Why is the insulated-tip solution important for the fin difficulties?

The heat transfer surface area of a fin is equal to the sum of all surfaces of the fin exposed to the surrounding medium, including the surface area of the fin tip. Under what conditions can we neglect heat transfer from the fin tip?

The heat transfer surface area of a fin is equal to the sum of all surfaces of the fin exposed to the surrounding medium, including the surface area of the fin tip. Under what conditions can we neglect heat transfer from the fin tip?

Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at $t_{\infty}$ with a heat transfer coefficient of h. The nodal network of the fin consists of nodes 0 (at the base) , 1 (in the middle) , and 2 (at the fin tip) with a uniform nodal spacing of $\Delta x.$ Using the energy balance approach, obtain the finite difference formulation of this problem to determine 7, and $T_2$ for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in $^\circ C.$

Consider steady one-dimensional heat conduction in a pin fin of constant diameter D with constant thermal conductivity. The fin is losing heat by convection to the ambient air at $T_{\infty};$ with a convection coefficient of h, and by radiation to the surrounding surfaces at an average temperature of $T_{surr}$ The nodal network of the fin consists of nodes 0 (at the base) , 1 (in the middle) , and 2 (at the fin tip) with a uniform nodal spacing of $\Delta x.$ Using the energy balance approach, obtain the finite difference formulation of this problem to determine $T_1$ and $T_2$ for the case of specified temperature at the fin base and negligible heat transfer at the fin tip. All temperatures are in $^\circ C.$

Derive a differential equation (do not solve) for the temperature distribution in a straight triangular fin. For convenience, take the coordinate axis and suppose a one-dimensional heat flow.

Circular fins of uniform cross section, with diameter of 10 mm and length of 50 mm, are attached to a wall with surface temperature of $350^\circ C.$ The fins are made of material with thermal conductivity of $240 W/m \cdot K,$ and they are exposed to an ambient air condition of $25^\circ C$ and the convection heat transfer coefficient is $250 W/m^2 \cdot K.$ Determine the heat transfer rate and plot the temperature Variation of a Single fin for the following boundary conditions: (a) Infinitely long fin (b) Adiabatic fin tip (c) Fin with tip temperature of $250^\circ C.$ (d) Convection from the fin tip

Circular fins of uniform cross section, with diameter of 10 mm and length of 50 mm, are attached to a wall with surface temperature of $350^{\circ} \mathrm{C}.$ The fins are made of material with thermal conductivity of $240 \mathrm{W} / \mathrm{m} \cdot \mathrm{K},$ and they are exposed to an ambient air condition of $25^{\circ} \mathrm{C}$ and the convection heat transfer coefficient is $250 \mathrm{W} / \mathrm{m}^{2} \cdot \mathrm{K}.$ Determine the heat transfer rate and plot the temperature variation of a single fin for the following boundary conditions: (a) Infinitely long fin (b) Adiabatic fin tip (c) Fin with tip temperature of $250^{\circ} \mathrm{C}$ (d) Convection from the fin tip

Obtain a relation for the fin efficiency for a fin of constant cross-sectional area $A_c,$ perimeter p, length L, and thermal conductivity k exposed to convection to a medium at $T_{\infty}$ with a heat transfer coefficient h, Assume the fins are sufficiently long so that the temperature of the fin at the tip is nearly $T_{\infty}.$ Take the temperature of the fin at the base to be $T_b$ and neglect heat transfer from the fin tips. Simplify the relation for (a) a circular fin of diameter D and (b) rectangular fins of thickness t.

One side of a copper block $5 \mathrm{~cm}$ thick is maintained at $250^{\circ} \mathrm{C}$. The other side is covered with a layer of fiberglass $2.5 \mathrm{~cm}$ thick. The outside of the fiberglass is maintained at $35^{\circ} \mathrm{C}$, and the total heat flow through the copper-fiberglass combination is $52 \mathrm{~kW}$. What is the area of the slab?

An electric current is used to heat a tube through which a suitable cooling fluid flows. The outside of the tube is covered with insulation to minimize heat loss to the surroundings, and thermocouples are attached to the outer surface of the tube to measure the temperature. Assuming uniform heat generation in the tube, derive an expression for the convection heat transfer coefficient on the inside of the tube in terms of the measured variables: voltage $E$, current $I$, outside tube wall temperature $T_0$, inside and outside radii $r_i$ and $r_o$, tube length $L$, and fluid temperature $T_f$.

The heater wire of Example 2-7 is submerged in a fluid maintained at $93^{\circ} \mathrm{C}$. The convection heat-transfer coefficient is $5.7 \mathrm{~kW} / \mathrm{m}^2 \cdot{ }^{\circ} \mathrm{C}$. Calculate the center temperature of the wire.

A 3.2-mm-diameter stainless-steel wire $30 \mathrm{~cm}$ long has a voltage of $10 \mathrm{~V}$ impressed on it. The outer surface temperature of the wire is maintained at $93^{\circ} \mathrm{C}$. Calculate the center temperature of the wire. Take the resistivity of the wire as $70 \mu \Omega \cdot \mathrm{cm}$ and the thermal conductivity as $22.5 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$.

A plate having a thickness of $4.0 \mathrm{~mm}$ has an internal heat generation of $200 \mathrm{MW} / \mathrm{m}^3$ and a thermal conductivity of $25 \mathrm{~W} / \mathrm{m} \cdot{ }^{\circ} \mathrm{C}$. One side of the plate is insulated and the other side is maintained at $100^{\circ} \mathrm{C}$. Calculate the maximum temperature in the plate.