Dynamic T-History method - A dynamic thermal

Dynamic T-History method - A dynamic thermal resistance for the evaluation of the enthalpy-temperature curve of phase change materials I M. Brütting ∗, S. Vidi, F. Hemberger, H.P. Ebert Bavarian Center for Applied Energy Research, Magdalene-Schoch-Straße 3, 97074 Würzburg

Abstract Thermal energy storage with phase change materials (PCM) provides high storage capacities in small temperature intervals. For the design of storage systems, the enthalpy curve of the used PCM has to be known with high precision. The T-History method has evolved to a widely used method for the measurement of the enthalpy as a function of temperature of PCM because of its simplicity and the advantage to be able to investigate larger sample volumes than typically used for differential scanning calorimeters. In order to ensure isothermal specimen during the measurement a thermal insulation is often mounted around the sample holder, but this insulation material is not considered in the evaluation model. In this work a new evaluation model for insulated T-History setups is developed by an analytical heat balance. This model is validated by numerical simulations of insulated T-History measurements and experimentally. By the use of the new model for the evaluation of the enthalpy a significant increase of accuracy can be achieved. Keywords: Heat Capacity, Calorimetriy, T-History Method, Numerical Simulation, Phase Change Materials (PCM), Calibration

Nomenclature

ρ

Roman symbols

Subscripts / superscripts

heat capacity (J · K−1 )

C

specific heat capacity (J · kg

cp

−1

Density (kg · m−3 )

*

constant value after initial transient

ce

center

dyn

dynamic

eff

effective

enc

encapsulation

init

initial

−1

·K ) −1

G

inverse dynamic time constant (s )

i

Material section

L

latent heat (kJ · kg−1 )

l

length (m)

Q˙

heat flux (W)

R

thermal resistance (K · W−1 )

ins

insulation

r

radial position (m)

mean

mean

T

temperature (K)

m

melting

t

time (s)

pcm

phase change material

w

full width at half maximum (K)

ref

reference

s

sensible

stat

stationary

Greek symbols λ

thermal conductivity (W · m I©

−1

−1

·K )

2018. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/. DOI: https://doi.org/10.1016/j.tca.2018.10.030 ∗ Corresponding author. Email address: [email protected] (M. Brütting ) Preprint submitted to Thermochimica Acta

Abbreviations DSC

differential scanning calorimetry

PCM

phase change materials November 7, 2018

1. Introduction

with the temperature at the interface between specimen and surrounding T 1 and ambient temperatrue T 2 and where Q˙ is the heat flux through the specimen surface. The thermal resistance R is defined for stationary conditions as

Energy storage is a key to a sustainable energy supply, because the fluctuating sources of renewable energy can be buffered and the supply can be synchronized with the demand. Phase change materials (PCM) as latent heat storage materials gained a lot of interest in the last 20 years and are widely applied in differnt fields like building applications for example PCM cooling ceilings [1, 2], building walls [3, 4, 5, 6, 7, 8], wallboards with incoorporated PCM [9, 10, 11] or insulation layers [12]. The main advantage of PCM is their ability to store a large amount of heat in a small temperature interval. Therefore, a main aspect in PCM research is the determination of the heat capacity but intercomparisons between different laboratories show large deviations in the measurement of enthalpy (up to 20 %) as well as temperature (up to 4 K), even when well established methods like differential scanning calorimetry (DSC) are used [13]. This fact shows, that there is still more effort necessary to develop new methods and standards like [14] in order to get reliable and comparable material properties. Many different measurement setups and methods were developed to determine the enthalpy as a function of temperature [15]. One of these methods is the T-History method, which has evolved to a widely used method for the determination of the heat capacity of PCM, because it can be easily setup with standard laboratory equipment and the sample volume is larger than typically used for DSC. Two basic setups are shown in literature, a non-insulated and an insulated setup. In this work a new evaluation method for insulated T-history measurements is presented using a dynamic thermal resistance. The dependence of the dynamic thermal resistance on the heat capacity of the specimen is shown and a correlation between the dynamic thermal response of the specimen and the thermal resistance of the system is derived. The method is validated using finite element numerical simulations of insulated T-history measurements and experimentally.

