2.3. GENERAL MOLECULAR TRANSPORT EQUATION FOR MOMENTUM, HEAT, AND MASS TRANSFER

2.3A. General Molecular Transport Equation and General Property Balance

1. Introduction to transport processes

In molecular transport processes in general we are concerned with the transfer or movement of a given property or entity by molecular movement through a system or medium which can be a fluid (gas or liquid) or a solid. This property that is being transferred can be mass, thermal energy (heat), or momentum. Each molecule of a system has a given quantity of the property mass, thermal energy, or momentum associated with it. When a difference in concentration of the property exists for any of these properties from one region to an adjacent region, a net transport of this property occurs. In dilute fluids such as gases, where the molecules are relatively far apart, the rate of transport of the property should be relatively fast, since few molecules are present to block the transport or interact. In dense fluids such as liquids, the molecules are close together, and transport or diffusion proceeds more slowly. The molecules in solids are even more closely packed than in liquids and molecular migration is even more restricted.

2. General molecular transport equation

All three of the molecular transport processes of momentum, heat or thermal energy, and mass are characterized in the elementary sense by the same general type of transport equation. First we start by noting the following:

Equation 2.3-1

This states what is quite obvious—that we need a driving force to overcome a resistance in order to transport a property. This is similar to Ohm's law in electricity, where the rate of flow of electricity is proportional to the voltage drop (driving force) and inversely proportional to the resistance.

We can formalize Eq. (2.3-1) by writing an equation as follows for molecular transport or diffusion of a property:

Equation 2.3-2

where is defined as the flux of the property as amount of property being transferred per unit time per unit cross-sectional area perpendicular to the z direction of flow in amount of property/s · m², δ is a proportionality constant called diffusivity in m²/s, Г is concentration of the property in amount of property/m³, and z is the distance in the direction of flow in m.

If the process is at steady state, then the flux is constant. Rearranging Eq. (2.3-2) and integrating,

Equation 2.3-3

Equation 2.3-4

A plot of the concentration Г versus z is shown in Fig. 2.3-1a and is a straight line. Since the flux is in the direction 1 to 2 of decreasing concentration, the slope dГ/dz is negative, and the negative sign in Eq. (2.3-2) gives a positive flux in the direction 1 to 2. In Section 2.3B the specialized equations for momentum, heat, and mass transfer will be shown to be the same as Eq. (2.3-4) for the general property transfer.

EXAMPLE 2.3-1. Molecular Transport of a Property at Steady State A property is being transported by diffusion through a fluid at steady state. At a given point 1 the concentration is 1.37 × 10−2 amount of property/m3 and 0.72 × 10−2 at point 2 at a distance z2 = 0.40 m. The diffusivity δ = 0.013 m2/s and the cross-sectional area is constant. Calculate the flux. Derive the equation for Г as a function of distance. Calculate Г at the midpoint of the path. Solution: For part (a), substituting into Eq. (2.3-4), For part (b), integrating Eq. (2.3-2) between Г1 and Г and z1 and z and rearranging, Equation 2.3-5 Equation 2.3-6 For part (c), using the midpoint z = 0.20 m and substituting into Eq. (2.3-6),

3. General property balance for unsteady state

In calculating the rates of transport in a system using the molecular transport equation (2.3-2), it is necessary to account for the amount of this property being transported in the entire system. This is done by writing a general property balance or conservation equation for the property (momentum, thermal energy, or mass) at unsteady state. We start by writing an equation for the z direction only, which accounts for all the property entering by molecular transport, leaving, being generated, and accumulating in a system shown in Fig. 2.3-1b, which is an element of volume Δz(1) m³ fixed in space.

Equation 2.3-7

The rate of input is · 1 amount of property/s and the rate of output is · 1, where the cross-sectional area is 1.0 m². The rate of generation of the property is R(Δz · 1), where R is rate of generation of property/s · m³. The accumulation term is

Equation 2.3-8

Substituting the various terms into Eq. (2.3-7),

Equation 2.3-9

Dividing by Δz and letting Δz go to zero,

Equation 2.3-10

Substituting Eq. (2.3-2) for into (2.3-10) and assuming that δ is constant,

Equation 2.3-11

For the case where no generation is present,

Equation 2.3-12

This final equation relates the concentration of the property Г to position z and time t.

