 In the previous classes we have been learning about the molding sands, core sands, the patterns and also in the previous lecture we have seen the different steps involved in the making of a sand casting. Now in this lecture let us learn about an important topic that is the design of the razor. So this lecture will be on the design of the razor, first of all what is a razor. Now we can see this is a, this is the casting, this is the casting and this is the pouring cup. This whole thing is compacted inside the molding boxes. Now this is the pouring cup, the molten metal will be coming like this and this is this sprue and it will be passing through this sprue and this is the sprue well. Then the horizontal process is known as the runner, it passes through the runner. Then it is the, this is the gate, it through the gate it enters into the mold cavity. Once it is, once the molten metal is full with the molten metal it raises like this through the razor. So this is the razor and here we can see another razor is there. So here we can see a top razor and here we can see a side razor. So through the razor the molten metal raises. So once the molten metal raises through the razor we stop pouring the metal. So that is why because it is raising it is known as the razor. Now let us see what is the primary function of a razor. It acts as a reservoir of the molten metal in the mold to compensate for the shrinkage during solidification. We all know that all the metals when they solidify in the liquid level and when they are solidifying they undergo shrinkage. This is known as the solidification shrinkage. Now that time some molten metal will be inside the mold cavity and during the solidification because of the solidification shrinkage there will be shrinkage cavity will be there and the molten metal which is inside the razor comes and fills that gap and it compensates that gap not only during solidification during the liquid cooling of the molten metal. For example, let us take the aluminum, aluminum melts at about 660 degree centigrade but we used to pour it about 750 degree centigrade. From 750 degree centigrade to 660 degree centigrade there is liquid cooling there is no solidification but during this liquid cooling also there is a reduction in the volume inside the mold cavity but this reduction is compensated by the liquid metal which is inside the razor. So the razor or the liquid metal in the razor compensates the what say reduction in volume due to liquid cooling of the metal. It also compensates the molten metal due to solidification shrinkage. So that is the primary function of the razor and it has got secondary functions also. It gives us an indication that the cavity is full with the molten metal. Yes, we keep pouring the molten metal through the sprue and through the pouring cup and through the sprue and it passes through the runner and it fills the cavity. Once the cavity is full with the molten metal it raises up that is why it is known as the razor because it is feeding the casting it is also known as the feeder. Next one there is another secondary function. It also enables escape of hot gases during pouring of the molten metal. Yes, as we keep pouring the molten metal immediately the moisture in the moldings and comes in contact with the molten metal spontaneously this moisture turns into vapor and hot gases will be released and as we keep pouring the molten metal yes through the razor hole they will be escaping. But this escaping of the hot gases is possible as long as the razor is not filled with the molten metal. Once the razor is filled with the molten metal hot gases cannot escape through the razor any longer that time they will be escaping through the wind holes. They also escape due to the permeability of the molding sand. So these are the functions of the razor remember that the primary function of the casting is to feed the casting during the liquid cooling and also during the solidification process. So that is why it is known as the feeder and it has got the secondary function it raises and gives us an indication that the cavity is full with the molten metal because it is raises it is known as the razor. Now the question is why design of the razor cannot we make the razor with some size and do the casting there is a problem if the size of the razor is not adequate what will happen an undersized razor could lead to shrinkage defects and ultimately result in rejection of the casting. If the size of the razor or if the diameter of the razor is not adequate and the molten metal inside the razor is not adequate to feed the casting what will happen ultimately during the solidification there will be shrinkage cavity inside the casting then what happens the casting has to be rejected. Can't we keep the razor size too large so that this problem won't occur what happens an oversized razor requires excess molten metal and results in excess power or fuel consumption for melting for melting the metal what we do we use the electricity or if we use the cupola furnace we use the coke sometimes we use the oil fired furnaces so that is also a fuel. So if we are using if we are keeping a what say a very big razor what happens it requires excess of the molten metal and to fill this excess of the molten metal what we have to do we have to spend excess what say money and excess time for melting the metal. So that what say increases the cost of the production that is why hence the size of the razor must be optimized using some systematic methods so that is why we are going to learn about the what say design of the razor in this lecture. Now this is what can happen if the size of the razor is not adequate if the liquid metal in the razor is not