Fins MCQ Quiz - Objective Question with Answer for Fins - Download Free PDF

Explanation:

Fin Effectiveness

• Fin effectiveness is a measure of how well a fin enhances heat transfer compared to the situation without the fin. It is defined as the ratio of the heat transfer with the fin to the heat transfer without the fin. Fin effectiveness depends on various parameters such as the thermal conductivity of the fin material, the convective heat transfer coefficient, the geometry of the fin, and the temperature gradient.

Formula: The effectiveness of a fin (ε) is given by:

ε = (Heat transfer with fin) / (Heat transfer without fin)

Fin effectiveness depends on the fin parameter, defined as:

\( m = \sqrt{\frac{hp}{k A_c}} \)

Where:

• \( h \) = convective heat transfer coefficient
• \( p \) = perimeter of the fin
• \( k \) = thermal conductivity
• \( A_c \) = cross-sectional area

Explanation:

Heat Transfer Using Fins

• Fins are extended surfaces used to enhance the heat transfer rate between a solid surface and the surrounding fluid. They work by increasing the surface area available for heat exchange. Fins are widely used in applications such as radiators, heat exchangers, and cooling systems in electronics.

Working Principle: The effectiveness of fins depends on their ability to conduct heat from the base (where the heat is generated or absorbed) to the fin tip and transfer it to the surrounding fluid. This involves two key processes:

• Conduction: Heat is conducted through the fin material from the base to the tip.
• Convection: Heat is transferred from the fin surface to the surrounding fluid via convection.

Key Factors for Large Heat Transfer Rate Using Fins:

• Fins should be used on the side where the heat transfer coefficient is small, as fins enhance the heat transfer rate by increasing the effective surface area.
• The thermal conductivity of the fin material should be large to ensure efficient conduction of heat from the base to the tip of the fin.
• Short and thin fins are generally preferred for practical applications as they balance heat transfer effectiveness with material usage and structural considerations.

This option correctly identifies the factors that contribute to a large heat transfer rate using fins:

• Statement 1: Fins should be used on the side where the heat transfer coefficient is small. This is correct because fins significantly enhance heat transfer in areas where the heat transfer coefficient is small, such as in natural convection environments.
• Statement 3: Short and thin fins should be used. This is correct because short and thin fins reduce material usage and structural weight while maintaining effective heat dissipation. Additionally, excessively long or thick fins may not contribute proportionally to heat transfer due to diminishing returns caused by thermal resistance and reduced temperature gradients along the fin length.
• Statement 4: Thermal conductivity of fin material should be large. This is correct because a high thermal conductivity ensures efficient heat conduction from the base of the fin to its tip, maximizing the overall heat transfer rate.

Explanation:

Heat Dissipation in Fins

Definition: Fins are extended surfaces used in heat transfer systems to increase the rate of heat dissipation by increasing the effective surface area available for heat exchange. Fins are commonly employed in applications such as radiators, heat exchangers, and electronic cooling systems. The root side of the fin is typically the region in direct contact with the heat source, such as a heated surface or base. To achieve maximum heat dissipation, the fin should have maximum lateral surface area at the root side. This design ensures efficient heat conduction from the base to the fin and maximizes the heat transfer to the surrounding medium through convection.

Working Principle: The primary function of fins is to enhance heat transfer from a surface to the surrounding medium (air, liquid, etc.). Heat transfer occurs through conduction within the fin material and convection at the fin surface. The design and placement of fins significantly impact the efficiency of heat dissipation.

Heat Conduction at the Root: The root of the fin is the point of attachment to the heat source. The heat travels from the base of the fin to its tip through conduction. If the fin has a larger lateral surface area at the root, it facilitates better heat conduction and improves the overall performance of the fin.

Convective Heat Transfer: The lateral surface area of the fin plays a crucial role in convective heat transfer. By increasing the surface area at the root, the fin can dissipate more heat to the surrounding medium, as the root is the hottest part of the fin. This design helps in maintaining a higher temperature gradient, which is essential for effective heat transfer.

