A mixture-enthalpy fixed-grid model for temperature evolution and heterocyclic-amine formation in a frying beef patty

Food Research International 44 (2011) 789–797

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Food Research International j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / f o o d r e s

A mixture-enthalpy fixed-grid model for temperature evolution and heterocyclic-amine formation in a frying beef patty Michael A. Sprague ⁎, Michael E. Colvin Center for Computational Biology and School of Natural Sciences, University of California, Merced, CA 95343, USA

a r t i c l e

i n f o

Article history: Received 20 October 2010 Accepted 8 January 2011 Keywords: Cooking simulations Spectral finite-element methods Carcinogens Mutagens

a b s t r a c t The ideal cooking process would heat food to a sufficient temperature throughout to kill bacteria without heating the food to temperatures that promote formation of toxic or carcinogenic compounds. Experimentally validated computer models have an important role to play in designing cooking processes since they allow rapid evaluations of different conditions without the confounding effects of experimental variation. In this paper we derive a mathematical model governing the heat and water transport in a cylindrical pan-fried beef patty. The continuum temperature model stems from a mixture-enthalpy formulation that accommodates the liquid and vapor states of water along with fat and protein. The governing equations were spatially discretized with Legendre spectral finite elements. All but two of the model properties were taken from the literature, with the remaining two determined through a comparison of numerical and physical experiments. These parameters were shown to produce solutions in agreement with a different set of experimental results. The model was used to calculate the formation of heterocyclic-amine (HA) compounds (known DNA mutagens and carcinogens). Results provide an explanation based on patty temperature for previous experimental studies showing that frequent patty flipping yields a dramatic reduction in HAs. Published by Elsevier Ltd.

1. Introduction This paper is concerned with the mathematical and computational modeling of pan frying of thawed beef patties. We are interested in predicting the transport of heat and water, and the associated formation of heterocyclic-amine (HA) compounds, which have been shown to be mutagenic (Felton et al., 1981; Sugimura et al., 1977) and carcinogenic (Adamson et al., 1990; Ohgaki et al., 1991). While it is established that lower cooking temperatures (Knize et al., 1994) and frequent patty flipping (Salmon et al., 2000) correspond to fewer carcinogens, adequately high temperatures must be reached throughout the patty to kill harmful bacteria. Thus, there are the competing goals of heating the patty sufficiently well while minimizing the formation of carcinogens. Accurate computational modeling of a frying beef patty allows analysts to better understand the evolution of the patty during frying, which may provide clues to develop improved cooking methods. The physical and chemical processes inherent in beef-patty frying are numerous, complicated, and, in many cases, poorly understood. Beef-patty frying is seen as inherently nonlinear due to such issues as temperature-dependent material properties and phase change due to the vaporization of water. A number of computational models for pan

⁎ Corresponding author. Current address: National Renewable Energy Laboratory, 1617 Cole Blvd., MS 1608, Golden, CO 80401, USA. Tel.: +1 303 275 4367. E-mail address: [email protected] (M.A. Sprague). 0963-9969/$ – see front matter. Published by Elsevier Ltd. doi:10.1016/j.foodres.2011.01.011

and/or immersion frying of foods have been established, which can be categorized as either fixed-grid (Chen, Marks, & Murphy, 1999; Goñi & Salvadori, 2010; Ikediala, Correia, Fenton, & Ben-Abdallah, 1996; Obuz, Posell, & Dikeman, 2002; Ou & Mittal, 2006a; Ou & Mittal, 2006b; Ou & Mittal, 2007; Pan, Singh, & Rumsey, 2000; Shilton, Mallikarjunan, & Sheridan, 2002; van der Sman, 2007; Wang & Singh, 2004) or moving-interface (Farid & Chen, 1998; Farkas, Singh, & Rumsey, 1996a; Farkas, Singh, & Rumsey, 1996b; Vijayan & Singh, 1997; Zorrilla & Singh, 2000; Zorrilla & Singh, 2003). In the fixed-grid approach, the entire domain is represented by the same mathematical model. In the moving-interface or moving-boundary approach, interfaces separate the model domain into regions (e.g. core and crust), each treated by different models and/or properties. In this paper, we employ the former, in part because it is more amenable to problems in general geometry. Many models for frying processes rely on properties fit to problem-specific empirical data, which limits broader applicability. For example, Ikediala et al. (1996) employ a data-fit moisture-loss model and Tran, Salmon, Knize, and Colvin (2002) employ data-fit conductivity, density, and heat capacity. Few of the existing models have been successfully validated in the high-temperature and/or latetime regime, when significant mass loss has occurred due to highly nonlinear water vaporization, which also dominates temperature response. When the temperature reaches the water boiling temperature, added energy is devoted to the vaporization process and temperature remains approximately constant until the accessible liquid water is fully vaporized.

