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Phys. Med. Biol. 42 (1997) 2027–2039. Printed in the UK PII: S0031-9155(97)75049-8

A new ozone-based method for virus inactivation:

preliminary study

M M Kekez† and S A Sattar‡

† National Research Council of Canada, Ottawa, Ontario, K1A OR6, Canada

‡ The Faculty of Medicine, University of Ottawa, Ottawa, Ontario, K1H 8M5, Canada

Received 31 May 1996, in ﬁnal form 10 July 1997

Abstract. The nebulization technique reported here could be used to inactivate viruses with

ozone in large volumes of body ﬂuids, such as plasma, partial blood and perhaps whole blood

in a short time. Coliphage MS2 was used as a model because it is safe, easy to handle and

more resistant to chemical disinfections than viruses such as HIV. The theoretical curves and

experimental points, describing ozone inactivation of MS2, form a semi-sigmoid of congruent

data. There was a >7log

10

reduction in MS2 viability and the possibilities of minimizing the

ozone concentration required to kill viruses are indicated.

The analysis was expanded to account for the interaction of ozone with a virus suspension

in the shape of a thin ﬁlm from the experimental ﬁndings of Bolton et al. We again ﬁnd a semi-

sigmoid of congruent data for their case, i.e. describing ozone inactivation of the inﬂuenza A

virus (WSN strain) and the vesicular stomatitis virus versus time. For the method of nebulization,

the exposure time of droplets with ozone is a few seconds, whereas for the thin ﬁlm method the

exposure time is measured in hours.

1. Introduction

A consortium involving the Canadian Department of National Defence, Agriculture Canada

and the Canadian Red Cross Society has conducted preliminary studies on ozone inactivation

of viruses in blood. One of the authors (MMK) was a member of this consortium and was

asked to develop an ozone delivery system to better facilitate interactions between the gas

(ozone) and the body ﬂuids, because the two existing techniques used by the consortium,

(i) ‘hollow ﬁbre’ (Wells et al 1991) (US patent of the date 911205) and (ii) ‘conventional

bubbling’ by M

¨

uller Medical Inc. (West German patent 1068428), convey inconsistent ozone

transfer to test ﬂuids. M

¨

uller’s method also produces excessive bubbling and with both

methods the consistency of the treated ﬂuid is not satisfactory. In M

¨

uller’s system ozone

is administered extracorporeally, i.e. ozone is foamed through a small sample of venous

blood (10 ml) removed from a patient by phlebotomy, exposed for 3 to 30 min to ozone,

and immediately injected intramuscularly back into the patient. This technique is referred

to as autoheamotherapy.

In the proposed method, the ﬂuids are thoroughly nebulized or atomized, i.e. dispersed

into minute droplets to create a ﬁne ‘rain’ to fall through a controlled atmosphere of O

3

/O

2

and/or O

3

/inert gas mixture. Electric and magnetic ﬁelds can be superimposed over the

space through which the droplets are passed. The preliminary experimental work shows

that this method yields consistent results. By comparison with M

¨

uller’s method, the system

described in this paper would ozonate test ﬂuids at a rate of at least 20 ml min

−1

.

0031-9155/97/112027+13$19.50

c

1997 IOP Publishing Ltd 2027

2028 M M Kekez and S A Sattar

Many industrial applications (e.g. powdered milk, powdered eggs, instant coffee and

the carburation of liquid fuels in cars and jet planes) are based on the nebulization principle

described in this paper.

Our work is grouped into ﬁve parts: (i) theoretical analysis of the ozone penetration

into the biological ﬂuids, (ii) comparison between the nebulization and the thin ﬁlm method

of Bolton et al (1982), (iii) experimental examination of the nebulization method by

inactivating MS2 coliphage and Bacillus subtilis in the selected reactor conﬁguration as

a function of ozone dosage, (iv) theoretical examination of the ﬁndings by Bolton et al

(1982) and (v) formulation of the suitability of surrogate viruses for subsequent evaluation,

particularly the lentivirus group of retroviruses.

