Derive expression for average heat flux in heat exchanger


> 
> The single-tube heat exchanger is used to increase the temperature of fluid
>  I from $T\_{b1}$ to $T\_{b2}$. The heating oil (fluid II) used
>  for this task enters at temperature $T\_{02}$ and leaves at
>  temperature $T\_{01}$.
> 
> 
> a) Give the expression that describes the average heat flux from fluid II to fluid I.
> 
> 
> [![Heat exchanger](https://i.stack.imgur.com/stUmQ.png)](https://i.stack.imgur.com/stUmQ.png)
> 
> 
> 


I don't really know how to derive the *average* heat flux but the equation I immedietly think of using is: 
$$q = U(T\_{b1} - T\_{b2})$$ 


Still I am not sure how I should calculate the overall heat transfer coefficient and also I recall being told that when dealing with heat exchangers you should use the logarithmic temperature difference. So the expression should be:


$$q = UΔT\_{\ln}$$ 


where the logarithmic temperature difference is


$$ΔT\_{\ln} = \frac{(T\_{01} - T\_{b1}) - (T\_{02} -T\_{b2})}{\ln\frac{T\_{01} -T\_{b1}}{T\_{02} - T\_{b2}}}$$


And an attempted solution for the overall heat transfer coefficient is:


$$U = h\_1 + \frac{k}{R\_0\ln(R\_a/R\_0)} + \frac{h\_2R\_a}{R\_0}$$


but I feel like this isn't correct since it is used when calculating the heat transfer in a heat exchanger and not the heat flux. Is it correct or is the heat flux for a heat exchanger calculated in a different way?


All help is greatly appreciated!


