Heat transfer - Transient state involving a solid and a surface

[lingoo]

Homework Statement

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

Homework 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...

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 !

 

[KataKoniK]

Hello there,

Thank you for sharing your approach to solving this problem. It seems like you have a good understanding of the concepts and equations involved, but your calculations may be off. Let's go through the steps together to see if we can find the mistake.

First, let's calculate the thermal diffusivity, \alpha, using the given values for thermal conductivity (k) and specific heat (c). We can use the formula \alpha= k/(ρ*c) and plug in the values to get:

\alpha= 17.2/(451*10^-3 * 0.05) = 7.6*10^-6 m^2/s

Next, let's calculate the Biot number (Bi) using the given values for heat transfer coefficient (hc) and thermal conductivity (k). We can use the formula Bi= hc*ro/k and plug in the values to get:

Bi= (31.529 * 0.3)/(17.2) = 0.55

Since Bi is not equal to 0, we cannot assume that there is no convection and we cannot use the diagram for dimensionless transient temperatures and heat flow for a long cylinder. Instead, we will need to use the general solution for transient heat conduction in a cylinder, which is:

T(x,t)= T(∞) + (T(0)-T(∞))*erfc(x/(2*sqrt(\alpha*t)))

Where erfc is the complementary error function and x is the distance from the center of the cylinder.

Since we are interested in the temperature at the tip of the thermometer, we can use x= L/2 = 0.5 cm.

Now, let's plug in the values for T(0)= 25°C, T(∞)= 38°C, and \alpha= 7.6*10^-6 m^2/s into the equation to get:

T(0.5, t)= 38 + (25-38)*erfc(0.5/(2*sqrt(7.6*10^-6 * t)))

Next, we can solve for t by setting T(0.5, t) equal to the desired temperature of 37.95°C and solving for t:

37.95= 38 + (25-38)*erfc(0.5/(2*sqrt(7.6*10^-6 *