Calculate overall heat transfer coefficient - walls or heat exchangers Sponsored Links
The overall heat transfer coefficient for a wall or heat exchanger can be calculated as: 1 / U A = 1 / h 1 A1 + dx w w / k A + 1 / h 2 A2
(1)
where 2
U = the overall heat transfer coefficient (W/m K) A = the contact area area for each each fluid side side (m2 ) k = the thermal conductivity of the material (W/mK) h = the individual convection heat transfer coefficient for each fluid (W/m2 K) dx w the wall thickness (m) w = The thermal conductivity - k - for - for some typical materials (varies with temperature)
Polypropylene PP : 0.1 - 0.22 W/mK Stainless steel : 16 - 24 W/mK Aluminum Aluminum : 205 - 250 W/mK 1 W/(m2 K) = 0.85984 kcal/(h m2 oC) = 0.1761 Btu/(ft 2 h oF) More about conductive Heat Transfer Thermal Conductivity for Several Materials
The convection heat transfer coefficient - h - depends on
the type of fluid - gas or liquid the flow properties such as velocity other flow and temperature dependent properties
Heat transfer coefficient for some common fluids:
2
Air - 10 to 100 W/m K 2 Water - 500 to 10 000 W/m K
Thermal resistance The overall heat transfer coefficient can also be calculated by the view of thermal resistance - R. The wall is split in areas of thermal resistance where
the heat transfer between the fluid and the wall is one resistance the wall it self is one resistance the transfer between the wall and the second fluid is a thermal resistance
Surface coatings or layers of "burned" product adds extra thermal resistance to the wall decreasing the overall heat transfer coefficient.
Some typical heat transfer resistances
2
static layer of air, 40 mm (1.57 in) : R = 0.18 m K/W 2 inside heat transfer resistance, horizontal current : R = 0.13 m K/W 2 outside heat transfere resistance, horizontal current : R = 0.04 m K/W 2 inside heat transfer resistance, heat current from down upwards : R = 0.10 m K/W 2 outside heat transfere resistance, heat current from above downwards : R = 0.17 m K/W
Example - Heat Transfer in a Heat Exchanger A single plate exchanger with media A transfers heat to media B. The wall thickness is 0.1 mm and the material is polypropylene PP, aluminum or stainless steel. 2
Media A and B are air with a convection heat transfer coefficient of hair = 50 W/m K. The overall heat transfer coefficient U per unit area can be expressed as: U = 1 / (1 / h A + dx w / k + 1 / h B )
(1b)
Using the values from above the overall heat transfer coefficient can be calculated to:
2
Polypropylene PP : U = 24.5 W/m K Steel : U = 25.0 W/m2 K 2 Aluminum : U = 25.0 W/m K 2
1 W/(m K) = 0.85984 kcal/(h m
2 o
2
o
C) = 0.1761 Btu/(ft h F)
Typical Overall Heat-Transfer Coefficients
2
Free Convection Gas - Free Convection Gas : U = 1 - 2 W/m K (typical window, room to outside air through glass) 2 Free Convection Gas - Forced liquid (flowing) water : U = 5 - 15 W/m K (typical radiator central heating) 2 Free Convection Gas - Condensing Vapor Water : U = 5 - 20 W/m K (typical steam radiators) 2 Forced Convection (flowing) Gas - Free Convection Gas : U = 3 - 10 W/m K (superheaters) Forced Convection (flowing) Gas - Forced Convection Gas : U = 10 - 30 W/m2 K (heat exchangers gases) 2 Forced Convection (flowing) Gas - Forced liquid (flowing) water : U = 10 - 50 W/m K (gas coolers) 2 Forced Convection (flowing) Gas - Condensing Vapor Water : U = 10 - 50 W/m K (air heaters) 2 Liquid Free Convection - Forced Convection Gas : U = 10 - 50 W/m K (gas boiler) Liquid Free Convection - Free Convection Liquid : U = 25 - 500 W/m2 K (oil bath for heating) Liquid Free Convection - Forced Liquid flowing (Water) : U = 50 - 100 W/m2 K (heating coil in 2 vessel water, water without steering), 500 - 2000 W/m K (heating coil in vessel water, water with steering)
2
Liquid Free Convection - Condensing vapor water : U = 300 - 1000 W/m K (steam jackets 2 around vessels with stirrers, water), 150 - 500 W/m K (other liquids) 2 Forced liquid (flowing) water - Free Convection Gas : U = 10 - 40 W/m K (combustion chamber + radiation) 2 Forced liquid (flowing) water - Free Convection Liquid : U = 500 - 1500 W/m K (cooling coil stirred) 2 Forced liquid (flowing) water - Forced liquid (flowing) water : U = 900 - 2500 W/m K (heat exchanger water/water) 2 Forced liquid (flowing) water - Condensing vapor water : U = 1000 - 4000 W/m K (condensers steam water) 2 Boiling liquid water - Free Convection Gas : U = 10 - 40 W/m K (steam boiler + radiation) 2 Boiling liquid water - Forced Liquid flowing (Water) : U = 300 - 1000 W/m K (evaporation of refrigerators or brine coolers) Boiling liquid water - Condensing vapor water : U = 1500 - 6000 W/m2 K (evaporators steam/water)
Heat Exchangers The general function of a heat exchanger is to transfer heat from one fluid to another. The basic component of a heat exchanger can be viewed as a tube with one fluid running through it and another fluid flowing by on the outside. There are thus three heat transfer operations that need to be described: 1. Convective heat transfer from fluid to the inner wall of the tube, 2. Conductive heat transfer through the tube wall, and 3. Convective heat transfer from the outer tube wall to the outside fluid. Heat exchangers are typically classified according to flow arrangement and type of construction. The simplest heat exchanger is one for which the hot and cold fluids move in the same or opposite directions in a concentric tube (or double-pipe) construction. In the parallel-flow arrangement of Figure 18.8(a), the hot and cold fluids enter at the same end, flow in the same direction, and leave at the same end. In the counterflow arrangement of Figure 18.8(b), the
fluids enter at opposite ends, flow in opposite directions, and leave at opposite ends. [Parallel
flow]
[Counterflow] Figure 18.8: Concentric tubes heat exchangers
[Finned with both fluids unmixed.]