enough to feed the casting during the solidification process a shrinkage cavity will be developed on the surface sometimes inside the surface also it can develop. Now these are the types of the razors before we learn about the design of the razor let us see these are the types of the razor one is the blind razor right means it is inside the mould cavity it is not visible and it is not exposed to the atmosphere. Next one the second category is the open razors these open razors are visible outside and they are exposed to the atmosphere so these are the open razors again in the open razors there are side razors and top razors side razors we have seen it is on the side of the casting whereas the top razor is above the kept above the casting. So these are the types of the razors in general for a side razor height is equal to its diameter if the height is h and if the diameter is d h is equal to d and for a top razor height is half of its diameter if the height is h and if the d is the diameter h is equal to 0.5 d this is in general right. Next one guidelines for razor design and location how to what say design the razor and how to locate the razor the razor or the feeder must not solidify before the casting if it solidifies before the casting how can it feed the casting so that is why it is known as the feeder it is like mother feeding her children right when the mother makes an important dish a delicious dish it mother feeds the children only after something is left out mother will take it if the mother consumes all the what say delicious dish how she can feed the children she cannot be a mother she cannot be a feeder similarly a feeder or a razor of the casting must solidify only after the casting solidifies so that is the fundamental rule. Next one the volume of the razor or razors must be large enough to feed the entire shrinkage of the casting the first rule says that the razor must solidify after the casting solidifies but here the razor is no doubt the razor is solidifying after the casting solidifies but it is not large enough to feed the entire shrinkage of the casting it is able to feed part of the shrinkage of the casting only during the initial stage of the solidification and at the final stage of the solidification there is no sufficient metal in the liquid metal in the razor then how can it feed the casting so the quantity of the razor liquid metal in the razor must be large enough to feed the entire shrinkage of the casting. So that is the second guideline the third guideline is the pressure head from the razor should enable complete cavity filling right so the pressure head from the razor it should enable complete cavity filling no cavity or no part of the cavity of the mound should be what is left out without the molten metal. Next one razor must be placed so that it enables directional solidification the position of the razor must be such that there will be directional solidification. Now the question is what is this directional solidification let us see right so here we can see there are two cases in one case directional solidification is in what is a progress means it is dominating in another case progressive solidification is dominating means what is happening here this is the casting and this is the razor now here there is liquid metal is there so this portion this gray portion indicates the solidified casting now the solidification is gradually propagating towards the razor as it is propagating towards the razor liquid metal in the razor is able to feed any shrinkage that is taking place during the process of the solidification. So what is happening so the direction of the solidification is in this direction right with the solidification is progressing this way like this now let us take the second case let us assume the razor is like this somewhere here the razor is like this no doubt there will be directional solidification will be there but here we can see progressive solidification means what is happening here the solidification in the previous case the solidification is what is a what is a going towards the razor now here the solidification is propagating perpendicular to the what is a direction of the solidification. This is the direction of the solidification even the whats a solidification is going on in this direction now the solidification is going perpendicular now what happens the razor is somewhere here the razor is high and the what say solidified portion is becoming more and more and after sometime this what say there will be a very narrow region with liquid metal. Now, suppose this portion is blocked because it is solidifying perpendicular to the direction then what happens the liquid metal may not be in a position to what say compensate the shrinkage that is taking place of may be here there is a liquid portion is there but because of this progressive solidification this portion is already solidified and the razor is here and here there is liquid metal and there will be shrinkage cavity how this can be compensated by the razor it cannot be compensated. So, here we can see directional solidification and progressive solidification. So, in a solidification of the casting both these solidifications will be there but the thing is with the directional solidification is dominating that is good there would not be any shrinkage defect. But on the other hand compared to the directional solidification if the progressive solidification is dominating then what happens there will be say some portion with the in the liquid state and the razor cannot feed that liquid portion because somewhere the passage is blocked excessive of progressive solidification leads to shrinkage defect. So, the razor must be placed in