# Fluence Rate in UV Photoreactor for Disinfection of Water: Isotropically Radiating Cylinder

Authors: Roman Ilinsky; Andrey Ulyanov
Journal: International Journal of Chemical Engineering
 (2014)

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## Abstract

The calculation of fluence rate in the photochemical reactor using ultraviolet (UV) radiation for disinfection of water for the case, when a cylinder of infinite length is used as a light source, has been considered. Such a cylinder is filled with an isotropically radiating medium. The dependence of the fluent rate on the diameter of the radiating cylinder has been analytically analyzed. The limiting case
when the diameter of the radiating cylinder tends to zero has been considered and the notion of “effective interval” has been introduced. Based on this notion, the comparison of fluence rates for the cylinders of finite and infinite lengths has been performed. In the calculations of fluence rate, it is advisable to use the Chebyshev method for the operations of numerical integration.


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## Body

## 1. Introduction

In a normal design process, the last stage is the certification of the reactor for a certain set of operational parameters. In most situations, biodosimetric tests are used for this purpose [1, 2]. If such certification fails, a certain part of the design process has to be repeated. The certification procedure and the necessary construction of a prototype unit are costly and time-consuming. The available increasingly powerful numerical simulation techniques enable the designer to predict reactor performance under certain operating conditions without incurring the high cost of prototype construction and certification.Numerical models simulating the disinfection process in a UV reactor involve fluence rate models to determine the distribution of intensity of UV radiation, the hydrodynamic models to predict motion paths of flow particles, and kinetic models of microbial inactivation [2–5]. The model of ultraviolet source (usually a lamp) is the significant element in the simulation of intensity distribution of UV radiation in a photoreactor [4, 5]. Over the last few decades, several models for calculations of the distribution of lamp radiation intensity have been suggested [6–13]. In the simplest models, a lamp was represented by point sources which were arranged in a straight line. With the endless increase of a number of sources, one gets the model of a radiation source in a form of a line. In more complicated computer models, the radiation source is regarded as a three-dimensional body (usually such a body is a cylinder).The goal of this work was to study analytically the model of radiation source in a form of infinite cylinder with regard to the distribution of fluence rate in a photoreactor. The radiation source in a form of a line is considered as the limiting case of the radiation source in a form of cylinder of infinitely small radius. In this study, we consider the case when this cylinder is filled with an isotropically radiating medium. A more simple case when the surface of the cylinder represents the diffuse radiator will be analyzed in the posterior paper.


## 2. Nomenclature

E
i:
Fluence rate[W·m-2]
l:
z optical direction cosine of the ray (dimensionless)
n
W:
Refractive index of water (dimensionless)
n
Air:
Refractive index of air (dimensionless)
p:
x optical direction cosine of the ray (dimensionless)
q:
y optical direction cosine of the ray (dimensionless)
r
Air:
Radius of radiating cylinder[m]
r
W:
Radius of the water/air border[m]
s:
Length[m]
V:
Volume[m3]
x:
Rectangular coordinate[m]
y:
Rectangular coordinate[m]
z:
Rectangular coordinate[m].Greek Letters
α:
Absorption coefficient of water[m-1]
β
^:
Coefficient, which characterizes the volumetric density of the radiated flux[W·m-3]
β:
Radiated flux of radiating cylinder unit length[W·m-1]
Θ:
Angle between the portion of the ray path in water and axisz[rad]
ρ
A:
Radius of the sphere with the center in pointA[m]
σ
i:
Mean value of the area of cross section of the light tube[m2]
τ:
Transmittance coefficient (dimensionless)
Φ:
Radiation flux[W]
φ:
Angle between the projection of the portion of ray path in water on planexy and axis y[rad]
Ω:
Solid angle[sr⁡].


