GEORGE D. FULFORD DAVID C. T. PEI
A Unified Approach to the Study of Transfer Processes The close similarities between the processes of mass, momentum, and energy transfer make a general treatment of all three a powerful tool in both teaching and problem solving
he importance of rates of heat and mass transfer and T fluid flow in engineering processes is well known. T h e traditional approach to the study of such processes in which heat, massr and momentum transfers are considered separately is adequate in dealing with relatively simple processes. However, the modern chemical engineer has to deal increasingly with complicated problems which may involve the simultaneous transfer of energy, momentum, mass, and electric charge, often in the presence of chemical changes. A unified and generalized approach is useful (and sometimes essential) in dealing with such problems. T h e main advantages of a unified, generalized approach may be summarized as follows: T h e underlying aspects of the transfer processes for mass, energy, momentum, etc., are basically similar, so that a better over-all understanding and a saving of time and effort are possible through a unified approach, which also facilitates the drawing of analogies among the processes; Once the physical significance of each term is understood in the generalized derivation, the significance of particular equations derived from it should follow automatically; When equations for a particular case are obtained by simplifying the general equations, the progress from the too-complicated-to-solve equations at one extreme to trivial oversimplified cases at the other can be readily followed as t h e v a r i o u s c o n d i t i o n s a n d s i m p l i f y i n g assumptions are made. I n most cases, there will be a better understanding of the limitations of the final equations selected for solution than if the equations are derived by setting up simple balances for the particular case with all the assumptions made en masse a t the start. Unified generalized treatments of the transfer processes can be set up at many levels of sophistication, varying from highly complicated approaches based on statistical and irreversible thermodynamics to simple approaches in which heat, mass, and momentum transfer are first considered separately and are then combined. T h e first approach is not to the taste of many chemical engineers, and the second involves a great deal of repetition. Here we have attempted to derive a general relationship for the conservation of some intensive property in a flowing medium using a mainly physical approach in which the significance of each term in the conservation equation is made apparent. T h e derivation is carried out in Cartesian coordinates; vector notation is introduced subsequently to condense the equations and to permit generalization of the results to other coordinate systems. After deriving this general conservation equation, it is then a simple matter, as indicated earlier by Foust et al. (7), to obtain from it the equations of change for momentum, energy, mass, electric charge, etc. These general equations of change can then be simplified to apply to specific cases by discarding terms which have no physical significance in the particular situation; solutions of the resulting equations have to be sought with the appropriate initial and boundary conditions. I n this review, we have surveyed some of the main routes by which useful information about concrete probVOL.
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INFORMATION CONTENT OF SOLUTION USUALLY INCREASES DIFFERENTIAL
START -
DIFFERENTIAL
FOR ++CHANGE GIVEN SYSTEM
.
~
EQUATIONS “F
LEGEND =
0
“ R A W MATERIALS”
0 0
“PROCESS OPERATIONS”
“INTERMEDIATE
[cI1 ”FINAL
PRODUCTS”
PRODUCTS”
MOST GENERAL ROUTE SHOWN BY HEAVY LINE
Figure I . Flow sheet of general approach to transport operations
lems can be extracted from the rather formidable conservation equations, which are often insoluble in their general forms. While these main routes have long been known, we feel that their collection and inspection in a generalized context may be useful to many chemical engineers. A certain amount of mathematical manipulation is unavoidable in reducing the general conservation equations to particular forms and in seeking solutions of these by any of the main routes. I n carrying this out, there is often a real danger of failing to see how the particular approach being applied relates to the over-all scheme of transfer process studies. To avoid this, Figure 1 summarizes the various approaches in the form of a flow sheet. This “road map” will assist in identifying the “route” being used in seeking the solution of a particular type of problem and in identifying how far we have progressed toward the goal. Figure 1 shows that all the routes start from the general concepts of conservation. By generalizing these concepts further to the case of flowing systems, we obtain (below) a set of general differential equations of change (or rate equations) for the transfer of momentum, mass, energy, etc. Several alternative routes can be used from this point. T h e most general route (Route 1, Figure 1, top) consists of making such simplifications to the general equations as the physical conditions of the problem permit, and applying the physical and transport properties to derive the particular forms of the differential equations valid for the problem in question. Analytical solutions obtained by integrating these equations give the maximum amount of information about the problem. If the equations cannot be integrated, it is still often possible to branch off just before the final integration step and obtain a numerical solution. Of course, this is valid for only 48
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one set of conditions and, therefore, provides much less information, as indicated on Figure 1. When there are random short-term pulsations in the system it may not be useful to find general analytical solutions following Route 1. Here it is customary to make preliminary simplifications by time-averaging the equations (Route 2) to arrive at useful analytical or numerical solutions. For example, this approach is necessary for transfer processes in turbulent flows. I n still other cases it may not be necessary to have information about local values of the parameters, and volume-averaging may be permissible (Route 3) to simplify the general equations of change before seeking a solution. Finally, even if the equations of change cannot be integrated at all, it is still possible to obtain some useful information from them by inspectional analysis of the rate equations (Route 4). Derivation of General Transfer Equations
Similarity of Transfer Processes. As a preliminary illustration of the underlying similarity of the transfer processes, consider the simple case of unidimensional, steady-state “molecular” transfer of some general property P , which may represent the mass of a component, momentum, energy, etc., per unit volume of the fluid. We can write: (Flux of property) = -(diffusivity of property) X [gradient of concentration of property in direction of transfer (or “driving force”) ]
AUTHORS George D . Fulford is Assistant Professor and David
C. T’. Pei is Associate Professor in the Dekartment of Chemical Engineering, Universio of Waterloo, Ont., Canada.
~~
TABLE I . Property / Unit Vol
~~~~
~~
TRANSFER O F HEAT, MASS, MOMENTUM, AND ELECTRICAL CHARGE I N X DIRECTION
I
Dirusivity
Flux
Driving Force =
I
(BTU) ft3
Mass of Component A,
~
Heat flux, qz, BTU/hr ft2
I
I
Thermal
I
ft2/hr’
I Heat concn
~
I
dT dx Fourier’s law (heat conduction)
Mass concn gradient,
Wa - D Pd dx ’ Fick’s first law (molecular diffusion)
I d(lb/ft3)/dx
(.#I%
= A-+
dv
Newton’s law (viscous flow)
(Ib-ft/hr)
d[(lb)(ft)/(hr)(ft3)l/dx
hr ft2
m3
-k - ,
(J‘A)~ =
Momentum concn
Electrical charge, coulombs
Law (Constant Physical Properties) qz =
dx ’ Id(BTU/ft3)/dx
Molecular diffusivity,
Mass flux, ( , j ~ ) =lb/hr , ft2
Law (Variable Physical Properties)
Charge flux, (electrical current density) - coulombs (unit area - unit time [(amp)/(m2)1
Electrical Electric field conductivity, strength, E , * ucb,mho/m = voltage gradient, dV’ for uniform - dx stationary conductor W/m)
where the flux of the property is defined as the quantity of the property (mass, momentum, heat, etc.) transferred per unit time per unit area normal to the direction of transfer, and the “driving force” is expressed in terms of the gradient of the concentration of the property. Table I shows the forms assumed by the flux, the diffusivity, the driving force, and the general expression above when the quantity being transferred is taken to be heat, mass, momentum, or electric charge. T h e units given in this table are for illustration; any compatible system of units based on the fundamental quantities listed in the Nomenclature can be used. T h e familiar rate equations for molecular heat conduction (Fourier’s law), molecular diffusion (Fick’s first law), viscous flow (Newton’s law of viscosity), and electrical conduction (Ohm’s law) are merely special cases of the more general rate expression above. While the various conservation equations are similar in that the terms in them have the same physical significances, there are some real mathematical differences. These arise because heat, mass, and electric charge are scalar quantities [their fluxes are, therefore, vector (firstorder tensor) quantities], while momentum is a vector quantity and its flux is a second-order tensor quantity. I n Cartesian coordinates, the momentum transfer equation has x , y , and .z components, while each of the other conservation equations takes the form of a single equation. If these differences [discussed, for example, in (9) ] are borne in mind, it is still useful to regard each component of the momentum equation as being similar to the other conservation equations, as in Table I. Derivation of General Conservation Equation for
I
I = u,Es
d V’ dx Ohm’s law (electrical conduction in stationary uniform conductor)
-
-ue-,
Property P . T h e fundamental concept of conservation of an intensive property, P, can be expressed for unit volume of a system as: (Accumulation of P ) = (input of P ) - (output of P ) More generally, timewise changes can be taken into account by considering the rates of the various processes: (Rate of accumulation of P ) = (rate of input of P ) - (rate of output of P ) = (net rate of input of P ) I n setting up the general conservation equation for the property, P, we have to apply this last concept to flowing systems by allowing for the various ways in which property, P, may enter or leave a volume element in the flowing medium. Various methods of setting up the conservation equations are given in the standard texts on transfer processes, a selection of which is listed in Section A of the Bibliography. Here we will apply the method used by Bird, Stewart, and Lightfoot ( 3 ) to the general property, P. For a volume element dx dz, assumed to be fixed in space, as shown in Figure 2, the law of conservation for an intensive property given above may be written in terms of the various possible contributions: Rate of accumulation of property in element (I) Accumulation net rate of transfer of net rate of transfer of property into element property into element by bulk flow of med. by molecular trans. (11) Convection (111) Molecular trans. (1)
