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Available online at www.sciencedirect.com

Chemical Engineering and Processing 46 (2007) 1310–1316

Concentrating dilute sulfuric acid by spray evaporator

Zhiming Zhou a,∗, Zhonghai Liu b

a State Key Laboratory of Chemical Engineering, East China University of Science and Technology, Shanghai 200237, Chinab College of Chemical Engineering, Sichuan University, Chengdu 610065, China

Received 16 December 2005; received in revised form 13 October 2006; accepted 21 October 2006Available online 20 November 2006

bstract

Large quantities of dilute spent sulfuric acid are released in many chemical processes. Recovering the dilute acid is not only profitable to theanufacturer but also imperative to environmental protection. This paper proposes a spray evaporator with a Venturi-type nozzle to concentrate

he dilute sulfuric acid. Both hot air and dilute acid flow concurrently upwards through the nozzle. Water involved in the droplets is vaporized inhe chamber and the dilute acid is concentrated. The bench-scale experimental results show that the dilute acid with initial concentration 18 wt%an be easily concentrated to 40–75 wt%. The measured parameters, such as concentration of outlet sulfuric acid, outlet air temperature and total

ressure drop, are in accordance with those estimated from a mathematical model incorporating momentum, mass and heat transfer between thecid and air. The model is also applied to simulate the performance of the concentrator, including variations of droplet diameter, droplet velocity,roplet temperature, air temperature, air absolute humidity as well as pressure drop along the concentrator.

2006 Elsevier B.V. All rights reserved.

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eywords: Spray evaporation; Droplet; Concentration; Dilute sulfuric acid; Mo

. Introduction

Sulfuric acid is one of the most fundamental raw materials forany chemical processes. Large quantities are used for solvent

nd dehydrating actions in sulfonations, nitrations, and chlori-ations involved in the organic chemical industry [1]. However,nly a small amount of sulfuric acid takes part in the relevanteactions as a substituent, while most is discharged in the formf dilute waste. Recovering and reusing the waste dilute sulfuriccid is profitable to the enterprise. In addition, the increas-ng environmental awareness and more stringent legislation foraste liquid emissions have spurred research interest to recover

he spent sulfuric acid.Methods for recovery of spent sulfuric acid can be mainly

lassified into two kinds, regeneration and concentration. Regen-ration of the waste acid is a method that decomposes the spentulfuric acid into sulfur dioxide at high temperature, and then

ulfur dioxide is converted into strong sulfuric acid by contactethod. Almost all impurities in the waste acid are removed by

his method [2], nevertheless high energy cost and operating cost

∗ Corresponding author. Tel.: +86 21 6425 2230; fax: +86 21 6425 3528.E-mail address: zmzhou@ecust.edu.cn (Z. Zhou).

ceimsst

255-2701/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2006.10.008

g

ake this method seldom practically applied. The other methodhat concentrates the dilute acid preceded or followed by purifi-ation process is often utilized. The main body of this kind ofechnology is the concentrator, which can be sorted into severalypes such as submerged heater, drum concentrator, Pauling con-entrator and spray evaporator [3]. Each concentrator has bothdvantages and disadvantages. The submerged heater is char-cteristic of direct heat exchange between gas and liquid andeems appealing for dilute sulfuric acid containing metal salts,ut the tail gas entrains large amounts of heat as well as sul-uric acid. The drum concentrator is quite suitable for dilutepent acid produced by nitrations due to its large capacity andigh thermal efficiency. Unfortunately, severe acid mist carriedlong in the tail gas is an intractable problem to be solved. Theauling concentrator is composed of an evaporator and a recti-ying column so that acid mist can be reduced to a certain extent.owever, relatively small capacity and poor thermal efficiency

aused by indirect heat exchange limit its application. Sprayvaporator makes use of the atomizer to reduce the spent acidnto fine droplets, and then these droplets exchange momentum,

ass and heat with the ambient gas. Consequently, the dilute

pent acid is concentrated. Compared with other concentrators,pray evaporator appears to be the most suitable one for concen-ration of dilute acid because of its overwhelmingly advantages:

ing and Processing 46 (2007) 1310–1316 1311

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bTcitmcaarcd

2

tztatrdwsrcibrdtst

tisacip

Ffl(d

3

nada

(

((

(

ssexpressed as follows [5]:

dn

ddp= 0.54d2

p exp(−0.765d0.5p ), (1)

