Simulación de precipitación de fases

Every surface of the dish and therefore external boundary of the model was exposed to the air in the incubator. The condition at this boundary is described by Equation 4.8. Therefore, as discussed in Section 4.4, the convective heat transfer coefficient (h)

within the incubator must be defined. To define the heat transfer coefficient (h) the

experiment for model validation was carried out five times. Simulation outputs from models using the best estimate of each input parameter but varying h (incrementally

by 0.5 W.m-2K-1) were compared with each set of experimental data and the best fit

was estimated by sight (thus estimating h to the nearest 0.5 W.m-2K-1. The outcome

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4.5.4 Model validation

Figure 4.10 displays the data collected while equilibrating a dish to 37 °C in a dry Sanyo incubator. The measured incubator air temperature provided a temperature data set for the exposed boundary condition of the dish. Due to the air temperature of the incubator recovering to 99.5% of its initial temperature within 30 seconds after the door was opened, a single air temperature value of 36.9 °C (the average steady state incubator air temperature) was applied to the exposed boundary with a convective heat transfer coefficient of 12 W.m-2K-1 as defined in the previous section. The initial temperature of the dish model was set to 25.4 °C; the initial steady state temperature of the dish on the bench (where t<0).

Figure 4.10: Experimentally measured temperatures at the centre floor of a Petri dish and in an incubator as the dish (initially 25.4 °C) was placed into the incubator at t=0 (with opening and closing of the incubator door) and equilibrated to the incubator temperature (36.9 °C). Oil depth (Ho) = 3.06

mm, foot height (xb-xf) = 0.40 mm, thickness of dish floor (xf) =1.12 mm.

25 27 29 31 33 35 37 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 Time (minutes) Te m p er a tur e ( ºC) Incubator Air

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Monte Carlo simulations were run with the initial conditions and boundary conditions described above. The thermophysical properties and their standard deviations were used as defined in Table 4.6. The oil mass (5.48 g) was measured and its depth calculated (3.06 mm) and the thickness of the base of the dish (1.12 mm) and the foot height of the dish (0.40 mm) were measured.

Forty Monte Carlo simulations were run. After every simulation the standard deviation was calculated at every time step for all the existing simulations. The maximum standard deviation across time steps for any given run was determined. Figure 4.11 is a plot of the maximum standard deviation of the first ‘n’ samples vs. the number of simulations (n). After approximately 30 simulations the standard deviation levels out and stays relatively constant indicating that there is little change in the data spread. This analysis shows that additional Monte Carlo simulations were not required. A similar analysis was carried out in all Monte Carlo simulations for the remainder of this thesis.

113 Figure 4.11: The maximum standard deviation for the first ‘n’ simulations (over the simulation’s time steps) vs. the number of simulations run (n) for equilibration of a Petri dish.

The model can only be validated against a single experimental run since input parameters, such as dish dimensions, oil depth and initial conditions, are specific to an individual run. A good model fit was defined when the experimental data fits within the mean ± 2 standard deviations of this simulation since we would expect any random data set, such as an experimental run, to fall between these limits 95% of the time. This definition of model fit is only valid if the distribution of the Monte Carlo simulation output is normal (or at least approximately so). Figure 4.12 demonstrates that the distribution of the Monte Carlo simulations, while skewed slightly toward higher temperatures, is close to normal justifying this approach. It is worth noting that the model output being sufficiently normal is not a given. In general the distribution of the output of a model will be affected by the action of the model itself on the input parameters. Hence, as has been done here, the distribution of the model

0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 10 15 20 25 30 35 40 # of simulations S tand ar d D e v iat io n ( ºC)

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output should be checked if conclusions which rely on a certain output distribution are to be drawn.

Figure 4.12: A histogram displaying a typical example of the temperature frequency of 60 Monte Carlo Simulations (at t=5 minutes) for equilibration of a Petri dish.

The comparison between the mean simulation (± 2 standard deviations) and the experimental data is displayed in Figure 4.13. The experimental data remained within two standard deviations of the mean simulation output, validating the model. The width of this prediction band does not exceed 0.59 °C.

115 Figure 4.13: Measured temperature rise at the centre floor of a well of a Petri dish compared with the model simulation output (mean ± 2 standard deviation, 30 Monte Carlo simulation outputs) after placement into an incubator at t=0. Oil depth (Ho) = 3.06 mm, foot height (xb-xf) = 0.40 mm, thickness

of dish floor (xf) =1.12 mm.

While only a single validation experiment is presented here four were carried out for a Petri dish equilibrating in a dry incubator. The model displayed a good fit with the experimental data in every case and the difference between the model and the experimental data did not exceed 0.3 °C. The model is further validated in Chapter 5 through comparison with experimental data for a number of other steps in the culture process.

Validating the model against data collected during Petri dish pre-equilibration enabled information to be collected about Petri dish temperature pre-equilibration in the culture process (steps 5.5a and 9.2.8a in the process flow diagram, Figure 3.10). The

25 27 29 31 33 35 37 -5 0 5 10 15 20 25 30 35 Time (minutes) T e mpe rat ur e ( ºC ) ± 2 Standard deviations Experimental Data Mean Air

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temperature at the centre floor of a Petri dish, initially at room temperature, equilibrates to within 0.3 °C of the incubator temperature in 28 minutes. It is standard practice to pre-equilibrate dishes overnight prior to use to ensure the gaseous and thermal environment is optimal for embryos when they are transferred into a dish. However, shortening this equilibration time is potentially beneficial since degradation of amino acids to ammonium increases with temperature (As discussed in section 2.6.4). Limiting the exposure of the culture media to 37 °C will limit ammonium production which is potentially beneficial since ammonium is embryo toxic. Thermal pre-equilibration for <1 hour prior to embryo placement into the dish will enable embryo placement into media at 37 °C media while limiting the exposure of the culture media to 37 °C.

A <1 hour pre-equilibration time, while sufficient for thermal equilibration to 37 °C, may not be an adequate period for equilibration of oxygen and CO2 (thus pH). Before

any practical recommendations with respect to the period of pre-equilibration may be

made, the time required for equilibration of oxygen and CO2 (thus pH) must be

addressed.

4.6 Summary

A model of heat transfer within the Petri dish drop culture system was developed and validated within this chapter. This model may be applied for the following purposes:

• To investigate more complex thermal boundary conditions which are present

throughout the embryo culture process.

• To identify factors in the environment which have a significant impact on heat

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• To predict changes to an embryo’s temperature given changes in boundary

condition or dish geometry.

Validation of the model for pre-equilibration of a Petri dish identified that the temperature at the centre floor of a Petri dish (initially at room temperature) equilibrates to within 0.3 °C of the incubator temperature in 28 minutes. This suggests that shortening the current overnight pre-equilibration may be a possibility but no recommendations can be made until pre-equilibration of oxygen and carbon dioxide have been assessed.

For this model the thermophysical properties of air, culture media (water), paraffin oil and the polystyrene dish were defined. These properties may be applied when modelling other systems throughout the culture process such as the 4-well dish, which involves identical materials in a different geometry, and the pipettes, which can contain air, paraffin oil and culture media.

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