Excerpt from the Proceed**in**gs of the **COMSOL** Conference 2010 Paris

**Model ing**

V. Nicolas 1,2* , P. Salagnac 2 , P. Glouannec 1 , V. Jury 3 , L. Boillereaux 3 , J.P. Ploteau 1

1 Laboratoire d’Ingénierie des MATériaux de Bretagne – Equipe Thermique et Energétique,

Université Européenne de Bretagne,

2 Laboratoire d’Etudes des Phénomènes de **Transfer**t et de l’Instantanéité : Agro-**in**dustrie et

Bâtiment, Université de La Rochelle,

3 Laboratoire de Génie des Procédés, Environnement, Agroalimentaire, ENITIAA.

*Correspond**in**g author: Université de Bretagne-Sud, rue de Sa**in**t Maudé – BP 92116, 56321 Lorient

Cedex, France, v**in**cent.nicolas@univ-ubs.fr

Abstract: In this paper, we present a first

model carried out with Comsol Multiphysics®

to model bread bak**in**g, consider**in**g heat **and**

mass transfer coupled with the phenomenon of

swell**in**g. This model predicts the pressures,

temperatures **and** water contents evolutions **in**

the dough for different energy requests. First

results obta**in**ed are analyzed accord**in**g to

various physical parameters **in** order to better

apprehend **in**teractions between the various

mechanisms **in** the porous matrix.

Keywords: **Bread** bak**in**g, heat **and** mass

transfer, model.

1. Introduction

With energy consumption close to

300 000 tons-equivalent-oil per year, bread

bak**in**g represents a non negligible part of the

energy dem**and** of the French Food sector. An

improvement of the energy efficiency of the

oven would make it possible to reduce the

energy request for this sector **and** consequently

the CO 2 emissions. Before the optimization

phase, it is necessary to evaluate the energy

needs of the product. This stage needs an

**in**creased knowledge **and** a model**in**g of the

transfer mechanisms **in** the product, here, a

porous medium: bread. **Bread** is a **com**plex

medium **in** which occur of many physical

phenomena dur**in**g bak**in**g: heat **and** mass

transfers (CO 2 , liquid water, vapor water),

swell**in**g with the formation of a porous

structure **and** various physico-chemical

reactions (gelat**in**ization, surface brown**in**g:

Maillard reaction…).

In this paper, we present physical model

designed to describe mass **and** heat transfer

with**in** the porous material dur**in**g bak**in**g. The

second part describes the numerical model

implemented **and** the simulated results

obta**in**ed.

Figure 1. **Heat** **and** mass phenomena **in** bread.

2. Govern**in**g equations

A mathematical model based on heat **and**

mass transfer **in** porous media is used to model

the bak**in**g of bread. It is derived from mass

balance of constituents **in** different phases:

liquid water (l), water vapour (v), CO 2 gas

(CO 2 ). The state variables are the temperature

T, the water content W **and** the total gas

pressure P g . This conservation-based approach

was developed for dry**in**g theory by Whitaker

[1] **and** Philip **and** De Vries [2] **and** used by

another authors (Zhang et al. [3]).

2.1. Hypothesis

In this problem, the hypotheses used are:

• medium is homogeneous,

• local thermodynamic equilibrium is achieved,

• liquid phase (l) is not **com**pressible: ρ l = cst,

• the gaseous phase (g) consists of a perfect

blend of gas: carbon dioxide (CO 2 ) **and**

vapour (v), whose equations read as follows:

P M

i j j

ρ

g

= for j = ( v,

CO2)

with P = P + P

g v CO2

RT

i i i

**and** ρ = ρ + ρ (i: **in**tr**in**sic)

g v CO2

• radiation **and** convection are negligible

with**in** the material,

• moisture content is the fraction between

liquid water mass **and** dry solid mass:

ml

W = .

m

s

2.2. **Mass** conservation

Liquid water phase

∂ρ r

l

r r

+ ∇ ⋅ nl

= −K

∂t

∂ρ r

v

r r

Vapour phase

+ ∇ ⋅ nv

= K

∂t

∂ρ

r

CO

CO 2 phase

r

2

+ ∇ ⋅ nCO

= 0

2

∂t

These equations show the matter flows,

which are derived from the Fick’s law for

diffusion **and** by Darcy’s generalised equations

giv**in**g the mean filtration velocity fields of the

liquid **and** gaseous phases.

