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TRT: thermo racing tyre a physical model to predict the tyre temperature distribution

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Abstract

In the paper a new physical tyre thermal model is presented. The model, called Thermo Racing Tyre (TRT) was developed in collaboration between the Department of Industrial Engineering of the University of Naples Federico II and a top ranking motorsport team.

The model is three-dimensional and takes into account all the heat flows and the generative terms occurring in a tyre. The cooling to the track and to external air and the heat flows inside the system are modelled. Regarding the generative terms, in addition to the friction energy developed in the contact patch, the strain energy loss is evaluated. The model inputs come out from telemetry data, while its thermodynamic parameters come either from literature or from dedicated experimental tests.

The model gives in output the temperature circumferential distribution in the different tyre layers (surface, bulk, inner liner), as well as all the heat flows. These information have been used also in interaction models in order to estimate local grip value.

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Notes

  1. In Figs. 3 and 4 camber values A, B, C, vertical load values FzA, FzB, FzC and inflating pressure values A, B, C are inside typical working ranges of the considered tyres. Their relative order is specified in figure captions and they are not explicited for confidentiality reasons.

Abbreviations

T :

temperature [K]

T air :

air temperature [K]

T ∞ :

air temperature at an infinite distance [K]

T r :

road surface temperature [K]

t :

time [s]

\(\alpha=\frac{k}{\rho\cdot c_{v}}\) :

thermal diffusivity; α t tyre, α r road \([\frac{\mathrm{m}^{2}}{\mathrm{s}}]\)

\(\dot{q_{G}}\) :

heat generated per unit of volume and time \([\frac{\mathrm{J}}{\mathrm{s}{\cdot}\mathrm{m}^{3}}]\)

ρ :

density \([\frac{\mathrm{kg}}{\mathrm{m}^{3}}]\)

c v :

specific heat at constant volume \([\frac{\mathrm{J}}{\mathrm{kg}{\cdot}\mathrm{K}}]\)

c p :

specific heat at constant pressure \([\frac{\mathrm{J}}{\mathrm{kg}{\cdot}\mathrm{K}}]\)

k t , k r :

tyre and road thermal conductivity \([\frac{\mathrm{W}}{\mathrm{m}{\cdot}\mathrm{K}}]\)

H c :

heat transfer coefficient \([\frac{\mathrm{W}}{\mathrm{m}^{2}{\cdot}\mathrm{K}}]\)

h :

external air natural convection coefficient \([\frac{\mathrm{W}}{\mathrm{m}^{2}{\cdot}\mathrm{K}}]\)

h forc :

external air forced convection coefficient \([\frac{\mathrm{W}}{\mathrm{m}^{2}{\cdot}\mathrm{K}}]\)

h int :

internal air natural convection coefficient \([\frac{\mathrm{W}}{\mathrm{m}^{2}{\cdot}\mathrm{K}}]\)

x, y, z :

coordinates

F x , F y :

longitudinal and lateral tyre-road interaction forces [N]

F z :

normal load acting on the single wheel [N]

v x , v y :

longitudinal and lateral slip velocity \([\frac{\mathrm{m}}{\mathrm{s}}]\)

A :

tyre-road contact area [m2]

A tot :

total area of external surface [m2]

k air :

air thermal conductivity \([\frac{\mathrm{W}}{\mathrm{m}{\cdot}\mathrm{K}}]\)

V :

air velocity \([\frac{\mathrm{m}}{\mathrm{s}}]\)

ν :

air kinematic viscosity \([\frac{\mathrm{m}^{2}}{\mathrm{s}}]\)

μ :

air dynamic viscosity [\(\frac{\mathrm{kg}}{\mathrm{s}{\cdot}\mathrm{m}}\)]

\(L= \frac{1}{\frac{1}{D_{e}} + \frac{1}{W}}\) :

characteristic length of the heat exchange surface [m]

W :

tread width [m]

La :

contact patch length [m]

De :

tyre external diameter [m]

pi :

tyre inflating pressure [bar]

g :

gravity acceleration \([\frac{\mathrm{m}}{\mathrm{s}^{2}}]\)

