# Dielectric Heating Calculator

The Dielectric Heating Calculator provides engineers, scientists, and students with an online tool that can used to analyze Microwave Heating of an arbitrary Dielectric Material. This is a 1-D theoretical analysis for heating a Dielectric Material, which involves the use of Temperature-Dependent and Frequency-Dependent Material Properties as well as the Microwave Engineering aspects of the heating problem. Therefore, the solution for Dielectric Heating problems requires a knowledge of Material Science and Microwave Engineering to be successful. A brief tutorial is provided on how to use this calculator.

## The Dielectric Heating Calculator determines the following Microwave Heating parameters for a Dielectric Material:

*1*Calculate the RF Power Dissipation required in a Dielectric Material to obtain a defined Temperature Increase in a specified Time Increment.*2*Calculate the RF Electric Field required to yield a defined Temperature Increase in a specified Time Increment.*3*Calculate the RF Power Depth of Penetration, where the Power decreases to e^{-1}.*4*Calculate the RF Power Dissipation per Unit Volume for a defined Frequency, Electric Field, and Dielectric Properties.

## Dielectric Heating Calculator Tutorial

The Dielectric Heating Calculator provides users with an analytic tool to assess the heating of an arbitrary Dielectric Material with microwave energy. The Calculator is designed as a 4-Step process discussed in this Tutorial. The critical aspect of the 4-Step process is that it requires information to be provided by the user, which defines the Temperature-Dependent and Frequency-Dependent properties of the Dielectric Material. The Temperature-Dependence of the Dielectric Material requires the analysis to be broken into finite increments to permit the analysis to include the Temperature-Dependent material properties over the heating range. Sufficiently small temperature increments must be used in the analysis to minimize the errors in the analysis. Also, the frequency for heating the Dielectric Material will typically change as material properties change with temperature, which is commonly referred to as “frequency drift” for the applicator.

The analytic technique discussed here is based upon the mathematical derivations provided by A.C. Metaxas and R.J. Meredith, ** Industrial Microwave Heating**, Peter Peregrinus, Ltd., 1993. Regrettably, this brief tutorial cannot include a detailed discussion of the derivations provided by Metaxas. The Tutorial merely provides the user with the information necessary to use the Dielectric Heating Calculator provided here. Additional knowledge can be gained by reading the reference written by Metaxas. Finally, the Tutorial uses “water” as the Dielectric Material for this discussion, since Microwave Heating of water is well understood and widely discussed in the open literature.

### First Step – Determine the RF Power Dissipation Required to Enable a Temperature Change in a Defined Time Increment.

The First Step in using the Dielectric Heating Calculator is to calculate the RF Power Dissipation required in the defined Dielectric Material to achieve a specified Temperature Increase for the material in a Defined Time Period. This appears to be an extremely difficult problem to solve, but the equation provided by Metaxas shown below makes this a tractable problem. The results can be iterated by the user to obtain the desired Temperature Increase in a Defined Time Period.

The mass of the Dielectric Material (M_{a}) to be heated is proportional to the RF Power Dissipation required to enable the required change in the temperature of the material. This is similarly true for Specific Heat Capacity (c_{p}) and the required temperature change (ΔT). Material Properties for many typical household and industrial materials can be found in an online database at https://www.matweb.com/. The database also contains Temperature-Dependent material properties, which are an essential aspect of this analysis.

The Temperature-Dependent Specific Heat Capacity (c_{p}) of water is contained here, and can also be found online at https://www.engineeringtoolbox.com/specific-heat-capacity-water-d_660.html. The following graph illustrates how the Specific Heat Capacity of water changes over the 0º to 100º C temperature span.

The Temperature-Dependence for the Specific Heat Capacity is clearly evident from this graph. However, it should be noted that the Temperature-Dependence is minimal over this temperature span. This is material dependent and varies widely between different materials.

Methods for experimentally measuring Specific Heat Capacity for a material are discussed online in the open literature, and contract resources exist for this aspect of the problem. Laboratory equipment can be obtained for measuring Specific Heat Capacity from Mettler Toledo (https://www.mt.com/us/en/home/applications/Application_Browse_Laboratory_Analytics/Application_Browse_thermal_analysis/specific-heat-capacity-measurement.html#:~:text=Specific%20heat%20capacity%20refers%20to,processes%20or%20assess%20thermal%20risk.).

Our example for using the Dielectric Heating Calculator will be for 1 kg of water. The density of water is 997 kg/m^{3}, so the volume of 1 kg of water is 0.001003 m^{3}, which is 1003 cc (33.91546 fluid ounces, 1.003 Liters). This is slightly larger than 1 quart of water in the US.

