Innovative Theory of Heat and Mass Transfer in Dry Spinning Volume 62- Issue 5

Tatsuhiro Yamamoto*

• Faculty of Human–Environment Studies, Kyushu University, Japan

Received: July 17, 2025; Published: July 25, 2025

*Corresponding author: Faculty of Human–Environment Studies, Kyushu University, Japan

DOI: 10.26717/BJSTR.2025.62.009793

Abstract PDF

This study presents an innovative numerical analysis methodology for dry-spinning carbon fiber production. Unlike conventional derivations that rely on dimensionless numbers such as the Nusselt and Sherwood numbers— rooted in boundary‐layer theory for simple geometries (e.g., duct flows or basic building shapes)—our approach unifies thermal and mass transfer coefficients into a single, “thermal” metric. We construct a compact theoretical framework that captures the key physical phenomena of dry spinning and validates its plausibility through numerical simulations in simplified systems. Focusing further on fiber flutter within the spinning chamber, we draw an analogy to structural vibrations in mechanics to propose an effective strategy for exploiting thermal and mass transfer effects. Demonstration calculations illustrate this strategy, confirming the theory’s applicability within defined parameter ranges, even though they do not amount to exhaustive precision validation. Synthesized from a comprehensive review of existing patents and literature, our model suggests significant potential for energy savings and advanced design support in manufacturing carbon–fibre–reinforced hydrogen tanks for automotive and industrial applications. To maintain theoretical integrity, we deliberately refrain from introducing arbitrary fitting parameters, and we ask for the reader’s understanding of this principled choice (Graphical Abstract).

Keywords: Dry Spinning; Computational Fluid Dynamics; Numerical analysis; Heat and Mass Transfer Equation; Fiber

Abbreviations: CFD: Computational Fluid Dynamics; BEV: Battery Electric Vehicles; FCV: Fuel Cell Vehicles

Graphical Abstract

Currently, the automotive industry is accelerating the development of battery electric vehicles (BEVs) and hydrogen fuel cell vehicles (FCVs). In particular, the rise of Tesla [1] has made BEVs an attractive choice for consumers, yet battery performance in cold climates remains a concern despite ongoing improvements [2]. Charging infrastructure has expanded compared to the past but still falls short of demand. Moreover, major nations such as the United States have declared plans to ban gasoline vehicles by 2035 [3], and Japan is charting a similar phased elimination strategy [4]. Recently, Toyota Motor Corporation and others have advanced FCV development and brought models to market [5], though refueling stations remain scarce and new entrants face challenges in adoption. Nonetheless, hydrogen vehicles operate on a fundamentally sound principle and offer energy‐efficient, environmentally friendly performance [6]. Thus, the prevailing view holds that, alongside BEVs, FCVs should play a complementary role in future mobility. Hydrogen vehicles employ pressurized hydrogen tanks whose burst life is critically important [7]. Carbon fiber [8], long utilized in military applications, is a logical reinforcement material for these tanks. Carbon fibers exhibit diverse grades and material characteristics [9,10], and their production involves various processes. Among these, dry spinning [11,12] reached a technological plateau in the 1990s [13], resulting in relatively few academic publications, although empirical quality improvements continue at university laboratories [14]. Traditional dry‐spinning models focus on radial macroscopic solvent diffusion and vertical moisture reduction (in cases where water is the solvent), but techniques designed for single fibers do not scale to mass production.

Moreover, thermal and mass transfer coefficients are treated as empirical fitting parameters, assuming post‐hoc calibration, which undermines their utility in practical product development. To capture the essence of heat and mass transfer, a straightforward numerical solution on a three‐dimensional mesh is preferable. While this approach aligns with computational fluid dynamics, it relies on robust, evolving metrics such as the SI unit system [15] rather than attempting to resolve the formidable, unresolved Navier–Stokes equations [16,17]. The continued use of Nusselt [18] and Sherwood numbers [19] via conversion or semi‐empirical formulas introduces inconsistencies, hinders precision validation, and depends on undisclosed adjustment factors, rendering them impractical for engineering design. For example, calculating heat flux from tire vertical oscillations using the Nusselt number has been demonstrated [20], yet alternative, non‐experimental formulations are feasible. In building engineering, macro‐scale simulations invariably implement the Nusselt number [21], but these models do not demand high precision; still, global heat transfer must remain consistent with micro‐scale CFD results [22], warranting careful consideration. Against this backdrop, the present study aims to establish a systematic theoretical framework for numerical analysis of the dry‐spinning process used to manufacture carbon fiber for reinforcing hydrogen tanks in automotive applications. Furthermore, the methodologies developed here can be extended to assess comfort indices in vehicle interiors [23] and architectural spaces [24].

In dry spinning, capturing heat and mass transfer is important for reproducing the physical phenomena that occur when the solvent in the fiber is blown away by hot air. Additionally, the complicated handling of dry spinning equipment makes interpreting the analysis results extremely difficult.

