Understanding crust formation during baking
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Understanding crust formation during baking
D.R. Jefferson a, A.A. Lacey a,*
, P.A. Sadd b
a
School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK
b
RHM Technology Ltd, The Lord Rank Centre, Lincoln Road, High Wycombe, Bucks HP12 3QR, UK
Abstract
A mathematical model of crust formation in bread baking is outlined, and used to explore the effect of the model parameters on
crust thickness and density. Experimentation suggested that the interactions between model parameters were relatively weak, so it
was possible to present the model results in terms of the percentage change in crust thickness from a 10% shift in each parameter.
The results showed that crust thickness was particularly sensitive to the temperature at which bubbles collapsed and the doughÕs
vapour pressure, but relatively insensitive to pre-oven dough conditions and dry crust properties. The overall mass in the crust
showed similar dependencies, but with the addition that the thermal conductivity of the inner dough became significant.
Keywords: Bread-crust formation; Crust thickness; Crust density
1. Introduction on the modelling of crumb. Consideration of the crust
has been more limited. For example, a lumped model
The texture of baked bread, and the processes which aimed at representing moisture variation has been dis-
determine that texture, are important issues for manufac- cussed in Lostie, Peczalski, and Andrieu (2004), while
turers of such products. The work presented in this paper Zanoni, Peri, and Bruno (1995) and Zanoni, Peri, and
concerns bread texture, or, more specifically, the forma- Pierucci (1997) consider drying and browning near the
tion of a crust during baking. Here, crust is taken to refer surface of a baking loaf.
to that part of the bread near its surface, where the den- The present model is based on consideration of the
sity is significantly higher than elsewhere. To help inves- changes undergone during baking by the dough, which
tigate the factors which are important in determining the is a bubbly liquid when placed in the oven. Two key
extent of the crust, a mathematical model of crust forma- events relating to the formation of a crust are the frac-
tion has been developed in Jefferson, Lacey, and Sadd turing of the bubbles and the setting of the dough, the
(Submitted for publication). In this paper we present former being assumed to occur at a slightly higher tem-
the results of numerical simulations from the model. perature than the latter. When part of the dough matrix
Other studies, such as de Vries and Rask (1990), surrounding a bubble reaches a certain temperature (at
Hasatani et al. (1992), Zanoni, Peri, and Pierucci which it is already set) it fractures and the bubble be-
(1992), Zanoni, Peri, and Pierucci (1994), which form comes part of a network of pores reaching the bread sur-
just part of an extensive literature, have largely focused face. At this point the pressure inside the bubble
decreases to the ambient pressure. If a bubble is still
partly liquid when this occurs it will be subject to
squashing by the inner part of the dough where the as
yet unfractured bubbles have an internal pressure thatis higher than the ambient pressure. The more a bubble • The surface heat transfer coefficient h. This deter-
is set when it fractures, the less squashing takes place mines the heat flow at the bread surface through
and the lower the final density in that part of the bread, the boundary condition
so that the greater the separation of the setting and frac-
oT
turing isotherms, i.e. the lower the temperature gradient, KðT Þ ¼ hðT a T Þ; r ¼ R. ð1:1Þ
the less a bubble will change size. It is the levelling off of or
the temperature gradient inside the bread which gives Here r is the spatial (radial) coordinate, R is the loaf
rise to a decrease in the density of the bread towards radius, T the temperature, Ta the ambient (oven) tem-
its centre. perature and K the thermal conductivity.
The current model assumes that the setting and frac- • The loaf radius R.
turing occur at precise temperatures, and that when • The oven temperature Ta.
fracturing reaches a bubble, the part of the bubble adja- • The collapse temperature Tc. This is the temperature
cent to liquid dough collapses instantaneously, as shown half way between the setting temperature Ts and the
in Fig. 1. This collapse mechanism, with a change of fracturing temperature Tf described above; Tf and
density fixed by temperature gradient, is coupled with Ts can be fixed in terms of Tc by specifying the differ-
a nonlinear heat equation, with an experimentally de- ence between them (dT, see below). At any time
fined thermal conductivity and an evaporation bound- during baking bubble collapse is taking place approx-
ary at 100 C, and allows for convection due to the imately on the isotherm corresponding to Tc.
expansion of the inner (unset) part of the dough. Also • The difference between the setting and fracturing tem-
in Jefferson et al., Submitted for publication, the local peratures, dT = Tf Ts.
