Derive expression for heat flow from oil bath to metal sphere


> 
> A spherical ball of iron is put into a hot bath of oil. The iron ball is initially at a low temperature and heat transfer from the oil to the iron will increase the temperature of the iron ball.
> 
> 
> Use the concept of heat transfer coefficients to express the heat flow (i.e. the quantity with units J/s) from the oil to the ball. Denote your variables according to the notation in **Table 1**. Notice that not all variables in Tab. 1 might show up in your expression. Variables not listed in **Table 1** must not be used in your final expression.
> 
> 
> **Table 1**: Notations
> $$
> \begin{array}{ll}
> \hline
> c\_{p, f} & \text{ heat capacity oil} \\
> c\_{p, s} & \text{heat capacity iron} \\
> D & \text{diameter iron ball} \\
> h & \text{heat transfer coefﬁcient} \\
> k\_f & \text{thermal conductivity oil} \\
> k\_s & \text{thermal conductivity iron} \\
> Q & \text{heat ﬂow oil to iron (J/s)} \\
> R & \text{radius of the iron ball} \\
> T\_s & \text{temperature iron ball} \\
>  & \text{(can be assumed homogeneous)} \\
> T\_0 & \text{bulk temperature oil} \\
> ρ\_s & \text{density iron} \\
> ρ\_f & \text{density oil} \\
> \hline
> \end{array}
> $$
> 
> 
> 


I'm really unsure of how to solve this problem. I know that the heat flow for a sphere with an inner and outer radius is


$$Q = \frac{4πkR\_o}{1 - R\_o/R\_a}(T\_0 - T\_a),$$


but here I only have the radius of the whole sphere. So, how do I write the expression for the heat flow then?




Newton's law of cooling is


$$q = A h (T\_0 - T\_S )$$


With the area of the sphere being


$$A = 4 \pi R^2$$


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