Experimental values of heat capacities at various temperatures are presented in the form of tables, graphs and empirical functions.
There are true and average heat capacities.
True heat capacity C is the heat capacity for a given temperature.
In engineering calculations, the average value of heat capacity in a given temperature range (t1;t2) is often used.
The average heat capacity is denoted in two ways: ,.
The disadvantage of the latter designation is that the temperature range is not specified.
True and average heat capacities are related by the relation:
True heat capacity is the limit to which the average heat capacity tends, in a given temperature range t1…t2, at ∆t=t2-t1
Experience shows that the true heat capacities of most gases increase with increasing temperature. The physical explanation for this increase is as follows:
It is known that the temperature of a gas is not related to the vibrational motion of atoms and molecules, but depends on the kinetic energy E k of the translational motion of particles. But as the temperature increases, the heat supplied to the gas is more and more redistributed in favor of the oscillatory motion, i.e. The temperature increase with the same heat supply slows down as the temperature rises.
Typical dependence of heat capacity on temperature:
c=c 0 + at + bt 2 + dt 3 + … (82) 
where c 0 , a, b, d are empirical coefficients.
c – True heat capacity, i.e. value of heat capacity for a given temperature T.
For the heat capacity, the bitoproximate curve is a polynomial in the form of a series in powers of t.
The fitting curve is carried out using special methods, for example, the least squares method. The essence of this method is that when used, all points are approximately equidistant from the approximating curve.
For engineering calculations, as a rule, they are limited to the first two terms on the right side, i.e. assume the dependence of heat capacity on temperature is linear c=c 0 + at (83)
The average heat capacity is graphically defined as the middle line of a shaded trapezoid; as is known, the average line of a trapezoid is defined as half the sum of the bases.
Formulas are applied if the empirical dependence is known.
In cases where the dependence of heat capacity on temperature cannot be satisfactorily approximated to the dependence c=c 0 +at, you can use the following formula:
This formula is used in cases where the dependence of c on t is significantly nonlinear.
From the molecular kinetic theory of gases it is known
U = 12.56T, U is the internal energy of one kilomole of an ideal gas.
Previously obtained for an ideal gas:
,
,
From the obtained result it follows that the heat capacity obtained using MCT does not depend on temperature.
Mayer's equation: c p -c v =R ,
c p =c v +R =12.56+8.31420.93.
As in the previous case for MCT of gases, the molecular isobaric heat capacity does not depend on temperature.
The concept of an ideal gas most closely corresponds to monatomic gases at low pressures; in practice, we have to deal with 2, 3... atomic gases. For example, air, which by volume consists of 79% nitrogen (N 2), 21% oxygen (O 2) (in engineering calculations, inert gases are not taken into account due to their low content).
You can use the following table for estimation calculations:
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monatomic | ||
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diatomic | ||
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triatomic |
For real gases, unlike ideal gases, heat capacities can depend not only on temperature, but also on the volume and pressure of the system.
Considering that heat capacity is not constant, but depends on temperature and other thermal parameters, a distinction is made between true and average heat capacity. The true heat capacity is expressed by equation (2.2) for certain parameters of the thermodynamic process, that is, in a given state of the working fluid. In particular, if they want to emphasize the dependence of the heat capacity of the working fluid on temperature, then they write it as , and the specific heat capacity as. Typically, true heat capacity is understood as the ratio of the elementary amount of heat that is imparted to a thermodynamic system in any process to the infinitesimal increase in the temperature of this system caused by the imparted heat. We will assume that the true heat capacity of a thermodynamic system at the temperature of the system is equal, and the true specific heat of the working fluid at its temperature is equal. Then the average specific heat capacity of the working fluid when its temperature changes can be determined as follows:
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Typically, tables give average heat capacity values for various temperature ranges starting with. Therefore, in all cases when a thermodynamic process takes place in the temperature range from to in which, the amount of specific heat of the process is determined using tabulated values of average heat capacities as follows:
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.The values of average heat capacities and are found from the tables.