Z. Zhou, Z. Liu / Chemical Engineer

i) the dilute acid can be easily concentrated to high concentra-ion by one-step operation instead of circulation of the acid; (ii)he acid is atomized into droplets and the gas–liquid interface isorrespondingly increased, which is favorable for mass and heatransfer between the hot air and the droplets; and (iii) the acidist entranced in the tail gas can be reduced drastically because

he discrete phase is not the gas like that in the submerged heaterr the drum concentrator but the liquid, which results in suspen-ion of droplets in the air. Because the partial pressure of sulfuriccid on the surface of sulfuric acid solution is much less than thatf water in most cases [4], it is not sulfuric acid but water that isvaporated from liquid droplets, which consequently results iness entrained acid mist [5,6].

Concentration of dilute sulfuric acid in a spray evaporator haseen proved successfully by the bench-scale experiments [6].he dilute acid with concentration of 18 wt% in the feedstockan be concentrated up to 75 wt%. Unfortunately, a mathemat-cal model for the concentrator has not yet been established. Inhe present work, efforts are made to develop a one-dimensional

odel that can account for the performance of the bench-scaleoncentrator. Although this model contains some simplifyingssumptions, it represents a first step towards development ofgeneral model to account for the complex phenomena occur-

ing in the spray evaporator. It is believed that these efforts willonduce to the scale-up of the concentrator for concentration ofilute spent sulfuric acid.

. Experimental

Air is introduced by fan into the preheater and heated tohe preset temperature, and then enters the Venturi-type noz-le. Meanwhile, the dilute sulfuric acid is piped into the upperank by pump, and subsequently conducted to the nozzle afterccurately controlling the flow rate. The liquid is atomized byhe hot air with high velocity in the nozzle and flows concur-ently upwards with the air. In the evaporator chamber, the liquidroplets are suspended in the hot air and exchange mass and heatith the air. As a result, the diameters of the droplets are reduced

ince the water in the droplets is vaporized, and the dilute sulfu-ic acid is concentrated. In the upper part of the evaporator, theoncentrated droplets with smaller diameters collide with thenner wall of the evaporator and flow downwards into the jacketetween the wall and the nozzle. Finally, the concentrated sulfu-ic acid outflows from the concentrator and its concentration isetermined by acid–base titration. Effluent gas is released afterreated by sodium carbonate. A flow scheme of the experimentaletup is shown in Fig. 1. The total pressure drop of the concen-rator is measured by a U-shape manometer.

To facilitate the discussion in the flowing part, the structure ofhe Venturi-type nozzle is briefly described here, which is shownn Fig. 2. The nozzle consists of three sections, i.e., converging

ection, throat section and diverging section. The dilute sulfuriccid is conducted into the nozzle from a certain position in theonverging section, as shown by italic character a. The positionndicated by character b denotes the lower point from which theressure drop is measured. The distance between a and b is H0.

ig. 1. Schematic diagram of the experimental apparatus. (1) Fan, (2) valve, (3)owmeter, (4) temperature gauge, (5) preheater, (6) dilute sulfuric acid tank,7) pump, (8) tank, (9) flowmeter, (10) Venturi nozzle, (11) evaporator, (12)efoamer, (13) manometer and (14) recovery tank.

. Model development

Spray evaporation is a complicated process, which involvesot only momentum, mass and heat transfer between dropletsnd hot medium, but collision, breakup and coalescence ofroplets as well. For the sake of simplicity, the following basicssumptions are used in the modeling of the spray evaporator:

1) The droplet is of spherical shape and both temperature gra-dient and concentration gradient in the droplet are absent.

