Dynamic method for measuring thermal characteristics of heating devices

. The central aim of this work is to find a way to know out the heat capacity, which is an inertial characteristic of a heating device. The thermal lag in heating devices is a factor considered in automated systems of indoor climate control. The suggested method to calculate the heat capacity is based on the differential heat-balance equation for the heating device. According to the balance equation, the method comes down to the comparative analysis of thermal and power characteristics of the heating device at different values of the temperature drop (the difference between the average temperature on the surface of the heater and the average room air temperature). The work provides a description of the experimental equipment that can be used to implement the given method, as well as the technique to conduct experiments. Power and temperature characteristics were taken three times, and the results were averaged. The values of the device's heat capacity were obtained from the averaged characteristics against the six values of the temperature drop. The average heat capacity was calculated and compared with the theoretical value, the deviation was equal to 1.63 %. The method developed offers the advantage of being able to determine the heat capacity of any type of heating device, taking into account their inherent characteristics, including those gained during their lifetime.


Introduction
Heating devices of water heating systems (cast iron and aluminium heaters, convector heaters) vary significantly in both design and performance characteristics.These characteristics include the heat transfer efficiency and the thermal lag.In practice and in experimental studies, researchers focus mainly on the heat transfer efficiency.In particular, they take the heat transfer coefficient, the thermal conductivity or the radiation coefficient as the main thermal characteristic [1][2][3].Such works use the static method of determining thermal characteristics, where it is necessary to measure the thermal power feeding a heating device and the temperature drop (the difference between the average heating device temperature and the room air temperature).
Much less attention is paid to the inertial characteristics of such devices, despite the fact that the thermal lag should be necessarily accounted for in the indoor climate control systems.Obviously, the thermal inertiality (thermal lag) is determined by the heat capacity of devices.The introduction of automated systems of indoor climate control to adjust the heat flux in heating equipment helps solve the issue of energy saving, which is one of the top priority issues at government level in Russia.There are also systems for monitoring thermal conditions in different rooms, which adjust for the thermal lag in heating equipment.One of such systems is based on a set of nonlinear differential balance equations composed for average temperatures of the heating device, room air and inner surface of the room [4].It will be shown forwards that the heat capacity needs to be considered as part of both the dynamic description and the operation of heating systems.
According to the works [5][6][7], the mathematical model of the heater with respect to its thermal lag can be put in a form of a differential heat-balance equation ℎ (1) where ℎ -average heating device (heat source) surface temperature, -average room air temperature, -heat capacity of the heating device, -current time [s], -input power, -thermal conductivity, -rate of the temperature change on the surface of the heater over time.
This equation allows us to derive methods for measuring the basic thermal characteristics C and G, as well as the output thermal power.
There is a well-known method for heat metering [1][2][3] which bases on the Newton-Richmann equation and calculates the thermal power of a heating device by the formula: The thermal conductivity can be determined either from the data sheet for the heating device or in a laboratory setting by using (2) and having the value of the input power, i.e. by the following formula (a static method): In addition, a dynamic method has been developed to determine the thermal conductivity [8].The method allows the characteristics of a heating device to be determined under operating conditions, not only in specialised isothermal chambers or laboratories.The method bases on the differential heat-balance equation 1, where the thermal conductivity can be determined under condition of input power cut-off 0 .Here, the thermal conductivity can be calculated by considering the cooling process of the heater and applying the formula: The heat capacity of the heater comprises that of the heat transfer agent as well as that of the heater body (framing) itself.That is, it would be determined as follows: (5) where -is the heat capacity of the heater body material, J/K; -is the heat capacity of the heat-transfer agent used in the heating device, J/K.
Herewith, the calculation of and comes down to a simple multiplication of the specific heat capacity by the mass of a substance.The mass of the framing and the volume of the heat-transfer agent can be obtained from the data sheet for the heating device.In this case, it should be noted that the data sheet gives a certain margin of error for these values which has a direct effect on the total heat capacity of the heater (5) and, consequently, on the margin of error in the determination of the thermal conductivity by the dynamic method (4).It may also be difficult or sometimes impossible to identify the make of a heater.The mass of the body and the volume of the heat-transfer agent can be measured using tools, in which case the measurement errors are added in the calculation of the thermal conductivity by the formula (4).This method cannot be applied to heating devices that are already put in operation.Apart from that, in practice, it is almost always the case that the specific heat capacity of the heat-transfer agent is not known due to its unidentified or variable chemical makeup (this is usually water with various additional matters).
The traditional method for determining the heat capacity is known as the calorimetric method [9] and is based on the formula (6): where в -mass of water, g; -mass of the dry calorimeter, kg; -mass of the sample under study, kg; -specific heat capacity of the calorimeter, J/kg×K; в -specific heat capacity of water in the calorimeter, J/kg×K; 0 -initial temperature of water in the calorimeter, K; -steam point of water in the heater, K; 1 -temperature in the calorimeter at the point of thermal equilibrium, K.
Specialised calorimeters are used to determine the heat capacity of solids by the above formula.Before the measurement, the sample under study, the dry calorimeter and the calorimeter filled with water should be weighed.It is also necessary to record the average temperature of the water in the calorimeter.After that, the object under study is placed in a special vessel filled with water, the temperature is brought to the boiling point, and the object in the vessel is heated until it reaches the temperature of the boiling water.The heated sample is then quickly removed and put into the water of the calorimeter.Once the thermal equilibrium is reached, the temperature of water therein t1 is measured, and the calculations are carried out using the formula 6.In existing calorimetric units, the vessels of both the heater and the calorimeter are rather limited in size, which does not allow large objects, such as heating devices, to be placed inside.Increasing the size of the vessels will make it difficult to maintain and monitor the average temperature of the water in the vessel and that of the sample under study, and it will increase heat losses through the insulation layer of the calorimeter vessel.The issues described above will decrease the accuracy of measurement of the heat capacity.In addition, it is impractical from the economic point of view to set up calorimeters of such considerable size.

