Badr Alghamdi
العربية
Tools

Contacting stages: a second ideal contact can add nothing

Two vessels can leave exactly the same recovery as one. Reversing one phase's direction changes the answer with the same total agent.

Live result

With one total agent quantity and extraction factor 2.4, 85% recovery appears at crosscurrent stage 5 and countercurrent stage 2; repeating cocurrent contact stays at 70.5882%.
With one total agent quantity and extraction factor 2.4, 85% recovery appears at crosscurrent stage 5 and countercurrent stage 2; repeating cocurrent contact stays at 70.5882%.

First stage meeting the comparison

Ideal equilibrium stages, one total solute-free agent stream, a linear equilibrium relation and constant phase flows. Crosscurrent contact divides that total agent equally among its stages.

Extraction factor: 2.4

Repeated cocurrent

Unreachable by repetition

Both phases travel together after the first equilibrium contact.

Equal-split crosscurrent

Stage 5

Each contact receives a fresh equal share of one total agent quantity.

Recovery at first crossing
85.9171%
Recovery one stage earlier
84.7412%

Countercurrent

Stage 2

The full agent stream travels opposite to the feed through the stages.

Recovery at first crossing
89.0830%
Recovery one stage earlier
70.5882%

Agent-phase concentration divided by feed-phase concentration at equilibrium. Use compatible concentration bases; from 0 through 100.

One total agent flow divided by feed flow on the same basis, from 0 through 100. This total stays fixed as stage count changes.

A comparison you supply, from 0 through 100. Counts are checked from 1 through 20; equality meets the comparison.

Recovered share of incoming solute, %

Recovery at fixed total agent quantityThree recovery curves across 1 through 20 ideal stages at extraction factor 2.4, holding total agent quantity fixed. The comparison is 85 percent. At 2 stages the cocurrent, equal-split crosscurrent and countercurrent recoveries are 70.5882, 79.3388 and 89.0830 percent. First crossings are above the fields; every plotted value is in the table below.

Ideal stages, whole count

CocurrentCrosscurrentCountercurrentTyped comparison
Values at every checked stage
Recovery percentages from the last valid input record; all three arrangements use one fixed total agent quantity.
StagesCocurrentCrosscurrentCountercurrent
170.5882%70.5882%70.5882%
270.5882%79.3388%89.0830%
370.5882%82.8532%95.6491%
470.5882%84.7412%98.2194%
570.5882%85.9171%99.2636%
670.5882%86.7190%99.6941%
770.5882%87.3004%99.8727%
870.5882%87.7411%99.9470%
970.5882%88.0865%99.9779%
1070.5882%88.3646%99.9908%
1170.5882%88.5931%99.9962%
1270.5882%88.7843%99.9984%
1370.5882%88.9466%99.9993%
1470.5882%89.0861%99.9997%
1570.5882%89.2073%99.9999%
1670.5882%89.3135%100.0000%
1770.5882%89.4074%100.0000%
1870.5882%89.4910%100.0000%
1970.5882%89.5659%100.0000%
2070.5882%89.6333%100.0000%

The calculation record

  1. Multiply equilibrium slope 3 by total agent/feed ratio 0.8: extraction factor 2.4.
    E=3×0.8=2.4E=3\times0.8=2.4
  2. At 2 stages, the recovered shares are 70.5882%, 79.3388% and 89.0830% in the arrangement order shown.
    R2,1=70.5882%,R2,2=79.3388%,R2,3=89.0830%R_{2,1}=70.5882\%,\quad R_{2,2}=79.3388\%,\quad R_{2,3}=89.0830\%R2,1=70.5882%,R2,2=79.3388%,R2,3=89.0830%\begin{gathered}R_{2,1}=70.5882\%, \\ R_{2,2}=79.3388\%, \\ R_{2,3}=89.0830\%\end{gathered}
  3. The countercurrent and repeated-cocurrent readings differ by 18.4947 percentage points at the same 2-stage count and total agent quantity.
    89.0830−70.5882≈18.494789.0830-70.5882\approx18.4947
  4. Check every whole count from 1 through 20 against 85%. Report the first count meeting or exceeding it; an absent crossing within this range is not a claim about larger counts.
    N∈{1,…,20},RN≥85%N\in\{1,\ldots,20\},\quad R_N\ge85\%N∈{1,…,20},RN≥85%\begin{gathered}N\in\{1,\ldots,20\}, \\ R_N\ge85\%\end{gathered}

Here is the maths

One balance, three arrangements

F and S are constant feed and total agent flow rates on compatible bases. The equilibrium relation uses feed-phase concentration x and agent-phase concentration y, with slope m. E is their dimensionless extraction factor, q is the fraction of incoming solute left in the feed, and N is a positive whole stage count.