+
[
I= I[
1
+
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Z
Therefore, the total net rate of transfer of the property to the element by the bulk flow in all three directions is
A
[(~YAZ)UZ] [PIx- PI z + Pi y + +Ax
Figure 2. Imuginarq' elementjxed in space through which JuidJofiows
net rate of production of property at surface of volume element
I+[
net rate of production of property within volume element (IV) Generation
Consider each of these terms separately: RATEOF ACCUMULATIOK OF PROPERTY.T h e rate of accumulation of property in a volume AxAyAz, such as that shown in Figure 2,will be simply the rate of change of property per unit volume multiplied by the volume of the element, or, in mathematical terms,
[$]
+ [(AxAC)uv.][PI
Ax]
+
ayl i ( A x A ~ ) ~[Pi ~ le
y-
- Pi
+ As1
(11)
NET R A T E OF TRANSFER O F P R O P E R T Y BY MOLECULAR TRANSPORT. First we define the fluxes of the property due to molecular transfer as IT, which are the rates of transfer of property per unit area. [11 can be a secondorder tensor (nine elements) in dealing with momentum transfer, or a vector quantity (three elements) in energy and mass transport.] Again, following the same general approach as above, the amount of property across the face AyAz per unit time at x in the x direction will be (AJAZ)&>~ z. [rI(,) represents the component(s) of II in the x direction.] T h e net rate of transfer of the property to the element in the x direction by molecular transport is, therefore, (A?Ahp) Area
in(,)'z
-
~(z)I
2
+ A21
Quantity of P/ unit area/ unit time
Hence, the total net rate of transfer of the property to the volume element by molecular transport will be [AYAhpl
[n(z)'x -
+
[n(u)/ w
[AX&]
(AxAyAhp)
+
~ ( z ) ] z A z ~
~ O O I+ A v l + [ ~ ( z I I - n(e)iz + A t ]
-
Y
2
(111)
NET RATEOF GENERATION OF PROPERTY AT SURFACE VOLUMEE L E M E N T . This term represents the rate of production of property at the surface of the element owing to all surface forces. T h e obvious example of these ~YETRATEOF TRANSFER OF PROPERTY BY CONVECTION. is the pressure force, which is the only important surface force discussed. Therefore, it follows that F, defined as From Figure 2, suppose that P at the faces of area AyAz the rate of generation of property per unit area at the has a value of PI at x and PI + at x Ax. [Here we surface of a volume element, will only appear when we have assumed that the volume element is small enough are dealing with momentum and energy transfers bethat it is valid to take a constant value over the whole cause mass cannot be produced by pressure force. face and that the element is small enough that the over-all Here, remember that we are considering the "surfaces" mass density p and other physical properties can be asof the imaginary fixed volume element within the fluid, sumed constant within the element at any given instant and not necessarily an actual physical surface (interface). (though, of course, they may vary with time). Because Conditions at the interface type of surface have to be we shall later make the volume element vanishingly taken into account through the boundary conditions imsmall, these assumptions are perfectly valid, provided posed on the conservation equations being developed that the properties of the system can be considered conhere. tinuous.] Similarly, the property P at the faces of area I t follows that the total net generation of property P AxAhp will have a value PI at y and PI + av a t y Ay at the surfaces of the volume element will be: and, at the corresponding faces of area AxAy, it will be PI at z and PI + A , at .t Az. [AyAhpI [Fl - Fl + A z ] f AxAz[F/ Now if we consider the rate of supply of property F1 Y + A v l f AxAJ[FI z - FI z iA z ] (IVa) owing to bulk flow in the x direction across the face NET RATEOF G E N E R A T I O N OF P R O P E R T Y 'WITHIN V O L AyAz at x, this can be written as (AyAhp)~,PlZ. Similarly, the rate of removal of property at x Ax in the x direcUME ELEMENT. Let the rate of production of the property per unit volume within the element be G. (In principle, tion will be (AyAhp)~,Pl. + A ~ .T h e net rate of transfer of property to the volume element by bulk flow in the x the total source term G may consist of several parts, each representing a production of the property arising for a direction is then different reason, G = GI Gg . . . . .) Then the total [(AyAhp)Uzl [PIz z + Azl rate of generation will be Volume,' Net quantity of unit time Plunit volume [GI [AxAyAzl (1Vb) Rate of accumulation/ unit volume
\'olume
OF
+
+
+
+
+ +
50
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With all terms taken care of, we can now write the general law of conservation of property for a finite Cartesian volume element in the following mathematical form: aP - AxAyAz = at
(I) Accumulation [AyAzuz(Pi .--PI + A = ) AxAzu,(P/ u-Pl
.
+
+
A x A ~ u z ( pZ/ - P I 2 + A , ) ] (11) Convection {AY&[&)l
Z-%)/Z+AZ]
+
+ AxAZ[%)/u-%,/u+A 1+
aP _ at
Rate of accumulation of Pi unit volume -
4Y]
AXAY[%)I z - & z ) l z + A,] (111) Molecular transport
) AxAz(Fl ,- F1 21 + A ~ [AyAz(FI - F / + A ~ f Ax&(Fl z - FI z + A , ) ] (IVa) Generation at surface of volume element GAxAyAs (IVb) Generation within volume element
+
we obtain the differential equation expressing the conservation of the property in a flowing medium:
Net rate of addition of P/ unit volume by convection (bulk flow)
f
+ )
Net rate of addition of P/ unit volume due to molecular fluxes
Net rate of generation of P/ unit volume at surfaces
G Rate of generation of P/unit volume in bulk of system
If we now divide Equation 1 throughout by Ax4yA.z and make use of the definition of the derivative as the lengths of the sides of the volume element tend to zero-ie., as
D e r i v a t i o n of Specific C o n s e r v a t i o n E q u a t i o n s . The general conservation equation, Equation 2, can now
Quanti&
Total Mass
Mass of Component i in n-Component System
P
P
Pi
n
0
ii
F
0
0
Total mass (continuity equation)
Additional electrical relationships
ap at
[I
Momentum (PV=, PO,,
P
V
or ~
P(U
PV
+3 4
P
= u e {E,
7r *
v =
Pe
i
4
7
= -V
Electric Charge
Energy
(PGii
+ .v 7)
(3a)
(PV)
+ pe(v X He) } I ;
0
[I
=
v
General Ohm's law
X He;
ap H, = v e at
X Eel
Maxwell equations
=In many cases, fluid is highly electrically conducting, so there is negligible accumulation of electric charge and equations can be further simplified by substituting p. = 0. Here the last term in Equation 3d becomes Z2/ue,which represents the Joule heating effect. ~~
~
~~~
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be made specific for various cases of transport by substituting suitably for P, F, and G. When this is done, the various groups of terms retain their significance in each case--e.g., convection contribution, bulk generation contribution, etc. -and to save space this information will not be repeated for the specific conservation equations obtained below. T h e forms assumed by P, II, F , and G for the cases of transport of total mass, Inass of component i of a mixture, momentum, energy (kinetic plus internal energy), and electrical charge are summarized in Table 11. When these forms are substituted into Equation 2 , the specific general conservation equations for the five types of transport being considered are obtained directly, as shown in Table 111. These general conservation equations form the starting point for various studies of transfer processes, and are those mentioned in the first box in Figure 1. I n Tables I1 and 111, the contributions due to electric and magnetic effects are enclosed in square brackets, and all the terms so enclosed will disappear if magnetic and electric effects are assumed to be negligible. This assumption is almost always made in chemical engineering studies at present, but these effects are undoubtedly important in more complex processes; by the present approach they can be included in the general equations with no additional difficulty, though they will be omitted later to facilitate comparison with conventional relationships. Before these conservation equations can be used in practice, however, the molecular fluxes in each of them must be expressed in terms of the driving forces causing the fluxes. I n the most general case, this becomes complicated to do because it is possible, for instance, to have molecular energy and mass flux contributions arising as a result of several different driving forces. Fortunately, except under specially contriLred conditions, the energy flux due to temperature gradients, the mass flux due to concentration gradients, and the momentum flux due to velocity gradients are the only important molecular fluxes, and in the one-dimensional case. these have been ex-
n,
TABLE IV. Equation of Change f o r
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-
1. [a% 3
~
ax
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30, I
ai:]')
a>
az
T h e resulting expression in terms of the internal energy U has been converted to a form in\-olving the heat capacity C , and the temperature 1 by making use of some simple but lengthy thermodynamic manipulations. Kote that in arriving at Table 111, the main assumption made was that the medium can be considered a continuum. I n going from Table 111 to Table IV, the additional assumptions are made that the molecular flux depends only on the major driving force for the process and the fluid has Newtonian flow behavior and negligible bulk viscosity. T h e equatioiis of change contained in Table I\' are still very complicated, and usually cannot be solved except for problems with simple geometries and boundary conditions. These equations can be greatly simplified if it is assumed that the medium is incompressible and the transport properties (viscosity, thermal conductivity, heat capacity, diffusivity) are constant. T h e fourth assumption is not serious unless extremely high velocities (approaching the velocity of sound in the medium) are encountered. T h e last assumption is con-
FORMS OF EQUATION 3 I N TERMS OF DRIVING FORCES
Energy
52
pressed in Table I. If these are generalized to allow for changes in three dimensions (ignoring bulk viscosity), and are then substituted into the conservation equations in Table 111, the more useful forms gilren in Table I\- are obtained. I n deriving the form of the energy equation shown in Table 1L7, the kinetic energy contribution found by forming the scalar product of the velocity with the equation of motion has been subtracted, giving rise to the term p a , which represents the dissipation of flow energy to heat. I n Cartesian coordinates,
TABLE V.