2.3. Heat capacities at constant volume and pressure
Of particular interest are the average and true heat capacities in processes at constant volume ( isochoric heat capacity, equal to the ratio of the specific amount of heat in an isochoric process to the change in temperature of the working fluid dT) and at constant pressure( isobaric heat capacity, equal to the ratio of the specific amount of heat in an isobaric process to the change in temperature of the working fluid dT).
For ideal gases, the relationship between isobaric and isochoric heat capacities is established by the well-known Mayer equation.
From Mayer's equation it follows that the isobaric heat capacity is greater than the isochoric heat capacity by the value of the specific characteristic constant of an ideal gas. This is explained by the fact that in an isochoric process () external work is not performed and heat is spent only on changing the internal energy of the working fluid, while in an isobaric process () heat is spent not only on changing the internal energy of the working fluid, depending on its temperature, but also to perform external work.
For real gases, since when they expand, work is done not only against external forces, but also internal work against the forces of interaction between gas molecules, which additionally consumes heat.
In heat engineering, the ratio of heat capacities, which is called Poisson's ratio (adiabatic index), is widely used. In table Table 2.1 shows the values of some gases obtained experimentally at a temperature of 15 °C.
Heat capacities depend on temperature, therefore, the adiabatic index must depend on temperature.
It is known that with increasing temperature the heat capacity increases. Therefore, with increasing temperature it decreases, approaching unity. However, there is always more than one remaining. Typically, the dependence of the adiabatic index on temperature is expressed by a formula of the form
and since
Specific, molar and volumetric heat capacity. Although the heat included in the PZT equations can be theoretically represented as the sum of microwork performed during the collision of microparticles at the boundaries of the system without the occurrence of macroforces and macromovements, in practice this method of calculating heat is of little use and historically heat was determined in proportion to the change in body temperature dT and a certain value C of the body , characterizing the content of a substance in the body and its ability to accumulate thermal motion (heat),
Q = C body dT. (2.36)
Magnitude
Body C = Q/dT; = 1 J/K, (2.37)
equal to the ratio of the elementary heat Q imparted to the body to the change in body temperature dT is called the (true) heat capacity of the body. The heat capacity of a body is numerically equal to the heat required to change the body temperature by one degree.
Since the temperature of the body changes when work is performed, work, by analogy with heat (4.36), can also be determined through a change in body temperature (this method of calculating work has certain advantages when calculating it in polytropic processes):
W = C w dT. (2.38)
C w = dW/dT = pdV / dT, (2.39)
equal to the ratio of the work supplied (removed) to the body to the change in body temperature, by analogy with heat capacity, we can call the “work capacity of the body.” The term “work capacity” is as conventional as the term “heat capacity”. The term “heat capacity” (capacity for heat) - as a tribute to the real theory of heat (caloric) - was first introduced by Joseph Black (1728-1779) in the 60s of the 18th century. in his lectures (the lectures themselves were published only posthumously in 1803)..
Specific heat capacity c (sometimes called mass, or specific mass heat capacity, which is outdated) is the ratio of the heat capacity of a body to its mass:
c = Stele / m = dQ / (m dT) = dq / dT; [c] = 1 J /(kgK), (2.40)
where dq = dQ / m - specific heat, J / kg.
Specific heat capacity is numerically equal to the heat that must be supplied to a substance of unit mass in order to change its temperature by one degree.
Molar heat capacity is the ratio of the heat capacity of a body to the amount of substance (molarity) of this body:
C m = C body / m, = 1 J / (molK). (2.41)
Volumetric heat capacity is the ratio of the heat capacity of a body to its volume reduced to normal physical conditions (p 0 = 101325 Pa = 760 mm Hg; T 0 = 273.15 K (0 o C)):
c" = body C / V 0 , = 1 J / (m 3 K). (2.42)
In the case of an ideal gas, its volume under normal physical conditions is calculated from the equation of state (1.28)
V 0 = mRT 0 / p 0 . (2.43)
Molecular heat capacity is the ratio of the heat capacity of a body to the number of molecules of this body:
c m = C body / N; = 1 J/K. (2.44)
The connection between different types of heat capacities is established by jointly solving relations (2.40) - (2.44) for heat capacities. The relationship between specific and molar heat capacities is established by the following relationship:
c = C body / m = C m. m/m = C m / (m/m) = C m /M, (2.45)
where M = m / m - molar mass of the substance, kg / mol.
