Conjugate heat transfer study of incompressible turbulent offset jet flows

Abstract

In the present case, the conjugate heat transfer involving a turbulent plane offset jet is considered. The bottom wall of the solid block is maintained at an isothermal temperature higher than the jet inlet temperature. The parameters considered are the offset ratio (OR), the conductivity ratio (K), the solid slab thickness (S) and the Prandtl number (Pr). The Reynolds number considered is 15,000 because the flow becomes fully turbulent and then it becomes independent of the Reynolds number. The ranges of parameters considered are: OR = 3, 7 and 11, K = 1–1,000, S = 1–10 and Pr = 0.01–100. High Reynolds number two-equation model (k–ε) has been used for turbulence modeling. Results for the solid–fluid interface temperature, local Nusselt number, local heat flux, average Nusselt number and average heat transfer have been presented and discussed.

Appendix

1.1 Deriving the expression for heat flux in the fluid side

At the interface between the solid and fluid, the following conditions are applied.

• θ _s  = θ _f at the interface.

• Heat transfer across the interface must be equal.

Wall heat flux in the fluid side is given by:

$$ q_{f} = \frac{{\left( {\theta_{w} - \theta_{p,f} } \right)C_{\mu } k_{n}^{3/2} }}{{Pr_{t} \left( {\frac{1}{\kappa }\log (EY^{ + } ) + P_{f} } \right)}} $$

(11)

where P _f pee-function, which is given by:

$$ P_{f} = 9.24\left[ {\left( {\frac{Pr}{{Pr_{t} }}} \right)^{3/2} - 1} \right] \times \left[ {1 + 0.28\exp \left( { - 0.007\frac{Pr}{{Pr_{t} }}} \right)} \right] $$

(12)

Wall heat flux in the solid side is given by:

$$ q_{s} = - \frac{1}{Re \cdot Pr}\left( {\frac{{k_{s} }}{{k_{f} }}} \right)\frac{\partial \theta }{\partial Y} = - \frac{1}{Re \cdot Pr}\left( {\frac{{k_{s} }}{{k_{f} }}} \right)\frac{{\theta_{w} - \theta_{p,s} }}{\Updelta Y} $$

(13)

Interface temperature is calculated by equating Eqs. 11 and 13. Where θ_p,f, θ_p,s are neighbor temperatures in the fluid and solid regions.

1.2 Deriving the expression for Nusselt number calculation

We can write the above equation as:

$$ Nu_{x} = \frac{{h_{c} H}}{k} = h_{c} \left( {\overline{T}_{w} - T_{\infty } } \right) \times \frac{\upsilon }{\alpha } \cdot \frac{1}{{\rho C_{p} }} \cdot \frac{1}{{U_{0} (\overline{T}_{w} - T_{\infty } )}} \cdot \frac{{U_{0} H}}{\upsilon } $$

(14)

$$ Nu_{x} = \frac{{Q_{w} \cdot Re \cdot Pr}}{{\rho C_{p} \left( {\overline{T}_{w} - T_{\infty } } \right)}} $$

(15)

Finally,

We can write the above equation as:

$$ Nu_{x} = \frac{{Q_{w} \cdot Re \cdot Pr}}{{\rho C_{p} \left( {\overline{T}_{w} - T_{\infty } } \right)}} \cdot \frac{{\left( {\overline{T}_{h} - T_{\infty } } \right)}}{{\left( {\overline{T}_{h} - T_{\infty } } \right)}} $$

(16)

Finally,

$$ Nu_{x} = \frac{{Q_{w,n} \cdot Re \cdot Pr}}{{\overline{\theta }_{w} }} $$

(17)

Since θ_∞ = 0. Which is used for calculating the Local Nusselt number distribution. The average Nusselt number is calculated as:

$$ \overline{Nu} = \int\limits_{0}^{L} {Nu_{x} dx} $$

(18)