Consider one-dimensional steady state heat conduction, without heat generation, in a plane wall; with boundary conditions as shown in the figure below. The conductivity of the wall is given by $$k = {k_0} + bT;$$ where $${k_0}$$ and $$b$$ are positive constants, and $$T$$ is temperature.

As $$x$$ increases, the state temperature gradient $$(dT/dx)$$ will

Consider one-dimensional steady state heat conduction along x-axis $$\left( {0 \le x \le L} \right),$$ through a planewall with the boundary surfaces $$(x=0$$ and $$x=L)$$ maintained at temperatures of $${0^ \circ }C$$ and $${100^ \circ }C$$. Heat is generated uniformly throughout the wall. Choose the CORRECT statement.

A steel ball of diameter $$60$$ $$mm$$ is initially in thermal equilibrium at $${1030^ \circ }C$$ in a furnace. It is suddenly removed from the furnace and cooled in ambient air at $${30^ \circ }C$$ with convective heat transfer coefficient $$h = 20\,W/{m^2}K.$$ The thermo-physical properties of steel are: density $$\rho = 7800\,\,kg/{m^3},$$ , conductivity $$k = 40 W/mK$$ and specific heat $$c = 600 J/kgK.$$ The time required in seconds to cool the steel ball in air from $${1030^ \circ }C$$ to $${430^ \circ }C$$ is

Two large diffuse gray parallel plates, separated by a small distance, have surface temperatures of $$400$$ $$K$$ and $$300$$ $$K.$$ If the emissivities of the surfaces are $$0.8$$ and the Stefan-Boltzmann constant is $$5.67 \times {10^{ - 8}}$$ $$W/{m^2}{K^4},$$ the net radiation heat exchange rate in $$kW/{m^2}$$ between the two plates is