On thermal processes in the calibration and control of liquid calorimeters for measuring the energy of high-power microwave pulses

The heat transfer processes associated with the heating of the operating liquid and the calibration of the liquid calorimeters for measuring the energy of high-power microwave pulses are studied analytically and by numerical simulation. It is shown that convection significantly enhances heat transfer between the calibration and thermostabilization heaters and the operating liquid, hence, it helps to reduce the formation of bubbles in the volume of the operating liquid when controlling and calibrating the calorimeter. Heat transfer from tape heaters is noticeably more efficient than that from single wire ones. The results obtained are essential for the development of liquid calorimeters and the adjustment of their operating modes.


Introduction
The most reliable method for determining an amplitude of power of high-power microwave pulses [1] is associated with a simultaneous measurement of their energy and envelope shape (power versus time). To measure the pulse energy, liquid calorimeters are successfully used [2][3][4][5][6][7][8][9], in which the measured microwave energy is determined by an increase in the volume of the working liquid (most often ethyl alcohol) due to its heating upon absorption of a microwave pulse. In this case, microwave detectors can be used to measure the time dependence of the microwave radiation power [10][11]. To control the position of the working liquid boundary in the measuring tube of such calorimeters, initial heating of the working liquid and its subsequent heating during measurements are required. For this, there is a heating metal resistor (heater) in the volume of the working liquid, through which current is passed from a special source. Calibration of such calorimeters is performed by supplying an electric pulse to another similar heater (calibration heater) from a calibration source. In this case, it is very important that there are no bubbles in the volume of the working liquid that can be formed due to overheating of the working liquid. Thus, the analysis of thermal processes associated with the control and calibration of the liquid calorimeter is essential. This article is devoted to such an analysis.

Analytical estimations
As noted above, the calorimeter is calibrated by applying an electric pulse to the calibration heater from a source of calibration pulses, and the heat transfer processes between the operating liquid and the calibration heater are unsteady. A fairly reliable assessment of the characteristics of the calibration heater, which guarantees the absence of bubbles, can be obtained with the assumption that the energy of the electrical pulse is first rapidly released in the volume of the heater, and then heat exchange occurs between the heater and the operating liquid. The maximum operating temperature of the calorimeter is assumed to be Tmax = 303 K. The boiling point of alcohol is Tb = 351.3 K [12]. Therefore, the maximum allowable increase in the temperature of the calibration heater is equal to T = 48.3 K. Stainless steel of grade 12X18H10T (RU) is suitable as a material to manufacture the thermostabilization and calibration heaters. It is very similar to AISI 321 (US) by the chemical composition and physical properties. It possesses the required chemical resistance against aggressive media. Its physical properties are as follows: specific heat capacity c = 0.462 J/(g·К), density, = 7.9 g / cm 3 , and specific resistivity е = 0.725 Ohm·mm 2 /m [13]. According to previous experience [7][8][9], the resistance of the thermostabilization and calibration heaters, which is optimal for the electronic control system of the calorimeter, is R = e l / S ≈ 10 Ohm, where l is the length of the heater and S is its cross-sectional area. Taking into account the design of the absorbing load of the calorimeter [9], the length of the thermostabilization and calibration heaters can be about 2 m. Thus, one can find the cross-section of the resistor conductor S = e l / R = 0.145 mm 2 . This cross-section corresponds to a wire with a diameter of 0.43 mm or a tape 5 mm wide and 29 m thick. In the latter case, a more efficient heat transfer between the heater and the operating liquid can be expected due to the larger surface area and smaller conductor thickness. The mass of the heater is estimated as m = S l ≈ 2.3 g, and the maximum permitted calibration energy is W = c m T = 51.3 J without taking into account additional heat removal due to convection, when the alcohol does not boil yet.
The design of the thermostabilization and calibration heaters can have a rather complicated form even for numerical simulation of the processes of calorimeter calibration and heating of the operating liquid. However, it is possible to perform simple analytical calculations considering the simplified cylindrical geometry of the heater and liquid volume (figure 1). In this case, to analyze the heating of the operating liquid, it is possible to use the stationary heat conduction equation without taking into account convection to determine the temperature distribution in the liquid. This allows one to get some upper estimation of heating power as a function of the temperature difference between the heater and the absorbing load case sustained at the ambient temperature. From this relationship, it can be seen that the largest temperature change occurs near the heater. An estimate of the power P0 released per unit length of the heater and the total power of the heater P can be obtained by integrating the equation [14]: Let's take the thermostabilization heater similar to the calibration one: made of steel 12X18H10T, length ≈ 2 m, resistance R ≈ 10 Ohm, cross-section S = 0.145 mm 2 . We would consider the cross-section mentioned above: the thin tube with the diameter of 3.18 mm (r1 = 1.59 mm), the perimeter of which is equal to the perimeter of the tape 5 mm wide. Let us put the radius r2 = 23.85 mm. It is the order of the distance from the heater to the housing wall of the absorbing load of the calorimeter. Let us set the temperature of the body of the absorbing load T2 = 298 K, and the temperature of the heater -T1 = 340 K which is below the boiling point of alcohol. We take the thermal conductivity coefficient for a temperature of 323 K, = 0.160 W / (m·K) [15]. Then the power per unit length of the heater is: These estimates show that at the same temperature difference heat transfer for the wire is 1.7 times lower than that for the tape of the same cross-section.

