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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 energy for a curve that has a more complicated profile, such as this one:

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:

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?

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 energy for a curve that has a more complicated profile, such as this one:

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?

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?