Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea

LWT - Food Science and Technology xxx (2014) 1e7

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Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea Kenneth R. Davey* School of Chemical Engineering The University of Adelaide, 5005, Australia

a r t i c l e i n f o

a b s t r a c t

Article history: Received 13 March 2014 Received in revised form 14 May 2014 Accepted 20 August 2014 Available online xxx

The use of ice slurry at sea to cool fish is widespread. Here we develop and illustrate a convenient simulation of fish cooling in ice slurry based on the transient cooling equation together with practical simplifying assumptions. It is convenient because the key input is the mass of the fish, which is routinely measured, and because it can be solved in standard spread-sheeting tools as a portable App. It is planned this innovative App would be calibrated for use in particular fisheries. Illustrative simulations for Southern BlueFin Tuna (Thunnus maccoyii) (SBT) show that a 35 kg fish at harvest (Ti ¼ 28  C) will cool to 5  C (at thermal centre) in 10.23 h in slurry at a temperature maintained at 0  C. Because predictions are demonstrated to be in good agreement with published Heisler charts it is concluded the simulation is free of programming and computational errors. Importantly, simulations reveal a small rise in the temperature of the slurry significantly affects the time for cooling. For example, if the temperature of the slurry rises to 2  C, because of inadequate quantities of ice, the time to cool the SBT to 5  C will be 12.27 h. Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved.

Keywords: Ice slurry Transient cooling of fish Thunnus maccoyii

1. Introduction The cooling of freshly harvested fish in ice slurry at sea is almost universal in the fishing industry (Davey, 2012; Graham, Johnston, & Nicholson, 1992, chap. 3, 75 pp.; Granata, Flick, & Martin, 2012, pp. 251eff; Huss, 1995, chap. 7, 195 pp.; Shawyer & Medina Pizzali, 2003, chaps. 4, 7, 108, pp.). Ice slurry is a mix of crushed or flaked ice with a small amount of water to make the mixture just mobile and to provide intimate contact between the surfaces of the fish and pieces of ice (Bellas & Tassou, 2005; Huss, 1995, chap. 7, 195 pp.). The slurry is intended for rapidly cooling the fish but not as a means of storage. It is harmless to fish taste and texture (Graham et al., 1992, chap. 3, 75 pp.). When properly used, it can keep fish fresh for long periods so that they remain attractive to wholesalers. A freshly harvested fish is one that is has been caught, bled, gilled, gutted and cleaned (Davey, 2012). To ensure sufficient amounts of ice are on hand at sea to cool the day's fish catch a number of predictive approaches have been made. These include the calculation methods of Graham et al. (1992), chap. 3, 75 pp.; Huss (1995), chap. 7, 195 pp. and Shawyer and Medina Pizzali (2003), chaps. 4, 7, 108, pp. and, more recently, that of Davey (2012).

* Corresponding author. Tel.: þ61 8 8313 5457; fax: þ61 8 8313 4373. E-mail address: [email protected].

However a major need of fishermen at sea is to know the time it will take for the fish in the catch to be cooled in the ice slurry to an acceptable (regulatory) temperature (at the thermal centre) (Anon, 2005, 2006; Graham et al., 1992, chap. 3, 75 pp.; Venugopal, 2006). Clearly, in a mixed catch of fish, the bigger (greater mass) will take longer than the smaller to cool in the slurry. A number of methods have been reported to predict cooling of individual fish, for example, those of Lin, Cleland, Cleland, and Serrallach (1996), Zhao, Kolbe, and Craven (1998) and Jain, Ilyas, Pathare, Prasad, and Singh (2005). These can however suffer drawbacks because Newtonian cooling (that is “lumped thermal capacity” Holman, 2002, pp. 31eff, 511eff, 675eff) is assumed (Jain et al., 2005; Jain & Pathare, 2007), or, they need a high level of mathematical sophistication (Lin et al., 1996; Zhao et al., 1998) for solution and application. These therefore have not been widely taken up for use by fisherman at sea. 1.1. This research Against this background a generalised and computationally convenient simulation of the cooling time of fish has been developed for fishermen and boat owners who use ice slurry to cool fish at sea. It is based on a fundamental solution of the transient (unsteady) cooling equation together with practical assumptions. The simulation is convenient because the key input is the mass of the harvested fish, which is readily and routinely measured at sea, and

http://dx.doi.org/10.1016/j.lwt.2014.08.022 0023-6438/Crown Copyright © 2014 Published by Elsevier Ltd. All rights reserved.

Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022

2

K.R. Davey / LWT - Food Science and Technology xxx (2014) 1e7

Nomenclature a, b AB Bi CB Cf F Fo h

constants, Eq. 11 constant, dimensionless, Eq. 1 Biot number, dimensionless, Eq. 4 constant, dimensionless, Eq. 1 heat capacity of fish, J kg1 K1, Eq. 7 fat content, % w/w, Eq. 12 Fourier number, dimensionless, Eq. 6 heat transfer coefficient (convection resistance), W m2 K1, Eq. 4 J0(), J1() Bessel functions of the First Kind, dimensionless, Eq. 5 kf thermal conductivity of fish (conduction resistance), W m1 K1, Eq. 4 K characteristic constant for a particular fish species at harvest, m1/2, Eq. 8a Lf characteristic length of a particular fish species at harvest, m, Eq. 8a mf mass of fish at harvest, kg, Eq. 8

because the simulation is presented as an “App” (Anon, 2012b) in standard spread-sheeting tools to give an immediate quantitative output for a wide range of users. Illustrative simulations are then presented and discussed for Southern BlueFin Tuna (Thunnus maccoyii) (SBT), an economically important and premium fish grown in Australia for export. Predictions when compared with established Heisler charts (Heisler, 1947; Holman, 2002, pp. 31eff, 511eff, 675eff; Schneider, 1955, pp. 213, 1963, pp. 213) show good agreement and it is concluded therefore the App is free of computational and programmable errors. The established simulations are used to underscore the significant impact small rises in temperature of the slurry will have in increasing the time for cooling and to highlight the need to maintain the slurry at the lowest equilibrium temperature with adequate amounts of prepared ice. The research will be of immediate benefit to fishermen and boat owners and agents who use ice slurry to cool and preserve fish. 2. Model development 2.1. The equations Consider a freshly harvested fish of mass, mf, at a uniform initial body temperature, Ti, suddenly (Bairi & Laraqi, 2003) immersed in ice slurry at a bulk temperature, TS (All terms used are carefully defined in the Nomenclature at the end of this paper). It is assumed that: i. There is sufficient ice so that the slurry bulk temperature, TS, does not change with time i.e. sufficient ice is available to prevent it all from melting in the slurry (The ice-to-fish mass ratio required can be reliably obtained for example from the method of Davey (2012)) ii. The harvested fish can be modelled as a cylinder of radius, s, on which any end effects are negligible iii. There is adequate mixing of ice, water and fish. This type of heat problem is defined as transient conduction with surface convection (Holman, 2002, pp. 31eff, 511eff, 675eff), that is, heat energy moves from within the fish to the fish surface by conduction and is carried away to the bulk slurry by convection. The temperature-time profile of interest during cooling in ice slurry is that at the centre of mass of the fish, that is, the centreline

M s t T0 TS Ti Vf

moisture content, % w/w, Eq. 12 characteristic radius of a particular fish species at harvest, m, Eq. 1 time for fish in ice slurry to reach T0, h, Eq. 1 target (centreline) temperature of cooled fish,  C, Eq. 2 ice slurry (convection environment) bulk temperature,  C, Eq. 2 initial (conduction environment) uniform temperature of fish  C, Eq. 3 characteristic volume of fish at harvest, m3

Greek symbols af thermal diffusivity of fish, m2 s1, Eq. 1 w0 target temperature difference (T0  TS), K, Eq. 2 wi initial temperature difference (Ti  TS), K, Eq. 3 rf density of fish, kg m3, Eq. 7 Other SBT

Southern Bluefin Tuna (Thunnus maccoyii)

temperature, T0, of the assumed equivalent cylinder (radius, s). This will be the point of slowest cooling in the fish mass. The solution for the centreline temperature of a cylinder with transient cooling, to within one percent, is given by Schneider (1955), pp. 213, (1963), pp. 213 and Heisler (1947) as:


q0 at ¼ CB exp  A2B f2 qi s

(1)

where two temperature difference terms are defined as:

q0 ¼ T0  TS

(2)

qi ¼ Ti  TS

(3)

CB and AB are constants that are defined using Bessel Functions, J0() and J1(), of the First Kind (Abramowitz & Stegun, 1972, chap. 9eff; Heisler, 1947; Schneider, 1955, pp. 213, 1963, pp. 213) such that:

AB J1 ðAB Þ hs ¼ Bi ¼ J0 ðAB Þ kf

(4)

and

2 CB ¼ AB

J1 ðAB Þ

!

