Estimate the number of ideal stages needed in the butane-pentane splitter defined by the compositions given in the table below. The column will operate at a pressure of 8.3 bar, with a reflux ratio of 2.5. The feed is at its boiling point. Note: a similar problem has been solved by Lyster et al.

Question

Estimate the number of ideal stages needed in the butane-pentane splitter defined by the compositions given in the table below. The column will operate at a pressure of 8.3 bar, with a reflux ratio of 2.5. The feed is at its boiling point.

Note: a similar problem has been solved by Lyster et al. (1959) using a rigorous computer method and it was found that ten stages were needed.

	Feed (f)	Tops (d)	Bottoms(b)
[latex] ext { Propane, } C _{3}[/latex]	5	5	0
[latex] ext { i-Butane, } iC _{4}[/latex]	15	15	0
[latex] ext { n-Butane, } nC _{4}[/latex]	25	24	1
[latex] ext { i-Pentane, } i C _{5}[/latex]	20	1	19
[latex] ext { n-Pentane, } nC _{5}[/latex]	35	0	35
	100	45	55 kmol

Accepted Answer

The top and bottom temperatures (dew points and bubble points) were calculated by the methods illustrated in Example 11.9. Relative volatilities are given by equation 8.30:

[latex]\alpha_{i}=\frac{K_{i}}{K_{u}}[/latex]

Equilibrium constants were taken from the Depriester charts (Chapter 8).

Relative volatilities

[latex] ext { Temp. }{ }^{\circ} C[/latex]	Top	Bottom	Average
[latex] ext { Temp. }{ }^{\circ} C[/latex]	65	120
[latex]C _{3}[/latex]	5.5	4.5	5.0
[latex]iC _{4}[/latex]	2.7	2.5	2.6
[latex]( K ) nC _{4}[/latex]	2.1	2.0	2.0
[latex] ext { (HK) } iC _{5}[/latex]	1.0	1.0	1.0
[latex]nC _{5}[/latex]	0.84	0.85	0.85

Calculations of non-key flows

Equations 11.50, 11.51, 11.52, 11.53

[latex]\underline{l_{i}}=\frac{d_{i}}{\alpha_{i}-1}[/latex] (11.50)

[latex]\underline{v_{i}}=l_{i}+d_{i}[/latex] (11.51)

[latex]\underline{v}_{i}^{\prime}=\frac{\alpha_{i} b_{i}}{\alpha_{ LK }-\alpha_{i}}[/latex] (11.52)

[latex]\underline{l}_{i}^{\prime}=v_{i}^{\prime}+b_{i}[/latex] (11.53)

	[latex]\alpha_{i}[/latex]	[latex]d_{i}[/latex]	[latex]\underline{l_{i}}=d_{i} /\left(\alpha_{i}-1 ight)[/latex]	[latex]\underline{\underline{v_{i}}}=\underline{l_{i}}+d_{i}[/latex]
[latex]C _{3}[/latex]	5	5	1.3	6.3
[latex]iC _{4}[/latex]	2.6	15	9.4	24.4
		[latex]\Sigma \underline{l_{i}}=10.7[/latex]	[latex]\Sigma \underline{v_{i}}=30.7[/latex]
	[latex]\alpha_{i}[/latex]	[latex]b_{i}[/latex]	[latex]v_{i}^{\prime}=\alpha_{i} b_{i} /\left(\alpha_{ LK }-\alpha_{i} ight)[/latex]	[latex]\underline{v_{i}}=\underline{l_{i}}+d_{i}[/latex]
[latex]nC _{5}[/latex]	0.85	35	25.9	60.9
			[latex]\Sigma \underline{v_{i}^{\prime}}=25.9[/latex]	[latex]\Sigma \underline{l_{i}^{\prime}}=60.9[/latex]

Flows of combined keys

[latex]L_{e}=2.5 imes 45-10.7=101.8[/latex] (11.46)

[latex]V_{e}=(2.5+1) 45-30.7=126.8[/latex] (11.47)

[latex]V_{e}^{\prime}=(2.5+1) 45-25.9=131.6[/latex] (11.49)

[latex]L_{e}^{\prime}=(2.5+1) 45+55-60.9=151.6[/latex] (11.48)

Slope of top operating line

[latex]\frac{L_{e}}{V_{e}}=\frac{101.8}{126.8}=0.8[/latex]

Slope of bottom operating line

[latex]\frac{L_{e}^{\prime}}{V_{e}^{\prime}}=\frac{151.6}{131.6}=1.15[/latex]

[latex]x_{b}=\frac{ ext { flow LK }}{ ext { flow }( LK + HK )}=\frac{1}{19+1}=0.05[/latex]

[latex]x_{d}=\frac{24}{24+1}=0.96[/latex]

[latex]x_{f}=\frac{25}{25+20}=0.56[/latex]

[latex]y=\frac{2 x}{1+(2-1) x}=\frac{2 x}{1+x}[/latex] (11.23)

x	0	0.20	0.40	0.60	0.80	1.0
y	0	0.33	0.57	0.75	0.89	1.0

The McCabe-Thiele diagram is shown in Figure 11.10.

Twelve stages required; feed on seventh from base.

	$\alpha_{i}$	$d_{i}$	$\underline{l_{i}}=d_{i} /\left(\alpha_{i}-1\right)$	$\underline{\underline{v_{i}}}=\underline{l_{i}}+d_{i}$
$C _{3}$	5	5	1.3	6.3
$iC _{4}$	2.6	15	9.4	24.4
		$\Sigma \underline{l_{i}}=10.7$	$\Sigma \underline{v_{i}}=30.7$
	$\alpha_{i}$	$b_{i}$	$v_{i}^{\prime}=\alpha_{i} b_{i} /\left(\alpha_{ LK }-\alpha_{i}\right)$	$\underline{v_{i}}=\underline{l_{i}}+d_{i}$
$nC _{5}$	0.85	35	25.9	60.9
			$\Sigma \underline{v_{i}^{\prime}}=25.9$	$\Sigma \underline{l_{i}^{\prime}}=60.9$

Question 11.5: Estimate the number of ideal stages needed in the butane-pen...

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