Ind. Eng. Chem. Res. 2002, 41, 285-292
285
A New Method for Designing an Energy-Saving Tray and Its Hydrodynamic Aspects: Model Development and Simulation Qing Lin Liu,* Peng Li, Jian Xiao, and Zhi Bing Zhang Department of Chemical Engineering, Nanjing University, Nanjing, 210093 People’s Republic of China
A general thermodynamic model is presented for analyzing distillation processes. The mass transfer in these processes was assumed to be in an irreversible thermodynamic state during model development. The simulated results show that this model can be applied not only to an analysis of the entropy generation in distillation processes, but also to a comparison of the effects of the tray parameters on the entropy generation in a column. It is shown that the entropy generation rate (EGR) can be reduced by increasing the tray active area for mass transfer or the weir length. In terms of energy savings, it is plausible for a distillation column with a smaller diameter, a lower weir height, a longer weir length, and a larger active area to save energy. A new tray, “95 tray”, was thus designed on the basis of the numerical calculations. Introduction The increasing awareness of the necessity to protect our resources and the environment is a driving force in making effective use of our resources and reducing process entropy generation rates (EGRs). Distillation, a major energy-consuming unit operation in the chemical process industries, has been analyzed intensively in terms of its EGR. Many valuable results have been obtained by using various kinds of thermodynamic analysis methods.1-3 Recently, considerable advances have been made by applying nonequilibrium thermodynamic theories to distillation,4 and as a result, the process EGR is understood more deeply and accurately. Distillation processes with lower energy demands have been generated using these methods. For example, Tondeur and Kvaalen2 found that the total EGR could be reduced if the local rate of entropy generation was uniformly equipartitioned along the space and/or time variables. Ratkje and Arons5 studied the work loss in a reacting-diffusing mixture and found that work losses could be minimized, but not avoided, by a reduction of the driving forces. These authors, however, focused on different technological processes and various process conditions. The effect of a tray’s structural parameters on the process entropy generation has largely been neglected. Using the nonequilibrium method to calculate the EGR in a single distillation column, we have found that tray parameters can have a significant influence on the process EGR. On the basis of these analytical results, a new tray, the “95 tray”, that can provide high capacity and high efficiency (compared to those of the conventional sieve tray tower) was designed.
the EGR per unit volume, σ, can be written as (see Forland et al.6)
σ ) -Jq∇
() 1
() µi
N
-
T
Ji∇ ∑ T p i)1
(1)
If the subscript p denotes that the contribution to the chemical potential gradient of component i due to the temperature variation is not included, eq 1 can be simplified to eq 2 when the subscript p is omitted
∇T 1 N c Ji∇µi,T σ ) -Jq 2 Ti)1 T
∑
(2)
Here, the superscript c indicates a contribution to the chemical potential gradient of component i due to the concentration variation, and it is omitted hereafter. The mass flux Ji for the N species can be related, in the frame of the mechanical balance equation, by N
MiJi ) 0 ∑ i)1
(3)
or
JN ) -
i)N-1 M i
∑ i)1
MN
Ji
(3a)
For a similar reason, the mass diffusion forces, or chemical potential gradients for the gas phase, are related through the Gibbs-Duhem equation N
Model Development For the case of nonviscous fluids with no chemical reaction and no external force exerted on a system of N species at constant pressure, a basic equation for * Corresponding author: E-mail:
[email protected]. Present address: Department of Chemical Engineering, Xiamen University, Xiamen, 361005 P. R. China.
yi∇µi,T ) 0 ∑ i)1
(4)
or
∇µN,T ) -
N-1 y
∑ k)1 y
10.1021/ie0008880 CCC: $22.00 © 2002 American Chemical Society Published on Web 12/27/2001
k
N
∇µk,T
(4a)
286
Ind. Eng. Chem. Res., Vol. 41, No. 2, 2002
Substituting eqs 3a and 4a into eq 2, one obtains
can be expressed by
∇T 1 N-1 1 σ ) -Jq Ji∇µi,T - JN∇µN,T T T2 T i)1
∑
∇T
) -Jq
-
1 N-1
1 N-1
N-1
Miyk
Ji ∑ ∑ Ji∇µi,T - T ∑ i)1 k)1 M
T i)1
T2
∇T 1 - Lql ∇µl,T 2 m T hT
(13)
∇T 1 - Lll ∇µl,T 2 m T hT
(14)
Jq ) -Lqq and
∇µk,T (5)
NyN
Jl ) -Llq
For simplification, we introduce
{
0 for i * k δik ) 1 for i ) k
(6)
Then, eq 5 can be written as
σ ) -Jq
∇T
-
1 N-1
[ (
∑ k)1 ∑
T i)1
T2
N-1
Ji
δik +
Miyk MNyN
) ] ∇µk,T
(7)
A set of new thermodynamic forces, Xi, is defined as follows
Xi ) -
1 N-1
∑
T k)1
(
δik +
Miyk MNyN
)
∇µk,T
(8)
By introducing Xi of eq 8 into eq 7, the latter can be expressed in the simplified form
σ ) -Jq
∇T T2
(
+
JiXi ∑ i)1
)
Mlyl 1 1 1 1+ ∇µl,T ) ∇µ T M h yh T mh l,T
