A large slab of 4 cm-thick glass is to be cooled from an...

A large slab of $4 \mathrm{~cm}$-thick glass is to be cooled from an initial temperature of $400^{\circ} \mathrm{C}$ by an array of air jets impinging on each side of the slab. The air temperature is $20^{\circ} \mathrm{C}$, and the convective heat transfer coefficient on both sides is estimated to be $30 \mathrm{~W} / \mathrm{m}^{2} \mathrm{~K}$. The glass properties can be taken as $k=1.5 \mathrm{~W} / \mathrm{m} \mathrm{K}$, $\alpha=0.8 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}$. On the given mesh, use the explicit method to obtain the node temperatures for five time steps. Use a mesh Fourier number of $0.4$ but show that it satisfies the appropriate stability criterion.

Key Concepts

Convective Boundary Conditions

Convective boundary conditions are used to model the heat transfer between a solid surface and a surrounding fluid. In problems involving cooling or heating, these conditions account for the heat removed or supplied via convection by relating the heat flux at the surface to the difference between the surface temperature and the ambient fluid temperature through a convective heat transfer coefficient. This provides a more realistic description of heat exchange at the boundary compared to assuming a constant temperature or flux.

Stability Criterion for Explicit Methods

The stability criterion for explicit finite difference methods ensures that the numerical solution remains bounded as the simulation progresses. It typically involves a condition on the mesh Fourier number, such that it does not exceed a critical value. If the chosen time-step or spatial discretization violates this criterion, the solution may become unstable and contain unphysical oscillations or growth in error.

Mesh Fourier Number

The mesh Fourier number is a dimensionless parameter that arises in the discretization of the heat conduction equation. It represents the ratio of the thermal diffusivity over a discrete time step to the square of the spatial step, and plays a critical role in determining the stability and accuracy of the chosen time step in explicit finite difference schemes. A properly selected Fourier number ensures that the numerical solution accurately approximates the physical process.

Finite Difference Method

The finite difference method is a numerical technique used for solving differential equations by approximating derivatives with differences. In the context of heat conduction, both time and space are discretized into a grid or mesh, and the governing differential equations are transformed into algebraic equations that can be solved iteratively or directly. This method is widely used to analyze problems that are difficult to solve analytically.

Transient Heat Conduction

This concept deals with the process of heat transfer within a material where the temperature field changes with time. It involves solving the heat conduction equation, which is a partial differential equation that describes the temporal and spatial distribution of temperature within a body. Transient analysis is critical when the initial temperature distribution is not equal to the steady?state distribution, and time-dependent boundary conditions or internal heat generation are present.

Explicit Finite Difference Method

The explicit finite difference method calculates the state of a system at a future time using the known states at the current time without solving a system of equations simultaneously. It is straightforward to implement and computationally efficient for simple problems. However, its stability is conditional, meaning that the time step must be chosen small enough to ensure that numerical errors do not amplify during the computation.

Transcript

00:01 In this question we have given that there is a window fitted with a single glass of thickness 2 mm and we have to find the rate of heat flow through the window and it is given that inside temperature of room is 32 degrees celsius and outside temperature is 40 degrees celsius so this is our single glass window and we have given that the inside temperature of the room is that is 32 degrees celsius this is t2 is 32 degrees celsius and outside temperature is that is t1 is 40 degrees celsius we have given the thickness of this glass is 2mm so this is 2 mm so in the first part of the equation we have to find the heat flow that this is given by q by t so q by t is equal to this is k a into t1 minus t2 upon l so this is now put all the data in this equation so this is 1 into area is 2 and this is 40 minus 32 upon this is 2 into 10 to the power minus 3 so here we get the value that is 8 ,000 joule per second so this is the answer for our first part of the cushion now in the second part of the cushion it is told that this window is replaced by two glass panes and each having a thickness of 1 mm and separated by a distance of 1 mm so for second part it is given that this is the glass window it is separated by a distance of 1 meter and this there is a air in this gap so this is air and there is our glass window so first of all i'm finding the resistance of this glass window let is this is r1 for this this is r2 and this for this resistance is r3 so r1 can be written as l upon k a and this will equal to l1 upon k1a similarly r2 will be equal to l2 upon k2a and resistance r3 is given by l3 nat resistance is given by r is equal to r1 plus r2 plus r3 so this this is l1 upon k1a plus l2 upon k2a plus l3 upon k3a.

03:02 Now put all the data in this equation.

03:04 So this is 1 into 10 to the power minus 3 into 1 .05 upon 0 .05...

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