Constant volume heat flow

The first law of thermodynamics also tells you that if no work is done on or by the sample, that is, pressure and volume are held constant, any heat flow is counterbalanced by a change in internal energy. An exothermic reaction releasing heat to the surroundings, therefore, is accompanied by a decrease in internal energy, whereas an endothermic reaction has a concomitant increase in internal energy. 

From equation (14) a simple and explicit formula for V2 (and therefore for /gp) can be drived if a one-component gas with a constant specific heat flows through the nozzle. Since h = constant CpT in this case, /iq 2 = CpTo(l — T2/T0). In place of the parameter T2, it is generally more convenient to use P2 because the atmospheric pressure (to which P2 has been equated above) is usually specified in discussing propellant performance. For isentropic flow [equation (13)] of a one-component ideal gas with Cp — constant, it is well known and readily derived from equations (6), (7), and (1-9) that T2/T0 = (P2/PoY where the ratio of specific heats is y = Cplc (Cj, = specific heat at constant volume). Since... 

If 1 mole of a gas in a constant-volume system is heated, and both the heat flow and the gas tempera-ture are measured-as a function of time, the constant-volume heat capacity can be computed from... 

If heat flows into a gas kept at constant volume, the energy of the gas is increased by the amount of energy transferred by the heat flow. The ratio of the increase in energy to the increase in temperature of the system is the constant volume heat capacity, C . Thus, by definition,... 

The requirements with regard to a calorimeter can be derived on the basis of the above analysis of the measuring problem. The necessary operating conditions have to be defined first an isothermal, isoperibol, adiabatic, or a scanning calorimeter What temperature range What heating rate Any other boundary conditions a constant pressure, constant volume, gas flow rate, and so on ... 

The equation of state, Eq. (34), is also used to calculate several derivatives of the pressure with respect to temperature and density. These are required for several of the previous equations for describing the compressible flow inside and outside the expansion device. The constant-pressure and the constant-volume heat capacity at a given temperature and density are found in standard textbooks of thermodynamics [e.g., Sandler (28)] and read as follows after minor transformations ... 

Air control louvers or dampers, popular in the past for air flow control, are used primarily for only very low scale air flow control. Louvers are used in many winterized heat exchangers in extremely low ambient temperature locations to retain and recirculate warm air in completely enclosed heat exchangers, sometimes in very compHcated control schemes. The use of louvers on the discharge side of a fan to control air flow is inefficient and creates mechanical problems in the louvers because of the turbulence. A fan is a constant volume device, thus use of louvers to control flow is equivalent to... 

In this apparatus the plastic to be tested is heated in a barrel and then forced through a capillary die as shown in Fig. 5.16, Normally the ram moves at a constant velocity to give a constant volume flow rate, Q. From this it is conventional to calculate the shear rate from the Newtonian flow expression. 

Hence, Sensible heat = Air-mass flow rate (q , kg s ) x Specific heat capacity of humid air at constant volume (c ), which is 1.012 kj kg K. ... 

When steam at the saturation temperature contacts a surface at a lower temperature, and heat flows to the cooler surface, some of the steam condenses to supply the energy. With a sufficient supply of steam moving into the volume that had been occupied by the steam now condensed, the pressure and temperature of the steam will remain constant. Of course, if the condensate flows to a zone where it is no longer in contact with the steam it can cool below steam temperature while supplying heat to a cooler surface. 

As noted earlier, for a reaction at constant pressure, such as that taking place in an open coffee-cup calorimeter, the heat flow is equal to the change in enthalpy. If a reaction is carried out at constant volume (as is the case in a sealed bomb calorimeter) and there is no mechanical or electrical work involved, no work is done. Under these conditions, with w = 0, the heat flow is equal to the change in energy, AE. Hence we have... 