FORMS O F EQUATION 4 FOR INCOMPRESSIBLE FLUIDS W I T H CONSTANT TRANSPORT PROPERTIES Gravity forces only Simplified Equation of Change for
Mass of component A (binary mixture of A and B )
P
D WA Dt
Momentum (Navier-Stokes equation)
p
Dt =
Dv
=
+
PD~BV’WA
-0P
+ Pg +
TA
PV2V
Energy
Mass Transfer Equation
Equation of Motion
Case
p
Isothermal potential flow of pure fluid (viscosity effects neglected)
Dv
=
-vp
=
-vT
+
No.
Energy Equation
Pg
...
...
(64
(Euler’s equation) Isothermal creeping flow of pure fluid (inertia effects neglected)
P
-;ii
=
-vY
+ PV2V
(Stokes’ equation)
Forced convection without chemical reaction (negligible body force, heat of mixing, and viscous dissipation)
k I
As last, with negligible pressure gradient
1 !?-!? I
Dv = CL v z v Dt P
siderably more limiting because it is known, for instance, that viscosity, thermal conductivity, etc., depend on temperature, pressure, and composition. However, for a wide range of chemical engineering problems, the assumption of constant physical properties more than justifies itself by the great simplification which is possible; comparison of calculated and experimental results has indicated that the errors introduced are not too serious provided temperature and concentration differences are not too large. If, instead of assuming a constant value, the viscosity is expressed by the rheological equation for the material, it is possible to branch off and obtain the rate equations for transfer processes in non-Newtonian fluids. T o obtain useful relationships, further assume that electrical and magnetic forces can be ignored, mass transfer is limited to binary systems, and gravity is the only external force acting on the system. With these assumptions, the equations of change assumed the more familiar forms shown in Table V. I n Equation 5d (Table V), the last term represents the energy contribution due to the occurrence of chemical reactions and heating effects on mixing. Often both this and the viscous dissipation term are small compared to the heat conduction term and can be ignored. Having derived in a unified way the rate equations and simplified them somewhat to the forms in which they apply to many chemical engineering problems, we can
Dt
=
DABV’WA
I
v2T
P CP
~
1
(6~)
1
I
OT
=
k
-v2T pc,
(6d)
now review the four main ways they are used to obtain information about real problems involving the transfer of mass, momentum, and heat. Routes for Obtaining Information from General Equations
Route 1: Simplification of Equations a n d Analytical Solutions. T h e first route toward solving actual problems outlined in Figure 1 involves simplifying the equations of change as much as the physical nature of the problem will permit, and then solving the resulting equations analytically, or, if this is impossible, numerically, using the boundary conditions dictated by the problem. Even when all the permissible simplifications have been made, the equations often remain intractable, except in simple cases. When this occurs, it is often useful to make still further assumptions, not justified in actual fact, but which reduce the general problem to simpler limiting cases which can be solved. I n this way, information can usually be obtained about the transfer behavior under limiting conditions, and the actual behavior of interest should tend toward this. A few limiting forms of the equations are shown in Table VI. T h e first case considered is that of isothermal potential flow of a pure fluid. Here, the “pure fluid” is taken to mean a fluid without concentration gradients; the mixture air is, therefore, a pure fluid in this sense. T h e VOL.
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assumptions that the fluid is pure and isothermal automatically cause the mass and energy transfer equations to be satisfied if viscous dissipation of flow energy is negligible. I n potential flow, it is assumed that the convective contribution (also termed the inertia contribution) is much larger than the viscous contribution in the NavierStokes equation. This may occur in flow at high velocities or in fluids with low viscosities. IVith this assumption, the Navier-Stokes equation simplifies to the form shown in Equation 6a, which was discovered by Euler. This, of course, has to be solved with appropriate boundary conditions for the problem so as to simultaneously satisfy the continuity equation. Euler’s equation applies strictly only to inviscid “ideal” fluids (which do not really exist) with zero vorticity (V x v = 0), but solutions of it do provide information o n the beha\,ior which is approached by real systems at large Reynolds numbers, especially in gases, and in flows outside the boundary layers which form adjacent to interfaces. Many solutions of this equation have been obtained for ideal fluids, but here let us look a t only one general solution. In steady-state flow p(dv/dt) = 0, and Euler’s equation becomes
change assume the forms shown in Equation 6d. T h e heat, mass, and momentum transfer equations have the same mathematical form and, if the boundary conditions are also similar, a solution of one gives the solution to the other two cases also, saving a great deal of work. This similarity forms the basis for most of the analogies among mass, momentum, and heat transfer which have played so important a part in the development of chemical engineering; the many simplifying assumptions made in arriving a t these forms of the general rate equations will suggest some of the limitations of the analogy approach. -4fter the various simplifying assumptions hai-e been made to arrive at the forms sholvn in Equation 6d, the only terms in Equation 2 which survive are the time change, coniTection, and molecular transfer terms. Because all three general transfer equations were derived from this single conservation equation and because only the corresponding terms in each general equation have been retained, it should not be a matter of surprise that the resulting equations are analogous in form. I n the special case of a stationary medium, the continuity equation and equation of motion are satisfied automatically, and the others are simplified to
p v - v v = -0?
+
where ? = p pgh, h being the vertical height above some horizontal datum plane. How~ever, v*Vv
=
vgu2 - v x [v x
VI
Because the flow is assumed irrotational, v’ X v = 0, and v * v v = 037, 1 2 For the incompressible case, therefore, v+p7J2=
- V?
and by integrating,
qpa? = --? + const.
