Units, or how we choose to measure magnitudes of quantities such as size, temperature, or mass are not required for laws of nature to function properly. Since equations of momentum, energy, and mass conservation are statements of natural laws, they must remain valid regardless of the units employed. That means we can still solve these equations even if we remove all the units by making them dimensionless. Below we will look at some examples of dimensionless conservations laws.
Dimensionless Conservation Laws. First, we consider a dimensionless momentum balance. For simplicity we specialize to an incompressible, constant viscosity, Newtonian fluid, in the absence of body forces. The ith component of the Navier-Stokes equations can then be expressed as
2 2 2 v v v v 1 p v v v i v i v i v i i i i (1) 1 2 3 2 2 2 t x x x xi x x x 1 2 3 1 2 3
Equation (1) can be made dimensionless by dividing each variable by a reference, constant value with the same units. For instance, we divide all lengths by some constant reference length, all pressures by a constant reference pressure, all velocities by a constant reference velocity, etc. We will denote the reference quantities by a subscript "o". The reference quantities must provide a scale for the problem being considered. For example, say you are considering pipe flow. Then a valid reference velocity could be the average velocity of the fluid in the pipe, or the maximum value of the fluid velocity, or the velocity at a point halfway between the center and the wall of the pipe, etc. Those would all work, as they provide information on how fast the fluid is flowing. However, the reference velocity could not be the velocity of your car on the way to the supermarket, since that has no information on the pipe flow being considered. Historically, certain conventions have been adopted. For instance, for pipe flow the reference velocity Vo is usually the average velocity of the fluid in the pipe (Vo = volumetric flowrate / pipe cross- sectional area), while for flow around a sphere Vo is the unperturbed velocity far from the sphere. Using the reference quantities we can form dimensionless variables to substitute into (1)
vi* = vi / Vo xi* = xi / Lo p* = p / po t* = t (Vo / Lo) (2)
Here Lo is a reference length that indicates the physical scale of the problem (such as pipe radius for pipe flow), po a reference pressure quantity that sets the pressure scale (such as pressure difference between two ends of a pipe section), and Lo/Vo a reference time (Lo / Vo represents the time it takes to traverse the reference distance Lo if moving with the reference speed Vo). The resultant dimensionless variables are denoted with an asterisk. To convert back to dimensioned (regular) variables, equations (2) can be rearranged to
vi = vi* Vo xi = xi* Lo p = p* po t = t* (Lo / Vo) (2b)
Inserting expressions (2b) into the Navier-Stokes equation (1) and slightly rearranging yields
* * * * 2 * 2 * 2 * v v v v p p* v v v i v* i v* i v* i o i i i (3) * 1 * 2 * 3 * 2 * *2 *2 *2 t x x x V x V L x x x 1 2 3 o i o o 1 2 3
Every term in equation (3) is dimensionless. Furthermore, two dimensionless combinations (also called dimensionless groups or dimensionless numbers) have appeared:
Reynolds Number: Re = VoLo/ (4)
2 Euler Number: Eu = po / (Vo ) (5)
Using equations (3) to (5), the momentum balance can be rewritten
* * * * 2 * 2 * 2 * v v v v p* 1 v v v i v* i v* i v* i Eu i i i (6) * 1 * 2 * 3 * * *2 *2 *2 t x x x x Re x x x 1 2 3 i 1 2 3
We see that by taking differential equations and/or boundary conditions for a problem and making them dimensionless, as done above for the Navier Stokes equation, dimensionless groups will be generated. An analysis can then be made to deduce the physical interpretation of the dimensionless groups. For example, if Re is large it is clear from (6) that the last term, representing transfer of momentum due to viscous forces, can be neglected compared to the other terms. Being familiar with such interpretations can be helpful in using the magnitude of a dimensionless group to decide which physical mechanisms (e.g. in the case of Re, convective vs. viscous transport of momentum) are dominant in a problem. In turn, this information can be used to simplify modeling.