## 3. Basic Definitions

A typical element of photoreactors, which are employed for water disinfection, consists of a quartz tube wherein a cylindrical lamp is placed. From the outside, this quartz tube is surrounded with water. The axes of the lamp and the quartz tube coincide (Figure1). For the sake of simplifying the calculations of fluence rate, we will exclude from considerationFigure 1
Experimental reactor of the “Svarog” Company. The figure shows motion paths of particles.(i)
the quartz casing between air and water,
(ii)
the casing of the lamp,
(iii)
the reflection of radiation from the border between any two media with different refraction indexes,
(iv)
the radiation from neighbouring lamps (for a multilamp reactor),
(v)
the effect of reactor walls and of similar elements of the photoreactor.The lamp is regarded as an infinitely long cylinder. This cylinder is evenly filled with a uniform substance radiating the energy. The refractive index of a given substance is equal to that of air. The region of space, which is filled with this substance, evenly radiates the flux in all directions:
(1)Φ=4πβ^V,
where β^ is the coefficient, which characterizes the volumetric density of the radiated flux, and V is the volume of this region of space.The fluence rate is calculated for pointA which is located in water (the medium under irradiation). Let us introduce the Descartes coordinate system xyz (Figure 2). Axis z coincides with the axis of radiating cylinder. Point A lies on axis y. The direction of axis y is selected so that y coordinate of point A should be negative (yA<0).Figure 2
Explanatory draft of the calculations of fluence rate in pointA.In the selected coordinate system, the water/air border is represented by the corresponding cylindrical surface:(2)y2+z2=rW2,
where rW is the radius of the water/air border. The border of the radiating cylinder is described by equation
(3)y2+z2=rAir2,
where rAir is the radius of radiating cylinder. As follows from formula (1), the radiation flux from a portion of the radiating cylinder located between planes z=z1 and z=z2 is equal to
(4)Φcyl=βz2-z1,
where
(5)β=4π2rAir2β^.


