A mixture-enthalpy fixed-grid model for

temperature evolution and heterocyclic-amine formation

in a frying beef patty

Michael A. Spraguea,1,∗, Michael E. Colvina

aCenter for Computational Biology and School of Natural Sciences, University of

California, Merced, CA 95343, USA

Abstract

The ideal cooking process would heat food to a sufficient temperature through-out to kill bacteria without heating the food to temperatures that promoteformation of toxic or carcinogenic compounds. Experimentally validatedcomputer models have an important role to play in designing cooking pro-cesses since they allow rapid evaluations of different conditions without theconfounding effects of experimental variation. In this paper we derive amathematical model governing the heat and water transport in a cylindri-cal pan-fried beef patty. The continuum temperature model stems from amixture-enthalpy formulation that accommodates the liquid and vapor statesof water along with fat and protein. The governing equations were spatiallydiscretized with Legendre spectral finite elements. All but two of the modelproperties were taken from the literature, with the remaining two deter-mined through a comparison of numerical and physical experiments. Theseparameters were shown to produce solutions in agreement with a differentset of experimental results. The model was used to calculate the forma-tion of heterocyclic-amine (HA) compounds (known DNA mutagens andcarcinogens). Results provide an explanation based on patty temperaturefor previous experimental studies showing that frequent patty flipping yieldsa dramatic reduction in HAs.

∗Corresponding AuthorEmail address: michael.a.sprague@nrel.gov (Michael A. Sprague)

1Current Address: National Renewable Energy Laboratory, 1617 Cole Blvd., MS 1608,Golden, CO 80401

Preprint submitted to Food Research International January 11, 2011

Keywords: cooking simulations; spectral finite-element methods;carcinogens; mutagens

Nomenclature

A exponential prefactor (ng-HA g-uncooked-meat−1 s−1)cP specific heat (J kg−1 oC−1)De effective liquid water diffusivity (kg-water kg-solid−1)Dd constant in De expression, 1.43 × 10−6 m2 s−1

DT constant in De expression, 1.58 × 103 KDX constant in De expression, 6.72 × 10−2 kg-water kg-solid−1

Ea activation energy (cal mol−1)k thermal conductivity (W m−1 oC−1)h heat-transfer coefficient (W m−2 oC−1)H mixture enthalpy (J m−3)[HA] local concentration of heterocyclic amine (ng-HA g-uncooked-meat−1)

[HA] overall concentration of heterocyclic amine (ng-HA g-uncooked-meat−1)L latent heat of vaporization, 2.260 × 106 J kg−1

n boundary-normal coordinate (m)N polynomial-order of Legendre spectral finite-element basis functionsNR number of elements spanning radius RNZ number of elements spanning height Zr radial coordinate (m)R patty radius (m)Rg gas constant, 1.986 cal K−1 mol−1

t time (s)tmax final simulation time (s)t nondimensional timetcook,sing total cooking time for single-flip calculations (s)tcook,mult total cooking time for multi-flip calculations (s)T temperature (oC)Tboil boiling-water temperature, 102 oCTd water-diffusion temperature, 30 oCTK temperature in Kelvin (K)Tref reference temperature for mixture enthalpy (oC)

T nondimensional temperature

2

θ meridional coordinateV volume of patty (m3)X volume fractionXw liquid-water mass fraction on dry basis (kg-water kg-solid−1)z vertical coordinate (m)Z patty height (m)

Greek symbolsα scale factor for radius and height shrinkageδwe constant in Xw,eq expression, 0.0132 oC−1

∆t time-step size (s)Ω patty meridional-section domain∂Ω meridional-section boundaryρ density (kg m−3)τv vapor-generation decay-time (s)

Superscripts0 initial condition′ denotes dummy-variable of integration

Subscriptsair aireq equilibriumf fati material-component place holderp proteinpan pansym symmetryw liquid waterv vapor waterZ vertical direction

3

1. Introduction

This paper is concerned with the mathematical and computational mod-eling of pan-frying of thawed beef patties. We are interested in predicting thetransport of heat and water, and the associated formation of heterocyclic-amine (HA) compounds, which have been shown to be mutagenic (Sugimuraet al., 1977; Felton et al., 1981) and carcinogenic (Ohgaki et al., 1991; Adam-son et al., 1990). While it is established that lower cooking temperatures(Knize et al., 1994) and frequent patty flipping (Salmon et al., 2000) corre-spond to fewer carcinogens, adequately high temperatures must be reachedthroughout the patty to kill harmful bacteria. Thus, there are the competinggoals of heating the patty sufficiently well while minimizing the formation ofcarcinogens. Accurate computational modeling of a frying beef patty allowsanalysts to better understand the evolution of the patty during frying, whichmay provide clues to develop improved cooking methods.