such a way that there will be more directional solidification and less progressive solidification. So, these are the important methods of the razor design. One is the Keynes method, second one is the modulus method, third one is the naval research laboratory method. Now, we will see all these one by one, first we will see the Keynes method. Now JP Keynes has conducted extensive experience on razoring sometime during 1949. He examined the presence and absence of shrinkage defects in various castings. Then he developed a term called freezing ratio which is defined as follows freezing ratio is equal to surface area of the casting by volume of the casting whole divided by surface area of razor divided by volume of the razor. So, this was the definition given by Keynes for the freezing ratio. Based on this freezing ratio he has given some idea how to identify the what is a shrinkage defects in the castings. Now, let us see what is that procedure how to identify the shrinkage defects in the castings. Now here you can see now the Keynes has plotted a graph like this means he has conducted extensive experience on the castings and on several castings he got the shrinkage defects and in some castings he got the sound castings and for all the castings he has calculated the freezing ratio and he has also calculated the razor volume to casting volume. Now with the freezing ratio as the x axis and the razor volume to casting volume as the y axis he has plotted this graph. So, this is the what is a graph plotted by Keynes. Now what does this graph tell us? The graph plotted by Keynes predicts whether the casting would be a sound or defective one. Now let us see this graph here there is a curve. So when he has plotted taking freezing ratio on the x axis and razor volume to casting volume on the y axis so a curve was plotted. Now all the castings that fall on the right side of this curve will be sound castings without any shrinkage defects. Now all the castings which are falling on the left side of this graph curve will be defective castings with shrinkage defects. So this was the finding of the Keynes and he this he has plotted after conducting extensive investigations. Now the problem is what is the limitation definitely this was a very good work but what is the limitation the what say this process says that you design some razor and I will tell you whether you will get a sound casting or a shrinkage casting. You design a razor yes this is going to get a this is going to give a shrinkage defect you design another razor this is going to give a sound casting. Now my question is what is the size of the razor with which I can get a sound casting and that size of the razor must be optimum the Keynes method cannot give answer to this. So that is the small limitation of this Keynes method. Now there were this work was further extended. Now another definition was given for freezing ratio. Now here X is the freezing ratio now a freezing ratio X is equal to A divided by Y minus B plus C where X is the freezing ratio Y is equal to volume of the razor divided by volume of the casting ABC are constants and they depend upon the material of the casting. Now these are the values of ABC constants for different metals first steel A is equal to 0.1 for steel B is equal to 0.03 and C is equal to 1 for aluminum A is equal to 0.1 B is equal to 0.06 C is equal to 1.08 for cast iron A is equal to 0.04 B is equal to 0.07 C is equal to 1. So these are the values of the ABC for different metals. So he has given these values for steel, aluminum and cast iron. Now let us see a problem how to solve this problem or how to get the design of the razor using the Keynes method. Now let us take this problem calculate the size of a cylindrical razor necessary to feed a steel slab casting of size 25 into 25 into 5 centimeters. Height of the razor is its and its diameter are equal means it is a side razor. Now how to design the razor means how to get the dimensions of the razor. Now this is the solution. Now volume of the casting but this is the simple geometry 25 into 25 into 5 that is equal to 3125 cubic centimeters. Now surface area of the casting it has got the 6 surfaces right 25 into 25 surfaces there are 2 right. So 2 into 25 into 25 similarly 25 into 5 surfaces there are 2. So similarly actually there are 4 so 4 into 25 into 5. So the surface area of the casting is equal to 1750 square centimeters. Now let the diameter of the razor is equal to D. Now what is the volume of the razor, pi D cube divided by 4. So here we can see this is the volume of the razor. Now we have to find out the surface area of the razor. How to find out the surface area of the razor? So surface area of the razor means it is a cylindrical one right under the what is if we consider the development of the surfaces if we develop right it will become a rectangle. Now what is the length of that rectangle that is the pi D, pi D multiplied by height what is the height, height is the diameter. So pi D into diameter that is equal to pi D square right plus what about the top circle that is pi D square by 4 means the top circle we are considering the bottom circle we are ignoring why we are ignoring because compared to the surface area of the casting the bottom circle is very small that too it is not exposed to the atmosphere. So that is why we are excluding the bottom circle. So this pi D square indicates the surface area of the top circle. Now the total surface area of the razor is equal to 1.25 pi D square. So we have calculated the surface area of the razor. Now freezing ratio X is equal to so this is how cane has defined the freezing ratio. Surface area of the casting by volume of the casting whole divided by surface