SPT: Formula
What is the formula for the q-line?
y = q/(q-1) · x - 1/(q-1) · z
What is the formula for the fraction of absorption (in case of ideal equilibrium stage)?
α= (y₀ − y₁)/(y₀ − y₀*) (see also: Kremser equation)
What is the Kremser equation (stripping)?
σ = (x₀ − xₙ)/(x₀ - xₙ₊₁*) = ... = (Sⁿ⁺¹ - S)/(Sⁿ⁺¹ - 1) with xₙ₊₁* = yₙ₊₁/m
How is the height term of a packed column built up?
∫₀ᴸ dz = −G/(K_y · a) · ∫_(y₁)^(y₂) dy/(y−y*) HTU: G/(K_y · a) NTU: - ∫_(y₁)^(y₂) dy/(y−y*) = ∫_(y₂)^(y₁) dy/(y−y*)
What is the formula for (L/G)ₘᵢₙ in case of absorption?
(L/G)ₘᵢₙ = (yₙ₊₁ - y₁)/(x*ₙ₊₁ - x₀) for a non-linear y=f(x). For the simple case y = m·x: (L/G)ₘᵢₙ = m·(yₙ₊₁ - y₁)/(yₙ₊₁ - y*₀) = m·α
What is the formula for (L/G)ₘₐₓ in case of stripping?
(L/G)ₘₐₓ is defined as the slope for an infinite number of stages and therefore describes the maximum slope of the operating line. Its calculated value is achieved by dividing the slope of the equilibrium line by the fractional stripping factor: (L/G)ₘₐₓ = m/σ
How can the feed stage be determined using Fenske's equation?
- assume, that the composition of the feed is the same as the composition at the feed stage - use Fenske's method, replacing the compositions of the bottoms by the composition of the feed, to get N_(rect. min) - get the number of stages for the rectification section, and therefore the position of the feed stage with: N_(rect. min)/Nₘᵢₙ = N_(rect.)/N
What are the requirements of the solute analysis by the Kremser equation?
1. Systems that have a linear equilibrium 2. Systems that are isothermal 3. Systems that are isobaric 4. Systems that have negligible heat of absorption 5. Systems that have constant flow rates and additionally for multisolute cases: 6. Solutes are independent of each other (→ dilute solutions).
What is the (first) Underwood equation?
1−q = ∑(αᵢ,ᵣ · zᵢ_F)/(αᵢ,ᵣ − φ) The method is applied to all the i-components of a distillation, under the column's condition of minimum reflux; keyword: pinch point. As for the solution φ, it is the value of φ staying between the α_LK and α_HK
What is the formula of the Absorption factor (in case of ideal equilibrium stage)?
A = L/(G · m)
What is the formula for Aₘᵢₙ in case of absorption?
Aₘᵢₙ = (L/G)ₘᵢₙ · 1/m With (L/G)ₘᵢₙ = (yₙ₊₁ - y₁)/(x*ₙ₊₁ - x₀) = m·(yₙ₊₁ - y₁)/(yₙ₊₁ - y*₀) = m·α
What is the formula for Murphree efficiency?
Eᵢ = (yᵢ₊₁ − yᵢ)/(yᵢ₊₁ − yᵢ*) < 1 ⇒ yᵢ₊₁ − yᵢ = Eᵢ · (yᵢ₊₁ − yᵢ*)
What is the formula for the overall stripping column efficiency E₀?
E₀ = Ntheoretical/Nactual The overall efficiency E₀ is defined as the ratio between the theoretical number of equilibrium stages to the actual number of stages required. Even if the overall efficiency E₀ is conceptually different from the Murphree efficiency, most often it is assumed that both have the same value.
What is the formula for the mass balance on the feed stage of a distillation column?
F + V' + L = V + L'
What is the formula for the heat balance on the feed stage of a distillation column?