The physical and chemical processes inherent in beef-patty frying are nu-merous, complicated, and, in many cases, poorly understood. Beef-pattyfrying is seen as inherently nonlinear due to such issues as temperature-dependent material properties and phase change due to the vaporization ofwater. A number of computational models for pan and/or immersion fryingof foods have been established, which can be categorized as either fixed-grid(Ikediala et al., 1996; Chen et al., 1999; Pan et al., 2000; Obuz et al., 2002;Shilton et al., 2002; Wang and Singh, 2004; Ou and Mittal, 2006a,b, 2007;van der Sman, 2007; Goñi and Salvadori, 2010) or moving-interface (Farkaset al., 1996a,b; Vijayan and Singh, 1997; Farid and Chen, 1998; Zorrilla andSingh, 2000, 2003). In the fixed-grid approach, the entire domain is repre-sented 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 thispaper, we employ the former, in part because it is more amenable to problemsin general geometry.

Many models for frying processes rely on properties fit to problem-specificempirical data, which limits broader applicability. For example, Ikedialaet al. (1996) employ a data-fit moisture-loss model and Tran et al. (2002)employ data-fit conductivity, density, and heat capacity. Few of the existingmodels have been successfully validated in the high-temperature and/or late-time regime, when significant mass loss has occurred due to highly nonlinearwater vaporization, which also dominates temperature response. When the

4

temperature reaches the water boiling temperature, added energy is devotedto the vaporization process and temperature remains approximately constantuntil the accessible liquid water is fully vaporized.

In this paper we consider the pan frying of a thawed beef patty com-posed of water, fat, and protein. Our primary goal is to develop a time-dependent temperature model that is sufficiently accurate for long-durationfrying, which requires accurate representation water transport and energyloss associated with water vaporization. Our secondary goal is for the modelto be broadly applicable. To that end, the model employs tabulated prop-erty values where appropriate and has as few free parameters as possible.Also, we spatially discretize the mathematical model with Legendre spectralfinite elements, which are high-order finite elements well suited for generalgeometry problems. Finally, we employ the temperature model to predictthe formation of HAs, and compare those results with empirical data in theliterature.

The paper is organized as follows. In Section 2 we describe the problemand the mathematical models for heat and liquid water transport, vaporgeneration, and HA formation. In Section 3 we describe the discretization ofthe mathematical model with Legendre spectral finite elements. Validationof the numerical model and numerical studies are described in Section 4, andconcluding 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) andheight Z(t) at time t; initial values are R(0) = R0 and Z(0) = Z0. Theschematic in Fig. 1 shows a meridional section of the patty in polar coordi-nates (r, θ, z). Assuming axially symmetric behavior, temperature is denotedby T (r, z, t). We denote the section domain of the patty as Ω and the bound-ary as ∂Ω = ∂Ωpan∪∂Ωair∪∂Ωsym, where ∂Ωpan are the boundaries in contactwith the pan, ∂Ωair are the boundaries in contact with the surrounding air,and ∂Ωsym is the symmetry boundary. The patty is initially composed offat, protein, and liquid water with uniform volume fractions X0

f , 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 surfacewith known temperature behavior Tpan(t), whose preheated temperature isTpan(0) = T 0

pan. The air temperature Tair is constant. Below we describe

5

models governing the transport of heat and water, the generation of watervapor, and the formation of heterocyclic amines. The section concludes witha presentation of material properties taken from the literature.

