Heat Transfer during Steaming of Bread - Unika Repository

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  100% Heat Transfer during Steaming of Bread.pdf

  100% Victoria K. Ananingsih, Edda Y. L. Sim, Xiao Dong Chen and Weibiao Zhou* Heat Transfer during Steaming of Bread

  Abstract: Understanding of heat transfer during steaming is important to optimize the processing of steamed bread and to produce desired qualities in the final product. Physicochemical changes occur during steaming of the dough which might be impacted upon by the heat trans- fer system. In this study, a mathematical model was developed to describe the heat transfersystem in the bread being steaming throughout the heating process. The Forward Euler method was employed for solving the three-dimensional partial differential equation set for heat transfer to produce temperature profiles at a number of individual locations in the steamed bread during its steaming. All the comparisons between the model-predicted values and the experimental results pro- duced root mean square error values ranged from 1.391 to 3.545 and R

  2

  values of all greater than 0.93. Therefore, it is confirmed that the model has a good performance and can be used to predict temperature profiles in the bread during steaming.

  Keywords: heat transfer, mathematical model, steamed bread, finite difference method

• Corresponding author: Weibiao Zhou, Food Science and Technology Programme, c/o Department of Chemistry, National University of Singapore, 3 Science Drive 3,Singapore 117543; National University of Singapore (Suzhou) Research Institute, 377 Lin Quan Street, Suzhou Industrial Park, Jiang Su 215123, People s ’ Republic of China, E-mail: chmzwb@nus.edu.sg Victoria K. Ananingsih, Food Science and Technology Programme,

  c/o Department of Chemistry, National University of Singapore,

  3 Science Drive 3, Singapore 117543; Department of Food Technology, Soegijapranata Catholic University, Jalan Pawiyatan Luhur IV/1, Semarang 50234, Indonesia, E-mail: kristina@unika.ac.id Edda Y. L. Sim, Food Science and Technology Programme, c/o Department of Chemistry, National University of Singapore,

  3 Science Drive 3, Singapore 117543, E-mail: u0306245@nus.edu.sg Xiao Dong Chen, Department of Chemical and Biochemical Engineering, Xiamen University, Fujian, China, E-mail: xdc@xmu.edu.cn

  1 Introduction

  Steamed bread, or called “mantou”, is a traditional staple food in northern China (i.e. the region north of the Yangtze River) and represents around 70% of the end usage of flour produced in this region and a lesser proportion in the south [1]. Among similar products widely embraced by people in Korea, Japan and Southeast Asian countries are a variety of steamed breads with different fillings (or called steamed buns). The major steps for producing these pro- ducts include making dough of wheat flour of lower pro- tein content compared to baked bread, dividing the dough into small pieces and moulding them and finally steaming the dough pieces using only saturated steam in the absence of any dry heat.

  Steaming, which is not to be confused with the pro- cess of injecting some steam during baking, is a process that results in a series of physical, chemical and bio- chemical changes in a product, such as volume expan- sion, condensation of water on the product surface followed by the formation of an outer elastic skin (not crust), protein denaturation and starch gelatinization. The system is characterized by simultaneous heat and water transport processes within the product as well as between the product and the environment inside the steamer. At the beginning, heat transport from the envir- onment (i.e. saturated steam) to the product surface occurs mainly through convection accompanied by latent heat of condensation of steam. Afterwards, while conduc- tion at the surface is the major mode of heat transfer, for bread inside, conduction from the surface to the bread crumb centre is dominating and the latent heat of con- densation contributes along a cold front which moves towards the centre. Depending on the actual type of the product to be made and the type of steamer used, relative contributions by the various mechanisms of heat transfer can be adjusted to achieve a desired quality in the final product. Due to the high energy consumption of the steaming process, it is crucial that the operating condition of a steamer is optimized to reduce the energy consumption as well as to improve product quality, similar to baking systems [2], i.e. possible mini- mum amount of steam used per kg of dry material processed.