Rstat =

(2)

In the T-History method the thermal resistance is normally evaluated by a dynamic measurement of a temperature step response with a reference specimen with known thermal properties. The thermal resistance of the system is then calculated as T 2 − T 1 (t) (3) R(t) = ∂T 1 (t) mref c p,ref ∂t with the time t, the reference mass mref and the specific heat capacity of the reference material c p,ref . As the specimen normally is encapsulated, the capacity of the encapsulation has to be considered in the evaluation of the thermal resistance as T 2 − T 1 (t) R(t) =   ∂T 1 (t) . mref c p,ref + menc c p,enc ∂t

(4)

and in the calculation of the heat flux into the specimen ˙ = T 2 − T 1 (t) − menc c p,enc ∂T 1 (t) . Q(t) R ∂t

(5)

There have been several different experimental setups applied for the realization of the T-History method. Their main difference is the thermal resistance used for the evaluation of the heat flux into the specimen. The first setups considered the convective heat transfer coefficient of natural convection to be the dominant thermal resistance. This unknown thermal resistance is determined by a reference measurement [16, 17]. In these setups the linear temperature gradient between the sensors can be assumed, because the heat capacity of air is small compared to the specimen heat capacity. But in a recent study the convective heat transfer coefficient of a typical T-History setup was investigated and it was found that large fluctuations of the coefficient during the measurement can lead to considerable errors [18]. Also temperature gradients inside the specimen can lead to deviations, if the measurement parameters are not chosen carefully. Later forced convection heating and cooling of the ambient air was established and an insulation layer around the specimen was introduced to reduce temperature gradients inside the specimen and to assure its thermal equilibrium during the measurement [19, 20]. As a consequence, the heat conduction through the insulation layer is the dominant thermal resistance and has to be determined by calibration. Insulated systems have the drawback, that the heat capacity of the insulation is in the same order of magnitude as of the specimen. Therefore the temperature gradient inside the insulation is not linear during the measurement and deviations are caused when the formalism developed for linear temperature gradient is used without correction. This problem is already known and treated in literature. Tan

2. Methods 2.1. T-History Method During a measurement with the T-History method a specimen is subjected to an outside temperature jump and the temperature response of the specimen is recorded. With a heat transfer model it is possible to calculate the heat flux into the specimen and subsequently its enthalpy change by integration of the heat flux. With the use of the assumptions of • a lumped capacitance (grad T = 0) inside the specimen and • a linear temperature gradient between the temperature sensors the evaluation equation is T2 − T1 Q˙ = R

T2 − T1 . Q˙

(1) 2

showed numerically [19] and experimentally [21] and Rathgeber [22] showed experimentally that due to the heat capacity of the insulation the thermal resistance of the insulation is different for the sensible and latent heat parts of the measurement. They solve this problem by introduction of correction factors for the sensible and latent parts of the measurement. 2.2. Insulated T-History Method The insulated T-history method consists of a sample volume surrounded by an insulation, as shown in a principal sketch in Fig. 1. A temperature jump is applied on the outside and the temperatures over time are measured at two isothermal surfaces, one being the specimen surface and the other the insulation’s outer surface.

Figure 2: Schematic diagram of the temperature profile inside the insulation during the dynamic heating of the specimen. The secant between T 1 and T 2 has the same slope as the tangent through T 0 .

x T2

λ2 grad(T 0 ) · S 0 < λ2 grad(T 2 ) · S 2

x T1 V1

The heat flux in stationary conditions is calculated with the secant between T 1 and T 2 . Therefore, the stationary thermal resistance Rstat of the setup defined in equation 2 can be used and the equations can be rewritten as follows:

V2

S1 S2

T 2 − T 1 (t) > λ2 grad(T 1 ) · S 1 Rstat

Figure 1: Schematic of the insulated T-History measurement setup.

T 2 − T 1 (t) = λ2 grad(T 1 ) · S 1 + Q˙ ins (t). Rstat

(12)

2.2.2. Energy Balance - Specimen The heat flux going through the specimen surface into specimen equals the heat stored in the specimen. w ρ1 c p,1 T˙ (~r, t) dV = λ2 grad(T 1 ) · S 1 (13) | {z } V1 Heat flow into specimen | {z }

2.2.1. Energy Balance - Insulation An energy balance of the insulation can be written as: w ρ2 c p,2 T˙ 1 (~r, t) dV = λ2 grad(T 2 ) · S 2 − λ2 grad(T 1 ) · S 1 . | {z } | {z } V1 | {z } Heat flow into insulation Heat flow out of insulation

Heat stored in specimen

With the lumped capacitance assumption the left hand side can be integrated and written as: C1 T˙ 1 (t) = λ2 grad(T 1 ) · S 1 .

into specimen

(6)

(14)

Using equation 12 the temperature difference is introduced, which is the measured value: T 2 − T 1 (t) C1 T˙ 1 (t) = − Q˙ ins (t). (15) Rstat

The mean value theorem states, that there exists a point r0 in the insulation, where the slope of the tangent to the temperature function equals the slope of the secant through the points T 2 and T 1 . T 2 − T 1 (t) = grad(T (r0 )). (7) ~r2 − ~r1

This can finally be written as: T˙ 1 (t) 1 Q˙ ins (t) = − . (16) T 2 − T 1 (t) Rstat T 2 − T 1 The parameter Q˙ ins (t)/T 2 − T 1 (t) represents the time dependent change of the apparent thermal resistance of the insulation which deviates from the stationary thermal resistance Rstat by a portion of heat flux that is stored in the insulation material. C1

This relationship is illustrated in Fig. 2. As there also exists an isothermal surface S 0 at the point r0 we can write: λ2 grad(T 0 ) · S 0 > λ2 grad(T 1 ) · S 1

(10)

T 2 − T 1 (t) < λ2 grad(T 2 ) · S 2 . (11) Rstat This means that the heat flux calculated using T 2 −T 1 and Rstat is larger than the flux through the inner surface of the insulation entering the specimen, as it contains part of the heat flux that is stored in the insulation Q˙ ins .

