ARTICLE pubs.acs.org/IECR
Reconciliation of Flow Rate Measurements in the Presence of Solid Particles Andrea P. Reverberi,† Cristiano Cerrato,‡ and Vincenzo G. Dovi*,† † ‡
Department of Chemical Engineering and Processes (DICheP), University of Genova, via Opera Pia 15-16145 Genova, Italy Department of Engines, Energy Systems and Transports (DIMSET), University of Genova, Via Montallegro 1-16145 Genova, Italy ABSTRACT: The reconciliation of flow rates of fluids entraining and reacting with solid particles is considered in this article. It is shown that, in case the amount of solid particles is not high, the reconciliation problem can be addressed formally as a rectification that includes additional components and solved using the techniques proposed by earlier research [Crowe, C. M., Garcia Campos, Y. A, Hrymak, A. AIChE J. 1983, 29, 881-888]. Furthermore, it is shown that the introduction of the interaction law between the fluid and the solid phase can improve the overall reliability of the reconciliation procedure.
’ INTRODUCTION The increased computational power presently available in chemical process plants has prompted the development of several techniques for optimizing their performances. In particular, reliable process data have become a key issue in the efficient operation of chemical plants. They play a fundamental role in the enhancement of quality, the control of polluting substances emission, the maintenance of safety conditions, and the optimal management of inventories. However, errors in process data or unreliable data-handling methods can easily overestimate or underestimate actual changes in the process performance. This difficulty can be overcome using reconciliation techniques (i.e., rectification of the process data through a rigorous statistical analysis of the measurement errors). If the variables (q0i - qi) (the difference of measured values and unknown exact values) are assumed to be random variables and their probability distribution is available, the estimation of the unknown variables can be attained by maximizing the corresponding likelihood function (i.e., the probability distribution function regarded as a function of the unknown variables). In particular, if the probability distribution function of the random errors is assumed to be normal, the maximization of the likelihood function is given by maximization of the function exp[-(q0 - q)TV-1(q0 - q)] (or by minimization of the term (q0 q)TV-1(q0 - q)), where V is the variance-covariance matrix of the random variables considered. However, this would lead to setting all (q0i - qi) terms equal to zero (i.e., assuming that the exact values are equal to the measured ones), unless suitable constraints are introduced in the maximization problem. This can be accomplished if the relationships that connect the set of variables qi are considered. Supposing that the variables measured are the flow rates of different streams of a process, it is quite straightforward to introduce mass conservation laws as constraints to the maximization of the likelihood function by forcing the unknown exact values of the variables measured to satisfy them. There is a double advantage in using mass conservation constraints: they are rigorous laws (and, consequently, they do not introduce any r 2010 American Chemical Society
additional bias) and can be frequently written as linear constraints. This makes it possible to obtain an analytical, compact solution to the overall reconciliation problem. Rigorous reconciliation procedures for steady-state operations can be found in the fundamental papers by Crowe and coworkers1,2 in which only conservation laws are employed. Further developments, which address important special problems, have appeared in the literature in the past few years. For instance, the presence of censored measurements when data are subject to detection limits (see the work of Dovi and coworkers,3-5), time-dependent (batch) processes (see the work of Ungarala and Bakshi6 and Kong Mingfang et al.7), and the presence of fluctuations in steady-state processes (see the work of Dovi and Del Borghi8). However, if the number of measured streams is not sufficient, the rectified variables cannot be estimated accurately. In this case, the use of additional constraints provided by closure relations are occasionally used to improve the overall rectification procedure. In this paper, we consider a class of data reconciliation problems that has received little attention: the reconciliation of flow rates that include both a fluid and a solid phase. In particular, it is necessary to consider the amount of the fluid that has been adsorbed onto or has reacted with the solid particles, as well as the amount of particles entrained by the fluid phase. In this work, we present an algorithm that performs the reconciliation problem by considering the presence of two phases in the measured flow rates. While the entrainment of solid particles in fluid flow rates is determined basically by fluid dynamic laws, the uptake of fluid onto solid particles can follow a variety of mass exchange mechanisms, including complex chemical reactions.
Special Issue: Puigjaner Issue Received: June 29, 2010 Accepted: November 2, 2010 Revised: October 16, 2010 Published: November 23, 2010 5248
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First, it is shown that the linear algebra technique developed by Crowe can be used if the amount of particles in the fluid phase is small. Then, the importance of introducing closure relations that describe fluid-solid interactions for improving the rectification accuracy and/or reducing the number of measurements is shown and a simple example is considered for illustration purposes.
