Model for heat and mass transport during cooking of cod loin in a convection oven

Mathematical modeling of cooking of several muscle foods has been accomplished, however there are very few models predicting cooking of fish. In this study, a 3D mathematical model of transport phenomena describing baking of cod muscle in a convection oven has been developed from first principles. Several properties lacking in the literature were also determined. This includes measurements of storage modulus, water holding capacity, and the fraction of energy used for evaporation as functions of temperature. The model was validated using temperature measurements and average mass loss measurements, with good agreements.


Introduction
Cooking of cod in a convection oven is a popular process, especially in the industry and the hotels, restaurants, and catering market. During cooking, several changes occur in the muscle which affect the eating quality of cod. Cod consists mainly of protein (∼20%) and water (∼80%), and one hypothesis is that the changes during convective heating can be regarded mainly as consequences of protein denaturation and moisture migration.
During cooking of cod, the collagens, which in the native state "bind" the myotome muscle sheets of the fish together, denature first. The amounts of collagen are very small compared to the other proteins, so it is usually not possible to detect denaturation of collagen when analyzing the denaturation peaks of whole muscle. Collagen isolated from skin and bone of Pacific and Atlantic cod, respectively, showed denaturation around 14-16°C (Sun, Li, Song, Si, & Hou, 2017;Żelechowska, Sadowska, & Turk, 2010), however higher denaturation values have also been reported (Hastings, Rodger, Park, Matthews, & Anderson, 1985;Shu, Ren, Ao, Qi, & Zhang, 2017). Denaturation of collagens may lead to separation of the myotome sheets, referred to as flaking. The majority of muscle protein is made up of the fibrillary proteins myosin and actin. In cod, the myosin part of the myofibrillar proteins denature at 38.4-44°C (Skipnes, Van der Plancken, Van Loey, & Hendrickx, 2008). This change is correlated with whitening and softening of the muscle (Ovissipour, Rasco, Tang, & Sablani, 2017). Sarcoplasmic proteins, which constitute the soluble proteins of the sarcolemma (Tornberg, 2005), denature at 57.3-69.5°C (Skipnes et al., 2008). These protein solubilize in the liquid fraction of the fish, are exudated with water loss (Shibata-Ishiwatari, Fukuoka, & Sakai, 2015), aggregate and contribute to the white color of the cook loss. If heating is prolonged, the actin part of the myofibrillar proteins denature around 73.8-76.1°C (Poulter, Ledward, Godber, Hall, & Rowlands, 1985;Skipnes et al., 2008). This change is correlated with toughening of the structure and increased hardness (Ovissipour et al., 2017). The extent of protein denaturation depend on the amount of heat the sample is subjected to (i.e. temperature and exposure time), as well as the history of the specimen (eg. freezing and temperature abuse during storage; Hastings et al., 1985;Matos et al., 2011;Poulter et al., 1985). In meat, changes in the availability of water in the muscle tissue, as measured by NMR, correlated well with protein denaturation temperatures (Bertram, Wu, van den Berg, & Andersen, 2006;Micklander, Peshlov, Purslow, & Engelsen, 2002). For cod, there is a loss of juiciness, measured as water holding capacity (WHC), for each major group of denatured proteins (Skipnes, Johnsen, Skåra, Sivertsvik, & Lekang, 2011), which supports the claim that protein denaturation is correlated with moisture migration. Heat-induced shrinkage of muscle, which is also associated with protein denaturation (Tornberg, 2005), may exert a mechanical force on the tissue (van der Sman, 2007). The mechanical force may in turn contribute to expulsion of free water and water-soluble compounds as cook loss. However, the mechanism and cause-and-effect relationship of shrinkage and cook loss has not yet been firmly established. In addition to expelled liquid, evaporation leading to surface drying will be an important process for water loss during convection oven cooking at low relative humidity.
To minimize the weight loss and control the heat dependent quality changes, the cooking process can be optimized using mathematical modeling of heat and mass transfer. Several studies of heat transfer during cooking of fish were found in the literature, but none were coupled to mass transfer. In the physics based studies, Fourier's law of conduction was used to model heat transfer in the product. Convective boundary conditions were given by Newton's law of cooling, and, when relevant, in combination with other modes of heat transfer, such as thermal radiation. The available publications focusing on heat transfer in fish during heating consider steaming, immersion in water, and autoclaving of vacuum-packed product. Models where cooking of cod was considered were used to confirm measurements of thermal load and end-point temperature as performed by IR measurements (Stormo, Sivertsen, Heia, & Skipnes, 2012), and to generate mild heat processing regimes that would inactivate the bacteria on the surface of vacuumedpacked cod loins (Stormo et al., 2017). Other authors studied steaming of tuna, and used a 1-dimensional model to estimate the core temperature, with good agreements between measured and simulated values (Bell, Farkas, Hale, & Lanier, 2001).
To the best of our knowledge, no physics-based models for cooking of fish have been coupled with mass transfer phenomena. However, models of coupled heat and mass transfer during cooking have been established for other muscle-based foods. Datta (2007) suggested to use a porous media approach when modeling mass transfer of muscle foods. Since the pores in muscle foods are small, Darcy's law can be used to describe the velocity of the liquid, as formulated from conservation of momentum. van der Sman (2007) applied Flory-Rehner theory to estimate the swelling pressure, which he stated as the driving force in Darcy's law. The resulting swelling pressure was expressed as proportional to the difference between the moisture content and the water holding capacity. He proposed that the details of molecular processes (such as protein denaturation) was absorbed in the WHC term. The same approach has been used in other studies when expressing the velocity of the liquid during cooking of meat and chicken (eg. (Feyissa, Gernaey, & Adler-Nissen, 2013;Rabeler & Feyissa, 2018a)). An alternative approach is considering the internal mass transfer as a pure diffusion process (Fick's law), i.e., by neglecting the pressure driven transport of moisture in the muscle. However, this approach has been criticized since it is not able to predict moisture transport inside the muscle (Feyissa et al., 2013;van der Sman, 2007).
There is a difference in the heating process between lean fish and meat during cooking. Some of this can be ascribed to the difference in the muscle fiber macro-orientation in fish into myotomes, and a looser overall structure in fish muscle due to less connective tissue compared to mammalian muscle tissue. In addition, the muscle fibers of cod are much smaller than the muscle fibers of beef. Due to differences in muscle structure, it is expected that some properties and mechanisms will differ. Therefore, the aim of the present study was to determine the properties needed for prediction of heat and mass transfer during cooking of cod, and develop and validate a numerical model for coupled heat and mass transport during cooking of cod loin in a convection oven.