or
+
$pv2 + p pgh = const. which is a form of Bernoulli‘s equation. Kote the large number of simplifying assumptions made in going from the general momentum conservation equation to Bernoulli’s equation, and also that this equation is of rather restricted applicability. A superficially similar but less restricted equation will be obtained later by the volume averaging technique. If the opposite limiting assumption is made (that \-iscous forces predominate over inertia forces, which can occur in the slow flow of a viscous fluid), the equation of motion assumes the form shown in Equation 6b. If this is solved for the steady-state case with boundary conditions for flow around a sphere, for instance, the familiar Stokes equations for the terminal velocity and drag of a sphere moving slowly through a viscous liquid are obtained. If, on a more ambitious scale, we consider heat, mass, and momentum transfer together, and assume that body forces and pressure gradients are negligible, that there are no chemical reactions occurring, and that the heat of mixing and rate of viscous dissipation of the flow energy are small enough to neglect, the equations of 54
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This form of equation has been solved analytically for a number of geometries and boundary conditions, and a large store of ready-made solutions exists (5, 11,15). Route 2: Volume-Averaging of Equations. I n many cases, even when exact solutions can be obtained, we d o not really need the detailed (or “microscopic”) information about local velocities, temperatures, concentrations, and pressures produced by a complete solution to analyze the over-all effects of a process. Instead, it is often sufficient to average the equations of change over the entire volume of the system by a volume-averaging procedure before sol\ring them to obtain oi-er-all or “macroscopic” equations of change for the system as a whole. Philosophically, this involves regarding the process being studied as a “black box”: 11-e derive equations which will tell us what the outputs from the black box are for given inputs and operating conditions. T h e weakness of this approach is that the exact details of the transfer processes involved are not known. There is no way, using this approach, to peep inside the black box, while the analytic approach of Route 1, when it can be used, does enable us to see parts of the interior and so affords a better opportunity for improving the working of the process. T h e advantage of the volume-averaging procedure is that it yields useful over-all equations even for complicated problems. Let us look at the volumeaveraging process and indicate briefly how some of the results can be used. T h e volume-averaging procedure consists of simply averaging each term of the equation of change over the entire volume of the system and simplifying the result. I n carrying out the procedure, it is necessary to use the
well-known divergence theorem of Gauss, by means of which the integrals over the volume V of the system can be replaced by an integral over the surface of the system S enclosing V. For scalars, vectors, and tensors, this theorem may be written as
where n is the outwardly directed normal vector at the surface. I n this way, many of the volume integrals are conveniently expressed in terrns of surface integrals, evaluated over the entire surface S of the system, which may, for instance, represent the bounding surfaces of a flow system. I t is then useful to break down the total surface S of a general system into partial surfaces of special sorts: an inlet and outlet control surface, S, and So,across which material enters and leaves the process; a mass transfer surface S, at which heterogeneous--e.g., catalytic-reactions can take place, or through which mass may be added or removed by blowing or suction; a “movable” surface Sw--e.g., an impeller, piston, etc.-by means of which useful work can be removed from the system, and the rest of the fixed surfaces Sf,assumed to be merely wetted by the fluid. (Work removed from the systeni and heat added to the system are assigned positive values, in accordance with the usual thermodynamic convention.) Hence,
s = st
+ so + s, + s, + Sf
(8)
Here it is assumed that the properties of the fluid do not vary significantly over the inlet and outlet control crosssections, and that the transport of mass, momentum, and energy across these surfaces by molecular mechanisms is small compared with transport by bulk flow. Usually the positions of the surfaces Siand So can be selected so that these assumptions are justified. Finally, we will use the convention that the change in any quantity over the system is given by A (quantity) = value of quantity at outlet - value ae inlet (subscript 0) (subscript i)
Finally, the surface integrals can be broken up into their component parts over Si,So, S, S,, and Sf,giving the generalized form of the macroscopic conservation equation for the general property P:
I. Total rate of accumulation of P in system
11. Net rate of transport of P into system by bulk flow across inlet and outlet control surfaces
111. Rate of transport of P into system by bulk flow across mass transfer surfacee.g., by blowing
IV. Rate of transport of P into system by molecular processes across
s,
Jsl(nWS
s,,(WdS
-
V. Rate of transport of P into system by molecular processes across fixed surface
V I . Rate of transport of P into system by molecular processes across surface for removing work
VII. Net rate of transport of P into system by forces at inlet and outlet control surfaces
V I I I . Rate of transport of P into system by forces acting on mass transfer surface n
Ix. Rate of transport of P into system by forces acting on fixed surface
X. Rate of transport of P into system by forces acting on surface for removing work (1 1)
If each term of the general equation for conservation of the intensive property P (Equation 2) is integrated over the volume of the system, and it is assumed that the surface of the systeni is not itself moving, we can write
XI. Total rate of generation of P within system Here three of the surface integrals have been omitted because they usually equal zero:
By using the Gauss theorem and its relatives,
L f ( n * v P ) d s= 0 (rate of transport of P into system by bulk flow across fixed surface, which is assumed to be impermeable and inert, n * v = 0) L w ( n - v P ) d S= 0 ( n - v = 0 at surface for removing work also) VOL.
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I + I + Y
W
4
c
0
56
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CHEMISTRY
0
0
0
J
Sf
+ so
(n*B)dS
o (net rate of transport of P into system by molecular transport across the inlet and outlet control surfaces, earlier assumed negligible)
T h e forms assumed by the various terms in Equation 11 for total mass, mass of component A , momentum, and energy transfer are shown in Table VII. T h e integrals So represent the change in the over the surfaces Si quantity between the inlet and outlet control sections, as may be illustrated by considering term 11:
+
n
= A(p;S) =
AI’
where r is the mass flow rate. At this point we have assumed, as noted earlier, that the properties of the stream (here p ) are constant over the inlet and outlet crosssections, and we have assigned average values of v , over the inlet and outlet sections. Other integrals over S, So are treated similarly except that in the momentum transfer case, S and are vectors whose directions give the flow direction. T h e other integrals have been assigned symbols for sake of abbreviation. T h e significance of each symbol will be obvious by referring to the original terms in Equation 11. When these terms are substituted into Equation 11, the final forms of the macroscopic conservation equations shown in Table VI11 are obtained. Table VI11 also contains the macroscopic mechanical energy balance equation, which is obtained by forming the scalar product of the equation of motion (Equation 5c) and the velocity v and then volume-averaging, assuming that the fluid is incompressible, that only gravity acts as an external force, and that chemical equilibrium exists in the system. represents the total rate of transfer I n this equation, of mechanical energy to the surroundings by the action
+
wt
of viscous and pressure forces on the surfaces, and E , , represents the total rate of dissipation of mechanical energy (the “friction losses”). Other macroscopic balances can also be obtained for special applications. T h e significance of the macroscopic balances should now be examined. Equation 12a merely expresses the over-all mass balance on’ the system and, therefore, does not yield much exciting information. Equation 12b gives the unsteady-state mass balance for each component in a system which may have mass transfer and chemical reactions occurring. I t is a general statement of the equations so ,widely encountered in the “industrial stoichiometry” exercises given in first chemical engineering courses, and its application is familiar. These equations can be obtained directly by simplifying the general equations suitably. T h e macroscopic momentum and energy balances are likewise generalized balances written for unsteady-state situations in flowing systems. A macroscopic internal energy balance could easily be obtained by subtracting Equation 12c from Equation 12d, and this will yield the over-all heat balance equation for a system, used by all engineers. Volume-averaging of the equations yields immediately recognizable and practically useful relationships. However, because we cannot see into our “black-box” system, we have no way of predicting theoretically what some of the terms will be. For instance, the rate of viscous dissipation of mechanical energy E , , cannot be calculated a priori, but has to be expressed empirically in terms of an experimentally determined friction factor. Similarly, rates of heat transfer and mass transfer have to be expressed in terms of experimental heat and mass transfer coefficients, which attempt to comprehend a variety of different types of local behavior in a single empirical factor. T h e macroscopic balance approach is widely used in engineering practice, though not always under this name. Acrivos and Amundson ( I ) have shown how this type of relationship can be applied to the study of a family of stirred flow tank reactor problems, and many examples are given by Bird, Stewart, and Lightfoot ( 3 ) and by Fredrickson and Bird (8).
~~
TABLE V I I I .
VOLUME-AVERAGED FORMS OF EQUATION 5
Incompressible fluid Macroscopac Conservation Equation of
NO.
Mechanical energy
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Route 3: Time-Averaging of Equations. I n the case of turbulent flows, the flow field is characterized by frequent, rapid, random fluctuation of velocity. Small elements of the fluid appear to move about in random directions for random distances and at random speeds, superimposed on the over-all forward velocity. These elements or eddies may transport heat, mass, or momentum transverse to the flow. There is, therefore, an additional mechanism present for transporting these properties which is absent in laminar flows, in which interactions between layers of the fluid in the direction normal to the flow direction occur only on a molecular level through the viscous transport of momentum and the transport of heat and mass by conduction and molecular diffusion, as considered in Route 1. I n principle, the equations of change obtained earlier should also apply to turbulent transport, because the continuum assumption does not appear to be infringed in ordinary cases. However, even if the Navier-Stokes equations of motion, for instance, could be simplified for the particular situation being studied and solved with appropriate boundary conditions at some instant in time, the results would give an instantaneous picture of the velocity distribution at that time. T h e instantaneous velocity at each point would depend partly on the average forward velocity at the point, and partly on the direction and velocity of an eddy which happened to be passing the point. An instant later, another solution would give a different picture of the flow field. T h e same is true of concentration and temperature fields. Such information is not useful for engineering purposes, where the average forward velocities, the average temperatures, and average concentrations at each point are more desirable. Instead of solving the equations first for a large number of different instants of time and then averaging the instantaneous local values obtained, it is far more convenient to time-average the actual equations of change, then attempt to solve these with the correct boundary conditions to obtain time-averaged local values of the velocity, pressure, concentration, and temperature. I n carrying out the time-averaging process, it is necessary to select carefully the time period At over which the average is taken. This period has to be long enough that the time-average will eliminate the random turbulent pulsations, but not so long that the time-averaging will become involved with unsteady-state processes which may be occurring in the system. Except for problems involving very rapid transient phenomena, these considerations do not usually cause serious difficulties. T h e procedure in time-averaging the equations is simple. T h e local instantaneous value of the velocity or other fluctuating parameter is replaced by a time-averaged part and an instantaneous deviating part; for instance, v =
-
V+V'
vdt INDUSTRIAL AND
However, obviously the time-average of the square of a deviating part cannot be equal to zero
and, in principle, the time-averages of products of deviating parts of different quantities also cannot be equal to zero. If, in addition to the velocity v, we also replace the instantaneous values of the general property P,its molecular flux rI, and its rates of generation at the surfaces and in the bulk of the system, F,G, by their time-averaged and deviating parts, noting, however, that the density will be constant, according to the assumption made earlier, we can rewrite the general conservation Equation 2 for the property P as
1
At
+
ENGINEERING
CHEMISTRY
-S,
t
+AtdP'
- dt =
At
at
la
Ib
IIa
ijf
t+At
IIb
a
d + -(uylP') + aY
[$(uZ'P')
's, t
At
where v represents the instantaneous velocity a t the point in question, 5 represents its time-averaged value, defined as
58
and V' is the instantaneous deviation of v from its timeaveraged value. From these definitions, it follows that
+At=
Gdt
+ AtLJ
t
+ At G'dt
(13)
Va All the "a" terms in this equation involve taking timeaverages of quantities which are already time-averaged and can be treated as constants. All the "b" terms can be reduced to taking the time-averages of deviating parts
TABLE I X . TIME-AVERAGED FORMS OF EQUATION 5 Constant density and binary systems assumed
NO *
Turbulent Transport Equation of
-
Energy a
p&*
has similar form to that of
Turbulent mass flux
pepDi= = - v . 3 - v 3’
(154
pz.