Minimizing Heat Loss: If the lateral surface area is maximized at the root, the fin can transfer heat efficiently before significant thermal resistance builds up towards the tip. This ensures that heat dissipation remains optimal across the fin's length.

Thermal Gradient Optimization: A fin designed with maximum lateral surface area at the root side optimizes the thermal gradient along its length. This design promotes uniform heat dissipation, preventing localized overheating and ensuring effective cooling.

Concept:

For a very long rod, the heat loss is given by: \( Q = \sqrt{hPkA_c}(T_s - T_\infty) \)

Where P is the perimeter and A_c is the cross-sectional area of the rod.

Given:

Diameter, d = 25 mm = 0.025 m

Thermal conductivity, k = 400 W/m·K

Convective heat transfer coefficient, h = 9 W/m²·K

Surface temperature, T_s = 121°C

Ambient temperature, T_∞ = 25°C

π² = 10

Calculation:

Perimeter, \( P = \pi \times d = 0.025\pi \)

Cross-sectional area, \( A_c = \frac{\pi d^2}{4} = 0.00015625\pi \)

Heat loss, \( Q = \sqrt{9 \times 0.025\pi \times 400 \times 0.00015625\pi} \times (121 - 25) \)

= \(0.372 \times 96 = 36~W \)

Explanation:

Heat Transfer Through Fins

• Heat transfer through fins is a common method used to enhance the heat dissipation from a surface by increasing the surface area. Fins are extensions added to a surface to increase the rate of heat transfer to or from the environment. The primary mechanism through which fins enhance heat transfer is through convective heat transfer.

Convective Heat Transfer:

• Convective heat transfer is the process of heat transfer between a solid surface and a fluid (liquid or gas) in motion. The rate of convective heat transfer is governed by Newton's Law of Cooling, which states that the heat transfer rate (Q) is directly proportional to the surface area (A) and the temperature difference (ΔT) between the surface and the fluid. Mathematically, it is expressed as:

Q = h × A × ΔT

where h is the convective heat transfer coefficient.

Effect of Increasing Surface Area:

• By increasing the surface area of the fin, the heat transfer rate is enhanced because a larger area is available for convective heat transfer.
• This is particularly beneficial in applications where heat needs to be dissipated efficiently, such as in heat exchangers, radiators, and electronic cooling systems.
• The increased surface area allows for more effective interaction between the surface and the surrounding fluid, thereby improving the overall heat transfer rate.

Fins are the projections protruding from a hot surface into ambient fluid and they are meant for increasing the heat transfer rate by increasing the area of heat transfer convection area.

The effectiveness of fin decides whether adding the fin to the hot surface will necessarily increase the heat transfer rate.

Effectiveness of a fin is defined as the ratio of heat transfer with fin and without fin.

\({{\epsilon }_{fin}}=\frac{{{q}_{with~fin}}}{{{q}_{without~fin}}}\)

If effectiveness is greater than one then only added fin will increase the heat transfer rate otherwise it will have no meaning of adding it to the surface.

The most common fin used is the adiabatic tip (fin is finite in length)

\({{q}_{with~fin}}=\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml\)

\({{q}_{without~fin}}=hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)\)

Where h: Convective heat transfer coefficient A: Area P: Perimeter and k: Thermal conductivity of the material

\({{\epsilon }_{fin}}=\frac{\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml}{hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)}\)

\({{\epsilon }_{fin}}=\frac{~\tanh ml}{\sqrt{\frac{hA}{kp}}}\)

Explanation:

Fins are the projections protruding from a hot surface into ambient fluid and they are meant for increasing the heat transfer rate by increasing the area of heat transfer convection area.

The effectiveness of fin decides whether adding the fin to the hot surface will necessarily increase the heat transfer rate.

Effectiveness of a fin is defined as the ratio of heat transfer with fin and without fin.

