Modeling of Natural Convection Heating and

International Journal of Food Science and Nutrition Engineering 2013, 3(4): 71-79 DOI: 10.5923/j.food.20130304.04

Modeling of Natural Convection Heating and Biochemical Changes in a Viscous Liquid Canned Food Using Computational Fluid Dynamics Khaled Rawajfeh1 , A. Ghani Albaali2 , Motasem Saidan1,* , Salah Abureden3 1 University of Jordan, Chemical Engineering Department, Amman, Jordan Princess Sumaya University for Technology, Environmental Technology and M anagement Department, Amman, Jordan 3 University of Waterloo, Chemical Engineering Department, Waterloo, Canada

2

Abstract This study presented the effect of heating temperature on the biochemical changes during thermal sterilization

of concentrated cherry ju ice. Natural convection heating of a viscous liquid (concentrated cherry ju ice 74° Brix) in a cylindrical can heating fro m all sides has been simulated and analy zed. The governing equations are solved numerically together with the concentration equations of bacteria and Vitamin C, using the Co mputational Fluid Dynamics CFD code PHOENICS. Arrhenius equation is introduced to the existing software package using a FORTRAN code to describe the biochemical changes kinetics. The model liquid is assumed to have constant properties except for the viscosity (temperature dependent) and density (Boussinesq approximat ion). The simu lations highlight the formation of the slowest heating zone (SHZ) resulted fro m d ifferent periods of heating. The results showed that the SHZ covers the whole cross sectional area of the can at early stages of heating then migrates during heating and settles at the bottom of the can. The simu lation also shows the formation of secondary flow due to the reverse flow, which in turn affect on the shape of the SHZ. It also shows the dependency of the concentration of live bacteria and Vitamin C on both temperature distribution and flow pattern.

Keywords Biochemical Change, Natural Convection, CFD Modeling, Heat Transfer

1. Introduction Thermal sterilizat ion process is usually done using steam or water under pressure to heat the food in the can to a temperature and hold it for a period sufficient to kill the microorganis ms. Thermal processing of food not only extends shelf life of the product, but also affects its quality such as vitamin destruction. One o f the challenges to the food canning industry is to min imize these quality losses. The optimizat ion to this process is possible because of the higher temperature dependence of the bacterial spore inactivation as compared to the rate of quality destruction[1]. Food processes must ensure safety and high-quality products for a growing demand consumer creat ing the need for better knowledge of its unit operations. The Co mputational Fluid Dynamics (CFD) has been widely used for better understanding the food thermal processes, and it is one of the safest and most frequently used methods for food preservation[2]. Heat is transferred through the liquid food in the can by * Corresponding author: [email protected] (Motasem Saidan) Published online on August 28th 2013 at http://journal.sapub.org/food Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved

both conduction and natural convection. Natural convection causes the slowest heating zone (SHZ) to move towards the bottom of the can. For this reason, estimates of the heat transfer rates are required in order to obtain optimu m processing conditions and to improve product quality. A better understanding of the heating mechanisms involved will assist in improved control of the processes and an increased in energy saving. Basic principles for determining the thermal processing methods have been presented[3, 4]. A number of nu merical heat transfer studies have been conducted to determine the temperature distributions within the liquid food material[5, 6]. Sterilizat ion of a v iscous liquid food in a metal can sitting in an upright position and heated from all sides (T = 121℃) in a still retort has been simu lated[7]. They also presented a simulat ion for the same can when its bottom and top surfaces were insulated[7]. Solving the governing equations for sodium carbo xy-methyl cellu lose and water using a computation fluid dynamics (CFD) to simu late natural convection heating during sterilizat ion of canned food has been done[8]. Other simu lation showed that the presence of an air gap at the top of the can filled with a viscous liquid (CM C) had no significant effect on the rate of heating. This is due to the strong influence of natural convection current, even with such highly viscous liquid[9].

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Khaled Rawajfeh et al.: M odeling of Natural Convection Heating and Biochemical Changes in a Viscous Liquid Canned Food Using Computational Fluid Dynamics

