# 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?

• If you let the outer radius approach infinity, you will be able to estimate the heat flow. – Karsten Theis Apr 22 at 20:58

$$q = A h (T_0 - T_S )$$
$$A = 4 \pi R^2$$