2.2. Heat transport

We derive here a continuum model for heat transport in a beef pattycomposed of fat, protein, and liquid water that can vaporize. We employ theconcept of a local volume fraction, which is valid over a small isothermal rep-resentative elementary volume (REV) (See, e.g., Bachmat and Bear (1986);Voller et al. (1990)). A field value at a point in the resulting continuum-model domain is representative of the constant value over the REV centeredat that point. It is assumed that a REV can be composed of multiple mate-rials and/or phases. Following Voller et al. (1990) and neglecting advectioneffects, the time evolution of the local mixture enthalpy H(r, z, t) is governedin Ω as

∂H

∂t=

1

r

∂

∂r

(

kr∂T

∂r

)

+∂

∂z

(

k∂T

∂z

)

, (1)

where the mixture conductivity is defined as

k = Xpkp + Xfkf + Xwkw + Xvkv , (2)

and

H = Xp

∫ T

Tref

ρpcP,p dT ′ + Xf

∫ T

Tref

ρfcP,f dT ′

+Xw

∫ T

Tref

ρwcP,w dT ′ + Xv

∫ T

Tref

ρvcP,v dT ′ + ρvXvL . (3)

Here, Tref is an arbitrary reference temperature taken as T 0; Xf(r, z, t),Xp(r, z, t), Xw(r, z, t), and Xv(r, z, t) are local volume fractions for fat, pro-tein, liquid water, and water vapor, respectively. Volume-fraction initialconditions are assumed to be spatially uniform; at t = 0 s, Xf(r, z, 0) = X0

f ,Xp(r, z, 0) = X0

p , 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, andcP,i(T ) is specific heat; p denotes protein, f denotes fat, w denotes liquidwater, and v denotes water vapor. L = 2.260× 106 (J/kg) is the latent heatof vaporization. Experimentally established temperature-dependent modelsfor ρi, cP,i, and ki are discussed in Section 2.6. We note that the rightmost

6

term of (3) is positive as an increase in vapor content corresponds to an in-crease in enthalpy. It is this term that plays an important role in reducingtime-dependent temperature changes as liquid water is converted to vapor.

Differentiating (3) with respect to time and applying the chain-rule ofdifferentiation yields

∂H

∂t=

∂Xp

∂t

∫ T

Tref

ρpcP,pdT ′ + XpρpcP,p

∂T

∂t+

∂Xf

∂t

∫ T

Tref

ρfcP,fdT ′

+XfρfcP,f

∂T

∂t+

∂Xw

∂t

∫ T

Tref

ρwcP,wdT ′ + XwρwcP,w

∂T

∂t

+∂Xv

∂t

∫ T

Tref

ρvcP,vdT ′ + XvρvcP,v

∂T

∂t+

∂ρv

∂T

∂T

∂tXvL

+ρv

∂Xv

∂tL. (4)

We assume that protein content remains constant (no diffusion or loss), re-quiring ∂Xp/∂t = 0. While it is known that fat transport occurs in a fryingpatty, and can thus have an effect on temperature evolution, the behavior isnot well understood. We assume here that fat content is constant, requir-ing ∂Xf/∂t = 0, and leave the development and inclusion of a fat-transportmodel as subjects of future work. With these simplifications, (1) and (4) arecombined to yield a single equation governing temperature in Ω as

(

XpρpcP,p + XfρfcP,f + XwρwcP,w + XvρvcP,v +∂ρv

∂TXvL

)

∂T

∂t=

1

r

∂

∂r

(

kr∂T

∂r

)

+∂

∂z

(

k∂T

∂z

)

−ρv

∂Xv

∂tL −

∂Xw

∂t

∫ T

Tref

ρwcP,wdT ′ −∂Xv

∂t

∫ T

Tref

ρvcP,vdT ′ , (5)

where initial conditions are described above. Temperature-dependent ma-terial properties are one source of nonlinearity in (5). If constant materialproperties are assumed and mass transport and vapor generation is neglected,(5) reduces to a linear diffusion equation for temperature like that used byTran et al. (2002). We note that the last two terms in (5) are apparently ne-glected in other computational studies (see, e.g., Ikediala et al. (1996); Wangand Singh (2004)).