β :

coefficient of thermal air expansion [1/T]

\(\mathit{Gr}=\frac{\mathrm{g}\cdot\beta\cdot L^{3}\cdot(T-T_{\infty})}{\nu^{2}}\) :

Grashof number [–]

\(\mathit{Pr}= \frac{\mu\cdot c_{p}}{K_{air}}\) :

Prandtl number [–]

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Author information

Authors and Affiliations

  1. Dipartimento di Ingegneria Industriale, Università degli Studi di Napoli Federico II, Via Claudio 21, 80125, Naples, Italy

    Flavio Farroni, Daniele Giordano, Michele Russo & Francesco Timpone

Authors
  1. Flavio Farroni
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  2. Daniele Giordano
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  3. Michele Russo
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  4. Francesco Timpone
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Corresponding author

Correspondence to Flavio Farroni.

Appendix

Appendix

As an example, heat balance equation for node 2 along the x direction is reported, recalling that, for the performed discretization, the nodes adjacent to 2 are 6 and 58:

$$\begin{aligned} &{\frac{k_{1}}{\Delta X} \cdot ( T_{6} - T_{2} ) \cdot \Delta Y \cdot \frac{\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta X} \cdot ( T_{2} - T_{58} ) \cdot \Delta Y} \\ &{\quad{}\cdot \frac{\Delta Z_{1}}{2} = m_{2} \cdot c_{v 1} \cdot \frac{\Delta T_{2}}{ \Delta t}} \end{aligned}$$
(25)

Substituting the expression of the mass (2) (reminding that in this case C=1/2) leads to the equation:

$$ \frac{\Delta T_{2}}{\Delta t} = \frac{1}{\rho\cdot c_{v 1}} \cdot \biggl[ \frac{k_{1}}{\Delta X^{2}} \cdot T_{6} - \frac{2 \cdot k_{1}}{\Delta X^{2}} \cdot T_{2} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{58} \biggr] $$
(26)

Taking into account the exchanges along all directions and all the possible heat generations, the equation of node 2 can be written:

• in the case of contact with the road

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{1}}{\Delta X} \cdot ( T_{6} - T_{2} ) \cdot \Delta Y\cdot \frac {\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta X} \\ &\quad{} \cdot ( T_{2} - T_{58} ) \cdot \Delta Y\cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{\Delta Y} \cdot ( T_{1} - T_{2} ) \cdot \Delta X \\ &\quad{}\cdot \frac{\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta Y} \cdot ( T_{2} - T_{3} ) \cdot \Delta X\cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{\Delta Z_{1}} \\ &\quad{} \cdot ( T_{62} - T_{2} ) \cdot \Delta X\cdot \Delta Y + CR\cdot \frac{F_{x} \cdot v_{x} + F_{y} \cdot v_{y}}{A} \\ &\quad{} \cdot \Delta X\cdot \Delta Y + H_{c} \cdot ( T_{r} - T_{2} ) \\ &\quad{} \cdot \Delta X\cdot \Delta Y = m_{2} \cdot c_{v 1} \cdot \frac{\Delta T_{2}}{\Delta t} \end{aligned}$$
(A)

• in the case of contact with external air

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{1}}{\Delta X} \cdot ( T_{6} - T_{2} ) \cdot \Delta Y\cdot \frac {\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta X} \\ &\quad{} \cdot ( T_{2} - T_{58} ) \cdot \Delta Y\cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{\Delta Y} \cdot ( T_{1} - T_{2} ) \cdot \Delta X \\ &\quad{}\cdot \frac{\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta Y} \cdot ( T_{2} - T_{3} )\cdot \Delta X\cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{\Delta Z_{1}} \\ &\quad{}\cdot ( T_{62} - T_{2} ) \cdot \Delta X\cdot \Delta Y + h_{forc} \cdot ( T_{air} - T_{2} ) \\ &\quad{} \cdot \Delta X\cdot \Delta Y = m_{2} \cdot c_{v 1} \cdot \frac {\Delta T_{2}}{\Delta t} \end{aligned}$$
(B)

having denoted by \(\dot{Q}_{SEL}\) the power dissipated by cyclic deformation.

