What is the rate at which energy is transferred

1.2.3 Fundamental of radiation

In general, the thermal heat transfer which is carried out by electromagnetic waves is called radiation. Since these waves are transmitted at the speed of light, so the energy transfer rate in this case equals the speed of light. After nanofluids appeared in the scientific community, the strong absorption behavior of nanofluids, especially in the solar spectrum, was discussed. It is a different concept of using the small absorption capacity of small particles in the fluid flow. Fig. 1.7 shows the structure of an electromagnetic wave.

At first time, concept of energy transfer through electromagnetic waves was introduced by Scottish scientist James Clerk Maxwell. He showed that energy transmission occurs at light speed. Usually electromagnetic waves are classified based on their frequency and wavelength. The relationship between wavelength and frequency is as follows (Eq. 1.4):

(1.4)λ=cν

where λ and ν are wavelength and frequency, respectively. Moreover, c shows the speed of light in the vacuum. The above equation proves that wavelength and frequency have a tradeoff. In fact, if one of them increases, the other one decreases.

The speed of light varies in different environments. Consequently to obtain the speed of light in an environment other than the vacuum, Eq. (1.5) is used.

(1.5)c=c0n

where n is the environmental refractive index in which we want to find the speed of light. For air, this coefficient is approximately equal to 1 and for water equal to 1.5. It should be noted that the frequency of an electromagnetic wave depends only on its source of emission and is not related to the atmosphere in which the wave is released.

At the beginning of the 20th century, Einstein presented a new theory of the propagation of radiated waves. Based on this theory, energy transfer is the transfer of energy packs, called photons. For each of these packages, you can define a frequency equal to ν. With regard to the frequency assigned to them, the energy of each of these packets is equal to the following value (Eq. 1.6):

(1.6)e=h×ν=hcλ

where h has a constant value, equal to 6.626176 × 10−34 and is called Planck’s constant. Note that in this assumption, the values of c and h are constant numbers. Hence, it can be said that the energy of the packs, or the photons, depends only on their wavelength. From the above, it can be seen that the lower wavelength means more photon energy. For example, X-rays or gamma rays have very low wavelengths while they have high energy and can be very destructive.

Fig. 1.8 shows a spectrum of different wavelengths. As it can be seen, a wavelength can be a micrometer variable. It is interesting to know that the cosmic waves have the shortest wavelengths, whereas electric waves have the longest wavelengths.

The main reason of heat transfer through radiation is by rotational and vibrational movements of molecules, atoms, and electrons. In fact, the temperature is the result of these movements. Hence, increasing the temperature increases the radiation heat transfer rate. The concept we know as the “light” is, in fact, the visible part of the electromagnetic spectrum. Fig. 1.8 shows that the optical wavelength is a subset of the thermal wavelength.

Radiation heat transfer is a volumetric phenomenon. Of course, for matte objects such as metals, the radiation is superficial. Note that the radiant properties of a surface can be introduced by covering it with new layers.

One of the ways to improve the properties of fluids in the radiative base systems is to improve the absorption properties of fluids using nanoparticles. Researchers have done extensive research on the use of nanoparticles in fluids to improve the absorption coefficient of surfaces in equipments such as solar collectors and so on.

Application to closed system

Under the simplified condition discussed earlier, the first law of thermodynamics for a closed system is given by

(3.26)ΔU=Q−W

where Q and W are the energy transfer rates by heat and work, respectively. The energy transfer by mass flow is not considered since there is no mass flow crossing the boundary in the closed system. One of the most important systems in the field of mechanical engineering is the heat engine that receives energy in the form of heat and converts it to energy in the form of work. Thus, in the aforementioned equation, the sign of heat is supposed positive when the energy enters the system while the sign of work is supposed positive when the energy leaves the system.

