Thermal Properties of Cells
◈ Thermal transfer
- Thermal conductivity is important as it affects the cell’s temperature distribution and influences cooling. The anisotropic thermal conduction of the battery can be characterized by in-plane (axial) thermal conductivity parallel to the cell layers and cross-plane (radial) thermal conductivity perpendicular to the layers. In-plane conductivity is roughly one order of magnitude higher than cross-plane conductivity. With the increase in SOC and the operating temperature, thermal conductivity goes up.
- According to the Fourier’s Law of heat conduction with steady-state heat flow (∂T/∂t=0), it reads:
$$ {q = - \kappa\nabla{T}} $$ , where q is heat flux vector and κ is thermal conductivity, and T is temperature.
For the 1-d heat conduction, it is simply q = κ dx/dT or more generally it can be written as:
$$ {Q = - \kappa{A}\frac{dT}{dx}} $$ , where Q is heat and A is the cross-sectional area, and d is the legth of the heat conduction in the cartesian coordinate.
For the cylindrical case (relevant to the cylindrical cell), the equation becomes:
$$ {Q = - \kappa (2\pi RL)\frac{dT}{dR}} $$ , where R is radius of the cylinder.
- The whole cell radial thermal conductivity can be estimated for the cylindrical cell by using the thermocouples placed at the inside (spindle) and outside the cell to measure the temperature difference and the metal alloy wire to heat the cell’s center using a DC power supply for applying different currents. The effective radial thermal conductivity is derived based on following equation:
$$ {Q = \frac{2\pi\kappa_{eff}L(T_{1}-T_{2})}{ln(\frac{R_{2}}{R_{1}})}} $$ , where T1, R1 and T2, R2 are temperature and radius of the inner part and outer part of the cell, respectively.
- Thermal conductivity of the cell is dependent on the internal components and their configuration. Hence, the equations for in-plane and cross-plane conductivities are dependent on the effective thermal conductivity of each component layer and its respective thickness as follows:
$${\text{Radial thermal conductivity} = \frac{\Sigma_{j} L_{j}}{\Sigma_{j} \frac{L_{j}}{\kappa_{eff, j}}}} $$
$${\text{Axial thermal conductivity} = \frac{\Sigma_{j} L_{j}{\kappa_{eff, j}}}{\Sigma_{j}{L_{j}}}} $$
To measure the thermal conductivity of each material, a combination of methods is considered – such as laser flash method (LF) and differential scanning calorimetry (DSC) – to measure the transient response of the temperature vs time. Thermal diffusivity can be derived based on the sample thickness and half time measured by using following relation: $${a = 0.1388 * \frac{L^2}{t_{1/2}}} $$
The heat capacity is then extracted by applying the DSC method based on the following equation:
$${c_{p} = dQ / dT} $$
The thermal conductivity k can then be computed:
$${\kappa = a * \rho * c_{p}} $$
The discrepancies between the two methods can be attributed to the test setup including the position of the heating wire, temperature sensors, and the presence of thermal paste.
◈ Heat generation
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Heat generation inside the battery happens through both Joule heating and reversible heat generation at solid and electrolyte during the charge transport. Charge and discharge rate of the cell influences heat generated by the cell – higher rate means more heat dissipated. The equation is written as:
$$ {Q = I(V_{OC} - V) - I(T_{cell} \frac{dV_{OC}}{dT}) = Q_{joule} + Q_{entropy}}$$ ,where Q is the heat generated, I is the nominal current, VOC is the open circuit voltage, V is the nominal voltage, T is the cell temperature.
Qjoule defines electrical energy loss through irreversible joule heating and is is I(VOC – V) = I2Rint where Rint is internal resistance of the cell. The second term is reversible and represents the heat generated through entropy changes with electrochemical reactions. Heat generated becomes high with the discharge rate.
$$ {Q_{entropy} = -I(T_{cell} \frac{dV_{OC}}{dT}) = -IT_{cell} \frac{\Delta{S}}{nF}}$$ , where change in S = entropy change of the battery, n = number of electrons transferred during discharge, F = Faraday constant.