Question

A solution containing 35% by mass of Glauber salt, Na2SO4 is cooled down from 323 K to 298 K in a Swenson-Walker crystalliser to form crystals of Na2SO4•10H2O. The solubility of Na2SO4at 298 K is 13.9 kg/100 kg of water, and the required product rate of crystals is 0.1 kg/s. The molecular mass of the anhydrous and hydrated salt is 142 kg/kmol and 322 kg/kmol, respectively. The mean heat capacity Cp of the solution is 2 kJ/(kg K), and the heat of crystallisation is 150 kJ/kg. The cooling water enters and leaves at 285 K and 293 K, respectively, and the overall coefficient of heat transfer is 125 W/m2 K. Assume counter- current flow and that 3% of the initial solution is evaporated.

(i) State the general equation for the calculation of the crystal yield and define the variables used.

(ii) Calculate the crystal yield.
(iii) Calculate the required feed to produce 0.1 kg/s of crystals. (iv) Calculate the heat required to cool the solution.

(v) Calculate the heat of crystallisation and the total heat required. (vi) Calculate the required heat transfer area.

Answer 1

Part i

General equation of crystal yield

y = R * Xw * [C1 - C2(1 - E)] / [1 - C2(R - 1)]

Where y = crystal yield

R = molecular mass of hydrate / molecular mass of anhydrate

= 322/142

= 2.268

Xw = mass fraction of water = 1 - 0.35 = 0.65

C1 = 0.35 / (1 - 0.35) = 0.538

C2 = 13.9/100 = 0.139

E = 0.03

Part ii

y = 2.268 * 0.65 * [0.538 - 0.139(1 - 0.03)] / [1 - 0.139(2.268 - 1)]

y = 1.4742 * (0.40317) / (0.823748)

= 0.722 kg

Part iiii

Required feed = mass of solution*required product rate/y

= 1 x 0.1 / 0.722

= 0.139 kg

Part iv

Heat required to cool the solution = mass x Cp x (T2-T1)

= 0.139 kg/s x 2 kJ/kg-K x (323 - 298)K

Q1 = 6.93 kW

Part v

Heat of crystallization in kW = required product rate * heat of crystallization

= 0.1 kg/s x 150 kJ/kg

Q2 = 15 kW

Total heat required = Q1 + Q2

= 6.93 + 15

Q = 21.93 kW

Part vi

For counter flow

∆T1 = 323 - 293 = 30 K

∆T2 = 298 - 293 = 5 K

LMTD = (30 - 5) / ln (30/5)

= 13.95 K

Area required

A = Q/U*LMTD

= 21.93*1000W / (125 W/m2-K * 13.95K)

= 12.58 m2