An Analytical Solution for Mechanical Responses Induced by Temperature and Air Pressure in a Lined Rock Cavern for Underground Compressed Air Energy Storage

Abstract

Mechanical responses induced by temperature and air pressure significantly affect the stability and durability of underground compressed air energy storage (CAES) in a lined rock cavern. An analytical solution for evaluating such responses is, thus, proposed in this paper. The lined cavern of interest consists of three layers, namely, a sealing layer, a concrete lining and the host rock. Governing equations for cavern temperature and air pressure, which involve heat transfer between the air and surrounding layers, are established first. Then, Laplace transform and superposition principle are applied to obtain the temperature around the lined cavern and the air pressure during the operational period. Afterwards, a thermo-elastic axisymmetrical model is used to analytically determine the stress and displacement variations induced by temperature and air pressure. The developments of temperature, displacement and stress during a typical operational cycle are discussed on the basis of the proposed approach. The approach is subsequently verified with a coupled compressed air and thermo-mechanical numerical simulation and by a previous study on temperature. Finally, the influence of temperature on total stress and displacement and the impact of the heat transfer coefficient are discussed. This paper shows that the temperature sharply fluctuates only on the sealing layer and the concrete lining. The resulting tensile hoop stresses on the sealing layer and concrete lining are considerably large in comparison with the initial air pressure. Moreover, temperature has a non-negligible effect on the lined cavern for underground compressed air storage. Meanwhile, temperature has a greater effect on hoop and longitudinal stress than on radial stress and displacement. In addition, the heat transfer coefficient affects the cavern stress to a higher degree than the displacement.

Appendix: Detailed Derivations for the Temperature Field in a Lined Cavern for Underground CAES

1.1 Charging Stage in the First Cycle

For the charging stage in the first cycle, M ^* = T ^* − 1 and M ^* _j  = T ^* _j  − 1 (j = 1, 2, 3) are defined. Thus, following the premise that $\rho_{av}^{*}$ is used instead of ρ*, Eqs. (11) to (17), that is, the governing equations for the temperature solution, are converted to the following expressions:

$$ \begin{gathered} \frac{{t_{1}^{*} \rho_{av}^{*} }}{{m_{r} }}\frac{{dM^{*} }}{{dt^{*} }} = \gamma T_{i}^{*} + (R^{*} - \gamma )(M^{*} + 1) \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \rho_{av}^{*} U_{{\rho^{*} }}^{*} + q_{r} \left( {M_{1}^{*} (r_{0} ,t) - M^{*} } \right) \hfill \\ \end{gathered} , $$

(55)

$$ \frac{{\partial M_{j}^{*} }}{{\partial t^{*} }} = \frac{{F_{j} }}{{r^{*} }}\frac{\partial }{{\partial r^{*} }}\left( {r^{*} \frac{{\partial M_{j}^{*} }}{{\partial r^{*} }}} \right) \quad ( {j = \, 1, \, 2, \, 3}), $$

(56)

$$ \frac{{\partial M_{1}^{*} }}{{\partial r^{*} }} = B_{i1} (M_{1}^{*} - M^{*} ),\;r^{*} = 1, $$

(57)

$$ M_{j}^{*} = M_{j + 1}^{*} ,\frac{{\partial M_{j}^{*} }}{{\partial r^{*} }} = \lambda_{j + 1} \frac{{\partial M_{j + 1}^{*} }}{{\partial r^{*} }},r^{*} = r_{j}^{*}\;\left( {j = \, 1, \, 2} \right), $$

(58)

$$ M_{3}^{*} = 0,\;r^{*} = r_{3}^{*} , $$

(59)

$$ M^{*} = M_{j}^{*} = 0,\;t^{*} = 0\quad( {j = \, 1, \, 2, \, 3}). $$

(60)

Laplace transform is applied to Eq. (56) [note that $\tilde{f}(s)$ denotes the Laplace transform of the function f(t ^*)] and we obtain the following equation:

$$ s\tilde{M}_{j}^{*} = \frac{{F_{j} }}{{r^{*} }}\frac{\partial }{{\partial r^{*} }}\left( {r^{*} \frac{{\partial \tilde{M}_{j}^{*} }}{{\partial r^{*} }}} \right)\quad( {j = \, 1, \, 2, \, 3}). $$

(61)

The solution for Eq. (61) is:

$$ \tilde{M}_{j}^{*} = A_{j} I_{0} \left( {\sqrt {s/F_{j} } r^{*} } \right) + B_{j} K_{0} \left( {\sqrt {s/F_{j} } r^{*} } \right)\quad( {j = \, 1, \, 2, \, 3}), $$

(62)

where $I_{0} \left( \bullet \right)$ stands for the modified Bessel function of the first kind of order zero and $K_{0} \left( \bullet \right)$ represents the modified Bessel function of the second kind of order zero. A _j and B _j are coefficients to be determined by the boundary and continuity conditions, respectively.