7

Heat-flux boundary conditions are applied at bounding surfaces. On theaxis of symmetry there is no heat flux and thus

k∂T

∂n= 0 on ∂Ωsym , (6)

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

k∂T

∂n= hair(T − Tair) on ∂Ωair , (7)

where hair is the air-patty heat-transfer coefficient. We take hair = 40W/(m2 C), which is consistent with values used in other studies; Ikedialaet al. (1996) reported values of 10-30 W/(m2 C), Geankoplis (1993) indi-cated values of 10-60 W/(m2 C) as reasonable, Tran et al. (2002) employeda value of 7 W/(m2 C), and Zorrilla and Singh (2003) employed a value of60 W/(m2 C). At a side in contact with the pan

k∂T

∂n= hpan(T − Tpan) on ∂Ωpan , (8)

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

2.3. Liquid-water transport

We assume that liquid water is transported diffusively, but with the po-tential to convert to vapor when the local temperature exceeds the boilingtemperature Tboil = 102 C (Vijayan and Singh, 1997). We follow Marouliset al. (1995) and Wang and Singh (2004), and write the equation for thevolume fraction of liquid water in Ω as

∂Xw

∂t=

1

r

∂

∂r

(

Der∂Xw

∂r

)

+∂

∂z

(

De

∂Xw

∂z

)

−ρv

ρw

∂Xv

∂t, (9)

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

8

results of Pan and Singh (2001) indicating that water diffusion occurs onlywhen T ≥ Td, where Td = 30 C:

De =

0 , T < Td ,

Dd exp(

−DT

TK+

DX

Xw

)

, T ≥ Td ,(10)

where TK(r, z, t) is temperature in Kelvin and Xw is the amount of liquidwater on a dry basis (kg-water/kg-solid). Constants in (10) for ground beefare given by Shilton et al. (2002) as Dd = 1.43×10−6 m2/s, DT = 1.58×103

K, and DX = 6.72 × 10−2 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.,

Xw = Xw,eq on ∂Ωair ∪ ∂Ωpan . (11)

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 =

X0w , T < Td ,

X0w exp [−δwe (T − Td)] , T ≥ Td ,

(12)

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 assumethat the vapor-generation rate is proportional to (Xw − Xw,eq). This can bewritten

∂Xv

∂t=

0 , T < Tboil or Xw < Xw,eq ,τ−1v (Xw − Xw,eq) , T ≥ Tboil and Xw ≥ Xw,eq ,

(13)

where τv is the decay-time constant determined based on numerical experi-ments as described in Section 4.1.

In (5), the terms on the left-hand side with Xv can be neglected. For thematerial properties discussed below, it can be shown, for example, that

|ρvcP,v +∂ρv

∂TL|/|ρwcP,w| < 0.001 (14)

9

for 10 oC < T < 200 oC. Similarly for (2), it can be shown that kv ≪kw, kp, kf . By taking Xv(r, z, t) = 0 in (5) and (2), we are effectively assum-ing that vapor leaves the domain Ω immediately. This provides grounds fordetermining patty shrinkage as described in Section 3. We remark that termsin (5) with ∂Xv/∂t are, however, significant and should not be neglected (seeSection 4.1).

2.5. Heterocyclic-amine formation

The generation of heterocyclic-amine is assumed to follow a first-orderreaction equation with rate constant given by the Arrhenius equation (Tranet al., 2002). The time evolution of the local concentration of heterocyclic-amine [HA] (ng per g uncooked meat) is governed by

∂[HA]

∂t= A exp

(

−Ea

RgTK

)

, (15)

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

2.6. Material properties

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

cP,f =(

1984.2 + 1.4733 T − 4.8006 × 10−3 T 2)

J/(kg C) ,

cP,p =(

2008.2 + 1.2089 T − 1.3129 × 10−3 T 2)

J/(kg C) ,

cP,w =(

4128.9 − 9.0864 T + 5.4731 × 10−3 T 2)

J/(kg C) ,

kf =(

0.18071 − 2.7064 × 10−4 T − 1.7749 × 10−7 T 2)

W/(m C) ,

kp =(

0.17881 + 1.1958 × 10−3 T − 2.7178 × 10−6 T 2)