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Nomenclature A cp De Dd DT DX– Ea k h H [HA] ½HA L n N NR NZ r R Rg t tmax tˆ tcook,sing tcook,mult T Tboil Td TK Tref Tˆ θ V X Xw z Z

exponential prefactor (ng-HA g-uncooked-meat− 1 s− 1) specific heat (J kg− 1 °C− 1) effective liquid water diffusivity (kg-water kg-solid− 1) constant in De expression, 1.43 × 10− 6 m2 s− 1 constant in De expression, 1.58 × 103 K constant in De expression, 6.72 × 10− 2 kg-water kg-solid− 1 activation energy (cal mol− 1) thermal conductivity (W m−1 °C− 1) heat-transfer coefficient (W m−2 °C− 1) mixture enthalpy (J m− 3) local concentration of heterocyclic amine (ng-HA g-uncooked-meat− 1) overall concentration of heterocyclic amine (ng-HA g-uncooked-meat− 1) latent heat of vaporization, 2.260 × 106 J kg− 1 boundary-normal coordinate (m) polynomial-order of Legendre spectral finite-element basis functions number of elements spanning radius R number of elements spanning height Z radial coordinate (m) patty radius (m) gas constant, 1.986 cal K−1 mol− 1 time (s) final simulation time (s) nondimensional time total cooking time for single-flip calculations (s) total cooking time for multi-flip calculations (s) temperature (°C) boiling-water temperature, 102 °C water diffusion temperature, 30 °C temperature in Kelvin (K) reference temperature for mixture enthalpy (°C) nondimensional temperature meridional coordinate volume of patty (m3) volume fraction liquid water mass fraction on dry basis (kg-water kg-solid− 1) vertical coordinate (m) patty height (m)

Greek symbols α scale factor for radius and height shrinkage δwe constant in Xw,eq expression, 0.0132 °C− 1 Δt time step size (s) Ω patty meridional-section domain ∂Ω meridional-section boundary ρ density (kg m− 3) τv vapor-generation decay time (s)

Superscripts 0 initial condition ′ denotes dummy-variable of integration

Subscripts air air eq equilibrium f fat i material-component place holder p protein pan pan sym symmetry w liquid water v vapor water

In this paper we consider the pan frying of a thawed beef patty composed of water, fat, and protein. Our primary goal is to develop a time-dependent temperature model that is sufficiently accurate for long-duration frying, which requires accurate representation water transport and energy loss associated with water vaporization. Our secondary goal is for the model to be broadly applicable. To that end, the model employs tabulated property values where appropriate and has as few free parameters as possible. Also, we spatially discretize the mathematical model with Legendre spectral finite elements, which are high-order finite elements well suited for general geometry problems. Finally, we employ the temperature model to predict the formation of HAs, and compare those results with empirical data in the literature. The paper is organized as follows. In Section 2 we describe the problem and the mathematical models for heat and liquid water transport, vapor generation, and HA formation. In Section 3 we describe the discretization of the mathematical model with Legendre spectral finite elements. Validation of the numerical model and numerical studies are described in Section 4, and concluding remarks are given in Section 5. 2. Formulation 2.1. Problem description The beef patty is represented as a right cylinder with radius R(t) and height Z(t) at time t; initial values are R(0) = R0 and Z(0) = Z 0. The schematic in Fig. 1 shows a meridional section of the patty in polar coordinates (r, θ, z). Assuming axially symmetric behavior, temperature is denoted by T(r, z, t). We denote the section domain of the patty as Ω and the boundary as ∂ Ω = ∂ Ωpan ∪ ∂ Ωair ∪ ∂ Ωsym, where ∂Ωpan are the boundaries in contact with the pan, ∂Ωair are the boundaries in contact with the surrounding air, and ∂Ωsym is the symmetry boundary. The patty is initially composed of fat, protein, and liquid water with uniform volume fractions X0f , X0p, and X0w, respectively. At t = 0, the patty has uniform temperature T 0 and the boundary located at z = 0 is placed in contact with a frying-pan surface with known temperature behavior Tpan(t), whose preheated 0 temperature is Tpan(0) = Tpan . The air temperature Tair is constant. Below we describe models governing the transport of heat and water,

R z r

Ω

Z

∂Ω Fig. 1. Schematic of a meridional section Ω (grey) located at θ = 0 of a cylindrical beef patty (dashed lines).

