A mechanistic model for baking of leavened aerated food

Journal of Food Engineering 143 (2014) 80–89

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

A mechanistic model for baking of leavened aerated food Ganesan Narsimhan ⇑ Department of Agricultural and Biological Engineering, Purdue University, West Lafayette, IN 47907, United States

a r t i c l e

i n f o

Article history: Received 14 January 2014 Received in revised form 21 June 2014 Accepted 24 June 2014 Available online 2 July 2014 Keywords: Baking Aerated food Leavened baked food Dough rise Density profile Bubble expansion

a b s t r a c t A mechanistic model for baking of leavened aerated food is proposed. The model accounts for heat conduction, moisture diffusion, diffusion of CO2 that is produced by fermentation, and the resistance to bubble expansion by the viscoelasticity of the dough which is described by Oldroyd B constitutive equation. Unsteady state heat conduction and moisture diffusion equations are solved accounting for bubble expansion due to heating, evaporation of moisture as well as diffusion of CO2 that is produced by fermentation to obtain the evolution of temperature, moisture and air volume fraction profiles as well as dough rise. The bubble expansion due to CO2 diffusion is coupled to temperature and moisture profiles. The model predicts that the growth of bubble exhibits a lag time followed by an exponential growth phase consistent with experimental observations. Eventhough the surface region is more expanded initially, at longer times, it is found to be more dense. The calculated air volume fraction profiles at longer times indicated an expanded inner region of more or less uniform density and a denser surface crust region of decreasing air volume fraction (or increasing density). The thickness of the crust region is found to increase with baking time. The average air volume fraction is found to increase with time because of the combined effects of air expansion as well as CO2 diffusion. The model predictions of dough rise as well as force exerted by the rising dough compare well with the experimental data (Singh and Bhattacharya, 2005; Romano et al., 2007). Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Baking is a common operation that is employed to make a variety of food products. Examples of some baked foods include cakes, muffins, and bread. The common feature of these baked goods is that air or carbon dioxide (or a mixture of the two) are incorporated in the food matrix either by mechanical or other means before the dispersion is subjected to baking in an oven. These products can be subdivided into unleavened and leavened baked foods. In leavened foods, CO2 is produced either due to fermentation or by chemical reaction during proofing followed by baking. In unleavened foods, only air is incorporated by mechanical means before baking. Baking has been described by phenomenological models in which simultaneous heat and mass transfer during baking have been modeled by unsteady state heat conduction and diffusion (Fan et al., 1999; Zhang and Datta, 2006; Sakin et al., 2007; Feyissa et al., 2011) accounting for evaporation of moisture. A general phenomenological formulation (Lostie et al., 2002; Zhang and Datta, 2006) has been presented for the diffusion of liquid water and water vapor as well as pressure flow of air through a ⇑ Tel.: +1 765 430 8136. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jfoodeng.2014.06.030 0260-8774/Ó 2014 Elsevier Ltd. All rights reserved.

polyphasic material during baking. Bread is modeled to consist of a crumb and a crust region with a moving interface (Lostie et al., 2004; Purlis and Salvadori, 2009; Purlis and Salvadori, 2010) at which evaporation of moisture is assumed to occur. Salient features of various models that have been proposed can be found in comprehensive reviews on baking (Sablani et al., 1998; Mondal and Datta, 2008). In an earlier work (Narsimhan, 2013), a model for baking of unleavened food was developed. Unsteady state heat conduction and moisture diffusion equations were solved accounting for bubble expansion due to heating as well as evaporation of moisture to obtain the evolution of temperature, moisture and air volume fraction profiles as well as cake rise. Growth of bubbles during proofing and baking occurs as a result of diffusion of the CO2 that is generated by fermentation through the viscoelastic dough into the bubbles (Chiotellis and Campbell, 2003) and bubble expansion also occurs due to heating during baking. The resistance from the surrounding viscoelastic medium has to be overcome by the expanding bubble. Consequently, the bubble growth involves coupled momentum, heat and mass transfer leading to a highly non-linear moving boundary problem. Scriven (1959) was the first to describe the growth of bubbles in an infinite medium of Newtonian liquid. Two types of simplifications have been made in his analysis. In the first, mass transfer was replaced by the imposition of constant pressure difference across the bubble

G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

81

Nomenclature Amax cp,eff D Deff E h(t) H keff Ks NCO2 pa pCO2 pg R R0 R_ Rg T T0 Tb x x2 y2

aeff c2 / /0

preexponential factor for the fermentation reaction for the production of CO2 effective heat capacity of aerated food diffusion coefficient of CO2 effective moisture diffusivity through dough activation energy for the fermentation reaction for the production of CO2 height of sample at time t Henry’s law constant effective thermal conductivity of dough constant in Monod equation for the rate of production of CO2 Number of moles of CO2 in bubble atmospheric pressure partial pressure of CO2 inside the bubble gas pressure inside the bubble bubble radius initial bubble radius rate of growth of bubble radius gas constant temperature initial temperature of the food oven temperature distance from the bottom of the sample mole fraction of solvent in the dough mole fraction of moisture in the bubble effective thermal diffusivity activity coefficient of water in dough volume fraction of vapor in dough initial volume fraction of vapor in dough

