What is Fourier's law?
Fourier's law, named after the famous French Mathematician and Physicist Jean-Baptiste Fourier, states that the heat flux through a surface is proportional to the negative of the temperature gradient at that surface. The proportionality constant from this law then gives an important thermal transport property called the 'thermal conductivity' of the material, given the symbol 'k' with units of W/(m.K).
How are Partial Differential Equations (PDEs) different from Ordinary Differential Equations (ODEs) and how can they be solved?
PDEs are differential equations where the 'dependent' variable is a function of multiple 'independent' variables, while in an ODE, the dependent variable is only a function of a single dependent variable. For example, in the heat equation, Temperature T(x,y,z,t) is a function of three spatial (x,y,z) and one temporal (t) variable and hence the heat equation is a PDE while a mass-spring system is modeled by an ODE where the acceleration 'x double dot' is a function of time only, PDEs are most commonly solved using the technique of separation of variables where the solution is assumed to be the product of the independent variables.
Can the work output from a device or heat engine be greater than its work input?
No. As per the second law of thermodynamics, heat will travel from a source to a sink and if run through a heat engine, only a part of that initial heat (energy) can be converted to useful work and the rest of it will have to be discarded to the heat sink. Hence every heat engine will have an efficiency less than one (1) and the second law provides an upper limit to that efficiency, which is called Carnot efficiency and such a process is called a Carnot cycle. That is the physics behind it. As engineers, we are ever trying to engineer better work-output devices to increase their efficiency. However, the theoretical upper limit as set by the Carnot cycle cannot be exceeded.