Freezing time, equation

Fourier s equafion does nof fake into account convection, which in the case of food freezing governs heat transfer befween fhe food surface and fhe refrigerafing medium, for example cold air in the case of a fluidized bed. Second, and more importanfly, if cannof accounf for fhe removal of lafenf heaf and fhe resultant phase change. Of the models available to predict freezing time. Plank s equation (Plank, 1913, 1941) is one of fhe simplesf and most widely used and is derived in many standard texts (Singh and Heldman, 2001 Smith, 2003). The principal... 

The freezing time using Nagaoka s equation is then... 

Vazquez and Calvelo (1983b) presented a model for the prediction of the minimum residence time in a fluidized bed freezer which can then be equated to the required freezing time. The model is defined in terms of a longitudinal dispersion coefficient D, which is a measure of the degree of solids mixing within the bed in the direction of flow (and has the dimensions of a diffusivity, and hence units of m s ), a dimensionless time T... 

Pham, Q.T., Simplified equation for predicting the freezing time of foodstuffs, /. Food Tech., 21 (1986) 209-219. 

Solving for t, using equation (8.50), we find that t = 390, from which freezing time, t, can be calculated using equation (8.48) as follows ... 

Fig. 41.6 Initial temperature 0,- dependence of a the freezing time dotted line) (t) and the corresponding relaxation time solid line) (t), b in contrast to the measurements of the c initial EnC i) and [18]. The h( i) was derived from solving Lagrangian equation [33] based on...

This equation cannot be integrated as simply as before because h is now a function of time. Fig. 5.26 shows how the depth of the cavity changes with time as the melt flows. Barrie has investigated this situation and concluded that the freezing-off could be described by a relation of the form... 

However, in the non-isothermal case the pressure is also high at low injection rates. This is because slow injection gives time for significant solidification of the melt and this leads to high pressures. It is clear therefore that in the non-isothermal case there is an optimum injection rate to give minimum pressure. In Fig. 5.28 this is seen to be about 3.0 x 10 m /s for the situation considered here. This will of course change with melt temperature and mould temperature since these affect the freeze-off time, //, in the above equations. 

The results of Example 5.2 apply to a reactor with a fixed reaction time, i or thatch- Equation (5.5) shows that the optimal temperature in a CSTR decreases as the mean residence time increases. This is also true for a PFR or a batch reactor. There is no interior optimum with respect to reaction time for a single, reversible reaction. When Ef < Ef, the best yield is obtained in a large reactor operating at low temperature. Obviously, the kinetic model ceases to apply when the reactants freeze. More realistically, capital and operating costs impose constraints on the design. 

The author has used a model and an equation developed by Steinbach [1.511 for many years and for many experiments in a wide field of applications. The model, shown in Fig. 1.60, uses an infinitely expanded plate of the product with the thickness d. Equation (14) describes the time of the main drying part of the freeze drying cycle ... 

It is now time to draw all the threads together, and look at the temperature at which the thin lines intersect. It is clear from Figure 5.18 that the intersection temperature for the mixture occurs at a cooler temperature than that for the pure material, showing why the melting point temperature for a mixture is depressed relative to a pure compound. The depression of freezing point is a direct consequence of chemical potentials as defined in Equation (5.12). 

We can make further approximations to simplify the NRF of the Hamiltonian presented in equation (75) for non-dynamical properties. For such properties, we can freeze the nuclear movements and study only the electronic problem. This is commonly known as the clamped nuclei approximation, and it usually is quite good because of the fact that the nuclei of a molecule are about 1836 times more massive than the electrons, so we can usually think of the nuclei moving slowly in the average field of the electrons, which are able to adapt almost instantaneously to the nuclear motion. Invocation of the clamped nuclei approximation to equation (75) causes all the nuclear contributions which involve the nuclear momentum operator to vanish and the others to become constants (nuclear repulsion, etc.). These constant terms will only shift the total energy of the system. The remaining terms in the Hamiltonian are electronic terms and nuclear-electronic interaction contributions which do not involve the nuclear momentum operator. 

Nagaoka s equation (Nagaoka et al, 1955) is an extension of Plank s model and takes into account the time required to reduce the temperature from an initial temperature T, above the freezing poinf. The lafenf heat of fusion in equafion 3.3 is replaced by the total enthalpy change A/i which includes the sensible heat which must be removed in reducing the temperature from an initial T and in addition an empirical correction factor is included. Thus... 

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