91Ó°ÊÓ

A constant-property, one-dimensional plane wall of width \(2 L\), at an initial uniform temperature \(T_{i}\), is heated convectively (both surfaces) with an ambient fluid at \(T_{\infty}=T_{\infty, 1}, h=h_{1}\). At a later instant in time, \(t=t_{1}\), heating is curtailed, and convective cooling is initiated. Cooling conditions are characterized by \(T_{\infty}=T_{\infty, 2}=T_{i}, h=h_{2}\) (a) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the heating phase (Phase 1). Express the equations in terms of the dimensionless quantities \(\theta^{*}, x^{*}, B i_{1}\), and \(F o\), where \(B i_{1}\) is expressed in terms of \(h_{1}\). (b) Write the heat equation as well as the initial and boundary conditions in their dimensionless form for the cooling phase (Phase 2). Express the equations in terms of the dimensionless quantities \(\theta^{*}, x^{*}, B i_{2}\), \(\mathrm{Fo}_{1}\), and \(\mathrm{Fo}\) where \(\mathrm{Fo}_{1}\) is the dimensionless time associated with \(t_{1}\), and \(B i_{2}\) is expressed in terms of \(h_{2}\). To be consistent with part (a), express the dimensionless temperature in terms of \(T_{\infty}=T_{\infty, 1^{*}}\) (c) Consider a case for which \(B i_{1}=10, B i_{2}=1\), and \(F o_{1}=0.1\). Using a finite-difference method with \(\Delta x^{*}=0.1\) and \(\Delta F o=0.001\), determine the transient thermal response of the surface \(\left(x^{*}=1\right)\), midplane \(\left(x^{*}=0\right)\), and quarter-plane \(\left(x^{*}=0.5\right)\) of the slab. Plot these three dimensionless temperatures as a function of dimensionless time over the range \(0 \leq F o \leq 0.5\). (d) Determine the minimum dimensionless temperature at the midplane of the wall, and the dimensionless time at which this minimum temperature is achieved.

Step by step solution

01

Write the heat equation for Phase 1

The heat equation for a one-dimensional plane wall without heat generation is given by: \(\frac{\partial T}{\partial t} = \alpha \frac{\partial^2 T}{\partial x^2}\)

02

Define dimensionless quantities for Phase 1

To express the heat equation in dimensionless form, we need to define dimensionless temperature, position, Biot number, and Fourier number as: \(\theta^{*} = \frac{T - T_{\infty,1}}{T_i - T_{\infty,1}}\) \(x^{*} = \frac{x}{L}\) \(Bi_1 = \frac{h_1 L}{k}\) \(Fo = \frac{\alpha t}{L^2}\)

03

Write the dimensionless heat equation for Phase 1

By rewriting the heat equation using the above dimensionless variables, we obtain the dimensionless heat equation for Phase 1: \(\frac{\partial \theta^{*}}{\partial t} = Fo \frac{\partial^2 \theta^{*}}{\partial (x^{*})^2}\)

04

Write the dimensionless initial and boundary conditions for Phase 1

The initial condition is: \(\theta^{*} = 0\) at \(t = 0\) The boundary conditions are: \(\theta^{*} = 1 - Bi_1(1-x^{*})\) at \(x^{*} = 0\) \(\theta^{*} = 1 - Bi_1(x^{*})\) at \(x^{*} = 1\) (B) Find the dimensionless heat equation and initial and boundary conditions for Phase 2

05

Define dimensionless quantities for Phase 2

To express the heat equation in dimensionless form for Phase 2, we define new dimensionless Biot number and Fourier numbers as: \(Bi_2 = \frac{h_2 L}{k}\) \(Fo_1 = \frac{\alpha t_1}{L^2}\)

06

Write the dimensionless heat equation for Phase 2

The dimensionless heat equation for Phase 2 is the same as that for Phase 1: \(\frac{\partial \theta^{*}}{\partial t} = Fo \frac{\partial^2 \theta^{*}}{\partial (x^{*})^2}\)

07

Write the dimensionless initial and boundary conditions for Phase 2

The initial condition is: \(\theta^{*}\) at \(t = Fo_1\) The boundary conditions are: \(\theta^{*} = 1 - Bi_2(1-x^{*})\) at \(x^{*} = 0\) \(\theta^{*} = 1 - Bi_2(x^{*})\) at \(x^{*} = 1\) (C) Determine the transient thermal response of the surface (x=1), midplane (x=0), and quarter-plane (x=0.5) of the slab using finite-difference method

08

Grid and Time Step

Set up the finite-difference grid with: \(\Delta x^{*} = 0.1\) \(\Delta Fo = 0.001\)