2. Theoretical formulation

2.1. Nebulization

In general, ozone is transferred to the interior of the ﬂuid by diffusion. The rate of ozone

absorption in an aqueous solution of droplets must be compensated by the addition of ozone

from outside the sphere. Razumovskii and Zaikov (1982), have shown that in zero-order

approximation the kinetics of the ozone–liquid interaction follows an exponential law

∂q/∂t

q

≈−

W

H

.

W is the speciﬁc rate of gas-ﬂuid feed (litre gas per litre solution) and H is the coefﬁcient

of Henry’s law. Let us now apply their experimental observation to a droplet: the ozone

penetrates the surface area of the droplet with a diffusion velocity, v, and in time, 1t,it

will occupy the volume of 4πR

2

v1t. R is the radius of the droplet and the droplet has the

volume of (4/3)π R

3

. Since W is the rate of feed, we see that W = 3αv/R. By deﬁnition,

the diffusion velocity, v, is approximately D/1x

d

, where 1x

d

is the characteristic diffusion

length (= 2(Dt )

1/2

). If t →∞and O

3

becomes Q

0

, the total amount of ozone absorbed

according to the above equation, q(X),is

q(X)

Q

0

= 1−e

−

√

X

(1)

where X is the normalized time equal to 9α

2

Dt/R

2

and α = 1/H . As a numerical

example, the diffusion constant, D, for water is 2.14 × 10

−5

cm

2

s

−1

and for glucose

0.52×10

−5

cm

2

s

−1

. For 1% ozone concentration in oxygen at 20

◦

C for water, α is 0.386.

We ﬁnd that the observation of Razumovskii and Zaikov is the well known Fick’s

law, which satisﬁes the classical mass diffusion–heat conduction (transport) law. In one

dimension (of x), this is

∂q

∂t

=D

∂

2

q

∂x

2

. (2)

In zero-order approximation equation (2) implies that the absorption of ozone by the ﬂuid

corresponds to the same law as the ﬂow of heat from the surroundings into the biological

ﬂuid.

For a droplet (sphere), equation (2) was solved (see the appendix). The total amount of

ozone absorbed, q(X),is

q(X)

Q

0

= 1−e

X

Erf

√

X. (3)

Ozone-based method for virus inactivation 2029

Erf is the complementary error function. In ﬁgure 1, equation (3) is given as a full line. If the

droplet before landing meets the requirement that X<1, equation (3) can be approximated

by equation (1) (see ﬁgure 1). If X 1, equation (1) can be reduced further to

q(X)

Q

0

=

√

X =

3α

√

Dt

R

(4)

giving

R =

3αQ

0

q

0

√

Dt ∼ Q

0

√

D ∼ constQ

0

. (5)

When the time of ﬂight is ﬁxed, the time, t, becomes a constant quantity, t

0

, equal to

the droplet’s time of ﬂight. To inactivate a virus by ozone, we assume that the dosage

must exceed a certain minimum value. If a single species of viruses is used, q(t) becomes

q

0

= constant. Q

0

is the saturation concentration of ozone in the liquid in accordance with

Henry’s law. If Q is the concentration in the chamber, then Q

0

= αQ (for t →∞). In

equation (5), Q is the only variable. The biological ﬂuid is characterized by the diffusion

constant, D, and the absorption coefﬁcient, α. The virus is speciﬁed by q

0

.

Figure 1. The full curve is equation (3): q(X)/Q

0

= 1 − expX ErfX

1/2

. The broken curve is

equation (1): q(X)/Q

0

= 1 − exp(−X)

1/2

.

To relate this analysis to virus inactivation in the ﬂuid’s droplets, the distribution of the

droplets in a spray can be described by the Gaussian function (represented by the left-hand

side of equation (6), having mean drop size, m, and standard deviation, σ . When a virus

titre of 2 × 10

7

plaque forming units (pfu) per ml is added, on average, each droplet of

a mean size of 30 to 40 µm contains a single virus. When the droplets are subjected to

the ozone atmosphere, the viruses present in the smaller droplets will be the ﬁrst to be

2030 M M Kekez and S A Sattar

inactivated. The relative number of inactivated viruses is

1

σ

√

2π

Z

x

−∞

{exp[−(t − m)

2

/(2σ

2

)]}dt =

1

2

1 + Erf

x − m

σ

√

2

. (6)

From −∞ to +∞, the integral has the value of 1, and Erf is the error function (see

Abramowitz and Stegun (1968)). Therefore, the relative number of viruses surviving the

ozone treatment is

s =

1

2

1 − Erf

x − m

σ

√

2

=

1

2

1 − Erf

−m + constQ

0

σ

√

2

. (7)

Here x represents a large size droplet (containing a virus) in the distribution that is not

affected by the ozone treatment. R of equation (5) is substituted in the middle part of

equation (7) as x = 2R, and the theoretical survival curves are produced following the path

given by Kekez et al (1996).