Once developed, the two expressions lead respectively to:

• in the case of contact with the road

$$\begin{aligned} \frac{\Delta T_{2}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 1}} \cdot \biggl[ \frac{2 \cdot \dot{Q}_{SEL}}{ \Delta X \cdot \Delta Y \cdot \Delta Z_{1}} \\ &{} + \biggl( - \frac{2 \cdot k_{1}}{\Delta X^{2}} - \frac{2 \cdot k_{1}}{\Delta Y^{2}} - \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} - \frac{2 \cdot H_{c}}{\Delta Z_{1}} \biggr) \\ &{}\cdot T_{2} + \frac{k_{1}}{\Delta Y^{2}} \cdot T_{1} + \frac{k_{1}}{\Delta Y^{2}} \cdot T_{3} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{6} \\ &{}+ \frac{k_{1}}{ \Delta X^{2}} \cdot T_{58} + \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} \cdot T_{62} \\ &{} + \frac{2 \cdot FP}{\Delta Z_{1}} + \frac{2 \cdot H_{c}}{ \Delta Z_{1}} \cdot T_{r} \biggr] \end{aligned}$$
(27)

• in the case of contact with external air

$$\begin{aligned} \frac{\Delta T_{2}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 1}} \cdot \biggl[ \frac{2 \cdot \dot{Q}_{SEL}}{ \Delta X\cdot \Delta Y\cdot \Delta Z_{1}} \\ &{}+\biggl( - \frac{2 \cdot k_{1}}{\Delta X^{2}} - \frac{2 \cdot k_{1}}{\Delta Y^{2}} - \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} - \frac{2 \cdot h_{forc}}{\Delta Z_{1}} \biggr) \\ &{}\cdot T_{2} + \frac{k_{1}}{\Delta Y^{2}} \cdot T_{1} + \frac{k_{1}}{\Delta Y^{2}} \cdot T_{3} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{6} \\ &{} + \frac{k_{1}}{ \Delta X^{2}} \cdot T_{58} + \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} \cdot T_{62} \\ &{} + \frac{2 \cdot h_{forc}}{\Delta Z_{1}} \cdot T_{air} \biggr] \end{aligned}$$
(28)

The equations showed for node 2 are valid for all the nodes belonging to the surface layer, localized internally in lateral direction.

For a node still belonging to the surface layer, but external in lateral direction (C=1/4), for example node 1, the complete equations are:

• in the case of contact with the road

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{1}}{\Delta X} \cdot ( T_{5} - T_{1} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta X} \\ &\quad{} \cdot ( T_{1} - T_{57} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{ \Delta Y} \cdot ( T_{2} - T_{1} ) \cdot\Delta X \\ &\quad{}\cdot \frac{\Delta Z_{1}}{2} + \frac {k_{1}}{\Delta Z_{1}} \cdot ( T_{61} - T_{1} ) \cdot \Delta X\cdot \frac{\Delta Y}{2} + CR \\ &\quad{}\cdot \frac{F_{x} \cdot v_{x} + F_{y} \cdot v_{y}}{A} \cdot \Delta X \cdot \frac{\Delta Y}{2} + H_{c} \cdot ( T_{r} - T_{1} ) \\ &\quad{} \cdot \Delta X\cdot \frac{\Delta Y}{2} = m_{1} \cdot c_{v 1} \cdot \frac {\Delta T_{1}}{\Delta t} \end{aligned}$$
(C)

• in the case of contact with external air

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{1}}{\Delta X} \cdot ( T_{5} - T_{1} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{1}}{2} - \frac{k_{1}}{\Delta X} \\ &\quad{} \cdot ( T_{1} - T_{57} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{ \Delta Y} \cdot ( T_{2} - T_{1} ) \cdot\Delta X \\ &\quad{} \cdot \frac{\Delta Z_{1}}{2} + \frac{k_{1}}{\Delta Z_{1}} \cdot ( T_{61} - T_{1} ) \cdot \Delta X \cdot \frac{\Delta Y}{2} + h_{forc} \\ &\quad{} \cdot ( T_{air} - T_{1} ) \cdot\Delta X \cdot \frac{\Delta Y}{2} = m_{1} \cdot c_{v 1} \cdot \frac{\Delta T_{1}}{ \Delta t} \end{aligned}$$
(D)

leading, respectively, to:

• for the first case

$$\begin{aligned} \frac{\Delta T_{1}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 1}} \cdot \biggl[ \frac{4 \cdot \dot{Q}_{SEL}}{ \Delta X \cdot \Delta Y \cdot \Delta Z_{1}} \\ &{} + \biggl( - \frac{2 \cdot k_{1}}{\Delta X^{2}} - \frac{2 \cdot k_{1}}{\Delta Y^{2}} - \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} - \frac{2 \cdot H_{c}}{\Delta Z_{1}} \biggr) \cdot T_{1} \\ &{}+ \frac{2 \cdot k_{1}}{ \Delta Y^{2}} \cdot T_{2} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{5} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{57} \\ &{} + \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} \cdot T_{61} + \frac{2 \cdot FP}{\Delta Z_{1}} + \frac{2 \cdot H_{c}}{\Delta Z_{1}} \cdot T_{r} \biggr] \end{aligned}$$
(29)

• for the second case

$$\begin{aligned} \frac{\Delta T_{1}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 1}} \cdot \biggl[ \frac{4 \cdot \dot{Q}_{SEL}}{ \Delta X\cdot \Delta Y\cdot \Delta Z_{1}} \\ &{}+ \biggl( - \frac{2 \cdot k_{1}}{\Delta X^{2}} - \frac{2 \cdot k_{1}}{\Delta Y^{2}} - \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} - \frac{2 \cdot h_{forz}}{\Delta Z_{1}} \biggr) \\ &{} \cdot T_{1} + \frac{2 \cdot k_{1}}{ \Delta Y^{2}} \cdot T_{2} + \frac{k_{1}}{\Delta X^{2}} \cdot T_{5} + \frac{k_{1}}{\Delta X^{2}} \\ &{}\cdot T_{57} + \frac{2 \cdot k_{1}}{\Delta Z_{1}^{2}} \cdot T_{61} + \frac{2 \cdot h_{forc}}{\Delta Z_{1}} \cdot T_{air} \biggr] \end{aligned}$$
(30)

The equation relating to the bulk layer, for an internal node in the lateral direction (C=1), e.g. node 62, is:

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{2}}{\Delta X} \cdot ( T_{66} - T_{62} ) \cdot \Delta Y \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} \biggr) \\ &\quad{} - \frac {k_{2}}{\Delta X} \cdot ( T_{62} - T_{118} ) \cdot \Delta Y \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} \biggr) \\ &\quad{} + \frac{k_{2}}{\Delta Y} \cdot ( T_{61} - T_{62} ) \cdot \Delta X \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac {\Delta Z_{2}}{2} \biggr) \\ &\quad{} - \frac{k_{2}}{\Delta Y} \cdot ( T_{62} - T_{63} ) \cdot \Delta X \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} \biggr) \\ &\quad{} + \frac{k_{2}}{\Delta Z_{2}} \cdot ( T_{122} - T_{62} ) \cdot \Delta X \cdot \Delta Y - \frac{k_{1}}{\Delta Z_{1}} \\ &\quad{}\cdot ( T_{62} - T_{2} ) \cdot \Delta X \cdot \Delta Y = m_{62} \cdot c_{v 2} \cdot \frac{\Delta T_{62}}{ \Delta t} \end{aligned}$$
(E)

