Further minor deviations are visible in the third row of Fig. 7.25 when comparing the liquid mass flow rates m˙ l of the systems. The maximum absolute deviations of m˙ l are max (∆ ˙m) = 1.4 mg s−1 for Model C and max (∆ ˙m) = 1.7 mg s−1 for Model M. Due to the different condenser models used in SIM and in the complex models Model C and Model M, the over- and undershoots caused by the Q˙ cc and the Tsk steps are pronounced differently in contrast to the Q˙ ev step. As m˙ l results from the balanced accumulated mass change of all LHP components in SIM, Q˙ cc and Tsk have a greater direct impact on m˙ l of SIM than on m˙ l of the complex models. Their m˙ l is dependent on m˙ v instead (see Sec. 4.1.6), where m˙ v is calculated using the stationary energy balance equation of the evaporator and a fixed mean void fraction in the two-phase region of the condenser. Thus, the difference between the levels of m˙ l before and after the first two steps is greater with SIM than with the complex models. Accordingly, the balanced temperature levels of Tcc in the first row after the steps vary between the systems, as mentioned early.

The first peak of m˙ l after the rising flank of the Tsk step is lower for both complex models than for SIM. The subsequent undershoot of Model M, however, is similar to the one of SIM since the additional evaporator capacitance Cev,sf incorporates the thermal inertia of the evaporator mass into the model, which is considered for the evaporator fluid in SIM. The resulting manipulation of the net heat load Q˙ ev at the evaporator in Model M amplifies the discussed reaction of m˙ l to the Tsk step in contrast to Model C. The same applies inverted to the falling flank of the Tsk step. An amplification of the behavior of m˙ l is also visible after the Q˙ cc step. Similar to SIM, the over- and undershoots of m˙ l are more pronounced with Model M than with Model C.

To analyze the LHP mass flow dynamics, the difference between m˙ l and m˙ v is presented in Fig. 7.26.

Figure 7.26: Step responses of the difference between the mass flow rates m˙ l and m˙ v of SIM, Model C, and Model M to Test Profile 1 in Fig. 7.24

The three rows correlate with the divided time axis of Fig. 7.25 to take a closer look at the separate step responses. Since the mass flow rates are equal in an equilibrium point, the mass flow difference always converges back to zero after a change of the inputs. The qualitative reaction to the three input changes is similar between the models and SIM. However, a smaller displacement of the mass flow difference of the complex models from zero is visible compared to SIM due to the mentioned modeling mismatch. Only for the Q˙ cc step, Model M shows initial spikes due to the integrated evaporator mass, which quickly return to the curve of Model C.

For tracking the liquid-vapor interface in the condenser, the lengths of the condenser regions are calculated based on the condenser inlet in the complex models. Thus, the position of the liquid-vapor interface results from the sum of the lengths Lsh and L2ϕ. For the test profile in Fig. 7.24, the variations of the position of the liquid-vapor interface and the condenser outlet temperature Tco,o are depicted in Fig. 7.27.

Figure 7.27: Step responses of the liquid-vapor interface in the condenser of SIM, Model C, and Model M to Test Profile 1 in Fig. 7.24

The maximum absolute deviations of the sums toward SIM are max (∆L) = 8.2 mm for Model C and max (∆L) = 12.7 mm for Model M. Since Lsh is small compared to L2ϕ, the differences between the curves in the first row of Fig. 7.27 and in the second row of Fig. 7.25 are small as well. With a considered total length Lco = 1.85 m, the remaining length Lsc of the subcooled region is large enough to cool the liquid down to near sink temperature at the condenser outlet as can be seen by comparing Tco,o in the second row of Fig. 7.27 to Tsk in the third row of Fig. 7.24. The only deviation between the complex models and SIM is present during the Q˙ cc step when Tco,o of SIM with its numerical solution method falls below Tsk

which contradicts the second law of thermodynamics (cf. Sec. 3.1.2). Such a contradiction is not possible with the heat exchanger approach of the complex models (see Sec. 4.1.4).

All curves of the mass flows and the region lengths in the previous figures are plausible from the models' point of view. To determine the proximity to the real LHP systems on the test benches, appropriate measurements of the interior of the LHP are necessary, which is not in the scope of this thesis.

### 7.2.4 Validation of the Order Reduction of the Complex Dynamical Model

A reduction of the order of the four-dimensional state-space model (4.66), hereinafter denoted as Model 4D, to the final three-dimensional state-space model (4.84), i.e. Model C, has been performed and qualitatively analyzed in Sec. 4.1.7. By assuming a constant volume fraction ratio β op and thus a constant thermal capacitance C op cc , the model equations have been simplified, and the order of Model 4D has been reduced. Now, the simplification is quantitatively investigated by comparing the results of SIM and both models. For the comparison, the same parameter set is used as defined in Table 7.2 to set the focus on the differences of the model equations.

Figure 7.28: Test Profile 2 with variations in the disturbances across the LHP operating range

A test profile is chosen that includes the variation of the disturbances, i.e. the heat load Q˙ ev and the sink temperature Tsk, at a constant system input Q˙ cc, which is commonly used in qualification tests of LHPs. Their respective curves are depicted in Fig. 7.28.

The presented test profile expands over a great part of the operating range of the LHP to evaluate the performance of the models outside the OP as well. The corresponding state variables of both models in comparison to the state variables of SIM are presented in Fig. 7.29.

Figure 7.29: Results of SIM, Model 4D, and Model C to Test Profile 2 in Fig. 7.28

A similar curve of the three state variables Tcc, L2ϕ, and m˙ l results from the chosen test profile despite the differences in the state variable β. For the comparison of the models with SIM, β of SIM is calculated by solving (4.83) with the fixed thermal capacity of the two-phase CC given in [MKHW19], since no separate determination is included. Due to the different modeling approaches, a negligible stationary deviation between the β curves of SIM and Model C exists. Although β of Model 4D slightly adapts to the variations of the disturbances, the effect of a varying β on the overall LHP behavior is low. Hence, the state variables Tcc, L2ϕ, and m˙ l of Model 4D and Model C in Fig. 7.29 are similar, and the model simplification is valid. Furthermore, L2ϕ and m˙ l are proportional to Q˙ ev of Fig. 7.28, while Tcc decreases with increasing Q˙ ev. A decreasing Tsk reduces the rise and fall of the three state variables during the variation of Q˙ ev, respectively.

The results in Fig. 7.29 underline once again the very good agreement between the complex dynamical models and the experimentally validated numerical LHP simulation SIM over the entire operating range of the LHP.

### 7.2.5 Occurrence of Temperature Oscillations

The investigated conventional LHP systems SIM, LHP 1, and LHP 2 include minor attached masses at the evaporators. Thus, Model C without considering a thermal mass at the evaporator and Model M with an appropriate thermal capacitance for the mass show related results in the previous model validation when comparing the measured and simulated temperatures. As discussed in Sec. 2.4, temperature oscillations in the normal operation of the LHP within its operating range are traced back to a significant thermal evaporator mass, as theoretically shown in the literature. With the introduced Model M, it is now possible to analyze the stability of conventional LHPs with attached thermal mass and to simulate corresponding temperature oscillations. In addition, an adequate controller compensating these oscillations can be designed based on the introduced model in a subsequent step.

The curves of the measured temperatures of Model M parameterized in the OP of SIM (see Table 7.4) for three different thermal masses

$$C\_{ev,sf}^1 = 40\,\text{J}\,\text{K}^{-1},\tag{7.13}$$

$$C\_{ev,sf}^2 = 260 \,\text{J} \,\text{K}^{-1},\tag{7.14}$$

$$C\_{ev,sf}^3 = 280 \,\text{J} \,\text{K}^{-1} \tag{7.15}$$

are depicted in Fig. 7.30. For the investigation of the stability of the nonlinear system Model M, the stability of the chosen OP of the system is regarded. Based on a displacement from the equilibrium point, the stability of the OP is analyzed in dependence on Cev,sf . Due to the bidirectional connection between the LHP fluid cycle and the evaporator mass, which is situated between the LHP and the heat source, the net heat load Q˙ ev at the evaporator is manipulated both by the fluid temperature of the evaporator, i.e. Tev, and by the temperature Tev,sf of the evaporator mass near the heat source. The interaction between both subsystems can cause temperature oscillations when the difference between the dynamics of the related temperatures are sufficiently large depending on the value of Cev,sf .

Figure 7.30: Comparison of Model M in the OP of SIM with different thermal masses C1 ev,sf = 40 J K−1 , C2 ev,sf = 260 J K−1 , and C3 ev,sf = 280 J K−1

In the chosen OP, the applied initial offset of the CC temperature is T of f cc = 0.03 K. As expected, the temperatures of Model M with the smallest thermal mass (7.13) are stable and run into the balanced OP, as presented in Fig. 7.30. By increasing the thermal mass to the value of (7.14), decreasing temperature oscillations become visible, but the trajectories of the temperatures are still stable. The stability limit of the OP is crossed when the thermal mass is further increased to the value of (7.15). Then, increasing oscillations are present, and the OP is unstable.

The stability limit can be calculated by means of Theorems 4.1 and A.1. For this purpose, the characteristic polynomial of the Jacobian of (6.27a) is established and the necessary and sufficient conditions of Theorem A.1 are solved for the thermal mass Cev,sf . The most restrictive condition for a stable OP finally results from the second and third leading principal minors:

$$0 < C\_{ev,sf} < 269.1 \,\text{J} \,\text{K}^{-1}.\tag{7.16}$$

Thus, the upper limit in (7.16) lays between (7.14) and (7.15) which confirms the results of the simulative comparison in Fig. 7.30.