2.2. Thin ﬁlm method

The thin ﬁlm method of Bolton et al (1982) also produces consistent results. The biological

ﬂuid is suspended in the shape of a thin ﬁlm on the surface of a rotating culture bottle

and the ozone is drawn through the bottle at a constant rate. The total amount of ozone

absorption in the thin ﬁlm is also given by equation (3) but here X = α

2

Dt/(1x

2

), where

1x is the thickness of the ﬁlm. If X 1, equation (3) with new X becomes

q(X)

Q

0

= 1−

1

2

√

X

= 1−

1x

2α

√

Dt

(8)

giving

1x = 2α

1 −

q

0

Q

0

√

Dt. (9)

Equation (3) (approximation) and equation (8) (exact solution) are compared in ﬁgure 2. We

see that the approximation is valid only for X>1. In the Bolton et al (1982) experiments

the ozone concentration in the chamber, Q, is the parameter and the time, t, is the main

variable. Their time is measured in hours, hence X>1. For their experimental conditions,

the titre of viruses is large (10

8

pfu ml

−1

). It is useful to assume that the thickness of the

ﬁlm approaches that of a monolayer; therefore, 1x is also proportional to the diameter of the

virus, x, having its own mean size, m, and standard deviation, σ . Substituting equation (9)

in the second bracket of equation (7), the relative number of the viruses, s, surviving the

ozone treatment is obtained when the ozone exposure time, t, is a variable quantity:

s =

1

2

1 − Erf

−m + const

√

t

σ

√

2

. (10)

2.3. Comparison between the nebulization method and the thin ﬁlm method

To make comparison for a given (constant) volume, we consider that the cubic volume of

the thin ﬁlm (1x)

3

is equal to the equivalent volume of the sphere (4/3)π R

3

. This gives

R = 1x(3/4π)

1/3

. However, the normalized time, X, for the sphere has the multiplier

of 9 in its deﬁnition in comparison to X used in the thin ﬁlm case. This implies that the

nebulization method is faster by a factor of 9(3/4π)

1/3

= 5.58 in comparison to the thin

ﬁlm method.

Ozone-based method for virus inactivation 2031

Figure 2. The full curve is equation (3): q(X)/Q

0

= 1 − expX ErfX

1/2

. The broken curve is

equation (8): q(X)/Q

0

= 1 − 1/2X

1/2

. The comparison applies when X>1.

3. Basic methodologies and rationales for their selection

3.1. Nebulization

In our experimental work, the ﬂuid is atomized into small droplets by a nebulizer. The

droplets are injected into an ozonation chamber and allowed to fall through a controlled

atmosphere containing ozone as indicated in ﬁgure 3; therefore the exposure time is a

function of the height of the ozonation chamber. The properties and composition of the ﬂuid

affect the ability of the nebulizer to atomize. The method proposed is ﬂuid speciﬁc, requiring

an appropriate nozzle for each ﬂuid. Since the absorption is an exponential function of the

radius (see equation (1)), the dosage of ozone received by any viral contaminants is related

to the droplet size and the reactor design. To treat plasma and/or serum it is necessary

that the droplets are as small as possible, in order for the ozone to eliminate effectively the

viruses inside the droplet. To make the apparatus compact, it is also advisable to minimize

the time of ﬂight, i.e. to shorten the height of the inactivation chamber. Reduction of

the effective dosage will also minimize additional damage to desirable components of the

test ﬂuids. The experiment used a commercial ozone generator (model GTC-0.5, Grifﬁn

Technics, New Jersey).