For the evaluation of the measurement one assumption is made, the lumped capacitance assumption. This states that the temperature gradient inside the specimen is negligible. This is a good approximation, if the thermal resistance inside the specimen is small compared to the thermal resistance of the insulation. In the following, the energy balances for the insulation and the specimen are formulated. Then an equation is introduced, which accounts for the heat capacity of the insulation and a method is shown to calibrate the apparent thermal resistance as a function of the temperatures T 1 and T 2 as well as the time derivative of T 1 .

Heat stored in insulation

(9)

(8) 3

With equation 1 and R∗dyn (t) the time dependent heat flux can be calculated. Integration of the heat flux leads to the enthalpy. w ∆H = Q˙ dt (24)

2.2.3. Dynamic Parameters For the investigation of the dynamic temperature response of the system the dynamic thermal resistance Rdyn is defined as  −1  1 Q˙ ins (t)   . (17) Rdyn (t) =  − Rstat T 2 − T 1 (t)

2.3. Simulation setup 2.3.1. General properties In order to investigate the developed methodology of section 2.2, a numerical simulation of an insulated T-History setup was created. With these simulations the transient temperature field in the T-History setup is calculated for different measurement scenarios. The temperature and heat flux values at the desired positions are extracted from the simulation data and are used for testing the T-History evaluation equations. The advantage of finite element numerical simulations of measurement setups for material characterization is that the used material properties are known exactly and the deviations from these properties originating from improper assumptions in the evaluation model can be studied. All simulations were done using the finite element solution environment FlexPDE with adaptive mesh refinement and variable time step [23] solving the heat equation:

The dynamic thermal resistance is a parallel connection of the stationary thermal resistance Rstat and the parameter Q˙ ins (t)/T 2 − T 1 . The inverse dynamic time constant G is defined as G(t) =

T˙ 1 (t) . T 2 − T 1 (t)

(18)

It is the time dependent temperature change of the specimen divided by the temperature difference between inner and outer surface of the insulation. This represents the ratio between the thermal response of the specimen and the driving force of the experiment. The parameter consists solely of temperatures and can be determined easily. Rearanging equation 16, the dynamic thermal resistance is calculated as 1 . (19) Rdyn (t) = G(t) C1

ρi c p,i

In a reference measurement of a material with known constant heat capacity, the system reaches a dynamic equilibrium after an initial transient and the inverse dynamic time constant G(t) is constant: G(t) → G∗ . (20)

1 . G∗ C1

(25)

with the density ρ, specific heat capacity c p , thermal conductivity λ and internal heat sources Q for each material section i. The principal setup of the simulation is shown in Fig. 3. A

In the following a * is used to indicate that the constant values after the initial transient are used. Consequently, a dynamic thermal resistance in dynamic equilibrium R∗dyn is defined as R∗dyn =


∂T + Qi = div λi grad T ∂t

z dins dtube ds

(21)

dins

∗

The value of G depends on the heat capacity C1 . Additionally, in real setups the heat transfer is multi dimensional. Therefore form factors for the specific geometry have to be introduced. The relation between R∗dyn and G∗ for different heat capacities C1 for the specific geometry can be approximated with the reciprocal of a second order polynomial: 
−1 R∗dyn ≈ a + b G∗ + c G∗ 2 . (22)

l

x Tce

x T1

x T2

specimen dtube tube insulation 0

The polynomial coefficients a, b and c contain the form factors and the dependency of G∗ on the heat capacity. In practice this means, that the polynomial coefficients have to be determined with different specimen heat capacities C1 . This can be achieved by using reference materials with different constant specific heat capacities or by using different specimen masses. Once the polynomial coefficients are known, the dynamic thermal resistance during the measurement can be calculated with equation 22. For measurements of phase transitions the effective heat capacity Ceff of the specimen varies with temperature and therefore also with time. In consequence, the dynamic thermal resistance must be time dependent: 
−1 R∗dyn (t) ≈ a + b G∗ (t) + c G∗ (t) 2 . (23)

ri

ro

r

Figure 3: Schematic diagram of the simulation setup. The z-axis is treated as axis of rotational symmetry.

two dimensional coordinate system with cylindrical coordinates was used with the z-axis acting as axis of rotational symmetry and the r-axis as the radial distance. The simulation geometry consists of a cylindrical specimen surrounded by a closed tube and a thermal insulation. The initial temperature in the simulation regime is set to T init = 293 K. At the outer boundary of the insulation a temperature Dirichlet boundary condition is applied. Due to numerical stability the boundary value is ramped in 200 s from the initial temperature to T 2 = 313 K and then held constant. This temperature ramp also reflects the heat up process of a climatic chamber where the measurements are performed in reality. 4

3. Results and Discussion

The thermophysical properties and dimensions used for the simulations are shown in Tab. 1. The thermal properties of the tube and the insulation are based on typical values for copper and expanded polystyrene. The temperature values T 1 and T 2 are calculated at the inner and outer surface of the insulation r1 and r2 respectively in the middle of the specimen length l.