behaviour near the onsets of collapse and evaporation • The temperature of vaporisation of water, Tv (this
was analysed, and a numerical method for solving the can be altered by changing the composition of the
radially symmetric problem was developed. dough, e.g. the addition of sugar or salt will increase
The present paper now uses the model and numerical Tv).
method to try to identify how qualities of the crust, • The mean bubble diameter (bubble diameters are
namely its ‘‘size’’, ‘‘thickness’’ and ‘‘mass’’, the precise assumed to be distributed log-normally).
meaning of which will be made clear below, depend on • Two parameters j1 and j2 determining the thermal
various physical properties, which can either be con- conductivity, expressed as
trolled, such as the water content of the dough, or might kðT Þ ¼ j1 ej2 T ð1:2Þ
be determined by the qualities of the flour used. As our
work is aimed at crust and we wish to keep calculations in wet, uncollapsed dough (this form is suggested by
as simple as possible, the simulations discussed here are experimental data (Sadd, 2003)).
for a spherical loaf or roll. • Two parameters e j 1 and e
j 2 determining the thermal
The following model parameters are varied: conductivity written in the same form
e j 1 eej 2 T
kðT Þ ¼ e ð1:3Þ
• The initial porosity of the dough, i.e. the (uniform)
porosity of the dough when it is placed in the oven. for wet, collapsed dough.
Fig. 1. Diagram of the idealised bubble collapse mechanism.A sequence of runs was performed to determine 2. Numerical results the effect of varying each parameter by 10% above and below a standard value, and to determine the In this section we present the results of numerical change in each parameter required to produce a 10% experiments designed to determine the way the model change in the crust properties. These results are listed parameters affect crust properties and investigate the in Tables 1–3. Some runs involving simultaneous varia- sensitivity of crust properties to changes in the various tion of two parameters were carried out, but no signifi- parameters of the model. Typical temperature and cant coupling was identified. The main results and density profiles arising from the numerical calculations implications of the changes found are discussed in can be found in Jefferson et al., Submitted for Section 3. publication. Table 1 Parameter changes required to produce a 10% increase in crust thickness Parameter Units Original value New value Percentage change Initial porosity /0 0.8 0.54 32% Convection coefficient h W m1 K1 35 27 23% Loaf radius R cm 5 6.6 32% Oven temperature Ta C 220 190 14% Collapse temperature Tc C 80 76 5% Vaporisation temperature Tv C 100 103 3% Mean bubble diameter m mm 1 1.37 37% Bubble mean/standard deviation 0.5 0.59 18% Fracturing – setting temperature dT C 8 5.8 27% Inner conductivity parameter j1 W m1 K1 0.04 0.029 28% Inner conductivity parameter j2 K1 0.0256 0.0214 16% Outer conductivity parameter ej1 W m1 K1 0.04 0.047 17% Outer conductivity parameter ej2 K1 0.0256 0.0273 7% Table 2 Parameter changes required to produce a 10% increase in the proportion of the mass in the crust Parameter Units Original value New value Percentage change Initial porosity /0 0.8 0.77 4% Convection coefficient h W m1 K1 35 59 65% Loaf radius R cm 5 4.3 14% Oven temperature Ta C 220 310 41% Collapse temperature Tc C 80 77 4% Vaporisation temperature Tv C 100 103 3% Mean bubble diameter m mm 1 1.12 12% Bubble mean/standard deviation 0.5 0.55 11% Fracturing – setting temperature dT C 8 7.1 11% Inner conductivity parameter j1 W m1 K1 0.04 0.035 11% Inner conductivity parameter j2 K1 0.0256 0.0241 6% Outer conductivity parameter ej1 W m1 K1 0.04 0.047 17% Outer conductivity parameter ej2 K1 0.0256 0.0273 7% Table 3 Parameter changes required to produce a 10% increase in final surface density Parameter Units Original value New value Percentage change Initial porosity /0 0.8 0.85 6% Convection coefficient h W m1 K1 35 27 23% Oven temperature Ta C 220 190 14% Collapse temperature Tc C 80 76 5% Mean bubble diameter m mm 1 1.13 13% Bubble mean/standard deviation 0.5 0.58 16% Fracturing setting temperature dT C 8 7.1 12% Inner conductivity parameter j1 W m1 K1 0.04 0.035 12% Inner conductivity parameter j2 K1 0.0256 0.0240 6%
We characterise the crust profile using three quanti- Percentage change in crust thickness
ties. The first is the crust thickness, defined as the dis-
tance from the surface at which the final density is 30 0 40
equal to the original dough density (see Fig. 2). The sec- Collapse temperature
ond quantity is the proportion of the loaf mass in the Oven temperature
crust between this point and the bread surface. The third Initial porosity
quantity is the final density at the bread surface, which Inner conductivity (b)
in this model can be calculated directly from the model Fracturing temp. setting temp.