surface thus reducing the problem to that of momentum transfer in a viscoelastic medium (Scriven, 1959; street, 1968; Papanastasiou et al., 1984; Kim, 1994; Huang and Kokini, 1999). In the second, the growth of bubbles was assumed to be sufficiently slow so that the viscoelasticity of the medium could be neglected. Consequently, the resistance to the growth of bubbles was considered to be only due to surface tension (Shimiya and Yano, 1987, 1988; Shah et al., 1998). These coupled momentum and mass transfer equations were solved for an isolated bubble surrounded by a Newtonian fluid (Venerus and Yala, 1997) and for power law (Ramesh et al., 1991), viscoelastic (Venerus et al., 1998) and Oldroyd-B model (Feng and Bertelo, 2004) polymeric melts. The evolution of bubble size distribution during proofing of bread was predicted for a Newtonian liquid accounting for the production of CO2 by yeast (Chiotellis and Campbell, 2003). Nucleation of bubbles was accounted for in the analysis of bubble growth in the foaming of Oldroyd-B polymeric melt (Feng and Bertelo, 2004) and in the extrusion of wheat dough described by the Lodge model (Hailemariam et al., 2007). In an earlier work (Narsimhan, 2012), we have developed a model for growth of bubbles during proofing in an infinite viscoelastic dough that is described by Oldroyd-B model. The production of CO2 by yeast was described by a Monod model. This treatment was also extended to bubbles of finite volume fraction through a shell model. In this manuscript, a mechanistic model for baking of leavened aerated food is proposed. The viscoelastic dough is described by Oldroyd-B model. The model predicts bubble expansion during baking in aerated food as a result of heating, moisture evaporation as well as diffusion of CO2 produced by fermentation. The model is also able to predict the evolution of rise as well as density profile.

/⁄ k

Ca ¼ rlRD0

dimensionless vapor volume fraction relaxation time for viscoelastic dough maximum growth rate dough viscosity solvent viscosity surface tension bulk density of aerated food radial normal stress normal stress along tangential direction on a radial plane Capillary number

De ¼ kD R2

Deborrah number

Re ¼ qlD

Reynolds number

lmax lp l r qeff srr shh

0

l b ¼ ls

viscosity ratio

c ¼ cD=lmax R20 dimensionless CO2 concentration k⁄ = kRgT dimensionless Henry’s law constant pCO2 ¼ R20 pCO2 =lD dimensionless partial pressure of CO2 r⁄ = r/R0 dimensionless radial coordinate R⁄ = R/R0 dimensionless bubble radius _ dimensionless rate of growth of bubble radius R_  ¼ ðR0 R=DÞ t  ¼ Dt=R20 dimensionless time 4

lmax ¼ lmaxlRD02Rg T dimensionless maximum growth rate srr ¼ R20 srr =lD dimensionless radial normal stress shh ¼ R20 shh =lD dimensionless normal stress along tangential direction on a radial plane

The formulation of the model is given in the next section. The effects of different variables on the evolution of moisture and density profile as well as rise are discussed in the subsequent section. This is followed by comparison of model predictions with experimental data.

2. Model formulation Baking of leavened food (bread dough for example) will lead to an increase in the temperature of the dispersion due to heat transfer (either by natural or forced convection, depending on the type of oven) from the oven. This, in turn, lead to (a) increase in the viscosity of the matrix due to starch gelatinization and/or gelation of protein. (b) expansion of incorporated air bubbles, (c) evaporation of moisture from the batter into the bubbles as temperature increases, (d) production of CO2 due to fermentation of leavened food, (e) diffusion of CO2 from the medium into the air bubbles by diffusion and (f) loss of moisture from the dispersion due to evaporation from the surface. Let us consider a rectangular aerated food consisting of equal sized bubbles of radius R and volume fraction /. It is assumed that the bubbles are distributed uniformly. If the volume fraction of the bubbles is less than 0.74, the liquid food can be considered as air–liquid dispersion. On the other hand, if the volume fraction of air in the aerated food is greater than 0.74 so that the air bubbles are deformed into polyhedra, the liquid food is a foam. As will be seen later, even if the liquid food is a dispersion to begin with, in the course of baking, the air volume fraction increases because of (i) bubble expansion due to increase in temperature and (ii) loss of liquid during baking. This liquid food of uniform initial temperature T0 is exposed to an ambient

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temperature Tb in an oven during baking. The length, width and thickness of the sample are L, W and h0 respectively. It is assumed that h0  W and h0  L so that heat transfer can be considered one dimensional along x direction. x = 0 refers to the axis of symmetry with x = h0/2 referring to the surface. This will also be applicable to situation in which the second surface is insulated (instead of surface of symmetry). Following assumptions are made in the model: 1. The aerated food is leavened and initially consists of spherical air bubbles of the same size that are uniformly distributed. As pointed out below, this assumption can easily be relaxed to extend the current analysis to account for bubble size distribution. 2. The viscoelasticity of the medium is described by Oldroyd B constitutive equation. 3. The gas phase consisting of air, water vapor and CO2 is always dispersed consisting of spherical bubbles, i.e. bubble coalescence is neglected. Consequently, the formation of interconnected channels and the resulting transport of air and water vapor through these channels to the atmosphere is neglected. 4. The water vapor in the gas phase (bubbles) is assumed to be in equilibrium with the liquid phase. In the present analysis, the partial pressure (water activity) of water in the vapor phase is given by Raoults law for binary liquid solution consisting of water and a low molecular weight solute. This can be generalized by assuming appropriate moisture adsorption equilibrium. 5. The moisture diffusion coefficient through the liquid medium is inversely proportional to its viscosity. 6. Mass transfer from dissolved CO2 in the dough to the bubble is by diffusion. 7. The interface CO2 concentration is in equilibrium with the bubble and is given by Henry’s law. 8. The nucleation of water vapor bubbles in the liquid medium is neglected. This assumption is reasonable so long as the temperature of aerated food during baking does not approach the boiling point of water, i.e. the oven temperature is not very high. 9. The bubble is surrounded by infinite medium of dough, i.e. the bubbles in the dough are assumed to be infinitely dilute. Unsteady state heat conduction equation through the sample is described by (Narsimhan (2013)),

