Chapter 3, Problem 3.9P

A technique for measuring convection heat transfer coefficients involves bonding one surface of a thin metallic foil to an insulating material and exposing the other surface to the fluid flow conditions of interest.

By passing an electric current through the foil, heat is dissipated uniformly within the foil and the corresponding flux, $P_{elec}^{"},$ may be inferred from related voltage and current measurements. If the insulation thickness L and thermal conductivity k are known and the fluid. foil, and insulation temperatures $\left(T_{\infty },T_s,T_b\right)$ are measured. the convection coefficient may be determined. Consider conditions for which $T_{\infty }=T_b=25°\text{C,}{P}_{elec}^{"}=2000{\text{W/m}}^{\text{2}},L=10\text{mm,}$ and $k=0.040\text{W/m}\cdot \text{K}\text{.}$

With water flow over the surface, the foil temperature measurement yields $T_s=27°\text{C}\text{.}$

1. Determine the convection coefficient. What error would be incurred by assuming all of the dissipated power to be transferred to the water by convection?
If, instead, air flows over the surface and the temperature measurement yields $T_s=125°\text{C,}$ what is the convection coefficient? The foil has an emissivity of 0.15 and is exposed to large surroundings at 25°C. What error would be incurred by assuming all of the dissipated power to be transferred to the air by convection? Typically. heat flux gages are operated at a fixed temperature $\left(T_s\right)$ in which case the power dissipation provides a direct measure of the convection coefficient. For $T_s=27°\text{C,}$ plot $P_{elec}^{"}$ as a function of $h_o$ for $10\le h_o\le 1000{\text{W/m}}^{\text{2}}\cdot \text{K}\text{.}$ What effect does $h_o$ have on the error associated with neglecting conduction through the insulation?