M.A. Sprague, M.E. Colvin / Food Research International 44 (2011) 789–797

the generation of water vapor, and the formation of heterocyclic amines. The section concludes with a presentation of material properties taken from the literature. 2.2. Heat transport We derive here a continuum model for heat transport in a beef patty composed of fat, protein, and liquid water that can vaporize. We employ the concept of a local volume fraction, which is valid over a small isothermal representative elementary volume (REV) (see, e.g., Bachmat and Bear (1986) and Voller, Swaminathan, and Thomas (1990)). A field value at a point in the resulting continuum-model domain is representative of the constant value over the REV centered at that point. It is assumed that a REV can be composed of multiple materials and/or phases. Following Voller et al. (1990) and neglecting advection effects, the time evolution of the local mixture enthalpy H(r, z, t) is governed in Ω as

 ∂H 1∂ ∂T ∂ ∂T = kr + k ; r ∂r ∂t ∂r ∂z ∂z

ð1Þ

With these simplifications, Eqs. (1) and (4) are combined to yield a single equation governing temperature in Ω as
 ∂ρv ∂T Xp ρp cP;p + Xf ρf cP;f + Xw ρw cP;w + Xv ρv cP;v + Xv L = ∂T ∂t

 1∂ ∂T ∂ ∂T kr + k r ∂r ∂r ∂z ∂z −ρv

ð5Þ

∂Xv ∂X T ∂X T L− w ∫Tref ρw cP;w dT ′ − v ∫Tref ρv cP;v dT ′ ; ∂t ∂t ∂t

where initial conditions are described above. Temperature-dependent material properties are one source of nonlinearity in Eq. (5). If constant material properties are assumed and mass transport and vapor generation is neglected, Eq. (5) reduces to a linear diffusion equation for temperature like that used by Tran et al. (2002). We note that the last two terms in Eq. (5) are apparently neglected in other computational studies (see, e.g., Ikediala et al. (1996) and Wang and Singh (2004)). Heat-flux boundary conditions are applied at bounding surfaces. On the axis of symmetry there is no heat flux and thus

where the mixture conductivity is defined as k k = Xp kp + Xf kf + Xw kw + Xv kv ;

791

ð2Þ

∂T =0 ∂n

on ∂Ωsym ;

ð6Þ

where ∂T / ∂n is the derivative in the direction normal-out from the patty boundary. On sides in contact with the surrounding air

and H = Xp ∫TTref ρp cP;p dT ′ + Xf ∫TTref ρf cP;f dT ′ +

T Xw ∫Tref

ρw cP;w dT ′ +

T Xv ∫Tref

ð3Þ

ρv cP;v dT ′ + ρv Xv L: 0

Here, Tref is an arbitrary reference temperature taken as T ; Xf (r, z, t), Xp(r, z, t), Xw(r, z, t), and Xv(r, z, t) are local volume fractions for fat, protein, liquid water, and water vapor, respectively. Volume-fraction initial conditions are assumed to be spatially uniform; at t = 0 s, Xf (r, z, 0) = X0f , Xp(r, z, 0) = X0p, Xw(r, z, 0) = X0w, and Xv(r, z, 0) = 0. For a pure quantity of material component i, ki(T) is thermal conductivity, ρi(T) is density, and cP, i(T) is specific heat; p denotes protein, f denotes fat, w denotes liquid water, and v denotes water vapor. L = 2.260 × 106 (J/kg) is the latent heat of vaporization. Experimentally established temperature-dependent models for ρi, cP, i, and ki are discussed in Section 2.6. We note that the rightmost term of Eq. (3) is positive as an increase in vapor content corresponds to an increase in enthalpy. It is this term that plays an important role in reducing time-dependent temperature changes as liquid water is converted to vapor. Differentiating Eq. (3) with respect to time and applying the chainrule of differentiation yields ∂Xp T ∂Xf T ∂H ∂T ∫ ρ c dT ′ + Xp ρp cP;p ∫ ρ c dT ′ = + ∂t Tref p P;p ∂t Tref f P;f ∂t ∂t ∂T ∂Xw T ∂T ∫ ρ c dT ′ + Xw ρw cP;w + + Xf ρf cP;f ∂t Tref w P;w ∂t ∂t ∂Xv T ∂T ∂ρv ∂T ∫ ρ c dT ′ + Xv ρv cP;v + + X L ∂t Tref v P;v ∂T ∂t v ∂t ∂X + ρv v L: ∂t

k

∂T = hair ðT−Tair Þ on ∂n

∂Ωair ;

ð7Þ

where hair is the air-patty heat transfer coefficient. We take hair = 40 W/(m2 °C), which is consistent with values used in other studies; Ikediala et al. (1996) reported values of 10–30 W/(m2 °C), Geankoplis (1993) indicated values of 10–60 W/(m2 °C) as reasonable, Tran et al. (2002) employed a value of 7 W/(m2 °C), and Zorrilla and Singh (2003) employed a value of 60 W/(m2 °C). At a side in contact with the pan

k

  ∂T = hpan T−Tpan ∂n

on ∂Ωpan ;

ð8Þ

where hpan is the pan-patty heat-transfer coefficient. There is a wide range of values for hpan reported in the literature. Example values are 8268 W/(m2 °C) (Tran et al., 2002), 900 W/(m2 °C) (Zorrilla & Singh, 2003), 250 W/(m2 °C) (Ikediala et al., 1996), and a piecewise-linear variation over 208–1250 W/(m2 °C) (Wang & Singh, 2004). We choose a value for hpan based on numerical experiments as described in Section 4.1. 2.3. Liquid water transport

ð4Þ

We assume that protein content remains constant (no diffusion or loss), requiring ∂Xp / ∂t = 0. While it is known that fat transport occurs in a frying patty, and can thus have an effect on temperature evolution, the behavior is not well understood. We assume here that fat content is constant, requiring ∂Xf / ∂t = 0, and leave the development and inclusion of a fat-transport model as subjects of future work.

