Entropy Balance Equation for Open Systems

Ch 8, Lesson B, Page 4 - Entropy Balance Equation for Open Systems

• Here it is. The equation we have been building up to.  Let’s take a look at each term in the equation.
• The left-hand side of the equation is the transient or entropy accumulation term.
• It is the rate at which the entropy of the system changes with respect to time.  Like all of the other terms in this equation, it has units of Watts per Kelvin.
• The first two terms on the right-hand side are the new terms.  They are the rate at which entropy flows into the system and out of the system due to the mass flows.  The summations take into account the fact there may be multiple inlets and outlets, each of which may have a different specific entropy.
• The remaining two terms are familiar.  In the last term, all we did was put a dot over Sgen to get the rate at which entropy is generated.
• The rate at which entropy enters or leaves the system due to heat transfer can be complicated or simple.
• If the system is at a uniform and constant temperature, the integral simplifies to just Qdot over T.  If the system exchanges heat with just one reservoir, the summation goes away as well.
• That isn’t so bad, is it ?
• Now, let’s make everyone’s favorite assumption.  Let’s assume that the process is operating at steady-state.
• This can give us two simplifications.  For starters, the left-hand side of the equation is zero.
• But we can also eliminate the integral in the heat transfer term because T is constant.
• I must caution you, however, that the summation in the heat transfer term can become an integral !
• Consider a simple process in which a fluid is flowing inside a pipe and it is heated as it flows at steady-state.
• Although T is not a function of time at any position along the pipe, T is higher at the outlet than it is at the inlet.
• So, we would be forced to break the pipe into infinitesimal lengths and sum the heat transfer that occurs in each infinitesimal length.
• That is to say, we would have to integrate along the length of the pipe.
• The good news is that, in this course, we will generally take advantage of the isothermal characteristics of phase changes to give up or absorb heat.  In this way, we can avoid even this integral because as long as the system remains in the two phase region, the temperature remains constant.
• Don’t worry, we will come back to this subtle point later. For now, let’s press on with further simplifications of this equation.