2) Collision, breakup and coalescence of droplets are ignored.3) Radial gradients of air temperature and velocity in a cross-

section of the evaporator are negligible, which proved cor-rect in a previous study [5].

4) Evaporation of sulfuric acid from the droplet is negligible,which is justifiable on the ground that the partial pressure ofsulfuric acid on the surface of sulfuric acid solution is muchless than that of water when the concentration of solutionis not higher than 80 wt% [4], and the concentration of theconcentrated acid ranges from 40 to 75 wt% in this study.

The droplet size distribution induced by atomization of diluteulfuric acid in the Venturi-type nozzle is measured by the micro-copic analysis method with manual counting, which can be

Fig. 2. Structure of the Venturi nozzle.

1 ing and Processing 46 (2007) 1310–1316

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A

B

d

wdop

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wpp

Table 1Heat balance over an elementary part (z, z + �z)

Enthalpy enteringat z

Enthalpy of air q1 = FgI1

Enthalpy of feed q2 =N∑

i=1

Fl1,iCl1,i(Tl1,i − 273.2)

Enthalpy leavingat z + �z

Enthalpy of air q3 = FgI2

Enthalpy of feed q4 =N∑

i=1

Fl2,iCl2,i(Tl2,i − 273.2)

lp

q

wr

I

wto

X

d

T

aeo

vot

t

−

where m is the liquid-to-gas mass flow ratio, m = Fl/FG, whichis a variable considering evaporation of droplets. f is the fric-

312 Z. Zhou, Z. Liu / Chemical Engineer

here n is the number of droplets with diameter dp. Based onhe force balance, the motion of each individual droplet alonghe axial direction is described by

dul

dz= −g(ρl − ρg)

ρlul+ 3CDρg(ug − ul)|ug − ul|

ρldpul, (2)

here ug, ul, dp and CD represent air velocity, droplet velocity,roplet diameter and the drag coefficient, respectively. For aingle sphere, CD is given by [7]

D =

⎧⎪⎨⎪⎩

24/Rer, Rer < 1

22.4(1 + 0.15Re0.687r )/Rer, 1 < Rer < 1000

0.44, Rer > 1000

, (3)

here Rer is the droplet Reynolds number,er = dp|ug − ul|ρg/μg. Based on Ranz–Marshall equation

8], the heat transfer between droplet and air is described by

dQ

dz= A · 12(1 + 0.3Re1/2Pr1/3)λ(Tg − Tl)

ρld2pul

, (4)

here Tg and Tl are air temperature and droplet temperatureespectively. A is a correction factor of the thermal conductiv-ty of air around the droplet on account of the effect of highemperature, which is calculated as follows [9]:

= B

eB − 1, (5)

= ln

[1 + Nu

2· CV(Tg − Tl)

γ

]. (6)

Similar with the heat transfer, the mass transfer betweenroplet and air is given by

dW

dz= 12(1 + 0.3Re1/2Sc1/3)

DgM(pd − pa)

RgTlρld2pul

, (7)

here pd and pa represent the partial pressure of water vapor atroplet wet surface and in the ambient gas individually. Basedn the heat balance between droplets and air, the droplet tem-erature can be calculated by

dQ

dz= (γ + CV �Tl)

dW

dz+ dQc

dz+ (1 − �W)Cl

dTl

dz, (8)

here �Tl is the temperature difference of droplet over a contrololume of differential length and �W the mass of evaporatedater per unit mass of droplet. The first term of the right-hand

ide of Eq. (8) is due to evaporation of water, the second tooncentration of dilute sulfuric acid and the third to an increase inemperature of droplet. The concentration heat can be calculatedy the following equation (see Appendix B):

dQc = C20

2 · dW, (9)

dz (1 − W)(5.416 − 5.416W − 0.033C0) dz

here C0 is inlet concentration of dilute sulfuric acid. Air tem-erature can be derived from the heat balance over an elementaryart (z, z + �z), which is listed in Table 1.

tawa

Enthalpy ofconcentration

q5 = Qc

For no heat accumulation in this zone, heat in = heat out + heatoss. Heat loss is negligible in this study on account of good heatreservation of the chamber. Thus

