Consider the behaviour of the process over a period of time when the two flowrates do not change, and any disturbances to the process resulting from previous changes in the flows have settled out.

*
F _{1} + F_{2} - F = 0*

- The condition of no change, etc, means that the process may be
said to be at
**steady state**. - The properties of the process variables, i.e. the flows, are
associated with one value at one point in the process, e.g.
an inlet stream. There are a finite
number of such points and in particular, not a continuum. This
is called a
**lumped**system. - The equation is an
**algebraic**equation, i.e. no integrals or derivatives. - The solution is a number.

Liquid flows at a rate ** F** kg/s into a tank. The mass of
material in the
tank initially, at say time

- This is a
**dynamic**situation becausechanges with time.*M* - This is a
**lumped**system because the property, holdup, is associated with a single point and is not distributed in space. That is, we can talk about*the mass of material in the tank*but not*the mass of material at point*.**x**in the tank - The equation is an
**o.d.e.**whose independent variable is**time**. - The solution is functional variation of
.*M(t)*

Heat flows at a constant steady rate ** Q** watts across a
uniform slab. Each
side of the slab is held at constant temperature.
The temperature of the slab, of cross sectional area

- This is called a
**distributed system**because a property, the temperature of the slab, is distributed in space. - There is one spatial dimension.
- There is no change in time because the heat flow is constant,
thus this is once again a
**steady state**situation. - The equation describing the system is an
**ordinary**differential equation. - The
**independent**variable in the o.d.e. is**distance**. - The solution is the
functional variation of the distributed property with distance,
i.e.
.*T(x)*

This is described by a p.d.e. whose independent variables
are time and distance along the pipe. Consider the situation
where the temperature `T` is changing with distance
` x ` down a pipe of total length ` L`
where the velocity of flow is `u`.

This is described by the convective flow terms of the Navier-Stokes equation:

Although this is a complicated looking equation it turns out
to have a simple solution. The temperature of the fluid leaving
the pipe at time `t` is exactly the same as
the temperature entering the pipe at a previous time
(`t - L/u`) since `L/u` is the residence
time for the fluid in the pipe.

- Steady state systems in two or more dimensions are described by p.d.e.s whose independent variables are all the spatial dimensions involved.
- The unsteady state of distributed systems involves equations containing p.d.e.s with independent variables of both space and time.
- The solutions of p.d.e.s are functions or functionals.
- Distributed problems may contain integrals, leading to
**integral**equations.

- Steady state lumped systems are described by
**algebraic**equations - Steady state distributed systems by
**o.d.e.s in distance** - Unsteady state lumped systems by
**o.d.e.s in time** - Anything more complicated by p.d.e.s.
**All**these are different kinds of equations, requiring different solution methods.- Algebraic equations also describe dynamic systems with no capacity
and thus
*instant*response.

Move back to Section 2 Index