W/(m C) ,

kw =(

0.57109 + 1.7625 × 10−3 T − 6.7063 × 10−6 T 2)

W/(m C) ,

ρf =(

925.59 − 0.41757 T)

kg/m3 ,

ρp =(

1329.9 − 0.51840 T)

kg/m3 ,

ρw =(

997.18 + 3.1439 × 10−3 T − 3.7574 × 10−3 T 2)

kg/m3 , (16)

10

where corrections to kf and cP,w have been applied (Okos, 2010), and T =T/(1 C) is a dimensionless temperature. For water vapor, material proper-ties are given by Beyler et al. (1995) as follows:

cP,v =(

2.4240 − 5.5636 × 10−3 T + 2.3502 × 10−5 T 2

−3.9028 × 10−8 T 3 + 2.3888 × 10−11 T 4)

J/(kg C) ,

kv =(

0.015258 + 8.3154 × 10−5 T)

W/(m C) ,

ρv =(

0.80844 − 2.6559 × 10−3 T + 6.1309 × 10−6 T 2

−7.9326 × 10−9 T 3 + 4.1920 × 10−12 T 4)

kg/m3 . (17)

3. Numerical method

The diffusion equations governing temperature and mass, (5) and (9),respectively, are spatially discretized (in r and z) with a Legendre spec-tral finite-element (LSFE) method (Ronquist and Patera, 1987). This isa high-order Bubnov-Galerkin finite-element method where the N th-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 et al. (2002)). LSFEs combine the geo-metric flexibility of low-order finite elements with the potential for achievingspectral-convergence rates. Matrix-vector products are evaluated efficientlythrough tensor-product factorization (Orszag, 1980). Additional efficiencyis accomplished with GLL nodal quadrature for evaluation of element-levelinner products without a singular term at the axis of symmetry. GLL nodalquadrature yields diagonal mass matrices and additional savings in tensor-product factorization. For inner products with a singularity at r = 0, thesingularity is avoided with (N + 1)× (N + 1)-point Gauss-Legendre quadra-ture. The domain Ω is discretized with NR ×NZ elements, and each elementhas (N +1)× (N +1) nodes. Figure 2 shows an illustrative coarse mesh withNR = 6, NZ = 2, and N = 6.

Explicit time integration of the semi-discrete equations is accomplishedwith a low-storage third-order Runge-Kutta scheme (Williamson, 1980) withtime step ∆t. Explicit integration is preferred over implicit integration due tothe constitutive and forcing nonlinearities. However, ∆t must be sufficientlysmall to ensure numerically stable solutions.

11

During simulation, after completion of each time step, R(t) and Z(t) arescaled uniformly to accommodate mass loss due to liquid and vapor waterleaving the patty volume. To this end, the new volume V (t + ∆t) is firstdetermined from the updated volume fractions as

V (t + ∆t) = 2π

∫ Z(t)

0

∫ R(t)

0

[Xf(r, z, t + ∆t)

+Xp(r, z, t + ∆t) + Xw(r, z, t + ∆t)] rdrdz , (18)

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

α =

(

V (t + ∆t)

V (t)

)1

3

. (19)

The solution code was written in FORTRAN and was solved on a 2.66GHz Intel Core i7 processor. Time step sizes and total CPU solution timesfor 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 un-specified (aside from problem-specific initial conditions) are the pan-pattyheat-transfer coefficient hpan and the vapor-generation time constant τv. Weemploy the experimental results of Tran et al. (2002) in our determinationof these parameters and in preliminary validation of our model. Tran et al.(2002) examined the single-sided pan frying of thawed 20% fat-by-weight beefpatties. Beef patty initial conditions are listed in Table 1 along with LSFEnumerical-model parameters. Here, ρ0, k0, and c0

P are the initial mixture val-ues of density, conductivity, and specific heat. Numerical experiments wereperformed to verify that the model was sufficiently refined in spatial andtemporal resolution. Figure 3 shows the experimentally determined pan-temperature history for an initial pan temperature T 0

pan = 146.9 C, along

12

with the data-fitted model used in our simulations, which is given by

Tpan =

135.2 − 0.01036 t + 11.74 exp(−0.06445 t), t < 216133.0 + 0.02166(t− 216) − 3.93 × 10−6(t − 216)2, t ≥ 216

C ,

(20)where t = t/(1 s) is a dimensionless time. As shown, when the relatively coolpatty first contacts the pan there is an abrupt and significant drop in Tpan

over 0 < t < 250 s followed by a gradual increase that exceeds the originalpreheated temperature for t > 975 s.

