Optimization of Daniel Cam Engines

Abstract

In this chapter the piston motion of a Daniel cam engine is optimized. Piston acceleration is taken as a control. The objective is to maximize the net output work during the compression and power strokes. A thermally insulated cylinder is considered and a realistic model taking into account the cooling system is developed. The sinusoidal approximation of piston motion in the classical rod-crank system overestimates the engine efficiency. The exact description of the piston motion in rod-crank system is used here as a reference. The net output work is much larger (by 12–13%) for the optimized system than for the classical rod-crank system, for similar thickness of cylinder walls and thermal insulation. The optimized cam is smaller for a cylinder without thermal insulation than for an insulated cylinder (by up to 8%, depending on the local polar radius).

Appendices

Appendix 21A

1.1 21.A.1 Combustion

The combustion process in CIE is described in details in many papers. Reasonably accurate results may be obtained by using simpler fixed (frozen) composition analysis in the range of pressure 1–100 bar, temperature 800–2500 K and relative air-fuel ratio 0.7–3.35 (Ghojel and Honnery 2005). Also, single-zone models are much simpler and easier to run as compared with multi-zone models.

The simple single-zone combustion model proposed by Burzler et al. (2000) has been used by other authors (Xia et al. 2012) and is used here. The fuel is usually injected during the final stage of the compression stroke. It evaporates in the hot air and after a short delay ignites and starts to burn rapidly. The moment when the fuel ignites, measured from the start of the compression stroke is called auto-ignition moment and denoted $t_{z}$. The reaction coordinate $\xi (t)$ describes the extent of the combustion. Burzler et al. (2000) used the heat production curves of real CIEs investigated by Kleinschmidt (1993) and the following approximation has been obtained:

$$ \xi (t) = \left\{ {\begin{array}{*{20}l} {0\quad for\;0 \le t \le t_{z} } \hfill \\ {1 + \left( {\frac{{t - t_{z} }}{{t_{b} }} - 1} \right)\exp \left( {\frac{{t_{z} - t}}{{t_{b} }}} \right)\quad for\;t_{z} < t \le t_{z} + t_{b} } \hfill \\ {1\quad for\;t_{z} + t_{b} < t \le t_{tot} } \hfill \\ \end{array} } \right. $$

(21.A.1)

with a characteristic combustion duration $t_{b}$. Before the auto-ignition moment $t_{z}$ the reaction coordinate is $\xi (t \le t_{z} ) = 0$ while $\xi (t) = 1$ means total combustion. Since t_tot denotes the time for compression and power stroke, the position of the crank angle A associated with the auto-ignition moment $t_{z}$ is simply approximated by $A = (t_{z} /t_{tot} ) \cdot 360^{ \circ }$. Note that A = 180° when the piston is closest to the fire deck.

The charge consists of combustion gases and is treated as an ideal gas whose time variation of the mole number N and heat capacity C depend on the reaction coordinate :

$$ \begin{aligned} N(t) & = N_{i} + (N_{f} - N_{i} )\xi (t) \\ C(t) & = C_{i} + (C_{f} - C_{i} )\xi (t) \\ \end{aligned} $$

(21.A.2,A.3)

where the subscript i and f denote the initial and final combustion moments, respectively. The heating function $h(t)$ describes the rate of heat generated during combustion:

$$ h(t) = Q_{c} N_{i} \dot{\xi }(t) $$

(21.A.4)

Here $Q_{c}$ is the molar heat of the air-fuel mixture.

1.2 21.A.2 Heat Losses

During the compression and power strokes, heat losses occur through the cylinder walls. One denotes by T and $T_{wg}$ the working fluid temperature and the temperature at the inner surface of the cylinder wall, respectively. Both quantities are space averages. Two different approaches are considered to calculate the heat flux $q_{c}$ transferred from the charge towards the cylinder walls. The first approach is called Newton-type heat transfer and is based on the relationship:

$$ q_{c}^{N} = Ak_{N} \left( {T - T_{wg} } \right) $$

(21.A.5)

where $k_{N}$ is an appropriate constant quasi-static heat transfer coefficient. Equation (21.A.5) is widely used in literature primarily due to its simplicity (Ge et al. 2011) and is used here as a reference.

