Heat Transfer and basic concepts analysis

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On this topic, we (wikitec Team) have tried to explore the complete analysis of the Heat Transfer portion and its basic concepts with different modes of heat transfer with all of its applied formulas in competitive or semester-wise examinations. Dear aspirant, please take a moment to read the full articles, which will be  your appreciation and trust in us as we help you achieve your goals.

Table of Contents:

Heat Transfer: Basic concepts: Heat Transfer Modes: Conduction: Fourier’s law of heat conduction: Thermal Conductivity: Thermal Resistance: Thermal Diffusivity: Convection: Forced Convection: Free Convection: Newton’s Law of Cooling: Radiation: Stefan-Boltzmann: Wien’s Law: Kirchhoff's Law: Applications:  Conclusion:

Heat Transfer:

Here, we have discussed about the basic concept of this topic. It is very interesting chapter. No doubt, everything in this chapter, we will talk about the basic concepts with its analysis. It will be very helpful for your revision as well as exam preparation.

So, let’s start the basic concepts about Heat and Heat Transfer.

Basic Concepts:

Actually, what do we mean about heat? The heat is nothing but it’s transit form of the energy. The nature of heat always flows from higher grade to lower grade medium or you may say temperature gradient. We will discuss about later.

The heat is such a factor, where it has more importance during the phase changes like solid, liquid and gases. Without heat, nothing is possible. During heating or cooling process, the body weight does not change.

 

If we will recall the heat theory such as modern or dynamic, then we must get to know about the molecules are in continuous or parallel in motion. The kinetic energy per each molecule of the substance is always proportional to the absolute temperature profile.

Here, we should know about the energy transmission through the various mediums; the result should be the temperature gradient. Basically, the heat transfer occurs due to the following purposes such like the energy flow in the form of heat through a particular boundary of the system during steady or transient conditions, which helps to know about the temperature profile.

Also, we have already discussed about the laws governing heat transfer for the first law and second law of thermodynamics. Here, in both cases, the heat transfer has vital role during the cycle or process.

 

We have discussed about the various modes of heat transfer here.

Heat Transfer Modes:

We have already known that the transient form of energy as heat occurs through these three modes like Conduction, Convection and Radiation.

Here, if we take an example of water boiler shell, which will place on fire bed. The boiler will receive the heat from fire bed by conduction, convection and radiation process, then the heat will transfer from the fire bed to the shell, and heat will conduct through the boiler shell, now this heat will transfer to the water from the inner shell by the process of conduction and convection.

So, we conclude that the nature of heat always transfers from the high-grade medium to low-grade medium.

Conduction:

The heat will conduct from the part of a substance to other. But it will execute within same substance or vice-versa by introducing the physical contact, without considerable displacement of molecules forming the substance. 

Now, we will discuss the conduction mechanism occurs during solids, liquids and gases phases.

 

For the solids, the heat conduction executes by two mechanisms like lattice vibration and transport of free electrons. For the lattice vibration, the molecules or atoms are fast in motion during the heat transfer from the hottest body to its nearest molecules by colliding due to the energy impact.

Thus, we conclude that the combination of vibrations of molecules and the energy transport by the free electrons.

Let’s take an example for this, eventually a cold canned drinking bottle in a warm room temperature. That’s mean the heat transfer from the room to the drink through aluminium executes by the conduction process. Here, you can see it below.

Heat conduction takes place through aluminium wall to cold drink

     

From this above figure, the rate of heat conduction Q_cond. through a layer of aluminium thickness dx, which is proportional to its temperature difference between inner thickness of aluminium to outer thickness of aluminium is dT under area A normal to the direction of heat transfer and inversely proportional to the thickness of the layer.

Or we can express this Q_cond. ∝ A. (dT/ dx)

Also, we can write this Q_cond. = kA (dT/ dx)

Where, Q_cond. = the rate of heat conduction

A = The surface area of heat flow

dT = The temperature difference between inner and outer thickness of the layer

dx = The thickness of the body in the heat flow direction.

K = constant proportionality or thermal conductivity of the material such as copper and silver are good heat conductors and also have high thermal  conductivity.

Similarly, materials like rubber, wood and Styrofoam having poor conductors and thermal low conductivity.  