Such expression, suitably developed, leads to:

$$\begin{aligned} \frac{\Delta T_{62}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 2}} \cdot \biggl[ \frac{\dot{Q}_{SEL}}{\Delta X\cdot \Delta Y\cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \\ &{} + \biggl( - \frac{2 \cdot k_{2}}{\Delta X^{2}} - \frac{2 \cdot k_{2}}{\Delta Y^{2}} - \frac{k_{2}}{\Delta Z_{2} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \\ &{} - \frac{k_{1}}{\Delta Z_{1} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \biggr) \cdot T_{62} + \frac{k_{2}}{\Delta Y^{2}} \cdot T_{61} \\ &{}+ \frac{k_{2}}{ \Delta Y^{2}} \cdot T_{63} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{66} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{118} \\ &{}+ \frac{k_{2}}{\Delta Z_{2} \cdot ( \frac {\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \cdot T_{122} \\ &{}+\frac{k_{1}}{\Delta Z_{1} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \cdot T_{2} \biggr] \end{aligned}$$
(31)

Similarly, relatively to a bulk external node in the transverse direction (C=1/2), it results:

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{2}}{\Delta X} \cdot ( T_{65} - T_{61} ) \cdot \frac{\Delta Y}{2} \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{ 2} \biggr) \\ &\quad{} - \frac{k_{2}}{\Delta X} \cdot ( T_{61} - T_{117} ) \cdot \frac{\Delta Y}{2} \cdot \biggl( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} \biggr) \\ &\quad{} + \frac{k_{2}}{\Delta Y} \cdot ( T_{62} - T_{61} ) \cdot \Delta X\cdot \biggl( \frac {\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} \biggr) \\ &\quad{} + \frac{k_{2}}{ \Delta Z_{2}} \cdot ( T_{121} - T_{61} ) \cdot \Delta X\cdot \frac{\Delta Y}{2} - \frac {k_{1}}{\Delta Z_{1}} \\ &\quad{} \cdot ( T_{61} - T_{1} ) \cdot \Delta X\cdot \frac{\Delta Y}{2} = m_{61} \cdot c_{v 2} \cdot \frac{\Delta T_{61}}{\Delta t} \end{aligned}$$
(F)

that becomes:

$$\begin{aligned} \frac{\Delta T_{61}}{\Delta t} =& \frac{1}{\rho\cdot c_{v 2}} \cdot \biggl[ \frac{2 \cdot \dot{Q}_{SEL}}{ \Delta X\cdot \Delta Y\cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \\ &{} + \biggl( - \frac{2 \cdot k_{2}}{\Delta X^{2}} - \frac{2 \cdot k_{2}}{\Delta Y^{2}} - \frac{k_{2}}{\Delta Z_{2} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \\ &{}- \frac{k_{1}}{\Delta Z_{1} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \biggr) \cdot T_{61} + \frac{2 \cdot k_{2}}{\Delta Y^{2}} \cdot T_{62} \\ &{} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{65} + \frac{k_{2}}{ \Delta X^{2}} \cdot T_{117} \\ &{}+ \frac{k_{2}}{\Delta Z_{2} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \cdot T_{121} \\ &{}+ \frac{k_{1}}{\Delta Z_{1} \cdot ( \frac{\Delta Z_{1}}{2} + \frac{\Delta Z_{2}}{2} )} \cdot T_{1} \biggr] \end{aligned}$$
(32)

As concerns the innermost layer, the inner liner, the equation of exchange for an internal node in the transverse direction (C=1/2), e.g. 122, is:

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{2}}{\Delta X} \cdot ( T_{126} - T_{122} ) \cdot \Delta Y \cdot \frac{\Delta Z_{2}}{2} - \frac{k_{2}}{\Delta X} \\ &\quad{}\cdot ( T_{122} - T_{158} ) \cdot \Delta Y \cdot \frac{\Delta Z_{2}}{2} + \frac{k_{2}}{\Delta Y} \\ &\quad{} \cdot ( T_{121} - T_{122} ) \cdot \Delta X \cdot \frac{\Delta Z_{2}}{2} - \frac{k_{2}}{\Delta Y} \\ &\quad{} \cdot ( T_{122} - T_{123} ) \cdot \Delta X \cdot \frac{\Delta Z_{2}}{2} + \frac{k_{2}}{ \Delta Z_{2}} \\ &\quad{} \cdot ( T_{62} - T_{122} ) \cdot \Delta X \cdot \Delta Y + h_{int} \cdot ( T_{air \_ int} - T_{122} ) \\ &\quad{}\cdot\Delta X \cdot \Delta Y = m_{122} \cdot c_{v 2} \cdot \frac{\Delta T_{122}}{\Delta t} \end{aligned}$$
(G)

