Report Forced Convection Heat Transfer PDF

# Forced Convection Heat Transfer to a Tube in Cross Flow ## Location: Heat Transfer Laboratory (80 points for the report and 20 points for the preparation) ## A. Objectives 1. To understand the difference between natural and forced convection. 2. To measure the heat transfer coefficient on a tube...

# Forced Convection Heat Transfer to a Tube in Cross Flow ## Location: Heat Transfer Laboratory (80 points for the report and 20 points for the preparation) ## A. Objectives 1. To understand the difference between natural and forced convection. 2. To measure the heat transfer coefficient on a tube in a cross flow as a function of flow velocity. 3. To present the experimental measurements in dimensionless form as Nusselt number (Nu) versus Reynolds number (Re) and to compare the measured Nu-Re relation with existing correlations in the textbooks. 4. To repeat the measurements correlation and comparison for a tube bundle. ## B. Background Convection is the mode of energy transfer between a solid surface and the adjacent liquid or gas that is in motion, and it involves the combined effects of conduction and fluid motion. The faster the fluid motion, the greater the heat transfer by convection. In the absence of any bulk fluid motion, heat transfer between a solid surface and the adjacent fluid is by pure conduction. The presence of bulk motion of the fluid enhances the heat transfer between the solid surface and the fluid, but it also complicates the determination of heat transfer rates. Energy is first transferred to the air layer adjacent to the block by conduction. This energy is then carried away from the surface by convection, that is, by the combined effects of conduction within the air that is due to random motion of air molecules and the bulk or macroscopic motion of the air that removes the heated air near the surface and replaces it by the cooler air. Convection is called forced convection if the fluid is forced to flow over the surface by external means such as a fan, pump, or the wind. In contrast, convection is called natural (or free) convection if the fluid motion is caused by buoyancy forces that are induced by density differences due to the variation of temperature in the fluid. Heat transfer between the egg and the surrounding air will be by conduction if the temperature difference between the air and the block is not large enough to overcome the resistance of air to movement and thus to initiate natural convection currents. **Figure 10.1: The cooling of a boiled egg by forced versus by natural convection** Despite the complexity of convection, the rate of convection heat transfer is observed to be proportional to the temperature difference, and is conveniently expressed by Newton's law of cooling as $Q_{conv} = h*A*(T_s - T_A)$ where *h* is the convection heat transfer coefficient in W/m². ## Table 1: Typical values of convection heat transfer coefficient | Type of convection | h, W/m² °C | |---|---| |Free convection of gases|2-25| |Free convection of liquids|10-1000| |Forced convection of gases|25-250| |Forced convection of liquids|50-20000| |Boiling and condensation| 2500-100000| Many heat exchanger designs depend upon external convective heat transfer from cylinders or multiple rows of tubes (tube bundle) subjected to cross flow. The heat transfer characteristics of these cross flow configurations are determined by thermal boundary layer development on the cylinders. Since the simplest configuration usually involves boundary layer separation, the time-dependent phenomena, it is not possible to calculate analytically the average heat transfer coefficients in cross flow. Even with the great progress made in numerical methods, calculations at arbitrary Reynolds number is not yet possible, because of the 3-D and time dependent flow instabilities after separation. As a consequence, heat transfer calculations for problems of this type are based upon correlations of experimental data which relate the Nusselt number to Reynolds number and Prandtl number. Since the Prandtl number is practically constant for gases under a wide range of conditions, the rate of heat transfer for gases is effectively a function of just Reynolds number. ## C. Experimental Setup It is intended for the study of heat transfer phenomena associated with flow past cylindrical tubes arranged either in isolation or in banks of various configurations. The apparatus consists of Perspex working section through which air may be drawn by a centrifugal fan. Perspex rods may be inserted into the working section with their axes at right angles to the direction of flow, thus simulating a typical