From the equation, we have  Q_cond. = kA (dT/ dx) 

If we take dx -> 0, then our equation reduces to its differential form

Q_cond. = - kA (dT/ dx)

Here, the -ve sign of k indicates the temperature decreases along with the increases the thickness of the materials.

dT/ dx is the temperature gradient and its value is always -ve along the +ve x direction; thus, Q is +ve.

Such above relation is also known as Fourier’s law of heat conduction.

Similarly, by the transporting of free electrons, it will execute an energy flux in the direction of decreasing temperature, especially for the metals.

 

For the liquids, the molecules are more closely spaced, where the intermolecular force executes. Here, the conduction takes place due to collisions between the molecules. 

 

For the gases, especially, heat conduction process is very simple and convenient. Here, we should know the kinetic energy of the molecule is a function of temperature and due to the temperature, the molecules are supposed to be in continuous random motion by exchanging their energy and momentum.

Fourier’s law of heat conduction:

Let’s take some assumptions, when we go for the analysis of Fourier’s law.

Basically, during steady conditions, the heat conduction takes place. The flow of heat is unidirectional.

The temperature gradient (dT/ dx) is constant and the temperature profile is linear. Where, no internal heat generation exists.

The surface boundary is in isothermal in nature.

The material is homogeneous and isotropic. So, for such condition, the thermal conductivity (k) is constant.

Now, we have discussed about the thermal conductivity of material (k)

Thermal Conductivity(k):

 We have already known that Q= -k A dT/ dx

Or we can write k = Q/ A. (dx/ dT)

If we consider the value of k=1 and Q=1, A=1, also dT/ dx = 1; then the above equation will reduce to k = Q/1. dx/ dT or you may write this like

[W x (1/m² ) x ( m/K)=W/mK]

The energy transition occurs through a material per unit area and unit thickness in unit time period, when the temperature difference between the faces results the flow of heat is the unit temperature difference.

Here, we have taken a table for thermal conductivity(k) of different materials.

Materials	Thermal Conductivity(k)
Steel	15-35 W/mK
Copper	385 W/mK
Asbestos	0.2 W/mK
Cast Iron	55-65 W/mK
Glass	1.2 W/mK
Aluminium	225 W/mK

N.B.:- k_(AIR) = 0.024 W/mK < k_(WATER) =0.6 W/mK < k_(ICE) = 2.25 W/mK

For your better understanding, we have taken a numerical question related thermal conductivity of steel.

Let suppose, the rate of heat transfer per unit area through a steel plate of thickness is 30mm, where it’s one side face maintained the temperature of 250 ⁰ C and the other side temperature maintained 50⁰C. Here, we consider thermal conductivity of steel 25 W/mK.

 

Now, we take a solution upon this question;

First, we take a steel plate figure and put all parameters relevant to this steel plate.

Heat Transfer through the steel plate 

From this above figure, we take t₁ and t₂ as both sides face temperature of the plate. So, we have t₁ = 250⁰ C & t₂ = 50⁰ C and also thickness of the steel plate (dx) = 30mm = 0.03m, thermal conductivity(k) value for steel plate= 25 W/mK

Now, we have already known from Fourier’s law that

Q = -k A (dT/ dx)

= -k A (t₂ - t₁/ dx); where dT= t₂ - t₁

Or we can write q=Q/A= -k(dT/dx)

= -25 x (50-250)/0.03

=1.6 x 10⁵ MW/m²  (solved).

N.B.: - The materials having high thermal conductivities are considered as good conductors and the low thermal conductivities are considered as good insulator.

Here, we have taken some considerations upon these factors for thermal conductivity. These are structure of materials, moisture deposition, density of the material and pressure as well as temperature effect on the materials.

Now, we have talked about the more important points related to thermal conductivity of the materials.

• The thermal conductivity of a gas depends on temperature and independent on molecular weight i.e. the value of k for gas increases with increase in temperature and decrease in molecular weight k ∝ (√T/√M).

• The thermal conductivity of a pure metal is high in value, but if we increase the impurity then ultimately it decreases.

• Thermal conductivity is independent of pressure exclude vacuum. Similarly, for liquid case, it will decrease with temperature exclude water and glycerine.

• The thermal conductivity of metal decreases with increase in temperature exclude mercury, aluminium and uranium.

• In case of pure metals, the heat conduction takes place due to the electron contribution prominently. But for the non-conductors and semiconductors, the lattice vibration takes place prominently.

• For the crystalline solid non-metals, such as diamond, the thermal conductivity by lattice vibration is quietly large value related to good conductor materials.