Liquid water flux

r

n

l

= D

W

l

r

∇W

Liquid water diffusion **in** bread is only due

to capillary diffusivity. The measurement of

capillary pressure is difficult to obta**in**, so we

have used an expression of the diffusion

coefficient obta**in**ed by Ni et al. [4], with the

parameters of Zhang **and** Datta [5].

W

Dl = Ck

2

ρs

exp( − 2.8 + 2W

)ε

with ε, the porosity **and** C k2 = 10 -6

coefficient.

Vapor flux

r kk

i rg

nv

= −ρ

v

µ

g

r

i

r r

i

( ∇P

)

g

− ρ

gg

− ρ

gDeff

∇ωv

ω v be**in**g the mass fraction of the vapour **in** the

gaseous phase, given by:

CO 2 flux

r

n

= −ρ

kk

m

ω

v

=

m

v

g

r

i

r r

i

( ∇P

)

g

− ρ

gg

− ρ

gDeff

∇ω

2

i rg

CO2 CO2

CO

µ

g

Moisture content equation

This equation is established by liquid

water **and** vapour mass conservation equation

sum.

∂W

r

ρ

s

+ ∇ ⋅

∂t

⎛ ∂T

∂W

= −⎜

β 1

+ β 2

⎝ ∂t

∂t

r r r

W W

T

Pg

[( Dl

+ Dv

) ∇W

+ Dv

∇T

+ Dv

∇Pg

]

⎞

⎟

⎠

Terms with beta coefficient are obta**in**ed

from vapor phase conservation equation. By

a

**in**clud**in**g the gradients of the state variables **in**

the expressions of mass flow, the diffusion

coefficients appear (see appendix). This model

is based on Salagnac et al. [6] developments.

Energy conservation equation

**Heat** transfer occurs **in** three forms:

conduction, convection **and** latent heat moved

outward by the vapor diffusion. The

convective term is negligible **com**pared to the

latent heat:

∂T

r r r r r

ρ C

p

+ ρ

gC

p,

gv

g∇T

= ∇ ⋅ ( λ∇T

) − K Lv

∂t

with

ρ = + **and** λ,

C

p

ρ

sC

p, s

+ ρlC

p,

l

ρ

gC

p,

g

the effective thermal conductivity. The

necessary energy for the water vaporisation is

obta**in**ed by the product of the phase change

rate K **and** the latent heat of vaporisation L v .

The evaporation rate, K, is given by Zhang **and**

Datta [3] **and**, obta**in**s with the liquid water

conservation equation:

∂W

r r

W

K = −ρ

s

+ ∇ ⋅( Dl

∇W

)

∂t

Gas pressure equation

The equation for the total pressure of the

gaseous phase P g is obta**in**ed from the read**in**gs

of the mass balance on CO 2 .

v r r r

W

T

Pg

⎡− ∇ ⋅( D ∇W

+ D ∇T

+ D ∇P

) ⎤

CO2

CO2

CO2

g

∂Pg

1 ⎢

⎥

=

∂t

γ ⎢ ∂T

∂W

⎥

3

⎢

−γ

−γ

1

2

⎣ ∂t

∂t

⎥

⎦

2.3. Boundaries conditions

The boundaries conditions on air/bread

**in**terface are for:

**Heat** equation

− r

n ⋅ ( ∇

r

λ T ) = h ( T − T )

air

with h, the heat coefficient (convection **and**

radiation phenomena).

Water content equation

Evaporated mass flux is equal to the sum

of liquid water **and** vapor water flux.

r

r

W W

− n ⋅ D + D ∇W

= F

by:

F

(

l v

)

m

Evaporated mass flux on surface is given

m

= k

m

⎛

⎜

Pt

M

⎝ RT

v

film

⎞ ⎛ ⎛

⎜

Pv

, surf

− Pv

⎟ln

1+

⎜

⎜

⎠ ⎝ ⎝ Pt

− Pv

,

surf

,**in**f

⎞⎞

⎟⎟

⎟

⎠⎠

Gas pressure equation

Atmospheric pressure is considered on

bread surface.

P = P

g atm

3. Physical properties **and** parameters

Physical properties are chosen for typical

French bread.

Vapor pressure **and** water activity

Vapor pressure is obta**in**ed by an equilibrium

approach.

