Question 1.5: Estimate ρ and c_p of steam at 100 lbf/in^2 and 400°F, in En...

• Approach (a)—the perfect-gas law: Although steam is not an ideal gas, we can estimate these properties with moderate accuracy from Eqs. (1.10) and (1.17). First convert pressure from 100 lbf/in^2 to 14,400 lbf/ft^2, and use absolute temperature, (400°F + 460) = 860°R. Then we need the gas constant for steam, in English units. From Table A.4, the molecular weight of H_2O is 18.02, whence

p = ρRT                          R = c_p – c_v = gas constant                     (1.10)

c_v = \frac{R}{k – 1} \approx 4293  ft^2/(s^2 \cdot ^\circ R) = 718  m^2/(s^2 \cdot K)                    (1.17)

^*The power-law curve fit, Eq. (1.27), µ/µ_{293K} \approx (T/293)^n, fits these gases to within ±4 percent in the range 250 ≤ T ≤ 1000 K. The temperature must be in kelvins.

Then the density estimate follows from the perfect-gas law, Eq. (1.10):

ρ ≈ \frac{p}{RT} = \frac{14,400  lbf/ft^2}{[2758  ft \cdot lbf/(slug \cdot ^\circ R)](860  ^\circ R)} \approx 0.00607 \frac{slug}{ft^3}

At 860°R, from Fig. 1.5, k_{steam} = c_p/c_v \approx 1.30. Then, from Eq. (1.17),

• Approach (b)—tables or software: One can either read the steam tables or program a few lines in EES. In either case, the English units (psi, Btu, lbm) are awkward when applied to fluid mechanics formulas. Even so, when using EES, make sure that the Variable Information menu specifies English units: psia and °F. EES statements for evaluating density and specific heat of steam are, for these conditions,

Rho = DENSITY(steam, P = 100, T = 400)
Cp = SPECHEAT(steam, P = 100, T = 400)

Note that the software is set up for psia and °F, without converting. EES returns the curve-fit values

Rho ≈ 0.2027 lbm/ft^3 ; C_p ≈ 0.5289 Btu/(lbm-F)

As just stated, Btu and lbm are extremely unwieldy when applied to mass, momentum, and energy problems in fluid mechanics. Therefore, either convert to ft-lbf and slugs using your own resources, or use the “Convert” function in EES, placing the old and new units in single quote marks:

Rho2 = Rho*CONVERT(‘lbm/ft^3’,‘slug/ft^3’)
Cp2 = Cp*CONVERT(‘Btu/lbm-F’,‘ft^2/s^2-R’)

Note that (1) you multiply the old Rho and Cp by the CONVERT function; and (2) units to the right of the division sign “/” in CONVERT are assumed to be in the denominator. EES returns these results:

Rho2 = 0.00630 slug/ft^3  Cp2 = 13,200 ft^2/(s^2-R)

• Comments: The steam tables would yield results quite close to EES. The perfect-gas estimate of ρ is 4 percent low, and the estimate of c_p is 9 percent low. The chief reason for the discrepancy is that this temperature and pressure are rather close to the critical point and saturation line of steam. At higher temperatures and lower pressures, say, 800°F and 50 lbf/in^2, the perfect-gas law yields properties with an accuracy of about ±1 percent.

Once again let us warn that English units (psia, lbm Btu) are awkward and must be converted in most fluid mechanics formulas. EES handles SI units nicely, with no conversion factors needed.