Then, referring to Eq. (62), the first derivative of $\tilde{M}_{j}^{*}$ with respect to r ^* is expressed as follows:

$$ \frac{{\partial \tilde{M}_{j}^{*} }}{{\partial r^{*} }} = A_{j} \sqrt {s/F_{j} } I_{1} \left( {\sqrt {s/F_{j} } r^{*} } \right) - B_{j} \sqrt {s/F_{j} } K_{1} \left( {\sqrt {s/F_{j} } r^{*} } \right), $$

(63)

where $I_{1} \left( \bullet \right)$ is the modified Bessel function of the first kind of order one and $K_{1} \left( \bullet \right)$ denotes the modified Bessel function of the second kind of order one.

In reference to the Laplace transform of Eq. (55), we obtain the following equation:

$$ \alpha_{1} \tilde{M}^{*} = \alpha_{2} /s + q_{r} \tilde{M}_{1}^{*} (1,s), $$

(64)

where:

$$ \alpha_{1} = \frac{{t_{1}^{*} \rho_{av}^{*} s}}{{m_{r} }} - (R^{*} - \gamma ) + q_{r} ,\;\alpha_{2} = \gamma T_{i}^{*} + (R^{*} - \gamma ) + \rho^{*} U_{{\rho^{*} }}^{*} . $$

Thus, based on Eq. (64) and applying the Laplace transform of the boundary condition Eq. (57), the following equation is obtained:

$$ \begin{gathered} A_{1} \sqrt {s/F_{j} } I_{1} \left(\sqrt {s/F_{j} } \right) - B_{1} \sqrt {s/F_{j} } K_{1} \left(\sqrt {s/F_{j} } \right) \hfill \\ = \frac{{B_{i1} (\alpha_{1} - q_{r} )}}{{\alpha_{1} }}\left[ {A_{1} I_{0} \left(\sqrt {s/F_{j} } \right) + B_{1} K_{0} \left(\sqrt {s/F_{j} } \right)} \right] - \frac{{B_{i1} \alpha_{2} }}{{s\alpha_{1} }} \hfill \\ \end{gathered} . $$

(65)

The Laplace transform of the second boundary condition Eq. (59) leads to the following expression:

$$ A_{3} I_{0} \left( {\sqrt {s/F_{3} } r_{3}^{*} } \right) + B_{3} K_{0} \left( {\sqrt {s/F_{3} } r_{3}^{*} } \right) = 0. $$

(66)

The resulting Laplace transforms of the continuity conditions (58) are expressed as follows:

$$ \begin{gathered} A_{j} I_{0} \left( {\sqrt {s/F_{j} } r_{j}^{*} } \right) + B_{j} K_{0} \left( {\sqrt {s/F_{j} } r_{j}^{*} } \right) \hfill \\ = A_{j + 1} I_{0} \left( {\sqrt {s/F_{j + 1} } r_{j + 1}^{*} } \right) + B_{j + 1} K_{0} \left( {\sqrt {s/F_{j + 1} } r_{j + 1}^{*} } \right) \hfill \\ \end{gathered} , $$

(67)

$$ \begin{gathered} A_{j} \sqrt {s/F_{j} } I_{1} \left( {\sqrt {s/F_{j} } r_{j}^{*} } \right) - B_{j} \sqrt {s/F_{j} } K_{1} \left( {\sqrt {s/F_{j} } r_{j}^{*} } \right) \hfill \\ = \lambda_{j + 1} \left[ {A_{j + 1} \sqrt {s/F_{j + 1} } I_{1} \left( {\sqrt {s/F_{j + 1} } r_{j + 1}^{*} } \right) - B_{j + 1} \sqrt {s/F_{j + 1} } K_{1} \left( {\sqrt {s/F_{j + 1} } r_{j + 1}^{*} } \right)} \right] \;\left( {j = \, 1, \, 2} \right) \hfill \\ \end{gathered} . $$