Students often ask, What is enthalpy The answer is simple. Enthalpy is a mathematical function defined in terms of fundamental thermodynamic properties as H = U+pV. This combination occurs frequently in thermodynamic equations and it is convenient to write it as a single symbol. We will show later that it does have the useful property that in a constant pressure process in which only pressure-volume work is involved, the change in enthalpy AH is equal to the heat q that flows in or out of a system during a thermodynamic process. This equality is convenient since it provides a way to calculate q. Heat flow is not a state function and is often not easy to calculate. In the next chapter, we will make calculations that demonstrate this path dependence. On the other hand, since H is a function of extensive state variables it must also be an extensive state variable, and dH = 0. As a result, AH is the same regardless of the path or series of steps followed in getting from the initial to final state and... 

The flow through the valve may be taken as isentropic and the expansion in the cylinder as isothermal. The ratio of the specific heats at constant pressure and constant volume is 1.4. 

In these equations x and y denote independent spatial coordinates T, the temperature Tib, the mass fraction of the species p, the pressure u and v the tangential and the transverse components of the velocity, respectively p, the mass density Wk, the molecular weight of the species W, the mean molecular weight of the mixture R, the universal gas constant A, the thermal conductivity of the mixture Cp, the constant pressure heat capacity of the mixture Cp, the constant pressure heat capacity of the species Wk, the molar rate of production of the k species per unit volume hk, the speciflc enthalpy of the species p the viscosity of the mixture and the diffusion velocity of the A species in the y direction. The free stream tangential and transverse velocities at the edge of the boundaiy layer are given by = ax and Vg = —ay, respectively, where a is the strain rate. The strain rate is a measure of the stretch in the flame due to the imposed flow. The form of the chemical production rates and the diffusion velocities can be found in (7-8). 

Figure 6-17 illustrates a constant-volume calorimeter of a type that is often used to measure q for combustion reactions. A sample of the substance to be burned is placed inside the sealed calorimeter in the presence of excess oxygen gas. When the sample bums, energy flows from the chemicals to the calorimeter. As in a constant-pressure calorimeter, the calorimeter is well insulated from its surroundings, so all the heat released by the chemicals is absorbed by the calorimeter. The temperature change of the calorimeter, with the calorimeter s heat capacity, gives the amount of heat released in the reaction. 

An ideal gas flows in steady state adiabatic flow along a horizontal pipe of inside diameter d, = 0.02 m. The pressure and density at a point are P = 20000 Pa and p = 200 kg/m3 respectively. The density drops from 200 kg/m3 to 100 kg/m3 in a 5 m length. Calculate the mass flux assuming that the Fanning friction factor /= 9.0 x 10 3 and the ratio of heat capacities at constant pressure and constant volume y = 1.40. 

The adsorption of carbon dioxide or oxygen on praseodymium samples was measured by a constant-volume method using a calibrated Pirani vacuum gauge. Praseodymium oxide was heated in oxygen (4 kPa) at 775°C for 1 h, then evacuated at 750°C for 0.5 h just before the measurement. The sample of praseodymium oxychloride was prepared from praseodymium chloride by heating under oxygen flow... 

The first law of thermodynamics simply says that energy cannot be created or destroyed. With respect to a chemical system, the internal energy changes if energy flows into or out of the system as heat is applied and/or if work is done on or by the system. The work referred to in this case is the PV work defined earlier, and it simply means that the system expands or contracts. The first law of thermodynamics can be modified for processes that take place under constant pressure conditions. Because reactions are generally carried out in open systems in which the pressure is constant, these conditions are of greater interest than constant volume processes. Under constant pressure conditions Equation 3 can be rewritten as... 