\({{\epsilon }_{fin}}=\frac{{{q}_{with~fin}}}{{{q}_{without~fin}}}\)

If effectiveness is greater than one then only added fin will increase the heat transfer rate otherwise it will have no meaning of adding it to the surface.

The most common fin used is the adiabatic tip (fin is finite in length)

\({{q}_{with~fin}}=\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml\)

\({{q}_{without~fin}}=hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)\)

Where, h: Convective heat transfer co-efficient, A: Area, P: Perimeter and k: Thermal conductivity of the material

\({{\epsilon }_{fin}}=\frac{\sqrt{hpkA}~\left( {{T}_{o}}-{{T}_{\infty }} \right)\tanh ml}{hA~\left( {{T}_{o}}-{{T}_{\infty }} \right)}\)

\({{\epsilon }_{fin}}=\frac{~\tanh ml}{\sqrt{\frac{hA}{kp}}}\)

Explanation: 

Fins:

Fins are the projections protruding from the hot surface into the ambient fluid and they are meant for increasing the heat transfer by increasing the heat transfer area.

Temperature distribution within the fin is given by:

Case (i)

When the fin is finite in length and the tip is insulated

At x = l 

\( \frac{\theta}{\theta_o} = \frac{1}{{\cosh ml}}\) ⇒ \(\theta = \frac{\theta_o}{cosh\;ml}\)

Additional Information   

Case (ii)

Fin is infinitely long or very long fin

The temperature at the fin tip is equal to the ambient temperature i.e. at x = ∞ , T = T∞ and then θ = 0

then temperature distribution within the fin is given by

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\)

Which is exponential

Case (iii)

When the fin is finite in length but losing heat through the tip

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = \frac{{\cosh m\left( {{l_c} - x} \right)}}{{\cosh m{l_c}}}\)

where lc is the corrected length of in

\({l_c} = l + \frac{t}{2}\;\)for rectangular fiin

\({l_c} = l + \frac{d}{4}\) for pin fin

Concept:

\(\eta = \frac{{Actual\;heat\;transfer}}{{Heat\;transfer\;keeping\;the\;entire\;fin\;at\;base\;temp}}\)

Calculation:

Heat loss (Q) = 6 W

Effectiveness (ε) = 3

Efficiency (η) = 0.75

\(\eta = \frac{{Actual\;heat\;transfer}}{{Heat\;transfer\;keeping\;the\;entire\;fin\;at\;base\;temp}}\)

\(0.75 = \frac{6}{{Heat\;transfer\;keeping\;the\;entire\;fin\;at\;base\;temp}}\)

Concept:

The relation between the efficiency of the fin and the effectiveness of the fin is given by,

\(\frac{\eta }{\varepsilon } = \frac{{Cross~Section~area~of~the~fin~\left( {Ac} \right)}}{{Surface~~area~~of~the~fin~\left( {As} \right)}}\)

Calculation

Given

Length of the fin ,L = 3 cm = 30 mm, Side of the square cross-section, a = b = 2 mm, Efficiency of the fin, η = 65% = 0.65

A_s = p × L = 2 × (a + b) × L = 2 × (2 + 2) × 30 = 240 mm²

A_c = a × b = 2 × 2 = 4 mm²  

\(Thermal~conductivity,k = 237~\frac{W}{{m^\circ C}}\)

Therefore, 
\( ⇒ \frac{{0.65}}{{\rm{\varepsilon }}} = \frac{4}{{240}}\)

⇒ ε = 39

Important Points

• The Biot number of the fin with good effectiveness should be less than 1.
• Fin material should have convection resistance higher than conduction resistance.

Explanation:

Fins are the projections protruding from the hot surface into the ambient fluid and they are meant for increasing the heat transfer by increasing the heat transfer area.

