Quantitative fit of a model for proving of bread dough and determination of dough properties

Journal of Food Engineering 96 (2010) 440–448

Contents lists available at ScienceDirect

Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Quantitative fit of a model for proving of bread dough and determination of dough properties Andrés Córdoba * Departamento de Ingeniería Química, Universidad de los Andes, Bogotá, Carrera 1 No. 18A 10, Bogotá, Colombia

a r t i c l e

i n f o

Article history: Received 26 January 2009 Received in revised form 17 August 2009 Accepted 20 August 2009 Available online 28 August 2009 Keywords: Dough proving Diffusion Newtonian viscosity Modeling Nonlinear constrained optimization Leavened products

a b s t r a c t The early stages of bread dough proving were modeled following the diffusion theory and using a newtonian constitutive equation to account for dough rheology. The model was fitted, by constrained minimization of the sum of squared errors function, to experimental growth curves of common dough formulations. Values for the dough viscosity, a first order fermentation rate constant and the bubble size at the beginning of proving were chosen as the degrees of freedom in the fitting procedure. The viscosity value obtained for a plain dough formulation was 30  106 Pa s, this value is in the range of viscosity values determined by Rouillé et al. (2005), for wheat dough through creep-recovery and lubricated squeezing flow tests. The results obtained were plotted as a function of dough formulation composition and the trends were observed and analyzed. The observed tendencies and correlations between dough properties and dough formulation follow the trends observed in experimental data reported in the literature for dough systems. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Dough is a multiphase and multicomponent system mainly composed of proteins, lipids, carbohydrates, water and air. The dough ingredients, as well as the processing conditions, determine the macroscopic structures of the baked dough which, in turn, is responsible for its appearance, texture, taste and stability. An important process of breadmaking is the creation of an aerated structure in the baked loaf. Different types of breads are characterized by distinctive aerated structures. This is achieved through the incorporation of bubbles in the dough during mixing and proving, and their transformation into interconnected gas cells in the crumb during baking. An important operation of breadmaking is proving (rising), in which yeast metabolizes flour sugars into carbon dioxide gas, which diffuses into bubbles incorporated during mixing, causing the bubbles to inflate and the dough to raise. Other operations that considerably modify bubbles size and distribution are forming, sizing and shaping. Bubble growth in liquid systems has been widely studied in the polymer processing field, due to the importance of this process in the production of polymer foams. Research on bubble growth in liquids has been traditionally conducted with two distinct modeling approaches. One approach has focused on the mass and heat transfer phenomena, which might drive bubble growth in liquid * Tel.: +17732513514. E-mail addresses: [email protected], [email protected], andcorduri@ gmail.com 0260-8774/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jfoodeng.2009.08.023

systems. This approach generally neglected surface tension and momentum transfer phenomena (Epstein and Plesset, 1950; Scriven, 1959). These models have been found to be inadequate for the description of the initial stages of bubble growth. The second approach focuses on momentum transport and thus on the viscoelastic properties of the liquid that surrounds the bubble. Amon and Denson (1984) developed a growth model which considered a polymer melt to be apportioned into ‘‘cells” within which a specified pressure difference between the melt and the bubble drives growth. In the Amon and Denson (1984) model the polymer melt rheology was described through a Newtonian constitutive equation. Favelukis and Albalak (1996) presented the modeling of the hydrodynamically controlled spherical bubble growth in quiescent power law type fluids. More recently, models that take into account both mass transfer and momentum transfer have been developed, Venerus et al. (1998) modeled the diffusion-induced bubble growth in viscoelastic liquids using the Phan-Thien Tanner (PTT) constitutive model for the fluid rheology. Some work has been done on bubble dynamics in wheat dough. Highly comprehensive models have been presented (Bikard et al., 2008; Hailemariam et al., 2007; De Cindio and Correra, 1995), which describe growth of bubbles in wheat dough, including mass transfer phenomena, viscoelatsic, surface tension and coalescence effects, and bubble size distribution. Models like the ones mentioned above are among the most complete available in the description of the relevant phenomena in dough systems. Rheological data, growth data, fermentation kinetics, diffusivities, among other experimental data are required to fit or validate

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

441

Nomenclature C* C1 R R0 Pb P1 n V Rg T t Q KL kL H kG D 2 PCo b

solute concentration at interface, kmol m3 solute concentration in dough, kmol m3 bubble diameter, m initial bubble diameter, m solute pressure in bubble, Pa pressure in dough, Pa number of moles of nitrogen in a bubble, kmol bubble volume, m3 universal gas constant, J kmol1 K1 absolute temperature, K time, s molar rate of transfer, kmol s1 overall mass transfer coefficient, m2 s1 liquid phase mass transfer coefficient, m2 s1 Henry’s law constant, J kmol1 gas phase mass transfer coefficient, m2 s1 coefficient of diffusivity, m2 s1 carbon dioxide pressure in bubble, Pa

