Theoretical Modelling of Heat Flux Measurement

1.2) Considering the substrate to be a semi-infinite solid maintained at an initial temperature T i and is

3.3 Theoretical Modelling of Heat Flux Measurement

independent on the polarity of the material i.e. selection of outer material as chromel or alumel.

Lastly, it is seen from Fig. 3.5, that the temperature-time history trend clearly depicts a parabolic behaviour, which is also in agreement with literature [Holman, 2001].

2

1 1 1 1

2 1

 

 

 

  

  

T c T

k t

x



(3.1) 2

2 2 2 2

2 2

T c T

k t

x



 

 

 

  

  

(3.2) The suffix 1 and 2 in Eq. (3.1) and Eq. (3.2) depicts the sensing surface and the substrate material of the heat flux gauges.

In order to have uniform initial conditions, if a heat load of qS

 

t



is applied instantly, the transient temperature of the coaxial sensor will beT tS

 

. Therefore, referring to Fig (3.1) initial and boundary condition can be writtenas follows:

Initial conditions: t0and 0 x l₂; T x

 

^{, 0} Tamb ³⁰⁰K Top wall: x^0,t ^0;q

 

^0,t qS

 

t

Interface: xl t₁, 0;q_S1q_S2 q_S (3.3) and, T1T2 T tS

 

Bottom wall: xl t2, 0;T l t

 

2, 300K

Now, solving the Eq. (3.1) and Eq. (3.2), considering the thermal properties of the coaxial sensors as constant, the heat flux^q_S

 

^t passing through the surface xl₁is calculated using the Duhamel’s superposition integral as given below [Taler, 1996; Carslaw and Jaeger, 1959].

     

0

1 ;

t

s S

q t d T d ck

t dt

    

 

 



 (3.4)

There are certain assumptions for evaluating the heat flux from the governing equation as given in Eq. (3.4). The assumptions hold good only if,

(i) the temperature measured by the sensing element/film is identical to the substrate surface temperature,

(ii) the substrate of the sensor does not undergo any lateral heat conduction and the heat is only conducted in the normal direction of the substrate, and

(iii) due to very small experiment time scale, the thermal properties of the substrate are treated as constant.

All the calorimetric gauges rely on transient surface temperature histories for prediction of surface heat flux by using Eq. (3.4). In many instances, the “Thermal Product (TP)” values are generally assumed from theoretical estimates, which are either thermal properties of the substrate (in the case of thin film gauges) or the surface junction (for CSJTs). In any case, the accuracy of surface heat flux predications mainly depends on the correctness of TP values as well as the acquisition of transient surface histories from CSJTs.

For using the above-mentioned Eq. (3.4), it is desirable to have a closed form solution of transient temperature data and the estimation of the value of the thermal product ( ). In the present case, three different discretization techniques namely, linear fit, polynomial fit and cubic-spline based techniques have been employed to study further and estimate a closed form solution [Sahoo and Peetala, 2010; Schultz and Jones, 1973].

       

 

0

1 ;

2

t L

T t T t T

q t ck

t t

   

 

  

  



   (3.5) Since, the thickness of the surface junction is small and further the same heat flux must leave the substrate, so the surface heat flux (i.e. at x=0) can be assumed asqL

 

t qS

 

t .

The Eq. (3.5) is the most effective form of temperature data analysis when the heat transfer rate is not constant. In many practical cases, the function in the Eq. (3.5) cannot be described by a simple expression; so, it is essential to perform numerical integration by discretizing the temperature data. The obtained temperature-time history out of the thermal sensors can be utilized to smoothen the data using three different curve fitting techniques namely, piecewise linear, least square and cubic spline.

The piecewise linear function is assumed for temperature if the surface temperature between successive times is assumed to vary linearly with time data [Sahoo and Peetala 2011;

Sahoo and Peetala, 2010; Cook and Felderman, 1966].



