análisis térmico del instrumento tirs (meda) del rover de...

Análisis térmico del instrumento TIRS (MEDA) del Rover de la misión MARS

2020Autor:

Adrián Chamorro [email protected]

Instituto IDR/UPMUniversidad Politécnica de Madrid (Spain)

2

Descripción del sistema

• MEDA (Mars Environmental DynamicAnalyzer)

• TIRS (Thermal InfraRed Sensor) es uno de los sensores del instrumento MEDA

• Dimensiones: 57 x 62.4 x 57 mm³; masa: 97 g

2

CarcasaPlaca aislante

Placa trasera

Placa delantera

Placa soporte

Placa de calibración

PCB

Termopilas

Objetivos del instrumento e implicaciones en el diseño térmico

• Objetivos científicos:• Caracterización del intercambio radiativo en la superficie (土5%)• Medición de la radiación solar reflejada (土5%)• Medición de la temperatura de la superficie y de la atmósfera

baja (土5K)

• Implicaciones en el diseño térmico:• Necesidad de alta estabilidad térmica

• Gradiente espacial máximo en las termopilas < 22 mK

• Imposibilidad de comprobar estos requisitos mediante ensayos• Validación mediante análisis • Necesidad de aplicación de métodos de reducción de

incertidumbre• Ensayo• Análisis de incertidumbre

Credit: E.Sebastián et al.

Modelo térmico detalladoModelización del instrumento:• Alta discretización del modelo

– 1281 nodos planos y 5 nodos nogeométricos

• Modelo simplificado del rover– Ambiente radiativo mas representativo– Sombras

Carcasa

Placa aislante

Placa soporte

Rover

Suelo marciano

TIRS

Termopilas

Placa delantera

Modelización del ambiente térmico:• Cargas solares directas y difusas

– Excentricidad de la orbita marciana; dispersión de laradiación solar por la atmosfera

• Convección:– Convección externa: natural y forzada.– Conducción a través del CO2 interno.

• Deposición de polvo sobre las superficieshorizontales y verticales

Test y correlación del modelo• Ensayo térmico para reducir la incertidumbre del modelo

• Montaje del ensayo:

– Cámara de CO2 presurizada

– Se realizaron dos ensayos con distintos perfiles de consumo de potencia

– Estimación del gradiente de las termopilas a partir de la medida de la termopila 3 (filtro paso banda en visible)

5

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

0 20,000 40,000 60,000

Te

mp

era

ture

(°C

)

Time (s)

TIRS Test 1 Temperature measurements

Support plate

Calibrationplate

Chamber

Case

6

• Correlación del estado estacionario (estado cuasiestacionario en los ensayos):• Acoplamientos conductivos (contacto), acoplamientos convectivos (internos),

conductividades térmicas

Model results before

correlation

(temperature (°C))

CaseCalibration

plate

Support

plate

Test 1Measured 31.79 46.98 43.43

Model 31.85 53.8 40.8

Test 2Measured 26.39 28.60 35.64

Model 26.4 27.2 33.8

Results after

steady state

correlation (ºC)

Case

temperature

Calibration

plate

temperature

Support plate

temperature

Test 1Measured 31.79 46.98 43.43

Model 31.85 47.56 43.26

Test 2Measured 26.39 28.60 35.64

Model 26.40 28.21 35.58

Error

Test 1 0.06 0.58 -0.17

Test 2 -0.05 -0.39 0.01• Correlación del ensayo transitorio:• Fuerte dependencia de los gradientes con la capacidad térmica de las termopilas.

-35.00

-25.00

-15.00

-5.00

5.00

15.00

25.00

3500 5500 7500

ΔT

(mk

)

Tiempo (s)

-100.0-75.0-50.0-25.0

0.025.050.075.0

100.0

0 20,000 40,000 60,000

ΔT

(mk

)

Tiempo (s)

TIRS Test 1: Gradiente de la termopila 3

Análisis realizados

• Análisis estacionarios:• Temperaturas máximas/mínimas• Convección natural• Condiciones de iluminación mas

desfavorables• Análisis de incertidumbre sobre

casos estacionarios

• Análisis transitorios:• Análisis mas relevantes desde el

punto de vista de los requisitos• Análisis de varios soles para

alcanzar una solución periódica• Análisis de sensibilidad frente a

ráfagas de viento-70.00

-50.00

-30.00

-10.00

10.00

30.00

0 20000 40000 60000 80000

Te

mp

era

ture

(°C

)

