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Abstract

The mathematical model of heat transfer phenomena is considered at the combustion chamber wall in an internal combustion (IC) engine. The mathematical model of proposed phenomena is established with respect to the crank angle. An inverse heat conduction problem is derived at the cylinder wall, and this problem is investigated numerically using Alifanov's regularization method. This problem studied as an optimization problem in which a squared residual functional is minimized with the conjugate gradient method. To show the ability of the proposed method, some test problems are considered.

1. Introduction

The solution of inverse heat transfer problems is becoming an essential part in the development of several relevant applications in engineering, such as design of thermal equipment, systems, and instruments. The use of inverse methods represent a new research direction, where the results obtained from numerical simulation and from experiments are not simply compared a posteriori, but a closed synergism exist between experimental and theoretical researches during a course of study, in order to obtain the maximum of information regarding the physical problem under consideration. Most of the methods for the solutions of inverse problems, which are currently in common use, were formalized in the last four decades in terms of their capabilities to treat ill-posed unstable problems. The basis of such formal methods resides on the idea of reformulating an inverse problem in terms of an approximate well-posed problem, by utilizing some kind of regularization techniques [1–10].

In the inverse analysis, most studies employed the nonlinear least-squares method [10–12] to determine the inverse problem. This method minimizes the formulation from the sum of the squares of the difference between the experimental measurements and the calculated response of the system. Based on the nonlinear least-squares method, various researchers have put their efforts in the field of inverse problems. In solving the problems, different algorithms have been adopted such as the conjugate gradient method, the Davidson-Fletcher-Powell method, the Monte-Carlo technique, the covariance analysis, and the dynamic programming. More sophisticated methods also have been developed such as the nonlinear least-squares formulation modified by the addition of a regularization term, the sequential estimation approach, and the adjoint equation approach coupled to the conjugate gradient method [1–7].

Motivation for the problem investigates here arises from the area of modeling of heat transfer phenomenon at the combustion chamber wall in an internal combustion (IC) engine. Heat transfer between the working fluid and the combustion chamber in an IC engine is one of the most important parameters for cycle simulation and analysis. High temperatures request to combust the fuel, so it is necessary to keep the temperature at a controllable level in order to operate the engine safely. Once the temperature in the engine has reached intolerable values, the engine blocks and components may suffer damage. Therefore it is essential to have a heat removal process which will maintain the engine at a safe operating condition. Heat transfer measurement by thermal sensors such as thermocouples lead to poor bandwidths and large uncertainties [1–4]. After the compression stroke and during combustion stroke, there is heat transfer to the surroundings from the hot gas through the cylinder walls. The peak gas temperatures of combustion are the order of 3000 (k) and that is why the cylinder walls of the chamber overheat. The only way in which energy can be transferred away from the combustion chamber is through convection and conduction. The heat transfer phenomena in internal combustion engines has been investigated by many authors [1–6].

In comparison with previous works, The heat transfer problem in the cylinder wall is studied by using the crank angle information. Heat transfer and cylinder pressure are two closed problems in IC engines. Cylinder pressure can be analyzed as a function of crank angle, so for this reason, the amount of temperature in the cylinder chamber can be state as a function of crank angle [1, 2, 7]. The goal of this study is estimation of internal temperature as a function of crank angle by solving an inverse heat conduction problem (IHCP). It is assumed that internal temperature are unknown function of crank angle and at the external surface heat flux are known. In addition we suppose that temperature at the some angles are available as overspecified data.

The inverse problem considered in this work is solved by using a function estimation approach [4–6], where no information is a priori available regarding the forms of the unknown function, except for the functional space that they belong to. It is assumed that the unknown function belong to Hilbert space of square integrable function in the spatial domain of interest ( $L_{2} [D]$ ).

The solution of inverse problems by using the conjugate gradient method with adjoint problem for function estimation consists of the following basic steps: (i) direct problem formulation, (ii) inverse problem formulation, (iii) sensitivity problem formulation, (iv) adjoint problem formulation, (v) gradient equations, (vi) iterative solution procedure, (vii) iterative procedure stopping criterion, and (viii) computational algorithm. Highlights of such steps are presented below as applied to the inverse problem of interest.