iA* =
___ v’pA/
(164
pv,/v,’
pv,‘/v,/
P uz/u./
P
Reynolds stresses (turbulent momentum flux)
I
+ pT- + p-5 * + QA?A
Dt
pvJ/v,‘
T-2
Turbulent energy flux
of quantities, which are equal to zero by definition and disappear. T h e “c” terms involve products of deviating parts, and consequently do not disappear. By making these simplifications, the equation reduces to
This is the “turbulent” general conservation equation for property F. Comparing this equation with the original general conservation equation in terms of the instantaneous values of the parameters, we see that the new expression is obtained by rewriting the original one in terms of the more useful time-averaged quantities, except that three additional terms appear which involve products of the fluctuating parts. I t is through these three terms that turbulence makes itself felt in transfer operations. By substituting into Equation 14 the forms of the terms shown in Table I1 (now using the time-averaged values), the generalized turbulent conservation equation can be converted into the turbulent conservation equations for total mass (continuity equation), mass of component A , momentum, and energy, which are shown in Table IX. T h e additional turbulent fluxes denoted by quantities with an asterisk are defined in Table X for the case of the fluid of constant density.
Before the turbulent transfer equations shown in Tables IX and X can be used, it is necessary to express the various fluxes in terms of appropriate driving forces or other measurable-ie., time-averaged-quantities. I t is assumed that the time-averaged fluxes-e.g., 5.can be defined in the usual way, except that now gradients of the time-averaged concentrations, temperatures, and velocities are used as the driving forces. Thus, these time-averaged fluxes can be expressed in terms of the usual transport properties (again assumed constant) without serious difficulty by a n extension of the procedure used for ordinary laminar or stagnant cases.T h e turbulent flux contributions, jA*,3*,and i*, however, present a more serious problem. At present, there is no method available for predicting these quantities theoretically as functions of the time-averaged properties of the stream; it is always necessary to express them empirically or semiempirically, and this ultimate total dependence on previous experimental data has long been a serious stumbling block in turbulent transfer process studies. Because of the important effects of turbulence on transfer processes in fluid phases, a n enormous amount of work has been done on studying the characteristics of turbulence, on trying to predict the nature of the turbulent behavior which will occur under specified conditions, and on the manner in which this will affect momentum, heat, and mass transfer. While some understanding has been achieved of some of the mechanisms involved, there is still no satisfactory theory for predicting the turbulent fluxes. At present it is only possible to calculate turbulent transfer rates on the basis of generalizations of experimental results. T h e main approaches are given here. VOL.
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T h e earliest approach to dealing with the turbulent fluxes involved Boussinesq’s concept of eddy kinematic viscosity, later extended to eddy thermal diffusivity and eddy diffusivity. This approach assumes that the turbulent fluxes will be proportional to the appropriate driving forces, expressed in terms of the gradients of time-averaged quantities. T h e proportionality constant is the eddy transport property. While this approach is pleasantly simple and provides a convenient way of thinking about the problem, it was soon realized that eddy transport properties are not constant, but depend markedly on the Reynolds number of the turbulent stream as well as on the distance from interfaces or solid objects, and this variation has to be determined experimentally. T h e approach does, however, permit experimental data on one of the turbulent transport processes to be used for predicting the other turbulent fluxes under the same geometric and flow conditions. Because heat, mass, and momentum are all transported in a turbulent stream by the motion of the same eddies, it is reasonable to expect that the three eddy transport properties mentioned above will be approximately numerically equal. Hence, if the eddy kinematic viscosity
TABLE X I .
is known by calculating back from the measured profiles of the time-averaged velocities in a particular turbulent stream, this information can be used to predict turbulent heat or mass transfer. T h e next approach to the turbulent flux problem was based on Prandtl’s mixing-length concept. Prandtl originally thought of the mixing length of an eddy as a quantity somewhat analogous to the mean free path of a molecule in a gas. After traveling for a distance equal to the mixing length, the eddy was assumed to merge its contents with the surrounding fluid, which might have different properties from those of the eddy. Again this concept suffered from the difficulty that the mixing length was a marked function of Reynolds number, and in a pipe flow, for instance, it varied from zero at the wall, passed through a maximum, and tended to zero again at the pipe axis. A vast amount of time-averaged velocity distribution data were obtained for various types of turbulent flows in channels and around solid objects. Such velocity data can be generalized into the so-called universal velocity profile, giving a distribution of the time-averaged velocity which is valid for boundary layers in fully developed
DIMENSIONAL ANALYSISa
A. Mass Transfer Equation
Equation:
Significance of term b
Rate of change (with time)
Pseudodimensions
c,!t
Rate of change by convection (inertia)
I
Rate of change by molecular transfer (diffusion)
1 1 r a t e of change by homogeneous chemical reaction
D A Bc / L 2 vciL
Groups for a general case
vt
tE ( T
h
)
d
Other mass transfer groups
60
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I
I
/I
kLc,/L Specimen boundary condition, mass transfer coefficient type
I
1'-
z
--
/I
-t-
4
\
k
h, lg 2 h
I II 0
c
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61
i
I
i * 1
- -
I
I *
------+-I I
-i*
+
i g
v
i + I
W
*
I -
62
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-
ENGINEERING
CHEMISTRY
- -
- -
T
- I5 II
i-)
-t
I
1
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63
turbulent flows. Based on this convenient generalization of the available experimental data, expressions have been obtained for the eddy transport quantities which can be used for calculating the turbulent fluxes appearing in the equations of Table I X . Perhaps the most widely used generalization of this type is Deissler’s ( 6 ) , which has been used with success in a variety of turbulent situations. Still another approach to the turbulence problem has been to study the detailed turbulent structures of flows under various conditions. A large part of this work has involved statistical studies of the fluctuations; in particular, studies have been made of how the turbulent fluctuations a t different times and positions correlate with each other. This statistical approach has suggested mechanisms by which turbulence is generated and decays. However, these studies have not yet led to results which are immediately useful in practice. T o sum up, then, the general conservation equations can be easily extended to cover transport in turbulent streams by rewriting the conservation equations obtained earlier for local instantaneous quantities in terms of the time-averaged values, which are the quantities of practical interest in such cases, and adding the nondisappearing contributions due to the products of fluctuating quantities (the turbulent fluxes). Although the latter cannot be expressed theoretically at present, means do exist for using generalized experimental data to predict them. T h e turbulent transfer equations can then be simplified for the particular situation being considered, and a solution can be sought which will satisfy the boundary conditions of the problem. I n principle, turbulent transfer processes can, therefore, be attacked in much the same way as the corresponding processes in laminar streams. Route 4: Inspectional Analysis of Equations. I t often happens that the equations of change applicable to a given case cannot be solved with the appropriate boundary conditions by any of the above routes even after all the simplifying assumptions that are permitted by the problem are made. T h a t the problem can be described by a set of (insoluble) differential equations and boundary conditions, however, means that a certain amount of information is available about it, and useful information can be obtained in the form of the governing dimensionless groups by an inspectional analysis of the transfer equations. This approach should not be confused with the Rayleigh indicia1 procedure and the 9-theorem procedure of dimensional analysis, which are used when no information about the problem except a list of important parameters is available. Once the dimensionless groups of importance are known, it is possible to plan efficient experimental model tests to study the problem, to analyze the experimental data, to obtain generalized correlations, and to scale the information to practical systems. Of course, the amount of information available in this way is small compared with that from a complete analytical solution of the equations. 64
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T h e main method of carrying out inspectional analyses of the rate equations consists of normalizing the equations and boundary conditions. This procedure is well known, as are its advantages and disadvantages. Instead, we will use the flexible procedure used by Klinkenberg and Mooy ( I Z ) , which seems to have some real advantages when complex transfer equations are handled. I n this procedure, the equation of change is written and the “pseudodimensions” of each term are derived by replacing velocities, lengths, etc., by general velocities, lengths, etc., and remembering that the operators v and V2 have the dimensions 1/L, 1/L2, respectively. This procedure will be clear from the examples shown in Table XI (third row in headings). Because the true dimensions of each term in the equation of change (which must be dimensionally homogeneous) are the same, it follows that dividing each of the pseudodimensions of the other terms by the pseudodimension of one of them will produce a set of dimensionless groups for the problem. I n this procedure, boundary conditions can be treated easily, provided a little care is used. This is best shown by an example. Suppose we are dealing with the flow equations for a situation in which surface tension at the free surface enters as a boundary condition. T h e true dimensions of each term in the flow equation as written in Table X I C (top row of heading) are force/unit mass. T h e dimensions of surface tension are force/length, and these can be brought to the same dimensions as the main terms by dividing by L2 and p , so that the pseudodimensions of this term become u / L 2 p . This is then treated exactly as the pseudodimensions of the main terms in forming dimensionless groups. Initial conditions give rise to additional groups, usually in the form of ratios of the initial to the subsequent or steady value of the parameter. For instance, in a n unsteady-state heat transfer problem with an initial temperature condition To,a simplex To/?-would be added to the dimensionless groups found by the method above. T h e dimensionless groups have been obtained in Table XI for the complete mass, heat, and momentum transfer equations for incompressible fluids with constant physical properties, each with at least one common type of boundary condition. T o illustrate the versatility of the method, the electric and magnetic terms have been included, making the assumption that the fluid is a good conductor. I n the tables, the convention is used that the pseudodimension being divided appears at the head of an arrow, while the pseudodimension by which it is divided appears at the tail of the arrow. I n other words, an arrow drawn from the column for the molecular transport term to the column for the inertia term represents a dimensionless group whose significance is the ratio of inertia/molecular transport. T h e various groups formed are identified on the arrows. Table XI1 gives the additional groups which arise when more than one of mass, energy, and momentum is being transferred simultaneously. I n an actual problem,
TABLE X I I.
