Immersion Battery Cooling: Flow, Pressure Drop and Cell Spread

The hydraulic model discretizes the dividing and collecting header momentum balances of Bajura and Jones (1976) at every lane. With $v$ the local header velocity, the momentum contribution is $\Delta p=\theta\rho(v_1^2-v_2^2)/2$; wall friction is integrated over the intervening header segments. U and Z layouts use different collector flow directions and cumulative flows. The common inlet pressure and all positive lane flows are solved together by a damped Newton iteration, subject to total-flow conservation. The Jacobian includes header momentum exchange, header friction and each lane's nonlinear resistance. The default coefficients, 1.00 and 2.65, follow the baseline discussed in the attached Energies 2021 paper, p. 16.

For each rectangular equivalent gap, $D_h=2gw/(g+w)$, $Re=uD_h/\nu$ and $x^+=L/(D_h Re)$. The Shah–London (1978) apparent Fanning-friction correlation includes velocity-profile development:

The rectangular-duct Poiseuille polynomial is used for $Po$; $K_\infty$ and $C$ are interpolated between the published aspect ratios. Lane pressure drop is $(4f_{app}L/D_h+K)\rho u^2/2$. Across-cell operation adds the entered row form losses. Smooth round headers use Churchill's 1977 all-regime Darcy friction correlation.

Across cylinders, the Churchill–Bernstein (1977) Nusselt correlation is evaluated at the minimum-gap velocity. Along flat cells, the Edwards–Denny–Mills (1979) thermal-entrance term is added to Shah–London's rectangular isothermal fully developed limit. Each cell's mean film coefficient comes from the difference of the integrated entrance conductance at its two ends, preserving thermal development downstream.

Each cell is an isothermal heated surface coupled to a finite-capacity coolant stream. Define $C_i=\rho q_i c_p$, $NTU_{ij}=h_{ij}A/C_i$ and $\varepsilon_{ij}=1-\exp(-NTU_{ij})$. Integrating $C_i\,dT_b=h_{ij}(T_s-T_b)\,dA$ gives
$T_s=T_{b,in}+\frac{Q_j}{C_i\varepsilon_{ij}},\qquad T_{b,out}=T_{b,in}+\frac{Q_j}{C_i},\qquad T_{core}=T_s+Q_jR_{core}.$
This is the constant-wall effectiveness relation documented by ChEDL ht.hx (and Bergman et al., sixth edition, 2011). It applies to both flow orientations and every sweep point. The implementation evaluates $1-\exp(-NTU)$ with expm1 to retain precision at small NTU. For positive heat, the outlet fluid approaches the cell surface from below, including at very low flow. The mixed outlet follows total energy conservation. Pump head is total loop pressure divided by $\rho g$; electrical input is $q\Delta p/\eta$.

Steady incompressible Newtonian fluid, constant properties, smooth equal-area round headers, equally spaced identical forward-flow lanes and a fixed common supply temperature. Heat leaves only through the assigned wetted area. All lanes share the same serial heat-load profile; its mean remains the entered cell heat. Each cell has a spatially uniform surface temperature found from its specified heat and finite coolant capacity. The cell resistance represents a lumped core-to-surface temperature rise. Header axial heat transfer and external heat gains are neglected. Header pressures are relative to the pack outlet static pressure; loop pressurization, elevation changes in open circuits and pump suction/NPSH are outside this model.

The gap-pressure model is restricted to $Re\leq2300$ in every lane. The flat-cell thermal entrance approximation is intended for wide slots (gap/width ≤0.1) and preferably Pr ≥5; its finite-aspect correction is an engineering extension of the parallel-plate entrance term. The local coefficient treatment does not solve the full conjugate Graetz problem for changing wall temperatures.

For crossflow, a rectangular equivalent gap plus user-entered row form loss represents the hydraulic path. The isolated-cylinder heat-transfer law is applied at the gap velocity; shielding, wake interactions, tube-bank enhancement and jets are not resolved. The default row loss of 1 is an illustrative design input, not a validated battery-array drag coefficient. Calibrate both row losses and header momentum coefficients against measurements or CFD. The reference cases verify correlations and limiting values, not a complete battery pack.

Use actual fluid properties near the operating film temperature. The illustrative Shell defaults mix density at 15 °C with viscosity, heat capacity and conductivity at 40 °C. No buoyancy, viscosity-temperature coupling, boiling, transient cell storage, runaway, electrical compatibility or heat-rejection-system model is included.

Every lane and cell is solved at the entered operating point. The flow sweep uses 61 solved points in crossflow and at the defaults. Large flat-cell cases use 13–61 points, scaled by the lane × cell count to stay near a three-second value calculation; a narrow flow optimum may then be missed. Charts use PCHIP interpolation to 121 points, so narrow extrema between sweep points may be missed. Header pressure curves interpolate between solved branch stations. Sweep portions above the laminar Reynolds limit are omitted. The first drawing is schematic and shows at most 16 lanes and 10 cells per lane; the temperature map includes every cell.