(68)

Let $\alpha_{0} = - \frac{{B_{i1} \alpha_{2} }}{{s\alpha_{1} }}\sqrt {F_{1} /s} ,\;h_{0} = \frac{{B_{i1} (\alpha_{1} - q_{r} )}}{{\alpha_{1} }}\sqrt {F_{1} /s} ,\;h_{1} = \lambda_{2} \sqrt {F_{1} /F_{2} } ,\;h_{2} = \lambda_{3} \sqrt {F_{2} /F_{3} } ,\;\eta_{j} = \sqrt {s/F_{j} } r_{j - 1}^{*}$ and $\varepsilon_{j} = \sqrt {s/F_{j} } r_{j}^{*} ,\left( {j = \, 1, \, 2, \, 3} \right),$ then Eqs. (65) to (68) can be expressed as the following matrix form:

$$ \left\{ {\left[ {\begin{array}{*{20}c} {I_{1} (\eta_{1} ) - h_{0} I_{0} (\eta_{1} )} & { - K_{1} (\eta_{1} ) - h_{0} K_{0} (\eta_{1} )} & 0 & 0 & 0 & 0 \\ {I_{0} (\varepsilon_{1} )} & {K_{0} (\varepsilon_{1} )} & { - I_{0} (\eta_{2} )} & { - K_{0} (\eta_{2} )} & 0 & 0 \\ {I_{1} (\varepsilon_{1} )} & { - K_{1} (\varepsilon_{1} )} & { - h_{1} I_{1} (\eta_{2} )} & {h_{1} K_{1} (\eta_{2} )} & 0 & 0 \\ 0 & 0 & {I_{0} (\varepsilon_{2} )} & {K_{0} (\varepsilon_{2} )} & { - I_{0} (\eta_{3} )} & { - K_{0} (\eta_{3} )} \\ 0 & 0 & {I_{1} (\varepsilon_{2} )} & { - K_{1} (\varepsilon_{2} )} & { - h_{2} I_{1} (\eta_{3} )} & {h_{2} K_{1} (\eta_{3} )} \\ 0 & 0 & 0 & 0 & {I_{0} (\varepsilon_{3} )} & {K_{0} (\varepsilon_{3} )} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {A_{1} } \\ {B_{1} } \\ {A_{2} } \\ {B_{2} } \\ {A_{3} } \\ {B_{3} } \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\alpha_{0} } \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right\}} \right\} . $$

(69)

Coefficients A ₁ to B ₃ seem to be capable of being directly determined from Eq. (69). However, in fact, the difference between the coefficients in the first matrix of Eq. (69) are large in magnitude, as a result of the different properties of the modified Bessel functions of the first second kinds. The singularity in the coefficient matrix would bring a great error in the solution for A ₁ to B ₃ if solving for Eq. (69) directly.

To overcome the drawback mentioned previously, some new functions (I ₀, K ₀, I ₁ and K ₁) are defined as follows:

$$ \left\{ {\begin{array}{*{20}c} {{\mathbf{I}}_{0} (x) = I_{0} (x)/\exp (x)} \\ {{\mathbf{K}}_{0} (x) = K_{0} (x) \bullet \exp (x)} \\ {{\mathbf{I}}_{1} (x) = I_{1} (x)/\exp (x)} \\ {{\mathbf{K}}_{1} (x) = K_{1} (x) \bullet \exp (x)} \\ \end{array} } \right., $$

(70)

where x is an argument.