Time (s)

TIRS Case Temperatures (Worst Hot Case)

Y+

Z+

Foot1

Foot2

Ground

External Atm

RSM

Ejemplo de resultados caso estacionario caliente

Ejemplo de resultados caso transitorio caliente

Concepto experimental: control PI de los heaters para mejorar el comportamiento del instrumento

• Primera aproximación: Aplicación de un bucle de control simple con el fin de evaluar la viabilidad de reducir el gradiente de las termopilas

8

0.00

2.00

4.00

6.00

8.00

10.00

60000 62000 64000 66000

Tiempo (s)

TIRS: Temperatura de la placa soporte

MS1

Pt1000

Pt1000 PI

MS1 PI

-65.00

-62.00

-59.00

-56.00

-53.00

-50.00

18000 20000 22000 24000

Te

mp

era

tura

(°C

)

Tiempo (s)

TIRS: Temperatura de la placa soporte

MS1

Pt1000

Pt1000 PI

MS1 PI

Resultados (activación del control al atardecer)

9

• Control activado durante 30 minutos

• Estado estacionario del control alcanzado

• El gradiente espacial de las termopilas no se ve

reducido• Se puede usar la correlación potencia-gradiente para

eliminar esa fuente de error

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

60000 62000 64000

ΔT

(K

)

Time (s)

TIRS – Gradiente en las termopilas

ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

MAX

MIN

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

60000 61500 63000 64500

Time (s)

PI controller operation

Power (W) Grad TP3 (K)

∆T = -0.08 Qi R² = 1.00E+00

-0.030

-0.015

0.000

0.015

0.030

0.00 0.05 0.10 0.15 0.20 0.25

∆T

(K

)

Qi heaters (W)

Correlation ∆T-Qi

Grad Tp3

Steady State

Resultados (activación del control al amanecer)

10



• El gradiente espacial de las termopilas se reduce• No se puede aplicar la correlación entre potencia y

gradiente debido a la baja potencia aplicada en este

caso

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

19000 21000 23000 25000

ΔT

(K

)

Tiempo (s)


ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

MAX

MIN

ΔT S1 PI

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

19000 20000 21000 22000 23000

Time (s)


Qi heaters (W) ΔT S1 PI

-0.350

-0.300

-0.250

-0.200

-0.150

-0.100

-0.050

0.000

0.050

0.000 0.200 0.400 0.600 0.800 1.000

∆T

(K)

Qi heaters (W)

Correlation ∆T-Qi

Resultados (activación del control al amanecer)

11



• El gradiente espacial de las termopilas se reduce• No se puede aplicar la correlación entre potencia y

gradiente debido a la baja potencia aplicada en este

caso

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

19000 21000 23000 25000

ΔT

(K

)

Tiempo (s)


ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

MAX

MIN

ΔT S1 PI

-0.400

-0.200

0.000

0.200

0.400

0.600

0.800

1.000

19000 20000 21000 22000 23000

Time (s)


Qi heaters (W) ΔT S1 PI

∆T = -0.06 Qi + 0.002R² = 1.00E+00

-0.010

-0.005

0.000

0.005

0.010

0.015

0.020

0.000 0.050 0.100

∆T

(K)

Qi heaters (W)

Correlation ∆T-Qi

ΔT S1 PI

SteadyState

Conclusiones• Se ha mostrado el impacto de los requisitos térmicos en el ciclo de desarrollo del

instrumento, tanto en diseño como en validación y verificación.

• El comportamiento térmico del instrumento TIRS ha sido modelizado y se ha realizado un análisis completo de los escenarios más desfavorables para el instrumento.

• Se ha analizado y mostrado la viabilidad de aplicar un sistema de control para mejorar las prestaciones del instrumento.

12

Trabajos futuros• Los resultados de estos análisis y el modelo térmico desarrollado podrán dar

soporte a las operaciones en vuelo.• Prueba de nuevos modos de operación una vez en Marte.• Prueba del comportamiento térmico ante condiciones degradadas (mitigación

ante posibles incidentes de la misión).