Equations (2.3-11) and (2.3-12) are general equations for the conservation of momentum, thermal energy, or mass and will be used in many sections of this text. The equations consider here only molecular transport occurring and not other transport mechanisms such as convection and so on, which will be considered when the specific conservation equations are derived in later sections of this text for momentum, energy, and mass.

2.3B. Introduction to Molecular Transport

The kinetic theory of gases gives us a good physical interpretation of the motion of individual molecules in fluids. Because of their kinetic energy the molecules are in rapid random movement, often colliding with each other. Molecular transport or molecular diffusion of a property such as momentum, heat, or mass occurs in a fluid because of these random movements of individual molecules. Each individual molecule containing the property being transferred moves randomly in all directions, and there are fluxes in all directions. Hence, if there is a concentration gradient of the property, there will be a net flux of the property from high to low concentration. This occurs because equal numbers of molecules diffuse in each direction between the high-concentration and low-concentration regions.

1. Momentum transport and Newton's law

When a fluid is flowing in the x direction parallel to a solid surface, a velocity gradient exists where the velocity ν_x in the x direction decreases as we approach the surface in the z direction. The fluid has x-directed momentum and its concentration is ν_xρ momentum/m³, where the momentum has units of kg · m/s. Hence, the units of ν_xρ are (kg · m/s)/m³. By random diffusion of molecules there is an exchange of molecules in the z direction, an equal number moving in each direction (+z and −z directions) between the faster-moving layer of molecules and the slower adjacent layer. Hence, the x-directed momentum has been transferred in the z direction from the faster- to the slower-moving layer. The equation for this transport of momentum is similar to Eq. (2.3-2) and is Newton's law of viscosity written as follows for constant density ρ:

Equation 2.3-13

where τ_ZX is flux of x-directed momentum in the z direction, (kg · m/s)/s · m²; v is μ/ρ, the momentum diffusivity in m²/s; z is the distance of transport or diffusion in m; ρ is the density in kg/m³; and μ is the viscosity in kg/m · s.

2. Heat transport and Fourier's law

Fourier's law for molecular transport of heat or heat conduction in a fluid or solid can be written as follows for constant density ρ and heat capacity c_p:

Equation 2.3-14

where q_z/A is the heat flux in J/s · m², α is the thermal diffusivity in m²/s, and ρc_pT is the concentration of heat or thermal energy in J/m³. When there is a temperature gradient in a fluid, equal numbers of molecules diffuse in each direction between the hot and the colder region. In this way energy is transferred in the z direction.

3. Mass transport and Fick's law

Fick's law for molecular transport of mass in a fluid or solid for constant total concentration in the fluid is

Equation 2.3-15

where is the flux of A in kg mol A/s · m², D_AB is the molecular diffusivity of the molecule A in B in m²/s, and c_A is the concentration of A in kg mol A/m³. In a manner similar to momentum and heat transport, when there is a concentration gradient in a fluid, equal numbers of molecules diffuse in each direction between the high- and low-concentration regions and a net flux of mass occurs.

Hence, Eqs. (2.3-13), (2.3-14), and (2.3-15) for momentum, heat, and mass transfer are all similar to each other and to the general molecular transport equation (2.3-2). All equations have a flux on the left-hand side of each equation, a diffusivity in m²/s, and the derivative of the concentration with respect to distance. All three of the molecular transport equations are mathematically identical. Thus, we state that we have an analogy or similarity among them. It should be emphasized, however, that even though there is a mathematical analogy, the actual physical mechanisms occurring may be totally different. For example, in mass transfer two components are often being transported by relative motion through one another. In heat transport in a solid, the molecules are relatively stationary and the transport is done mainly by the electrons. Transport of momentum can occur by several types of mechanisms. More-detailed considerations of each of the transport processes of momentum, energy, and mass are presented in this and succeeding chapters.