The presented system stability analysis can be performed with any other LHP parameterization. For the other two investigated LHP systems LHP 1 and LHP 2, the respective upper limits in (7.16) are Cev,sf = 864.3 J K−1 and Cev,sf = 1844.7 J K−1 , respectively. The different sizes of the Cev,sf limits match the rising order of the sizes of the thermal masses of the respective CC (cf. Table 7.4). The dependence of the operating stability of an LHP on the relation between the thermal masses of the CC and the evaporator thus coincide with the theoretical considerations in [HB12]. Thanks to the introduced Model M in this thesis, these findings can now be reproduced in simulations for the first time.

## 7.3 Analysis of the State-Space Model for Parallel Loop Heat Pipes

The combination of two LHPs across a thermal mass, as given by the heat transport system of an LCT, poses additional challenges to the modeling and control process. Besides the exact modeling of the heat flows in the attached LCT structure and between the LHPs, the manipulation of two controllers on one system must be coordinated to ensure overall stability. In this section, the focus is therefore set again on the dynamics of parallel LHPs under variation of the system inputs.

In Fig. 7.31, another 1W Q˙ cc step at a different equilibrium point of LHP 3 is measured and compared to the results of Model P. The offset adjustment of the system inputs, as explained in Sec. 7.2.1, is applied to evaluate the validity of the fixed OP values over the operating range of LHP 3. In comparison to the 1W Q˙ cc step in Fig. 7.10, both the heat load Q˙ hs and the sink temperature Tsk are reduced. Here as well, the dynamics of the CC temperature Tcc in the first row of Fig. 7.31 are closely reproduced by Model P despite the fixed model parameters. Only a small stationary deviation between the modeled and the measured Tcc remains after the step (∆T a cc(2200 s) = 0.35 K, ∆T b cc(2200 s) = 0.17 K).

For the evaporator temperature Tev in the second row, a similarly precise behavior of Model P is shown where the simulated temperatures follow the measured temperatures closely. The largest deviation is present in the third row with the condenser inlet temperatures Tco,i. The reasons are connected to the unmodeled influences of the close condenser system on the pipes causing parasitic heat flows between the condenser and the VL. Due to the sensor

Figure 7.31: Comparison of 1W Q˙ cc step responses of LHP a and LHP b of LHP 3 and Model P (Q˙ hs = 126W, Tsk = 10 ◦C)

placement on the condenser mentioned in Sec. 7.1.1, the measured and simulated condenser outlet temperatures Tco,o are very close to the sink temperature Tsk and coincide with each other. Overall, Model P with fixed parameters keeps a satisfactory behavior toward variations of the system inputs under a balanced heat distribution between both LHPs.

The analysis of the oscillation behavior of Model P with regard to the thermal masses of the evaporators yields a different result compared to Model M with a single LHP. According to the procedure in Sec. 7.2.5, the upper limits of C a ev,sf and C b ev,sf for LHP 3 are

Cev,sf = 149,835 J K−1 . This would be equivalent to an aluminum mass of approximately m = 166 kg. Since such a value is hardly ever reached or desired in any application, a connection between the operating stability and the thermal masses of the evaporator can not be confirmed for Model P. Instead, instability is reached in simulations due to an unbalanced heat distribution between both LHPs and thus an unequal operation of the coupled LHPs. The heat distribution in a heat transport system depends on the individual heat source structure. Based on an exact investigation of the heat flow paths of each distributed heat source instead of the total heat load, a more detailed thermal network of the heat source structure can be constructed to evaluate the stability limit of each heat transport system separately.

## 7.4 Simulative Validation of the Controller Performances

The objective of this section is the evaluation of the proposed controllers concerning various aspects of an effective LHP-based heat transport system, i.e. its stability, accuracy, and disturbance rejection. The operating conditions are varied to test the controller performance over the entire operating range of the respective applications at a constant setpoint temperature. Two quantitative validation metrics are chosen to evaluate both the static and the dynamic performance of the controllers:

In the following analysis of the controller performances, the model-based designed PI controller piAW/S is comparable with the heuristic state-of-the-art PI controller in the literature. The setpoint temperatures Tset for the CC temperature Tcc and Tset,ev for the evaporator temperature Tev of the controllers are fixed on the following values, which are within the specifications of the considered applications:

$$T\_{set} = 27 \, ^\circ \text{C},\tag{7.17}$$

$$T\_{set,ev} = 29\,^\circ \text{C}.\tag{7.18}$$

First, the temperature control of single LHPs is evaluated taking into account the complexity, computational effort, and implementation requirements of the respective controllers. Subsequently, the temperature control of advanced LHP-based heat transport systems including the single LHP with attached thermal mass and the parallel LHPs are investigated and analyzed.

### 7.4.1 Temperature Control of a Single Loop Heat Pipe

A typical benchmark for the performance of an LHP-based heat transport system is the variation of the disturbances along its operating range. In the application, these variations correspond to the changing dissipated heat of the cooled object in accordance with its operating status and the variability of the heat sink. In Fig. 7.32, such a benchmark profile is presented which is used for the subsequent validation of the controller performances.

Figure 7.32: Benchmark profile of a single LHP

Both the CC controllers in Table 7.6 and the evaporator controllers in Table 7.7 are tested in SIM with the benchmark profile in Fig. 7.32.

#### Compensation Chamber Control

The comparison of the controlled CC temperatures Tcc of the different control loops is presented in Fig. 7.33.

The largest deviation from the setpoint temperature Tset is reached by piAW/S and thus by the state-of-the-art controller. All other controllers proposed in this thesis achieve a better performance result. While a heat load change yields an over- or undershoot dependent on the movement on the U-shaped curve of the SSOT (see Fig. 3.9) with a subsequent convergence to the setpoint, a change of the sink temperature results in an offset of Tcc. The qualitative results are underlined by the quantitative metrics in Table 7.12.

Table 7.12: Evaluation of the proposed CC controllers based on Model S and Model C

The lowest over- and undershoots are visible in Tcc controlled by nMIAC/S. Accordingly, its MAD is the smallest. However, the convergence to Tset is slower for nMIAC/S and is exceeded by piDF/S and nSF/C. Measured by the RMSE, the deviation error for nMIAC/S is smaller than for piDF/S, but nSF/C achieves the smallest RMSE and thus the best disturbance rejection. Although piOF/S has a similar MAD like nSF/C, its overall performance measured by the RMSE is the second worst.

The good performance of piDF/S is traced back to the good agreement between the stationary and the transient LHP simulation of SIM. Thus, the controller performance of piDF/S is also significantly dependent on the agreement between the stationary model and the real system in contrast to the dependence of the performances of the other controllers on their dynamic models. The improved performances of the proposed controllers are bought by the additionally required knowledge of the system's operating state. Therefore, additional sensors are necessary which come along with elevated costs. As presented in the third and fourth column of Table 7.12, piDF/S, nMIAC/S, and nSF/C with the lowest RMSE require the availability of the disturbances to include the directly connected dynamics of the mass flow rates. In contrast, piOF/S achieves an improved performance compared to piAW/S by two additional temperature sensors. Due to the proximity of the sensors to the impact points of the disturbances, a faster reaction to disturbance changes is possible.

#### Computational Time

In order to evaluate the different controllers with respect to the necessary computing power of the target processor hardware, the computational time of each controller step is averaged over the total time frame of the benchmark profile. The CPU used for the simulations is an Intel Core i7-7700 at 3.60 GHz and with 16 GB of RAM. The results of the different controllers are given in Table 7.13.

Table 7.13: Evaluation of the computational time of the proposed CC controllers based on Model S and Model C

As expected, piAW/S being based on the state-of-the-art PI controller has the lowest mean computational time and thus the lowest processor requirements. Similarly small is the mean computational time of piOF/S due to both their simple controller structures. The disturbance feedforward control implemented as lookup table requires additional computational time in order to interpolate the output value of the table. A significantly larger distance exists to the next fastest controller, piDF/S. In contrast, the mean computational time of nMIAC/S is already above 1 ms which is traced back to the optimization method to determine the identified model parameters for the subsequent state estimation. The largest mean computational time, however, is necessary for nSF/C based on Model C. Since the state feedback controller is based on the state estimation via an SRUKF, additional computational time is needed for the discrete transformation of the sigma points through the stiff system equations (see Sec. 4.2.3). The computational time of nSF/C in Table 7.13 is based on the proposed implementation of Matlab's ode15s solver. However, a mean computational time of 595 ms is measured when the multi-rate framework and Euler's forward method is implemented. Such a high computational time is already close to the sampling time Tst = 1 s commonly used for the control cycles of LHPs considering the additional computational time required for processing the data of sensors and actors. Hence, the choice of the controller also depends on the available processor hardware of the developed LHP-based heat transport system. However, the realtime capability of all introduced controllers has been verified by assuring that the respective maximal computational time over all simulation steps is smaller than Tst.

#### Anti-Windup Strategy

For the verification of the proposed AW strategy, the controlled CC temperatures Tcc and the manipulated variables Q˙ cc of all controllers are depicted in Fig. 7.34.

Due to a reduction of the heat load Q˙ ev during the saturation of the control heater, Tcc rises again, and a lower Q˙ cc is sufficient to keep Tcc at Tset. The controller piAW/S leaves the saturation the latest since the output xaw of the AW structure is the largest, and the overshoot is most distinct. Both the controllers with integral action and AW strategy, namely piAW/S, piDF/S, piOF/S, and nSF/C, as well as nMIAC/S leave the saturation of the control heater smoothly without major overshoots.