Table 1: Problem-specific conditions and numerical parameters for comparison with ex-perimental data in the literature.

Tran et al. Ikediala et al. Salmon et al.(2002) (1996) (2000)

hair (W m−2 C−1) 40 40 40hpan (W m−2 C−1) 150 150 150τv (s) 0.1 0.1 0.1R0 (m) 0.0475 0.0450 0.0450Z0 (m) 0.0250 0.0150 0.0150Tair (C) 20 20 20T 0 (C) 13.5 7.0 10.6T 0

pan (C) 146.9 180.0 variableX0

f 0.238 0.121 0.238X0

p 0.139 0.169 0.139X0

w 0.623 0.710 0.623ρ0 (kg m−3) 1024 1044 1024k0 (W m−1 C−1) 0.4390 0.4672 0.4358c0P (J kg−1 C−1) 3330 3513 3329

NR 14 14 14NZ 9 5 5N 6 6 6tmax (s) 2500 500 360–523∆t (s) 0.046 0.051 0.051CPU time (s) 420 41 31–44

Figure 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

13

center (r ≈ 0 m) for two or three thermocouple readings at each point. Alsoshown 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 experimentsat z = 0.009 m and 0.012 (over 0 ≤ t ≤ 1500 s), the FE model significantlyover predicts temperature at z = 0.006 m at late time due, in part, to itslack of a vaporization model.

For the LSFE model, it was determined through numerical experimentsthat hpan = 150 W/(m2 C) provides good agreement with the average of thetwo experimental histories at z = 0.006 m for early time (t < 250 s) beforevapor-generation effects were evident. Unless noted otherwise, all subsequentnumerical simulations employ this value. Numerical experiments were per-formed with several τv values. Temperature response histories were largelyinsensitive over the range 0.05 s ≤ τv ≤ 0.2 s, exhibiting a correspondingfinal temperature range (at t = 2500 s) of 95.7–98.7 oC at z = 0.006 m.However, τv = 0.1 s exhibited the best agreement with the experimentaldata; unless otherwise noted, all subsequent simulations employ this value.Figure 4 shows temperature histories at the three depths calculated with theLSFE model with vapor-generation time constants τv = 0.1 s and τv → ∞.For the solutions produced with τv → ∞, for which vaporization is effectivelyneglected, poor agreement with the experimental results is exhibited.

Figure 5 shows the temperature and water volume fractions at the endof the LSFE simulation (tmax = 2500 s). The outer black boxes show thedomain at t = 0; there was significant shrinkage over the simulated fry-ing. Figure 5(a) shows significant temperature variation in the radial di-rection in a majority of the domain and serves to emphasize the need fora two-dimensional axisymmetric model, as opposed to a one-dimensionalthrough-thickness model (e.g. Pan et al. (2000)), for accurate representationof the overall temperature field, which is necessary for accurate calculation ofheterocyclic-amine formation. Figure 5(b) shows less radial variation in wa-ter 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 thechoice of hpan. Figure 6 shows the baseline results of Fig. 4 with hpan = 150W / (m2 oC) and with hpan values 20% greater and smaller. These variationsin hpan provide histories that differ from the baseline histories by a maximumof 3.5 oC at 366 s and with final-time results differing by approximately 1oC. We also examine the importance of including the last two terms in (5),