F · h_f + V' · H + L · h = V · H + L · 'h (the CMO allows us to assume: H_V = H_V ′ and h_L = h_L′)
What is the formula for the Height Equivalent to a Theoretical Plate (HETP)?
HETP = (Packed height)/(Number of equivalent equilibrium stages)
What is the formula for the NTU of a structured packed (or counter-current) column?
NTU = A/(A−1) · ln((1 − α/A)/(1 − α)) = n · (A · ln(A))/(A - 1)
How do Murphree efficiency for liquid side differ from the one for vapor side?
Normally, they can be assumed the same, and the one of the reboiler ≈ 1
How is (L/V)ₘᵢₙ related to Rₘᵢₙ in a distillation column?
Thanks to the relationshop with the operating line of the rectifying section: (L/V)ₘᵢₙ = Rₘᵢₙ/(Rₘᵢₙ + 1) Rₘᵢₙ = (L/V)ₘᵢₙ/(1- (L/V)ₘᵢₙ)
What is the Fenske equation?
The Fenske equation in continuous fractional distillation is an equation used for calculating the minimum number of theoretical plates required for the separation of a binary feed stream by a fractionation column that is being operated at total reflux (i.e., which means that no overhead product distillate is being withdrawn from the column).
What is the (second) Underwood equation?
Vₘᵢₙ = ∑(Dxᵢ^D · αᵢ)/(αᵢ−ϕ) With ϕ=L/(V·Kᵣ)
How is the relative average volatility calculated?
for the sake of simplicity, it is possible to use a constant value valid for all the column stages. When this is not possible, an average value of αLK,HK among all the (N = Nmin) values must be then calculated, e.g. average between the value at the first stage and at the reboiler: α_average = √(α₁αᵣ)
What is the number of theoretical equilibrium stages n, which can be calculated with the Kremser Equation?
n = 1/ln(A) · ln((1 − α/A)/(1 − α))
What is the formula for the feed quality?
q = (L' - L)/F = (H - h_F)/(H - h) q - 1 = (V' - V)/F
What is the formula for the stripping line in a binary distillation?
y = L'/(L' - B) · x - B/(L' - B) · xᴮ y = L'/V' · x - (L'/V' - 1) · xᴮ, L′/V′ >1 Most often, it is just constructed visually as the connection between the intersection between the q-line and the rectifying line and the intersection of equilibrium line with the bottom composition.
What is the formula for the rectifying line in a binary distillation?
y = R/(R + 1) · x + 1/(R + 1) · xᴰ y = L/V · x + (1 - L/V)· xᴰ, 0 < L/V < 1
What is the equilibrium condition in ideal equilibrium stages?
yᵢ = m·xᵢ m = H(T)/P
What is the general difference equation and its solution?
yᵢ = γyᵢ₋₁ + δ has the solution: yᵢ =y₀γⁱ + δ·(1 - γⁱ)/(1 - γ)
What is the column/overall efficiency for real stages?
α = (yin − yout )/(yin − y₀*) (= amount removed/max. amount removed)
What is the Kremser equation (counter current absorption)?
α= (yₙ₊₁ − y₁)/(yₙ₊₁ − y₀*) = ... = 1 − (A − 1)/(Aⁿ⁺¹ − 1) Kremser equation cannot be used for A = 1 or nonlinear equilibria
How does the formula for the fractional absorption (see Kremser) look like in case of cross-current absorption?
α= (yₙ₊₁ − y₁)/(yₙ₊₁ − y₀*) = ... = 1 − 1/(1 + A/N)ⁿ
What is the Kremser equation (counter current absorption) in case of A = 1?
α= (yₙ₊₁ − y₁)/(yₙ₊₁ − y₀*) = 1 - 1/(1 + N)
What is the formula for fractional recovery?
ξᵢˢ = xᵢˢ·S/(zᵢ · F) with S being a flow rate/phase, e.g. B or D
What is the formula of the Stripping factor 𝑆?
𝑆 = m·G/L