Figure 7.34: Performance comparison of the proposed CC controllers when leaving the saturation of the control heater (Tsk = −25 ◦C)

#### Integrated State Estimation Methods

In order to analyze the satisfactory performance of nMIAC/S and nSF/C in Fig. 7.33 closer, their integrated state estimation methods are evaluated in this section separately.

In Fig. 7.35, the results of the state prediction for the nonlinear Lyapunov-based controller of nMIAC/S are presented in comparison with the corresponding temperatures of SIM. The two state variables in the first two rows are closely reproduced by the state prediction based on Model S since the two model parameters m˙ and Ulk are optimized by the parameter identification with respect to both measured state variables. As it can already be seen in the validation of Model S in Sec. 7.2.1, a major deviation is visible when comparing the third state variable of Model S, the condenser temperature Tco, with the condenser outlet temperature Tco,o of SIM. As expected, this mapping is indeed imperfect from the model's point of view, but the measured temperature Tco,o comes closest to the effective model temperature Tco, which is the overall temperature of the three subsystems VL, condenser, and LL. Since the other model parameters of Model S are fixed in the OP based on the mapping of Tco and Tco,o in Sec. 7.1.2, a close agreement between all three curves of Fig. 7.35 is achieved in the proximity of the OP at approximately t = 5900 s.

Figure 7.35: Comparison of the predicted states of nMIAC/S with the corresponding temperatures of SIM

The integrated parameter identification based on the measured temperatures Tcc and Tev is responsible for the high agreement between Tcc and Tev of Model S and SIM. The optimized model parameters m˙ and Ulk enable the adaption of Model S to the varying operating conditions. A comparison of the identified parameters of nMIAC/S with the corresponding variables of SIM is presented in Fig. 7.36.

Figure 7.36: Comparison of the identified model parameters of nMIAC/S with the corresponding parameters of SIM

The simulated mass flow rates m˙ l,SIM and m˙ v,SIM show a similar behavior confirming the remarks in Assumption 5.1. Thus, a single mass flow rate m˙ is sufficient to approximate the LHP fluid dynamics in Model S closely. Due to the online identification of m˙ in dependence on the heat load Q˙ ev, the variation of the mass flow rate can be considered in the state prediction based on Model S. Besides the mass flow rate, the CC dynamics are influenced by the heat leak between the CC and the evaporator as part of the heat load. Furthermore, the temperature of the liquid entering the CC is governed by the other disturbance, the sink temperature Tsk. By identifying the thermal conductance Ulk in nMIAC/S, the adaption of the temperature difference between Tcc and Tev to the changing operating conditions including the variation of both disturbances is considered. The energy balance in the CC is set by adapting the heat leak from the evaporator through Ulk to the incoming liquid flow from the condenser subsystem at Tco. In comparison to the fixed U SIM lk of SIM, where the heat leak is fitted to experimental data in dependence on the disturbances, the identified Ulk changes over time but within a reasonable order of magnitude close to the constant U SIM lk . Thus, the deviation between both Ulk is caused by the adaption of the CC energy balance to the measured temperatures besides the adaption to the varying disturbances. In summary, the performance of nMIAC/S is very satisfactory due to the adaption of the controlled CC dynamics based on the parameter identification.

The reason for the spikes of Tcc in Fig. 7.33 and 7.35 at approximately t = 6100 s can be explained using Fig. 7.36. A spike is also visible in m˙ l,SIM at the same time which is traced back to numerical inaccuracies in the balancing solution method of the numerical LHP simulation SIM for the mass flow rates m˙ l,SIM and m˙ v,SIM. This numerical inaccuracy of the mass flow rates reproduces in the simulated temperature Tcc and thus is not related to the action of the controller.

The combined state and parameter estimation of nSF/C is based on an SRUKF for the states Tcc, L2ϕ, and m˙ l as well as an SRUKF for the parameter Rsh, which include the influences of the disturbances on the LHP dynamics through the complex equations of Model C. Due to the considered phase transition in the condenser, an adaption of the CC energy balance is not necessary compared to the nMIAC algorithm. The results of the combined state and parameter estimation of the nSF/C in the closed control loop are depicted in Fig. 7.37.

In the third row, the estimated liquid mass flow rate m˙ l follows the simulated m˙ l of SIM closely. Despite the different modeling approaches of the mass flow in Model C and SIM, a high agreement is achieved. Another good result is obtained for the length L2ϕ of the two-phase condenser region. While a high agreement is present at the lower sink temperatures, the already mentioned deviation at high sink temperatures due to the fixed OP parameters is also the reason for a slightly higher deviation in L2ϕ. The biggest deviation, however, is visible in the controlled CC temperature Tcc. Compared to the estimated Tcc, a small deviation of less than 0.55 K remains because of the discrepancy between the reached temperature equilibria for the same value of the input variable Q˙ cc (see Sec. 7.2.1). The integrated parameter estimation with another SRUKF contributes to the decrease of this discrepancy besides an improved general estimation of the states. As depicted in the fourth row of Fig. 7.37, the agreement between the estimated Rsh and the equivalent resistance of SIM is high. Thus, the influence of the flow dynamics in the evaporator on the states under varying operating conditions are correctly reproduced.

With regard to the CC temperature control in Fig. 7.12, the PI controller of nSF/C takes over the adjustment of the controller output subject to the state deviations in comparison with the adaptive function of nMIAC/S. By using the measured Tcc instead of the estimated Tcc inside the nSF/C, its disturbance rejection is further improved due to the relief of the PI controller.

Figure 7.37: Comparison of the estimated variables of nSF/C with the corresponding variables of SIM

#### Evaporator Control

With the benchmark profile in Fig. 7.32, the evaporator controllers based on a cascade control are evaluated next. The results of the controllers in SIM are presented in Fig. 7.38.

All introduced evaporator controllers show a good disturbance rejection considering the small y-axis range of 0.06 K. Although the controlled Tev of nSFc/C shows the largest deviation from the setpoint temperature Tset,ev at the highest sink temperature Tsk, the performance of nSFc/C improves with falling Tsk resulting in a decreasing control error. The larger control error of nSFc/C at higher Tsk is related to the fact that Model C with fixed model parameters yields the largest model error compared to SIM at these operating conditions due to unconsidered changing heat transfer processes and fluid kinematics. In contrast to nSFc/C, the other cascade controllers based on Model S show increasing over- and undershoots with decreasing Tsk. It is noticeable that piDFc/S and nMIACc/S show a similar curve of the controlled evaporator temperature Tev because of their good CC control characteristics in the inner control loop. Thus, the setpoint temperature Tset dictated by the outer control loop is closely realized by both controllers. In contrast, piAWc/S and piOFc/S have the worst performance under disturbance variation since they do not consider the disturbances explicitly in their control law.

The quantitative analysis of the performance of the evaporator controllers is performed with the metrics in Table 7.14.

Table 7.14: Evaluation of the proposed evaporator controllers based on Model S and Model C

Once again, the cascade control piAWc/S with the state-of-the-art PI controllers achieves the worst overall performance based on the RMSE. For changes of both the heat sink and the heat source, the control error of piAWc/S is the largest. However, its MAD is slightly smaller than the MAD of the other controllers due to a better reaction to the dropping dissipated heat of the heat source after every operating cycle (see Fig. 7.32). The best overall result based on the RMSE is achieved by nSFc/C due to the exact state and parameter estimation based on Model C.

### 7.4.2 Compensation of Temperature Oscillations

In this section, the behavior of the introduced controllers for a single LHP with a relevant thermal mass at the evaporator is investigated. Furthermore, the controller performances for a stable and for an unstable thermal mass setting are compared and evaluated.

As elucidated in Sec. 7.2.5, the oscillatory behavior of a single LHP depends on the size of the attached thermal mass at the evaporator. At a thermal mass of Cev,sf = 40 J K−1 , the behavior of Model C and Model M is similar and stable. Thus, all introduced controllers, namely piAW/M, piDF/M, piOF/S, nMIAC/S, nSF/C, and nSF/M, are able to stabilize the CC temperature Tcc and keep the desired setpoint temperature Tset as depicted in Fig. 7.39.

The largest deviations from Tset occur for piAW/M while the smallest control errors are achieved by the nonlinear PI state feedback controllers nSF/C and nSF/M. Both controllers benefit from the good agreement between Model C and Model M for the exact input-output linearization.

With increasing thermal mass, the LHP temperatures start to oscillate until the stability limit is reached according to (7.16). As simulations of the proposed control approaches show, not all of the controllers are able to stabilize Tcc and compensate for the temperature oscillations despite their adaption to the increasing thermal mass. With decreasing sink temperature Tsk, the amplitudes of the oscillations increase until the control loops become unstable. Furthermore, the stability limit is reached with piAW/M, piDF/M, and nSF/C for lower Cev,sf than with piOF/S. Only nMIAC/S and nSF/M enable a stable temperature control due to their accurate adaption to the disturbance impacts. The successful performances of both controllers at Cev,sf = 280 J K−1 above the calculated stability limit of Model M is presented in Fig. 7.40.

Figure 7.40: Disturbance behavior of the oscillation-compensating CC controllers for a single LHP with Cev,sf = 280 J K−1 with respect to the benchmark profile in Fig. 7.32

For a stable state estimation of nSF/M with an SRUKF based on Model M, the sampling time must be reduced to Tst = 0.1 s in the simulations compared to the SRUKF based on Model C with Tst = 1 s. This leads to a slightly increased computational time but is still within the requirements of the LHP control cycle. However, without considering the thermal mass in the nonlinear PI state feedback control, as it is the case with nSF/C based on Model C, a stable temperature control without oscillations cannot be achieved. While nMIAC/S achieves the desired compensating performance due to its parameter identification, although it is based on Model S, nSF/M rebuilds the modulation of the net heat load physically with the attached

thermal network of the evaporator. Thus, the disturbance rejection of nSF/M yields clearly smaller over- and undershoots compared to nMIAC/S in Fig. 7.40.