(9)
(10)
Here, yl and yh are the mole fractions of the light and the heavy components in the gas phase; mh denotes the mass fraction of the heavy component and is given by
mh )
M hy h Mhyh + Mlyl
(11)
For a binary system, by introducing Xl from eq 10 into eq 9, the EGR can be written as
σ ) -Jq
(
Jq ) - Lqq - Lql
∇T 1 - Jl ∇µl,T 2 m T hT
(12)
According to the linear phenomenological relationship of nonequilibrium thermodynamics, these flux equations
)
Llq ∇T Lql + J Lll T2 Lll l
(15)
The EGR per unit volume is obtained by introducing Jq of eq 15 into eq 12
(
N-1
A distillation column for separating a binary mixture is taken as an example for analysis. There are three fluxes across the vapor-liquid phase boundary: the heat flux, Jq, and two mass fluxes, Jl and Jh. These fluxes are arbitrarily chosen to be positive when transport takes place from the liquid to the vapor. The following assumptions are made for simplification: (1) There is no significant pressure gradient along or across the interface film and the liquid phase is well-distributed on a stage. (2) The temperature gradients and the chemical potential gradients are constant in the gas phase on each stage, and the temperature gradient along the column is small. (3) The thermal contribution to the mass flux is so small that it can be neglected. Then, one can conclude from eq 8 that
Xl ) -
Here, Llq indicates the contribution to the mass flux of the light component due to heat transfer or temperature variation, Lql indicates the contribution to the heat flux due to the concentration variation, and the relationship Lql ) Llq holds in terms of the Onsager reciprocity relations. Comparing Fourier’s law of heat conduction with eq 13 in the absence of mass flux, we obtain the heat conductivity λ ) Lqq/T2; in a similar way, from Fick’s law of diffusion and eq 14, we have Fick’s diffusion coefficient D ) Lll(1/mhT)(∂µl/∂cl)T. A convenient expression for the heat flux is obtained by introducing the term ∇µl,T/Tmh of eq 14 into eq 13
σ ) Lqq - Lql
)( )
Llq ∇T Lll T2
2
+
Jl2 Lll
(16)
The first term on the right is the dissipation of energy due to ∇T, while the last term is that due to the mass transfer. It is reasonable to neglect the thermal contribution to the energy dissipation, so eq 16 can be simplified to
σ)
Jl2 Lll
(17)
Then, the total rate of entropy generation, P, for a stage that transfers mass and heat over the transport path can be written as
P≈
Jl2 dv + interface L ll
∫V
∫V-V
Jl2 dv interface L ll
(18)
Here, Vinterface ) A∆x is the volume of the interface; A is the area of the interface; and ∆x is the thickness of the interface film, typically in micrometers, which can be calculated from ∆x ) D/kc. If the dissipation of energy is attributed mainly to the mass transfer across the interface from the concept of the thermodynamic interface layer, then the last term on the right is neglibible. We thus obtain eq 18a, if the phase contact area, A, is kept constant
∫0
P≈A
∆x
Jl2 dx Lll
(18a)
Generally, the contribution to the concentration gradient due to temperature variation is negligible (assumption 3) in the case where the temperature gradient is small (assumption 2), so the impact of ∇T on Jl can
Ind. Eng. Chem. Res., Vol. 41, No. 2, 2002 287
be neglected, and Jl can be expressed in the form
1 ∇µ Jl ) -Lll mhT l,T
Table 1. Column Specifications for the Separation of Benzene-Toluene
(19)
Introducing Jl of eq 19 into eq 18a and omitting the subscript T, we obtain
P≈A
∫0∆x Lllm 12T2(∇µl)2 dx
(20)
h
The process of vapor enrichment during bubble ascent on a stage is assumed to be in a quasi-steady state. Setting ∇cl ) ∆cl/∆x and ∇µl ) ∆µl/∆x by assumption 2 and using the assumption that yl and T are approximately constant on a given stage, we obtain the total EGR for a stage as
P≈
( )
∇µl 2 A Tmh
∫0∆x Lll dx )
( )
∆µl 2 A Tmh (∆x)2
∫0∆x Lll dx
(21)
The average coefficient Lll is defined as
Lll )
A (∆x)2
∫0∆x Lll dx
(22)
and Lll can be obtained from eq 19 as
( )
Lll ) -
JlTmh ∇µl
(23)
∇T)0
Thus, by using Fick’s Law and the assumption of constant forces, Lll can be obtained, by introducing eq 23 into eq 22, as
∫0∆x -
( ) JlTmh ∇µl
Lll )
A (∆x)2
)
A (∆x)2
)
A D ∆cl Tmh ∆x ∆µl
∫0∆x
dx ∇T)0
D‚∆cl‚T‚mh dx ∆µl (24)
In the end, the total EGR for a stage can be expressed as
∆µl D∆cl P≈A Tmh ∆x )
∆µl Ak ∆c Tmh c l
)
∆µl k a∆cl Tmh c
(25)
Calculation Procedures The distillation column for the separation of benzene (I) and toluene (II) studied in this work is specified in Table 1, which gives the column stage numbers, feed stages, stage efficiencies, etc. The stage compositions, temperature, and liquid and vapor streams for each stage were first obtained from the standard commercial simulation package Aspen Plus, version 10.1. The
stage numbers
18-28
diameter (m)
1.2
feed stages X1 at the feed stage efficiency X1 at the distillate X1 at the bottom
10, 12, 15 0.44 0.8 >0.98