parameters (without numerical simulations). Surface heat transfer coefficient
Figs. 3–5 show the percentage change in crust thick- Inner conductivity (a)
ness, mass proportion and final surface density resulting Latent heat of vaporisation
from a 10% increase in each parameter from its value Mass fraction of water
Density of gas free dough
given in Table 4 (with the other parameters set at the
Initial temperature
values given there). These central values are suggested
Conductivity of dry crust
by data from Rask (1989) and Sadd (2003). That chang-
Specific heat of water
ing the (bubble-free) dough matrix density has no effect
Specific heat of dry dough
on the crust profile is a consequence of the fact that
Loaf radius
changing this parameter is equivalent to rescaling time. Mean bubble diameter
Tables 1–3 show the parameter changes necessary to Bubble standard deviation
produce a 10% change in the crust thickness, mass pro- Outer conductivity (a)
portion and surface density respectively. In the parame- Outer conductivity (b)
ter ranges considered here, the model does not give rise Vaporisation temperature
to any complicated interactions between parameters.
Fig. 3. Bar chart showing the percentage change in crust thickness
Examples of the effect of varying two parameters to- resulting from a 10% increase in the numerical parameter values given
gether are shown in, Figs. 6 and 7. in Table 4. Here inner conductivity (a), inner conductivity (b), outer
In producing these results, each parameter was varied conductivity (a) and outer conductivity (b) refer to the parameters j1,
independently. In practice, various parameters will be j 2, e
j 1 and e
j 2 respectively (see (1.2) and (1.3)).
linked together. For example, we have seen that raising
the vaporisation temperature gives rise to a thicker crust
(due to the fact that vaporisation hinders the supply of
heat from the oven). However, raising the vaporisation Percentage change in proportion of mass in crust
temperature will have the effect of decreasing heat trans-
20 0 30
fer due to moisture migration at a given temperature,
resulting in a lower effective thermal conductivity. This Collapse temperature
would tend to counteract the Ôlevelling outÕ of tempera- Inner conductivity (b)
Fracturing temp. setting temp.
Loaf radius
Inner conductivity (a)
2.5
Initial temperature
Density as a proportion of original density
Latent heat of vaporisation
Mass fraction of water
2
Density of gas free dough
Conductivity of dry crust
Specific heat of water
1.5
Specific heat of dry dough
Oven temperature
Surface heat transfer coefficient
1
Initial porosity
Outer conductivity (a)
Mean bubble diameter
0.5
Crust thickness Bubble standard deviation
Outer conductivity (b)
Vaporisation temperature
0
0 2 4 6 8 10
Distance from surface (mm) Fig. 4. Bar chart showing the percentage change in the proportion of
mass in the crust resulting from a 10% increase in the numerical
Fig. 2. Crust thickness is defined as the distance from the bread parameter values given in Table 4. Here inner conductivity (a), inner
surface at which the density is equal to the original density before conductivity (b), outer conductivity (a) and outer conductivity (b) refer
baking. to the parameters j1, j2, ej 1 and e
j 2 respectively (see (1.2) and (1.3)).Percentage change in surface density Lines of equal crust thickness (thicknesses shown in mm)
75
2.5
30 0 30
74 3
Initial porosity 3.5
Collapse temperature (°C)
Collapse temperature 73 4
Inner conductivity (b)
4.5
Inner conductivity (a) 72
Fracturing temp. setting temp. 5
Bubble standard deviation 71
5.5
Mean bubble diameter
6
Surface heat transfer coefficient 70
Oven temperature
6.5
69
Fig. 5. Bar chart showing the percentage change in final surface
density resulting from a 10% increase in the numerical parameter
68
values given in Table 4. Here inner conductivity (a) and inner
conductivity (b) refer to the parameters j1 and j2 respectively (see (1.2) 90 95 100 105 110
and (1.3)). Vaporisation temperature (°C)
Fig. 6. Contour plot showing the effect on crust thickness of varying
both vaporisation and collapse temperature. Other parameters have
the values given in Table 4. Note that the contours are fairly straight.