  Both experimental and mathematical modelling approaches are often used for this purpose. In the former approach, the process is simulated in a laboratory or pilot-scale set-up to generate valuable data on the system behaviour [3]. While careful experimental data are indis- pensable in validating the process, however, unlike knowledge obtained through mathematical models, they cannot be easily generalized. Mathematical modelling, doi 10.1515/ijfe-2013-0023 International Journal of Food Engineering 2014; 10(4): 613 623 –

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V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread

  Thermophysical properties are essential to develop- ing models for heat transfer. Thermophysical properties of dough and bakery products for baked goods were reported by Baik et al. [9], Marcotte [10] and Zuniga and Le-Bail [11]. Thermal conductivity, specific heat and mass diffusivity are some of the important thermophysi- cal properties essential for the modelling of heat and mass transfer during both baking and steaming processes.

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  The determination of bread moisture distribution during steaming at different times was achieved by sampling at the locations of points 2 & 3, points 4 & 5, bottom, centre and surface. The steamed bread samples were divided into several portions, and their moistures were

  2.3 Moisture measurements

  As shown in Figure 1, the thermocouples were speci- fically placed at eight different locations of the dough/ steamed bread, namely the centre (point 1), the bottom, four other different locations in the crumb of the dough (points 2–5), directly above the dough (i.e. surrounding) and just under the bread skin (i.e. surface). It was earlier observed that the temperature profiles yielded from the points on and under the bread skin were very similar due to the rapid thermal conduction through the skin. Hence, the points were taken as one (i.e. the surface point).

  which is often based on known physical and chemical principles, is important in reducing the time and costs involved in experimentation. Models can also be used for the design, optimization and control of the process. Paulus [4] described transport models as one of the main characteristics of the modelling of industrial cook- ing. In recent years, there has been growing interest for the modelling of simultaneous heat and water transfers in baking [5–7]. Zhou [8] presented several heat and mass transfer models during baking which involved thermo- physical properties of product that are changing during the baking process, e.g. density, specific heat, thermal conductivity and thermal diffusivity.

  2.2 Steaming and temperature measurements

  Ms01T03-2, France) for 30 min at 40°C and a relative humidity of 80%. The hemispherical dough pieces after proofing were steamed for various pre-determined steaming periods. At each steaming operation, eight pieces of proofed samples were fitted into the steamer each time.

  ®

  Netherlands). The moulded dough pieces were weighed to be 50 g to ensure consistency. The hemispherical dough pieces were placed on the plastic trays of a stea- mer (Phillips, H-RC 19, Singapore), and then the trays were placed into a proofing cabinet (EUROFOURS

   Dough was let to rest for 10 min at ambient temperature of around 22°C and was then sheeted and formed into hemispherical pieces using a moulder (Dr ROBOT II,

  WAG-RN20, Europe) com- prising one low-speed mixing step (6 min, 100 rpm). The temperature of the dough after kneading was 23 0.5°C.

  ®

  The study was carried out on northern-style steamed bread. Soft or low-protein “mantou” flour (Bake King, Hong King Extra Special Flour, Singapore) was purchased from a local supermarket. The dough formulation in per- centage of flour weight was 58% water, 1% yeast (Dry baker’s yeast, Saf-instant, France), 1% salt (Pagoda, China) and 1% sugar (Sis, Australia). Mixing was done in a spiral mixer (Varimixer

  So far, relatively few studies on steamed bread have been reported in the literature [1, 12]. Theoretical model- ling of the heat transfer system for steaming bread has not been reported, in contrast to many studies on baked bread [10, 13, 14]. This study aimed to develop a mathe- matical model for the heat transfer in steamed bread during its steaming process.