The Whole Idea Behind the Dry Spinning Revolution

In evaluating the dry‐spinning process, it is instructive to consider the system in a two‐ or three‐dimensional plane as illustrated in Figure 1. In this representation, the heat supply strategy is critical: near the nozzle exit, one increases the volumetric flow of hot air while concurrently lowering its temperature to prevent excessive fiber sticking. At the final winding section, the hot‐air flow rate is then reduced and the temperature further decreased as a fine‐tuning step. However, any nonuniformity in the hot‐air distribution can induce diameter variations in the fibers. If such effects can be faithfully reproduced in numerical simulations, process control and equipment development for dry spinning would be greatly enhanced. For example, when using a water‐soluble solvent such as water, the moisture content at the winding stage becomes a key quality metric. Achieving close agreement between experimental measurements and simulation results is therefore imperative, yet in industrial practice experiments typically lead the development effort and numerical analyses are relegated to post-hoc validation.

Figure 1

Confidentiality restrictions on proprietary process data further hinder academic researchers’ access to real‐world conditions. Conventionally, the extent of fiber “dryness” is governed by coupled heat‐ and mass‐transfer equations. In the present study, however, we adopt the simplifying assumption that heat and mass transfer within the fiber are perfectly mixed. This choice is justified by the considerable difficulty of constructing a fully detailed theory of intrafiber transport phenomena. Frankly, we observe that by applying classical heat‐conduction equations within a CFD framework, one can achieve sufficiently accurate results for engineering purposes.

Derive Formulas Using a Simple Model

Equation (1) shows the new heat transport coefficient. It is characterised using the mass flow rate, and the dissipation mechanism from the fiber surface should be determined through experimentation. Additionally, using the average air velocity of cells near the fiber makes it possible to understand the appearance of “heat” transport, which makes the heat transfer coefficient highly versatile.

Equation (2) shows the mass transfer coefficient. It differs from equation (1) in that water vapor is in the first cell near the fiber. This location is also involved in the mass flow rate and includes a large component of “heat” and convection. This is why the mass transfer coefficient differs from previous coefficients, such as those related to texture.

The symbols are as follows:
q: Heat value of the first cell near the fiber [J or W]
ρ: Air density[kg/m3]
F: Macroscopic mass flow rate at the fiber surface[kg/s]
T_s : Fiber surface temperature[K]
T_fluid : Macroscopic fluid temperature[K]

w: Average wind speed in the first cell near the fiber [m/s]
H_c : Convective heat transfer [J or W/m2K]
M_c: convective mass transfer[kg/m2K]
For example, in Equations (1) and (2), non-stationarity can be associated with the forward difference of q or g by converting to per unit time. Figure 2 shows a schematic diagram of the fundamental reformulation of the heat and mass transfer coefficients based on “heat” in a two-dimensional model. Although the heat and mass transfer coefficients per unit of space are necessary, this paper focuses on heat and mass transport in the vicinity of fibers. While the heat and mass transport coefficients per unit space are required, this paper focuses on heat and mass transport in the vicinity of the fiber. If heat flow per unit m³ exists, then precise heat transport could theoretically be reproduced by combining the order of the unit m³ and the mesh division volume. This is merely an idea, but it is demonstrated in equation (3).

Here, W_r has units of m³/s, and the heat and mass transfer coefficients for a unit spatial cell are defined accordingly. In the present calculation, we assume a mesh width of 1 m; by scaling the volumetric term (m³) to match this resolution, it becomes possible to represent heat and mass transport between adjacent cells. In other words, by formulating balance equations at each cell interface, the incoming heat flux (and mass flux) into neighbouring cells can be computed directly.

H_c: Convective heat transfer for units [W/K] M_c: Convective mass transfer unit [kg/K]

Figure 2

The objective is to examine the usefulness of the constructed theory by means of a simple model system. The objective is to demonstrate how the theoretical equations can be applied by performing realistic calculations.

Description of the Analysis Model

Figure 3 illustrates the analytical conditions of the unit mesh for a dry-spinning fiber mesh division. The fiber can be conveniently modelled in two dimensions. Although polyhedral or tetrahedral meshes have been available in recent years, they differ in volume, making the calculation complicated. Therefore, a square mesh is used. It is possible to calculate the temperature of the fiber and the moisture content if water is used as the solvent. Assuming complete mixing, the dimensionless contribution in each direction can be calculated based on the ratio of the behaviour of the heat or mass transfer coefficient to the raw dryness, after complete mixing and diffusion. Equation (5) provides the mixing temperature inside the cell. The water content can be easily calculated from equation 2 since equation 6 can be obtained from equation 2. Depending on the location of the unknowns, it may require some creativity and ingenuity to perform the calculation.