Now if we want to increase the temperature of 1 kg of water by 1.5º C (20º C to 21.5º C) in 1 second, the Calculator tells us that 6.276 watts must be dissipated in the water to achieve this change in temperature in the defined time period. A larger temperature increase requires more RF power to dissipate in the water in the defined period. The required power dissipation in the water increases as the time period decreases for heating the water.

This calculation provides us with a measure of the microwave power that must be dissipated in the water for a defined time period to achieve the specified temperature increase. The terms RF and Microwave are used interchangeably in this discussion, and RF is commonly used as an abbreviation for Microwave to conserve space.

The Second Step of this analysis uses all of the information provided for the analysis in the First Step, but adds the Frequency-Dependent loss of the material. This permits the Electric Field to be determined to provide the temperature increase for the Dielectric Material in the designed time period.

### Second Step – Determine RF Electric Field Required to Achieve Temperature Increase in Defined Time Period

The Second Step of the Dielectric Heating Calculator determines the RF Electric Field that must be uniformly impressed upon the Dielectric Material to provide the required temperature increase in the designated time period. This must be done in one of the designated Industrial, Scientific, and Medical (ISM) frequency bands, which would commonly include:

- 902 to 928 MHz
- 2.400 to 2.4835 GHz
- 5.725 to 5.875 GHz

Other ISM frequency bands are available, but these are the most commonly used ISM bands in the US. Generally, ISM bands are country-specific, so this frequency assignment for the analysis may vary depending on your location.

The following equation is provided by Metaxas for calculating the Electric Field strength required at a defined frequency (f) to heat the Dielectric Material. The information used in the First Step is also used for this equation in the Second Step.

A new parameter is introduced in the equation shown above, ε”, which defines the imaginary part of the Frequency-Dependent Complex Permittivity for the Dielectric Material to be heated in this part of the analysis. Our analysis shall be conducted at 2.45 GHz, which is the frequency commonly associated with home microwave ovens.

Complex Permittivity is both Frequency-Dependent and Temperature-Dependent for Dielectric Materials, so our analysis must include this aspect. Complex Permittivity can be measured for a Dielectric Material by the user or by a contract service. An invaluable reference for the laboratory techniques available to measure Complex Permittivity is: L.F. Chen, C.K. Ong, C.P. Neo, V.V. Varadan, and V.K. Varadan, * Microwave Electronics: Measurement and Materials Characterization*, Wiley, 2004.

Next, the Temperature-Dependent Complex Permittivity of water was reported on by A. von Hipple, * Dielectric Material and Applications*, Artech House, 1995, p. 361. The Temperature-Dependent Complex Permittivity data must be used for this part of the analysis, and the Temperature-Dependence for the Real (ε’) and Imaginary (ε”) parts can be seen in this graph of the data provided by von Hipple.

Finally, our analysis makes use of the Imaginary part of the Complex Permittivity provided by Metaxas at 2.45 GHz, where ε” = 11.7. This calculation tells us that the 1 kg sample of water must be uniformly immersed in an Electric Field strength of 62.66 V/m to achieve the 1.5º C temperature increase in a 1-second time period.

The Third Step of our analysis looks at whether a 1 kg sample of water can be uniformly immersed in an Electric Field strength of 62.66 V/m. This aspect of the analysis determines the Depth of Penetration for the Microwave Power into the Dielectric Material before the power is attenuated to e^{-1} of its original value. The Depth of Penetration is used to define the optimal size and shape for a Dielectric Material to be heated in a yet to be designed Microwave Applicator.

### Third Step – Determine the Power Depth of Penetration Where the Power Decreases to e^{-1}

The Depth of Penetration into a Dielectric Material is an essential parameter since it is commonly used for optimizing the design of a Microwave Applicator, which is the apparatus used for Microwave Heating the designated Dielectric Material. The Electric Field is attenuated by the loss associated with the Complex Permittivity of the Dielectric Material, so uniformly heating a Dielectric Material is dependent upon the size of the material to be heated. If the material is too big, then cold spots exist inside it when heated. Many of you have experienced this problem when heating food in a microwave oven.

Therefore, the Depth of Penetration into a Dielectric Material is an important parameter to calculate in the Third Step of our analysis. This parameter is Frequency-Dependent, and also changes as the Complex Permittivity of the material changes with temperature. Our prior analysis used a frequency of 2.45 GHz, and we will use the Complex Permittivity for water defined by Metaxas as ε’ = 78 and ε” = 11.7 at room temperature. This yields a Depth of Penetration into the water of 1.47 cm (0.579-inches) using the following equation.

