Capillary Length & Response Time Calculator

This industrial-grade calculator simulates the performance of Remote Seal Systems. It calculates the Combined Zero Shift Error (Head + Stiffness) and the dynamic Time Constant ($\tau$) to predict sluggishness in control loops caused by viscous fill fluids in long capillaries.

1. Fill Fluid Properties

Fluid Type
Physical Data

2. Capillary & Seal Geometry

Dimensions
Diaphragm
Temperatures

Engineering Insights: The Three Errors of Remote Seals

1. Head Error (Density Effect)

This is the most common error in level applications. The oil in the vertical section of the capillary has weight. As temperature rises, the oil expands and becomes less dense. This reduces the hydrostatic head exerted on the transmitter.

$$ \Delta P_{head} = \Delta \rho \cdot g \cdot H_{vertical} $$

For a 10m capillary with Silicone oil, a 40°C temperature rise can cause a zero shift of over 50 mmH2O. This error is purely a function of vertical height, not total length.

2. Stiffness Error (Volumetric Expansion)

Fluids expand with temperature. In a closed capillary system, this expanding volume ($\Delta V$) has nowhere to go but to push against the sensing diaphragm. The diaphragm acts like a spring with a stiffness constant ($K$).

$$ \Delta P_{stiff} = \frac{\Delta V_{fluid}}{K_{diaphragm}} $$

Small Diaphragms = Big Errors: A 2-inch diaphragm is much stiffer than a 4-inch diaphragm. It resists the expanding oil more, creating a much larger back-pressure error. This is why 3" or 4" seals are preferred for low-pressure measurements. This error depends on Total Length (Total Volume), not just vertical height.

3. Response Time (Viscosity Drag)

Pushing oil through a tiny tube takes time. The friction of the fluid against the capillary walls creates a time delay ($\tau$).

Viscosity is King: Viscosity is highly temperature-dependent. Silicone oil that flows like water at 25°C turns into honey at -40°C.
The Diameter Law: Flow resistance is proportional to $1/d^4$. Reducing the capillary ID from 2mm to 1mm increases the response time by a factor of 16!
Design Trade-off: Larger ID improves response time but increases fluid volume, which worsens the Stiffness Error (Temperature Drift). It is a balancing act.

4. Cold Ambient Effects

In winter, outdoor transmitters can become extremely sluggish. If a control loop relies on a fast pressure reading (e.g., surge control), a cold capillary can cause the loop to go unstable or fail to react in time. Heat tracing capillaries is often required not to prevent freezing, but to maintain a constant, low viscosity for speed.

5. Hydrogen Permeation (Gold Plating)

In certain processes (Refining, Hydrotreating), atomic hydrogen can diffuse through standard SS316 diaphragms. It accumulates in the fill fluid, forming hydrogen gas bubbles. This destroys the vacuum and creates massive drift. Solution: Use Gold-Plated diaphragms which act as a hydrogen barrier.

6. The Myth of "Balanced" Systems

Many engineers assume equal capillary lengths cancel all errors. This is false. Equal lengths cancel Stiffness Error (Volume expansion matches), but they do NOT cancel Head Error unless the vertical elevation is also identical. If the LP leg drops 5m and the HP leg drops 1m, the LP leg will have 5x the density shift error, regardless of total length.