Mass Input, Output, and Control

This can be factored because the coefficient of t is two times the square root of the coefficient of Therefore, we find  

A comparison of the function to the data points gives an excellent fit  

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This can be done with an MFC (mass flow controller). An MFC will measure the flow of gas, as well as control the flow when set up with attendant circuitry. The particular model shown in the photo has a 0-5 input/output and is powered by 15 volt DC source. It has a maximum operating pressure of 1500 psi and the valve is NC (normally closed). 

FI) Continuous-flow system There is at least one input stream and one output stream of material the mass inside the control volume may vary. 

In equation 14.3-8, subscript o represents an inlet condition, cP is the specific heat of the (total) system as indicated, and mt is the total mass contained in the control volume at time f the interpretation of the various quantities is shown in Figure 14.3. The first term on the left side is the input of enthalpy by flow, the second term is the output of enthalpy by flow, and the third and fourth terms represent heat transfer and enthalpy generation or consumption by reaction, respectively. 

Here A represents the cross-sectional area of the inlet and outlet, z their elevations, u the fluid s mean velocity and p its density, nip is the total mass within the control volume, q rate of heat input to the control volume per unit mass and wx shaft work output from the control volume per unit mass. Note this is the Engineers convention for shaft work. In the Chemists convention, positive shaft work is an input to the control volume, i.e. the direction of positive shaft work is opposite. 

Next we consider the rate of energy input and output associated with mass in the control volume. The mass added or removed from the system carries internal, kinetic, and potential energy. In addition, energy is transferred when mass flows into and out of the control volume. Net work is done by the fluid as it flows into and out of the control volume. This pressure-volume work per unit mass fluid is pV. The contribution of shear work is usually neglected. The pV term and U term are combined using the definition of enthalpy, H. 

In many cases the relationships between plant inputs and outputs may be derived in the form of a mathematical model, which can then be used to understand and control the plant. For the tank example in Figure 12.17 the dynamic mass balance becomes Eq. (38), where p is the liquid density, A is the cross-sectional area of the tank, h is the height of liquid in the tank, and q, qd arc the inlet and outlet volumetric flow rates [7, 22]. If we assume constant density this becomes Eq. (39). 

The name material balance control was introduced by Shinkey (1984). The different control schemes that the author developed were based on the concept of relative gains (= power of control) of the different input-output combinations. Speed of control was only considered as a secondary factor. A simple explanation is given by Ryskamp (1980). Also Van der Grinten (1970) presented a nrrmber of common control schemes for distillation colnmns. The latter author used behavioral models in the eontrol scheme selection procedure. None of the mentioned references takes inverse responses into accoimt when X > 0.5. In the case of the more traditional approach, the energy balanee eontrol, the reflux ratio and/or vapor flow is used to eontrol the top product qrrality, while the distillate and bottom flow are nsed to maintain the mass balanee. In the ease of the material balance control, one of the prodnct flows is used to control product qrrahty, while the other product flow maintains the material balance. 

We have previously discussed outputs and inputs for process models we now introduce more precise working definitions. The word output generally refers to a controlled variable in a process, a process variable to be maintained at a desired value (set point). For example, the output from the stirred blending tank just discussed is the mass fraction x of the effluent stream. The word input refers to any variable that influences the process output, such as the flow rate of the stream flowing into the stirred blending tank. The characteristic feature of all inputs, whether they are disturbance variables or manipulated variables, is that they influence the output variables that we wish to control. 

In principle, any type of process model can be used to predict future values of the controlled outputs. For example, one can use a physical model based on first principles (e.g., mass and energy balances), a linear model (e.g., transfer function, step response model, or state space-model), or a nonlinear model (e.g., neural nets). Because most industrial applications of MPC have relied on linear dynamic models, later on we derive the MPC equations for a single-input/single-output (SISO) model. The SISO model, however, can be easily generalized to the MIMO models that are used in industrial applications (Lee et al., 1994). One model that can be used in MPC is called the step response model, which relates a single controlled variable y with a single manipulated variable u (based on previous changes in u) as follows ... 

The construction of a mass balance model follows the general outline of this chapter. First, one defines the spatial and temporal scales to be considered and establishes the environmental compartments or control volumes. Second, the source emissions are identified and quantified. Third, the mathematical expressions for advective and diffusive transport processes are written. And last, chemical transformation processes are quantified. This model-building process is illustrated in Figure 27.4. In this example we simply equate the change in chemical inventory (total mass in the system) with the difference between chemical inputs and outputs to the system. The inputs could include numerous point and nonpoint sources or could be a single estimate of total chemical load to the system. The outputs include all of the loss mechanisms transport... 

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