## 4. Calculations of Fluence Rate

Every rayRA starting from point A may be set by two angles, namely, by angle Θ between the portion of the ray path in water and axis z and by angle φ between the projection of the portion of ray path in water on plane xy and axis y. The calculation of the ray path RA(Θ,φ) is given in Appendix A. The conditions, under which the ray path RA(Θ,φ) intersects the radiating cylinder, are stated in Appendix B. If at given angles Θ′,φ′ the ray path RA(Θ′,φ′) intersects the radiating cylinder, then in the system of Θ-φ coordinates point (Θ′,φ′) belongs to region ΓA. Figure 3 shows the exemplary view of region ΓA in the Θ-φ coordinate system.Figure 3
RegionΓA in the Θ-φ coordinate system.Note thatΓA is the closed and continuous region. Let us select a small uniform subregion ΔΓA inside ΓA. The point with coordinates (Θ0,φ0) is the center of subregion ΔΓA. Angles Θ0 and φ0 define ray RA(Θ0,φ0).The path of rayRA(Θ0,φ0) starts from point A and intersects the water/air boundary (the surface (2)) in point B. Further, the path of ray RA(Θ0,φ0) intersects the radiating cylinder (Figure 2). Let us divide the portion of ray path RA(Θ0,φ0) in the cylinder into ν-1 pieces of small lengths C1C2,C2C3,…,Cν-1Cν where C1 and Cν are the points of intersection of the reference ray with the radiating cylinder, C1Cν=C1C2+C2C3+⋯+Cν-1Cν.Now, let us regard the light tubeA. This light tube represents the region of space in the system of xyz coordinates. It involves all the points, through which the paths of rays RAΘ0+ΔΘ,φ0+Δφ pass, when the condition Θ0+ΔΘ,φ0+Δφ∈ΔΓA is fulfilled. As far as all rays RAΘ0+ΔΘ,φ0+Δφ start from one point A, the initial portion of light tube A is a cone (Figure 4). The corresponding conical surface is the border of solid angle ΩA with a vertex in point A. The value of this solid angle is
(6)ΩA=∬ΔΓA‍sin⁡ΘdΘdφ.
Taking into account that region ΔΓA has small sizes, the value of solid angle ΩA may be assumed to be infinitesimal.Figure 4
The portions of light tubesA and Ci in the radiating cylinder.Let us draw planes through pointsC1,C2,C3,…,Cν-1,Cν (Figure 5). These planes are perpendicular to straight line C1Cν. Let σi be the mean value of the area of cross section of the light tube A between planes Ci and Ci+1. As ΩA→0, then σi→0. At CiCi+1→0 and σi→0, the volume of the portion of light tube A between planes Ci and Ci+1 is equal to
(7)ΔVi=σiCiCi+1.
As follows from relations (7), (1), and (5), the element of volume ΔVi radiates evenly in all directions the flux
(8)ΔΦi=4πβ^ΔVi=βΔViπrAir2.
Let us circumscribe sphere (A,ρA) with the center in point A and infinitely small radius ρA. The radiation flux propagates from the portion of light tube A between planes Ci and Ci+1 toward sphere (A,ρA) through a light tube Ci. The light tube Ci represents the region of space. It includes all the points through which the rays from the portion of light tube A between planes Ci and Ci+1 pass to sphere (A,ρA).Figure 5
The portions of light tubesA and Ci near sphere (A,ρA).AtCiCi+1→0 and σi→0, the radiating region, which is limited by planes Ci and Ci+1 and light tube A, may be considered as the point source. It is located in point Ci. The radiation flux from this point source is equal to ΔΦi and is evenly distributed in all directions. In this case, the portion of light tube Ci in air is the cone with vertex in point Ci. The given conical surface is the border (the limiting element) of solid angle Ωi′ with the vertex in point Ci. As the radiation flux ΔΦi is evenly distributed over all directions (i.e., within solid angle 4π), the value of radiation flux within solid angle Ωi′ is equal to
