Principles of Heat Flow in Fluids
Introduction:
1)Tubular condenser:
Temperature profiles in a condenser:
The temperatures of the condensing vapor and of the liquid are plotted against the tube length as shown in fig. The horizontal line represents the temperature of the condensing vapor, and the curved line below it represents the rising temperature of the tube-side fluid.
The inlet and outlet temperatures of cold fluid are Tca and Tcb respectively,and the temperature of the vapor is Th.The temperature difference at the inlet of the tubes is
Th-Tca,denoted by ΔT1, and that at the exit end is Th-Tcb,denoted by ΔT2.The terminal point differences ΔT1 and ΔT2 are called approaches.
Double pipe heat exchanger:
It is the simplest type heat exchanger.The function of a heat exchanger is to increase the temperature of a cooler fluid and decrease that of a hotter fluid. It is used when required heat transfer area required is small. It consists of two concentric pipes with standard return bends. One fluid flows through inside inner pipe and the second fluid flows through the annulus between the outside pipe and inside pipe. The flow directions may be either parallel or counter current.. These exchangers are useful when heat transfer area required is not more than 100 to 150 ft2.In a typical exchanger, the inner pipe may be 1- 1/4 inch and the outer pipe is 2-1/2 inch., both IPS.
Advantages;
1. Simple in construction
2. Cheap
3. Very easy to clean
4. Very attractive when required Heat transfer areas are small.
Disadvantages:
1. The simple double pipe heat exchanger is inadequate for large flow rates
2. If several double pipes are used in parallel, the weight of metal required for the outer tubes becomes so large.
3. Smaller heat transfer area in large floor space as compared to other types
4. Dismantling requires more time
5. Leakage are more.
Temperature profiles: /
Types of flows in heat exchangers:
1)Countercurrent flow :
The two fluids enter at different ends of the exchanger and pass in opposite directions through the heat exchanger. This type of flow is that commonly used and is called counter current flow.The temperature-length curves are shown in the Fig. The four terminal temperatures are denoted as follows:
Temperature of entering hot fluid Tha
Temperature of leaving hot fluid Thb
Temperature of entering cold fluid Tca
Temperature of leaving cold fluid Tcb
In this flow, it is possible to bring the outlet temperature of the cold fluid nearly to the inlet temperature of the hot fluid. This type of heat exchanger needs a small heat transfer area for a given heat duty than parallel flow because of more driving force available. So, it is widely used in practice.
Parallel flow:
If the two fluids enter at the same end of the exchanger and flow in the same direction to the other end, the flow is called parallel.The temperature-length curves for parallel flow are shown in Fig.
Parallel flow is rarely used in a single-pass exchanger because it is not possible with this method of flow to bring the exit temperature of one of the fluid nearly to the temperature of the other, and the heat that can be transferred is less than that possible in countercurrent flow.
Parallel flow is used in special situations where it is important to change the temperature of one fluid very rapidly, such as when quenching a hot fluid from a chemical reactor to stop further reactions.
Cross Flow:
In some exchangers one fluid flows across banks of tube at right angles to the axis of the tubes. This is known as crossflow. An automobile radiator and the condenser in a home refrigerator are examples of crossflow heat exchangers.
Energy balances in heat exchangers:
Energy balances and estimations of rates of heat transfer are considered for heat exchangers operating under steady-state conditions.
Enthalpy balances in heat exchangers:
Assuming that there is no heat transfer with ambient air,
For Hot fluid ,Rate of heat loss = mh(Hhb-Hha)=qh ------(1)
And for the cold fluid ,rate of heat gained= mc(Hcb-Hca)=qc ------(2)
Where mc,mh=mass flow rates of cold fluid and warm fluid,respectively
Hca,Hha=enthalpy per unit mass of entering cold fluid and entering hot fluid, respectively
qc,qh=rates of heat addition to cold fluid and warm fluid, respectively
Rate of heat lost by the hot fluid = Rate of heat gained by the cold fluid
Therfore, from Eqs.(1) and(2),
mh(Hha-Hhb)=mc(Hcb-Hca)=q ------(3)
Equation (3) is called the overall enthalpy balance.
