A heating cable is embedded in a concrete slab for snow melting. The heating cable is heated electrically with joule heating to provide the concrete slab with a uniform heat of $1200\frac{W}{m^2}.$ The concrete has a thermal conductivity of $1\mathrm{.}4\frac{W}{m}*K.$ To minimize thermal stress in the concrete, the temperature difference between the heater surface $(T_1)$ and the slab surface $(T_2)$ should not exceed $18C(2015$ ASHRAE Handbook$$HVAC Applications, Chap. $51).$ Formulate the temperature profile in the concrete slab, and determine the thickness of the concrete slab $(L)$ so that $T_1-T_{2}18C.$

Is the temperature profile in the concrete slab $T(x)=\frac{q_0^{}}{k}(L-x)+T_2?$

$(),$ the temperature profile$,T(x)=\frac{q_0^{}}{k}(L-x)+T_2$

The thickness of the concrete slab so that $T_1-T_{2}18C$ is mm $.$