Muchas gracias por su atención, ¿alguna pregunta?

“Two possibilities exist: either we are alone in the Universe or we are not. Both are equally terrifying.” Arthur C. Clarke

Adrián Chamorro [email protected]

Source: NASA/JPL

EXTRA

14

Thermal requirements: drivers of the thermal analysis

•Thermal requirements have guided the thermal analysis process

–Discretization of the model–Transient analyses required (temporal and spatial gradients)

–Needed of performing uncertainty analyses–Dedicated thermal test

15

0.02

0.03

0.04

0.05

0.06

0.07

69000 70000 71000 72000

ΔT

(K

)

Time (s)

Example of sensitivity analysis to wind gust

ΔT S1

ΔT S2

ΔT S3

ΔT S4

-120-100-80-60-40-20

02040

0 25000 50000 75000

Te

mp

era

ture

(°C

)

Time (s)

Environment temperature(Martian summer)

Atmosphere Ground Sky

Thermal environment modelling•Modelling of Martian thermal environment

–Undefined landing site: worst landing site selected (Holden crater)

–Solar loads•Winter and summer (high eccentricity of the Martian orbit)

•Direct and diffuse

•Rover orientation (shadows)

–Boundary temperatures:•Ground, sky, atmosphere and rover temperatures provided as inputs

–Natural and forced convection•External surfaces and atmosphere heat exchange

•Heat exchanged between internal parts because of the presence of the atmosphere

–Dust deposition on surfaces•Effect on the optical finish of the external surfaces 16

0

50

100

150

200

250

300

0 25000 50000 75000

Hea

t fl

ux

(W

/m2)

Time (s)

Direct Ground Solar heat flux(Martian Winter)

Test and model correlation• Dedicated test has been performed to reduce

the uncertatinty of the model.

• Conductive couplings between the componentsof the instrument and the capacitance of thethermopiles have been correlated with themeasurements of the test

• Test setup:

–CO2 pressurized chamber

–Performed in EM of the TIRS (the differences betweenthe EM and FM have been taken into account in theTMM of the test)

–Two test have been performed with different powerconsumption strategy.

17

-200.0

-100.0

0.0

100.0

200.0

0 20,000 40,000 60,000

Th

erm

op

ile

3 g

rad

ien

t (m

K)

Time (s)

TIRS Test 1 Thermopile 3 gradient

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

0 50,000Tem

pera

ture

(°C

)

Time (s)


Supportplate

Calibration plate

Chamber

Case

Test and correlation• Two test performed with different power profile

• Gradient in thermopiles estimated from voltage measurements of thermopile 3 during the test

18

15.0

20.0

25.0

30.0

35.0

40.0

0 20,000 40,000 60,000 80,000 100,000Tem

pera

ture

(°C

)

Time (s)


-300.0

-200.0

-100.0

0.0

100.0

200.0

0 20,000 40,000 60,000 80,000

Th

erm

op

ile

3

gra

die

nt

(mK

)

Time (s)

TIRS Test 2 Thermopile 3 gradient

Test and correlation

19

• Steady state correlation (quasy-steady state in the tests):

• Conductive couplings (contact), convective couplings ( internal) and thermal

conductivityModel results

before correlation

(temperature (°C))

CaseCalibration

plate

Support

plate

Test 1Measured 31.79 46.98 43.43

Model 31.85 53.8 40.8

Test 2Measured 26.39 28.60 35.64

Model 26.4 27.2 33.8

Results after

steady state

correlation (ºC)

Case

temperature

Calibration

plate

temperature

Support plate

temperature

Test 1Measured 31.79 46.98 43.43

Model 31.85 47.56 43.26

Test 2Measured 26.39 28.60 35.64

Model 26.40 28.21 35.58

Error

Test 1 0.06 0.58 -0.17

Test 2 -0.05 -0.39 0.01

-35.00

-25.00

-15.00

-5.00

5.00

15.00

25.00

3500 5500 7500

ΔT

(mk)

Time (s)

Grad TP3 Model C+40%(mk)

Grad Tp3 Test (mK)

Grad TP3 Model C+20%(mk)

Grad TP3 Model C+30%(mK)