the aforementioned models for bubble growth in wheat dough. Therefore, a major challenge with these models has been the experimental verification of bubble dynamics. Experiments from where the parameters are obtained in many occasions do not reproduce with precision typical proving conditions. Such is the case of rheological measurements which are usually made in equipment that do not necessarily produce deformation modes typical of dough proving. Therefore if pertinent and sufficient data are not available the values given to the model parameters will only be speculative, and the good macroscopic predictions of the model will not necessarily be a reflection the underlying phenomena. Measuring the dough’s properties while it is undergoing physical and chemical changes is difficult due to the dynamic and fragile nature of dough. Invasion during the expansion process could cause gas cell leakage which would reduce expansion. Older techniques used interval measurements, but this yielded inconsistent, time intensive results that were not fully accurate or representative of on-line changes. Modern microscopic measuring techniques may be applied for dynamic measurements of dough properties during proving (Babin et al., 2006). However the microscopic techniques pose a challenge for the development of a dynamic method because the material structure is fixed either through heat or freezing before analysis. This may destroy the structure, the entity to be measured, and is not fully representative of the dynamic system (Bache and Donald, 1998). To overcome the challenge of measuring bubble growth in situ, an experimental method to study microstructure growth through macroscopic investigation (Penner et al., 2009; Romano et al., 2007) was used in this work. The proving process was followed by means of only one experimental measurement (volume). Dough expansion was constrained to one direction (vertical) and volume change was measured directly from the experimental set-up. Recently Penner et al. (2009) have shown similar direct growth measurements in radial expanding dough (vertically constrained) to be in close agreement with data obtained through image analysis software. The aim of this work is to use a model based on diffusion theory and a Newtonian approach for the dough rheology to model and simulate the growth of four different types of common breads. Volume change of four types of dough formulations will be monitored during the early stages of the leavening process of the dough, and the data will be used to fit and calibrate the proposed model, by constrained minimization a nonlinear squared error function.

YMax KM Nb VE ti to

terminal value of CO2 production rate by the yeast, kmol m3 s1 time when Y equals YMax/2, s number of bubbles per volume of gas-free dough, m3 experimental measurement of dough volume, m3 time at which an experimental volume measurement was made, s time at which the first experimental volume measurement was made, s

Greek letters viscosity of the dough, Pa s surface tension as the bubble–liquid interface, N m1 /0 initial void fraction of the dough (q0/qgas-free dough) q0 dough density after mixing and kneading qgas-free dough gas-free dough density

l c

Important characteristic parameters, such as CO2 concentration in the dough at the beginning of leavening, viscosity and fermentation kinetics, of the different doughs will be calculated. Hypothesis about the relations between dough composition and the parameters calculated will be stated and analyzed, both quantitatively and qualitatively.

2. Mathematical modeling Some authors (Amon and Denson, 1984) have hypothesized that, in the bubble growth process in viscous liquids, the viscous effects are predominant over surface tension effects. However in these models a constant pressure gradient, between the bubble and the liquid, was used as the driving force for bubble growth. However during proving of wheat dough the pressure gradient is not constant due to the continuous production of carbon dioxide, caused by the fermentation process. At the beginning of the proving process the concentration carbon dioxide in the liquid phase is low, and consequently the driving force for the mass transfer into the bubbles is initially low. The increasing concentration of the carbon dioxide causes an increase in the rate of transfer into the bubbles as proving progresses and therefore an increment in the pressure gradient between the bubble and the liquid phase. Chiotellis and Campbell (2003) and Shah et al. (1998) have modeled the bubble growth process in dough considering diffusion controlled growth and including surface tension as the only hydrodynamic contribution to the process. At the start of bubble growth, when the pressure difference between the bubble and the liquid is small, the dynamics are known to be controlled by surface tension (Hailemariam et al., 2007). However once the pressure difference reaches a certain threshold the process becomes dominated by viscous/viscoelastic effects. Hailemariam et al. (2007) have proposed that, in modeling bubble growth in wheat dough, no effect should be given priority before results are obtained. The basic assumptions of the model at the microscopic level are that the bubbles are spherical, CO2 is the only diffusing species and mass transfer of CO2 occurs in thin shell around the bubble. Momentum transport is accounted for by modeling dough viscosity with a Newtonian constitutive model. At the macroscopic level, the major assumptions are that the bubbles are homogeneously distributed inside the dough, no coalescence effects are considered,

442

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

the concentration of CO2 in the liquid phase is uniform, and that gas does not diffuse out of the dough. The initial mixing and kneading of the dough creates the first air (nitrogen) bubbles inside the dough, this bubbles are of radius R = R0. The bubbles formed during dough mixing then act as nucleation sites for CO2 produced by the yeast during subsequent proving stages (Mills et al., 2003). It is assumed that new bubbles are not created during the proving process. Mixing may also bring both soluble and insoluble components to the bubble surface, the movement of which be otherwise hindered by the structure and viscosity of the gluten starch matrix of dough (Örnebro et al., 2000). The development of the model equation that follows will be carried out for a single bubble. It was assumed that the fractional change in volume of a single bubble of initial radius R0 is proportional to the fractional change of volume of the dough, this approach is generally called homogenization approach. To derive the momentum transfer equations for the proving process, the equations of continuity and motion in spherical coordinates, for the incompressible liquid phase, are used to obtain the pressure gradient between the liquid bulk and the liquid at the bubble surface. The relation between the pressure inside bubble and the pressure in the liquid at the bubble surface is obtained through a mass balance at the bubble surface (see Favelukis and Albalak, 1996 for complete derivation of the momentum transfer equation). The derived relation is known as the extended Rayleigh equation. For the case of a Newtonian fluid and if inertial effects in the bubble are neglected, which implies that the pressure within the bubble is uniform, the momentum transfer equation for bubble growth in a viscous incompressible liquid is:

2c 4l dR Pb ¼ P1 þ þ R R dt

ð1Þ

where the total pressure inside the bubble of radius R is Pb, the surface tension c and dough viscosity is l. Therefore, assuming ideal gas behavior for the gas inside the bubble, the number of moles of CO2 present in a bubble of radius R can be written as:

  P1 þ 2Rc þ 4Rl dR pð2RÞ3 dt Pb V n¼ ¼ Rg T 6Rg T

ð2Þ

Initially, when R = R0, there is nitrogen but no CO2 in the bubble, therefore a concentration gradient for mass transfer exists allowing CO2 to diffuse into the bubble, causing the bubble to grow. The following model describes the rate of growth, by considering the rate of mass transfer into the bubble. The transfer of CO2 into the bubble causes a change in the total number of moles inside the bubble

dn ¼Q dt

ð3Þ

As stated above the mass transfer rate, Q, into a single bubble is related to the CO2 concentration gradient, between the liquid and the bubble, by a standard mass transfer equation:

Q ¼ K L 4pR2 ðC 1  C  Þ

ð4Þ

where C* is the concentration of carbon dioxide in the dough that would be in equilibrium with the partial pressure of carbon dioxide in the bubble, KL is the overall mass transfer coefficient and 4pR2 is the surface area of a sphere of radius R. From the two film theory of mass transfer at a gas–liquid interface:

1 1 Rg T ¼ þ K L kL HkG

ð5Þ

where kL and kG are the individual mass transfer coefficients for the liquid and gas phases, respectively. Assuming mass transfer is liquid side controlled (i.e. H is relatively large), then:

KL  KL

ð6Þ

and for pure diffusion:

Sh ¼

kL R ¼2 D

ð7Þ

Therefore, for this case, the mass transfer rate simplifies to,

dn ¼ 4DpRðC 1  C  Þ dt

ð8Þ

The previous second order differential equation (with respect to R) requires two initial conditions which are, a known initial radius of the nitrogen bubbles:

Rðt ¼ 0Þ ¼ R0

ð9Þ

and the assumption that bubbles do not start growing until the start of the leavening process

dR ðt ¼ 0Þ ¼ 0 dt

ð10Þ

The initial radius of the bubbles and the number of bubbles are related to the gas-free dough density and to the density of the dough after mixing and kneading, this density can be written as,

q q0 ¼  gas-free dough3  1 þ 34 Nb pR0

ð11Þ

Another important parameter commonly used in bread dough proving is the volumetric void fraction. By algebraic manipulation an expression, for the volumetric fraction, which is only a function of the model parameters is obtained,

 /0 ¼ 

4 N 3 b

4 N 3 b

pR30





pR30 þ 1

ð12Þ

Using Eqs. (11) and (12) the following relation, between the densities and the volumetric void fraction is obtained,

q0 ¼ qgas-freedough ð1  /0 Þ

ð13Þ

A constitutive model for the equilibrium concentration of CO2 is also required, assuming that Henry’s law applies (Shah et al., 1998),

 3  2 !! 2 PCo p1 R0 8c R0 b C ¼ þ ¼ 1 1 H H R HR R 

ð14Þ

The rate of change of dissolved CO2 concentration in the liquid phase of the dough depends on the fermentation kinetics and on the mass flux that diffuses out of the liquid phase and into the bubble. Also the model assumes that no diffusion of CO2 into the atmosphere takes place.

dC 1 Y Max t ¼  4Nb DpRðC 1  C  Þ KM þ t dt C 1 ðt ¼ 0Þ ¼ 0

ð15Þ ð16Þ

Finally the model consist of one ordinary differential equation (15), with its initial condition (16), one second order differential equation (8), with two initial conditions ((9) and (10)), which must be solved simultaneously for the bubble radius and the concentration of CO2 in the liquid phase of the dough. 3. Materials and methods 3.1. Dough formulations Four distinctive dough formulations were chosen with the aim of assessing the capacity of the model to represent significant differences in composition, one plane dough (PD), two rich doughs (RD1 and RD2) and one spicy dough (SP) were prepared. PD is a

443

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

plain dough formulation, one of its most common uses being pizza crusts. RD1 is a typical dough formulation for doughnuts and sweet rolls, RD2 is a Brioche formulation and SD is a Foccacia formulation. All doughs were made using different proportions of commercial white flour (Harina Haz de OrosÒ), warm water (30 °C), salt, brown sugar, canola oil, and active dry yeast (LevapanÒ). RD1 and RD2 formulas also included milk, shortening and eggs. The SD dough included dry species (oregano, thyme, garlic and pepper). Table 1 shows the specific composition (in grams per hundred grams of flour) of the bread dough formulations used. In all the cases the warm water with the sugar dissolved was used to activate the dry yeast, the activation process lasted 10 min. The basic breadmaking procedure involved pre-mixing of the dry ingredients then the activated yeast mixture was added and the dough was kneaded manually, at 19 °C, for 10 min. 3.2. Bread proving and volume change measurement After mixing and kneading the dough was gently rounded by hand into a cylindrical shape, and 47 mL of dough were inserted into a lubricated (vegetable oil) glass or polypropylene calibrated mold (cylindrical). The geometry and set-up of the mold allowed the dough to expand only vertically. The samples were incubated in a closed chamber at 40 ± 2 °C. Volume expansion was monitored during the proving process. The volume data was obtained using the calibrated marks on the mold, with a precision of ±1 mL. Data was taken until cessation of growth was observed. Once the dough

was put inside the mold, no additional manipulation of the dough was made. 3.3. Model calibration The model was fitted to the experimental data by fixing different parameters as constants, and leaving at least three parameters to be determined by a least squares fit (degrees of freedom). The parameters that were finally chosen as degrees of freedom were R0, l and YMax, the other model parameters were fixed according to the values shown in Table 2. Because the proposed model is better solved by a numerical method no explicit formulation of the least squares objective function can be done. The model was written, solved and optimization executed in Wolfram’s MathematicaÒ version 6. The differential equations that constitute the model where solved, for each iteration of the optimization algorithm, using the function NDSolve. The experimental and model data were related according to the following relation (this quantity will be called growth in this text) 4 3

pRðtÞ3 Volume of dough at time t  4 Initial volume of dough pR30 3

ð17Þ

Therefore the least squares objective function used to fit the model to the experimental data was

f ðR0 ; l; Y Max Þ ¼

# data Xpoints i¼1

V E ðt i Þ ðRðti ÞÞ3  V E ðt 0 Þ ðR0 Þ3

!2 ð18Þ

Subject to the restrictions:

Table 1 Composition of dough formulations. Weight content

PD

RD1

RD2

SD

Flour Water Yeast Sucrose Sodium chloride Vegetable oil Shortening Eggs Milk Solid species

100.0 85.5 2.2 0.5 1.4 6.7 0.0 0.0 0.0 0.0

100.0 9.6 2.2 16.0 1.0 0.0 11.2 16.0 52.0 0.0

100.0 18.2 2.3 2.3 1.4 0.0 51.1 45.5 0.0 0.0

100.0 68.1 2.9 1.2 1.7 4.0 0.0 0.0 0.0 2.0

R0 P 0;

l P 0; Y Max P 0

ð19Þ

where VE(t0) and VE(ti) are the initial volume and the volume at a given time of the dough measured experimentally, and R(ti) and R(t0) are the initial average bubble radius and the average bubble radius at a time t, on the dough as given by the proposed model. The function was minimized using the global optimization algorithm RandomSearch of Mathematica 6Ò. This method uses gradient based methods to find local minimums starting from randomly generated initial points, and at the end chooses the best found (global minimum) finite difference approximation for derivatives was used

Table 2 Constant parameters used in the model. Parameter

Value

Reference

Diffusivity of carbon dioxide in the dough, D Henry’s law constant, H Ambient pressure in dough, P1 Universal gas constant, R Proving temperature, T Surface tension, c Number of bubbles, Nb Time when Y equals YMax/2,KM

7.44  1010 3.30  106 100,000 Pa 8314 J kmol1 K1 313 K (40 °C) 0.04 N m1 1011 m3 0s

De Cindio and Correra (1995) Bloksma (1990) Shah et al. (1998)

Kokelaar and Prins (1995) Chiotellis and Campbell (2003) Chiotellis and Campbell (2003)

Table 3 Values of the parameters obtained by fitting the model to the experimental data of different doughs. The reported values have an average coefficient of variation of 18%. Dough PD RD2 RD1 SD Romano et al. (2007) data Babin et al. (2006) data

R0 (m)

/0 (%) 4

3.14846  10 3.16285  104 3.25145  104 3.08251  104 8.89309  105 6.87489  105

92.8943 92.9841 93.5059 92.4635 22.7566 11.9803

YMax (kmol m3 s1) 4

1.62229  10 9.52567  105 1.01171  104 8.87672  105 7.22461  105 1.68239  105

l (106 Pa s) 27.2559 34.0786 22.4251 29.9734 386.93 2.0109

444

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

in the optimization technique. In all the fits made the Karun–Kuhn– Tucker Conditions were satisfied with a precision of 4  106. The model was fitted for a minimum of two growth curves for each formulation the values reported in Table 3 had an average coefficient of variation of 18%. The apparently large value of the variation coefficients for the fitted model parameters is more likely related to a poor reproducibility of the mixing operation, than to the manipulation done to put the dough inside the calibrated mold. While the mixing and kneading operation is complex and requires strict control to be reproducible, putting the dough inside the mold is a much simpler process. 3.4. Contour plots and properties variation analysis To better understand the effect of composition on the dough properties contour plots of dough viscosity, YMax, and initial void fraction as a function of composition were constructed. The effect of sucrose and sodium chloride on yeast activity and dough rheology was evaluated by constructing a contour plot of the obtained YMax and viscosities as a function of the concentration of sucrose and sodium chloride in each formulation. The viscosity and initial void fraction of the formulations were plotted against the polar liquids to flour ratio and added fats to flour ratio. 4. Results and discussion The physical constants of the model to be determined by fitting the model to the experimental data were chosen by trial and error, with the purpose of obtaining the lower values of the objective function, but also of maintaining the values of the fitted parameters within a coherent physical order of magnitude. The choice of the parameters to be used as degrees of freedom, to fit the model to the experimental data, was made after several trials which shown a clear tendency to obtain smaller residuals when R0, l and YMax were chosen. The values for these parameters obtained for different doughs are shown in Table 3. In Fig. 1 the experimental data and the model obtained by fitting the parameters are shown for comparison. The values obtained are in the range of typically reported values for these quantities in dough systems (Shah et al., 1998; Chiotellis and Campbell, 2003; Epstein and Plesset, 1950). As was mentioned above, only three model parameters were finally used as degrees of freedom to fit the model. With these three parameters the smallest sum of residuals was obtained for all the formulations studied. Other parameters such as the surface tension and the diffusivity coefficient were also tested as degrees of freedom; however the fitting algorithm usually converged to values of these parameters highly divergent from the values reported in the literature and the minimum of the least

