In an experiment, the local heat transfer over a flat plate were correlated in the form of local Nusselt number as expressed by the following correlation $$ \mathrm{Nu}_{x}=0.035 \mathrm{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3} $$ Determine the ratio of the average convection heat transfer coefficient \((h)\) over the entire plate length to the local convection heat transfer coefficient \(\left(h_{x}\right)\) at \(x=L\).

We start this exercise by recalling the definitions of Nusselt number, average Nusselt number, and their connections to the convection heat transfer coefficients: $$ \text{Nusselt number (Nu)} = \frac{h_x \cdot x}{k} $$ $$ \text{Average Nusselt number (Nu}_{avg}) = \frac{h \cdot L}{k} $$ where \(h_x\) is the local convection heat transfer coefficient, \(h\) is the average convection heat transfer coefficient, \(k\) is the thermal conductivity of the fluid, and \(x\) or \(L\) are the distance along the flat plate.

Based on the definition of Nusselt number and average Nusselt number, we can express the convection heat transfer coefficients \(h_x\) and \(h\) in terms of the Nusselt numbers as follows: $$ h_x = \frac{\mathrm{Nu}_x \cdot k}{x} $$ $$ h = \frac{\mathrm{Nu}_{avg} \cdot k}{L} $$

The given correlation is: $$ \mathrm{Nu}_{x}=0.035 \mathrm{Re}_{x}^{0.8} \operatorname{Pr}^{1 / 3} $$ At \(x = L\), the value of the local Nusselt number would be: $$ \mathrm{Nu}_{L}=0.035 \mathrm{Re}_{L}^{0.8} \operatorname{Pr}^{1 / 3} $$ where \(\mathrm{Re}_L\) is the Reynolds number at \(x=L\).

Now that we know the local Nusselt number value at \(x = L\), let's create an expression for the ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient. Using the expressions derived in step 2, the ratio can be written as: $$ \frac{h}{h_x} = \frac{\mathrm{Nu}_{avg} \cdot k / L}{\mathrm{Nu}_L \cdot k / L} $$ We can simplify this expression by cancelling out the \(k\) and \(L\) terms: $$ \frac{h}{h_x} = \frac{\mathrm{Nu}_{avg}}{\mathrm{Nu}_L} $$ So the ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient at \(x = L\) is equal to the ratio of the average Nusselt number to the local Nusselt number at the same position. To get the numerical value of this ratio, we would need to find the values for \(\mathrm{Nu}_{avg}\) and \(\mathrm{Nu}_L\). This information is not provided in the problem statement, and thus we cannot compute an exact value but we have derived the relationship between the average and local coefficients.