Materials and Methods
This paper discusses an experimental method to determine the heat capacity.To do this, it is necessary to know the thermal conductivity values obtained by the static and dynamic methods.By equating the right-hand parts of the equations 2 and 4 to each other, we come to the following: Given the equal temperature drop, C can be found as: To solve the task of determining the heat capacity using the dynamic method, it is necessary to carry out 2 series of experiments: 1) Find out the thermal power of the heating device in its steady-state mode at different values of the temperature drop.
2) Turn the heating device into dynamic mode (cooling mode) and obtain the characteristic curve of dependency of the surface temperature from time .As a subject of study, an eight-section aluminium Alleator heater was chosen.The mass of the framing and the volume of water in the heater were measured, and the heat capacity was calculated using formula 4 C=24193 J/K.
The calculation of the heat capacity using the formula 8 was carried out in a special test bench in a laboratory setting.The hydraulic part of the test bench consists of a closed heating system including pipes, a tubular heating element, a recycling pump, an expansion vessel, a shunt and the heater under study itself being connected to this system.Platinum resistance thermometers are used to measure the inlet and outlet temperatures of the heater.Electromagnetic transducers measure the flow rate of the heat-transfer agent in the heating device.Based on the difference in temperatures between the heater's inlet and its outlet as well as on the heat flow rate, the Heat Quantity Calculator HQC-7 allows to find the thermal power P in the steady-state mode.
The unit allows maintaining the preset difference between the surface temperature of the heater and the room air temperature (i.e. the temperature drop).The stabilisation of the temperature drop comes down to the stabilisation of the heat power.The amount of heat is controlled by means of PWM modulation, which is implemented by connecting a relay in a circuit with the heating element.
Digital thermometers DS18B20 with the One-Wire interface are used to monitor the surface temperature of the heater as well as the air temperature in the room.The average room air temperature was measured at a distance of 1.5 m from the heater at a height of 1 m above the floor, as recommended by the measurement guidelines of the Association of HVAC Engineers [10].In order to monitor the average surface temperature of the heater, it was necessary to determine the point where the temperature sensor should be mounted.This feature point was found empirically using a set of temperature sensors mounted equidistantly from each other.During the experiment, the temperature distribution was recorded both during the dynamic mode of the system (cooling and heating) and during the steady-state mode (maintaining a stable temperature for 4 hours).It was found that the feature point representing the average temperature in each mode was slightly above the geometric centre.
Data from the temperature sensors was recorded using a system based on a personal computer equipped with software for automatic data collection and processing.Temperature readings were taken every 10 seconds to an accuracy of 0.5 degrees Celsius.Based on the data obtained and using an algorithm that takes into account the thermal lag of the system, the amount of heat was controlled to maintain the required temperature drop.

Results
In the first series of experiments, which aimed to determine the thermal power, the static characteristic was taken three times.For each characteristic, the thermal power was determined based on six values of the temperature drop ranging from 10 to 35 with a 5-degree step.The time taken to record each point of temperature drop was 4 hours.The resulting characteristic curves were averaged to produce a single curve, which was then approximated by a linear equation.The results of the experiment are given in Figure 1.
In the second series of experiments, the heater was put into the dynamic mode, i.e. the supply of heat-transfer agent to the studied object was cut off.While the device was cooling down, three dynamic characteristics were recorded, i.e. the rate of temperature change on the heater surface over time .In order to obtain values of at the same points of temperature drop as were used in the static mode, we limited the temperature recording to the range between 10 and 35 degrees Celsius.The dependencies obtained are linear in nature, so they were averaged and approximated by a linear equation.The results obtained are summarised in Figure 2. The two series of experiments carried out allow us to calculate the heat capacity of the heater at six points of temperature drop using the values obtained for the thermal power and the cooling rate and applying the formula (7).We calculated the mean value of the heat capacity as well as the deviation of the heat capacity from the mean (24588.66691J/K) at each point.The result is given in Table 1.The deviation of the value of the heat capacity found experimentally (24588.66691J/K) from the value calculated by the formula 4 (24193 J/K) was 1.63 %.

Conclusions
The thermal conductivity of heating devices can be found by the dynamic method when they are cooling down during their normal operation.In this case, the heat capacity of the heater must be known.A method was developed which can be used to determine the heat capacity of a heater from the known power consumption and the cooling curve of the heater.The method developed allows determining the heat capacity of any type of heating device, taking into account their inherent characteristics, including those gained during their operational lifetime.The error of the experimental method in comparison to the calculation method is 1.63%.Disadvantages of this method may include the considerable time required to conduct the experiment, which includes the time needed to record the cooling of the heater and to determine the thermal power at several points of temperature drop.

Table 1 .
Heat capacity and its deviation from the mean as a function of temperature drop