The following section discusses estimation of tower height.
Height Equivalent to a Theoretical Plate (HETP)
This is an old method for estimation of tower height. But it ignores the difference between stage-wise and continuous contact. In this method, the number of theoretical plates or trays required for a given change in concentration is first determined by using the method of computation for stage-wise contact.
This number is then multiplied by a quantity known as the HETP to get the required height of packing to achieve the same change in concentration
Z = (HETP) # N (5.46)
where Z and N represent the total height of packing and number of theoretical or equilibrium stages respectively. Many designers prefer to use the HETP approach because it provides a comparison with the number of theoretical stages determined with tray column.
The HETP is evaluated simply as the ratio of packed height used for a certain degree of separation to the theoretical number of stages. Its relation to the fundamental quantity, HTU- the height of a transfer unit, is
HETP = HTU (5.47)
w her e, m is the slope of the equilibrium curve. In distillation, the equilibrium and operating lines diverge below the feed point and converge above it. As a result the value of (mGM/LM) averages approximately unity for distillation so that HETP and HTU become essentially equal. However, this is not true in absorption and stripping processes. Figure 5.15(a) shows that the magnitude of number of theoretical stages per metre depends on the size of packing as well as on the system conditions. The variation of HETP value as shown in Figure 5.15(b), indicates its dependency on the geometry of packing and also on the gas flow rate.
Figure 5.15(a) Number of theoretical stages per m. (1/HETP), L/G = 1 for (i) Methanol/Ethanol, and (ii) Ethylbenzene/Styrene system.
This method has also two fundamental drawbacks. Firstly, it ignores the basic difference between stage-wise and continuous contact. Secondly, the HETP must be an experimentally determined quantity which strongly depends on several factors like the system and the concentration changes involved, the flow rates of the fluids and the type and size of packing. Enormous quantity of data must be accumulated to make use of this method. HETP values for a wide range of packing used in industrial separations are available in literature (Kister 1992, Peters and Timmerhaus 1991). A glimpse of some data is given Table 15.1.
Figure 5.15(b) HETP as function of geometry of packing and gas rate.
Table 5.1 Characteristics of some packings
Type of pack ing Void fraction Surface area per unit volume, m2/m3 Approx. HETP, m Random packings
Ceramic Raschig rings, 25 mm 0.73 190 0.6-0.12
Ceramic Intalox saddles, 25 mm 0.78 256 0.5-0.9
Ceramic Berl saddles, 25 mm 0.69 259 0.6-0.9
Plastic Pall rings, 25 mm 0.90 267 0.4-0.5
Metal Pall rings, 25 mm 0.94 207 0.25-0.3
Structured packings
Intalox 2T (Norton) 0.96 213 0.2-0.3
Flexipac® 1 (Koch) 0.91 558 0.2-0.3
Flexipac® 2 (Koch) 0.93 249 0.3-0.4
Gempak® 4A (Glitsch) 0.91 525 0.2-0.3
Gempak® 2A (Glitsch) 0.93 262 0.3-0.4
Sulzer BX (Sulzer) 0.90 499 0.2-0.3
If such data are not available, Kister has provided helpful rules of thumb for predicting HETP values with random packings in terms of column diameter, D:
HETP = D for D ≤ 0.5 m
= 0.5D0.3 for D > 0.5 m
= D0.3 for absorption columns with D > 0.5 m.
For vacuum distillation, it is recommended that an extra 0.15 m be added to these predicted values of HETP.
A rule of thumb for quick estimation of HETP for structured packing has also been presented (Harrison and France 1989):
HETP = + 0.10(5.48)
where ap is the packing surface area per unit volume, m2/m3 and HETP in metres. A more accurate approach would however be to use the interpolation or extrapolation of packing efficiency data presented by Kister.
HETP correlations: The correlation has been developed based on primarily laboratory data which may not be suitable for understanding the large scale behaviour. Some of values pertaining to large scale behaviour are given as (Frank 1977)
Types of pack ing/Application HETP, m
25 mm dia packing 0.46
38 mm dia packing 0.66
50 mm dia packing 0.90
Absorption duty 1.5-1.8
Small diameter columns (< 0.6 m dia.) Column diameter Vacuum columns Values as above + 0.1 m.