ADDITIONAL GROUPS ARISING IN PROBLEMS W I T H SIMULTANEOUS MOLECULAR MASS, HEAT, AND M O M E N T U M TRANSFER
I
I
Mass
I
Heat
Momentum
Pseudodimensions
I I
Groups: Heat
+ mass
Mass
+ momentum
+ momentum Mass + heat + momentum
-I
I I
k
(Le)+
&&AB
Heat
I
TABLE X I I I .
I
I
Any two of Le, Sc, Pr.
DIMENSIONLESS GROUPS USED I N TABLES X I AND XI1
V = general velocity; L = general length dimension Name Brinkman No Capillary No. Damkahler > Nos. Eckert No. Elsasser No. Euler No. Fourier No. (heat transfer) Fourier No. (mass transfer) Froude No. Grashof No. Joule No. Lewis No. Hartman No. Magnetic force parameter
the starting differential equations on which the inspectional analysis is performed would be simplified in advance as much as possible, so that some of the terms present in the general equations might disappear. T h e procedure is the same, however. An example of this is shown in Table X I C , where the terms of importance for a steady, horizontal, free-surface wave-formation problem a r e shown in the third block of the table, and a correspondingly reduced number of groups is obtained. Table XI11 gives the names and definitions of the dimensionless groups obtained in Tables X I and X I I . A
Symbol
Ne Nu Peh Pe, Pr Re
RM S sc Sh St Th We
Name
Dejinition
Newton No. Nusselt No. Ptclet No. (heat transfer) Peclet No. (mass transfer) Prandtl No. Reynolds No. Magnetic Reynolds No. Magnetic Pressure No. Schmidt No. Sherwood No. Stanton No. Thomson No. Weber No. -
a Reduces to Fanning Friction Factor for fully developed channel flow.
system of units based on mass, length, time, temperature degree, and magnetic permeability as the fundamental units is used here. As a result, the dimensional constant g c and mechanical equivalent of heat do not appear. Some of the advantages of this procedure are shown in the tables. First, it is a simple matter to obtain a complete set of dimensionless groups for the problem (and appropriate boundary conditions) by starting all the arrows in one column and making sure that one ends in every other column. There is no need to worry about whether all the important parameters have been inVOL.
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cluded, as in the r-theorem procedure. If all the arrows were started in some other column, an apparently different set of groups would be obtained, but these could still be expressed in terms of the first set by taking products or ratios of the first groups. Also, if two or more arrows cover the same distance as other arrows, then all the groups represented by these arrows cannot be independent. For instance, in Table XII, only two of the three groups Le, Sc, and Pr can be independent for a given problem. Extraneous groups are, therefore, easily avoided. Even more important, however, is that in using this procedure, the significance of each group is immediately clear; this is not always easy to deduce using the Rayleigh or a-theorem methods. From the significances and sizes of the various groups it is often possible to make important simplifications to the original equations, which might then assume a form which can be solved. For instance, Table X I B shows that the Brinkman number represents the ratio of heat generation by viscous energy dissipation to the rate of heat conduction. Should the Brinkman number be very small for a given problem, it could be deduced that it would be satisfactory to simplify the heat transfer equation by discarding the complicated term pa. Similarly, under Route 1 it was noted that if the Reynolds number is large enough, the Navier-Stokes flow equation may be replaced by the simpler Euler equation by omitting the viscous transfer contribution. T h e basis for this should now be clear. T h e “largeness” or ‘%mallness” of the various groups needed to permit such simplifications to be made has to be based largely on practical experience. T h e use of these procedures for initially analyzing and interpreting transfer equations should become increasingly valuable as more and more complicated problems have to be investigated.
assume in the usual way that the flow is irrotational= 0. Then, by using the vector identity
i.e., (V X v)
v v = v+u2 - v
v
I t is impossible to consider in detail the possible applications of the general procedures outlined above for examining transfer processes. T w o widely different examples are given below, and it is hoped that these will provide some idea of the usefulness of the general approach as well as the scope of its applicability. Derivation of Bernoulli Equation W h e n Transverse Magnetic Fields A r e Present. T h e Bernoulli equation is widely used in fluid flow studies. I n view of the increasing importance of processes occurring in transverse magnetic fields, let us derive a n analog of the simple Bernoulli equation for this type of situation. Start with the equation of motion (Equation 4c) and assume that viscous forces can be ignored to obtain the analog of Euler’s equation which, for the steady-state case, is =
-Vy
+
PeEe
+
Pe(1
X He)
(Ia)
Assume that the electrical field effect is small (highly conducting fluid), so that p e E , can bc ignored. Next, 66
INDUSTRIAL
x
VI
V$pv2
=
-V?
+ p e ( I X He)
(Ib)
Consider that flow occurs only in the x direction. Then,
Finally, assume that the magnetic field is applied transversely to the flow direction--i.e., H , = 0, H , = 0 , H , # 0. Because from Maxwell’s relationships
I
=
1 -(V
X B)
Pe
where B
=
peHc
or
equation I b may be written for the x direction as
By integrating,
B + P + pgh + 7 = const. ‘P Z2
e
Sample Examples
Vv)
[v
write Equation I a as
+pu2
P(V
x
AND
ENGINEERING
CHEMISTRY
which is the analog of Bernoulli’s equation for this situa, the significance of an tion. T h e new term, B Z 2 / 2 p e has additional pressure arising in the fluid as a result of the application of a transverse magnetic field of field strength H , = B Z / p L ethis ; may clearly give rise to a number of interesting effects. Recently Bopp ( 4 )showed how the equation of motion, including the electrical and magnetic terms, may also be readily applied to the study of various magnetohydrodynamic flow processes and to the analysis of such phenomena as electroosmosis. Simplified Analysis of Vortex T u b e of R a n q u e and Hilsch. T h e vortex tube is an interesting device discovered by Ranque (17)and first studied in detail by Hilsch ( 1 0 ) . Basically, it consists of an empty cylindrical tube into which a gas stream is introduced tangentially at one end at high speed. A fraction, $, of the gas is removed as a cold stream from near the axis a t the injection end, the remainder at the far end of the tube from near the periphery as a hot stream (Figure 3a).