  Steaming was carried out in a conventional, electrically operated steamer (3 tier steamer, Tefal, Berkshire, UK) at an equilibrium surrounding temperature of 100°C in the steamer. The steamer had three trays which were stacked on top of each other. Due to the high power efficiency and the small size of the steamer, the temperature of the top tray reached an equilibrium temperature as fast as the bottom tray. In the lid of the steamer, three small holes were present which served to regulate the inside pressure of the steamer. These holes were also used to insert up to nine K-type thermocouples of 0.3 mm in diameter through the lid and place them at different positions in the dough pieces inside the steamer during steaming, through the entire stipulated steaming process.

2 Materials and methods

2.1 Bread for steaming

V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread 615

  Heat transfer inside the dough can be treated as a heat conduction process despite that conduction was not the only mechanism for heat transfer in the dough.

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  Figure 1 Top: Schematic diagram of the cross-section of steamed bread and labelled locations for temperature measurements. Bottom: Mechanism of heat and mass transfer in the steamed bread

  In this study, two major assumptions were made: (1) In the model, only heat conduction was taken into consideration neglecting the effects of mass transfer due to the computational efficiency and the complexities involved in modelling them together. Heat transfer was carried out by conduction due to the temperature the surface. (2) The total apparent volume was taken as constant during steaming (i.e. material expansion was ignored). The validity of these assumptions for the system will be evaluated by examining how well the model out- puts are agreeable to the experimental data.

  Another mechanism was through water vapour diffusion from outer layers towards dough centre (Figure 1). Water vapour can diffuse from the outside environment, which consists of 100% water vapour, towards the centre of the dough; meanwhile water vapour also condenses at the surface of steamed bread from where moisture is trans- ferred either via liquid water diffusion or water vapour diffusion towards the centre. However, it was noted that the formation of skin might have significantly restricted the diffusion of water vapour from outside to bread. This was similar to the function of crust observed by various investigators during their study of baked bread, which restricted the diffusion of water vapour from bread inside towards environment [16, 17].

  determined using a gravimetric method [15] by weighing a sample before and immediately after an oven drying. Moisture measurements were done immediately after steaming without cooling the final product. This was to ensure that the moisture contents of the bread samples during the steaming process were more accurately deter- mined, as the gravimetric analysis was done without cooling when water loss by evaporation was still com- paratively small. After drying, the samples were cooled in a desiccator and weighed by an analytical balance (sen- sitivity 0.001 g). The wet-based moisture content (M%) of a sample was calculated by the difference between the mass before (m

  i

  2.4 Mathematical modelling of heat transfer during steaming

  i .

  )  100/m

  i – m f

  ) drying using the equation: M m (%) ¼ (

  f

  ) and after (m

  The dough/steamed bread piece is modelled as a three- dimensional hemisphere of radius R. The dough pieces at a were taken out of the proofer and placed into the steamer which was at T and its relative humidity (RH) was taken as 100% (i.e. full steam) at room pressure (i.e. 1 atm).

V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread

  T ðt þ Δ Δ t Þ  TðtÞ

  2

  h x

  þ1; ;

   2T i j ; ; k þ T i j k

  ¼ α T i 1;j;k

  Δ Δ t j i; ; j k

  where T(t) and T (t þ Δ t) are temperature at time and t time ( is time step. Substituting eqs t þ Δ t), respectively. Δ Δ t (2) and (3) into eq. (1) results in (for simplicity, only x-direction derivative is shown for illustration purpose):

  4 Equation (4) is subsequently solved for T (t þ Δ t) to obtain the T i,j,k value at a future time step. This scheme is called an explicit method because, if T i,j,k at t are given for all the grid points, T i,j,k for the new t þ Δ t for any grid point can be immediately calculated by simply using eq. (4).

  Δ Δ t Þ  TðtÞ Δ Δ t

  t ¼ T ðt þ

  T @

  time step (t þ Δ t) using the following equation: @

  i j,k , at a future

  The Forward Euler method evaluates T

  The difference equation for a grid point ( ) i, j, k located inside the boundary is derived by considering the grid point (i, j, k) and six surrounding grid points.