Thus, Eq. (69) is changed to the following form:

$$ \left[ {\begin{array}{*{20}c} {{\mathbf{a}}_{1} } \\ {{\mathbf{b}}_{1} } \\ {{\mathbf{a}}_{2} } \\ {{\mathbf{b}}_{2} } \\ {{\mathbf{a}}_{3} } \\ {{\mathbf{b}}_{3} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {A_{1} \exp (\eta_{1} )} \\ {B_{2} /\exp (\eta_{1} )} \\ {A_{2} \exp (\eta_{2} )} \\ {B_{2} /\exp (\eta_{2} )} \\ {A_{3} \exp (\eta_{3} )} \\ {B_{3} /\exp (\eta_{3} )} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\alpha_{0} } \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \end{array} } \right\}, $$

(71)

where:

$$ \begin{gathered} {\mathbf{a}}_{1} = \left\{ {{\mathbf{I}}_{1} (\eta_{1} ){ - }h_{0} {\mathbf{I}}_{0} (\eta_{1} ),{ - }{\mathbf{K}}_{1} (\eta_{1} ){ - }h_{0} {\mathbf{K}}_{0} (\eta_{1} ),0,0,0,0} \right\}, \hfill \\ {\mathbf{b}}_{1} = \left\{ {{\mathbf{I}}_{0} (\varepsilon_{1} )\exp (\varepsilon_{1} { - }\eta_{1} ),{\mathbf{K}}_{0} (\varepsilon_{1} )\exp (\eta_{1} { - }\varepsilon_{1} ),{ - }{\mathbf{I}}_{0} (\eta_{2} ),{ - }{\mathbf{K}}_{0} (\eta_{2} ),0,0} \right\}, \hfill \\ {\mathbf{a}}_{2} = \left\{ {{\mathbf{I}}_{1} (\varepsilon_{1} )\exp (\varepsilon_{1} { - }\eta_{1} ),{ - }{\mathbf{K}}_{1} (\varepsilon_{1} )\exp (\eta_{1} { - }\varepsilon_{1} ),{ - }h_{1} {\mathbf{I}}_{1} (\eta_{2} ),h_{1} {\mathbf{K}}_{1} (\eta_{2} ),0,0} \right\}, \hfill \\ {\mathbf{b}}_{2} = \left\{ {0,0,{\mathbf{I}}_{0} (\varepsilon_{2} )\exp (\varepsilon_{2} { - }\eta_{2} ),{\mathbf{K}}_{0} (\varepsilon_{2} )\exp (\eta_{2} { - }\varepsilon_{2} ),{ - }{\mathbf{I}}_{0} (\eta_{3} ),{ - }{\mathbf{K}}_{0} (\eta_{3} )} \right\}, \hfill \\ {\mathbf{a}}_{3} = \left\{ {0,0,{\mathbf{I}}_{1} (\varepsilon_{2} )\exp (\varepsilon_{2}{ - }\eta_{2} ),{ - }{\mathbf{K}}_{1} (\varepsilon_{2} )\exp (\eta_{2} { - }\varepsilon_{2} ),{ - }h_{2} {\mathbf{I}}_{1} (\eta_{3} ),h_{2} {\mathbf{K}}_{1} (\eta_{3} )} \right\}, \hfill \\ {\mathbf{b}}_{3} = \left\{ {0,0,0,0,{\mathbf{I}}_{0} (\varepsilon_{3} )\exp (\varepsilon_{3} - \eta_{3} ),{\mathbf{K}}_{0} (\varepsilon_{3} )\exp (\eta_{3} { - }\varepsilon_{3} )} \right\}. \hfill \\ \end{gathered} $$

(72)

We define the matrix ${\mathbf{S}} = [\begin{array}{*{20}c} {{\mathbf{a}}_{1} } & {{\mathbf{b}}_{1} } & {{\mathbf{a}}_{2} } & {{\mathbf{b}}_{2} } & {{\mathbf{a}}_{3} } & {{\mathbf{b}}_{3} } \\ \end{array} ]^{T} ,$ which has a determinant ΔS = |S|, and assume that:

$$ \begin{gathered} \Delta S_{j1} = \left| {\begin{array}{*{20}c} S & {{\text{row}}\;1\;{\text{is}}\;{\text{deleted}}} \\ {{\text{column}}\;2j - 1\;{\text{is}}\;{\text{deleted}}} & \cdots \\ \end{array} } \right| \\ \Delta S_{j2} = \left| {\begin{array}{*{20}c} S & {{\text{row}}\;1\;{\text{is}}\;{\text{deleted}}} \\ {{\text{column}}\;2j\;{\text{is}}\;{\text{deleted}}} & \cdots \\ \end{array} } \right|\quad( {j = \, 1, \, 2, \, 3}) \\ \end{gathered} . $$

(73)