We assume that liquid water is transported diffusively, but with the potential to convert to vapor when the local temperature exceeds the boiling temperature Tboil = 102 °C (Vijayan & Singh, 1997). We follow Maroulis, Kiranoudis, and Marinos-Kouris (1995) and Wang and Singh (2004), and write the equation for the volume fraction of liquid water in Ω as

 ∂Xw 1∂ ∂X ∂ ∂X ρ ∂X = De r w + De w − v v ; r ∂r ρw ∂t ∂t ∂r ∂z ∂z

ð9Þ

where De is an effective diffusivity. The effective diffusivity is based on a model proposed by Maroulis et al. (1995), but with a modification

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based on results of Pan and Singh (2001) indicating that water diffusion occurs only when T ≥ Td, where Td = 30 °C: 8 0; > > < De =

Temperature-dependent material properties for fat, protein, and water are given by Choi and Okos (1986) as follows:

T b Td ;

! DT DX– > exp − + ; T ≥ Td ; D > : d TK Xw

ð10Þ

where TK(r, z, t) is temperature in Kelvin and X w is the amount of liquid water on a dry basis (kg-water/kg-solid). Constants in Eq. (10) for ground beef are given by Shilton et al. (2002) as Dd = 1.43 × 10− 6 m2/s, −2 DT = 1.58 × 103 K, and DX– = 6:72 × 10 kg-water/kg-solid. At external domain boundaries, the liquid water volume fraction is taken equal to the equilibrium water-holding capacity Xw,eq, i.e., on ∂Ωair ∪ ∂Ωpan :

Xw = Xw;eq

ð11Þ

There is zero water flux at the symmetry boundary; ∂Xw / ∂n = 0 on ∂Ωsym. Based on the experimentally determined model of Pan and Singh (2001), ( Xw;eq =

0 Xw ;

T b Td ;

ð12Þ

0

Xw exp½−δwe ðT−Td Þ; T ≥ Td ;

where δwe = 0.0132(°C)− 1. 2.4. Water-vapor generation and transport We assume that water-vapor generation occurs only when T(r, z, t) ≥ Tboil, and when water content is greater than Xw,eq. Further, we assume that the vapor-generation rate is proportional to (Xw – Xw,eq). This can be written 8 < 0;

∂Xv   = : τ−1 ∂t Xw −Xw;eq ; v

T b Tboil T ≥ Tboil

or

2.6. Material properties

Xw b Xw;eq ;

and Xw ≥ Xw;eq ;

  2 ˆ cP;f = 1984:2 + 1:4733 T−4:8006 J = ðkg ∘ CÞ; × 10−3 Tˆ   2 ˆ J = ðkg ∘ CÞ; × 10−3 Tˆ cP;p = 2008:2 + 1:2089 T−1:3129   2 −3 ∘ J = ðkg CÞ; Tˆ cP;w = 4128:9−9:0864 Tˆ + 5:4731 × 10   2 ˆ W = ðm ∘ CÞ; × 10−7 Tˆ kf = 0:18071−2:7064 × 10−4 T−1:7749   2 ˆ × 10−6 Tˆ kp = 0:17881 + 1:1958 × 10−3 T−2:7178 W = ðm ∘ CÞ;   2 ˆ W = ðm ∘ CÞ; × 10−6 Tˆ kw = 0:57109 + 1:7625 × 10−3 T−6:7063   ρf = 925:59−0:41757 Tˆ kg =m3 ;   3 ρp = 1329:9−0:51840 Tˆ kg=m ;   2 ˆ kg= m3 ; × 10−3 Tˆ ρw = 997:18 + 3:1439 × 10−3 T−3:7574 ð16Þ where corrections to kf and cp,w have been applied (Okos, 2010), and Tˆ = T = ð1 ∘ CÞ is a dimensionless temperature. For water vapor, material properties are given by Beyler et al. (1995) as follows:  −3 ˆ −5 ˆ 2 cP;v = 2:4240−5:5636 × 10 T + 2:3502 × 10 T  3 4 J = ðkg ∘ CÞ; −3:9028 × 10−8 Tˆ + 2:3888 × 10−11 Tˆ   kv = 0:015258 + 8:3154 × 10−5 Tˆ W = ðm ∘ CÞ;  −3 ˆ −6 ˆ 2 ρv = 0:80844−2:6559 × 10 T + 6:1309 × 10 T

