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Update April 15, 2024 – Visit my new site FFOS Software http://www.modelingatffos.com to view my portfolio and pricing for the Molecular Modeling Pro Flavor Plus and the ChemicaElectrica Gateway programs. http://www.modelingatffos.com

In the next few months I will be posting chapters here of my AICHE article on a predictive model for calculating odor thresholds: (see below)

2RM Technology started in 2006 as a consulting company for the flavor and fragrance industry. Currently, a program Molecular Modeling Pro Flavor Plus is available to researchers and developers in the food, fragrance, and environmental arena. 

Issue 5 – Dimensional analysis of Thresholds

From- A Flavor-Receptor Ellipsoid Model for the Prediction of Flavor Sensory Thresholds

Author:                  Richard S. Turk – Senior Research Scientist

                              2RM Technology

Issue 5

Dimensional Analysis

Given the number of fundamental dimensional variables and the number of variables describing a system, it is possible to predict important groupings associated with these variables.  The normal dimensional variables used are mass, length, and time.  Assuming that olfaction occurs at constant temperature, temperature will not be a variable.  What are the important physical and chemical variables that describe the system?  At this point in developing a theory it is important to choose these variables based on several criteria.  First, the variables must have some fundamental reason for being considered.  Second, values should be assignable to these variables.  It would be meaningless to use a variable that cannot be measured or calculated.

In the analysis of the volatility of flavor compounds several variables are found to be important.  First, the vapor pressure, P_v, of a flavor is an important measure of its volatility.  In most foods and beverages, the concentration of the flavor is very low, and the vapor-liquid diagram is non-ideal.  The relative vapor pressure is given by the product of an activity coefficient,g, the mole fraction in solution, x, and the pure component vapor pressure.  A second important factor for volatility is the diffusivity or the flavor in the vapor phase.  This diffusivity D_ab is a binary diffusion of the flavor in a solvent vapor, usually water being the solvent.

Other important variables to consider are the molecular weight M_w of the flavor and its density r_l, r_v in the liquid and vapor state.  The final variable to consider is some energy parameter in the liquid state.  This energy parameter is not a mechanical energy but can be related to either the enthalpy of vaporization or the enthalpy of dissolution.  This later enthalpy may be important at least in the original binding of  a flavor with a hydrophobic receptor or membrane.  As flavors range from low to high water soluble and high to low organic solubile,  the affinity or interaction of a particular flavor with a receptor can be related to its solubility in water and organic solvents.  The energy difference e₁₂ = e_12w – e_12o is proportional to the partition coefficient of a flavor in water and a water immiscible solvent.  The energy e₁₂ is called the partition energy.

The five variables D_ab, P_v, M_w, r_v , and e₁₂ and the three-dimensional variables indicate that there can be 5-3 = 2 dimensional variables which describe Newtons law.  The easiest separation is to break the acceleration vector into two components: acceleration has dimensions length/time², or (1/time)*(length/time).  The two-dimensional quantities represent a frequency or first order rate constant (1/time) and a velocity (length/time).

The velocity group can be formed from the vapor pressure and density.  Since pressure has units of M/Lt² (M, L,t is mass, length, time respectively) and density has units of M/L³, the ratio pressure/density has unit L²/t², or velocity squared.  The square root gives the desired velocity.  Since the actual pressure in the vapor is g*x*P_v, where g is an activity coefficient (no dimension) and x is a mole fraction in solution (no dimension) the vapor pressure can be multiplied by these factors with no dimension change.  The density is the gas phase density.

The frequency or first order rate constant can be formed from the variables M_w, ,D_ab , and e₁₂ .  Since e₁₂  has dimensions of calories/mole ( M*L²/mole *t²) and D_ab has dimensions of  L²/t, the ratio is M/mole which is exactly a molecular weight unit.  At this point it can also be realized that the frequency can be given by other combinations.  The energy can be made dimensionless by dividing by the gas constant k and temperature T.  The remaining variable D_ab can be made to a frequency by dividing by a cross sectional area (L²).  This area could be represented by a molecular area calculated from the liquid molar volume. For spherical packing in a liquid the molecular area can be given as p*(s/2)² where s/2 is:

(11)                 s/2 = (3*0.7405*(M_w/r_l)/(4*p*N_o))^1/3

where N_o is Avogadro’s number.

The exact form of the dimensional frequency will be discussed below.  The two groupings important to the olfactory sensory process are then:

(12a,b)            velocity –                     (g*x*P_v / r_v )^1/2

                        frequency –                  (e₁₂/ M_w *D_ab )  or  (D_ab / p*(s/2)²)

The threshold concentration (M/L³) can be made dimensionless by an equivalent unit such as density or molar volume.  If a concentration unit is used for m in equation (1) then the force is a volumetric force (Newtons/volume).  The threshold concentration is usually given as ppm (parts per million) or mg/ (1000 cm³) = (microgram/cm³) which can be scaled as a density unit.  If the concentration is converted to moles/cm³ then the scaling factor is the molar volume.  Alternatively, this concentration factor can be scaled by its own threshold concentration c_T with the group m/c_T being a dimensionless variable. For consistency of equation(3) [F_t = m_t * g_f] the mass m will be identified as the threshold concentration in ppm units.

Dimensional Analysis

• Define dimensional variables describing the system from the set of five system variables{partition energy e₁₂, vapor pressure P_v, diffusivity D_ab, vapor density r_v, molecular weight M_w }

velocity group-                       (g*x*P_v / r_v )^1/2

frequency group –                   (e₁₂/ M_w *D_ab )

Conclusion –

g_f  = G =product of velocity and frequency groups

Next – Threshold G factors and Rotation

Previous Issues:

Issue 1 Introduction

Issue 2 Threshold Energy

issue-3-threshold-force

Issue 4 Treshold Response and Force Model

Please contact me if any of these research areas are of interest to you or your organization.

Richard Turk

Senior Research Scientist
2RM Technology

rsturkg@gmail.com

or visit Flavor, Fragrance and Odor Simulator at http://www.modelingatffos.com U.S.

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