Process description
During cooking of lean fish muscle in a convection oven, heat is transferred from the hot air to the surface of the loin by convection and radiation, and from the baking plate by conduction (Fig. 1). The heat is transported from the surface into the core of the loin by conduction and convection. Mass transport through the loin is driven by diffusion and convection. At the surface, mass is lost through evaporation and as exudate. In this study, it is assumed that all the liquid exudate evaporated from the fish surface. This can be argued for as follows: No visible amounts of cook loss was expelled to the baking tray during the first phase of heating. Only after cooking for 5-6 min, when the core temperature of our small samples was approaching 80°C, bubbles of cook loss emerged from the sample surface (video recorded observations). For larger pieces of fish this does not hold true due to longer heating times and smaller surface area compared to the total volume, and this will therefore be dealt with in later research. Since the amount of fat in the cod was measured to < 0.35% (Section 3.8), no fat transport is a justified assumption. The energy that was consumed during denaturation of proteins, as measured previously using differential scanning calorimetry (eg. Skipnes et al., 2008) is very small compared to the energy transferred from the oven and was therefore neglected.
During cooking of normal sized cod loins, an unpredictable dimensional change was encountered, which made validating the heat and mass transfer model difficult. By reducing the sample size, the muscle was detached from its original macro-structure, and this resulted in a different kind of dimensional change, which occurred to a much lesser extent. The smaller samples were also much quicker to cook, which also contributed to limited dimensional change. This phenomena was therefore not taken into account in this study, but during cooking of cod loins with larger dimensions, this will have to be considered.

