Heat capacity, constant

A number of properties can be computed from various chemical descriptors. These include physical properties, such as surface area, volume, molecular weight, ovality, and moments of inertia. Chemical properties available include boiling point, melting point, critical variables, Henry s law constant, heat capacity, log P, refractivity, and solubility. 

Fig. 3. Temperature—enthalpy representation of stream where A represents a pure component that is condensiag, eg, steam B and C represent streams having constant heat capacity, that are to be heated or cooled, respectively and D represents a multicomponent mixture that changes phase as it is...

Adiabatic process with AC = 0 and with constant heat capacities AH,iqs(n o - n, )... 

Amplitude of controlled variable Output amplitude limits Cross sectional area of valve Cross sectional area of tank Controller output bias Bottoms flow rate Limit on control Controlled variable Concentration of A Discharge coefficient Inlet concentration Limit on control move Specific heat of liquid Integration constant Heat capacity of reactants Valve flow coefficient Distillate flow rate Limit on output Decoupler transfer function Error... 

For heat exchangers in true counter-current (fluids flowing in opposite directions inside or outside a tube) or true co-current (fluids flowing inside and outside of a tube, parallel to each other in direction), with essentially constant heat capacities of the respective fluids and constant heat transfer coefficients, the log mean temperature difference may be appropriately applied, see Figure 10-33. ... 

In the next chapter, we will return to the Carnot cycle, describe it quantitatively for an ideal gas with constant heat capacity as the working fluid in the engine, and show that the thermodynamic temperature defined through equation (2.34) or (2.35) is proportional to the absolute temperature, defined through the ideal gas equation pVm = RT. The proportionality constant between the two scales can be set equal to one, so that temperatures on the two scales are the same. That is, 7 °Absolute) = T(Kelvin).r... 

Example 3.8 Show that for the reversible adiabatic expansion of ideal gas with constant heat capacity... 

Integration assuming constant heat capacities gives C... 

Figure 7.3 shows how the entropy of a substance changes as it is heated through a temperature range in which it has a constant heat capacity. 

FIGURE 7.3 The change in entropy as a sample is heated for a system with a constant heat capacity (O in the range of interest. Here we have plotted AS/C. The entropy increases logarithmically with temperature. 

Change in entropy when a substance of constant heat capacity, C, is heated from T, to T2 ... 

Assuming constant heat capacity, the following balance equations apply For the tubeside fluid... 

A pump contains 4 kg of water. If the pump is rated at 1 HP, what is the maximum temperature increase expected in the water in °C/hr Assume a constant heat capacity for the water of 1 kcal/kg°C. What will happen if the pump continues to operate ... 

Adiabatic process with ACp =0 and with constant heat capacities... 

Figure 2-13 Schematic drawing of (a) density as a function of temperature, and (b) entropy as a function of temperature for glasses with different cooling rates and hence different glass transition temperature (Martens et al., 1987). The entropy of the undercooled liquid is estimated assuming constant heat capacity.

Principles of the adiabatic reactor method have been discussed elsewhere [67,68], Under adiabatic conditions, assuming constant heat capacity, constant heat of reaction, and homogeneous reaction, temperature rise data yields fractional conversion, X [68] ... 

The terms in Equation 1.2 are described in Nomenclature. The condition of constant heat capacity can be relaxed if accurate data is available for heat capacity as a function of both conversion and temperature. 

The heat absorbed on heating a sample with constant heat capacity, Cp, between temperatures T0 and T is, by definition,... 

Then, assume that the reaction takes place in a fixed bed of 1.61 m diameter and 16.1 m height, under contact time of 5 min, and the inlet temperature of gas being 50 °C, for different CO inlet concentration (several runs). Estimate the conversion of CO in an isothermal and adiabatic fixed-bed reactor and under the following assumptions isobaric process, negligible external mass transfer resistance, and approximately constant heat capacity of air (cp = 1 kJ/kg K) and heat of reaction (AH = -67,636 cal/mol). The inlet temperature of the reaction mixture is 50 °C and its composition is 79% N2 and approximately 21% 02, while the inlet CO concentration varies from 180-4000 ppm (mg/kgair) (for each individual ran). 