  ð Þ

  It was noteworthy that the time step Δ t of the explicit

  Heat transfer equation in Cartesian coordinates is given by @

  . This also means that

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  616

  x is reduced.

  become increasingly smaller if h

  Δ Δ t must

  2 α

  Δ Δ t 

  0:5h x

  be equal to or smaller than

  Δ Δ t must

  5 Otherwise, instability of the solution will happen. Hence, no matter how slow the physical change of the system is,

  ð Þ

  2 α

  0:5h x

  ð2Þ where h x is taken to be the step-size of x. Similar approx- imations can be done for the y- and z-direction second- order derivatives.

  The complete description of a mathematical model for the above-mentioned system is presented in the following sections.

2.4.1 Governing equation

  h x

  @ y

    ¼ αÑ

  2

  @ z

  2 T

  þ @

  2

  þ @

  2 T

  1; ; j k  2T i j ; ; k þ T i þ1;j k ;

  2

  @ x

  2 T

  @

  t ¼ α

  T @

  2 T ð Þ

  1 α

  ¼ λ

  ρ C Cp p

   T i

  2

  @ x

  2 T

  follows: @

  2 can be approximated by eq. (2) as

  2 T @ x

  @

  Under the central difference scheme, the -direction x second-order derivative term in the right-hand side of eq. (1), i.e.

  The numerical method used to solve the partial differential equation, i.e. eq. (1) was the Forward Euler method. It was applied via a finite difference analysis. The finite differ- ence scheme was to approximate the derivatives using changes that occur at finite intervals. To derive the finite difference equations, the dough was split into grid points with equispaced intervals that were imposed on a rectan- gular domain. Each temperature reading on the grid is denoted by T i,j,k . The grid spacing sizes in the x-, y- and z-directions are denoted by h x , h y and h z respectively. The grid points are numbered i, j and k labelling the positions on the grid in the x-, y- and z-directions, respectively, according to Cartesian coordinates.

  There were two boundaries taken into consideration in this study. The temperature of the dough/bread sur- roundings was taken to be 100°C throughout the steam- ing process. Since the sample was placed on a perforated solid plate which allowed a continuous access of steam to the bottom of the dough/bread, the bottom and top sur- faces of bread were also taken to be at 100°C. This was due to condensation of vapour at dough surface at the beginning so that the temperature differences between them were close to zero at all times during steaming. In other words, the temperature at the surroundings and the dough surfaces (i.e. top and bottom) was the same (100°C). This assumption was subsequently verified by the experimental results.

  minimizing the root mean square error (RMSE) between the experimental and modelled values. It was found to be around 0.80 W m ⁻¹ °C ⁻¹ .

  λ was obtained through optimization of themodel by

  s ⁻¹ ). Assume that ρ and Cp were constant during the steaming process whose values are 544 kg m ⁻³ and 2,310 J kg ⁻¹ °C ⁻¹ . Then,

  2

  where T is temperature (°C), λ is thermal conductivity (W m ⁻¹ °C ⁻¹ ), ρ is density (kg m ⁻³ ), Cp is specific heat (J kg ⁻¹ °C ⁻¹ ), and α is thermal diffusivity (m

  2

2.4.2 Boundary conditions

V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread 617

  Figure 2 presents the moisture profiles at different points of measurement in the steamed bread. Moisture contents throughout the bread increased with steaming time. Also, the moisture content became relatively lower as the measurement point moved further away from the surface. Hence, it can be suggested that water diffusion occurred from the outside towards the inside of the bread to its centre. As shown in Figure 2, the moisture contents during the entire bread steaming process fell within a relatively narrow range, hence moisture transfer and its impact on heat transfer were assumed to be negli- gible in the current model developed for heat transfer.