The same stability results as before with the CC controllers apply to the evaporator controllers of a single LHP with attached thermal mass. Thus, only the respective cascade controllers based on nMIAC/S and nSF/M are able to compensate the temperature oscillations. The performance of nSFc/M is verified as well in simulations and compared to nMIACc/S. Both controllers are parameterized based on Model M as given in Table 7.9. The results of the simulations are presented in Fig. 7.41.

Figure 7.41: Disturbance behavior of the oscillation-compensating evaporator controllers for a single LHP with Cev,sf = 280 J K−1 with respect to the benchmark profile in Fig. 7.32

Over the operating range of the LHP, both controllers nMIACc/S and nSFc/M keep the controlled evaporator temperature Tev in a narrow corridor around the setpoint temperature Tset,ev. Although both controllers are based on different evaporator models within Model S and Model M, a stable control loop is achieved in both cases due to the parameter adaption and the physical foundation, respectively.

### 7.4.3 Temperature Control of Parallel Loop Heat Pipes

For the verification of the performances of piAW/P and nSF/P, the test profile in Fig. 7.42 is used. The test profile considers the increased heat transport capability of the parallel LHPs by means of an increased dissipated heat of the heat source compared to Fig. 7.32.

Figure 7.42: Test Profile 3 with variations in the disturbances across the operating range of LHP 3

The corresponding comparison of the performances of both controllers is depicted in Fig. 7.43. To set the focus on the controllers, the state input of nSF/P is connected with the state feedback of the system. For the realization of nSF/P, the state estimation based on the SRUKF in Sec. 4.2.3 for a single LHP can be extended accordingly.

The controlled CC temperatures T a cc and T b cc of both LHPs are given in the first and the second row of Fig. 7.43, respectively. Both controllers stabilize the system and keep T a cc and T b cc near the setpoint temperature Tset. A smaller deviation from Tset is achieved by the introduced nSF/P compared to the state-of-the-art PI controllers of piAW/P. During the sink temperature change, a larger deviation between Tset and Tcc results for piAW/P than for nSF/P. The deviations caused by heat load changes turn out to be smaller than in the case of a single LHP due to the low disturbance dynamics based on the increased delay of the thermal mass of the LCT dummy (cf. Table 7.1). As a consequence of the same parameterization of the controllers for LHP a and LHP b, the heat transport performance of both LHPs is comparable and thus leads to similar Tcc curves during the modeled balanced heat distribution.

Figure 7.43: Disturbance behavior of the proposed CC controllers for parallel LHPs based on Test Profile 3 in Fig. 7.42

### 7.4.4 Numerical Stability Analysis of the Control Loops Based on the Complex Model

In this section, the numerical results of the stability analysis of the control loops with the controllers based on the complex models, i.e. nSF/C, nSF/M, and nSF/P, are presented. While the other proposed controllers based on Model S are stable due to their individual control designs and parameterizations, the stability of the internal dynamics of Model C, Model M, and Model P must be evaluated specifically according to Sec. 4.2.1, 6.2.1 and 6.2.2, respectively.

#### Model C

For the stability of the zero dynamics of Model C in dependence on the OP, its Jacobian (4.114) is evaluated according to Theorem 4.1. In order to consider the multiple equilibrium points of the LHP, (4.114) is analyzed for a number of sample OPs which are determined at equilibrium over the entire operating range. Furthermore, the external state variable x op ex is fixed in the setpoint temperature Tset according to (7.17) in order to calculate the different internal state vectors x op in in the balanced OP in dependence on the varying disturbance vector d op. The numerical solution of the stationary state equations (7.2) to (7.4) with the model parameters of SIM in Table 7.2 is obtained by minimizing the squared sum of the equations for x op in and the input variable u op with the direct search method applied in Sec. 5.2.4. For the analysis of the eigenvalues of the resulting Jacobians, the Hurwitz criterion in Appendix A.2 is applied which refers to the characteristic polynomial of a Jacobian. Since the first condition in Theorem A.1 is a necessary and sufficient condition for the second-order characteristic polynomial (4.115), the corresponding coefficients γ0 and γ1 must be positive for stable eigenvalues and thus stable OPs of the zero dynamics. The calculated coefficients in dependence on the determined OPs are depicted in Fig. 7.44 as surface plots above d op .

Figure 7.44: Surface plots of the coefficients γ0 and γ1 of the characteristic polynomial (4.115) of the zero dynamics of Model C in different OPs within the LHP operating range at x op ex = Tset

The slopes of the surfaces in Fig. 7.44 deliver insight into the dynamics of a controlled LHP. The highest values of the coefficients are reached for the highest heat load Q˙ ev and the highest sink temperature Tsk. With falling Q˙ ev as well as falling Tsk, the values of the coefficients decrease. As coefficients of the characteristic polynomial of the zero dynamics, the absolute values of the eigenvalues become smaller as the coefficients decrease. Hence, the zero dynamics become slower. Transferred to the overall dynamics of the LHP, the lower both disturbances the slower the system reacts to input changes. This explains the decreasing performance of the controllers with falling Tsk resulting in increasing over- and undershoots, e.g. from the left to the right of Fig. 7.33.

Considering the exponents of the z-axes in Fig. 7.44, both surfaces are situated above zero and do not cross the xy-plane. Therefore, the coefficients γ0 and γ1 in (4.115) are positive over the LHP operating range. This means that the eigenvalues of the Jacobian (4.114) are negative, and thus, the OPs of the zero dynamics of Model C are locally exponentially stable according to Theorems 4.1 and A.1, i.e. nSF/C stabilizes Model C over the entire operating range.

#### Model M

Due to the extension of the state vector x of Model M in (6.27) by Tev,sf compared to Model C, the Jacobian of the zero dynamics of Model M in Theorem 4.1 is four-dimensional. Hence, the corresponding characteristic polynomial is three-dimensional, and both the coefficients γ0 to γ2, since γ3 = 1, and the minors M1 to M3 has to be evaluated according to the previously described procedure. The resulting surface plots of the calculated coefficients and minors are presented for Cev,sf = 280 J K−1 in Fig. 7.45 and 7.46.

As depicted in both figures, the slopes of the surfaces are similar to the ones in Fig. 7.44, since the fundamental behavior of Model C and Model M is similar as well. They also explain the varying controller performances in Fig. 7.39 as mentioned before. All surfaces are fully situated again above zero, and the coefficients and minors are positive. Hence, the OPs of the zero dynamics of Model M are locally exponentially stable according to Theorems 4.1 and A.1, i.e. nSF/M stabilizes Model M over the entire operating range of the running LHP.

Figure 7.45: Surface plots of the coefficients γ0, γ1, γ2 of the characteristic polynomial of the zero dynamics of Model M in different OPs within the LHP operating range at x op ex = Tset

Figure 7.46: Surface plots of the minors M1, M2, and M3 of the characteristic polynomial of the zero dynamics of Model M in different OPs within the LHP operating range at x op ex = Tset

#### Model P

The stability of the OP of the zero dynamics of Model P in dependence on the disturbances further increases the complexity of the calculation according to the previous procedure due to the increased dimension of thirteen of the state vector (6.39). While the operating range with the respect to the heat sink stays the same, the range of the dissipated heat of the heat source increases due to the double heat transfer by the parallel LHPs. Thus, Q˙ hs ranges from 40W to 200W. Because of the resulting high number of coefficients and minors and the connected high values, the illustration of all corresponding surface plots is omitted here. Instead, the minima of both are listed in the following, whereby the inequalities of Theorem A.1 can be verified:

Since the listed minima of the coefficients and minors are positive, all their values are positive. Thus, the OPs of the zero dynamics of Model P are locally exponentially stable according to Theorems 4.1 and A.1, i.e. nSF/P stabilizes Model P over the entire operating range of the running LHP.

## 7.5 Discussion

In order to discuss the performance of the models and corresponding controllers in the previous validation, the differences as well as their advantages and disadvantages are highlighted in this section. Finally, answers to the initial research questions in Sec. 2.5 are found based on the discussion.

The benefit of a complex model with and without additional thermal mass and its impact on the corresponding controller performance has been clearly shown in the previous sections. Combining the mass flow dynamics with the thermodynamics of an LHP enables the complex state-space modeling of a correct disturbance behavior of the measured temperatures on a physical basis. The corresponding nonlinear PI state feedback control law is thus able to consider these influences during the control of the CC or evaporator temperature. By simplifying the state-space model around an OP, the linear and nonlinear controllers based on this model are restricted to that OP and their performances vary in other equilibrium points over the entire operating range. Nevertheless, the adaption of the relevant model parameters in the nonlinear state feedback controller of nMIAC related to the mass flows and the heat leak let it reach almost the same performance as being based on the complex model. Thus, both the nMIAC and the nonlinear PI SF are able to compensate oscillations and achieve the best disturbance rejection as verified in Sec. 7.4.2.

For the review of the proposed controllers for a single LHP, the different controller types are assessed qualitatively with respect to five categories. These categories include the implementation effort concerning the complexity of the respective algorithms, the general performance of the controllers based on the relevant disturbance rejection, the computational effort for the target processor dependent on the computational time of the implemented controllers in Table 7.13, the ability to compensate temperature oscillations due to an attached thermal mass (see Sec. 7.4.2), and the number of required sensors of each controller regarding the effort and costs of the application. The grading distinguishes between four grades: ++, +, -, --. These symbols correspond to the rating of each category between low and high or bad and good, respectively. The results of the assessment are recorded in Table 7.15.