Some physical properties of water are shown in Table 7.2. Water has higher melting and boiling temperatures, surface tension, dielectric constant, heat capacity, thermal conductivity and heats of phase transition than similar molecules (Table 7.3). Water has a lower density than would be expected from comparison with the above molecules and has the unusual property of expansion on solidification. The thermal conductivity of ice is approximately four times greater than that of water at the same temperature and is high compared with other non-metallic solids. Likewise, the thermal dif-fusivity of ice is about nine times greater than that of water. 

Hence, for temperatures very close to the boiling point, we integrate Eq. 4-7 by assuming that Avap//,(T) = AvapH,(Tb) = constant (see Section 4.2). However, in most cases, one would like to estimate the vapor pressure at temperatures (e.g., 25°C) that are well below the boiling point of the compound. Therefore, one has to account for the temperature dependence of Avap// below the boiling point. A first approximation is to assume a linear temperature dependence of Avap/7, over the temperature range considered, that is, to assume a constant heat capacity of vaporization, A Cpi (the difference between the vapor and liquid heat capacities). Thus, if the heat capacity of vaporization, AvapCpi(Tb), at the boiling point is known, Avap/7,(7) can be expressed by (e.g., Atkins, 1998) ... 

Theorem 1 (Corner Point Theorem). Assume the following (1) constant heat capacities and no phase change, (2) supply and target temperature uncertainties only (no uncertainties in flow rates or heat transfer coefficients), (3) constant stream split fractions (Saboo et al., 1987b), and... 

Saboo and Morari (1984) give the general test to determine whether a problem is class 1 or class 2. In their test the uncertainty range can include supply temperatures, target temperatures, and flow rates. (It would not matter whether the uncertainty range included heat transfer coefficients.) In this chapter, we restrict their test to problems with constant or piecewise constant heat capacities. 

Supply Temperatures, Target Temperatures, and Flow Rates Corresponding to Maximum Heating and Maximum Cooling (When All Streams Have Constant Heat Capacity and No Phase Change... 

Heat exchanger network resilience analysis can become nonlinear and nonconvex in the cases of phase change and temperature-dependent heat capacities, varying stream split fractions, or uncertain flow rates or heat transfer coefficients. This section presents resilience tests developed by Saboo et al. (1987a,b) for (1) minimum unit HENs with piecewise constant heat capacities (but no stream splits or flow rate uncertainties), (2) minimum unit HENs with stream splits (but constant heat capacities and no flow rate uncertainties), and (3) minimum unit HENs with flow rate and temperature uncertainties (but constant heat capacities and no stream splits). 

Resilience Analysis with Piecewise Constant Heat Capacities... 

Floudas and Grossmann (1987b) have shown that for HENs with any number of units, with or without stream splits or bypasses, and with uncertain supply temperatures and flow rates but with constant heat capacities, the active constraint strategy decomposes the resilience test (or flexibility index) problem into NLPs which have a single local optimum. Thus the resilience test (or flexibility index) also has a single local optimum solution. 

Different algorithms are required to solve these three basic resilience analysis problems depending on whether the problem is linear, nonlinear, or class 2. A HEN resilience problem is linear under the following conditions (corner point theorem, Saboo and Morari, 1984) (1) constant heat capacities and no phase change, (2) temperature uncertainties only... 

Different algorithms are required if the HEN resilience problem is nonlinear. Special algorithms were presented for testing the resilience of minimum unit HENs with piecewise constant heat capacities, stream splits, or simultaneous flow rate and temperature uncertainties. A more general algorithm, the active constraint strategy, was also presented which can test the resilience or calculate the flexibility index of a HEN with minimum or more units, stream splits and/or bypasses, and temperature and/or flow rate uncertainties, but with constant heat capacities. 

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