Analysis of Propagation of Freezing and Thawing Fronts YESHATAHYU TALMON and H. TED DAVIS ABSTRACT The modified isotherm migration method (MIMM) is used to calcu- late the temperature profiles and freezing (thawing) times for sym- metrically cooled (heated) slabs with meat-like thermal properties. MIMM, which is reviewed briefly is not restricted to the usual ideali- zations such as zero surface resistance,uniform critical temperature distribution, and infinitely large sample. Dimensional analysis is used to identify the dimensionlessgroups controlling freezing and thawing front propogation in slabs, cylinders, and spheres having uniform thermal properties and constant freezing temperature. The MIMM resultsprovidea basis for evaluating the qualitativeoriginsand the quantitative extent of the success of Plank’s (1913) old model. The freezing times predicted by the moving front model used here agree with those predicted by the empirical correlation found by Cleland and Earle for the Karlsruhe test substance. INTRODUCTION THE FREEZING AND THAWING of food products is an old problem, one which is of obvious importance and is reasonably well understood (Fennema and Powrie, 1964; Fennema et al. 1973; Heldman, 1975). Although the water in biological tissue freezes over a range of temperatures (Riedel, 1956, 1957a,b), it is useful to treat the freezing or thawing process as propagation of a constant tempera- ture phase transition front (Heldman, 1975; Cleland and Earle, 1976, 1979a,b). The thawing region will be a narrow zone for those systems iri which the freezing range is narrow. In lean beef, for example, the majority of the latent heat in freezing is accounted for in the temperature range 25-30’F (Riedel, 1957a). In canned or packaged foods containing large amounts of bulk water, much of the water will freeze or thaw at an almost constant temperature. ‘The present paper focuses on those systems in which the freezing point range of the aqueous phase is sufficiently narrow to be approximated by a single freezing tempera- ture TF. The oldest solution to the problem of freezing front propogation is that of Plank (19 13). He derived the freez- ing time of a slab having constant thermal properties which was initially at its freezing point and which was cooled symmetrically from each side under conditions of pseudo- steady state (i.e., the temperature profile between the surface temperature and the freezing front was assumed to be linear). Owing to its simplicity and its ease of generali- zation to other geometries (Ede, 1949; Heldman, 1975), Plank’s theory has provided the most frequently used formulas for estimating freezing times. The estimates are usually low, since the theory is based on pseudo-steady state and ignores the sensible heat of the system. The theory does, however, correctly account for the thermal resistance between the sample and its environment. This is accomplished through a “radiation” boundary condition Author Davis is Head of the Dept. of Chemical Enginnering & Materials Science, Univ. of Minnesota, Minneapolis, MN 55455. Author T&non is affiliated with the Dept. of Chemical Engineering, Technion-Israel Institute of Technology, Haifa, Israel 32000. 1478-JOURNAL OF FOOD SCIENCE-Volume 46 (1981) (also known as a boundary condition of the third kind) which ‘introduces a heat transfer coefficient h between the environment and the sample. Nagaoka et al. (1955) and Cleland and Earle (1979a) have improved somewhat Plank’s original model by including, approximately, sen- sible heat contributions. Plank’s theory also provides the skeleton for the very useful and quite accurate empirical correlation formulas determined by Cleland and Earle (9 176; 1979a,b) from experimental studies of the Karls- ruhe test substance, a material used for its reproducibility and the nearness of its freezing characteristics to those of real foods. Another idealization, whose attractiveness lies in the fact that it can be solved analytically (Carslaw and Jaeger, 1959) the transient heat equation for a moving front in a semi-infinite slab having uniform thermal properties in each phase, a fixed surface temperature (which implies h = “) and a uniform initial temperature. The solution to this problem provides a rough estimate of initial heat penetra- tion rates of samples suddenly fixed against a cold solid surface. However, the problem is too idealized for practical predictions, ignoring as it does sample geometry and finite size, nonuniform thermal properties and initial temperature distribution, and the finite thermal resistance between the sample and its environment. Most of the research efforts in freezing and thawing heat transfer have been aimed at eliminating some or all of the limitations of the above two idealizations. We shall not present an extensive literature review here. The published works of Hayakawa (1977) and of Cleland and Earle (1977a,b; 1979a,b) serve this purpose adequately. In gen- eral finite difference and finite element numerical tech- niques have been introduced to relax one or more of the undesirable model assumption. One approach, typified by the work of Tao (1967, 1968) is to retain the assumption that freezing takes place at a single, fixed temperature but to handle correctly the sample, geometry, the initial tem- perature distribution (superheat), and the radiation boun- dary condition. Application of a new version of this so- called moving front approach to the freezing and thawing problem is the purpose of the present paper. Another ap- proach to the modelhng of food systems is to introduce an effective heat capacity to account for the fact that water in a food system freezes over a range of temperatures. In this approach (Hohner and Heldman, 1970; Comini et al., 1974; Cleland and Earle, 1979a,b) one sidesteps the problem of a propagating phase transition front and solves the heat equation for a substance with a strongly tempera- ture dependent heat capacity. This approach, when the temperature dependences of the thermal conductivity and density are included, although presumably the most accur- ate one, has the disadvantages of requiring a great deal of experimental data and of demanding a very find grid for discretization of temperature in the vicinity of the specific heat maximum if finite difference or finite element nu- merical techniques are used. Hayakawa and Bakal’s (1973) model in which it is assumed that a partially frozen zone exists between a totally thawed and totally frozen zone incorporates in a moving front analysis the idea of a tem- perature range over which freezing occurs.