T h e vortex tube has been proposed as a cheap and simple refrigeration device, and several detailed studies of it have been made (2, 14, IS, 18, 19). By using a simplified model proposed by Kowalewicz (13) and applying the basic conservation equations, a simplified analysis of its operation may be carried out which yields results in good qualitative agreement with the experimental observations. MODEL.For a simplified model of the process, let us assume that the motion can be separated into a rotary motion, with an outer velocity equal to the injection velocity V , at the tube radius R, and no axial velocity, on which we will subsequently superimpose an axial velocity. Obviously, this involves severe oversimplification of the real situation. Assume, also, that physical properties other than density are constant, and that the gas obeys the ideal gas law. First, consider the steady-state rotary motion. Here
a a _ -0 at ’ ae
= 0
a -
’ at
= 0,v =
UB
= u,v, =
v, = 0
T h e conservation equations are then written in cylindrical ( r , 8, t) coordinates with the simplifications above. T h e continuity equation is automatically satisfied. T h e r component of the equation of motion ( p # const.) gives
9= p dr
T h e boundary conditions are v at tube center), and
v = V,, j~ = Pa, p = p o and
r dr
=
0 (symmetry
dT = 0 (insulated wall) at dr
Substitution of this value in Equation I I d indicates that
Wf)
d T / d r = 0, or T = const.
From Equations Ile, IIf, and the ideal gas law, Equation I I a can be written
2 -- 1 v*2 rdr 2-
P
or
r2
0 at r
Voy/R0
v =
r
1 dv -- -v = o
=
r = Ro-i.e., inlet injection conditions prevail at the periphery. If Equation I I b is first solved with these boundary conditions, the velocity distribution is
V2
&component:
d2v -+ dr2
When the viscous dissipation term is written in cylindrical coordinates and simplified for the present case,
Inp =
~ T R ,
1
;j2+
V,2%T(
const.
and because p = Po when r = R,, we have finally
and the energy conservation equation gives
Feed
Even though the static temperature is constant, remember that in a high velocity gas stream it is the total or stagnation temperature which is important--i.e.,
Is
=
T
section A-A L
A
a b
C
TS,tn
6 . Predictions bysimpliJed
(IIh)
and because the velocity increases with r (Equation IIe), it follows that the peripheral parts of the stream will carry much more energy per unit mass than those near the axis, explaining why the inner stream is “cold” and the outer part “hot”. Approximate expressions for the temperatures of the streams can be obtained now by superimposing uniform axial velocities on the flow considered above. If the heat capacity of the medium is constant, if no heat is gained or lost by the tube surfaces, and if we consider that a mass fraction $ of the input stream is removed in the “cold” stream, we can write a n over-all heat balance as
Figure 3. Vortex tube. a. Vortex tubecooler. model. c. Trends of some experimental results
+ (V)2/2C,
=
fiT8.G
+ (1 -
(11.i)
+.)T,,h
where T 8represents the stagnation temperature, and the VOL.
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67
subscripts in, h, c denote the feed and hot and cold streams. For simplicity, assume (though this is not necessary) that the radius of the "cold" stream outlet, R,, is equal to the inside radius of the annular outlet for the hot stream, so that the hot stream can be assumed to occupy a uniform annulus between r = R , and r = R,, while the cold stream occupies the cylindrical space from 7 = 0 to r = R,. Using Equation IIh, we can now apply the velocity profile (Equation IIe) and write expressions for the mean stagnation temperatures of the two streams, assuming a tube length L and a mass flow rate (total) of M . For the cold stream:
CA
= =
d
= = =
c*
DAB E,, E, E1 E,
hut
=
F
=
$ :? Flm)
= = =
g G
=
h h/
=
7
+
(W
T d , c
where T d , represents the dynamic temperature of the cold stream averaged with respect to mass, as expressed by the integral term; this will obviously be a function of the tube geometry, total mass flow rate, etc. For the hot stream:
=
T -k
(IIni)
T d , h
similarly represents the mass-averaged dywhere 7 h, namic temperature of the hot stream. When Equations IIj, IIk, and I I m are used, the static temperature T (previously shown to be constant) is
T
= Ts,in
-
[ # y d , c
+ (1 -
$)Td,h]
and on substituting for T in Equations IIk and IIm, we get: -
7 3 * h
-
= Ts.in
+
$ [ T d , h
-
-
T d , c l
(114
If the factor in the last brackets remains approximately constant, we can predict from Equations I I n and I I p that the vortex tube will behave approximately as sketched in Figure 3b as the fraction $ is varied. Some sample experimental results obtained in tubes of different geometries by Hilsch (10)and by Spahiu and Cserveny (20) are sketched in Figure 3c for comparison. I n view of the simplifications which have been made, the agreement of the approximate theory with the experimental results is rather good. Other deductions that can be made from this type of simple analysis of the vortex tube cooler are discussed by Kowalewicz (13). NOMENCLATURE Dimensions of the quantities are based on the fundamental , quantities mass [MI or [moles], length [ L ] ,temperature [ I ]time [e]: and electric charge [Q]. Dimensionless groups are defined separately in Table XIII. A
B
= =
c
=
68
component A (binary system) magnetic induction (B = p L e H c )[MjQO] molar concentration, molesjunit volume [moles/L3]
INDUSTRIAL
AND
ENGINEERING
CHEMISTRY
= =
=
-
H i%H; He i
= = = = = =
1
=
I
=
molar concentration of component A [moles/L3] heat capacity per unit mass a t constant pressure [L2/0*T] diameter [ L ] binary diffusion coefficient [L*/O] electrical field strength [ML/Qf12] total energy in system [ M L 2 / e 2 ] total energy dissipated per unit mass flowing = &,/r
[~*je*]
total rate of dissipation of mechanical energy in system ("friction losses") [ML2/O31 rate of generafion of property P per unit surface area of element [ (P) /L28] total viscous plus form drag force (Table V I I I ) [ML/O2] total external body forces acting on system [ML/eZ] force due to mass transfer processes [ M L / 0 2 ] gravity acceleration (gravity force per unit mass) [L,w] rate of generation per unit volume of property P i n bulk of system [(P)/L3e] vertical height above datum plane [ L ] heat transfer coefficient [ M / T 0 3 ] enthalpy per unit mass [L2/O*] partial molal enthalpy of component i [ L 2 / 0 2 ] magnetic field strength [Q/LO] general component in multicomponent system electric charge flux with respect to mass average velocity
[Q/L~~I
electric charge flux with respect to stationary coordinates [ Q / L 2 8 ]( = i pev) = mass flux with respect to mass average velocity [M/L*B] i* = turbulent mass flux [ M / L 2 0 ] = molar flux with respect to mass average velocity J [moles!L*O] k = thermal conductivity [ M L / I O 3 ] = mass transfer coefficient [L/O] kL = total kinetic energy in system [ML2/OZ] Ki 1 = mixing length [ L ] L = characteristic length [ L ] m = mole fraction [ - ] M = mass flow rate [M/O] M A , M ~= molecular weights of components A , i [M/mole] = total moment;m in system [ M L I B ] = number of components in mixture = outwardly directed normal unit vector at system surface = molar flux relative to stationary coordinates [mole / L 201 = static pressure [:zii/L02] = intensive property = modified pressure = p gph [M/L021 molecular heat flux [Mje3] heat of mixing effect, defined in Table VI11 turbulent heat flux [ M / 0 3 ] heat liberated in chemical reaction per mole reacting [ M L2 / B *mole] heat liberated in chemical reaction per unit mass of A reacting [ LZ / e * I total rate of heat conduction across S,, and S f [ M L ~ I ~ ~ I total rate of heat generation in system [ M L 2 / S 3 ] total rate of heat transfer occurring as a result of mass transfer [1ML2/83] radial coordinate [ L ] rate of generation of component A per unit volume in bulk of system [ M / B L 3 ] total rate of generation of mass of component A in system [ M j B ] pipe radius [ L ] molar rate of generation of component A per unit volume in bulk of svstem imoles/flL31 = gas constant [ML2/e2Tmole] = general scalar quantity = surface enclosing volume V [L2] = surface area vector [ L * ] = fixed (impermeable) surface [ L 2 ] = inlet control surface [ L * ] = mass transfer surface [ L 2 ] = outlet control surface [ L 2 ] = movable surface through which work can be removed from system [ L z ] = time [e]
i,
+
+
T T, U
= temperature [ T ] = stagnation temperature [TI, defined in Equation = internal energy per unit mass [Lz/Bzl
V
=
V
IIh
=
voltage [MLZ/QO2]
= mass fraction of component A [ -1 = total rate of transfer of mechanical energy to sutround-
ings by action of viscous and pressure forces [ML2/8s] = Cartesian coordinates [L] = thermal diffusivity [L2/8] r = mass flow rate [ M p l r = mass flow rate vector [M/B] Til = mass flux of component A [MI01 r(nr)= total mass flus due to processes occurring a t mass transfer surface [MI01 = mass flux of component A due to processes occurring a t mass transfer surface of system [M/O] = Kronecker delta (Sg = 1 if E = J ; ,6, = 0 if i Z J ) .6, A = finite increment of quantity (general) [-I; value of parameter at outlet - value at inlet (Route 2) [ -1 e = dielectric constant [Q20z/ML3] e = angular coordinate [-I P = dynamic viscosity [M/LO] Pe = magnetic permeability [ML/Q21 V = kinematic viscosity [L2/O] ?r = pressure tensor (T = p6,, T ) [MILPI II = molecular flux of property P = mass density of pure fluid or mixture [M/L3] P = electrical charge density [Q/L3] PC = mass concentration of components A , i [M/L8] PA, p, = total mass in system [MI PI = total mass of component A in system [MI p , ~ U = surface tension [M/e21 ce = electrical conductivity [Q20/L3M] 7 = general tensor quantity; molecular momentum flux (shear stress) [M/LB2] ;* = turbulent momentum flux (Reynolds stress) [ M / L 0 2 ] = generalized nonelectromagnetic body force per unit mass [ L / 0 2 ] = potential energy per unit mass [L2/Ozl cp = total potential energy in system [ M L 2 / 8 2 ] ‘p1 = viscous dissipation function (defined on page 52 for the case of Cartesian coordinates) [e-z] = fraction of total gas stream removed in cold stream of vortex tube [ -1 x,y, z a
+
+
+
Superscripts = contribution due to mass transfer (m) = instantaneous value of parameter averaged over cross section - = local time-averaged value of parameter I = local instantaneous deviating part of variable * = turbulent flus Subscripts
A e
. .