[Unfinned with one fluid
mixed and the other unmixed] Figure 18.9: Cross-flow heat exchangers.
Alternatively, the fluids may be in cross flow (perpendicular to each other), as shown by the finned and unfinned tubular heat exchangers of Figure 18.9. The two configurations differ according to whether the fluid moving over the tubes is unmixed or mixed. In Figure 18.9(a), the fluid is said to be unmixed because the fins prevent motion in a direction ( ) that is transverse to the main flow direction ( ). In this case the fluid temperature varies
with and . In contrast, for the unfinned tube bundle of Figure 18.9(b), fluid motion, hence mixing, in the transverse direction is possible, and temperature variations are primarily in the main flow direction. Since the tube flow is unmixed, both fluids are unmixed in the finned exchanger, while one fluid is mixed and the other unmixed in the unfinned exchanger. To develop the methodology for heat exchanger analysis and design, we look at the problem of heat transfer from a fluid inside a tube to another fluid outside.
Figure 18.10: Geometry for heat transfer between two fluids
We examined this problem before in Section 17.2 and found that the heat transfer rate per unit length is given by
(18..21)
Here we have taken into account one additional thermal resistance than in Section 17.2, the resistance due to convection on the interior, and include in our expression for heat transfer the bulk temperature of the fluid, than the interior wall temperature,
.
, rather
It is useful to define an overall heat transfer coefficient
per unit length as (18..22)
From (18.21) and (18.22) the overall heat transfer coefficient,
, is (18..23)
We will make use of this in what follows.
Figure 18.11: Counterflow heat exchanger
A schematic of a counterflow heat exchanger is shown in Figure 18.11. We wish to know the temperature distribution along the tube and the amount of heat transferred.
18.5.1 Simplified Counterflow Heat Exchanger (With Uniform Wall Temperature)
To address this we start by considering the general case of axial variation of temperature in a tube with wall at uniform temperature inside the tube (Figure 18.12).
and a fluid flowing
Figure 18.12: Fluid temperature distribution along the tube with uniform wall temperature
The objective is to find the mean temperature of the fluid at case where fluid comes in at
with temperature
,
, in the
and leaves
at with temperature . The expected distribution for heating and cooling are sketched in Figure 18.12. For heating (
), the heat flow from the pipe wall in a length
is
where is the pipe diameter. The heat given to the fluid (the change in enthalpy) is given by
where is the density of the fluid, is the mean velocity of the fluid, is the specific heat of the fluid and is the mass flow rate of the fluid. Setting
the last two expressions equal and integrating from the start of the pipe, we find
Carrying out the integration,
i.e., (18..24)
Equation (18.24) can be written as
where
This is the temperature distribution along the pipe. The exit temperature at is (18..25)
The total heat transfer to the wall all along the pipe is (18..26)
From Equation (18.25),
The total rate of heat transfer is therefore
or
(18..27)
where
is the logarithmic mean temperature difference, defined as
(18..28)
The concept of a logarithmic mean temperature difference is useful in the analysis of heat exchangers. We will define a logarithmic mean temperature difference for the general counterflow heat exchanger below.
18.5.2 General Counterflow Heat Exchanger We return to our original problem, to Figure 18.11, and write an overall heat balance between the two counterflowing streams as
From a local heat balance, the heat given up by stream is
. (There is a negative sign since
up by stream because
is
in length
x
decreases). The heat taken
. (There is a negative sign
decreases as
increases). The local heat balance is (18..29)
Solving (18.29) for
and
, we find
where . Also, coefficient. We can then say
where
is the overall heat transfer
Integrating from
to
gives (18..30)
Equation (18.30) can also be written as (18..31)
where
We know that
(18..32)
Thus
Solving for the total heat transfer:
(18..33)
Rearranging (18.30) allows us to express parameters as
in terms of other
(18..34)
Substituting (18.34) into (18.33) we obtain a final expression for the total heat transfer for a counterflow heat exchanger :
(18..35)
or
(18..36)
This is the generalization (for non-uniform wall temperature) of our result from Section 18.5.1.