@T @2T ¼ aeff 2 @t @x

ð1Þ

where T is the sample temperature at any location x at time t and

  @x2 @ @x2  R_ ev  R_ gel ¼ Deff @x @t @x

where x2 is the mole fraction of water in the solution and Deff is the effective diffusion coefficient for the solvent, R_ ev refers to the rate of moisture evaporation per unit volume of the liquid phase and R_ gel refers to the rate of water uptake (moles of water per mole of continuous phase) by starch granules due to starch gelatinization. Therefore, the rate of depletion of solvent (moisture) from the solution to the bubbles in moles/s is given by,

  @ / c2 x2 Psat ðTÞ R_ ev ¼ @t ð1  /Þ Rg T

t ¼ 0 T ¼ T 0 8x @T x¼0 ¼ 0 symmetry @x hðtÞ @T x¼  keff ¼ hc ðT  T b Þ 2 @x

ð6Þ

One can assume that the vapor is an ideal gas in equilibrium with the solution so that the partial pressure of solvent (water) in the vapor p2(T) at temperature T is given by,

p2 ðTÞ ¼ y2 P0 ¼ c2 x2 Psat ðTÞ

ð7Þ

In the above equation, c2 is the activity coefficient of water in the liquid containing solutes. The activity coefficient will be a function of mole fraction of water as well as temperature. Eventhough the continuous phase is a multicomponent system consisting of water, low molecular weight solute (salt or sucrose), starch, and protein (glutenin or gliadin), it can be considered to be a two component sucrose solution because of high molecular weight of protein and low solubility of starch in water. Because of swelling of starch granules, the amount of free solute that is available will decrease with temperature. One can assume that there is negligible partitioning of solute inside and outside the starch granules, i.e. the solute concentration inside and outside the swollen starch granules are equal. The water activity coefficient in sucrose solution can be expressed by generalized Marguelles equation as,

ln c2 ¼ aðhÞ

4 X bk2 xk1

ð8Þ

k¼2

where x1 is the mole fraction of sucrose, h = T/298 and

aðhÞ ¼

a0 þ a1 þ a2 ln h þ a3 h þ a4 h2 h

ð9Þ

The values of the constants ak and bk can be found elsewhere (Starzak and Mathlouthi, 2006). As pointed out earlier, the above equation for partial pressure of water vapor (water activity) can be replaced by an appropriate moisture adsorption isotherm equation. The thermal conductivity will depend on the air volume fraction of the sample and is defined as,

aeff is the thermal diffusivity of the sample. The initial and boundary

aeff ¼

conditions are given by,

ð5Þ

keff

qeff cp;eff

; keff ¼ kð1  /Þ; qeff ¼ qð1  /Þ; cp;eff ¼ cp ð1  /Þ

ð2Þ

ð10Þ

ð3Þ

where k, q and cp refer to the thermal conductivity, density and heat capacity of liquid respectively. The variation of effective thermal conductivity with air volume fraction and temperature during baking is neglected and an average value is taken. Psat(T) can be found elsewhere (http://www.engineeringtoolbox.com/watervapor-saturation-pressure-air-d_689.html). The rate of water uptake by starch granules due to gelatinization can be expressed as (Zanoni et al. (1995)),

ð4Þ

where keff is the effective thermal conductivity of the sample, hc is the heat transfer coefficient. It is to be recognized that the thickness of the sample h(t) will change with time as a result of (i) expansion of the bubbles resulting from heating and (ii) evaporation of moisture from the surface as discussed below. As pointed out above, during baking of air–liquid dispersion, there is a loss of water from the surface due to evaporation. In addition, water will evaporate into the bubbles the extent of which will depend on the temperature and moisture content. Since the latter occurs throughout the volume of the dispersion, it can be considered as a volumetric rate of depletion of liquid water. An unsteady state mass balance for the solvent yields (Narsimhan, 2013),

  Eg M av SðT  T gel Þmg ws R_ gel ¼ kg0 exp  Rg T Mw

ð11Þ

where Tgel is the starch gelatinization temperature, S is a step function, kg0 and Eg are preexponential factor and activation energy respectively for starch gelatinization kinetics, mg is the water uptake per unit weight (wt/wt) of starch granules, Mw is the molecular weight of water, ws and Mav are the weight fraction of starch

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G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

and average molecular weight respectively. The effective diffusion coefficient through voids in a medium consisting of random packing of spherical particles (air bubbles) is given by (Weisenberg (1963)),

Deff ¼ Dl ðTÞStð/Þ ¼

Dl ðTÞð1  /Þ 1  12 lnð1  /Þ

ð12Þ

where Dl is the diffusion coefficient of solvent (water) in the batter and / is the air volume fraction and the structure factor  Stð/Þ ¼ ð1  /Þ= 1  12 lnð1  /Þ . The diffusion coefficient of moisture in the dough Dl varies inversely with the viscosity. Consequently, the variation of rheological properties of the dough with temperature should be known in order to evaluate Dl at different temperatures. Consequently, the dimensionless diffusion coefficient D⁄ = D/D35 is related to g⁄ = g/g35 via

D ¼

1

ð13Þ

g



As a result of change in rheology of batter during baking, the diffusion coefficient of water through the batter is a function of temperature and therefore a function of position. Also, because of moisture profile within the product during baking (with the surface being drier than the interior), the number of moles that would evaporate into the bubble will be higher in the interior as can be seen from Eq. (6). As a result, as will be discussed below, the bubble expansion will be more pronounced for interior bubbles thus resulting in larger bubble size and hence larger air volume fraction. This, in turn, will lead to a gradient of air volume fraction within the sample. Consequently, Eq. (5) can be recast as,

! # @ 2 x2 @Dl @T @x2 @x2  Dl ðTÞ Dl ðTÞ 2 þ @x @T @x @x @x   sat @ @ / c2 x2 P ðTÞ  kg0 Stð/Þ   @x @t ð1  /Þ Rg T    Eg M av S T  T g mg ws  exp  Rg T Mw

@x2 ¼ Stð/Þ @t

"