Taylor (Ref 5) obtd a transient flow behind a C-J discontinuity using Riemann equations for polytropic gases. A plot of u/u2 vs x/Ut shown in Fig 12 of Ref 6 (See here Fig 1) is for u2 =U/3, c2 = 2U/3 and y=1.3, where u is material velocity in x direction, u2 is material velocity immediately behind the discontinuity at Ut (U = velocity of C-J wave t = time coordinate) C2 = sound velocity and y = Cj/cv (cp=specific heat at constant pressure and cv = sp heat at constant volume), Taylor calculated pressure in the rarefaction wave behind C-J point and plotted it in F ig given as Fig 12 of Ref 6 (Our Fig 2)... 

Constant-volume calorimetry Constant-volume calorimetry directly measures a change in internal energy (A , not A/ for a reaction because it monitors heat flow at constant volume. Often, A and A//are very similar values. 

A common variety of constant-volume calorimetry is bomb calorimetry, a technique in which a reaction (often, a combustion reaction) is triggered within a sealed vessel called a bomb. The vessel is immersed in a water bath of known volume. The temperature of the water is measured before and after the reaction. Because the heat capacity of the water and the calorimeter are known, you can calculate heat flow from the change in temperature. 

With these stipulations, the model reduces to one more variable than there are relationships. Therefore, if temperature is considered constant, the model would become invariant otherwise we may consider a dynamic situation with temperature varying independently (as a function of meteorological conditions, heat flow, etc.). The actual temperature distribution of the Great Lakes (except for Lake Erie and shallow areas of other lakes) is quite limited relative to volume distribution, and to a first approximation it approaches a constant of about 5°C. This conclusion may be readily ascertained by realizing the typical depth of water above the thermocline (where there is a temperature gradient) is from 10-20 meters, whereas depth equivalent to water below the thermo-... 

How big a difference is there between AE, the heat flow at constant volume, and AEf, the heat flow at constant pressure Let s look again at the reaction of propane, C3H8, with oxygen as an example. When the reaction is carried out at constant volume, no PV work is possible and all the energy is released as heat AE = —2045 kj. When the same reaction is carried out at constant pressure, however, only 2043 kj of heat is released (AH = —2043 kj). The difference, 2 kj, occurs because at constant pressure, a small amount of expansion work is done against the atmosphere as 6 mol of gaseous reactants are converted into 7 mol of gaseous products. 

A gas cylinder containing air discharges to atmosphere through a valve whose characteristics may be considered similar to those of a sharp-edged orifice. If the pressure in the cylinder is initially 350 kN/m2, by how much will the pressure have fallen when the flowrate has decreased to one-quarter of its initial value The flow through the valve may be taken as isentropic and the expansion in the cylinder as isothermal. The ratio of the specific heats at constant pressure and constant volume is 1.4. 

The home-made heat-flow calorimeter used consisted of a high vacuum line for adsorption measurements applying the volumetric method. This equipment comprised of a Pyrex glass, vacuum system including a sample holder, a dead volume, a dose volume, a U-tube manometer, and a thermostat (Figure 6.3). In the sample holder, the adsorbent (thermostated with 0.1% of temperature fluctuation) is in contact with a chromel-alumel thermocouple included in an amplifier circuit (amplification factor 10), and connected with an x-y plotter [3,31,34,49], The calibration of the calorimeter, that is, the determination of the constant, k, was performed using the data reported in the literature for the adsorption of NH3 at 300 K in a Na-X zeolite [51]. 

For a non-flow reaction proceeding at constant volume the heat added is equal to the gain in the internal energy of the system ... 

The dimensions of this constant are clearly [Lz T l] = rate of flow, q/t = 0. Gardner regards this constant as equivalent to the capillary conductivity on the basis of heat and electrical analogies. This result is remarkable considering the assumptions involved. Eq (15-30) indicates that Xfl, the capillary conductivity, may be determined directly from measurement of the volume moisture-content V, the actual volume of flow per unit cross section Vv, and the gradient moisture-content dV/dx at the point x. These quantities are all experimentally observable. 

A homogeneous, constant-volume chemical reaction taking place in a tubular reactor is influenced by mass and heat transfer processes. The flow condition is described by... 

---