ð17Þ

 3 4 −7:9326 × 10−9 Tˆ + 4:1920 × 10−12 Tˆ kg=m3 :

ð13Þ 3. Numerical method

where τv is the decay-time constant determined based on numerical experiments as described in Section 4.1. In Eq. (5), the terms on the left-hand side with Xv can be neglected. For the material properties discussed below, it can be shown, for example, that jρv cP;v +

∂ρv L j = jρw cP;w j b 0:001; ∂T

ð14Þ

for 10 °C b T b200 °C. Similarly for Eq. (2), it can be shown that kv ≪ kw, kp, kf. By taking Xv(r, z, t) = 0 in Eqs. (5) and (2), we are effectively assuming that vapor leaves the domain Ω immediately. This provides grounds for determining patty shrinkage as described in Section 3. We remark that terms in (5) with ∂Xv / ∂t are, however, significant and should not be neglected (see Section 4.1). 2.5. Heterocyclic-amine formation The generation of heterocyclic amine is assumed to follow a firstorder reaction equation with rate constant given by the Arrhenius equation (Tran et al., 2002). The time evolution of the local concentration of heterocyclic amine [HA] (ng per g uncooked meat) is governed by ! ∂½HA Ea ; = A exp − Rg TK ∂t

ð15Þ

where Ea is the activation energy, Rg = 1.986 cal/(K mol) is the gas constant, and A is the unknown exponential prefactor.

The diffusion equations governing temperature and mass, Eqs. (5) and (9), respectively, are spatially discretized (in r and z) with a Legendre spectral finite-element (LSFE) method (Ronquist & Patera, 1987). This is a high-order Bubnov–Galerkin finite-element method where the Nth-order-polynomial basis functions are Lagrangian interpolants (see, e.g., Hughes (1987)), but element nodes are located at the Gauss–Legendre–Lobatto (GLL) quadrature points (see, e.g., Deville, Fischer, and Mund (2002)). LSFEs combine the geometric flexibility of low-order finite elements with the potential for achieving spectral-convergence rates. Matrix–vector products are evaluated efficiently through tensor-product factorization (Orszag, 1980). Additional efficiency is accomplished with GLL nodal quadrature for evaluation of element-level inner products without a singular term at the axis of symmetry. GLL nodal quadrature yields diagonal mass matrices and additional savings in tensor-product factorization. For inner products with a singularity at r = 0, the singularity is avoided with (N + 1) × (N + 1)-point Gauss–Legendre quadrature. The domain Ω is discretized with NR × NZ elements, and each element has (N + 1) × (N + 1) nodes. Fig. 2 shows an illustrative coarse mesh with NR = 6, NZ = 2, and N = 6. Explicit time integration of the semi-discrete equations is accomplished with a low-storage third-order Runge–Kutta scheme (Williamson, 1980) with time step Δt. Explicit integration is preferred over implicit integration due to the constitutive and forcing nonlinearities. However, Δt must be sufficiently small to ensure numerically stable solutions. During simulation, after completion of each time step, R(t) and Z(t) are scaled uniformly to accommodate mass loss due to liquid and

M.A. Sprague, M.E. Colvin / Food Research International 44 (2011) 789–797

793

sided pan frying of thawed 20% fat-by-weight beef patties. Beef patty initial conditions are listed in Table 1 along with LSFE numericalmodel parameters. Here, ρ0, k0, and c0ρ are the initial mixture values of density, conductivity, and specific heat. Numerical experiments were performed to verify that the model was sufficiently refined in spatial and temporal resolution. Fig. 3 shows the experimentally determined pan0 temperature history for an initial pan temperature Tpan = 146.9 °C, along with the data-fitted model used in our simulations, which is given by

vapor water leaving the patty volume. To this end, the new volume V(t + Δt) is first determined from the updated volume fractions as Z ðt Þ

Rðt Þ

V ðt + Δt Þ = 2π∫0 ∫0

h

Xf ðr; z; t + Δt Þ

i + Xp ðr; z; t + Δt Þ + Xw ðr; z; t + Δt Þ rdrdz;

ð18Þ

which is calculated from element-level calculations with GLL quadrature. Volume fractions are then normalized at each node such that relative values are maintained and Xf (r, z, t + Δt) + Xp(r, z, t + Δt) + Xw(r, z, t + Δt) = 1. Patty dimensions (and all node locations) are then updated as R(t + Δt) = αR(t) and Z(t + Δt) = αZ(t), where α=


 V ðt + Δt Þ 13 : V ðt Þ

ð19Þ

The solution code was written in FORTRAN and was solved on a 2.66 GHz Intel Core i7 processor. Time step sizes and total CPU solution times for the cases studied are included in Table 1. 4. Model validation 4.1. Comparison with Tran et al. (2002) As discussed above, the two model parameters whose values are unspecified (aside from problem-specific initial conditions) are the pan-patty heat transfer coefficient hpan and the vapor-generation time constant τv. We employ the experimental results of Tran et al. (2002) in our determination of these parameters and in preliminary validation of our model. Tran et al. (2002) examined the singleTable 1 Problem-specific conditions and numerical parameters for comparison with experimental data in the literature. Tran et al. (2002) Ikediala et al. (1996) Salmon et al. (2000) −2

