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Experimental design for estimation of the distribution of the convective heat transfer coefficient for a bubbly impinging jet

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Abstract

In this study, a two-dimensional inverse algorithm is developed to determine the heat transfer coefficient distribution of a two-phase air–water bubbly jet impinging on a steel cylindrical thermal mass. All procedures of thermal mass heating and cooling are simulated by solving two-dimensional, transient heat conduction equations using finite difference method. Afterward, the nonlinear inverse heat conduction problem is implemented to directly predict the local convective heat transfer coefficient of the bubbly jet. The sum of squared differences between calculated and measured temperature data is the objective function. Conjugate gradient method is employed sequentially in every time step to optimize the objective function at four gas Reynolds numbers which represent four different two-phase jets. The inverse scheme is validated using exact temperature data without noise. Local heat transfer coefficients are then estimated by inverse technique at five data acquisition times and four initial temperatures of thermal mass in the presence of noise. Furthermore, the effects of uncertainties due to indefinite lateral boundary conditions, temperature dependency of thermal conductivity, and the non-uniformity of the initial temperature distribution are investigated. A satisfactory agreement between exact and estimated heat transfer coefficients is achieved. However, the results show a greater sensitivity to the highest value of initial temperature, the shortest data acquisition time, and the lowest gas Reynolds number allowing a better estimation of heat transfer distribution for the bubbly jet.

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Abbreviations

CGM:

Conjugate gradient method

\(C_{\rm P}\) :

Heat capacity (J kg−1 K−1)

\(d\) :

Nozzle tube diameter (m)

D :

Thermal mass diameter (m)

DAT:

Data acquisition time (s)

E :

Error

\(h\) :

Heat transfer coefficient (W m−2 K−1)

\(H\) :

Nozzle-to-target spacing (m)

\(H_{\rm h}\) :

Heater thickness (m)

\(H_{_{\rm ins}}\) :

Insulation thickness (m)

\(H_{\rm s}\) :

Steel thermal mass thickness (m)

IHCP:

Inverse heat conduction problem

\(k\) :

Thermal conductivity (W m−1 K)

\(Nu\) :

Nusselt number (hd/kL)

\(q^{\prime\prime}\) :

Heat flux (W m−2)

\(r\) :

Radial distance from stagnation point (m)

\(Re\) :

Reynolds number (ρvd/µ)

S:

Sensor

\(S\) :

Sum of squared errors (K2)

\(\vec{S}\) :

Search direction

\(t\) :

Time (s)

\(T\) :

Temperature (K)

\(T_{\rm i}\) :

Initial temperature (K)

\(T_{\infty }\) :

Surrounding temperature (K)

\(Y\) :

Measured temperature (K)

\(z\) :

Vertical distance from stagnation point (m)

\(Z\) :

Sensitivity coefficient (m2 K2 W−1)

b:

Bottom

e:

End

f:

Final

G:

Gas

i:

Number of iterations

ins:

Insulation

j:

Jet

L:

Liquid

meas:

Measured

p:

Peripheral

rms:

Root mean square

s:

Steel

t:

Top

\(\alpha\) :

Thermal diffusivity (m2 s−1)

\(\gamma\) :

Conjugate coefficient

\(\lambda\) :

Step size

\({\rho }\) :

Density (kg m−3)

\(\sigma\) :

Standard deviation

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Authors and Affiliations

  1. School of Mechanical Engineering, University College of Engineering, University of Tehran, Tehran, 1439957131, Iran

    Farshad Kowsary, Honeyeh Razzaghi & Mehdi Ashjaee

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  1. Farshad Kowsary
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  2. Honeyeh Razzaghi
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Kowsary, F., Razzaghi, H. & Ashjaee, M. Experimental design for estimation of the distribution of the convective heat transfer coefficient for a bubbly impinging jet. J Therm Anal Calorim 140, 439–456 (2020). https://doi.org/10.1007/s10973-019-08819-z

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