Conservation of Mass


From relativistic considerations that mass and energy are related by the well-known equation:

E = mc2

where c = velocity of light and E = energy. It may be concluded from this equation that the mass of a control mass changes when its energy changes. The magnitude of this change of mass for a typical problem may be calculated and it may be determined whether this change in mass is significant.

Consider a rigid vessel that contains a 1-kg stoichiometric mixture of a hydrocarbon fuel (such as gasoline) and air. From knowledge of combustion, it is known that after combustion takes place it will be necessary to transfer about 2900 kJ from the system to restore it to its initial temperature. From the first law:

1Q2 = U2U1 + 1W2

It may then be concluded that since 1W2 = 0 and 1Q2 = −2900 kJ, the internal energy of this system decreases by 2900 kJ during the heat transfer process. Now the decrease in mass during this process will be calculated. The velocity of light, c is 2.9979 × 108 m/s. Therefore,

2900 kJ = 2900000 J = m (kg) × (2.9979 × 108 m/s)2

and so:

m = 3.23 × 10−11 kg

Thus, when the energy of the control mass decreases by 2900 kJ, the decrease in mass is 3.23 × 10−11 kg.

A change in mass of this magnitude cannot be detected by even the most accurate chemical balance. Certainly, a fractional change in mass of this magnitude is beyond the accuracy required in essentially all engineering calculations. Therefore, if the laws of conservation of mass and conservation of energy are used as separate laws, significant error will not be introduced into most thermodynamic problems and the definition of a control mass as having a fixed mass can be used even though the energy changes.

Now consider the conservation of mass law as it relates to the control volume. The physical law concerning mass says that mass cannot be created or destroyed. This law will be expressed in a mathematical statement about the mass in the control volume. To do this all the mass flows into and out of the control volume and the net increase of mass within the control volume must be considered. As a somewhat simpler control volume a tank with a cylinder and piston and two pipes attached as shown, are considered. The rate of change of mass inside the control volume can be different from zero if a flow of mass is added or removed, represented by:

Rate of change = +in − out

Rewritten for several possible flows:

(Eq1)    
dmCV
dt
  =  ΣiΣe

which states that the mass inside the control volume changes with time if mass is added or removed. There are no other means by which the mass inside the control volume can change. Eq1, which expresses the conservation of mass, is also termed the continuity equation. While this form of the equation is sufficient for the majority of applications in thermodynamics, it is frequently rewritten in terms of the local fluid properties in the study of fluid mechanics and heat transfer. When concern is mostly focused on the overall mass balance, Eq1 is considered as the general expression for the continuity equation.

Because Eq1 is written for the total mass (lumped form) inside the control volume, several contributions to the mass may be considered:

mCV  =   ρ dV  =  (1/v) dV  =  mA + mB + mC + ⋅⋅⋅

Such a summation is needed when the control volume has several accumulation units with different states of the mass.

Now consider the mass flow rates across a control volume surface in a little more detail. For simplicity, assume that fluid is flowing in a pipe or duct as illustrated:



It is desired to relate the total flow rate that appears in Eq1 to the local properties of the fluid state. The flow across the control volume surface can be indicated with an average velocity shown to the left of the valve or with a distributed velocity over the cross section as shown to the right of the valve.

The volume flow rate is then:

  =  uA  =  ulocaldA

so the mass flow rate becomes:

(Eq2)    
= ρavg  =  /v  =  (ulocal / v) dA = uA/v

where often the average velocity is used. It should be noted that Eq2 has been developed for a stationary control surface and it was tacitly assumed the flow was normal to the surface. This expression for the mass flow rate applies to any of the various flow streams entering or leaving the control volume, subject to the assumptions mentioned.