cross flow heat exchanger of the type used in many branches of engineering. One of the spaces provided in the working section for the rods is occupied by an element consisting of a rod of pure copper approximately 10 cm in length carried between two extension rods of fabric-based plastic compound. Arrangements are made for heating this copper element in isolation from the working section, replacing it in the section and then recording its rate of cooling as indicated by thermocouple imbedded at its center. A semi-logarithmic plot of rate of cooling together with knowledge of the thermal capacity and surface area of the copper then permits a direct calculation of the heat transfer coefficient between the copper element and the air flowing past it. The element is heated by withdrawing it from the working section and placing it in a cylindrical electric heater. The heater is supplied with current at a low voltage from a rectifier and raises the temperature of the element to a maximum of about 80 °C. The element temperature is indicated by a thermocouple potentiometer, either a standard laboratory instrument or supplied as an extra. The potentiometer records the temperature difference between the hot junction embedded in the element and a cold junction in the air stream at the inlet to the working section. The initial temperature of the air is indicated by a mercury-in-glass thermometer at the air inlet. The apparatus includes a centrifugal fan driven by a 1 HP electric motor and having its inlet connected to the working section. Air enters the apparatus by way of a bell-mouth. A transition pieces is connected after the working section and leads to the fan inlet and carries a honeycomb flow straighter intended to prevent the transmission of swirl from the fan back into the working section. The fan discharges to graduated throttle valve by means of which the air velocity through the apparatus may be regulated. To permit exploration of the flow pattern upstream of the tube bank a total head tube is provided, and this may be traversed in a direction perpendicular both to the air flow and to the axes of the element. Five traversing stations are provided at 2.5 cm centers, as shown in the following figure, permitting a complete survey of the cross section. In addition, a traversing position is provided downstream of the tube bank, permitting investigation of the flow pattern in the wake. Associated static tapings are provided so that the velocity head may be recorded by means of the manometer. The velocity distribution upstream of the tube bank is sensibly constant and may be established by a single measurement of the static wall pressure downstream of the bell-mouth. When all the tube elements are in position, the static pressure drop across the four tube banks is about four times the velocity head, and once the relation between velocity head and pressure drop has been established it is preferable to observe the pressure drop rather than the velocity head as an indication of the air velocity past the tube bank. Pressure difference and velocity heads in the apparatus reach a maximum of about 75 mm water gauge. The inclined manometer provided has a range of this magnitude; for greater precision an electronic micro manometer, available as an extra, may be used. The thermocouples in the element and at the air inlet are of copper and constantan to British Standard Reference. Within the small range 0-50°C temperature difference, the temperature characteristic of the thermocouple is approximately linear and: 1 °C = 0.041 mV. The apparatus is mounted on a tubular steel bench carried on castors and an integral cabinet carries an isolator, starter, and control switch for the element heater. **Figure 10.2: Schematic of the experimental setup** ## Table 2: Nominal dimensions (in mm) | Dimension | Value | |---|---| | Width of the working section | 125 | | Height of the working section | 125 | | Diameter of the elements | 12.5 | | Transverse pitch of the elements | 25 | | Longitudinal pitch of the elements | 18.75 | ## Table 3: Specimen specifications | Dimension | Value | |---|---| | Material | Copper | | Diameter, d | 12.43 mm | | Length, l | 95.08 mm | | Weight, m | 107.3 gm | ## Table 4: Nomenclature | Notation & Symbol | Units | |---|---| | Diameter of element, d | m | | Length of element, l | m | | Effective Length of element, l₁ | m | | Surface area of element, A | m² | | Effective surface area of element, A₁ | m² | | Mass of element, m | kg | | Specific heat of copper element, C (=380) | J/kg.K | | Barometric pressure, p_a | N/m² | | Temperature of air, T_A | K | | Velocity head, upstream, H₁ | cm H₂O | | Velocity