A correlation for Raschig rings and Berl saddles (Murch 1953) covers columns up to 30 in. diameter and 10 ft high is as follows:
HETP = C1G′C2
deC3Z1/3 (5.49) where,
G′ = mass velocity of vapour per unit tower area, kg/m2$s de = column diameter, m
Z = packed height, m a = relative volatility
nL = liquid viscosity, N s/m2 tL = liquid density, kg/m3
Values of the constants Ci are as follows:
Types of pack ing Size, mm C1(# 10-5) C2 C3
Ellis (1953) developed a correlation for 25 and 70 mm raschig rings with HETP (m)
Ellis (1953) developed a correlation for 25 and 70 mm raschig rings with HETP (m) HETP = 18dr + 12 m (5.50)
where
dr = diameter of the rings, m
m = average slope of equilibrium curve G′M = vapour mass flow rate
L′M = liquid mass flow rate
Method based on individual phase mass transfer coefficients
Limiting our discussion to transfer of a single component from a binary gas mixture, the required height of packing for a desired change in composition may be estimated by equating the differential form of material balance with the point value of the rate equation.
Figure 5.16 Flow diagram of a packed tower showing differential height (dZ).
Considering the differential height dZ of a packed tower as shown in Figure 5.16, let dy be the change in gas composition and let (NAa) be the point value of mass flux in mole/(unit time) (unit volume). A material balance for the absorbed component then becomes
-d(GM y) = -GMdy - ydGM = NAa dZ(5.51)
Since for transfer of only one component, dGM = -NAa dZ, Eq. (5.51) may be written as -GMdy - y (-NAa dZ) = NAa dZ(5.52)
Equation (5.52) may be rearranged as
dZ = - (5.53)
The required height of packing may be determined by integration of Eq. (5.53) over the tower after substituting for NAa from Eq. (3.19)
Z = (5.54)
Equation (5.54) is valid for those cases where the gas-phase resistance controls, i.e. where the gas is highly soluble in the solvent. With a slightly soluble gas, the corresponding equation for liquid-phase resistance controlling is
Z = (5.55)
Equations (5.54) to (5.55) may be integrated graphically by evaluating the constituent terms at a series of points on the operating line. Alternatively, the integration may be done numerically by a digital computer. The use of either of the above two equations requires the value of individual mass transfer coefficient which can be determined experimentally by the methods described in Chapter 6. However, the same may not be possible for some systems. In such situation values can be obtained form the literature. Figure 5.17 for instance depicts a relationship between the volumetric mass transfer coefficient and liquid flow rate while gas flow rate was kept constant for absorption of CO2 in NaOH using various packings.
Figure 5.17 Volumetric mass transfer coefficient (Kya) for absorption of CO2 in NaOH solution with gas flow rate = 1.22 kg/m2$s.
Also, a number of empirical and semiempirical equations are available in the literature (Coulson et
al. 1991, Geankoplis, 2005). One such method (Bravo and Fair 1982) involves estimating the individual mass transfer coefficients and an effective interfacial area for mass transfer. The mass transfer coefficients in m/s established by these investigators are obtained from
ky = (5.56) and
kx = 3.72 # 10-5 ScL-0.5 (apdp)0.4 (5.57)
where, = ReG =
WeL = , ScL =
where, subscripts L and G refer to the liquid-phase and vapour-phase, respectively, GM and LM are the superficial velocity of gas-phase and liquid-phase, respectively, ap is the packing surface area, m2/m3
D represents the diffusivity, m2/s dp means the packing diameter, m
aw refers to the area of the wetted packing, m2 t refers to the density, kg/m3
n refers to the viscosity, c.p
v represents the surface tension, dyne/cm and vc refers to the critical surface tension which is assumed to be 61 dynes/cm for ceramic packing, 75 dynes/cm for structured packing and 33 dynes/cm for PEB packing
Fr, We and Sc refer to Froude, Weber and Schmidt numbers, respectively.
Determination of the individual mass transfer coefficients permits evaluation of the effective area a for mass transfer using the relation
a = ap v0.5 (CaLReG)0.392(HTU)0.4(5.58)
where the dimensionless capillary number, CaL is obtained from CaL =
The mass transfer coefficients based on gas phase (kya) and liquid phase (kxa) may also be approximately estimated from the following correlations (Semmelbaur 1967):
Gas-film controlling:
For 100 < ReG < 10,000 and 0.01 m < dp < 0.05 m ShG = bReG0.59 ScG0.33(5.59)
where, b = 0.69 for Raschig rings and 0.86 for Berl saddles, dp = packing size.