Thermal processing of liquid model food pasteurization in two commercial bottles using CFD has been studied[10]. The model validation was carried out by comparing the obtained thermal profile with experimental data. The flu id behavior during heating and cooling was detailed. The data obtained using CFD described well the experimental processing, which suggest the possibility of using CFD for temperature profile evaluation in thermal processing of packaged convective foods. In reality the death of microorganisms is expected to begin at an early stage of heating especially at locations near the wall, where the temperature approaches the retort temperature very quickly. Hence, it is necessary to solve the partial differential equation governing the live bacteria concentration, coupled with the equations of continuity, mo mentu m and energy, and followed this approach to study sterilizat ion of liquid food in a non-agitated cylindrical enclosure heated from all sides[5]. In this work, the computational procedure required to determine most of the biochemical changes during processing has been described but without explicitly fo llo wing the liquid elements. The deactivation of bacteria in a canned liquid food during sterilizat ion was investigated and found that the concentration of the live bacteria depended on both temperature distribution and flo w pattern. The effect of diffusion of bacteria on the rate of sterilization was found to be negligible in the cases simulated in the study[11, 12]. As mentioned above, thermal processing of liquid food materials always results in biochemical changes, depending on sterilization time and temperature. To estimate these changes the liquid food need to be tagged and monitored, which is co mputationally difficult for most flow situation of interest. Our developed method, shows a computational procedure for describing the biochemical changes during sterilizat ion process of viscous liquid (concentrated cherry juice - 74o Brix), in a non-agitated cylindrical can heated fro m all sides. A co mputational fluid dynamics software package (CFD) is used to study the effect of mediu m heating temperature. The partial differential equations describing the conservation of mass, momentu m, and energy (governing equations), are solved numerically together with those for bacteria and vitamin C concentrations. Saturated steam at 121℃ was used as a heating media. The effects of increasing temperature on the temperature of the slowest heating zone SHZ, bacteria deactivation and vitamin destruction are presented in this paper for different periods of heating.

2. Numerical Solutions 2.1. Computati onal Gri d Temperature and velocit ies have their largest variations in the boundary layer. To adequately resolve this boundary layer flow i.e. to keep discretizat ion error small, the mesh should be optimized and a large concentration of grid points is needed in this region. If the boundary layer is not resolved

adequately, the underlying physics of the flow is lost and the simu lation will be erroneous. On the other hand, in the rest of the domain where the variations in the temperature and velocity are small, the use of a fine mesh will lead to increases in the co mputation time without any significant increase in accuracy. Thus a non-uniform g rid system is needed, to resolve the physics of the flow properly. A non-uniform grid system was used in the simulat ions with 3500 nodal points: 70 in the axial direct ion and 50 in the radial direction, g raded in both directions with a finer grid near the wall. The co mputations are performed for a can with a radius of 0.040 m and height of 0.11m. The can outer surface temperature (top, bottom and side) is assumed to rise instantaneously and remained at constant temperature of 121 ℃ throughout the heating period. The natural convection heating of the liquid food was simulated for 2600s. It took 100 steps to achieve the first 200s of heating, another 100 steps to reach 1000 s and 100 steps for the rest of heating. This required 63 hr of CPU time on the UNIX IBM RS6000 workstations at the University of Auckland. Solutions have been obtained using a variety of grid sizes and time steps and the results show that the solutions are time-step independent and weekly dependant on grid variation. 2.2. Convection and Temporal Discretization An important consideration in CFD is the discretizat ion of the convection terms in the fin ite volume equations. The accuracy, numerical stability and the boundedness of the solution depends on the numerical scheme used for these terms. The central issue is the specification of an appropriate relationship between the convected variables, stored at the cell center, and its value at each of the cell faces[13]. The convection discretization scheme used for all variab les in our simu lations is the Hybrid-Differencing scheme (HDS). The HDS used in PHOENICS, switches the discretization of the convective terms between Central Differencing Scheme (CDS) and Upwind Differencing Scheme (UDS) according to the local cell Peclet nu mber[14]. The cell Peclet nu mber Pe (ratio of convection to diffusion) for bacteria and vitamins within the flow do main in the z-d irection is:

Pe =

u ∆z

α

(1)

where u and ∆z is the typical velocity and typical distance of the cell in the z-direct ion (vert ical). The thermal diffusivity α of the bacteria and the vitamins in the flu id is given by the Stockes-Einstein equation:

α=

kT T 6πµ a

(2)

where k T is the reaction rate constant for particle at temperature T, T is the temperature , µ is the apparent viscosity and a is the radius of the particle. The calculated cell Peclet numbers with in the flow do main in this study are of the order of 104 , and so the diffusion of

International Journal of Food Science and Nutrition Engineering 2013, 3(4): 71-79

bacteria and vitamins has been ignored. Within PHOENICS, the temporal d iscretizaion is fully exp licit.

