# Ice melting equation units don't match?

I am working on a model to simulate algal growth in Antarctica, and want to incorporate surface ice melting. I found this equation for melting: $$\frac{dV}{dt} = \frac{-kAT}{\rho L_f}$$ where $V$ is the melted volume in cubic meters, $k$ is thermal conductivity in Watt per degree Kelvin per meter, $A$ is the melting area in square meters, $T$ is the temperature difference between the two surfaces in degrees kKelvin, $\rho$ is the density of the ice in kilograms per cubic meter and $L_f$ is the heat of fusion in Joule per kilogram.

The problem is: if I add all the units up, I get $m^4s^{-1}$ in stead of the $m^3s^{-1}$ that I was expecting. Is there something wrong in the formula, or is the unit of the melted water realy $m^4s^{-1}$?

I know it's a simplified formula, but I don't want it to be too complicated. All i need is a rough water input from the ice into the ocean.

Edit: The formula comes from this article. http://home2.fvcc.edu/~dhicketh/DiffEqns/spring2014projects/AudreyJones/Icecube.pdf I have changed it, because I am using a 2D surface rather than a 3D cube, which might be where the problem comes from.

• What is the reasoning behind your equation? Where did you get if from? – sammy gerbil Feb 24 '17 at 14:26
• Ok, so in the article you are citing something is not right in the first equation of section 2.4: The dimension of the heat transfer $Q'$, which is supposed to have reasonable SI units of $\frac{\text{J}}{\text{s}}$, does not match the dimension of the expression $-kAT$, which has units of $\frac{\text{J}}{\text{s}}\text{m}$. Unfortunately, "After much research and consultation with experts the following general melting equation was developed." does not tell us how the author arrived at that equation. – Wojciech Morawiec Feb 24 '17 at 14:46

@WojciechMorawiec is correct. The $k$ in the referenced article is not thermal conductivity. What would thermal conductivity have to do with the transfer of energy into of the object?
$k$ in that formula is the heat transfer coefficient which has units of W/m${}^2$/ K