Calculation of Replacement Rate for Electrophoretic Coatings during Tank Mixing

After the initial filling of an electrophoretic tank, a large quantity of electrophoretic coating is present in the tank. Following tank filling, aged electrophoretic coating is consumed regularly on a daily basis, and fresh electrophoretic coating is replenished to maintain stable solids content and other bath parameters. The electrophoretic coating is thus subject to continuous dynamic renewal and replacement.

For commissioned electrophoretic coating lines, switching to electrophoretic coatings from different suppliers or of different grades is often required due to evolving quality requirements or cost considerations. From environmental and economic perspectives, tank mixing is widely adopted in practical production. During on-site mixed-tank production, fresh coating can only be replenished after a portion of the existing bath is consumed. Therefore, theoretical calculations are needed to correlate renewal time, coating replenishment quantity and replacement rate (proportion).

A consumption-renewal model is established to simulate actual production conditions with the following approximations:

• The volume of the electrophoretic tank remains constant, so the total mass of electrophoretic coating in the tank M_n is fixed at M.
• The bath solids content remains stable after replenishment of fresh electrophoretic coating. The daily consumed mass of electrophoretic coating m_n (consumption) equals the daily replenished mass m_n (replenishment), denoted as m. The daily consumption rate and replenishment rate are constant, defined as V = m/M.
• After replenishment, fresh coating is thoroughly mixed by continuous tank circulation, such that aged coating and fresh coating are uniformly distributed. During production consumption, aged and fresh coating are consumed via film formation proportionally to their respective fractions.

Based on the definition of renewal period:
Renewal period T.O (d) = M/m, and V = m/M = 1/T.O.

Derived from mass fraction definition:
On day n:
Fraction of aged coating component a: P_a,n = (1 − m/M)ⁿ = (1 − V)ⁿ = (1 − 1/T.O)ⁿ …… ①
Fraction of fresh coating component b: P_b,n = 1 − P_a,n = 1 − (1 − m/M)ⁿ = 1 − (1 − 1/T.O)ⁿ …… ②
Total cumulative fresh coating mass: ΣM_b = nm …… ③

Using exponential and logarithmic operation rules, Equation ① can be rearranged to:
n = log_(1−m/M)P_a,n = log_(1−V)P_a,n = log_(1−1/T.O)P_a,n …… ④

In Equations ①, ②, ③ and ④, n represents time in days. Time conversion can be performed between renewal period and renewal cycles. For example, at a renewal of 1/T.O, n = M/m days can be substituted directly into Equation ① to calculate P_a(1/T.O) = P_a,n = P_a(M/m).

The above model is built on a daily consumption basis. If the on-site consumption and replenishment model is differentiated, the variation law of aged coating in the electrophoretic tank can be derived using multi-component mass balance theory[2]. This model treats consumption and feeding as a continuous dropwise addition process: m approaches an infinitesimal value while n approaches infinity.

Fraction of aged coating: P_a,x = e^−x …… ⑤
Fraction of fresh coating: P_b,x = 1 − e^−x …… ⑥
where e is Euler’s number, and x is the renewal cycle, x = nm/M.

When the renewal reaches x_T.O on site, the formula can be applied directly to calculate the fractions of aged and fresh coating.

The fresh coating replacement rate rises rapidly at the initial stage, yet the incremental rate of fresh coating proportion slows down as time elapses.

Two calculation scenarios are commonly encountered in practical mixed-tank work:

1. Calculate the fresh coating proportion (renewal replacement rate) by first computing the renewal cycles from the total fresh coating added.
2. Specify the target fresh coating proportion (renewal replacement rate), then calculate the required renewal cycles and total fresh coating dosage.

Key representative datasets can be tabulated for both scenarios (see Table 1 and Table 2).

Case Study

Take a body cathodic electrophoretic line as an example: tank volume ≈ 200 m³, typical bath solids content 20%, total mass of electrophoretic coating in tank M = 100 t. Daily coating consumption and replenishment m = 2 t.
Renewal period T.O = M/m = 100/2 = 50 d.

Table 3. Correlation of feeding days, aged/fresh coating fraction, renewal cycles and cumulative coating addition

This paper aims to help practitioners master the calculation methods for renewal period and replacement rate, guide laboratory simulated mixed-tank test design, and ultimately support on-site bath renewal control and electrophoretic coating switching via tank mixing. Quality control and risk assessment can thus be implemented both theoretically and practically.