18.5.3 Efficiency of a Counterflow Heat Exchanger Suppose we know only the two inlet temperatures find the outlet temperatures. From (18.31),
,
, and we need to
or, rearranging,
(18..37)
Eliminating
from (18.32), (18..38)
We now have two equations, (18.37) and (18.38), and two unknowns,
and
. Solving first for
,
or (18..39)
where
is the efficiency of a counterflow heat exchanger:
(18..40)
Equation 18.39 gives in (18.38) to find
in terms of known quantities. We can use this result
:
We examine three examples. 1. can approach zero at cold end.
as
, surface area,
.
Maximum value of ratio
Maximum value of ratio
.
2.
is negative,
as
Maximum value of ratio
Maximum value of ratio
.
3.
temperature difference remains uniform,
.
overall heat transfer coefficient is influenced by the thickness and thermal conductivity of the mediums through which heat is transferred. The larger the coefficient, the easier heat is transferred from its source to the product being heated. In a heat exchanger, the relationship between the
overall heat transfer coefficient (U) and the heat transfer rate (Q) can be demonstrated by the following equation:
where Q = heat transfer rate, W=J/s [btu/hr] A = heat transfer surface area, m2 [ft2] U = overall heat transfer coefficient, W/(m2°C) [Btu/(hr-ft2°F)] ΔTLM = logarithmic mean temperature difference, °C [°F] From this equation we can see that the U value is directly proportional to Q, the heat transfer rate. Assuming the heat transfer surface and temperature difference remain unchanged, the greater the U value, the greater the heat transfer rate. In other words, this means that for a same kettle and product, a higher U value could lead to shorter batch times. Several equations can be used to determine the U value, one of which is:
where h = convective heat transfer coefficient, W/(m2°C) [Btu/(hr-ft2°F)] L = thickness of the wall, m [ft] λ = thermal conductivity, W/(m°C) [Btu/(hr-ft°F)]
Heat transfer through a metal wall
The convective heat transfer coefficient (h), sometimes referred to as the film coefficient, is often used when calculating heat transfer between a fluid and a solid. In the case of a heat exchanger, heat transfer basically occurs from fluid 1 (source of heat) to solid (metal wall) to fluid 2 (product being heated). In the event that heat transfer occurs through several solids, the above equation can be adapted by supplementing the solid's thickness (L) divided by its thermal conductivity (λ).
To simplify the calculation, the following values may be used as a reference for the convective heat transfer coefficients: Fluid Convective heat transfer coefficient (h) 2
2
Water
about 1000 W/(m °C) [176 Btu/(hr-ft °F)]
Hot Water
1000 – 6000 W/(m )°C [176 - 1057 Btu/(hr-ft °F)]
Steam
6000 – 15000 W/(m °C) [1057 - 2641 Btu/(hr-ft °F)]
2
2
2
2
Example Two jacketed kettles made of carbon steel (λ = 50 W/(m°C) [28.9 Btu/(hr-ft°F)] ) with an inner wall thickness of 15mm [0.049 ft] are used to heat water. One uses hot water as the heat source, while the other uses steam. Assuming heat transfer coefficients of 1000 W/m 2°C [176 Btu/(hr-ft2°F)] for the water being heated, 3000 W/m2°C [528 Btu/(hr-ft2°F)] for hot water, and 10000 W/m 2°C [1761 Btu/(hr-ft2°F)] for steam, let's calculate the U values for both heating processes. Carbon Steel Jacketed Kettle Hot water:
U = 612 W/(m2°C) Steam:
U = 714 W/(m2°C) In this case, steam could theoretically improve the U-value by 17%. Let's now imagine the same kettle is lined with glass 1mm [0.0033 ft] thick (λ = 0.9 W/(m°C) [0.52 Btu/(hr -ft°F)]). Including these values into the above U-value equation gives the following: Glass-lined Jacketed Kettle Hot water:
U = 364 W/(m2°C) Steam:
U = 398 W/(m2°C) In this case, the U-value is only improved by 9%, which shows how a poor thermal conductor such as glass can greatly interfere with heat transfer. So in a carbon steel kettle, for example, changing the heat source from hot water to steam can potentially improve the U-value by several 10’s of percent. However, the same effect would not be expected in a glass-lined kettle. Nevertheless, certain circumstances require that a kettle remain unchanged. For example, some processes require kettles made of a certain material to prevent reactivity with the product. If such is the case and the heat transfer rate needs to be improved, changing the heat source from hot water to steam may provide the needed solution.
