contact and friction is relatively constant; the mixed regime (b), in which the sliding surfaces are mostly separated by the lubricating material and friction decreases to a minimum value with increased sliding speed; and the hydrodynamic regime (c), where the sliding surfaces are fully separated by the lubricating material. Friction behavior in the mouth is generally in the boundary to mixed regime, with oral sliding speeds estimated at about 10–30 mm/s.

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig17_HTML.jpg — figure 29.17
Tribological testing geometries (Courtesy of TA Instruments, New Castle, DE)

/epubstore/N/S-S-Nielsen/Food-Analysis/OEBPS/images/104747_5_En_29_Chapter/104747_5_En_29_Fig18_HTML.png — figure 29.18
Stribeck curve

As a system property rather than a physical property, friction is affected by many aspects of the system. For foods, friction can be impacted by the food composition and physicochemical properties, food viscosity, food particle size, oral surface composition and conditions, and saliva composition and amount. For example, whole wheat bread has a rougher mouthfeel (higher friction) than bread made with refined flour because the large, irregularly shaped bran particles increase the friction of the whole wheat bread as it moves against the oral surfaces during consumption.

29.6 Summary

Rheological testing is simple in that it only requires the measurement of force, deformation, and time. To convert these measurements into fundamental physics-based rheological properties requires an understanding of the material and testing method. Materials should be homogeneous and isotropic for most fluid foods and many solid foods. Fundamental rheological properties are determined based on knowledge of the stress or strain applied to the sample and the geometry of the testing fixture. Once rheological properties are determined, they can be described by physical or mathematical models to gain a more complete understanding of the rheological properties. The advantage of determining fundamental, rather than empirical, rheological properties is the use of common units, independent of the specific instrument, to determine the rheological property. This approach not only allows for comparison among values determined on different instruments but it also permits comparisons like the flow behavior of honey vs. the flow of paint. Through rheological methods, food scientists have the ability to relate theoretical and experimental information from a range of disciplines, including polymer chemistry and materials sciences, to gain a greater understanding of the quality and behavior of food materials.

Glossary

Bostwick consistometer: A rheological device used to measure viscosity in the food industry
Boundary regime: Tribological behavior characterized by high, constant friction due to surface-surfacecontact
Brookfield viscometer: A rheological device used to measure viscosity in the food industry
Compression: A force acting in a perpendicular (normal) direction toward the body
Concentric cylinder: A test fixture for rotational viscometry frequently called a cup and bob
Cone and plate: A test fixture for rotational viscometry
Constitutive equation: An equation relating stress with strain and sometimes other variables including time,temperature, and concentration
Dilatent: Shear-dependent thickening
Empirical test: Simple tests measuring poorly defined parameters but typically found to correlatewith textural or other characteristics
Fundamental test: A measurement of well-defined, physically based rheological properties
Homogeneous: Well mixed and compositionally similar regardless of location
Hydrodynamic regime: Tribological behavior characterized by complete surface-surface separation by thelubricating material
Incompressible: No change in material density
Isotropic: The material response is not a function of location or direction
Kinematic viscosity: The viscosity divided by the density of the material
Laminar flow: Streamline flow
Mixed regime: Tribological behavior characterized by a reduction in friction to a minimum as surface-surface contact decreases
Modulus: A ratio of stress to strain
Newtonian fluid: A fluid with a linear relationship between shear stress and shear rate without ayield stress
Non-Newtonian fluid: Any fluid deviating from Newtonian behavior
No slip: The fluid velocity adjacent to a boundary has the same velocity as the boundary
Oscillatory rheometry: Dynamic test using a controlled sinusoidally varying input function of stress or strain
Parallel plate: A test fixture for rotational viscometry
Pseudoplastic: Shear thinning
Rheogram: A graph showing rheological relationships
Rheology: A science studying how all materials respond to applied stresses or strains
Rheometer: An instrument measuring rheological properties
Rheopectic: Time-dependent thickening of a material
Shear (strain) rate: Change in (shear) strain with respect to time
Simple shear: The relative motion of a surface with respect to another parallel surface creating ashear field within the fluid contained between the surfaces
Simple shear approximation: A prediction technique for shear rate estimation of fluids within a narrow gap
Steady shear: A flow field in which the velocity is constant at each location with time
Steady state: Independent of time
Strain: Relative deformation
Stress: Force per unit area
Tension: A force acting in a perpendicular direction away from the body
Test fixture: A rheological attachment, sometimes called a geometry, which shears thesample material
Thixotropic: Time-dependent thinning of a material
Torque: A force-generating rotation about an axis, which is the product of the force and theperpendicular distance to the rotation axis
Torsion: A twisting force applied to a specimen
Tribology: A branch of rheology involving friction, lubrication, and wear
Viscoelastic: Exhibiting fluid-like (viscous) and solid-like (elastic) behavior simultaneously
Viscometer: An instrument measuring viscosity
Viscosity: An internal resistance to flow
Yield Stress: A minimum stress required for flow to occur
Zahn cup: A rheological device used to measure viscosity in the food industry