Geometry
The 3D model consisted of two domains: A rectangular cod sample and a baking tray (Fig. 2). The cod sample was built with a surface area of 24 × 33 mm and height of 13 mm. During validation, the average sample dimensions used for each experiment was substituted. The fish sample had six boundary surfaces: The top, back, and right surfaces were external, and exposed to the oven air (Fig. 2). The bottom surface was in direct contact with the baking tray. To reduce the computational burden, we took advantage of the symmetries of the geometry. Two symmetry planes (left and front) were used as shown in Fig. 2, reducing both the length and the width of the studied sample to ½, and reducing the full sample volume to ¼. To yield the full rectangular geometry during computation, the solution was reflected along the lines of symmetry. Similar computational approaches taking advantage of symmetric boundary conditions were applied in other studies using the same software (Feyissa et al., 2013;Rabeler & Feyissa, 2018a). The core (1) and bottom surface (2) positions used for temperature validation are also shown in Fig. 2. The baking tray was constructed as a 1.4 mm thick 200 × 400 mm steel plate, upon which the cod sample was positioned in the middle. The baking plate was built with the same internal symmetry as the fish sample.

Heat transfer
The heat transfer in the fish (Equation (1a)) and baking tray (Eq. (1b)), is described by: where the material properties are given by , c p , and k -the density (kg/m 3 ), specific heat (J/(kg K)), and thermal conductivity (W/(m K)) of the materials (cod and baking tray), respectively. Similarly, w and c p w , are the thermophysical properties of the water transported within the fish sample, respectively. The thermophysical properties used and the input parameters are given in Table 1. is the three-dimensional del operator i.e. partial derivative in x, y, and z direction ( = + + / x / y / z). The flow velocity of the liquid (m/s) is denoted by u w , and T is the temperature (K).

Mass transfer
The mass transfer within the fish sample is based on the conservation mass (Bird, Stewart, & Lightfoot, 2002), and given by Eq. (2): where c is the moisture concentration (mol/m 3 ) and D w is the moisture diffusion coefficient (m 2 /s) in the sample. A porous media approach to determine the velocity inside the fish, where the driving force is the pressure gradient in the sample (Datta, 2007). The velocity of the water inside the fish sample, u w , was described using Darcy's law (Eq. (3)): In Eq. (3), is the permeability of cod (m 2 ) and µ w is the dynamic viscosity of water (Pa s). The swelling pressure vector, p, is proportional to the excess moisture concentration within the fish (Barrière & Leibler, 2003;van der Sman, 2007), and is given by Eq. (4): where G' is the storage modulus of the cod as a function of temperature (kPa), which is given in Eq. (9) (see Section 4.4), and C T ( ) eq is the water   Combined convective and radiative flux boundary conditions were applied to all air-exposed external surfaces (back, top, and right; Fig. 2) of the fish sample and baking tray (Fig. 2). The boundary condition (Eq. (6)) has a convective heat flux term given by a modified Newton's law of cooling (Feyissa et al., 2013), and a radiative term given from the Stefan-Boltzmann law (Isleroglu & Kaymak-Ertekin, 2016): where h c is the convective heat transfer coefficient (W/(m 2 K)), T oven is the measured average temperature of the oven (K), and T s is the surface temperature (K). f h is a step function turning the heat transfer off when the surface temperature approaches 100°C (Feyissa et al., 2013). For the radiative part of the equation, is the measured emissivity of the oven at room temperature (−), is Stefan Boltzmann's constant (W/ (m 2 K 4 )), and T wall is the wall temperature in the oven (K).

Mass transfer boundary conditions.
The mass transfer boundary condition at the fish sample was applied to all external surfaces (back, top, and right; Fig. 2). The evaporative flux was modeled as described by Feyissa et al. (2013), and is the diffusive flux relative to the convective flux (Eq. (7)): where f evap is the measured fraction of the internal energy used for evaporation (Eq. (10); Section 4.5). The nominator, Newton's law of cooling, represents the energy needed for evaporation, and the denominator is given by the latent heat of evaporation, H evap . The concentration gradient, C C s air , accounts for the difference between the mass fraction of water at the surface of the sample (C s ) and the mass fraction of water in the oven air as calculated from the relative humidity (C air ). The latter term is the driving force of the equation. It is divided by the molecular weight of water, M w , for conversion of units.

