## Design of Experiments

In engineering, experimental verification of analysis or computational results is extremely important. For that, it is necessary to design experiments. Experiments can only be designed for those parameters for which we can make measurements. As was discussed before, the measurable parameters in thermodynamics are p, T, V, cp, cv, a, and kt. Therefore, the designed experiment can only involve these parameters.

As an example of this important issue, consider the Clausius' equation which is f dL) = I Q12 I By considering the T-s diagram, it is ob-

vious that the heat transferred is equal to Q12 = T(s2 - st). From the Gibbs' equation for enthalpy, {dh = Tds + v dp}, and since at saturation the pressure and temperature are constants, then {dh = T ds} giving \(h2 - h1) = T(s2 - s1)]. Upon substitution, i^P.^ = J (h_h) I, where 2 refers to the saturated vapor

state, g, and 1 to the saturated liquid state, f. Therefore, the Clausius equation be comes —— [ dT

) saturation TVfg

Clapeyron modified this equation by making certain assumptions. The first assumption is that vg >> vf which is an acceptable one since the vapor specific volume is much greater numerically than the liquid specific volume, as can be seen in the Steam Tables, Qengel and Boles (2008). The second assumption is that of an ideal gas for the vapor, i.e., v s v = | |l T j which, upon substitution, g VMJ (p)

makes the Clausius equation ( dp dT

Collecting like terms, this equation reduces to dT

"fg

Considering that h is a constant, an assumption that needs to be verified for the design of the experiment, and integrating and simplifying, the Clausius-Clapeyron equation is obtained as d (ln p )

which is only valid if v >> vf and we have an ideal gas for constant ht • All of these three assumptions must be satisfied before the results of the experiment can have any significance. Therefore in the design of the experiment we must consider these facts very carefully.

In designing an experiment, the first assumption for the Clausius-Clapeyron equation is satisfied when one looks at the Steam Tables, Qengel and Boles (2008), to compare the numerical values of the specific volume as a vapor and a liquid. The second assumption, an ideal gas, requires some more discussion. When we look at the compressibility diagram for substances, we see that the compressibility for all substances approach unity, meaning they approach the characteristics of an ideal gas, as the reduced pressure of the substance approaches zero,

1 as preduced

This result signifies that the experiment

Pcritical must take place at low real pressures, i.e., below atmospheric, since the critical pressure for steam is 22.09 MPa to make sure that we are approaching zero for the reduced pressure to guarantee ideal gas situation. Finally, constant ht assumption requires that the measurements must take place at a pressure an increment above and an increment below the saturation pressure under investigation. By taking this increment small and splitting it up for as many precise measurements as possible, all that is needed to do is to plot (ln p) versus 11

using the absolute temperature.

The slope of the line is negative, as the Clausius-Clapeyron equation requires, and will give the value ht once the molecular mass of water is used. Other fluids can also be used in this experiment; the only requirement is to make sure that Z ^ 1

Preduced

. 0 for the liquid that is used.

### Pcritical

For the speed of sound or the Joule-Thomson coefficient similar experiments can be designed, constructed, and experimental results can be obtained once their measurable forms are derived, as was done above. This methodology is good for any quantity for which an experiment is to be designed; the important thing to keep in mind is the assumptions under which the result is obtained. If any one of the assumptions is not met, then the experiment will not give the expected results.

This, of course, is not different from any scientific/engineering analysis or experiment. The results obtained are only as good as the assumptions made and the results are only valid under those assumptions. This may sound trivial but we will show that it is a mistake made quiet commonly by researchers and/or authors of textbooks. 