Dimensionless volumetric flow rate

In order to perform order-of-magnitude scaling for this problem, we need a dimensionless distance, dimensionless time, and dimensionless volumetric flow rate, which we define as... 

The dimensionless volumetric flow rates through the rectangular channel can be defined by qi = + =... 

Combining Eqs (9b)- 9d) gives the final expression for the dimensionless volumetric flow rate as... 

Fig. 4 Dimensionless volumetric flow rate in axial generalized annular Couette flow as a function of A and n for k = 0.5.

Tadmor and Klein [10] classified the solution into six separate cases at this juncture, depending on the sign of A and the value of This is not necessary because a single-case approach [as used in Eqs. (8) through (10)] leads to the following exact expression for the dimensionless volumetric flow rate Q(A, n, k) on analytically integrating by parts twice ... 

The dimensionless volumetric flow rate is then easily evaluated using Eq. 

The flooding point diagram, developed by Mersmann, uses the falling or rising velocity ws of an individual droplet as a parameter. Acc. to Fig. 7-14, the ordinate axis shows the dimensionless volumetric flow rate of the continuous phase... 

Impellers are sometimes viewed as pumping devices the total volumetric flow rate Q discharged by an impeller is made dimensionless in a pumping number ... 

The conclusion that dimensionless numerical values are universal is valid only if a consistent system of units is used for all quantities in a given equation. If such is not the case, then the numerical quantities may include conversion factors relating the different units. For example, the velocity (F) of a fluid flowing in a pipe can be related to the volumetric flow rate (Q) and the internal pipe diameter (D) by any of the following equations ... 

We will use the Bernoulli equation in the form of Eq. (6-67) for analyzing pipe flows, and we will use the total volumetric flow rate (Q) as the flow variable instead of the velocity, because this is the usual measure of capacity in a pipeline. For Newtonian fluids, the problem thus reduces to a relation between the three dimensionless variables ... 

Oi This dimensionless parameter involves ko, the factor preceding the exponential term of the Arrhenius reaction formula (2.1) or (2.2), called the preexponential factor or the frequency factor q, the volumetric flow rate and V, the volume of the reactor, which are related via the formula a = ko V/q. Here ko is usually quite large. 

Hints Start your investigation of part (a) with the adiabatic case of Kc = 0.0. You can use dimensionless units in your dynamics and stability part of the investigation by using the dimensionless time t = t/r, where t is the real time, r = V/q is the residence time, V is the active volume of the reactor, and q is the constant volumetric flow rate.]... 

This differential equation can be put into dimensionless form by introducing one simple assumption such as a constant volumetric flow rate q, i. e.,... 

Next we use the dimensionless parameter fo = Qo to/(A H (1 — 6 )) and assume a constant bed height H so that the volumetric flow rate Qo of product removal equals the rate of increase in bed volume due to polymerization. This gives us... 

Consequently, it was prudent to acknowledge that the above regression equation is not dimensionless because for all practical purposes, the Reynolds number Re was replaced by T Re, what the authors called a pseudo Reynolds number with the dimensions [L- T]. This predicament did not deter a plethora of other studies in the same line of reasoning to be published in recent years. Note that this pseudo Reynolds number has a physical meaning it is a reciprocal of volumetric flow rate. 

Because the extraction factor is a dimensionless variable, its value should be independent of the units used in Eq. (15-11), as long as they are consistently applied. Engineering calculations often are carried out by using mole fraction, mass fraction, or mass ratio units (Bancroft coordinates). The flow rates S and F then need to be expressed in terms of total molar flow rates, total mass flow rates, or solute-free mass flow rates, respectively. In the design of extraction equipment, volume-based units often are used. Then the appropriate concentration units are mass or mole per unit volume, and flow rates are expressed in terms of the volumetric flow rate of each phase. 

The [4 — 4]/4 max is the corrected height at the peak maximum and is dimensionless. The measured peak width at corrected half-height is converted to halftime by dividing by chart speed. The volumetric flow rate or total flow rate through the detector is measured with a bubble flow meter and is typically 50 to 150 mL/min. 

To reduce the design equation to a dimensionless form, we select a reference stream that combines the inlet steam and the injection stream. Hence, its total volumetric flow rate is... 

Lapidus and Amundson (L2) have solved this problem for a constant volumetric flow rate. For r = 1, the result is again given by Eq. (103).6 The dimensionless parameters have their usual values, with v calculated as... 

In this problem, the pressure-gradient is fixed at a dimensionless value of—4, and any changes in the flow may produce changes only in the volumetric flow rate. Hence, c)pH/c)z = 0 for all n > 1. A closely related problem would have the volumetric flow rate fixed, and, in this case, changes in the flow would produce changes in the required pressure-gradient so that c)pH/c)z 0 for / > 1. 

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