3.2. Selection of test ﬂuid

A major concern with the ozone treatment process is virus inactivation without unacceptable

damage to desirable components of the treated product(s). Since the likely damage to

whole blood is greater and more difﬁcult to assess because of its cellular components, in

this study we propose to test only plasma, a major source for blood products. Although

2032 M M Kekez and S A Sattar

Figure 3. The ultrasonic nebulizer system.

it is recognized that the possibility of damage to proteinaceous material as well as the

possibility of generating toxins during the process exists, we consider that the assessment

of these factors is beyond the scope of this exploratory study.

3.3. Selection of nebulizer

In this work, the nebulizers considered were: a compressed gas atomizer (or pressure jet

atomizer) and twin ﬂuid atomizer; an ultrasonic nebulizer and a rotary nebulizer. Of these,

the ultrasonic nebulizer is preferable due to the low velocity of the incoming spray, and the

experimental chamber with such a nebulizer appears to yield the most compact design for

an equivalent ozone dose. Very ﬁne atomization is achieved at low velocity (1–3 m s

−1

),

making this device ideal for use with the ﬂow rates of 10 to 400 ml min

−1

.

The nebulizer system in ﬁgure 3 consists of a piezoelectric resonant device (19), whose

transducer discs are energized by high-frequency electrical signals applied to the terminals

(20). This results in the propagation of pressure waves in both directions along the nozzle

(21). The liquid inlet is at point (22) and the spray originates at the atomizing surface (23).

The droplets are collected at a dish (24), and a peristaltic pump (5) removes the liquids

from this dish (24). The inactivation chamber is formed between the two ﬂanges (25 and

26) and a glass cylinder (27). Ozone is injected into the chamber via an annular ring (28)

through 24 small-diameter openings to ensure that the ozone stream is slow-moving. The

ozone outlet (29) feeds the used gas to the vent decomposer (11).

Ozone-based method for virus inactivation 2033

3.4. Application of electric and magnetic ﬁelds

The ﬂight characteristics of the droplets can be augmented and controlled electronically.

Each droplet is charged by an electrostatic ﬁeld. A high voltage is applied to the dish (24)

and the nozzle (21) acts as a ground electrode, ensuring that the droplets do not fuse in

ﬂight. To further control the speed of the droplets, another electrode in the form of a ring

could be placed in the space between the nozzle (21) and the dish (24) to create a classical

triode structure. If this process is used for whole blood, a magnetic ﬁeld could be applied

to increase the ﬂight time of the droplets by making them follow a helical path, due to the

iron in red blood cells. The magnetic ﬁeld is produced by putting a coil around the glass

cylinder (27).

3.5. Separation of ozone from oxygen

When ozone is generated by electrical discharges, a gas mixture (O

3

–O

2

or O

3

–air) is

produced in which the ozone concentration is typically 1 to 5% by volume. Organic

matter is prone to oxidation and it may be important in some applications to minimize the

oxidative stress caused by oxygen radicals. Work is in progress to develop a low-temperature

distillation (adsorption–desorption) technique to attain an ozone-to-oxygen ratio of about 1

to 5 (i.e. 20 to 30% by volume over silica gel with a low impurity level of Fe) during

the adsorption cycle. Later, during the desorption phase (when ozone is separated from

oxygen), an inert gas (e.g. N

2

, He, etc) will carry ozone to the inactivation chamber.

3.6. Control of droplet size

The size and distribution of the droplets are determined by a Doppler velocity apparatus.

This apparatus can also help measure the time of ﬂight, optimize the ﬂow rate and determine

the droplet’s evaporation rate.

4. Experimental results

4.1. Whole blood

Preliminary experiments were conducted to demonstrate some aspects of the proposed

concept. A drop of blood was released from a 10 ml syringe with a needle (gauge 14–20)

by knocking the needle with a ﬁnger, and these drops were allowed to fall freely over a

distance of 5 to 10 cm until enough blood collected in a Petri dish for the analysis. There

was minimal (<2%) or no haemolysis of the red blood cells due to free fall and exposure

to air. With the ﬁrst (compressed gas) atomizer built, 87% of the total number of the red

blood cells present in whole blood survived the nebulization treatment.

4.2. Tests with bacteriophage MS2

An ultrasonic nebulizer was used in this study. A bacteriophage, the tailless icosahedral

RNA-coliphage MS2, suspended in bovine serum, was used to optimize a ‘viral inactivator’.

