4.2.1. Solution T1(x, t) for Air Temperature

The following differential equation describes the air temperature variation in the accumulator:

$$
\pi\_{\mathcal{S}} \frac{\partial T\_1}{\partial t} + \frac{1}{N\_{\mathcal{S}}} \frac{\partial T\_1}{\partial \mathbf{x}^+} = -(T\_1 - T\_{\mathbf{w}}) \tag{31}
$$

where *T*<sup>1</sup> is the excess temperature over the ambient temperature *T*0. The following differential equation describes the variation of the packing temperature over the time:

$$
\tau\_w \frac{\partial T\_{w1}}{\partial t} = T\_1 - T\_{w1} \tag{32}
$$

The time changes of the air inlet temperature are described by the boundary condition (Figure 7b):

*T*1|*x*=<sup>0</sup> = Δ*Tg*, 0 < *t* (33)

The initial conditions have the following form:

$$T\_1 = 0\tag{34}$$

$$T\_{w1} = 0\tag{35}$$

The solution of the initial-boundary problem (31)–(33) is defined as:

$$\frac{T\_1 - T\_{\mathbb{S}, 0}}{\Delta T\_{\mathbb{S}}} = \mathcal{U}(\mathbb{S}, \eta) \exp[- (\xi + \eta)], \ t \ge t\_{pr} \tag{36}$$

$$\frac{T\_{w1} - T\_{\xi,0}}{\Delta T\_{\xi}} = \left[ L(\tilde{\xi}, \eta) - I\_o \left( 2\sqrt{\xi \eta} \right) \right] \exp\left[ - (\xi + \eta) \right], \ t \ge t\_{pr} \tag{37}$$

4.2.2. Solution T2(x, t) for Air Temperature

The differential equations describing the changes in air temperature and packing have the following forms:

$$
\pi\_{\mathcal{S}} \frac{\partial T\_2}{\partial t} + \frac{1}{N\_{\mathcal{S}}} \frac{\partial T\_2}{\partial \mathbf{x}^+} = -(T\_2 - T\_w) \tag{38}
$$

$$
\tau\_w \frac{\partial T\_{w2}}{\partial t} = T\_2 - T\_{w2} \tag{39}
$$

The air temperature *T*<sup>2</sup> at the inlet to the accumulator increases linearly with time and the initial air and packing temperatures are equal to zero. The boundary condition and initial conditions have the form:

$$\left.T\_{2}\right|\_{x=0} = f\_{\mathbf{l}}(t),\ 0 \le t \le t\_{\mathrm{cn}}\tag{40}$$

$$T\_2|\_{t=0} = 0\tag{41}$$

$$T\_{\mathbf{w2}}|\_{\mathbf{x}=\mathbf{0}} = \mathbf{0} \tag{42}$$

where *f* I(*t*) = *vT t*. The symbol *vT* denotes the rate of change of the air temperature at the inlet to the accumulator (Figure 7c).

The solution of the initial problem (38)–(42) has the following form:

$$T\_2 = v\_T \text{tr}\_w \left\{ e^{-(\frac{\tau}{\xi} + \eta)} \left[ (\eta - \xi) \mathcal{U} + \xi I\_0 \left( 2\sqrt{\xi \eta} \right) + \sqrt{\xi \eta} I\_1 \left( 2\sqrt{\xi \eta} \right) \right] \right\}, \ 0 \le t \le t\_{cn}, \tag{43}$$

$$T\_{w2} = T\_2 - \upsilon\_T \tau\_w e^{-(\xi + \eta)} \left[ \mathcal{U} - I\_0 \left( 2\sqrt{\xi \eta} \right) \right], \ 0 \le t \le t\_{cn\nu} \tag{44}$$

where the function *U* is defined as follows:

$$\mathcal{U}I = \mathcal{U}(\tilde{\xi}, \eta) = e^{\tilde{\xi} + \eta} - \sum\_{n=1}^{\infty} \left(\frac{\tilde{\xi}}{\eta}\right)^{\frac{\eta}{2}} I\_n\left(2\sqrt{\tilde{\xi}\eta}\right) \tag{45}$$

where *I*0(*x*), *I*1(*x*), and *In*(*x*) denote modified Bessel functions of the zero, first, and *n*-th order, respectively.
