11.5. DISTILLATION AND ABSORPTION EFFICIENCIES FOR TRAY AND PACKED TOWERS

11.5A. Tray Efficiencies

In all the previous discussions of theoretical trays or stages in distillation, we assumed that the vapor leaving a tray was in equilibrium with the liquid leaving. However, if the time of contact and the degree of mixing on the tray are insufficient, the streams will not be in equilibrium. As a result the efficiency of the stage or tray will not be 100%. This means that we must use more actual trays for a given separation than the theoretical number of trays determined by calculation. The discussions in this section apply to both absorption and distillation tray towers.

Three types of tray or plate efficiency are used: overall tray efficiency E_O, Murphree tray efficiency E_M, and point or local tray efficiency E_MP (sometimes called Murphree point efficiency). These will be considered individually.

11.5B. Types of Tray Efficiencies

1. Overall tray efficiency

The overall tray or plate efficiency E_O concerns the entire tower and is simple to use, but it is the least fundamental. It is defined as the ratio of the number of theoretical or ideal trays needed in an entire tower to the number of actual trays used:

Equation 11.5-1

For example, if eight theoretical steps are needed and the overall efficiency is 60%, the number of theoretical trays is eight minus a reboiler, or seven trays. The actual number of trays is 7/0.60, or 11.7 trays.

2. Murphree tray efficiency

The Murphree tray efficiency E_M is defined as follows:

Equation 11.5-2

where y_n is the average actual concentration of the mixed vapor leaving the tray n as shown in Fig. 11.5-1, y_n+1 the average actual concentration of the mixed vapor entering tray n, and the concentration of the vapor that would be in equilibrium with the liquid of concentration x_n leaving the tray to the downcomer.

The liquid entering the tray has a concentration of x_n−1; as it travels across the tray, its concentration drops to x_n at the outlet. Hence, there is a concentration gradient in the liquid as it flows across the tray. The vapor entering the tray comes in contact with liquid of different concentrations, and the outlet vapor will not be uniform in concentration.

3. Point efficiency

The point or local efficiency E_MP on a tray is defined as

Equation 11.5-3

where is the concentration of the vapor at a specific point in plate n as shown in Fig. 11.5-1, the concentration of the vapor entering the plate n at the same point, and the concentration of the vapor that would be in equilibrium with at the same point. Since cannot be greater than , the local efficiency cannot be greater than 1.00 or 100%.

In small-diameter towers, the vapor flow sufficiently agitates the liquid so that it is uniform on the tray. Then the concentration of the liquid leaving is the same as that on the tray. Then , and . The point efficiency then equals the Murphree tray efficiency, or E_M = E_MP.

In large-diameter columns, incomplete mixing of the liquid occurs on the trays. Some vapor will contact the entering liquid x_n−1, which is richer in component A than x_n. This will give a richer vapor at this point than at the exit point, where x_n leaves. Hence, the tray efficiency E_M will be greater than the point efficiency E_MP. The value of E_M can be related to E_MP by integration of E_MP over the entire tray.

11.5C. Relationship Between Tray Efficiencies

The relationship between E_MP and E_M can be derived mathematically if the amount of liquid mixing is specified together with the amount of vapor mixing. Derivations for three different sets of assumptions are given by Robinson and Gilliland (R1). However, experimental data are usually needed to obtain amounts of mixing. Semitheoretical methods for predicting E_MP and E_M are summarized in detail by Van Winkle (V1).

When the Murphree tray efficiency E_M is known or can be predicted, the overall tray efficiency E_O can be related to E_M by several methods. In the first method, an analytical expression is as follows when the slope m of the equilibrium line is constant as well as the slope L/V of the operating line:

Equation 11.5-4

If the equilibrium and operating lines of the tower are not straight, a graphical method employing the McCabe–Thiele diagram can be used to determine the actual number of trays when the Murphree tray efficiency is known. In Fig. 11.5-2 a diagram is given for an actual plate as compared with an ideal plate. The triangle acd represents an ideal plate and the smaller triangle abe the actual plate. For the case shown, the Murphree efficiency E_M = 0.60 = ba/ca. The dashed line going through point b is drawn so that ba/ca for each tray is 0.60. The trays are stepped off using this efficiency, and the total number of steps gives the actual number of trays needed. The reboiler is considered to be one theoretical tray, so the true equilibrium curve is used for this tray, as shown. In Fig. 11.5-2, 6.0 actual trays plus a reboiler are obtained.

