# Natural convective heat transfer from a vertical plate with constant surface heat flux and boundary layer

Ajinkya Zalte

Sardar Vallabhbhai National Insitute of Technology, Surat, Gujarat 395007

K.R. Sreenivas

Professor, Engineering Mechanics Unit, Jawaharlal Nehru Center for Advanced Scientific Research, Bangalore 560064

## Abstract

Heat transfer is a ubiquitous physical phenomenon. Natural Convection (one of the mechanism of heat and mass transfer) is responsible for the natural phenomenon of land and sea breeze to the phenomenon of maintaining the core body temperatures of all warm-blooded creatures. Now considering natural convection has wide application in industries like heat exchangers, etc. and other wide range of application like hair blowers, etc. it is essential to understand and investigate the phenomenon.

The goal of this experiment is to study the natural convective heat transfer phenomenon over a vertical flat hot plate with uniform heat flux. Due to natural convection the resultant boundary layer formation over the plate and its overall profile will be studied along with the numerical solution to this phenomenon. The main objective was to measure the temperature profile using non-contact measurement and visualize the thermal boundary layer. Air cannot be directly images using infrared imaging and hence an alternative arrangement was used to measure the temperature indirectly using horse-hair strings.

Keywords: Natural Convection, IR camera , Boundary layer

## Background

Many of the heat transfer phenomenon happening around us involve natural convection as the primary mechanism. From the cooling of electronic equipments such as power transistors to the heat transfer from the refrigeration coils and power transmission lines and the heat transfer from the body of animal and human beings natural convection plays primary role. The cooling by natural convection is considered as the most effective and efficient cooling mechanism. Natural convection works due to a temperature difference existing between body and the surrounding air with causes difference in the density of air giving rise to density difference and hence bouyant force, which the primary cause of natural convection currents.

Infrared imaging had been used for decades in industries and research works, but new innovations have reduced the costs and newer advances have increased their reliability and hence have resulted in infrared sensors offering better accuracy and increased sensitivity. Due to these advancements the technology is finding its place in new applications and uses. Infrared imaging is a non-contact measurment technique which is used to obtain information about parameters such as temperature, pressure, velocity distribution, etc. occuring in a control volume which in this case is the measurement of thermal boundary layer.

## Statement of the Problems

The phenomenon of Natural Convective heat transfer over a vertical flat plate with constant surface heat flux and the thermal boundary layer formed due to convection is to be studied. This study will be conducted via an IR camera and results obtained will be compared with numerical solution of the phenomenon.

## Information

The problem of laminar free-convection heat transfer from a heated verical plate in still air was first considered by Lorenz (1) in 1881, by assuming that the temperature and velocity at any point of th eflow field depends only on the distance from the plate.

The first exact solution of the free convecive problem for the vertical flat plate was developed by Pohlhausen (3) in 1921. The basic boundary layer equation was consiered and by introduction of stream function and a suitable similarity transformation, he converted partial differential equation to an ordinary differential equation and then intergrated the eqation to find the solution.

Ostrach (2) performed numerical solution of the free-convection boundary layer equations for Prandtl numbers of 0.01, 0.72, 0.733, 1, 2, 10, 100 and 1000 using computer technique.

Later Sparrow and Gregg (4) analyzed the problem of laminar free-convection from a vertical plate with uniformsurface heat flux, by integrating the transformd moemtum and energy equations. The modified Grashof number $Gr^*_x = \frac{g\beta qx^4}{k\nu^2}$, was introduced to replace the conventional Grashof number $Gr_x=\frac{g\beta(T_w-T_\infty) x^3}{\nu^2}$. the surface temperatue variation and local Nusselt numbers were calculated for Prandtl number 0.1, 1, 10, 100. Results of the numerical calculation of $\frac{Nu_x}{ (Gr^*_x )^{1/5}}$ vs Pr were extrapolated to a Prandtl number of 0.01.

The integral method most often employed in free-convection boundarylayer flows is attributed to Squire. He analyzed the problem under consideration by assuming a polynomial for the velocity and temperature profiles which could be made to satisfy the boundary conditions.

Luciano Peera and Benjamin Gebhart (5) analyzed the laminar natural convection boundary-layer dlow abouve horizontal and slightly inclined surfaces. The effect of a small surface inclination on flow and transport is studied and solution of the higher approximations have been numerically deermined for the velocity, pressre and temperature fieds for a Prandtl number of 0.7. This study was done for both uniform temperature and heat flux cases and Mach-Zehnder interferometer was used to study the nature of laminar boundary layer.

