1.6. ENERGY AND HEAT UNITS

1.6A. Joule, Calorie, and Btu

In a manner similar to that used in making material balances on chemical and biological processes, we can also make energy balances on a process. Often a large portion of the energy entering or leaving a system is in the form of heat. Before such energy or heat balances are made, we must understand the various types of energy and heat units.

In the SI system, energy is given in joules (J) or kilojoules (kJ). Energy is also expressed in btu (British thermal units) or cal (calories). The g calorie (abbreviated cal) is defined as the amount of heat needed to heat 1.0 g water 1.0°C (from 14.5°C to 15.5°C). Also, 1 kcal (kilocalorie) = 1000 cal. The btu is defined as the amount of heat needed to raise 1.0 lb water 1°F. Hence, from Appendix A.1,

Equation 1.6-1

1.6B. Heat Capacity

The heat capacity of a substance is defined as the amount of heat necessary to increase the temperature by 1 degree. It can be expressed for 1 g, 1 lb, 1 g mol, 1 kg mol, or 1 lb mol of the substance. For example, a heat capacity is expressed in SI units as J/kg mol · K; in other units as cal/g · °C, cal/g mol · °C, kcal/kg mol · °C, btu/lb_m · °F, or btu/lb mol · °F.

It can be shown that the actual numerical value of a heat capacity is the same in mass units or in molar units. That is,

Equation 1.6-2

Equation 1.6-3

For example, to prove this, suppose that a substance has a heat capacity of 0.8 btu/lb_m · °F. The conversion is made using 1.8°F for 1°C or 1 K, 252.16 cal for 1 btu, and 453.6 g for 1 lb_m, as follows:

The heat capacities of gases (sometimes called specific heat) at constant pressure c_p are functions of temperature and for engineering purposes can be assumed to be independent of pressure up to several atmospheres. In most process engineering calculations, one is usually interested in the amount of heat needed to heat a gas from one temperature t₁ to another at t₂. Since the c_p varies with temperature, an integration must be performed or a suitable mean c_pm used. These mean values for gases have been obtained for T₁ of 298 K or 25°C (77°F) and various T₂ values, and are tabulated in Table 1.6-1 at 101.325 kPa pressure or less as c_pm in kJ/kg mol · K at various values of T₂ in K or °C.

Table 1.6-1. Mean Molar Heat Capacities of Gases Between 298 and TK (25 and T°C) at 101.325 kPa or Less (SI Units: c_p =×kJ/kg mol · K)

T(K)	T(°C)	H2	N2	CO	Air	O2	H2O	CO2	CH4	SO2
298	25	28.86	29.14	29.16	29.19	29.38	33.59	37.20	35.8	39.9
373	100	28.99	29.19	29.24	29.29	29.66	33.85	38.73	37.6	41.2
473	200	29.13	29.29	29.38	29.40	30.07	34.24	40.62	40.3	42.9
573	300	29.18	29.46	29.60	29.61	30.53	34.39	42.32	43.1	44.5
673	400	29.23	29.68	29.88	29.94	31.01	35.21	43.80	45.9	45.8
773	500	29.29	29.97	30.19	30.25	31.46	35.75	45.12	48.8	47.0
873	600	29.35	30.27	30.52	30.56	31.89	36.33	46.28	51.4	47.9
973	700	29.44	30.56	30.84	30.87	32.26	36.91	47.32	54.0	48.8
1073	800	29.56	30.85	31.16	31.18	32.62	37.53	48.27	56.4	49.6
1173	900	29.63	31.16	31.49	31.48	32.97	38.14	49.15	58.8	50.3
1273	1000	29.84	31.43	31.77	31.79	33.25	38.71	49.91	61.0	50.9
1473	1200	30.18	31.97	32.30	32.32	33.78	39.88	51.29	64.9	51.9
1673	1400	30.51	32.40	32.73	32.76	34.19	40.90	52.34

