3.1. Starting Point
To grow an optimized model, the system (1) is used as a “seed”. To simplify the explanation and reduce the size of the article, consider only the first equation of this system.
Coefficients
C0 and
a01 are calculated from known temperatures
T0,
T1, and power
P0 by using the “least squares” method for the best match between the left and right sides of Equation (5). In other words, minimizing the square of the residual,
where both the derivative and the integral are calculated numerically. A reduced set of only three measurements (
Figure 4) is used here as an example.
In addition to the values of the coefficients, their standard deviations (SD) were also calculated and used as an error estimation (
Table 1).
To assess how well the model reflects the real processes, standard deviations of the heater power, SD(P0) = 2.72 W, and the residual, SD(δP) = 0.57 W, were calculated. These give the relative error for the starting point, δ = 21%.
3.2. Generative Design. Generalization of Dependencies and Structure
A simple model (1) assumes the constancy of the coefficients. To build a more realistic model, we need to consider the dependence of heat capacity on temperature and the presence of various heat exchange mechanisms. Moreover, the assumption of the interaction of only neighboring structural elements may not correspond to the actual installation.
Sets of heat capacities and heat flow
can contain physically meaningful functions or just simply polynomial approximations. In our case, we obtain
where linear regression gives the following coefficients (
Table 2).
This reduces the standard deviation of the residual by about two-thirds, to SD(δP) = 0.18 W, and the relative error to δ = 6.6%, less than one third of the starting point.
In addition, at this stage, there are no new components corresponding to heat exchange with structural elements other than the inner shell. Thus, if it is present, its role is insignificant.
3.3. Generative Design. Expansion
The equations of the model are essentially the equations of thermal balance, the law of conservation of energy. Thus, a residual other than 0 means a thermal imbalance. The set of random bursts in
Figure 4 can be made more convenient for analysis by going from power to enthalpy and using the integral residual:
Just like the peaks in
Figure 4c and
Figure 5 show, the discrepancy is because the simple model (1) does not describe the fast transient processes in the system since it reduces the entire structure to several large elements. The relationship between the spatial and temporal resolution of the model can be demonstrated by the example of constructing a conservative numerical scheme for the equation of thermal conductivity:
where
c,
ρ,
T, and
p are the specific heat, density, temperature, and volumetric heating power, respectively, all dependent on the coordinates and time, and
g is the heat flow. At any given time, (7) can be linearized by a Jacobian matrix
Jnm.The solution of (8) can be described using eigenvalues and eigenvectors of
Jnm.According to (9), the solution contains components with a time constant of . Due to physical reasons, all eigenvalues are not positive, and when. . In other words, the greater the spatial detail, the better the model describes transients.
To build an optimal model, we will consistently add new thermal elements to it in those places where the residual is maximum or close to it. To determine the coefficients, we will use the results of experiments with heating in the form of a Heaviside function:
and we will limit the area of linear regression calculation to the duration of the transient process. We will use the same Equation (5) as a starting point to simplify the presentation. In this case, new calculation gives coefficients (
Table 3) that are slightly different from those calculated earlier (
Table 1) due to the different regression base.
The maximum residue—the peak in
Figure 6a (step in
Figure 6b)—corresponds to the fastest process in the system, turning on the cell heater. Adding an intermediate thermal element within the cell, which can be identified as the actual heater with its own temperature and heat capacity, gives the system (15). With the coefficients of
Table 4, this can reduce the residual power by 5 times (
Figure 7) and significantly improve the enthalpy residue (
Figure 8).
To determine what other terms and equations should be added to the system (15), we will perform an approximation of the residual enthalpy with powers of temperatures
T0 and
T1. The coefficients and their standard deviations are shown in
Table 5.
Residual enthalpy decreases by more than 10 times, as shown in
Figure 9.
Obviously, the approximation of
Table 5 contains an excessive number of approximating terms. This is evidenced by a significant relative (about 10%) error of the coefficients. The magnitude of this error can serve as a criterion for choosing the final approximation. After excluding several terms, the relative error of the remaining coefficients is less than 1% (
Table 6). Reducing the number of terms practically did not worsen the approximation—the graph of residual enthalpy has no visible differences from
Figure 9.
Since residual enthalpy is an integral characteristic, the equation for
T0 must contain derivatives
. These terms look like heat capacity correction and appear on the right side of equation for
T0 in the system (16) as an additional heat transfer channel.
The dependence on
T1 can be understood if we add the last two Equations in (16):
where
C0+1 and
C1 are both linear functions of the temperature. Due to the large number of coefficients, we did not list them in the table.
This process continues to sequentially remove the main part from the residue step-by-step until the required accuracy is achieved.
The constructed model is conveniently represented as an undirected graph showing the thermal relationship between the installation elements. For example, for system (16), we have
Figure 10.
Figure 10 shows that in addition to the thermal conductivity along the air gap, there is a thermal contact between the cell and the inner shell along the suspension and the supply wires. The heater temperature during rapid heating will differ from the cell’s temperature.
The use of the model in the control loop made it possible to maintain the temperature difference between the cell and the inner shell on the measuring heaters with an error of less than 0.01 °C.