General conduction equation for 1D using Fourier's law A product is coming out of a single hole die plate at a rate of 2g/s from a temperature of 98C to 10C. I want to know how long it takes to cool below 43C. The product is a cylindrical tube or radius of 7mm and can be broken into elements with length 10cm. The specific heat= 2 kJ/kg.k, thermal conductivity=0.4  and density=890kg/m3.
How do I derive the general conduction equation using fourier law of thermodynamics?
 A: Besides the 1D Fourier conduction approach, there is another, simpler (but also more approximate) method: lumped thermal analysis.
Here we consider an infinitesimal element $\text{d}z$ to be thermally uniform, that is:
$$\frac{\partial T}{\partial x}=\frac{\partial T}{\partial y}=\frac{\partial T}{\partial z}$$
This means of course by Fourier that there's no conduction going on, only convection:

The 'go to' equation in the case of pure convection is of course Newton's Law of Cooling, generically:
$$Q=hA\Delta T$$
Applied to the infinitesimal element, which we'll follow as it travels down the $z$-axis:
$$\frac{\text{d}q}{\text{d}t}=2\pi R\text{d}zh(T-T_0)$$
with $T_0$ the surrounding temperature (constant).
$$\text{d}q=-\text{d}mc_p\text{d}T$$
Eliminating $\text{d}z$:
$$\text{d}m=\rho \text{d}V=\rho \pi R^2 \text{d}z$$
$$-\rho \pi R^2 c_p\text{d}z\frac{\text{d}T}{\text{d}t}=2h\pi R\text{d}z(T-T_0)$$
$$-\rho  R c_p\frac{\text{d}T}{\text{d}t}=2h(T-T_0)$$
$$\frac{\text{d}T}{T-T_0}=-\frac{2h}{\rho  R c_p}\text{d}t$$
$$\alpha=\frac{2h}{\rho  R c_p}$$
with $T_i$ the temperature after exiting the die plate.
$$\int_{T_i}^T\frac{\text{d}T}{T-T_0}=-\alpha \int_0^t\text{d}t$$
$$\frac{T-T_0}{T_i-T_0}=-\alpha t$$
$$\boxed{T=T_0+(T_i-T_0)\exp{(-\alpha t)}}$$
Time can be converted to length by means of the mass throughput.
Hope this helps!
