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Chapter 3: Problem 4

Can we define the convection resistance for a unit surface area as the inverse of the convection heat transfer coefficient?

Short Answer

Expert verified
Answer: Yes, the convection resistance for a unit surface area can be defined as the inverse of the convection heat transfer coefficient. This relationship is derived from the equations of convection heat transfer coefficient (h) and convection resistance (R_conv), where h = 1 / R_conv.

Step by step solution

01

Understand the concepts involved

In order to answer this question, we need to comprehend two main topics: 1. Convection Resistance: In heat transfer, resistance represents the opposition to heat transfer. For convection, this opposition occurs due to the fluid flow over a surface. 2. Convection Heat Transfer Coefficient (h): This is a measure of the efficiency at which energy is transferred due to convection per unit area. It depends on the nature of the fluid and its flow properties.
02

Define the equations related to convection resistance and heat transfer coefficient

The convection heat transfer coefficient is typically represented as: h = Q / (A * 鈭員) Where: - Q is the convective heat transfer (in Watts or W) - A is the surface area over which convection is occurring (in square meters or m^2) - 鈭員 is the temperature difference between the solid surface and the fluid (in Kelvin or K) Now, the convection resistance (R_conv) can be defined as the opposition to heat transfer per unit area. Mathematically, R_conv = 鈭員 / (Q/A)
03

Manipulate the heat transfer coefficient equation to establish the inverse relationship

Now, we will manipulate the equation we obtained for the convection heat transfer coefficient (h): h = Q / (A * 鈭員) And, we will also rewrite the convection resistance equation: R_conv = 鈭員 / (Q/A) Notice that: Q / (A * 鈭員) = 1 / (R_conv) Comparing these equations, we can see that: h = 1 / R_conv
04

Conclude the relationship between convection resistance and heat transfer coefficient

Based on the derived relationship h = 1 / R_conv, we can conclude that the convection resistance for a unit surface area can indeed be defined as the inverse of the convection heat transfer coefficient.

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Key Concepts

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

Convection Resistance
Convection resistance is a concept used in heat transfer to describe how much a fluid opposes the transfer of heat as it flows over a surface. When you think of resistance, consider it as a barrier that makes heat transfer harder. This resistance affects how quickly energy moves from a hot surface to a cooler fluid.
To understand convection resistance, first, imagine a warm surface that wants to transfer its heat to a fluid (like air or water) flowing over it. The convection resistance, often given the symbol \( R_{\text{conv}} \), represents how difficult it is for this heat transfer to occur.
Lower convection resistance means easier heat transfer and vice versa. It's a key player in determining how efficiently thermal energy is transferred between surfaces and fluids.
Convection Heat Transfer Coefficient
The convection heat transfer coefficient, denoted as \( h \), is a measure of how effectively heat is transferred due to convection per unit surface area.
Think of it as the rate of heat flow per unit area per degree of temperature difference between a surface and a fluid.
The equation representing the convection heat transfer coefficient is \( h = \frac{Q}{A \cdot \Delta T} \), where:
  • \( Q \) is the heat transferred in watts (W)
  • \( A \) is the surface area in square meters (m虏)
  • \( \Delta T \) is the temperature difference in Kelvin (K)
This coefficient depends on the properties of the fluid, like its viscosity and temperature, as well as the nature of the flow, whether it's turbulent or laminar.
In general, higher \( h \) values signify more efficient heat transfer from the surface to the fluid.
Thermal Resistance Concepts
The concept of thermal resistance is crucial when studying heat transfer. It helps us understand how different materials and situations oppose the flow of heat.
In essence, thermal resistance is like a roadblock to heat flow.
When dealing with convection, thermal resistance gives us insight into how "tough" it is for a surface to pass its heat to a fluid moving over it.
In technical terms, for convective heat transfer, thermal resistance \( R_{\text{conv}} \) is the inverse of the convection heat transfer coefficient \( h \).
The magic formula is \( R_{\text{conv}} = \frac{1}{h} \).
  • Lower thermal resistance means better heat transfer.
  • Higher thermal resistance indicates more hindrance to heat transfer.
By analyzing thermal resistance, engineers can predict and manage the performance of heat transfer systems effectively.

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Most popular questions from this chapter

Steam at \(320^{\circ} \mathrm{C}\) flows in a stainless steel pipe \((k=\) \(15 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) ) whose inner and outer diameters are \(5 \mathrm{~cm}\) and \(5.5 \mathrm{~cm}\), respectively. The pipe is covered with \(3-\mathrm{cm}\)-thick glass wool insulation \((k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Heat is lost to the surroundings at \(5^{\circ} \mathrm{C}\) by natural convection and radiation, with a combined natural convection and radiation heat transfer coefficient of \(15 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). Taking the heat transfer coefficient inside the pipe to be \(80 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine the rate of heat loss from the steam per unit length of the pipe. Also determine the temperature drops across the pipe shell and the insulation.

Steam at \(235^{\circ} \mathrm{C}\) is flowing inside a steel pipe \((k=\) \(61 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) whose inner and outer diameters are \(10 \mathrm{~cm}\) and \(12 \mathrm{~cm}\), respectively, in an environment at \(20^{\circ} \mathrm{C}\). The heat transfer coefficients inside and outside the pipe are \(105 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\) and \(14 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), respectively. Determine ( \(a\) ) the thickness of the insulation \((k=0.038 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) needed to reduce the heat loss by 95 percent and \((b)\) the thickness of the insulation needed to reduce the exposed surface temperature of insulated pipe to \(40^{\circ} \mathrm{C}\) for safety reasons.

A 0.3-cm-thick, 12-cm-high, and 18-cm-long circuit board houses 80 closely spaced logic chips on one side, each dissipating \(0.04 \mathrm{~W}\). The board is impregnated with copper fillings and has an effective thermal conductivity of \(30 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\). All the heat generated in the chips is conducted across the circuit board and is dissipated from the back side of the board to a medium at \(40^{\circ} \mathrm{C}\), with a heat transfer coefficient of \(40 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\). (a) Determine the temperatures on the two sides of the circuit board. (b) Now a \(0.2\)-cm-thick, 12-cm-high, and 18-cmlong aluminum plate \((k=237 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\) with 864 2-cm-long aluminum pin fins of diameter \(0.25 \mathrm{~cm}\) is attached to the back side of the circuit board with a \(0.02\)-cm-thick epoxy adhesive \((k=1.8 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K})\). Determine the new temperatures on the two sides of the circuit board.

What is the reason for the widespread use of fins on surfaces?

The unit thermal resistances ( \(R\)-values) of both 40-mm and 90-mm vertical air spaces are given in Table 3-9 to be \(0.22 \mathrm{~m}^{2} \cdot \mathrm{C} / \mathrm{W}\), which implies that more than doubling the thickness of air space in a wall has no effect on heat transfer through the wall. Do you think this is a typing error? Explain.
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