## Numerical Solution

Introduction

Natural convection is a phenomenon in which hydrodymanics and heat transfer are coupled together, here the energy equations drive the momemtum. In the case of vertical plate gravity plays an important part as the Bouyant force acting due to the density difference between the hot sir close to the plate and the ambient cool air is the driving force, so basically the boundary layer forms due the temperature gradient between the ambient surrounding and the plate. As the air close to the plate heats up, its density decreased and hence the air rises along the plate froming the boundary layer. So the momentum and energy equations in the Navier-Stokes equations are to be coupled to get the solutions.

3.1.1 Governing Equations

The 2-Dimensional steady-state flow continuity, momentum and energy equation are shown below respectively:

$\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0$

$u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y} = g\beta(T-T_{\infty}) + \nu\frac{\partial^2u}{\partial y^2}$

$u\frac{\partial T}{\partial y}+v\frac{\partial T}{\partial y}= \alpha\frac{\partial^2T}{\partial y^2}$

where ν = Kinematic viscosity , β = Coefficient of thermal expansion, α = Thermal diffusibility

$\alpha= \frac{k}{\rho C_p}$ and $\beta = - \frac{1}{\rho}\frac{\partial\rho}{\partial T}$

(x is along the plate and y is perpendicular to the plate)

Boundary layer with velocity and temperature profile

3.1.2 Similarity solution

The above eqautions were solved by Sparrow and Gregg by introducing similaritity variable. In similaritysolution by the introduction of the stream function and a suitable similarity transformation, the partial differential equations are reduced to ordinary differential equation which can be integrated to obtain the solutions.

The similarity variable used to solve the equations are:

$Gr_x^\ast=\frac{g\beta Qx^4}{k\nu^2}$, ​,

$u_{ref}=2\sqrt{\frac{xgQy\beta f’}k}$, $\theta = \frac{(T-T_{\infty})k}{Qy}$

where, $Gr^*_x$= Modified Grashoff's no., $u_{ref}$= reference velocity, T = Temperature of plate at some point, $T_{\infty}$= Temperature of surrounding air, Q = Constant heat flux of the plate

Using these variables the stream function is found and then by substituting and modifying variables the governing equation transform into following momentum and energy equation respectively:

$\frac{d^3f}{d\eta^3}-3(\frac{df}{d\eta})^2 + 4f\frac{d^2 f }{\partial \eta^2}- \theta = 0$

$\frac{d^2 \theta}{d\eta^2}- Pr(4f\frac{d\theta}{d\eta}+\theta\frac{df}{d\eta}) = 0$

where, Pr = Prandtl no.

For solving the above equations boundary conditions are given below:

At y = 0 → η = 0 : u = 0 and v = 0; $\frac{df}{d\eta}=0$; $\frac{d\theta}{d\eta}=-1$

At η = ∞ : $\frac{df}{d\eta}=0$; θ = 0​

The equations are solved using the above boundary conditions and the values for velocities and temperatures were calculated for various Prandtl no. The approximate nusselt no. was calculated to be:

$Nu_x\;\approx\;\;(\frac{Gr_x^\ast Pr^2}{4+9Pr^{1/2}+10Pr})^{1/5}$

This nusselt no. can be used to calculate the mean nusselt no. over the surface and hence can be used to find the convective heat transfer co. (h) as $Nu_x = \frac{hx}{k}$

Velocity profile inside the boundary layer
Temperature profile inside the boundary layer

3.1.3 Approximate Solutions

This is the second method to solve the equation by assuming a suitable profile. The governing equations are first converted to their integral form. The integral momentum and energy equations are shown below:

$\frac{d}{dx}(\int_{0}^{\delta} u^2dy )= \nu\frac{\partial u}{\partial y}_{y=0} + \int_{0}^{\delta} g\beta(T – T_{\infty})dy$

$\frac{d}{dx}(\int_{0}^{\delta}u(T – T_{\infty})dy) = -\alpha\frac{\partial T}{\partial y}_{y=0}$

Assume the velocity profile to be cubic: $\frac{u}{u_{ref}}=$
A+B\eta+C\eta^2+D\eta^3

and temperature profile to be quadratic: $\theta = A+B\eta+C\eta^2$

where, $\eta$
= \frac{y}{\delta}
(here δ = δt.....assumption)

Now applying the boundary conditions to the equation:

At y = 0 → u = 0, η = 0 and $Q$
= -k\frac{\partial T}{\partial y}_{y=0}

At η = 1 → u = 0, $\frac{\partial$
u}{\partial \eta}=0
, T = T∞ and $\frac{\partial$
T}{\partial y}=0