area of the razor by volume of the razor. So that comes to be 0.11 to D. Now there is another definition for Y right what is Y? Y is equal to volume of the razor divided by volume of the casting. So that is equal to 0.000251 D cube it comes to be that much say it seems to be a complex value right it comes to be that much. Now so far we have calculated two freezing ratios right using one is the using the freezing ratio this was the first definition of the cane given by cane. So using this definition we got the freezing ratio as 0.11 to D and using the another definition right. So this is the other definition for the freezing ratio using this definition yes we have substituted these are the values for the A, B and C we are substituting those values right. So when we substitute those values it will become this much. So the previous what say the what say this one X the previous freezing ratio and the present freezing ratio we are equating we are equalizing. Then by we get an expression in that expression say D is unknown. So we cannot calculate this D directly one thing is we have to try some value put some value and try whether it satisfies both sides. So by trial and error method we have to substitute several values finally one value will be what say satisfying both the sides that value is the correct value of the diameter of the razor. So by trial and error method we get diameter of the razor is equal to 11.44 centimeters or it is almost equal to 12 centimeters. Now we know the diameter of the razor diameter and height are equal. So when the diameter is 12 centimeters height is also equal to 12 centimeters. So this is how we have to design the razor using Keynes method drawbacks of the Keynes method. For each material the constants A, B and C keep changing for steel it is different for aluminum it is different likewise for the he has mentioned for the three materials right only for these three materials we can what say design the razor using Keynes method. Now it is what say every time it is difficult for us to remember these values so that is another drawback. Next one calculation of freezing ratio is difficult if the surface of the casting is complex right that is another drawback. Next one the solution is to be obtained by trial and error method right directly we cannot solve we have to substitute some value and try whether it satisfies both sides we have to keep on what say changing till both the sides are equal. So this time this takes time and this process is tedious. Next one so we have seen the Keynes method now let us see the modulus method this in this modulus method was based on the Nikolas Chorino's work right Nikolas Chorino has conducted again investigation in the year 1939 and he found that solidification time was directly related to Casting's volume to surface area ratio. Volume to surface area ratio right means V by SA Chorino's volume to surface area ratio was termed as solidification modulus or simply modulus. Now this became the basis for the modulus method. So Chorino has developed a term called modulus or the solidification modulus just like Keynes has developed the freezing ratio Chorino has developed a term called the solidification modulus which is equal to the ratio of the volume to surface area. Now this was the Chorino's rule right in the Chorino's rule TST is equal to CM multiplied by V by A to the power of N, where TST is the total solidification time, V is the volume of the casting, A is the surface area of the casting, N is an exponent and it is usually taken as 2, CM is a constant which depends upon the moulding materials right. So this is the Chorino's rule, TST is equal to CM multiplied by V by A to the power of N. So here we can see that volume means volume of the casting, A means surface area of the casting right. So it applies to the casting, this also applies to the razor means if we apply this rule to the razor what these terms will become, TST means total solidification time of the razor, V means volume of the razor, A means surface area of the razor and N is the component and CM it depends upon the moulding materials right. What does Chorino's rule tell us? TST is equal to CM multiplied by V by A to the power of N, what does this rule tell us? A casting with a higher modulus, what is modulus? Volume to surface area ratio V by SA or V by A ratio right with a casting with a higher modulus ratio cools and solidifies more slowly than the casting with a lower modulus. Here the modulus when the modulus is very high the casting cools down very slowly. High means the V by A ratio, V by A ratio is high means what? The A area is less because area is less V by A ratio is high means when area is less it is less exposed to the mould wall means less heat is dissipated to the mould wall. That is how it takes more time for solidification compared to a casting or compared to a razor with a lower modulus. Now to feed the molten metal to the casting total solidification time of the razor must be greater than the total solidification time of the casting right. The total solidification time of the razor right must be greater naturally if the razor solidifies before the casting it cannot feed the casting. So the razor must solidify after the casting solidifies means the total solidification time of the razor must be greater than the total solidification time of the casting. So that is the second what say information we can get from the Juvenile's rule. Next one since the mould constants of razor and casting will be equal razor should be designed to have a larger modulus so that the main casting solidifies first. So the third rule says that so if we want the what say razor to have what say longer solidification