3.1. Determination of polynomial coefficients For the calibration of the thermal resistance, the polynomial coefficients of equation 22 have to be determined by varying the specimen heat capacity C1 and measuring G∗ . Therefore, measurements with different heat capacities of the specimen C1 were simulated. From these simulations different temperatures T 1 were extracted as function of time and different G and Rdyn were calculated. The values are shown as a function of time in Fig. 4.

C

3 1 5 T

3 1 0

-1

/ J K s

2

T

3 0 5

(C 1

T

3 0 0 T T

2 9 0 T

= 4 1 .4 ) s

(C 1

2 9 5

= 2 5 .7 ) s

(C 1

= 5 7 .1 ) s

(C 1 1

(C

= 8 8 .5 ) s s

= 1 5 1 .4 )

2 8 5 0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 .0

0 .6 0 .4 s p e c im e n C

G

/ s

-1

x 1 0

-3

0 .8

0 .2

/J K

0 .0 0

2 0 0 0

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

4 5 4 0

s

-1

2 5 .7 4 1 .4 5 7 .1 8 8 .5 1 5 1 .4 s ta tio n a r y

-1

3 5

R

d y n

/ K W

3 0 2 5 2 0 1 5 1 0 5 0

T im e t / s

Figure 4: Simulated temperatures, inverse dynamic time constant G and dynamic thermal resistance Rdyn as a function of time during a T-History measurement for different specific heat capacities of the specimen.

2.3.3. Testing the method with PCM For testing this determined relation between R∗dyn and G∗ measurements of a PCM are simulated. The latent heat is modeled as a temperature dependent specific heat capacity with a gaussian peak function:    1 T − T m !2  L   . c p,pcm (T ) = c p,s + √ · exp −  2 w w · 2π

3 2 0

T e m p e ra tu re T / K

2.3.2. Validation of the dynamic parameters The focus of this investigation is on the dynamic behavior of the insulation layer and its influence on the dynamic parameters. In order to suppress other effects causing deviations in the evaluation of the heat capacity of the specimen, a very high thermal conductivity of the specimen material is chosen to minimize temperature gradients inside the specimen and to satisfy the lumped capacitance approach. In section 3.3 simulations with properties typical for PCM are discussed causing temperature gradients inside the specimen. For the investigation of the dependence of G∗ on the heat capacity of the specimen C1 , the value is varied starting from 25.7 J · K−1 heat capacity representing an empty tube and then increasing the value in several steps from 41.4 J · K−1 to 151.4 J · K−1 . The highest value represents a tube filled with water. The smaller values are chosen arbitrarily to cover the range between the empty tube and the water filled tube. These values cover a range of heat capacities, which can be realized with easily availabe materials like water. In a real experiment the capacities can be varied by changing the material or by simply changing the filling level. In the simulations this is realized by varying the specific heat capacity of the specimen material starting from 0 J · kg−1 · K−1 and then doubling the value from500 J · kg−1 · K−1 to 4000 J · kg−1 · K−1 . This could also be achieved by varying the specimen density without other effect. From these simulations the polynomial coefficients a, b and c of equation 22 are determined. The stationary thermal resistance is calculated with equation 2 by applying a constant heat source of 1 W inside the specimen and evaluation of the resulting temperature difference T 1 − T 2 in stationary state.

In the beginning of the measurement the temperature wave has to propagate through the insulation material and heat is accumulated in the insulation. After this initial transient a dynamic equilibrium is reached and the inverse dynamic time constant G reaches a constant value. The duration of this process depends on the heat capacity and the thermal conductivity of the insulation material. The settling time until a constant value of G is reached is independent from the specimen capacity. In the simulations a heat capacity Cs = 25.7 J · K−1 of the specimen represents an empty tube and Cs = 151.4 J · K−1 is roughly the capacity of a water filled tube. With increasing heat capacity, the value of G decreases. With equation 21 the dynamic thermal resistance is calculated and shown in Fig. 4 in the lower diagram. In addition, the stationary thermal resistance is shown. With rising heat capacity the constant value of Rdyn

(26)

with L as the latent heat and w as the full width at half maximum. Material properties typical for PCM were taken and are shown in Tab. 2. Thermal conductivity is set to a very high value, in order to avoid temperature gradients inside the specimen. An evaluation with realistic values is shown in section 3.3 5

Table 1: Thermophysical properties and dimensions used for the simulations.

