Question 47 of 50

What is the fundamental assumption of the lumped capacitance method used for solving transient conduction problems?

The use of the lumped capacitance method for transient conduction analysis is considered to introduce only a small error if which condition is satisfied?

What is the physical interpretation of the Biot number (Bi)?

What physical interpretation is given for the Fourier number (Fo)?

A thermocouple junction, approximated as a sphere, has a thermal conductivity k = 20 W/m K, specific heat c = 400 J/kg K, and density ρ = 8500 kg/m3. It is exposed to a gas stream with a convection coefficient h = 400 W/m2 K. What is the required junction diameter for the thermocouple to have a thermal time constant of 1 second?

A thermocouple junction is initially at 25°C and is placed in a hot gas stream at 200°C. The junction diameter is 0.706 mm, and the thermal time constant is 1 second. How long will it take for the junction to reach 199°C?

When analyzing transient conduction in a plane wall, infinite cylinder, or sphere where spatial effects are important, the one-term approximation of the infinite series solution is valid for which condition?

How can the transient conduction solution for a plane wall of thickness 2L with symmetric convection on both sides be applied to a plane wall of thickness L that is insulated on one side and has convection on the other?

A steel pipeline (AISI 1010) with a wall thickness of 40 mm is initially at -20°C. Hot oil at 60°C is pumped through it, creating a convective condition with h = 500 W/m^2·K. The properties of the steel are k = 63.9 W/m·K and α = 18.8 × 10^−6 m^2/s. Assuming the pipe wall can be approximated as a plane wall, what is the Biot number for this process?

For a semi-infinite solid initially at temperature Ti, whose surface is suddenly brought to a constant temperature Ts, what mathematical function describes the dimensionless temperature distribution?

Soil, initially at a uniform temperature of 20°C, is subjected to a constant surface temperature of -15°C for 60 days. The soil has a thermal diffusivity of 0.138 x 10^-6 m^2/s. What is the minimum burial depth to avoid freezing (reaching 0°C)?

What is the primary advantage of the implicit finite-difference method for transient conduction analysis compared to the explicit method?

In the explicit finite-difference method for a 1-D interior node, what does the stability criterion Fo <= 0.5 physically require?

A thick slab of copper (α = 117 x 10^-6 m^2/s) is analyzed using an explicit finite-difference method with a space increment of Δx = 75 mm. To simplify the finite-difference equations, the maximum allowable value of the Fourier number (Fo = 1/2) is chosen. What is the corresponding time step Δt?

For a plane wall of thickness 2L symmetrically cooled by convection, the total energy transferred from the wall up to time t, Q, can be expressed relative to the maximum possible energy transfer, Qo. For the one-term approximation, how is this ratio Q/Qo related to the midplane temperature?

A sphere of radius ro = 5 mm, initially at 400°C, is cooled in air at 20°C with a convection coefficient ha = 10 W/m^2·K. The sphere's properties are k = 20 W/m·K, ρ = 3000 kg/m^3, and c = 1000 J/kg·K. Using the lumped capacitance method, how long does it take for the center temperature to reach 335°C?

For the same sphere from the previous question, after it reaches 335°C, it is plunged into a water bath at 20°C with a convection coefficient hw = 6000 W/m^2·K. What is the Biot number for this second cooling step?

In general lumped capacitance analysis including radiation, such as in Equation 5.15, why is the governing differential equation considered nonlinear?

What is the purpose of nondimensionalizing the governing equations for transient conduction with spatial effects, as done in Section 5.4?

For a semi-infinite solid subjected to a constant surface heat flux (q_s'' = q_o''), how does the surface temperature T(0,t) change with time?

In the explicit finite-difference method for transient conduction, why is there a stability criterion that limits the size of the time step Δt?

A fuel element shaped as a plane wall of thickness 2L = 20 mm is convectively cooled (h = 1100 W/m^2·K, T∞ = 250°C). For an explicit finite-difference analysis with a space increment Δx = 2 mm, what is the most restrictive stability criterion on the Fourier number (Fo)? The material properties are k = 30 W/m·K and α = 5 x 10^-6 m^2/s.

For a transient conduction problem, if the one-term approximation is valid (Fo > 0.2), what is the relationship between the dimensionless temperature anywhere in the object (θ*) and the dimensionless center temperature (θo*) for a plane wall?

What is the characteristic length (Lc) customarily used in the definition of the Biot number (Bi = hLc/k) for the purpose of checking the validity of the lumped capacitance method?

A 3-mm-thick panel of aluminum alloy (k = 177 W/m·K) is being heated in an oven. The convection coefficient is ho = 40 W/m^2·K. What is the Biot number for the heating process, which is needed to assess the validity of the lumped capacitance approximation?