t,J
t
lam mas n 0
t turb x,
y, e
= component A = electrical or magnetic quantity = general components in multicomponent system = = = = = =
= =
(Table V) inlet value (Route 2) laminar maximum normal component outlet value total in system turbulent x , y, z directions
O t h e r Symbols
Dt
=
= operator del. [L-’] = Laplacian operator [L-2]
general vector quantity
= velocity vector [ L / e ] VYl = outward normal velocity at system surface [L/O] a=, uy, vz = velocity components in x, y, z, directions [ L / e ] V = volume of system (Route 2), [Lsl V = characteristic velocity (Route 4), [LIB] V
V’
V2
BIBLIOGRAPHY Specific References in T e x t (1) Acrivos, A., and Amundson, N. R., IND.ENO.CHEM.,47,1533 (1955). (2) Alekseev, T. S., Inth.-Fiz. Zh., 7 (4), 121 (1964). (3) Bird, R. B., Stewart, W. E., and Lightfoot, E. N., “Transport Phenomena,” Wiley, New York, 1960. (4) Bopp, G . R., Chcm. Eng. Progr., 63 (lo), 74 (1967). (5) Carslaw, H . S.,. and Jaeger, J. C., “Conduction of Heat in Solids,” 2nd ed, Oxford Univ. Press, 1959. (6) Deissler, R. G., N A C A Repl., 1210 (1955). (7>‘Foust, A. S., Wenzel, L. A. Clump, C. W., Maus, L., and Anderson, L. B., Principles of Unit Operations),” Wiley, New York, 1960. (8) Fredrickson, G., and Bird R B. “Trans ort Phenomena in Multicomonent Systems Sect. 6 in “Hahdbook’of Fluid gynamies,” (V. L. Sweeter, Ed.), LcGraw-Hill, h e w York, 1961. (9) Garner, F. H., Jenson, V. G . , and Keey, R. B., Trans. Ins!. Chem. Eng. (London), 37, 191 (1959). (10) Hilsch, R., .Z. Nnlutforsch., 1, 208 (1946). (1 :b$Ft, W., ‘‘Diffusion in Solids, Liquids, and Gases,” Academic Press, New York,
e;
(12) Klinkenberg, A., and Mooy, H . H., Chcm. Eng. Progr., 44, 17 (1948). (13) Kowalewicz, A., Arch. Budowy Mnszyn, 13 (4), 447 (1966). (14) Linderstrom-Lang, C. U.,, Nulurforsch., A, 22 (5), 835 (1967). (15) Luikov, A. V., and Mikhailov, Yu. A,, “Theory of Energy and Mass Transfer,” Pergamon Press, Oxford, 1965. (16) Martynov, A. V., and Brodyanskii, V. M., Inzh.-Fir. Zh., 12 (15), 639 (1967). (17) Ranque, G. J a , Bull. Bimcnruel SOL.Fruncnisc Phys., S.115, 112 (June 2, 1933). (18) Scheller, W. A., and Brown, G. M., I N n . ENO.CHEM.,49,1013 (1957). (19) Sibulkin, M., 3. Fluid Mech., 12,269 (1962). (20) Spahiu, I., and Cserveny, I., Rev. Elecfrotcch. Eneye!., Ser. B., 7 (2), 201 (1962).
z.
Additional Reading GENERAL A N D ROUTE1 (1A) Bennett, C. O., and Myers, J. E.,“Momentum, Heat and Mass Transfer,” McGraw-Hill, New York, 1962. (2A) Bird, R . B. “Theory of Diffusion ” p 155-239 in “Advances in Chemical Engineering,” Go]. 1 (Ed. T. B. Drew’et uf), Academic Press, New York, 1956. (3A) Bosworth, R . C. L., “Transport Processes in Applied Chemistry,” Wiley, New York, 1956. (4A) Eckert, E. R . G., and Drake, R. M., Jr., “Heat and Mass Transfer,” McGrawHill, New York, 1959. (5A) Gold, R: R., “Magnetohydrodynamic Channel Flow,” pp 353-417 in “Progress in Aeronautical Sciences,” Vol. 8, (Ed. D. Kuchemann e! ai,), Pergamon Press, Oxford, 1967. (6A) Kay, J. M., “An Introduction to Fluid Mechanics and Heat Transfer,” 2nd ed., Cambridge Univ. Press, 1963. (7A) Kays, W. M., “Convective Heat and Mass Transfer,” McGraw-Hill, New York, 1966. (8A) Knudsen, J. G., and Katz, D. L., “Fluid Dynamics and Heat Transfer,” McGraw-Hill, New York, 1958. (9A) Lamb, H., “Hydrodynamics,” 6th ed., Dover Publications, New York, 1945. (10A) Luikov, A. V., and Mikhailov, Yu. A,, “Theory of Energy and Mass Transfer,” Pergamon Press, Oxford, 1945. (11A) Pai, S. I., “Magnetogasdynamics and Plasma Dynamics,” Springer-Verlag, Vienna, 1962. (lZA),,Rohsenow W. M . and Choi, H. Y., “Heat, Mass and Momentum Transfer, Pretrtice-hall, Enilewood Cliffs, 1961. (1 3A) Schlichting, H., “Boundary Layer Theory,” 4th ed., McGraw-Hill, New York, 1960. (14A) Spalding, D . B., “Convective Mass Transfer,” McGraw-Hill, New York, 1963. (15A) Treybal, R. E., “Mass Transfer Operations,” 2nd ed., McGraw-Hill, New York, 1968. See also References 3 and 7. ROUTE2 (1B) Bird, R. B., Chcm. Eng. Progr., Symp. Ser., 61 (58), 1 (1965) (2B) Bird, R. B., Chem. Eng. Sci., 6, 123 (1957). (3B) Gaggioli, R. A,, Chem. Eng. Sci., 13, 167 (1961). (4B) Slattery, J. C., and Gaggioli, R. A,, ibid., 17, 893 (1962). See also References 3, 8 , and 3A. ROUTE3 (1C) Brodke R. S., “Phenomena of Fluid Motions,” Addison-Wesley, ’Reading, Mass., 1967: (2C) Goldstein, S., Ed., “Modern Developments in Fluid Dynamics,” Oxford Univ. Press, 1938. (3C) Hinze, J. O., “Turbulence,” McGraw-Hill, New York, 1959.. (4C) Menkes and Tchen C. M “Fundamentals of Turbulence,” Inst. for Defense AiarI;fes, Res. Papdr P.28< (AD.659997) (October 1967). (5C) Schubauer, G . B., and Tchen, C. M., “Turbulent Flow,” Princeton University Press, 1961. See also References 3, 6 , 6A, 8A, and 13A. ROUTE4 (1D) Catch ole, J. P., and Fulford, G . D., IND. ENO.CHEM.,58 (3), 46 (1966); 60 (3). . ,, 71 (1668,. . (2D) Langhaar, H. L., “Dimensional Analysis and Theory of Models,” Wiley, New York, 1951. (3D) Stewart W. F Chcrn. Eng. Progr., Symp. Scr., 61 (58), 16 (1965). See also RGerenc;; 72, BA, and 73A. I
a 4- (V * V) [1/8]
substantial derivative operator = at
VOL.
61
NO.
5
MAY
1969
69
