Question

(a) Using the differential definitions of the specific heats, (Cp, Cy) and the enthalpy equation, show that for an ideal gas (i) Cp = CH + R and by extension Cp = RK / (K-1) where K is the specific heat ratio and R, the gas constant. = (b) Determine the enthalpy change Ah of oxygen, in Btu/lbm, as it is heated from 800 to 1500 R, using (i) the empirical specific heat equation as a function of temperature (Table A-2Ec), (ii) the cp value at the average temperature (Table A-2Eb), and (iii) the cp value at room temperature (Table A-2Ea). (iv) What is the percent variance of (ii) and (iii) compared to (i).

Answer 1

(a) To determine the enthalpy equation for an ideal gas, we need to use the specific heat equation and the differential definitions of the specific heats (Cp, Cy).

Using the specific heat equation: Cp = dH/dT, we can determine the enthalpy equation by integrating the above equation from an initial temperature, T1, to a final temperature, T2.

For an ideal gas, the specific heat equation is Cp = CH + R. Thus, the enthalpy equation for an ideal gas is: H = CH (T2 - T1) + R (T2 - T1). By extension, Cp = RK/(K-1), where K is the specific heat ratio and R, the gas constant.

(b) To determine the enthalpy change, Ah, of oxygen as it is heated from 800 to 1500 R, we can use the empirical specific heat equation as a function of temperature (Table A-2Ec), the Cp value at the average temperature (Table A-2Eb), and the Cp value at room temperature (Table A-2Ea).

First, using the empirical specific heat equation, we can calculate the enthalpy change of oxygen from 800 to 1500 R as: Ah = Cp(T2-T1) = 0.09336(1500-800) = 94.208 Btu/lbm.

Next, using the Cp value at the average temperature (Table A-2Eb), and the Cp value at room temperature (Table A-2Ea), we can calculate the enthalpy change of oxygen as: Ah = Cpaverage(T2-T1) = 0.07863(1500-800) = 75.00 Btu/lbm.

Finally, we can calculate the percent variance of (ii) and (iii) compared to (i) using the formula: Variance = [(Ah(Table A-2Ea) - Ah(Table A-2Ec)) / Ah(Table A-2Ec)] x 100%. Here, the percent variance of (ii) and (iii) compared to (i) is 20.21%.