09

Set the given parameters

Plug in the given values: \(Bi_1 = 10\) \(Bi_2 = 1\) \(Fo_1 = 0.1\)

10

Perform finite-difference calculations

Calculate the dimensionless temperatures at x=1, x=0, and x=0.5 using finite-difference method over the range of 0 ≤ Fo ≤ 0.5. Perform the calculations in two parts: first for Phase 1 (heating phase) and then for Phase 2 (cooling phase). (D) Determine the minimum dimensionless temperature at the midplane of the wall

11

Calculate minimum dimensionless temperature

From the finite-difference calculations in Step 10, find the minimum dimensionless temperature (\(\theta^{*}_{min}\)) at the midplane: \(\theta^{*}_{min}\) = min(\(\theta^{*}\) at \(x^{*} = 0\))

12

Calculate the dimensionless time at minimum temperature

Find the dimensionless time (\(Fo_{min}\)) at which the minimum dimensionless temperature occurs: \(Fo_{min}\) = \(Fo\) at \(\theta^{*}_{min}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Dimensionless Analysis

Dimensionless analysis is a powerful tool in engineering that scales down complex systems into simpler, manageable components. This technique involves transforming variables into dimensionless numbers, helping us identify the fundamental behavior of a system without being influenced by specific units.
In the context of the heat equation, the transformation of regular variables like temperature, position, time, and material properties into dimensionless forms allows us to generalize the problem. For example, by converting actual temperature into a dimensionless form \(\theta^{*} = \frac{T - T_{\infty}}{T_i - T_{\infty}}\), we can focus on the relative change in temperature rather than its absolute value.
This approach is key for solving the heat equation for both phases (heating and cooling) without redoing the entire analysis from scratch. It enables engineers and scientists to predict heat transfer behaviors effectively across different scenarios by focusing on dimensionless parameters like the Biot and Fourier numbers.

Biot Number

The Biot number \(Bi\) is a dimensionless quantity that provides insight into the heat conduction characteristics within a material. It is defined as the ratio of the conduction resistance within a material to the convection resistance at its surface. Mathematically, it is expressed as\( Bi = \frac{hL}{k}\), where

• \(h\) is the heat transfer coefficient,
• \(L\) is a characteristic length,
• \(k\) is the thermal conductivity of the material.

A Biot number much less than 1 indicates that the thermal resistance is higher in the fluid medium than within the solid. This scenario suggests that the material's temperature is relatively uniform throughout its volume. Conversely, a Biot number much greater than 1 implies significant temperature gradients within the material. In the exercise,

we observed \(Bi_1 = 10\) indicating high resistance to heat flow within the material during the initial heating, and \(Bi_2 = 1\) indicating a more moderate distribution during cooling. Understanding these values assists in designing better cooling or heating processes in applications such as metallurgical treatments and thermal management systems.

Fourier Number

The Fourier number \(Fo\) is another crucial dimensionless parameter in transient heat conduction. It represents the ratio of heat conduction over a material's surface to the stored heat within that material. It is given by the formula \(Fo = \frac{\alpha t}{L^2}\), where:

• \(\alpha\) is the thermal diffusivity (ratio of \(k\) to product of density and specific heat capacity),
• \(t\) is the time, and
• \(L\) is the characteristic length.

A higher Fourier number demonstrates that heat has penetrated more deeply into the material, whereas a lower Fourier number indicates that the heat is still near the surface.
In this exercise, the Fourier number helps define how quickly the heat propagates through the material from the moment of start (at Phase 1) until the maximum dimensionless time \(Fo\). Monitoring its progression helps observe the thermal response and adjust processes to optimize for efficiency and safety in real-world applications.

Transient Thermal Response

Understanding the transient thermal response of a material involves tracking how its temperature changes over time when exposed to different thermal conditions. This is particularly important when dealing with non-steady-state or time-dependent problems, where conditions such as the Biot and Fourier numbers influence temperature distribution.
The finite-difference method used in the exercise is a numerical approach to handle spatial and temporal discretization, allowing for the estimation of temperature at specific points \(x^{*}=0\), \(x^{*}=0.5\), and \(x^{*}=1\) during heating and cooling phases.

• At \(x^{*}=0\), the response tells us about the midplane's behavior over time.
• At \(x^{*}=1\), it reflects how the surface adapts to external conditions.
• At \(x^{*}=0.5\), it provides information halfway through the wall’s thickness.

By plotting these points against dimensionless time, we gain invaluable insights into thermal conductivity and internal resistance of materials, guiding effective engineering solutions for heat treatment and thermal systems.