Lumped Model for Target Temperature Prediction

Because we can generally ignore thermal gradients in PSII targets, a lumped capacitance model is sufficient for studying target temperatures. Thus, Equation 1 becomes:

where is the volume of the body considered and is the net heat flow into the body. That is:

where is the heated target area, is the radiating target area, and is the rate at which heat is lost due to conduction down the stalk.

Our first task in applying this equation to PSII implants will be to determine the importance of the the duty cycle of the PSII process. In other words, we must determine whether the analysis must consider the power input over the full cycle, or whether the cycle-averaged power can be used in a constant-power analysis. The semi-infinite body analysis in a prev section indicates that the cycle-averaged analysis is sufficient, since the surface temperature rise is small during a single pulse, but it is instructive to apply the lumped capacitance model to this problem.

In order to obtain an analytical solution to the problem, we consider a linear problem of the following type:

where is the applied heat flux and all thermal losses are modeled using some heat transfer coefficient which can be considered as representative of any appropriate heat loss. Only the radiative heat loss is not easily cast in this form, but we can use the concept of an effective heat transfer coefficient (Equation 16) to estimate the impact of pulsing on our analysis. In addition, a change of variables can remove the existence of the bulk temperature, so we can, without generality, solve the following equation:

We begin solution of this equation by rewriting it as:

where

and

For the PSII model, we adopt a time-dependent applied heat flux of the form:

We also adopt the initial condition . The solution to this system can be obtained by separately solving for the temperatures during the power-on and power-off phases of operation and coupling the solutions by requiring that the temperature be continuous.

The solution to this equation is shown schematically in Figure 5. The temperature ratchets up, increasing during the power-on period and decreasing during the power-off period. At some point, the target will reach a quasi-steady state in which the temperature at the end of the cycle is identical to the temperature at the beginning of the cycle. This case can be studied analytically. Assuming that the temperature at the beginning of the cycle is , the temperature at the end of the power-on period is:

where is the length of the power-on period. This is then used as the initial temperature for the calculation of the temperature during the power-off period. The temperature at the end of the power-off period then becomes:

where is the length of the power-off period. In quasi-steady state, this last result must be identical to , thus allowing us to solve for the value of during this state. That is:

where is the temperature at the end of a quasi-steady cycle. The maximum temperature during the quasi-steady state is that given by Equation 27, while the minimum temperature is . Hence, the ratio of the maximum temperature to the minimum temperature is:

Substituting for in this equation gives the following equation for ratio of the maximum temperature state to the minimum temperature during the quasi-steady state:

If this ratio is near 1, then the temperature swings during a cycle are small and can be ignored. Hence, we can ignore temperature swings when

This can be simplified by converting the ratio of the target volume to the target area to an equivalent target thickness. Hence, the criterion becomes

where is the volume-to-area ratio (). This quantity (symbolized as ) is shown in Figure 6. For this figure, the sample thickness was chosen to be 1cm, the power-off period length was chosen to be 0.01 seconds, and the chamber temperature was chosen to be 0 K. As seen in this figure, the ratio in the equation above is much less than 1 for typical PSII conditions. This is further evidence that we can ignore the details of the cycle in analyzing PSII target temperature histories. This leads us to a lumped capacity, cycle-averaged thermal model that is capable of modeling nearly all PSII target conditions.