squares function increased. Therefore no further attempts were made in using other model parameters as degrees of freedom for the fitting procedure. The model was also fitted to the early stages of an experimental dough leavening curve reported by Romano et al. (2007), the obtained values of R0, l and Ymax are shown in the last row of Table 3. The selected data corresponds to a dough with the following composition (weight percentage), 2.9% yeast, 57% water, 2.9% NaCl and 1.14% Sugar. The minimum value of the squared error function obtained when the model was fitted to this data was 0.003. This composition is similar to the PD formulation, however important differences exist such as significantly lower water content, no presence of added lipids, and a higher content of sodium chloride. These differences in formulation are clearly reflected on the values of the dough properties shown in Table 3, especially in dough viscosity where the lower water content and higher NaCl content causes an important increase in this property for the Romano et al. (2007) formulation. In the modeling approach where mass transfer phenomena has been considered to be dominant in the bubble growth process during dough proving (Shah et al., 1998; Chiotellis and Campbell, 2003; Epstein and Plesset, 1950) the viscosity effects have been neglected because of their low significance to bubble pressure calculation during the early stages of proving. However the results of this work suggest that the inclusion of the viscosity term in the extended Rayleigh equation, significantly improves the fitting of the model to the experimental data of all the different baking products studied. The final value of the objective function could be reduced to values as low as 0.001 when the viscous effects were included in the model, obtaining coherent values for the physical parameters used as degrees of freedom. When viscosity was not included in the models the objective functions could not be reduced to values below 0.03, and this lower values were attained only with interfacial tensions in the order of hundreds, which is physically incoherent. This could indicate that actually the growth of the doughs is not slow enough, during the proving time monitored, to make the viscous effects negligible compared with the surface tension effects. At the early stages of proving, the concentration of CO2 in the liquid phase increases faster than the concentration of CO2 in the bubbles. This is because at the beginning of proving the concentration of CO2 in the liquid phase is low, and consequently the driving force for the mass transfer into the bubbles is initially low. The increasing concentration of the CO2 causes an increase in the rate of transfer into the bubbles as proving progresses. When the dough becomes saturated all the CO2 that the yeast produces is used to inflate the bubbles. However in practice bubbles cannot grow indefinitely, but will coalesce.

0.02 2

RD2

PD

1.6

SD

1.4

Kmol m3

Growth

RD1

RD2

PD

0.015

1.8

RD1

0.01

SD

0.005

1.2 0 1 0.

900. 0.

PD 1800. 2700. 3600. 1600. 3200. 0. 1150. 2300.

0.

1000.

RD2 4800. 6400. RD1 3450. 4600. SD 2000. 3000. 4000.

Time s Fig. 1. Experimental data (points) and fitted model for different dough formulations. Growth ¼ VVEEðtðtÞ0 Þ.

PD 0.

900.

1800. 2700. 3600.

0.

1600. 0.

3200.

RD2 4800.

6400. RD1 1150. 2300. 3450. 4600. SD 0. 1000. 2000. 3000. 4000.

Time s Fig. 2. Carbon dioxide concentration as a function of proving time, calculated with the fitted models for different dough formulations.

445

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

The effects of coalescence were not included in the model used in this work. However, in experimental observations, these effects became evident after the volume of the dough doubled with respect to the initial volume. In fact it has been demonstrated by X-ray tomography experiments (Babin et al., 2006) that coalescence may occur much sooner, during dough proving. Fig. 3 shows that the residuals of the fit for all breads had the same tendency to become larger for data points corresponding to times were dough volume had already doubled. This tendency was observed for all the doughs studied, this indicates that the model presented here will simulate accurately the leavening of dough until a doubling in volume occurs, however will produce inaccurately results after that. Real dough systems are not saturated with carbon dioxide for at least the first half of proving, and possibly never reach saturation (Shah et al., 1998). Under the proving conditions used in this work saturation occurs when C1 = 0.0303 kmol m3. The simulations of carbon dioxide concentration in the liquid phase (Fig. 2), obtained using the fitted models, clearly show that the models accurately predict the expected undersaturated conditions, during the proving times monitored. Gas also is lost to the atmosphere, but this has been considered negligible for the early stages of proving (Chiotellis and Campbell, 2003) and was not included in the model. This simplification of the mass balances did not seem to significantly affect the capacity of the model to fit the experimental data accurately. The exclusion of inertia and buoyancy effects from the momentum transfer equation did not seem to affect in a significant extent, the capacity of the model to describe the proving of the studied doughs. Most models (Venerus et al., 1998; Favelukis and Albalak, 1996; Amon and Denson, 1984) for describing growth of bubbles in viscous liquid media neglect inertia and buoyancy effects. Inertia controlled growth of bubbles occurs only when the pressure inside the bubbles is nearly constant and the bubble size is large enough so the surface tension is negligible (Robinson and Judd, 2004). Neither of these two conditions is likely to occur at the same time during bread dough proving. Rouillé et al. (2005) have shown through experimental measurements the existence of a Newtonian plateau in doughs at shear rates below 1  106 s1, at higher shear rates the viscosity has a shear thinning behavior. Hailemariam et al. (2007) and other authors have included non-Newtonian constitutive equations in dough growth models. However under traditional proving conditions, as the shear rate is very low (about 103 s1) and does not vary much in the proofing step (Babin et al., 2006), the Newtonian behavior may well be assumed. Rouillé et al. (2005) report experimental values of viscosities of bread doughs under large bi-extensional strain obtained through lubricated squeezing flow tests at low shear rates (103–101 s1). These types of strains are typical during proofing. The reported results vary between 106 and

