# Characteristic dimension for combining free and forced convection?

I have a situation that I have a combination of free and forced convection over a cylinder. Remembering back from heat transfer, to combine these I use the formula

$$Nu_{\mathrm{comb}}^n = Nu_{\mathrm{forced}}^n + Nu_{\mathrm{free}}^n$$

And that's fine. However, what dimension do I use to find $h_{\mathrm{comb}}$? Because

$$Nu_{\mathrm{forced}} = \frac{h_{\mathrm{forced}} k}{D}$$

while

$$Nu_{\mathrm{free}} = \frac{h_{\mathrm{free}} k}{L}$$

• Can you provide more details? Are you trying to calculate an effective heat transfer coefficient $h_{comb}$ that will be valid for both forced and free? Or you have two separate regions that are exposed to different regimes? Feb 3 '17 at 22:07
• And by the way, the formulas are wrong. L is in the numerator Feb 3 '17 at 22:14
• Sorry, it was a typo. $${Nu}_{free}=\frac{h_{free} L}{k}$$ and $${Nu}_{forced}=\frac{h_{forced} D}{k}$$ Feb 4 '17 at 5:04
• I'm trying to calculate the combined heat transfer coefficient Feb 4 '17 at 5:05
• Okay. Then my answer should be okay Feb 4 '17 at 8:47

Okay, I'm not 100% sure if I got the question right but, to combine natural and forced convection for flow over a cylinder (I assumed horizontal) you have the following formula: $$Nu_{comb}=(Nu_{free}^n+Nu_{forced}^n)^{1/n}\tag{1}$$ where $n$ depends on your cylinder type (horizontal or vertical). Next you need to calculate the Nu for each regime (free and forced). For the natural convection over a horizontal cylinder, you have this correlation: $$Nu_{free}=\left(0.6+\frac{0.387Ra^{1/6}}{[1+(0.559/Pr)^{9/16}]^{8/27}}\right)^2\tag{2}$$ You can find whatever correlation you find better for your situation.
For the forced convection, you have this correlation: $$Nu_{forced} = 0.989Re^{0.330}Pr^{1/3}\tag{3}$$ (for 0.4 < Re < 4) For other type of flows you can find different Nusselt correlations. You can find them on the internet.
Then you replace them in Eq. 1 and finally you get: $$h_{comb}=\frac{k_{fluid}\cdot Nu_{comb}}{D}\tag{4}$$ where D is the diameter of your cylinder.