In the absence of an universally applicable heat transfer model for CIEs most authors prefer simple methods. For instance, the Wiebe metho d was adopted by Ghojel and Honnery (2005) while the Whitehouse and Way method was adopted by Rakopoulos et al. (2008). However, many authors are using the model of Annand (1963) [see e.g. Burzler et al. (2000); Murthy et al. (2010)]. The Annand model takes account on both conduction and convection and is based on the relationship:

$$ q_{c}^{A} = Ak_{A} (T,x)(T - T_{wg} ) $$

(21.A.6)

where $k_{A}$ is a time-variable heat transfer coefficient given by:

$$ k_{A} (T,x) = a\frac{\kappa (T)}{d}\text{Re}^{b} (T,x) $$

(21.A.7)

where $\kappa (T)$ is charge’s thermal conductivity, $\text{Re}$ is Reynolds number while a and b are empirical coefficients. Relationships used to compute $\kappa (T)$ and $\text{Re} (T,x)$ are shown in Table 21.8.

Table 21.8 Relationships used to compute $k(T)$ and $\text{Re} (T,x)$ in Eq. (21.A.7)

During the power stroke solid incandescent carbon particles appear as intermediate combustion products and radiate heat towards the cylinder’s walls. The following relationship is used to estimate the heat flux $q_{r}$ lost by radiative transfer (Annand 1963):

$$ q_{r} = Ac\sigma (T^{4} - T_{wg}^{4} ) $$

(21.A.8)

Here $\sigma$ is Stefan-Boltzmann constant while c is an empirical coefficient. During the compression stroke $c = 0$ while during the power stroke c ranges between 0.04 and 0.32 for CIEs (Annand 1963).

The total heat flux $q_{lost}$ lost by the charge may be obtained simply by summing the losses by conduction and convection and by radiation, respectively:

$$ q_{lost} = q_{c} + q_{r} $$

(21.A.9)

where $q_{c}$ is a shorthand for $q_{c}^{N}$ or $q_{c}^{A}$, depending on the approximation adopted.

1.3 21.A.3 Frictional Losses

The frictional loss rate of mechanical energy $w_{f}$ is proportional to the square of piston velocity v (Burzler et al. 2000):

$$ w_{f} = \alpha v^{2} $$

(21.A.10)

where $\alpha$ is a coefficient of constant value during the compression and power stroke. The assumption that the heat generated by friction does not contribute to the engine heat production as a low-grade heat source is adopted.

Appendix 21B

1.1 21.B.1 Classical Rod-Crank System

In case of the classical rod-crank system the piston movement is obtained by using Eqs. (21.2.3, 21.2.4). Equation (21.2.5) is not needed. In the following, the specific form of Eqs. (21.2.3, 21.2.4) is derived. The movement of the rod-crank system is ideally described in Fig. 21.19 where r and l are the length of the crank and rod, respectively. O and P denote crank and piston position, respectively, while M is the fire deck position. Also, A is crank angle while $x_{0}$ and $x_{f}$ denote the minimum and maximum distance between piston and fire deck. Note that A = 180° when the piston is closest to the fire deck .

Fig. 21.19

Movement of the rod-crank system. P denotes the piston while r and l are the lengths of the crank and rod, respectively. A is crank angle (=180° at the top-dead-center, i.e. the position when the piston is closest to the fire deck)

Piston position is described in terms of r and l by the following equation:

$$ y = - r\cos A + \sqrt {l^{2} - r^{2} \sin^{2} A} $$

(21.B.1)

where:

$$ A \equiv \frac{2\pi t}{{t_{tot} }} $$

(21.B.2)

Here t denotes the time while $t_{tot}$ denotes the duration of the compression and power stroke together. Equation (21.B.1) is now used to derive the piston motion in x coordinate. One easily sees that:

$$ \begin{aligned} y_{\hbox{max} } & \equiv y\left( {\cos A = - 1} \right) = l + r \\ y_{\hbox{min} } & \equiv y(\cos A = 1) = l - r \\ \end{aligned} $$

(21.B.3a,b)

Geometry constraints yield:

$$ x + y = x_{0} + y_{\hbox{max} } = x_{f} + y_{\hbox{min} } \equiv D $$

(21.B.4a–d)

where the constant D is still to be prescribed. Usage of Eqs. (21.B.3a, b) and (21.B.4a, b) yields

$$ x_{0} + r + l = x_{f} + l - r = D $$

(21.B.5a,b)

From Eq. (21.B.5a) one finds:

$$ r = \frac{{x_{f} - x_{0} }}{2} $$

(21.B.6)

