Simple heat conduction problem setup

[mahdert]

Homework Statement

a thermometer wall mounted through the wall of a steam pipe is a steel tube with 0.1 in wall thickness, 0.5 in outer diameter, 2 in length and k=26W/(mK). The flow produces an h value of 100W/(m^2K) on the outside surface of the well. If the thermometer reads 149 deg C, and the temp. of the pipe is 65 deg C, estimate the average steam temp.

Homework Equations

Fouriers law of cunduction: q = -k A $$\frac{\partial T}{\partial x}$$
Newtons law of cooling: q = hA(T$$_{\infty}$$ - T)

The Attempt at a Solution

Just need help with the problem setup:
A heat balance between the heat convected into the well from the steam is equal to the heat conducted through the thermometer well to the outside surface of the pipe. Heat conduction occurs axially through the pipe, in other words, there is no temperature difference inside the pipe in the radial direction. (is this a correct assumption).

The convection occurs throught the outside surface of the well and the bottom surface that is exposed. (is this a correct assumption).

 

[LightHero]

Therefore, the heat balance equation is: hA_{1}(T_{steam} - T_{pipe}) = kA_{2} \frac{\partial T}{\partial x} + hA_{3}(T_{infty} - T_{pipe})where A_{1} = circumference*length, A_{2} = pi*2in*0.1in, A_{3} = area of bottomI am not sure how to solve this equation for T_{steam}.