Grad TP3 Model C estimated(mk)

-45.00-30.00-15.00

0.0015.0030.0045.00

3800 8800 13800 18800

Err

or

in t

he

th

erm

op

ile

3 g

rad

ien

t (m

K)

Time (s)

Error in the thermopile 3 gradient for the first and second activation of the heaters

Error Test 1C+20%

Error Test 1C+40%

Error Test 1C+30%

• Transient correlation:

• Strong dependency of the gradients with thermal capacity of the thermopiles

Worst case Analyses in operative environment

•Steady Analyses–To set the thermal envelope of the problem and check the model

–Summer day analysis•14:00 LST and highest summer boundary conditions (BC) temperatures

–Winter night analysis•No Sun and lower winter BC temp.

–Qualification analysis•Hot qualification case has been performed

•Transient Analyses–Nominal operation analyses:

•WHC, no heaters, no wind (natural convection)

•WCC, no heaters, no wind (natural convection)

–Calibration modes•Activation of calibration plate heaters and support plate heaters

–Sensitivity to wind analysis•Change in steady wind

•Gusts 20

Results

•Summer day steady results:–Power: 0 W

Temperature Results (°C)

Part Max Min Average Max ΔT

Case 12.3 10.1 11.7 2.2

Calibration Plate 13.1 13.0 13.1 0.1

Sensors support plate 12.0 12.0 12.0 0.0

Sensors cover plate 12.1 12.0 12.1 0.0

Insulation Plate 14.0 12.2 13.0 1.8

Back Plate 12.9 10.7 11.7 2.2

PCB 12.1 12.3 12.1 -0.2

Thermopiles 12.03 12.02 12.02 0.01

21

Results

•Summer day steady results

22

Transient analysis

•Example of results of a nominal operationcase

23

-0.150

-0.100

-0.050

0.000

0.050

0.100

0.150

0 20000 40000 60000 80000

ΔT

(K

)

Time (s)

TIRS Calibration Plate Gradients (WHC)

ΔT Pt1000-A1

ΔT Pt1000-A1

ΔT Pt1000-A2

ΔT Pt1000-A2

ΔT Pt1000-A2-70.00-60.00-50.00-40.00-30.00-20.00-10.00

0.0010.0020.00

0 50000Te

mp

era

ture

(°C

)

Time (s)

TIRS Calibration Plate Temperatures (WHC)

Pt1000 A2 A2 A1 A1 A2

-0.050

-0.030

-0.010

0.010

0.030

0.050

0.070

0 50000

ΔT

(K

)

Time (s)

TIRS Thermopiles Temperature Gradient (WHC)

ΔT S1ΔT S2ΔT S3ΔT S4ΔT S5 -0.014

-0.010

-0.006

-0.002

0.002

0.006

0.010

0.014

0 20000 40000 60000 80000

dT

/dt

(K/s

)

Time (s)

TIRS Thermopiles Temperature temporal gradient (WHC)

dT/dt S1

dT/dt S2

dT/dt S3

dT/dt S4

dT/dt S5

-70.00

-50.00

-30.00

-10.00

10.00

30.00

0 20000 40000 60000 80000

Te

mp

era

ture

(°C

)

Time (s)

TIRS Case Temperatures (WHC)

Y+

Z+

Foot1

Foot2

Ground

ExternalAtm

Experimental concept: PI heaters control to improve instrument performance

• First approach: Simple control loop in order to assess the

feasibility of the PI control loop to reduce the gradients or

lead to a known gradient in the thermopiles

• Control loop: • PI control loop with Kp = 0.5 and Ki =5/512

• The control loop has been applied where the maximum

gradients were reached on the transient analyses

24

0.00

2.00

4.00

6.00

8.00

10.00

60000 65000

Tem

pera

ture

(°C

)

Time (s)

TIRS Support Plate Temperatures (WCC)

MS1

Pt1000

Pt1000PI

Results (PI activation at 60000 s)

25

• PI controller activated during 30 minutes.