Numerical simulation
Convection of the operating liquid should be considered as a factor additional to its thermal conductivity, significantly increasing heat transfer, and contributing to an additional heat removal from the heater. It results in an increase in the maximum calibration energy for a calibration heater and a decrease in the required temperature of the thermostabilization heater at a given power or an increase in the heat removal power at a given temperature. Heat transfer between the heater and the liquid in our case is described as natural convective heat transfer with laminar flow. An analytical solution to this problem in the case of a large number of heating elements distributed in the calorimeter's operating liquid volume which has a rather complex design is practically impossible.
Computational Fluid Dynamics (CFD) methods are now widely developed to solve such problems numerically. We have two multiphysics computer simulation software packages licensed with CFD modules: ANSYS Fluent [16]  packages were used to solve the problem of heat transfer between heating elements and alcohol as an operating liquid taking into account natural convection.
The test calculations two-dimensional problems were considered with heaters made of wire 0.3 mm in diameter and tape with a cross section of 5 mm × 30 m, which were placed in a tube with the diameter of 50 mm. Thus, the volume of the liquid was significantly greater than the volume of the heater in both cases. The problem geometries are shown in figure 2. a b Figure 2. Scheme of the numerical experiment: (a)with a wire heater with a diameter of 0.3 mm, (b)with a tape heater with a cross section of 5 mm × 30 μm. A is the outer boundary of the liquid volume, B is the vertical axis of symmetry of the problem, C is the heater.
In figure 2a, the center area is zoomed for clarity. An isothermal boundary condition was set on the outer diameter of the liquid (A). The temperature was maintained at T0 = 298 K. Heating elements (C) are located in the center. Free fall acceleration is directed vertically downward. In this formulation of problems, the maximum power of the heating elements per unit length was calculated.
It is known that the use of the Boussinesq approximation [18] is justified when solving problems of natural convection with a little temperature difference. In this case, the system of equations is greatly simplified, which makes it possible to obtain a sufficiently accurate solution with fewer calculations. The vertical symmetry condition of the problem (B) was used to reduce the number of calculations also. "No slip" boundary conditions with the zero velocity of the liquid at the boundary were set at the boundaries of the liquid adjacent to the heater and the outer wall. Ethyl alcohol was considered as the operating liquid, as mentioned above. In nonstationary simulations, steel was used as the heater material. Thermal and mechanical characteristics of the simulated materials were taken from the databases of materials present in each specific software product. It should be noted that quantitatively these characteristics differ little from those values that were used in analytical calculations. At first, non-stationary problems were solved in both packages. We used iterative solvers with an adaptive time step. At each time step, the iterations were repeated until the specified accuracy was achieved, which was set by the relative error in the components of the fluid velocity and the accuracy of compliance with the continuity condition. In the initial state, the entire system was at the temperature T0 = 298 K. A two-second power pulse is generated in the volume of the heater, the amplitude of which was selected in such a way that the temperature T of the liquid on the heater surface would increase by about 40 degrees relative to the initial one. The pulse duration was chosen this way because by the end of the second second the system reaches an almost stationary state, which will allow comparing the obtained data with the stationary solution later. The applied pulse has sharp edges to facilitate the convergence of the solvers. As an example, figure 3 shows the time dependences obtained using ANSYS Fluent: the power generated in the volume of the wire heater, the energy of the supplied pulse, the power integrated over the surface of the heater, the calculated energy introduced into the liquid, and the maximum liquid temperature increase T -T0. It is possible to make sure that the energy conservation