Bottom surface
Conductive heat transfer (Eq. (1b)) was applied from the baking tray to the bottom fish surface (Fig. 2). A no flux mass transfer condition was also applied.

Internal surfaces
Along the internal boundaries of the sample, namely the left and front surfaces (Fig. 2), symmetry boundary conditions were assigned to yield a solution for each element in the full geometry (see Section 2.2). Symmetry was also applied along the internal surfaces of the baking tray.

Model solution
The mathematical model was solved using the Finite Element Method (FEM) in the software COMSOL Multiphysics ® version 5.4.
All domains (fish sample and baking tray) were meshed using the free tetrahedral method. A free tetrahedral distribution with 50 number of elements were used to increase the resolution along the edges of the fish (Fig. 3). For the remaining geometry, the predefined "finer" setting in COMSOL Multiphysics was applied (maximum element size: 0.88 mm; minimum element size: 0.064 mm; maximum element growth rate: 1.4; curvature factor: 0.4; resolution of narrow regions: 0.7).

Raw material
Cod (Gadus morhua) from the Aquaculture research station in Tromsø was used. The fish was 2 years old, had an average weight of 3.75 kg, and was starved for 9 days before slaughter in December 2017. The fish was sacrificed by a blow to the head, followed by bleeding in seawater at 5.7-6.0°C for 25 min. The fish was then put on ice until direct filleting and deskinning to a filet weight of 373 ± 75.8 g, numbered, followed by drying lightly with a paper towel, weighting, photographing, individual packaging in plastic bags, before packaging into Styrofoam containers with ice and absorbent, and transportation overnight to our lab in Stavanger. The next morning the temperature in the boxes were still 0°C, and the fish was put in a storage room at 0°C to undergo rigor mortis. After 5 days, the filets were cut into 2-4 pieces of depending on the size of the filet. The cod pieces used for analysis in this study were quick frozen directly in a freezing chamber at −60°C, vacuum packed at 92.2% vacuum to avoid thawing, and stored at −80°C until analysis to maintain the freshness and avoid major changes in water state.

Sample preparation
Cod loins were selected from −80°C storage and placed in a −30°C freezer overnight. The loins were allowed to warm at 0°C for 30-90 min prior to cutting to form a smooth top and bottom surface using a meat slicer, while also cutting away any brown muscle, gaping, or blood stains. The samples were still frozen when cut, but at a temperature high enough not to force expulsion of exudate when cutting. From each loin, 2-3 rectangular specimen were cut using a 20 × 30 mm stencil. The exact dimensions of each sample was measured using a caliper, prior to thawing on an aluminum tray under plastic film at 0-2°C over night. From each fish, the water content was gravimetrically determined to 77.3 ± 0.62% from 4.0 ± 0.2 g of finely chopped sample.

Calibration of thermocouples
The temperature probes used were calibrated at 0°C in equilibrated ice water, and 30, 60, 90, 120 and 150°C in a LiquiCal-HM oil bath (Ellab Validation Solutions, Hillerød, Denmark) using a recently calibrated ETS20 (Ellab Validation Solutions, Hillerød, Denmark) as a standard.

Temperature measurements
For temperature measurements, 11 samples were used. Thermocouples (SSA-TF, Ellab Validation Solutions, ± 0.2°C) were inserted into the geometric core of the samples and put in a central position between the sample and baking tray, hence referred to as the "bottom surface", and 15 cm above the fish in the oven, referred to as the oven temperature. The samples were allowed to equilibrate under plastic film in room temperature for 1 h, then heated one-at-a-time for 12 min in a Metos System Rational oven (MSCC 61, Kerava, Finland). Prior to analysis, the oven was pre-heated for a minimum of 45 min at 148.2 ± 1.59°C, with a fan speed of 3/5. The humidity of the oven was measured to 6-9% during cooking. During experiments, the baking tray with the fish sample was placed in position 2 from the bottom (approx. 1/3 up), with direct exposure from the oven fan from one side.