2. Direct Problem Formulation

The physical model is presented in Figure 1. The following hypotheses have been taken into account.

A cylinder of internal radius $R_{1}$ and external radius $R_{2}$ is considered.

The thermophysical properties are constant, and heat transfer is one-dimensional.

Thermal and physical properties are considered constant.

The cylinder wall is treated as thin rectangular sheet.

Figure 1:

A simple view of cylinder chamber of an IC engine.

Under these conditions, the mathematical model of heat transfer process in cylinder wall is given by the following system of equations:

\begin{matrix} \frac{\partial^{2} T}{\partial r^{2}} = K \frac{\partial T}{\partial t}, R_{1} < r < R_{2}, t > 0, \\ T (r, 0) = h_{0} (r), \\ T (R_{1}, t) = f_{0} (t), \\ k \frac{\partial T}{\partial r} (R_{2}, t) = g_{0} (t), \end{matrix}

(1)

where $K = (ρ C_{p}) / k, C_{p}$ is the specific heat, ρ is density, k is hermal conductivity,

$h_{0} (r)$ is the initial temperature, $f_{0} (t)$ is the temperature of internal surface, and $g_{0} (t)$ is the heat flux at the external surface of the combustion chamber wall. In an combustion chamber let α show thecrank angle, TDC is top dead center (piston position), BDC is bottom dead center (piston position), DC is dead center, and V_c is combustion chamber volume. One may see that the crank angle varies with time; that is, $α = α (t)$ , so we have

\begin{matrix} \frac{\partial T}{\partial t} = \frac{\partial T}{\partial α} \frac{d α}{d t} . \end{matrix}

(2)

In the engine $d α / d t$ is

\begin{matrix} \frac{d α}{d t} = W_{e} = \frac{2 π π}{60}, \end{matrix}

(3)

where W_e is the engine angular speed, and N is the engine speed Therefore, for the one cycle, the problem (1) can be rewritten as follows:

\begin{matrix} \frac{\partial^{2} T}{\partial r^{2}} = K W_{e} \frac{\partial T}{\partial α}, R_{1} < r < R_{2}, - π < α < π, \\ T (r, - π) = h (r), \\ T (R_{1}, α) = f_{1} (α), \\ k \frac{\partial T}{\partial r} (R_{2}, α) = g_{1} (α) . \end{matrix}

(4)

Now, suppose

\begin{matrix} x = \frac{r - R_{1}}{R_{2} - R_{1}}, \\ τ = \frac{α + π}{{(R_{2} - R_{1})}^{2} K W_{e}}, \\ U (x, τ) = T ((R_{2} - R_{1}) x + R_{1}, {(R_{2} - R_{1})}^{2} K W_{e} τ - π) . \end{matrix}

(5)

Then, the problem (4) is reduced to the following standard heat conduction problem

\begin{matrix} \frac{\partial^{2} U}{\partial x^{2}} = \frac{\partial U}{\partial τ}, 0 < x < 1, 0 < τ < τ_{f}, \end{matrix}

(6)

\begin{matrix} U (x, 0) = h (x), \end{matrix}

(7)

\begin{matrix} U (0, τ) = f (τ), \end{matrix}

(8)

\begin{matrix} \frac{\partial U}{\partial x} (1, τ) = q (τ), \end{matrix}

(9)

where

\begin{matrix} τ_{f} = \frac{2 π}{{(R_{2} - R_{1})}^{2} K W_{e}}, \\ f (τ) = f_{1} ({(R_{2} - R_{1})}^{2} K W_{e} τ - π), \\ q (τ) = \frac{(R_{2} - R_{1})}{k} g_{1} ({(R_{2} - R_{1})}^{2} K W_{e} τ - π) . \end{matrix}

(10)

3. Inverse Problem

In the problem (6)–(9) suppose that $h (x)$ and $q (τ)$ are known $L_{2}$ functions and $f (τ)$ is an unknown function. In addition as an overspecified data, suppose that the transient measurements of $U (x, τ)$ taken at the location x = 1, and $τ = τ_{i}, i = 1,2, \dots, N$ are available as $Y (τ_{i})$ .

