Heating rate, 93 independent variable

The basis for the calculations will be L = 100m. Because the insulation comes in 1-cm increments, let us calculate the net present value of insulating the pipe as a function of the independent variable jc vary x for a series of 1-, 2-, 3-cm (etc.) thick increments to get the respective internal rates of return, the payback period, and the return on investment. The latter two calculations are straightforward because of the assumption of five even values for the fuel saved. The net present value and internal rates of return can be compared for various thicknesses of insulation. The cost of the insulation is an initial negative cash flow, and a sum of five positive values represent the value of the heat saved. For example, for 1 cm insulation the net present value is (r = 0.291 from Table 3.1)... 

In order for a process to be controllable by machine, it must represented by a mathematical model. Ideally, each element of a dynamic process, for example, a reflux drum or an individual tray of a fractionator, is represented by differential equations based on material and energy balances, transfer rates, stage efficiencies, phase equilibrium relations, etc., as well as the parameters of sensing devices, control valves, and control instruments. The process as a whole then is equivalent to a system of ordinary and partial differential equations involving certain independent and dependent variables. When the values of the independent variables are specified or measured, corresponding values of the others are found by computation, and the information is transmitted to the control instruments. For example, if the temperature, composition, and flow rate of the feed to a fractionator are perturbed, the computer will determine the other flows and the heat balance required to maintain constant overhead purity. Economic factors also can be incorporated in process models then the computer can be made to optimize the operation continually. 

Experimentally observed quantities pertaining to the whole surface, such as the amount of adsorbed substance, heat of adsorption, reaction rate, are sums of contributions of surface sites or, since the number of sites is extremely great, the respective integrals. As increases monotonously with s, each of them can be taken as variable for integration both methods of calculation are used. If is chosen as an independent variable, a differential function of distribution of surface sites with respect to desorption exponents, [Pg.211]

A situation sometimes occurs in which the maximum temperature of a utility coolant fluid is fixed, thereby setting the coolant flow rate. In other cases, the operating circumstances may arbitrarily set the utility-fluid flow rate as well as the process-fluid flow rate. Under these conditions, At, is constant, and only two independent variables are involved in the optimization procedure. Since At, was held constant in the development of Eqs. (45c), (46b), and (47), these equations can be used for determining the optimum overall heat-transfer coefficient. This coefficient, combined with the known values of At, and At, is sufficient to permit calculation of the optimum heat-transfer area by Eq. (39), and the rest of the design variables are then established. 

Extended nonequilibrium thermodynamics is not based on the local equilibrium hypothesis, and uses the conserved variables and nonconserved dissipative fluxes as the independent variables to establish evolution equations for the dissipative fluxes satisfying the second law of thermodynamics. For conservation laws in hydrodynamic systems, the independent variables are the mass density, p, velocity, v, and specific internal energy, u, while the nonconserved variables are the heat flux, shear and bulk viscous pressure, diffusion flux, and electrical flux. For the generalized entropy with the properties of additivity and convex function considered, extended nonequilibrium thermodynamics formulations provide a more complete formulation of transport and rate processes beyond local equilibrium. The formulations can relate microscopic phenomena to a macroscopic thermodynamic interpretation by deriving the generalized transport laws expressed in terms of the generalized frequency and wave-vector-dependent transport coefficients. 

The main variables associated with phase relationships include the overall composition, Z , temperature, pressure, liquid composition, X , vapor composition, F, vapor mole fraction, /, and heat transferred, Q. A process in which Z, and two other independent variables are set, and equilibrium separation of the phases is allowed to take place, is called a flash operation. A general flash operation is shown in Figure 2.4. A feed stream initially at conditions T, and P, is controlled so that its final conditions satisfy two specifications. The feed is of fixed rate and composition, F and Z . A heat duty, Q, may be added to or removed from the system as required. The feed is flashed to generate a vapor product with flow rate Ft r and a liquid product with flow rate F(1 -1 /), where / is the vapor mole fraction at flash conditions and P. In general, tj/ may be equal to zero or one or any value in between. The enthalpies of the vapor and liquid products are H2 and /Z2> respectively. The type of flash operation... 

If one independent variable is the temperature or pressure and the only product is a saturated liquid (i.e., the other independent variable is the vapor fraction set at the zero limit), then the operation is a bubble point flash. The dependent variables are the pressure or temperature, the heat duty, and the liquid and vapor compositions. Since the liquid is the only product, its composition is identical to the feed composition. The vapor composition refers to the composition of a vapor at equilibrium with the liquid. Although its rate is zero, its composition is determinable and is equal to the composition of the first vapor bubble resulting from infinitesimal vaporization. Bubble points are not feasible at all temperatures and pressures. As discussed in... 

An existing column has a reboiler, no condenser, and two feeds, one of which is at the top of the column. The top feed is of fixed temperature, pressure, and composition, but its rate is variable. The other feed is of fixed rate, composition, and pressure, and flows through a heat exchanger upstream of the column so that its temperature may be controlled before entering the column. How many independent variables are required to define this process If the key components are ethane and propane, describe conceptual control loops to control the separation and performance of this process. 

