Solving Heat Transfer Coefficient Error: Lab Report Analysis

[thepatient]

Homework Statement

This is a lab report that I'm working on. I'm almost done, except for the last part which consists of finding the error in the result for our heat transfer coefficient h using Newton's law of cooling. I've tried this in different ways but I can't seem to get it right.

We have 51 data points for temperature and 51 data points for their respective times.

Homework Equations

ln( (T-T(room))/(T(initial) - T(room)) = -hAt/(C*p*V)

Where T is the variable of temperature of the metal sphere whose temperature is being measured, T(room) is room temperature, T(initial) is initial temperature of sphere, h is the heat transfer coefficient, A is the surface area, t is the time, C is the specific heat, p is the density and V is the volume.

The root sum square method says, that if we have h(T,t), then the uncertainty of h:

Uh = +/- [ (T/h * ∂h/∂T * U_T)^2 + (t/h * ∂h/∂t * U_t)^2 ] ^(1/2)

U_T is the uncertainty of the measurement device, which is 0.5 degrees C. U_t I assume will be 1 second, since the timer was read to the nearest second.

The Attempt at a Solution

First, solving for h(T,t):

h = - [CpVln((T-T(room))/(T(initial)- T(room)) ]/(At)

Partial derivatives:
∂h/∂T = -CpV/(A*t*(T-T(room)))

∂h/∂t = [CpVln((T-T(room))/(T(initial)- T(room)) ]/(At^2)

Multiplying by T/h and t/h for the respective components above:

∂h/∂T *T/h = 1/(T-T(room)) * (1/ln((T-T(room))/(T(initial)- T(room))

∂h/∂t * t/h = -1

But then when placing these two in equation for U_h, I'm not sure which T to use for the portion of ∂h/∂T *T/h*U_t. Another way that I tried solving this was by letting f(T) = ln(T-T(room)/(Tin - T(room)).

But then the function will be come h= h(f(T), t), so the uncertainty would be:

U_h = +-[ (∂h/∂f(T) *f(T)/h*U_f(T))^2 + (1)^2] ^(1/2).

∂h/∂f(T) * f(T)/h would reduce down to -1, so I would get:

U_h = +-[ (-U_f(T))^2 +1 ]^(1/2).

I wasn't sure if this second method was right, and if it is, what would I use for the uncertainty of the f(T) function U_f(T)?

I hope this post makes sense. I've been working on this particular part of the lab for so long. Any help would be greatly appreciated.

 

[KataKoniK]

Hello,

Thank you for sharing your lab report and explaining your thought process for finding the error in the heat transfer coefficient using Newton's law of cooling. It seems like you have been working hard on this and have tried different methods, but are still struggling to find the correct solution. I will try my best to help you with this problem.

First, I want to make sure that we are on the same page with the equations and variables. From your post, it seems like you are using the following equation to calculate the heat transfer coefficient:

ln((T-T(room))/(T(initial)-T(room))) = -hAt/(C*p*V)

Where T is the temperature of the metal sphere, T(room) is the room temperature, T(initial) is the initial temperature of the sphere, h is the heat transfer coefficient, A is the surface area, t is the time, C is the specific heat, p is the density, and V is the volume.

Now, let's look at the equation you provided for calculating the uncertainty in h:

Uh = +/- [ (T/h * ∂h/∂T * U_T)^2 + (t/h * ∂h/∂t * U_t)^2 ] ^(1/2)

Here, U_T and U_t are the uncertainties in the temperature and time measurements, respectively. It is important to note that these values are not the same as the uncertainties in the measurement device. These values represent the precision of your measurements, which is typically determined by the instrument used and the experimental setup.

Now, let's move on to your attempts at solving this problem. In your first attempt, you correctly calculated the partial derivatives of h with respect to T and t. However, when it comes to using these values in the equation for U_h, you need to use the uncertainties in the measurements, not in the variables themselves. So, for the portion of ∂h/∂T * T/h * U_t, you would use U_T instead of T. Similarly, for ∂h/∂t * t/h * U_t, you would use U_t instead of t.

In your second attempt, you are using a different method, which is also correct. You are trying to calculate the uncertainty in h by using the uncertainty in the function f(T). Here, you are assuming that the uncertainty in the heat transfer coefficient is solely dependent on the uncertainty in the function f(T). This approach