Nomenclature

Symbol	Name	Units
A	Area	m²
A _i	Initial sample area	m²
E	Modulus of elasticity	Pa
F	Force	N
G	Shear modulus	Pa
h	Height	m
K	Consistency coefficient	Pa sⁿ
L	Length	m
L _i	Initial length	m
ΔL	Change in length	m
M	Torque	N m
n	Flow behavior index	Unitless
r	Radial distance	m
R	Radius	m
R _i	Initial radius	m
R _b	Bob radius	m
R _c	Cup radius	m
t	Time	s
U	Velocity	m s^-1
ε	Normal strain	Unitless
γ	Shear strain	Unitless
γ	Angle of shear	Radians or degrees
$\dot{\gamma}$	Shear (strain) rate	s⁻¹
η	Apparent viscosity	Pa s
θ	Cone angle	Radians or degrees
μ	Newtonian viscosity	Pa s
μ _pl	Plastic viscosity	Pa s
σ	Stress	Pa
σ _b	Shear stress at the bob	Pa
σ _o	Yield stress	Pa
Ω	Angular velocity	Radians s⁻¹

29.7 Study Questions

1.

How is stress different from force?
2.

What is the difference between shear stress and normal stress?
3.

What is the definition of apparent viscosity? How does apparent viscosity differ from Newtonian viscosity?
4.

Pure maple syrup is a Newtonian fluid and imitation maple syrup is a Power Law fluid. What are the differences in flow behavior of these fluids and how do these differences alter the processing and final texture of these foods?
5.
The stress response of applesauce at 26 °C may be described by the following mathematical expression:

$\sigma =5.6{\dot{\gamma}}^{0.45}$

The stress response of honey at 26 °C obeys a Newtonian model:

$\sigma =8.9\dot{\gamma}$
In both equations, stress is in Pascals and shear rate is in s^-1.
1. (a)
  
  Which rheological model is described by the equation for applesauce? What are the consistency coefficient and flow behavior index? Include the units on these quantities.
2. (b)
  
  Calculate the apparent viscosities of applesauce and honey at a shear rate of 0.25, 0.43, 5.10, and 60.0 s⁻¹.
3. (c)
  
  Compare how the viscosities of the two food products change with shear rate. Which food has a higher viscosity?
4. (d)
  
  Explain the importance of multipoint testing when measuring viscosity.
6.

You are designing a new chip dip. Describe at least three rheological behaviors that you would like your dip to display.
7.
Rheological tests can be empirical or fundamental.
1. (a)
  
  What are the differences between empirical and fundamental rheological tests?
2. (b)
  
  Develop two empirical tests for comparing viscosities of different tomato sauce formulations.
3. (c)
  
  Identify at least one fundamental rheological test that could be used to determine similar properties from your empirical tests.
4. (d)
  
  Explain the advantages of using fundamental rheological tests instead of empirical tests.
8.

What sort of tribological differences might you expect between a full-fat product and a low-fat product?

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1.

Steffe JF, Daubert CR (2006) Bioprocessing pipelines: rheology and analysis. Freeman, East Lansing, MI
2.

Steffe JF (1996) Rheological methods in food process engineering, 2nd edn. Freeman, East Lansing, MI
3.

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4.

Rao MA (1999) Rheology of fluid and semisolid foods: principles and applications. Aspen, Gaithersburg, MD
5.

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Hamann D, Zhang J, Daubert CR, Foegeding EA, Diehl KC (2006) Analysis of compression, tension and torsion for testing food gel fracture properties. J Texture Stud 37: 620–639Crossref
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Bourne MC (1982) Food texture and viscosity: concept and measurement. Academic, New York
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Macosko CW (1994) Rheology: principles, measurements, and applications. VCH, New York
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Barnes HA, Hutton JF, Walters K (1989) An introduction to rheology. Elsevier Science, New York