Engineering
Kremser Gas Absorption Factor (A) & Theoretical Scrubber Stages
Calculate the absorption factor and ideal equilibrium-stage estimate for a dilute countercurrent absorber, with input validation, a pinch check and stable evaluation near A = 1.
Use an ideal absorber model, not a complete equipment design
This example assumes dilute solute, constant liquid and gas molar flow rates on a consistent basis, and a straight equilibrium relation y* = m x with a supplied positive slope m. Inputs L and V must use the same molar-flow units; x and y are mole fractions, not percentages. Numerical fraction bounds alone do not establish that a mixture is dilute or that m is valid.
The result estimates equilibrium contacts. It does not determine real tray count, efficiency, packing height, column diameter, pressure drop or operating cost. Reactive, concentrated or strongly non-isothermal systems may need a different model. The absorption formula below must not be applied unchanged to stripping.
Validate the six inputs
Place this in B8. Require positive L, m and V; fractions from zero up to but excluding one; inlet gas concentration at least the target; and target gas concentration strictly above m times entering-solvent concentration. Equality with that solvent equilibrium level is a limiting case, not a finite result from this worksheet.
=IF(COUNT(B2:B7)<>6,"Enter six numeric inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<0,B5>=1,B6<0,B6>=1,B7<0,B7>=1),"Check flows and mole fractions",IF(OR(B5<B6,B6<=B3*B7),"Check absorption driving forces","OK")))Calculate the absorption factor and driving-force ratio
Place A = L/(mV) in B9 using this formula. Then place =(y_in − m x_in)/(y_out − m x_in) in B10 using the next formula. For the example A = 1.5 and R = 25. Keep raw values; do not round intermediate factors.
=IF(B8="OK",B2/(B3*B4),B8)Driving-force ratio R in B10
This ratio includes nonzero solute in the entering solvent. When inlet and target are equal, R = 1 and the modeled removal requirement is zero.
=IF(B8="OK",(B5-B3*B7)/(B6-B3*B7),B8)Evaluate the stage expression stably
The usual expression is N = LN(R(1−1/A)+1/A)/LN(A). At A = 1 its limit is R−1. Define z = (R−1)(A−1)/A in B11 using this formula. If z is at most −1, there is no finite solution for the specified target under this model; review the inputs and model rather than treating this as an equipment instruction.
=IF(B8="OK",(B10-1)*(B9-1)/B9,B8)Raw ideal stages in B12
This algebraically equivalent expression evaluates N = (R−1)/A × g(z)/g(A−1), where g(t)=LN(1+t)/t and g(0)=1. For |t| below 0.0001 it uses 1−t/2+t²/3−t³/4+t⁴/5 to avoid subtractive cancellation near A=1. This is a numerical approximation in that small interval, not unlimited precision.
The initial raw result is about 5.419022583 ideal stages. With L=160 it is about 3.700439718; with L=80 it is 24. With L=40 the target has no finite-stage solution. The displayed decimal precision does not establish engineering accuracy.
=IF(B8<>"OK",B8,IF(B11<=-1,"No finite stage solution",(B10-1)/B9*IF(ABS(B11)<0.0001,1-(B11)/2+(B11)^2/3-(B11)^3/4+(B11)^4/5,LN(1+(B11))/(B11))/IF(ABS((B9-1))<0.0001,1-((B9-1))/2+((B9-1))^2/3-((B9-1))^3/4+((B9-1))^4/5,LN(1+((B9-1)))/((B9-1)))))Whole ideal contacts, if needed
B13 rounds the raw estimate upward, rather than rounding to two decimals first. This gives 6 for the initial example. It remains an ideal-contact count, not a specification for six physical trays.
=IF(ISNUMBER(B12),ROUNDUP(B12,0),B12)Try the example
Use the original values or change them to test your understanding.