1 + q2 = q3 + q4 + q5, (10)

here Fg and Fl are the dry air and liquid mass flow rates,espectively. I is enthalpy of air, which is expressed as

= (Cg + CVX)(Tg − 273.2) + 2491.3X, (11)

here X is the absolute humidity of air, which presents the mois-ure content per unit weight of dry air. Based on moisture balancever the same part (z, z + �z), the absolute humidity is

2 = X1 +∑N

i=1Fl1,i �Wi

Fg. (12)

Combining Eqs. (10) and (11), the air temperature can beerived as

g2 = 273.2 + q1 + q2 − q4 − q5 − 2491.3FgX2

Fg(Cg + CVX2). (13)

For the sulfuric acid droplet, its diameter reduces graduallys a result of evaporation of water. The diameter change can beasily expressed according to the mass balance over the dropletn the basis of unchanged mass of sulfuric acid.

The total pressure drop in the chamber consists of many indi-idual losses. They are due to friction, air and feed contractionr expansion. The following part will derive the expression ofhe pressure drop.

For a control volume of differential length in the chamber,he momentum balance is given by [10]

dp

ρg= ug dug + mug dul + (m + 1)fu2

g dz

2De, (14)

ion factor. The last term of the right-hand side of Eq. (14) ismomentum loss caused by the friction of air/droplet and theall. Integration of Eq. (14) from the point b shown in Fig. 2 topoint z yields

ing and Processing 46 (2007) 1310–1316 1313

−

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−

dip

mdtttnu

4

aptic1

otFfvdmi

TS

T

566

Fig. 3. Effect of inlet air temperature on concentration of outlet sulfuric acidFg,in = 48.2 kg h−1.

FT

a4ift

rator, more simulation results are itemized as follows. Exceptwhere specified otherwise, the aftermentioned discussion isbased on the same operating condition, viz. Tg,in = 573 K,

Z. Zhou, Z. Liu / Chemical Engineer

�pz

ρg= u2

g − u2g,b

2+

∫ z

0mug

(dul

dz

)dz

+∫ z

0

(m + 1)fu2g

2Dedz +

∫ 0

−H0

fu2g

2Dedz, (15)

here ug,b is the air velocity at the point b. For the droplets withsize distribution, the size distribution can be described by N

ntervals, and then, Eq. (15) becomes

�pz

ρg= u2

g − u2g,b

2+

N∑i=1

∫ z

0miug

(dul

dz

)i

dz

+∫ z

0

(m + 1)fu2g

2Dedz +

∫ 0

−H0

fu2g

2Dedz. (16)

The friction factor f can be determined using the total pressurerop by a numerical technique known as the shooting technique,.e., the model shows good agreement with the observed totalressure drop with the suitable f.

The aforementioned equations constitute the mathematicalodel for the spray evaporator that is used to concentrate the

ilute sulfuric acid. The fourth-order Runge–Kutta numericalechnique is used to integrate the differential equations alonghe length of the evaporator. The Newton–Raphson techniqueogether with adaptive Simpson quadrature is used to solve theonlinear algebraic equations. The physical properties that aresed in solving the model are provided in the literature [4,11,12].

. Results and discussion

Experiments are performed to study the influence of inletir temperature, inlet air flow rate and inlet liquid flow rate onroduct quality. The inlet air temperature are varied from 573o 633 K over inlet air flow rate range of 39.3–61.7 kg h−1 andnlet liquid flow rate range of 3.17–6.34 kg h−1. The inlet con-entration of the dilute sulfuric acid is unchangeably equal to8 wt%.

