med simplified model

8/10/2019 MED Simplified Model

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Multi Effect Evaporative desalination (FF) MODEL

Brine

mf, Ti-1, Xcwmf, Ti, Xcw

Di, Tvi

Bi, Ti, Xi

Bi, Ti, Xi

di, Tvidi, Tvi

di+di, Tci

Bi-1, Ti-1, Xi-1

Di-1, Tc,i-1

Simplified mathematical model will be covered in this session. The detailed model will be

worked out as a course project (Constant HT rate per effect).

Data generated are related to:

Brine and distillate flow rates.

Brine concentration.

Temperature.

Heat transfer area.

Heat and material balances for flash boxes and preheaters are excluded in this model .

Assumptions:

Constant specific heat, Cp, for the seawater at different temperature and concentration.

Constant thermodynamic losses in all effects.

Constant heat transfer area in all effects.

No vapor flashing takes place inside the effects.

Feed seawater is at the saturation temperature of the first effect.

Equal thermal loads in all effects.

The formed vapors are salt free.

The driving force for heat transfer in the effect is equal to the difference of the condensation and

evaporation temperatures.

Energy losses to the surroundings are negligible.

The number of material and energy balance equations, which can be written for each effect, is three.

Seawater is modeled as a binary mixture of fresh water and salt.


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There are n equation for the heat transfer rate in each effect, which relates the effect thermal load to the

area, overall heat transfer coefficient, and temperature driving force.

Brine flow rates, B1, B2, ..., Bn-i, Bn n unknowns

Brine concentration, X1, X2, ... , Xn-1 n-1 unknowns

Distillate flow rate, D1, D2, ..., Dn-1, Dn n unknowns

Effect temperature, T1, T2, ..., Tn-1 n-1 unknowns

Steam flow rate 1 unknowns

Heat transfer area 1 unknowns

Total 4n unknowns

Parameters to be specified before solution:

a.

Temperature of the motive steam, Tg.

b. Vapor temperature in effect n, Tn.

c.

Salt concentration in the brine stream leaving effect n, Xn.d.

Salt concentration in the feed stream, Xf.

e.

Total distillate flow rate, Md.

Brine Brine Brine Brine

From

boiler

To

boiler

Rejected brine

Product line

Down Condenser

Feed seawater Rejected seawater

Seawaterin

Model Equations

Mass Balance: f= d+ Bn (1)

Xff= XnBn (2)

Then, ( ) d (3)

Parameters at RHS are known, then we can calculate Bnand then Mf(using eq. 1). This process is

not included in the iterative process.

Temperature Profile

q1= q2=..=qn-1=qn (4)

Note q1= shfg,s (5)

Brine

Di-1, Tci-1

Di, Tvi

Di-1, Tci-1

Bi,Xi, Ti

Bi-1,Xi-1, Ti-1

Effect i


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and qi= Dihfg,vi (i = 2 to n) (6)

Note that

q is the thermal load, qi= AiUiTi (7)

S is the mass flow rate of the motive steam,

Di is the distillate flow rate at effect i,

hfg,s is the steam latent heat at Ts,

hfg,vi is the steam latent heat at temperature (TiTloss)

Subscripts:

i: effect

s: steam

v: formed vapor

Remember that the heat transfer rate and the area are equal in each effect (for practicality), then

q1/A1= q2/A2= ..= qn-1/An-1=qn/An (8)

Therefore

U1T1= U2T2=..= Un-1Tn-1= UnTn (9)

Total temperature drop: T = Ts- Tn (10)

Tn: temperature of formed vapor at the last stage (n).This drop is equal for each effect;

T =T1+T2+ ..+Tn-1+ Tn (11)

From equation (9):

T2= (U1/ U2) T1 (12)

T3= (U2/ U3) T2= (U2/ U3) (U1/ U2) T1 = (U1/ U3) T1 (13)

Generally;

Ti= (U1/ Ui) T1 (14)

Substitute in (11)

T= T1U1(1/ U1+1/ U2+ 1/Un-1+1/Un) (15)


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Or: 1 () (16)

Actual Temperature profiles:

T1= TsT1 (17)

Ti= Ti-1 - (U1/ Ui) T1 (18)

Salt Concentration Profile

Distillate flow rate:

d= D1+ D2+ ..+Dn (19)

Dihfg,vi= Di-1hfg,v,i-1 ( i = 2,..,n) (20)

Then, we can write

D2= D1hfg,v1/ hfg,v2 (21)

D3= D2hfg,v2/ hfg,v3= D1hfg,v1/ hfg,v2(hfg,v2/ hfg, v3) = D1hfg, v1/ hfg, v3 (21a)

Di= D1hfg,v1/ hfg,vi (I = 2, .,n) (22)

Substitute in 19

d= D1+ D1hfg,v1/ hfg, v2+..+D1hfg, v1/ hfg, vn-1+ D1hfg,v1/ hfg, vn (23a)

d= D1hfg,v1(/hfg, v1+/ hfg,v2+.+/ hfg,vn) (23b)

D1=d/( hfg, v1(/hfg, v1+/ hfg, v2+.+/ hfg,vn)) (23)

Accordingly, we can write:

D2= D1hfg, v1/ hfg, v2 , D3= D1hfg,v1/ hfg, v3 , Dn= D1hfg, v1/ hfg,n

Brine flow rate at the first effect:

B1= fD1 (24)

Bi= i-1Di (25)

Similar Salt balances are obtained

X1= Xff/B1 (26)


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Xi= Xi-1Bi-1/Bi (27)

Heat Transfer Area

For the first effect:

1 hfg,vi (28)

hfg,vi (29)

Tlossrepresents the thermodynamic losses in each effect (0.5 to 3oC)

Convergence criteria and setting up new iteration

It is based on maximum difference in heat transfer area

Amax= max (Ai+1Ai) with i =1, n-1 (30)

IF the error is higher than the tolerance, a new estimate ofTiis made.

Ti= TiAi/Am (Amaverage heat transfer area) (31)

(32)

Then iterations continue by calculating:

The temperature profile, Ti, in effects 1 to n from Eqs. 17 and 19.

The distillate flow rate in the first effect, D1, Eq. 23.

The distillate flow rates in effects 2 to n, D i, Eq. 21.

The brine flow rate in the first effect, B 1, Eq. 24.

The brine flow rates in effects 2 to n, B i, Eq. 25.

The salt concentration in the first effect, X1, Eq. 26.

The salt concentration effects 2 to n, Xi, Eq. 27.

The heat transfer area in effects 1 to n, A i,Eqs. 28 and 29.

Then, convergence criteria is checked and iterations are made till the convergence criteria is realized.

Performance ParametersPerformance ratio PR = d/s (33)

Steam flow rates = D1hfg,v1/ hfg,s (34)

Specific heat transfer area +d (35)

Where Ac= qc/(UcLMTD) (36)


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(37)

Condenser thermal load in the condenser , Qc

qc= Dnhfg,vn (38)

Specific cooling water flow rate

scw= cw/d (39)

Cooling water mass flow rate is obtained from condenser heat balance

qc= Dnhfg,vn = (f+cw) cP(TfTcw) (40)

It should be noted that Tfis the feed seawater temperature leaving the condenser into the first effect.

1

=1 , ,

1

=2

med simplified model

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