The Law of Conservation of Matter

The law of conservation of matter or principle of matter conservation states that the mass of an object or collection of objects never changes over time, no matter how the constituent parts rearrange themselves.

The mass can neither be created nor destroyed.

The law requires that during any nuclear reaction, radioactive decay or chemical reaction in an isolated system, the total mass of the reactants or starting materials must be equal to the mass of the products.

The concept of mass conservation is widely used in many fields such as chemistry, mechanics, and fluid dynamics. In chemistry the law of conservation of matter may be explained in the following way (see the picture of combustion of methane). The masses of a methane and oxygen together must be equal to the masses of carbon dioxide and water. In other words, during a chemical reaction, everything you start with, you must end up with, but it might look different.
Law of Conservation of Matter

Historically, already the ancient Greeks proposed the idea that the total amount of matter in the universe is constant. The principle of conservation of mass was first outlined by Mikhail Lomonosov in 1748. However, the law of conservation of matter (or the principle of mass/matter conservation) as a fundamental principle of physics was discovered in by Antoine Lavoisier in the late 18th century. It was of great importance in progressing from alchemy to modern chemistry. Before this discovery, there were questions like:

  • Why a piece of wood weighs less after burning?
  • Can a matter or some of its part disappear?

In the case of burned wood the problem was the measurement of the weight of released gases. Measurements of the weight of released gases was complicated, because of the buoyancy effect of the Earth’s atmosphere on the weight of gases. Once understood, the conservation of matter was of crucial importance in the progress from alchemy to the modern natural science of chemistry.

In this example, we will calculate the flow rate through a reactor core from continuity equation. It is an illustrative example, following data do not represent any reactor design.

in = ṁout 

(ρAv)in = (ρAv)out 

____________________________

Chart - density - water - temperature

Density as a function of temperature of water

Pressurized water reactors are cooled and moderated by high-pressure liquid water (e.g. 16MPa). At this pressure water boils at approximately 350°C (662°F).  Inlet temperature of the water is about 290°C (⍴ ~ 720 kg/m3). The water (coolant) is heated in the reactor core to approximately 325°C (⍴ ~ 654 kg/m3) as the water flows through the core.

The primary circuit of typical PWR is divided into 4 independent loops (piping diameter ~ 700mm), each loop comprises a steam generator and one main coolant pump. Inside the reactor pressure vessel (RPV), the coolant first flows down outside the reactor core (through the downcomer). From the bottom of the pressure vessel, the flow is reversed up through the core, where the coolant temperature increases as it passes through the fuel rods and the assemblies formed by them.

Calculate:

  • the primary piping volumetric flow rate (m3/s),
  • the primary piping flow velocity (m/s),
  • the core inlet flow velocity (m/s),
  • the core outlet flow velocity (m/s)

when

  • the mass flow rate in the hot leg of primary piping is equal to 4648 kg/s,
  • Reactor core flow cross-section is equal to 5m2,
  • Primary piping flow cross-section (single loop) is equal to 0.38 m2

Results:

Continuity Equation - Flow Rates through Reactor

Example of flow rates in a reactor. It is an illustrative example, data do not represent any reactor design.

Cold leg volumetric flow rate:

Qcold = ṁ / ⍴ = 4648 / 720 = 6.46 m3/s = 23240 m3/hod

Cold leg flow velocity:

A1 = π.d2 / 4

vcold = Qcold / A1 = 6.46 / (3.14 x 0.72 / 4) = 6.46 / 0.38 = 17 m/s

Hot leg volumetric flow rate:

Qhot = ṁ / ⍴ = 4648 / 654 = 7.11 m3/s = 25585 m3/hod

Hot leg flow velocity:

A = π.d2 / 4

vhot = Qhot / A1 = 7.11 / (3.14 x 0.72 / 4) = 7.11 / 0.38 = 18,7 m/s

or according to the continuity equation:

1 . A1 . v1 = ⍴2 . A2 . v2

vhot =  vcold . ⍴cold / ⍴hot = 17 x 720 / 654 = 18.7 m/s

Core inlet flow velocity:

Acore = 5m2

Apiping = 4 x A1 = 4 x 0.38 = 1.52 m2

inlet = ⍴cold

according to the continuity equation:

inlet . Acore . vinlet = ⍴cold . Apiping . vcold

vinlet =  vcold . Apiping / Acore = 17 x 1.52 / 5 = 5.17 m/s

Core outlet flow velocity:

inlet = ⍴cold

outlet = ⍴hot

according to the continuity equation:

outlet . Acore . voutlet = ⍴inlet . Acore . vinlet
voutlet =  vinlet . ⍴inlet / ⍴outlet = 5.17 x 720 / 654 = 5.69 m/s

The Law of Conservation of Matter in Special Relativity Theory

At the beginning of the 20th century, the notion of mass underwent a radical revision. Mass lost its absoluteness. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc2. In words, energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen to be equivalent to energy, to be interconvertible with energy, and to increase significantly at exceedingly high speeds near that of light. The total energy of an object was understood to comprise its rest mass as well as its increase of mass caused by increase in kinetic energy.

In special theory of relativity certain types of matter may be created or destroyed, but in all of these processes, the mass and energy associated with such matter remains unchanged in quantity. It was found the rest mass an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E = mc2) this binding energy is proportional to this mass difference and it is known as the mass defect.

Calculate the mass defect of a 63Cu nucleus if the actual mass of 63Cu in its nuclear ground state is 62.91367 u.

63Cu nucleus has 29 protons and also has (63 – 29) 34 neutrons.

The mass of a proton is 1.00728 u and a neutron is 1.00867 u.