14

including temperature dependence in the material properties, and includingshrinkage. Figure 7 shows the temperature histories for the properties ofTable 1 with hpan = 150 W/(m2 C) and τv = 0.1 s for four variations of(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(5) neglected. These histories are distinctly lower than the those calculatedwith all terms. The dashed lines show histories calculated with materialproperties taken as constant at their initial conditions. Again, these aresignificantly different than the baseline solutions. Finally, the dot-dashedlines show results of the full model but with shrinkage neglected; significantdifferences are seen. These results illustrate that these various terms andeffects 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 andnumerical values reported in the single-sided beef-patty frying experiments ofIkediala et al. (1996). The initial conditions are listed in Table 1 along withthe 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 throughoutthe domain with an experimentally fit decay function. Empirical relationsfor thermal properties reported by Dagerskog (1979) were used. Their pan-patty heat transfer coefficient was 250 W/(m2 C), which is greater thanthat used in our LSFE model. Patty shrinkage was neglected and a constantpan temperature was assumed. Figure 8 shows the experimental and finite-element results of Ikediala et al. (1996) for temperature histories at threeelevations near the patty radial center (z = 0.0035 m, 0.0055 m, and 0.0095m).

Figure 9 shows the measured pan-temperature history (Ikediala et al.,1996) along with a least-squares-fitted approximation given by

Tpan(t) =(

180 − 0.11117 t + 0.00044487 t2 − 4.7894 × 10−7 t3)

C. (21)

Temperature histories produced with the proposed LSFE model with initialconditions listed in Table 1 and pan temperature given by (21) are shownin Fig. 8. All other models and material properties are the same as those

15

employed in the previous section. Here, variation in pan temperature is seento be significantly smaller than that in the previous problem. LSFE numericalsolutions calculated with a constant pan temperature corresponding to theaverage of the data in Fig. 9 are virtually indistinguishable from the historiesshown in Fig. 8. While the proposed model had only problem-specific initialconditions, the numerical temperature histories agree with the experimentalhistories at least as well as those of Ikediala et al. (1996), which employed aproblem-specific empirically determined mass-loss model.

4.3. HA formation comparison with Salmon et al. (2000).

We examine here the ability of the proposed model to predict the for-mation of heterocyclic amines. We compare our model results with the ex-periments of Salmon et al. (2000). In those experiments, beef patties withinitial conditions listed in Table 1 were pan fried with various initial pan tem-peratures until the center temperature reached 70.6 C (on average). Theyexamined patties that were either flipped once at a frying time of 300 s, orflipped multiple times every 60 s. Experimental results are reproduced inFig. 10 where [HA] is the overall concentration of HAs. It is clear that theincreased flipping frequency yields dramatically fewer HAs. This reductionhas been attributed to lower maximum patty temperatures and the increasedloss of meat drippings, which are known to contain a significant portion ofHAs (Gross et al., 1993). The relative contributions of these two reductionmechanisms are not fully understood. Tran et al. (2002) examined numeri-cally the HA formation by integrating (15) with Ea = 18×103 cal/mol chosento best fit the data and cooking times were closely related to those of Salmonet al. (2000). Their normalized FE results are included in Fig. 10. Normal-ization was accomplished by scaling all data by the same constant such thatthe single-flip HA concentration for T 0

pan = 160 oC matched the average ofthe Salmon et al. (2000) data at that initial pan temperature. While themodel captured well the rate of increase of HAs with pan temperature forsingle-flip frying, it greatly under predicted the HA reduction accomplishedby frequent flipping.

In our numerical experiments, the overall HA concentration is calculatedby time-integrating the local first-order reaction equation (15) at all nodesin the LSFE model. Simulation-time durations were identical to the averagevalues reported by Salmon et al. (2000) as a function of initial pan temper-ature. For patties cooked with single and multiple flips, the total cooking

16

times were given by Salmon et al. (2000) as

tcook,sing = 3600

[

18.21 − 0.1289 T 0pan + 3.630 × 10−4

(

T 0pan

)2]

−1

s, (22)

and

tcook,mult = 3600

[

18.59 − 0.1144 T 0pan + 3.130 × 10−4

(

T 0pan

)2]

−1

s, (23)

respectively, where T 0pan = T 0

pan/1oC. The numerical parameters used in themodel are listed in Table 1, and the pan temperature was taken as (20) withthe appropriate offset. Ea = 25.3 × 103 cal /mol was chosen to best fit thesingle-flip data.