f ðR0 ; l; Y Max Þ ¼

# data Xpoints



0 @/E ðti Þ  

4 N 3 b

4 N 3 b

i¼1

pRðti Þ3 

 12 A

ð20Þ

pRðti Þ3 þ 1

where /E(ti) are the experimental data points of volumetric void fraction as a function of time reported by Babin et al. (2006). The minimum sum of residuals that was obtained for this case was 0.005, which is very close to the ones obtained when the growth curves for the formulations prepared in this work were used to fit the model. The values of the fitted parameters are shown in Table 3. The ranges of initial values used for the minimization algorithm were the same for all the data sets. Therefore the differences observed in the fitted model parameters are entirely due to the minimization of the least square function and not to differences in the initial values. The large initial volumetric void fractions obtained, when the data from the formulations prepared in this study was used to fit the model, is probably caused by the mixing and kneading procedure. Differently from the procedures that were used by Romano et al. (2007) and Babin et al. (2006) in which the mixing times were

2.0

0.10

Error Function

RD2 PD RD1 SD

0.05

Residuals

107 Pa s the specific values depending on the chemical composition of the flour used to prepare the dough. The viscosities obtained through the procedure proposed in the present work are between 22  106 and 34  106 Pa s. There is a significant concordance between the viscosity values obtained in this work and the previously reported ones. However viscosity results should not be compared to literature data in a straightforward way, because dough rheology depends greatly on flour prolamin content, on glutein/gliadin ratio, on dough hydration, and on testing conditions (Rouillé et al., 2005). Also it should be noted that rheological methods are used to determine constitutive laws whereas the model fitting proposed in this paper will only give one value of viscosity for each composition, at an assumed constant strain and strain rate. A sensitivity analysis of the objective error function with respect to the three fitted parameters, R0, l and YMax, was carried out and the one corresponding to PD is illustrated in Fig. 4. For all the dough formulations considered the error function is affected in a much more significant extent by the initial size of the bubbles (or initial void fraction), followed by the reaction rate parameter, the dough viscosity has little influence over the error function value. To further asses the capacity of the model to represent experimental data obtained using different formulations and experimental techniques the model was also fitted to the void volumetric fraction curve for a standard French baguette recipe reported by Babin et al. (2006). This curve was obtained using X-ray tomography. The least square function used to fit the model proposed in this work to the aforementioned data was,

0.00

0.05

0.10 0

1000

2000

3000

4000

5000

6000

Time s Fig. 3. Residuals of the fit for the different dough formulations.

R0

1.5

YMax

1.0

0.5

0.0 0.0

µ 0.5

1.0

1.5

2.0

Ratio of the considered parameter to its optimal value Fig. 4. Parametric sensitivity of the error function, for the PD formulation.

446

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

short (3 min), the mixing and kneading stage in this study was carried out for about 10–15 min. The values are also much larger than the values reported by Bellido et al. (2006) for void fractions measured by gravimetric and X-ray tomography techniques. The maximum void fraction after mixing reported by Bellido et al. (2006) is 10.41%, however in that study the doughs were prepared without yeast an were mixed for 4 min, whether in our study the doughs were prepared with a pre-activated yeast, and were mixed and kneaded for 10 min. The void fraction obtained when the data by Romano et al. (2007) was used to fit the model is much closer to the values reported by Bellido et al. (2006). The mixing procedure of the doughs used in the works by Romano et al. (2007) and Bellido et al. (2006) are very similar. This later fact may indicate that the model is very sensitive to the condition of the dough after mixing and kneading. As can be seen in Eq. (12) the only two model parameters that affect the initial volumetric void fraction are the average initial bubble radius (R0) and the number of bubbles per volume of gas free dough (Nb). This last parameter was set to be 1011 m3 for all the formulations. Therefore the only fitted parameter that affects the calculation of the initial void volumetric fraction is the initial bubble radius, which was always an order of magnitude larger for the formulations prepared in this work, compared to the formulations of Romano et al. (2007) and Babin et al. (2006). With the purpose of obtaining model fits with lower initial volumetric fractions trials were made changing the parameter Nb to values below 1011. However when this was done the fitting procedure converges to even larger values of initial bubble diameter and therefore the initial void volumetric fraction remains practically unchanged. For the data of the formulations prepared in this work no minimums of the least square functions could been found that resulted in lower volumetric fractions than the ones reported in Table 3. The fitted models were used to assess the effect of formulation on the rheological and fermentation rate behavior of the doughs. In Fig. 5A the combined effect of salt concentration and sucrose on the dough viscosity is shown. The lower viscosities correspond to the formulations with lower NaCl content but higher sucrose content, whether a clear maximum in viscosity is observed for the formulation with a high NaCl concentration but low sucrose concentration. Chloride anions are known to stiffen wheat flour doughs when used at higher concentrations by promoting the aggregation of gluten proteins (Preston, 1981; Bellido et al., 2006). The RD1 formulation which has the lowest NaCl content and the highest sucrose content produces the dough with lowest

