# Calculate energy from Temperature - Time curve

I've seen many times people estimating the energy just by looking at the temperature curve, but for me, it's a bit hard to understand how can they do it so fast and efficient. Therefore I would like to ask if anyone can help me to figure out how to get the total energy from a Temperature - Time plot. For this, I have formulated a simple example where 1 kg of water is heated with 50 [kW].

Parameters:
m = 1 kg
P = 50 kW
Cp = 4.186 kJ/(kg.K)
Tinit = 20 C


So by using:

$$m C_p \dfrac{dT}{dt}=P \tag{1}$$

One ends up with a graph like this:

And the values can be put in table:

t   T
----------
0   20.00
1   31.94
2   43.89
3   55.83
4   67.78
5   79.72
6   91.67
7   103.61
8   115.56
9   127.50
10  139.45


Now the question. Is there any way to look at this plot and tell/calculate the total amount of energy put in for the entire time interval? What would be the methods for that?

EDIT: Can the same method be used to find the net energy ($E_{in} - E_{out}$) for a curve that has a more complicated profile, such as this one:

It would just be $mC_p(T_{final}-T_{initial})$, assuming that the heat capacity doesn't vary much with temperature over the temperature range of interest.
• To get the temperature variation you get in your second figure, you must be removing heat from the material as well as adding heat. In the final state, the amount of heat you have removed just matches the amount of heat you have added, so here again, the net energy change is zero $\Delta E =0$. The only way you can determine the specific amount of heat added for a curve like this is to know the amount of heat that was removed. This requires a heat transfer calculation involving heat conduction to the cold reservoir. – Chet Miller Feb 3 '17 at 15:16