Body that both cools (proportionally to temperature difference) and heats with constant power

Question

This is almost more of a math question, and a simple one, but I can't figure it out, and I haven't been able to find a solution online since I don't really know what words to search for.

There is a body of some temperature, the starting temperature is 20deg, and it is both cooling to the surroundings (power is proportional to delta T) and being heated by an electric heating element with a constant power.

How need a function that describes the temperature at some time t.

I have abstracted things like mass, heat capacity, power, etc away in the example.

• heating: adds 1 degree per minute
• cooling: subtracts 10% of the difference in temperature between the element and the surroundings (always at 0deg). Since the surrounding temperature is 0deg, the delta temperature is the same as the temperature of the body.

In case the above can be misunderstood (my physics or math language is very limited) I have written a piece of pseudo code below that shows the development (is that the right word) over time:

HEATING = 1 # degree per minute
COOLING_RATIO = 0.2 # 20% per minute
START_TEMP = 20

function nextTemp(prevTemp):
    cooling = prevTemp * COOLING_RATIO
    newTemp = prevTemp - cooling + HEATING
    return newTemp

function printVals(): 
    print "t= %2d   T= %5.2f" % (time, temp)

temp = START_TEMP
time = 0
printVals()
for i in range(15):
    temp = nextTemp(temp)
    time += 1
    printVals()

This produces

    t=  0   T= 20.00
    t=  1   T= 17.00
    t=  2   T= 14.60
    t=  3   T= 12.68
    t=  4   T= 11.14
    t=  5   T=  9.92
    t=  6   T=  8.93
    t=  7   T=  8.15
    t=  8   T=  7.52
    t=  9   T=  7.01
    t= 10   T=  6.61
    t= 11   T=  6.29
    t= 12   T=  6.03
    t= 13   T=  5.82
    t= 14   T=  5.66
    t= 15   T=  5.53

Answer 1

I'll assume this problem is continuous as it makes sense on physics, and not discrete as you wrote on your pseudocode. Its worth pointing out, it makes a huge difference if the problem is discrete or continuous, as I have explained in the comments. However, one can also write a pseudocode for the continuous problem as well:

HEATING = 1 # degree per minute
COOLING_RATIO = 0.2 # 20% per minute
START_TEMP = 20

function nextTemp(prevTemp, timeElapsed):
    cooling = prevTemp * COOLING_RATIO * timeElapsed
    heating = HEATING * timeElapsed
    newTemp = prevTemp - cooling + heating
    return newTemp

function printVals(): 
    print "t= %2d   T= %5.2f" % (time, temp)

temp = START_TEMP
time = 0
dt = 0.00001;   # The smaller the better.
printVals()
for i=0 until i=15
    temp = nextTemp(temp, dt)
    i += dt
    time += dt
    if (isInteger(time)) printVals()    # e.g.: If time=1.0000, then printVals()

Well, it is likely you seems not to know calculus, so, I'll "solve" without it. According to what you said: Each minute, it raises 1 degree, and it decreases the temperature of the body. So, the rate of increase of temperature: $$ \frac{\Delta T}{\Delta t} = 1 - T $$

We have to compute this in the tiniest possible $\Delta t$. Mathematically, we say that $\Delta t$ is so small that it approach zero: $\Delta t \to 0$. In that case, it becomes a differential and I can write $dt$. A tiniest difference in time will mean a tiniest difference in temperature $dT$. Thus: $$ \frac{dT}{dt} = 1 - T $$

We identify a differential equation of first order. We sick a function $T(t)$ dependent on time, such that its derivative is equal to $1-T(t)$. The derivative of the function (the left part) can also be written in this notation: $T'(t)$. With a prime on it. Our equation then: $$ T'(t) = 1 - T(t), \quad\mbox{ for }\quad T(0) = 20 $$

Now we copy paste it on wolframalpha.com, and got a solution: $$ T(t) = 10\left(1 + e^{-0.1t}\right) $$

Note that, in the continuous solution, as t gets larger and larger, that is, as $t\to\infty$, then $T(t) = 10$. So, the temperature of the body will estabilize at 10 degree. Also, notice that in the solution to the continuous problem, $T$ is very different from your solution (the discrete problem).

   t=0,   20.000
   t=1,   19.048
   t=2,   18.187
   t=3,   17.408
   t=4,   16.703
   t=5,   16.065
   t=6,   15.488
   t=7,   14.966
   t=8,   14.493
   t=9,   14.066
   t=10,  13.679
   t=11,  13.329
   t=12,  13.012
   t=13,  12.725
   t=14,  12.466
   t=15,  12.231