As mentioned before, the peak gas temperatures of combustion are the order of 3000 (k) and for this resason, the exact measurement of internal wall temperature is impossible. To this end, for the estimation of internal temperature, that is, $f (τ)$ , we make use of a minimization procedure involving the following objective functional:

\begin{matrix} J [f] = \frac{1}{2} \sum_{i = 1}^{N} ‍ {[U (1, τ_{i}; f) - Y (τ_{i})]}^{2}, \end{matrix}

(11)

where $U (1, τ_{i}; f)$ is the temperature at x = 1 and $τ = τ_{i}, i = 1,2, \dots, N$ computed from the solution of the direct problem (6)–(9), by using the estimated values for $f (τ)$ .

This inverse heat conduction problem (IHCP) is recast as an optimum control problem of finding the unknown control function $f (τ)$ such that it minimizes the functional (11) [3, 8–11].

4. Sensitivity Problem

The sensitivity function, solution of the sensitivity problem, is defined as the directional derivative of $U (x, τ)$ in the direction of the perturbation of the unknown function [12]. The sensitivity problem is obtained by assuming that the dependent variable $U (x, τ)$ is perturbed by $ε Δ U (x, τ)$ when the function $f (x)$ is perturbed by $ε Δ f (x)$ . Here, ε is a real number. The sensitivity problem for $U (x, τ)$ is then obtained by applying the following limiting process:

\begin{matrix} \lim_{ε \to 0} \frac{L_{ε} (f) - L (f)}{ε} = 0, \end{matrix}

(12)

where $L_{ε} (f)$ and $L (f)$ are the direct problem formulations written in operator form for perturbed and unperturbed quantities, respectively. The application of the limiting process given by (12) results in the following sensitivity problem:

\begin{matrix} \frac{\partial^{2} Δ U}{\partial x^{2}} = \frac{\partial Δ U}{\partial τ}, 0 < x < 1, 0 < τ < 1, \\ Δ U (x, 0) = 0, 0 < x < 1, \\ Δ U (0, τ) = Δ f, 0 < τ < 1, \\ \frac{\partial Δ U}{\partial x} (1, τ) = 0, 0 < τ < 1 . \end{matrix}

(13)

5. Adjoint Problem

A Lagrange multiplier $λ (x, τ)$ is utilized in the minimization of the functional (11), because the estimated dependent variable $U [1, τ_{i}; f (x)]$ appearing in such functional needs to satisfy a constraint, which is the solution of the direct problem. Such Lagrange multiplier, needed for the computation of the gradient equations (as will be apparent below), is obtained through the solution of problems adjoint to the sensitivity problem (13) [2, 3]. In order to derive the adjoint problem, the governing equation of the direct problem (6) is multiplied by the Lagrange multiplier $λ (x, τ)$ , integrated in the space and time domains of interest and added to the original functional (11). The following extended functional is obtained:

\begin{matrix} S [f (τ [f = \frac{1}{2} \int_{0}^{τ_{f}} ‍ \int_{0}^{1} ‍ \sum_{i = 1}^{N} ‍ [U - Y] δ] δ - 1, τ - τ_{i}) d x d τ + \int_{0}^{τ_{f}} ‍ \int_{0}^{1} ‍ [\frac{\partial^{2} U}{\partial x^{2}} - \frac{\partial U}{\partial τ}] λ (x,, τ) d x d τ . \end{matrix}

(14)

Directional derivatives of $S [f (τ)]$ in thedirections of perturbations in $f (τ)$ is defined by

\begin{matrix} Δ S [f] = \lim_{ε \to 0} \frac{S [f_{ε}] - S [f]}{ε}, \end{matrix}