Regression analysis is often employed to fit experimental data to a mathematical model. The purpose may be to determine physical properties or constants (e.g., rate constants, transport coefficients), to discriminate between proposed models, to interpolate or extrapolate data, etc. The model should provide estimates of the uncertainty in calculations from the resulting model and, if possible, make use of available error in the data. An initial model (or models) may be empirical, but with advanced knowledge of reactors, distillation columns, other separation devices, heat exchangers, etc., more sophisticated and fundamental models can be employed. As a starting point, a linear equation with a single independent variable may be initially chosen. Of importance, is the mathematical model linear In general, a function,/, of a set of adjustable parameters, 3y, is linear if a derivative of that function with respect to any adjustable parameter is not itself a function of any other adjustable parameter, that is. 

W2) is the Hildebrand solubility parameter, which is calculated from the molar heat of vaporization. The coefficients of the independent variables are understood in the present case to represent the sensitivity of the rate to the associated variables. 

If we consider the fluid outlet temperatures or heat transfer rate as dependent variables, they are related to independent variable/parameters of Fig. 17.21 as follows. 

The calibration coefficient is also dependent on other instrumental variables such as furnace atmosphere. David (104) determined X as a function of temperature while varying the composition of the furnace atmosphere (air, N2, or He) and also the pressure (5 x 10 6 Torr to 147 atm). The effect of different gaseous atmospheres on the value of K is shown in Figure 5.40 (104). The difference between the curves can be related to the different thermal conductivities of the gases studied. It was found, as expected, that K was independent of heating rate in the range from 2-40°C/min. 

This problem requires an analysis of coupled thermal energy and mass transport in a differential tubular reactor. In other words, the mass and energy balances should be expressed as coupled ordinary differential equations (ODEs). Since 3 mol of reactants produces 1 mol of product, the total number of moles is not conserved. Hence, this problem corresponds to a variable-volume gas-phase flow reactor and it is important to use reactor volume as the independent variable. Don t introduce average residence time because the gas-phase volumetric flow rate is not constant. If heat transfer across the wall of the reactor is neglected in the thermal energy balance for adiabatic operation, it... 

If convergence problems arise in the numerical solutions, especially for hand Calculations, it is often useful to use conversion as the independent variable. Thus, increments of conversion give increments of time from the mass balance, and these give increments of temperature from the heat balance iterations on the evaluations of the rates are also often required. 

Many industrial processes take place in open systems in which material enters and leaves the system through process streams and in which energy can cross system boundaries as heat and work. At any instant, a complete identification of the state requires specification of values for such variables as temperatures, pressures, compositions, and flow rates. However, because of the stuff equations in 3.6.1, not all of these quantities are independent. So we have here the same kinds of questions addressed in 3.1 How many interactions are available to change the state How many independent variables must be specified to identify the state of an open steady-flow system The discussion here extends that in 3.1 from closed systems to open ones however, the discussion remains limited to systems composed of a single homogeneous phase with no chemical reactions. The extensions to multiphase systems are given in 9.1 and to those having chemical reactions in 10.3.1... 

Theoretically the enthalpy of vaporization is independent of the heating rate. In practice it is not independent, but shows a variable increase at increasing heating rates. However, the maximum temperatures of the evaporation peaks do shift depending on the heating rates. This fact permits the ealculation of the aetivation energy of the evaporation process, which does not exhibit any temperature dependence. Theoretical considerations lead to the eonclusion that this aetivation energy is equal to the molar enthalpy of vaporization. 

The design of a distillation column involves many parameters product compositions, product flow rates, operating pressure, total number of trays, feed-tray location, reflux ratio, reboiler heat input, condenser heat removal, column diameter, and column height. Not all of these variables are independent, so a degrees of freedom analysis is useful in pinning down exactly how many independent variables can (and must) be specified to completely define the system. 

Under this category, three different variables are considered heating rate, as well as time and temperature of heat treatment. These parameters affect the activation process independently of the activating agent (KOH or NaOH) and the precnrsor used. 

Equation 17 can be solved numerically for any specified temperature history. However, the mesophase temperature history is a function of both time and position (see Fig. 4), and so the transient temperature distribution T(x, t) must be specified in order to obtain an analytic solution for equation 17. If the temperature distribution is static during steady burning of a thick sample as the surface x = 0 recedes at constant velocity v = (dx/df)r and the mesophase is a thin surface layer, then the rate of temperature rise of the mesophase is constant, (dT/df)a =o = -v(dT/dx)t = The assumption of a constant heating rate for the mesophase transforms the independent variable in equation 17 from time to temperature and allows for a solution in terms of the mass fraction of polymer remaining at temperature T... 

Most FCC units only have a few independent variables. Typically, these independent variables are the feed rate, feed preheat temperature, reactor/riser temperature, air flow rate to the regenerator, and catalyst activity. The feed rate and air flow rate to the regenerator are set by flow controllers. The feed temperature is set by the feed temperature controller. Catalyst activity is set by catalyst selection and fresh catalyst addition rate. Reactor temperature is controlled by the regenerator slide valve that regulates the catalyst circulation rate. The catalyst circulation rate is not directly measured or controlled. Instead, the unit relies on the heat balance to estimate the catalyst circulation rate. Except for these independent variables, other variables, such as regenerator temperature, degree of conversion, and carbon-on-catalyst, etc., will vary accordingly to keep the FCC unit in heat balance. These variables are dependent variables. 

---