# Gas Calculations

(Unless otherwise stated use mass flow rates in kg/hr and absolute pressures in Pa and show temperature to two decimal places) The three conservation laws discussed in the calculation methods section apply equally to compressible as well as incompressible flows. However, a more rigorous form of the Conservation of Energy law must be applied if the fluid is a gas and the momentum equation is only applicable in a differential form. First, let's consider a simple case of flow of air down a single pipe. Develop the system shown in Figure 1 below. Figure 1. As we can see the velocity increases, the density and temperature decreases. What are the implications of the changing conditions as the fluid flows down the pipe? The implications are that the simplifying assumptions we used to derive the energy and momentum equations for a liquid, i.e., Bernoulli and Darcy Weisbach are not justified if we wish maximum accuracy in the simulation. If we use the same simplifying approach for compressible flow, we will obtain less accurate results, with the accuracy decreasing as temperature, velocity and density changes become more significant. So if we can't use Bernoulli and Darcy equations what do we do? There are 2 different approaches to overcoming this increasingly complex problem: 1.The first approach is to obtain an analytical solution to the fundamental equations by making simplifying assumptions. Nearly all text books follow this approach and usually make the assumptions of Ideal Gas Laws (or use simple equations of state) and assume adiabatic flow. Using a simple gas law provides the relationship between pressure, temperature and density and so explicit analytical equations can be derived. Even so, we cannot apply these equations to inlet conditions and ignore the physical property changes as the fluid flows down the pipe. So we must split the pipe into fixed length segments and march down the pipe solving for downstream conditions as we go. 2.We accept that it is not possible to obtain a complete analytical solution without making simplifying assumptions. We also accept that we need an incremental (marching) approach to solve for the changing conditions as we flow down the pipe. If we use a more complex equation of state we do not need to make simplifying assumptions and we can use this together with differential forms of the continuity and momentum equation which we can solve numerically. There are 2 drawbacks to the first approach, using simplifying assumptions and determining the pipe length to use. FluidFlow uses the second approach. This means that we can rigorously calculate adiabatic, isothermal or flow with heat transfer. It allows for the increase in kinetic energy as the gas accelerates with pressure drop. It also allows the changing non-ideal behavior of the gas as it flows along the pipe by using a more complex equation of state to calculate the thermodynamic properties of the gas, and thereby use actual enthalpy and density at each increment. For a complete explanation of the method used see calculation methods. The advantage of this approach is to obtain the most accurate solution possible. Leaving the complexities of the theory for now, we will focus on the practical implications of compressible flow. If we go back to the initial example, one implication that we should understand is that if we increase the supply pressure the gas will keep accelerating until we reach the maximum limit, which is the speed of sound. It is not possible to obtain a velocity of greater than the speed of sound in a pipe system. So using the same example, increase the supply pressure and watch what happens.
Initial Pressure P 1 in atm | Downstream Velocity V 2 in m/s | Downstream static temperature in °C | Downstream static pressure in Pa
1.1 | 66 | 12.8 | 98682
1.4 | 145 | 4.4 | 89482
1.6 | 188 | -2.8 | 82370
1.8 | 229 | -11.4 | 74407
2.0 | 269 | -21.5 | 65692
2.1 | 315 | -34.8 | 59126
2.5 | 315 | -34.7 | 78954
You should note the following points; the reasoning should become clear if you refer to the energy equation, shown below. In this system at pressures 2.1 atm and above the exit velocity reaches a maximum. This is called sonic velocity. Sonic velocity is a thermophysical property and is a function of temperature and pressure, note that the sonic velocity decreases slightly as we increase the supply pressure. If you watch the static temperature at the outlet you should also note that this decreases as we increase the supply pressure. The mass flow keeps increasing as we increase the supply pressure, as we would expect, but perhaps not by the amount we would expect. The other important observation you should note is that after we reach sonic velocity what happens to the downstream pressure? It no longer is at the specified stagnation value of 1 atm. Why? The energy equation must always hold true and so if the velocity has reached a maximum we cannot increase the kinetic energy any further and so the static pressure, p2 must increase. Consider the energy equation below and you can see why this happens. Δ P tot = (ρ 1 U 1+ z 1 ρ 1 g + p 1 + 0.5N 1 ρ 1 v 1 2) - (ρ 2 U 2+ z 2 ρ 2 g + p 2 + 0.5N 2 ρ 2 v 2 2) ρ 1/ρ 2 For reference the corresponding energy terms are (internal energy + potential energy + pressure energy + kinetic energy). When sonic velocity is reached at the end of a pipeline this is often referred to as choking. Sonic choking can also occur at other positions in gas networks and we will consider this later. There are a few more terms that we should be aware of which are useful when considering gas flow problems. Mach No is defined as the ratio of actual velocity to sonic velocity. You should note that once Mach No reaches above about 0.8 any small increase in upstream pressure has a very significant effect on downstream velocity and is likely to result in a downstream Mach 1. Stagnation Conditions: We have already illustrated that the pressure and temperature of a gas falls as it flows along a pipe (assuming no heat transfer input). This is due to the combined effects of friction and the need to maintain an energy balance. Friction reduces pressure, causing the velocity to increase. The increased velocity means that kinetic energy has increased and so the temperature must be reduced to maintain the energy balance. As the lower temperature tends to increase the density, it will tend to reduce the increase in velocity. These effects make it difficult to visualize the pressure change brought about by friction and how much was changed by the need to maintain the energy balance. The two effects can be separated by using the concept of stagnation conditions. If at any point in the pipe the gas was brought to rest isentropically then the kinetic energy would be converted back to enthalpy and the resulting conditions would be the stagnation pressure and temperature. This means that we can see the effects of pressure loss with the energy balance removed if we consider stagnation pressure between any two points. To reinforce the concept of static and stagnation pressures and their effects on flow and conceptualisation, take a look at the following example, which is a very simple steam distribution network with virtually no resistances. (Saved as "[FluidFlow3 folder]\QA CompressibleFlow\Distribution\Steam Distribution Question.ff3"). Figure 2: Example steam distribution network. In this system we are using 4" pipes for the headers and 2" pipes for the laterals (pipe node -13 & -14). After the first pipe take off (-13) what happens to the velocity in the header downstream of the take off (5) ? It reduces which means that because of the energy balance the static pressure must increase. This is called pressure recovery and often occurs in distribution systems. Friction tends to reduce this pressure, but in this case the effect of velocity is greater. Take a look at the static pressure at the inlet of the two laterals (pipes -13 and -14) and you should see that this is greater in the most remote pipe. In effect this means a greater pressure driving force is available in the most remote pipe and hence the greater flow. Don't believe this? Change the header pipe to be 2", the same size at the laterals, and see what happens. In this case the decrease in velocity downstream of the first branch still occurs but the effect of increased friction more than cancels out this effect and so we get more flow through the first pipe. The design of even flow distribution systems for gases can be problematic because of the pressure recovery effect, sizing of headers is often critical.