Therefore, based on Eqs. (62), (69), (71) and (73), we obtain the following equation:

$$ \begin{gathered} \tilde{M}_{j}^{*} = \alpha_{0} \frac{{\Delta S_{j1} }}{\Delta S}{\mathbf{I}}_{0} \left( {\sqrt {s/F_{j} } r^{*} } \right)\exp \left( {\sqrt {s/F_{j} } r^{*} - \eta_{j} } \right) \hfill \\ \;\;\;\;\;\;\; - \alpha_{0} \frac{{\Delta S_{j2} }}{\Delta S}{\mathbf{K}}_{0} \left( {\sqrt {s/F_{j} } r^{*} } \right)\exp \left( {\eta_{j} - \sqrt {s/F_{j} } r^{*} } \right). \hfill \\ \end{gathered} $$

(74)

From Eqs. (65) and (74), the following equation can be derived:

$$ \begin{gathered} \tilde{M}^{*} = \frac{{\alpha_{2} }}{{s\alpha_{1} }} + \frac{{q_{r} \alpha_{0} }}{{\alpha_{1} \Delta S}}\left[ {\Delta S_{11} {\mathbf{I}}_{0} \left( {\sqrt {\frac{s}{{F_{1} }}} r^{*} } \right)\exp \left( {\sqrt {\frac{s}{{F_{1} }}} r^{*} - \eta_{1} } \right)} \right. \hfill \\ \left. {\;\;\;\;\;\;\;\; - \Delta S_{12} {\mathbf{K}}_{0} \left( {\sqrt {\frac{s}{{F_{1} }}} r^{*} } \right)\exp \left( {\eta_{1} - \sqrt {\frac{s}{{F_{1} }}} r^{*} } \right)} \right]. \hfill \\ \end{gathered} $$

(75)

Finally, the inverse Laplace transform is performed to obtain the results shown in Eqs. (33) and (34), in which:

$$ M^{*} = \frac{\ln 2}{{t^{*} }}\sum\limits_{i = 1}^{{n_{\text{cal}} }} {V_{i} \tilde{M}^{*} \left(\frac{\ln 2}{{t^{*} }}i\right)} , $$

(76)

$$ M_{j}^{*} = \frac{\ln 2}{{t^{*} }}\sum\limits_{i = 1}^{{n_{\text{cal}} }} {V_{i} \tilde{M}_{j}^{*} \left(r^{*} ,\frac{\ln 2}{{t^{*} }}i\right)}\; (j = 1,2,3). $$

(77)

1.2 Other Stages

Taking the first storage stage of cycle n (n ≥ 1) for example, the new functions T ^* _a2 (t ^*) _n and T ^*₁₂ (r*, t*) _n meet the following condition:

$$ \begin{aligned} \frac{{t_{1}^{*} \rho_{av}^{*} }}{{m_{r} }}\frac{{dT_{a2}^{*} (t^{*} )_{n} }}{{dt^{*} }} =& - \gamma T_{i}^{*} - (R^{*} - \gamma )T_{c}^{*} (t^{*} + t_{1}^{*} )_{n} \hfill \\ & - \rho_{av}^{*} U_{{\rho^{*} }}^{*} + q_{r} \left( {T_{12}^{*} (1,t^{*} )_{n} - T_{a2}^{*} (t^{*} )_{n} } \right). \hfill \\ \end{aligned} $$

(78)

After some simple derivations, it is found that T ^* _a2 (t ^*) _n and T ^* _j2 (r ^*, t ^*) _n (j = 1, 2, 3) meet the solution-determining conditions represented by Eqs. (56) to (60). Here, for convenience, the following approximation is adopted:

$$ T_{av}^{*} = {{\int\limits_{0}^{{t^{*} }} {T_{c}^{*} (t^{*} + t_{1}^{*} )_{n} dt} } \mathord{\left/ {\vphantom {{\int_{0}^{{t^{*} }} {T_{c}^{*} (t^{*} + t_{1}^{*} )_{n} dt} } {t^{*} }}} \right. \kern-0pt} {t^{*} }}. $$

(79)