If only sensible heat is transferred and constant specific heats assumed, the overall enthalpy balance for a heat exchanger becomes
mhCph(Tha-Thb)=mcCpc(Tcb-Tca)=q ------(4)
Where Cpc=specific heat of cold fluid
Cph=specific heat of warm fluid
Rate of heat lost by the hot fluid = Rate of heat gained by the cold fluid =heat transferred from hot fluid to cold fluid
mhCph(Tha –Thb) =mcCpc(Tcb-Tca) =UAΔTL
U=overall heat transfer coefficient,A=heat transfer area, =LMTD
ENTHALPY BALANCES IN TOTAL CONDENSERS
mhλ=mcCpc(Tcb-Tca)=q ------(5)
Where mh=rate of condensation of vapour
λ =latent heat of vapourization of vapour
Equation (5) is based on the assumption that the vapour enters the condenser as a saturated vapour (no superheat) and the condensate leaves at the condensing temperature without being further cooled.. .
Overall Heat-Transfer Coefficient:
In a heat exchanger the driving force is taken as Th-Tc, where Th is the local average temperature of the hot fluid and Tc is that of the cold fluid. The quantity Th-Tc is called Overall local temperature difference ΔT, the heat flux also varies with tube length. It is necessary to start with a differential equation by focusing attention on a differential area dA through which a differential heat flow dq occurs under the driving force of a local value of ΔT by the equation
The quantity U, is a proportionality factor between dq/dA and ΔT, is called the local Overall heat-transfer coefficient.
To complete the definitions of U in a given case, it is necessary to specify the area. If Ais taken as the outside tube area A0, U becomes a coefficient based on that area and is written U0. Likewise, if the inside area Ai is chosen, the coefficient is also based on that area and is denoted by Ui.
Since ΔT and dq are independent of the choice of area, it follows that
-
Where Di and Do are the inside and outside diameters, respectively.
Derivation for LMTD (Logarthemic mean temperature difference):
To apply the equation
----(1)
to the entire area of a heat exchanger, the equation must be integrated. This can be done using the following assumptions .
1.The overall coefficient U is constant,
2.The specific heats of the hot and cold fluids are constant,
3.Heat exchange with the ambient is negligible, and
4.The flow is steady and either parallel or countercurrent
The overall coefficient does in fact vary with the temperatures of the fluids, but its change with temperature is gradual, so that when the temperature ranges are moderate, the assumption,of constant U is not seriously in error.
Assumptions 2 and 4 imply that Tc and Th are plotted against q, as shown in Fig. below straight lines are obtained. Since Tc and Th vary linearly with q, ΔT does likewise and d(ΔT)/dq, the slope of the graph of ΔT vs q, is constant.
Therefore
------(2)
Where ΔT1 , ΔT2 =approaches
qT=rate of heat transfer in entire exchanger
elimination of dq from Eqs.(1) and (2) gives
------(3)
The variables ΔT and A can be separated, and if U is constant, the equation can be integrated over the limits AT and 0 for A and ΔT2 and ΔT1, where AT is the total area of the heat-transfer surface. Thus
------(4)
Equation (13) can be written
------(5)
where
= ------(6)
Equation (6) defines the logarithmic mean temperature difference (LMTD).
When ΔT1 and ΔT2 are nearly equal, their arithmetic average can be used for
Calculation of overall heat transfer coefficient from individual heat transfer coefficients:
Consider in a double pipe heat exchangerin which warm fluid flows through the inside pipe and that the cold fluid is flows through the annular space. Both surfaces of the inside tube are clear of dirt or scale.The metal wall of the tube separates the warm fluid on the right from the cold fluid on the left.
The change in temperature with distance is shown by the line Ta Tb TWh TWc Te Tg. The temperature profile is thus divided into three separate parts, one through each of the two fluids and the other through the metal wall.
The average temperature of the warm fluid and cold fluid are Th and Tc, .Inner and outer surface temperatures of metal wall are Twh, Twc respectively.
The overall resistance to the flow of heat from the warm fluid to the cold fluid is a result of three separate resistances operating in series. 1) Warm fluid resistance 2) metal wall resistance 3) cold fluid resistance.
