How energy saving reduced head pressure operation effects TEV Capacity

For the general purposes of refrigeration, the principle function of a TEV is to provide a metered quantity of refrigerant to affect the required refrigeration duty. TEV's use small orifice, a small orifice is one in which the variation of pressure or head across the orifice hole, when fed horizontally, is small enough to be neglected. The thermo-physical properties of the refrigerant either side of the TEV dictate its performance. In reduced head pressure scenarios we need only concern ourselves with the refrigerant conditions ahead of the TEV since suction pressures should remain constant per the systems designed evaporator temperature and load specifications. The main thermo-physical contributors are Pressure (P), Net Refrigeration Effect (NRE) and Density ( r). Where small orifice used with refrigerants are concerned, viscosity is generally not considered contributory since it has little effect on small orifice flow rates, if anything an increase in viscosity will very slightly increase small orifice flow rates, the opposite of what one expects to see with tubes, including capillary tubes, which are different from an orifice in that they have wall or boundary length.

To understand the extents to which energy-saving reduced head pressures affect TEV performance it is perhaps necessary to study some aspects of the derivation of the orifice flow formula, which I decided should make up the bulk of this particular article.

In essence, the volume flow of fluid through an orifice is a function of the fluid velocity and the orifice free cross sectional area, put simply; it is the product of fluid velocity and orifice area:

Q = v A

Where:
Q = volume flow
v = velocity
A = flow area.

Since orifice area is already available from, say, valve production data tables, the first trick is in calculating the velocity at which the fluid would approach the orifice. This is a concept I have found many tech's have difficulty with, as important I think it is, so I decided I would expand on the velocity aspect here a little more than I otherwise would.

Lets consider an orifice at the bottom of a 10m high fluid tank. If we were to hold a sample of fluid up at the brim of the tank, 10m from the bottom, the potential energy contained by that sample would be:

PE = r g h

Where:
PE = Potential Energy
r = density
g = gravitational acceleration
h = height.

The Kinetic Energy that a fluid sample would have attained on approaching the tank base after being dropped from 10m above would be precisely equal to the Potential Energy originally possessed by the sample prior to release, 10m above. So, to calculate the final velocity we can borrow from a formula that equates the initial potential energy with the attained kinetic energy, KE = PE, which would be:

½ r v² = r g h

The above formula, when rearranged to determine velocity, will take the form:

v = (2 g h)^½

The final velocity of our fluid sample would then be v(2x9.81x10) = 14m/s being the arrival velocity at which the sample would pass through the orifice. Putting numbers to these assumptions will demonstrate the validity of the premise, if our fluid sample were water then the pressure exerted at the base of the tank would be (PE = r g h): 1000kg/m³ x 9.8m/s² x 10m = 98,000Pa

If instead a cubic meter of water were dropped into the tank from 10m then the velocity pressure immediately approaching the tank base would be (KE = ½ r v²): ½ x 1000kg/m³ x 14m/s x 14m/s = 98,000Pa

Interestingly, we can see that whether a sample of our fluid is dropped from a given height or a fluid column stands the same height, the pressures and resulting fluid velocities amount to the same value. We also see confirmation of the results of Galileo's leaning tower experiments, that final velocity is independent of density i.e. weight.

The theoretical volume of fluid passing through the orifice can then be calculated by multiplying the theoretical fluid velocity by the orifice area.

Q = A (2 g h)^½

The fluids mass flow would then simply be the product of fluid volume flow and fluid density:

kg/s = A r (2 g h)^½ = Q

Which can be rearranged as:

kg/s = A (2 r² g h)^½

However, when considering refrigerant properties immediately surrounding a TEV, we don't use fluid height, instead we talk in terms of fluid pressure. If we rearrange an appropriate formula for pressure, the one presented previously, h = P/(rg), in order that we may substitute it for the height component, in the last formula above, we are left with the final product:

kg/s = A (2r P)^½

There are coefficients of orifice dynamics to take into consideration when true volume or mass flow is required but these are outside the scope of this article.

The capacity (W = Watts) of the TEV orifice is then simply the product of mass flow and the refrigerants NRE.

W = NRE A (2r P)^½

Returning to the fluid tank, an interesting observation occurs when the fluid's density is doubled while the pressure is kept constant. Ordinarily, if the density is doubled then so would the pressure, however, if at the same time the fluid height in the tank is halved, then the pressure at the tank bottom will remain unchanged. The observation to note then is that the modified orifice approaching velocity is not half of the original velocity, an earlier formula demonstrated that the velocity changes as the root of the height, meaning if the height is halved, reduced by factor of 0.5 of the original, then the velocity is only reduced to a factor of 0.71 of the original. Of course, the same principle will be seen when the pressure is halved by means of halving the fluid height, again the orifice approach velocity is only reduced to a factor of 0.71 rather than the more intuitive guess of 0.5. All this is summarised in the expression: v = v(2rP)^½.

As mentioned in a previous article, when it comes to working with formulae in the analysis of varying scenarios, I prefer to keep them, as best I can, in the proportional expression form. This form doesn't always allow for the calculation of specific unknowns but does help predict the values that would occur in scenarios different from any known base scenario.

In a speculative discussion with Andy Schoen, of Sporlan Valve Company, USA, we developed a useful proportional expression that can be used to determine by how much a valves' capacity would change when, on the high side only, system refrigerant pressures and temperatures are changed, as would occur when head pressures are reduced in an attempt to affect energy savings.

The formula has other uses, for instance as a tool to predict by how much a valves capacity would change when the system refrigerant itself is changed altogether, as occurs during ozone friendly retrofits, for example.

The formula borrows from all of the previously discussed principles and demonstrates very neatly the major high side thermo-physical players in TEV capacity.

When the formula is applied to the R22 system originally discussed in this article series, where the saturated discharge conditions are allowed to drop from 45°C to some 23°C in response to the lower UK average 10°C ambient temperatures, it predicts a 25% drop in TEV capacity meaning the TEV would only match compressor capacity any time the compressor was 75% loaded or less.