Thermal conductiviy λ Specific heat capacity c p Density ρ Length l Thickness d

W · m−1 · K−1 J · kg−1 · K−1 kg · m−3 m m

Specific heat (sensible) Latent heat Melting temperature Full width at half maximum Thermal conductivity Density (mean)

c p,s L Tm w λ ρ

Value

J · kg−1 · K−1 J · kg−1 K K W · m−1 · K−1 kg · m−3

4000 105 303 0.1 500 1000

4 4

R e c ip r o c a l p o ly n o m ia l fit

-1

0.03644 W · K−1 −11.28156 W · K−1 · s −1791.69317 W · K−1 · s2

During a PCM measurement with the T-History method, the thermal response of the specimen changes over time due to the phase change. Therefore a PCM measurement with high thermal conductivity is simulated, in order to get idealized temperature values T 1 and T 2 without influence of temperature gradients inside the specimen. The temperatures during a measurement are shown in Fig. 7 in the upper diagram. The middle diagram shows the inverse dynamic time constant G over time during the measurement. After a settling period the temperature of the specimen rises due to the storage of sensible heat. During that part the inverse dynamic time constant G reaches a constant value. When the melting temperature is reached, the heat capacity of the specimen is very large due to the latent heat. The temperature remains almost constant and the inverse dynamic time constant G drops to almost zero. When the material is molten, the temperature rises again asymptotically to thermal equilibrium and the inverse dynamic time constant G rises again to the constant value of the sensible part at the beginning of the measurement. With the empirical relation of equation 22, the determined polynomial coefficients of section 3.1 and the inverse dynamic time constant G during the measurement a time dependent dynamic thermal resistance Rdyn (t) is calculated. The dynamic thermal resistance profile of a PCM measurement is shown in Fig. 7 in the lower diagram. After the initial settling time the dynamic thermal resistance is constant. During the melting

3 8

/ K W

0.045 1600 150 0.0175

3.2. Application to the measurement of idealized PCM R

4 0

d y n

401 390 8960 0.001

*

4 6

4 2

*

500 c p,s 1000 0.1 0.01

are determined for different specimen with different known heat capacities. From these time dependent parameters the quasi stationary values are extracted and the parameters of system specific empirical relation between R∗dyn and G∗ are determined by curve fitting. Thus, it is possible to derive a dynamic thermal resistance for the setup from its thermal response by calibrating it with different heat capacities. This means, that the dynamic thermal resistance depends on the total heat capacity of the specimen. Therefore, not only the specific heat capacity of the reference material but also the specimen mass has an influence on the calibration of a T-History setup.

4 8

R

Insulation

a b c

gets closer to the stationary value, which represents the limit for an infinite heat capacity and therefore stationary heat transfer. Taking the constant values R∗dyn and G∗ for different heat capacities of the specimen, the system specific parameters of equation 22 can be determined. In Fig. 5 can be seen, that R∗dyn is not directly proportional to G∗ but has a light curvature. This

3 6 3 4 3 2 3 0 2 8 2 6 2 4 -0 .0 0 0 2

Tube

Table 3: Polynomial coefficients for the empirical relation between R∗dyn and G∗ .

Table 2: Material properties of the artificial PCM used for the numerical simulations.

Unit

Specimen

0 .0 0 0 0

0 .0 0 0 2

0 .0 0 0 4

G

*

0 .0 0 0 6

/ s

0 .0 0 0 8

0 .0 0 1 0

-1

Figure 5: System specific empirical relation between the constant values of the dynamic thermal resistance Rdyn and the inverse dynamic time constant G for different specimen heat capacities. The quasi stationary values of both dynamic quantities are plotted against each other and fitted with a reciprocal polynomial fit.

dependence can be fitted with equation 22. The polynomial coefficients for this setup are shown in Tab. 3. The calibration of the system is performed by evaluation of the polynomial coefficients of equation 22. This process is shown in a sequence diagramm in Fig. 6 in the upper part. For calibration of the system the dynamic values of Rdyn and G 6

Calibration

Measure Rdyn and G over t with different heat capacities

Determine the system specific empirical relation between R*dyn and G*

Extract the quasi stationary values * and G* of Rdyn after initial settling

G

* Rdyn

C1

C2

R*dyn=(a+bG*+cG*²)-1

t

C3

Rdyn G

*

t

Measurement

Measure G over t

. Integrate Q and plot H against T

. Calculate Q

Calculate Rdyn with the system specific empirical Rdyn relation

G

. Q

H

Cx t

Rdyn=(a+bG+cG2)-1

t

t

T

T e m p e ra tu re T / K

Figure 6: Sequence diagram for the calibration process and the measurement process of the dynamic T-History method.