time it is modulus should be smaller than the modulus of the casting. When it is modulus will become smaller right it is sorry it is the modulus should be larger right that is what the rule says the razor should be designed to have a larger modulus when the surface area is minimum the surface area of the razor should be minimum then only it will have a larger modulus then only it will take longer time for solidification. Then that be the case what happens the Chironov rule says that the razor should have the minimum surface area then only it will have the larger modulus. Now if we take a what say particular what say mass in different shapes may be in a what say square prism and a triangular prism and a spear and a what say cylinder in all the cases the mass is same the which one which geometry has the least surface area spear has got the least surface area. So Chironov suggested that the ideal shape of the razor is a spear but spear spherical razor yes ideal it is very good it has the least surface area thus it will have a larger modulus and it will take longer time for solidification. But incorporating a spherical razor inside the moulding boxes would be a tough task that is why instead of a spherical razor a cylindrical razor would be adopted. So these are the things which we can learn from the Chironov's rule. Now Chironov has suggested that the modulus of the razor must be larger than the modulus of the casting then only the razor will take longer time for solidification then only it can feed the casting is it not. So then that be the case how much the modulus of the razor should be larger than the modulus of the casting. By experiments see the further investigators found that the modulus of the razor is equal to 1.2 times the modulus of the casting MR is equal to 1.2 times MC where MR is the modulus of the razor MC is the modulus of the casting how much time it should be larger it is enough if the modulus of the razor is 1.2 times the modulus of the casting then it is large enough to what say take longer time for solidification and to feed the casting perfectly. So this is the relation MR is equal to 1.2 into MC remember this. So these are the moduli of simple geometric shapes. So this is a plate a long plate its thickness is t and its length is a and its width is b then the modulus is 0.5 into t. So you can see the modulus now it is no way connected with the length and the what say width it is only dependent on the thickness modulus is equal to 0.5 t where a is less than 5 t. Next one let us consider another simple shape that is the long bar where a is the height b is the width then modulus is equal to a b divided by 2 into a plus b. Next one let us take the cube another simple geometrical shape this is a cube right. So where right one side is equal to d modulus is equal to d by 6. So calculation I am not showing but even if you calculate you will get the same what say figures same values. Next one let us consider a cylinder where diameter is equal to d and if you try to get the modulus right modulus means volume by surface area you calculate the volume and you calculate the surface area and the ratio of the volume to surface area and if you calculate it comes to be d by 6. Next one a sphere and for sphere the modulus means the ratio of the volume to surface area again it is equal to d by 6. Next one let us consider a hollow cylinder and here we can see this is a cylinder right. So inside there is a hole axially you can see and the height of the cylinder is h and here we can see the thickness of the solid portion is r. Now in such a case what is the modulus is the same expression modulus is equal to volume divided by surface area and if you calculate the volume and if you calculate the surface area and you if you calculate the ratio of the volume to surface area you will get this thing m means modulus is equal to r h divided by 2 multiplied by r plus h. So these are the moduli of some simple geometrical shapes. Now let us take a problem determine the size of a side razor for a casting of dimensions 25 into 25 into 5 cm using modulus method. Now previously we have designed the razor using Keynes method now we are going to design the razor using modulus method this is the solution. Now in the modulus method what we have to do we have to find out the rule says that the modulus of the razor MR is equal to 1.2 times the modulus of the casting. So we have to initially find out the modulus of the casting then we have to find out the modulus of the razor based on the modulus of the razor we have to find out its diameter so that is the procedure. So first we are going to find out the modulus of the casting. So volume of the casting VC is equal to just multiply this 25 into 25 into 5 that is equal to 3125 cubic centimeters. Next we have to find out the surface area of the casting surface area of the casting means again so this 25 into 25 surfaces there are 2 so 2 into 25 into 25 similarly this 25 into 4 there are 4 surfaces 4 into 25 into 25 the total surface area of the casting is equal to 1750 square centimeters. Now the modulus what is modulus volume by surface area so MC is equal to VC by SCSC VC means volume of the casting SCSC means surface area of the casting that is equal to 3125 divided by 1750 that comes to be 1.7857. So far we have found out the modulus of the casting now based on this we have to find out the modulus of the razor the rule says that the modulus of the razor MR is equal to 1.2 multiplied by modulus of the casting. So this is the modulus of the razor MR is equal to 1.2 into MC that is equal to so 1.2 into MC what is MC we got 1.7857 so 1.2 into 1.7857 so this is the modulus of the razor. Now but previously we have