(9)ΔΦi′=Ωi′4πΔΦi.
As follows from relations (8) and (9),
(10)ΔΦi′=β4π2rAir2Ωi′σiCiCi+1.In order to determine solid angleΩi′, we shall use the theorem Kirchhoff-Clausius-Straubel [14–17]. Based on this theorem, the relationship for light tubes A and C is as follows:
(11)πρA2nW2ΩA=σiΩi′.
As a result of the substitution of relationship (11) into expression (8), we obtain
(12)ΔΦi′=βCiCi+1ρA2nW2ΩA4πrAir2.
Taking into consideration the absorption of radiation in water, the flux of radiation falling onto sphere (A,ρA) is expressed as
(13)Φi′′=τABΦi′,
where τAB is the transmittance coefficient for the portion of light tube Ci which is located in water. According to the Bouguer law [18],
(14)τAB=exp⁡-αAB,
where α is the absorption coefficient of water. (As a matter of convention, the transmittance of water, which fills a reactor, is characterized by transmittance τ• of a water layer of known thickness s• (usually, s•=10mm); then, α=-ln⁡τ•(s•)/s•.)The element of volumeΔVi creates the fluence rate Ei in point A. Taking into account that, in the calculation of fluence rate Ei, radius ρA should be taken as infinitesimal, the following expression should be valid:
(15)Ei=lim⁡ρA→0Φi′′πρA2,
where πρA2 is the area of the principle cross section of sphere (A,ρA). As follows from relationship (15),
(16)Ei=βexp⁡-αBACiCi+1nW2ΩA4π2rAir2.The fluence rate in pointA from the region of space, whose points simultaneously belong to the radiating cylinder and light tube A, comprises the value
(17)EΔΓA=∑i=1ν-1‍Ei=βexp⁡⁡-αBAnW2ΩA4π2rAir2∑i=1ν-1‍CiCi+1=βexp⁡-αBAnW2ΩAC1Cν4π2rAir2.
As follows from relationships (6) and (17),
(18)EΔΓA=βexp⁡-αBAnW2C1Cν4π2rAir2∬ΔΓA‍sin⁡ΘdΘdφ.
Based on relationship (A.4), we arrive at
(19)BA=sW⊳Θ=Θ0,φ=φ0=-yAcos⁡⁡φ0-rW2-yA2sin2φ0sin⁡Θ0.
From relation (A.12), we get
(20)C1Cν=ΔsAir⊳Θ=Θ0,φ=φ0=2rAir21-nW2cos⁡2Θ0-yA2nW2sin2φ0sin2Θ01-nW2cos⁡2Θ0.
The substitution of expressions (19) and (20) into (18) gives
(21)EΔΓA=nW2β2π2rAir2exp⁡-α-yAcos⁡⁡φ0-rW2-yA2sin2φ0sin⁡Θ0×rAir2(1-nW2cos⁡2Θ0)-yA2nW2sin2φ0sin2Θ01-nW2cos⁡2Θ0×∬ΔΓA‍sin⁡ΘdΘdφ.As far as regionΔΓA has small sizes and the coordinates of its center are (Θ0,φ0), formula (21) may be transformed to the following view:
(22)EΔΓA=nW2β2π2rAir2∬ΔΓA‍exp⁡-α-yAcos⁡⁡φ-rW2-yA2sin2φsin⁡Θ×rAir2(1-nW2cos⁡2Θ)-yA2nW2sin2φsin2Θ1-nW2cos⁡2Θ×sin⁡ΘdΘdφ.
The relationship for calculations of the complete fluence rate E in point A may be derived from (22) through the substitution of the integration field ΔΓA for ΓA. Taking into account inequalities (B.5), which define the borders of region ΓA, we obtain
(23)E=βnW22π2rAir2×∫(π/2)-Θlim⁡(π/2)+Θlim⁡‍∫-φlim⁡+φlim⁡‍exp⁡-α-yAcos⁡⁡φ-rW2-yA2sin2φsin⁡Θ×rAir2(1-nW2cos⁡2Θ)-yA2nW2sin2φsin2Θ1-nW2cos⁡2Θ×sin⁡ΘdφdΘ.
Let
(24)φ=arcsint1+t2,Θ=π-arccosunW.
Then, (23) takes the following view:
(25)E=nWβ2π2rAir×∫-1+1‍∫-tlim⁡+tlim⁡‍1-t2tlim⁡211+t23/21-u2×exp⁡-α-yA-rW2+rW2t2-yA2t2nW1+t2nW2-u2×dtdu,
where
(26)tlim⁡=1-u2rAirnW2-u2yA2-1-u2rAir2.
Let us use the expansion in the Taylor series:
(27)(-yA-rW2+rW2t2-yA2t2)nW1+t2nW2-u2≈-yA-rWnWnW2-u2+nWrWyA+yA22rWnW2-u2t2.
After the substitution of expression (27) into (25), we obtain
(28)E=nWβ2π2rAir×∫-1+1‍∫-∞+∞‍exp⁡-α-yA-rWnWnW2-u2×exp⁡-αnWrWyA+yA22rWnW2-u2t2httlim⁡×11+t23/21-u2dtdu,
where function
(29)h(ξ)=0,atξ>1,1-ξ2,atξ≤1
describes the Wigner semicircle distribution.