The differential rate of heat transfer from inside fluid(warm fluid) to the inside surface of the wall is
------(1)
The differential rate of heat transfer through the metal wall is
------(2)
The rate of heat transfer from inside fluid(warm fluid) to the inside surface of the wall is
------(3)
We know that ------(7)
Comparing eqs (6) and (7)we have
Fouling factors:
In actual service, heat-transfer surfaces do not remain clean. Scale, dirt, and other solid deposits form on one or both sides of the tubes, provide additional resistances to heat flow, and reduce the overall coefficient. The effect of such deposits is taken into account by adding a term 1/(dA hd) to the term in parantheses in Eq.(4) for each scale deposit. Thus, assuming that scale is deposited on both the inside and the outside surface of the tubes, Eq.(4) becomes, after correction for the effect of scale,
Where hdi and hdo are the fouling fectors for the scale deposits on the inside and outside tube surfaces, respectively. The following equations for the overall coefficients based on the outside and inside areas, respectively , follows from the Eq.(9)
------(11)
The actual thickness of the deposits are neglected in the Eqns(11) and (12)
Numerical values of fouling factors are given corresponding to satisfactory performance in normal operation, with reasonable service time between cleanings. They cover a range of approximately 600 to 11,000 W/m2.0c (100 to 2,000 Btu/ft2.h.0F). Fouling factors for ordinary industrial liquids fall in the range of 1,700 to 6,000 W/m2.0c (300 to 1000 Btu/ft2.h.0F). Fouling factors are usually set at values that also provide a safety factor for design.
Determination of Nusselt number (Heat transfer coefficient):
Consider the local overall coefficient at a specific point in the double-pipe exchanger shown in the fig. For definiteness, assume that the warm fluid is flowing through the inside pipe and that the cold fluid is flowing through the annular space. Assume also that the Reynolds numbers of the two fluids are sufficiently large to ensure turbulent flow and that both surfaces of the inside tube are clear of dirt or scale. The change in temperature with distance is shown by the line Ta Tb TWh TWc Te Tg. The temperature profile is thus divided into three separate parts, one through each of the two fluids and the other through the metal wall.
It was already known that that in turbulent flow through conduits three zones exist, even in a single fluid, so that the study of one fluid is itself complicated. In each fluid shown in the Fig there is a thin sub layer at the wall, a turbulent core occupying most of the cross section of the stream, and a buffer zone between them. The velocity gradient is large near the wall, small in the turbulent core, and in rapid change in the buffer zone. Similarly the temperature gradient is large at the wall and through the viscous sub-layer, small in the turbulent core, and in rapid change in the buffer zone. Basically, the reason for this is that heat must flow through the viscous sub-layer by conduction, which calls for a steep temperature gradient in most fluids because of the low thermal conductivity, whereas the rapidly moving eddies in the core are effective in equalizing the temperature in the turbulent zone.
The film coefficient for the warm fluid is defined by the equation
For the cold fluid the terms in the denominator are reversed to make h positive
Where dq/dA = local heat flux, based on area in contact with fluid
Th = local average temperature of warm fluid
Tc = local average temperature of cold fluid
Tw = temperature of wall in contact with fluid
An another expression for h is derived from the assumption that heat transfer very near the wall occurs only by conduction, and the heat flux is given by,
The subscript w calls attention to the fact that the gradient must be evaluated at the wall. Eliminating dq/dA from the above equations gives
Here, T is the average fluid temperature, which is Thfor the warm side and Tcfor the cool side. The above equation can be put into a dimensionless form by multiplying by the ratio of an arbitrary length to the thermal conductivity. The choice of length depends on the situation. For heat transfer at the inner surface of a tube, the tube diameter (D) is the usual choice.
The dimensionless group hD/k is called a Nusselt number Nu. The above one is a local Nusselt number based on diameter.
The physical meaning of the Nusselt number : The ration of temperature gradient at the wall, (dT/dy)w to the average temperature gradient across the entire pipe, (T — Tw)/D
Another interpretation of the Nusselt number can be obtained by considering the gradient that would exist if all the resistance to heat transfer were in a laminar layer of thickness δin which heat transfer was only by conduction.
Generally δis slightly greater than the thickness of the laminar boundary layer because there is some resistance to heat transfer in the buffer zone.