At η = 0 → $\frac{\partial$
^2u}{\partial y^2}=\frac{-g\beta(T-T_{\infty})}{\nu}

So, the profiles comes out to be:

$\frac{u}{u_{ref}} = \eta(1-\eta)^2$ and $\theta = (1-\eta)^2$ where $\eta = \frac{y}{\delta}$

so from the equations we can vsualize the velocity distribution in the boundary layer

Now substitute these equations into the integral equations to get following equations:

$\frac{1}{105}\frac{d}{dx}(u_{ref}^2\delta) = \frac{\delta^2 g\beta Q}{6k} - \nu \frac{u_{ref}}{\delta}$ and

$\frac{d}{dx}(\frac{u_{ref}Q\delta^2}{60k}) = \frac{\alpha Q}{k}$

Assume $u_{ref} = C_1x^m$ and $\delta = C_2x^n$, on substituting we get $\delta = C_2x^{1/5}$ which is similar to the solution obtained in similarity solution.​

## Experimental Setup

The setup consists of a flat plate which is to be uniformly heated to obtain uniform heat flux. This surface is to be subjected to natural convection forming a boundary layer over the surface. The basic idea is to provide a heat source within two parallel flat plates to observe the phenomenon and then make observations using an IR camera.

3.2.1 Approach

Firstly, the main part of the setup was to think about the heat source within the plate, and we worked out three different method with which we could work this out:

1.      Nichrome coil heat source.

2.      Aluminium foil

3.      Heat transfer via fluids.

Heat Transfer via fluids

The idea was to pass fluid (water) in between the two plates, this fluid will have higher temperature than the plates and hence the fluid would give its heat to the plates by conduction and convection phenomenon.

Aluminium foil

The idea was to make a long coil of aluminium foil and place it between the two plates. As we know that when voltage is applied between the two ends of the foil, a current will be induced which heats up the coil and hence can be used as a heat source.

Nichrome Wire Heating Coil

The idea is to used coiled nichrome wire, when voltage is provided across the two ends it will heats up and hence acting as a heat source. This coil is to be placed them between the two plates.

Nichrome wire Coiled around the Hylam sheet

3.2.2 Final Design

The nichrome heating source was selected for the final design (Nichrome wire is used widely for heating coils and can provide wide range of heat flux depending on the voltage applied). Now the most important part was to design and select appropriate material for the heating coil and the flat plate respectively. The main output that was expected was to obtain a uniform heat flux so the parallel plates were decided to be made up of Copper (Copper has thermal conductivity (K) = 401 Wm^-1K^-1while Aluminium has K= 237 Wm^-K^-1). Copper has a high thermal conductivity, so it will reduce minor variation in the flux output from the plate and the plate will have a uniform heat flux. The dimensions of the plate are 155 mm × 300 mm.

Top view of the copper plates

The base of the heating coil was selected to be Hylam, which is a non-conducting, insulation material used widely in electrical applications. The design of the nichrome wire coil was also important as it would affect the thickness of the setup and the overall heat output. The nichrome wire was coiled in the form of a helix around the hylam plate, which would run on both the sides of the plate. The plates had grooves machined around its length as helix on a screw or a bolt (figure), this would flush the wire to the surface.

Hylam sheet having grooves holding the Nichrome wire

The machining of the grooves on the hylam sheets was carried out using CNC milling machining, which took 30 hours to machine. Machining accuracy would be achieved using the CNC milling and the grooves were too small to machine manually. The nichrome wire was wound and then copper plates were stuck to either side of the hylam via silicone RTV glue (RTV silicone stands for Room Temperature Vulcanizing silicone which has the property to withstand high temperatures without losing its properties and affecting the heat transfer between the plates and heating source).

The coil and copper plates are insulated from each other as copper has high electrical conductivity and any contact would short circuit the connection and would render the setup ineffective. Multiple layer of insulating varnish was applied on the coil and the copper plates, a layer of Mica sheets was also added to provide absolute insulation between the coil and plates (Mica sheets are stable to high temperatures and are good electrical insulators). The copper plates used are 3 mm in thickness, thus the whole setup is about 10 mm in thickness.

Aluminium frame used to hold the plate vertically

For the observing the boundary layer forming over the vertical plate a support frame was made out of aluminium extrusions so the plate was held perfectly vertical. The voltage was provided from a Variac (Voltage Regulator) and the voltage used for the experiment was 70 Volts, thus power can be worked out to be P = 33 Watts.