5.3.4 Energy flow analysis

Table 5.3 summarized the energy flow analysis on three types of Cymbal transducers: endcap thickness of 0.3 and 0.4 mm with and without bias force under various cyclic vibration levels and drive durations [28, 29]. Note first that the mechanical-to-mechanical energy transfer rate is good for 0.3-mm thick cymbals (83%–87%), while it is rather low for 0.4-mm thick cymbal (46%). This is related to the mechanical impedance matching; 0.4 mm endcap seems to be too rigid (effective stiffness is too high) to match the vibration source shaker.

Table 5.3. Energy flow analysis on three types of Cymbal transducers: endcap thickness of 0.3 mm and 0.4 mm with and without bias force under various cyclic vibration levels and drive durations [29]

Second, the received mechanical energy to electrical energy transduction rate can be evaluated from the effective electromechanical coupling factor. Because the value of keff of the cymbal is around 25%–30%, the energy transduction rate keff2 can be evaluated around 6.25%–9%, which agrees well with the experimental results illustrated in Table 5.3. Thus, we obtained the conversion rate from the vibration source energy to the stored electric energy in the cymbal transducer as 7.5%–7.8% for the 0.3-mm thick endcap cymbals, and 2.9% for the 0.4-mm thick endcap cymbal. The reduction of the conversion rate is primarily originated from the mechanical/acoustic impedance mismatch.

1.31.2.1.2 Excitation energy transfer

The second length scale is associated with excitation energy transfer. The excitation energy is transferred primarily via dipole–dipole Coulomb interactions in a Förster mechanism according to

[2]kEET=kf(r0r)6

with kEET being the excitation energy transfer rate, kf the radiative constant, and r0 the half-radius of the Förster process. Since the length scale of exciton migration in the antenna is large, the exciton lifetimes should be long enough to allow for photons striking any part of the antenna to reach the RC. Typical r0 values for photosynthetic pigments are 5–10 nm.

Once excitons are trapped in the RC, the process of charge separation should take place faster compared to back transfer to the antenna units. This is achieved by a relatively large antenna–RC distance (∼2 nm) versus a short distance for the redox pigments in the RC. Furthermore, surrounding of the RCs by assemblies of antenna pigments gives a spatial arrangement in which energy transfer is optimized by multiple entries to the RC. The combination of constraints (spatial distance between antenna–RC and multiple entries to the RC) is achieved by circular-like arrangements of antenna pigments around the RCs, with a ‘cordon sanitaire’ zone (see Noy et al. [33]) of ∼2 nm around the RC where no antenna pigments occur.

III.C.1 Total Energy (First Law of Thermodynamics)

When the various energy quantities used in arriving at Eq. (23) are introduced into Eq. (46), we obtain

(55)∂E∂t+〈ρe′v⋅n〉1A1+〈ρe ′v⋅n〉2A2=Q.−W.+QCR',

in which E = u + v2/2 + Φ is the total energy content of the fluid,e′=e+p/ρ,Q. is the total thermal energy transfer rate,

(56)Q˙=−∬S q⋅nds,

Q′CR is the total volumetric energy production rate due to chemical reactions or other such sources, andW. is the total rate of work done or power expended against the viscous stresses,

(57)W˙=∬S(v⋅T) ⋅nds.

In common engineering practice Eq. (55) is applied to steady flow in straight pipes and is divided by the mass flow ratem.=ρ〈υ〉 to put it on a per unit mass basis,

(58)Δu+Δ〈υ〉2/2+gΔz+Δp/ρ=qˆ−wˆ+qˆ′ ,

where the operator Δ implies average quantities at the downstream point minus the same average quantities at the upstream point. The terms on the right-hand side of Eq. (58) are just those on the right-hand side of Eq. (55) divided by ρ〈v〉A. In Eq. (58) z is vertical elevation above an arbitrary datum plane.