period the dynamic thermal resistance drops to the stationary value and subsequently rises to the constant value of the sensible part at the beginning of the measurement. With the dynamic thermal resistance the time dependent heat flux into the specimen can be calculated. Integration of the heat flux according equation 24 and plotting it against the specimen temperature T 1 yields the enthalpy temperature curve. For comparison with the conventional T-History evaluation method as described in section 2.1, a measurement with a reference specimen was simulated and the simulation data were used for the determination of R according equation 4. For the reference material a c p of 2000 J · kg−1 · K−1 and a density of 1000 kg · m−3 was used. These thermal properties are comparable to an organic material for example a liquid oil. A comparison between the resulting enthalpy temperature curves is shown in Fig. 8. The black curve is the integrated c p gauss curve from equation 26 used as input material parameter for the simulation. The red curve is evaluated with a constant thermal resistance like in conventional T-History measurements and the green curve is evaluated with the developed dynamic thermal resistance. As can be seen in the figure, the use of the conventional THistory evaluation leads to considerable deviations. Both the slope in the sensible parts and the rise in the latent part deviate from the real values. It is worthwhile to mention, that using a material for evaluation with comparable thermal properties, deviations in the latent heat up to 10 % are obtained supposing equal specimen volume. In contrast to that, the evaluation with the dynamic thermal resistance yields the correct enthalpy curve after the initial settling period with deviations below 1 % in the evaluated temperature range. The slope in the sensible parts is calculated correctly as well as the amount of latent heat

p h a s e c h a n g e

3 1 5 3 1 0 3 0 5

T

3 0 0

T

1 2

2 9 5 2 9 0 0

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

0

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

0

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

2 .0 1 .5 1 .0

G

/ s

-1

x 1 0

-4

2 .5

0 .5 0 .0

3 5

R

d y n

/ K W

-1

3 0 2 5 2 0 1 5 1 0 5 0

T im e t / s

Figure 7: Temperatures of the inner and outer insulation surface and the derived values of G and Rdyn during a PCM measurement. G is calculated from the temperature profile of T 1 and T 2 . Rdyn is calculated from G with the system specific empirical relation determined by calibration with different heat capacities.

7

1 8 0

s im u la tio n in p u t c u r v e c o n s ta n t th e r m a l r e s is ta n c e d y n a m ic th e r m a l r e s is ta n c e

1 4 0 1 2 0 1 0 0 8 0

3 1 2

6 0

3 1 0

4 0

3 0 8

2 0

T e m p e ra tu re T / K

S p e c i f i c e n t h a l p y ∆h / J k g

-1

x 1 0

3

1 6 0

0 -2 0 -4 0 2 9 0

2 9 5

3 0 0

3 0 5

3 1 0

3 1 5

T e m p e ra tu re T / K

T

3 0 6

1

T

c e

3 0 4

T 3 0 2

m e a n

3 0 0 2 9 8 2 9 6 2 9 4

Figure 8: Comparison of the specific enthalpy derived with different thermal resistances. The simulation input curve is compared to a measurement curve evaluated with a constant thermal resistance determined by a calibration with an organic liquid. These curves are compared to a measurement evaluated with the derived dynamic thermal resistance. All enthalpy curves are set to zero at 296 K.

0

5 0 0 0

1 0 0 0 0

1 5 0 0 0

2 0 0 0 0

T im e t / s

Figure 9: Temperatures at different positions during a simulation of a T-History measurement with realistic thermal properties.

during the melting period. The process of this new evaluation method is shown in Fig. 6 in the lower part. 3.3. Application to measurement of PCM with realistic properties 3.3.1. Handling temperature gradients inside the specimen In the previous section a very high thermal conductivity of the PCM was used in order to avoid temperature gradients inside the specimen and deviations in the measurement result arising from that. Temperature gradients inside the specimen cause deviations in the evaluation of specimen temperature and the measured enthalpy is shifted to higher temperatures in case of heating and to lower temperatures in case of cooling. This phenomenon is well known and treated in literature for DSC measurements [24, 25, 26, 27] and standards were developed to encounter this problem [14, 28]. To quantify the deviations resulting from temperature gradients inside the specimen of a T-History experiment, realistic thermal properties are used. For the following simulations a thermal conductivity of 0.3 W · m−1 · K−1 was used and the full width at half maximum w was set to 1 K. These values are typical for many organic PCM. The other thermal properties remain the same as in the previous section. In Fig. 9 the simulated temperatures at different positions are shown during a measurement. A temperature difference of about 1 K can be observed between the center temperature of the PCM T ce and the interface temperature between the specimen and the insulation T 1 . In Fig. 10 the evaluated enthalpy is plotted against the different temperatures and compared to the integrated gauss curve used as simulation input parameter. The evaluation of the enthalpy is performed like in section 3.2. The only difference between the different curves is the used temperature for plotting. Due to the temperature gradient inside the specimen, the

s im u la tio n in p u t c u r v e

1 6 0

T

S p e c i f i c e n t h a l p y ∆h / J k g

c e

T

-1

x 1 0

3

1

T

1 4 0 1 2 0

m e a n

1 0 0 8 0 6 0 4 0 2 0 0 -2 0 2 9 4

2 9 6

2 9 8

3 0 0

3 0 2

3 0 4

3 0 6

3 0 8

3 1 0

3 1 2

3 1 4

T e m p e ra tu re T / K

Figure 10: Enthalpy curve plotted against different temperatures and compared to the simulation input curve.