The experimental measurements, such as concentration ofutlet sulfuric acid, total pressure drop and outlet air tempera-ure, are compared with simulation results, which are shown inigs. 3–6 and Table 2. As can be seen from the data, the model isound to simulate the performance of the bench-scale evaporator

ery well in the range of operating conditions studied. It is evi-ent that increasing inlet air temperature and inlet air flow rateake for concentration of dilute sulfuric acid, while increasing

nlet liquid flow rate plays a negative effect. The dilute sulfuric

able 2imulation of outlet air temperature vs. experimental dataa

g,in (K) Fg,in = 39.3 kg h−1 Fg,in = 48.2 kg h−1

Tg,out,exp (K) Tg,out,cal (K) Tg,out,exp (K) Tg,out,cal (K)

73 346 338 346 34603 349 344 347 35233 358 363 372 372

a Fl,in = 5.28 kg h−1.FF

ig. 4. Effect of inlet air flow rate on concentration of outlet sulfuric acid

g,in = 573 K.

cid with concentration of 18 wt% can be easily concentrated to0–70 wt%, and further up to 75 wt%. The total pressure dropn the concentrator is low and varies from 400 to 1700 Pa. Theriction factor f equals to 0.032 because it gives the best fit inhe shape of the total pressure drop curve.

In order to illustrate the performance of the spray evapo-

ig. 5. Effect of inlet liquid flow rate on concentration of outlet sulfuric acid

g,in = 39.3 kg h−1.

1314 Z. Zhou, Z. Liu / Chemical Engineering and Processing 46 (2007) 1310–1316

Fig. 6. Model predictions of total pressure drop compared to the experimentaldata Tg,in = 573 K, Fl,in = 5.28 kg h−1.

Table 3Droplet size distribution of liquid droplets

Diameter range (�m) Average diameter (�m) f(dp)a (%)

10–30 20 24.130–50 40 23.450–70 60 17.770–90 80 12.690–110 100 8.8

110–130 120 6.1130–150 140 4.21

FdT

oddFciTta

dottv

ooTthe droplet size distribution at the inlet and the outlet of theconcentrator. The vertical axis represents the cumulative masspercentage of the droplets with diameter less than dp. It can be

50–170 160 3.0

a f (dp) = nidp,i/∑N

i=1nidp,i, i denotes the ith group of droplets.

l,in = 5.28 kg h−1 and Fg,in = 39.3 kg h−1. We describe theroplet size distribution by eight intervals, which is listed inable 3.

The variation of the liquid droplet diameter along the lengthf the concentrator is presented in Fig. 7. The diameter of smallerroplets is found to decrease faster than that of larger dropletsue to much higher specific surface area of the smaller droplet.ig. 8 shows the simulation of the droplet velocity along theoncentrator. It is shown that the droplet velocity drastically

ncreases firstly, and then decreases after arriving at the apex.his can be explained by the drag force acting on the droplet. In

he beginning, ul < ug, the drag force is an impetus for the dropletnd accelerate its movement. On the contrary, when ul > ug, the

Fig. 7. Variation of the droplet diameter along the concentrator. F

Fig. 8. Variation of the droplet velocity along the concentrator.

rag force represents a resistance and decelerates the movementf the droplet correspondingly. Moreover, it can be deducted thathe residence time of the smaller droplet is shorter than that ofhe larger one because the inertia of larger droplets keep theirelocity for a longer time.

Fig. 9 presents the variation of the proportion of water evap-rated for each group of droplets along the concentrator. Obvi-usly, the smaller the droplet is, the faster evaporation rate is.his result is consistent with that given by Fig. 7. Fig. 10 shows

Fig. 9. Percent evaporation along the concentrator for various droplets.

ig. 10. Droplet size distribution at the inlet and the outlet of the concentrator.

Z. Zhou, Z. Liu / Chemical Engineering and Processing 46 (2007) 1310–1316 1315

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ca

thdpca7

atbutbuaIh

ccs(c

tohfc

odiiT(eihiprl

5

ig. 11. Variation of air temperature and absolute humidity along the concen-rator.

learly seen that the droplet size distribution becomes narrowfter water is evaporated from the droplet.

From the simulation results mentioned above, it is knownhat the smaller droplet has the faster evaporator rate, and henceas the higher concentration in the end. However, the smallerroplet is apt to be entrained by air. Moreover, sulfuric acid isrobably evaporated from the smaller droplet due to its higheroncentration. Therefore, preferred droplet sizes are between 40nd 100 �m, corresponding to the final droplet concentration of7 and 46 wt%, individually.