In order to find the temperature distribution along the fin surface, let us consider a small differential element of the fin at a distance x from the root of the fin

Let q_x is the heat conducted to the element then,

\({q_x} = \; - kA\frac{{dT}}{{dx}}\)

and q_x+dx is the heat conducted out of the element then,

\({q_{x + dx}} = \;{q_x} + \left( {\frac{{\partial {q_x}}}{{\partial x}}} \right)dx\)

Heat convected from the surface of the element to the ambient fluid = \(h\; \times p \times dx\;\left( {T - {T_\infty }} \right)\)

Now from the conservation of energy

Heat conducted into the element = Heat conducted out of the element + Heat convected from the surface of the element to the fluid

\({q_x} = {q_{x + dx}} + hpdx\left( {T - {T_\infty }} \right)\)

\({q_x} = {q_x} + \left( {\frac{{\partial {q_x}}}{{\partial x}}} \right)dx + hpdx\left( {T - {T_\infty }} \right)\)

\(0 = \frac{\partial }{{\partial x}}\left( { - kA\frac{{dT}}{{dx}}} \right)dx + \;hpdx\left( {T - {T_\infty }} \right)\)

\(\frac{{{d^2}T}}{{d{x^2}}} - \frac{{hp}}{{kA}}\left( {T - {T_\infty }} \right) = 0\)

Let T – T_∞ = θ = f(x)

\(\frac{{dT}}{{dx}} = \frac{{d\theta }}{{dx}}\)

\(\frac{{{d^2}T}}{{d{x^2}}} = \frac{{{d^2}\theta }}{{d{x^2}}}\;\) and Putting \({m^2} = \frac{{hP}}{{kA}}\) 

\(\frac{{{d^2}\theta }}{{d{x^2}}} - {m^2}\theta = 0\)

The solutuion for above differential equation is given by

\(\theta = {C_1}{e^{ - mx}} + {C_2}{e^{mx}}\)

C₁ and C₂ are the constants of integration and are determined by the boundary conditions

One boundary condition is at x = 0, T = T₀  and θ = θ₀ = T₀ – T_∞ the second boundary condition depends upon three different cases

Case (i)

Fin is infinitely long or very long fin

The temperature at the fin tip is equal to the ambient temperature i.e. at x = ∞ , T = T_∞ and then θ = 0

then temperature distribution within the fin is given by

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = {e^{ - mx}}\)

Case (ii)

When the fin is finite in length and the tip is insulated

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = \frac{{\cosh m\left( {l - x} \right)}}{{\cosh ml}}\)

Case (iii)

When the fin is finite in length but losing heat through the tip

\(\frac{{T - {T_\infty }}}{{{T_0} - {T_\infty }}} = \frac{{\cosh m\left( {{l_c} - x} \right)}}{{\cosh m{l_c}}}\)

where l_c is the corrected length of in

\({l_c} = l + \frac{t}{2}\;\)for rectangular fiin

\({l_c} = l + \frac{d}{4}\) for pin fin

Explanation:

Effectiveness (ϵ)

Effectiveness is defined as the ratio of heat transfer rate with a fin to heat transfer rate without fin.

\(\epsilon = \frac{{{{\dot Q}_{with\;fin}}\;}}{{{{\dot Q}_{without\;fin}}}}\)

For a long fin

\(\epsilon = \frac{{{{\dot Q}_{with\;fin}}\;}}{{{{\dot Q}_{without\;fin}}}} = \sqrt {\frac{{KP}}{{h{A_c}}}} \)

where K = thermal conductivity of fin, P = perimeter of fin

h = heat transfer coefficient, A_c = cross-section area of fin

From the above formula, we can conclude:

Case 1: If \(\frac{P}{{{A_c}}}\) increases, the effectiveness of fin increases.

• A_c must be small (thin fin) and fins must be closed (not too close).
• If fins are too closed it will abstract flow of air which will decrease the effectiveness of fin.

Case 2: if K increases, the effectiveness of fin increases.

• If K is large, the temperature drop along the length will be less.
• And the temperature drop between fin and surrounding will be more and heat transfer will be more.

Case 3: if h decreases, the effectiveness of fin increases.

• Fins are more effective under free convection ( h is less). 