(15)

where $S [f_{ε}]$ denote the extended functional (14) written for perturbed $f (τ)$ . After performing some lengthy but straightforward manipulations and letting the directional derivatives of $S [f (τ)]$ go to zero, which is a necessary condition for the minimization of the extended functional (14), the following adjoint problem for the Lagrange multiplier $λ (x, τ)$ is obtained:

\begin{matrix} \frac{\partial^{2} λ}{\partial x} (x, τ) + \frac{\partial λ}{\partial τ} (x, τ) = \sum_{i = 1}^{N} ‍ [U (x, τ_{i}; f) - Y (1, τ_{i})] \times δ (x - 1, τ - τ_{i}), 0 < τ < τ_{f}, 0 < x < 1, \\ λ (0, τ) = 0, \\ λ (x, τ_{f}) = 0, \\ \frac{\partial λ}{\partial x} (1, τ) = 0 . \end{matrix}

(16)

6. Conjugate Gradient Method

In this section, we consider that the unknown function $f (τ)$ is approximated in the form of cubic B-spline as follows:

\begin{matrix} f (τ) = \sum_{i = 0}^{m} ‍ c_{j} ϕ_{j} (τ), \end{matrix}

(17)

where m is the number of approximation parameters, c_j are unknown approximation parameters, and $ϕ_{j} (τ)$ are given basis cubic B-spline.

As a result the IHCP is reduced to the estimation of a vector of parameters $P = [c_{1}, \dots, c_{m}]$ . Now, we consider the conjugate gradient iterative procedure to obtain the estimated $f (τ)$ that minimizes the functional (11). Unknown function is considered as an element of $L_{2} [0, τ_{f}]$ space.

The iterative procedure of the conjugate gradient method is written as

\begin{matrix} c_{j}^{n + 1} = c_{j}^{n} - γ^{n} d_{i}^{n}, i = 1, \dots, m, \end{matrix}

(18)

where n is the iteration index, $γ^{n}$ is descent parameter and a positive scaler, and $d^{n}$ is the descent direction that can be found as follows:

\begin{matrix} d_{i}^{n} = - g_{i}^{n} + β^{n} d_{i}^{n - 1}, i = 1, \dots, m, \end{matrix}

(19)

where the parameter $β^{n}$ is computed as follows:

\begin{matrix} β^{0} = 0, \\ β^{n} = \frac{〈 g^{n} - g^{n - 1}, g^{n} 〉}{{‖ g ‖}^{2}}, \end{matrix}

(20)

where $〈 \cdot, \cdot 〉$ and $∥ \cdot ∥$ respectively are the usual scaler product and norm in the space $R^{m}$ of approximation parameters. The realization of the iterative procedure (18) is based on the computing the vector $g = [g_{1}, \dots, g_{m}]$ at each iteration. This vector is determined by the relationship for the residual functional variation:

\begin{matrix} Δ J [f] = {〈 J_{c}^{'}, Δ c 〉}_{R^{m}} = {〈 J_{f}^{'}, Δ f 〉}_{L_{2}}, \end{matrix}

(21)

where $J_{c}^{'}$ is the residual functional gradient in $R^{m}$ space of approximation parameters and $J_{f}^{'}$ is the gradient in $L_{2}$ space of parameterized functions. The vector g is determined by the following equation:

\begin{matrix} G g = J_{f}^{'}, \end{matrix}

(22)

where G is the Gram's matrix for basis functions

\begin{matrix} G = {G_{i j} = {〈 ϕ_{i}, ϕ_{j} 〉}_{L_{2}}, i, j = 1, \dots, m} . \end{matrix}

(23)

This matrix is positive definite and symmetric. The most effective method for calculating the adjoint $J_{c}^{'}$ in $R^{m}$ space is based on introducing an adjoint problem [8, 11].

