A very large metal plate of thickness 𝑑 and thermal conductivity 𝑘 is cooled by a stream of air at temperature 𝑇_∞ = 300 K with a heat transfer coefficient ℎ, as shown in the figure. The centerline temperature of the plate is T_P. In which of the following case(s) can the lumped parameter model be used to study the heat transfer in the metal plate?

The correct answer is:
ℎ = 100 Wm^-2K^-1, k = 1000 Wm^-1K^-1, 𝑑 = 1 mm, T_P = 325 K

1. Physical Model and Image Analysis
From the provided diagram, we observe a flat metal plate of total thickness d. The centerline of the plate is marked at temperature T_P, and the distance from the centerline to either of the exposed outer surfaces is labeled as d/2.
The plate is symmetrically cooled on both the top and bottom surfaces by a stream of air at an ambient temperature of T_∞ = 300 K with a convective heat transfer coefficient h, indicated by the arrows representing the air flow above and below the plate.

2. Criteria for Lumped Parameter Model
The lumped parameter model assumes that the temperature within the solid is spatially uniform at any given instant during a heat transfer process. This assumption is valid when the internal conductive resistance to heat transfer is much smaller than the external convective resistance.
The relative magnitude of these resistances is measured by the dimensionless Biot number (Bi), defined as:

$$Bi=\frac{h{L}_c}{k}$$

where:
• h is the convective heat transfer coefficient.
• k is the thermal conductivity of the solid plate.
• L_c is the characteristic length of the solid.

The lumped parameter model can be reliably used when:

$$Bi<0.1$$

3. Calculation of Characteristic Length ($L_c$)
For a solid body, the characteristic length is the ratio of its volume (V) to its surface area participating in convective heat transfer (A_s).
For a very large metal plate of area A and thickness d cooled symmetrically from both sides, the heat transfer occurs through both the top and bottom faces:

$$L_c=\frac{V}{A_s}=\frac{A\cdot d}{2A}=\frac{d}{2}$$

4. Verification of the Correct Option
Let us evaluate the Biot number using the parameters specified in the correct option:
• Convective heat transfer coefficient, $h=100{\text{W m}}^{\mathrm{-2}}{\text{K}}^{\mathrm{-1}}$
• Thermal conductivity, $k=1000{\text{W m}}^{\mathrm{-1}}{\text{K}}^{\mathrm{-1}}$
• Thickness of the plate, $d=1\text{mm}=1\times {10}^{\mathrm{-3}}\text{m}$

First, compute the characteristic length:

$$L_c=\frac{1\cdot {10}^{-3}\text{ m}}{2}=0.5\cdot {10}^{-3}\text{ m}$$

Now, compute the Biot number:

$$Bi=\frac{100\cdot (0.5\cdot {10}^{-3})}{1000}=\frac{0.05}{1000}=5\cdot {10}^{-5}$$

Comparing with the criterion:

$$5\cdot {10}^{-5}<0.1$$

Because the Biot number is significantly less than 0.1, the internal thermal resistance is negligible compared to the external convection resistance. Thus, the lumped parameter model can be applied successfully.