Slider Crank Model

Piston Cylinder Volume Applet

Link to: Piston Cylinder Surface Area Applet

Derivation of Slider-Crank Model

The volume of the piston cylinder can be determined as a function of crank angle from the compression ratio, the stroke, bore and connecting rod length. The geometric parameters of the piston cylinder are represented in Figure 1.

Figure 1. Piston Cylinder

b = bore

s = stroke

l = connecting rod length

a = crank radius (= ½ s)

theta = crank angle

TDC = top dead center

BDC = bottom dead center

Top dead center refers to the position of the crank shaft at an crank angle of 0^o. This position is otherwise known as the clearance volume, V_c. At bottom dead center the crank angle is at 180 ^o. In this position the cylinder volume is at its maximum, V₁. The difference between the maximum and minimum volume, V₁- V_o, is defined as the displacement volume, V_d. The displacement volume can also be represented as a function of the bore and stroke:

(1)

At a given crank angle the volume is given by : V = V_c + p /4 b² (2)

Figure 2. Piston Cylinder Geometry

Again using geometry, a relationship for x can be developed:

Figure 3. Geometric solution for x

(3)

₁

r = 1 + V_d/V_c(4)

V_o = V_d / r-1 (5)

Substituting equations (3) and (5) into equation (2), results in the following relationship for the cylinder volume.

(6)

R = l/a

The final form of the Slider-Crank model is given as a non-dimensional relationship by dividing both sides of equation (6) by V_d.

(7)

Volume Applet

Derivation of Surface Area of the Piston Cylinder

In order to study the effect of heat transfer within the piston cylinder, the surface area of the cylinder has to be evaluated. The combustion chamber surface area is given by:

(8)

Where:

A_ch = cylinder head surface area

A_p = piston head area

The equation can be further approximated by:

(9)

Surface Area Applet

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