Essential Formulas for Thermal Processes and Metal Casting

Thermal Expansion and Heat Energy Calculations

Section 4.1-2: Thermal Properties

The change in length due to thermal expansion is calculated using the following formula:

L₂ - L₁ = αL₁(T₂ - T₁)

• α (Alpha): Coefficient of thermal expansion, measured in °C^-1 or °F^-1.
• L₁ & L₂: Initial and final lengths, measured in mm (in), corresponding to temperatures T₁ & T₂.

Heat Energy Required for Temperature Change

The amount of heat energy ($H$) required to change the temperature of a material is given by:

H = C W (T₂ - T₁)

• H: Amount of heat energy, measured in J (Btu).
• C: Specific heat of the material, measured in J/Kg°C (Btu/lb°F).
• W: Weight of the material, measured in kg (lb).
• (T₂ - T₁): Change in temperature, measured in °C (°F).

Volumetric Specific Heat and Thermal Diffusivity

Volumetric Specific Heat (V_c):

V_c = ρC

• V_c: Volumetric specific heat, measured in J/mm³°C (Btu/in³°F).

Thermal Diffusivity (K):

K = k / (ρC)

Where $K$ (thermal diffusivity) is the ratio of $k$ (thermal conductivity) to $ ho C$ (volumetric specific heat).

Formulas for Metal Casting Operations

Section 10.1-3: Heat Required for Casting

The total heat ($H$) required to raise the temperature of the metal from the starting temperature ($T_o$) to the pouring temperature ($T_p$) is calculated by summing the heat required for solid heating, fusion, and liquid heating:

H = ρV { C_S(T_m - T_o) + H_f + C_l(T_p - T_m) }

Where:

• H: Total heat required, J (Btu).
• ρ: Density, g/cm³ (lbm/in³).
• V: Volume of metal being heated, cm³ (in³).
• C_S: Weight specific heat for the solid metal, J/g°C (Btu/lbm°F).
• T_m: Melting temperature of the metal, °C (°F).
• T_o: Starting temperature (usually ambient), °C (°F).
• H_f: Heat of fusion, J/g (Btu/lbm).
• C_l: Weight specific heat of the liquid metal, J/g°C (Btu/lbm°F).
• T_p: Pouring temperature, °C (°F).

Fluid Dynamics in Gating Systems (Bernoulli's Equation)

Bernoulli's equation is crucial for analyzing the flow of molten metal in the gating system. The general form relates head ($h$), pressure ($P$), velocity ($v$), and friction loss ($F$):

h + P/(ρg) + v²/(2g) = Constant + F

Variable definitions:

• h: Head, cm (in).
• P: Pressure on the liquid, N/cm² (lb/in²).
• ρ: Density, g/cm³ (lbm/in³).
• v: Flow velocity, cm/s (in/sec).
• g: Gravitational acceleration constant, 981 cm/s² (386 in/sec²).
• F: Head loss due to friction, cm (in).

Simplified Bernoulli's Equation

For conditions assuming no friction loss and constant atmospheric pressure:

Velocity at the Sprue Base

If we assume $h_2 = 0$ at the base of the sprue, the velocity ($V$) of the liquid metal is related to the height ($h$) of the sprue:

Where $V$ is the velocity of the liquid metal at the base of the sprue, and $h$ is the height of the sprue.

Volumetric Flow Rate and Mold Filling Time

Continuity Equation (Volumetric Flow Rate Q):

Q = v₁A₁ = v₂A₂

• Q: Volumetric flow rate, cm³/s (in³/sec).
• A: Cross-sectional area of the liquid, cm² (in²).

Mold Filling Time (T_MF):

T_MF = V / Q

• T_MF: Mold filling time, s (sec).
• V: Volume of mold cavity, cm³ (in³).

Casting Solidification Time (Chvorinov's Rule)

The total solidification time ($T_{TS}$) is often estimated using Chvorinov's Rule:

T_TS = C_m (V/A)ⁿ

• T_TS: Total solidification time, min.
• V: Volume of the casting, cm³ (in³).
• A: Surface area of the casting, cm² (in²).
• n: An exponent, usually taken to have a value of 2.
• C_m: Mold constant (for n=2), min/cm² (min/in²).

Specialized Casting Calculations

Section 11.1-7: Core Buoyancy Force

The buoyancy force ($F_b$) acting on a core placed within molten metal is the difference between the weight of the displaced molten metal and the weight of the core itself:

F_b = W_m - W_c

• F_b: Buoyancy force, N (lb).
• W_m: Weight of molten metal displaced, N (lb).
• W_c: Weight of the core, N (lb).

True Centrifugal Casting

The centrifugal force ($F$) exerted on a mass ($m$) rotating at velocity ($v$) at radius ($R$) is calculated as:

F = mv² / R

• F: Force, N (lb).
• m: Mass, kg (lbm).
• v: Velocity, m/s (ft/sec).
• R: Inside radius of the mold, m (ft).

The G-factor (GF) relates the centrifugal force to gravity, based on rotational speed ($N$) and mold diameter ($D$):

• GF: G-factor.
• N: Rotational speed, rev/min.
• D: Inside diameter of the mold, m (ft).

Vertical Centrifugal Casting

When casting vertically, the shape of the resulting parabolic cavity is determined by the rotational speed, height ($L$), and radii ($R_t, R_b$):

• L: Vertical length of the casting, m (ft).
• R_t: Inside radius at the top of the casting, m (ft).
• R_b: Inside radius at the bottom of the casting, m (ft).