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  Figure 2 Moisture profiles of steamed bread at different measurement points

  Point 2 Point 3 Point 4 Point 5 Bottom Centre Surface

  M o is tu re c o n te n t, % M o is tu re c o n te n t, %

  20 Steaming time, min Steaming time, min

  18

  16

  14

  12

  10

  8

  6

  A program based on the above-mentioned Forward Euler method was implemented using Matlab, and the model was simulated for a period of 20 min. The RMSE between the modelled results and the experimental results was minimized through adjusting model para- meters. In particular, λ value in eq. (1) was usedas an input in the optimization system to minimize the output, i.e. the RMSE value.

3 Results and discussion

3.1 Moisture profiles

  2

  3.3 Temperature profiles

  The increase in moisture content with steaming time was due to condensation of water at the surface of bread.

  The saturated steam in the surroundings also prevented evaporation of water from the bread, resulting in a sur- face of high moisture content. The high water content at the bottom of the steamed bread was due to the fact that the sample was resting on athin, wet piece of cloth on the top of a stainless steel tray during steaming. The cloth was to prevent the bread from adhering to the tray, a common practice in industrial steam bread production. Since the cloth was able to absorb more water, it pro- duced a high moisture gradient between the cloth and the bottom of the steamed bread, thus causing high moisture contents at the bottom.

  3.2 Numerical results

  Forward Euler method was implemented using Matlab, and the model was simulated for a period of 20 min. Figure 3 shows some snap-shots of the temperature pro- file at the steaming times of 0, 1, 2, 5, 10 and 20 min. The gradual change in temperature after 5 min accurately depicted and matched the characteristics of the experi- mental results obtained.

  Figure 4 shows that at all measurement points during the steaming process, the temperature eventually reached the heating medium temperature of 100°C. All locations in

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  V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread _{t t}

  Temperature profile of cross-section of mantou during steaming at = 0 Temperature profile of cross-section of mantou during steaming at = 1 min

  100 100

  90

  90 C

  C ,° ,°

  80

  80

  re re tu tu

  70

  70

  ra ra e e p p

  60

  60

  m m e e T T

  50

  50

  40

  40

  5

  5

  5

  5

  4

  4

  3

  3

  2

  2

  1 x , cm –5

  1 x z , cm –5 z

  , cm _t , cm _t

  Temperature profile of cross-section of mantou during steaming at = 2 min Temperature profile of cross-section of mantou during steaming at = 5 min

  100 100

  90

  90 C

  C ,° ,°

  80

  80

  re re tu tu

  70

  ra ra

  70

  e e p p

  60

  m m

  60

  e e T T

  50

  50

  40

  40

  5

  5

  5

  5

  4

  4

  3

  3

  2

  2

  1 x , cm

• –5

  1 x –5 z , cm , cm _t z , cm _t

  Temperature profile of cross-section of mantou during steaming at = 10 min Temperature profile of cross-section of mantou during steaming at = 20 min

  100 100

  90

  90 C

  C ,° ,°

  80

  80

  re re tu tu

  70

  70

  ra ra e e p p

  60

  60

  m m e e T T

  50

  50

  40

  40

  5

  5

  5

  5

  4

  4

  3

  3

  2

  2

  1 x , cm

  1

• –5 z

  , cm x , cm –5 z , cm

  Figure 3 Snap-shot of the temperature profile through steaming at times 0, 1, 2, 5 and 20 min Brought to you by | Cornell University Library

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V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread 619

  The initial lower temperature for the surrounding (i.e. less than 100°C) was due to the opening and closing of the steamer lid prior to the steaming process, resulting in a loss of steam and a dip in the temperature. For simpli- city in the modelling, the surrounding temperature was taken to be 100°C at all times. The rapid heat transfer through convection of steam from the boiling water at the bottom reservoir to the other parts of the steamer justifies such an assumption.