A

viscosity. Fig. 5B shows the combined effect of NaCl and sucrose content of the formulation on the first order fermentation rate constant in the dough. It is clearly observed that NaCl negatively affects the fermentation rates, for the contrary higher concentrations of water soluble carbohydrate as sucrose maximizes the fermentation rate. The hydrolysis of starch molecules is slow process, and therefore, the amount of a readily soluble available carbon source is expected to accelerate the fermentation kinetics during the early stages of proving. However higher amounts of sucrose could produce osmotic stress, and therefore the existence of a threshold concentration value where sucrose begins to be detrimental to the fermentation process is expected to exist. The amount of sodium chloride added to common dough formulations for organoleptic reasons is known to have a detrimental effect on yeast growth kinetics. During optimal fermentation conditions NaCl inhibits Saccharomyces cerevisiae at all concentrations (Almagro et al., 2000). Increasing concentrations of NaCl progressively decrease the growth kinetics of yeast, at a concentration of 0.5 M, growth is completely inhibited (Almagro et al., 2000). The SD formulations, which has the highest NaCl composition and lowest sucrose composition produces the dough with lowest fermentation rate constant. Fig. 6A shows the relation between the Newtonian viscosity of the doughs (l) with the amount of polar liquids (water + milk + eggs) and fats (shortening + oil) added to the formulations. The viscosity has a minimum for the composition ratios corresponding to low fats/flour ratios, and to intermediate values of polar liquids/ flour ratios, high lipids content leads to a significant increase on the viscosity of the proving dough. In Fig. 6B the void fraction of the doughs after mixing and kneading is shown as a function of polar liquids and fats added to the formulations. The highest void volume corresponds to a formulation with low fats/flour ratios, and to intermediate values of polar liquids/flour ratios. Important to note is the fact that the composition with the lowest viscosity is also the one with the highest initial void fraction. The incorporation of bubbles in the dough during mixing and kneading is easier in doughs with lower viscosities. Martin et al. (2004) observed that aeration of a dough increased with an increase in the Reynolds number. In the same way, the fewer bubble numbers and lower void fraction of the stiffer doughs in the present study may be explained in terms of the Reynolds numbers, as this number decreases when the viscosity increases. The experimental formulation with the lowest viscosity and highest initial void fraction was RD1 this is a typical formulation for doughnuts. The formulation with the higher viscosity was RD2, a typical formulation for the traditional

B

0.017

Max

Max

0.016

0.015 0.014 Min

0.013 0.012 0.011

Sodium Chloride Flour

Sodium Chloride Flour

0.016

0.017

0.015 0.014 Min

0.013 0.012 0.011

0.010

0.010 0.05

0.10

Sucrose Flour

0.15

0.05

0.10

0.15

Sucrose Flour

Fig. 5. (A) Effect of sucrose and salt on dough viscosity (l) (Max = 34.0786  106 Pa s; Min = 22.4251  106 Pa s). (B) Effect of sucrose and salt on the CO2 production rate (YMax) during proving (Max = 1.62229  104 kmol m3 s1; Min = 8.87672  105 kmol m3 s1).

447

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

A 0.5

Max

B 0.5 0.4

0.3 Min

Fats/ Flour

Fats/ Flour

0.4

Max

0.3 Min

0.2

0.2

0.1

0.1

0.65

0.70

0.75 0.80 Polar liquids/ Flour

0.85

0.65

0.70

0.75 0.80 Polar liquids/ Flour

0.85

Fig. 6. (A) Contour plots of the effect of polar liquids and fats content on dough viscosity (l) (Max = 34.0786  106 Pa s; Min = 22.4251  106 Pa s). (B) Contour plots of the effect of polar liquids and fats content on initial void fraction of the dough (/0) (Max = 92.9841%; Min = 92.4635%).

brioche bread. Brioche bread is characterized by high amounts of lipids on its composition. Pure liquids are incapable of supporting a foam, requiring additional surface active molecules to form a film around the bubbles, stabilizing them against coalescence and hence foam collapse (Mills et al., 2003). It is thought that the liquid film is formed from the aqueous phase of dough, when doughs are mixed above a certain critical moisture content of around 0.3 g g1 dry flour (Gan et al., 1990). The drainage, surface properties and composition of such a thin film (lamella) would play an important role in determining the stability of adjoining bubbles during proving. It is well known that the protein content positively affects dough volume (Frgestad et al., 2000). It has been found that dough liquor is not able to form a foam unless prepared from defatted flour (Dubriel et al., 1998). However recent studies (Mills et al., 2003) have lead to the postulate that liquid films lining the bubbles comprise both soluble surface active proteins and lipids. The results presented in this work (Fig. 6) suggest that a specific relation between protein and lipid content exists in which dough properties benefit the growth process. In the formulations with higher protein (RD1) content and lower lipid content (PD) lower viscosities and higher initial void fractions are obtained, both dough qualities that benefit the growth process. 5. Conclusions In this work a model based on diffusion theory and a Newtonian approach for the dough rheology was used to describe and simulate the growth of four different types of common leavened products. The model was fitted to experimental growth curves of the formulations, using constrained global minimization of the sum of squared errors function. The model achieved a good quantitative fit to experimental growth data during the early stages of proving. Additionally the model parameters were related to the composition of common leavened products formulations. Usually modeling work on dough proving has been carried out with a basic water–flour–yeast–NaCl formulation. This is one of the first attempts of studying the proving of traditional dough formulations through mathematical modeling. The experimental characterization, of the doughs, used in this work was straight forward and can be easily carried out in industrial settings, without the need of complex experimental techniques. The proving process was monitored by means of only one experimental measurement and was measured directly from the experimental set-up. The

dough properties calculated through the experimental, modeling and fitting methodologies used in this work were shown to have an acceptable reproducibility. Although the dough properties obtained in this work are not directly comparable to previously reported data, precisely because the rheological data, bubble size and distribution, and fermentation kinetics have usually been studied with basic water–flour– yeast–NaCl formulations, the obtained values fall inside the range of values reported in the literature, for dough systems. Correlations between the composition of common dough formulations and the dough properties and growth kinetics during typical proving conditions were also established in this work. The observed tendencies and correlations between dough properties and dough formulation follow the trends observed in experimental data reported in the literature for dough systems.