11.5D. Efficiency of Random-Packed and Structured Packed Towers

In tray towers, a theoretical tray is defined as a tray in which equilibrium is attained between the gas or vapor leaving and the liquid leaving the tray. In random-packed or structured packed towers, the same approach is used, where the HETP in m (ft) is defined as the height of the packed column necessary to give a separation equal to one theoretical plate. The design of mass-transfer towers requires evaluation of the number of theoretical stages or transfer units. Hence, the height of packing H in m (ft) required to perform a given separation is

Equation 11.5-5

where n is the number of theoretical stages needed. Using the number-of-transfer-units method, the height H is

Equation 11.5-6

where H_OG is the overall height of a transfer unit in m (ft) and N_OG is the number of transfer units.

The HETP and overall height of a transfer unit H_OG can be related by the following equation, which is the same as Eq. (10.6-55):

Equation 11.5-7

where m is the slope of the equilibrium line in mole fraction units and L and V are molar flow rates. If the operating line and equilibrium line are parallel, then HETP = H_OG. In distillation, the operating and equilibrium lines diverge below the feed point and converge above, so that the average mV/L is about 1.0 and HETP ≅ HTU for distillation. This is usually not true for absorption or stripping towers. Although the HETP concept lacks a sound theoretical basis, unlike the mass-transfer and H_OG concepts, it is simple and easy to use in computer equilibrium stage-to-stage calculations in distillation and is widely used to estimate packing height (S3). It is especially useful in multicomponent systems.

For a tray tower, the HETP can be defined as

Equation 11.5-8

where E₀ is the overall tray efficiency and T is the tray spacing, which varies from about 0.3 m for a 0.5-m-diameter tower, 0.6 m for a 1.0-m-diameter tower, to about 0.8 m (2.6 ft) for much larger towers over 4 m in diameter.

11.5E. Estimation of Efficiencies of Tray and Packed Towers

In order to design the number of trays or packing heights needed, efficiencies must be obtained.

1. Efficiency of tray towers

For estimating the overall tray efficiency of bubble-tray towers for distillation, the O'Connell (O1) correlation can be used (K2) with about a ±10% error. The following equation for these data from Lockett (L1) can be used for sieve and valve trays as well, but predictions will be slightly conservative:

Equation 11.5-9

where E₀ is fraction efficiency, α is relative volatility of the two key components at the average tower temperature, and μ_L is the molar average viscosity in cp of the liquid feed at the average tower temperature of the top and the bottom. Most typical efficiencies are between 40 and 80%.

To estimate the overall tray efficiency for absorption towers, the O'Connell correlation (O1) can be used. This correlation is represented by the equation (S3)

Equation 11.5-10

where E₀ is overall average % tray efficiency, M_L is the average molecular weight of the liquid, m is the slope of the equilibrium line in mole fraction units, ρ_L is the liquid density in lb_m/ft³, and μ_L is molar average liquid viscosity in cp at the average tower temperature. The average deviation of the data is ±16.3% and the maximum is 157%. This can be used for bubble-cap, sieve, and valve-type trays. Note that these tray efficiencies are quite low and in the range of 1% to 50%, with typical values of 10 to 30%.

2. Efficiency of random-packed towers

For estimates for random packing, Eq. (11.5-11) can be used to determine the HETP for second- and third-generation packings only (S2):

Equation 11.5-11

where HETP is in m and d_p is packing diameter in mm. In English units, HETP is in ft and d_p is in in. Also, for small-diameter towers, where the tower diameter D is less than 0.60 m (2 ft), HETP = D, but not less than 0.3 m (1 ft).

For vacuum service (S3),

Equation 11.5-12

The above equations are useful for low-viscosity liquids. For absorption with high-viscosity liquids, the values of HETP are much greater, with values of 1.5 to 1.8 m (5 to 6 ft).

3. Efficiency of structured packing in towers

For approximate estimates of the efficiency of structured packings at low to moderate pressure with low viscosity liquids (K4),

Equation 11.5-13

where HETP is in m and a is surface area in m²/m³. In English units, HETP is in ft and a in ft²/ft³. The value added of 0.10 m gives the estimate a conservative bias (K4). Values of HETP range from about 0.3 to 0.6 m (1.0 to 2.0 ft). For wire mesh (gauze), structured-packing values range from 0.1 to 0.3 m (0.3 to 1.0 ft).