The simulation of air temperature and absolute humiditylong the spray evaporator is shown in Fig. 11. Air tempera-ure decreases at first, and then it remains constant. This cane explained by the fact that, the hot air transfers heat to liq-id droplets for their evaporation and heating-up, while its ownemperature accordingly decreases. When mass and heat balanceetween air and droplets are attained, air temperature appearsnchanged. It should be noted that the absolute humidity ofir presents an opposite trend compared with air temperature.ts value at the outlet is 0.0887, and the corresponding relativeumidity can be further calculated, which is equal to 33.5%.

Fig. 12 presents the variation of droplet temperature along theoncentrator. The faster temperature rise of the smaller dropletompared with the larger one can be attributed to the higher

pecific surface area of the former according to Eqs. (8) and9). It is found that the decrease part of the droplet temperatureurve shown in Fig. 11 is quite similar with the counterpart of

Fig. 12. Variation of droplet temperature along the concentrator.

za4tiamtvamaa

A

CC

Fig. 13. Variation of pressure drop along the concentrator.

he gas temperature curve in Fig. 12, and the outlet temperaturesf air and droplets are identical. This indicates that mass andeat balance between air and droplets are attained in successionor different groups of droplets at different locations along theoncentrator.

Fig. 13 depicts the pressure drop along the concentrator. Theperating condition is the same as that of Fig. 6. The pressurerop of the spray evaporator shows the similar profile as it doesn the Venturi-type nozzle [10], i.e., the pressure drop increasesn the converging section and decreases in the diverging section.hough the pressure drop in the converging section seems high

caused by an increase in air velocity), it can lower to a largextent in the diverging section (caused by a decrease in air veloc-ty), and so, the total pressure drop of the spray evaporator is notigh actually. This is attractive from the point of power savingn plant operation. Compared with the converging section, theressure drop in the throat section increases very slowly. This iseasonable considering the unchanged air velocity and the shortength of the throat section.

. Conclusions

In this work, a spray evaporator with a Venturi-type noz-le is used to concentrate the dilute sulfuric acid. The dilutecid with initial concentration of 18 wt% can be concentrated to0–75 wt% in the bench-scale concentrator by one-step opera-ion. It is shown that either an increase in inlet air temperature andnlet air flow rate or a decrease in inlet liquid flow rate is favor-ble for concentration of dilute sulfuric acid. A one-dimensionalathematical model is developed to describe the performance of

his kind of concentrator. Variations of droplet diameter, dropletelocity, droplet temperature, pressure drop, air temperature andbsolute humidity along the concentrator are simulated by thisodel. Comparing the predictions with experimental data, such

s outlet concentration of sulfuric acid, outlet air temperaturend total pressure drop, a good agreement is observed.

ppendix A. Nomenclature

0 inlet sulfuric acid concentration (wt%)D drag coefficient

1 ing an

CCCdDDfFgMnNp�

QRTuW

Xz

Gγ

λ

μ

ρ

SgilopV

A

s

Q

wkse

n

S

Q

Tt

wTlkt

Sfi

R

[

[uids, fourth ed., McGraw-Hill, New York, NY, 1987.

316 Z. Zhou, Z. Liu / Chemical Engineer

g specific heat capacity of dry air (J kg−1 K−1)l specific heat capacity of sulfuric acid (J kg−1 K−1)V specific heat capacity of water vapor (J kg−1 K−1)p droplet diameter (m)e equivalent diameter of concentrator (m)g diffusion coefficient of water in air (m2 s−1)

friction factormass flow rate (kg h−1)gravitational constant (m2 s−1)molecular weight (g mol−1)number of droplets in size groupnumber of size groups for dropletspressure (Pa)

p pressure drop (Pa)enthalpy of air (J kg−1)

g universal gas constant (J mol−1 K−1)temperature (K)velocity (m s−1)weight of evaporated water per unit weight of sulfuricacid solution (kg kg−1)absolute humidity of air (kg kg−1)axial coordinate in the flow direction (m)

