Chapter 1, Problem 1.51P

Three electric resistance heaters of length $L=250\text{mm}$ and diameter $D=25\text{mm}$ are submerged in a 10-gal tank of water, which is initially at 295 K. The water may be assumed to have a density and specific heat of $\rho =990{\text{kg/m}}^{\text{3}}$ and $c=4180\text{J/kg}\cdot \text{K}\text{.}$

1. If the heaters are activated, each dissipating $q_1=500\text{W,}$ estimate the time required to bring the water to a temperature of 335 K.
2. If the natural convection coefficient is given by an expression of the form $h=370{\left(T_s-T\right)}^{1/3},$ where $T_s$ and T are temperatures of the heater surface and water, respectively, what is the temperature of each heater shortly after activation and just before deactivation? Units of h and $\left(T_s-T\right)$ are ${\text{W/m}}^{\text{2}}\cdot \text{K}$ and K, respectively.
3. If the heaters are inadvertently activated when the tank is empty, the natural convection coefficient associated with heat transfer to the ambient air at $T_{\infty }=300\text{K}$ may be approximated as $h=0.70{\left(T_s-T_{\infty }\right)}^{1/3}.$ If the temperature of the tank walls is also 300 K and the emissivity of the heater surface is $\epsilon =0.85,$ what is the surface temperature of each heater under steady-state conditions?