Balances

Equations for mass and energy balances. Macro and micro balances.

Balancing equations are a way of bookkeeping for the main quantities involved in a process: mass, energy, and momentum. They comprise everything that transports the quantity: processes through which the quantity enters the system and leaves it, through which the quantity is produced or absorbed, and which resistances there are to the transport of the quantity.

General balance of $x$ :

$$\frac{dx}{dt}=\Phi_{x,in} - \Phi_{x,out} + \text{production}(x)$$

Where:

$\frac{dx}{dt}$ is the rate of change of $x$ in the system as a function of time. In a steady state, $\frac{dx}{dt}=0$ .
$\Phi_{x,in}$ is the rate of $x$ entering the system.
$\Phi_{x,out}$ is the rate of $x$ leaving the system.
$\text{production}(x)$ includes production and consumption or loss of $x$ , and thus can be positive or negative.

Balances can be set up on different levels:

Macro balances describe the change of quantities over the whole system. They are set up on an integral balance.
Micro balances describe the change of quantities over a very small volume. They are set up on a differential basis. ( $\Phi_{x,in} - \Phi_{x,out}$ becomes $\Phi_x|_{t} - \Phi_x|_{t+dt}$ which is the same as $-d\Phi_x$ , because $\frac{df}{dx}=\text{lim}_{dx\to 0} \frac{f(x+dx) - f(x)}{dx}$ .) This provides information about temperature, concentration, velocity profiles within the system.

Mass balance

Total mass balance:

$$\frac{dm}{dt}=\Phi_{m,in} - \Phi_{m,out} + \text{production}(m)$$

With $\Phi_m$ having the standard unit kg/s.

Since mass is volume times concentration ( $m=Vc$ ) and volume times density ( $m=V\rho$ ), the mass flow rate $\Phi_m$ can be rewritten as $\Phi_{V}c$ or $\Phi_{V}\rho$ (with $\Phi_{V}$ having the standard unit m^3/s, and $c$ and $\rho$ having the standard unit kg/m^3). Further, if the volume does not change, this becomes an equation of concentration:

$$V\frac{dc}{dt}=\Phi_{V,in}c_\text{in} - \Phi_{V,out}c_\text{out} + \text{production}(m)$$

For more on volume flow in fluids, see fluid flow.

Energy balance

Total energy balance:

$$\frac{de}{dt}=\Phi_{m,in}ec_\text{in} - \Phi_{m,out}ec_\text{out} + \Phi_q + \Phi_W$$

Where:

$\Phi_W$ is the work done on the system.
$\Phi_q$ is the heat flow into the system.
$\Phi_{m,in}$ is the mass inflow and $\Phi_{m,out}$ is the mass outflow.
$ec$ is the energy concentration:

$$ec=u+gh+\frac{1}{2}v^2+\frac{p}{\rho}$$

With:

$u$ the internal energy
$gh$ the potential energy (per mass unit)
$\frac{1}{2}v^2$ the kinetic energy (per mass unit)
$\frac{p}{\rho}$ the pressure energy of an incompressible fluid (where $p$ the pressure and $\rho$ the density)

The total energy balance can be split into mechanical and thermal energy.

Mechanical energy

$$\frac{de}{dt}=\Phi_{m,in}(gh+\frac{1}{2}v^2+\frac{p}{\rho})_\text{in} - \Phi_{m,out}(gh+\frac{1}{2}v^2+\frac{p}{\rho})_\text{out} + \Phi_W$$ Minus energy lost from friction.

Neglecting friction, in a steady-state system with equal in- and outflow, where no work is performed, this can be simplified to Bernoulli’s equation: $$(gh+\frac{1}{2}v^2+\frac{p}{\rho})_\text{out}=(gh+\frac{1}{2}v^2+\frac{p}{\rho})_\text{in}$$

Thermal energy

$$\frac{de}{dt}=\Phi_{m,in} c_{p,\text{in}} T_\text{in} - \Phi_{m,out} c_{p,\text{out}} T_\text{out} + \Phi_q$$ Plus energy gained from friction.

For a primer on the different forms of energy, cf. the entry on energy.

Momentum balance

Momentum is mass times velocity: $\overrightarrow{p}=m\overrightarrow{v}$ . Note that while mass $m$ is a scalar, momentum $\overrightarrow{p}$ and velocity $\overrightarrow{v}$ are vectors, i.e. have a direction.

Momentum of a control volume is $\frac{\overrightarrow{p}}{V} = \frac{m\overrightarrow{v}}{V}$ , and since $\frac{m}{V}$ is density, the flow of momentum in or out of a control volume can be written as $\Phi_V\rho\overrightarrow{v}$ . The momentum balance thus is:

$$V\frac{d\overrightarrow{p}}{dt}=\Phi_{V,in}(\rho\overrightarrow{v})_\text{in} - \Phi_{V,out}(\rho\overrightarrow{v})_\text{out} + \sum\overrightarrow{F}$$

Where $\sum\overrightarrow{F}$ is the sum of all forces acting on the control volume (for example resulting from exerted pressure). One can say that force is the production of momentum, which can also be seen when formulating Newton’s 2nd law in terms of momentum:

$$\overrightarrow{F}=m\overrightarrow{a}=m\frac{d\overrightarrow{v}}{dt}=\frac{dm\overrightarrow{v}}{dt}=\frac{d\overrightarrow{p}}{dt}$$