The combined mass is: 29 protons x (1.00728 u/proton) + 34 neutrons x (1.00867 u/neutron) = 63.50590 u

The mass defect is Δm = 63.50590 u – 62.91367 u =  0.59223 u

Convert the mass defect into energy (nuclear binding energy).

(0.59223 u/nucleus) x (1.6606 x 10-27 kg/u) = 9.8346 x 10-28 kg/nucleus

ΔE = Δmc2

ΔE = (9.8346 x 10-28 kg/nucleus) x (2.9979 x 108 m/s)2 = 8.8387 x 10-11 J/nucleus

The energy calculated in the previous example is the nuclear binding energy.  However, the nuclear binding energy may be expressed as kJ/mol (for better understanding).

Calculate the nuclear binding energy of 1 mole of 63Cu:

(8.8387 x 10-11 J/nucleus) x (1 kJ/1000 J) x (6.022 x 1023 nuclei/mol) = 5.3227 x 1010 kJ/mol of nuclei.

One mole of 63Cu (~63 grams) is bound by the nuclear binding energy (5.3227 x 1010 kJ/mol) which is equivalent to:

  • 14.8 million kilowatt-hours (≈ 15 GW·h)
  • 336,100 US gallons of automotive gasoline
Calculate the mass defect of the 3000MWth reactor core after one year of operation.

It is known the average recoverable energy per fission is about 200 MeV, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The reaction rate per entire 3000MWth reactor core is about  9.33×1019 fissions / second.

The overall energy release in the units of joules is:

200×106 (eV) x 1.602×10-19 (J/eV) x 9.33×1019 (s-1) x 31.5×106 (seconds in year) = 9.4×1016 J/year

The mass defect is calculated as:

Δm = ΔE/c2

Δm = 9.4×1016 / (2.9979 x 108)2 = 1.046 kg

That means in a typical 3000MWth reactor core about 1 kilogram of matter is converted into pure energy.

Note that, a typical annual uranium load for a 3000MWth reactor core is about 20 tonnes of enriched uranium (i.e. about 22.7 tonnes of UO2). Entire reactor core may contain about 80 tonnes of enriched uranium.

Mass defect directly from E=mc2

The mass defect can be calculated directly from the Einstein relationship (E = mc2) as:

Δm = ΔE/c2

Δm = 3000×106 (W = J/s) x 31.5×106 (seconds in year) / (2.9979 x 108)= 1.051 kg

Nuclear binding energy curve.

Nuclear binding energy curve.
Source: hyperphysics.phy-astr.gsu.edu

During the nuclear splitting or nuclear fusion, some of the mass of the nucleus gets converted into huge amounts of energy and thus this mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. The nuclear binding energies are enormous, they are of the order of a million times greater than the electron binding energies of atoms.

Generally, in both chemical and nuclear reactions, some conversion between rest mass and energy occurs, so that the products generally have smaller or greater mass than the reactants. Therefore the new conservation principle is the conservation of mass-energy.

See also: Energy Release from Fission

Mass Defect

The Law of Conservation of Matter in Fluid Dynamics

The mass can neither be created nor destroyed.
Continuity Equation - Definition

Continuity Equation – Definition

This principle is generally known as the conservation of matter principle and states that the mass of an object or collection of objects never changes over time, no matter how the constituent parts rearrange themselves. This principle can be use in the analysis of flowing fluids. Conservation of mass in fluid dynamics states that all mass flow rates into a control volume are equal to all mass flow rates out of the control volume plus the rate of change of mass within the control volume. This principle is expressed mathematically by following equation:

in = ṁout +∆m∆t

Mass entering per unit time = Mass leaving per unit time + Increase of mass in the control volume per unit time

Continuity Equation - Flow Rates through Reactor

Example of flow rates in a reactor. It is an illustrative example, data do not represent any reactor design.

This equation describes nonsteady-state flow. Nonsteady-state flow refers to the condition where the fluid properties at any single point in the system may change over time. Steady-state flow refers to the condition where the fluid properties (temperature, pressure, and velocity) at any single point in the system do not change over time. But one of the most significant properties that is constant in a steady-state flow system is the system mass flow rate. This means that there is no accumulation of mass within any component in the system.

See also: Continuity Equation

Continuity Equation

The continuity equation is simply a mathematical expression of the principle of conservation of mass. For a control volume that has a single inlet and a single outlet, the principle of conservation of mass states that, for steady-state flow, the mass flow rate into the volume must equal the mass flow rate out.

in = ṁout 

Mass entering per unit time = Mass leaving per unit time

This equation is called the continuity equation for steady one-dimensional flow. For a steady flow through a control volume with many inlets and outlets, the net mass flow must be zero, where inflows are negative and outflows are positive.

This principle can be applied to a streamtube such as that shown above. No fluid flows across the boundary made by the streamlines so mass only enters and leaves through the two ends of this streamtube section.

When a fluid is in motion, it must move in such a way that mass is conserved. To see how mass conservation places restrictions on the velocity field, consider the steady flow of fluid through a duct (that is, the inlet and outlet flows do not vary with time).

Differential Form of Continuity Equation

A general continuity equation can also be written in a differential form:

∂⍴∂t + ∇ . (⍴ ͞v) = σ

where

  • ∇ . is divergence,
  • ρ is the density of quantity q,
  • ⍴ ͞v is the flux of quantity q,
  • σ is the generation of q per unit volume per unit time. Terms that generate (σ > 0) or remove (σ < 0) q are referred to as a “sources” and “sinks” respectively. If q is a conserved quantity (such as energy), σ is equal to 0.
Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.