Pressure screen Mass balance and Rejects ratio calculations

 Objective:

To Calculate “Pressure Screening system mass balance” and “Rejects ratio” for multi-stage cascaded screening system.

 
Introduction:

The Primary purpose of Pressure screens is to remove contaminants from pulp stock such as shives, knots, flakes, plastics, adhesives etc. Pressure screen are used for fractionating the fibers as well based on the fiber length like long fiber and short fiber. Every source of pulp has its own characteristic type of contaminants; hence, Pressure screens are applied in Pulp mill, Secondary fiber treatment, Paper machine stock and approach flow system and in Broke handling system. 

Pressure screen process parameters:

Parameter		Unit	Feed	Accepts	Rejects
Volumetric flow rate	V	l/min	VF	VA	VR
Consistency	C	%	CF	CA	CR
Mass flow rate	M	kg/min	MF	MA	MR

 

Formula for various process control parameters:

In order to calculate the mass balance of a multi stage pressure screening system, we need to establish the following parameters and the same will be used to control the quality of pulp and runnability of the system.

Formula-1 (Stock Consistency)

Consistency (%) is given by = (mass flow rate x 100) ÷ (volumetric flow rate)

C = M x 100 ÷ V
 

Formula-2 (Volumetric Rejects ratio)

“Volumetric Rejects ratio” is defined as the ratio between “Volumetric flow rate of Rejects” and the “Volumetric flow rate of Feed”

RR_v = V_R ÷ V_F

Formula-3 (Mass Rejects ratio)

“Mass Rejects ratio” is defined as the ratio between “Mass flow rate of Rejects” and the “Mass flow rate of Feed”

RR_M = M_R ÷ M_F

Formula-4 (Rejects thickening factor)

“Rejects thickening factor” indicates how rejects consistency is thickened compared to the feed consistency; it is an important factor which affects the cleanliness, stock freeness and runnability of the screen. It is calculated by dividing “Mass rejects ratio” by “Volumetric rejects ratio”

TF = RR_M ÷ RR_v

also can be calculated by

TF = C_R ÷ C_F

Formula-5 (Passing ratio)

“Passing ratio” indicates the performance of the screen; the quality of pulp can be controlled by mass and volumetric reject ratio as both the Rejects thickening factor and Passing ratio are the functions of these rejects ratios.

P = log (RR_M) ÷ log (RR_V)

Formula-6 (Rejects ratio of Multi-stage screening system)

For multi-stage screening system which is connected in series, the total system rejects ratio is calculated by multiplying the reject ratio of each individual screen. This is applicable for both Volumetric and Mass reject ratios.

RR (total) = RR₁ x RR₂ X RR₃ …

Formula-7 (Rejects thickening factor of Multi-stage screening system)

The 'total rejects thickening factor' for a multi-stage screening system (connected in series) is calculated by dividing the 'total mass rejects ratio' by 'total volumetric rejects ratio'.

TF (total) = RR_M (total) ÷ RR_V (total)

Formula-8 (Passing ratio of Multi-stage screening system)

It is calculated as follows

P (total) = log [RR_M (total)] ÷ log [RR_V (total)]

Mass balance calculation:

In general mass balance calculation starts from the demand from the system accepts side (output) which will be the feed to the next process module. In this case, we consider the pressure screen system primary stage fed at 100 bdtpd at 1.1% consistency rate, and we can calculate the system balance as follows

Calculation for Primary stage screen:

Let us convert bdtpd rate to kg/min rate
Feed of Primary stage 100 bdtpd = (100 x 1000 ÷ 1440) = 69.44 kg/min

So using the above kg/min and consistency data of feed, we can calculate the volumetric flow rate of feed using Formula-1

V_F = M_F x 100 ÷ C_F

V_F = 69.44 x 100 ÷ 1.0 = 6944.44 l/min

So the ‘Feed’ parameters are

M_F = 69.44 kg/min
C_F = 1.1 %
V_F = 6944.44 l/min

In order to control the system, the operator gives few set points mainly on reject ratios as at optimum reject ratios the screen gives optimum accepts pulp freeness and cleanliness; when the rejects ratio lowered the cleanliness of accepts pulp will become poor and when the rejects ratio is more than optimum, the screen and rejects line will get plugged and further stage screen load will increase. Pressure screen vendor normally supply the rejects ratio ranges and this is set during the operation by ‘trial-and-error correction’ method to achieve better runnability.