that simplified returns:

$$\begin{aligned} \frac{\Delta T_{122}}{\Delta t} =& \frac{1}{\rho\cdot c_{v2}} \cdot \biggl[ \frac{2 \cdot \dot{Q}_{SEL}}{ \Delta X \cdot \Delta Y \cdot \Delta Z_{2}} \\ &{}+ \biggl( - \frac{2 \cdot k_{2}}{\Delta X^{2}} - \frac{2 \cdot k_{2}}{\Delta Y^{2}} - \frac{2 \cdot k_{2}}{\Delta Z_{2}^{2}} - \frac{2 \cdot h_{int}}{\Delta Z_{2}} \biggr) \\ &{}\cdot T_{122} + \frac{k_{2}}{\Delta Y^{2}} \cdot T_{121} + \frac{k_{2}}{\Delta Y^{2}} \cdot T_{123} \\ &{}+ \frac{k_{2}}{\Delta X^{2}} \cdot T_{126} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{158} + \frac{2 \cdot k_{2}}{\Delta Z_{2}^{2}} \\ &{}\cdot T_{62} + \frac{2 \cdot h_{int}}{\Delta Z_{2}} \cdot T_{air\_int} \biggr] \end{aligned}$$
(33)

Finally, for an external node in the transverse direction belonging to the Inner liner (C=1/4), it is:

$$\begin{aligned} &\dot{Q}_{SEL} + \frac{k_{2}}{\Delta X} \cdot ( T_{125} - T_{121} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{2}}{2} - \frac{k_{2}}{\Delta X} \\ &\qquad{} \cdot ( T_{121} - T_{157} ) \cdot \frac{\Delta Y}{2} \cdot \frac{\Delta Z_{2}}{2} + \frac {k_{2}}{\Delta Y} \\ &\qquad{} \cdot ( T_{122} - T_{121} ) \cdot \Delta X \cdot \frac{\Delta Z_{2}}{2} + \frac{k_{2}}{\Delta Z_{2}} \\ &\qquad{}\cdot ( T_{61} - T_{121} ) \cdot \Delta X \cdot \frac{\Delta Y}{ 2} \\ &\qquad{} + h_{int}\cdot ( T_{air\_int} - T_{121} ) \cdot \Delta X \cdot \frac{\Delta Y}{2} \\ &\quad{} = m_{121} \cdot c_{v2} \cdot \frac{\Delta T_{121}}{\Delta t} \end{aligned}$$
(H)

which, simplified, provides:

$$\begin{aligned} \frac{\Delta T_{121}}{\Delta t} =& \frac{1}{\rho\cdot c_{v2}} \cdot \biggl[ \frac{4 \cdot \dot{Q}_{SEL}}{ \Delta X \cdot \Delta Y \cdot \Delta Z_{2}} \\ &{}+ \biggl( - \frac{2 \cdot k_{2}}{\Delta X^{2}} - \frac{2 \cdot k_{2}}{\Delta Y^{2}} - \frac{2 \cdot k_{2}}{\Delta Z_{2}^{2}} - \frac{2 \cdot h_{int}}{\Delta Z_{2}} \biggr) \\ &{} \cdot T_{121} + \frac{2 \cdot k_{2}}{\Delta Y^{2}} \cdot T_{122} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{125} \\ &{} + \frac{k_{2}}{\Delta X^{2}} \cdot T_{157} + \frac{2 \cdot k_{2}}{\Delta Z_{2}^{2}} \cdot T_{61} \\ &{}+ \frac{2 \cdot h_{int}}{ \Delta Z_{2}} \cdot T_{air\_int} \biggr] \end{aligned}$$
(34)

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Farroni, F., Giordano, D., Russo, M. et al. TRT: thermo racing tyre a physical model to predict the tyre temperature distribution. Meccanica 49, 707–723 (2014). https://doi.org/10.1007/s11012-013-9821-9

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