• Steady state of the controller achieved

• The temporal gradient in the thermopiles is

reduced

• The spatial gradient does not achieve a steady

state (variation of the boundary temperatures)

-0.030

-0.020

-0.010

0.000

0.010

0.020

0.030

0.040

0.050

60000 62000 64000

ΔT

(K

)

Time (s)


ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

MAX

MIN

-0.015

-0.010

-0.005

0.000

0.005

0.010

0.015

60000 62000 64000

dT

/dt

(K/s

)

Time (s)


dT/dt S1

dT/dt S2

dT/dt S3

dT/dt S4

• Linear correlation appears between power in the heaters during control loop and spatial gradient in the thermopiles

• Usable to improve the accuracy under certain circumstances (more effective in the afternoon)

• Deeper analysis needed to calibrate the parameters of the correlation in different situations

26

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

60000 61000 62000 63000 64000

Time (s)


Power (W)

∆T = -7.95E-02 Qi R² = 1.00E+00

-0.030

-0.015

0.000

0.015

0.030

0.000 0.100 0.200 0.300

∆T

(K

)

Qi heaters (W)

Correlation ∆T-Qi (t=61000 s) Grad Tp3

Steady State

Conclusions

•Detailed thermal analysis of the TIRS has been performed.

•Thermal behavior of the TIRS has been modeled and worst case scenarios have been analyzed.

•The strict thermal requirements have driven the activities and extra activities to increase the reliability and the accuracy of the results have been performed.

•The control of the heaters of the support plate does not lead to a steady spatial gradient in the thermopiles, but it could be used to improve the accuracy in some circumstances.

27

Future work• Thermal model and analyses could support in flight operations.

• For example new operation modes could be tested in this model

or the thermal performance of the instrument under degraded

conditions could be assessed.

EXTRA 2

28

Test and correlation

29

• The temperature gradient in the thermopiles have been correlated with the estimated temperature in the

test.

TestTest 1

measurement

Test 2

measurement

Gradient in

thermopile 3 (K)0.003 -0.020

Max

gradient

Min

gradient

Average

gradient

Gradient in

Thermopile 3

Test 1 -0.004 -0.002 -0.003 0.003

Test 2-0.026 -0.020 -0.025 -0.020

• The results after the correlation of the gradients in the thermopiles are shown in the following table. Max,

min and average gradient correspond to the five thermopiles in each one of the tests. The values which

has been correlated corresponds to the gradient in the thermopile 3.

Transient analysis•Example of results of a nominal operationcase

30

-120.00

-100.00

-80.00

-60.00

-40.00

-20.00

0.00

20.00

40.00

0 20000 40000 60000 80000

Te

mp

era

ture

(°C

)

Time (s)


Y-

Y+

Z+

Z-

Foot1

Foot2

Foot3

Foot4

Sky

Ground

External Atm

RSM

Transient analysis

•Example of results for a non-operational case

31

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0 20,000 40,000 60,000 80,000

Po

wer

(W)

Time (s)

Conductive heat transfer

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.0 50,000.0

Film

Co

eff

icie

nt

h (

W/(

m^2

·K))

Time (s)

Convective heat transfer

hconvnatTopFW

hconvnatBotFW

hconvnatLatFW

hconvnatFcFW

Transient analysis (II)

•Example of results for calibration mode (heaters of the calibration plate ON)

32

-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

19500.00 21500.00 23500.00

ΔT

(K

)

Time (s)


ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

-0.002

0.000

0.002

0.004

0.006

0.008

19500 21500 23500

dT

/dt

(K/s

)

Time (s)


dT/dt S1

dT/dt S2

dT/dt S3

dT/dt S4

dT/dt S5

Transient Analysis (III)• Example of results for WHC wind sensitivity analysis: steady wind

change

33

-30.00

-25.00

-20.00

-15.00

-10.00

-5.00

68500 69500 70500 71500 72500

Te

mp

era

ture

(°C

)

Time (s)


Y-

Z+

Foot1

Foot2

Y- no wind

Foot1 no wind

Z+ no wind

Transient Analysis (III)• Example of results for WHC wind sensitivity analysis: steady wind

change

34

0.02

0.03

0.04

0.05

0.06

0.07

69000 70000 71000 72000

ΔT

(K

)

Time (s)


ΔT S1

ΔT S2

ΔT S3

ΔT S4

ΔT S5

ΔT S5 no wind

-0.014

-0.010

-0.006

-0.002

0.002

0.006

69000 70000 71000 72000 73000

dT

/dt

(K/s

)