Knowing the Depth of Penetration at a frequency for a Dielectric Material assists with defining the dimensions for the Dielectric Material in a microwave applicator. The microwave applicator is commonly designed to provide uniform heating of the material, and the microwave applicator can be designed to be a batch system, or a continuous system. A batch system is designed to heat a defined volume of material, whereas a continuous system uses a conveyor belt to move material through an applicator, or perhaps the material is moving in a pipeline or in sheet form. As you can see, every microwave heating problem is unique, and a custom microwave applicator is required to heat the material for the designated application.

For instance, the Depth of Penetration into the water at room temperature at 2.45 GHz is 1.47 cm (0.579-inches). An efficient system for heating water at 2.45 GHz may use a 0.25-inch ID tube, thereby providing a pipeline system for heating the fluid.

### Fourth Step – Determine the RF Power Dissipation Per Unit Volume

The Fourth and final Step in this analysis process determines the RF Power Dissipation per Unit Volume for the volume of Dielectric Material to be processed. This is important since it gives the Microwave Applicator design engineer an estimate of the power required to heat a defined volume of material in the Applicator.

We can look at the example for heating 1 kg of water at 2.45 GHz. We have already calculated the required Electric Field (E) strength required to increase the temperature of the water by 1.5º C in 1-second, and we know the imaginary part of the Complex Permittivity (ε”) for water at 2.45 GHz, so Metaxas provides us with the following equation for calculating the RF Power Dissipation per Unit Volume for a Dielectric Material.

When f = 2.45 GHz, ε” = 11.7, E = 62.66 V/m, then P_{v} = 6,260.8 watts/m^{3} as shown in the Dielectric Heating Calculator. We know that the volume of 1 kg of water is 0.001003 m^{3}, so we only require 6.28 watts to be dissipated in the water to achieve a 1.5º C temperature increase in a 1-second time period. The designer can use this power calculation for the load to design the Microwave Applicator while increasing the power required to account for system losses and inefficiencies.

The calculation in the Fourth Step agrees with the calculation previously performed in the First Step, which tends to validate our work with the Dielectric Calculator. However, this is an analytic solution, and a lot of work remains to be performed to successfully design a Microwave Heating System.

The design for the Microwave Applicator can now commence using the knowledge gained from the Microwave Heating Calculator. The design process must include the Temperature-Dependent Complex Permittivity and Specific heat Capacity, which is necessary to calculate the temperature increase for the Dielectric Material. These parameters can be expressed as polynomial equations, or data tables, in microwave heating analysis programs such as Comsol (https://comsol.com/). It is important to realize that the final design yielded from a microwave heating analysis program is strongly dependent upon the accuracy of the Temperature-Dependent material properties provided to the analysis program. Hence, minimize the error in the Microwave Applicator design by thoroughly understanding the material properties! Contact **SiberSci, LLC** to help with your design problem to minimize time, cost, and errors in the project.

## Dielectric Heating Calculator

Finally, **SiberSci, LLC **also provides visitors with access to Electromagnetic Engineering Calculators that are used for designing circuits, or exploring solutions to problems. Three transmission line calculators are provided that include a Coax TEM Mode Calculator, a Rectangular Waveguide TE_{m,n} Calculator, and to complete this set a Rectangular Waveguide TM_{m,n} Calculator. The remaining three Electromagnetic Engineering Calculators are general purpose analysis tools. For instance, the Wavelength Calculator can be used to determine the length of a Monopole Antenna, while the Skin Depth Calculator is commonly used to determine the metal thickness for Microstrip, Stripline, and CPW transmission lines. The Skin Depth Calculator is also used to determine the metal thickness for electroplating cavity assemblies and waveguide assemblies. The final calculator is the Dielectric Heating Calculator, which is used to estimate the frequency-dependent power and electric field to heat a defined volume and mass of dielectric material in a defined time period. If you do not see the calculator that you need for your problem, then Contact Us to determine if we can provide you with this calculator.

## Do you need another Calculator for your design analysis, then chose one from the following list:

- Coax TEM Mode Calculator - https://sibersci.com/coax-tem-mode-calculator/
- Dielectric Heating Calculator - https://sibersci.com/dielectric-heating-calculator/
- Rectangular Waveguide TEm,n Calculator - https://sibersci.com/rectangular-waveguide-temn-calculator/
- Rectangular Waveguide TMm,n Calculator - https://sibersci.com/rectangular-waveguide-tmmn-calculator/
- Skin Depth Calculator - https://sibersci.com/skin-depth-calculator/
- Wavelength Calculator - https://sibersci.com/wavelength-calculator/