J20 ðAB Þ þ J21 ðAB Þ

(5)

where Bi ¼ Biot number, h ¼ convective heat transfer coefficient from the fish to the bulk slurry and kf ¼ thermal conductivity of the fish. The Bi number defined in Eq. (4) is seen to compare the surfaceconvection with internal-conduction resistances to heat transfer (Holman, 2002, pp. 31eff, 511eff, 675eff). A low value of Bi (typically < 0.1) means that the internal conduction resistance is negligible compared with the surface convection resistance. In Eq. (1) the term:

af t ¼ Fo s2

(6)

where Fo ¼ Fourier number, t ¼ time elapsed in the slurry, and af ¼ thermal diffusivity of the fish, which is defined by:

Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022

K.R. Davey / LWT - Food Science and Technology xxx (2014) 1e7

af ¼

kf rf Cf

(7)

where respectively, rf ¼ density and Cf ¼ heat capacity, of the fish. From the practical standpoint, the radius of equivalent cylinder of the harvested fish needs to be expressed in terms of the more convenient fish mass, mf, which is readily and routinely measured. Given the mass is the product of density and volume: mf ¼ rf Vf ¼ rf (p s2 Lf) where Lf ¼ the characteristic length of a particular fish species at harvest; it can be readily shown therefore that the fish radius (s) will be reasonably approximated for a particular fish species at harvest over a defined range of size by:

s¼K

rffiffiffiffiffiffiffiffiffiffiffiffiffiffi . ffi mf rf

(8)

where K ¼ a characteristic constant for the particular fish species at harvest (m1/2) is;

K¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi . 1 pLf

(8a)

Substitution of Fo as defined in Eq. (6) into Eq. (1) and rearranging gives:


 q0 ¼ CB exp  A2B Fo qi

(9)

Equations 2 through 9 can be used to estimate the time (t) for cooling of a fish of size mf from an initial uniform temperature (Ti) to a target temperature at the centre of mass (slowest cooling point) (T0) in ice slurry of small-size particles (Huss, 1995, chap. 7, 195 pp.) and constant temperature (TS). 2.2. The assumptions The assumptions made in the model development can be readily implemented. This is because there are quantitative methods available for fisherman to make certain that sufficient quantities of ice will be on hand to both cool down the expected day's catch (Davey, 2012; Graham et al., 1992, chap. 3, 75 pp.; Huss, 1995, chap. 7, 195 pp.; Shawyer & Medina Pizzali, 2003, chaps. 4, 7, 108, pp.) and to maintain a constant slurry bulk temperature (TS) with time (Davey, 2012). Also, the use of equivalent shapes for cooling and heating calculations without sacrificing significant accuracy is well established (Abramowitz & Stegun, 1972, chap. 9eff; Holman, 2002, pp. 31eff, 511eff, 675eff; Heisler, 1947; Schneider, 1955, pp. 213, 1963, pp. 213). Given that there is a mixing action occasioned by the rocking of a boat at sea, and that the ice is chipped or flaked to small pieces of about 2 cm square (Davey, 2012; Huss, 1995, chap. 7, 195 pp.), and that it is a simple matter to visually check that ice particles do remain in the slurry, there will be a practically adequate mixing of ice and water. The assumptions in the method development are therefore seen to be practically valid.

3

3. Bi is calculated from the RHS of Eq. (4), and; an iterative procedure is used to calculate AB and the Bessel functions J0() and J1() 4. CB is obtained from Eq. (5), and w0 and wi from, respectively, Eqs. (2) and (3) 5. Fo is calculated from a rearranged Eq. 9 6. af is obtained from Eq. 7 7. Cooling time t is obtained from a rearranged Eq. (6). The calculations can be readily performed in Microsoft Excel™. The Solver function is used for step 3 such that the target cell is set to 0that is:

Bi 

AB J1 ðAB Þ ¼0 J0 ðAB Þ

(10)