seen we have found out the moduli of some simple geometrical shapes among them cylinder was there do you remember just to go back so what was the modulus of the cylinder d by 6. So now we do not have to calculate again we need not calculate the volume and we need not calculate the surface area again straight away we take the modulus of the cylinder as d by 6 now the modulus of the razor that is a cylinder is equal to d by 6. Now this d by 6 is equal to 2.1429 yes from this we can find out the d d is equal to 6 into 2.1429 that comes to be 12.6 centimeters so the diameter of the razor is equal to 12.5 centimeters similarly the height of the razor is equal to 12.6 centimeters so this is a side razor so this is the way to calculate or to design the razor using modulus method. Now let us see one more problem during the casting of a certain alloy using a sand mould it took 155 seconds for a cube shaped casting to solidify the cube was 50 millimeters on each side. Now there are two questions the first question is determine the value of the mould constant in Chernow's rule and the second question is for the same alloy and mould determine the total solidification time for a cylindrical casting whose diameter is 30 millimeters and the length is 50 millimeters. Now let us see the solution to determine the mould constant that is the first question volume of the cube B is equal to 50 cube that is equal to 125,000 cubic millimeters. Now area of the cube A is equal to 6 into 50 square that is equal to 15,000 square millimeters. Now we have to find out the V by A ratio so that is equal to 125,000 divided by 15,000 that is equal to 8.333. Now total solidification time TST is equal to 155 seconds that is given. Now assumed value of the N is equal to 2. Now what is the mould constant? Now this is the Chernow's rule TST is equal to cm multiplied by V by A to the power of N where TST is equal to total solidification time that is equal to 155 seconds, V is equal to volume of the casting that is equal to 125,000 cubic millimeters, A is equal to surface area of the casting that is equal to 15,000 square millimeters, N is equal to exponent usually taken as 2 and we have taken as 2 and cm is a constant which depends upon the mould material. Now from Chernow's rule cm is equal to TST divided by V by A square so that is equal to 155 divided by 8.333 square and if we simplify this it will become 2.232. So the mould constant in this problem is 2.232 seconds per square millimeters. Next problem is to determine the total solidification time. For cylindrical casting with diameter 30 millimeters and length 50 millimeters, volume V is equal to pi d square L by 4 that is equal to 35,343 cubic millimeters, area A is equal to 2 pi d square divided by 4 plus pi dL that is equal to pi multiplied by 30 square divided by 2 plus pi 30 into 50 and if we simplify this it becomes 6126 square millimeters. So this is the area. Now V by A is equal to 35,343 divided by 6126 that is equal to 5.77. TST is equal to cm multiplied by V by A to the power of N that is equal to 2.232 multiplied by 5.77 to the power of 2 means that is the constant. So if we simplify this it is 74.3 seconds or if we convert into minutes it becomes 1.24 minutes. So the casting solidifies in a time span of 1.24 minutes. So in this lecture we have seen the what is the design of the casting. Now we will see design of the riser. Now we will see the merits of the modulus method. Now these are the merits of the modulus method. The method is independent of the material of the casting whereas in the case of the Keynes method for each material there were different constants A, B, C. So for each material the values of these constants were different. So there was a lot of calculation. So whereas in the case of the modulus method these constants A, B, C do not come into picture. So that is the merit of the modulus method. Next one method is simple and not like tedious like Keynes method. In the Keynes method we got a what say higher order equation and by trial and error we had to solve it. So that was a tedious process. So such a process is not this one. This is a simple process. But it has got limitation to what are its limitations. So this is the demerit of the modulus method. The modulus means the volume to surface area ratio of the casting depends upon the surface area of the casting is it not? We have considered in this problem just for the sake of what say exercise we have taken simple shape a rectangular block. But in the real world in the practical applications what would be the geometry of the casting? Most complex geometries would be there. The most complex surfaces would be there. Then how to find out the surface area of such surfaces? It is very difficult. That is why in many cases determination of surface area of the casting becomes difficult due to its complex geometry. No problem as long as the what say geometry of the casting is simple we can very well find out the surface area and we can find out the modulus. But once the casting has a complex surfaces then the determination of the surface area would become a tough task. At such times use of this method would be a difficult task. So friends in this lecture we have seen the purposes of the riser, the functions of the riser, types of the riser and different methods that have been developed for designing the riser. They are the Keynes method, modulus method and the Neville research laboratory method. In this lecture we have seen and we have learnt about the Keynes method and the modulus method. We have seen the merits and the limitations of both these methods and in the next lecture we will be learning about the Neville research laboratory method. Thank you.