Table 7.15: Evaluation table of the introduced controller types for a single LHP

The grades in this evaluation table are the results of the analysis in the previous sections in this chapter. The table facilitates the choice of the correct controller for the specific application by an LHP end user being a tradeoff between the desired performance and hardware requirements. While the PI controller, as commonly used in the literature, has an advantage over the more complex controllers regarding their implementation and computational efforts as well as their sensor requirements, an improved disturbance rejection and oscillation compensation comes along with a higher degree of complexity. A better disturbance rejection of the PI controller with DF is outweighed by a worsening in all other categories. In contrast, the PI OF achieves

this advantage at a similar computational effort with a better sensor requirement since it does not need knowledge about the disturbances. The good performance of the PI controller with DF, however, is based on the good agreement between the stationary and the transient model of SIM. On the test bench, a higher mismatch between model and reality may be expected, as a lower benefit compared to the PI controller appears in the test bench results presented in [GMK+18]. The best performances are achieved by the nMIAC and the nonlinear PI SF. For such a performance, the nMIAC needs one more temperature sensor than the nonlinear PI SF for the CC temperature control. However, the computational effort is higher for the nonlinear PI SF due to the more elaborate state estimation. In contrast to the other controllers, their implementation efforts are clearly increased due to the integrated optimization method and unscented Kalman filter, respectively.

The parallel LHPs are modeled by connecting their evaporators via the thermal network of the supporting structure with a single heat source. The investigated Q˙ cc step responses are closely reproduced by the introduced model. However, the sensibility of such a system with respect to its stability can already be experienced when determining the equilibrium states for the parameterization and numerical stability analysis as well as in the subsequent simulation. Minor changes to the model parameters may already result in infinite solutions. Furthermore, an imbalance between LHP a and LHP b due to the supposed heat distribution promote unstable temperature curves. The impact of the evaporator masses on the stability of the system cannot be verified. Instead, a reconstruction of the heat flow paths with a more detailed thermal network could clarify the situation of the imbalance, which is an application-specific challenge to the LHP end user. For now, the proposed MIMO controller shows the potential of model-based designed advanced controllers for the improved performance of heat transport systems with multiple LHPs.

Following the initial research questions in Sec. 2.5, an answer to each question can now be found:

test the compensation ability of LHP controllers besides their disturbance rejection. As it has been shown in this chapter, the model extension can be transferred to the subsequent model-based control design smoothly.

4. In the end, two proposed controllers have been identified which are able to compensate the temperature oscillations in contrast to the state-of-the-art PI controller. Besides the aforementioned nonlinear state feedback controller based on the extended accurate Model M, the unstable LHP system can be controlled by the adaptive controller based on the simplified Model S, which both enable the smooth control of the CC and evaporator temperature.

## 7.6 Conclusion

This chapter has delivered the evaluation of the introduced state-space models and the proposed linear and nonlinear controllers. The dynamical models have been thoroughly validated with measurements from two different single LHPs on a test bench and a numerical LHP simulation confirming the validity and the transferability of the introduced models. The controllers have proven their stable performance and improved disturbance rejection compared to the state-of-the-art controller, and their advantages and disadvantages have been listed. The choice of the correct controller for the specific application is finally a tradeoff between performance and hardware requirement.

The investigation of the Q˙ cc step responses has shown the weaknesses of Model S compared to Model C and Model M. However, Model S has achieved a good model accuracy in certain operating areas sufficiently representing the dynamics of relevant temperatures of an LHP for a subsequent model-based design of simple controllers with low processor requirements as preferred in aerospace systems.

The strongest disturbance rejection of a Model S-based controller is achieved by the proposed nMIAC that adapts the disturbance-related model parameters to the mass and fluid dynamics through the model identification adaptive structure. In contrast, the physical modeling of the disturbance behavior with Model C and Model M, mainly based on incorporating the relevant mass flow dynamics and phase transitions, has enabled a precise reproduction of the LHP temperatures subject to disturbance changes. The corresponding model-based designed controllers have gained even further performance advantages compared to the state-of-the-art controllers.

The correct agreement between Model P and the parallel LHPs of the heat transport system of an LCT has been validated with measured data of another test bench. For a balanced heat distribution, Model P with its simplified thermal network representation of an LCT models the advanced LHP-based heat transport system precisely as it has been shown by the comparison of the control heater steps. Accordingly, the corresponding MIMO controller has shown an improved performance compared to the separate PI controllers of the state of the art.

In the case of parallel LHPs, these temperature oscillations, however, have not been reproducible with Model P. Instead, temperature oscillations during the normal LHP operation of

parallel LHPs are more likely to be caused by the individual heat distribution of the connecting thermal mass of the heat source and not by the specific LHP designs.

Besides the desired model-based control designs, the simulation of the LHP behavior with an attached thermal mass and its impact on the LHP dynamics is now possible with Model M. Thus, the theoretical arguments in [HB12] have been reproduced in simulations for the first time. Furthermore, the simulations have confirmed the cause for temperature oscillations during the normal operation of an LHP when a significant evaporator mass is present. Thus, a quantitative answer to the question when a running LHP starts to oscillate is possible. As previously shown, these temperature oscillations are compensable by two of the proposed controllers in contrast to the common state-of-the-art PI controller.

## 8 Conclusions

The increasing number of temperature-sensitive electronic components in aerospace and terrestrial systems comes along with a need for more effective thermal control systems. LHPs as passive, two-phase heat transport systems are in the focus of the industry for the construction of successful thermal management systems. In addition to ensuring the multiphase heat transfer from the heat source to the heat sink, adequate control methods and algorithms are necessary to keep the cooled objects at optimal operating conditions. Control heaters provide a potent solution with a low risk of failure for the operating temperature (OT) control of LHPs. So far, simple controllers with a heuristic parameterization are commonly used in the literature to control the heater output showing moderate performances with a high design effort at the same time. For a structural approach to the design of control algorithms with improved performance for the control heaters of current and future LHP-based systems, a model-based control design is pursued in this thesis for the first time.

The first complex dynamical model of a conventional LHP is based on a system of differential equations in the desired state-space form. The decomposition of the multiphase system is chosen according to its components. It simplifies the derivation of the required energy balance equations from the respective conservation laws to form the final three-dimensional state-space model with lumped parameters. A major emphasis is set on integrating the mass flow dynamics into the model. The mass flows are connected to the phase transition in the two-phase regions of the LHP and govern the disturbance behavior of the LHP. The resulting nonlinear model is able to reproduce the measured temperatures of an LHP subject to changes in the operating conditions. For the corresponding nonlinear model-based control design, a PI state feedback controller is designed based on the exact input-output linearization that fits to the present model structure. The classical approach to govern the OT is the control of the compensation chamber (CC) temperature. Owing to the complex model, a corresponding cascade control design is now possible to achieve smoother temperature curves of the cooled object. This is achieved by keeping the evaporator temperature, i.e. the temperature closest to the object, near its constant setpoint temperature. The required feedback of the unmeasured states in both controllers is provided by a nonlinear state estimation with a square-root unscented Kalman filter (SRUKF).

Based on the detailed investigation of a single LHP with the complex state-space model, the model-based control design approach is extended to advanced LHP systems including LHPs with attached thermal mass and parallel LHPs. For these complex thermodynamic systems, the introduced model of a single LHP is adapted based on the thermal analogy of electrical networks. Analogously, the state feedback controllers are adjusted to the new models. Besides the consideration of extended state vectors in both cases, the MIMO form of the controller is designed to include the coupled dynamics of parallel LHPs. The modular model-based control

design approach in this thesis provides access to further investigations and improvements of advanced heat transport systems with multiple components or LHPs.

Taking into account the complexity of the previous LHP state-space model and the increased hardware requirements of the corresponding controllers, a simplified second state-space model of a conventional LHP is derived. This model includes less model parameters and fully measurable states. It enables the model-based design of various controllers that consider different LHP-specific extensions to improve the disturbance rejection. The proposed linear controllers include an additional disturbance feedforward control based on a stationary LHP model, which is commonly established for the characterization and design of LHPs. Furthermore, an output feedback control based on the fully measurable states of the linearized model is designed, accordingly. In addition to the linear controllers, a nonlinear model identification adaptive control is proposed based on the simplified model. This approach directly includes the influence of the disturbances in the controller by optimizing the related model parameters with regard to the measured temperatures. For the alternative control of the evaporator temperature, all introduced controller types are extended to cascade controllers. Thus, a range of controllers with different hardware requirements is made available for single LHPs to address the various constraints of power demand and mass in aerospace and terrestrial applications.

The practical relevance and transferability of the proposed models is verified with measurement data from different LHP designs on various test benches. The improved performance of the controllers compared to the state of the art is verified in simulations showing the tradeoff between complexity and performance. The stability of the multiple operating points (OPs) of the controlled nonlinear systems is proven numerically based on Lyapunov's indirect method. A critical stability factor of single LHPs is the attached thermal mass at the evaporator that can have a negative impact on the controller performance. The simulated temperature oscillations of single LHPs are consistent with the theoretical findings in the literature. In contrast to the state-of-the-art PI controller, two of the proposed controllers are able to compensate these oscillations and stabilize the LHP in the desired OP. The same type of oscillation cannot be reproduced in parallel LHPs. However, the heat distribution between both LHPs is more likely to cause instabilities which can only be considered by adequate measurements of the heat flows and a corresponding detailing of the thermal network of the laser communication terminal (LCT) structure.

