Volume derived units expression

Units may be combined together into derived units to express a property more complicated than mass, length, or time. For example, volume, V, the amount of space occupied by a substance, is the product of three lengths therefore, the derived unit of volume is (meter)3, denoted m3. Similarly, density, the mass of a sample divided by its volume, is expressed in terms of the base unit for mass divided by the derived unit for volume—namely, kilogram/(meter)3, denoted kg/m3 or, equivalently, kg-m-3. The SI convention is that a power, such as the 3 in cm3, refers to the unit and its multiple. That is, cm3 should be interpreted as (cm)3 or 10-6 m3 not as c(m3), or 10 2 m3. Many of the more common derived units have names and abbreviations of their own. 

Describe the various mass transfer and reaction steps involved in a three-phase gas-liquid-solid reactor. Derive an expression for the overall rate of a catalytic hydrogenation process where the reaction is pseudo first-order with respect to the hydrogen with a rate constant k (based on unit volume of catalyst particles). 

In physiological terms we can also define clearance as the volume of blood cleared of the toxicant by an organ or body per unit time. Therefore, as the equations above indicate, the body clearance of a toxicant is expressed in units of volume per unit time (e.g., L/h), and can be derived if we know the volume of distribution of the toxicant... 

The fundamental units of dynamics are those of mass, length, and time. These units are denoted by the symbols [M], [L], and [T], respectively. The magnitude of these units may be fixed arbitrarily but all other units are derived from them, and depend upon them alone. The various derivative units are developed simply. For example, unit-density is unit-mass contained in a unit-volume. In terms of the fundamental units it is therefore [M/L8] or [ML-8]. Again, unit-velocity is unit-distance divided by unit-time, or [L/T]. By means of these units the terms in a given equation may be checked, inasmuch as all terms added to a given expression must be of the same kind and therefore of the same dimensions. If, for illustration, it is desired to determine the units of surface tension, a, from the capillary equation... 

A plane wall of thickness 2L has an internal heat generation which varies according to q = q0 cos ax, where q0 is the heat generated per unit volume at the center of the wall (x = 0) and a is a constant. If both sides of the wall are maintained at a constant temperature of Tw, derive an expression for the total heat loss from the wall per unit surface area. 

By considering the radiation as such a gas, the principles of quantum-statistical thermodynamics can be applied to derive an expression for the energy density of radiation per unit volume and per unit wavelength ast... 

The flow method that has been briefly discussed sometimes offers special advantages in kinetic studies. The basic equations for flow systems with no mixing may be derived as follows let us consider a tubular reactor space of constant cross-sectional area A as shown in Fig. 7.4 with a steady flow of u of a reaction mixture expressed as volume per unit time. Now we will select a small cylindrical volume unit dV such that the concentration of component i entering the unit is C(- and the concentration leaving the unit is C,- + dC-,. Within the volume unit, the component is changing in concentration due to chemical reaction with a rate equal to r(. This rate is of the form of the familiar chemical rate equation and is a function of the rate constants of all reactions involving the component i... 

For gases and liquids the liter is a much more practical measure of volume, especially in concentration expressions, than the derived unit of cubic meter, m3. Usage therefore has established the liter (L) as an accepted named unit, even though cubic decimeter (dm3) is the correct SI designation. The only prefix to be used with liter is milli, that is milliliter (ml). The symbol L is used for liter to prevent confusion with the letter I and the number 1. 

From the discussion it also follows that there Is a certain freedom of choice regarding the units Into which measured quantities are expressed. Recall, however, the restriction that if cross-relationships are to be derived on the basis of Onsager s reciprocal relationships (secs. 1.6.2b and c) the fluxes J. resulting from forces X, must have such dimensions that the product XJ has the dimensions of entropy production per unit time and unit volume (SI units J m K" s ). 

You will find that the use of SI units simplifies mathematical manipulations and ensures that you obtain the correct units for the parameter being calculated. Remember that you must convert all units into the appropriate SI units, e.g. masses must be expressed as kg, volumes as m and concentrations as kgm or molm etc., and that you may need to use alternatives in derived units (Table 9.2). The application of these principles is shown in Box 9.2. 

For simplicity, in this equation, we have assumed that activities are equal to concentrations and brackets refer to activities. C is a units conversion constant = Vy m relating void volume Vy (mL) in the porous media and the mass m (g) of the aquifer material in contact with the volume Vy, is the formation constant for an aqueous uranyl complex, and the superscripts i, j, k describe the stoichiometry of the complex. The form that the sorption binding constant takes is different for the different sorption models shown in Figure 4 (e.g., see Equation (5)). Leckie (1994) derives similar expressions for more complex systems in which anionic and cationic metal species form poly dentate surface complexes. Equation (7) can be derived from the following relationships for this system ... 

Derive an expression for the heat capacity of electromagnetic radiation and determine its value for unit volume at 1 K, 300 K, 30,000 K. Compare these values with those of a monatomic ideal gas ate room temperature. At what temperature do these quantities become equal ... 

The mass and volume of a sample are physical properties that can be determined without changing the substance. But each of these properties changes depending on how much of the substance you have. The density of an object is another physical property the mass of that object divided by its volume. As a result, densities are expressed in derived units such as g/cm or g/mL. Density is calculated as follows ... 