Measurement of average mass loss
For measurements of average mass loss, 12 cod loins and two samples per loin were used. The 24 samples had heights of 12.8 ± 1.22 mm (as measured after thawing), widths of 33.2 ± 0.45 mm, and depths of 23.9 ± 0.40 mm. After cutting, the samples were divided into three batches of 8, 7, and 9 specimen, respectively, according to height. The samples were heated individually using the same oven and set-up as described above, for 2-10 min. The heating time was evenly distributed according to the number of samples in the batch, to spread the effect of dimensional differences. Care was also taken to disperse samples from the same fish throughout the heating time range. The samples were equilibrated for 30-45 min prior to cooking to minimize temperature gradients within the fish. Before and after heating, the samples were weighed to 4 decimals on an analytical scale, prior to gravimetric analysis (18 h, 105°C) of the remaining fraction of water. The measured temperature profiles were divided into two groups based on the height of the fish sample prior to cooking, and the average dimensions, initial temperatures and oven temperatures measured for each group was used for the simulations.

Water holding capacity
Prior to analysis, eight frozen pieces of cod loins were cut horizontally into thin slices with a meat slicer. Any brown muscle was carefully cut away, and the remaining muscle was finely chopped. The small pieces were distributed into sealed plastic bags and thawed over night at 0-2°C. Analysis of water holding capacity (WHC) was performed the following two days, using the methodology described by Skipnes, Østby, and Hendrickx (2007), with some alterations. Briefly, 4.2 ± 0.13 g of fish pieces were weighed into cooled, pre-weighed steel sample cups. The cups had an adjustable, central filter, making the fish sample closely situated to the top of the cup, and the expelled liquid was allowed to exit to the removable bottom. For analysis of cooked samples, the filled sample cup was isothermally heated in a water bath (GR150, Grant Instruments, Cambridge, UK) at 25, 30, 35, 40, 50, 70 and 90°C for 10 min, before cooling in ice water for at least 5 min. The exudate was removed to determine the amount of cook loss, before centrifugation (Rotina 420R, Hettich, Tuttlingen, Germany) at 4°C for 15 min at 528 g. The water holding capacity was determined as the remaining mass after centrifugation as a fraction of the original, raw mass. The gravimetrically determined (18 h, 105°C) water content of each individual fish was used in the calculation.

Determination of storage modulus
Prior to analysis, five frozen pieces of cod were cut horizontally into 3 mm thick slices using a meat slicer. Circles of 30 mm were then cut from the slices using a sharp edged pipe, while avoiding any brown muscle and uneven areas. The discs were singly put in small plastic bags, vacuum packed at 92.2% vacuum, and stored at −80°C until analysis. To avoid thawing of the samples, the preparation was performed in a chill room with circulating air at 0-2°C.
A Discovery hybrid rheometer-2 from TA Instruments (New Castle, DE, USA) with a 20 mm cross-hatched parallel plate and temperature control connected to a heat exchanger (P/N 953260.901 TGA, TA Instruments) was used for the analysis. Fish samples were collected one-at-a-time from storage, thawed in ice water while still in the vacuum bag (< 5 min), and put on the 0°C Peltier plate. Amplitude sweeps were run at 1 Hz/0.01-100% strain at 25°C, as well as 40, 60, and 80°C after preheating for 10 and 60 min to ensure that strain in the linear viscoelastic region was applied in later testing. Temperature ramps were performed at 0.05% strain, and 1.0 Hz frequency, from 0 to 100°C, with a constant heating rate of 2.5°C/min (n = 9). Prior to all testing, a conditional step was included to lower the geometry to 0.25 ± 0.1 N axial force. A solvent trap was placed around the sample and geometry to prevent heat loss and drying of the sample, and aluminum foil was placed around the solvent trap for additional prevention of heat loss.

Evaluating the fraction of energy used for evaporation
Since water evaporates from the cook loss after it leaves the fish, the original amount of water leaving the fish as cook loss was calculated from the original dry matter content of the cook loss (Section 3.5.1). From this and the weight loss of the fish sample (Section 3.5.2), the weight loss due to evaporation was determined. This value was used together with the core temperatures, heat capacity, sample mass, and time between reaching the various core temperatures, to determine the fraction of energy used for evaporation, as theoretically described by Feyissa et al. (2013).