In summary, this thesis contributes to the model-based control design for LHP-based heat transport systems. Besides the detailed consideration of single LHPs, the procedure is also adapted to the control design of parallel LHPs in advanced LHP systems. The introduced models are able to simulate the significant dynamics of an LHP. Furthermore, the proposed controllers are successfully applied in simulations showing an improved performance and accuracy. This thesis can be seen as a foundation for the dynamical modeling and model-based control design for future LHP-based heat transport systems. Its results may be a guideline to update current control parameters, improve control designs, and efficiently adapt controllers to new applications.

## A Appendix

In this appendix, the physical properties of the considered working fluid inside the LHP and their mathematical approximations are stated, which are used throughout this thesis. Furthermore, the algorithms of the state and the parameter estimation based on unscented Kalman filtering are described in detail, before the implemented startup strategy of a controlled LHP is shortly presented.

## A.1 Physical Properties of the Working Fluid

The working fluid of the considered LHP is ammonia. The corresponding values of the physical properties are taken from the tables in [WKSK10] with respect to the reference temperature Tref = 0 ◦C. In the following, all temperatures are expressed in ◦C.

The density ρ l of the saturated liquid is approximated with the polynomial

$$
\rho^l(T) = -4 \cdot 10^{-5} \cdot T^3 - 0.0027 \cdot T^2 - 1.3522 \cdot T + 638.57.\tag{A.1}
$$

The density ρ v of the saturated vapor is approximated with the polynomial

$$
\rho^v(T) = 1 \cdot 10^{-5} \cdot T^3 - 0.0017 \cdot T^2 + 0.1229 \cdot T + 3.4553. \tag{A.2}
$$

The specific heat capacities are

$$c\_p^l(T) = 5 \cdot 10^{-4} \cdot T^3 + 3 \cdot T^2 + 5.6 \cdot T + 4616.5 \tag{A.3}$$

for the saturated liquid and

$$c\_p^v(T) = 0.1 \cdot T^2 + 15.1 \cdot T + 2680.8\tag{A.4}$$

for the saturated vapor.

The heat of evaporation ∆h v is given by

$$
\Delta h^{v}(T) = -3 \cdot 10^{-2} \cdot T^{3} - 11.5 \cdot T^{2} - 3572.3 \cdot T + 1262300. \tag{A.5}
$$

The dynamic viscosity µ l of the liquid is approximated by the polynomial

$$
\mu^l(T) = \left(-2 \cdot 10^{-8} \cdot T^5 + 10^{-6} \cdot T^4 - 10^{-4} \cdot T^3\right.
$$

$$
+ 0.0151 \cdot T^2 - 1.8665 \cdot T + 170.1 \right) \cdot 10^{-6} \,. \tag{A.6}
$$

The Antoine parameters are

$$A\_{wf} = 21.633,\tag{A.7}$$

$$B\_{wf} = 2026.1,\tag{A.8}$$

$$C\_{wf} = 235.00.\tag{A.9}$$

The surface tension σ of the saturated liquid is

$$
\sigma(T) = 0.10175 \cdot \left(1 - \frac{T + 273.15}{405.50} \right)^{1.21703} \tag{A.10}
$$

according to [KJS10].

## A.2 Hurwitz Criterion

### Theorem A.1 (Hurwitz criterion [Lun20a, p. 434]) Consider the polynomial p(λ) = γn · λ n + γn−1 · λ n−1 + ... + γ1 · λ + γ0. (A.11)

All zeros of (A.11) have a negative real part if both following conditions hold:

1. All coefficients γi are positive:

$$
\gamma\_i > 0, \quad i = 0, 1, 2, \dots, n. \tag{A.12}
$$

2. The n leading principal minors Mi of the (n × n)-matrix

$$\mathbf{H} = \begin{bmatrix} \gamma\_1 & \gamma\_3 & \gamma\_5 & \gamma\_7 & \dots \\ \gamma\_0 & \gamma\_2 & \gamma\_4 & \gamma\_6 & \dots \\ 0 & \gamma\_1 & \gamma\_3 & \gamma\_5 & \dots \\ 0 & \gamma\_0 & \gamma\_2 & \gamma\_4 & \dots \\ \vdots & \vdots & \vdots & \vdots & \ddots \\ & & & & \gamma\_n \end{bmatrix} \tag{A.13}$$

are positive:

$$M\_i > 0, \quad i = 1, 2, \ldots, n. \tag{A.14}$$

## A.3 Discrete Implementation

The implementation of the continuous controllers on electronic hardware requires a discrete representation of the corresponding equations. By choosing sufficiently high sampling rates compared to the maximal frequency of the signal, both a quasi-continuous reconstruction of the controlled temperature signal and a quasi-continuous control signal are realized. According to the Nyquist-Shannon sampling theorem, the sampling rate must be greater than twice the maximum frequency of the signal (see [Lun20b, p. 430]). Due to the slow temperature processes of the LHP, such a realization is possible without increased hardware requirements. Thus, a simple discretization method is sufficient for the proposed control law. For the discrete representation in this thesis, Euler's backward method is used to perform the transformation between the continuous and the discrete time domain. In the case of the state feedback controllers, the structure of the differential equations of the resulting state variables can lead to increased hardware requirements based on the required state estimation, which makes it necessary to use different discretization methods (see Sec. 4.2.3).

## A.4 Numerical Analysis of the Rank of the Observability Matrix

The rank of the observability matrix Qobs in (4.150) is analyzed for a number of sample values for the states Tcc(t), L2ϕ(t), and m˙ l (t) as well as the inputs Q˙ cc(t), Q˙ ev(t), and Tsk(t) distributed over the entire LHP operating range. Since Qobs is a quadratic matrix, its determinant must be nonzero for Qobs to be regular and thus full rank. Due to the dependence on six parameters, some example values of the calculation are given in Table A.1:

Table A.1: Example values of the determinant of the observability matrix Qobs in (4.150)

## A.5 Square-Root Unscented Kalman Filtering

The algorithm of the square-root unscented Kalman filter (SRUKF) used in this thesis is based on [vdMW01]. The advantage of a direct time update of the matrix square root of the state covariance matrix results from the application of the Cholesky factorization. At the same time, this factorization guarantees the calculation of positive semidefinite state covariances in each time step for unique matrix square roots [vdMW01]. For more information about the algorithm and the underlying linear algebra techniques, the reader is referred to [vdMW01] and [vdM04].

### A.5.1 State Estimation

For the discrete state estimation, a nonlinear system extended with additive process and measurement noise is considered:

$$x(k) = F(x(k-1), u(k-1)) + \nu\_{pn}(k-1),\tag{A.15a}$$

$$\mathbf{y}(k-1) = \mathbf{G}(x(k-1), \mathfrak{u}(k-1)) + \nu\_{mn}(k-1),\tag{A.15b}$$

with the vector functions F and G as well as the process noise νpn(k) and the measurement noise νmn(k) being Gaussian. The estimated state vector xˆ(k) is initialized by the expectation E{·} of the initial state vector x0:

$$
\hat{x}\_0 = \mathbb{E}\left\{x\_0\right\}.\tag{A.16}
$$

Accordingly, the initialization of the Cholesky factor S(k) is done via the Cholesky factorization of the initial state covariance matrix denoted by the function chol() (see also [Mat20c, p. 1-1489]):

$$\mathbf{S}\_0 = \text{chol}\left(\mathbb{E}\left\{ \left( \mathbf{z}\_0 - \hat{\mathbf{x}}\_0 \right) \cdot \left( \mathbf{z}\_0 - \hat{\mathbf{x}}\_0 \right)^\top \right\} \right). \tag{A.17}$$

For the calculation of the sigma points x(k), the previous state estimates xˆ(k−1) are weighted with S(k − 1):

$$\mathbf{x}(k-1) = \begin{bmatrix} \hat{\mathbf{x}}(k-1) & \hat{\mathbf{x}}(k-1) + \eta\_{ukf} \cdot \mathbf{S}(k-1) & \hat{\mathbf{x}}(k-1) - \eta\_{ukf} \cdot \mathbf{S}(k-1) \end{bmatrix},\tag{A.18}$$

with the scaling parameters ηukf and λukf :

$$
\eta\_{ukf} = \sqrt{n + \lambda\_{ukf}},\tag{A.19}
$$

$$
\lambda\_{ukf} = n \cdot \left(\alpha\_{ukf}^2 - 1\right). \tag{A.20}
$$

The spread of the sigma points (A.18) is set by the SRUKF parameter αukf , for which usually holds [vdMW01]:

$$10^{-4} \le \alpha\_{ukf} \le 1.\tag{A.21}$$

The next step is the transformation of the sigma points x(k) with the nonlinear function F in (A.15a):

$$\mathbf{x}(k) = F\left(\mathbf{x}(k-1), \mathbf{u}(k-1)\right). \tag{A.22}$$

By weighting the transformed x(k), the updated state vector xˆ −(k) is determined:

$$
\hat{x}^-(k) = \sum\_{i=0}^{2n} Z\_i^a \cdot \mathbb{x}\_i(k), \tag{A.23}
$$

with Z a being the weight of a sigma point. The first weight Z a 0 in (A.23) is given by

$$Z\_0^a = \frac{\lambda\_{ukf}}{n + \lambda\_{ukf}},\tag{A.24}$$

while the rest of the weights are calculated by

$$Z\_i^a = \frac{1}{2\left(n + \lambda\_{ukf}\right)}, \quad i = 1, \ldots, 2n. \tag{A.25}$$