## 5. The Radiation Source in a Form of Line

In the calculations of photoreactors, the cylindrical radiation source of finite sizes is very often substituted for the radiation source in a form of infinitely thin line. With that, the radiation fluxes from the portions of equal length of the cylindrical source and of the source in a form of infinitely thin line are regarded as equal.The fluence rate in pointA from the source in a form of line represents the limiting value of expression (28) at rAir→0:
(30)Eline=lim⁡rAir→0βnW2π2rAir×∫-1+1‍∫-∞+∞‍exp⁡-α-yA-rWnWnW2-u2×exp⁡-αnWrWyA+yA22rWnW2-u2t2httlim⁡×11+t23/21-u2dtdu.
Since h(ξ) function is related to the Dirac delta function δ(ξ) by the relationship
(31)δ(ξ)=lim⁡rAir→02πrAirhξrAir,
then
(32)Eline=βnW4πyA×∫-1+1‍exp⁡-α-yA-rWnWnW2-u2×1nW2-u2du.


## 6. Comparison of Values of Fluence Rate from the Radiation Source in a Form of Line and from the Cylindrical Radiation Source

The integral in formula (25)(33)I=∫-tlim⁡+tlim⁡‍exp⁡-αnWrWyA+yA22rWnW2-u2t2×httlim⁡11+t23/2dt
may be calculated only by numerical methods. One of such methods was suggested by Chebyshev [19]. By calculating integral (33) through the Chebyshev method, we get
(34)I≈Cexp⁡-αnWrWyA+yA22rWnW2-u2t1211+t123/2+exp⁡-αnWrWyA+yA22rWnW2-u2t2211+t223/2+exp⁡-αnWrWyA+yA22rWnW2-u2t32×11+t323/2-αnWrWyA+yA22rWnW2-u2t12,
where
(35)C=13∫-tlim⁡+tlim⁡‍httlim⁡dt=πtlim⁡6,t1=-t3=12C∫-tlim⁡+tlim⁡‍httlim⁡t2dt=38tlim⁡,t2=0.
Let us expand expression (34) into series in powers of rAir. If the terms beginning from rAir4 are rejected, then
(36)I≈π2yA·1-u2nW2-u2rAir+0.2rWnW2-u2-αnWyA2-αnWyArW+1-u23/2yA3rWnW2-u2rAir3.
The substitution of formula (36) into formula (25) allows us to get the expression which approximates the value of fluence rate in point A:
(37)E≈Eline+Er,
where Eline is the value of fluence rate in point A from the source in a form of infinitely thin line (see formula (32))
(38)Er=0.2βnWrAir22π2yA3×∫-1+1‍1-αnWrWyA+yA2rWnW2-u2×exp⁡-α-yA-rWnWnW2-u21-u2nW2-u23du.
The ratio Er/Eline characterizes the relative accuracy of the fluence rate value under the substitution of the source in a form of infinitely thin line for the radiating cylinder. As follows from these formulas, the ratio takes the maximal value at rAir→rW and yA→-rW. In this case,
(39)ErEline=0.4π·∫-1+1‍1-u2nW2-u2-3/2du∫-1+1‍nW2-u2-1/2du.
Based on formula (39), at nW=1.373, which is the typical value for water, we have Er/Eline≈0.05.In the calculations of integrals in expressions (32) and (38), it is convenient to use the Chebyshev numerical method [19]. All formulas required for these operations are provided in Appendix C. It is easy to verify that, at α(yA+rW)=0, the Chebyshev method gives the exact values of integrals in expressions (32) and (38).


