Heat Balance Equations

Derivation of the working equations of upwinded schemes for heat transport in a polymeric flow is similar to the previously described weighted residual Petrov-Galerkm finite element method. In this section a basic outline of this derivation is given using a steady-state heat balance equation as an example. 

The rate constant k is expressed as k = exp(15.32 - 7,550/T), sec Determine the operating points for both the mass and heat balance equations. 

The first order reaction is represented by (-r ) = kC, and applying the mass balance Equation 6-120 and the heat balance Equation 6-121, respectively, gives the fractional conversion in terms of the mass balance equation ... 

The fractional conversions in terms of both the mass balance and heat balance equations were calculated at effluent temperatures of 300, 325, 350, 375, 400, 425, 450, and 475 K, respectively. A Microsoft Excel Spreadsheet (Example6-ll.xls) was used to calculate the fractional conversions at varying temperature. Table 6-7 gives the results of the spreadsheet calculation and Eigure 6-24 shows profiles of the conversions at varying effluent temperature. The figure shows that die steady state values are (X, T) = (0.02,300), (0.5,362), and (0.95,410). The middle point is unstable and die last point is die most desirable because of die high conversion. 

A second order reaction is performed adiabatically in a CESTR. Use die data in Example 6-11 to plot bodi conversions for die mass and heat balance equations. The second order rate constant k is... 

As has been mentioned earlier, within each temperature interval, it is ther modynamically as well as technically feasible to transfer beat from a hot process stream to a cold process stream. Moreover, it is feasible to pass heat from a hot process stream in an interval to any cold process stream in a lower interval. Hence, for the zth temperature interval, one can write the following heat-balance equation ... 

For the zth temperature interval, one can write the following heat-balance equation (see Fig. 9.14) ... 

The value used in ISO 7933, required sweat rate, SW is based on the heat balance equation (6.1). Assuming the heat storage is equal to 0, the necessary evaporation from the skin, > to ensure a heat balance is calculated as foil... 

The air exchange rate, G , required for temperature control in the occupied zone can be calculated from the room heat balance equation ... 

Substituting this into the heat balance equation for sensible heat only gives 0),(kW) = X p2 X Cp X (e, - 6,)... 

The air temperature of a room at any given time is given hy a heat-balance equation which includes the heat flux exchanged by convection at each wall element (A q ), the heat flow exchanged by ventilation (4>j,), the convective part of heat flow due to internal heat gains the convective part of heat flow due to the HVAC system (< >[,(.), and the variation of energy in the room air(r M ... 

The hunnidity ratio oJ a rotjm at any given time is given by a Jatent heat balance equation including the water vapor flows due to infiltration to ventilation to moisture transport through envelope elements... 

Applying the heat balance equation, AH = 0 = AH - AH -I-AHp, we realize that AHjj = 0 because the feed gases are supplied at the reference temperature of 25°C = 298°K. 

Parallel dispersions can be introduced by the mass and heat balance equations... 

N = L/d, or that each row of pellets amounts to a mixing cell. This is intuitively reasonable. However, the parallel dispersion for heat has a Peclet number equal to 0.5, which would argue that four rows of pellets should be considered to be a mixing cell. The heat balance equation for cell i in a cascade of N cells is... 

It can be seen that k in Eq. (10) replaces the system-describing parameters L and Ah in Eq. (1). A direct test of the hypothesis is therefore to plot (j> against k for fixed values of P, G, and d, with L and Ah varying. For the hypothesis to be correct, the data points must all lie on a smooth curve. Experience shows, however, that plotting (f> against k often produces an undue amount of scatter which may obscure and distort any true relationship existing. This enhanced scatter is caused by the cumulative effect of experimental errors in the various terms in the heat-balance equation from which the quality k is derived. 

The heat-balance equation in its general form applicable to any uniformly heated channel ... 

In Fig. 26, the dotted lines correspond to 100% quality at the tube exit and represent the 0/G ratios obtained from the following heat-balance equation for a uniformly heated tube (with k = 1) ... 

Mass and heat balance equations for typical gas-liquid reactors in heterogeneous systems at steady state... 

Correlate temperature-versus-time (distance) data with an appropriate function T = 71(f), replace T with 7(f) in heat balance equation (5.4-130) and minimize SSres defined as... 

Regions of stable and unstable operation determined by numerical simulation of mass and heat balances equations first- and second-order, autocatalytic, and product-inhibited kinetics graphically presented boundaries in co-ordinates in practice. safe operation if l/5e>2. Equality of heat generation and heat removal rates Semenov approach modified for first-order kinetics. 

Liquid is fed continuously to a stirred tank, which is heated by internal steam coils (Fig. 1.21). The tank operates at constant volume conditions. The system is therefore modelled by means of a dynamic heat balance equation, combined with an expression for the rate of heat transfer from the coils to the tank liquid. 

With no heat of reaction and neglecting any heat input from the agitator, the heat balance equation becomes... 

Liquid flows continuously into an initially empty tank, containing a full-depth heating coil. As the tank fills, an increasing proportion of the coil is covered by liquid. Once the tank is full, the liquid starts to overflow, but heating is maintained. A total mass balance is required to model the changing liquid volume and this is combined with a dynamic heat balance equation. 

For complex reaction systems, the heats of reaction of all individual reactions have to be estimated and the dynamic heat balance equations must include the heats of all the reactions. 

Since the pocket temperature, Tp, is now a variable in the system, an additional heat balance equation is now required for the pocket. This is of the same form as the bulb, except that heat is now transferred both to the pocket from the surrounding and from the pocket to the bulb. Thus... 

Assuming the liquid in the jacket is well-mixed, the heat balance equation for the jacket becomes... 

This gives the resultant heat balance equations as... 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

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