Table 4
The ÔcentralÕ model parameter values used
Initial porosity /0 = 0.8
Density of gas-free dough qm = 1250 kg m3 Lines of equal crust thickness (thicknesses shown in mm)
Dry specific heat c ¼ 1680 J kg1 K1 82
Dry thermal conductivity k ¼ 0.055 W m1 K1
Convection coefficient h = 35 W m1 K1 80
Bread radius R = 0.05 m 4.5
Collapse temperature (°C)
Oven temperature Ta = 200C
78
Collapse temperature Tc = 80C
Vaporisation temperature Tv = 100C
76
Initial temperature T0 = 40C 5
Mean bubble diameter m = 0.001 m
Bubble mean/standard deviation br = 1/2 74
Fracturing temperature – setting dT = 8C
temperature 72
Latent heat of vaporisation L = 2.272 M J kg1 K1
Initial mass fraction of water mw = 0.45 70 5.5
Specific heat water cw = 4187 J kg1 K1
Conductivity parameters j 1 ¼ 0.04 W m1 K1
j1 ¼ e
68
(see (1.2) and (1.3)) j 2 ¼ 0.0256 K1
j2 ¼ e
7.2 7.4 7.6 7.8 8 8.2 8.4 8.6 8.8
Difference between fracturing and setting temperatures (°C)
Fig. 7. Contour plot showing the effect on crust thickness of varying
ture in the inner part of the dough, and make the crust both the difference between the setting and fracturing temperatures
thinner than it would otherwise be (the results show that and the collapse temperature (the meaning of these temperatures is
a higher inner thermal conductivity gives rise to a thick- explained in Section 1). Other parameters have the values given in
er crust). To better account for these interactions, more Table 4. Note again that the contours are fairly straight.
detailed modelling of moisture migration and its rela-
tionship with heat transfer is required.
volume – it is a function of the bubble diameter distribu-
A feature of this model is that the collapsed density at
tion, see Jefferson et al., Submitted for publication – and
the bread surface is given immediately by the model
parameters and no numerics are necessary to obtain it. dT ðj1 ej2 T c Þ
f0 ¼ .
The expression for this density is hðT a T c Þ
qm ð1 /0 Þ For the derivation of these expressions see Jefferson
^0 ¼
q et al., Submitted for publication. The way that the final
1 /0 m0
surface density varies as these parameters are varied is
where qm is the density of gas-free dough, /0 is the shown in Fig. 8.
porosity at the start of baking, m0 = m(f0) with m the That the collapsed density at the surface can be calcu-
volume of bubble collapse per unit uncollapsed bubble lated directly from the model parameters without the4 5
4.5
3.5
Final surface density/original density
Final surface density/original density
4
3
3.5
2.5 3
2.5
2
2
1.5
1.5
1 1
50 100 150 200 250 300 350 400 0 20 40 60 80 100 120 140 160
Oven temperature (°C) Collapse temperature (°C)
4 5.5
5
3.5
Final surface density/original density
Final surface density/original density
4.5
3 4
3.5
2.5
3
2 2.5
2
1.5
1.5
1 1
0 10 20 30 40 50 60 70 0 2 4 6 8 10 12 14 16
Convection coefficient h (W m k ) Difference between fracturing and setting temperatures (°C)
3 4
2.8
3.5
Final surface density/original density
Final surface density/original density
2.6
2.4
3
2.2
2 2.5
1.8
2
1.6
1.4
1.5
1.2
1 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2
Initial porosity Mean bubble diameter (mm)
Fig. 8. Graphs showing how final surface density changes when model parameters are varied (each parameter being varied independently with the
others being fixed at the values given in Table 4). In the last graph the mean bubble diameter is varied while the ratio between the mean and the
standard deviation is fixed.
need for numerical simulations is a consequence of the face reaches the collapse temperature Tc will be higher,
fact that the temperature gradient at the bread surface resulting in a higher crust density at the surface.
is determined by the boundary condition (1.1) with
T = Tc. As explained in Section 1, a higher temperature
gradient gives rise to a higher collapsed density. For 3. Discussion
example, it is apparent from (1.1) that if the surface heat
transfer coefficient h or the oven temperature Ta are in- The results presented here give a strong indication of
creased, the surface temperature gradient when the sur- the relative importance of the various parameters of thebread baking process in determining crust properties. the relationship between heat transfer and moisture
For example, they suggest that the collapse temperature migration is needed.
(the temperature characteristic of the setting and frac-
turing of the dough) has a greater effect on the extent
and density of the crust (a lower collapse temperature References
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