Let us assume the RR_V and RR_M for Primary stage as 15% and 22% respectively

So with the above we can calculate the Mass and volumetric flow rate of Primary stage Rejects by multiplying the feed flow rates with corresponding reject ratios as follows

M_R = M_F x RR_M

M_R = 69.44 x 22% = 15.28 kg/min

V_R = V_F x RR_V

V_R = 6944.44 x 15% = 1041.67 l/min

And calculating Rejects consistency using Formula-1

C_R = M_R x 100 ÷ V_R

C_R = 15.28 x 100 ÷ 1041.67 = 1.47

So the ‘Rejects’ parameters are

M_R = 15.28 kg/min
C_R = 1.47 %
V_R = 1041.67 l/min

The accepts parameters can simply be calculated by subtracting the Rejects parameters from the Feed

M_A = M_F – M_R = 69.44 – 15.28 = 54.17 kg/min

V_A = V_F - V_R = 6944.44 – 1041.67 = 5902.78 l/min

So the C_A = M_A x 100 ÷ V_A = 54.17 x 100 ÷ 5902.78 = 0.92

So the ‘Accepts’ parameters are

M_A = 54.17 kg/min
C_A = 0.92
V_A = 5902.78 l/min

Rejects Thickening factor is calculated as follows using Formula-4

TF = RR_M ÷ RR_v

TF = 22% ÷ 15% = 1.5

Passing ratio is calculated as follows using Formula-5

P = log (RR_M) ÷ log (RR_V)

P = log (0.22) ÷ log (0.15) = 0.8

Similarly, the Secondary and Tertiary stage screen Feed, Accepts and Rejects parameters are calculated by assuming the rejects ratios for each stage and finally the total Reject ratios and total Thickening factor and passing ratio are calculated using the formula given above. When the system is cascaded, the accepts from the Secondary will be fed back to Primary and its rejects to Tertiary; and Tertiary accepts will be fed to Secondary as shown in the Block flow diagram

The following table is prepared to show a typical three stage cascaded screening system. The highlighted parameters are assumed for the calculation

Code	Stream	Volumetric flow rate (l/min)	Consistency (%)	Mass flow rate (kg/min)	Rejects rate by Volume	Rejects rate by mass	Rejects thickening factor	Passing ratio
		V	C	M	RRV	RRM	TF	P
	Primary
F	Feed from previous process module	5242.10	1.06	55.56
A2	Accepts from Secondary	1702.34	0.82	13.89
F1	Total Feed	6944.44	1.00	69.44
A1	Accepts to next process module	5902.78	0.92	54.17
R1	Rejects to Secondary	1041.67	1.47	15.28	15%	22%	1.5	0.80
	Secondary
R1	Rejects from Primary	1041.67	1.47	15.28
A3	Accepts from Tertiary	484.93	0.67	3.24
D2	Dilution	500.00
F2	Feed	2026.60	0.91	18.52
A2	Accepts to Primary	1702.34	0.82	13.89
R2	Rejects to Tertiary	324.26	1.43	4.63	16%	25%	1.6	0.76
	Tertiary
R2	Rejects from Secondary	324.26	1.43	4.63
D3	Dilution	260.00
F3	Feed	584.26	0.79	4.63
A3	Accepts to Secondary	484.93	0.67	3.24
R3	Rejects	99.32	1.40	1.39	17%	30%	1.8	0.68
	Total				0.4%	1.7%	4.0	0.75