References Almagro, A., Prista, C., Castro, S., Quintas, C., Madeira-Lopes, A., Ramos, J., LoureiroDias, M.C., 2000. Effects of salts on Debaryomyces hansenii and Saccharomyces cerevisiae under stress conditions. International Journal of Food Microbiology 56, 191–197. Amon, M., Denson, C.D., 1984. A study of the dynamics of the growth of closely spaced spherical bubbles. Polymer Engineering Science 24, 1026–1034. Babin, P., Della Valle, G., Chiron, H., Cloetens, P., Hoszowaska, J., Pernot, P., Réguerre, A.L., Salvo, L., Dendeviel, R., 2006. Fast X-ray tomography analysis of bubble growth and foam setting during breadmaking. Journal of Cereal Science 43, 393–397. Bache, I.C., Donald, A.M., 1998. The structure of the gluten network in dough: a study using environmental scanning electron microscopy. Journal of Cereal Science 28, 127–133. Bikard, J., Coupez, T., Della Valle, G., Vergnes, B., 2008. Simulation of bread making process using a direct 3D numerical method at microscale. Part I: Analysis of foaming phase during proofing. Journal of Food Engineering 85, 259–267. Bloksma, A.H., 1990. Rheology of the breadmaking process. Cereal Foods World 35, 228–236. Chiotellis, E., Campbell, G.M., 2003. Proving of bread dough I: Modelling the evolution of the bubble size distribution. Transactions of the Institution of Chemical Engineers, Part C, Food and Bioproducts Processing 81, 194–206. De Cindio, B., Correra, S., 1995. Mathematical modelling of leavened cereal goods. Journal of Food Engineering 24, 379–403. Dubriel, L., Mélinole, S., Chiron, H., Compoint, J.P., Quillen, L., Branlard, G., Marion, D., 1998. Effect of puroindolines on the bread making properties of wheat flour. Cereal Chemistry 75 (2), 222–229. Epstein, P.S., Plesset, M.S., 1950. On the stability of gas bubbles in liquid–gas solutions. Journal of Chemical Physics 18 (11), : 1505–1509. Frgestad, E.M., Molteberg, E.L., Magnus, E.M., 2000. Interrelationships of protein composition, protein level, baking process and the characteristics of hearth bread and pan bread. Journal of Cereal Science 31, 309–320. Favelukis, M., Albalak, R.J., 1996. Bubble growth in viscous Newtonian and nonNewtonian liquids. The Chemical Engineering Journal 63, 149–155.

448

A. Córdoba / Journal of Food Engineering 96 (2010) 440–448

Gan, Z., Angold, R., Williams, M.R., Ellis, P.R., Vaughn, J.G., Galliard, T., 1990. The microstructure and gas retention of dough. Journal of Cereal Science 12, 15–24. Hailemariam, L., Martin, O., Campanella, O., 2007. A mathematical model for the isothermal growth of bubbles in wheat dough. Journal of Food Engineering 82, 466–477. Kokelaar, J.J., Prins, A., 1995. Surface rheological properties of bread dough components in relation to gas bubble stability. Journal of Cereal Science 22, 53–61. Martin, P.J., Chin, N.L., Campbell, G.M., Morrant, C.J., 2004. Aeration during bread dough mixing. III. Effect of scale-up. Transactions of the Institution of Chemical Engineers, Part C, Food and Bioproducts Processing 82, 282–290. Mills, E.N.C., Wilde, P.J., Salt, J., Skeggs, P., 2003. Bubble formation and stabilization in bread dough. Transactions of the Institution of Chemical Engineers, Part C, Food and Bioproducts Processing 81, 189193. Örnebro, J., Nylander, T., Eliasson, A.C., 2000. Critical review – Interfacial behaviour of wheat proteins. Journal of Cereal Science 31, 195–221. Penner, A., Hailemariam, L., Okos, M., Campanella, O., 2009. Journal of Cereal Science 49, 65–72.

Preston, K.R., 1981. Effects of neutral salts upon wheat gluten protein properties. I. Relationship between the hydrophobic properties of gluten proteins and their extractability and turbidity in neutral salts. Cereal Chemistry 58, 317–324. Robinson, A.J., Judd, R.L., 2004. The dynamics of spherical bubble growth. International Journal of Heat and Mass Transfer 47, 5101–5113. Romano, A., Toraldo, G., Cavella, S., Masi, P., 2007. Description of leavening of bread dough with mathematical modeling. Journal of Food Engineering 83, 142–148. Rouillé, J., Della Valle, G., Lefebvre, J., Sliwinski, E., van Vliet, T., 2005. Shear and extensional properties of bread doughs affected by its minor components. Journal of Cereal Science 42, 45–57. Scriven, L., 1959. On the dynamics of phase growth. Chemical Engineering Science 10, 1–13. Shah, P., Campbell, G.M., McKee, S.L., Rielly, C.D., 1998. Proving of bread dough: modelling the growth of individual bubbles. Transactions of the Institution of Chemical Engineers, Part C, Food and Bioproducts Processing 76, 73–79. Venerus, D.C., Yala, N., Bernstein, B., 1998. Analysis of diffusion-induced bubble growth in viscoelastic liquids. Journal of Non-Newtonian Fluid Mechanics 75, 55–75.