## 7. Effective Interval

A lamp in a photoreactor cannot be of infinite length. Nevertheless, the formulas for the calculations of fluence rate at infinitely long cylindrical source of radiation are of practical importance. Let us regard the rays which start from pointA and form one and the same angle Θ~ with axis z. The portions of paths of these rays in air lie on surface S. The surface S′ is symmetric with respect to surface S relative to plane xy (Figure 6).Figure 6
Explanatory draft for the definition of effective interval.Atπ/2-Θlim⁡<Θ~<π/2, the maximal coordinate z of the intersection of surface S with the radiating cylinder is equal to coordinate z of the second (along the ray path) point of the intersection of ray RA(Θ~,0) with the cylindrical surface (3). As follows from formulas (A.5), (A.10), and (A.12), this coordinate is equal to
(40)zmax⁡=nWcos⁡⁡Θ~-yA-rWnW2-nW2cos⁡2Θ~+nWcos⁡⁡Θ~rAir+rW1-nW2cos⁡2Θ~.
Let the radiating volume V be limited by the cylindrical surface (3), surface S, and surface S′. To calculate the fluence rate in point A from the radiating volume V, one may use formula (23) under the condition of substitution of integration limits
(41)E*=βnW22π2rAir2×∫+Θ~π-Θ~‍∫-φlim⁡+φlim⁡‍exp⁡-α-yAcos⁡⁡φ-rW2-yA2sin2φsin⁡Θ×rAir2(1-nW2cos⁡2Θ)-yA2nW2sin2φsin2Θ1-nW2cos⁡2Θ×sin⁡ΘdφdΘ.
Similar to the procedure described in Section 6, it is possible to derive the approximate expression based on formula (41) for the calculation of fluence rate in point A from the radiating volume V:
(42)E*≈Eline*+Er*,
where
(43)Eline*=βnW4πyA×∫-nWcos⁡⁡Θ~+nWcos⁡⁡Θ~‍exp⁡-α-yA-rWnWnW2-u2×1nW2-u2du,Er*=0.2βnWrAir22π2yA3×∫-nWcos⁡⁡Θ~+nWcos⁡⁡Θ~‍1-αnWrWyA+yA2rWnW2-u2×exp⁡-α-yA-rWnWnW2-u2×1-u2nW2-u23du.
Let us find such value of Θ~, at which the following condition is fulfilled:
(44)E-E*≤μE,
where μ is a given value. Based on expressions (37) and (42), inequality (44) may be presented in the form:
(45)Eline+Er-Eline*-Er*≤μEline+Er.
At small values of μ, condition (44) is fulfilled in the case when nWcos⁡⁡Θ~≈1. With an accuracy of values of the second order of smallness relative to (1-nWcos⁡⁡Θ~), formula (43) acquires the view:
(46)Eline*=Eline-exp⁡-α-yA-rWnWnW2-1×βnW22yAπnW2-1(1-nWcos⁡⁡Θ~),Er*=Er.
It is seen from these formulas that inequality (45) is transformed into equality Eline+Er-Eline*-Er* = μEline+Er at
(47)cos⁡⁡Θ~=1nW-2μEline+ErπyAnW2-1nW2βexp⁡-α(-yA-rWnW/nW2-1).
Upon the substitution of the obtained value of cos⁡Θ~ into formula (40), we may find the value of distance zmax⁡.The portion of radiating cylinder, which is limited by planesz=zmax⁡ and z=-zmax⁡, creates the fluence rate Ezz*=E*+2ESz*, in point A, where ESz* is the fluence rate from the portion of radiating cylinder, which is limited by plane z=zmax⁡ and surface S. As ESz*>0, then Ezz*>E*. It is evident that Ezz*<E. Hence, E>Ezz*>E*, or
(48)E>Ezz*>(1-μ)E.
As follows from inequality (48), the fluence rate from the portion of radiating cylinder, which is limited by planes z=zmax⁡ and z=-zmax⁡, and the fluence rate from the infinite radiating cylinder differ by most μE. Based on this fact, we will name the interval [-zmax⁡;zmax⁡] as the effective interval.Let us use the concept of the effective interval for the estimation of the fluence rate along the linem-m, which is parallel to the axis of the radiating cylinder of finite length (Figure 7), and calculate the values of Eline, Er, and zmax⁡ by means of formulas (32), (38), and (40). Let point M be located on the axis of radiating cylinder. Point M⊥ is the projection of point M onto the axis of the cylinder. If each of the distances from point M⊥ to the face planes of the cylinder is larger than a half length of the effective interval (i.e., zmax⁡), then the fluence rate in point M cannot be greater than E and cannot be smaller than (1-μ)E. If the length of radiating cylinder is greater than the effective interval, then the plot of fluence rate along the line m-m has the form of trapezium with curvilinear lateral sides. With that, the length of the plot portion where the fluence rate is greater than (1-μ)E and smaller than E is equal to the difference between the length of the radiating cylinder and the length of effective interval.Figure 7
The fluence rate from the cylindrical source of finite length. The plot is given without scale compliance.The data of calculations and experiments presented in studies [3–6] confirm that the above described plot of the fluence rate is characterized by the trapezium form with curvilinear lateral sides.