8

enthalpy of the phase change is shifted to higher temperatures, when the interface temperature between specimen and insulation T 1 is used. In contrast to that, the enthalpy is shifted to lower temperatures, when the PCM center temperature T ce is used. Additionally, the arithmetic mean T mean of T ce and T 1 is calculated. Plotting the enthalpy against the mean temperature T mean leads to a nearly perfect match of input enthalpy curve and calculated measurement result. This fact recommends the use of an additional sensor for recording the PCM center temperature. Then the correct enthalpy can be evaluated with the temperatures T 1 and T 2 and be plotted against the mean PCM temperature T ce .

2 4 0

T

m e a n

2 1 0

T

1 8 0

T

S p e c i f i c e n t h a l p y ∆h / J k g

m e a n

1 5 0

T

1 5 0

9 0

1 2 0

6 0

9 0

3 0 6 0 0

3 0

-3 0

3 0 2

1 8 0

T

3 1 0

3 2 0

3 3 0

1 5 0 1 2 0 1 2 0 9 0

3 0 6 0

3 0 0 0 3 0 -1

-3 0 -6 0

3 0 2

3 0 4

3 0 6

3

-9 0 2 9 0

3 0 0

3 1 0

3 2 0

3 3 0

S p e c i f i c e n t h a l p y ∆h / 1 0

2 8 0

2 5 0

J k g

x 1 0

3

1 5 0

6 0

3 0 0

Therefore an insulated T-History setup is calibrated with specimens with different known heat capacities and the parameters of equation 22 are determined by curve fitting. For the comparison n-octadecane (Parafol 18-97 from Sasol Germany GmbH) is measured in the T-History setup. The gathered temperature values are then evaluated according to the different methodologies. In Fig. 13 the specific enthalpy curves of one melting cycle are shown exemplarily. As reference curve the data from a

(2 0 K )

9 0

2 9 0

Figure 12: Enthalpy temperature curves for different outside temperature step widths ∆T 2 ranging from 20 K to 40 K symmetrically around the melting temperature T m . The enthalpy is plotted against the respective mean specimen temperature T mean .

s im u la tio n in p u t c u r v e

-1

S p e c i f i c e n t h a l p y ∆h / J k g

1

3 0 6

T e m p e ra tu re T / K

(3 0 K ) 1

3 0 4

-9 0

(4 0 K ) 1

T

(2 0 K )

-6 0

3.3.2. Influence of outside temperature step width In order to evaluate the influence of different temperature gradients inside the specimen different outside temperature step widths were applied ranging from 20 K to 40 K symmetrically around the melting temperature T m . In Fig. 11 the evaluated enthalpy temperature curves plotted against the respective values of T 1 are shown. With larger temperature difference be-

2 1 0

(3 0 K )

1 2 0

2 8 0

2 4 0

(4 0 K )

s im u la tio n in p u t c u r v e

-1

x 1 0

3

m e a n

T e m p e ra tu re T / K

Figure 11: Enthalpy temperature curves for different outside temperature step widths ∆T 2 ranging from 20 K to 40 K symmetrically around the melting temperature T m . The enthalpy is plotted against the respective interface temperature between the insulation and the specimen T 1 .

2 0 0

1 5 0

1 0 0

5 0

D S C r o u n d r o b in te s t d y n a m ic T - H is to r y c o n v e n tio n a l T - H is to r y 0

tween T m and T 2 the gradient inside the specimen during the phase change increases and the specimen surface temperature T 1 tends to higher values for the respective enthalpy during phase change. In Fig. 12 the enthalpy is plotted against the mean specimen temperature T mean . For all step widths no deviations from the simulation input curve are observable. That means, independent from the outside temperature step width the enthalpy can be evaluated correctly, if a second temperature sensor is used.

-5 0 2 9 0

2 9 2

2 9 4

2 9 6

2 9 8

3 0 0

3 0 2

3 0 4

3 0 6

3 0 8

3 1 0

3 1 2

T e m p e ra tu re T / K

Figure 13: Specific enthalpy curves of a melting experiment with n-octadecane evaluated by the dynamic T-History method and the conventional T-History method compared to results from a DSC round robin test.