1.6.2.1 The first law of thermodynamics

This is a law of energy conservation. When applied to a process we write

where Q is the heat transfer (kJ) or q is the specific heat transfer (kJ/kg), W is the work transfer (kJ) or w is the specific work transfer (kJ/kg), and ΔE is the energy change (kJ) or Δe is the specific energy change (kJ/kg).

The change symbol Δ means final value minus initial value. ΔE embraces all forms of energy but in the non-flow process it is usual to find that the only significant change is in the internal energy (U, u) and we write the non-flow energy equation

For the steady flow system we write

Q−W˙ x=m˙Δ(h+V22+gz) or q−wx=Δ(h+V22+gz)

where Q˙ and W˙x are the energy transfer rates and m is the steady mass flow rate across the boundary (in and out), Δh is the change in specific enthalpy (h = u + pv), ΔV2/2 is the change in specific kinetic energy and Δgz is the change in specific potential energy. The suffix x is used on the work transfer to denote that this is the useful work from the system as the flow work is included in the enthalpy term. In flow problems it will also be necessary to use the continuity equation

where p is the density and A is the area normal to the velocity V. Analysis of non-steady flow may also be made, in which case energy terms to allow for the storage of energy in the system will be added.

Warning: A sign convention for work and heat is built into the equations above. Positive work means work obtained from the system and positive heat means heat put into the system. Care should be taken to be clear about the symbol V, which may appear as velocity or volume in many equations.

In order to allow continuous energy transfers a cycle is defined in which a series of processes brings the working substance back to the initial state so that the cycle can be repeated continuously. If we apply the first law to a cycle it follows that ΔE is zero and

1.3.4 Why the standard mixture model does not apply to plasmas

In plasmas, electrons have a much smaller mass than ions. Because of this very small mass, electron–ion collisions are very efficient to the relax the electron velocity toward the ion one, but very inefficient to relax the temperatures of the two species toward each other. The reason is that collisions between two particles of very different masses result in large momentum changes for the light particle (but very small momentum changes for the heavy one), and almost no energy transfers between the particles. Therefore, two typical collision time scales occur, respectively related with momentum and energy transfer rates.

For this reason, in plasmas or disparate gas binary mixtures, the collision time scales for the two species of particles are not of the same order of magnitude. More specifically, the light species collision time scale is smaller by a factor me /mi than that of the heavy species. These considerations will be detailed below (see Section 1.5).

Therefore, the heavy species collision time scale is already a macroscopic time scale for the light species. At this scale, it can be shown that the light species can be described by a set of macroscopic equations coupled with a kinetic equation for the heavy species. This model has been developed in Ref. [2]. In the present chapter, however, we shall rather consider a different scaling, where both species can be described by macroscopic equations.

The time-scale that we shall be interested in is a macroscopic time scale relative to the heavy species, which corresponds to an even longer time-scale, the diffusion time-scale, for the light species. Therefore, this approach will give rise to a system of hydrodynamic equations for the ions (or the heavy species) coupled with diffusion equations for the electrons (or the light species).

As a by-product, this separation of scales between the light and heavy species opens the possibility of constructing macroscopic models where the velocities and temperatures of the two species are different. Therefore, the dynamics of a plasma or a disparate mass binary mixture leads to a more complex physics than that of ordinary mixtures.

To achieve our goal, we have to go back to the kinetic level and insert the relevant scaling of the masses into the kinetic equations. The best way to perform this scaling in a systematic way is to use dimensionless variables. We shall perform this task in Section 1.5. In Section 1.4 below, we summarize the plasma fluid model obtained from this scaling.

III.C Rate Models

Traditionally, absorbers and strippers were described as stagewise contactors. Krishnamurthy and Taylor developed a new rate (nonequilibrium stage) approach for modeling absorbers and strippers. This approach describes an absorber as a sequence of nonequilibrium stages. Each stage represents a real tray in the case of a tray tower or a section of a continuous contacting device such as a packed column. For each nonequilibrium stage, the mass, component, and energy balance equations for each phase are solved simultaneously, together with the mass and energy transfer rate equations, reaction rate equations, and the interface equilibrium equations. Computation of stage efficiencies is thus avoided altogether and is, in effect, substituted by the rate equations.