ð14Þ

ð15Þ ð16Þ

ð17Þ

srr  shh r

! þ

2r 4ls ðTÞR_ þ 2 R R

Z

1

R

dr

ð19Þ

_ are the instantaneous pressure inside the where pg(t) , R(t) and RðtÞ bubble, bubble radius and rate of growth of the bubble radius respectively, ls(T) is the Newtonian viscosity at temperature T, r is the surface tension, and srr and shh are the viscoelastic normal stress components respectively. The last term in the above equation accounts for the resistance due to viscoelasticity. The viscoelasticity of the dough is expressed by Oldroyd B model. Of course, in case of Newtonian behavior, the last term in the above equation is zero. The change in the pressure inside the bubble is due to evaporation of moisture as well as diffusion of CO2, i.e.

pg ðtÞ  pg ð0Þ ¼ fpH2 O ðtÞ  pH2 O ð0Þg þ fpCO2 ðtÞ  pCO2 ð0Þg þ fpair ðtÞ  pair ð0Þg

ð20Þ

fpH2 O ðtÞ  pH2 O ð0Þg þ fpCO2 ðtÞ  pCO2 ð0Þg þ fpair ðtÞ  pair ð0Þg !   Z 1 dR_ 3 _ 2 1 1 4l ðTÞR_ þ s þ R þ 2r  2 ¼q R dt 2 R R R0 R

srr  shh r

dr

The bubbles would expand when heated. The expansion would also occur due to moisture evaporation and diffusion of CO2 produced by fermentation. The fermentation of dough can be expressed by Monod type of equation as (Bailey and Ollis (1977)),

ð18Þ

where s is the substrate (dough) concentration, lmax is the maximum growth rate and Ks is a constant. Usually s  Ks so that Eq. (18) becomes,

ð18aÞ

ð21Þ

The mass balance for the bubble yields,

pH2 O ¼ c2 x2 Psat ðTÞ

 dNCO2 4 p d

@c pCO2 R3 ¼ DCO2 4pR2 ¼ 3 Rg T dt @r r¼R dt

ð22Þ ð23Þ

In the above equation, NCO2 is the number of moles of CO2 inside the bubble, Rg is the gas constant, T is the temperature, DCO2 is the diffusion coefficient of CO2 through the leavened food and c is the concentration of CO2 in the food with the last term on the right hand side of Eq. (23) referring to the rate of diffusion of CO2 from the food into the bubble. Eq. (23) can be recast as,

dpCO2 3Rg T pCO2 @T @c 3 dR DCO2  p þ ¼ R @r r¼R R dt CO2 dt T @t

3. Bubble expansion

d½CO2   lmax dt





In the above equation, km is the mass transfer coefficient, RH is the relative humidity of the oven and Psat(T) is the vapor pressure of water at temperature T.

d½CO2  lmax s ¼ dt Ks þ s

dR_ 3 _ 2 pg ðtÞ  pa ¼ q R þ R dt 2

where pH2 O ðtÞ and pCO2 ðtÞ refer to the partial pressure of water vapor and CO2 at time t respectively. Since, pg ð0Þ  pa ¼ 2Rr0 , Eq. (20) can be rewritten as,

The initial and boundary conditions are given by,

t ¼ 0 x2 ¼ x02 @x2 x¼0 ¼0 @x h ð1  /Þ @x c 2 x¼  Dl ðT sur Þ  2 1  12 lnð1  /Þ @x   c x2 Psat ðT sur Þ Psat ðT b Þ  RH ¼ km 2 RT sur RT b

CO2 that is produced will diffuse through bread (a viscoelastic medium) into the bubbles that are aerated thus resulting in growth of these bubbles over time. Of course, the rate of production of CO2 lmax will depend on the temperature and more importantly on the concentration of the leavening agent in the dough. The temperature dependence of lmax can be expressed in terms of activation energy E as lmax = Amaxexp(E/RT). The momentum equation for growth of the bubble is given by (Feng and Bertelo (2004)),

ð24Þ

Recognizing that the number of moles of air within the bubble remains constant and from ideal gas law, one obtains,

pair ¼ patm

 3   R0 T T0 R

ð25Þ

The mass balance equation for CO2 in the food is given by,

  @c R2 R_ @c DCO2 @ @c þ lmax ðTÞ r2 þ 2 ¼ 2 @t @r r @r r @r

ð26Þ

The normal stresses are governed by the following Oldroyd-B constitutive equations (Feng and Bertelo, 2004),

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G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

!



4lp ðTÞR2 R_ ¼ r3 ! 2lp ðTÞR2 R_ @ shh R2 R_ @ shh 2R2 R_ shh þ k þ 2 þ 3 shh ¼ r r @t @r r3 @ srr R2 R_ @ srr 4R2 R_ srr þ k þ 2 þ 3 srr r r @t @r

ð27Þ ð28Þ

The viscosity of food lp(T) = Apexp(Ep/RgT) is a function of temperature where Ep is the activation energy and Ap is a constant. The initial and boundary conditions for the above equations are given by,

