**4.1 Air gap measurement technique**

This method calculates the IHTC based on entrapped gas properties present at the interface. The thermal conductivity of the air between the cast mold interface and the distance of air gap measured as x with the LVDT [7]. The formula used for IHTC calculation is, h = k/x, W/m2 K. The mode of heat transfer assumed in this method is conduction at the interface, but the other modes of heat transfer are also practically possible as we have discussed in the above section. Hence this method is not widely accepted by the researchers.

## **4.2 Numerical approaches (inverse method)**

In this approach, experimental cooling curves were obtained at certain locations of the cast surface and on the mold to estimate the IHTC. The IHTC is calculated based on measured cast temperature, estimated mold surface temperature and estimated mold surface heat flux. Generally solidification heat transfer problems as shown in **Figure 5** were categorized as


In the DHCP the boundary conditions were known at the metal mold interface (which is a moving boundary problem and is difficult to acquire the parameters at the interface) and the effects were determined, mathematically it is known as a well posed problem. But in solidification of casting, knowing the boundary condition is very difficult because of its high transient nature, moving boundary problem, high temperature region, combination of all modes of heat transfer, etc., at the interface. So the inverse heat conduction problem is used to approach the problem. In order to calculate the boundary condition at the interface as a surface heat flux and surface temperature of the mold, experiments were carried out to determine temperatures in the mold to get the input data. This leads to a method of adoption of an ill-posed problem or the inverse heat conduction problem (IHCP) [8]. This ill-posed nature makes IHCP conduct experimentation to determine the boundary conditions at the interface before it has to be solved from the available data rather than using a DHCP approach.

**49**

*Heat Transfer Studies on Solidification of Casting Process*

The interfacial heat transfer coefficient at the cast mold interface can be calculated based on Eq. (1), requiring the transient surface heat flux. Cast and mold surface temperatures are measured using thermocouples during solidification regardless of its uncertainty in the physical measurements. The pure analytical or other methods mentioned above are unable to determine the surface heat flux at the interface. This leads to the numerical approaches and their formulation of inverse heat conduction problem (IHCP) at the interface to determine the boundary conditions. The boundary conditions at the interface are explored or determined by the IHCP. This has been studied by various techniques like FDM, FEM, FVM and CV methods. One of the common and mostly used method is mainly based on the function minimiza-

tion technique based on the numerically calculated and measured data [6].

( ) ( )

Where, F(h) is the minimization function, Ti, Yi are calculated and measured transient temperatures at the same locations, i= 0 to N, nodal point. The errors in the temperature measurement may also lead the IHCP into ill-posed. This problem leads the researchers to propose many techniques to solve for IHCP to determine boundary conditions at the interface with the measured temperature histories.

−1 = − ∑ *N*

*i*

1.Polynomial extrapolation method: The temperature at the interface was

2.Regularization method: In order to minimize the error from the measurement obtained a sensitivity analysis can be carried out using the Tikhonov regularization theory. This was used to regularize some function to relate the measured data and this was improving the accuracy and stability of the results obtained. This method could achieve an excellent solution and could be applied to any complex geometry, but the computation takes a very long time.

3.Boundary element method and Laplace transform: the unknown temperature were transformed into equations as well as written as matrix format. This could be easily solved and written into a computer program. But it has some restrictions. It was an effective method to solve a simple linear problem. But the measured temperature data always has more noise (disturbances) in the

4.Beck's function specification with finite difference method (implicit & explicit): It was another minimizing error technique used based on heat flux, where sum of squares of assumed and calculated data are used into the function. This method could be used for linear or nonlinear problems. Also, it has long computation time and also could achieve an accurate solution with efficient computation.

5.Control volume method: This method works, based on energy balance applied over a control volume drawn on each nodal point. The next one is the governing equation for the transient heat conduction written as a partial transient heat conduction equation changed into an ordinary differential transient equation. This involves both energy and mass conservation on each node, leads to a complex formulation equation containing up to 4th order, which may be difficult to program using computer languages, and can only be applied to simple

This mathematical tool failed to minimize measurement errors.

data, this could fluctuate the result obtained as heat flux.

geometrical shapes and one dimension.

deduced by extrapolating any one of the polynomial curve fitting techniques. This method needed many measurements inside the cast and mold surfaces.