Experimental Setup

3.2.3 IR camera

IR camera measures the infrared radiations emitted by a body, it consists of an optical system that focuses infrared energy onto a sensor array to produce a signal, which is then processed to create a colour map of the apparent temperature of a body. The main advantage of visualization by infrared thermography is its non-invasive character.

IR camera infront of the plate

3.2.4 Imaging

During the experimentation various precautions had to be taken while measuring to get good images. The camera was highly sensitive thus care was taken not to use body with high thermal reflectivity in and around the setup which would affect the readings. The setup was enclosed so that it is not disturbed by the surrounding factors as shown in figure 9, cardboard was used as it has low reflectivity and hence the readings are not affected.

The most difficult part was the measurement of air temperature as air cannot be directly imaged using an infrared imaging technique and hence alternative arrangements are to be made to measure the temperature indirectly, hence horse-hair strings were used as they show quick response to change in tempratures due to its high thermal diffusivity and the available surface area is also large and so the string attains the local temperature of air very quickly.

The strings are placed parallel to the flat plate at regular intervals i.e 1mm to 13mm at every 2 mm, so a total of 7 threads were used.

• 1

## RESULTS AND DISCUSSION

One of the main objective of the experiment was to achieve uniform heat flux on the surface, which was achieved during testing.

An almost uniform heat flux

In the Figure 8 one can see the non-uniformity in the temperature (colour variation) at the surface due to the presence of varnish on the surface. The varnish layer was later removed and a paper was stuck to the surface to provide uniformity and the final result is shown in Figure 11.

IR image of boundary layer
Contouring of the above IR image

From the temperature profiles shown in figure the experimental data is in accordance to the numrical results obtained. The image below shows the temperature of the threads at midpoint of the plate. The peaks seen in the image are the temperatures of the threads and the general behaviour can be seen in agreement to the temperture profile as shown in Figure 14.

Temperature of threads at mid-point of plate

The graph in Figure 15 shown below is the mean temperature of the threads which also follows the general behaviour.

Mean Temperature of plate

## Conclusion

From the results obtain we can say that the results are in agreement with the numerical available.

From the experimentations I learned the techniques to use thermal imaging, its advantages and limitations. Got a deeper understanding about the reflectivity and emissivity of objects and how they affect the thermal imaging. Understood methods for numerical solution like similarity solutions, approximate solution.

From experimentation detail study of the thermal boundary layer so carried out successfully.

## References

1. Lorenz, H. H. "Die Varmeubertragung an einer ebenen senkrechten Platte an 01 bei natiirlicher Konvektion." Z. Techn, Physik (1934), 362.

2. Ostrach, S. "An Analysis of Laminar Free Convection Flow and Heat Transfer about a Flat Plate Parallel to the Direction of the Generating Body Force." NACA TR 1111 (1953).

3. Pohlhausen, E. "Der Warmeanbtausch Zwichen festen Korpern und Fl'ussigkeiten mit kleiner Reiburg und kleiner Warmeleitung. " ZAAM, 1 (1921), 115.

4. Sparrow, E. M. , and Gregg, J. L. "Laminar Free Convection from a Vertical Plate with Uniform Surface Heat Flux." ASME Trans., 78, U''eb. 1956), 435-440.

5. Pera, L., & Gebhart, B. (1973). Natural convection boundary layer flow over horizontal and slightly inclined surfaces. International Journal of Heat and Mass Transfer, 16(6), 1131–1146.doi:10.1016/0017-9310(73)90126-9

6. A. Bejan "Convective Heat Transfer", Wiley Publication

7. "Viscous Fluid Flow" by Frank M. White, McGraw Hill Publication

8. "Heat Transfer" by Yonus A. Cengel

9. "Laminar Free-Convection On a Vertical Flat Plate With Unifrom Surface Temperature or Uniform Suraface Heat flux" by Jerome Chieh-Jan Lee, 1950

## Acknowledgements

I would like to thank my guide Prof. K.R Sreenivas for giving me an opportunity to work in under his guidance and for using his lab facilities. I would also like to thank the institute Jawaharlal Nehru Center for Advanced Scientifivc Research, Bengaluru for providing me the opportunity to experience such a great research environment, one which I have never had before and for allowing me to work with the lab equipments and other facilities.

I would like to thank Dr Diwakar V, for allowing me to use his lab facilities and helping me out in the experimentation.

I express my deepest thanks to our fellow lab members Saifuddin V, Rafiuddin Mohammad, Vybhav Rao, Pravesh Shukla, Amit Mishra for providing their valuable time and guidance in the project work.

I would like to thank Authorcafe for providing a good platform for writing the reports.

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