Rð0Þ ¼ R0 _ Rð0Þ ¼0

ð29Þ

pCO2 ð0Þ ¼ pCO2 ;0

ð31Þ

cðr; 0Þ ¼ c0 ðrÞ r P R0

ð32Þ

srr ð0; rÞ ¼ srr;0 r P R0 shh ð0; rÞ ¼ shh;0 r P R0

ð33Þ ð34Þ

cðR; tÞ ¼ kðTÞpg ðtÞ

ð35Þ

cðr; tÞ ¼ cb ðtÞ at r ! 1

ð36Þ

ð30Þ

dsrr ¼ 0 at r ! 1 dr dshh ¼ 0 at r ! 1 dr dcb  lmax ðTÞ ¼ A expðE=Rg TÞ dt

ð37Þ ð38Þ ð39Þ

dR _  ¼ R dt

ð43Þ

  1 dR_  pCO2 l Pl Tr 3 R_ 2 psat þ 3  1   ¼  þ    2 Re R Re R DCO2 M dt 2M 2 R R   R_  l 2l 1 4b     1  M Re R2 DCO CaRe R DCO2 R 2 Z 1  2l srr  shh  þ dr Re R DCO2 R r  3pCO2 R_  lmax @c pCO2 @T  1 dpCO2 ¼  þ þ     M dt M R R @r r ¼R MT  @t Psat ¼

c2 x2 Psat ðTÞ

ð44Þ

ð45Þ ð46Þ

Patm

 R2 R_  DCO2 @c 1 @c @ 2 c 2DCO2 @c  þ D þ þ1 CO  ¼  2 M @t @r  @r 2 r  @r  Mr2

ð47Þ

 1 @ srr R2 R_  @ srr 4R2 R_   4DeDCO2 b R2 R_     DeD s  s  CO rr rr  ¼  2 3 2 M @t M r 3 Mr @r  Mr ð48Þ

2DeDCO2 b R2 R_  1 @ shh R2 R_  @ shh 2R2 R_     ¼   DeD s  s þ hh CO2 hh M @t  M r 3 Mr 2 @r  Mr 3 ð49Þ With the initial and boundary conditions,

4. Dimensionless equations In order to non-dimensionalize the above equations, we choose two length scales, namely, (i) h0/2 for heat conduction and moisture diffusion equations and (ii) R0 for bubble growth due to 2 leavening, the timescale as h0 =4aeff , normalize partial pressure of CO2 and stress with respect to l0 DCO2 =R20 ; l0 ¼ ls0 þ lp0 , and CO2 concentration with respect to lmax R20 =DCO2 to yield the following dimensionless variables,

T ¼

Tb  T ; Tb  T0



R ¼ R=R0 ;

t ¼

4aeff t 2 h0

;

x ¼

 2_ R ¼ h0 R=4R 0 aeff ; _

2x ; h0 pCO2



h ðt Þ ¼ ¼

hðtÞ ; h0

R20 pCO2 =

r  ¼ r=R0 ;

lDCO2 ;

c ¼ cDCO2 =lmax R20 ;  rr

s ¼

R20 rr =

 hh

s lDCO2 ; s ¼

t ¼ 0 T ¼ 1

ð50Þ

t ¼ 0 x2 ¼ x20

ð51Þ



R20 hh =

s lDCO2 ð40Þ

The dimensionless equations are,

@T  @ 2 T  ¼ 2 ð41Þ @t  @x   / c2 p2 / x2 p2 @ c2 @x2 1 þ Mv þ Mv ð1  /Þ ðT=T b Þ ð1  /Þ ðT=T b Þ @x2 @t  " 2 Dl0 @ x2 @D @T  @x2 Dl ðT  ÞStð/Þ 2 þ Stð/Þ l ¼ aeff @x @T @x @x      @ @T @x2 Dl0 / c2 x2 @p2 @T  Mv þ  fStð/ÞgDl    @x ð1  /Þ ðT=T b Þ @T  @t @x @x aeff   / c2 x2 p2 @T 1 c2 x2 p2 @/    þ  2 2  ð1  /Þ ðT Þ ð1  T 0 =T b Þ @t ð1  /Þ f1  T ð1  T 0 =T b Þg @t   / x2 p2 @ c2 @T þ ð1  /Þ f1  T  ð1  T 0 =T b Þg @T  @t

  Eg 1 1 M av SðT  T g Þmg ws ð42Þ  SGexp    Rg T b  T ðT b  T a Þ T g Mw

@T  ¼0 @x @x2 ¼0 x ¼ 0 @x   @T  h0    hT ¼ Nu x ¼ h ð t  Þ @x 2L   sat    @x2 P ðT sur Þ Tb   R ¼ Pe M c x x ¼ h v H 2 2 @x T sur Psat ðT b Þ x ¼ 0

R ð0Þ ¼ 1 R_  ð0Þ ¼ 0

ð52Þ ð53Þ ð54Þ ð55Þ ð56Þ ð57Þ

pCO2 ð0Þ ¼ 0

ð58Þ

c ðr ; 0Þ ¼ 0 srr ð0Þ ¼ 0

ð59Þ ð60Þ

shh ð0Þ ¼ 0 

c ðR ; t  Þ ¼



k



lmax

 pCO2 ðt Þ

ð61Þ ð62Þ



@c ¼ 0 at r ! 1 @r  @ srr ¼ 0 at r  ! 1 @r  @ shh ¼ 0 at r  ! 1 @r

ð63Þ ð64Þ ð65Þ

In the above equations, the dimensionless groups are defined as, qDCO2 ;0 lDCO2 ;0 kDCO2 ;0 l Re ¼ ; Ca ¼ ; De ¼ ; b ¼ s ; Nu ¼ hc L=k; l0 rR 0 l R20

Pe ¼ km h0 =2Deff ðT b ; /0 Þ; M¼



R0 h0

2

lmax ¼

lmax R40 Rg T   ; k ¼ kRg T; h ¼ h=h0 ; l0 D2CO2 ;0

  2 4aeff p R2 h kg0 Eg ; Mv ¼ v m =v ; P ¼ atm 0 ; SG ¼ 0 exp  DCO2 ;0 l0 DCO2 4aeff Rg T g ð66Þ