EXAMPLE 11.5-1. Estimation of Tray and Packing Efficiencies and Tower Height A liquid feed of benzene–toluene is being distilled under the same conditions as in Ex. 11.4-1. The number of theoretical steps needed is calculated as 7.6. The feed composition xF = 0.45, xD = 0.95, and xW = 0.10. Do as follows: Using valve trays, calculate the overall efficiency E0 and the tower height. Assume a tray spacing T of 0.6 m. Using 2-in. metal Pall rings, calculate the HETP and the tower height. Using Flexipac No. 2 structured packing, calculate the tower height. Solution: For part (a), from Fig. 11.1-1 at 1 atm abs and xW = 0.10, the boiling temperature is 106.5°C. At the top, for xD = 0.95, the dew point is 82.3°C. The average is (106.5 + 82.3)/2 or 94.4°C. Using Table A.3-12 and Fig. A.3-4 at 94.4°C, the viscosity of benzene is 0.26 cp, and of toluene, 0.295 cp. The molar average viscosity is μL = 0.45(0.26) + 0.55(0.295) = 0.279 cp. From Table 11.1-1, the vapor pressure of benzene PA at 94.4°C is 153.3 kPa, and for toluene, PB = 62.2 kPa. Hence, the average relative volatility α = 153.3/62.2 = 2.465. Using Eq. (11.5-9), E0 = 0.492(μLα)−0.245 = 0.492 (0.279 × 2.465)−0.245 = 0.539. Using Eq. (11.5-8), HETP = T/E0 = 0.6/0.539 = 1.113 m/theoretical steps. The number of tray theoretical steps = 7.6 − 1 step for the reboiler, or 6.6 steps. Hence, tower tray height H = 1.113(6.6) = 7.35 m. For part (b), for 2-in. (25.4 × 2 mm) Pall rings, Eq. (11.5-11) gives For 6.6 theoretical steps, tower height H = 0.914(6.6) = 6.03 m. For part (c), for Flexipac No. 2 structured packing and using Table 10.6-1, a = 223 m/m2. From Eq. (11.5-13),

11.5F. Flooding Velocity and Diameter of Tray Towers

In tray towers the maximum vapor velocity can be limited either by entrainment of small liquid droplets or by the liquid handling capacity of the tray downcomer whereby the liquid in the downcomer backs up to the next tray. The general basis for design uses the allowable-vapor-velocity concept. The equation by Fair (F1) for estimating the maximum allowable vapor velocity for sieve, bubble-cap, or valve trays is

Equation 11.5-14

where ν_max is the allowable vapor velocity in ft/s based on the total cross-sectional area minus the area of one downcomer, σ is the surface tension of the liquid in dyn/cm (mN/m), and ρ_L and ρ_V are liquid and gas densities, kg/m³ or lb_m/ft³. The value of K_ν in ft/s is obtained from Fig. 11.5-3, where L and V are total flow rates in kg/h or lb_m/h. As a rule-of-thumb, the value of K_ν should be multiplied by a factor of 0.91 to account for the downspout area of 9% of the tray (T2).

The Eq. (11.5-14) holds for nonfoaming systems. For many absorbers the K_ν should be multiplied by 0.90 or so to account for foaming (S3). For the final design the above ν_max should be multiplied by 0.80 to be 20% below flooding. For organic liquids a typical value of σ is about 20–25 dyn/cm. For water with a surface tension of about 72 dyn/cm, the flooding velocity is larger by a factor of about (72/22.5)^0.20, or 1.26/1.

EXAMPLE 11.5-2. Tray Diameter of a Valve-Tray Absorption Tower A valve-tray absorption tower is being used to absorb and recover ethyl alcohol vapor in air by pure water. The tower operates at 30°C and a pressure of 110 kPa abs. The entering gas contains 2.0 mol % ethyl alcohol in water, and 95% of the alcohol is recovered in the outlet water. The inlet pure aqueous flow is 186 kg mol/h and the inlet gas flow is 211.7 kg mol/h. Calculate the required tower diameter. A tray spacing of 0.610 m (2.0 ft) will be assumed. Use a factor of 0.95 for possible foaming. Solution: The molecular weight of the entering air at the tower bottom where the flow rates are the greatest is The density of the vapor is The mass flow rate of the gas is V = 211.7(29.34) = 6211 kg/h. Moles ethyl alcohol absorbed = 0.95(0.02)(211.7) = 4.02. The outlet aqueous flow rate L = 186.0 + 4.02 = 190.02 kg mol/h. The liquid mass flow rate is The outlet water density, assuming a dilute solution at 30°C, from Appendix A.2-3 is ρL = 0.995(1000) = 995 kg/m3. Then, Using Fig. 11.5-3 for a 24-in. tray spacing, Kν = 0.385 ft/s. Substituting into Eq. (11.5-14) and using a surface tension of 70 (slightly less than water), Multiplying the above by 0.91 to account for downspout area, by 0.95 for foaming, and by 0.80 for 80% of flooding, The tower cross-section equals Solving for the diameter, πD2/4 = 0.464. Then, D = 0.7686 m (2.522 ft).