Liquid-film controlling:
For 3 < ReL < 3000 and 0.01 m < dp < 0.05 m
ShL = b′ ReL0.59 ScL0.50(5.60)
where, b′ = 0.32 for Raschig rings and 0.25 for Berl saddles.
Method based on overall mass transfer coefficient
It is often inconvenient to use single phase mass transfer coefficient because of the difficulty in measuring the interfacial concentration. As a result, overall coefficients are frequently used.
The following similar procedure as in previous section, i. e. by integration of Eq. (5.53) over the tower after substituting the expression for NAa from Eq. (4.5), the equations for estimation of height of a tower can be developed in terms of overall gas-phase mass transfer coefficient and overall gas-phase driving force
Z = (5.61)
Similar expression in terms of overall liquid-phase mass transfer coefficient and overall liquid-phase driving force, can be obtained as follows:
andZ = (5.62)
Equation (5.61) is based on overall mass transfer coefficient and overall driving force, both in terms of gas-phase mole fraction of solute while in case of Eq. (5.62), both are in terms of liquid-phase mole fraction of solute.
For gas-film controlled processes, the following simple correlation may be used for estimation of overall mass transfer coefficient based on gas-phase concentration difference (Kowalke et al. 1925):
Kya = aG′0.8(5.63) where,
Kya is in kmol/m3$s (unit partial pressure difference in kN/m2) G′ = mass velocity of gas, kg/m2$s
and a varies from 0.00048 for 25 mm spheres to 0.001 for 6.4 mm crushed stones.
Method based on height of a transfer unit
Mass transfer data are not usually available in a form suitable for use in Eqs. (5.54), (5.55), (5.61) and (5.62). Some suitable method has to be developed to utilize these data.
Since for transfer of one component the term [GM/(kya)(yBM)] is theoretically independent of concentration and total pressure (Sherwood and Pigford 1952), Eq. (5.54) may be multiplied and divided by yBM, then we get
Z = (5.64)
w h e r e , yBM = (5.65)
The left-hand term under the integral has the dimension of length (or height) and is designated as the Height of a Transfer Unit or HTU, HtG.
HtG = (5.66)
As mentioned above, HtG is independent of pressure and concentration. Mass transfer data are often reported in the form of HtG. Unless the changes in flow rate or specific volume are large over the tower, an average value of HtG may be used and it may be taken out of the integration sign, so that Eq.
(5.64) becomes
Z = HtG (5.67)
The integral in Eq. (5.67) is called the Number of Transfer units of NTU, NtG.
NtG = (5.68)
The tower height thus becomes the product of the HtG and NtG.
Z = HtG$NtG(5.69)
Similar equations can be developed by using liquid-phase mass transfer coefficient when the tower height becomes
Z = HtL$NtL(5.70)
The number of transfer units represents the difficulty encountered in the separation. A high degree of separation or a small available driving force requires large number of transfer units. The height of a transfer unit, on the other hand, represents inversely the relative ease with which the transfer can be achieved. As may be seen from Eq. (5.66), large mass transfer coefficient or large interfacial area per unit volume will give lower value of HtG which means lower packing height for the same separation.
For dilute systems for which the equilibrium and operating lines are straight and parallel so that (y -yi) is approximately constant, Eq. (5.68) gives
NtG ≈ (5.71)
From Eq. (5.71), one transfer unit is equivalent to the height of packing over which the composition of the stream changes by an amount equal to the average driving force.
The following similar procedure as has been made in cases of individual phase coefficients in Eqs.
(5.64) to (5.70), overall height of a transfer unit and overall number of transfer unit for gas-phase may be obtained as
HtoG = (5.72) NtoG = (5.73)
Z = HtoG$NtoG (5.74)
In terms of overall mass transfer coefficients based on liquid phase, ‘height of an overall liquid-phase transfer unit’ and ‘number of overall liquid-phase transfer units’ are:
HtoL = (5.75)
NtoL = (5.76)
Z = HtoL. NtoL(5.77)
Correlation for HTU: A number of design procedures for evaluating HTU for packed columns are available in the literature. The values of HTU for certain types of packing for absorption of ammonia from air in water can be obtained from Figure 5.18.
Figure 5.18 HTU of some packing for absorption of NH3 from air by water.
Approximate range of values of HTU for packings of some common sizes are as follows:
Packing size, mm HTU, m
25 0.30-0.60
38 0.50-0.75
50 0.60-1.00
A number of empirical and semiempirical correlations are also available in the literature (Fair and Bolles 1979, Geankoplis 2005).