Original dry matter content in cook loss
The original moisture content in the cook loss was determined by heating vacuum packed (92.2% vacuum) rectangular specimen (17 × 20 × 30 mm) of cod together with a 30 mm glass tube in a water bath (GR150, Grant Instruments, Cambridge, UK). The glass tube was included to lead the liquid away from the specimen, allowing for easier collection. The amount of dry matter in the cook loss was determined gravimetrically, by weighting the cook loss into pre-weighed aluminum cups with a thin layer of pre-dried sea sand (pro analysis, Merch KGaA, Darmstadt, Germany), and drying in a heating cabinet at 105°C for 16-18 h before weighting again (ISO 6496, 1999).

Measurement of evaporation
From four cod loins, three 16 × 20 × 30 mm samples per loin were cut as described in Section 3.2.1. One sample per loin was used for each temperature. The samples were weighed in pre-weighed aluminum cups with height 1-2 mm around the edge before and after heating. Mineral isolated thermocouples (Testo, West Chester, PA, USA, ± 1°C) were inserted in the core position and centered between the sample and the cup at the bottom. The samples were heated one-at-a-time in the oven as described in Section 3.2.3 until a core temperature of 50, 70, or 90°C was reached. After heating, the weight of the samples and cook loss was recorded, before gravimetric determination of the fraction of water in the cook loss accompanying each sample. The water concentration in the cook loss was determined to 90-92%.

Estimation of the emissivity in the oven
The emissivity was estimated using an IR thermal camera (MobIR ® M8, PAL/NTSC, 9V) from Wuhan Guide Infrared (Wuhan, P. R. China). The camera was secured to a tripod 1 m from the oven wall. A triple layer of black plastic was used to cover the oven door opening, and a hole just large enough for the camera lens was cut out. The camera was then taped to the plastic around the opening, allowing no light to escape into the oven. The relative humidity setting used was calculated from the wet and dry bulb temperature in the oven. The emissivity setting in the camera was then changed until the temperature in the oven as measured by the camera was in accordance with the temperature measured using the calibrated thermocouples.

Determination of the heat transfer coefficient
The heat transfer coefficient was determined as described by Ghisalberti and Kondjoyan (1999) and summarized by Skåra et al. (2014), using the same aluminum cylinder (Ø = 30 mm, h = 30 mm) as described. Thermocouples type K (PR Electronics Inc., San Diego, CA) were used for measurement of the geometrical center of the cylinder. Prior to measurements, the accuracy was determined to ± 0.37°C between 30 and 150°C. The oven temperature was recorded using a high temperature thermocouple (STC25012E700KT, Ellab Validation Solutions, Hillerød, Denmark) calibrated to an accuracy of ± 0.2°C. The oven was pre-heated and allowed to equilibrate for 20-30 min prior to analysis, using the same settings and an empty baking tray in the same position as described above (Section 3.2.3). The cylinder was hanged centrally in the oven from a cooling rack, within 5-10 cm of the thermocouple, and within 2-3 cm of the baking plate. Temperatures were recorded for 40 min (n = 2), and the total surface, radiative and convective heat transfer was determined using the lumped capacity method as described by Isleroglu and Kaymak-Ertekin (2016).

Chemical analysis of fat
The amount of fat in the muscle was analyzed using ethyl acetate extraction (NS 9404, Nofima BioLab) from a sample of 10.4 ± 0.5 g from 6 fish loins, and reported as 0.298 ± 0.0305%.

Statistics
Statistical analysis was performed using Minitab ® 18.1. One-way ANOVA with 95% confidence interval and Tukey post-hoc test was performed for analysis of significant difference. Analysis of outliers was performed using Dixon's Q test with 0.05% significance level. Analysis of linear correlation coefficients (R 2 ) was performed using Microsoft ® Excel ® 2013.