E=mSF,y=mxE=m\frac{S}{F},\qquad y=mxE=mSF,y=mx\begin{gathered}E=m\frac{S}{F}, \\ y=mx\end{gathered}

For cocurrent contact, the first equilibrium split leaves one part in the feed for every E parts in the agent. A second stage receives those same equilibrated phases and repeats exactly the same outlet. This establishes the reported unreachability for any larger positive count.

qN=11+E(N≥1)q_N=\frac{1}{1+E}\quad(N\ge1)qN=11+E(N≥1)\begin{gathered}q_N=\frac{1}{1+E} \\ (N\ge1)\end{gathered}

For N fresh crosscurrent contacts, each receives only SN\frac{S}{N} of the total agent. Every contact multiplies the remaining fraction by the same factor. Keeping the full S at every contact would change the agent budget and answer a different question.

qN=(1+EN)−Nq_N=\left(1+\frac{E}{N}\right)^{-N}

For countercurrent contact, stage i receives feed from the preceding stage and agent from the following stage. Normalize incoming feed concentration to 1 and entering agent concentration to 0. The stage balances give the geometric sum below, including the non-singular case E=1E=1.

xi−1+Exi+1=(1+E)xi,x0=1,xN+1=0x_{i-1}+E x_{i+1}=(1+E)x_i,\qquad x_0=1,\quad x_{N+1}=0xi−1+Exi+1=(1+E)xi,x0=1,xN+1=0\begin{gathered}x_{i-1}+E x_{i+1}=(1+E)x_i, \\ x_0=1, \\ x_{N+1}=0\end{gathered}
qN=1∑j=0NEjq_N=\frac{1}{\sum_{j=0}^{N}E^j}
RN=100(1−qN)%R_N=100(1-q_N)\%

At E=1E=1, countercurrent recovery is NN+1\frac{N}{N+1}. At finite E, no finite count reaches exactly 100%, even when displayed rounding reads 100.0000%. At E=0E=0 the recovery is zero. A zero comparison first meets the first plotted count, 1.

E=1⇒RN=100NN+1%E=1\quad\Rightarrow\quad R_N=100\frac{N}{N+1}\%E=1⇒RN=100NN+1%\begin{gathered}E=1 \\ \Rightarrow \\ R_N=100\frac{N}{N+1}\%\end{gathered}

The opening record

The typed slope is 3 and the total agent/feed ratio is 0.8, giving E=2.4E=2.4. At 2 stages: cocurrent 70.588235294118%, crosscurrent 79.338842975207%, countercurrent 89.082969432314%.

For the 85% comparison, crosscurrent stage 4 gives 84.741210937500% and stage 5 gives 85.917121804860%. Countercurrent stage 1 gives 70.588235294118% and stage 2 gives 89.082969432314%. These adjacent values prove the first crossings.

E=3×0.8=2.4,R2=100(1−11+2.4+2.42)%=89.082969…%E=3\times0.8=2.4,\qquad R_2=100\left(1-\frac{1}{1+2.4+2.4^2}\right)\%=89.082969\ldots\%E=3×0.8=2.4,R2=100(1−11+2.4+2.42)%=89.082969…%\begin{gathered}E=3\times0.8=2.4, \\ R_2=100\left(1-\frac{1}{1+2.4+2.4^2}\right)\%=89.082969\ldots\%\end{gathered}

Source basis

Authored solute mass balances with a linear equilibrium relation. The independent reference calculation solves the countercurrent balance equations as an exact tridiagonal system; the browser port uses the closed-form geometric sum. No equipment table or published design correlation is used.

Assumptions
Stage count moves from 1 through 20. The typed equilibrium slope, one total agent/feed ratio and recovery comparison stay fixed. Curves connect whole-stage values; they do not compute fractional stages.
Model version
1.0.0

These ideal-stage balances describe the typed arrangements; they do not size or select equipment, establish real stage efficiency, or recommend a plant arrangement.

Related

Ideal contacting-stage comparison

Contacting stages: a second ideal contact can add nothing

Text as entered

Equilibrium slope, m
3Opening text
Total agent/feed ratio, S/F
0.8Opening text
Comparison recovery, %
85Opening text

First checked crossings

Extraction factor
2.4
Repeated cocurrent
Unreachable by repetition
Equal-split crosscurrent
Stage 5 · 85.9171%
Countercurrent
Stage 2 · 89.0830%

The calculation record

1
Multiply equilibrium slope 3 by total agent/feed ratio 0.8: extraction factor 2.4.
2
At 2 stages, the recovered shares are 70.5882%, 79.3388% and 89.0830% in the arrangement order shown.
3
The countercurrent and repeated-cocurrent readings differ by 18.4947 percentage points at the same 2-stage count and total agent quantity.
4
Check every whole count from 1 through 20 against 85%. Report the first count meeting or exceeding it; an absent crossing within this range is not a claim about larger counts.

The record uses the typed text shown here. Its flows and equilibrium assumptions are those stated on this sheet.

One balance, three arrangements
Ideal equilibrium stages, one total solute-free agent stream, a linear equilibrium relation and constant phase flows. Crosscurrent contact divides that total agent equally among its stages.

These ideal-stage balances describe the typed arrangements; they do not size or select equipment, establish real stage efficiency, or recommend a plant arrangement.

Model version 1.0.0 · 7 September 2026

Badr Alghamdi · badralghamdi.com/tools/contacting-stages