Since tabular values for molar heat capacities are more often given, relation (2.45) should be used to calculate the values of specific heat capacities through molar heat capacities.
The relationship between volumetric and specific heat capacities is established by the relation
c" = body C / V 0 = cm / V 0 = c 0 , (2.46)
where 0 = m / V 0 - gas density under normal physical conditions (for example, air density under normal conditions
0 = p 0 /(RT 0) = 101325 / (287273.15) = 1.29 kg / m 3).
The relationship between volumetric and molar heat capacities is established by the relation
c" = C body / V 0 = C m m / V 0 = C m / (V 0 / m) = C m /V m0, (2.47)
where V 0 = V 0 / m = 22.4141 m 3 / kmol - molar volume reduced to NFU.
In the future, when considering general provisions for all types of heat capacities, we will consider specific heat capacity as the initial one, which, to shorten the notation, we will simply call heat capacity, and the corresponding specific heat - simply heat.
True and average heat capacity. The heat capacity of an ideal gas depends on the temperature c = c (T), and that of a real gas also depends on the pressure c = c (T, p). Based on this criterion, true and average heat capacity are distinguished. For gases with low pressure and high temperature, the dependence of heat capacity on pressure turns out to be negligible.
True heat capacity corresponds to a certain body temperature (heat capacity at a point), since it is determined with an infinitesimal change in body temperature dT
c = dq / dT. (2.48)
Often in thermotechnical calculations the nonlinear dependence of the true heat capacity on temperature is replaced by a linear dependence close to it
c = b 0 + b 1 t = c 0 + bt, (2.49)
where c 0 = b 0 - heat capacity at Celsius temperature t = 0 o C.
Elementary specific heat can be determined from expression (4.48) for specific heat capacity:
dq = c dT. (2.50)
Knowing the dependence of the true heat capacity on temperature c = c(t), we can determine the heat supplied to the system in a finite temperature range by integrating expression (2.53) from the initial state 1 to the final state 2,
In accordance with the graphical representation of the integral, this heat corresponds to an area of 122"1" under the curve c = f(t) (Fig. 4.4).
Figure 2.4 - To the concept of true and average heat capacity
The area of a curved trapezoid 122"1", corresponding to heat q 1-2, can be replaced by the equivalent area of a rectangle 1"342" with base DT = T 2 - T 1 = t 2 - t 1 and height: .
The value determined by the expression
and will be the average heat capacity of the substance in the temperature range from t 1 to t 2.
If dependence (2.52) for the true heat capacity is substituted into expression (2.55) for the average heat capacity and integrated over temperature, we obtain
Co + b(t1 + t2) / 2 = , (2.53)
where t cp = (t 1 + t 2)/2 is the average Celsius temperature in the temperature range from t 1 to t 2.
Thus, in accordance with (2.56), the average heat capacity in the temperature range from t 1 to t 2 can be approximately determined as the true heat capacity calculated from the average temperature t cp for a given temperature interval.
For the average heat capacity in the temperature range from 0 o C (t 1 = 0) to t, dependence (2.56) takes the form
C o + (b / 2)t = c o + b"t. (2.54)
When calculating the specific heats required to heat a gas from 0 o C to t 1 and t 2, using tables where each temperature t corresponds to the average heat capacity, the following relationships are used:
q 0-1 = t 1 and q 0-2 = t 2
(in Fig. 4.4 these heats are depicted as the areas of figures 0511" and 0522"), and to calculate the heat supplied in the temperature range from t 1 to t 2, the relation is used
q 1-2 = q 0-2 - q 0-1 = t 2 - t 1 = (t 2 - t 1).
