Thermodynamic Equations
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Thermodynamics Nomenclature:
| T = temperature (oK) |
| V = volume of system (cubic metres) |
| P or p = pressure at the boundary of the system and its environment, (in pascals) |
| W = work done by or on a system (joules) |
| Q = heat transfer in or out of a system (joules) |
| q = specific heat transfer in or out of a system (joules per kg) |
| U = internal energy of a system mainly contained in solid and liquid components (joules) |
| u = specific internal energy of a system (joules per kg) |
| H = enthalpy of a system (joules) |
| h = specific enthalpy (joules per kg) |
| S = entropy (joules per oK) |
| s = specific entropy (joules per kg per oK) |
| n = heat capacity ratio (= Cv/Cp) |
Thermodynamics Equations:
Thermodynamics equations can be difficult to understand. The following is a simpified summary where the term “system” can be equated to a steam locomotive’s cylinder:
The First Law of Thermodynamics (conservation of energy) can be expressed as “The increase in internal energy of a system = the heat supplied to the system minus the energy that flows out in the form of Work that the system performs on it environment” [ref Wikipedia]. In this case, the external “environment” is the locomotive’s piston, hence the definition can be formulated by the equation:
δU = Q – Wpiston ………… (1)
which may also be written:
dU = dQ – dWpiston ………. (1a)
However, the work done on a system (locomotive cylinder) by changing its volume is dW = p.dV, hence:
dU = dQ – p.dV …………. (1b)
If the process (steam expansion) is assumed to be adiabatic – i.e. with no heat transfer in or out, then dQ = 0, whence
dU = – p.dV ………………. (1c)
However Enthalpy (see separate page) is defined as the sum of a system’s internal energy plus the product of its pressure and volume – i.e.
H = U + P.V ……………..(2)
from which a change in enthalpy can be defined (by differentiation) as
dH = dU + p.dV + V.dp ……………..(2a)
Thus by combining equations (1c) and (2a) we get (for adiabatic expansion): dH = -p.dV + p.dV + V.dp, or
dH = V.dp ……………..(3)
Combining eqns (1c) and (3) gives:
dH/dU = – V.dP / P.dV …………….. (4)
The Second Law of Thermodynamics (heat always flows to regions of lower temperature) can be expressed as “a change in the entropy (S) of a system is the infinitesimal transfer of heat (Q) to a closed system driving a reversible process, divided by the equilibrium temperature (T) of the system” [ref Wikipedia]. This definition is formulated by the equation:
dS = δQ/T or dQ = T.dS ………………(5)
By combining Eqn (4) with (1b), a change in internal energy is given by:
dU = T.dS – p.dV ………………………….(6)
Ideal Gas Laws (from physics): The ideal gas law is defined by the equation:
pVn = k …………………………. (7)
where n is the “heat capacity ratio“: n = Cp / Cv = – V.dp / p.dV [ref Wikipedia]
Thus from equation (4):
n = dH/dU
———— page in progress as at 10th Mar 2011 ———–