EXAMPLE 1.6-1. Heating of N2 Gas The gas N2 at 1 atm pressure absolute is being heated in a heat exchanger. Calculate the amount of heat needed in J to heat 3.0 g mol N2 in the following temperature ranges: 298–673 K (25–400°C) 298–1123 K (25–850°C) 673–1123 K (400–850°C) Solution: For case (a), Table 1.6-1 gives cpm values at 1 atm pressure or less which can be used up to several atm pressures. For N2 at 673 K, cpm = 29.68 kJ/kg mol · K or 29.68 J/g mol · K. This is the mean heat capacity for the range 298–673 K: Equation 1.6-4 Mean Molar Heat Capacities of Gases Between 25 and T°C at 1 atm Pressure or Less (English Units: cp =×btu/lb mol · °F) T(°C) H2 N2 CO Air O2 NO H2O CO2 HCl Cl2 CH4 SO2 C2H4 SO3 C2H6 25 6.894 6.961 6.965 6.972 7.017 7.134 8.024 8.884 6.96 8.12 8.55 9.54 10.45 12.11 12.63 100 6.924 6.972 6.983 6.996 7.083 7.144 8.084 9.251 6.97 8.24 8.98 9.85 11.35 12.84 13.76 200 6.957 6.996 7.017 7.021 7.181 7.224 8.177 9.701 6.98 8.37 9.62 10.25 12.53 13.74 15.27 300 6.970 7.036 7.070 7.073 7.293 7.252 8.215 10.108 7.00 8.48 10.29 10.62 13.65 14.54 16.72 400 6.982 7.089 7.136 7.152 7.406 7.301 8.409 10.462 7.02 8.55 10.97 10.94 14.67 15.22 18.11 500 6.995 7.159 7.210 7.225 7.515 7.389 8.539 10.776 7.06 8.61 11.65 11.22 15.60 15.82 19.39 600 7.011 7.229 7.289 7.299 7.616 7.470 8.678 11.053 7.10 8.66 12.27 11.45 16.45 16.33 20.58 700 7.032 7.298 7.365 7.374 7.706 7.549 8.816 11.303 7.15 8.70 12.90 11.66 17.22 16.77 21.68 800 7.060 7.369 7.443 7.447 7.792 7.630 8.963 11.53 7.21 8.73 13.48 11.84 17.95 17.17 22.72 900 7.076 7.443 7.521 7.520 7.874 7.708 9.109 11.74 7.27 8.77 14.04 12.01 18.63 17.52 23.69 1000 7.128 7.507 7.587 7.593 7.941 7.773 9.246 11.92 7.33 8.80 14.56 12.15 19.23 17.86 24.56 1100 7.169 7.574 7.653 7.660 8.009 7.839 9.389 12.10 7.39 8.82 15.04 12.28 19.81 18.17 25.40 1200 7.209 7.635 7.714 7.719 8.068 7.898 9.524 12.25 7.45 8.94 15.49 12.39 20.33 18.44 26.15 1300 7.252 7.692 7.772 7.778 8.123 7.952 9.66 12.39               1400 7.288 7.738 7.818 7.824 8.166 7.994 9.77 12.50               1500 7.326 7.786 7.866 7.873 8.203 8.039 9.89 12.69               1600 7.386 7.844 7.922 7.929 8.269 8.092 9.95 12.75               1700 7.421 7.879 7.958 7.965 8.305 8.124 10.13 12.70               1800 7.467 7.924 8.001 8.010 8.349 8.164 10.24 12.94               1900 7.505 7.957 8.033 8.043 8.383 8.192 10.34 13.01               2000 7.548 7.994 8.069 8.081 8.423 8.225 10.43 13.10               2100 7.588 8.028 8.101 8.115 8.460 8.255 10.52 13.17               2200 7.624 8.054 8.127 8.144 8.491 8.277 10.61 13.24               Source: O. A. Hougen, K. W. Watson, and R. A. Ragatz, Chemical Process Principles, Part I, 2nd ed. New York: John Wiley & Sons, Inc., 1954. With permission. Substituting the known values, For case (b), the cpm at 1123 K (obtained by linear interpolation between 1073 and 1173 K) is 31.00 J/g mol · K: For case (c), there is no mean heat capacity for the interval 673–1123 K. However, we can use the heat required to heat the gas from 298 to 673 K in case (a) and subtract it from case (b), which includes the heat to go from 298 to 673 K plus 673 to 1123 K: Equation 1.6-5 Substituting the proper values into Eq. (1.6-5),

On heating a gas mixture, the total heat required is determined by first calculating the heat required for each individual component and then adding the results to obtain the total.

The heat capacities of solids and liquids are also functions of temperature and independent of pressure. Data are given in Appendix A.2, Physical Properties of Water; A.3, Physical Properties of Inorganic and Organic Compounds; and A.4, Physical Properties of Foods and Biological Materials. More data are available in (P1).

EXAMPLE 1.6-2. Heating of Milk Rich cows' milk (4536 kg/h) at 4.4°C is being heated in a heat exchanger to 54.4°C by hot water. How much heat is needed? Solution: From Appendix A.4, the average heat capacity of rich cows' milk is 3.85 kJ/kg · K. Temperature rise ΔT = (54.4 − 4.4)°C = 50 K.

The enthalpy, H, of a substance in J/kg represents the sum of the internal energy plus the pressure–volume term. For no reaction and a constant-pressure process with a change in temperature, the heat change as computed from Eq. (1.6-4) is the difference in enthalpy, ΔH, of the substance relative to a given temperature or base point. In other units, H = btu/lb_m or cal/g.

