In this screencast we are going to take a look at an example of sensible heat and use
physical property data to help us perform some sensible heat calculations. For example
in this case we want to calculate the heat that is required to increase the temperature
of 100 moles of water vapor per second. From 100 C to 300 C at atm pressure. So that is
shown in the flow chart below and the heat that is added to the system is raising the
temperature from the inlet to the outlet, while keeping the phase and the pressure constant.
So anytime that we have heat added to the system that elicits a change in temperature, it
is called sensible heat. So sensible heat does not refer to a specific type of energy.
It simply means that when we are adding heat to the system, we have a change in temperature.
If we want to calculate the magnitude of the sensible heat in this case, we can start with
the appropriate form of the energy balance. So for an open system the energy balance reduces
to Q is equal to delta H. This assumes there is no work done by or on the system. It also
neglects any change in potential or kinetic energy. The enthalpy leaving and entering
the system is extensive. Another words it is proportional to the amount of material
that we are moving through the system. If we want to account for how much water we have
we can rewrite the energy balance as Q is equal to n dot , which is the molar flow rate
of water. Times the change in specific enthalpy, which is in intensive property. So the units
of specific enthalpy are in kJ/mol. So if we can calculate the change in specific enthalpy
on our per mole basis. We can scale it by how much water we have, which is 100 mol/s.
To determine the sensible heat. To visualize how the specific enthalpy of water changes
with temperature. We can plot a schematic of H vs T, and for a gas we would see that
specific enthalpy increases with temperature. However the temperature dependence is non-linear.
The slope of this non-linear curve will tell us how the specific enthalpy changes at any
specific temperature. Mathematically we can represent this as the partial derivative of
the specific enthalpy with respect to temperature at constant pressure, which we have in this
case. This has a physical significance. The change in enthalpy with respect to temperature
is the heat capacity, and here we are noting the heat capacity with a subscript of P to
indicate that we are at constant Pressure. If we look back at the diagram as we increase
the temperature. We can see that we are also increasing the slope of this line. Which indicates
that heat capacity is a function of T. If we want to calculate a finite change in specific
enthalpy over our temperature interval we can rewrite this expression as a differential
equation. So we can write dH is equal to the heat capacity, which is a function of T, times
dT, and if we integrate this expression from the inlet to the outlet condition we can obtain
a change in specific enthalpy, which is equal to the integral from Tin to Tout, and we are
integrating the heat capacity as a function of temperature with respect to T. To actually
solve this integral we need an expression for the heat capacity, that expresses how
the heat capacity changes with temperature. Generally the heat capacity is expressed as
a Polynomial function of temperature. Specific functions for the heat capacity can be obtained
from a variety of different text books or chemical engineering hand books. For example
the expression I have given bellow was obtained from F&R in table B2, and this particular
expression applies to an ideal gas. So for an ideal gas, which if we are operating at
atm pressure. That assumptions is valid. We can write that the heat capacity at constant
pressure is equal to a plus bT, plus cT^2 plus dT^3, and the units for the heat capacity
are in kJ/mol*degC, and A, B,C, and D represent temperature independent coefficients that
are specific to a particular species. So for example for water the table bellow provides
a valid value for a through d, and again these are also obtained from table B2 from F&R.
So with the appropriate polynomial expression for the heat capacity and the coefficients
that go with that. We can use this information to calculate the change in specific enthalpy
over our temperature, integral. So here we are integrating from our temperature at the
inlet, which is 100 degree C to the temperature of the outlet, which is 200 degC, and we are
integrating the heat capacity function with respect to T. If we solve this integral we
would have aT, plus b/2T^2, plus c/3T^3, plus d/4T^4, and we are evaluating that over our
limits of integration, which are 300 and 100. I have substituted in all the coefficients
a through d. As well as the temperature limits. Keep in mind that we are substituting the
temperature limits into each temperature term. So for example we have 300^2 minus 100^2.
We are not squaring the difference between our two temperatures, and that leads to the
following 4 terms if we treat them each independently, and if we look at these 4 terms independently.
We see that the majority of the change in enthalpy is derived from the first term. So
often the change in enthalpy are the heat capacity is assumed to be constant, and this
demonstrates that the assumptions of a constant heat capacity will get us pretty close. The
higher order terms with the B coefficient and c and d respectively kind of fine tone
that first term. So if we combine all 4 of our terms. We find that the change in specific
enthalpy is equal to 7.03 kJ/mol. So we are one step closer in finding the sensible heat.
If we go back to our energy balance. We show that Q is equal to n dot times the change
in specific enthalpy on a per mole basis. So we have 100 mol/s times our change in specific
enthalpy, which gives us a sensible heat of 703 kJ/s. So when shown an example of sensible
heat and a source of physical property that allows us to calculate a sensible heat. If
we could calculate the change in that specific enthalpy using heat capacity functions. We
can scale it by the material that we have and calculate the heat associated with that
process.