Dimensional Analysis

Dimensional Analysis. 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.

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