2

*Fh T Y* (2)

*i i*

*DOI: http://dx.doi.org/10.5772/intechopen.95371*

**Figure 5.** *Schematic diagram for DHCP and IHCP conditions.*

*Heat Transfer Studies on Solidification of Casting Process DOI: http://dx.doi.org/10.5772/intechopen.95371*

*Casting Processes and Modelling of Metallic Materials*

**4.1 Air gap measurement technique**

IHTC calculation is, h = k/x, W/m2

not widely accepted by the researchers.

shown in **Figure 5** were categorized as

**4.2 Numerical approaches (inverse method)**

• Direct Heat Conduction Problem (DHCP)

• Indirect Heat Conduction Problem (IHCP)

listed above.

The following section explains the detailed procedure of these methods

This method calculates the IHTC based on entrapped gas properties present at the interface. The thermal conductivity of the air between the cast mold interface and the distance of air gap measured as x with the LVDT [7]. The formula used for

method is conduction at the interface, but the other modes of heat transfer are also practically possible as we have discussed in the above section. Hence this method is

In this approach, experimental cooling curves were obtained at certain locations of the cast surface and on the mold to estimate the IHTC. The IHTC is calculated based on measured cast temperature, estimated mold surface temperature and estimated mold surface heat flux. Generally solidification heat transfer problems as

In the DHCP the boundary conditions were known at the metal mold interface (which is a moving boundary problem and is difficult to acquire the parameters at the interface) and the effects were determined, mathematically it is known as a well posed problem. But in solidification of casting, knowing the boundary condition is very difficult because of its high transient nature, moving boundary problem, high temperature region, combination of all modes of heat transfer, etc., at the interface. So the inverse heat conduction problem is used to approach the problem. In order to calculate the boundary condition at the interface as a surface heat flux and surface temperature of the mold, experiments were carried out to determine temperatures in the mold to get the input data. This leads to a method of adoption of an ill-posed problem or the inverse heat conduction problem (IHCP) [8]. This ill-posed nature makes IHCP conduct experimentation to determine the boundary conditions at the interface before it has to be solved from the available data rather than using a DHCP approach.

K. The mode of heat transfer assumed in this

**48**

**Figure 5.**

*Schematic diagram for DHCP and IHCP conditions.*

The interfacial heat transfer coefficient at the cast mold interface can be calculated based on Eq. (1), requiring the transient surface heat flux. Cast and mold surface temperatures are measured using thermocouples during solidification regardless of its uncertainty in the physical measurements. The pure analytical or other methods mentioned above are unable to determine the surface heat flux at the interface. This leads to the numerical approaches and their formulation of inverse heat conduction problem (IHCP) at the interface to determine the boundary conditions. The boundary conditions at the interface are explored or determined by the IHCP. This has been studied by various techniques like FDM, FEM, FVM and CV methods. One of the common and mostly used method is mainly based on the function minimization technique based on the numerically calculated and measured data [6].

$$F(h) = \sum\_{i=1}^{N} \left(T\_i - Y\_i\right)^2 \tag{2}$$

Where, F(h) is the minimization function, Ti, Yi are calculated and measured transient temperatures at the same locations, i= 0 to N, nodal point. The errors in the temperature measurement may also lead the IHCP into ill-posed. This problem leads the researchers to propose many techniques to solve for IHCP to determine boundary conditions at the interface with the measured temperature histories.


**Figure 6.** *IHTC variation for the rectangular aluminum casting with sand mold.*

A sample of a rectangular geometry with an aluminum (Al6061) cast volume of 45 cm3 was solidified and the IHTC was calculated as shown below in **Figure 6**. Here the IHTC curve was calculated using the control volume method and it shows a gradual increase. Various characteristics of the IHTC and the heat transfer can be discussed [9].
