Question

For laminar flow over a flat plate, the local heat transfer coefficient varies as \(h_{x}=C x^{-0.5}\), where \(x\) is measured from the leading edge of the plate and \(C\) is a constant. Determine the ratio of the average convection heat transfer coefficient over the entire plate of length \(L\) to the local convection heat transfer coefficient at the end of the plate.

Short answer

Answer: The ratio between the average convection heat transfer coefficient over the entire plate length and the local heat transfer coefficient at the end of the plate is 2.

Step-by-step solution

Determine the average heat transfer coefficient over the entire plate length

First, we need to determine the average convection heat transfer coefficient (\(\overline{h}\)) over the entire length of the plate, which can be found by integrating the given function of \(h(x)\) with respect to \(x\), and then dividing this result by the length of the plate. This can be expressed as: $$ \overline{h} = \frac{1}{L} \int_0^L h_x dx = \frac{1}{L} \int_0^L C x^{-0.5} dx $$ We can now find this integral and obtain the value for the average heat transfer coefficient.

Integrate the heat transfer coefficient function

Integrating the given function of \(h_x\) with respect to \(x\) gives the following result: $$ \int_0^L C x^{-0.5} dx = C \int_0^L x^{-0.5} dx = 2C \left[x^{0.5}\right]_0^L = 2CL^{0.5} $$

Calculate the average heat transfer coefficient

Plugging the result of the integral back into the equation for the average heat transfer coefficient, we can find the value for \(\overline{h}\): $$ \overline{h} = \frac{1}{L} (2CL^{0.5}) = 2C L^{-0.5} $$

Find the local heat transfer coefficient at x=L

The local heat transfer coefficient at the end of the plate, \(h_L\), is given by the same function \(h_x\), and we need to plug in \(x=L\) to find its value: $$ h_L = C L^{-0.5} $$

Determine the ratio of average to local heat transfer coefficients

Now that we have calculated both the average and local heat transfer coefficients, we can find the ratio of \(\overline{h}\) to \(h_L\). We can do that by dividing \(\overline{h}\) by \(h_L\) as follows: $$ \frac{\overline{h}}{h_L} = \frac{2C L^{-0.5}}{C L^{-0.5}}=2 $$ Thus, the ratio between the average convection heat transfer coefficient over the entire plate length and the local heat transfer coefficient at the end of the plate is 2.