Disturbance Condition Number

Consider a single (scalar) disturbance and let the vector represent its effect on the outputs. The disturbance condition number or DCN is defined as ... 

Calculate the frequency-dependent disturbance condition number, and compare with the plant condition number. 

All of the linear C R measures use the approximations, F s and which describe the effects of the control variables and disturbances, respectively, on the process outputs. A commonly used controllability measure is the relative-gain array (RGA Bristol, 1966), which relies only on F s. The disturbance condition number (DCN Skogestad and Morari, 1987) and the disturbance cost (DC Lewin, 1996) are resiliency measures that require a disturbance model, in addition to P s. These C R measures are especially useful in... 

Optimising polymerisation reactors can cause major controllability problems. Lewin and Bogle [28] showed how optimising a methylmethacrylate polymerisation reactor brought the operating conditions much closer to the bifurcation point which causes major problems with controlling to this set point. The use of the disturbance condition number indicated this problem. The effects are nonlinear which is reflected in different linearised measures at different operating points. The problem also exhibits input multiplicity conditions. 

The issue of this kind of control configuration has been investigated using frequency dependent formulations of measures such as the condition number, the Relative Gain Array by Bristol (1966) and the Relative Disturbance Gain by Stanley et al (1985). This paper will focus on discussing the dynamic control structure on the heat pump section and how each dynamic control structure affects on the stability of the integrated distillation column. 

Laminar flow ceases to be stable when a small perturbation or disturbance in the flow tends to increase in magnitude rather than decay. For flow in a pipe of circular cross-section, the critical condition occurs at a Reynolds number of about 2100. Thus although laminar flow can take place at much higher values of Reynolds number, that flow is no longer stable and a small disturbance to the flow will lead to the growth of the disturbance and the onset of turbulence. Similarly, if turbulence is artificially promoted at a Reynolds number of less than 2100 the flow will ultimately revert to a laminar condition in the absence of any further disturbance. 

Similarly it may be shown that, at the critical conditions, the flowrate is a maximum for a given value of the specific energy J. At the critical velocity, (ir/gD) is equal to unity. This dimensionless group is known as the Froude number Fr. For velocities greater than the critical velocity Fr is greater than unity, and vice versa. It may be shown that the velocity with which a small disturbance is transmitted through a liquid in an open channel is equal to the critical velocity, and hence the Froude number is the criterion by which the type of flow, tranquil or rapid, is determined. Tranquil flow occurs when Fr is less than unity and rapid flow when Fr is greater than unity. 

The ratio of t/t, which is characteristic of the possibility of vortices, does not depend on the micro-channel diameter and is fully determined by the Reynolds number and L/d. The lower value of Re at which f/fh > 1 can be treated as a threshold. As was shown by Darbyshire and Mullin (1995), under conditions of an artificial disturbance of pipe flow, a transition from laminar to turbulent flow is not possible for Re < 1,700, even with a very large amplitude of disturbances. 

Mast cells express high-affinity IgE Fc receptors (FceRI) on their surface, contain cytoplasmic granules which are major sources of histamine and other inflammatory mediators, and are activated to release and generate these mediators by IgE-dependent and non-IgE-dependent mechanisms [1]. Disturbances either in the release of mast cell mediators or in mast cell proliferation are associated with clonal mast cell disorders including monoclonal mast cell activation syndrome (MMAS) and mastocytosis respectively, which are in turn associated with some cases of anaphylaxis [2], Molecular mechanisms have been identified which may link increased releasability of mast cell mediators and conditions leading to increased mast cell numbers [3]. Patients with mastocytosis have an increased risk to develop anaphylaxis [4, 5] and those with anaphylaxis may suffer from unrecognized mastocytosis or may display incomplete features of the disease [6-8]. 

Microbial communities can respond to disturbances, such as contamination, in many different ways any of these responses may result in perceived stability or the continuation of essential soil functions [80]. The key species may show resistance to perturbation, meaning that the pollutants have no negative (or positive) effect on them. If the initial reaction is negative but the key species are able to regain their numbers and functionality, the community is said to be resilient. If the key species are irreversibly affected but are replaced by other indigenous species that are able to perform the same task under the new conditions, we see redundancy. Only if all these backup strategies fail will the deleterious effects of contamination on soil functions be observed. 

Careful study of various fluids in tubes of different sizes has indicated that laminar flow in a tube persists up to a point where the value of the Reynolds number (NRt = DVp/n) is about 2000, and turbulent flow occurs when NRe is greater than about 4000, with a transition region in between. Actually, unstable flow (turbulence) occurs when disturbances to the flow are amplified, whereas laminar flow occurs when these disturbances are damped out. Because turbulent flow cannot occur unless there are disturbances, studies have been conducted on systems in which extreme care has been taken to eliminate any disturbances due to irregularities in the boundary surfaces, sudden changes in direction, vibrations, etc. Under these conditions, it has been possible to sustain laminar flow in a tube to a Reynolds number of the order of 100,000 or more. However, under all but the most unusual conditions there are sufficient natural disturbances in all practical systems that turbulence begins in a pipe at a Reynolds number of about 2000. 

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