Optimization of Pin Fin Profiles

Abstract

Direct and indirect optimal control methods are used. Two different objective functions are considered. First, the transferred heat flux is maximized and the resulting optimal pin fin shape is a cylinder. Second, the pin volume is minimized for given value of the heat flux. Then, the optimum pin fin profile consists of two regions. In the first region, close to the basis, the pin thickness decreases linearly. In the second region the pin thickness is constant or may decrease, depending on thermal loads and operation. The optimal control solution is usually singular but may be very well approximated by a bang-bang solution. The technology and design constraints have important effects on pin fin profile.

Appendix 13A

The objective is to optimize the pin shape for the maximization of the heat flux Q. For simplicity Eq. (13.1.5) with $k = 1$ is adopted. The following notation is used:

$$ \begin{aligned} A_{0} \equiv A(z = 0) \hfill \\ P_{0} \equiv P(z = 0) \hfill \\ \end{aligned} $$

(13.A.1a, b)

Usage of Eqs. (13.1.2)–(13.1.5) and (13.A.1a, b) gives the dependence of the cross section area and perimeter, respectively, on $z$:

$$ \begin{aligned} A(z) = u^{2} (z)A_{0} \hfill \\ P(z) = u(z)P_{0} \hfill \\ \end{aligned} $$

(13.A.2a, b)

From Eq. (13.A.2a, b) one finds:

$$ P(z) = P_{0} \left[ {\frac{A(z)}{{A_{0} }}} \right]^{1/2} $$

(13.A.3)

The procedure adopted here is based on Pontryagin’s principle. The same procedure has been used by Maday (1974), Razelos and Imre (1983) and Natarajan and Shenoy (1990). The quoted authors used two state functions and the control was the pin diameter. Thus, the Hamiltonian was non-linear in the control and the solution did not contain singular arcs (Natarajan and Shenoy 1990). Here the heat flux Q given by Eq. (13.1.8) is maximized by using Eq. (13.1.7) as a constraint. The derivative:

$$ \tilde{A} \equiv \frac{dA}{dz} $$

(13.A.4)

is the control while a new variable, $\tilde{Q}$, is introduced in order to transform the Bolza optimal control problem into a standard Mayer problem. Then, the following equations system is to be solved:

$$ \begin{aligned} \frac{dT}{dz} & = \tilde{T} \\ \frac{{d\tilde{T}}}{dz} & = - \frac{{\tilde{A}}}{A}\tilde{T} + \frac{h}{\lambda }\frac{{P_{0} }}{{A_{0}^{1/2} }}\frac{1}{{A^{1/2} }}(T\text-T_{\infty } ) \\ \frac{{d\tilde{Q}}}{dz} & = \frac{{hP_{0} }}{{A_{0}^{1/2} }}A^{1/2} (T - T_{\infty } ) \\ \end{aligned} $$

(13.A.5a–c)

Equation (13.A.5a, b) comes from Eq. (13.A.7) by using Eqs. (13.A.3) and (13.A.4) while Eq. (13A.5c) comes from Eqs. (13.1.8) and 13.A.3). The objective is to maximize $\tilde{Q}(z = L)$.

The ordinary differential Eq. (13.A.5a–c) can be solved by using appropriate boundary conditions. However, there is no need to do this here. The Hamiltonian $H$ of the system is built as described by Tolle (1975) and Chap. 5 of this book:

$$ \begin{aligned} & H \equiv f_{T} \frac{dT}{dz} + f_{{\tilde{T}}} \frac{{d\tilde{T}}}{dz} + f_{{\tilde{Q}}} \frac{{d\tilde{Q}}}{dz} = \\ & f_{T} \tilde{T} + f_{{\tilde{T}}} \left[ { - \frac{{\tilde{A}}}{A}\tilde{T} + \frac{h}{\lambda }\frac{{P_{0} }}{{A_{0}^{1/2} }}\frac{1}{{A^{1/2} }}(T - T_{\infty } )} \right] + f_{{\tilde{Q}}} \left[ {\frac{{hP_{0} }}{{A_{0}^{1/2} }}A^{1/2} (T\text{ - }T_{\infty } )} \right] \\ \end{aligned} $$

(13.1.26)

where $f_{T} ,f_{{\tilde{T}}} ,f_{{\tilde{Q}}}$ are adjoint functions. The Hamiltonian is linear in the control $\tilde{A}$. Therefore, the solution is non-regular and the control is not uniquely determined for all values of state and adjoint functions [Oberle and Grimm (2001, p. 22)]. In some common cases depending on constraints, the non-regular solution is of bang-bang type, i.e. the control jumps from the minimum to the maximum allowed value. The maximum value of $\tilde{A}$ is zero and this means that $A(z)$ is a constant (= $A_{0}$). From Eq. 13.A.3) one sees that $P(z)$ is also constant. Thus, the optimal pin shape which maximizes the heat flux transferred to the fluid is a cylinder whose basis is delimitated by the contour curve C.