During the limiting processes used to obtain the adjoint problem, applied to the directional derivatives of $S [f (τ)]$ in the directions of perturbations in $f (τ)$ , the following integral term is obtained:

\begin{matrix} Δ S [f] = \int_{0}^{τ_{f}} ‍ Δ f (τ) \frac{\partial}{\partial x} λ (0, τ) d τ . \end{matrix}

(24)

The gradient of the residual functional is determined by solving adjoint problem. The following expression for the gradient components can be derived:

\begin{matrix} J_{c_{j}}^{'} = \int_{0}^{τ_{f}} ‍ λ_{x} (0, τ) ϕ_{j} (τ) d τ . \end{matrix}

(25)

Determination of descent parameter γ in each iteration deals with solution of sensitivity problem.

Descent parameter is determined minimizing the residual functional (11) with respect to γ. By using a Taylor series expansion, $γ^{n}$ can be derived as follows:

\begin{matrix} γ^{n} = \frac{\sum_{i = 1}^{N} ‍ [U (x, τ_{i}; f) - Y (1, τ_{i})] Δ U (1, τ_{i}; d^{n})}{\sum_{i = 1}^{N} ‍ Δ U {(1, τ_{i}; d^{n})}^{2}} . \end{matrix}

(26)

The iterative regularization method is used here to obtain stable solution of the proposed inverse heat transfer problem. The main idea is to terminate the iterative procedure with the residual criterion:

\begin{matrix} J (f^{n}) \leq δ^{2}, \end{matrix}

(27)

where $δ^{2}$ is defined by

\begin{matrix} δ^{2} = \sum_{i = 1}^{N} ‍ δ (1, τ_{i}), \end{matrix}

(28)

where $δ (1, τ_{i})$ is an estimate of the standard deviation for temperature measured at points $(1, τ_{i}), i = 1,2, \dots, N$ .

7. Results and Discussion

In order to examine the ability and accuracy of this model and the inverse analysis presented here for the estimation of the internal temperature, in problem (6)–(9) we consider that $τ_{f} = π / 60, h (x) = 100,$

\begin{matrix} q (τ) = {\begin{cases} 100 e^{τ}, & 0 < τ \leq \frac{π}{120} \\ 100 e^{(π / 60) - τ}, & \frac{π}{120} < τ < \frac{π}{60} \end{cases} \end{matrix}

(29)

Here we try to recover the function $f (τ)$ as follows

\begin{matrix} f (τ) = {\begin{cases} \begin{matrix} 100 e^{130 τ}, \end{matrix} & 0 < τ \leq \frac{π}{120} \\ 100 e^{130 ((π / 60) - τ)}, & \frac{π}{120} < τ < \frac{π}{60} . \end{cases} \end{matrix}

(30)

Since experimental data were not available we generate simulated transient temperature data $Y (1, τ)$ , by using the computed exact temperature U_exact, as follows:

\begin{matrix} Y (1, τ) = U_{exact} (1, τ) (1 + 0.02 σ), \end{matrix}

(31)

where $- 0.5 < σ < 0.5$ is a random number.

Figures 1, 2, 3, and 4 show the results after 30 iterations when we use m = 10, m = 20, and m = 30 basis functions for estimating $f (τ)$ as an unknown function. Figure 5 shows the behavior of the functional J when the number of basis functions is increased.

Figure 2:

The comparison between exact and approximate solutions using m = 10 cubic B-spline basis functions.

Figure 3:

The comparison between exact and approximate solutions using m = 20 cubic B-spline basis functions.

Figure 4:

The comparison between exact and approximate solutions using m = 30 cubic B-spline basis functions.

Figure 5:

The behavior of the functional J with respect to the number of basis functions.

8. Conclusion

This paper presents a new differential model of heat conduction in the cylinder wall of internal combustion engine based on crank angle. Since a lot of information about cylinders problem such as in-cylinder pressure and temperature amount could be found by using crank angle, it is convenient to use crank angle to estimate temperature at the internal surface of cylinder wall. This model can be used to estimate heat flux and temperature at the other parts of engine that deal with combustion chamber as a function of crank angle.

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