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3.10

## University of Liverpool

Skip to 0 minutes and 4 seconds So I’d like to show you an overview of the first law of thermodynamics for a system. This is the expression that was developed in an earlier article. And I just like to go through each term in this and remind you what each of them means and the ways in which we can evaluate each of these terms. So I’m going to start on the right-hand side of the equation with the terms that represent the change in the energy content of our system, and our system could be anything we like that we want to define as being separate from the rest of the universe. So starting on the extreme right, Δ here means a change in and PE stands for potential energy.

Skip to 0 minutes and 55 seconds And from mechanics we know that potential energy is equal to mass times gravitational constant, g times z, the height of the object above the ground. or the datum. And so in this case, it’s the mass of the system times the gravitational acceleration, which we normally take as being 10 metres per second per second, times z height above a datum. And so if we have a change in the height of the system, we’ll have a change in the potential energy of the system. And then moving leftwards the next term is the change in the kinetic energy. So Δ, again, means change in, KE stands for kinetic energy.

Skip to 1 minute and 35 seconds And again, from mechanics, we know that kinetic energy is equal to mass times velocity squared divided by 2. So in this case, it’s the mass of the system, and if the velocity of the system changes, then we’ll get a change in the kinetic energy. And then continuing leftwards to ΔU, this is really where the thermodynamics starts. So Δ, again, means change in. Capital U means internal energy. The capital indicates that it’s a total internal energy of the system. When I use lowercase symbols, it means it’s the energy per unit mass.

Skip to 2 minutes and 19 seconds And we’ve defined enthalpy as h - lowercase h is enthalpy per unit mass - is equal to the internal energy u plus the product of the pressure times the volume of the matter inside the system. And we looked at the definition of the specific heat capacity at constant volume. That’s equal to the gradient of the internal energy as a function of temperature. And so that’s what I’ve expressed in the middle of the series of three equations in the yellow box here. Small change in internal energy d u is equal to the specific heat capacity at constant volume - that’s why there’s a little lowercase subscript v on it - times small change in the temperature.

Skip to 3 minutes and 11 seconds And there’s a corresponding definition for the specific heat capacity at constant pressure. That’s related to the gradient of enthalpy plotted with temperature. And so we can say a small change in enthalpy, d-h, is equal to the specific capacity at constant pressure - subscript p indicates it’s constant pressure multiplied by dT, a small change in temperature. So those three terms equate to the changes in the energy of the system. And that happens as a consequence of energy flows in and out of the system, which are described on the left-hand side of the equation. And so the first of those is the energy that flows across the system boundaries as a consequence of matter or mass flowing across the system boundary.

Skip to 4 minutes and 3 seconds So we have the energy transfer as a consequence of mass flowing in minus the energy transfers occurring as a consequence of mass flowing out. And we can equate those energy transfers as being equal to the mass flowing times the flow energy, which is given the symbol Θ. And the flow energy theta is the sum of the enthalpy, the kinetic energy, and the potential energy of the flow. And so that’s sort of expressed in the bottom equation there, that the energy transfer as a consequence of the mass flow is equal to the mass flowing times h plus v squared upon 2 plus gz.

Skip to 4 minutes and 47 seconds And that definition allows us to get at a special case of the first law of thermodynamics, which is the steady flow of energy equation. And its derivation is described in an earlier article that you could go back and read it if you wish. So moving leftwards to the next term, this is the work term. This describes the work done to the system - that’s the work in - minus the work done by the system– that’s the work out. And that work can occur in several different ways. It could simply be by a change in the pressure and volume of the matter contained in the system, which case the work done is the area under the pressure-volume graph.

Skip to 5 minutes and 34 seconds And so we need to integrate underneath the graph in order to arrive at the work done. And that’s the first equation listed underneath the work term. It could be as a consequence of some mechanical work. For instance, force on a spring, in which case, again, we need to plot the force versus the distance that it moves and integrate underneath it to get the work done. And that’s just the second little expression there and W mechanical. It could be as a consequence of power being supplied or taken away by a shaft, and so ‘W dot’ shaft is the power. It’s work per unit time supplied by or taking away by shaft.

Skip to 6 minutes and 18 seconds And that’s equal to 2 pi times ‘m dot’ which is the rate of rotation times the torque involved in doing that. And then the final equation relates to electrical power or rate of doing work, ‘W dot.’ And it’s equal to the electrical current squared multiplied by the resistance.

Skip to 6 minutes and 43 seconds The final term is the heat transfer term. This is the heat flowing into the system - ‘Q in’ - minus the heat going out - ‘Q out.’ And that heat transfer can occur in a number of ways, by conduction, convection, or radiation. So I’ve listed here the three equations describing the rate of heat transfer that’s got a dot over top of my Q, so that’s rate of heat transfer occurring. In the case of conduction, it’s equal to the heat transfer coefficient multiplied by the cross-section over which the conduction is occurring multiplied by the temperature gradient along which the heat transfer is occurring. So dT is the temperature change and dx is the distance along which the heat transfer is occurring.

Skip to 7 minutes and 35 seconds In this case, the heat transfer is occurring in the x direction, and our cross-section is perpendicular to the x-direction. The middle equation of this set is the convection equation. And here it’s equal to transfer coefficient multiplied by the area over which the heat transfer is occurring, multiplied by the temperature difference between the two points the heat transfer is occurring to and from. And then the last equation is the rate of radiation heat transfer.

Skip to 8 minutes and 8 seconds This is equal to the emissivity which characterises the surface from which the radiation is occurring or to which is occurring, multiplied by Stefan Boltzmann’s constant, and then multiplied by the area involved in the radiation heat transfer, finally multiply by the temperature of the surface to the power 4.

Skip to 8 minutes and 31 seconds So these three large terms on the left-hand side of the main equation describe the energy flows into and out of our system, and of course the net change in those energy flows must be equal to the change in the energy content of the system, which is described on the right-hand side of the equation So I hope that laying out all of these relationships that we’ve covered so far in this single graphic help to place into context for you and gives you an overview and summary that’s useful.

# Summary: first law analysis

Listen to Eann provide an overview of the first law of thermodynamics applied to a system. He starts from the fundamental energy balance equations and discusses how each term in the equation can be evaluated.