1.6C. Latent Heat and Steam Tables

Whenever a substance undergoes a change of phase, relatively large amounts of heat change are involved at a constant temperature. For example, ice at 0°C and 1 atm pressure can absorb 6013.4 kJ/kg mol. This enthalpy change is called the latent heat of fusion. Data for other compounds are available in various handbooks (P1, W1).

When a liquid phase vaporizes to a vapor phase under its vapor pressure at constant temperature, an amount of heat called the latent heat of vaporization must be added. Tabulations of latent heats of vaporization are given in various handbooks. For water at 25°C and a pressure of 23.75 mm Hg, the latent heat is 44 020 kJ/kg mol, and at 25°C and 760 mm Hg, 44 045 kJ/kg mol. Hence, the effect of pressure can be neglected in engineering calculations. However, there is a large effect of temperature on the latent heat of water. Also, the effect of pressure on the heat capacity of liquid water is small and can be neglected.

Since water is a very common chemical, the thermodynamic properties of it have been compiled in steam tables and are given in Appendix A.2 in SI and in English units.

EXAMPLE 1.6-3. Use of Steam Tables Find the enthalpy change (i.e., how much heat must be added) for each of the following cases using SI and English units: Heating 1 kg (lbm) water from 21.11°C (70°F) to 60°C (140°F) at 101.325 kPa (1 atm) pressure. Heating 1 kg (lbm) water from 21.11°C (70°F) to 115.6°C (240°F) and vaporizing at 172.2 kPa (24.97 psia). Vaporizing 1 kg (lbm) water at 115.6°C (240°F) and 172.2 kPa (24.97 psia). Solution: For part (a), the effect of pressure on the enthalpy of liquid water is negligible. From Appendix A.2, In part (b), the enthalpy at 115.6°C (240°F) and 172.2 kPa (24.97 psia) of the saturated vapor is 2699.9 kJ/kg or 1160.7 btu/lbm. The latent heat of water at 115.6°C (240°F) in part (c) is

1.6D. Heat of Reaction

When chemical reactions occur, heat effects always accompany these reactions. This area where energy changes occur is often called thermochemistry. For example, when HCl is neutralized with NaOH, heat is given off and the reaction is exothermic. Heat is absorbed in an endothermic reaction. This heat of reaction is dependent on the chemical nature of each reacting material and product and on their physical states.

For purposes of organizing data, we define a standard heat of reaction ΔH⁰ as the change in enthalpy when 1 kg mol reacts under a pressure of 101.325 kPa at a temperature of 298 K (25°C). For example, for the reaction

Equation 1.6-6

the ΔH⁰ is −285.840 × 10³ kJ/kg mol or −68.317 kcal/g mol. The reaction is exothermic and the value is negative since the reaction loses enthalpy. In this case, the H₂ gas reacts with the O₂ gas to give liquid water, all at 298 K (25°C).

Special names are given to ΔH⁰ depending upon the type of reaction. When the product is formed from the elements, as in Eq. (1.6-6), we call the ΔH⁰ the heat of formation of the product water, . For the combustion of CH₄ to form CO₂ and H₂O, we call it heat of combustion, . Data are given in Appendix A.3 for various values of .

EXAMPLE 1.6-4. Combustion of Carbon A total of 10.0 g mol of carbon graphite is burned in a calorimeter held at 298 K and 1 atm. The combustion is incomplete, and 90% of the C goes to CO2 and 10% to CO. What is the total enthalpy change in kJ and kcal? Solution: From Appendix A.3 the for carbon going to CO2 is −393.513 × 103 kJ/kg mol or −94.0518 kcal/g mol, and for carbon going to CO it is −110.523 × 103 kJ/kg mol or −26.4157 kcal/g mol. Since 9 mol CO2 and 1 mol CO are formed,

If a table of heats of formation, of compounds is available, the standard heat of the reaction, ΔH⁰, can be calculated by

Equation 1.6-7

In Appendix A.3, a short table of some values of ΔH_f is given. Other data are also available (H1, P1, S1).

EXAMPLE 1.6-5. Reaction of Methane For the following reaction of 1 kg mol of CH4 at 101.32 kPa and 298 K, calculate the standard heat of reaction ΔH0 at 298 K in kJ. Solution: From Appendix A.3, the following standard heats of formation are obtained at 298 K:   (kJ/kg mol) CH4(g) −74.848 × 103 H2O(l) −285.840 × 103 CO(g) −110.523 × 103 H2(g) 0 Note that the of all elements is, by definition, zero. Substituting into Eq. (1.6-7),