DSC round robin test are used [14]. The value of the latent heat and the shape of the curve can be reproduced almost exactly with the dynamic T-History method compared to the values of the DSC round robin test. The deviation of enthalpy change between 298 K and 306 K is below 1 %. In contrast to that, the evaluation with the conventional T-History method with a fixed copper reference material shows a deviation in enthalpy change

3.4. Experimental validation In a validation experiment the new evaluation method of the dynamic T-History method described in section 2.2 is compared to the conventional T-History method described in section 2.1. 9

in the same temperature interval of about 16 %. This means, with the dynamic T-History method an increase of accuracy of about 15 percentage points was achieved, which is a relative increase in accuracy of more than 90 % between the two compared methods.

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4. Conclusion In this work a new evaluation method for insulated T-History measurements is developed, which takes into account the heat capacity of the insulation material. The new evaluation method is validated by 2-D numerical simulations of a T-History setup. With the dynamic temperature response of different heat capacities, a correlation between the dynamic thermal resistance and the inverse dynamic time constant can be found. By means of the correlation, the dynamic thermal resistance during a PCM measurement is derived. A comparison of the enthalpy curves derived with constant thermal resistance and with the dynamic thermal resistance is presented. By the use of this new evaluation, the accuracy of the calculated enthalpy is increased drastically. The applicability to real measurements is shown by simulations with realistic thermal properties and an experimental validation. It is shown, that it is possible to determine the correct mean specimen temperature by using two temperature sensors, one at the outside of the specimen and one in the center of the specimen. The results show good agreement and recommend the method for practical use. In further studies a detailed experimental investigation and validation will be carried out. Acknlowledgments This work has been carried out with support from the Bavarian Ministry of Economic Affairs and Media, Energy and Technology. References [1] H. Weinläder, F. Klinker, M. Yasin, PCM cooling ceilings in the Energy Efficiency Center—passive cooling potential of two different system designs, Energy and Buildings 119 (2016) 93–100, ISSN 0378-7788. [2] H. Weinläder, F. Klinker, M. Yasin, PCM cooling ceilings in the Energy Efficiency Center–Regeneration behaviour of two different system designs, Energy and Buildings 156 (2017) 70–77, ISSN 0378-7788. [3] K. Kant, A. Shukla, A. Sharma, Heat transfer studies of building brick containing phase change materials, Solar Energy 155 (2017) 1233–1242, ISSN 0038092X, doi:10.1016/j.solener.2017.07.072. [4] M. A. Izquierdo-Barrientos, J. F. Belmonte, D. Rodríguez-Sánchez, A. E. Molina, J. A. Almendros-Ibáñez, A numerical study of external building walls containing phase change materials (PCM), Applied Thermal Engineering 47 (2012) 73–85, ISSN 13594311, doi: 10.1016/j.applthermaleng.2012.02.038. [5] X. Jin, M. A. Medina, X. Zhang, On the importance of the location of PCMs in building walls for enhanced thermal performance, Applied Energy 106 (2013) 72–78, ISSN 03062619, doi: 10.1016/j.apenergy.2012.12.079. [6] X. Kong, S. Lu, Y. Li, J. Huang, S. Liu, Numerical study on the thermal performance of building wall and roof incorporating phase change material panel for passive cooling application, Energy and Buildings 81 (2014) 404–415, ISSN 03787788, doi:10.1016/j.enbuild.2014.06.044.

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[24] G. Albright, M. Farid, S. Al-Hallaj, Development of a model for compensating the influence of temperature gradients within the sample on DSC-results on phase change materials, Journal of Thermal Analysis and Calorimetry 101 (3) (2010) 1155–1160, ISSN 1388-6150 1572-8943, doi: 10.1007/s10973-010-0805-x. [25] E. Franquet, S. Gibout, J.-P. Bédécarrats, D. Haillot, J.-P. Dumas, Inverse method for the identification of the enthalpy of phase change materials from calorimetry experiments, Thermochimica acta 546 (2012) 61–80, ISSN 0040-6031. [26] S. Gibout, E. Franquet, D. Haillot, J.-P. Bédécarrats, J.-P. Dumas, Challenges of the Usual Graphical Methods Used to Characterize Phase Change Materials by Differential Scanning Calorimetry, Applied Sciences 8 (1) (2018) 66, ISSN 2076-3417, doi:10.3390/app8010066. ˇ [27] J. Foˇrt, Z. Pavlík, A. Trník, M. Pavlíková, R. Cerný, Effect of the mode and dynamics of thermal processes on DSC-acquired phase-change temperature and latent heat of different kinds of PCM, Materiali in tehnologije 51 (6) (2017) 919–924, ISSN 15802949 15803414, doi: 10.17222/mit.2017.026. [28] Gütegemeinschaft PCM e.V. RAL-GZ 896, URL http://www.pcm-ral.de, 2018 (date accessed 1.10.2018).

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