The Mass Balance or Continuity Equation

Using the notation used in Fig. 5.1, we can perform a volumetric balance in and out of the differential element as, 

All balance equations are put in a more amenable form by dividing them by the differential volume, AxAyAz, and putting them in terms per unite volume, 

Letting the size of the differential element go to zero results in, 

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Equation (3.90) is the mass balance or continuity equation, (3.91) the momentum balance or Cauchy s equation of motion and (3.92) is the energy balance. As a momentum balance exists for each of the three coordinate directions, j = 1, 2, 3, there are five balance equations in total. The enthalpy form (3.83) is equivalent to the energy balance (3.92). 

The principles of physics most useful in the applications of fluid mechanics are the mass-balance, or continuity, equations the linear- and angular-raomentum-balance equations and the mechanical-energy balance. They may be written in differential form, showing conditions at a point within a volume element of fluid, or in integrated form applicable to a finite volume or mass of fluid. Only the integral equations are discussed in this chapter. 

To obtain an estimate of the maximum thickness of a tissue engineered implant based on transport considerations, we must balance metabolite delivery with consumption by the cells in the construct. For this purpose, we use the mass balance or continuity equation, which is generally written as ... 

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