’ THEORETICAL DEVELOPMENT Let us first consider the rectification of fluid flow rates. Typically, we must solve a problem of the type ðq0 - qÞT V -1 ðq0 - qÞ ¼ min
ð1aÞ
subject to Aq ¼ 0
ð1bÞ 0
where q is the vector of unknown theoretical flow rates, q the vector of the corresponding experimental values, V the variancecovariance matrix of the experimental observations, and A the incidence matrix (i.e., the matrix that provides the mass conservation laws among the flow rates). However, in case solid particles interacting with the fluid phase are present, this relationship must be modified. This is due to two factors: (1) The measured flow rates differ from the theoretical values not just by a random (typically normal) error, but by a systematic bias related to the presence of solid particles. In other words, constraint 1b (which follows from the maximum likelihood theory, in the case of a Gaussian distribution) is to be modified accordingly. (2) The mass balances must take into consideration the amount of fluid that has been adsorbed by or has reacted with the solid phase. In other words, constraint 1b also must be modified. The bias introduced by the presence of entrained solid particles in the measurements of fluid flow rates can be taken into account by taking into consideration the head losses caused by them. For this purpose, we have considered the relationship q0 ¼ q þ ε þ f ðdÞ
ð2Þ
where ε is the random Gaussian noise and f(d) is a suitable function of the amount of solid particles d present in the flow (expressed in units of g/s). Although there is no general physical law for f(d), it can be assumed that a linear approximation is resonable if d is not large. Thus, eq 2 reduces to q0 ¼ q þ ε þ ξd
T
ðq - q - ξdÞ V
-1
0
ðq - q - ξdÞ
ð3Þ
Setting p = q - ξd, we obtain the new minimization problem: ðq0 - pÞT V -1 ðq0 - pÞ ¼ min
Ap þ Bξd ¼ 0
ð4bÞ
where B is an incidence matrix that includes only the streams affected by the presence of solid particles. Thus, formally, taking the presence of entrained particles into consideration is equivalent to increasing the dimension of the vector of flow rates using several unmeasured variables (d). If the parameter ξ is known, the solution of this problem can be expressed, because of its linear structure, in a closed form by premultiplying (constraint 4b) with the matrix YT, such that YTB = 0 (from the work of Crowe et al.1). However, this implies a reduction of the number of equations contained in constraint 4b and, consequently, a reduction in the amount of information that can be used in the reconcilation task. This is frequently offset by the availability of some data on the flow rate of the solid particles. This makes it possible to rewrite constraint 4a4 as ðq0 - pÞ Vq -1 ðq0 - pÞ þ ðd0 - dÞ Vd -1 ðd0 - dÞ ¼ min ð5aÞ T
T
subject to Ap þ B1 ξd þ B2 ξd ¼ 0
ð5bÞ
where the original matrix B has been split into two submatrices to take measured and unmeasured flow rates of the solid particles into consideration (for the sake of simplicity, all the flow rates of the fluid phase are assumed to have been measured). If the parameter ξ has been estimated in a preliminary tuning procedure, the linear solution previously referenced can be used; otherwise, the more-complex procedure proposed by Crowe2 for the bilinear case must be applied. Again, the minimization problem (expression 5) can be solved using the procedure proposed by Crowe et al.,1 i.e., premultiplying (constraint 5b) by the matrix Y2T such that Y2TB2 = 0. As mentioned previously, the interaction between the fluid and the solid phase can be taken into consideration by suitably modifying the constraints. Following Crowe et al.,1 we can rewrite constraint 5b as Ap þ B1 ξd þ B2 ξd þ ST ε ¼ 0 where S is a master stoichiometric matrix with a nonzero block Sk for reaction nodes (units where interactions with the solid phase occur). Thus, the final reconciliation problem can be written as ðq0 - pÞ Vq -1 ðq0 - pÞ þ ðd0 - dÞ Vd -1 ðd0 - dÞ ¼ min ð6aÞ T
T
subject to
where ξ is a suitable parameter that can be estimated together with the theoretical values or through a suitable tuning procedure. Thus, the objective function (expression 1a) should be changed to 0
subject to
ð4aÞ
Ap þ B1 ξd þ B2 ξd þ ST ε ¼ 0
ð6bÞ
YTBS by which constraint 6b is to be multiplied now
The matrix must satisfy the condition
Y TBS ½B2 : ST ¼ 0 or, in other words, YBS is the matrix whose columns span the null space of [B2:ST]. Thus, the general problem of data reconciliation in the presence of fluid-solid interactions can be cast, formally, into the well-known procedure developed by Crowe et al.,1 provided the bias introduced by solid particles in the measurement of fluid 5249
