Question

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 $${\mathrm{Nu}}_x=0.035{\mathrm{Re}}_x^{0.8}{\mathrm{Pr}}^{1/3}$$ 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 $x=L$.

Short answer

Answer: The ratio of the average convection heat transfer coefficient to the local convection heat transfer coefficient at x=L is given by: $$\frac{h_{avg}}{h_x}=\frac{N{u}_{avg}}{N{u}_L}$$ Find the values of $N{u}_{avg}$ and $N{u}_L$ by integrating the given Nusselt number correlation and using the resulting expressions for local and average convection heat transfer coefficients.

Step-by-step solution

Understanding Nusselt number

The Nusselt number (Nu) is a dimensionless number that represents the ratio of convective heat transfer to conductive heat transfer. The higher the Nusselt number, the better the heat transfer performance. In this exercise, we have the local Nusselt number $N{u}_x$ given as a function of the local Reynolds number, $R{e}_x$, and the Prandtl number, Pr: $$N{u}_x=0.035R{e}_x^{0.8}P{r}^{1/3}$$

Express local convection heat transfer coefficient in terms of Nusselt number

To find the local convection heat transfer coefficient ($h_x$) in terms of the Nusselt number, we can use the following formula: $$h_x=\frac{N{u}_x\cdot k}{x}$$ Where $k$ is the thermal conductivity and $x$ is the distance from the leading edge of the flat plate.

Calculate the average Nusselt number using the given correlation

To find the average Nusselt number ($N{u}_{avg}$) for the entire flat plate length, we need to integrate the given Nusselt number correlation over the plate length and divide by the plate length: $$N{u}_{avg}=\frac{1}{L}{\int }_0^{L}N{u}_{x}dx$$ Substitute the given correlation for $N{u}_x$ into the formula above: $$N{u}_{avg}=\frac{1}{L}{\int }_0^L(0.035R{e}_x^{0.8}P{r}^{1/3})dx$$

Express average convection heat transfer coefficient in terms of Nusselt number

Similar to the local convection heat transfer coefficient, we can express the average convection heat transfer coefficient ($h_{avg}$) in terms of the average Nusselt number using the following formula: $$h_{avg}=\frac{N{u}_{avg}\cdot k}{L}$$

Find the ratio of $h_{avg}$ to $h_x$

We now have the expressions for both the local and average convection heat transfer coefficients, so we can calculate the desired ratio: $$\frac{h_{avg}}{h_x}=\frac{\frac{N{u}_{avg}\cdot k}{L}}{\frac{N{u}_x\cdot k}{x}}$$ Simplify the equation by canceling out $k$ and then substitute $x=L$: $$\frac{h_{avg}}{h_x}=\frac{N{u}_{avg}}{N{u}_L}$$ Determine the ratio after integrating in step 3 and substituting the results into this equation.