Because Excel has nearly universal use this is computationally convenient and makes for streamlined and efficient communication of results. 2.4. Illustrative example Southern BlueFin Tuna (Thunnus maccoyii) (SBT) are an economically important fish exported from Australia (Phua, 2008; Phua, Davey, & Daughtry, 2007). Some 9133 tonne is exported annually, principally to Japan for the sushi market, and is valued at about AUD$150,000,000 (Anon. 2011). These are wild-caught and grown in sea-cages anchored to the sea bed about 16 km off the southern coasts and can be assumed to have an initial uniform body temperature at harvest of Ti ¼ 28  C (Davey, 2012) (This will be subject to the level of capture stress however; the more stress the higher this body temperature). A mean value of specific heat of Cf ¼ 3.35 kJ kg1  C1 (Anon, 2002, chap. 8, 2012a; Radhakrishnan, 1997, pp. 20; Zhang, Farkas, & Hale, 2001) and density of rf ¼ 1040 kg m3 (Davey, 2012). Regulation (Anon, 2005, 2006) requires that the thermal centre temperature of the fish T0  5  C after 12 h in the slurry. A harvested SBT has a typical mean mass of mf ¼ 35 kg with an equivalent diameter of 0.2 m (s ¼ 0.1) (J. Carragher, personal communication). Substitution into Eq. (8) for s, rf and mf gives K ¼ 0.545 m1/2. Because the range of mass of the SBT is small around the mean (with few fish 40 kg or more, or 30 kg or less) (J. Carragher, personal communication) this value of K serves to define this species catch. From Eq. (8a) the length of the SBT as an equivalent cylinder is therefore a realistic, Lf ¼ 1/ ((0.545)2p) ¼ 1.072 m (J. Carragher, personal communication). A value for the thermal conductivity kf ¼ 0.36 W m1 K1 is assumed (Radhakrishnan, 1997, pp. 20; Zhang et al., 2001). A practical value of the heat transfer coefficient for the ice slurry h ¼ 3010 W m2 K1 (Meewise & Infante Ferreira, 2003) is assumed. In practice, a period of about 12 he15 h will elapse between the time the harvested fish are placed in ice slurry aboard boats at sea, and being brought to shore for packing into refrigerated trucks for transport to wholesalers (J. Carragher, personal communication). 3. Results and discussion

2.3. The simulations 3.1. The simulation as a transportable App Simulations of the time needed for the fish to cool to the target temperature at the slowest point of cooling are carried out as follows: 1. Each of TS, Ti and T0, and; Cf, rf, kf, h, and; K and mf, must be specified 2. s is obtained from Eq. 8

Table 1 presents a summary simulation of the illustrative example carried out in Excel. It is seen from the table (row 22, column 3) the time required for cooling at the centre of mass (centreline) of an SBT of mf ¼ 35 kg in ice slurry at a constant temperature of TS ¼ 0  C from an initial uniform temperature of the fish of Ti ¼ 28  C to a target temperature of T0 ¼ 5  C is t ¼ 10.23 h.

Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022

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K.R. Davey / LWT - Food Science and Technology xxx (2014) 1e7

From inspection of Table 1 it can be seen that the calculation steps involved in the simulation have been constructed as a convenient and generalised “App” (Anon, 2012b; see Davey, 2012 for an example) in which input lines can be readily manipulated; this underscores the advantage of standard spread sheeting tools. A generalised App made widely available would benefit commercial fishermen and boat owners and agents because immediate quantitative output is available for a wide range of users of differing sophistication. Further, a transportable App might be programmed to permit a running simulation of how much cooling has actually occurred, and; with accumulated experience be used to quantitatively plan around any breakdown in continuous ice cooling at sea by manipulating slurry bulk temperature for example. 3.2. Effect of slurry bulk temperature Repeat simulations were carried out to investigate the effect of the bulk slurry temperature on time for cooling. This is important because if all the ice in the slurry melts it will manifest as a rise in slurry temperature (above 0  C) reducing the cooling effect and thereby increase the cooling time needed to the target temperature. Table 2 is a summary of these simulations for a wide range of values of SBT mass 25 kge45 kg with an ice slurry bulk temperature maintained at each of 2  C, 0  C and 2  C for cooling from an initial uniform temperature of 28  C. From Table 2 it is seen that the bigger the SBT (greater the mass) the longer the cooling time to the target temperature. For a typical mean mass of an SBT at harvest of 35 kg the estimated cooling time in slurry at an equilibrium temperature of 0  C is shown as 10.23 h. For the smallest fish of 25 kg the cooling time is reduced to 7.31 h. Importantly, the table highlights that these fish will cool to the Australian standard (Anon, 2006) centre temperature of 5  C in a typical 12 h cooling period. An unexpected and large SBT of 45 kg however would not be expected to be cooled in this time as is seen from the table. Significantly, Table 2 highlights that some 20% additional time is required for a slurry temperature that is raised only 2  C above the equilibrium ice slurry temperature of 0  C. For a typical harvested SBT of mean mass of 35 kg this is an increase of (12.27  10.23 ¼ Table 1 The simulation constructed as an App in Microsoft Excel™ spread sheeting tools. The particular output (bold text) is presented in Table 2. Inputs Row 1 2 3 4 5 6 7 8 9 10 Calculations 11 12 13 14 15 16 17 18 19 20 Outputs 21 Row 22