condition is satisfied sufficiently accurate using the energy values by the end of the calculation. Figure 4 shows the distributions of the temperature increase relative to the initial one for the wire (1) and tape (2)   Distributions are presented in the form of a three-dimensional surface, in which the XY plane corresponds to the geometry of the problem (the Y axis is directed upward), and the temperature is plotted along Z and additionally has a color scale. These results are shown "as is" from the COMSOL software interface. It can be noted that in the case of a tape heater, the temperature and velocity of the rising up liquid is higher due to the higher power introduced into the liquid.
Stationary problems were formulated as follows. Constant temperature conditions were set on the surfaces of the liquid touching the heater and the outer wall. The temperature of 340 K was set as the maximum temperature at the heater-liquid interface. The temperature of the outer wall was 298 K. In these problems, the calculations were performed only for the volume of the liquid. The task was to determine the power flow taken from the heater by the emerging convective fluid flow at the given temperature difference. The heater material did not matter in this formulation of the problem. The simulation results obtained in both programs showed good qualitative and quantitative agreement. Figure 5 shows temperature distributions for wire and tape heaters simulated in ANSYS Fluent and COMSOL. The temperature color scale in figures a and b, representing the simulation results in ANSYS Fluent, is truncated at the top to increase resolution at the bottom of the scale. Figure 5c shows the simulation result in COMSOL as a 3D surface for the increase of liquid temperature comparing to the wall ambient temperature T -T0. The fluid paths shown as lines in figure 6 are practically indistinguishable from each other for both wire and tape heater. As in the non-stationary problem, the fluid transfer rate represented as color scale in figure 6 is slightly higher in the case of a tape heater. The   The linear powers obtained from the results of stationary and non-stationary problems simulations in the Fluent and COMSOL programs are tabulated in table 1.
The power values for both problems coincide with an accuracy of a few percent. The main reason for the error is the accuracy in the maximum heater temperature obtained during non-stationary simulations. In addition, when simulating in COMSOL, the ethanol model took into account the dependences of density and specific heat on temperature, while these values were constant in ANSYS Fluent. Calculations performed without taking into account the convection show that the linear power for both types of heaters is more than 5 times less. It clearly indicates the need to consider the natural convection as an essential process in the problem of heat transfer between heaters and an operating liquid in liquid calorimeters. a b Figure 6. Results of the stationary problem simulation. Liquid trajectories (lines) and velocity amplitude (color scale) in the case of wire (a) and tape (b) heaters. For a practical use of the simulation results, the dependence of the temperature on the surface of the tape heater with the cross section of 5 mm × 30 m on the input power per unit length of the heater is of interest. This dependence is shown in figure 7. It was obtained by the stationary problem simulations at the temperature on the outer wall (environment) of 298 K. One can shift the curve vertically as a whole to the desired ambient temperature in parallel to have approximated curve for another initial condition. Doing this, one should avoid the exceeding of the boiling point of ethyl alcohol.

Conclusion
Comparison of the results of analytical estimates and the results of numerical simulations shows that natural convection plays a significant role in the processes of heat transfer between the operating liquid and heaters when controlling and calibrating liquid calorimeters. It provides significantly more heat removal from heaters than thermal conductivity. This reduces the risk of bubbles forming in the volume of the operating liquid when calibrating the calorimeter and reduces the power required to heat the liquid when controlling the calorimeter. Tape heaters provide more efficient heat transfer compared to wire heaters.
The results of the calculations performed must be taken into account for the design of heaters for liquid calorimeters and their operating modes.