Equations (21.B.5b) and (21.B.6) yield:

$$ l = D - \frac{{x_{f} + x_{0} }}{2} $$

(21.B.7)

Equations (21.B.4a–c) shows that $x + y = D$ and usage of Eq. (21.B.1) gives:

$$ x = D + r\cos A - \sqrt {l^{2} - r^{2} \sin^{2} A} $$

(21.B.8)

From Eqs. (21.B.6), (21.B.7) and (21.B.8) one finds after some algebra:

$$ x(t) = \frac{{x_{f} - x_{0} }}{2}\cos A + D - \left( {D - \frac{{x_{f} + x_{0} }}{2}} \right)\sqrt {1 - \left( {\frac{{x_{f} - x_{0} }}{{2D - (x_{f} + x_{0} )}}} \right)^{2} \sin^{2} A} $$

(21.B.9)

This is the piston movement law in $x(t)$ coordinates.

In case the distance between the crank shaft and the cylinder is large, the approximation often used is $D \to \infty$. Then, Eq. (21.B.9) yields:

$$ x = \frac{{x_{f} - x_{0} }}{2}\cos A + \frac{{x_{f} + x_{0} }}{2} $$

(21.B.10)

The sinusoidal piston law Eq. (21.B.1) is an approximation often used in practice. Here the more realistic Eq. (21.B.9) is implemented for the particular case:

$$ D = x_{f} + x_{0} $$

(21.B.11)

Equations (21.B.9) and (21.B.11) give:

$$ x(t) = \frac{{x_{f} - x_{0} }}{2}\cos A + \frac{{x_{f} + x_{0} }}{2}\sqrt {1 - \left( {\frac{{x_{f} - x_{0} }}{{x_{f} + x_{0} }}} \right)^{2} \sin^{2} A} $$

(21.B.12)

The speed and acceleration laws $\dot{x}(t)$ and $\ddot{x}(t)$, respectively, are given by:

$$ \dot{x}(t) = \frac{2\pi }{{t_{tot} }}\left[ { - r\sin A - \frac{{r^{2} \sin A\cos A}}{{\sqrt {l^{2} - r^{2} \sin^{2} A} }}} \right] $$

(21.B.13)

$$ \ddot{x}(t) = \left( {\frac{2\pi }{{t_{tot} }}} \right)^{2} \left[ { - r\cos A - \frac{{r^{2} \left( {\cos^{2} A - \sin^{2} A} \right)}}{{\sqrt {l^{2} - r^{2} \sin^{2} A} }} - \frac{{r^{4} \sin^{2} A\cos^{2} A}}{{\left( {\sqrt {l^{2} - r^{2} \sin^{2} A} } \right)^{3} }}} \right] $$

(21.B.14)

where r and l are given by Eqs. (21.B.6) and (21.B.11), respectively.

Note that in some works the classical rod-crank system under the approximation of a sinusoidal piston movement has been used as a reference (Hoffmann and Berry 1985; Burzler et al. 2000). In this chapter the “exact” solution Eq. (21.B.12) is used as a reference. The approximate sinusoidal solution Eq. (21.B.10) is considered only in Sect. 21.3.1.1.

Appendix 21C

See Figures  21.20, 21.21, 21.22 and 21.23; Tables 21.9, 21.10, 21.11, 21.12 and 21.13.

Fig. 21.20

Pressure–volume diagram for optimally controlled cam-lever system and classical rod-crank system. Newton and Annand heat transfer models are shown

Fig. 21.21

Pressure–volume diagram for optimally controlled cam-lever system. Two cases have been considered: no thermal insulation and a PSZ layer of 1.5 mm thickness, respectively

Fig. 21.22

Pressure–volume diagram for optimally controlled cam-lever system, for two values of the auto-ignition moment $t_{z}$

Fig. 21.23

Pressure–volume diagram for optimally controlled cam-lever system. Two values of the heat convection coefficient h _c have been considered

Table 21.9 Results obtained by using the optimized cam-lever system and the classical rod-crank system

Table 21.10 The dependence of the net output work W on the module of the maximum acceleration

Table 21.11 The same as Table 21.5 but the heat transfer model from cylinder wall to the cooling fluid is not taken into consideration

Table 21.12 The same as Table 21.6 but results for the classical rod-crank system are shown

Table 21.13 The same as Table 21.7 but results for the classical rod-crank system are shown