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Chapter 9: Problem 100

When is natural convection negligible and when is it not negligible in forced convection heat transfer?

Short Answer

Expert verified
Answer: Natural convection can be neglected in forced convection heat transfer if the Grashof number is much smaller than the Reynolds number squared (Gr/Re^2 << 1). Under these conditions, forced convection is the dominant mode of heat transfer, and the effect of natural convection can be disregarded in the analysis.

Step by step solution

01

Understanding Natural and Forced Convection

Natural convection is the heat transfer process that occurs due to the movement of fluid caused by buoyancy forces generated by temperature differences within the fluid. In forced convection, heat transfer takes place due to the forced movement of a fluid, which is usually caused by an external force like a pump, fan, or a moving object.
02

Grashof Number and Reynolds Number

To determine whether natural convection can be neglected or not, we need to compare the relative importance of natural and forced convection. We can do this using two dimensionless numbers: the Grashof number (Gr) and the Reynolds number (Re). The Grashof number is a measure of the importance of buoyancy forces due to natural convection relative to viscous forces, and it is given by: Gr = (g * beta * (T_s - T_inf) * L^3) / (nu^2) Where g is the acceleration due to gravity, beta is the thermal expansion coefficient, T_s is the surface temperature, T_inf is the ambient temperature, L is a characteristic length, and nu is the kinematic viscosity of the fluid. The Reynolds number is a measure of the ratio of inertial forces to viscous forces and is mainly used to determine the flow regime (laminar or turbulent) in forced convection. It is given by: Re = (rho * U * L) / mu Where rho is the fluid density, U is the velocity of the fluid, and mu is the dynamic viscosity of the fluid.
03

Comparing Grashof Number and Reynolds Number

To compare the importance of natural and forced convection in a situation, we can use the ratio of the Grashof number to the Reynolds number squared (Gr/Re^2), which is also known as the Richardson number. If this ratio is much less than 1, the forced convection is dominant, and natural convection can be neglected. If the ratio is greater than 1, natural convection is more significant, and both natural and forced convection need to be considered in the analysis.
04

When is natural convection negligible?

Natural convection is negligible in forced convection heat transfer if the Grashof number is much smaller than the Reynolds number squared (Gr/Re^2 << 1). Under these conditions, forced convection is the dominant mode of heat transfer, and the effect of natural convection can be disregarded in the analysis.
05

When is natural convection not negligible?

Natural convection is not negligible in forced convection heat transfer if the Grashof number is comparable to or greater than the Reynolds number squared (Gr/Re^2 鈮� 1). In this situation, both natural and forced convection are significant in the heat transfer process, and both factors must be considered in the analysis. In conclusion, the importance of natural convection relative to forced convection in a heat transfer situation can be assessed using the ratio of the Grashof number to the Reynolds number squared (Gr/Re^2). If this ratio is much smaller than 1, natural convection can be neglected. If it is greater than or equal to 1, both natural and forced convection need to be considered.

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Key Concepts

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

Forced Convection
Forced convection occurs when a fluid, such as air or water, is forced to flow over a surface or through a duct by an external means such as a fan, pump, or blower. In these instances, the movement of the fluid is independent of the natural buoyancy-driven motions that arise from differences in temperature.
This system is crucial for heat transfer in many practical applications. For example, in car radiators, industrial cooling systems, and HVAC systems, forced convection provides efficient heat removal or distribution.
Some significant features of forced convection include:
  • It requires external energy to move the fluid.
  • It can be controlled and manipulated easily to enhance heat transfer rates.
  • It's generally dominant when the Reynolds number ( Re ) is high.
When dealing with forced convection, engineers must carefully consider the fluid's velocity, viscosity, and density to optimize the system efficiency.
Grashof Number
The Grashof number (Gr) is a dimensionless value that quantifies the buoyancy's effect in natural convection situations. It helps determine how significant natural convection is compared to other forces. A high Grashof number indicates a strong influence of natural convection.
The formula for calculating the Grashof number is:
\[ Gr = \frac{g \cdot \beta \cdot (T_s - T_{inf}) \cdot L^3}{u^2} \]

Where:
  • \( g \) is the acceleration due to gravity.
  • \( \beta \) is the thermal expansion coefficient.
  • \( T_s \) is the surface temperature.
  • \( T_{inf} \) is the ambient temperature.
  • \( L \) is a characteristic length.
  • \( u \) is the kinematic viscosity of the fluid.
In the context of forced convection, the Grashof number helps decide if natural convection can be ignored. If the Grashof number is small, the buoyancy forces are not significant, and forced convection predominates.
Reynolds Number
The Reynolds number (Re) is another important dimensionless quantity in fluid mechanics and heat transfer. It indicates whether the flow of fluid will be laminar or turbulent, influencing the efficiency of forced convection.
The equation for Reynolds number is as follows:
\[ Re = \frac{\rho \cdot U \cdot L}{\mu} \]