• Thermal conductivity of non-metallic liquids becomes decrease with increase in temperature.

• Thermal conductivity of liquids is usually insensitive to pressure except critical point.

• According to the Wiedemann and Franz law, the ratio between the thermal and electrical conductivities is same for all metals at the same temperature. And their ratio is directly proportional to the absolute temperature of the metal. Here, we can express this mathematically like k/σ ∝ T Or we can also write this k/σT = C; where σ = electrical conductivity of metal, C = Constant for all metals.

• Thus, we conclude that the materials having good conductor of electricity are also good conductor of heat.

Thermal Resistance (R_th):

Form the equation of Fourier’s law, the heat flow rate (Q) = kA(dT/dx)

Here, we can also write heat flow rate (Q) = Temperature difference (dT) / (dx/kA), now we should know that as per the ohm’s law V=IR

Or we can write Current (I) = dV/R; where dV = potential difference & R = Electrical resistance

Now, we have two equations in above, we can compare these equations the value of I & Q, we get the thermal conduction resistance (R_th)_cond.  = dx/kA.

N.B.:- The reciprocal of the thermal resistance is called as thermal conductance.

Thermal Diffusivity(α):

Here, we should know that the larger thermal diffusivity, the propagation of heat wave into the mediums will be very fast.

For the thermal diffusivity, α= heat conducted through a medium/heat capacity

Also, we can express this, α = k/𝛒Cp, here, 𝛒Cp = the storage of energy of a material per unit volume and k = nature of heat conducts and heat capacity

Next, we will discuss about the Convection. It’s also a very interesting topic.

Convection:

The convection is basically the mode of energy transfer occurring between a solid surface and its adjacent liquid and gas which is in motion condition. It deals with the combined effects of both conduction as well as fluid motion. Here, one thing we should remember that the faster the fluid motion the result will be the greater the convection rate of heat transfer.

Especially, the heat transfer rate within the fluid particles by mixing of one portion of fluid to another portion.

If we consider the heat transfer rate between the solid surface and its adjacent fluid layer in the absence of bulk fluid motion then the result will be the pure form of conduction. If it is presence i.e. bulk fluid motion then the result will be the heat transfer rate.

Here, we have discussed about the heat transfer occurs from a hot surface to air by convection.

Heat Transfer from hot surface to air

Let’s take a cooling of a hot block surface (see the above figure here) by supplying of cool air to its top surface. Here, the energy will be transferred to the corresponding air layer adjacent to the block surface by conduction. There are the combined effects of conduction within the air, due to this random motion of air particles and the bulk motion of the air, it will remove the hot air near the surface and will replace this by cooling air.

Here, some important tag lines

N.B.:- 

• The convection is only possible in fluid particle motion and having the direct relation with the transport of a medium.
• The convection relates to the macroform of heat transfer, so due to the heat exchange, the microscopic particles of the fluid moving in space. Thus, the effectiveness of heat transfer by convection depends only on the combined motion of fluid layers.
• Now, we should know about the type of convection mode over fluid particle’s shape or nature.
• Here, we have discussed one by one.

Forced Convection:

The term ‘Forced’ mean something push or pump hardly, that’s mean the fluid is forcefully pumped to flow over a surface by the help of some external agents like fan, pump or wind.

For forced convection: N_u = f (R_e, P_r); where N_u = Nusselt’s number, R_e = Reynold’s number, P_r = Prandtl number

Similarly,

Free Convection:

Cooling Process by Forced and Natural Convection

The free convection or you may natural convection. Here, if the fluid is caused by buoyancy forces imposed by the density differences due to the temperature variation in the fluid.

Please See the above figure of hot block surface case, if we remove fan or absence of fan, then the heat transfer occurs from the surface of hot block by the natural convection. However, any motion in the air occurs due to hot air near the surface and fall the cooling air to fill its place. The heat transfer occurs between hot block and surrounding air due to conduction only if the temperature difference between the air and block is not so high to overcome the movement of air resistance, then result will be natural convection.

Newton’s Law of Cooling:

The process of heat transfer occurs due to change in phase of fluid. This happens due to the fluid motion induced during the process of which bubbles are vaporizing during boiling and are falling in the form of liquid droplets during condensation. Hence, the rate of heat transfer by convection Q_conv.. The rate of heat transfer can be calculated from Newton’s law of cooling.

