# Governing Equations and Dimension less Numbers

Similarity principles for a physical process can either be derived from an assumption of the number of physical parameters involved (Buckingham’s

7R-theorem) or from the governing equations of the flow. The latter is to be preferred because this method will give a sufficient amount of dimension — less numbers. Furthermore, it will connect the numbers to the physical process via the equations and give important information in cases where it is necessary to make approximations.

The governing equations for mass flow, energy flow, and contaminant flow in a room will be the continuity equation, Navier-Stokes equations (one in each coordinate direction), the energy equation, and the mass transport equation, respectively.

The continuity equation for an incompressible flow is given by the follow­ing expression:

 (12.40)

+ , Dw = n

3* Dy dz

Where U , V , and W are the velocities in the three coordinate directions X, y, And Z. The symbol A indicates that the variables are instantaneous values. For example, U is the sum of a mean value U and a turbulent fluctuation U’.

The Navier-Stokes equation in the direction of gravity (y-direction) is given by the expression

 Dd
 Dv Dt
 P^ + uTx + i^ + wTz
 >dv
 (12.41)

 Mo

 Dx2 dy2 dz2

~PoPg(T-T0) -+

Where P, T, and T are pressure, temperature, and time, respectively, and T0 is a reference temperature (supply temperature). p0 6, /x0, and G are density, volume expansion coefficient, viscosity, and gravitational acceleration, respec­tively. The density and the viscosity are in principle functions of the instanta­neous temperature T, but except for the gravitational term the effect is ignored due to the level of temperature differences that occur in practice. (T — T0 )expresses the influence of the temperature in the gravitational term in a formulation called the Boussinesq approximation.

The energy equation is given by the expression

 /a a a D2T d2T d2T ydx2 dy2 Dz2
 = A
 PoCp
 / A A a a DT. *dT *dT z‘dT ^ + v-^— + w^r- Dt Ox ay Dz
 (12.42)

Where Cp and А Are specific heat and thermal conductivity.

The mass transport equation for gas in air (binary mixture) has a similar structure:

 /
 Ґl + Ґз_ ,d^C_ dx2 dy2 dz2
 De ^«3c ~dc ^ «de _ n Tt + UDп + + WD^ ~ °AB
 (12.43)

Where C is the instantaneous concentration and DAB is the binary mass diffu­sion coefficient.

Equation (12.43) is called an Euierian approach because the behavior of the species is described relative to a fixed coordinate system. The equation can also be considered to be a transport equation for particles when they are

Small, without any influence from gravity. Flow of larger particles Can Be sim­ulated by the following transport equation if the influence of Gravity is intro­Duced in the Y Convection term:7’8

 (12.44)

Dx~ dy2 dz1

Where Vs is the settling velocity in air. Settling velocity versus particle Size is Given by

Vs = 3 x 10~5d2 (m s-1), (12.45)

Where D is the diameter in |xm. The density of the particles is 1 g cm — ’ and the particles are assumed to be of spherical shape in Eq. (12.45). Simulation of concentration distribution given by Eq. (12.44) is restricted to situations where initial particle velocity and particle inertia can be ignored.

It might be necessary to describe a particle contaminant source as a distri­bution of particle sizes with initial velocity different from the local air velocity. A typical example of this situation is the particle distribution and particle tra­jectory from a grinding wheel. The particle inertia is important in this situa­tion, which makes it necessary to work with a model where the particles are treated individually through the solution of a particle motion equation. Refer­ence 7 gives the simplified equations that govern the motion of a spherical particle (inertia equals friction and gravity forces) in a flow field. This ap­proach, called the Lagrangian approach because the concentration changes are described relative to the moving fluid, will not be considered further.

Equations (12.40) to (12.45) describe the velocities U, v, w, the tempera­ture distribution T, the concentration distribution C (mass of gas per unit mass of mixture, particles per volume, droplet number density, etc.) and pressure distribution P. These variables can also be used for the calculation of air vol­ume flow, convective air movement, and contaminant transport.

It should be noted that radiant heat transfer, which can be an important part of the heat flow in a building, has not been considered.