P = a P

v

The water activity (a w ) have been

determ**in**ed by L**in**d **and** Rask [7], Van**in** [8],

Jury [10], Zhang **and** Datta [5] with different

models.

Activity

1

0,9

0,8

0,7

0,6

0,5

0,4

0,3

0,2

0,1

0

w

0 0,5 1 1,5

vs

Water content, kg/kg

Osw**in**(Zhang)

Van**in**

Osw**in** (Jury)

GAB (Jury)

Figure 2. Activity models for bread.

The Osw**in** model fitted by Zhang **and**

Datta [5] is used.

a w

⎛

⎜⎛

=

⎜

⎜

exp

⎝

⎝

100W

Thermal conductivity

( − 0.0056T

+ 5.5)

⎞

⎟

⎠

− 1

0.38

⎞

+ 1

⎟

⎟

⎠

−1

**Heat** transfer **in** the porous media is

described by two phenomena, conduction **and**

evaporation-condensation. Two solutions are

developed **in** bibliography, some authors use

multiphase model of conductivity **and** others

an experimental effective conductivity. In this

paper, an effective conductivity, tak**in**g **in**to

account evaporation-condensation **and**

conduction phenomena has been used. The

values of thermal conductivity **com**e from

experimental data of Jury et al. [10] **and** have

been fitted by Purlis **and** Salvadori [11].

⎧

⎪

λ = ⎨1+

exp

⎪

⎩0.2

0.9

( − 0.1( T − 353.16)

)

+ 0.2 if T ≤ T

if T > T

Figure 3. Effective thermal conductivity.

f

f

− ∆T

+ ∆T

Diffusivity of vapor water **in** CO 2 is given by:

D

eff

= D

vc

[( 1−1.11S)

ε ] 4 / 3

Table 1: Input parameters.

Parameters Values Units

Initial moisture content, W 0 0.54 kg of water/kg of dry solid

Initial temperature, T 0 27 °C

Initial pressure, P g0 1.013 10 5 Pa

Initial dough density, ρ

0

305.4 kg/m 3

i

Intr**in**sic density of solid matrix, ρ

s

705 kg/m 3

Initial porosity 0.72 -

Oven gas temperature, T air 190 °C

heat transfer coefficient, h 10 W/(m 2 .K)

Convective mass transfer coefficient, k m 0.01 m/s

Vapor pressure **in** surround**in**g air, P v, **in**f 0 Pa

Gas **in**tr**in**sic permeability, k 2.5 10 -12 m 2

1−1.1S

for S ≤ 0.9

Gas relative permeability, k rg -

0 for S > 0. 9

St**and**ard b**in**ary diffusivity, D vc 2 10 -5 m 2 /s

**Bread** swell**in**g

To simulate the volume expansion of the

bread, different mechanical models exist (Zhang

[13], Van**in** [8]). In first approximation, we

**in**troduced **in**to the model a deformation of the

bread **com****in**g from numerical results (Zhang

[13]). In this case, the volume expansion is a

function of time **and** is given by a radius

expression:

α

( t)

⎛V0 α(

t)

⎞

R ( t)

= ⎜ ⎟ with:

⎝ π ⎠

5

4

⎧

⎫

-4⎛

t ⎞

-3⎛

t ⎞

⎪- 210 ⎜ ⎟ + 510 ⎜ ⎟ ⎪

⎪ ⎝ 60 ⎠ ⎝ 60 ⎠ ⎪

⎪

3 ⎪

⎪

-2⎛

t ⎞

- 4.4910 ⎜ ⎟ ⎪

⎪ ⎝ 60 ⎠ ⎪

⎪

⎬ for t ≤ 360 s

=

2

⎨

-1⎛

t ⎞ ⎪

⎪+1.51710

⎜ ⎟ ⎪

⎪ ⎝ 60 ⎠ ⎪

⎪

-3⎛

t ⎞

⎪

⎪+ 4.810 ⎜ ⎟ + 0.9968 ⎪

⎪ ⎝ 60 ⎠

⎭

⎪

⎩1.7132

for t > 360 s

4. Numerical model

0.5

equilibrium approach **in** **com**mercial software

due to divergence of heat source term

correspond**in**g to phase change. The choice of

general form makes it possible to **in**troduce this

approach.