Q_conv. = hA(T_s-T_f)

Where, Q_conv. = the rate of conductive heat transfer

h= Co-efficient of convective heat transfer

T_s = Surface temperature

T_f = Fluid temperature

Or we can write, h= Q_conv. / A (T_s - T_f) = W/m² degree centigrade

             Or =W/m²K Kelvin

Here, we can talk about the co-efficient of convective heat transfer(h) that the quantity of heat supply for a unit temperature difference between the fluid and the unit area of surface in unit time.

The value of h depends on these following factors like thermodynamic properties like Specific heat, viscosity & density as well as the nature of fluid flow and shape of the surface.

N.B.:- NU= f (Gr.Pr) for free convection. Where NU= Nusselt’s number, Pr= Prandtl’s number, Gr= Grashof number

Nusselt Number:

The Nusselt number basically redefine the heat transfer rate within the same fluid layer which will be introducing the convection to conduction across the fluid. If we increase the Nusselt number then result will be the more effective convection. There is a relation between convection and conduction with Nusselt number.

Which can be expressed as Nu >>>1 then thermal convection mode >>>thermal conduction mode. 

Next is Stanton number, there is a relation with Nusselt number.

Stanton Number:

We should know that the Stanton number is also called as modified Nusselt or we can express it like (St) = NU/R_e.P_r.

Here, we have discussed about the radiation.

Radiation:

Here, before we will discuss about the radiation, try to understand the below figure carefully.

Radiation between two bodies by separate medium

Form this above figure, we have discussed about the thermal radiation which is in the form of radiation emitted due to their temperature.

Radiation takes place due to the energy emission by the matter in the form of electromagnetic waves, which results electronic configuration of the atoms or molecules presence. Suck line conduction and convection, there is no required of intervening mediums, you may observe it by the above figure here.

For the case of heat transfer, we always talked about the thermal radiation. We should know that the thermal radiation emits in the form of electromagnetic radiation like X-rays, gamma rays, radio waves and micro waves. Whatever the bodies at temperature above the absolute temperature emit thermal radiation.

Also, radiation is the volumetric mechanism. For the solids, liquids and gases cases, the emission, absorption and transmission take place due to radiation of varying degrees.

Now, let’s consider the surface phenomena for the solids, these are opaque in nature to thermal radiation. Such like metals, woods and stones. Even the emission of radiation cannot reach the surface.

Thus, the rate of radiation takes place from the surface at absolute temperature T_s. According to the law of Stefan-Boltzmann statement

Q_emt.max=σAT_s⁴

Where, A= surface area and the value of σ=5.67 x 10^-8 W/m³

Now, we can also express Q∝ T⁴

That’s mean the emissivity of black body is directly proportional to the fourth power of its absolute temperature.

Again, we can write here, Q = FσA(T₁⁴-T₂⁴)

Where, F = factor depends on geometry and surface properties.

T₁ & T₂ = Temperatures in degree kelvin(K).

Some laws here, we have discussed about.

Let’s take a look,

Wien’s Law:

This law is especially dealing with the wavelength λ_m to the maximum energy is inversely proportional to the absolute temperature T of the hottest part.

That’s why, we can express it λ_m ∝ 1/T or λ_mT = Constant

 

  Kirchhoff's Law:

The emissivity of the body at specific temperature limit is numerically equal to its absorptivity from the radiant energy from the body at same temperature limit.

Some important properties relate with radiation:

• The transmission take place without presence of the material medium.
• The radiant energy can be reflected from the surfaces and valid for the laws of reflection.
• The wavelength of heat radiation is longer than that of light waves; which are invisible to the necked eye.

Applications:

The area of applications under the disciplines of heat transfer.

We have broad application in the fields of thermal and nuclear power plants including heat engines, steam generators, condensers, and some heat exchanger like furnaces and catalytic converters etc.

Also, we have in the fields of IC engine, refrigeration and air-conditioning, design of cooling systems for electric motors, generators and transformers, the heat treatment of metals, thermal control of space engines and construction of dams, and structures.

 Conclusion:

In this post-discussion, we are emphasizing the point-to-point bullet lines, or, concisely, the topic of Heat Transfer with its basic concepts and various modes of heat transfers, only with most relevant and important explanations, as well as its formulas to help our readers with their revision and enable them to move confidently towards success. We wish you to ace your path significantly.

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