Mathematical modeling and experimental validation of heat transfer during upward subcooled flow boiling in a vertical annulus

Abstract

In this paper, subcooled flow boiling heat transfer was experimentally and numerically investigated to determine the temperature distribution, outlet temperature, and heat transfer coefficient in a vertical annulus with laminar upward flow of water as a working fluid. Experiments were developed in a way that the effect of different parameters such as fluid mass flux (26, 35 and 44 kg m⁻² s⁻¹), fluid inlet temperature (333 and 343 K), and heat flux (40–90 kW m⁻²) to be clarified. A mathematical model was then developed to determine the temperature distribution (in r and z directions), and the heat transfer coefficient along the heat transfer surface with the help of the asymptotic model, predicting more accurate results in comparison with Chen’s model. The advantage of this model over the empirical correlations is that it can predict the surface and bulk temperatures according to the operational conditions of the problem and use the obtained temperatures to calculate the thermal parameters of the system along the heat transfer surface, so, there is no need to obtain time-consuming and expensive experimental data. The maximum values of the contribution of nucleate boiling heat transfer mechanism to the total heat transfer were determined about 0.5 to 0.7 with the concept of nucleate boiling fraction. Moreover, the thermal boundary layer thickness was theoretically calculated over the heat transfer surface. The results showed that the maximum boundary layer thickness developed from 0.003 to 0.01 m when the mass flux decreases from 44 to 2.6 kg m⁻² s⁻¹. Effect of increasing mass flux and decreasing heat flux on the location of onset of nucleate boiling (ONB) demonstrated that ONB location shifted to the right from 0.017 to 0.043 m in z direction which implied the subcooled flow boiling incipience at the distance lower than 0.05 m from the beginning of the heat transfer surface.

Appendices

Appendix A

Wilson plot technique for calculation of $\frac{\Delta x}{{\uplambda }_{\mathrm{w}}}$

The actual surface temperature can be calculated using Fourier’s conduction equation, as follows:

$$ T_{{\text{w}}} = T_{{{\text{th}}}} - q\frac{\Delta x}{{\lambda_{{\text{w}}} }} $$

(41)

The value of $\Delta x/\lambda_{{\text{w}}}$ is obtained for thermocouples using Wilson plot technique. The main goal of Wilson plot technique is calculation of convection coefficient of the fluid or any other thermal resistance participated in the overall thermal resistance. The overall thermal resistance of the system can be expressed as the sum of two thermal resistances corresponding to the stainless steel wall and convection from surface to the fluid bulk.

$$ q = U\left( {T_{{{\text{th}}}} - T_{{\text{b}}} } \right) = \alpha \left( {T_{{\text{w}}} - T_{{\text{b}}} } \right) = \frac{{\lambda_{{\text{w}}} }}{\Delta x}\left( {T_{{{\text{th}}}} - T_{{\text{w}}} } \right) $$

(42)

Equation (42) can be simplified to Eq. (43):

$$ \frac{1}{U} = \frac{1}{\alpha } + \frac{\Delta x}{{\lambda_{{\text{w}}} }} $$

(43)

where U denotes the overall thermal heat transfer coefficient, and $\alpha$ is heat transfer coefficient. For estimating $\frac{\Delta x}{{\lambda_{{\text{w}}} }}$, the value of heat transfer coefficient is necessary. Mathematical relationship of heat transfer coefficient to some measurable values like friction factor (f) and velocity (v) can be presented as follows [37]:

$$ \alpha \propto f.{\text{Re}} $$

(44)

The friction factor related to Reynolds number according to Blasius’s correlation [52].

$$ f \propto \frac{1}{{{\text{Re}}^{0.25} }} $$

(45)

Combination of Eqs. (44) and (45) yields Eq. (46).

$$ \alpha \propto {\text{Re}}^{0.75} $$

(46)

where

$$ {\text{Re}} = \frac{{\rho \cdot v \cdot d_{{\text{h}}} }}{\mu } $$

(47)

By combining Eqs. (46) and (47) and inserting into Eq. (43) as well as bringing together all the constants as “C₁”, Eq. (48) is obtained:

$$ \frac{1}{U} = \frac{{C_{1} }}{{v^{0.75} }} + \frac{\Delta x}{{\lambda_{{\text{w}}} }} $$

(48)

Equation (48) shows that the plot of 1/U versus 1/v^0.75 for each thermocouple gives the values of $\frac{\Delta x}{{\lambda_{{\text{w}}} }}$ as the intercept of the line. A sample figure, Fig. 

Fig. 16

Wilson plot line for the calculation of $\frac{\Delta x}{{\uplambda }_{w}}$ value for a thermocouple

16 is plotted for one of the thermocouples to show the variation of overall thermal resistance (U⁻¹) against ν^−0.75. As can be seen from Fig. 16, the value of $\frac{\Delta x}{{\lambda_{{\text{w}}} }}$ is estimated about 6 $\times$ 10^–4 m² K W⁻¹ and applied to modify the surface temperature using Eq. (41).

Appendix B

The temperature dependent physical properties of water used in Eq. (10):

$$ C_{{\text{p}}} = 4.21 - 0.005T + 0.0012T^{1.5} - 0.00011T^{2} + 0.000004T^{2.5} $$

$$ \rho = 999.8 + 0.06T - 0.01T^{2} + 0.0008T^{2.5} - 0.000023T^{3} $$

$$ \mu = \frac{1}{{\left( {557.8 + 19.4T + 0.13T^{2} - 0.00031T^{3} } \right)}} $$

Appendix C

Algorithm used for calculation of tuning parameters of α₀ and n based on the experimental data applied in Chen and asymptotic models.

Appendix D

Algorithm used in this research to solve the equation in the subcooled flow boiling region.