**Heat** source term has been modified to

correspond to the PDE general form:

∂T

r r

ρ C

p

+ ∇ ⋅

l

∂t

r r

W

a ∂W

− Dl

∇W

⋅∇Lv

+ ρ

s

Lv

∂t

5. Results

W

( − λ∇T

+ D L ∇W

)

v

= −ρ

C

g

r r

⋅∇T

p, gv g

Simulation has been realised for a 15 m**in**

bak**in**g **in** an oven at 190°C. Simulated results are

**com**pared with Zhang **and** Datta [13]

experimental data.

Figure 5 presents the evolution of

temperature obta**in**ed **in** the center **and** at 1.5 mm

of the surface.

140

120

100

r

The numerical model was programmed with

Comsol Multiphysics®. The geometry is 2D

cyl**in**drical. The **in**itial radius of bread is

36.5 mm. A mobile triangular mesh**in**g (ALE)

with 548 elements is used.

Temperature, °C

80

60

40

20

0

Experimental surface

Experimental Center

Simulation surface

Simulation center

0 5 10 15

Time, m**in**

Figure 5. Temperatures **in** bread.

Figure 4. Geometry **and** mesh.

The equations are simultaneously resolved

with a free step time by the solver UMFPACK.

A bak**in**g of 15 m**in** is calculated **in** 47 s. All

equations are implemented with PDE

formulations **in** general form time dependant:

∂u

d a

+ ∇ ⋅ Γ = F **in**side doma**in**

∂t

− n ⋅ Γ = G⎫

⎬ on doma**in** boundary

0 = R ⎭

with u the variable correspond**in**g to T, W **and**

P g . Equations have to be identified **in** different

PDE terms. It is difficult to implement an

Temperature evolutions of numerical model

are **in** good agreement with experimental data.

The surface temperature **in**creases until the end

of bak**in**g. At 8 m**in**, the slope break of the curve

shows the phenomenon of evaporation. In the

center of bread, the temperature **in**creases but

stay under 100°C.

Figure 6 presents the evolution of mean

moisture content.

Moisture content, kg/kg

0,6

0,5

0,4

0,3

0,2

0,1

0

0 5 10 15

Time, m**in**

Figure 6. Mean moisture content **in** bread.

Moisture content evolution corresponds very

well to experimental data. The quantity of liquid

water decreases almost l**in**early with time dur**in**g

bak**in**g.

Local moisture content, kg/kg

0,60

0,50

0,40

0,30

0,20

0,10

0,00

0 5 10 15

Time, m**in**

Surface

Figure 7. Local moisture content **in** bread.

As for the local moisture contents, one

notices a light **in**crease **in** moisture content **in** the

center of the bread (dough) dur**in**g bak**in**g. This

phenomenon is caused by the evaporationcondensation

phenomenon **in** crumb. In surface

(crust), moisture content decreases **in** few

m**in**utes. This evolution corresponds well to

typical bak**in**g evolution (Wagner [14]).

6. Conclusion

A mathematical model has been developed

for bread bak**in**g. The numerical model has been

**com**puted with Comsol Multiphysics® **in**

deformed mesh. Deformed mesh provide to

model bread deformation dur**in**g bak**in**g. Some

modifications have been used to **com**pute phase

change **in** heat transfer equation. Temperature

**and** moisture content evolution are **in** good

agreement with experimental data.

7. References

1. S. Whitaker, Simultaneous **Heat**, **Mass**, **and**

Momentum **Transfer** **in** Porous Media: A Theory

of Dry**in**g, Elsevier, 13, pp. 119-203 (1977)

2. J.R. Philip, D.A. De Vries, Moisture

movement **in** porous materials under temperature

gradient, Trans. Amer. Geophys. Union, 38,

pp. 222-232 (1957)

3. J. Zhang, A.K. Datta, Some Considerations **in**

**Model ing** of Moisture Transport

Hygroscopic Materials, Dry**in**g Technology, 22,

(8), pp. 1983-2008 (2004).