Some properties, such as time or length, can be expressed in terms of SI base units. Other properties, such as volume or density, are expressed in SI derived units, which are really made by combining the SI base units. Following are some examples of physical properties and the Si-derived units, which can be used to measure them. 

The combined gas law equation is derived by combming Boyle s and Charles s Laws, so the comments in earlier Problem-Solving Tips also apply to this equation. Remember to express all temperatures in kelvins. Volumes can be expressed in any units as long as both are in the same units. Similarly, any pressure units can be used, so long as both are in the same units. Example 12-4 uses torr for both pressures Example 12-5 uses kPa for both pressures. 

A certain chemical species is adsorbed by the particles of an aerosol. The mass adsorbed is proportional to the surface area of the particle. Derive an expression for the distribution of the species with respect to particle size expre.ssed as mass of the species per unit volume of gas in the size range u to u + ttv. Express your answer in terms of it and n(tt). Define any constants you introduce. 

We consider a polydisperse aerosol growing by gas-to-particle conversion. The system is spatially uniform in composition—growing aerosol in a box. As growth occurs, the size distribution function changes with time we wish to derive an expression for dn/di. Let /(u, /) be the particle current or number of particles per unit time per unit volume of gas passing the point u. The rate at which particles enter the small element of length in a space (Fig. 10.4) is given by... 

Problem 8.5 Consider the flow tube for rapid polymerization reactions shown schematically in Fig. 8.1. Let V be the volume of the tube (distance between the mixing jets) and V be the volume of the total liquid flowing through in time t. Denoting the total concentration (constant) of polymer chain ends by Eq. (8.29) and the concentration of monomer units in polymer by [Mjp derive an expression for monomer conversion as a function of t. 

The ceU density (N) is defined as the number of cells or channels per unit of cross-sectional area perpendicular to the axis of the channel. This is usually expressed in units of cells per square inch, and abbreviated cpsi. The open frontal area (OFA) is equal to the open area of an individual channel multiphed by the cell density and is usually expressed as a percent. The geometric surface area (GSA) of a cellular structure is derived by establishing the surface area per unit length of an individual channel that is then multiphed by the cell density. This represents a surface area per unit volume and is expressed as cm /cm, m /hter, or some other appropriate set of units. The total surface area (TSA) of a structure is then the geometric surface area multiphed by the volume (V) of the structure under consideration. 

We can also derive an expression on the basis of the Mooney-Rivlin strain energy function for swollen elastomers. A dry elastomer sample will undergo two types of deformation one due to swelling and the other due to extension. The strain energy function per unit volume of swollen elastomer is related to that of the dry sample by... 

The best inemals and the optimum values of pressure, vapor velocity, and reboil vapor ratio are those that permit production of heavy water at minimum cost. The initial cost of the plant depends on a number of factors including the total number of towers, the total amount of reboiler and condenser surface, and the total volume of tower internals. The principal operating cost is for power, which is proportional to total loss in availability of steam as it flows through the towers. A complete minimum-cost analysis requires knowledge of the unit cost of all the important cost components and is beyond the scope of this book. Design for minimum volume of tower internals or minimum loss in availability due to tower pressure drop and for minimum cost of these two important contributors to total cost can be carried out without complete unit-cost data and will be discussed. Because the same choice of reboil vapor ratio minimizes the number of towers, their volume, and the loss of availability within them, this reboil vapor ratio is close to that which leads to minimum production cost. An equation for this optimum reboil vapor ratio will now be derived, and expressions will be developed for the total volume of towers and the total loss in availability in towers designed for the optimum ratio. 

SI units consist of seven base units and numerous derived units. Exponential notation and prefixes based on powers of 10 are used to express very small and very large numbers. The SI base unit of length is the meter (m). Length units on the atomic scale are the nanometer (nm) and picometer (pm). Volume units are derived from length units the most important volume units are the cubic meter (m ) and the liter... 

The Kinetic Limit. To initiate homogeneous nucleation, a vapor embryo is required that has the critical radius r. In principle, the formation of just one such embryo would be sufficient to initiate the nucleation process, but in practice it is found that conditions must be such that J, the number of vapor embryos formed in a unit volume per unit time, has a high value (typically J > 1012). Carey [4] derives the following expression for J ... 

The diffusion equation. The general problem of unsteady-state diffusion within a solid involves the prediction of the concentration distribution C(x,y,z) within a solid as a function of the space coordinates and time, t. To derive an equation that can be solved for C(x,y,z,t), conservation of mass and Pick s first law (i.e., the rate of transfer of mass per unit area is proportional to the concentration gradient, see Pick 1855) are applied to a differential control volume. The resulting expression is the diffusion equation... 

To derive an expression for nucleation rate per particle we use a procedure analogous to that for homogeneous nucleation discussed in the previous section. The main difference is that, instead of calculating a nucleation rate per unit volume, we calculate a rate per unit area of foreign substrate and then introduce the surface area of the particle, assumed to have homogeneous properties, to calculate the nucleation rate per particle. For a spherical particle of radius R cm the result is... 

It flows through a ( lindrical tube of radius R. Derive an expression for the volume flow-rate Q which results from a pressure drop per unit length P/AL. 

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