Heat transfer prediction and validation
The developed model of heat transfer predicts the temperature at each point in space and time. As shown in Fig. 4, the temperature distribution in a fish sample can be obtained at specific cooking times. After 1 min, a gradient is forming throughout the fish, with the highest temperature on the surface and the lowest near the geometric core. After 4.5 min of cooking, the small (13.4 × 24.4 × 33.4 mm) piece of fish has already reached a temperature of 70°C in the coldest spot.
The effect of the conductive heating from the baking tray to the fish can also be seen from the model prediction since the coldest spot is raised from the geometric core. In Fig. 4, this is easiest to see after 3 min of cooking, when the temperature at the bottom surface is red and has surpassed 90°C, whereas the temperature at the central top surface remains at the color orange under 80°C. Thus, in the portrayed scenario, the conductive heating from the baking tray is quicker than the convective and radiative heating from the surrounding air.  The model prediction at the core and bottom surface positions (Fig. 2) were compared to the measured data from the same positions, as seen in Fig. 5. There is a generally good agreement between the measured and the predicted temperatures, and the curves show a similar tendency in slope and lag. For the core temperature prediction, the model is within 2°C in the critical range between 40 and 80°C. After 80°C, the average measured core temperature rises somewhat quicker than the prediction. This is probably attributed to shrinkage in height during this region, which would lead to quicker heating of the sample.
The fit in the range 40-80°C is made even better when including the geometry and parameters of the thermocouple in the model simulation (Fig. 6). The thermocouple was modeled as a 40 mm long hollow stainless steel (316L) cylinder using the measured diameter and wall thickness of the electrode, 1.2 and 0.08 mm, respectively, with thermophysical properties from Kim (1975). The volume inside was assumed to have the same insulating properties as air. When the sample size is increased to 80 × 60 × 25 mm, which is a normal size for a cod loin, the effect of the thermocouple on the core temperature diminishes to under 0.8°C. During cooking of samples of normal dimensions, the effect can therefore be neglected.

Mass transfer prediction and validation of average mass loss
The predicted moisture profile shows that there is a gradual drying of the sample surface, and a drying front is slowly moving inwards during cooking (Fig. 7). According to our model, the change in the water profile in the central positions is very slow.
The predicted average mass fraction of water in the sample is very dependent on the relative humidity of the oven, since the evaporative (mass transfer) boundary condition is driven by the difference between the moisture concentration on the sample surface and the relative humidity of the oven air. Fig. 8 shows the effect of raising the humidity of the oven on the predicted average evaporation rate. Increasing the humidity to 50% reduces the evaporative weight loss after 4.5 min by 3.5% compared to a relative humidity of 10%. Like the figure shows, humidity can be an important parameter in process optimization to minimize the weight lost as evaporation.
In this study, mass transfer was only validated using average mass loss data. While this method has limitations, including not showing the moisture profile within the sample during cooking, it is able to give a gross overview of how the reality fits the prediction on a macro-scale. However, in order to truly validate the mass transfer model, information about the distribution and movement of moisture during cooking must be acquired.
Our measurements showed that during cooking of 12.8 × 33.2 × 23.9 mm samples, the mass fraction of water decreased linearly from the initial 77.5% until it reached 68.9 ± 0.491% after 10 min. For the first 8.5 min of the process, there was good agreement between the measured and the predicted water concentration in the samples (Fig. 9). After cooking for 10 min, all measured data values were below the model prediction, with 1.3% between average measured and predicted values. During the last stage of heating (8-10 min), cook loss was expelled from the sample surface in addition to evaporation (video recorded observations). Since the mass transfer boundary condition only included evaporation and not mass lost in liquid form as cook loss, the deviation can be accounted for in the future by adding equations of cook   loss to the model. This is especially important when predicting the mass loss during cooking from larger samples.