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  Figure 4 Experimental results of temperature profiles in the steamed bread during steaming. Each temperature profile is a cumulative representation of three or more profiles of the same point in three or more different experiments

  Surroundings Bottom Points 2 & 3 Points 4 & 5 Centre Directly under surface Direct on surface

  T e m p e ra tu re , °C T e m p e ra tu re , °C

  28 Steaming time, min Steaming time, min

  24

  20

  16

  12

  8

  4

  80 100 120

  the bread samples had an initial temperature of ~42°C. It was also observed that the temperature rise at the centre of the sample was slowest due to its longest distance away from the surface. The temperature rise at a point of measurement became greater as its distance from the surface of the bread decreased.

  40

3.4 Optimization of values λ

  ) values between the two sets of data using the optimized λ values, as listed in Table 1. Figure 5shows the comparison between the modelled results and the experimental results. Based on the small RMSE values and high R

  The optimized λ values which resulted in the lowest RMSE value between the modelled and experimental values are points directly under and on the surface were observed to be almost the same for every experiment, these tempera- ture profiles at all times of the steaming process were taken to be the same in this work (i.e. “surface” measure- ment point). The effective thermal conductivity (λ) values determined in this study were in the range of 0.725–0.855 W m ⁻¹ °C ⁻¹ (Table 1), which were close to the values reported in the literature for baked bread, e.g. 0.72

  W m ⁻¹ °C ⁻¹ for white bread after 8 min of baking [18].

  3.5 Verification of model using optimized λ values

  The obtained model was verified with the experimental temperature profiles at the internal five points of mea- surement during the steaming. The verifications (R

  2

  2

  20

  values, the model verification results suggest that the model is acceptable. The model has a good performance and could be used to reproduce Table 1 Optimized λ values at five different locations in the steamed bread with the corresponding RMSE and R

  2

  values Measurement points λ RMSE

  R

  2 Surface 0.740 1.676 0.953

  Centre 0.835 1.478 0.997 Points 2 & 3 0.810 1.640 0.987 Points 4 & 5 0.855 2.967 0.980 Bottom 0.725 2.139 0.912

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V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread

  Temperature profile of model and experiment (surface), correlation coefficient = 0.95308 Temperature profile of model and experiment (centre), correlation coefficient = 0.99702

  100 100

  Model Model Experiment Experiment

  90

  90

  80

  80

  °C °C , , re re tu tu

  70

  70

  ra ra e e p p m m e e T T

  60

  60

  50

  50 (b)

  (a)

  40

  40

  5

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  20

  25

  30

  5

  10

  15

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  30 Time, min Time, min

  Temperature profile of model and experiment (points 2 & 3), correlation coefficient = 0.98662 Temperature profile of model and experiment (points 4 & 5), correlation coefficient = 0.97993

  100 100

  Model Model Experiment Experiment

  90

  90

  80

  80

  °C °C , , re re tu tu

  70

  70

  ra ra e e p p m m e e T T

  60

  60

  50

  50 (c) (d)

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  25

  30

  5

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  30 Time, min Time, min

  Temperature profile of model and experiment (bottom), correlation coefficient = 0.91165

  100

  Model Experiment

  90

  80

  °C , re tu

  70

  ra e p m e T

  60

  50 (e)

  40

  5

  10

  15

  20

  25

  30 Time, min Figure 5 Verifications of the developed model using experimental data of temperature profiles taken at (a) the surface, (b) the centre, (c) points 2 & 3, (d) points 4 & 5 and (e) the bottom of the bread

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V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread 621

  temperature profiles in the steamed bread during steam- with two additional sets of independent experimental ing process. runs. The validations also produced low RMSE and

  2

  high R values between the two sets of data using the optimized λ values, which are listed in Table 2. Figure 6

3.6 Validation of model

  shows the validation results of the obtained model using the independent experimental data. A good agree- The model was validated with the experimental tempera- ment between the modelled values and experimental ture profiles at the five points of temperature measurement results is evident in Figure 6. Together with the low

  2 RMSE values and high R values shown in Table 2, it

  Table 2 Validations of the developed model using independent

  2

  confirms that the model has a good quality and can be experimental data with the corresponding RMSE and R values used to predict temperature profiles and thus energy con-

  Measurement points RMSE (validations) (validations) R

  2 sumption in the steamed bread during steaming.