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flow rates is approximately proportional to the amount of particles, as assumed above. However, the dimension of the vector d is generally small, with respect to the dimension of vector d. In other words, there are not many streams of solid particles measured in actual industrial plants, because of the difficulty of this type of measurement. This can lead to a low dimensional vector YTBS[B2:ST], which generally implies an unsatisfactory reconciliation, as will be shown in the simple example considered in the next section. The introduction of closure relations can improve the overall procedure, provided reliable information on the physical-chemical parameters generally contained in them is available. In particular, the following assumptions are generally made: • Size distributions of solid particles in fluid streams can be assumed to have the following probability distribution function: " # 1 -ðln μ - RÞ2 ð7Þ ψðμÞ ¼ pffiffiffiffiffiffi exp 2β2 μβ 2π
Similarly, the amount of solid particles being segregated is equal to Z μ ψðμÞ dμ ¼ k1 R00 ð13Þ Nss ¼ k1 0
μ π k1 ψðμÞ Fμ3 dμ ¼ k1 β00 ¼ dss 6 0 Z
and, consequently, Nss ¼
μ
where θ is a constant that is supposedly known. • The overall efficiency of a cyclone can be assumed to be equal to 8" !#0:5 9 < FF 4 μ H= ð16Þ -1 ηc ¼ 1 - exp : 15 Dcol CD Fg γ; where Dcol is the column separation diameter, H the column separation height, FF the particle density, Fg the gas density, γ a constant parameter, and CD the drag coefficient. As for the streams exiting a screening device, we can write Nce ¼ γ000 dce Ncs ¼ γ0000 dcs
0
Z
¥
0
π ψðμÞ Fμ3 dμ ¼ k1 β ¼ d 6
ð9Þ
with obvious meanings of the symbols. Consequently, R d ¼ γd N ¼ β • If screening equipment is present, its efficiency ηsc(μ) can be assumed equal to ηsc ðμÞ ¼ 1 ηsc ðμÞ ¼ 0
μ g μ μ < μ
Thus, the number of solid particles entrained in the fluid phase is given by Z ¥ ψðμÞ dμ ¼ k1 R0 ð10Þ Nse ¼ k1 μ
Because
Z k1
¥ μ
ψðμÞ
π Fμ3 dμ ¼ k1 β0 ¼ dse 6
it follows that Nse ¼
! R0 de ¼ γ0 dse β0
ð11Þ
ð12Þ
ð15Þ
with the obvious meanings of the symbols. • The mass transfer from the fluid to the solid phase, or from the solid phase to the fluid (f), is assumed to be proportional to the overall surface of the solid particles: Z ¥ 0 4πμ2 ψðμÞ dμ ¼ K k1 ¼ θd f ¼ Kk1
where μ is the particle diameter and R and β are, respectively, the logarithms of the mean and standard deviation of the particle distribution. Consequently, the number of particles whose diameter is included between μ and μ þ dμ is given by k1 ψ(μ), where the constant k1 is related to the overall number of particles (and, consequently, to the flow rates of the solid particles) through the following relationships: Z ¥ ψðμÞ dμ ¼ k1 R ð8Þ N ¼ k1
k1
! R00 dss ¼ γ00 dss β00
ð14Þ
’ A NUMERICAL EXAMPLE Let us consider, for example, the process described by Figure 1. We shall refer to the set of measurements {q1,q3, q6,d2}. A gas stream (#1) and a solid stream (#2) are fed countercurrent to an adsorption column whose top output (#3) goes to a cyclone. In the column, there is a screen that separates the largest solid particles coming out through a drain located at a convenient column height. The cyclone removes solid particles from the gas stream. The bottom output (#5) is collected, together with the output coming from the column drain device (#4), in a storage tank. The main goal of the reconciliation procedure in this process is the evaluation of solid particles released to the atmosphere, i.e., an estimation of the solids content of stream #6, which cannot be measured directly. The solid streams measured are streams #2 and #4, whereas the flow rates of all gas streams are measured. There is only one unit where mass transfer between the two phases is possible (the reactor) and, consequently, the vector εB consists of one unknown element ε. It is straightforward to verify that, if closure relations are not used, the previously described procedure (i.e., premultiplying the equations that represent mass balances by the relevant YTBS matrix) simply cancels the constraints and leads to the 5250
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even introduce an additional bias. Thus, the decision of whether or not to use them should be made from case to case, depending on the degree of confidence in the values of these parameters.