The next step toward the similarity principles is to develop the governing equations in a nondimensional form. The equations are normalized by first defining the dimensionless, independent variables as

X"=x/hQ, y" = Y/h0, and Z" = Z/b0, (12.46)

Where is a characteristic length of interest in the problem. A typical charac­teristic length in ventilation problems (general ventilation) is the height of the

Supply slot H0 or the square root of the supply area Joq. The height of a hot or a cold surface Ј can also be used as characteristic length in situations where free convection is the most important problem as, for example, in the case of cold downdraft in an atrium at low outdoor temperatures.

The velocities

U’—u/un, v* = v/u0, and U>’ = u)/w0 (12.47)

Are normalized by the supply velocity U0 found from

U0 = q0/a0, (12.48)

Where QO is the volume flow rate to the diffuser and A0 is the supply area of the diffuser.

 Temperature and concentration are normalized by the following expressions: ‘ T-T0 ‘
 And Ј* = _ЈzЈ! L
 (12.49)
 T
 Tr-T0 cr-c0 Where T0, TR, c0, and CR are supply temperature, return temperature, supply concentration (if any), and return concentration, respectively. Pressure P and the independent variable T time are normalized by the fol­lowing expressions:
 -v A a, * -»A* . + ^d^ + ^dv_ + ^dv_ =
 Dt* dx‘ 3y* 5? (TR-Tn)^ , Mo f32Ј* , a2?“’, a2^
 (12.52)
 Po^oMo A / Ar a
 2 «b
 3x 2 3v 1 32
 3T* . ^*3T* . **3T~ a. + M “ “ + ^ TT—T + If
 32r 32r. 32T* 3x*2 3v’2 3z’2
 Dz C„p0b0
 3x‘
 3 Y’
 Df
 P* = P/tfQp and T’ = tu0/h0.

 (12.50)

 Equations (12.46), (12.47), (12.49), and (12.50) are substituted into Eqs. (12.40) to (12.44), and the following nondimensionalized governing equa­tions are obtained:

 Du , Dt* , Dw _ Q t).v’ Dy’ dz*

 ; 12.51’

 (12.53)

 (12.54)

-tDab

-z~T + K ^—r + ^ rr + UJ rr~T — 7———————————

Dt ox ay dz n0u0 /

32c. 32C* D23* L^x*2 3y*2 3z’2

 3c, a,3c -DAB Ay dz h0u o
 V, A + V " Un
 (12.55)
 ^ y * T-v ( y. » » 32C* + 3Јcl + 32c
 3x’2 dy*2 3«*2
 7
 DA » A *. Df ox

It is observed that the following dimensionless numbers appear in the equations:

 (12.56) (12.57) (12.58) (12.59) (12.60)

Ar = PskoiTg. — T0) «2

Re=^ Mo

 A ’ _ Mo Po-Dab’

Pr =, ^°ct>

Sc

Ar, Re, Pr, Sc, and V’s are called the Archimedes number, Reynolds number, Prandtl number, Schmidt number, and the settling velocity ratio, respectively.

The Archimedes number may be considered as a ratio of thermal buoy­ancy force to inertial force, while the Reynolds number may be looked upon

As a Ratio of inertial force to viscous force. The Prandtl number Is the ratio of Momentum diffusivity to thermal diffusivity, while the Schmidt number is the ratio of momentum diffusivity to mass diffusivity.

Ar and 1/Re appear in the Navier-Stokes equation (Eq. (12.52) j and l/(Re Pr) in the energy equation (Eq. (12.53)). l/(Re Sc) appears in the transport equations for contaminants (Eqs. (12.54) and (12.55)), and the settling velocity ratio : ap­

Pears in the contaminant transporr equation for large particles (Eq. (12. >5)).

The dimensionless numbers are important elements in the performance of model experiments, and they are determined by the normalizing procedure of the independent variables. If, for example, free convection is considered in a room without ventilation, it is not possible to normalize the velocities by a supply velocity «0. The normalized velocity can be defined by U’ — uf. p0 //uq where Ј, is the height of a cold or a hot surface. The Grashof number, Gr, will then appear in the buoyancy term in the Navier-Stokes equation (1 /’ Is The temperature difference between the hot and the cold surface):

Gr = (12.61)

Mo

The Grashof number can be expressed as

Gr = Ar Re2. (12.62)