73

in a cylindrical space are given below[15]: Continuity equation

∂ 1∂ (rρv ) + ( ρu) = o ∂z r ∂r

2.3. Governing Equati ons Heat transfer in a can with 80 mm d iameter and 111 mm height is studied. The partial differential equations governing natural convection with constant thermal physical properties

(3)

Energy conservation

∂T ∂T ∂T k +v +u = ∂t ∂r ∂z ρC p

 1 ∂  ∂T  ∂ 2 T  + r  2   r ∂r  ∂r  ∂z 

(4)

Momentum equation in the vertical direction

 1 ∂  ∂u  ∂ 2 u  ∂u ∂u  ∂p  ∂u = − + µ +v +u + ρg +  r  2  ∂r ∂z  ∂z  ∂t  r ∂r  ∂r  ∂z 

ρ

(5)

Momentum equation in the radial direction 2 ∂ 1 ∂  ∂v ∂v  ∂p  ∂v (rv ) + ∂ 2v  +v +u  = − + µ  ∂r ∂z  ∂r  ∂t  ∂z   ∂r  r ∂r

ρ

(6)

The partial d ifferential equations of continuity, mo mentu m and energy described earlier in this section were solved together with that for bacteria deactivation and Vitamin C concentration given below: Concentration equation (Mass Balance) for bacteria deactivation

 1 ∂  ∂C b ∂C b ∂C b ∂C +v b +u = D r r r ∂ ∂t ∂r ∂z  ∂r 

2  ∂ Cb  +   − k b Cb 2 z ∂  

(7)

Concentration equation (Mass Balance) for Vitamin C destruction

 1 ∂  ∂C v  ∂ 2 C v  ∂C v ∂C v ∂C v +v +u = D r +  − k v Cv ∂t ∂r ∂z ∂z 2   r ∂r  ∂r 

(8)

2.4. Boundary and Initi al Conditions At the can boundary, r = R,

T = Tw ,

u = 0,

v = 0,

fo r

u = 0,

v = 0,

fo r

∂u = 0, ∂r

v = 0,

At the bottom of the can, z = 0,

T = Tw ,

At symmetry, r = 0,

∂T = 0, ∂r

At the top of the can, z = H,

T = Tw

u = 0,

v = 0,

fo r

fo r

0≤ z≤ H 0≤r ≤ R

0≤ z≤ H

0≤r ≤ R

(9) (10) (11) (12)

Initially, the fluid is at rest and is at a uniform temperature. The in itial conditions used are: T = To, Cb = Cbo , Cv = Cvo, u = 0 and v = 0 (13) Where, Cbo and Cvo are the initial relat ive concentration of bacteria and Vitamin C respectively. The properties of concentrated cherry juice used in the current simu lations are: ρ =1052 kg m-3 , CP = 3500 Jkg -1 K-1 , k = 0.554 W m-1 K-1 and β = 0.0002 K-1 [16]. The governing equations with the conditions described above were solved using the fluid dynamics analysis program PHOENICS software package, which is based on a finite volume method. The details of this code can be found in the PHOENICS manuals, especially the PHOENICS input language manual (PHOENICS Reference Manual). 2.5. Assumptions Made for Si mplicity

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Khaled Rawajfeh et al.: M odeling of Natural Convection Heating and Biochemical Changes in a Viscous Liquid Canned Food Using Computational Fluid Dynamics

(1) A xisymmetry, wh ich reduces the problem fro m 3-D to 2-D. (2) Heat generation due to viscous dissipation is negligible. (3) Boussinesq approximation is valid. (4) Specific heat (Cp ), thermal conductivity (k), and thermal expansion coefficient ( β) are constants. (5) The assumption of no-slip condition at the inside wall of the can is valid. (6) The condensing steam maintains a condition of constant wall temperature. (7) The thermal boundary conditions are applied to liquid boundaries rather than the outer boundaries of the can. (8) The effect of mo lecular diffusion of bacteria and Vitamin C can be neglected co mpared to the strong influence of the convection current. 2.6. Viscosity Model Food materials are in general h ighly viscous and have properties, wh ich are temperature dependent. In natural convection heating of viscous liquid, shear rate is s mall[8] and hence the viscosity may be assumed function of temperature only (Newtonian fluid). This Newtonian approximation may be valid for most liquid food materials such as tomato puree, carrot puree, green bean puree, apple sauce, apricot puree, and banana puree, which are regularly canned and usually preserved by thermal sterilization[17]. In the simulat ion presented here, the viscosity is assumed to be a function of temperature. For co mputational convenience non-linear curve fitting is used to the availab le data for the v iscosity of concentrated cherry ju ice (74o Brix) as reported[18] to obtain the values of second order polynomial constants (a, b and c) of the form:

µ = a + bT + cT 2

(14) -2

-1

Which are found to be 1.47 Pa.s, -4.21x10 Pa.s K and 3.15x10-4 Pa.s K-2 respectively. 2.7. Bacteria Deacti vation and Vitamin C Destruction Ki netics Thermal treat ment of food causes some b iological reaction that affects its quality. These changes may include positive impacts such as the reduction of microbial populations (i.e. bacteria deactivation). Ho wever, it may also include a negative impact on the nutrient concentration and taste (i.e. vitamin destruction). The rates of bacteria deactivation and vitamin destruction are usually assumed to follow first order kinetics[19]. It is known that the reaction rate constants (kb and kv) are function of temperature and usually described by Arrhenius equation: − Eb / Rg T (15) b b

k =Ae

k v = Av e

− Ev / Rg T

(16)

-1

where Ab , Av = reaction frequency factor[s ] Eb, Ev = activation energy[kJ (kg mo l)-1 ] Rg = universal gas constant[kJ (kg mo l)-1 K-1 ] T = temperature[K] In food process engineering context, the decimal reduction time (D) is used more. The relat ionship between the reaction rate constant and the decimal reduction time is[20]:

k b ,v =

2.303 DT

(17)

The activation energy for bacteria deactivation (Clostridium botilinium) used in our simu lation is 300 kJ/ mol and the corresponding value for Vitamin C is 100 kJ/ mol[21]. Decimal reduction time at 121℃ (D121 ) for Clostridium botilinium is 0.2 min [20] and for Vitamin C is 245 min[21]. The reaction rate constants kb and k v are calcu lated using equation (17). Equations (15) and (16) are then used to calculate the constants of Arrhenius equation Ab and Av giving values of 2.5 x 1011 s -1 for bacteria and 2.85 x 109 s -1 for Vitamin C. The concentrations Cb and Cv in equations (7) and (8) are taken as a dimensionless concentration defined as the ratio of the real time concentration to the initial concentration (mult iplied by 100).

3. Result and Discussion The objective of th is study is to obtain improved quantitative understanding of effect of the natural convection upon biochemical changes in a viscous liquid canned food during thermal sterilizat ion. At the early stages of heating, the results of the study[11] showed that the process of heating is governed by conduction only. As time progresses, heating is dominated by natural convection, as shown in figures 1, 2 and 3. These figures show the results of the simu lation after relat ively longer periods of 1000s, 1650s and 2450s for a metal can filled with concentrated cherry juice (74° Brix), steam and heated fro m all sides (at 121℃) in a still retort. Figures 1, 2 and 3 show clearly, the comb ined effects of the temperature distribution and velocity profile on the shape and location of the high bacteria concentration zone HBCZ and high vitamin concentration zone HVCZ. The streamline shown in these figures also shows the effect of flu id circulat ion that leads to two stagnant zones, which affect on HBCZ and HVCZ. These results lead to the conclusion that, the bacteria and Vitamin C profiles depend not only on the temperature distribution, but also on the velocity profiles in the can. The locations of the HBCZ and HVCZ occur almost within the stagnant zones, which belong to minimu m liquid velocity.

International Journal of Food Science and Nutrition Engineering 2013, 3(4): 71-79

Temperature

Vitamin C destruction

75

Bacteria deactivation

Streamline

Figure 1. T emperature, bacteria deactivation, Vitamin C destruction and streamline contours in a can filled with concentrated cherry juice (74o Brix) and heated by conducting steam after 1000s

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Khaled Rawajfeh et al.: M odeling of Natural Convection Heating and Biochemical Changes in a Viscous Liquid Canned Food Using Computational Fluid Dynamics

Temperature

Vitamin C destruction

Bacteria deactivation

Streamline

Figure 2. Temperature, bacteria deactivation, Vitamin C destruction and streamline contours in a can filled with concentrated cherry juice (74o Brix) and heated by conducting steam after 1650s

International Journal of Food Science and Nutrition Engineering 2013, 3(4): 71-79

Temperature

Vitamin C destruction

77

Bacteria deactivation

Streamline

Figure 3. Temperature, bacteria deactivation, Vitamin C destruction and streamline contours in a can filled with concentrated cherry juice (74o Brix) and heated by conducting steam after 2450s

Figure 1 shows that the slowest heating zone (SHZ) migrates and settles down at the bottom of the can due to the effect of natural convection. The slowest heating zone reaches 93℃ after 1000s of heating while it rises to 106℃ and 113℃ when the sterilizat ion time is progressed to 1650s

and 2450s respectively. Th is has caused the relative bacteria concentration in the slowest heating zone (SHZ) to drop down fro m 67% for the sterilization time of 1000s to 3.6 x 10-3 and 3.8 x 10-9 for the sterilizat ion times of 1650s and 2450s respectively. Co mp lete destruction of bacteria has