−1

hair (W m °C ) hpan (W m− 2 °C− 1) τv (s) R0 (m) Z0 (m) Tair (°C) T 0 (°C) 0 Tpan (°C) X0f 0 Xp X0w ρ0 (kg m− 3) k0 (W m− 1 °C− 1) c0p (J kg− 1 °C− 1) NR NZ N tmax (s) Δt (s) CPU time (s)

40 150 0.1 0.0475 0.0250 20 13.5 146.9 0.238 0.139 0.623 1024 0.4390 3330 14 9 6 2500 0.046 420

40 150 0.1 0.0450 0.0150 20 7.0 180.0 0.121 0.169 0.710 1044 0.4672 3513 14 5 6 500 0.051 41

40 150 0.1 0.0450 0.0150 20 10.6 Variable 0.238 0.139 0.623 1024 0.4358 3329 14 5 6 360–523 0.051 31–44

Tpan

  9 8 <135:2−0:01036 tˆ + 11:74 exp −0:06445 tˆ ; tˆ b 216 = ∘    2 = C; −6 ˆ :133:0 + 0:02166 t−216 ˆ −3:93 × 10 t−216 ; tˆ ≥ 216 ;

ð20Þ where tˆ = t = ð1 sÞ is a dimensionless time. As shown, when the relatively cool patty first contacts the pan there is an abrupt and significant drop in Tpan over 0 b t b 250 s followed by a gradual increase that exceeds the original preheated temperature for t N 975 s. Fig. 4 shows experimental temperature histories of Tran et al. (2002) at three elevations (z = 0.006 m, 0.009 m, and 0.012 m) near the patty center (r ≈ 0 m) for two or three thermocouple readings at each point. Also shown are the simulation histories produced by the FE model of Tran et al. (2002), whose system properties were tuned to match experimental data. While the FE model provides histories in good agreement with experiments at z = 0.009 m and 0.012 (over 0 ≤ t ≤ 1500 s), the FE model significantly over predicts temperature at z = 0.006 m at late time due, in part, to its lack of a vaporization model. For the LSFE model, it was determined through numerical experiments that hpan = 150 W/(m2 °C) provides good agreement with the average of the two experimental histories at z = 0.006 m for early time (t b 250 s) before vapor-generation effects were evident. Unless noted otherwise, all subsequent numerical simulations employ this value. Numerical experiments were performed with several τv values. Temperature response histories were largely insensitive over the range 0.05 s ≤ τv ≤ 0.2 s, exhibiting a corresponding final temperature range (at t = 2500 s) of 95.7–98.7 °C at z = 0.006 m. However, τv = 0.1 s exhibited the best agreement with the experimental data; unless otherwise noted, all subsequent simulations employ this value. Fig. 4 shows temperature histories at the three depths calculated with the LSFE model with vapor-generation time constants τv = 0.1 s and τv → ∞. For the solutions produced with τv → ∞, for which vaporization is effectively neglected, poor agreement with the experimental results is exhibited. Fig. 5 shows the temperature and water volume fractions at the end of the LSFE simulation (tmax = 2500 s). The outer black boxes show the domain at t = 0; there was significant shrinkage over the simulated frying. Fig. 5(a) shows significant temperature variation in the radial direction in a majority of the domain and serves to emphasize the need for a two-dimensional axisymmetric model, as opposed to a onedimensional through-thickness model (e.g. Pan et al. (2000)), for

160

T (°C)

Fig. 2. Representative mesh of Legendre spectral finite elements with NR = 6, NZ = 2, and N = 6. Nodes (shown as diamonds) are located at the (N + 1) × (N + 1) GLL quadrature points of each element.

150 experiments

140

Eq. (20) 130

0

500

1000

1500

2000

2500

t (s) Fig. 3. Pan-temperature history (unpublished) measured by Tran et al. (2002) and the approximate model Tpan(t) used in the simulations below.

794

a

M.A. Sprague, M.E. Colvin / Food Research International 44 (2011) 789–797

120 100

T (°C)

80 60 40 20 0

b

100

T (°C)

80

60

40

20

0

c

100 Fig. 5. (a) Temperature and (b) water volume fraction at t = 2500 s for the Tran et al. (2002) parameters listed in Table 1. The outer boxes indicate the domain of the patty at t = 0 s and the solid squares indicate the locations of the temperature probes used to produce Fig. 4.