  Furthermore, the model may aid to optimizing/minimizing Surface 2.037 0.964 the steaming time that assures a satisfactory product,

  Centre 1.391 0.997 which enables the minimization of energy consumption

  Points 2 & 3 1.970 0.987 by the steaming process. Points 4 & 5 3.545 0.980 Bottom 2.551 0.934

  Temperature profile of model and experiment (surface), correlation coefficient = 0.96378 Temperature profile of model and experiment (centre), correlation coefficient = 0.99716

  100 100

  Model Model Experiment Experiment

  90

  90 C

  C

  80

  80

  ,° ,° re re tu tu

  70

  70

  ra ra e e p p m m e e

  60

  60 T

  T

  50

  50 (a)

  (b)

  40

  40

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  5

  10

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  25 Time, min Time, min

  Temperature profile of model and experiment (points 4 & 5), correlation coefficient = 0.9795 Temperature profile of model and experiment (points 2 & 3), correlation coefficient = 0.98704

  100 100

  Model Model Experiment Experiment

  90

  90 C

  80 C

  80

  ,° ,° re re tu tu ra

  70 ra

  70

  e e p p m m e e

  60 T

  T

  60

  50

  50 (d)

  (c)

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  5

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  25 Time, min Time, min Figure 6 Validations of the developed model using independent experimental data of temperature profiles taken at (a) the surface, (b) the centre, (c) points 2 & 3, (d) points 4 & 5 and (e) the bottom of the steamed bread

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V. K. Ananingsih et al.: Heat Transfer during Steaming of Bread

  Temperature profile of model and experiment (bottom), correlation coefficient = 0.93423

  verification and validation datasets that were independently 100

  Model

  generated through experiments. This indicates the good

  Experiment

  quality of the model. The results demonstrated that the

  90 temperature profiles in steamed bread during steaming can

  C

  80 be satisfactorily modelled and predicted by considering only

  ,° re

  heat transfer through optimized effective thermal conductiv-

  tu

  70

  ra

  ity (λ) values. Moisture contents throughout the steamed

  e p

  bread increased progressively with steaming time, albeit

  m e

  60 T varying in a narrow range, which indicated that water diffu- sion occurred from the outside towards the inside of the

  50 (e) steamed bread to its centre. Future works should be aimed to study the process of simultaneous heat transfer and mass

  40

  5

  10

  15

  20

  25 transfer during the steaming process to further evaluate if

  Time, min the contribution from the mass transfer is significant or not to the quality of a coupled model.

  Figure 6 (continued) Acknowledgements: Financial supports from Singapore Ministry of Education through Academic Research Fund

4 Conclusions

  Tier 1 grant R-143-000-404-112 and the National University of Singapore (Suzhou) Research Institute

  A three-dimensional partial differential equation heat trans- under the grant number NUSRI2011-007 are acknowl- fer model for the steaming process of steamed bread has edged. The first author is grateful to the financial support been developed and numerically solved by employing the from the Directorate General of Indonesian Higher Forward Euler method in Cartesian coordinates. Adequately 2 Education in sponsoring her PhD study. low RMSE and high R values were obtained for both

  References

  1. Huang S, Betker S, Quail K, Moss R. An optimized processing

  8. Zhou W. Baking process: mathematical modeling and procedure by response surface methodology (RSM) for analysis. In: Farid MM, editor. Mathematical modeling of food northern-style Chinese steamed bread. J Cereal Sci processing. Boca Raton, FL: CRC Press, Taylor & Francis Group, – 1993;18:89 102.

  2010:357–372).

  2. Carvalho MG, Martins N. Mathematical modeling of heat