Equation (6) is an example of a dimensionless momentum balance for Newtonian fluids. We can also make other conservation laws dimensionless. As another example, the unsteady state energy balance for incompressible, constant k materials in the presence of conduction (only) can be written
T k = 2T (7) t ρcˆP
We can then define
2 2 2 T* = (T – T1)/(T2 – T1) * = Lo t* = t (Vo / Lo) (8)
Here, the range T2 - T1 sets the scale of the temperature difference in the system (T1 and T2 could come from boundary conditions, for example). Also, we note that the Laplacian operator 2 has 2 2 units of inverse length squared, so we can make it dimensionless by writing Lo . Substituting (8) into (7) leads to
Vo T * k 2 = 2 *T * (9) Lo t * ρcˆP Lo
The factor (T2 – T1) used to make temperature dimensionless was present in each term and so was cancelled out. Rearranging,
T * k 2 1 2 = *T * = h *T * (10) t * ρcˆP LoVo Pe
ρcˆ L V Pe h P o o (11) k
Peh is called the Peclet number for heat transport, and is interpreted as representing ratio of convective to conductive heat transport. Although we did not include convection of heat in the starting equation (7), if we had we would have similarly derived Peh. In addition, we can take the ratio of Peclet and Reynolds numbers to derive the Prandtl number Pr, which is interpreted as representing ratio of viscous transport of momentum to conductive transport of heat,
Pe h ρcˆ L V cˆ Pr P o o P (12) Re k Vo Lo k
Equation (10) is a simple example of a dimensionless energy balance. We can keep going to also look at mass transport of a species in a multicomponent system. Taking the mass balance for species A in the absence of reactions, and assuming that the density and diffusion coefficient are constant, we previously derived
A 2 = DAB A – v ρ (13) t A
Proceeding similarly as for the energy balance, we next define dimensionless quantities
2 2 2 A* = (A – A 1)/(A 2 – A 1) * = Lo v* = v/Vo t* = t (Vo / Lo) (14)
The range A 2 – A 1 sets the scale of concentration differences in the system. Substitution of (14) into (13) leads to
* Vo A DAB 2* * Vo * = 2 A – v * * ρA (15) Lo t * Lo Lo
The factor (A 2 – A 1) used to make A dimensionless was cancelled out since each of the three terms in (15) had this factor. Rearranging,
* A DAB 2* * * 1 2* * * = A – v * * A = m A – v * * A (16) t * Vo Lo Pe
L V Pe m o o (17) DAB
The dimensionless group Pem is known as the Peclet number for mass transport, and is interpreted as representing ratio of convective to diffusive mass transport. The ratio of Peclet and Reynolds numbers is known as the Schmidt number Sc, another dimensionless group, which is interpreted as the ratio of viscous transport of momentum to diffusive transport of mass,
Pe m L V Sc o o (18) Re DAB Vo Lo DAB
The above examples illustrate how some of the more famous dimensionless groups arise by rewriting conservation laws into a dimensionless form. The Buckingham PI Theorem can be used for the same purpose. The advantage of generating dimensionless groups from the conservation laws that describe a problem, instead of the Buckingham Pi Theorem, is that the differential equations, boundary conditions, and other equations that form the mathematical statement of a problem are directly derived from its physical characteristics. Therefore, these equations will provide only those dimensionless groups relevant to the problem, so long the equations are correct to start with. In contrast, the Buckingham Pi Theorem approach requires that the set of parameters governing the problem of interest be guessed. The disadvantage is that it may not be easy to write down the full set of equations needed, especially for complex geometries or situations. In such situations the Buckingham PI Theorem can be much easier for deriving the relevant dimensionless groups.
Important: if two problems obey the same form of dimensionless differential equations and also any auxiliary dimensionless equations such as boundary or initial conditions, they are said to be geometrically similar. If in addition the two problems have identical values of the dimensionless groups (e.g. Re, Pr, Sc) found in the equations, then the problems are also said to be dynamically similar. Therefore, two problems that are geometrically and dynamically similar will possess identical dimensionless problem statements and, thus, their dimensionless solutions for v*, T*, p*, A* or other dependent variables of interest will also be identical. The solutions do not necessarily have to be calculated, but can also be obtained experimentally. In such an approach one uses a “model” system to measure the dependent experimental variables of interest; for example, the energy dissipated in a flow as a function of flow velocity. These results are then presented as the dimensionless solution; for example, dimensionless dissipation of energy as a function of a dimensionless flowrate. Friction factors for pipe flow, in which the dimensionless friction factor f is plotted as a function of the dimensionless Reynolds number Re, are an example of such a correlation. Once these dimensionless correlations are established, they can then be used to predict the behavior of other geometrically similar systems (e.g. pipe flow) when operated under condition of dynamic similarity (e.g. same Re numbers as for which the correlation was determined).
EXAMPLE: Dissipation in Pipe Flow. We will illustrate calculation of the energy dissipation associated with steady state, incompressible, parabolic (Hagen-Poiseuille) flow in pipes, using standard and dimensionless approaches. From this illustration we will motivate why dimensionless representation of a problem is useful. Consider the pipe flow depicted in Figure 1. The pipe diameter is D = 2R and the distance between the entry port 1 and the exit port 2 is L. We want to calculate how much mechanical energy is dissipated to internal energy per mass of fluid flowing from port 1 to port 2, we call this dissipation Wf. In other words, Wf is the work done to overcome retarding frictional forces per unit mass of fluid as it flows from port 1 to 2.