Explanation: 

Fins:

Fins are extended surfaces used to increase the heat transfer from a surface by increasing the effective surface area.

The effectiveness ϵ of a fin is given by -

\(ϵ = \frac{{{\rm{Heat\;transfer\;rate\;with\;fin}}}}{{{\rm{Heat\;transfer\;rate\;without\;fin}}}} = \sqrt {\frac{{{\rm{kP}}}}{{{\rm{hA}}}}} \)

where k = thermal conductivity, P = perimeter of fin, h = heat transfer coefficient and A = cross-sectional area.

For large heat transfer, ϵ should be as high as possible. It can be increased by - 

• Materials of high thermal conductivity (k) like copper and aluminium.
• Increasing the ratio of the perimeter to the cross-sectional area of the fin i.e. P/A which can be achieved by placing short, thin fins in closed spacing instead of thick fins.
• Placing fins on the gas side which has a lower heat transfer coefficient (h).
  • hgas << hliquid and h_{natural convection} << h_{forced convection}    

Explanation:

Efficiency of fin

It is defined as the ratio of actual heat transfer rate (taking place through the fin) and the maximum possible heat transfer rate (that could occur through the fin i.e. when the entire fin is at its root temperature or base temperature)

\(\eta_{fin}=\frac{Actual ~heat~ rate}{Maximum ~heat~ possible}\)

\(\eta_{fin}=\frac{\sqrt{hPkA}(T_o-T_{\infty})~tanh~mL}{hPL(T_o-T_{\infty})}\)

\(\eta_{fin}=\frac{tanh~mL}{\sqrt{\frac{hP}{kA}}L}\)

\(\eta_{fin}=\frac{tanh~mL}{mL}\)

Explanation:

Heat flow mainly depends on three factors

• area of the surface
• temperature difference and
• the convective heat transfer coefficient.

The rate of heat transfer can be increased by enhancing any one of these factors.

• Fins are thus used whenever the available surface area is found insufficient to transfer the required quantity of heat with the available temperature gradient and heat transfer coefficient.
• In the case of fins, the direction of heat transfer by convection is perpendicular to the direction of conduction heat flow.
• Fins are the extended surface protruding from a surface or body, and they are meant for increasing the heat transfer rate between the surface and the surrounding fluid by increasing the heat transfer area.

If the fin is finite in length and the tip is insulated, it means it is not losing the heat through the tip,

Heat conducted into the tip = 0

\(- \;kA\;{\left( {\frac{{dT}}{{dx}}} \right)_{at\;x = L}} = 0\)

\({\left( {\frac{{dT}}{{dx}}} \right)_{at\;x = L}} = 0\) , Boundary condition

The temperature distribution within the fin is given by

\(\frac{{T - {T_\infty }}}{{{T_o} - {T_\infty }}} = \frac{{\cosh m\left( {L - x} \right)}}{{\cosh mL}}\)

Heat transfer through fin = \(\sqrt {hpkA} \left( {{T_o} - {T_\infty }} \right)\tanh mL\) Watts

Where, h is the outside fluid heat transfer coefficient, p is the perimeter of the cross-section of the fin, A is the cross-sectional area of the fin and k is the thermal conductivity of the fin, L is the length of the fin and m is the slope of differential equation, T_o is the base temperature and T_∞ is the ambient temperature

Calculation:

We have been given with m i.e. slope of the differential equation and length of the fin

Therefore for other parameters to be constant we can see from the equation of heat transfer for this fin that

\(Q \propto \tanh mL\)

∴ We will check the value of tan h ⁡mL for tall the given condition and the condition which is having higher value will have the higher heat transfer.

First option: m = 0.75, L = 3, tan h⁡ mL = 0.9780

Second option: m = 1, L = 3, tan h⁡ mL = 0.9950

Third option: m = 3, L = 0.72, tan h ⁡mL = 0.9737

Fourth option: m = 2, L = 1.2, tan h⁡ mL = 0.9836