G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

where L is the length of the sample, vm and v are molar volumes of liquid water and water vapor at Tb respectively. 5. Evaporation of moisture A mass balance for water gives,



 h km vm 2 RT  sat   sat  Z t P ðT sur Þ P ðT b Þ  RH dt  c2 x2 jx¼h=2 RT sur RT b 0 / 1  v m c2 x2 Psat ðTÞ ð1  /Þ Rg T

v l ðx; tÞ ¼ 1  d

x

ð67Þ

where d is the direc delta function and the dimensionless volume fraction of liquid v l ðx ; t  Þ at x⁄ and t⁄ is defined as vl(x, t)/ (1  /0). In the above equation, vm is the molar volume of liquid, the second term on the right hand side refers to the loss of water from the surface due to evaporation and the third term refers to the evaporation of water into the air bubbles. Therefore, the dimensionless volume of liquid V l ðt  Þ defined as Vl(t)/Vl0 is given by,

V l ðt Þ ¼

Z

h

0

transfer coefficients for natural convection were evaluated using equations given elsewhere (Narsimhan, 2013). The partial differential equations were solved by the method of lines in which the unknown functions were expressed in terms of cubic Hermite polynomials with 21 collocation knots. These equations were solved with a dimensionless time step Dt⁄ = 0.01. The accuracy of the profiles were found to be maintained as the time step values were increased from 105 to 0.01. The solution of temperature and moisture profiles are then employed in Eqs. (43)–(49) to solve for the evolution of bubble radius at a fixed location. This is a mixture of ordinary differential equations and partial differential equations. The ordinary differential Eqs. (43)–(45) are solved by 5th order Runge Kutta and the partial differential equations (46)– (49) are solved by the method of lines. Because of the disparity in the two timescales (for ordinary and partial differential equations), the problem is stiff. Therefore, the discretization timestep for the ordinary differential equations is an order of magnitude (107) smaller than that for partial differential equations. The solution is then employed in the evaluation of air volume fraction profile /ðx ; t Þ as given by Eq. (69) which is then compared with the previous profile. If the two agree within a prescribed limit of the error err defined as

v l ðx ; t Þdx

km ¼1 vm RT  sat   sat  Z t  P ðT sur Þ P ðT b Þ  RH dt  c2 x2 jx¼h=2 RT sur RT b 0  Z h / 1  v m c2 x2 Psat ðTÞ dx ð1  /Þ Rg T 0

err ¼

ð68Þ

Therefore, the air volume fraction /(x, t) at position x and time t is given by, 3

/ðx ; t  Þ ¼ /0 / ðx ; t Þ ¼ fR ðx ; t  Þg /0

ð69Þ

85

 Z T 1=2 1  ð/ðt Þ  /pr ðt  ÞÞ2 dt T 0

ð72Þ

the calculations are stopped. However, if the calculated err is not within the prescribed limit, the updated air volume fraction profile /ðx ; t  Þ is then employed in Eq. (42) to recalculate the moisture profile x2 ðx ; t  Þ which is then used to resolve Eqs. (43)–(49). This iteration is continued until convergence. Typical evolution of air volume fraction at different dimensionless vertical locations is shown in Fig. 1. At small times, the surface region of the sample is more expanded than the inner region because of higher temperature. At larger times, however, the

The volume of air is therefore given by,

V air ðt  Þ ¼

Z

h ðt  Þ

/ðx ; t Þdx



ð70Þ

0

From a volume balance one obtains, 

h ðt Þ ¼ /0 V air þ ð1  /0 ÞV l

ð71Þ

The equations for the prediction of heat and mass transfer coefficients during baking for natural and forced convection ovens are given elsewhere (Narsimhan, 2013). 6. Results and discussion The evolution of dimensionless temperature and moisture profiles (as given by Eqs. (41) and (42)) are coupled with the growth of bubbles due to combined effects of fermentation and moisture evaporation (as given by Eqs. (43)–(49)). Eqs. (41) and (42) give evolution of temperature and moisture profiles respectively in terms of dimensionless vertical coordinate x⁄ whereas Eqs. (43)– (49) give the evolution of bubble radius and CO2 concentration profile (in terms of dimensionless radial coordinate r⁄). Therefore, there are two lengthscales involved in the calculations. The first length scale pertains to the sample dimension while the second depends on the bubble size. The set of equations are, therefore, stiff and very difficult to solve. In order to circumvent this difficulty, an iterative scheme is devised. The dimensionless temperature and concentration profiles were first obtained by solving Eqs. (41) and (42) with initial and boundary conditions (50)–(65) by neglecting the effect of CO2 fermentation of bubble growth. The air volume fractions were updated using Eqs. (67)–(70). The heat and mass

Fig. 1. The evolution of dimensionless air volume fraction profile at different dimensionless locations x, x = 0 and x = 1 denoting the insulated bottom (or plane of symmetry) and the top respectively. Panel (b) refers to expanded plot for larger times (above 40 min). The parameters are: R0 = 5  103 m, /0 = 0.1, xw = 0.38, T0 = 308 K, Tb = 449.67 K, l = 1320 Pa s, k = 1.4115  104 s, lmax ¼ 3:096  108 , b = 0.1, h0 = 5  102 m, r = 30 mN/m.

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G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

Fig. 2. Dimensionless air volume fraction profile at different times. The parameter values are the same as that given in Fig. 1.

surface region becomes less expanded than the inner region because of lower moisture which in turn leads to less moisture vaporization into the bubbles (less expansion). This can be more clearly seen in Fig. 1b where the cross over in air volume fraction for the surface regions occurs. The cross over time is lower for surface region and occurs at increasingly longer times for inner regions. As a result, the surface crust region is built up only at longer times as can be seen from Fig. 2 which shows the air volume fraction profile within the sample at different times. At smaller times, the air volume fraction is more or less uniform within the sample though the value increases with time. At larger times, however, the surface dense layer is formed as evidenced by a sharp decrease in surface air volume fraction. The crust region can clearly be identified at 60 min baking time. At small times, the air volume fraction is highest at the surface increasing from the inner regions as a result of more bubble expansion at higher temperature. The surface air volume fraction increases much slower than the inner regions. Above a cross over time, however, the surface air volume fraction becomes smaller than the inner values as a result of buildup of denser surface crust. The integration of the profile gives the average air volume fraction in the sample at different times as shown in Fig. 3. As expected, the average air volume fraction increases with time due to the combined effects of moisture evaporation and CO2 diffusion. The evolution of average air volume fraction for different fermentation rates (values of dimensionless number lmax ) is shown in Fig. 4. The air volume fraction increases very slowly until one reaches a critical time beyond which the increase is dramatic as a result of expansion of air bubbles. This behavior is consistent with the experimental observations of Singh and Bhattacharya (2005) who monitored the expansion of dough during baking using flooded plane parallel plate geometry viscometer. As expected, the increase in air volume fraction is higher at higher fermentation rates (higher values of dimension-

Fig. 3. The evolution of average dimensinless air volume fraction. The parameter values are the same as that given in Fig. 1.