Hence, from Eq. (78), the new temperature functions are expressed as follows:

$$ \tilde{T}_{a2}^{*} (s)_{n} = \frac{{ - \left[ {\gamma T_{i}^{*} + \rho_{av}^{*} U_{{\rho^{*} }}^{*} + (R^{*} - \gamma )T_{av}^{*} } \right]}}{{\frac{{t_{1}^{*} \rho_{av}^{*} s}}{{m_{r} }} + q_{r} }} + \frac{{q_{r} }}{{\frac{{t_{1}^{*} \rho_{av}^{*} s}}{{m_{r} }} + q_{r} }}\tilde{T}_{12}^{*} (1,s)_{n} . $$

(80)

From what has been mentioned previously, a similar solution for $\tilde{T}_{a2}^{*} (s)_{n}$ and $\tilde{T}_{j2}^{*} (r^{*} ,s)_{n}$ to $\tilde{M}^{*}$ and $\tilde{M}_{j}^{*} \left( {j = \, 1, \, 2, \, 3} \right)$ can be acquired only if α ₁ is represented by t ^*₁ ρ ^* _av s/m _r  + q _r and α ₂ is changed to $- \left[ {\gamma T_{i}^{*} + \rho_{av}^{*} U_{{\rho^{*} }}^{*} + (R^{*} - \gamma )T_{av}^{*} } \right] .$ A similar process to that discussed in the previous section of this appendix is performed, and then the air temperature associated with the temperatures in the layers around the lined cavern can also be determined.

Likewise, for the purpose of determining $\tilde{T}_{a3}^{*} (s)_{n}$ and $\tilde{T}_{j3}^{*} (r^{*} ,s)_{n} \left( {j = \, 1, \, 2, \, 3} \right),$ α ₁ should be changed to t ^*₁ ρ ^* _av s/m _r  + CD R ^* + q _r . Meanwhile α ₂ is represented by $- {\mathbf{CD}}(\rho_{av}^{*} U_{{\rho^{*} }}^{*} + R^{*} T_{av}^{*} ) .$ Of course, T ^* _av also requires corresponding substitution, namely:

$$ T_{av}^{*} = {{\int\limits_{0}^{{t^{*} }} {T_{s1}^{*} (t^{*} + t_{2}^{*} - t_{1}^{*} )_{n} dt} } \mathord{\left/ {\vphantom {{\int_{0}^{{t^{*} }} {T_{s1}^{*} (t^{*} + t_{2}^{*} - t_{1}^{*} )_{n} dt} } {t^{*} }}} \right. \kern-0pt} {t^{*} }}. $$

(81)

Again, aimed at obtaining $\tilde{T}_{a4}^{*} (s)_{n}$ and $\tilde{T}_{j4}^{*} (r^{*} ,s)_{n} \left( {j = \, 1, \, 2, \, 3} \right),$ α ₁ is represented by t ^*₁ ρ ^* _av s/m _r  + q _r a, T ^* _av a is changed to ${{\int_{0}^{{t^{*} }} {T_{d}^{*} (t^{*} + t_{3}^{*} - t_{2}^{*} )dt} } \mathord{\left/ {\vphantom {{\int_{0}^{{t^{*} }} {T_{d}^{*} (t^{*} + t_{3}^{*} - t_{2}^{*} )dt} } {t^{*} }}} \right. \kern-0pt} {t^{*} }}$ and α ₂ is changed to ${\mathbf{CD}}(\rho_{av}^{*} U_{{\rho^{*} }}^{*} + R^{*} T_{av}^{*} ) .$

Finally, to acquire $\tilde{T}_{a1}^{*} (s)_{n}$ and $\tilde{T}_{j1}^{*} (r^{*} ,s)_{n}$ (j = 1, 2, 3 and the first cycle is excluded), α ₁ should be changed to t ^*₁ ρ ^* _av s/m _r  + q _r  − (R ^* − γ) and T ^* _av is represented by ${{\int_{0}^{{t^{*} }} {T_{s2}^{*} (t^{*} + 1 - t_{3}^{*} )_{n - 1} dt} } \mathord{\left/ {\vphantom {{\int_{0}^{{t^{*} }} {T_{s2}^{*} (t^{*} + 1 - t_{3}^{*} )_{n - 1} dt} } {t^{*} }}} \right. \kern-0pt} {t^{*} }} .$ Meanwhile α ₂ is changed to $\gamma T_{i}^{*} + \rho_{av}^{*} U_{{\rho^{*} }}^{*} + (R^{*} - \gamma )T_{av}^{*} .$