A condition in which the transport of material or energy in and out of a process is not balanced; instead there is either a loss or an accumulation over time. An unsteady-state mass balance involves the flow of materials into a process together balanced with the flow of materials out with any accumulation or loss. For example, the flow of a liquid into a tank, Q_in, with an open drain valve that has a flow out, Q_out, resulting in a change in capacity dV/dt, can be expressed mathematically as:

In an unsteady-state energy balance the same principle applies. The accumulation of energy within a process where all the energy forms are considered including kinetic, potential, heat flow rates, enthalpies, and stirrer works may result in an increase in the thermal energy and a rise in temperature. Unsteady-state heat transfer involves the transfer of heat under conditions where the temperature changes with time. For the simple case of one-dimensional conduction in a solid slab, the accumulation of heat is a product of the mass and specific heat of the material and the increase in temperature where:

where α is the thermal diffusivity of the material. General solutions of unsteady-state conduction for simple geometries are available such as for slabs, infinitely long cylinders, and spheres. For a semi-infinite slab, the integration for the heating or cooling from both sides by a medium of constant surface temperature is:

where T_S is the average temperature of the surface, T₁ is the initial temperature, T₂ is the temperature at time t, Fo is the Fourier number, and a is (π/2)². Unsteady-state heat transfer occurs where there is a change of material within a space with time. Similar one-dimensional mathematical principles apply.