## 8. Example

The calculations of fluence rate are performed atrW=16mm. The refractive index of water under irradiation depends on a wavelength, temperature, and admixtures dissolved in water [20]. In all calculations, the refractive index of water under irradiation is assumed to be equal to nW=1.373.For every set of parametersyA, rAir, and α, the values of fluence rate were calculated(i)
by means of the ray-trace procedure,(ii)
by means of formulas (37), (32), and (38).In the calculations of fluence rate by the method of ray tracing, the rectangular net was set in the Θ-φ coordinate system. The centers of net meshes were located in points (Θ^i,φ^j), where Θ^i=arccos(1/1024)j+1/2, i=0,±1,±2,±3,…, and φ^j=(2π/1024)j+1/2, j=0,±1,±2,±3,…. The sizes of the net knowingly exceeded region ΓA. Every net mesh had a corresponding ray RA(Θ^i,φ^j). The procedure of calculations of ray path RA(Θ^i,φ^j) is considered in Appendix A. If ray RA(Θ^i,φ^j) intersects the radiating cylinder (see Appendix A), then the mesh with center in point (Θ^i,φ^j) is regarded as subregion ΔΓA of region ΓA, and the fluence rate is calculated by formula (21). The fluence rate in point A is equal to the sum of fluence rates from all meshes belonging to region ΓA.In the calculations of the fluence rates according to formulas (37), (32), and (38), the integrals were determined(i)
with the use of a  Fortran  subroutine for the calculation of define integral according to the Simpson generalized quadrature rule with the specified relative accuracy of 0.01;(ii)
with the use of the Chebyshev formulas (C.2) and (C.3).The calculated values of fluence rate are summarized in Tables 1 and 2. There is a good agreement between all data of these calculations.Table 1
The values of fluence rate. The transmittance of 10 mm thick water layer is 60%.
y
A  [mm]
r
Air [mm]
The ray-trace procedure
The integration by the Simpson method
The integration by the Chebyshev method
E
/
β [m−1]
(
E
line
+
E
r
)
/
β [m−1]
E
line
/
β [m−1]
E
r
/
β [m−1]
E
line
/
β [m−1]
E
r
/
β [m−1]
−16.0
5.0
11.199
11.197
11.143
0.054
11.143
0.054
−16.0
10.0
11.373
11.358
11.143
0.215
11.143
0.215
−16.0
15.0
11.712
11.627
11.143
0.484
11.143
0.484
−21.0
5.0
6.370
6.370
6.359
0.011
6.371
0.012
−21.0
10.0
6.406
6.404
6.359
0.046
6.371
0.046
−21.0
15.0
6.466
6.462
6.359
0.103
6.371
0.104
−26.0
5.0
3.854
3.851
3.851
0.001
3.863
0.001
−26.0
10.0
3.854
3.854
3.851
0.003
3.863
0.003
−26.0
15.0
3.854
3.857
3.851
0.007
3.863
0.007
−31.0
5.0
2.424
2.422
2.424
−0.002
2.434
−0.002
−31.0
10.0
2.416
2.416
2.424
−0.008
2.434
−0.008
−31.0
15.0
2.404
2.406
2.424
−0.017
2.434
−0.017
−36.0
5.0
1.565
1.566
1.568
−0.002
1.575
−0.002
−36.0
10.0
1.559
1.559
1.568
−0.009
1.575
−0.009
−36.0
15.0
1.546
1.547
1.568
−0.021
1.575
−0.021
−41.0
5.0
1.033
1.033
1.035
−0.002
1.040
−0.002
−41.0
10.0
1.027
1.027
1.035
−0.008
1.040
−0.008
−41.0
15.0
1.016
1.017
1.035
−0.018
1.040
−0.018Table 2
The values of fluence rate. The transmittance of 10 mm thick water layer is 80%.
y
A [mm]
r
Air [mm]
The ray-trace procedure
The integration by the Simpson method
The integration by the Chebyshev method
E
/
β [m−1]
(
E
line
+
E
r
)
/
β [m−1]
E
line
/
β [m−1]
E
r
/
β [m−1]
E
line
/
β [m−1]
E
r
/
β [m−1]
−16.0
5.0
11.199
11.197
11.143
0.054
11.143
0.054
−16.0
10.0
11.373
11.358
11.143
0.215
11.143
0.215
−16.0
15.0
11.712
11.627
11.143
0.484
11.143
0.484
−21.0
5.0
7.499
7.499
7.482
0.018
7.489
0.018
−21.0
10.0
7.555
7.553
7.482
0.071
7.489
0.071
−21.0
15.0
7.653
7.641
7.482
0.160
7.489
0.160
−26.0
5.0
5.337
5.332
5.326
0.006
5.336
0.006
−26.0
10.0
5.352
5.350
5.326
0.024
5.336
0.024
−26.0
15.0
5.381
5.380
5.326
0.054
5.336
0.054
−31.0
5.0
3.944
3.940
3.938
0.002
3.948
0.002
−31.0
10.0
3.945
3.945
3.938
0.006
3.948
0.006
−31.0
15.0
3.951
3.952
3.938
0.014
3.948
0.014
−36.0
5.0
2.990
2.990
2.991
0.000
3.000
0.000
−36.0
10.0
2.990
2.990
2.991
−0.001
3.000
−0.001
−36.0
15.0
2.987
2.988
2.991
−0.002
3.000
−0.002
−41.0
5.0
2.314
2.315
2.316
−0.001
2.324
−0.001
−41.0
10.0
2.312
2.312
2.316
−0.004
2.324
−0.004
−41.0
15.0
2.307
2.307
2.316
−0.009
2.324
−0.009Tables3 and 4 present the values of the effective interval lengths.Table 3
The values ofzmax⁡. The transmittance of 10 mm thick water layer is 60%.
y
A [mm]
r
Air [mm]
2
z
max
⁡ (μ=0.15) [mm]
2
z
max
⁡ (μ=0.05) [mm]
−16.00
5.00
79.55
146.82
−16.00
10.00
97.64
180.40
−16.00
15.00
114.79
212.44
−21.00
5.00
84.02
150.40
−21.00
10.00
101.74
183.39
−21.00
15.00
119.09
215.79
−26.00
5.00
88.16
153.89
−26.00
10.00
105.27
185.87
−26.00
15.00
122.33
217.79
−31.00
5.00
91.97
157.33
−31.00
10.00
108.37
188.15
−31.00
15.00
124.93
219.23
−36.00
5.00
95.42
160.73
−36.00
10.00
111.07
190.33
−36.00
15.00
127.02
220.38
−41.00
5.00
98.47
164.07
−41.00
10.00
113.36
192.43
−41.00
15.00
128.65
221.37Table 4
The values ofzmax⁡. The transmittance of 10 mm thick water layer is 80%.
y
A [mm]
r
Air [mm]
2
z
max
⁡ (μ=0.15) [mm]
2
z
max
⁡ (μ=0.05) [mm]
−16.00
5.00
79.55
146.82
−16.00
10.00
97.64
180.40
−16.00
15.00
114.79
212.44
−21.00
5.00
86.29
153.98
−21.00
10.00
104.42
187.64
−21.00
15.00
122.07
220.52
−26.00
5.00
92.85
161.02
−26.00
10.00
110.81
194.39
−26.00
15.00
128.53
227.39
−31.00
5.00
99.26
167.98
−31.00
10.00
116.95
200.92
−31.00
15.00
134.56
233.73
−36.00
5.00
105.51
174.89
−36.00
10.00
122.90
207.33
−36.00
15.00
140.30
239.79
−41.00
5.00
111.61
181.76
−41.00
10.00
128.67
213.65
−41.00
15.00
145.81
245.67