MS2 was used as a model here because it is safe, easy to handle, and can grow to titres up

to 10

13

pfu ml

−1

in a suitable bacterial host. MS2 phage, being a small (27 nm in diameter),

icosahedral and non-enveloped virus, survives better in the environment and is generally

more resistant to inactivation by chemical agents when compared with larger, enveloped

viruses such as HIV. The titre of active MS2 in the serum/plasma sample can be accurately

2034 M M Kekez and S A Sattar

measured by counting the number of plaques produced in an Escherichia coli lawn. The

effectiveness of the ozone treatment for the inactivation of the test virus in treated serum

samples was indicated by the reduction in the number of plaques as a function of ozone

dosage.

The phage was diluted to a ﬁnal concentration of about 2 × 10

7

pfu ml

−1

for the data

shown in ﬁgure 4. Dose–response curves of MS2 inactivation by ozone were obtained

with the virus suspended in (i) Dulbecco’s phosphate buffered saline (PBS), (ii) a 10%

solution of bovine serum in PBS, and (iii) a 25% bovine serum in PBS. A comparison

of the experimental points and the theoretical curves for MS2 shows that the curves and

points form a semi-sigmoid of congruent data (ﬁgure 4). The increase in bovine serum

concentration only results in translating the curve to the right.

Figure 4. Dose–response curves of MS2 inactivation by ozone as a function of ozone

concentration using the nebulization method. MS2 was suspended in (i) Dulbecco’s phosphate

buffered saline (PBS), (ii) a 10% solution of bovine serum in PBS and (iii) a 25% bovine serum

in PBS. The experimental points are:

•, PBS; , 10% serum; , 25% serum. Data points are

the average of three experiments performed in triplicate. Theoretical curves are described by

the following expressions.

PBS: s = 0.5{1 − Erf[0.0241(−36.38 −0.7Q)]}

10% serum: s = 0.5{1 − Erf[0.0241(−36.38 −0.014Q)]}

25% serum: s = 0.5{1 − Erf[0.0241(−36.38 −0.008Q)]}

when the ozone exposure time used was about 1 s.

The theoretical curves were computed using equation (7) for the measured value of

m = 36 µm, σ = 29 (µm)

−1

. To get a good ﬁt with the experimental points, a set of

values for the ‘constant’ of equation (7), named here β, was carried out. We ﬁnd that β has

a value of 0.7, 0.014 and 0.008 for PBS, 10% serum and 25% serum respectively. From

equation (5) we see that β is proportional to the square root of the diffusion constant. The

Ozone-based method for virus inactivation 2035

ratio of β for 10% serum to β for 25% serum is 1.75, implying that the diffusion constant at

25% serum is decreased by a factor of 3 with respect to the diffusion constant at 10% serum

concentration. To get a more accurate evaluation of the change in the diffusion constant,

it is necessary to use equation (1) instead of equation (5) in determining the value of R.

Hence, we have estimated that q

0

(for an air-borne virus to be inactivated) has a value of

20 ppm, when the time of ozone exposure is a few seconds.

4.3. Tests with the spores of Bacillus subtilis

An ultrasonic nebulizer was used in this study. The system was examined for its ability to

inactivate the spores of Bacillus subtilis (ATCC 19659). Such spores are routinely used to

assess the sporicidal activity of liquid and gaseous chemicals and as biological indicators to

validate the performance of steam and gas sterilizers. The bacterium was grown in Columbia

Broth (Difco, Detroit, Michigan, USA) diluted ten-fold in deionized water. In this medium,

there was nearly a 100% sporulation after 48 h at 37

◦

C. The spores were washed three

times in deionized water and suspended in the test medium to a ﬁnal concentration of 10

8

colony forming units/ml. When B. subtilis spores were suspended in PBS and nebulized in

the presence of ozone, there was >5log

10

reduction in their viability.