TS Ti T0 Cf



kf h

0 28 5 3350 1040 0.36 3010

C C  C J kg1 K1 kg m3 W m1 K1 W m2 K1

K mf

0.545 35

m2 Kg

s Bi AB J0 J1 CB

0.10 836.4 2.4019521 0.001492642 0.51976641 1.6019681 5 28 0.380 1.033E-07

M Dimensionless 2.48382  107 Dimensionless Dimensionless Dimensionless  C  C dimensionless m2 s1

36,828.9 10.23

S h

rf

q0 qi

Fo

a t



Eq. 8 Eq. 4 Solver

Eq. Eq. Eq. Eq. Eq.

5 2 3 9 7

Eq. 6

Table 2 Summary table of estimates for the time taken (t) to cool a freshly harvested Southern BlueFin Tuna to a centre temperature of 5  C from an initial uniform temperature of 28  C for a range of values of fish mass 25e45 kg with an ice slurry bulk temperature (TS) maintained at each of 2  C, 0  C and 2  C (The bold text is the particular simulation illustrated in Table 1). mf (kg)

T time taken to cool to 5  C from 28  C (h) TS ( C)

25 35 45

2

0

2

6.42 8.98 11.55

7.31 10.23 13.15

8.77 12.27 15.77

2.04 h) necessary in the ice slurry. At a total of 12.27 h needed in the slurry at this slightly elevated temperature the fish would not be cooled to Australian standards (Anon, 2006). It is crucial therefore that sufficient ice is prepared in advance and made available to maintain the minimum equilibrium ice slurry temperature of 0  C (Davey, 2012). In commercial practice at sea however, the assumption that ice slurry with observable pieces of ice in it will be at a bulk temperature of 0  C may not always hold true. This is simply because with too much melt water the ice floats on top. In this way commercial fish slurries can have a temperature of plus 2  C in volumes of the slurry. The impact of this effect on cooling time is highlighted by the quantitative estimates of the method presented in Table 2. This insight provided by the method underscores the need for removing excess water from the slurry. Adequate drainage is also important so that the fish do not become immersed in dirty water (Graham et al., 1992, chap. 3, 75 pp.). If there is too much ice however, there can be a tendency for it to freeze together and reduce mixing and cooling of the fish (Shawyer & Medina Pizzali, 2003, chaps. 4, 7, 108, pp.). Table 2 highlights also that a reduction of some 12% in the time for cooling can be achieved if the slurry bulk temperature can be  maintained at 2 C. This is because the lower the slurry bulk temperature the greater is the temperature gradient between the fish and the slurry for cooling. For an SBT of mean mass of 35 kg this is a reduction of (10.23  8.98 ¼ 1.25 h) in cooling time from an equilibrium bulk slurry temperature of 0  C in which particles of solid ice would be seen. In practice values of TS lower than 0  C can be realised if brine is frozen as ice, rather than (fresh) water. Brine slurry with a freezing point of 2  C can be readily made with a salt concentration of 62.82 g NaCl (molecular weight 58.44) per 1000 g water (Tinoco, Sauer, Wang, & Puglisi, 2002, p. 233). However, there are practical risks with using salt-water ice slurries. Tuna fish can become partly frozen and can absorb salt. These effects impact negatively on fish quality (Starling & Diver, 2005, 80 pp.).  A slurry temperature of 2 C, or less, could be met practically without brine however since fishing vessels often store ice brought on board from shore in a freezer that can result in ice at tempera tures as low as 20 C. Under these circumstances the fish will cool quickly to the target temperature. The model could be used to simulate a range of possible scenarios. 3.3. Effect of fish mass Table 3 is a summary of cooling time estimates for a range of values of SBT mass but with an initial uniform temperature of the  harvested fish of 26 C. From Tables 2 and 3 a direct comparison of the times to cool to the target temperature reveals that fish at an  initial temperature of 28 C will take about 3.5% longer to cool in ice  slurry (2  TS  2 C) over that from an initial temperature of  26 C. The initial harvest temperature of the fish over this range

Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022

K.R. Davey / LWT - Food Science and Technology xxx (2014) 1e7 Table 3 Summary table of estimates for the time taken (t) to cool a freshly harvested Southern BlueFin Tuna to a centre temperature of 5  C from an initial uniform temperature of 26  C for a range of values of fish mass 25e45 kg with an ice slurry  bulk temperature (TS) maintained at each of 2  C, 0 C and 2  C. mf (kg)

T time taken to cool to 5  C from 26  C (h) 

TS ( C)

25 35 45

2

0

2

6.19 8.66 11.13

7.06 9.88 12.71

8.50 11.19 15.29

therefore does not appear to significantly affect the time for cooling. 3.4. Validation In the absence of experimental field data for cooling of SBT in ice slurry at sea, estimates from the model were tested against widely established and appropriate Heisler charts (Bairi & Laraqi, 2003; Cengel, 2008, chap. 11, pp. 487; Heisler, 1947; Holman, 2002, pp. 31eff, 511eff, 675eff). For example, from the data of Table 1: at/ s2 ¼ 1.033  107  36828.9/0.12 ¼ 0.380 and k/hs ¼ 0.36/ 3010  0.10 ¼ 0.0012. From Fig. 4(a) of Bairi and Laraqi (2003), which conveniently has an expanded area of the chart, we get: q0/  qi ¼ 0.185. Using Eqs. (2) and (3) this gives T0 ¼ 5.18 C which is in  very good agreement with the value of 5 C of Table 1. Other calculations were made and shown to further verify the estimates of the method over the range of values of interest. It was concluded therefore that the method was free from programming and computational errors. 3.5. Effect of heat transfer coefficient To investigate any effects of the value of the heat transfer coefficient, h, repeat simulations were carried out for a range of values that cover those reported in the literature from fundamental experimental studies, Table 4. It can be seen in this table that the value of the heat transfer coefficient is reported as low as 1200 W m2 K1 (Bedecarrats, Strub & Peuvrel, 2009). The value of 3010 W m2 K1 from Meewise & Infante Ferreira (2003) however has been used and is readily justified because it was determined from experiments directly with ice slurry. Table 5 presents a summary of simulations from an initial uniform temperature of the  fish of Ti ¼ 28 C with a mean mass of mf ¼ 35 kg in a slurry at a bulk temperature of TS ¼ 0  C. It can be seen from Table 5 that there is no real change in the time required for cooling of 10.23 h. That is, the estimates from the method are largely insensitive to the value of the heat transfer coefficient over the range 500e3500 W m2 K1. Given this, it seems evident that more complex models (for example finite element heat transfer packages) may not in fact provide additional significant information for commercial management and operation of cooling practices by fisherman and wholesalers.

5

Table 5 Summary table of estimates for the time taken (t) to cool a 35 kg harvested Southern BlueFin Tuna to a centre temperature of 5  C from an initial uniform temperature of 28  C in ice slurry at 0  C for a range of values of the heat transfer coefficient (h). h (W m2 K1)

t (h)

500 1000 1500 2000 2500 3010 3500

10.35 10.28 10.25 10.24 10.24 10.23 10.23

3.6. Continuous cooling curves Fig. 1 presents a graphical summary of simulations for the centreline temperature (T0) of a 35 kg freshly harvested SBT with an initial uniform temperature of Ti ¼ 28  C immersed in ice slurry maintained at a bulk temperature of each of TS ¼ 2, 0 and 2  C. The value of this presentation of the simulations is that the lag that occurs at the centreline with transient cooling is clearly seen. The curves show that following sudden immersion there will be a lag of about 132 min before logarithmic cooling begins at the centre of mass of the fish. This lag is not apparent from the tabulated outputs for cooling. Clearly, the lag will be shorter with smaller fish. From the data of Table 1 it can be seen Bi ¼ 836.4 for a 35 kg SBT. Because this value and all others in the simulations for 25  mf  45 kg, respectively, 706.9  Bi  948.4, are significantly different >0.1 (Holman, 2002, pp. 31eff, 511eff, 675eff). It reinforces the fact that Newtons cooling law, that is lumped thermal capacity, cannot be reasonably applied with fish of typical mean size of 35 kg. In any event the generalised transient method developed here can be readily solved as illustrated without recourse to such simplifying assumptions.