From this expression we can find the average heat capacity of the gas in the temperature range from t 1 to t 2:
= = (t 2 - t 1) / (t 2 - t 1). (2.55)
Therefore, in order to find the average heat capacity in the temperature range from t 1 to t 2 using formula (2.59), one must first determine the average heat capacity and using the corresponding tables. After calculating the average heat capacity for a given process, the supplied heat is determined by the formula
q 1-2 = (t 2 - t 1). (2.56)
If the range of temperature changes is small, then the dependence of the true heat capacity on temperature is close to linear, and the heat can be calculated as the product of the true heat capacity c(t cp), determined for the average gas temperature? t cp in a given temperature range, by temperature difference:
q 1-2 = = . (2.57)
This calculation of heat is equivalent to calculating the area of the trapezoid 1"1""22" (see Fig. 2.4) as the product of the midline of the trapezoid c(t cp) and its height DT.
The true heat capacity at the average temperature t cp in accordance with (4.56) has a value close to the average heat capacity in this temperature range.
For example, in accordance with table C.4, the average molar isochoric heat capacity in the temperature range from 0 to 1000 o C = 23.283 kJ / (kmol.K), and the true molar isochoric heat capacity corresponding to the average temperature of 500 o C for this temperature interval is C mv = 23.316 kJ/(kmol.K). The difference between these heat capacities does not exceed 0.2%.
Isochoric and isobaric heat capacity. Most often in practice, the heat capacities of isochoric and isobaric processes are used, occurring at constant specific volume x = const and pressure p = const, respectively. These specific heat capacities are called isochoric c v and isobaric c p heat capacities, respectively. Using these heat capacities, any other types of heat capacities can be calculated.
Thus, an ideal gas is an imaginary gas (gas model), the state of which exactly corresponds to the Clapeyron equation of state, and the internal energy depends only on temperature.
In relation to an ideal gas, instead of partial derivatives (4.66) and (4.71), one should take total derivatives:
c x = du/dT; (2.58)
c p = dh / dT. (2.59)
It follows that c x and c p for an ideal gas, just like u and h, depend only on temperature.
In the case of constant heat capacities, the internal energy and enthalpy of an ideal gas are determined by the expressions:
U = c x mT and u = c x T; (2.60)
H = c p mT and h = c p T. (2.61)
When calculating the combustion of gases, volumetric enthalpy, J/m 3, is widely used,
h" = H/V 0 = c p mT/V 0 = c p c 0 T = c" p T, (2.62)
where c"p = cp c0 - volumetric isobaric heat capacity, J/(m 3 .K).
Mayer's equation. Let us establish a connection between the heat capacities of an ideal gas c x and c p. To do this, we use the PZT equation (4.68) for an ideal gas during an isobaric process
dq p = c p dT = du + pdх = c x dT + pdх. (2.63)
Where do we find the difference in heat capacities?
c p - c x = pdx / dT = p (x / T) p = dw p / dT (2.64)
(this relation for an ideal gas is a special case of relation (2.75) for a real gas).
Differentiating the Clapeyron equation of state d(pх) p = R dT under the condition of constant pressure, we obtain
dx / dT = R / p. (2.65)
Substituting this relation into equation (2.83), we obtain
c p - c x = R. (2.66)
Multiplying all the quantities in this relation by the molar mass M, we obtain a similar relation for the molar heat capacities
cm p - cm x = Rm. (2.67)
Relations (2.65) and (2.66) are called Mayer’s formulas (equations) for an ideal gas. This is due to the fact that Mayer used equation (2.65) to calculate the mechanical equivalent of heat.
The ratio of heat capacities c p / c x. In thermodynamics and its applications, not only the difference in heat capacities c p and c x, determined by the Mayer equation, is of great importance, but also their ratio c p / c x, which in the case of an ideal gas is equal to the ratio of heat to the change in HE in an isobaric process, i.e. the ratio is a characteristic of an isobaric process:
k p = k X = dq p / du = c p dT / = c p dT / c x dT = c p / c x.