head, downstream of element, H₂ | cm H₂O | | Static pressure drop across element, H₃ | cm H₂O | | Velocity upstream of working section, V₁ | m/s | | Mean velocity past element, V | m/s | | Velocity downstream of element, V₂ | m/s | | Density of air, p | kg/m³ | | Specific heat of air at constant pressure, c_p | J/kg.K | | Viscosity of air, µ | kg/m.s | | Thermal conductivity of air, k | W/m.K | | Temperature of element, T | K | | Slop of cooling cure, M | - | | Rate of heat transfer to air, q | W | | Coefficient of heat transfer, h | W/m².K | ## Table 5: Dimensionless groups | Dimensionless group | Formula | |---|---| | Nusselt number, Nu | $Nu = \frac{hd}{k}$ | | Prandtl number, Pr | $Pr = \frac{μC_p}{k}$ | | Reynolds number, Re | $Re = \frac{ρVd}{μ}$ | ## D. Assumptions & Theoretical Background It is assumed that the whole of the heat lost from the cylindrical copper element is transferred to the air flowing past it. It is further assumed that temperature gradients within the element are negligible, so that the thermocouple embedded at the center gives a true indication of the effective surface temperature. A certain amount of heat is conducted from the element into the plastic extension pieces. The extent of this effect has been determined by making comparative tests using copper elements of identical diameter but varying length. From these tests the equivalent additional surface area represented by the plastic extensions has been calculated and is allowed for by making an addition to the true length of the element to give an effective length that is used in the calculations. This correction amounts to 8.4 mm, where: $l = l_1 + 0.0084mm$ From the definition of the heat transfer coefficient, the rate of transmission of heat from the element to air is given by: $q=h*A*(T-T_A)$ (1) In a time period (dt) the fall in temperature (dT) is given by: $-q.dt = m.c.dT$ (2) Combining equations (1) and (2) and eliminating (q): $\frac{dT}{dt} = \frac{hA_1}{m.c}(T-T_A)$ Integrating: $log_e(T-T_A)-log_e(T_o-T_A) = -\frac{hA_1t}{m.c}$ (3) $log_e\frac{T-T_A}{T_o-T_A} = -\frac{hA_1t}{m.c}$ (4) or $log_e\frac{T-T_A}{T_o-T_A} = -\frac{hA_1t}{m.c}$ (4a) where $(T_o)$ is the element temperature at (t) = 0 This equation suggests that a plot of loge [(T-TA)/(To-TA)] against (t) should yield a straight line of slope (-hA₁). Since the other factors in this expression are known, the heat transfer m.c coefficient (h) may be calculated. Therefore, the heat transfer coefficient is estimated by measuring how fast the element is cooled. In practice, you can plot log_e(T-T_A)/(T_o-T_A) against (t) on a linear axis and estimate the slope of the best straight line that fits the data; or you can plot [(T-TA)/(To-TA)] on a semi-log paper (linear on the horizontal axis (t) and log₁₀ on the vertical axis), but in this case make sure you plot [(T-TA)/(To-TA)], not log[(T-TA)/(To-TA)] and notice that the original expression had log_e, not log₁₀, and the semi-log paper is using log₁₀, not loge. Therefore, if you use a semi-log paper, knowing that log_e N = 2.3026 log₁₀ N, the heat transfer coefficient is related to the slope (M) of this line by the expression: $h = -2.3026\frac{m.c}{A_1}M$ (5) In order to establish the effective velocity of the air passing the element it is necessary to calculate the velocity upstream from the velocity head downstream. The velocity of the air, V₁, is measured with a Pitot tube. Neglecting the compressibility effects (since the Mach number is very small), the dynamic pressure measured, ΔP, is given by: $ΔP = \frac{ρV_1^2}{2}$ (6) where ΔP is given by p_water g ΔH, where ΔH is the head difference measured by the manometer. Since, 1 cm H₂O = 98.1 N.m², Equation (7) becomes: $ρV_1^2 = 98.1 H$ (6a) The density of under pressure (p_A) and at temperature (T_A) is given by: $p_A = \frac{R.T_A}{P}$ (7) where the gas constant R = 287 J/kg.K. Combining equations (6a) and (7): $V_1 = 237.3\sqrt{\frac{H.T_A}{p_A}}$ (8) It is usual, when calculating the effective velocity through a bank of tubes to base this on the minimum flow area. When all the tubes are present this minimum area occurs in a transverse plan including a row of 5 tubes. Since the tubes have a diameter of 12.5 mm and the width of the working section is 125 mm the effective area is one half that of the working section, and we may write for this case: $V = 2V_1$ (9a) when a single element is being studied isolation, the minimum flow area is (9/10) of the full working section area, and we may write: $V = \frac{10}{9}V_1$ (9b) Substituting (H₂) for (H₁). The theory of convective heat transfer leads us to expect that the heat transfer coefficient, (h) will be some function