Although the rate model can be applied to any separation, it has become most popular in absorption and stripping. Reported case studies demonstrated that, in at least some situations, a rate model can more closely approximate absorber performance than can an equilibrium stage model. The success of rate models in absorption is largely a result of the difficulty of reliably predicting stage efficiencies in absorbers. The presence of many components, low stage efficiencies, significant heat effects, and chemical reactions are commonly encountered in absorbers and difficult to accommodate in stage efficiency prediction.

Figure 12 is a schematic diagram of a nonequilibrium stage n in an absorber. The equations applying to this stage are described below. A more detailed description is given by Krishnamurthy and Taylor.

Component balances for component j on stage n are given in Eqs. (31a–c) for the vapor phase, the liquid phase, and the interface, respectively:

(31a)νj,n−νj,n+1+Nj,nV=0,

(31b)lj,n−lj,n−1−Nj,nL=0,

(31c)Nj,nV=Nj,nL

Energy balances on stage n are given in Eqs. (64a–c) for the vapor phase, the liquid phase, and the interface, respectively:

(32a)VnHnV−Vn+1Hn+1V+QnV+EnV =0

(32b)LnHnL−Ln−1Hn+1L+QnL−EnL=0

(32c)EnV=EnL

The interface equilibrium is written at the interface.

(33)yj,nI=mj,nIxj,n I

In Eq. (31c), NVj,n and NLj,n are the mass transfer rates. These are calculated from multicomponent mass transfer equations. The equations used take into account the mass transfer coefficients and interfacial areas generated in the specific contactor, reaction rates, heat effects, and any interactions among the above processes.

The above equations, including those describing the mass transfer rates on each stage, are solved simultaneously for all stages. Solution of these nonlinear equations is complex and usually requires a computer. Newton's numerical convergence technique, or a variant of it, is considered to be most effective in solving these equations.

1 Introduction

Distillation is among the most important operations for the separation and purification of multicomponent mixtures in the chemical industry. This process is based on the differences on relative volatilities and boiling points between the components. Distillation can be done either in a continuous operational regime or a discontinuous regime (batch distillation). Batch processes are commonly used in pharmaceutical, food, and specialized products as alcoholic beverages, essential oils, perfume, pharmaceutic and petroleum products (Lucet et al., 1992). The advantages of batch distillation over a continuous distillation relies on its capacity to produce many different products.

Two models are commonly used to represent the phenomena occurring during distillation: the equilibrium model and nonequilibrium (rate-based) model. The first one assumes that phase equilibrium occurs in all the stages of the column, making use of efficiencies to predict the deviation from the actual performance of the column; the second one avoids the use of efficiencies, instead, it involves calculations for the mass transfer rates between the phases. Rate-based model consist of material and energy balances for each phase, models for mass and energy transfer rates across the interface, and equilibrium relations for the interface compositions. An important requirement of the rate-based model is an adequate mass transfer model, which is related with the kind of mixture and the trays/packing geometry. Trays are used to contact the liquid and vapor phase and through these the mass transfer takes place. Trays are widely used in distillation columns because they are easy to design and have a relatively low cost. Correlations of mass transfer coefficients for a wide variety of contacting devices have been published in the literature. The mass transfer coefficients depend on properties such as viscosity and diffusivity, as well as on operating and design parameters such as flowrates and column diameter, tray or packing type and so on. Khrishnamurthy and Taylor (1985) published a mathematical model for nonequilibrium stages in multicomponent separation processes. The model includes the mass balance of each component in each phase, the heat balances, the mass transfer rate, energy transfer equations and the equilibrium equations at the interface. Gorak et al. (1991) found that the compositions predicted by the nonequilibrium model are closer to the experimental data than those predicted by an equilibrium model. Kooijman and Taylor (1995) presented a nonequilibrium model for the dynamic simulation of distillation columns. Nada et al. (2009) presented two models (equilibrium and nonequilibrium) to study the dynamic behavior of multicomponent azeotropic system in a batch column with bubble-cap trays. Taylor y Krishna (1993) used a model in which they consider the bubbling and jet regime for a sieve tray in a distillation column.