Time (s)


dT/dt S1

dT/dt S2

dT/dt S3

dT/dt S4

dT/dt S5

Uncertainty analysis

• Uncertainty analysis has been performed taking into account the

guidelines provided in ECSS-E-HB-31-03A

• First, a sensitivity analysis is performed by changing the

parameters of the model taking into account the following table:

• Then, the uncertainty in the different components of the model has

been calculated by quadrature:

𝜙𝑖 = 𝑗=1𝑛 (Δ𝜙𝑟,𝑗)𝑖

2 + (Δ𝜙𝑠)𝑖

where, Δ𝜙𝑖 is the overall uncertainty on model output i, Δ𝜙𝑟,𝑗 is the

uncertainty due to statistical parameters j on model output i, and

(Δ𝜙𝑠)𝑖 is the systematic uncertainty on model output i. 35

Parameter Inaccuracy

Thermal conductivity (homogeneous material) ±10%

Thermal conductivity (composites) ±30%

Contact resistance (by similarity) ±50%

Emissivity ±0.03

Emissivity (<0.2) ±0.02

Absorptance ±0.1

Absorptance (<0.2) ±0.03

-115.00

-105.00

-95.00

-85.00

-75.00

0.6 0.8 1 1.2 1.4Te

mp

era

ture

(°C

)

k FR4 (W m-1 K-1)

Sensitivity Analysis k FR4 (cold case)

Uncertainty analysis• Uncertainty analysis has been performed to

increase the knowledge about the thermal behavior and the sensitivity to the unknowns of the problem.

• First, it is needed to perform a sensitivity analysis:

36

Parameter Inaccuracy

Thermal conductivity

(homogeneous material) ±10%

Thermal conductivity

(composites)±30%

Contact resistance (by

similarity)±50%

Emissivity ±0.03

Emissivity (<0.2) ±0.02

Absorptance ±0.1

Absorptance (<0.2) ±0.03

𝛥𝜙𝑖 =

𝑗=1

𝑛

Δ𝜙𝑟,𝑗 𝑖2+ (Δ𝜙

𝑠 𝑖

-3.00

-2.00

-1.00

0.00

1.00

2.00

3.00

4.00

0.7 1 1.3

ΔTe

mp

era

ture

(°C

)

k FR4 (W m-1 K-1)

Sensitivity Analysis k FR4 (cold case)T Case 1

T Case 2

T Case 3

T Case 4

T CALP

T SP

T IP JCALP

T IP JCASE

T IP Centre

• Once the sensitivity analysis has been performed, the

uncertainty associated to the studied parameters is

computed as:

Uncertainty analysis• The results of all the analyses performed can be found on the

report

• A summary of the results is shown in the following table:

• Taking into account the results of the sensitivity analysis, the results are especially sensitive to

FR4 conductivity and contact resistance. Therefore, it is shown the importance of the dedicated

test that was performed because it has helped to correct the nominal values of the parameters of

the model, what leads to reduce the modelling uncertainty of these parameters.

37

Summary of the results of the sensitivity analysis

Analysis Case Support plateCalibration

plateInsulation plate

Δ𝑘𝐴𝑤6082Max |Δ𝑇| cold case 0.20 0.19 0.21 0.23

Max |Δ𝑇| hot case 0.48 0.45 0.45 0.52

Δ𝑘𝐹𝑅4Max |Δ𝑇| cold case 0.14 0.13 2.54 3.02

Max |Δ𝑇| hot case 0.10 0.26 2.28 2.80

Δ𝑅𝑐Max |Δ𝑇| cold case 0.55 2.09 3.91 2.39

Max |Δ𝑇| hot case 1.57 7.39 5.44 3.76

Δ𝜀Max |Δ𝑇| cold case 0.12 0.02 0.1 0.08

Max |Δ𝑇| hot case 0.16 0.23 0.36 0.40

Δ𝛼 Max |Δ𝑇| cold case - - - -

Max |Δ𝑇| hot case 1.14 0.99 1.34 1.79

Results Δ𝜙𝑟,𝑗

2cold 0.61 2.10 4.67 3.86

hot 2.00 7.48 6.08 5.06

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