Figure 4 describes the calculation method when the condition of the nearby air cell is given. First, the room temperatures in regions 1 and 2 are calculated using Equation (7). In this case, however, the heat transfer coefficients in region 4 are used for the heat transfer coefficients, assuming a convergent calculation. This is somewhat difficult, but since the purpose is to construct a theory, we will not discuss it extensively.

The Results of the Calculation and the Ensuing Discussion

Table 1 shows the calculation conditions and results for Figures 3 & 4. Although the formulas have many theoretical unknowns, the calculations generally converge within a few hundred iterations, as is typical in steady-state analyses, when certain common-sense assumptions are made. In this study, we will leave the pros and cons of convergence calculations for a future issue. To this end, several numerical analysis trials based on detailed experiments will be necessary. It would be absurd to attempt this without prior research, so this study will be limited to applying and confirming the new theory. It is particularly desirable to measure the surface temperature, mass flux, etc., so the introduced formulas will aid the new fluid theory in the future, even though it cannot be said that a single boundary condition can completely solve the boundary conditions. The heat and mass transfer coefficients were confirmed to be within the range of common knowledge. It should be noted that, given a hypothesis of the conditions, the temperature and water content inside the fiber can be determined. Since the formula is simple, it would be easy to implement a convergence calculation algorithm.

Table 1: Conditions Presented and Achieved Outcomes.

Figure 3

Figure 4

Approach to the Structure of Dry Spinning Equipment

Dry spinning equipment originally used a combination of convection and radiation to reduce moisture or water content and perform flame-retardant treatment. However, fire-quenching may be necessary. Using a simple hot-air generator to prepare a heat source for fire-quenching is more likely to improve fiber quality. This is why high strength can be expected by passing fire through the fiber. It is also beneficial in creating a stable dry spinning device. With a structure where the fire flows in from the bottom and exits at the top, it is possible to supply the same amount of heat in both directions.

In theory, the only way to address fiber sway is to fly the beam, adjust the stiffness, and observe the outcome. However, this is not easily achieved in practice. The coefficients of heat and mass transport are greatly affected by fiber sway. In other words, the influence of average wind velocity on the first cell near the fiber, the movement of water content from inside the fiber to the surface layer, and other phenomena must be examined in detail.

An Approach to Adjusting Moisture Content Through Efficient Fiber Shaking During Dry Spinning

As shown in Figure 5, if the hot air in the dry spinning equipment is uneven, the fibers will oscillate. If the oscillation is periodic and stationary, it can be expressed physically in a manner like a static structure in structural mechanics.

A Method for Calculating Fiber Shaking That Occurs During the Hot Air Drying of Dry Spinning

Similarly, the distance to the fiber windings changes concerning fiber shaking. Additionally, the averaging process must be considered under the assumption of periodic stationarity since the nearby wind speed will be inaccurate as displacement occurs. While it is possible to easily calculate the averaged wind velocity by applying the principle of virtual work, the focus of this paper is a new derivation and verification of the heat and mass transfer equation. Therefore, I will only propose the theory (Figure 6).

Figure 5

Figure 6

In this study, we examined the design of a dry spinning apparatus for producing carbon fiber wound around hydrogen tanks and developed a novel formulation for heat and mass transfer in the fiber’s surroundings. As a result, the following insights were obtained:

1. We proposed a new theory of heat and mass transfer around a unit fiber region. Certain terms depend on parameters such as mass flow rate and the air flow velocity in the first computational cell adjacent to the fiber, indicating behaviours distinct from conventional models.

2. By using temperature as the unifying variable, we established a framework for understanding fiber surface temperature, the surrounding cell temperature, and mass transport. This theory is highly beneficial, but validation against experimental data remains a future challenge due to the added complexity in theoretical analysis.

3. We extended our formulation to account for heat and mass transfer induced by fiber oscillation. While the principle of virtual work is broadly applicable, incorporating this approach into the design of a periodic steady-state dry spinning apparatus requires detailed structural mechanics knowledge. Deformation-driven variations in internal moisture content introduce further complexity and demand a cautious methodology.

Our primary focus was on theoretical development, and we simplified the problem to a two-dimensional plane to assess the impact of the new derivation process. Although not entirely unprecedented, this approach captures the physical phenomena more accurately than directly applying standard convection and mass transport equations without a dedicated derivation. We also outlined a theoretical vision of fiber oscillation, but manufacturing practices often govern the actual design of dry spinning apparatuses. To achieve technological breakthroughs, researchers must align theoretical advances with the rapid pace of product development. Therefore, we stress that these theories should ultimately be implemented at the forefront of manufacturing.

Our derivation of heat and mass transfer in dry spinning not only leverages the clarity of the SI unit system but also incorporates meticulous dimensional checks. We expect that this novel theoretical framework will prove highly useful in the practical manufacturing environment of dry spinning apparatuses.

There are no conflicts of interest to declare.

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