3.7. Customisation Because in a catch of SBT the variance about the mean fish mass of 35 kg is small, the value of the characteristic constant at harvest K ¼ 0.545 m2 and hence s ¼ 0.1 m (Eq. (8)) used in the illustrative simulations was proximate for the whole catch (Substitution into

Table 4 Reported values of an appropriate heat transfer coefficient (h) for cooling of fish in ice slurry. Researchers

h (W m2 K1)

Bedecarrats et al. (2009) Bellas, Chaer, and Tassou (2002) Meewise & Infante Ferreira (2003)

1200 2400 3010

Fig. 1. Simulated cooling curve for the centreline temperature (T0) of a 35 kg freshly harvested fish with an initial uniform temperature of Ti ¼ 28  C in ice slurry maintained at a bulk temperature of TS ¼ 2, 0 and 2  C.

Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022

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K.R. Davey / LWT - Food Science and Technology xxx (2014) 1e7

Eq. (8) shows the variance about this parameter to be about 7.2% to define the whole catch). However, in customised applications to other single- or mixed-species in which the variance about the mean mass of the catch may be large, this parameter may need to be determined concisely. This could be done based on a number of limited bench-scale measurements of fish from a typical catch and, where appropriate for particular commercial fleets, could be determined selectively for a range of only the larger fish, since the smaller fish will cool to the target temperature in a shorter time. In this way, with experience the App could become customised for particular fisheries, boats and catches. 3.8. Further generalisation Mathematically, the simulation might be further generalised and extended. For example, the heat capacity (specific heat) of the fish could be mathematically defined in terms of mass (see for example Siebel, 1982) such that:

Cf ¼ a þ bmf

(11)

where a and b are empirical constants. However, small changes in the value of Cf can readily be shown to not appreciably impact estimates of cooling time. Similar empirical equations exist for kf and rf. A significant number of these empirical equations for seafood are reviewed by Radhakrishnan (1997), pp. 20 who experimentally developed the following equation for thermal conductivity:

kf ¼ 0:223  0:0036F þ 0:0035M

(12)

where F ¼ % fat content (w/w) and M ¼ % moisture content (w/w). However, a trade off needs to be made between increasing the complexity of the method through incorporating these, and any practical benefits in the resulting estimate that could be demonstrated. 3.9. Single fish and bulk numbers The method, strictly speaking, has been developed from considerations of an individual fish. However, providing fish immersed in the ice slurry are not so tightly packed as to prevent movement of ice and water (Graham et al., 1992, chap. 3, 75 pp.) it is evident that estimates will apply to each fish in the bulk numbers. Because the method is based on a fundamental solution to the transient cooling equation together with readily implemented practical assumptions, and predictions are not sensitive to a range of values of the heat transfer coefficient as demonstrated in extensive analyses, and because computations have been demonstrated to be free programming and computational errors, and the spreadsheet computations are stable, it is believed the method provides reliable estimates of the time for cooling of a fish in ice slurry at sea. It will therefore be of immediate practical use to fishermen, boat owners and agents who use ice to cool and preserve a range of fish species at sea. 4. Conclusions The transient (unsteady) cooling equation for a cylinder together with practical assumptions can be used to simulate the time for cooling of harvested fish in ice slurry at sea. A key input is the mass of the fish, which is readily and routinely measured. Illustrative simulations for Southern BlueFin Tuna (Thunnus maccoyii) show that a typical fish of 35 kg at harvest will cool to a centre mass temperature of 5  C in 10.23 h in slurry in equilibrium at 0  C.