2

  

2

^{ }

1 ²

^{ }

²

^{  }

¹ 1

^

i i

i i

linear

T t T t

T T t t

  _  ^t ^   _

 (3.6) Where,  t_i i t n

 

 i t i^; 0,1, 2,3,...,n (3.7)

The factor ‘n’ is the number of temperature data points recorded by the acquisition system, during the experiment. The simplified expression for Eq. (3.5) is given as below;

    ^{   }



²

 

^{2 1}



2 2 2

1 2 1 2

1 1

2

n

i i

L linear

i n i n i

T t T t

q t c k

t t t t







 

 

  



(3.8) Further, polynomial-based data discretization techniques can also be utilized for obtaining the heat flux:



2

  

0 1 2 ² 3 ³

0

...

m

m i

m i

poly i

T t A A t A t A t A t A t



     



(3.9)

Additionally, the heat flux value can be evaluated using the following equation

   

^{  }

^{ }  

   

2 1 2

2 2 2 1

1

2

1

1 1 !

2 1

2 1 ! 1 !

m

i i k

L poly i

i

k

i

q t c k A t iA t

k k i k





 





    

  

     

 

  

     

 

 

(3.10)

With the help of regression analysis and matrix inversion technique, the coefficients

0, 1, 2, 3,... _m

A A A A A can be obtained using the relation as given by the researcher Taler (1996).

The analysis deals with polynomial fitting for estimation of surface temperature and heat flux. The major flaw of Eq. (3.10) lies in the accuracy of the polynomial fitting; as only, the calculation of high-order derivatives are allowed using the polynomial fitting of higher order, which may not reproduce thereal data points, especially when taken into consideration that the time spread of the fitted data is large. In addition, one more technique is utilized to fit the experimental data points given by the cubic-spline method, the mathematical expression of which follows:

         

 

2 3

1, 2, 3, 4,

1

1 1

2 6

, 1, 2,3,...,

S spline i i i i i i i

i i

T a a a a

for i M

      

  _

      

  

(3.11)

where, ^T_s

 

^ is the surface temperature history.

The constants in the Eq. (3.11) can be evaluated using the following expression:

 

^'

 

^''

 

^'''

 

1,_i 2 _i ; 2,_i 2 _i ; 3,_i 3 _i ; 4,_i 4 _i

a T  a T  a T  a T  (3.12) The surface heat flux for the spline technique can be discretized in the following form:

   



^{1/ 2 1/ 2}



i i i 1

3 3 2

M 1 M M

2 2 2 i 2 2 2 2 2

3 5

i i t

M 4,i

i 1 2 2

M M

5 5

4,i 2 2

i i

V P R

V P

ρ c k W ρ c k

t 2 P R 2 S

W a

π 3 π

P P

3 10

a P R

10

L spline

q





   

    

    

        

 

        



(3.13)

2

3,i 2 4,i 3 i i

i M 1 i i M 1 1 i i 1,i 2,i i i i i i 2

M 1 M 1

t

a a dF d F

P τ τ ;  R τ τ ;  F a a P P P ;  V ;  W   ;

2 6 dτ dτ

S time scaling factor

   

 

        



(3.14)

A numerical algorithm is developed in-house using MATLAB® for discretization of temperature data (Eq. (3.11)) and subsequent computation of surface heat flux using the numerical algorithm (Eq. (3.13)) from the known values of thermal properties [Sahoo and Peetala, 2010].

Fig. 3.8: Variation of heat flux extracted from the CFD simulation using the in-house developed code Furthermore, the obtained temperature signal from the CFD simulation as shown in section 3.2, i.e. for both 2-D and 3-D geometry under a step input time of 1 s, are processed through the in-house code for checking the validity of the ANSYS model. A cubic spline discretization technique as per Eq. (3.11), were considered to discretize the present set of data (Fig. 3.8). The results produced a heat flux value, which is approximately equal to 1000 W/m², same as the input value given to the CFD with an uncertainty of ±5 %. The correctness of simulation is thus validated

nutshell, it validates the assumption of choosing the dimension of the coaxial surface junction thermocouple to be appropriate. For all the fabrication process, a length of 10 mm is considered for the thermal probe.