Figure 1
Wf is given by integrating the previously introduced dissipation function , representing rate of dissipation of mechanical to internal energy per volume, over the volume of interest between ports 1 and 2, and then dividing this integral by the rate of mass flow through the pipe:
L 2 R r drddz 0 0 0 W f 2 R (19) v r drd Z 0 0 where the dissipation function for this flow is given by (see earlier handout on differential energy balances)
2 = (dvZ/dr) (20)
The top integral in (19) is the total rate of dissipation of mechanical to internal energy in the pipe volume between the two ports, while the denominator is the mass flow rate through the pipe. The ratio gives the desired dissipation per mass Wf. The parabolic flow is given by
2 2 vZ = 1/(4) dp/dz (r - R ) (21) where the pressure gradient dp/dz responsible for the flow is taken as constant. Inserting (20) and (21) into (19) leads to
L 2 R 2 R r dp 3 r drddz r dr 0 0 0 2 dz L dp 0 W f 2 R R 1 dp 2 2 dz 2 2 (r R ) r drd (r R ) r dr 0 0 4 dz 0 R 4 L dp 4 L dp W (22) f dz R 4 R 4 dz 4 2
Equation (22) is somewhat inconvenient to use since dp/dz is not always known. On the other hand, viscosity and flowrate are more readily accessible than dp/dz. Recalling that the average velocity for parabolic pipe flow is given by
2 R v rdrd z volumetric flowrate 0 0 2 Vo = = -R /(8) (dp/dz) (23) area of flow R2
(22) can be rewritten by solving (23) for dp/dz and substituting the result into (22)
8LV W o (24) f R 2
We can further simplify (24) by writing Wf per unit length of pipe. We will call this Ŵf, and it will express energy dissipated per mass of fluid as it travels along a unit length of pipe (rather than all the way from port 1 to 2),
32Vo Ŵf = Wf /L = (25) D 2 where we also replaced R with D/2 to connect back to Figure 1. Formula (25) was able to be calculated because we had an expression for the velocity profile in the pipe. However, in more complex situations, say for turbulent flow, we would not be able to do that as easily and may need to resort to experimental measurements of Ŵf. Examination of the parameters in equation (25) suggests that such experiments should be performed as a function of the viscosity, density, and flowrate of the fluid, as well as the diameter of pipe; in other words,
Ŵf = g(, , Vo, D) (26) where the unknown function g would be determined experimentally.
Now let’s re-express Ŵf in a dimensionless format. Since Ŵf has units of energy per mass per 2 length, one way to make it dimensionless is to divide it by Vo /2D since this quantity has the 2 same units as Ŵf. Vo /2 provides a scale of the kinetic energy of the flow per mass, while D provides a size scale of the flow. The new dimensionless quantity will be denoted by f. For parabolic flow, f follows from (25)
32V f = Ŵ /(V 2/2D) = o (27) f o V 2 D 2 o 2D
64 64 f = = (28) DVo Re where the Reynolds number Re = DVo/. Note that, because f is dimensionless, it can only depend on dimensionless combinations of parameters, in this case Re.
Formula (28) applies to parabolic flows through pipes. For more complex flows, such as turbulent flows, we may need experiments to determine an expression for f. Importantly, (28) suggests that only the single parameter Re would need to be varied,
f = h(Re) (29) where h is the function to be determined experimentally. This is a great simplification over (26); indeed, by expressing the problem in a dimensionless form it was clarified that only a single parameter, Re, needs to be varied to establish the correlation with f. In contrast, equation (26) suggested that four parameters would need to be varied independently. The simplification realized in (29) thus greatly reduces the number of experiments that would be needed to characterize how energy dissipation depends on characteristics of the flow.
In practice, measurements of f would be performed on a model pipe flow as a function of Re. These data would then be used to generate a plot of f vs Re. Once the plot is established, it can then be used to determine f for geometrically similar systems (meaning: other pipe flows) under conditions of dynamic similarity (meaning: look up f from the plot for the Re value calculated for the geometrically similar flows of interest). As stated earlier, when two problems are geometrically and dynamically similar, they will have the same dimensionless solution; in other words the same value of f will apply. The actual dissipation in the geometrically similar flow can be calculated from the looked up f value by re-arranging (27)
2 Ŵf = (Vo /2D) f (30) where Vo and D refer to values for the geometrically similar flow of interest.
In summary, benefits of dimensionless representation include:
(1). Dimensionless representation allows looking up solutions for geometrically and dynamically similar systems based on dimensionless solutions obtained (often experimentally) on a model system.
(2). Dimensionless representation reduces the total number of parameters describing a problem. For the above example we went from four (equation 26) to just one (equation 29) parameter. This is of tremendous help especially if the solution needs to be obtained experimentally, as many fewer parameters need to be varied.
(3). Dimensionless representation helps identify how correlations should be presented, e.g. as plots of f vs Re.
In the following handouts we will encounter other examples in which a dimensionless dependent variable, such as f in the above example, is expressed in terms of dimensionless independent variables, such as Re. Although we will not repeat a detailed discussion of each such case, the motivation for these dimensionless correlations (such as equation (28) for f) remains the same, namely to facilitate calculations for geometrically and dynamically similar problems.