# Heat transfer - Transient state involving a solid and a surface

1. Nov 17, 2012

### lingoo

1. The problem statement, all variables and given/known data

A thermometer for medical applications to determine the time required for the tip to reach a temperature of 37.95 C when in contact with the skin temperature of 38 C, starting from an initial value of 25 ° C.

Admit that the tip of the thermometer is completely metallic (all sides except the bottom) in contact with the skin of a person.

The tip material is stainless steel with a thermal diffusivity of 0.05 cm2 / s, the specific heat at constant pressure of 451 J / (kg.K) and thermal conductivity of 17.2 W / mK.
The contact resistance between the skin and the metal end is estimated at 31.529 K / W.

Assume that the metallic end of the thermometer can be considered a system with internal resistance to heat conduction negligible.

Suppose that the tip of the thermometer is a cylinder of 1 cm in length and 0.6 cm diameter

2. Relevant equations

k -> thermal conductivity
c-> specific heat
$\alpha$ -> thermal diffusivity

Fo = $\alpha$ * t/r^2

$\alpha$= k/(ρ*c)

Bi= hc*ro/k

Diagrams for dimensionless transient temperatures and heat flow

and many more...

3. The attempt at a solution

At first, there's no convection in the problem, so Bi = 0, in the diagram, the curves 1/Bi would be equal to 0 because hc -> ∞. Okay, and making the temperatures difference
(T-T(∞))/(T(0)-T(∞)) = -0.05/-13 = 0.00385
So looking into the diagram for dimensionless transient temperatures and heat flow for a long cylinder, I will find Fo = 1
When I put the values in formula -> Fo = $\alpha$*t/r^2 it doesn't seem to work (I have converted $\alpha$(thermal diffusivity) to m^2/s, so it will be 5*10^-6 m2/s).
After that, I tried making ρ*c*V*dT/dt = Q, making Q equal to the (T-T(∞))/(Rk+Rcontact)
I'm really lost in this problem, because I get used to solve problem that involved a solid and a fluid, this problem is related to fins and transient state, but I can't find a way through in this problem.

Well, the answer should be around t= 170,52 seconds as it says on the answer.

I would appreciate a help for tackling out this problem, any sugestions, ideas, solutions are welcome. Thanks !