WHC
The water holding capacity (WHC) of the raw fish was measured to 81.3 ± 0.822% (Fig. 10). From 25 to 40°C, the WHC decreased until reaching a local minimum of 67.0 ± 4.83%, which is a similar trend to what has been found previously for post-rigor fileted farmed cod (Skipnes et al., 2011). In this study, consistently higher values were found throughout the temperature range for pre-rigor fileted fish than was found for post-rigor fileted fish in the aforementioned study. This may be a consequence of increased loss of free water during the rigor mortis stage when fileting prior to this biological process (Kristoffersen, Vang, Larsen, & Olsen, 2007), leaving the fish with less water available for loss during cooking and centrifugation. From 40 to 90°C, there were no significant differences between the measured temperature points. It therefore seems that the water holding capacity reached a plateau after the proteins were denatured. The average value obtained after cooking at 40-90°C was used as the final value in the proposed equation. There were large standard deviations accompanying the data on account of large biological differences between each fish. The low or high WHC behavior seen in each fish specimen was consistent along the temperature range applied.
A function for the change in water holding capacity with temperature has previously been formulated by van der Sman (2007). The experimental data found for farmed cod was fitted to this relation to give a function for WHC of cod muscle (Eq. (8)): eq eq,0 1 2 3 where C eq,0 is the initial water holding capacity of raw sample, 0.82, T is the temperature in°C, T is the center of a logistic curve, 25°C, and a 1 , a 2 , and a 3 are fitting parameters set to 0.12, 23.0, and 0.42 by trial-anderror. It should be noted that the water holding capacity is specific to the raw material, and affected by conditions before, during, and after sacrificing the fish. Especially, differences are expected between wild and farmed cod (Rustad, 1992), between pre-and post rigor fileted cod, and as a consequence of freezing regimes (Schubring, 2005). When differences in WHC is found, the parameters of Eq. (8) should be adjusted accordingly.

Rheological analysis
The rheological measurements (Fig. 11) showed that the storage modulus (G′) of cod decreased initially from 0 to 37°C, before an increase between 50 and 75°C, and eventually a plateau was reached after heating to 80-100°C. The change in G' as a function of temperature was fitted to a sigmoidal curve, as outlined in Feyissa et al. (2013) for meat and in Rabeler and Feyissa (2018b) for chicken breast (Eq. (9)): where G max is the mean maximum elastic modulus, 48.2 ± 8.55 kPa, G min is the lowest measured elastic modulus, 14.2 ± 3.20 kPa, T is the temperature (°C), and g 1 and g 2 are fitting patterns with values 64 and 5, respectively, determined by trial-and-error. The data, especially after 70°C, was accompanied by large standard deviations, which yields an uncertainty for the equation. This is not unexpected, since measurements of cod texture have been correlated with high standard deviations also previously, when using a compression test to measure hardness (Skipnes et al., 2011). When statistically analyzing the storage modulus at critical points (0, 37, and 76°C), no correlation between sample height and storage modulus was found (R 2 < 0.28).

Fraction of internal energy used for evaporation
The fraction of internal energy used for evaporation, calculated from the measured data and plotted against the average core temperature of the time intervals used for the calculation (0-50°C, 50-70°C and 70-90°C), is shown in Fig. 12. In the case of samples cooked from 0 to 50°C, the fraction of internal energy used for evaporation had already reached 0.19 ± 0.05. For samples reaching a core temperature of 70 and 90°C, the fraction increased to 0.78 ± 0.02 and 0.86 ± 0.01, respectively. The fitted sigmoidal function describing the fraction of internal energy used for evaporation, f (T) evap , is given by Eq. where f max is the maximum fraction of internal energy that can be used for evaporation, 1, which is achieved at the boiling point of water, 100°C. The fraction of evaporation at the freezing point, 0°C, given by f 0 , has a value of 0. T is the surface temperature (°C) of the sample, and f 1 and f 2 are fitting parameters with values 47 and 15, respectively, determined by trial-and-error and visual inspection of the curves. In practice, this function gradually turns up the fraction of internal energy used for evaporation at the surface.  For clarity, standard deviation is included each 10°C.

Conclusion
A comprehensive model of heat and mass transport during baking of small pieces of cod on a baking tray in a convection oven was formulated. The model showed the ability to predict the temperature and moisture concentration during cooking as functions of time and space. Empirical relations for storage modulus and water holding capacity of cod were developed, as well as a relation for the fraction of internal energy used for evaporation. The model does not consider dimensional change and expulsion of water as cook loss, which are processes of great importance when heating larger pieces of cod. In order to make a more general model with a larger specimen validity range, these processes will be considered in later studies. In addition, quality and safety parameters can be added, so that the model may assist industry and the hotels, restaurants, and catering businesses in process optimization to minimize the liquid loss and optimize quality, while maintaining safety.