reek letterslatent heat of vaporization of water (J kg−1)thermal conductivity of air (J m−1 s−1 K−1)viscosity (Pa s)density (kg m−3)

ubscriptsgas (air)

n inlet of the concentratorliquid (sulfuric acid)

ut outlet of the concentratordropletvapor

ppendix B. The concentration heat of sulfuric acid

The dilution heat of sulfuric acid was experimentally mea-ured as follows [13]:

d = 83809n

n + 2.17, (A.1)

here Qd is the dilution heat of sulfuric acid,J (kmol sulfuric acid)−1. n is the molar ratio of water toulfuric acid in the sulfuric acid solution, which can be

xpressed in terms of the mass concentration of sulfuric acid

= 98

18· 100 − C

C. (A.2)

[

[

d Processing 46 (2007) 1310–1316

ubstitution of Eq. (A.1) into Eq. (A.2) gives

d = 83809 × 98 × (100 − C)

98 × (100 − C) + 39.06C. (A.3)

herefore, variation of the dilution heat of sulfuric acid alonghe axial direction is given by

dQd

dz= dQd

dC.dC

dz= −83809 × 98 × 3906

(9800 − 58.94C)2 · dC

dz, (A.4)

here the unit of dQd/dz is kJ m−1 (kmol sulfuric acid)−1.o facilitate computer programming for numerical simu-

ation, it would be better for the unit of dQd/dz to beJ m−1 (kg sulfuric acid solution)−1. Based on this considera-ion, Eq. (A.4) is rearranged as follows:

dQd

dz= −83809 × 98 × 3906

(9800 − 58.94C)2 .C

98 × 100.dC

dz

= − C

(5.416 − 0.033C)2 .dC

dz. (A.5)

ubstituting dQc/dz = −dQd/dz and C = C0/(1 − W) in Eq. (A.5)nally gives

dQc

dz= C2

0

(1 − W)(5.416 − 5.416W − 0.033C0)2 · dW

dz. (A.6)

eferences

[1] K. Bodenbenner, H.V. Plessen, R. Steiner, Regeneration of spent sulfuricacid, Int. Chem. Eng. 20 (3) (1980) 343–351.

[2] G.M. Smith, E. Mantius, The concentration of sulfuric acid, Chem. Eng.Prog. 74 (9) (1978) 78–83.

[3] J.Z. Shan, Y.J. Chen, Technologies of concentration of dilute sulfuric acidin China, Sulfur Acid Ind. (5) (1983) 38–43 (in Chinese).

[4] R.H. Perry, Perry’s Chemical Engineers’ Handbook, sixth ed., McGraw-Hill, New York, NY, 1984.

[5] X.R. Fu, Y.J. Fu, Study on the evaporation mechanism and mathematicalmodel for a new fashioned spray evaporator, J. Chem. Eng. Chin. Univ. 13(1) (1999) 17–24 (in Chinese).

[6] Z.M. Zhou, Z.H. Liu, D.W. Zhao, Experimental study on the concentrationof dilute sulfuric acid by spray evaporator, Chem. Ind. Eng. 18 (2) (2001)115–119 (in Chinese).

[7] G.B. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York,NY, 1969.

[8] W.E. Ranz, W.R. Marshall, Evaporation from drops, Chem. Eng. Prog. 48(3) (1952), pp. 141–146, 173–180.

[9] W.R. Marshall, Heat and mass transfer in spray drying, Trans. ASME 77(1955) 1377–1385.

10] R.H. Boll, Particle collection and pressure drop in Venturi scrubbers, Ind.Eng. Chem. Fundam. 12 (1) (1973) 40–50.

11] R.C. Reid, J.M. Prausnitz, B.E. Poling, The Properties of Gases and Liq-

12] C.L. Yaws, Chemical Properties Handbook, McGraw-Hill, New York, NY,1999.

13] Z.L. Cai, Computation of thermal effect for sulfuric acid, Sulfur Acid Ind.(2) (1981) 13–14 (in Chinese).