Determine heat loss rate

Question

For my master thesis I am researching the heating and cooling down of a system with oil as thermal energy storage. I'll explain what and how I calculated what I have before I form my question: I have the temperature wrt time when heating up and cooling down. I can calculate how much heat is added to the system with the mass, specific heat capacity and change in temperature compared to the starting temperature with following formula: \begin{equation} Q=m \int_{T1}^{T2}c(T) \end{equation} For materials with a constant c this is just the normal formula, but the oil has a variable specific heat so this is how I calculate the heat added. Now to calculate the amount of heat that is added/removed per time unit (in J/s), I use following formula: \begin{equation} {\dot Q}=\frac{Q}{dt}=m \frac{\int c(T_1)}{dt} \end{equation} So I first calculate the integral implicitly, then fill in the temperature, and then calculate the derivative of the specific heat capacity compared to the previous value (on previous measured point). Now how does this formula compare to the following, which is the only one that I have found to calculate ${\dot Q}$. \begin{equation} {\dot Q}=hA\Delta T(t) \end{equation} For the materials with a constant c I just calculated the derivative of the temperature wrt time, then multiplied the derivative with m and c: \begin{equation} {\dot Q}=mc\frac{dT}{dt} \end{equation} I also plotted this derivative and determined a trendline which has the equation: \begin{equation} \large \frac{dT}{dt}=-0.0115\cdot e^{-t \cdot 5.43\cdot 10^{-05}} \end{equation} Now my questions:

1. How do I relate the previous equation to the folowing equations and can I make any conclusions from the coefficients? \begin{equation} \displaystyle T(t)=T_{\text{env}}+(T(0)-T_{\text{env}})e^{-rt} \end{equation} \begin{equation} \displaystyle {\dot {T}}=r\left(T_{\text{env}}-T(t)\right) \end{equation}

2. Do you guys think the formulas I used to calculate the heat loss rate are correct and will give the results I expect?

Answer 1

For a body of spatially uniform temperature, the correct equations are $$Q=m\int_{T(0)}^T{C(T')dT'}$$and$$\frac{dQ}{dt}=mC(T)\frac{dT}{dt}$$where T' is a dummy variable of integration. The rate of heat loss through the surface of the body is $hA(T-T_{env})$. So we have $$mC(T)\frac{dT}{dt}=-hA(T-T_{env})$$If C is independent of T, the solution to this equation is $$\frac{T-T_{env}}{T(0)-T_{env}}=\exp{\left(-\frac{hAt}{mC}\right)}$$