## 9. Conclusions

The calculations of fluence rate in the ultraviolet photoreactor for water disinfection have been considered for the case when a cylinder of infinite length is the radiation source. This cylinder is filled with an isotropically radiating medium. The dependence of fluence rate on the diameter of radiating cylinder has been analytically analyzed. The radiation source in a form of line is regarded as the limiting case when the diameter of radiating cylinder tends to zero. The comparison of values of the fluence rate from the radiation source in a form of line and from the cylindrical source shows that, at the value of refractive index of the medium under irradiation equal to 1.373, the difference between fluence rates from these sources is smaller than 1%. The notion of “effective interval” has been introduced. Based on this notion, the conditions, under which the fluence rate from the infinitely long cylinder and from the cylinder of finite length differs by indefinitely small preliminary specified value, have been formulated. It has been shown that in the calculations of fluence rate the use of the Chebyshev method for numerical integration is expedient.This paper is the result of investigations which have been performed and sponsored by the JSC “Svarog” with the purpose of creating high-performance photochemical reactors of “Lasur” series designed for water disinfection (see Figure8). The results of investigations reported in the present paper are actively employed in the JSC “Svarog.” Thus, taking into account that the diameter of radiation source slightly affects the fluence rate, the ratio (radiation flux)/(length of lamp) has been accepted as the main criterion for the of photoreactor lamps. The estimation of a variation in the fluence rate for the direction parallel to the lamp made on basis of the “effective interval” notion is useful for the choice of proper places for devices of water input and output and arrangement of the sources of ultrasound in the photoreactor.Figure 8
Multilamp reactors of “Lasur” series.For more accurate computation of the operational characteristics of photoreactors, the JSC Svarog uses the complicated computer simulation. The models are based on the calculations of fluence rate in points of particle pathways (see Figure1). These calculations take into account such factors as the impact of quartz casing between air and water, the reflection of radiation from a border between two media with different refractive indexes, the mutual effect of neighbouring lamps (for multilamp reactors), and the effects of walls and similar design elements of photoreactor. With that, several lamp models are used. For the final evaluation of operating performance of photoreactors, the JSC “Svarog” uses biodosimetric tests [21].



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*Source: 310720-2014-11-10.xml*