Consequently, if in the process of changing the state of an ideal gas, the ratio of heat to the change in HE is equal to the ratio c p /c x, then this process will be isobaric.
Since this ratio is often used and is included as an exponent in the equation of the adiabatic process, it is usually denoted by the letter k (without an index) and called the adiabatic exponent
k = dq p / du = c p / c x = C m p / Cm x = c" p /c" x. (2.68)
The values of true heat capacities and their ratio k of some gases in an ideal state (at p > 0 and T C = 0 o C) are given in Table 3.1.
Table 3.1 - Some characteristics of ideal gases
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Chemical formula |
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kJ/(kmolK) |
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water vapor |
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Carbon monoxide |
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Oxygen |
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Carbon dioxide |
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Sulfur dioxide |
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Mercury vapor |
On average for all gases of the same atomicity, it is generally accepted that for monatomic gases k? 1.67, for diatomic k ? 1.40, for triatomic k ? 1.29 (for water vapor the exact value k = 1.33 is often taken).
Solving (2.65) and (2.67) together, we can express the heat capacities in terms of k and R:
Taking (2.69) into account, equation (2.50) for the specific enthalpy takes the form
h = c p T = . (2.71)
For diatomic and polyatomic ideal gases, k depends on temperature: k = f(T). According to equation (2.58)
k = 1 + R / c x = 1 + Rm / Cm x. (2.72)
Heat capacity of the gas mixture. To determine the heat capacity of a mixture of gases, it is necessary to know the composition of the mixture, which can be specified by mass g i , molar x i or volume r i fractions, as well as the values of the heat capacities of the mixture components, which are taken from the tables for the corresponding gases.
The specific heat capacity of a mixture consisting of N components for isoprocesses X = x, p = const is determined through mass fractions according to the formula
cXcm = . (2.73)
The molar heat capacity of a mixture is determined in terms of mole fractions
The volumetric heat capacity of the mixture is determined through the volume fractions according to the formula
For ideal gases, the molar and volume fractions are equal: x i = r i.
Calculation of heat through heat capacity. Here are the formulas for calculating heat in various processes:
a) through the average specific heat capacity and mass m
b) through the average molar heat capacity and the amount of substance m
c) through the average volumetric heat capacity and volume V 0 reduced to normal conditions,
d) through the average molecular heat capacity and the number of molecules N
where DT = T 2 - T 1 = t 2 - t 1 - change in body temperature;
Average heat capacity in the temperature range from t 1 to t 2;
c(t cp) - true heat capacity, determined for the average body temperature t cp = (t 1 + t 2)/2.
Using Table C.4 of the heat capacities of air, we find the average heat capacities: = = 1.0496 kJ / (kgK); = 1.1082 kJ / (kgK). The average heat capacity in this temperature range is determined by formula (4.59)
= (1.10821200 - 1.0496600) / 600 = 1.1668 kJ / (kgK),
where DT = 1200 - 600 = 600 K.
Specific heat through average heat capacity in a given temperature range = 1.1668600 = 700.08 kJ/kg.
Now let’s determine this heat using the approximate formula (4.61) through the true heat capacity c(t cp), determined for the average heating temperature t cp = (t 1 + t 2)/2 = (600 + 1200) / 2 = 900 o C.
The true heat capacity of air c p for 900 o C according to table C.1 is equal to 1.1707 kJ/(kgK).
Then the specific heat through the true heat capacity at the average heat supply temperature
q p = c p (t cp) = c p (900) DT = 1.1707600 = 702.42 kJ/kg.
The relative error in calculating heat using an approximate formula through the true heat capacity at an average heating temperature is e(q p) = 0.33%.
Therefore, if you have a table of true heat capacities, the specific heat is most easily calculated using formula (4.61) through the true heat capacity taken at the average heating temperature.