4.4. Thin ﬁlm method of Bolton et al (1982)

4.4.1. Test with vesicular stomatitis virus. If equation (10) is to be applied to these results,

detailed knowledge of the virus distribution is required. From Wagner (1975) and Fields

et al (1996) we see that there is considerable heterogeneity in particle size. The typical

infectious B virion is a bullet-shaped (B particle) cylinder, 180 ± 10 nm in length and

65±10 nm in diameter at the blunt end. If this volume is approximated by a sphere, it will

have a radius of 50.65 nm. For the predominant defective truncated (T) virions having about

one-third the length (65 nm) of the infectious B virions, we ﬁnd the equivalent radius to be

36 nm. The molecular weight of the infective VS B-virion was estimated to be 4.4 × 10

6

daltons at the upper extreme and 3.2 × 10

6

daltons at the lower end. For the density of

1.31 g cm

−3

, this corresponds to the equivalent radius of 9.9 nm at the lower end. Defective

VS virions have a molecular weight of (0.7–1.2) × 10

6

daltons which corresponds to the

equivalent radius of 5.9 to 7.1 nm.

If half of all the particles are occupied by the infectious B viruses in the Gaussian type

distribution curve we have from the deﬁnition of the error function that (x − m)/(σ

√

2) =

0.477 as Erf (0.477) = 0.50. For the mean radius of the infectious B virion, m, of 50.6 nm,

we get x −m = 14.8 nm. Here, x corresponds to the mean radius of the defective truncated

(T) virion of 36 nm. This gives 1/(σ

√

2) = 0.032 nm

−1

. Putting the values for m and

1/(σ

√

2) into equation (11), the theoretical curves for ozone inactivation versus time are

obtained. The results are given in ﬁgure 5, again showing a semi-sigmoid of congruent

data.

4.4.2. Test with inﬂuenza A virus (WSN strain). The information about the distribution of

this virus can be obtained from the work by Compans and Choppin (1975) and Fields et al

(1996). The majority of particles are spherical with a diameter of 80–100 nm, making the

average radius 45 nm. The purity, homogeneity and approximate composition of population

of virus particles were not conclusively established. However, to derive the theoretical

curves, we assume that the relative distribution for inﬂuenza A virus is the same as for

2036 M M Kekez and S A Sattar

Figure 5. Ozone inactivation of vesicular stomatitis virus versus time. Experimental points

obtained by Bolton et al (1982) were replotted on the log–log scale. The thin ﬁlm method was

used. The virus suspension was exposed to: , 0.64 ppm of ozone; ◦, 0.16 ppm; M, 0.00 ppm.

Full curves are the theoretical curves for:

0.00 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +20(time)

1/2

)]}

0.16 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +26(time)

1/2

)]}

0.64 ppm: s = 0.5{1 − Erf[0.0323(−50.65 +33(time)

1/2

)]}.

vesicular stomatitis virus. This means that 1/(σ

√

2) equals 0.032 nm

−1

; the results are

given in ﬁgure 6. We suggest that the slightly higher resistance (manifested by higher value

of q

0

used in equation (9)) to inactivate inﬂuenza A viruses in comparison with vesicular

stomatitis viruses, may be attributed to the fact that they have a segmented genome. There

are eight separate segments which make up the full genetic complement of the inﬂuenza

virus.

5. Selection of test viruses for future study

To assess the ability of ozone applied by the nebulization method for inactivating viruses in

biological ﬂuids, it is necessary to spike the test samples with relevant viruses and examine

their inactivation as a function of ozone dosage. Relevant human viruses may include the

hepatitis viruses, parvoviruses (including B19), the herpes viruses (including herpesvirus 2

and cytomegalovirus) and retroviruses (including type C and HIV). However, safety issues

as well as practical difﬁculties of working with some of these agents precluded their use at

the preliminary stages of such a study. It is also difﬁcult at this stage to reliably extrapolate

to mammalian viruses from studies on bacteriophage(s). Therefore, experiments will be

performed using a panel of surrogate viruses in proper containment facilities.

Ozone-based method for virus inactivation 2037

Figure 6. As in ﬁgure 5, but for inﬂuenza A virus (WSN strain). Full curves are the theoretical

curves for:

0.00 ppm: s = 0.5{1 − Erf[0.0323(−45 +15(time)

1/2

)]}

0.16 ppm: s = 0.5{1 − Erf[0.0323(−45 +20(time)

1/2

)]}

0.64 ppm: s = 0.5{1 − Erf[0.0323(−45 +27(time)

1/2

)]}.