Predictions are in good agreement with published Heisler (1947) charts. The simulations can be used to provide important quantitative insight into the impact of bulk temperature of the slurry on cooling times. If for example the bulk temperature of the ice slurry is  allowed to rise to 2 C, because of inadequate quantities of prepared  ice, the time taken to cool to 5 C will be 12.27 h. Simulations underscore that for best cooling practice, freshly harvested fish should be placed in ice slurry that is as close as possible to the theoretical minimum temperature of 0  C (not withstanding any acceptable use of brine slurries). The method is generalised and can be readily applied to a wide range of fish species. It can be solved computationally in standard spread-sheeting tools as a convenient App. It could be further generalised and extended to include effects of fish fat and moisture content. With experience it could be customised for particular fisheries and boats and catches. It will be of immediate practical use. Acknowledgement The author gratefully acknowledges industry assistance in data for the simulations from John Carragher, Food Innovation and Safety Research, SARDI, South Australia. References Abramowitz, M., & Stegun, I. A. (1972). Handbook of mathematical functions with formulas, graphs, and mathematical tables. with corrections (10th ed.). Washington DC: US Govt. Printing Office. Anon. (2002). ASHRAE handbook, refrigeration (SI ed.). Atlanta: American Society of Heating, Refrigerating and Air-conditioning Engineers Inc. Anon. (2005). AQIS (Australian Quarantine Inspection Service) Export orders, fish and fish products, Part 2, temperature Controls, chilling (Clause 18.1a). Canberra, Australia: Commw. Govt. Printer. Anon. (2006). Food Standards Australia and New Zealand, Standard 4.2.1, primary production and processing standard for seafood (Issue 78). Canberra, Australia: Commw. Govt. Printer. Anon. (2011). Australian fisheries statistics, Australian bureau of agricultural and resource economics statistics (ABARES), Dec. 2012. Canberra, Australia: Commw. Govt. Printer. Anon. (2012a). Engineering tool box Accessed 08.08.12 http://www. engineeringtoolbox.com/specific-heat-capacity-food-d_295.html. Anon. (2012b). Oxford English Dictionary (3rd ed.) Accessed 06.08.13 http://www. oed.com/view/Entry/249075. Bairi, A., & Laraqi, N. (2003). Diagrams for fast transient conduction in sphere and long cylinder subject to sudden and violent thermal effects on its surface. Applied Thermal Engineering, 23(11), 1373e1390. http://dx.doi.org/10.1016/ S1359-4311(03)00086-3. Fig. 4 (a). Bedecarrats, J. P., Strub, F., & Peuvrel, C. (2009). Thermal and hydrodynamic considerations of ice slurry in heat exchangers. International Journal of Refrigeration, 32, 1791e1800. http://dx.doi.org/10.1016/j.ijrefrig.2009.04.002. Fig. 5. Bellas, J., Chaer, I., & Tassou, S. A. (2002). Heat transfer and pressure drop of ice slurries in plate heat exchangers. Applied Thermal Engineering, 22, 721e732. http://dx.doi.org/10.1016/S1359-4311(01)00126-0. Fig. 7. Bellas, J., & Tassou, S. A. (2005). Present and future applications of ice slurries. International Journal of Refrigeration, 28(1), 115e121. http://dx.doi.org/10.1016/ j.ijrefrig.2004.07.009. Cengel, Y. A. (2008). Introduction to thermodynamics and heat transfer (2nd ed.). (p. 487). New York, NY: McGraw-Hill Higher Education, ISBN 0073380172. Fig. 1116. Davey, K. R. (2012). Calculating quantities of ice for cooling and maintenance of freshly harvested fish at sea. Journal of Food Science, 77(11), e335e341. http:// dx.doi.org/10.1111/j.1750-3841.2012.02963.x. FAO Fisheries Technical Paper. No. 331 Graham, J., Johnston, W. A., & Nicholson, F. J. (1992). Ice in fisheries (p. 75). Rome, FAO, ISBN 92-5-103280-7 http://www.fao. org/DOCREP/T0713E/T0713E00.HTM (1 of 3)5.7.2006 14:55:13. Granata, L. A., Flick, G. J., & Martin, R. E. (2012). The seafood industry: Species, products, processing, and safety (2nd ed.). (p. 251). Sussex, UK: Wiley-Blackwell, ISBN 978-0-8138-0258-9. Heisler, M. P. (1947). Temperature charts for induction and constant temperature heating. Transactions American Society of Mechanical Engineers, 69, 227e236. Holman, J. P. (2002). Heat Transfer (9th ed.). (p. 31). New York, NY: McGraw-Hill Higher Education, 511 ff, 675 ff. ISBN 13 978 0 07 240655 9. Huss, H. H. (1995). Quality and changes in fresh fish. FAO Fisheries Technical Paper No. 348. Rome, FAO (p. 195), ISBN 92-5-103507-5.

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Please cite this article in press as: Davey, K. R., Development and illustration of a computationally convenient App for simulation of transient cooling of fish in ice slurry at sea, LWT - Food Science and Technology (2014), http://dx.doi.org/10.1016/j.lwt.2014.08.022