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作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org
Thermodynamics is a branch of physics that deals with the energy and work of a system. As aerodynamicists, we are most interested in thermodynamics in the study of propulsion systems and understanding high speed flows. The first law of thermodynamics indicates that the total energy of a system is conserved. Total energy includes the potential and kinetic energy, the work done by the system, and the transfer of heat through the system. The second law of thermodynamics indicates that, while many physical processes that satisfy the first law are possible, the only processes that occur in nature are those for which the entropy of the system either remains constant or increases.
Entropy, like temperature and pressure, can be explained on both a macro scale and a micro scale. Since thermodynamics deals only with the macro scale, the change in entropy delta S is defined here to be the heat transfer delta Q into the system divided by the temperature T:
delta S = delta Q / T
During a thermodynamic process, the temperature T of an object changes as heat Q is applied or extracted. A more correct definition of the entropy S is the differential form that accounts for this variation.
dS = dQ / T
The change in entropy is then the inverse of the temperature integrated over the change in heat transfer. For gases, there are two possible ways to evaluate the change in entropy. We can develop equations for a constant volume process or for a constant pressure process. For the constant volume process, the equations are formulated in terms of the internal energy. For the constant pressure process, the equations are formulated in terms of the enthalpy of the gas.
Considering the constant volume process, the first law of thermodynamics may be written:
dE = dQ - dW
where E is the internal energy and W is the work done by the system. Substituting for the definition of work for a gas.
dQ = dE + p dV
where p is the pressure and V is the volume of the gas. The equation of state is:
p * V = R * T
where R is the gas constant. We can substitute for p in the energy equation:
dQ = dE + R * T dV / V
The heat transfer of a gas is equal to the heat capacity times the change in temperature; in differential form:
dQ = C * dT
If we have a constant volume process, the formulation of the first law gives:
dE = dQ = C (constant volume) * dT
Combining with the first law equation:
dQ = C (constant volume) * dT + R * T dV / V
Then the change in entropy is given by:
dS = C (constant volume) * dT / T + R * dV / V
The "specific" form of this equation is found by dividing by the mass of the gas:
ds = cv * dT / T + R * dv / v
Integrating this equation, we obtain:
s2 - s1 = cv * ln ( T2 / T1) + R * ln ( v2 / v1)
where cv is the specific heat capacity at constant volume and ln is the symbol for the logarithmic function.
The derivation is quite similar if we assume a constant pressure process. Beginning with the first law of thermodynamics:
dE = dQ - dW
dQ = dE + p dV
From the definition of enthalpy H:
H = E + p * V
Then:
dH = dE + p dV + V dp
Substitute into the first law equation:
dQ = dH - V dp - p dv + p dV
dQ = dH - V dp
Substitute for the volume from the equation of state:
dQ = dH - R * T dp / p
For a constant pressure process, the formulation of the first law gives:
dH = dQ = C (constant pressure) * dT
Substitute into the first law equation:
dQ = C (constant pressure) * dT - R * T dp / p
Then the change in entropy is given by:
dS = C (constant pressure) * dT / T - R * dp / p
The "specific" form of this equation is found by dividing by the mass of the gas:
ds = cp * dT / T - R * dp / p
Integrating this equation, we obtain:
s2 - s1 = cp * ln ( T2 / T1) - R * ln ( p2 / p1)
where cp is the specific heat capacity at constant pressure. Depending on the type of process we encounter, we can now determine the change in entropy for a gas.
作者：Anonymous 在 罕见奇谈 发贴, 来自 http://www.hjclub.org |
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