6. Conclusions

The technique reported could be used to inactivate viruses with ozone in large volumes of

body ﬂuids such as plasma, partial blood and perhaps whole blood in a short period of time.

To enhance the absorption of ozone in the ﬂuid, the method atomizes the ﬂuid into small

droplets that are sprayed into an atmosphere of ozone which kills viruses. It is shown by

Carpendale and Freeberg (1991) that ozone inactivates HIV at non-cytotic concentrations.

The current work offers many opportunities to minimize the ozone concentration

required to kill viruses. It is necessary to decrease the droplets size as much as possible

in order that the virus inactivation is done at low ozone concentration; however, if a very

small droplet size is chosen, the use of hydrocyclone technology to collect droplets becomes

necessary.

An objection to our approach has been raised that we have neglected the chemical

reaction of ozone with liquid. For example, when ozone diffuses in a liquid and reacts

with it irreversibly, equation (2) must be modiﬁed according to a ﬁrst-order reaction. This

means that, on the left hand side of equation (2) we must add the term βq to account for

this reaction. Here, β is a constant greater than zero. After solving this new expression, the

total amount of ozone absorbed versus normalized time again reaches the semi-saturation

for X>1 in a similar fashion as in ﬁgure 2, but at smaller value of q(X). The fundamental

relationship of both equations (4) and (8) is maintained and the validity of equation (4) is

expanded over the large domain of X.

2038 M M Kekez and S A Sattar

Acknowledgments

Interest shown by DrALVanKoughnett, NRC, and Dr G Adams, NRC, is appreciated.

The authors wish to thank Mrs V S Springthorpe of the Faculty of Medicine, University

of Ottawa for useful ideas and helpful discussions. The authors are indebted to Dr P

Savic, Researcher Emeritus, NRC, for his help in the theoretical formulation. The technical

assistance of Mrs H Rahman and Mr R Sansom is gratefully acknowledged.

Appendix

Carslaw and Jaeger (1989, p 349 (III)) quote the problem of a sphere of radius a and perfect

conductor of speciﬁc heat c

1

, surrounded by an inﬁnite region of diffusivity D, speciﬁc heat

c and contact resistance (1/h). When the heat source Q is absent, the problem becomes

that given on p 350 (IV). With the conductor at the initial temperature T

0

, and with contact

resistance approaching zero, h →∞, the integral #21 on p 350, describing the cooling of

the sphere, reduces to

T =

2kT

0

π

Z

∞

0

[exp(−u

2

Dt/a

2

)]u

2

(u

2

− k)

2

+k

2

u

2

du

where k = 4πa

3

ρc/M

1

c

1

. Here, M

1

is the mass of the sphere. This integral can be evaluated

by partial fraction expansion of the polynomial part of the integrand. The denominator is a

quadratic in u

2

with two roots:

v = u

2

= k

1 −

k

2

±

k

2

2

r

1 −

4

k

.

For large values of k, the square root can be expanded to ﬁrst order, giving the following

value for the two roots: v

1

= 0 and v

2

=−k

2

. Hence, the integral becomes

T =

2kT

0

π

Z

∞

0

exp(−u

2

Dt/a

2

)

u

2

+k

2

du. (A1)

From Gradshteyn and Ryzhik (1965, p 338, #3.466 (1)), we have

Z

∞

0

exp(−µ

2

x

2

)

x

2

+ β

2

dx = Erf(βµ)

π

2β

exp(β

2

µ

2

)

and the solution of equation (A1) is

T = T

0

e

X

Erf

√

X (A2)

where

√

X = βµ µ =

r

Dt

a

2

β = k =

3ρc

ρ

1

c

1

since

D

1

D

=

ρ

2

ρ

2

1

c

c

1

= α

we get

X =

9α

2

D

1

t

a

2

.

Ozone-based method for virus inactivation 2039

Here, D

1

is the diffusion constant inside the sphere. If the sphere were to be be heated,

equation (A2) would read

T

T

0

= 1 − e

X

Erf

√

X.

For the analogies between heat and mass transfer, we need to substitute T by Q. Hence,

the equation above becomes equation (3) given in the main text.

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