Fig. 4. The evolution of dimensionless air volume fraction for three different values of lmax . Other parameter values are the same as that given in Fig. 1.

Fig. 5. Evolution of bubble size for different initial bubble sizes as predicted by the model. Other model parameters are the same as those given in Fig. 1.

less number lmax ) (see Fig. 4). The effect of initial bubble size on the growth of bubbles during baking at the center of the sample is shown in Fig. 5. Consistent with reported experimental findings and the results on bubble growth during proofing (Narsimhan, 2012), there is a lag time during which an insignificant increase in bubble size is observed, followed by exponential growth. This lag time was found to be smaller for larger initial bubble sizes. Unlike the model predictions during proofing, a cross over in bubble growth was not observed. Such a behavior is believed to be due to the combined effects of diffusion of CO2 produced by fermentation and temperature increase due to heating. The effects of De and l on bubble growth were found to be similar to earlier reported results during proofing (Narsimhan, 2012) and therefore are not presented here. The effect of oven temperature on bubble growth is shown in Fig. 6. As expected, the lag time for bubble growth is smaller for higher oven temperature. In addition, the rate of bubble growth in the second exponential phase was also higher at higher

Fig. 6. Evolution of dimensionless bubble size for three different baking temperatures in K as predicted by the model. Other model parameters are the same as those given in Fig. 1.

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G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

oven temperature. This can be explained as due to faster rate of fermentation and diffusion of CO2 at higher temperatures.

7. Comparison with experiments Romano et al. (2007) measured the height vs time during proofing of wheat dough at different yeast concentrations. They also reported the Micheales-Menton rate constant at different yeast concentrations. Rheological parameters (G0 and tan d) of wheat dough as reported by Upadhyay et al. (2012) were employed in the model. G0 = 2.38  105 Pa and tand = 0.2 at a frequency of 10 rad/s gave a complex viscoisity of 2.427  104 Pa s. The other parameters used in the model are given in the caption to Fig. 7. However, Romano et al. (2007) did not report the bubble size. The present model was applied to proofing using the experimental parameters for different initial bubble sizes. Eventhough there will be a distribution of bubble sizes in the real system, the model predictions are made for a single bubble size. The extension of the current model for a bubble size distribution is discussed in the subsequent section. Since temperature was maintained constant during proofing, only Eqs. (43)–(49) were solved with the initial and boundary conditions (56)–(65). The experimental volume expansion plateaued at sufficiently long times. Such a behavior was attributed by Romano et al. (2007) to loss of gas from the dough. The present model does not account for this loss. Consequently, the model could not be compared with the experimental data for sufficiently long times. The comparison of the predicted evolution of volume change with the experimental data for different yeast concentration at smaller times is shown in Fig. 7. As pointed out above, the average bubble size was a parameter that was assumed in the model for best fit of the experimental data. The agreement between the experimental data and model predictions is found to be good. The bubble sizes for which the model predictions agree with the data decreases at higher yeast concentrations. This may be because of different bubble size distributions at different yeast concentrations with larger bubbles at lower concentrations. Singh and Bhattacharya (2005) monitored the expansion of dough during baking between two parallel plates by measuring the displacement of movable top plate which was spring loaded. They also measured the normal force exerted by the expanding dough. The normal force was found to increase almost linearly

until a certain time beyond which the force exerted decreased precipitously. The dough expansion, however, was found to increase dramatically around the starch gelatinization temperature eventually leveling off to a constant value at large times. They attributed the sudden drop in normal force to cell opening as a result of formation of open cells at high air volume fractions. They correlated the dough expansion to measured G0 and G00 during baking. Typical parameters for wheat dough (Chiotellis and Campbell, 2003) are assumed for model predictions. Singh and Bhattacharya (2005) did not report bubble size or air volume fraction in their experiments. In addition, their experiments were conducted in a flooded parallel plate geometry which allowed for some moisture evaporation during baking. Since heat transfer to the dough during baking was from the top and bottom plates, a heat transfer coefficient was fitted by minimizing the error between the calculated and measured (reported by Singh and Bhattacharya (2005)) temperature vs time data at midpoint between two plates. Also, the dough expansion was calculated for different assumed values of initial air volume fraction. It was found that the results were not sensitive to assumed bubble size. These calculations assumed that the top surface of the dough was free to expand which was not the case in the actual experiments since there was some resistance to dough expansion by the restraining spring loaded top plate. In the experiments, the dough expansion is restricted by the presence of spring loaded top plate. The effect of spring loaded top plate on dough expansion is accounted for as follows. A force balance for the top plate due to expanding dough is written as, 2

m

d y dt

2

¼ F n ðtÞ  kspring y

ð73Þ

where m is the mass of dough between the two plates, y is the displacement of the top surface from its initial position at time t as measured by the experiment, Fn(t) is the normal force exerted by the dough and kspring is the spring constant. The second term is the restraining force due to spring. The spring constant is usually very small in order to provide free movement of the dough. Now, the normal force Fn(t) is given by, 2