of fluid velocity (V) and of the independent variables: D, p, c_p, p, k. Dimensional analysis shows that the relationship between (h) and the independent variables may be expressed in the form: $\frac{hd}{k} = f( \frac{pVd}{μ},\frac{μC_p}{k})$ or $Nu = f(Re.Pr)$ In practice the Prandtl (Pr) number is constant for gases under a wide range of conditions, and the rate of heat transfer is effectively a function of Reynolds number (Re). ## E. Experimental Procedure i.e. $Nu = f(Re)$ The purpose is to produce cooling curves for the element used under various flow conditions, in order to measure the heat transfer coefficient, h. Follow up the following procedure: 1. The apparatus should be set up with the heated element in the desired position. 2. The manometer should be connected to the total head tube, which should be located in the center upstream position, with the tube itself on the horizontal centerline of the working section and facing upstream. 3. The other leg of the manometer should be connected to the static tapping at the upstream end of the working section. 4. The throttle valve should be closed, and the fan stared up. 5. Open the throttle to give the desired flow rate. 6. Switch on and standardize the thermocouple potentiometer in accordance with the instructions provided with the instrument. 7. It will be found that the velocity head upstream of the working section is, in fact, equal to the pressure drop between atmosphere and the upstream static pressure tapping. 8. Once this has been established the depression at the static tapping may be employed as a measure of (H₁). 9. Reading of air inlet temperature and total head tube reading should be recorded. 10. The element heater switched on and the element removed from the working section and inserted in the heater. 11. When the temperature of the element reaches to about 70°C, corresponding to a thermocouple voltage of about 2.8 mV, replace the element in the working section. 12. Now, set the potentiometer to reading rather lower than that corresponding to the temperature of the element, observe the galvanometer needle and start a stopwatch when the needle passes through the zero position. 13. Re-set the potentiometer to a lower value and observe the stopwatch reading when the galvanometer needle again passes the zero mark. 14. Repeat this operation for a series of diminishing potentiometer settings: the resulting information enables a cooling curve to be plotted. 15. It will be found instructive to plot cooling curves for a range of different air velocities and also with the element in each of the four ranks of the heat exchanger and an isolation. 16. As a preliminary experiment when using the full tube bank it is useful to determine the relation between upstream velocity head and pressure drop across the tube bank. The latter pressure drop may then be used as a more accurate measure of flow velocity. 17. As a further test, the velocity distribution upstream of the working section and also at various positions in the wake may be explored by means of total head tube. ## F. Test Sheet | Measurement | Value | Units | |---|---|---| | Date | .../.../20 | - | | Ambient temperature | °C | - | | Voltage | 220 V | - | | Element dimensions: Length (l) | 1 phase | - | | Element dimensions: Length (l₁) | | | | Element dimensions: Diameter (d) | 50 Hz | mm | | Element dimensions: Mass (m) | - | kg | | T/°C, mV | - | - | | Log₁₀ (T-T_A) | | - | | Velocity head, upstream the element H₁ cm H₂O | | - | | Slope M | | - | | L₁=1+0.0084 | | m | | A₁ = π.δ. 11 | | m² | | h=874.988 (m.M/A₁) | | W/m².K | | T_A | | K | | p_A 133.322 x mm Hg | | N/m² | | V₁=237.3H₁.T_A/p_A | | M/s | | Re = 815 V | | - | | Nu h.d/0.0259 | | - | ## G. Required Work 1. Fill-in the table above. (35 points) 2. Using heat transfer textbooks (or the internet) find the relationship between the Nusselt number and the Reynolds number for the case studied and compare it to the obtained relationship in the lab (state your references). (10 points) 3. Calculate the heat transfer coefficient for the case studied in the lab using the relationship obtained in the lab. (10 points) 4. Calculate the heat transfer coefficient for the case studied in the lab using the relationship obtained from any heat transfer textbook (10 points) 5. The radiator in your car is not functioning properly. When you drive at medium speed, the temperature gage in your car does not show dangerous temperature level (red range). However, when you stop in the traffic signal, the temperature, as shown in the gage, shows dangerous level (red). Explain why. (10 points) 6. Give three examples of forced convection from industry. (5 points)