T (°C)

80

60

40

20

0

0

500

1000

1500

2000

2500

shrinkage. Fig. 7 shows the temperature histories for the properties of Table 1 with hpan = 150 W/(m2 °C) and τv = 0.1 s for four variations of Eq. (5). The solutions to the full equation are shown by the solid lines (c.f. Fig. 4). The dotted lines show histories calculated with the last two terms of Eq. (5) neglected. These histories are distinctly lower than those calculated with all terms. The dashed lines show histories calculated with material properties taken as constant at their initial conditions. Again, these are significantly different than the baseline

t (s) 100

Fig. 4. Temperature histories as measured experimentally and numerically (FE model) by Tran et al. (2002) at three elevations, (a) z = 0.006 m, (b) 0.009 m, and (c) 0.012 m, for beef patties with initial conditions listed in Table 1. Also shown are numerical results of the LSFE model calculated with hpan = 150 W/(m2 °C) and two values of τv.

80 0.012 m 60

T (°C)

accurate representation of the overall temperature field, which is necessary for accurate calculation of heterocyclic-amine formation. Fig. 5(b) shows less radial variation in water volume fraction, but shows the significant water loss near the pan surface, and large gradients near edges. We examine here the sensitivity of calculated temperature histories on the choice of hpan. Fig. 6 shows the baseline results of Fig. 4 with hpan = 150 W/(m2 °C) and with hpan values 20% greater and smaller. These variations in hpan provide histories that differ from the baseline histories by a maximum of 3.5 °C at 366 s and with final-time results differing by approximately 1 °C. We also examine the importance of including the last two terms in Eq. (5), including temperature dependence in the material properties, and including

z = 0.006 m

0.009 m

40 hpan = 150 W/(m2 °C), c.f. Fig. 4 hpan = 180 W/(m2 °C)

20

hpan = 120 W/(m2 °C) 0 0

500

1000

1500

2000

2500

t (s) Fig. 6. Temperature histories at three elevations as predicted by the LSFE model for the Tran et al. (2002) properties listed in Table 1 with three values of hpan.

M.A. Sprague, M.E. Colvin / Food Research International 44 (2011) 789–797

Ikediala et al. (1996) for temperature histories at three elevations near the patty radial center (z = 0.0035 m, 0.0055 m, and 0.0095 m). Fig. 9 shows the measured pan-temperature history (Ikediala et al., 1996) along with a least-squares-fitted approximation given by

100 z = 0.006 m 80 0.012 m

60

T (°C)

795

  2 −7 ˆ3 ∘ Tpan ðt Þ = 180−0:11117 tˆ + 0:00044487 tˆ −4:7894 × 10 t C:

0.009 m

ð21Þ

40 full model, c.f. Fig. 4 last two terms in (5) neglected

20

constant material props. 0

shrinkage neglected 0

500

1000

1500

2000

2500

t (s) Fig. 7. Temperature histories at three elevations as predicted by the LSFE model for the Tran et al. (2002) properties listed in Table 1 with (i) temperature-dependent material properties (c.f. Fig. 4), (ii) temperature-dependent material properties but with the last two terms of Eq. (5) neglected, (iii) temperature-independent (constant) material properties, and (iv) the full model but with shrinkage neglected.

solutions. Finally, the dot-dashed lines show results of the full model but with shrinkage neglected; significant differences are seen. These results illustrate that these various terms and effects should be included in any rigorous modeling effort. 4.2. Comparison with Ikediala et al. (1996) In the previous section, the two free model parameters were “matched” to the results of Tran et al. (2002) as τv = 0.1 s and hpan = 150 W/(m2 °C). Here, we test the model with these values against the experimental and numerical values reported in the singlesided beef-patty frying experiments of Ikediala et al. (1996). The initial conditions are listed in Table 1 along with the numerical parameters used in the LSFE model. Ikediala et al. (1996) solved the diffusion equation for temperature using bilinear finite elements. Their model treated mass loss due to vaporization uniformly throughout the domain with an experimentally fit decay function. Empirical relations for thermal properties reported by Dagerskog (1979) were used. Their pan-patty heat transfer coefficient was 250 W/(m2 °C), which is greater than that used in our LSFE model. Patty shrinkage was neglected and a constant pan temperature was assumed. Fig. 8 shows the experimental and finite-element results of 100 z = 0.0035 m 80 0.0055 m