m

d y0 dt

2

¼ F n ðtÞ

ð74Þ

where y0 is the displacement of unrestrained expanding dough which is predicted by the model. Recasting the above two equations in terms of dimensionless variables, y⁄ = y/yinit, t⁄ = t/s, s being the characteristic time and small perturbation parameter e ¼ kspring s2 =m, one obtains, 2

d y dt

2

2

¼

d y0 dt

2

 ey

ð75Þ

Expanding the displacement in terms of perturbation parameter

e as y ¼ y0 þ ey1 , substituting and equating terms of the same order, one obtains the following for the first order perturbation, 2

d y1 dt

2

¼ y0

ð76Þ

whose solution is given by Fig. 7. Comparison of experimental data of dimensionless change in bread volume vs time during proofing as obtained by Romano et al. (2007). The parameters that were employed in the model as well as the bubble sizes /0 = 0.1, xw = 0.4, T0 = 309 K, l = 2427 Pa s, k = 0.5 s, b = 0.1, r = 30 mN/m. Yeast concentrations corresponding to different data points are: j 3.4%; r 1.7%; N 1.1%; and 0.60%. The values of lmax for different yeast concentrations as obtained from the data of Romano et al. (2007) are: 3.4% – 1.16  103 s1; 1.7% – 8.6  104 s1; 1.1% – 6.38  104 s1; 0.60% – 3.47  104 s1. The fitted average bubble sizes for the model at different yeast concentrations are: 3.4% – 6  104 m; 1.7% – 1.2  103 m; 1.1% – 1.5  103 m; 0.60% – 2.0  103 m

y1 ¼

Z 0

t

Z

t 1 0





y0 dt2 dt 1

ð77Þ

As a result, the measured displacement is given by,

y ¼ y0  e

Z 0

t

Z 0

t 1





y0 dt 2 dt 1

ð78Þ

where the first order displacement y0 is given by the model. As expected, the measured displacement decreases as the spring

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G. Narsimhan / Journal of Food Engineering 143 (2014) 80–89

constant (or equivalently, the perturbation parameter) increases. The model prediction for a fitted perturbation parameter value of 5  105 is compared with the experimental data of in Fig. 8. The model predictions agree fairly well with the data. The calculated force as given by Eq. (75) at different times is also compared with the experimental measurement of force vs time in Fig. 9. Consistent with the experiments, the force increases, reaches a maximum and decreases dramatically at subsequent times and agrees reasonable well with experimental data. The mechanistic model that is presented above is an idealized model in that it does not consider bubble size distribution and bubble coalescence. As bubbles expand due to the combined effects of heating and an increase in the number of moles of water vapor as a result of evaporation, the air volume fraction increases and reaches a critical value of 0.74, volume fraction corresponding to close packed spheres, beyond which the dispersion becomes a foam. The viscosity of the continuous phase in the thin film separating neighboring air bubbles is extremely high as a result of gelatinization. It has been observed that growth of thermal perturbations in such viscous thin films leads to formation and growth of holes to a certain extent leading to partial opening of thin films leading to an interconnected network of channels thus

forming a porous structure (Masson and Green, 2002; Rathfon et al., 2011). Since foam structure is believed to be a precursor to the formation of pores, the proposed model is likely to be a good approximation at smaller air volume fraction (denser formulations) and lower baking temperatures. At other conditions, however, the model has to be improved by accounting for the formation of porous structure as a result of bubble coalescence. The model can be generalized for any initial bubble size distribution by considering the effect of initial bubble size distribution on growth of bubbles as described by the following population balance equation,

@f ðR; tÞ @ _ þ ½Rf ðR; tÞ ¼ 0 @t @R

ð79Þ

where f(R, t)dR refers to the number fraction of bubbles in the size range R, R + dR at time t. The above equation is a number balance for bubbles in the size range R, R + dR assuming that no breakage or coalescence of the bubbles occurs. From the solution of bubble _ growth rate RðR; tÞ as a function of bubble size and time as evaluated from the proposed model, the population balance equation can be solved by the method of characteristics to give the evolution of bubble size distribution as explained elsewhere (Narsimhan, 2012). 8. Conclusions

Fig. 8. Comparison of experimental data of evolution of normalized deformation of bread dough during baking as obtained by Singh and Bhattacharya (2005) using flooded plate rheometry with model predictions. The model parameters are: /0 = 0.1, T0 = 298 K, Tb = 373 K, h0 = 4.5  103 m, b = 1.032, lmax ¼ 1:7  1011 , De = 3.8  102, Re = 5.095  1011, Ca = 5.92  104. The other parameters are the same as that given in Fig. 1.

A mechanistic model for baking of leavened aerated food is proposed. The model accounts for heat conduction, moisture diffusion, diffusion of CO2 that is produced by fermentation, and the resistance to bubble expansion by the viscoelasticity of the dough which is described by Oldroyd B constitutive equation. The bubble expansion due to CO2 diffusion is coupled to temperature and moisture profiles. The model predicts that the growth of bubble exhibits a lag time followed by an exponential growth phase consistent with experimental observations. Eventhough the surface region is more expanded initially, at longer times, it is found to be more dense. The calculated air volume fraction profiles at longer times indicated an expanded inner region of more or less uniform density and a denser surface crust region of decreasing air volume fraction (or increasing density). The thickness of the crust region is found to increase with baking time. The average air volume fraction is found to increase with time because of the combined effects of air expansion as well as CO2 diffusion. The model predictions of dough rise as well as force exerted by the rising dough compare well with the experimental data (Singh and Bhattacharya, 2005; Romano et al., 2007). References

Fig. 9. Comparison of experimental data of evolution of normalized force for bread dough during baking as obtained by Singh and Bhattacharya (2005) using flooded plate rheometry with model predictions. The parameters are the same as those given in Fig. 8.

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