What are the examples of combined mode of heat transfer?

It is true that heat flow occurs mainly in three ways that is conduction, convection, and thermal radiation. In industrial engineering designs sometimes the combined heat flow are utilised to meet the requirement of cooling in the system. For example in electronic circuits some power transistors emanate a lot of heat from their body and heat sinks are provided to conduct the heat away to bigger metal plates and then radiate it in the surrounding. This is a simultaneous process and conduction and radiations are taking place continuously. If heat generated is high sometimes instrument cooling fans are also mounted to cool the heat sinks. Nowadays miniature liquid cooled heat sinks are also made to cool the circuit elements that give rise to heat. This aspect was discussed some time back in a relevant question in this section. Other example of combined heat flow is the internal cooling obtained for the turbine blade in a gas turbine where the conduction and convections both are used by cooling the turbine through a water jacket and releasing the heat with convection.

1. A hot cup of coffee conducts heat in all three forms.
A. Hot coffee heats up the cup material via conduction. Because of this, only the outside of the cup will get heated up. If you use a ceramic or clay cup this transfer will be less and if you use a metallic cup it may be more. That is why you will feel hot when you hold a steel glass with hot coffee.
B. There will be natural convection of heat through the liquid. This is due to the density difference in the liquid you have in the cup and outside.
c. The heat will get radiate in the form of electromagnetic waves to the surroundings.

2. The water in the sea will be heated up by radiation of sunlight. Conduction and radiation will help in heating up the interiors of the ocean. Convection will happen through the surface of the ocean where the water gets boiled and will get lifted.

3. The work carried out in a boiler will have all the three ways of heat transmission.
The water in a boiler will get heat from the fire bed by conduction, convection, and radiation. The heat from the fire to the shell conducted heat through the shell and conducted and convected heat from the inner shell to the water.

The three modes of heat transfer are through conduction, convection and through thermal radiation. The best example is when we place a can with water on a burner. Normally the flame gives radiation which heats the can. It then transfers the heat to water through conduction. In the process of convection, the hot water will rise to the top.
Another example will be the atmosphere. The atmosphere gets the heat through radiation from sun and atmosphere gets hot air which is near the equator and it produces the winds and at last, there is conduction which happens through air molecules and from some amounts of air land conduction.
Next best example is an electric oven and in that the metal gets red hot which emits good light like black body radiation. Normally there is air in the oven and we will feel that there is opening up the oven for more time and the gases are not storing for more energy. Also, there will be conduction of heat and we can feel it when we touch the inside surface of the oven.

As you rightly mentioned, there are three kinds of heat transfer, namely: conduction, convection, and radiation. Some examples are given below for these activities.

For Conduction:
1)When you touch the surface of a burning stove or a charges iron box, you will take back the finger suddenly due to the heat and maybe there is a chance to burn your finger
2)We can boil the water by immersing a red hot piece of iron into it.
3) If you press an ice bar in your hand, you will feel utmost cold inside your hand.

Convection:
1)A good example would be heating a tin can of water using a Bunsen burner. Initially, the flame produces radiation which heats the tin can. The tin can then transfers heat to the water through conduction. The hot water then rises to the top, in the convection process.
2)An old-fashioned radiator where it creates a convection cell in a room by emitting warm air at the top and drawing in cool air at the bottom.
3) The human heart is a pump and blood circulation in the human body is an example of forced convection. The heat which is generated by the cells in the body is transferred to air or water which is flowing over the skin.

For Radiation:
The heat from the sun warming your face is the best example of Radiation. Same as heat from a lightbulb also due to the radiation process. In the same manner, the heat from a fire and heat from anything else which is warmer than its surroundings can take best examples for Radiation process.

Let us understand the mechanism of evaporation of sea - water where the radiation plays a significant role causing the following changes -
1) Evaporation of water
2) Heating up the interior of the ocean by means of both conduction and radiation.
3) Heating the surrounding adjacent to surface of ocean resulting in a chance of escape of air because of presence of heat and the same gets lifted due to lesser density.
If the vacuum is of high order, it can cause depression and as a result it my lead to storm formation due to the presence of forced convection.
Here the entire cycle is controlled by all modes of heat transfer simulteneously. By removing any such parameter, process will proceed further. This offers an example of heat transfer where all the three process are present in the process of heat transfer.