4. H. Ni, A. Datta, K. Torrance, Moisture

transport **in** **in**tensive microwave heat**in**g of

biomaterials: A multiphase porous media model,

International Journal of **Heat** **and** **Mass**

**Transfer**, 42, (8), pp. 1501-1512 (1999)

5. J. Zhang, A. Datta, Mathematical model**in**g of

bread bak**in**g process, Journal of Food

Eng**in**eer**in**g, 75, (1), pp. 78-89 (2006)

6. P. Salagnac, P. Glouannec, D. Lecharpentier,

Numerical model**in**g of heat **and** mass transfer **in**

Center

porous medium dur**in**g **com**b**in**ed hot air, **in**frared

**and** microwaves dry**in**g, International Journal of

**Heat** **and** **Mass** **Transfer**, 47, (19), pp. 4479-4489

(2004)

7. I. L**in**d, C. Rask, Sorption isotherms of mixed

m**in**ced meat, dough, **and** bread crust, Journal of

Food Eng**in**eer**in**g, 14, (4), pp. 303-315 (1991).

8. F. Van**in**, Formation de la croûte du pa**in** en

cours de cuisson, propriétés rhéologiques et

séchage en surface : une approche expérimentale

et de modélisation, PhD, Institut des sciences et

**in**dustries du vivant et de l’environnement

(AgroParisTech) (2010).

9. V. Jury, **Transfer**ts couplés masse chaleur

d'une matrice alveolée. Application à la

décongélation-cuisson du pa**in** précuit surgelé,

PhD, ENITIAA de Nantes (2007).

10. V. Jury, J. Monteau, J. Comiti, A. Le-Bail,

Determ**in**ation **and** prediction of thermal

conductivity of frozen part baked bread dur**in**g

thaw**in**g **and** bak**in**g, Food Research

International, 40, (7), pp. 874-882 (2007)

11. E. Purlis, V.O. Salvadori, **Bread** bak**in**g as a

mov**in**g boundary problem. Part 2: Model

validation **and** numerical simulation, Journal of

Food Eng**in**eer**in**g, 91, (3), pp. 434-442 (2009)

12. A. Ousegui, C. Moresoli, M. Dostie, B.

Marcos, Porous multiphase approach for bak**in**g

process - Explicit formulation of evaporation

rate, Journal of Food Eng**in**eer**in**g, pp. 535-544

(2010)

13. J. Zhang, A.K. Datta, S. Mukherjee,

Transport processes **and** large deformation

dur**in**g bak**in**g of bread, AIChE Journal, 51, (9),

pp. 2569-2580 (2005)

14. M.J. Wagner, T. Lucas, D. Le Ray, G.

Trystram, Water transport **in** bread dur**in**g

bak**in**g, Journal of Food Eng**in**eer**in**g, 78, pp.

1167–1173 (2007)

8. Acknowledgements

The authors want to thank the National

Research Agency of France (ANR) for its

f**in**ancial contribution (ANR ALIA-BRAISE).

9. Appendix

Vapor diffusion coefficients:

D

D

W

v

⎛ M M ⎞

eff v CO Pv

D ⎜

∂

2

= −

⎟

M RT

⎝ ∂W

g ⎠

−D

⎛ M M ⎞

v CO

⎜ ⎟

∂Pv

M RT

⎝ ∂T

g ⎠

T

eff

2

=

v

⎡ k ⎛ M M

i g eff v CO

= −⎢ρ

− D ⎜

v

⎢

⎣

µ

g ⎝ M RT

g

P

D g 2

v

⎞ ⎤

⎟

Pv

⎥

⎠ Pg

⎥⎦

CO 2 diffusion coefficients:

D

D

W

CO2

T

CO2

P

D g

CO

= −D

= −D

= −ρ

W

v

T

v

i

CO

k

2 2

µ

Other coefficients:

γ =

1

γ 3

g

g

− D

eff

⎡ Pv

− Pg

∂Pv

⎤

⎢ − ⎥

⎣ T ∂T

⎦

⎛ M M

v CO2

⎜

⎝ M RT

g

⎟ ⎞

⎠

P

P

v

g

( P − P )

⎡ ∂P

ρ

( ) ⎥ ⎤

v s g v

γ

2

= −γ

3 ⎢ −

i

⎣∂W

ρlε

1−

S ⎦

ε M CO

( 1− S)

2

γ

3

=

RT

ε M

v

( − S ) ⎡∂Pv

Pv

⎤

β =

1

1

⎢ −

RT

⎥

⎣ ∂T

T ⎦

ε M

v

( − S ) ⎡ ∂Pv

⎤

β =

1

2

RT

⎢ ⎥

⎣∂W

⎦