’ CONCLUSIONS Two different approaches have been considered for the reconciliation of process data when solid particles are entrained in fluid streams. The first approach considers only rigorous mass balances, whereas the second makes use of closure relations, to increase the amount of information employed for the reconciliation task. No general recommendations about which of the two methods provides the best offset is possible. The best strategy will vary from case to case, according to the amount of plant measurements available and the reliability of the closure relationships used. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
Figure 1. Process scheme to which the present reconciliation technique is applied.
trivial solution d ¼ d0 p ¼ q0 for the streams measured, whereas the unmeasured flow rates can be calculated from the constraints, i.e., 0
0
d ¼ d4 - d2 3 d6 ¼ 1ðp0 - p0 þ ξd3 Þ 3 ξ 6 d5 ¼ d3 - d6 ε ¼ p3 - p1 - ξd3 0
0
On the other hand, if closure relations are included, the following linear constraints are to be considered: p3 - p1 þ θd2 þ ξd3 ¼ 0 p3 - p6 þ ξd3 - ξd6 ¼ 0 d2 - d3 - d4 ¼ 0 N2 - N3 - N4 ¼ γd2 - γ0 d3 - γ00 d4 ¼ 0 d3 - d6 - d5 ¼ 0 N3 - N6 - N5 ¼ γ0 d3 - γ0000 d6 - γ000 d5 ¼ 0 Using the maximum likelihood condition, subject to these constraints, it is straightforward to (i) determine the rectified flow rates and (ii) verify that the uncertainty in their estimation has been reduced, with respect to the previous procedure. However, if the parameters {γ,γ0 ,γ00 ,γ000 ,γ0000 ,θ} are not known with a sufficient degree of accuracy, the use of closure relations could improve the rectification procedure only marginally or
’ NOMENCLATURE A = incidence matrix B = incidence matrix for solid-carrying flows B1, B2 = submatrices of incidence matrix CD = drag coefficient d = mass flow rate of solid phase (kg/s) D = separation diameter (m) f = gas-solid mass transfer (kg/s) H = column separation height (m) K = constant parameter in the relation for gas-solid mass transfer k1 = constant parameter in the relation for overall number of particles N = number of particles p = modified flow rate (m3/s) q = flow rate (m3/s) S = stoichiometric matrix V = variance-covariance matrix of the experimental observations Y = matrix whose columns span the null space of the matrix with which its transpose is premultiplied Greek Letters
r, β = logarithms of the standard deviations of particles γ = constant parameter η = overall cyclone efficiency ε = flow rate random noise (m3/s) μ = particle diameter (m) F = density (kg/m3) σ = flow rate standard deviation (m3/s) θ = constant parameter ξ = solid-phase transport parameter ψ = probability distribution function of particle diameters Subscripts
BS = related to the matrix [B2:ST] c = cyclone ce = entrained particles from a cyclone cs = segregated particles from a cyclone col = column 5251
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d = dust g = gas s = solid se = entrained particles from a screening device ss = segregated particles from a screening device Superscripts
T = transpose 0 = experimental value Accent & = theoretical value
’ REFERENCES (1) Crowe, C. M.; Garcia Campos, Y. A.; Hrymak, A. Reconciliation of process flow rates by matrix projection. Part 1: Linear case. AIChE J. 1983, 29, 881–888. (2) Crowe, C. M. Reconciliation of process flow rates by matrix projection. Part 2: The nonlinear case. AIChE J. 1986, 32, 616–623. (3) Dovi, V. G.; Reverberi, A. P.; Maga, L. Reconciliation of process measurements when data are subject to detection limits. Chem. Eng. Sci. 1997, 52, 3047–3050. (4) Dovi, V. G.; Del Borghi, A. Reconciliation of process flow rates when measurements are subject to detection limits: The bilinear case. Ind. Eng. Chem. Res. 1999, 38, 2861–2866. (5) Dovi, V. G.; Solisio, C. Reconciliation of censored measurements in chemical processes: an alternative approach. Chem. Eng. J. 2001, 84, 309–314. (6) Ungarala, S.; Bakshi, B. R. A multiscale, Bayesian and error-invariables approach for linear dynamic data rectification. Comput. Chem. Eng. 2000, 24, 445–451. (7) Kong, M.-F.; Chen, B.-Z.; Li, B. Simultaneous gross error detection and data reconciliation based on the robust estimation principle. Qinghua Daxue Xuebao 2000, 40, 46–50. (8) Dovi, V. G.; Del Borghi, A. Rectification of flow measurements in continuous processes subject to fluctuations. Chem. Eng. Sci. 2001, 56, 2851–2857.
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