4.3. HA formation comparison with Salmon et al. (2000) We examine here the ability of the proposed model to predict the formation of heterocyclic amines. We compare our model results with the experiments of Salmon et al. (2000). In those experiments, beef patties with initial conditions listed in Table 1 were pan-fried with various initial pan temperatures until the center temperature reached 70.6 °C (on average). They examined patties that were either flipped once at a frying time of 300 s, or flipped multiple times every 60 s. Experimental results are reproduced in Fig. 10 where ½HA is the overall concentration of HAs. It is clear that the increased flipping frequency yields dramatically fewer HAs. This reduction has been attributed to lower maximum patty temperatures and the increased loss of meat drippings, which are known to contain a significant portion of HAs (Gross et al., 1993). The relative contributions of these two reduction mechanisms are not fully understood. Tran et al. (2002) examined numerically the HA formation by integrating Eq. (15) with Ea = 18 × 103 cal/mol chosen to best fit the data and cooking times were closely related to those of Salmon et al. (2000). Their normalized FE results are included in Fig. 10. Normalization was accomplished by scaling all data by the same constant such that the single-flip HA 0 concentration for Tpan = 160 °C matched the average of the Salmon et al. (2000) data at that initial pan temperature. While the model captured well the rate of increase of HAs with pan temperature for single-flip frying, it greatly under predicted the HA reduction accomplished by frequent flipping. In our numerical experiments, the overall HA concentration is calculated by time-integrating the local first-order reaction Eq. (15) at all nodes in the LSFE model. Simulation-time durations were identical

T (°C)

60

Temperature histories produced with the proposed LSFE model with initial conditions listed in Table 1 and pan temperature given by Eq. (21) are shown in Fig. 8. All other models and material properties are the same as those employed in the previous section. Here, variation in pan temperature is seen to be significantly smaller than that in the previous problem. LSFE numerical solutions calculated with a constant pan temperature corresponding to the average of the data in Fig. 9 are virtually indistinguishable from the histories shown in Fig. 8. While the proposed model had only problem-specific initial conditions, the numerical temperature histories agree with the experimental histories at least as well as those of Ikediala et al. (1996), which employed a problem-specific empirically determined mass-loss model.

180

0.0095 m

40

experiments

T (°C)

178 experiments

20

FE model LSFE model 0

0

100

200

300

Tpan(t) (21)

176 174 172

400

500

t (s)

170

0

100

200

300

400

500

t (s) Fig. 8. Temperature histories as measured experimentally and numerically (FE model) by Ikediala et al. (1996) at three elevations in beef patties with initial conditions listed in Table 1. Also shown are numerical results of the LSFE model.

Fig. 9. Pan-temperature history as measured by Ikediala et al. (1996) and the approximate model Tpan(t) used in the LSFE simulations of Fig. 8.

796

M.A. Sprague, M.E. Colvin / Food Research International 44 (2011) 789–797

distributions in the flipped versus non-flipped patties. This work shows that more complex models of cooking processes, including water transport and evaporation, yield more accurate results and open the door to fully predictive simulations to develop safer methods for food preparation. The model would likely be improved by including fat-transport effects, which is the subject of future work.

[HA] (ng/g)

10

1 single flip (exp.)

Acknowledgments

multi flip (exp.) single flip (FE)

0.1

multi flip (FE) single flip (LSFE) multi flip (LSFE) 0.01

160

180

200

220

240

260

Tpan (°C) Fig. 10. Overall concentration of heterocyclic amine [HA] as a function of initial pan temperature as measured experimentally by Salmon et al. (2000). Also shown are the predictions from the FE model of Tran et al. (2002) and the proposed LSFE model. Numerical-model data were normalized to the single-flip experimental average at 0 Tpan = 160 °C.

to the average values reported by Salmon et al. (2000) as a function of initial pan temperature. For patties cooked with single and multiple flips, the total cooking times were given by Salmon et al. (2000) as   2 −1 0 −4 ˆ 0 tcook;sing = 3600 18:21−0:1289 Tˆ pan + 3:630×10 T pan s; ð22Þ and   2 −1 0 −4 ˆ 0 tcook;mult = 3600 18:59−0:1144 Tˆ pan + 3:130×10 s; T pan ð23Þ 0 where Tˆ pan

0 = Tpan = 1o C. The numerical parameters used respectively, in the model are listed in Table 1, and the pan temperature was taken as Eq. (20) with the appropriate offset. Ea = 25.3 × 103 cal/mol was chosen to best fit the single-flip data. Fig. 10 includes results predicted from our LSFE model, where again, data was normalized to the single-flip experimental results at T0pan = 160 °C. Unlike the FE model employed by Tran et al. (2002), our model captures a significant fraction of the HA reduction accomplished in the multi-flipping experiments. This is largely because patty temperatures are better modeled, and significantly lower, than those in the model of Tran et al. (2002). The differences between the LSFE results and the experimental data for the multi-flip patties can be attributed, in part, to the absence of HA loss through drippings in the numerical model.

5. Conclusion This paper presents a new formulation for simulating the pan frying of beef patties that includes the transport of water in the meat as well as the energy and mass loss due to water evaporation during cooking. This formulation requires only a few empirical parameters, but yields good agreement with experimental time-temperature data from two independent studies. Importantly, our method accurately models the long-duration temperature distributions in the meat, which is necessary to predict cooking times and HA formation. This model was used to investigate the experimentally observed result that frequent flipping of beef patties during cooking reduces the formation of HA mutagens; a result that was not explained by earlier modeling studies. These results show that a significant fraction of the observed reduction in HA formation is due to the different temperature

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