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Khaled Rawajfeh et al.: M odeling of Natural Convection Heating and Biochemical Changes in a Viscous Liquid Canned Food Using Computational Fluid Dynamics

occurred in most of the other locations. Most of Vitamin C destruction has occurred at locations near the wall and especially at the top of the can, which are the locations of high temperature. The figures also show that Vitamin C is concentrated at the SHZ, and the destruction of the vitamin has occurred mostly in other locations. The liquid and thus the vitamin carried with it are exposed to much less thermal treatment at the SHZ than the rest of the product. As the sterilizat ion time is increased fro m 1000s to 1650s and 2450s, the relative concentration of Vitamin C at locations near the wall has dropped fro m 86% to 76% and 65% for the same reason explained earlier. Figures 2 and 3 show the time at wh ich the bacteria concentration drops down to low practical level, and hence it represents an important illustration of what is likely to occur in practice. The influence of the SHZ on the relat ive bacteria concentrations is very significant due to the exponential effect of temperature on bacteria deactivation rate constant, as defined by Arrhenius equation. The corresponding relative Vitamin C concentration at locations near the wall has dropped to 76% to 65% for the same reason. Both concentrations of bacteria and vitamin depend not only on the temperature, but also on the flow pattern. This has been discussed in detail for the bacteria concentration only[11]. Figures 1, 2, and 3 show that, SHZ and HBZ occur at a similar location in the can. However, the HBZ is clearly influenced by the flow pattern as shown in the figures, which illustrates the existence of two stagnant regions. The influence of flow pattern on vitamin C concentrations is even stronger as the locations of the low vitamin C concentration occur exactly at the stagnant zones.

4. Conclusions Transient temperature, velocity profile, bacteria deactivation and Vitamin C destruction evolving during natural convection heating of viscous liquid (concentrated cherry juice - 74° Brix) in a cylindrical can have been simu lated. This has been done by solving the governing equations for continuity, mo mentum and energy conservation together with that for bacteria and Vitamin C concentrations. Arrhenius equation is used to describe the temperature dependency of bacteria deactivation and Vitamin C destruction kinetics. The result of the simulat ion shows the occurrence of secondary flow at the bottom of the can due to reverse flow in addition to the slowest heating zone and its migration towards the bottom of the can. The concentrations of bacteria and Vitamin C have been seen to depend on both temperature distribution and flo w pattern. It is shown clearly that the SHZ and HBCZ occur at similar locations in the can. Vitamin C concentrations is even stronger influenced by the flow pattern as the locations of the low Vitamin C concentration occur exactly at the stagnant zone.

Nomenclature A reaction frequency factor[s -1 ] Ab bacteria reaction frequency factor[s -1 ] Av vitamin reaction frequency factor[s -1 ] a radius of the particle[mm] Cb concentration of the bacteria in food liquid[nu mber of bacteria m-3] Cv concentration of the vitamin in food liquid[kg m-3] Cbo initial concentration of bacteria[nu mber of bacteria m-3 ] Cvo initial concentration of vitamin[kg m-3 ] CP specific heat of liquid food[J kg -1 K-1 ] D diffusion coefficient[m2 s -1 ] DT decimal reduction time[min] activation energy for bacteria deactivation[kJ (kg Eb mo l)-1 ] activation energy for vitamin destruction[kJ (kg Ev mo l)-1 ]

Gr

Garshof number, Gr =

gβ∆Tx 3 ρ 2

µ2

acceleration due to gravity[ms -2 ] height of the can[m] thermal conductivity of liquid being heated[W m-1 K-1 ] k T reaction rate constant T[s -1 ] kb reaction rate constant of bacterial deactivation at temperature T[s -1 ] k v reaction rate constant of vitamin destruction at temperature T[s -1 ] pressure[Pa] p Pe Peclet number r radial position fro m center line[m] ro radius of the can[m] Rg universal gas constant[kJ (kg mo l)-1 K-1 ] t heating time[s] T temperature[°C] Tw wall temperature[°C] Ti initial temperature[°C] To reference temperature[°C] u velocity in vertical d irection[ms -1 ] v velocity in radial direction[ms -1 ] z distance in vertical direct ion fro m the bottom[m] α thermal diffusivity[m2 s -1 ] β thermal expansion coefficient[K-1 ] µ apparent viscosity[Pa.s] ρ density[kg m-3 ] g H k

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