Ambient air flows over a heated slab having flat, top surface at 𝒚 = 𝟎. The local temperature (in Kelvin) profile within the thermal boundary layer is given by 𝑻(𝒚) = 𝟑𝟎𝟎 + 𝟐𝟎𝟎 𝐞𝐱𝐩 (−𝟓𝒚), where 𝒚 is the distance measured from the slab surface in meter. If the thermal conductivity of air is 1.0 W/m.K and that of the slab is 100 W/m.K, then the magnitude of temperature gradient |𝒅𝑻/𝒅𝒚| within the slab at 𝒚 = 𝟎 is _______________ K/m (round off to the nearest integer).

The correct answer is 10.

Based on the provided schematic, we analyze the thermal boundary layer developing as ambient air flows over a heated slab with a flat top surface at y = 0. The image displays the given parameters:
- Thermal conductivity of air: $k_{\text{air}}=1.0\text{W/m·K}$
- Thermal conductivity of the slab: $k_{\text{slab}}=100\text{W/m·K}$
- Local temperature profile in the air thermal boundary layer: $T\left(y\right)=300+200e^{-5y}$

At the interface ($y=0$), the heat flux leaving the slab by conduction must equal the heat flux entering the air boundary layer. Applying Fourier's Law of conduction at the interface yields:
$-k_{\text{slab}}{\left(\frac{dT}{dy}\right)}_{\text{slab,}y=0}=-k_{\text{air}}{\left(\frac{dT}{dy}\right)}_{\text{air,}y=0}$

First, we determine the temperature gradient in the air boundary layer by differentiating $T\left(y\right)$ with respect to $y$:
${\left(\frac{dT}{dy}\right)}_{\text{air}}=\frac{d}{dy}\left(300+200e^{-5y}\right)=200·\left(-5\right)e^{-5y}=-1000e^{-5y}$

Evaluating this gradient at the interface surface ($y=0$):
${\left(\frac{dT}{dy}\right)}_{\text{air,}y=0}=-1000e^0=-1000\text{K/m}$

Next, we substitute the interface heat flux balance equation with the known values:
$-100·{\left(\frac{dT}{dy}\right)}_{\text{slab,}y=0}=-1·\left(-1000\right)$
$100·{\left(\frac{dT}{dy}\right)}_{\text{slab,}y=0}=-1000$
${\left(\frac{dT}{dy}\right)}_{\text{slab,}y=0}=-10\text{K/m}$

Taking the absolute value, the magnitude of the temperature gradient within the slab at $y=0$ is:
$\left|{\left(\frac{dT}{dy}\right)}_{\text{slab,}y=0}\right|=\left|-10\right|=10\text{K/m}$