Figure 10 includes results predicted from our LSFE model, where again,data was normalized to the single-flip experimental results at T 0

pan = 160 oC.Unlike the FE model employed by Tran et al. (2002), our model captures asignificant fraction of the HA reduction accomplished in the multi-flippingexperiments. This is largely because patty temperatures are better modeled,and significantly lower, than those in the model of Tran et al. (2002). Thedifferences between the LSFE results and the experimental data for the multi-flip patties can be attributed, in part, to the absence of HA loss throughdrippings in the numerical model.

5. Conclusion

This paper presents a new formulation for simulating the pan-frying ofbeef patties that includes the transport of water in the meat as well as theenergy and mass loss due to water evaporation during cooking. This formu-lation requires only a few empirical parameters, but yields good agreementwith experimental time-temperature data from two independent studies. Im-portantly, our method accurately models the long-duration temperature dis-tributions in the meat, which is necessary to predict cooking times and HAformation. This model was used to investigate the experimentally observedresult that frequent flipping of beef patties during cooking reduces the for-mation of HA mutagens; a result that was not explained by earlier modelingstudies. These results show that a significant fraction of the observed re-duction in HA formation is due to the different temperature distributions

17

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

6. Acknowledgments

This work was funded by National Cancer Institute grant CA55861 and bythe US Department of Energy, Office of Science, Offices of Advanced ScientificComputing Research, and Biological & Environmental Research through theU.C. Merced Center for Computational Biology #DE-FG02-04ER25625.

18

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∂Ω

Ω

R

Zr

z

Figure 1: Schematic of a meridional section Ω (grey) located at θ = 0 of a cylindrical beefpatty (dashed lines).

Figure 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) GLLquadrature points of each element.

0 500 1000 1500 2000 2500130

140

150

160

T(

C)

experiments

Eq. (20)

t (s)

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

23

0

20

40

60

80

100

120

T(

C)

experiments

FE model

LSFE model; τv = 0.1 s

LSFE model; τv → ∞

(a)

0

20

40

60

80

100

T(

C)

(b)

0 500 1000 1500 2000 25000

20

40

60

80

100

T(

C)

(c)

t (s)

Figure 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 resultsof the LSFE model calculated with hpan = 150 W/(m2 C) and two values of τv.

24

(a)

(b)

Figure 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 pattyat t = 0 s and the solid squares indicate the locations of the temperature probes used toproduce Fig. 4.

25

0 500 1000 1500 2000 2500

0

20

40

60

80

100

z = 0.006 m

0.009 m

0.012 mT

(C

)

hpan = 150 W/(m2 oC), c.f. Fig. 4

hpan = 120 W/(m2 oC)

hpan = 180 W/(m2 oC)

t (s)

Figure 6: Temperature histories at three elevations predicted by the LSFE model for theTran et al. (2002) properties listed in Table 1 with three values of hpan.

0 500 1000 1500 2000 2500

0

20

40

60

80

100

z = 0.006 m

0.009 m0.012 m

T(

C)

full model, c.f. Fig. 4

last two terms in (5) neglected

constant material props.

shrinkage neglected

t (s)

Figure 7: Temperature histories at three elevations predicted by the LSFE model for theTran et al. (2002) properties listed in Table 1 with (i) temperature-dependent materialproperties (c.f. Fig. 4), (ii) temperature-dependent material properties but with the lasttwo terms of Eq. (5) neglected, (iii) temperature-independent (constant) material proper-ties, and (iv) the full model but with shrinkage neglected.

26

0 100 200 300 400 5000

20

40

60

80

100

z = 0.0035 m

0.0055 m

0.0095 mT(

C)

experimentsFE modelLSFE model

t (s)

Figure 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 inTable 1. Also shown are numerical results of the LSFE model.

0 100 200 300 400 500170

172

174

176

178

180

T(

C)

t (s)

experiments

Eq. (21)

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

27

160 180 200 220 240 2600.01

0.1

1

10

[HA

](n

g/g)

T 0pan (C)

single flip (exp.)multi flip (exp.)single flip (FE)multi flip (FE)single flip (LSFE)multi flip (LSFE)

Figure 10: Overall concentration of heterocyclic amine [HA] as a function of initial pantemperature as measured experimentally by Salmon et al. (2000). Also shown are thepredictions 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 T 0

pan =160 C.

28