To propagate S(k) in time, the QR decomposition (see also [Mat20c, p. 1-10168]) is first used to calculate the updated Cholesky factor S −(k):

$$\mathbf{S}^{-}(k) = \text{qr}\left( \left[ \sqrt{Z\_1^b} \cdot \left( \mathbf{x}\_{1:2n}(k) - \hat{\mathbf{x}}^{-}(k) \right) \quad \sqrt{R\_{pn}} \right] \right), \tag{A.26}$$

with the process noise covariance matrix Rpn and the weights Z b , where

$$Z\_1^b = Z\_1^a. \tag{A.27}$$

The compound matrix in (A.26) consists of the weighted state error of the sigma points and the matrix square root of Rpn [vdMW01]. Second, consecutive updates with the transition error denoted by the function cholupdate() are applied to the upper triangular part of S −(k) that has the necessary square matrix form (see also [Mat20c, p. 1-1499]):

$$\mathbf{S}^{-}(k) = \text{chulupdate}\left(\mathbf{S}^{-}(k), \mathbf{x}\_{0}(k) - \hat{\mathbf{x}}^{-}(k), \sqrt{Z\_{0}^{b}}\right),\tag{A.28}$$

with

$$Z\_0^b = \frac{\lambda\_{ukf}}{n + \lambda\_{ukf}} + \left(1 - \alpha\_{ukf}^2 + \beta\_{ukf}\right). \tag{A.29}$$

The positive SRUKF parameter βukf is chosen to βukf = 2 assuming Gaussian distribution of the states x(k) [vdM04, p. 56]. Depending on the sign of Z b 0 , either an update or a downdate is performed in (A.28) with the absolute value of Z b 0 (see [QSWL18]).

For the prediction of the system's outputs, the sigma points x(k) are also transformed with the nonlinear function G of (A.15b) to form the sigma points y(k) of the output:

$$\mathbf{y}(k) = \mathbf{G}\left(\mathbf{x}(k-1), \mathbf{u}(k-1)\right). \tag{A.30}$$

Accordingly, the updated output vector yˆ −(k) is given by

$$\hat{\mathbf{y}}^{-}(k) = \sum\_{i=0}^{2n} Z\_i^a \cdot \mathbf{y}\_i(k). \tag{A.31}$$

Thus, the correction of the state estimates with the measurements follows in the filter process. Similar to (A.26) and (A.28), the Cholesky factor Sy(k) is calculated and updated subsequently:

$$\mathbf{S}\_{y}^{-}(k) = \text{qr}\left( \left[ \sqrt{Z\_{1}^{b}} \cdot \left( \mathbf{y}\_{1:2n}(k) - \hat{\mathbf{y}}^{-}(k) \right) \quad \sqrt{R\_{mn}} \right] \right), \tag{A.32}$$

$$\mathcal{S}\_y^-(k) = \text{chulupdate}\left(\mathcal{S}\_y^-(k), \chi\_0(k) - \hat{\mathfrak{y}}^-(k), \sqrt{Z\_0^b}\right),\tag{A.33}$$

with Rmn being the measurement noise covariance matrix. The cross-covariance matrix P xy(k) is then calculated by

$$\mathbf{P}\_{xy}(k) = \sum\_{i=0}^{2n} Z\_i^b \cdot \left( \mathbf{x}\_i(k) - \hat{\mathbf{x}}^-(k) \right) \cdot \left( \mathbf{y}\_i(k) - \hat{\mathbf{y}}^-(k) \right)^\top. \tag{A.34}$$

With (A.34), the Kalman gain matrix Kukf (k) is derived:

$$\mathbf{K}\_{ukf}(k) = \left(\mathbf{P}\_{x,y}(k) / \mathbf{S}\_y^-(k)^\top\right) / \mathbf{S}\_y^-(k),\tag{A.35}$$

where the nested divisions are efficiently solved by a QR decomposition with pivoting [vdMW01]. Thus, the state estimates xˆ(k) result from the correction of xˆ −(k):

$$
\hat{\mathbf{x}}(k) = \hat{\mathbf{x}}^{-}(k) + \mathbf{K}\_{ukf}(k) \cdot \left(\mathbf{y}(k) - \hat{\mathbf{y}}^{-}(k)\right). \tag{A.36}
$$

Finally, another time update is required for the correction of the Cholesky factor S(k):

$$\mathbf{S}(k) = \text{cholupdate}\left(\mathbf{S}^{-}(k), \mathbf{K}\_{ukf}(k) \cdot \mathbf{S}\_{y}^{-}(k), -1\right). \tag{A.37}$$

### A.5.2 Parameter Estimation

The discrete parameter estimation with an SRUKF is directly derived from Sec. A.5.1 by describing the nonlinear output equation (A.15b) in dependence on the parameter vector p to be estimated:

$$y(k) = H(x(k), \mathbf{p}),\tag{A.38}$$

with the nonlinear function H. Thus, a new state-space model can be established for the stationary process of p(k) [vdMW01]:

$$\mathbf{p}(k) = \mathbf{p}(k-1) + \nu\_{ppn}(k-1),\tag{A.39a}$$

$$\mathbf{y}\_d(k-1) = \mathbf{H}(\mathbf{z}(k-1), \mathbf{p}(k-1)) + \mathbf{e}\_{ukf}(k-1),\tag{A.39b}$$

with the parameter process noise νppn(k) and the residual output error eukf (k) between the desired output vector yd (k) and the output of H. The convergence and tracking performance of the parameter estimation is determined by the chosen variances in νppn(k) [vdMW01].

Due to the stationary state equation (A.39a), the time update of the state covariance is much simpler. As a result, the prediction part of the SRUKF algorithm for the parameter estimation is shorter than in the previous section. After the initialization of the estimated parameter vector pˆ(k) and its corresponding Cholesky factor Sp(k) in accordance with the approaches in (A.16) and (A.17), the updated parameter vector pˆ −(k) is directly given by the previous estimated parameter vector pˆ(k − 1):

$$
\hat{\mathbf{p}}^{-}(k) = \hat{\mathbf{p}}(k-1). \tag{A.40}
$$

The updated Cholesky factor S − p (k) is calculated by weighting past data exponentially [vdMW01]:

$$\mathbf{S}\_p^-(k) = \gamma\_{ukf}^{-\frac{1}{2}} \cdot \mathbf{S}\_p(k-1),\tag{A.41}$$

with the weighting factor γukf < 1. Thus, the sigma points p(k) are determined by

$$\mathfrak{p}(k) = \begin{bmatrix} \hat{\mathbf{p}}^{-}(k) & \hat{\mathbf{p}}^{-}(k) + \eta\_{ukf} \cdot \mathbf{S}\_{p}^{-}(k) & \hat{\mathbf{p}}^{-}(k) - \eta\_{ukf} \cdot \mathbf{S}\_{p}^{-}(k) \end{bmatrix}. \tag{A.42}$$

Subsequently, the sigma points p(k) are transformed by H similar to (A.30):

$$\mathbf{y}\_d(k) = \mathbf{H}\left(x(k), \mathfrak{p}(k)\right). \tag{A.43}$$

With (A.43), the updated desired output vector yˆ − d (k) is given by

$$
\hat{\mathbf{y}}\_d^-(k) = \sum\_{i=0}^{2n} Z\_i^a \cdot \mathbf{y}\_{d,i}(k). \tag{A.44}
$$

For the correction step with measurements, the parameter estimation follows the same procedure as for the state estimation starting from (A.32). In the procedure, the variables of (A.15) are replaced by their corresponding variables of (A.39), which finally yields the estimated parameter vector pˆ(k) and the Cholesky factor Sp(k).

## A.6 Startup Strategy for Actively Controlled Loop Heat Pipes

When the controller of the control heater of an actively controlled LHP is turned on, a high deviation between the controlled temperature and the desired setpoint temperature may exist dependent on the operating status of the LHP. Based on a high control error, a high control heater output would be dictated by a control algorithm to quickly raise the OT toward its setpoint temperature. As a result, the LHP would shutdown and stop the heat transfer from the heat source to the heat sink (see Sec. 3.2.2). In order to achieve a gentle setpoint approach, an adequate startup process for the controller of the control heater is required.

For the discrete OT control of an LHP, the actual setpoint temperature Tset,act for the controller is initialized at the controlled temperature T and increases continuously with a fixed value, until it reaches the desired setpoint temperature Tset. The fixed value is calculated by the maximum setpoint rate Kr and the sampling time Tst. The variable Kr is chosen in accordance with the permitted maximum temperature gradient of the LHP, e.g. the experimentally determined operation limit of ∆Tcc/∆t = 0.07 K s−1 in this thesis. The setpoint ramp allows the LHP to reach its desired OP smoothly without interrupting its operation. The relevant controller startup is tracked by the boolean variable reset. The overall startup strategy is implemented as follows:

Algorithm 1 Controller startup

1: if Tset − Tset,act > Tst · Kr and reset is true then 2: Tset,act ← Tst · Kr + Tset,act 3: else 4: reset ← false 5: Tset,act ← Tset 6: end if

## References

## Public References

[ZSC+08] W. Zimbeck, G. Slavik, J. Cennamo, S. Kang, J. Yun, and E. Kroliczek. Loop heat pipe technology for cooling computer servers. In 11th Intersociety Conference on Thermal and Thermomechanical Phenomena in Electronic Systems. IEEE, 2008.

## Own Publications and Conference Contributions

## Supervised Theses

[Ge21] F. Ge. Model predictive control of loop heat pipes in satellites. Master's thesis, Department of Electrical Engineering and Information Technology, Karlsruhe Institute of Technology (KIT), 2021.

