Ratio of the Heat Capacities

As can be seen from the relatively simple design equations, the ratio of the heat capacities, k, is an important physical property 

The heat capacities of real gases are a function of the pressure and thus may differ from the ideal gas case shown in the plot. However, the author s experience is that using the ideal gas k is sufficient for most engineering applications. 

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The heat capacity of a subshince is defined as the quantity of heat required to raise tlie temperature of tliat substance by 1° the specific heat capacity is the heat capacity on a unit mass basis. The term specific heat is frequently used in place of specific heat capacity. This is not strictly correct because traditionally, specific heal luis been defined as tlie ratio of the heat capacity of a substance to the heat capacity of water. However, since the specific heat of water is approxinuitely 1 cal/g-°C or 1 Btiiyib-°F, the term specific heal luis come to imply heat capacity per unit mass. For gases, tlie addition of heat to cause tlie 1° tempcniture rise m iy be accomplished either at constant pressure or at constant volume. Since the mnounts of heat necessary are different for tlie two cases, subscripts are used to identify which heat capacity is being used - Cp for constant pressure or Cv for constant volume. Tliis distinction does not have to be made for liquids and solids since tliere is little difference between tlie two. Values of heat capacity arc available in the literature. ... 

R = Ratio of the heat capacities of tube-side to shell-side fluid, dimensionless... 

The ratio of the heat capacity at constant pressure to that at constant volume is... 

The Nusselt number with respect to the tube Nu(= hdt/k) is expressed as a function of four dimensionless groups the ratio of tube diameter to length, the ratio of tube to particle diameter, the ratio of the heat capacity per unit volume of the solid to that of the fluid, and the tube Reynolds number, Rec = (ucdtp/p,). However, equation 6.59 and other equations quoted in the literature should be used with extreme caution, as the value of the heat transfer coefficient will be highly dependent on the flow patterns of gas and solid and the precise geometry of the system. 

Cp - Cy equals [P + (dU/dV)T](dV/dT)p. The dUldV term is often referred to as the internal pressure and is large for liquids and solids (See Internal Pressure). Since ideal gases do not have internal pressure, Cp - Cy = nP for ideal gases. The ratio of the heat capacities, Cp/Cy, is commonly symbolized by y. 

Recall (see any standard physical chemistry text) that for adiabatic expansions or compressions of an ideal gas, there are several relationships between P, V, and T that hold e.g., PVy = constant, where y is the ratio of the heat capacities at constant pressure and volume, i.e., y = cp/cv. Most useful in the context of potential temperature is TPy/y 1 = constant. Applying this latter relationship,... 

Another useful term is specific heat, which is the ratio of the heat capacity of one substance to the heat capacity of a reference material. The heat capacity of water is approximately unity in cgs and American engineering units. 

In these equations, e represents the relative volume increase due to the feed and Rh the ratio of the heat capacities of both liquid phases. By representing the reactivity number as a function of the exothermicity number (Figure 5.3), different regions are obtained. The region where runaway occurs is clearly delimited by a boundary line. Above this region, for a high reactivity, the reaction is operated in the QFS conditions (Quick onset, Fair conversion and Smooth temperature profile) and leads to a fast reaction with low accumulation and easy temperature control (see Section 7.6). 

The ARC analysis has been extended to determine the conditions that may lead to thermal runaway in reactors or storage vessels [10]. The equipment timeline is defined by the ratio of the heat capacity of the reactor and its contents to the reactor heat transfer area and heat transfer coefficient. This is expressed by... 

When the exponent on the ratio of the pressures is other than the ratio of the heat capacities, this is a general expression for a polytropic expansion. 

The temperature changes in both fluids are linked to each other due to the first law of thermodynamics. They are related inversely to the ratio of the heat capacity flow rates. 

For certain monatomic gases, such as helium, neon, argon, and mercury and sodium vapors, the ratio of the heat capacities at moderate temperatures has been found to be very close to 1.67, as required by equation (15.6). The values of the individual heat capacities at constant pressure and constant volume are 5.0 and 3.0 cal. deg. mole , respectively, in agreement with equations (15.5) and (15.4). It appears, therefore, that for a number of monatomic gases the energy of the molecules, at least that part which varies with temperature and so affects the heat capacity, is entirely, or almost entirely, translational in character (see, however, 16f). 

Specific heat. The ratio of the heat capacity of a substance to the heat capacity of water, or the quantity of heat required for a 1 degree temperature change in a unit weight of material. Commonly expressed in Btu/lb/degree F or in cal/g/degree. For gas, the specific heat at constant pressure is greater than that at constant volume by the amount of heat needed for expansion. 

Here P indicates the effectiveness of the heat exchanger (to be elaborated in Section 7.4) and R (from its definition) is the ratio of the heat-capacity flow rates Note the change in nomenclature from subscripts h and c to t and s, the latter two referring to tube and shell, respectively. An important fact is that whether the hot (or cold) fluid is flowing in the shell side or in the tubes has no effect on F as long as the heat transfer to the ambient is negligible. Otherwise, the cold fluid should be in the shell side to reduce heat losses Combination of Eqs (7.28) and (7.29) gives... 

Some texts use the term specific heat in place of specific heat capacity. This usage is very common but somewhat incon ect. Specific heat is the ratio of the heat capacity of 1 g of a substance to the heat capacity of 1 g of H2O and therefore has no units. 

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