## Static Dissipator Performance

- Published: December 24, 2013, By Kelly Robinson

**1. SUMMARY**

We will find that (1) the better static bars produce more ions, and (2) that better installation exposes a longer web distance to the corona ions from our dissipator. The key quantitative result is Equation 8.

**Equation 8: For better static dissipator efficiency η _{DISSIPATOR}, the time T_{WEB} that the web is exposed to the dissipator should be bigger and the dissipator time constant τ_{DISSIPATOR} should be smaller.**

The neutralization efficiency η_{DISSIPATOR} is determined by the electric Reynolds number Re in Equation 9 that includes the web speed U_{WEB}, the number density of ions n_{ION} produced by our static dissipator, and the web distance L_{ION} that is exposed to corona ions from our dissipator.

**Equation 9: The electric Reynolds number Re determines dissipator performance.**

The quantities in parentheses are constants. Following are the design parameters that we control that determine the dissipator neutralization efficiency.

**2. GEOMETRY**

**Figure 1. The static dissipator generates positive and negative corona ions. The negative ions move across the gap G to neutralize the positive static charges on the web.**

Let’s analyze the performance of a static dissipator used to neutralize the charges on the web in Figure 1. The web enters with static charge density σ_{WEB} and moves past the static bar at speed U_{WEB}. The static bar on the grounded metal machine frame generates both positive and negative corona ions near the tips of each pin. The electric field E from the positive static charges on the web within the ion field length L_{ION} reach the static bar and attract negative corona ions from the active static bar across gap G. The positive corona ions are repelled by the electric field and move towards the grounded plate.

**3. ANALYSIS**

To estimate the neutralization efficiency, let’s look at the flow of corona ions from the static dissipator to the charged web. The current density J_{ION} in Equation 2 is proportional to the electric field E.

**Equation 2: The current density J _{DISSIPATOR} is proportional to the electric field E.**

The static dissipator generates ions with a number density n_{ION} each having charge e and electric mobility b. The number of ions n_{ION} is the key performance metric for the static dissipator. More ions are better. Most corona ions carry one electronic charge (e=1.6×10^{-19} C). Corona ions in air have a mobility of about 3×10^{-4} m^{2}/(V-sec).

We can find the effective resistance of the ion flow by estimating the electric field using Equation 3.

**Equation 3: The electric field E is estimated to be the web voltage V _{WEB} divided by the gap G.**

Use Equation 2 and Equation 3 to find the dissipator resistance R_{DISSIPATOR} in

Equation 4.

**Equation 4: The dissipator resistance R _{DISSIPATOR} is the ratio of web voltage V_{WEB} to dissipator current I_{DISSIPATOR}.**

Equation 4 is sensible. The dissipator resistance R_{DISSIPATOR} is larger when the gap G is larger. And R_{DISSIPATOR} is smaller when our dissipator produces more ions n_{ION}.

To find the time required to neutralize the web, we need to know the web capacitance. The capacitance between the web and the static dissipator is estimated in Equation 5 using a parallel plate approximation.

**Equation 5: The capacitance C _{DISSIPATOR} between the web and the static dissipator is estimated using a parallel plate approximation.**

Equation 5 is also sensible. The capacitance is smaller when the gap G is bigger.

Finally, the time constant τ governing the dissipation of the web static is found in Equation 6.

**Equation 6: Time constant τ _{DISSIPATOR} governs the dissipation of web static.**

Finding the dissipator time constant τ_{DISSIPATOR} in Equation 6 is a key result. The time constant τ_{DISSIPATOR} governing static dissipation by a static dissipator depends only on the number of ions n_{ION} generated by the dissipator.

Wow! The dissipation time constant is determined solely by the design of the static dissipator and does not depend on how the dissipator is installed. Specifically, the gap G and web length L_{ION} do not appear on Equation 6.

Web static σ_{WEB} in Equation 7 dissipates exponentially with time.

**Equation 7: Web static σ _{WEB} dissipates exponentially with time.**

**4. NEUTRALIZATION EFFICIENCY**

When the web exits with zero static in Equation 8 (σ_{WEB-OUT}=0), the static dissipator efficiency n_{DISSIPATOR} is 100%.

**Equation 8: For better static dissipator efficiency n _{DISSIPATOR}, the time Τ_{WEB} that the web is exposed to the dissipator should be bigger and the dissipator time constant τ_{DISSIPATOR} should be smaller.**

The electric Reynolds number Re in Equation 9 determines how much charge is dissipated.

Equation 9. The electric Reynolds number Re determines performance.

The quantities in parentheses are constants. So, the neutralization efficiency is determined by the web speed U_{WEB}, the number density of ions n_{ION}, and the web length L_{ION} that is exposed to the flow of ions.

Equation 9 is sensible. For higher efficiency n_{DISSIPATOR}, the electric Reynold's number Re should be smaller. So, static neutralization efficiency is better when the web speed U_{WEB} is slower, when the number of ions n_{ION} from our dissipator is higher, and when the web length L_{ION} exposed to ions is larger.

**5. DISSIPATOR INSTALLATION**

Clearly, installing the static dissipator is important. A better static dissipator installation makes L_{ION} larger. Distance L_{ION} in Figure 10 depends on the installation geometry. To determine L_{ION}, in your mind's eye, draw a pencil line across the web. This line is the center of a cylinder that expands radially. The two lines in Figure 10 are the centers of cylinders that touch the tips of the dissipator pins and also just touch the idler rollers.

Figure 10. Install static dissipators to maximize the distance L_{ION}

that the web is exposed to corona ions.

**6. CONCLUSIONS**

Our analysis provides two key insights.

- Better static bars produce more ions. Look for static bars that have more pins, that have sharper pins, and that have higher operating voltages.
- Better installations expose longer web spans to corona ions. A better installation has a longer L
_{ION}, which is the web span exposed to corona ions from the static dissipator.

And, our analysis provides one key quantitative result. The neutralization efficiency n_{DISSIPATOR} in Equation 8 depends on the electric Reynolds number Re in Equation 9. In summary, the neutralization efficiency is determined by the web speed U_{WEB}, the number of ions N_{ION} generated by the static dissipator, and the distance L_{ION} that the web is exposed to corona ions.

I invite you to ask questions about this column and to suggest future topics. My e-mail address is This email address is being protected from spambots. You need JavaScript enabled to view it..