### **Karlsruher Beiträge zur Regelungs- und Steuerungstechnik (ISSN 2511-6312) Institut für Regelungs- und Steuerungssysteme**

INSTITUT FÜR REGELUNGS- UND STEUERUNGSSYSTEME 20

Effective heat transport systems in aerospace and terrestrial applications are based on multiphase loop heat pipes (LHPs). For a precise thermal control of the integrated electronics, electrical heaters are additionally used to control the operating temperature of the LHP. Current state-of-the-art controllers for the control heaters, however, achieve moderate control results and are unable to compensate nonlinear phenomena like temperature oscillations. Furthermore, these controllers reach their limits in advanced heat transport systems with multiple LHPs or components. In order to eliminate these shortcomings, this work focusses on the dynamical modeling and model-based control design for LHP-based heat transport systems. Based on the one-dimensional, two-phase heat and mass transfer, a nonlinear state-space model of a conventional LHP is introduced which covers the fundamental dynamic behavior of the measured temperatures to input and disturbance changes. Nonlinear state feedback controllers are designed based on the model with proven stability and accurate control results. To consider the strict requirements in aerospace and terrestrial applications, simpler controllers, such as the state-of-the-art PI controller, are designed based on a simplified model with less model parameters. For the modeling of LHPs with attached thermal mass or parallel LHPs, the complex model is modularly extended, and the proposed controller is adapted, accordingly. Thus, undesired temperature oscillations during the continuous operation of an LHP with attached thermal mass are reproduced and compensated by the corresponding controller. The introduced models are thoroughly validated with measurements of three different LHP systems on two different test benches. A numerical LHP simulation is used to verify the behavior of internal variables and the performance of the proposed LHP controllers. The results of this work can be used for the optimization of current control parameters, the improvement of control designs, and the efficient adaption of controllers to future LHP applications.

ISSN 2511-6312 ISBN 978-3-7315-1231-8


*2.2. Study Design, Treatments, and Endpoints*

This single-arm-multicenter study involved daily oral administration of osimertinib (80 mg/day). Osimertinib had to be started at 80 mg/day, and if adverse events (AEs) occurred, dose reduction was performed according to the dose reduction criteria. Administration of osimertinib was continued until the patient met the discontinuation criteria or disease progression. Tumor assessments were performed at baseline, every 6 weeks (± 2 weeks) for 6 months, and then every 9 weeks (± 2 weeks) until disease progression. Baseline brain imaging was performed on a similar schedule. Among patients with T790M mutations, the objective response rate (ORR) was 62% (95% confidence interval [CI]: 54–68) in the AURA extension study (201 patients). In the AURA2 study (210 patients) the ORR was 70% (95% CI: 64–77) and the median PFS was 9.9 months (95% CI: 9.5–12.3) [21–23]. Docetaxel is the standard treatment for elderly patients based on the Japanese guidelines, as it provided an ORR of 22.7% in a study that compared docetaxel to vinorelbine [24]. Another recent study evaluated carboplatin plus pemetrexed for elderly Japanese patients and revealed an ORR of 41.2% [25]. Based on these findings, a required sample size of 31 patients was calculated according to the normal approximation method, with an expected response rate of 60%, a threshold response rate of 35%, two-sided alpha = 0.05, and 1 – beta = 0.8. However, the target sample size was increased to 35 patients to account for potential dropout cases. The primary endpoint for the trial was the overall response rate (ORR), while the secondary endpoints were PFS, OS, disease control rate (DCR), and safety events.


**12. Fracture resistance**

*Oral Diseases*

in the root canals at 2 months.

endodontic re-interventions.

**11. Endodontic reintervention**

completely removed from the root canal system.

Shokouhinejad et al. [45] evaluated the bond strength of EndoSequence BC endodontic cement (Brasseler USA, Savannah, GA) when used with gutta-percha in the presence or absence of moisture within root canals. The mean bond strength of EndoSequence BC sealant and filler in the wet channels was significantly higher than that of the dry 1 week. In contrast, there was no significant difference between dry and wet root canals at 2 months. In the dry channels, the adhesion strength increased significantly over time, while in the wet, the difference was not significant. The presence of moisture inside the root canals increased the bond strength of EndoSequence BC cement in 1 week. However, no difference was found between the bond strength of EndoSequence BC cement in the presence or absence of moisture

According to Oltra et al. [16], recently new bioceramic sealant cements have been marketed and are being used in endodontic practice. However, these bioceramic cements have limited research related to their removal ability during

Uzunoglu et al. [46] evaluated the removal capacity of three different endodontic cements iRoot SP (bioceramic cement), MTA Fillapex (sealant based on MTA), and AH-26 (epoxy resin) from the root canal system. Channel filler was removed with ProTaper Universal Retreatment PTR. The time to reach the working length has been recorded. The roots were sectioned longitudinally and each half was evaluated using a stereomicroscope. Three observers scored every third of all specimens. In the GP/MTA Fillapex single cone group, the time required to reach working length was significantly shorter. The remaining filler material in the apical and middle thirds of the groups was similar. None of the tested cements can be

Oltra et al. [16] analyzed the ability to remove two BC sealer endodontic sealants compared to AH Plus using microcomputer tomographic analysis. Computed tomography was performed before and after obturation and retreatment and then analyzed for residual material volume. The specimens were sectioned longitudinally and the digitized images obtained with the microscope. In the present experiment, significant differences were found, since less root canal filling material was associated with the AH Plus group when using chloroform as a solvent when compared with the others. BC Sealer samples represented using chloroform as the solvent had better results than those removed without chloroform. The results of the present experiment demonstrate that the BC Sealer group presented significantly more residual obturator material than the AH Plus group, regardless of whether the two

Zuolo et al. [17] evaluated the effectiveness of the TRUShape system (Dentsply

Tulsa Dental Specialties, Tulsa, OK) compared to Reciproc (VDW, Munich, Germany) in unblocking channels filled with two different sealants and the working time required to achieve working length. A tomographic microcomputer was used to evaluate the removal of obturation material. The average volume of remaining obturator material was similar when comparing the two file systems. However, in the groups filled with bioceramic, the percentage of remaining obturator material was higher than in the groups filled with Pulp Canal Sealer. The clearance was faster in the groups that were filled with Pulp Canal Sealer when compared with bioceramics. There was no difference in the percentage of remaining shutter material when comparing file systems. However, Reciproc was faster

cements were associated with the use of solvent for their removal.

**186**

than TRUShape.

It is commonly believed that endodontically treated teeth are more fragile and more prone to fracture than vital teeth [47]. There are several factors that affect the strength of endodontically treated teeth: excessive tooth loss due to caries or trauma, dentin dehydration, access cavity preparation, instrumentation, excessive pressure during root filling, and preparation of intraradicular pins [48]. Reinforcement of the remaining tooth structure after endodontic procedures is one of the main objectives of rehabilitation. It has been suggested that the bioceramic cements may adhere to the dentinal surface of the root canal, strengthening the remaining dental structure, contributing to the long-term success of an endodontically treated tooth [49].

Topçuoğlu et al. [15] analyzed the strength of the values necessary to induce root fracture of teeth filled with three different endodontic sealants. Each specimen was then subjected to fracture testing using a universal testing machine at a speed of 1.0 mm/min (−1) until the root was fractured. The force required to fracture each specimen was recorded, and the data were statistically analyzed. The fracture values of the groups filled with bioceramic and gutta percha, and sealant based on epoxy resin and gutta percha, were significantly higher than those of group MTA and gutta percha. No significant differences were found between the bioceramic and epoxy resin groups, the Endosequence BC and AH Plus cements increased the strength of the values required to induce a root fracture of uniradicular premolars.


*5.2. Considerations Regarding Nonlinear Refraction*

It should be mentioned that both model approaches allowed for estimating the nonlinear refractive index as well, and such like calculations could be compared to the reported experimental data collected in Table 2. This is shown in Figure 10 (excluding the extraordinarily large value from Reference [24]). The predictions of the Sheik–Bahae model (assuming the input data from Table 1) almost coincided with the chosen ordinate scaling and were in reasonable agreement with the experimental data from References [20–23,26]. The ß\_do model delivered results in the infrared that practically coincided with the Sheik–Bahae model predictions up to the TPA threshold. At higher wavenumbers, both approaches delivered divergent results. Note that none of the models was able to reproduce the strongly negative *n*2 value from Reference [19], but the Sheik–Bahae model came at least close to this value. This is not astonishing, because in the ß\_do approach (Equations (3) and (4)), only the TPA resonant contribution to *n*2 is taken into account, while nonresonant contributions are not considered at all.

**Figure 10.** Measured nonlinear refractive indices from different TiO2 modifications compared to simulations in terms of the Sheik–Bahae approach as well as Equations (3) and (4).

The relative stability of the Sheik–Bahae approach with respect to correlated changes in the optical gap and the refractive index makes it difficult to understand density-related differences in the nonlinear response of different TiO2 modifications. Nevertheless, a density dependence of *n*2 and β is physically reasonable and expected. It seems to be an advantage of the ß\_do model that differences in the density are explicitly taken into account in terms of the model parameter *J*; consequently, the modeled *n*2 data for the IBS film is larger than those of the PIAD film. This way, the ß\_do model might provide access to explaining the significant scatter in measured nonlinear constants, as has been reported in the different studies cited. For future practical modeling, it might therefore be useful to merge both models in a suitable way.


**Author details**

Kerimcan Apak Adana Alparslan Türkeş Science and Technology University, Adana, Turkey

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*Morphology of the Vernacular Masonry Structure in Adana City and Its Qualitative Analysis… DOI: http://dx.doi.org/10.5772/intechopen.108692*