A spherical tumor of diameter 3 mm is to be treated by heating it to a steady-state surface temperature of 55°C. The surrounding healthy tissue has a thermal conductivity of 0.5 W/m·K and is at a body temperature of 37°C. What is the steady-state heat loss rate from the tumor?

What is the thermal penetration depth, δp, an indication of?

In the implicit finite-difference method, how are the unknown nodal temperatures at a new time (p+1) determined?

An infinite cylinder is subjected to a sudden change in convective conditions. For Fo > 0.2, the dimensionless centerline temperature (θo*) is given by θo* = C1 * exp(-ζ1^2 * Fo). How is the dimensionless temperature at any other radius (θ*) related to the centerline temperature?

Two semi-infinite solids, A and B, initially at uniform temperatures TA,i and TB,i, are placed in contact. The equilibrium surface temperature, Ts, is a weighted average of the initial temperatures. What property grouping acts as the weighting factor?

Why are solutions for transient conduction in a plane wall of thickness 2L also applicable to a short cylinder (length 2L, radius ro) or a rectangular parallelepiped as part of a product solution?

In the transient finite-difference equations presented in Table 5.3, what is the primary difference between the Explicit Method (a) and the Implicit Method (b) formulations for an interior node?

The heat equation for transient conduction in its most general form is a partial differential equation. Under what specific condition does it reduce to the simpler form used for lumped capacitance analysis?

A thick copper slab (k = 401 W/m·K, α = 117 x 10^-6 m^2/s) is initially at 20°C. It is suddenly exposed to a constant surface heat flux of 3 x 10^5 W/m^2. Using the analytical solution for a semi-infinite solid, what is the surface temperature after 2 minutes (120 s)?

For the same copper slab from the previous question (k = 401 W/m·K, α = 117 x 10^-6 m^2/s, initial temperature 20°C, constant heat flux 3 x 10^5 W/m^2), what is the temperature at an interior point x = 0.15 m after 2 minutes (120 s)?

For very small values of the Fourier number (Fo < 10^-3), the transient thermal responses of various geometries (plane walls, cylinders, spheres) subjected to a step change in surface temperature all collapse to a single curve when plotted as dimensionless heat rate (q*) versus Fo. What solution does this curve represent?

In transient conduction analysis, which dimensionless number provides a measure of the temperature drop within a solid relative to the temperature difference between the surface and the surrounding fluid?

For a long cylinder or sphere, what characteristic length is typically used for a conservative check on the validity of the lumped capacitance method (Bi < 0.1)?

A steel strip (thickness δ = 12 mm) is being heated in a furnace. The strip's properties are ρ = 7900 kg/m^3, cp = 640 J/kg·K, k = 30 W/m·K. The furnace provides a uniform convection coefficient of h = 100 W/m^2·K. What is the thermal time constant for the strip, treating it as a lumped capacitance?

What is the key difference in the definition of the Nusselt number (Nu) and the Biot number (Bi)?

Which of the following scenarios is most likely to be accurately modeled using the lumped capacitance method?

In the context of the explicit finite-difference method, if a 1D surface node at x=0 is subjected to convection, what is the stability criterion?

What does a Fourier number of Fo = 0.2 signify in the context of transient heat transfer in a plane wall?

The one-term approximation for the total energy transferred from an infinite cylinder is given by Q/Qo = 1 - (2θo*/ζ1)J1(ζ1). What does the term J1 represent?

A steel pipeline wall with thickness L=0.04 m is approximated as a plane wall. It is initially at -20°C and is suddenly exposed to 60°C oil. After 8 minutes, the Fourier number is Fo=5.64 and the Biot number is Bi=0.313. The corresponding coefficient C1 is 1.047 and eigenvalue ζ1 is 0.531 rad. What is the temperature of the outer, insulated surface after 8 minutes?

In transient conduction, how can the solution for a body subjected to a sudden change in surface temperature be obtained from the solutions for surface convection?

An initial steady-state temperature distribution for a fuel element with generation rate q_dot_1 = 1x10^7 W/m^3 is T(x) = 16.67 * (1 - x^2/L^2) + 340.91°C, where L=0.01 m. What is the initial temperature at the centerline (x=0)?

For the transient fuel element problem in Example 5.9, the explicit finite-difference solution is marched in time from t=0. After 1.5 seconds (5 time steps of 0.3 s), what is the temperature at the surface (node 5)?

Comparing the explicit and implicit finite-difference solutions in Example 5.10 for a copper slab, which method provided results that were closer to the exact analytical solution?