Where:
  • \( \rho \) is the fluid density.
  • \( U \) is the velocity of the fluid.
  • \( L \) is a characteristic length.
  • \( \mu \) is the dynamic viscosity of the fluid.
When the Reynolds number is high, the flow is typically turbulent, which enhances the heat transfer rate substantially. On the other hand, a low Reynolds number indicates a laminar flow, which is less efficient for heat transfer.
In forced convection scenarios, achieving a high Reynolds number can be essential to ensuring effective heat dissipation or distribution.
Richardson Number
The Richardson number (Ri) is used to determine the relative importance of natural convection versus forced convection. It is defined as the ratio of the Grashof number to the Reynolds number squared, linking buoyancy to inertial forces in a system.
Its formula is:
\[ Ri = \frac{Gr}{Re^2} \]

If the Richardson number is:
  • Less than 1 (Ri < 1), forced convection is dominant, and natural convection effects can be neglected.
  • Greater than or equal to 1 (Ri \geq 1), natural convection becomes significant and must be considered alongside forced convection.
This value is crucial for engineers and scientists when deciding whether to incorporate natural convection effects into their thermal system analysis or when focusing mainly on the forced convection dynamics.

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Most popular questions from this chapter

Skylights or "roof windows" are commonly used in homes and manufacturing facilities since they let natural light in during day time and thus reduce the lighting costs. However, they offer little resistance to heat transfer, and large amounts of energy are lost through them in winter unless they are equipped with a motorized insulating cover that can be used in cold weather and at nights to reduce heat losses. Consider a 1 -m-wide and \(2.5\)-m-long horizontal skylight on the roof of a house that is kept at \(20^{\circ} \mathrm{C}\). The glazing of the skylight is made of a single layer of \(0.5\)-cm-thick glass \((k=0.78 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\varepsilon=0.9)\). Determine the rate of heat loss through the skylight when the air temperature outside is \(-10^{\circ} \mathrm{C}\) and the effective sky temperature is \(-30^{\circ} \mathrm{C}\). Compare your result with the rate of heat loss through an equivalent surface area of the roof that has a common \(R-5.34\) construction in SI units (i.e., a thickness-to-effective-thermal- conductivity ratio of \(\left.5.34 \mathrm{~m}^{2} \cdot \mathrm{K} / \mathrm{W}\right)\). Evaluate air properties at a film temperature of \(-7^{\circ} \mathrm{C}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Show that the volume expansion coefficient of an ideal gas is \(\beta=1 / T\), where \(T\) is the absolute temperature.

Consider an industrial furnace that resembles a 13 -ft-long horizontal cylindrical enclosure \(8 \mathrm{ft}\) in diameter whose end surfaces are well insulated. The furnace burns natural gas at a rate of 48 therms/h. The combustion efficiency of the furnace is 82 percent (i.e., 18 percent of the chemical energy of the fuel is lost through the flue gases as a result of incomplete combustion and the flue gases leaving the furnace at high temperature). If the heat loss from the outer surfaces of the furnace by natural convection and radiation is not to exceed 1 percent of the heat generated inside, determine the highest allowable surface temperature of the furnace. Assume the air and wall surface temperature of the room to be \(75^{\circ} \mathrm{F}\), and take the emissivity of the outer surface of the furnace to be \(0.85\). If the cost of natural gas is \(\$ 1.15 /\) therm and the furnace operates \(2800 \mathrm{~h}\) per year, determine the annual cost of this heat loss to the plant. Evaluate properties of air at a film temperature of \(107.5^{\circ} \mathrm{F}\) and \(1 \mathrm{~atm}\) pressure. Is this a good assumption?

Consider a thin 16-cm-long and 20-cm-wide horizontal plate suspended in air at \(20^{\circ} \mathrm{C}\). The plate is equipped with electric resistance heating elements with a rating of \(20 \mathrm{~W}\). Now the heater is turned on and the plate temperature rises. Determine the temperature of the plate when steady operating conditions are reached. The plate has an emissivity of \(0.90\) and the surrounding surfaces are at \(17^{\circ} \mathrm{C}\). As an initial guess, assume a surface temperature of \(50^{\circ} \mathrm{C}\). Is this a good assumption?

Exhaust gases from a manufacturing plant are being discharged through a \(10-\mathrm{m}-\) tall exhaust stack with outer diameter of \(1 \mathrm{~m}\). The exhaust gases are discharged at a rate of \(0.125 \mathrm{~kg} / \mathrm{s}\), while temperature drop between inlet and exit of the exhaust stack is \(30^{\circ} \mathrm{C}\), and the constant pressure-specific heat of the exhaust gases is \(1600 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}\). On a particular calm day, the surrounding quiescent air temperature is \(33^{\circ} \mathrm{C}\). Solar radiation is incident on the exhaust stack outer surface at a rate of \(500 \mathrm{~W} / \mathrm{m}^{2}\), and both the emissivity and solar absorptivity of the outer surface are \(0.9\). Determine the exhaust stack outer surface temperature. Assume the film temperature is \(60^{\circ} \mathrm{C}\).
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