# Time in the simple Heat Equation

I was teaching the simple Heat equation to a pharmacy student, namely the equation

$$Q = m\mathcal{C} \Delta T$$

And I was thinking about: this equation gives us the amount of energy necessary to raise the temperature of a mass $m$ of a particular substance from $T_1$ to $T_2$.

If I know the density and the volume of the substance I can always find the mass, but let's suppose I know it's a liter of water inside a pot. Ok being water I already know the volume and the mass and the specific heat.

What about a formula (maybe a modification of this?) that tell me how much time it will take me to make a liter of water to boil? I know it depends on how powerful is the flame of the stove but.. is there some equation as a function of the time?

• For boiling water in a pot, you need to know a heat transfer coefficient and a pot surface area. The area is easy, but the heat transfer coefficient is not. Do a Google search and see if you can find the heat transfer coefficient for this particular case. If you can find this information, use the equation Q = UA(Delta-T), where U is the overall heat transfer coefficient. – David White Oct 23 '16 at 22:33
• This might be a duplicate, assuming STP, I think physics.stackexchange.com/questions/263981/… – user108787 Oct 23 '16 at 22:35

The 'go to' time dependent equation for solving transient heat problems is Fourier's heat equation (here in 3D and Cartesian coordinates):

$$\frac{\partial T}{\partial t}=\kappa\Big(\frac{\partial^2 T}{\partial x^2}+\frac{\partial^2 T}{\partial y^2}+\frac{\partial^2 T}{\partial z^2}\Big)$$ Where $\kappa$ is the thermal diffisivity: $$\kappa=\frac{k}{\rho c_p}$$

Fourier's equation is derived from Fourier's law:

$$q=-k\Big(\frac{\partial T}{\partial x}+\frac{\partial T}{\partial y}+\frac{\partial T}{\partial z}\Big)$$

Where $q$ is the heat flux: heat flow per unit of area through a surface.

In the absence of work done, a change in internal energy per unit volume in the material, $\Delta Q$ becomes (here in 1D):

$$\Delta Q=\rho c_p\Delta T$$

What about a formula (maybe a modification of this?) that tell me how much time it will take me to make a liter of water to boil? I know it depends on how powerful is the flame of the stove but.. is there some equation as a function of the time?

Application of Fourier's law or the heat equation depends highly on the actual real world problem you're trying to solve, so a 'one-fits-all' formula does not exist.

In the case of the simple heating problem of a kettle on a stove we can use the following simple derivation.

We'll assume there is a contact area $A$ between the kettle and stove and that the stove temperature is at a constant $T_{\infty}$. We'll neglect heat losses (convection, radiation) from the kettle.

The rate of heat transfer between stove and kettle is then given by Newton's cooling/heating law:

$$\frac{dQ}{dt}=uA(T_{\infty}-T)\tag{1}$$

Where $T$ is the kettle temperature (assumed homogeneous) and $u$ the overall heat transfer coefficient.

As the kettle heats up by an infinitesimal amount $dT$, by differentiating:

$$Q = mc_p \Delta T$$

$$\implies dQ=mc_pdT$$

Divide both sides by $dt$:

$$\frac{dQ}{dt}=mc_p\frac{dT}{dt}\tag{2}$$

With the identity $(1)=(2)$ we get the differential equation:

$$mc_p\frac{dT}{dt}=uA(T_{\infty}-T)\tag{3}$$

If we integrate $(3)$ between $(0,T_1)$ and $(t,T_2)$ we get:

$$-\frac{uA}{mc_p}t=\ln\Big[\frac{T_{\infty}-T_2}{T_{\infty}-T_1}\Big]\tag{4}$$

$(4)$ then allows to calculate the time $t$ to heat from $T_1$ to $T_2$.

Note that for $t \to \infty$ then $T_2 \to T_{\infty}$.

• Wow thank you for all the derivation! Really interesting! – Les Adieux Oct 23 '16 at 23:15
• Thanks. "[...] and that the stove temperature is at a constant $T_{\infty}$" – Gert Oct 23 '16 at 23:19
• $T_1$ is the initial temperature of the kettle+liquid, $T_2$ is its temperature after some time $t$. $T_{\infty}$ is the temperature of the stove. How can $T_2=666.3C$ when the boiling point of milk is $100C$!?! This equation cannot be solved w/o reasonable estimates of $C$ and $u$. The derivation presented is box-standard: you'll find it easily on some specific web pages/engineering books on heating/cooling. – Gert Oct 28 '16 at 19:34
• Added a figure, hope that clarifies things. – Gert Oct 28 '16 at 19:49
• You'll find the same derivation here (p.2 and 3.): web2.clarkson.edu/projects/subramanian/ch330/notes/… – Gert Oct 28 '16 at 22:04