The batch column involves a time-dependent process, therefore requires differential equations for the molar and energy balances and algebraic equations for all other relationships; i.e. a differential-algebraic (DAE) system. There are methods for solve the DAE system: solver than employ backwards differentiation formulas (BDF) and one-step implicit methods (such as Runge-Kutta methods). In this work, a pilot-scale batch distillation column is modelled through a rate-based approach. A mass transfer model is proposed for the sieve trays on the column. The mass transfer model is incorporated to the rate-based model, and the whole set of equations is solved using a specialized software (Matlab R2013a).

2.7.2 Fluorescence resonance energy transfer

Fluorescence (or Förster) resonance energy transfer (FRET) [26,27,28] is a non-radiative transfer of the excitation energy from a donor to an acceptor chromophore that involves a distance-dependent interaction between the emission and the absorption transition dipole moments of the donor and acceptor, respectively. The rate of energy transfer depends on the spectral overlap of the donor emission and acceptor absorbance, the donor fluorescence quantum yield, the relative orientation of their transition dipole moments, and the distance between donor and acceptor molecules. The energy transfer rate, kET, is given by

(19)kET=1τd( R0r)6

where τd is the decay time of the fluorophore in the absence of an acceptor, and r is the distance between the donor and the acceptor. R0 is the Förster radius (typically 2 to 9 nm) characterizing the donor/acceptor pair. It is defined as the distance at which the efficiency of resonance energy transfer is 50% and can be estimated as

(20)R0(in nm)=979(κ2n4Φ0J)1/6

where n is the refractive index of the medium, Φ0 is the fluorescence quantum yield of the donor, J is the spectral overlap integral, and κ2 is the orientation factor. The overlap integral, J, expresses the extent of overlap between the donor emission and the acceptor absorption.

(21)J=∫0∞FD(λ)εA(λ)λ4dλ

where FD(λ) is the normalized fluorescence spectrum of the donor and εA(λ) is the molar extinction coefficient of the acceptor as a function of wavelength, λ. The rate of energy transfer varies linearly with the overlap integral [29]. A large value of overlap integral ensures high sensitivity for imaging and sensing as well as high selectivity.

Resonance energy transfer provides an additional deactivation pathway for the excited fluorophore and results in reduced excited state lifetime of the donor fluorophore. The Förster distances are comparable to the size of biological macromolecules, and thus, FRET is extensively used as a “spectroscopic ruler” for measuring distances between sites on interacting proteins [30–36].

The magnitude of kET can be determined from the efficiency of energy transfer, ET, using the relation

(22)kET=1τd(ET1−ET )

and ET can be determined experimentally by measuring the decrease in the intensity F or the lifetime τ of the donor in the presence of the acceptor,

(23)ET=1−FFd=1−ττd

Hence, the change in observed fluorescence intensity or the excited state lifetime of the donor fluorophore, due to resonance energy transfer, can be used to determine the distance between the donor acceptor pair. However, the distances so estimated are also influenced by the orientation factor, κ2, which depends on the relative orientation of donor emission transition moment and acceptor absorption transition moment. The value of κ2, in general, varies from 0 (perpendicular orientation) to 4 (for parallel orientation of transition moments). For fast and freely rotating fluorophores, its value may be isotropically averaged over all possible orientations, and is often taken to be 2/3. However, this assumption may not be valid for immobilized fluorophores. For a given FRET pair, under a given set of physical conditions, the value of κ2 can be estimated by polarization measurements [37].