Modern Engineering Thermodynamics

220 CHAPTER 7: Second Law of Thermodynamics and Entropy Transport and Production Mechanisms
or
s ½in kJ/ ðkg.°CÞŠ≡ s ½in kJ/ ðkg.KÞŠ
This does not mean that the °F andR(or°C and K) scales are equal but only that their degree sizes are equal.
Therefore, when you have units like Btu/(lbm·ºF), you need not use any mathematical formula to convert °F to
R in order to write this grouping as Btu/(lbm·R). This is a simple but often confusing point.
However, the temperature unit you choose to place in the denominator of a term’s units grouping may depend
on how the term is to be used in relation to other temperature terms in the equation. An example of where this
occurs is in the use of specific heats. Equation (3.15) is
c v =
∂u
(3.15)
∂T v
where c v is the constant volume specific heat.
We discussed reversible processes briefly in Chapter 4 and noted that there are few reversible processes in the
real world. In fact, every heat transport of energy through a finite temperature difference is irreversible. We are
able to write Eqs. (7.21) through (7.24) as reversible heat transfers only because we created a very special
situation, in which the heat transport of energy was assumed to take place through an infinitesimal temperature
difference. But, in the real world, it would require an infinite amount of time to transport a finite amount of
energy by this method. If we try to alter the results of Eq. (7.22) by considering only real irreversible heat
transports of energy, we immediately realize that the amount of work done by the cyclic heat engines must be
less than in the reversible case. Then, for an actual heat engine,
W actual < W reversible
and using the first law of thermodynamics, we conclude that, since the system total energy E is a point function
and therefore independent of whether the process path is reversible or irreversible,
dE = dQ rev − dW rev = dQ act − dW act
WHEN DO WE USE ºF (OR ºC) AND WHEN DO WE USE R (OR K)?
Whether you use relative or absolute temperature units in an equation depends on whether the temperature appears in an
equation as a difference or stands alone. For example, assuming c v is a constant, integrating Eq. (3.15) gives u 2 − u 1 = c v (T 2 − T 1 ),
and since the temperature appears as a difference here, we can use either ºF or R (or ºC or K) temperature units, because
(T 1 in ºF + 459.67 R) − (T 2 in ºF + 459.67 R) = (T 1 in ºF − T 2 in ºF), as the conversion from ºF to R cancels out. You
can use either ºF or R and you get the same answer in each case.
Also, the numerical value of c v in Btu/(lbm·ºF) has the same value in Btu/(lbm · R). For example, for air, c v = 0.172 Btu/
(lbm·ºF) = 0.172 Btu/(lbm · R). That is because the temperature unit appears in the denominator as “per degree,” and the
Fahrenheit degree is the same size as a Rankine degree (only their zero point is different). Similarly, c v = 0.718 kJ/(kg·K) =
0.718 kJ/(kg·ºC) for the same reason. 8 This also applies to numerical values of c p , entropy s, and the gas constant R = R/M.
However, in Eqs. (7.18) and (7.20), the temperature stands alone in the numerator, but the denominator has a temperature
difference. What do you do now?
T
COP Carnot
= H
(7.18)
T H − T L
heat pump
T
COP Carnot = L
(7.20)
refrig: or
T H − T L
air cond:
Theruleisthat,wheneveryouhaveanequationinwhichthetemperatureT stands alone (and not as a temperature difference),
the temperature must always be in an absolute unit (R or K). So the numerators on Eqs. (7.18) and (7.20) must be in
absolute temperature units (R or K), but since the denominator has a temperature difference, the temperatures here can be in
either relative or absolute temperature units.Ifyouareunsurewhethertouseabsoluteorrelativetemperatureunitsinan
equation, use absolute temperature units, since they always give the correct answer.
8 But be careful if you use a table with c v or s in Btu/(lbm ·ºF) because you might be tempted to use T in °F to cancel the temperature unit. This would be
incorrect. You have to understand that c v or s in Btu/(lbm ·ºF) has the same numerical value as c v or s in Btu/(lbm · R).